The Effect of Hydrogen and Hydrides on the Integrity of Zirconium Alloy Components: Delayed Hydride Cracking

Engineering Materials

For further volumes: http://www.springer.com/series/4288

Manfred P. Puls

The Effect of Hydrogen and Hydrides on the Integrity of Zirconium Alloy Components Delayed Hydride Cracking

123

Manfred P. Puls MPP Consulting Oakville, ON Canada

ISSN 1612-1317 ISBN 978-1-4471-4194-5 DOI 10.1007/978-1-4471-4195-2

ISSN 1868-1212 (electronic) ISBN 978-1-4471-4195-2 (eBook)

Springer London Heidelberg New York Dordrecht Library of Congress Control Number: 2012940465  Springer-Verlag London 2012 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To my family: Wife Gloria; children Larissa and Nicholas; grandchildren Alexander, Jessica, Pascale, Benjamin and twins Ashton and Brayden

Preface

Zirconium alloys are widely used in nuclear reactors, their most challenging applications being as cladding for nuclear fuel in Pressurized and Boiling Water Reactors (PWR and BWR, respectively) and as pressure tubes in CANDUTM and Pressurized Heavy Water (PHW) reactors. The possible failure of these components has consequences to the safe and economic operation of these reactors. Data and understanding to prevent any such failures is, therefore, of great importance to the nuclear industry. It was recognized early on in the development of the design of nuclear reactors that these alloys have an affinity for absorbing hydrogen both during their manufacture and operation. When the hydrogen solubility limit in these alloys is exceeded, their excess amount results in the formation of zirconium hydride precipitates. These precipitates have been shown to be less ductile than the surrounding zirconium alloy matrix. In the 1960s when the first commercial nuclear reactors were being designed and built, the initial concern regarding the presence of these precipitates was their potential for decreasing the fracture toughness of zirconium alloy components. This led to an initial intensive study of the physical and mechanical properties of zirconium hydride in its bulk form and as precipitates uniformly distributed in the alloys. Somewhat later, in the early 1970s, as a result of leaks found in some over-rolled pressure tubes in two of the earliest commercial CANDU reactors, a time-dependent failure mechanism associated with hydrogen and hydrides in the pressure tube alloy was discovered and subsequently extensively studied. This time-dependent failure mechanism was initially referred to by a variety of different names but is now universally called Delayed Hydride Cracking (DHC). A vast body of knowledge has since appeared in the open literature and in internal reports on this topic including some review articles. To acquaint oneself with this knowledge, however, a thorough review of the literature can be an arduous and sometimes confusing task since models, measurement methods, and data have changed over the years, been revised or updated, taking on different forms over time. This book, then, is an attempt to provide the reader with a more coherent account of this topic than is obtainable through a reading of the extant literature. Emphasis in this book is placed on showing how the fundamental aspects of DHC vii

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have informed the methods—and provided the underpinning—for its practical applications. In so doing, the book provides more detailed descriptions of the current theoretical and experimental foundations of the properties and behavior of hydrogen, hydrides, and DHC in zirconium and its alloys than is usually found in individual papers in the literature. Although emphasis is on zirconium alloys, this book has been presented in such a way that it could also serve as a basis for the treatment of other hydride forming metals. In fact, it is noted that the reverse has frequently been the case with the knowledge produced concerning the properties and behavior of hydrogen and hydrides in other hydride forming material providing starting points for quantifying and understanding these in zirconium alloys. The book is not meant to be an exhaustive source of the documented work involving DHC. Nevertheless, the author has attempted to give as complete an account of the present state of knowledge of this field as possible. To make the derivations of the theoretical models reproduced in this book more understandable, background information is provided to show how these derivations follow from the general theories of the diffusion of atoms in crystalline solids, the macroscopic theory of irreversible thermodynamics, thermodynamics of coherent misfitting phases in crystalline solids containing mobile interstitial atoms, self and interaction energies of point and line defects in crystalline solids, and continuum fracture and solid mechanics. This book could also be useful as a teaching tool in that it provides an illustration of a specific application of these general theories to a practical engineering problem. Although the primary emphasis in this book is on zirconium alloys used in pressure tubes of CANDU and other PHW reactors, the treatment is sufficiently general for applications to other zirconium alloys and components used in nuclear reactors, particularly those for nuclear fuel cladding.

Acknowledgments

The author is indebted to his former colleagues in Atomic Energy of Canada Ltd. (AECL), Ontario Power Generation, and Kinectrics Inc. (previously named, successively, Ontario Hydro Research Division and Ontario Hydro Technologies) for sharing their thoughts, insights, and data with him and for collaboration on joint publications. Particular acknowledgments go to the following: (1) Roger Dutton for introducing the author to the theory of subcritical crack growth by diffusion creep mechanisms (which formed the precursor of the DHC growth rate model) and his collaboration and support associated with the development of the various versions of the DHC crack growth models; (2) Leonard Simpson for his pioneering experimental work on DHC growth rate and the determination of embedded hydride fracture strengths; (3) Brian Leitch for the support provided through his theoretical calculations of the accommodation energies and stress states in and around misfitting hydride precipitates to solvus and hydride fracture models; (4) San-Qiang Shi for his seminal work on the theory of KIH; (5) Ian Ritchie and Zheng-Liang Pan for their outstanding experimental work in the development of techniques using various internal friction methods for solvus and hydride property determinations; (6) Stefan Sagat, Max Resta Levi and Gordon (GK) Shek for their mastery and inventiveness in the selection and execution of all aspects of DHC experimentation, (7) Richard Sauvé and Don Metzger for their outstanding modeling of DHC behavior based on their own versatile finite element program. The author would also like to acknowledge the important contributions of Marc Léger to the DHC research community overall through his insights, the generous sharing of his own work, and his technical leadership. Also acknowledged is the ongoing leadership provided by Doug Scarth for the development of flaw assessment methodologies for pressure tubes, both through his own individual contributions and through his expert guidance as chair of various technical committees connected with this effort. The author thanks AECL and the CANDU Owners Group (COG) for permission to use, where indicated in the text, hitherto externally unpublished data.

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Reviews of parts of draft versions of Chap. 3 by Vuko Perovic, of Chap. 9 by Axel Steuwer and Mark Daymond and Chap. 11 by Doug Scarth are also gratefully acknowledged.

Contents

1

Introduction . . . . . . . . . . . . . . . 1.1 Overview. . . . . . . . . . . . . 1.2 Delayed Hydride Cracking. 1.3 Objectives and Outline . . .

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Properties of Bulk Zirconium Hydrides. . . . . . . . . . . . . . . . . 2.1 Hydride Phase Compositions, Lattice Structure and Parameter Determinations . . . . . . . . . . . . . . . . . . . . 2.1.1 Crystallographic Properties of the c-Hydride Phase . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Phase Relationships, Phase Stability, and Hydrogen Compositions in the d- and c-Hydride Phases at the (a + d)/d Phase Boundary. . . . . . . 2.1.3 Crystallography of the d-Hydride Phase. . . . . . . 2.2 Mechanical Properties of Bulk Zirconium Hydrides . . . . . 2.2.1 Yield Strength . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Fracture Toughness . . . . . . . . . . . . . . . . . . . . . 2.2.3 Microhardness, Elastic Moduli, Internal Friction. 2.2.4 Summary of Mechanical Strength Results . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Hydride Phases, Orientation Relationships, Habit Planes, and Morphologies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Hydride Precipitation in a-Zr Alloys: Early Analyses Derived from Observations of c-hydride Precipitates. . 3.3 Hydride Precipitation in a–Zr Alloys: Determinations of Lattice Transformation Relationships of c-Hydride Precipitates . . . . . . . . . . . . . . . . . . . . . 3.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .

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3.3.2 f1 0 1 ‘g Habit Planes. . . . . . . . . . . . . . . . . . 3.3.3 f1 0 1 0g Habit Planes . . . . . . . . . . . . . . . . . 3.4 Hydride Precipitation in a/b-Zr Alloys . . . . . . . . . . . . . 3.5 Hydride Nucleation Studies in a/b Zr-2.5Nb Pressure Tube Material . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Microstructure of Zr-2.5Nb Alloys . . . . . . . . . 3.5.2 Hydride Precipitation in a/bZr Microstructures . 3.5.3 Hydride Precipitation in a/b/x Microstructures. 3.5.4 Influence of Prior Deformation on Hydride Precipitation . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

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Solubility of Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Solubility in the Dilute Phase of Single Phase Metal–Hydrogen Systems . . . . . . . . . . . . . . . . . . . 4.2 Solubility of Hydrogen in the Dilute Phase of a/b Zirconium Alloys . . . . . . . . . . . . . . . . . . . . 4.3 Effect of Stress on Hydrogen Chemical Potential in the Dilute Phase . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Interaction Energy Expressions . . . . . . . . . . . . . . . 4.4.1 Partial Molar Volume . . . . . . . . . . . . . . . 4.4.2 Size–Effect Interaction . . . . . . . . . . . . . . 4.4.3 Diaelastic Polarizability . . . . . . . . . . . . . . 4.4.4 Paraelastic Polarizability . . . . . . . . . . . . . 4.4.5 Interactions Between Hydrogen Atoms in Solution . . . . . . . . . . . . . . . . . . . . . . . 4.5 Interaction of Hydrogen Atoms in Solution with Internal Defects. . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Density of Site Energies (DOSE) and Fermi–Dirac Statistics . . . . . . . . . . . . 4.5.2 Interaction of Hydrogen with Dislocations: DOSE Method . . . . . . . . . . . . . . . . . . . . 4.5.3 Formation of Hydrides at Dislocations: Thermodynamic Method . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Diffusion of Hydrogen. . . . . . . . . . . . . . . 5.1 Phenomenological Flux Equations . . 5.2 Diffusivity—Theory . . . . . . . . . . . . 5.3 Diffusivity in Dilute Phase—Results . 5.4 Thermal Diffusion—Results. . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

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Characteristics of the Solvus . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 General Considerations Concerning Hysteresis in Phase Transformations. . . . . . . . . . . . . . . . 6.3 Theories of Solvus Hysteresis Based on Accommodation Energy . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theories of Coherent Phase Equilibrium . . . . . . 7.1 General Features. . . . . . . . . . . . . . . . . . . . 7.2 Polymorphic Phase Transformation. . . . . . . 7.3 Isomorphic Phase Transformation . . . . . . . . 7.4 Stability Conditions and Path Dependences for Coherent Phase Transformations . . . . . . 7.4.1 Stability Conditions for Closed Polymorphic Systems . . . . . . . . . 7.4.2 Stability Conditions for Open Isomorphic Systems . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Experimental Results and Theoretical Interpretations of Solvus Relationships in the Zr–H System. . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Application of Coherent Phase Stability Analysis to the Zr–H System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Evaluation of Models of Hysteresis for the Zr–H System . . . . 8.4 Summary of Results of Experimental Solvus Determinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Effect of Experimental Methods . . . . . . . . . . . . . . . 8.4.2 Differences Between Solvi in a and a/b Zr Materials: Effect of b-Zr Phase . . . . . . . . . . . . . . . . . . . . . . . 8.4.3 Effect of Cold Work and Irradiation . . . . . . . . . . . . 8.4.4 Combined Effect of Thermal Aging and Irradiation in Zr–2.5Nb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.5 Effect of Manufacturing Variables, Microstructure, and/or Composition. . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fracture Strength of Embedded Hydride Precipitates in Zirconium and its Alloys . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Early Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Fracture Strength of Radial Hydrides: Rising Load Tensile Tests . . . . . . . . . . . . . . . . . . . . . .

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9.4

Mechanism for Fracture of Embedded Radial Hydride Clusters in Rising Load Tests . . . . . . . . . . . . . . . . . . . . . 9.5 Hydride Stress State Determinations in Tensile Tests Observed Under Synchrotron X-ray Irradiation . . . . . . . . . 9.6 Fracture Strength of Radial Hydrides: Constant Load Tests . 9.7 Comparison of Rising Load and Constant Load Hydride Fracture Strength Results . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10 Delayed Hydride Cracking: Theory and Experiment . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Historical Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 General Features of DHC . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Theory of DHC Growth Rate. . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Test Temperature Approached from Below . . . . . . 10.4.2 Test Temperature Approached from Above . . . . . . 10.4.3 Dependence on Direction of Approach to Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.4 Dependence on KI and Activation Energy . . . . . . . 10.4.5 Dependence on Yield Strength . . . . . . . . . . . . . . . 10.4.6 Dependence on Total Hydrogen Content . . . . . . . . 10.5 General Theory of KIH . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.2 Fracture Condition of Crack Tip Hydrided Region: KI Versus Lc Relationship . . . . . . . . . . . . . . . . . . 10.5.3 Limiting Values of Lc and KIH . . . . . . . . . . . . . . . 10.5.4 KI Versus Lc Relationship: Characteristics and Comparison with Experimental Data . . . . . . . . 10.5.5 Comparison of KI Versus Lc Relationships from Different Models. . . . . . . . . . . . . . . . . . . . . 10.5.6 Limits to Crack Tip Hydrided Region Growth . . . . 10.5.7 Summary of General Limiting Conditions for DHC Initiation at Cracks . . . . . . . . . . . . . . . . 10.6 Analysis of Some Limiting Conditions for DHC . . . . . . . . . 10.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6.2 High Temperature DHC Limit: Experimental Results. . . . . . . . . . . . . . . . . . . . . . 10.6.3 High Temperature DHC Limit: Theoretical Analysis . . . . . . . . . . . . . . . . . . . . . . 10.6.4 DHC Arrest Temperature: Test Temperature Approached from Below . . . . . . . . . . . . . . . . . . . 10.6.5 DHC Limiting Conditions: Summary Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

Initiation at Volumetric Flaws . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Early Models of Blunt Flaw Assessment . . . . . . . . . . . . . . Hydride Process Zone Approach to Volumetric Flaw Assessment: General Considerations . . . . . . . . . . . . . . . . . 11.4 Hydride Process Zone Model: Closed Form Solution . . . . . 11.5 Hydride Process Zone Model: Effect of Flaw Tip Plasticity. 11.6 Engineering Process Zone Model . . . . . . . . . . . . . . . . . . . 11.7 Validation of the Engineering Process Zone Model . . . . . . 11.7.1 Flaw Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7.2 Flaw Root Radius . . . . . . . . . . . . . . . . . . . . . . . 11.7.3 Flaws with Depth Greater than 1 mm . . . . . . . . . 11.7.4 Flaw Surface Roughness and Secondary Flaw Significance . . . . . . . . . . . . . . . . . . . . . . . 11.7.5 Use of Flaw Tip Plasticity and Creep in the Engineering Process Zone Model Application . . . 11.7.6 Accuracy of the Cubic Polynomial Expression in the Engineering Process Zone Model Application . . . . . . . . . . . . . . . . . . . . . . 11.7.7 Scatter and Material Variability . . . . . . . . . . . . . 11.7.8 Hydrogen Isotope Content and Number of Reactor Cooldown/Heatup Cycles . . . . . . . . . . 11.7.9 Above Threshold Conditions . . . . . . . . . . . . . . . 11.7.10 Effect of Irradiation . . . . . . . . . . . . . . . . . . . . . 11.7.11 Cyclic Loading and Overload Conditions. . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 DHC 11.1 11.2 11.3

12 Applications to CANDU Reactors . . . . . . . . . . . . . . . . . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Overview of Assessment Approach. . . . . . . . . . . . . . . . . . 12.2.1 DHC Initiation at Planar Flaws: KIH . . . . . . . . . . 12.2.2 DHC Initiation at Volumetric Flaws: TSSD . . . . . 12.2.3 Planar Flaw Growth to the End of an Assessment Period: DHC Growth Rate . . . . . . . . . . . . . . . . . 12.2.4 Reactor Core Assessment: Leak Before Break Analysis . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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List of Abbreviations and Acronyms1

hcp fcc bcc fct bct cw wppm TEM AECL PT DOSE SANS TSS TSSP TSSPI TSSD DSC DEM IF ND DHCi CSA EB HVEMS NBS SEM AE

hexagonal close packed face centred cubic body centred cubic face centred tetragonal body centred tetragonal cold worked weight parts per million Transmission Electron Microscope Atomic Energy of Canada Limited Pressure Tube Density Of Site Energy Small Angle Neutron Scattering Terminal Solid Solubility Terminal Solid Solubility for Precipitation Terminal Solid Solubility for Precipitation Isothermal Terminal Solid Solubility for Dissolution Differential Scanning Calorimetry Dynamic Elastic Modulus Internal Friction Neutron Diffraction Delayed Hydride Cracking initiation Canadian Standards Association Electron Beam Hot Vacuum Extraction Mass Spectometry National Bureau of Standards Scanning Electron Microscope Acoustic Emission

1

The abbreviations and acronyms used and their meanings are listed in approximately the order that they appear in the text, starting with Chap. 1.

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RD TD ND FE SEN CB ID CCT CT IAEA HRR SMiRT DBCS

List of Abbreviations and Acronyms

Rolling Direction Transverse Direction Normal Direction Finite Element Single Edge Notched Cantilever Beam Inside Diameter Curved Compact Toughness Compact Toughness International Atomic Energy Agency Hutchinson-Rice-Rosengren Structural Mechanics in Reactor Technology Dugdale-Bilby-Cottrell-Swinden

Nomenclature1

KIH a, b, c (a0, b0, c0) Xchyd Zr chyd Zr V Xdhyd Zr dhyd Zr V XaZr Zr aZr Zr V Xchyd H Hchyd ; V Hh V dhyd XH Hdhyd ; V Hh V r, rH bpha

hpha T xi edhyd ¼ eT11 ¼ eT22 a

Threshold stress intensity factor for delayed hydride cracking Lattice parameters (notation in brackets for room temperature values) Atomic volume (Wigner–Seitz cell volume) of Zr in the c-hydride phase Molar volume of Zr in the c-hydride phase Atomic volume (Wigner–Seitz cell volume) of Zr in the d-hydride phase Molar volume of Zr in the d-hydride phase Atomic volume (Wigner–Seitz cell volume) of Zr in the a-Zr phase Molar volume of Zr in the a-Zr phase Atomic volume of hydrogen in the c-hydride phase Molar volume of hydrogen in the c-hydride phase Atomic volume of hydrogen in the d-hydride phase Molar volume of hydrogen in the d-hydride phase Ratio of hydrogen to metal (zirconium) atoms Ratio of hydrogen to zirconium atoms for the stoichiometric composition of the phase ‘pha’ (e.g., c- or d-hydride) ¼ rH =bpha Temperature Orthogonal coordinate axes in the i = 1, 2, 3 directions Lattice transformation strain for the formation of d-hydride from a-zirconium along the a directions of the a-Zr lattice; equivalent to the transformation strains,

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The symbols used and their meanings are listed in approximately the order that they appear in the text, starting with Chap. 1.

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edhyd ¼ eT33 c

KIc KIchyd E m E0 Hv s S1,S2

cH pH 2 KS ScH R, or kB NH NM lH loH loH lH2 =2 loH2 =2 DGoH DHHo DSoH 1 o 2 SH 2

Nomenclature

eT11 and eT22 , along the orthogonal coordinate directions, x1 and x2 Lattice transformation strain for the formation of d-hydride from a-zirconium along the c direction of the a-Zr lattice; equivalent to the transformation strain, eT33 , along the orthogonal coordinate direction, x3 Stress intensity factor for fracture toughness (unstable fracture) of the material Stress intensity factor for fracture toughness (unstable fracture limit) of bulk zirconium hydride phase material Elastic (Young’s) modulus Poisson’s ratio Elastic (Young’s) modulus in plane strain; ¼ E=ð1  mÞ Vicker’s hardness number Magnitude of shear strain in lattice transformation strain matrix for a-Zr to hydride transformation (Chap. 2) Transformation strain matrices making up the lattice transformation strains from a-Zr to hydride transformation Concentration of hydrogen in a metal lattice (any units) Partial pressure of hydrogen gas Sievert’s constant Ideal partial configurational entropy of hydrogen in a metal lattice Universal gas, or Boltzman constant Number of hydrogen atoms in the solid Number of metal atoms (M) in the solid Chemical potential of hydrogen that is dissolved in a metal A standard value of the chemical potential of hydrogen dissolved in a metal A standard value of the chemical potential of hydrogen dissolved in a metal Hydrogen chemical potential in the gas phase Chemical potential of hydrogen gas at standard pressure (0.1 MPa (1 atmosphere)) and temperature (298.15 K) Gibbs free energy change for an ideal solution of hydrogen in a metal lattice Enthalpy change for an ideal solution of hydrogen in a metal lattice Enthalpy change for an ideal solution of hydrogen in a metal lattice Entropy of dissociation of hydrogen in the standard state of the gas

Nomenclature

SoH 1 o 2 HH2

HHo lid H lEH HHE HH SEH SH b=a

b=a

rH ðNbÞ; rH cTH wb n nb , na U V Ni F G P lk Ji Lik Wk k w I V sij  V xI

xxi

Non-configurational component of entropy of the hydrogen atom in the metal lattice Energy of dissociation of hydrogen in the standard state of the hydrogen gas at 298.15C and 0.1 MPa Formation enthalpy of hydrogen in the metal lattice at infinite dilution rH ! 0 Chemical potential of hydrogen in the metal lattice in its ideal state Excess chemical potential of hydrogen in the metal  lattice ¼ lH  lid H Excessenthalpy of formation of hydrogen in the metal lattice ¼ HH  HHo Actual enthalpy of formation of hydrogen in the metal lattice Excessentropy of formation of hydrogen in the metal lattice ¼ SH  SoH Actual entropy of formation of hydrogen in the metal lattice Partitioning ratio of hydrogen between the Zr-20Nb b-Zr phase and the a-Zr phase ð¼ cZr20Nb =caZr Þ Total concentration of hydrogen in solution in the alloy Weight fraction of the b-Zr phase in an a/b Zr alloy Total weight percent of Nb in the of a/b Zr alloy Nb weight percent in the b- and a-Zr phases, respectively, of an a/b Zr alloy Total energy of the thermodynamic system Total volume of the thermodynamic system Number of component (mass) with i = 1, 2, ….. Helmholtz free energy Gibbs free energy or shear modulus, depending on the context Pressure of the thermodynamic system Chemical potential of the kth component in the thermodynamic system Diffusion flux for the ith component in a multi-component fluid Phenomenological mobility coefficients Work done per mole addition of component, k Partial molar strain energy of component, k Molar volume of component, I Elastic compliances of the solid Molar volume of the solid solution Mole fraction of component, I

xxii

ei (i = 1,…,6) ri (i = 1,…,6)

r l H V r and r0 Gik fik si(r) Pnk Cij0 P0 s1 sI

R Dv Dv1 DvI K c

XM NA kij k1 ; k2 ; k3 k0 fi(r), f

Nomenclature

The six components of the elastic strains (in the Voigt formulation) The six components of elastic stresses corresponding to the six components of elastic strains (in the Voigt formulation) Externally applied uniaxial stress (Chaps. 4 and 5) Length of the solid along the direction of applied stress (Chap. 4) Partial molar volume of hydrogen in the solid Position vectors Static Green’s function kth component of a force acting in or on a solid Displacement field produced by a defect (misfitting atom or precipitate) Dipole tensor representing the source strength of a defect (misfitting atom or precipitate) Elastic constants Isotropic dipole tensor (: Pii/3 (repeated indices imply summation)) Displacement produced by a defect (misfitting atom or precipitate) in a solid of infinite size Additional displacement produced by a defect (misfitting atom or precipitate) in a solid of finite size (‘I’ stands for ‘image’) Outer radius of a solid sphere Total volume change in a finite solid produced by the introduction of a defect Volume change in an infinite solid produced by the introduction of a defect Volume change additional to that given by Dv1 produced by the introduction of a defect in a finite solid Bulk modulus Ratio of total volume change produced by a defect in a finite solid to the volume change produced in an infinite solid ¼ Dv=Dv1 Atomic (Wigner-Seitz cell) volume of the stress-free (defect-free) metal (M) lattice Avogadro’s number k-tensor: strains produced in a solid per defect (i = 1, 2, 3) Principal components of k-tensor Isotropic strain per solute atom given by Eq. 4.47 for the isotropic (cubic) case for which k1 = k2 = k3 Source force of defect and/or externally applied surface and body forces

Nomenclature

s Eint eaij ðr0 Þ dij ph ðr0 Þ eTij ; or eTij d V

DPdij adij;kl df a ; df b dCij;kl Wd ; Ws

inh Eint rii ; rij

DPpij apij;kl Es _ ph ðcI Þ T ^ int E EsT or ET

c0 NC Eint

xxiii

Lattice displacement derived from an exact solution for a given force, f Interaction energy: energy change (work) created when one defect is moved in the presence of another defect and/or other external or internal forces Strain imposed by internal or external stresses at the location of the defect in the absence of the defect Kronecker delta function   Hydrostatic stress at r0 ¼ raii ðr0 Þ=3 for an applied stress of raii ðr0 Þ Stress-free misfit (transformation or eigen) strains of lattice defect or misfitting precipitate where i, j are coordinate directions Molar volume of the defect site in a perfect, unstressed crystal lattice; in the case of hydrogen in a-zirconium, aZr Zr this is given by V or XaZr Zr Induced diaelastic dipole tensor change (¼ Pij ðea Þ Pij ð0Þ) produced by an imposed strain from another source of stress Diaelastic polarizability (fourth-ranked tensor having the symmetry of the elastic constant tensor) Changes (induced) on defect source forces of defects identified by ‘a’ and ‘b’ Change in elastic constants when adding a small concentration of atoms to the solid Interaction energy densities for a single defect as a result of, respectively, the dilatational and the deviatoric part of the induced strain Inhomogeneity interaction energy Dilatational and deviatoric ði 6¼ jÞ components, respectively, of the stress tensor Induced paraelastic dipole tensor (¼ Pij ðea Þ  Pij ð0Þ) Paraelastic polarizability tensor (fourth ranked) Self strain energy of a defect Average hydrostatic stress produced by a concentration, cI , of interstitial atoms in a finite lattice Average total elastic image interaction energy per defect Total misfit strain (interaction) energy of a collection of defects in the solid Defect volume ratio defined in terms of a similar factor, c, given by Eshelby (¼ ð1  1=cÞ Non-configurational average interaction energy per defect pair

xxiv

Nomenclature

ENC

Total non-configurational energy ENC when there are N defects (interstitials or solute atoms) in the host lattice Non-configurational component of the chemical potential of hydrogen in a lattice containing N hydrogen (defect) atoms Normalized number of sites in a given interval of energy between E and E þ dE Thermal occupancy distribution of energy level, Ei Lattice site energy in a perfect lattice containing no defects Volume fraction of b phase (Chap. 3); fraction of defects in the solid (Chap. 4); correlation factor for an atom jump in a lattice (Chap. 5) Concentration of hydrogen atoms in a lattice not bound to lattice defects Magnitude of the Burger’s vector of a dislocation Cylindrical coordinates defined in Fig. 4.3 Isobaric stress locus centred on the tensile region of the dislocation core forming the boundary inside of which hydride is expected to be formed Hydride formation enthalpy Average hydrogen to metal atom ratio locked up in hydrides at the dislocation cores Ratio of hydrogen to H atoms in the hydride (Eq. 4.101) Dislocation density in the crystal Total number of hydrogen atoms per unit volume inside a cylinder of unit height and radius, Rhyd Number of metal (Pd) sites per unit volume in the hydride Solvus composition of hydrogen in a metal Total hydrogen content in the solid Cartesian components of the thermodynamic forces appearing in the entropy production rate expression Entropy production rate Indices referring to heat flux and particle flux, respectively (Chap. 5) Partial specific enthalpy of component k Constrained force conjugal to Jk ; ( rðlk  ln ÞT ) Heat of transfer of component, k Reduced heat of transfer obtained from thermal diffusion experiments n P hk Jk Heat flux defined by: Jq 

lNC H ðN Þ nð E Þ oðEi Þ Eo f

cfrH b r; H Rhyd

Ehyd rHhyd ðdislocÞ a qd NHhyd hyd NPd s cH ctotal H Xi

DS_ q; k Fk ðXk ÞT Qk Q0 k Jq0

k¼1

aH ; c H

Activity and activity coefficient, respectively, of hydrogen in a solid

Nomenclature

Dchem rm D C ‘ d Qo Do a0 , c 0 md , mp a=b

DH DaH DbH db=a

DT cT ðx; tÞ s a, s b ta,, tb tt and tr Q c0H dS, deS

diS dQ0

Wirrev , Wrev , Wloss

Tf , Td , Teq

xxv

Chemical diffusion coefficient Hydrostatic (mean) stress Tracer diffusion coefficient Jump frequency of an atom in a lattice Jump distance of an atom in a lattice Dimensionality of the lattice Activation energy of diffusion in a reference (generally perfect) lattice Tracer diffusion coefficient in a reference (generally perfect) lattice with sites of energy Eo Activity and activity coefficients, respectively, for a reference (generally perfect) lattice Isotopic mass of deuterium and protium, respectively Diffusion coefficient of hydrogen in an a/b Zr alloy Diffusion coefficient of hydrogen in a Zr Diffusion coefficient of hydrogen in b Zr Ratio of the thickness of the b–grains to that of the a–grains in an a/b Zr alloy Tritium tracer diffusion coefficient Tritium concentration at a distance, x, and time, t Total times a hydrogen atom spends in the respective a and b phases Total times to corresponding sa and sb in these phases resulting in net diffusion Volume fractions of the b phase aligned in the transverse and radial pressure tube directions, respectively Heat of transport; equivalent to the phenomenological constant, Q0H  , defined by Eq. 5.15 Hydrogen concentration in the bulk of the material away from regions of elevated stress or reduced temperature Entropy changes of the system and of an associated quantity of heat transferred to or from a surrounding heat bath, respectively Internal entropy production change (¼ dS  de S) Change in ‘‘uncompensated heat’’ that is not detected by heat transfer and corresponding changes to the surroundings Work terms representing irreversible and reversible work, respectively, the difference between which results in work lost (Wloss ) Temperatures of formation, and dissolution, respectively, of a second phase that results in irreversible work. Teq is the temperature at which the phase transformation occurs if it could be done reversibly

xxvi

Nomenclature

Tcool , Theat

, cs;heat cs;cool H H 1 or a, b or cs;f H 2 s;d (and cs;f H ), cH cs;eq H Tav

DHplat DHðcsH Þ *

(

pf , pd or p, p *

peq , ~ p

DHsolv or DHeq

cs;incoh ð½a þ d=dÞ H

cs;incoh ða=½a þ dÞ H

Temperatures at which the (e.g., hydride) phase transformation starts or terminates during cooling and heating, respectively, at a given constant total (hydrogen) composition; these temperatures are equivalent to Tf and Td , respectively Solvus compositions for first formation and last dissolution of a second (e.g., hydride) phase during cooling and heating, respectively, corresponding to Tcool and Theat ; compositions with superscripts f1 and f2 represent upper and lower bounds over which these can range Solvus composition for first formation or last dissolution of a second (e.g., hydride) phase if there were no hysteresis in the phase transformation Average temperature between second (e.g., hydride) phase formation and dissolution temperatures that serves as an estimate of the physically unrealizable equilibrium phase transformation temperature, Teq Plateau enthalpy for solution of hydrogen in a metal lattice such as a-Zr Enthalpy for solution of hydrogen in a metal lattice, e.g., a-Zr Plateau pressures for isothermally adding or removing hydrogen, respectively, during external gaseous hydrogenation. Equilibrium and average pressures, respectively, during the plateau stage of the phase transformation (direction of arrow is not significant). The average pressure is an approximate estimate of the equilibrium pressure for phase transformations exhibiting hysteresis Enthalpies for the phase transformation that are unaffected by hysteresis (i.e., it consists only of the chemical energy of bonding when hydride is formed from hydrogen within the solid. Experimentally, this enthalpy can be obtained from calorimetry Composition for formation or dissolution of the a from the d phase on the dilute a phase side of the phase diagram if this were to occur incoherently in which case this composition is the true equilibrium phase transformation composition Composition for formation of the d from the a phase on the dilute d phase side of the phase diagram when this occurs incoherently, in which case this composition is the true equilibrium phase transformation composition

Nomenclature p DHtotal

p p DHa!b , DHb!a

D wel inc e DHa!b

z b, z a a ls;eq H ðTeq ; r Þ

2 a ls;f H ðTf 2 ; r Þ

1 a ls;f H ðTf 1 ; r Þ

a ls;d H ðTd ; r Þ

 ee Dw inc

 pe Dw inc  es Dw inc  ps Dw inc

xxvii

Total unrecoverable plastic work (enthalpy) which is lost during the formation and dissolution stages of the hysteresis cycle Unrecoverable plastic work (enthalpy) which is lost during the formation and dissolution stages, respectively, of the hysteresis cycle Total pure elastic strain energy for misfitting second (hydride) phase formation Elastic strain energy component produced by the formation of a misfitting second (hydride) phase resulting in both elastic and plastic work Phase fraction of b hydride phase formed from the a phase containing a dilute concentration of hydrogen in solution and vice versa Equilibrium chemical potential for hydride formation or dissolution at the solvus composition (temperature) under an external stress, ra Chemical potential for hydride formation at the solvus temperature—including the effect of an external stress, ra—affected by hysteresis that experimentally represents the upper-bound for solvus formation designated as TSSP2 Chemical potential for hydride formation at the solvus temperature—including the effect of an external stress, ra—affected by hysteresis that experimentally represents the lower-bound value for solvus formation designated as TSSP1 Chemical potential for hydride dissolution at the solvus temperature, including the effect of an external stress, ra, affected by hysteresis that experimentally represents the solvus formation temperature designated as TSSD Molar or Partial molar (without or with bar, respectively) elastic hydride-matrix coherency energy difference during hydride formation (expansion) for the case where some of the misfit strain has been plastically relaxed Molar or partial molar (without or with bar, respectively) plastic work done during hydride formation (expansion) stage Molar or partial molar (without or with bar, respectively) elastic hydride-matrix accommodation energy change during hydride dissolution (shrinkage) Molar or partial molar (without or with bar, respectively) plastic work done during hydride dissolution (shrinkage)

xxviii

 int Dw inc N NHa , NHb Na, Nb N ¼ Na þ Nb

caH ; cbH

zb , za   DGel zb W Gcoh gachem , gbchem ^cH

L Ka, Kb

bo cao H , cH bc cac H , cH T Es;incoh

Esa=b T Es;coh

Gaincoh , Gbincoh A

Nomenclature

Partial molar interaction energy of a defect (misfitting atom (H) or precipitate (hydride)) with a source of stress that is external to it Total number of hydrogen atoms in the solid (Chap. 7) Total numbers of hydrogen atoms in the a and b phases, respectively Total numbers of equivalent interstitial lattice sites in the a and b phases, respectively Total number of equivalent interstitial sites in the solid, which for this case of an interstitial solution, is the total number of interstitial sites that could, for a given phase, be occupied by hydrogen atoms Fractional interstitial (H) site concentrations (¼ NHa =N a ; ¼ NHb =N b ). These concentrations are equivalent to h ¼ rH =bpha Phase fractions of b and a phases, respectively (¼ NHb =N; ¼ NHa =N) Total elastic coherency energy for a two-phase mixture ¼ D wel inc (in Eq. 7.1) Total Gibbs free energy of the coherent solid Molar chemical Gibbs free energies of the a and b phases, respectively, in their single-phase states Average hydrogen concentration in the solid (¼ NH =N) where NH is the total number of hydrogen atoms in the solid Lagrangian function Constants of proportionality between the chemical Gibbs free energies of the a and b phases, respectively and their variations with composition from their equilibrium values (Chap. 7) Compositions for incoherent equilibrium of the a and b phases, respectively Compositions for coherent equilibrium of the a and b phases, respectively Total strain energy of the incoherent (i.e. elastically decoupled, unconstrained) phase mixture Strain energy caused by the elastic constraint of the b phase when the a and b phases are coherently in contact with each other Total strain energy of the coherent two-phase mixture Gibbs free energies per host atom of the a and b single phase states A constant defined by Eq. 7.28

Nomenclature

  gchem P; T; caH ; cbH ; zb zbc Y a, Y b W K, d, f

G, C , v q00 lrH Xð^cH ; lrH Þ cb;end H

Qd!a mH  d!a DH X DSd!a ðAÞ

 DH d!a

xxix

Total chemical Gibbs free energy of the coherent or incoherent two-phase mixture Lever law relationship for the phase fraction of the coherent b phase given by Eq. 7.32 Dimensionless phase compositions; see eqs. 7.44 and 7.45 for their definitions Dimensionless average composition (Chaps. 7 and 8 only); see Eq. 7.46for its definition Dimensionless parameters used in phase transformation stability analyses by Johnson and Voorhees ([12], Chap. 7); see Eqs. 7.47 to 7.51 for their definitions. Physically these parameters represent, respectively, the ratios of the coherency energy to the chemical driving force, the degree of asymmetry of the variation with composition of the elastic moduli and of the chemical free energies of the two single phase states, a and b. Note that Ga is the shear modulus of the a phase, elsewhere defined as, G. The latter parameter is here (and in other chapters) also used for the Gibbs free energy. The meaning of this parameter in any particular case should be clear from the context Parameters used in phase stability analyses defined by Eqs. 7.50 to 7.52, respectively Number of lattice points per unit volume in the reference state, (External) hydrogen chemical potential of the reservoir Grand thermodynamic potential of a thermodynamic system consisting of a solid surrounded by a reservoir of hydrogen gas that has a chemical potential of lrH Hydrogen composition of the b phase upon complete conversion by hydrogenation from an external source of hydrogen gas at fixed chemical potential, lrH , starting from the a single phase state Solvus dissolution heat determined by numeric integration of the DSC dqðtÞ=dt versus time curve (Chap. 8) Total number of moles of hydrogen in the sample (Chap. 8) Solvus dissolution enthalpy (affected by hysteresis) Dimensionless ratio of the total mass of hydrogen to the total mass of Zr in the solid Entropy of hydride dissolution at the solvus excluding the entropy of mixing Normalization factor for Qd?a obtained from a van’t Hoff plot of the solvus

xxx

k rhf rN , ehyd N ðnÞ

Epl

rys Lave d-spacing ‘t

pc

KI T1 to T6 D cD H , cH

DH lD H lBH cBH Hh V r = ‘ or L EL and E‘

UðL; ‘Þ qhyd

Nomenclature

Ratio of the semi-minor (c) to semi-major (a) axes of a spheroidal precipitate (Chap. 9) Fracture strength of a hydride cluster embedded in a zirconium alloy Applied net section stress and plastic strain in the tensile stress direction in the matrix at the location of those hydrides first detected to crack in the specimen (Chap. 9) ‘‘Plastic equivalent’’ Young’s modulus where n refers to the axiality of the stress state (n = 1 to 3) Uniaxial yield strength of the matrix Average length of an embedded hydride cluster in its longest direction Distance between equivalent lattice planes Stress transfer length of a precipitate platelet elongated in the applied stress direction; beyond this length there is no further increase in the induced stress for a given value of applied stress A strength parameter used in the hydride process zone model for blunt flaws. It is the fracture strength of an infinitely long hydrided region grown from a nominally smooth, planar surface Mode I stress intensity factor of a planar flaw Temperatures representing different DHC rate limits, as defined in Fig. 10.14 (Chap. 10) Number of hydrogen atoms per unit volume and atom fraction, respectively, free to diffuse in a-Zr Chemical diffusion coefficient of hydrogen in a-Zr (Chaps. 5 and 10) Chemical potential of the hydrogen in the lattice that is free to diffuse Chemical potential of hydrogen at a boundary where hydrogen can be added or removed Concentration of hydrogen at a boundary where hydrogen can be added or removed Partial molar volume of hydrogen in a particular hydride phase (as defined in Chap. 2) Locations at which a fixed value of diffusible hydrogen is assumed; see Fig. 10.15 for their definition Expressions in the solution of the steady-state diffusion equation for hydrogen diffusion to a crack tip; see Eqs. 10.15 and 10.16 for their definitions Integral in the expression for hydrogen diffusion to a crack tip, as defined in Eq. 10.14 Hydride density

Nomenclature

thyd a raeff rh rlocal Lc

1 KIH

a LPZ d 1 KIH ðembeddedÞ 1 ðtotalÞ KIH  eff  eff 1 KIH KIH [ KIH Lmax TDAT

TA

TC

vT

pH

xxxi

Thickness of hydrided region at the crack tip Length of macro-crack or flaw (Chaps. 10 and 11, respectively) Crack tip effective stress: stress normal to the crack plane acting on the region around the crack tip in the absence of any hydrides Crack plane normal stress in the embedded hydride cluster produced by its transformation strains alone Net normal stress inside the hydride resulting from the sum of raeff and rh Critical length of a platelet-shaped hydride cluster length at which the net normal stress inside the hydride cluster equals, at some location along its length, the embedded hydride’s fracture strength Theoretical lower limit for the stress intensity factor for DHC of a crack assuming that the hydride cluster’s length grown from the crack tip had extended to infinity A constant depending on the shape of the hydrided region grown at the crack tip (Chap. 10) Length of plastic zone along the crack plane Crack tip opening displacement (Chap. 10) 1 KIH given by Eq. 10.26 1 The sum of KIH ðembeddedÞ and KIchyd Stress intensity limit for DHC initiation resulting as a result of time or diffusion limited conditions Maximum calculated length to which a hydride cluster at a crack can grow (see Eq. 10.32) Limit temperature (equivalent to T6 in Fig. 10.14) at which there starts a sharp reduction in the linear, Arrhenius-type increase with decreasing inverse temperature of the DHC growth rate DHC growth rate arrest temperature (equivalent to T5 in Fig. 10.14) which applies when approaching the test temperature from below DHC growth rate starting temperature (equivalent to T5 in Fig. 10.14), which applies when approaching the test temperature from a sufficiently high temperature from above at which all hydrides in the bulk and at the crack tip to have been dissolved Process zone model crack tip opening; for hydride process zone model it is the relative flaw root displacement produced by the hydrided region alone Cohesive strength of the hydrided region as used in the hydride process zone model

xxxii

vc

rp rTH or rnTH ( rp )

s b kt q ¼ b2 =a w KEFF

KTH

vy

vT ðeÞ vT ðepÞ

KTH ðepÞ

CV

Nomenclature

Critical hydride process zone model flaw tip opening displacement produced by a hydrided region growing from the root of the flaw Peak normal stress at the flaw root Minimum normal stress at vT ¼ 0 imposed by the external nominal load, rn , for which the cohesive strength, pH , of the hydrided region reaches its critical values, vc and pc (see Fig. 11.4 for a schematic illustration) Length of process zone (Chap. 11) A volumetric flaw dimension as defined in Fig. 11.5 (Chap. 11) Volumetric flaw-tip elastic stress concentration factor Flaw root radius for the flaw shown in Fig. 11.5 Non-dimensional parameter defined by Eq. 11.8 Effective stress intensity factor for the blunt flaw assuming the crack to have the same planar dimensions as that of the flaw Applied threshold stress intensity for fracture of the hydrided region at a flaw; this threshold parameter is most suitable for a flaw with small root radius (Eq. 11.14) Flaw root opening displacement of a volumetric flaw with a plastically deformed process zone of uniform critical yield stress, ryc Elastic flaw root opening displacement Elastic plastic flaw root opening displacement of a volumetric flaw (defined by Eq. 11.16 in terms of vT ðeÞ and vy ) Applied threshold stress intensity for fracture of the hydrided region at a flaw when also accounting for crack tip plasticity An empirical correction factor (= 1.15) dividing vc for use in the elastic plastic Engineering Process Zone model to account for plasticity effects implicitly contained in KIH but not accounted for in the derivation of the hydride process zone model

Chapter 1

Introduction

1.1 Overview Zirconium alloys are used in nuclear reactors because of their combination of high strength, high corrosion resistance, and low neutron absorption cross-section. Their most demanding applications in nuclear reactors are as fuel cladding and in CANDU,1 RBMK, and other Pressurized Heavy Water (PHW) reactors as pressure tubes containing the fuel bundles. It is important for the safe and economic operation of these reactors that these components maintain their integrity throughout their design life. However, during their residence in the reactor these components are subject to aging mechanisms resulting from thermal- and pressuredriven changes, fast neutron bombardment, and corrosion at the water/metal interface, the latter resulting in a small fraction of the released hydrogen produced during the corrosion reaction being absorbed in the zirconium alloy. When the hydrogen concentration in the material exceeds the Zr–H solvus composition, zirconium hydrides are formed. These hydrides, which are less ductile than the surrounding metal matrix, can have deleterious effects on the mechanical properties of these components when present at sufficiently high volume fraction. Their deleterious effects are exacerbated by increases in yield strength and decreases in fracture toughness of the zirconium material. These changes are produced as a consequence of the production of dislocation loops and other microstructural changes during fast neutron bombardment in the reactor core. There were some early indications that at relatively low hydrogen content in the material there could exist a time-dependent mechanism of cracking (now called Delayed Hydride Cracking (DHC)) involving localized increase in hydrogen concentration at stress concentrators. This localized increase in hydrogen can lead to the formation of a dense hydrided region, causing subcritical crack initiation and

1

CANDU, CANada Deuterium Uranium, is a trademark of Atomic Energy of Canada, Ltd (AECL). M. P. Puls, The Effect of Hydrogen and Hydrides on the Integrity of Zirconium Alloy Components, Engineering Materials, DOI: 10.1007/978-1-4471-4195-2_1, Ó Springer-Verlag London 2012

1

2

1 Introduction

propagation. However, it was not until the discovery that this DHC initiation and growth mechanism was the cause of leaks in some pressure tubes in two of the earliest operating CANDU reactors that it was realized that this mechanism could potentially pose a greater threat to pressure tube integrity than the reduction in overall fracture toughness produced by the presence of hydride precipitates uniformly distributed throughout the material. As the early work had shown, the latter embrittlement mechanism might be of concern only near end of life of the pressure tubes when there could be sufficient increase in hydrogen in the alloy so that hydrides would be present during normal reactor operating temperatures. As this increase in hydrogen continues there would then be a threshold level of hydride volume fraction at which there is a fracture toughness reduction of the material as a result of these hydride precipitates that is unacceptable. This threshold level of hydrogen volume fraction for unacceptable fracture toughness reduction would be lower when the platelet-shaped hydrides are oriented in a direction that is not their usual one in pressure tube material, but this ‘‘reorientation’’ of the hydrides is generally not likely under normal reactor operating conditions. Only limited studies, sufficient for present reactor circumstances, have been carried out on this topic and these will not be considered in this book since the focus here, as discussed in more detail in the following, is on DHC. For zirconium alloy fuel cladding, on the other hand, the main concern for their integrity generally is the effect that the presence of high volume fractions of hydride precipitates have. This concern exists more for fuel cladding in Pressurized Water and Boiling Water Reactors (PWR and BWR, respectively) than it does in fuel cladding of RBMK, CANDU, and other PHW reactors, since the former operate at higher temperatures and the fuel resides longer in the reactor, the net effect of which is a greater pickup of hydrogen over the lifetime of the component compared to in the fuel cladding of RBMK, CANDU, and other PHW reactors. However, in recent years, it has been suspected that DHC may also have been a factor in some fuel cladding failures observed in the former reactors. This possibility and the need to predict the integrity of fuel cladding of spent fuel under long-term storage conditions has led to studies to determine whether DHC may be a failure mechanism under some reactor operating conditions and under the lowertemperature long-term storage conditions of the fuel, respectively. These two considerations have resulted in increased attention being paid by the nuclear research community to the study of DHC in zirconium alloy fuel cladding. In this book, emphasis is placed on the experimental and theoretical studies that have been carried out to understand and model the behavior of DHC in pressure tubes of CANDU reactors. Focus on the behavior of this phenomenon is mostly on pressure tube material because by far the greatest amount of effort has been expended by researchers on this material and, therefore, a very large and coherent body of work has been produced from which, it is shown in this book, one can develop a fairly comprehensive picture not only of the DHC phenomenon in pressure tubes, but also of the effect that hydrogen and hydrides have, overall, on the mechanical behavior of other zirconium alloys and components used in nuclear reactors or for other industrial applications.

1.2 Delayed Hydride Cracking

3

1.2 Delayed Hydride Cracking DHC is a subcritical crack growth mechanism in which hydrogen in solution in a component under tensile stress migrates from locations of lower to higher tensile stress. If the increase in hydrogen in the region at higher tensile stress is sufficient to exceed the solvus for hydride precipitation, hydrides will form and the resultant hydrided region could grow to a size and shape at which this localized hydrided region fractures, resulting in the formation of a crack that would then be able to continue to propagate in repeated steps by the same mechanism. The experimental evidence of DHC clearly indicates that fracture initiation occurs at localized regions containing a large volume fraction of hydrides. The fracture of such regions is evidently the result of the brittleness of these hydride precipitates. Thus it would appear, at first glance, to be a phenomenon much easier to understand (and hence, model) than similar subcritical crack growth processes observed in non-hydride forming metals susceptible to the deleterious effects of hydrogen absorbed in the material. In these latter materials, relatively small elevations of hydrogen in solution locally in the material can cause the initiation of subcritical crack propagation. The subcritical crack growth process initiated in these metals as a result of this local elevation of hydrogen in solution shares some similarities to DHC, the reason being that both are processes driven by hydrogen diffusion in a tensile stress gradient. On the other hand, understanding the underlying causes of hydrogen-induced subcritical crack propagation in the latter metals has been much more difficult, as seen by the many conferences and papers that have been and continue to be devoted to this topic over the years. A closer look at the DHC process shows, however, that even in this seemingly obvious case it is not entirely clear why and how fracture of a localized hydrided region under high tensile stresses occurs since the hydrides have positive transformation strains that create large compressive stresses inside them that could potentially shield them from any externally applied, high local tensile stresses. It is, in fact, the existence of these large transformation strains that is the cause of the considerable complexity of DHC. This has provided major challenges in measuring, understanding, and modeling DHC. Understanding and quantifying this complexity has required input from parallel advances ongoing at the time in many other fields, such as studies of the thermodynamics of phase transformations of metal–hydrogen systems, phase equilibrium of coherent crystalline metallic solids, the physics of point and line defects, diffusion of substitutional and interstitial atoms in crystalline solids, and continuum fracture and solid mechanics. The advances being made in these fields at the same time as the research on DHC was ongoing were important in providing many of the underlying theoretical frameworks and solution templates for the development of models and experimental methods to explain and quantify the process of DHC initiation and growth. It is one of the objectives of this book to show in more detail, while also providing a more cohesive account than can be obtained through a reading of the literature, the origins of the theoretical concepts and constructs that form the bases of our understanding and theoretical developments of the

4

1 Introduction

DHC process. In doing so, advantage has also been taken in this book of the overall advances that have been made up to the present day in all of these fields, both in theory and experimental observations, to update our present understanding and the theoretical models of DHC.

1.3 Objectives and Outline An important aspect of the study of DHC is in understanding and predicting its potential effect on the integrity and design life of components used in nuclear reactors susceptible to this phenomenon. The complexity of commercial zirconium alloys used in nuclear reactors, the lack of control over, and precise knowledge of some of the environmental conditions under which such alloys and components are manufactured and used, plus the commercial and safety implications connected with the reliable predictions of the performance of these components, has dictated that studies of the effect of hydrogen and hydrides focus on experiments and theoretical developments that mimic as closely as possible the exact materials and ranges of conditions existing in nuclear reactors. For practical reasons, it is rarely possible to carry out such engineering oriented studies on full-sized components or even parts of full-sized components in sufficient numbers to meet the statistical requirements needed to account for the observed variability in the behavior of these complex materials. As a result, most tests are carried out using small specimens, usually manufactured with material taken from actual reactor components, either pre- or ex-service, and tested under accelerated time and environmental conditions. To gauge the possible conservatism of the results of such tests, these were generally supplemented by others that focussed on the mechanistic aspects of the phenomenon through studies of the effects of hydrogen and hydrides at size scales ranging from the micro to the atomic level, supported by theoretical models. It is, therefore, another purpose of this book to show how these two types of approaches have supplemented and supported each other. A final motivation for this book has been this writer’s conviction—as someone who has been involved in and contributed to this field from the start of its discovery—that the time is well past due for a coherent account that documents and interprets the knowledge that has been collected over the almost 40 years of study of this subject. This book is organized as follows. Chapter 2 provides details on the crystallography and mechanical (including fracture) properties of bulk zirconium hydrides. The chapter highlights a conclusion by Beck concerning the role of the c -hydride phase as a suppository for excess hydrogen atoms when thermodynamic conditions drive the d-hydride phase to reduce its hydrogen composition. Chapter 3 deals with the morphology, habits, and orientation relationships of single, or groups of hydride precipitates in zirconium and its alloys in stressed and unstressed material. Chapter 4 is on the properties of hydrogen atoms dissolved in the a- and b-zirconium phases, including their interactions with each other and with dislocations. Chapter 5 deals with the diffusion properties of hydrogen in zirconium as

1.3 Objectives and Outline

5

driven by mass, stress, and thermal gradients and the effect that fast diffusing channels such as the b-zirconium phase in a/b zirconium alloys plus large numbers of strong trapping sites such as dislocation cores have on, respectively, speeding up or slowing down hydrogen diffusion. Chapters 6–8 deal with various aspects of the phase relationships between a-Zr and hydride phases. Chapter 6 deals with the dilute side of the phase diagram (referred to as the solvus). This chapter contains a review of the causes of hysteresis in first order phase transitions generally, but particularly those most relevant for interpreting the solvus relationships of metal– hydrogen systems. This chapter and the next two present some hitherto unexplored ideas as regards the Zr–H system concerning the physical origin of hysteresis in such systems. A general treatment of phase equilibrium in coherent solids is provided in Chap. 7 showing how such systems differ in their equilibrium behavior from those derived from the usual Gibbsian thermodynamic conditions. The ideas and concepts provided in Chaps. 6 and 7 are then quantitatively examined in Chap. 8 through their application to the Zr–H system. This chapter also gives a thorough review and critical assessment of experimental results of solvus relationships in zirconium alloys. Chapter 9 deals with the fracture strength of embedded hydride precipitates. The information concerning the conditions for fracture of embedded hydrided precipitates have been seminal in providing important insights in understanding DHC initiation and propagation, which is dealt with in Chap. 10. The overall objective of this chapter is to provide a comprehensive examination of experimentally determined DHC behavior in relation to the theoretical models developed to rationalize this phenomenon. Chapter 11 deals with the special case of DHC initiation at volumetric flaws, which requires a different treatment than is appropriate for DHC initiation at cracks. Finally, Chap. 12 gives examples of how the present data base and understanding of the behavior of hydrogen and hydrides in zirconium alloys is used in integrity assessments of pressure tubes of CANDU PHW reactors. In the process of collecting and critically reviewing the extant experimental data and theoretical models, including those developed, or codeveloped, by the author, new ways of understanding and describing these results have occurred to the author, which have been documented in this book. Hence, this book contains numerous instances where the existing experimental information and associated theoretical models given in the extant scientific literature have been reinterpreted in the light of the present state of knowledge. As well, this book contains some additional development not previously given in the published literature of theoretical models to which this author has contributed in the past. The largest of these new contributions are on the theoretical interpretation of solvus hysteresis (Chaps. 6–8) and on the threshold stress intensity factor for DHC, KIH (Chap. 10).

Chapter 2

Properties of Bulk Zirconium Hydrides

2.1 Hydride Phase Compositions, Lattice Structure and Parameter Determinations Bulk zirconium hydrides refer to hydrides of macroscopic dimensions in which the zirconium starting material has been completely converted to a single or a mixture of zirconium hydride phases. The need for the study of such bulk material is particularly important in the determination of lattice parameters to ensure that these are obtained for elastically unconstrained material and are, thus, representative of the stress-free state of the lattice. It turns out that there are large differences between the unconstrained lattice parameters of the parent a-Zr phase and a given hydride phase as well as between different hydride phases. These large differences can result in significant coherency stresses in and around hydride precipitates. Such coherency stresses have important effects on phase relationships. There have been various theoretical interpretations of the effects of these coherency strains on hydrogen solubility in the a-Zr phase at the a/(a ? d) solvus boundary (see Chap. 6). These interpretations have been used to rationalize the experimental observation that hydrogen solubility at this phase boundary is history dependent and differs substantially depending on whether the state of the system is one of formation or dissolution of the hydrides. Not quantitatively assessed for metal-hydrogen and particularly the Zr-H system before this publication is that these coherency effects manifest themselves also in differences in the (a ? d)/ d phase boundary composition, even when hydrogen absorption is carried out to complete conversion between parent and solid hydride phase and all of the coherency stress effects have disappeared. The reason is discussed in detail in Chap. 7. The constraint effects resulting from the large lattice mismatches between phases create large stresses in the hydride and matrix, well exceeding the elastic limits of the materials. These high stresses can be relieved through internal microscopic relaxation mechanisms such as those of thermal diffusion of the Zr

M. P. Puls, The Effect of Hydrogen and Hydrides on the Integrity of Zirconium Alloy Components, Engineering Materials, DOI: 10.1007/978-1-4471-4195-2_2,  Springer-Verlag London 2012

7

8

2 Properties of Bulk Zirconium Hydrides

atoms at the hydride-metal interface and/or of plastic deformation of the hydride and metal phases. If this is not the case, cracks or pores could form during the formation (or dissolution) stages of the hydrides. This could occur if the rate of hydrogen ingress and/or of cooling of the specimen is too rapid compared to the rate at which the foregoing internal relaxation processes would be able to relieve the associated internal stresses. These rather complex aspects of hydride-metal ‘‘equilibrium’’ are discussed in more detail in Chaps. 6–8 dealing with the a/(a ? d) and (a ? d)/d solvus relations where it will be shown that in this type of system, at temperatures where full or partial coherency stresses are generated and maintained between any two phases, only metastable equilibrium states are possible and these metastable states have compositions that are different from those expected if true thermodynamic equilibrium could have been achieved. Hence, the phase boundary compositions at the (a ? d)/d solvus given in this chapter—since they are taken from early published literature on this topic when workers were unaware of these hysteresis effects—do not explicitly account for any possible effect of hysteresis on their compositions. Studies of the phase constitution of the Zr-H system have identified three zirconium hydride phases, c, d and e, differing in their crystallography and hydrogen composition. These phases have, respectively, nominal stoichiometric compositions of ZrH, ZrH1.5 and ZrH2. It seems, however, that only the c-hydride phase is truly stoichiometric, having a one-to-one ratio of hydrogen to zirconium atoms while the other two hydrides can exist over a range of hydrogen compositions.

2.1.1 Crystallographic Properties of the c-Hydride Phase Most measurements have indicated that c-hydride has an ordered, tetragonal structure (space group P42/n) with room temperature lattice parameters, a0 = 4.596 Å and c0 = 4.969 Å, resulting in a unit cell volume, a2c, of 103.72 Å3 [24]. Since there are four Zr atoms in a unit cell this results in an atomic volume c Zr (Wigner–Seitz cell volume), XcZr or V , of 25.93 Å3/(atom Zr-c) or 15.62 9 10-6 3 m /(mol Zr–c), respectively. The hydrogen atoms in this structure occupy tetrahedral sites on alternate (1 1 0) planes. The Zr atoms are found at the (1/4 1/4 1/4) sites while the hydrogen atoms occupy the (0 0 0) or (0 0 1/2) sites. Results from inelastic neutron scattering studies of ZrD show that the c-phase structure actually has a small orthorhombic distortion, space group Cccm (No. 66) with a0 = 4.549 Å, b0 = 4.618 Å, and c0 = 4.965 Å [16]. The orthorhombic structure of c-ZrH(D) is consistent with that of c-TiH(D) obtained previously [2, 17, 19, 21]. The difference between the lattice parameters, a and b, is, however, small (1.5 %) and, as Kolesnikov et al. [16] point out, this small difference would not be detectable by conventional neutron diffractometers. It should also be noted that these results were obtained in specimens having an overall composition of ZrD0.28 consisting (according to the neutron scattering results) of a-, c-, and d-hydride and compositions given as follows:

2.1 Hydride Phase Compositions, Lattice Structure and Parameter Determinations

ZrD0:28 $ 0:718ða-ZrD0:001 Þ þ 0:269ðc-ZrD0:98 Þ þ 0:013ðd-ZrD1:2 Þ

9

ð2:1Þ

From the proportions and compositions of the phases in the specimen given by Eq. 2.1 it is evident that the proportion of the d-phase present is very small. However, given the much larger amount of a-Zr phase present compared to the c-phase and the very small amount of the d-phase, one can surmise that the c-phase would have been dispersed as precipitates in the a phase (unfortunately no metallographic evidence was provided to show the structure and orientation of these precipitates). This means that the c precipitates would be under some compression, possibly decreasing their lattice parameters relative to their values at zero stress (but not necessarily all of them equally). The magnitudes of the three orthogonal normal stresses in the hydride precipitates would depend on their shapes and transformation strains. It is shown in Chap. 3 that c-hydride formation in the a-Zr phase likely occurs via an invariant plane strain transformation. This implies that the net transformation involves only a dilatation normal to the invariant habit plane (except for possibly also a small uniform dilatation in the plane) plus shears parallel to this plane. The precipitate shape of minimum energy for such a transformation would be a plate-shaped precipitate having a small thickness-to-length ratio. For such a hydride shape and transformation strains, the stress in the plate-normal direction, c, would be negligible along most of the length of the plate-shaped precipitate. However, because of the considerable dilatational transformation strain of *0.12 in that direction, the Poisson’s contraction would generate large compressive stresses in the in-plate directions, a and b, of these c-hydride precipitates. The reduction in lattice spacing as a result of these compressive stresses may not be equal if the plate has dimensions that are different in the a and b direction, which may account for Kolesnikov et als’ finding of orthorhombic symmetry for the embedded c-hydride precipitates. Because of the foregoing concerns regarding the experimental results of Kolesnikov et al. and the small differences between the a and b lattice parameter values, we will continue to refer to the lattice structure of the c-hydride phase as being face centred tetragonal (fct) in the following. It should be noted that the foregoing considerations could also apply to the lattice parameter determinations of the c-hydride phase made by other workers, since this phase is almost always found in an embedded state, as discussed in more detail further on. However, using powder X-ray diffraction techniques the low penetrating power of the X-rays compared to that of neutrons, plus the use of powder specimens, which means that any c-hydride precipitates would not be far from a free surface, would mitigate these lattice constraint effects because the lattice spacing of only those c-hydride precipitates that are close to the surface would be measured, for which the constraint effects as a result of their embedment would be minimized. Before continuing with a description of the lattice properties of the c-hydride phase, it is convenient for the treatment of their thermodynamic properties further on to define a concentration variable, hpha ¼ rH =bpha , where bpha is the number of available (or preferred) interstitial sites per Zr atom sites in the phase ‘‘pha’’ and rH is the ratio of the number of hydrogen atoms to Zr atoms in the crystal in the phase

10

2 Properties of Bulk Zirconium Hydrides

being considered. (To avoid cumbersome notation in some cases, where the meaning is clear the subscript ‘‘H’’ is omitted from rH ). Hchyd or Xchyd , The hydrogen molar or atomic volume in the c-hydride phase,V H respectively, is defined as the difference between the molar or atomic volume of zirconium in the c-phase and in the a-Zr phase at infinite dilution of H, viz., aZr  chyd  V Zr Xchyd  XaZr Zr Hchyd ¼ VZr V or Xchyd ¼ Zr H bchyd bchyd

ð2:2Þ

and similarly for the d-hydride molar or atomic volumes. The value of bpha  bchyd in Eq. 2.2 depends on which hydride is being considered. In the case of the c-hydride crystal structure, experimental evidence shows that only half of the two available preferential tetrahedral interstitial sites per zirconium atom are actually ever occupied at any one time by hydrogen atoms. Hence, bchyd ¼ 1. On the other hand, for d-hydride it would appear that although there are again two available preferential tetrahedral interstitial sites per zirconium atom, the composition of this hydride seems centred at a stoichiometric composition of 1.5 suggesting that bdhyd ¼ 1:5. However, the equilibrium composition of the hydride, which is given by rH ; can vary somewhat in this case since rH decreases to values less than bdhyd ¼ 1:5 as the temperature is raised from ambient to the eutectoid temperature. Now, the room temperature lattice parameters of the hcp a phase are a0 = 3.231 Å and c0 = 5.146 Å [16]. The unit cell volume of the a-Zr phase, given by 3a2csin 60 (or H3/4a2c), is thus 139.57 Å3. There are six Zr atoms in a unit cell of a-Zr; therefore the atomic or molar volume of a-Zr (Wigner–Seitz cell aZr Zr volume), denoted by XaZr or V , is the unit cell volume divided by six, Zr 3 yielding 23.26 Å /(atom Zr) or 14.01 9 10-6 m3/(mol Zr), respectively. The experimental results of Kolesnikov et al. indicate that rH is close to unity in the Hchyd , of H in c-hydride given by Eq. 2.2, c-hydride phase. The molar volume,V -6 3 then, is 1.61 9 10 m /(mol H).

2.1.2 Phase Relationships, Phase Stability, and Hydrogen Compositions in the d- and c-Hydride Phases at the (a 1 d)/d Phase Boundary Over the years, there have been numerous experimental observations and theoretical arguments put forth in the literature concerning the nature of the c-hydride phase, particularly its thermodynamic stability. With reference to this debate, which is dealt with further in Chap. 6 in which the a/(a ? d) and (a ? d)/d solvus relations are explored, it should be noted that despite some recent concerted attempts toward this goal [6, 16], no one has, so far, been able to produce bulk specimens consisting entirely of the c-hydride phase, even when hydrogenating specimens to an overall composition equal to the observed stoichiometric composition of this phase,

2.1 Hydride Phase Compositions, Lattice Structure and Parameter Determinations

11

i.e., ZrHr or ZrDr, with r = 1. This inability to produce bulk specimens consisting entirely of the c-hydride phase appears to be an important indication that this phase is metastable throughout the temperature range below the eutectoid temperature. Generally, what has been observed in hydrogenating Zr metals to overall hydrogen compositions below a range from ZrH1.3 at the eutectoid temperature to ZrH1.6 at room temperature—at and above which composition only the single d-hydride phase is present until the e-hydride phase field is reached—has consisted of a mixture of either the a-Zr, c-, and d-hydride phases, or only of the c- and d-hydride phases. These results are described in more detail in the following. In neutron and X-ray diffraction studies of hydrogenated Zr metal prepared in the form of thin strips or filings from crystal bar purity stock, Sidhu et al. [24] listed the observed phases and their relative proportions in terms of diffraction intensity signals starting from an overall composition range from ZrH0 to ZrH0.11 and ending with the terminal composition range from ZrH1.70 to ZrH1.99. As the composition of hydrogen was increased, it was inferred from the signal strength data that the starting decomposition of the a-Zr phase resulted in a mixture of c-hydride and remaining a-Zr phases, with the c-hydride likely present initially as precipitates within the a-Zr matrix that, up to ZrH1.0, evolved into a mixture of predominantly c- and d-hydride phases in addition to a small amount of remaining a-Zr phase. At higher H/Zr ratios the a-Zr phase disappeared entirely and the c-hydride phase became a minority one, likely contained within the d-hydride phase. Between ZrH1.4 and ZrH1.6, only the d-hydride phase was found. Beyond that composition there was a mixture of d- and e-hydride phases over a narrow composition range from ZrH1.6 to ZrH1.70. Above these composition ratios only the e-hydride phase was observed. Beck [7] carried out a study of the Zr–H system with the main objective the determination of the hydrogen solubility limit in the d-hydride phase at the (a ? d)/ d phase boundary, since—particularly at room temperature—this composition had been the source of disagreement among investigators. Beck determined the room temperature composition to be at 61.4 ± 0.2 at % H (rH ¼ 1:59) with, as he stated, ‘‘…considerable certainty’’. Beck’s confidence in the correctness of this result was based, in part, on the results and interpretation of supporting experimental studies concerning the nature and origin of the c-hydride phase. All previous workers had found from room temperature measurements that there existed a third phase in the (a ? d) region which was identified as the c-hydride phase. To determine whether this phase was metastable, Vaughn and Bridge [28] had carried out high temperature diffraction studies by aging the material at 500 C for 48 h. They had found that the c-hydride phase could be eliminated. On the other hand, Eiler et al. [13], using only diffraction measurements taken at room temperature, had found the c-hydride phase in specimens that, prior to the room temperature diffraction measurements, had been aged for a similar period at 540 C. Similarly, Gulbransen and Andrew [14] had found evidence in room temperature diffraction measurements of the presence of the c-hydride phase in specimens that had been hydrogenated several hundred degrees below the eutectoid temperature. In Beck’s own studies [7], vycor-encapsulated specimens were aged for 5 days at 500 C followed by room temperature diffraction studies. These studies found no change in the room

12

2 Properties of Bulk Zirconium Hydrides

temperature intensities of the powder patterns taken subsequent to the aging treatment compared to those taken before this treatment. Since Vaughn and Bridge [28] were the only workers that had made diffraction measurements at the aging temperature, Beck suggested that a plausible explanation for these combined observations is that the c-hydride phase had actually disappeared at the aging temperature and then reappeared on cooling to room temperature. On the basis of these results, Beck concluded that the c-hydride phase must be a metastable phase that is formed on cooling. To determine the reason for its formation during cooling, Beck noted that in a powder X-ray diffraction pattern the relative amounts of phases present is qualitatively proportional to the observed signal strength of the strongest line for each phase. From the data listed in this author’s Table 1, he concluded, therefore, that the amount of the c-hydride phase was clearly proportional to the d-hydride and not the a-Zr phase. Beck pointed out that the likelihood of the c-hydride phase being an allotropic modification of the d-hydride phase is very low given its observed insensitivity to quenching and aging procedures and the relatively high rate of the transformation. Beck, therefore, concluded that c-hydride is a metastable decomposition phase of d-hydride rather than of the equilibrium a-Zr phase and which—on the basis of studies of the reaction kinetics—forms continuously as a function of temperature. Beck constructed a model to support this conjecture as follows. Beck assumed that the c-hydride phase must have a hydrogen composition intermediate between that of the d-hydride and a-Zr phases. Beck further conjectured that the actual composition of the c-hydride phase would have to be ZrHr, where r  rH ¼ 1, on the basis of its crystal structure, a conjecture that has since been verified by many workers [34]. According to Beck, the structure of the c-hydride phase is body-centered tetragonal (bct),1 which he claimed was additionally supported by theoretical intensity calculations showing that the Zr atoms are located at the center of the lattice points. Assuming that the density of the d-hydride phase must be intermediate between those of the d-hydride and a-Zr phases, the unit cell would have to be simple bct containing only two metal atoms, ignoring the small contribution that hydrogen would make to the density. As a result, the arrangement of Zr atoms in the d-hydride phase must be identical to that in the e-hydride phase. Beck noted that the bct cell can be converted to the face centered cubic cell simply by multiplying the a lattice parameter by H2. Using this lattice as the reference state for convenience, the fcc phase would be one having a : c. Conversion of the fcc d-hydride phase to the fct c-hydride phase, then, involves a contraction in the a directions and an expansion in the c direction of the d-hydride phase of magnitudes such that c/a [ 1. The reverse occurs for the conversion of the fcc d-hydride phase to the fct e-hydride phase, resulting in c/a \ 1. In this latter case, the increase in hydrogen composition of the e-hydride 1

It should be noted that Beck erred in describing the c- and e-hydride phases as having bct structures, which all other investigators have determined to have fct structures. However, this does not affect his arguments since Beck based his analysis on differences in the hydrogen composition in the transformation of the d-hydride phase to an fct one having either positive or negative c/a.

2.1 Hydride Phase Compositions, Lattice Structure and Parameter Determinations

13

phase from its value at the (d ? e)/e phase boundary to its terminal, stoichiometric composition, ZrH2, results in an increase in a and a corresponding decrease in c, with the latter decreasing less than the increase in a, thus resulting in a continuous reduction in c/a over the phase field composition, Fig. 2.1. As can be seen from this figure, this decrease flattens out as the stoichiometric composition of the hydride is approached and stops completely when this composition is reached and all the tetrahedral sites in this lattice are occupied by hydrogen atoms. Extrapolating this trend in the opposite direction for the c-hydride phase, starting from the fcc d-hydride phase, it is conceivable that the (notional) removal of a sufficiently large number of hydrogen atoms from this initially fcc lattice results in an fct c-hydride lattice that is exactly the reverse of the e-hydride phase, having an increase of the c/a ratio with (notional) decrease in hydrogen composition until a composition is reached at which the observed lattice parameters and the c/a ratio for this phase are obtained. Similar to the composition that stabilized the e-hydride phase, the only unique solution for the composition that could stabilize the c-hydride phase would seem to be a stoichiometric one, the most likely choice being ZrHr, with r ¼ 1, in which only alternate tetrahedral sites would be filled with hydrogen. This prediction for the stable composition of the c-hydride phase is also a consequence of the postulate that, with decreasing temperature, it is the precipitation product (instead of the ‘‘equilibrium’’ a-Zr phase) of the decomposition of the d-hydride phase and must have a lower hydrogen composition than that of the parent, d-hydride phase. This, then, leads to the conclusion that the c-hydride’s terminal composition should be saturated in hydrogen, which, on the basis of symmetry considerations and the available tetrahedral sites, again leads to the prediction that the only composition possible for this phase is ZrHr, with r ¼ 1. On the basis of the relative X-ray diffraction intensities obtained by Beck and of visual estimates of the volume fraction of c-hydride precipitates observed in metallographic samples, the maximum amount of c-hydride phase that can form under ordinary circumstances was estimated to be 30 or 40 %. Using the proposed model for c-hydride formation and assuming on the basis of the foregoing reasoning that c-hydride formation is stoichiometric, of composition ZrHr, with r ¼ 1, a shift in the composition of the d-hydride phase from r ¼ 1:3 at the eutectoid to 1.6 at room temperature would result in a maximum volume fraction of precipitated c-hydride phase of 44 % if no a-Zr phase were formed. This prediction is in good agreement with Beck’s corresponding estimate derived from a comparison between the relative X-ray diffraction intensities and also from the observed volume fraction of c-hydride precipitates as described in the foregoing. The foregoing proposed model for the formation of the c-hydride phase also provides an explanation for the disagreement among various investigators at the time as to the location of the (a ? d)/d phase boundary from the eutectoid to room temperature. In all samples containing an overall composition of less than r ¼ 1:3, appreciable amounts of the a-Zr phase were observed. This is as expected since this phase would be formed as an equilibrium product of the eutectoid reaction. However, once the total amount of hydrogen in the sample exceeds this composition, the amount of a-Zr phase present would be insufficient for its detection by

14

2 Properties of Bulk Zirconium Hydrides

Fig. 2.1 Hydrogen composition dependence of Zr–H lattice parameters (from left to right) of the d, d ? e, and e phase regions at room temperature (from Zuzek et al. [34]). The symbols refer to the following sources: [68 Moo]—Moore and Young [20]; [70 Bar]—Barraclough and Beevers [5]; [85 Bow]—Bowman and Clark [9]

X-rays, as was observed. Therefore, without the proposed model for c-hydride formation as a reaction product of the d-hydride decomposition, it would appear from these observations that above this overall composition the (a ? d)/d phase boundary would similarly be at r ffi 1:3 at room temperature. However, taking on board the foregoing proposal that the c-hydride phase is a metastable product of the decomposition of the d-hydride phase, the correct (a ? d)/d composition can be located quite easily by factoring in the volume fraction and terminal hydrogen composition, r ¼ 1 of the c-hydride phase. Following this insightful study by Beck [7], a series of bulk hydrogenated specimens in the composition range from ZrH1.27 to ZrH1.92 were prepared by Barraclough and Beevers [5] to determine their room temperature metallography, microhardness, and crystal structure properties. It is shown that the findings of these authors support those of Beck. Barraclough and Beevers [5] found that for specimens having overall compositions of ZrH1.47, ZrH1.52, and ZrH1.57, in addition to a strong signal indicating copious amounts of the d-hydride phase, there were also weak signals attributed to the c-hydride phase. Results from the powder pattern of ZrH1.27 also provided evidence of the presence of both the c- and d-hydride phases plus a small amount of a third phase that was not identified, but likely was the a-Zr phase on the basis of the results by Sidhu et al. [24] and microhardness measurements of Barraclough and Beevers [5]. Supporting metallographic studies of specimens in the composition range r = 1.47–1.57 showed that the c-hydride phase appeared as banded precipitates of lenticular shape embedded in the majority d-hydride phase (Figs. 2.2 and 2.3). Quantitative metallography showed the volume fraction of c-hydride precipitates

2.1 Hydride Phase Compositions, Lattice Structure and Parameter Determinations

15

Fig. 2.2 Banded c-hydride precipitates in d-hydride matrix of ZrH1.52 (from Barraclough and Beevers [5])

Fig. 2.3 The structure of a ZrH1.27 specimen showing banded c-hydride precipitates in a d-hydride matrix and a third phase at the grain boundary (from Barraclough and Beevers [5])

increased as the overall hydrogen content decreased. Lenticular, c-hydride precipitates within the d-hydride matrix were also observed for the ZrH1.27 specimens. The latter specimens showed some structural instability since re-examination of these specimens after 6 months revealed additional c-hydride precipitates formed at the a–d boundaries while small, globular precipitates of a-Zr were found at the d grain boundaries. These findings are in agreement with Beck’s proposed model for the decomposition of the d-hydride phase at temperatures below the eutectoid and at overall hydrogen compositions ranging from, r ffi 1:3 to 1:6 . The metallographic evidence obtained by Barraclough and Beevers indicates that the c-hydride phase forms in d-hydride by a shear mechanism, with the banded structure indicating that this shear is relieved by alternate twinning of the lattice in opposite directions. A similar conclusion was formed for the formation of the e-hydride phase in d-hydride in the narrow composition range from r ffi 1:6 to 1:7 (Fig. 2.4). Beyond this composition all specimens contained only the e-hydride phase, forming a completely

16

2 Properties of Bulk Zirconium Hydrides

Fig. 2.4 The banded ehydride phase and untransformed regions of the d-hydride phase in ZrH1.71 (from Barraclough and Beevers [5])

Fig. 2.5 The completely banded e-hydride structure of ZrH1.77 (from Barraclough and Beevers [5])

banded structure (Fig. 2.5). Barraclough and Beevers [5] found, in agreement with Beck [7], that in this phase field range the c lattice parameter and the c/a ratio of the e-hydride phase decreased with increase in hydrogen composition while the a lattice parameter increased. The suggestions by both Beck and Barraclough and Beevers that the formation of the c- and e-hydride phases in d-hydride occurs essentially by a martensitic type transformation are supported by the analyses and experimental results of Cassidy and Wayman [10, 11] noting, however, that in contrast to a strictly diffusionless martensitic transformation these invariant plane strain transformations must be associated with a corresponding change in hydrogen composition. The need for a composition change creates some uncertainty in their analyses to determine the habit planes of the product c- or e-hydride precipitates of these transformations as it is not certain whether this occurs before, during or after the transformation.

2.1 Hydride Phase Compositions, Lattice Structure and Parameter Determinations

17

Nevertheless, based on these observations, Barraclough and Beevers—taking on board the considerations of Beck regarding the nature of the d- to e-hydride phase transformation—suggest that the (d ? e) phase field is actually not a true equilibrium one, but represents the range of compositions over which the martensitic transformation is stabilized as a result of factors such as the low jump frequency of the hydrogen atoms at room temperature and the creation of plastic deformation caused by the large transformation strains of the e-hydride precipitates in the d-hydride phase (the same considerations would apply to the c-hydride phase). Barraclough and Beevers [5] note that the hydrogen jump frequency (Hon [15]) and dislocation mobility (Barraclough and Beevers [3]) are extremely temperature sensitive. Consequently, inhibition of growth of the e-hydride phase, either by the low jump frequency of hydrogen or by irrecoverable plastic deformation in the d-hydride matrix, would be expected to decrease as the martensitic start temperature is increased; i.e., as the hydrogen concentration increases, the transformation temperature range between start and finish of the martensitic transformation from d- to e-hydride would be expected to decrease. For the similar d- to c-hydride transformation there is only one composition over which this occurs, and thus there would be no variable composition range over which the transformation is stabilized. However, as with the d- to e-hydride transformation, the rate of conversion would obviously be a function of temperature, decreasing with decreasing temperature. Although the e-hydride phase is generally not formed for hydrogen compositions in zirconium alloys of practical interest in nuclear applications, and is therefore not of direct interest for the topic of this book, the transformation characteristics of this phase and its parallelism with that of the c-hydride phase provide useful insights into the behavior and prevalence of the latter phase. This is the reason for which the characteristics and the conditions leading to its formation have been presented here to the level of detail provided. A more recent study of the phases present in bulk specimens of the Zr–H system was that of Simpson and Cann [25] who hydrogenated compact toughness specimens to hydrogen contents ranging from ZrH0 to ZrH1.6 to measure the fracture toughness of bulk zirconium hydride material over this composition range. Both commercial-grade, cold-rolled Zr and flattened Zr-2.5 Nb pressure tube material was used as starting material. The main difference between the two materials after hydrogenation was in the grain size (the unalloyed Zr material had smaller grain size) and in the morphology and distribution of the hydride phases. (The presence of the alloying element, Nb, in the starting pressure tube material appears not to have a significant effect on the properties of the bulk hydrides produced.) A feature of their hydrogenation procedure was that some of their specimens were hydrogenated at 600 C and either furnace cooled or air quenched, while others were hydrogenated at 900 C and furnace cooled (1 C/minute). For compositions below r ¼ 1, the microstructures of the at 900 C consisted of d-hydride precipitates in an a-Zr matrix. In hydrogenated commercial grade Zr, the hydrides were very large and unconnected and contained c-hydride needles, Fig. 2.6. The hydrogenated Zr-2.5 Nb specimens had large a-Zr grain sizes, but the hydrides were finely dispersed within these grains. Specimens hydrogenated

18

2 Properties of Bulk Zirconium Hydrides

Fig. 2.6 Microstructure of ZrH1.1: a Large d-hydride grains containing c-hydride needles; grain boundaries contain a mixture of a-Zr and unidentified hydride (bright field). b Transmission micrograph of c-hydride needles, showing twinned structure (from Simpson and Cann [25])

at 600 C (in the b or (b ? d) region) had much finer grain sizes than those of the 900 C series specimens. A ZrH0.88 specimen was quenched from well within the (b ? d) region and had a dispersion of a-Zr grains in a d-hydride matrix. The ZrH0.45 specimen, quenched from the b region, resulted in a fine grain structure with a grain boundary network of hydrides. The slow cooled specimen of composition ZrH0.40 resulted in a similar structure, but on a considerably larger scale. Compositions between ZrH1.0 and ZrH1.5 resulted in mixtures of d- and c-hydride phases, the proportion of the d-hydride phase increasing with the hydrogen content. Some a-Zr phase was also observed at the low hydrogen content range of these compositions. The c-hydride had formed as heavily twinned, needle-like precipitates, the twinning being revealed in TEM examinations. All specimens with hydrogen content greater than ZrH1.5 consisted entirely of the d-hydride phase. It is evident that the types and proportions of phases observed in these hydrogenated fracture toughness specimens prepared by Simpson and Cann [25] were very similar to those obtained by previous workers such as Beck [7] and Barraclough and Beevers [5] for the same composition ranges. In general, all workers were able to produce crack-free hydrogenated specimens, although distortions were evident in specimens having plate or rectangular geometry. However, all of the workers also found that specimens in a narrow range from r ffi 1:5 to 1:66 either contained cracks or cracked during surface preparation, indicating exceptional brittleness for solid hydride specimens of that composition range, which is approximately the range over which the hydrogenated solid is made up solely of the d-hydride phase.

2.1 Hydride Phase Compositions, Lattice Structure and Parameter Determinations

19

2.1.3 Crystallography of the d-Hydride Phase The d phase is a disordered fcc lattice of the CaF2 prototype, space group ðFm 3mÞ with room temperature lattice parameter, a0 = 4.778 Å [7]. This results in a unit cell volume, a3, of 109.07 Å. Similar to the c phase, there are four Zr atoms in a unit cell of dhyd Zr d hydride. Therefore, the atomic or molar volume of d-hydride, Xdhyd or V , is Zr 3 6 3 27.27 Å /(atom Zr) or 16.42 9 10 m /(mol Zr), respectively. According to the studies of Simpson and Cann [25] presented in the foregoing, the r ratio in this phase varies from *1.6–1.64 at room temperature to *1.31–1.7 at the eutectoid temperature of *550 C. These phase boundary compositions differ slightly from those given by others [5, 7, 20] and others, as summarized by Zuzek et al. [34]. The lower numbers in each of the foregoing range intervals are those that apply at the (a ? d)/d phase boundary. At this composition the entire two-phase material has been converted to the d phase, above which there is a narrow range throughout which the d-hydride phase increases in composition until the d/(d ? e) phase boundary is reached. Although earlier results had found that the room temperature lattice parameter of the d phase is approximately constant with hydrogen composition over the composition range, r ffi 1:56 to 1:70 (see Fig. 2.1 and the references cited therein) the results of Yamanaka et al. [31–33] show that there is a slight increase while the lattice parameter of ZrDr over this composition range is slightly smaller than that of ZrHr and has a weaker composition dependence. However, in addition to the composition dependence, because the concentration of hydrogen in the hydride at the (a ? d)/d phase boundary varies with temperature, as shown in Fig. 2.7, so will the hydrogen molar volume. From Eq. 2.2 for r ¼ 1:5, which is at the midpoint in the temperature variation of the H/Zr ratio, the molar volume of hydrogen in Hdhyd , is 1.607 9 10-6 m3/mol H. It is interesting to note that the molar d-hydride,V volume of hydrogen in d-hydride is approximately the same as that in c-hydride for this hydrogen concentration even though a c-hydride precipitate has a smaller volume expansion in a-Zr than does the d-hydride precipitate. As is shown in Chap. 8, this demonstrates the importance of accounting for the hydrogen concentration, not only in the a-Zr phase, but also in the corresponding hydride phase in formulations of the chemical potential of hydrogen in a-Zr in local equilibrium at the a/(a ? d) and (a ? d)/d phase boundaries. Although it is evident from the foregoing that the molar volume of hydrogen in d hydride varies with temperature because its hydrogen composition, r, varies with temperature, there would also be a variation in this molar volume (as well as in the other molar volumes and transformation strains from which the hydrogen molar volume is derived) with temperature and composition if the differences in the lattice parameters between the a-Zr and d-hydride phases vary with temperature and composition. Recently, Singh et al. [26] evaluated the temperature and composition dependence of these differences using data provided by Douglass [12] for the temperature dependence of the lattice parameters in a-Zr and by Yamanaka et al. [31, 32] for the temperature dependence and hydrogen composition dependence of the lattice parameter in d-hydride. Singh et al. [26] focused on the

20

2 Properties of Bulk Zirconium Hydrides

Fig. 2.7 The high hydrogen composition region of the Zr-H phase diagram (from Libowitz [18])

temperature dependence of the lattice parameters for a single d-hydride phase composition of r ffi 1:66. Their results are reproduced in Fig. 2.8 showing that the variation with temperature of the difference in the lattice parameters of the two phases is not large. The largest difference in the temperature dependence is between the a lattice parameters, while the difference between the c and a lattice parameters of the a-Zr and d-hydride phases, respectively, is very small. Fig. 2.9 plots the variations in volumetric transformation strain and corresponding plate normal transformation strain as a function of temperature in which the variation with temperature of the composition of the d-hydride at the (a ? d)/d phase boundary, using the data summarized by Beck [7], is also included. The latter is obtained by combining the linear dependence on hydrogen composition determined by Yamanaka et al. [31, 32] of the d-hydride phase’s lattice parameter with that of the corresponding average thermal expansion coefficient that was determined by them over the temperature range from 298 to 573 K. In this calculation, it was assumed that the dependence of the average thermal expansion coefficient on hydrogen composition in d-hydride would apply outside the measured temperature range, up to 400 C. This exercise results in the following lattice parameter (in Å) dependencies on temperature and d-phase hydrogen or deuterium composition, rH or rD , respectively: aaZr ¼ 3:23118 þ 1:6626  105  ½TðKÞ  298

ð2:3Þ

2.1 Hydride Phase Compositions, Lattice Structure and Parameter Determinations

21

Fig. 2.8 Temperature dependence of the lattice parameters of hcp a-Zr, fcc d-hydride ZrH1.66 and fcc d-hydride ZrD1.66 (from Singh et al. [26])

Fig. 2.9 Temperature dependence of plate normal transformation strain and volumetric transformation strain for d-hydride at (a ? d)/d phase boundary composition (the latter corresponding to chosen temperature based on Zr–H phase diagram of Beck [7])

22

2 Properties of Bulk Zirconium Hydrides

caZr ¼ 5:14634þ47:413  106  ½TðKÞ  298

ð2:4Þ

adhydðHÞ ¼ 4:706þ4:382  102  rH ð2:5Þ  þ 2:475  105 þ6:282  105  rH þ5:8281  107  r2H ½TðKÞ  298 adhydðDÞ ¼ 4:738þ1:961  102  rD ð2:6Þ  þ 2:492  105 þ6:3119  105  rD þ2:6081  107  r2D ½TðKÞ  298 Table 2.1 summarizes the temperature and composition dependencies of these and other key parameters of potential use for applications in mechanistic models of hydride and DHC behavior described in subsequent chapters. The plot of Fig. 2.9 and the results summarized in Table 2.1 show that, in the temperature range of technological interest between room temperature and 400 C, the increases in the dilatational transformation strains of d-hydride are relatively small. However, for the molar volume of hydrogen in d-hydride it can be seen from Eq. 2.2 that the explicit introduction of the hydrogen concentration, rH, in the denominator results in a further variation in temperature dependence of the molar volume which is in the same direction as that of the lattice parameters, increasing its overall dependence on temperature in comparison to that of other parameters— such as the transformation volume and the individual dilatational transformation strain values—that depend only the lattice parameters’ variations with temperature. The overall result is that the molar volume of hydrogen in the d-hydride phase increases from 1.49 9 10-6 m3/mol H at room temperature to 1.77 9 10-6 m3/ mol H at 300 C, an increase of 0.28 9 10-6 m3/mol H.

2.2 Mechanical Properties of Bulk Zirconium Hydrides In this section, key results of elastic moduli, uniaxial deformation and fracture toughness parameters obtained from specimens of bulk zirconium hydride consisting of a single or a mixture of the a-Zr, c-, and d-hydride phases are presented.

2.2.1 Yield Strength An early study by Beck and Mueller [8] of the tensile properties of bulk zirconium hydride specimens within the composition range ZrH0.7 to ZrH1.9 and the temperature range from room temperature to *600 C showed that macroscopic plastic deformation was not observed in the single phase d– and e-hydride composition ranges. Moreover, they were unable to produce crack free specimens in the hydrogen composition range r = 1.53–1.70, which is an indication of the

1.61 1.5 1.47 1.42 1.37 1.33

25 225 300 400 454 504

3.2312 3.2345 3.2357 3.2374 3.2383 3.2391

5.1463 5.1552 5.1586 5.1635 5.1664 5.1693

4.7766 4.7958 4.8030 4.8114 4.8140 4.8167

23.266 23.354 23.387 23.433 23.460 23.485

27.245 27.575 27.699 27.845 27.891 27.937

2.4715 2.8144 2.9335 3.1070 3.2348 3.3473

1.488 1.695 1.767 1.871 1.948 2.016

0.1710 0.1808 0.1844 0.1883 0.1889 0.1896

0.072 0.074 0.075 0.076 0.076 0.076

0.0453 0.0484 0.0496 0.0509 0.0512 0.0515

Table 2.1 Lattice Properties of d-hydride at composition corresponding to the (a + d)/d phase boundary as a function of hydrogen composition. The values given are for the hydrogen isotope, protium (H), only aZr dhydðHÞ dhydðHÞ dhydðHÞ dhydðHÞ ðHÞ ðHÞ ca-Zr ad-hyd XZr aa-Zr rH T Zr edhyd edhyd ðXZr (XZr (XZr ) (V c a (Å) (Å) (C) (Å) aZr aZr aZr (Å3/atom  T T (Å3/atom H) XaZr V )/r XZr =XZr Þ H ¼ e33 ¼ e11 ¼ eT22 Zr Zr )/rH Zr) (910-6m3 (Å3/atom H) /mol H)

2.2 Mechanical Properties of Bulk Zirconium Hydrides 23

24

2 Properties of Bulk Zirconium Hydrides

extreme brittleness of such specimens in this composition range. Based on the phase diagram available at the time, the foregoing composition range is expected to produce specimens made up of only the single d-hydride phase. The authors speculated that the extreme brittleness of their specimens was a consequence of a very low solubility in the d-hydride phase of impurity elements present in the a-Zr starting material. According to Beck and Mueller [8], these impurity elements would be more soluble in the e- than in the d-hydride phase. As a result, these impurities would precipitate as small second phase particles in the d-hydride phase and act as internal stress centers, making the bulk material consisting of this phase more susceptible to fracture during formation or grinding. This would, then, also explain why above r ¼ 1:70 in the single phase e-hydride composition region no difficulties were encountered with premature cracking of this material. Nevertheless, results of tensile tests of specimens consisting only of the e-hydride phase always resulted in premature fracture at locations in the specimens other than the gauge region, indicating the extreme brittleness also of this phase, and making the testing of this material in tension highly sensitive to alignment problems. The next attempt at determining the tensile properties of the d- and e-hydride phases was carried out by Barraclough and Beevers [3]. For tensile tests on specimens of overall composition, ZrH1.66, they found that from room temperature to 500 C fracture always preceded measurable plastic deformation. The stress at fracture was very low, in the range of approximately 48–69 MPa, with little dependence on test temperature. However, some of the specimens fractured outside of the gauge region. This was likely the result of non-axial loading that was a consequence of the warped shape of the hydrogenated specimens. As a result of these findings, all other tests were carried out in compression. The results of the compression tests on specimens of composition ZrH1.66 are reproduced in Fig. 2.10. Most of the tests were interrupted prior to fracture for examination of the specimens. Only those tests where this is indicated on the figure were taken to fracture. Below *100 C these specimens were completely brittle, exhibiting no macroscopic plasticity. At temperatures from 100 to 120 C, small plastic strains of *3 % could be accommodated prior to transgranular fracture. Metallographic examination of the polished surfaces of these fractured specimens revealed cracks and planar slip lines. Above 120 C the intensity of slip markings increased and above 250 C wavy slip lines was a predominant feature of the test results. Analysis of the slip lines on a specimen tested at 109 C showed that the operative slip planes were of the {1 1 1} type. This type of slip was prevalent up to 250 C, above which the slip lines became too wavy to be analyzed by the authors’ microbeam X-ray and two-face stereographic analyses techniques. Compressive tests of specimens with compositions ZrH1.57, ZrH1.52, and ZrH1.47 were carried out between room temperature and 500 C. These specimens consisted of the d-hydride phase interspersed with an increasing volume fraction of c-hydride platelets. A set of specimens with composition ZrH1.27 were also tested over this temperature range. These specimens consisted of c-hydride platelets in a predominantly d-hydride matrix plus small amounts of a-Zr along the grain boundaries. All of these specimens were sufficiently ductile that they could be

2.2 Mechanical Properties of Bulk Zirconium Hydrides

25

Fig. 2.10 Nominal stress– strain curves for the compressive deformation of ZrH1.66 in the temperature range from 22 to 453 C (from Barraclough and Beevers [3] (10 kg mm2 ’ 98 MPa))

taken beyond their limit of proportionality before fracture. The results of the dependence of the limit of proportionality on temperature, including the results from the tests on the ZrH1.66 specimens, are plotted in Fig. 2.11. It is evident from this plot that the proportional limit in these specimens, consisting of two- or threephase mixtures, decreases rapidly from its value at room temperature, leveling off, or reaching a local minimum, at about 200 C. This means that above 150–200 C, hydrides consisting of predominantly d-hydride phase with an increasing proportion of c-hydride platelets would deform plastically under compression at fairly low applied stresses. An important feature to note in the plot of Fig. 2.11 is the increase in proportional limit with temperature from 200 C to a peak at 270 C, beyond which it decreases again with further increases in temperature. The height of this peak increases with decrease in hydrogen content of the specimens. Based on the model developed by Beck [7] for why the volume fraction of the c-hydride phase increases with decrease in overall hydrogen content, the peak likely reflects the changes in the deformation properties of the two-phase mixture brought about by the dissolution of the c-hydride phase as the temperature is increased. Hence, this effect would be greatest for the specimens with the lowest total hydrogen content, for which the volume fraction of the c-hydride phase would be largest. Supporting evidence for the dissolution of the c-hydride precipitates and their subsequent re-precipitation upon cooling was obtained from room temperature metallographic examinations of specimens tested above 280 C. These showed that the c-hydride platelets tended to be much smaller and were undistorted,

26

2 Properties of Bulk Zirconium Hydrides

Fig. 2.11 Temperature dependence of the limit of proportionality of bulk hydride specimens consisting of d-hydride, (d ? c)-hydride and (d ? c ? a)-hydride phases (from Barraclough and Beevers [3])

in contrast to the distorted c-hydride platelets observed in the interior of sectioned specimens that had been tested at lower temperatures. However, a specimen of composition ZrH1.27, tested at 360 C, still showed the presence of large, distorted c-hydride platelets, indicating that these precipitates had not been completely dissolved at that temperature. To obtain information on the microscopic fracture process in the specimens exhibiting some ductility at room temperature prior to macroscopic fracture, a series of interrupted room temperature tests were carried out by Barraclough and Beevers on specimens of compositions ZrH1.47 and ZrH1.52. Surface cracks were first observed at 189 and 186 MPa, respectively. The cracks were initiated in the d-hydride phase and not in the c-hydride needles. In fact, there was evidence that these precipitates could sometimes act as crack arresters, Figs. 2.12 and 2.13 shows the evolution of surface cracks in a specimen of composition ZrH1.47 which had a proportional limit of 311 MPa. The Burgers vectors of the dislocations responsible for the observed slip on {1 1 1} planes were not experimentally determined by Barraclough and Beevers, but are expected to be of a/2 \ 1 1 0 [ types, based on the lattice structure of the d-hydride phase (CaF2) and results in the literature for ionic crystals with such structure. However, the predominant slip planes in ionic crystals are {1 0 0}. This suggests that the occurrence of slip on {1 1 1} planes is indicative of considerable metallic bonding. Assuming, then, that the slip systems in the d-hydride phase are

2.2 Mechanical Properties of Bulk Zirconium Hydrides

27

Fig. 2.12 The surface of a ZrH1.52 specimen after 0.5 % strain at room temperature, showing a crack arrested in a c-hydride platelet (from Barraclough and Beevers [3])

of {1 1 1} \ 1 1 0 [ types, Barraclough and Beevers conjectured that the extreme brittleness of this phase could be the result of the influence of the hydrogen lattice on the movement of the Zr lattice glide dislocations. The authors suggest that it is the hydrogen vacancies (i.e., the unoccupied tetrahedral lattice sites in the CaF2-type lattice) that act as the primary source of lattice friction for dislocation movement. This suggestion is supported by the sharp decrease in microhardness between ZrH1.60 and ZrH1.66 (reproduced further on) and the results given in Fig. 2.11 showing that from 400 to 500 C—throughout which it is thought only the d-hydride phase would be present—the limit of proportionality decreases with an increase in hydrogen content. The rapid decrease in proportional limit above 100 C may then be the result of the reduction in influence of the hydrogen lattice because of the increased mobility of the hydrogen atoms in the d-hydride phase. However, the influence of the vacant hydrogen lattice sites is nevertheless still evident in affecting its ductility, as evidenced by the low values of the limits of proportionality plotted in Fig. 2.11 in the temperature range from 300 to 400 C. Over this temperature range, it is thought that the c-hydride precipitates would have been dissolved and the specimens would have consisted only of d-hydride phase (Fig. 2.11). Moreover, the limits of proportionality of the ZrH1.57, ZrH1.52, and ZrH1.47 specimens, which decreased rapidly with temperature from 100 to 200 C, were virtually independent of hydrogen content over this temperature range. This observation is consistent with the (d ? c) material having a d-hydride matrix that would retain its low temperature composition of *ZrH1.59 over this temperature range. Metallographic evidence showed that the limit of proportionality is primarily controlled by the deformation characteristics of the d-hydride matrix which, in turn—based on the foregoing reasoning—would be controlled by the number of vacant hydrogen lattice sites. Therefore the limit of proportionality would remain independent of overall hydrogen content, as a result of the conjectured constancy of the composition of the d-hydride phase in these (d ? c) alloys. Note also that the limits of proportionality of the (d ? c) materials were always

28

2 Properties of Bulk Zirconium Hydrides

Fig. 2.13 Surface cracks on a ZrH1.47 specimen loaded at room temperature to, a 189 MPa and, b 249 MPa (from Barraclough and Beevers [3])

greater at any given temperature than those of the single phase d-hydride materials, which is also consistent with the foregoing conjecture concerning the role of hydrogen vacancies in controlling the mobility of dislocations. Further, the foregoing observation of distorted c-hydride platelets in specimens containing (d ? c) material shows that these precipitates would not have had significant influence on the yield characteristics over the temperature range from 100 to 200 C. Based on these considerations, the peak at *270 C in the plots of Fig. 2.11 can be explained on the basis of the increasing dissolution of the c-hydride phase. Support for this comes from the findings of Sidhu et al. [24] who reported that a transformation from (d ? c) to (d ? a) occurs at *250 C and from Beck [7] who reported fast reaction kinetics for the dissolution of the c-hydride phase. The dissolution of this phase results in an increased hydrogen vacancy concentration in the d-hydride matrix since its overall hydrogen composition decreases. This is also consistent with the increase in the limit of proportionality from 200 to 270 C being greater for materials with lower hydrogen content, which is expected if the vacancy concentration in the d-hydride phase is responsible for controlling the deformation characteristics. Further support for Barraclough and Beever’s conjecture that there is considerable dissolution of the c-hydride phase above 200 C came from the observations that the c-hydride precipitates were not distorted in specimens tested above this temperature. As seen in Fig. 2.11, the dependence of the proportional limit of the ZrH1.27 specimens on temperature does not follow the trend with hydrogen composition observed for specimens with hydrogen contents, r ffi 1:47 to 1:57: However, the ZrH1.27 specimens contained large, distorted c-hydride platelets (Fig. 2.14)—even after testing at 360 C—indicating that not all of the c-hydride precipitates had been taken into solution. For such a case, the d-hydride matrix would have a composition that is dependent on the reaction kinetics of the three-phase material, which would be expected to decrease from its low temperature value of r ffi 1:59 on partial dissolution of the c-hydride precipitates, based on the two-phase composition of the (a ? d)/d phase boundary shown in Fig. 2.7.

2.2 Mechanical Properties of Bulk Zirconium Hydrides

29

Fig. 2.14 Large distorted chydride platelets in a ZrH1.27 specimen after 11 % strain (from Barraclough and Beevers [3])

An attempt was made by Barraclough and Beevers [3, 4] to explain the fracture characteristics of the various materials. As noted, the specimens of composition ZrH1.66 tested in tension were extremely brittle, in agreement with the results of Beck and Mueller [8]. Since the available {1 1 1} \ 1 1 0 [ slip systems would satisfy the von Mises requirement for polycrystalline ductility, the observed brittle behavior must come from the increased lattice friction imposed on the dislocation mobility by the hydrogen vacancies in the d-hydride matrix. From the results of Fig. 2.15 it is seen that cracks are formed in the ZrH1.66 and ZrH1.59 specimens at approximately the same applied stress value. The influence of the volume fraction of the c-hydride platelets is plotted in Fig. 2.16. From metallographic observations there was an increase in hydride precipitate size and interspacing. If these c-hydride platelets were to act as crack nucleation centers, then the fracture stress would be expected to decrease with increase in volume fraction of this phase, which is contrary to observations (Fig. 2.16). The results in Fig. 2.15 show that cracks appeared in the specimens at applied stresses well below those for final fracture, indicating that the increase in fracture stress could be associated with the restrictions on crack growth as a result of the c-hydride platelets as shown in Fig. 2.12. The c-hydride platelets were observed to have {1 0 0} habit planes in the d-hydride matrix, which means that if the cleavage planes in both hydrides are the same—of {1 1 1} type—then the cracks must change their directions to pass through the c-hydride platelets. These precipitates also exhibited a twinned micro-structure that, by further twinning as a result of the increasing external stress, could possibly offer further resistance to crack propagation. These considerations lead to the conclusion that the primary role of the c-hydride platelets during the deformation process was to inhibit crack propagation, rather than to act as crack nucleation centers. For the one set of specimens consisting of (d ? c ? a) material (ZrH1.27), the observation of a further increase in inhibition of crack propagation was probably the result of the additional presence of the more ductile a-Zr phase. Based on these considerations, it is likely that the deformation in region A-B in Fig. 2.15—throughout which micro-crack

30 Fig. 2.15 The fracture and deformation characteristics of d-hydride, (d ? c)-hydride and (d ? c ? a)-hydride phases (from Barraclough and Beevers [3])

Fig. 2.16 Influence of volume fraction of secondphase c-hydride platelets on the room temperature fracture stress of (d ? c)-hydride material (from Barraclough and Beevers [3])

2 Properties of Bulk Zirconium Hydrides

2.2 Mechanical Properties of Bulk Zirconium Hydrides

31

nucleation and propagation occurred—may be mainly the result of shape changes in the specimen associated with this process. Recently, Puls et al. [23] summarized results of additional compressive deformation tests on solid zirconium hydride specimens of hydrogen composition ranging from ZrH1.0 to ZrH1.9. Starting materials for these solid zirconium hydride specimens were either cold-worked Zr-2.5 Nb pressure tube or reactor grade unalloyed Zr material. In addition, two different hydrogenation temperatures, 850 or 600 C, were used. The results obtained by Barraclough and Beevers [3, 4] have shown that specimens consisting of single phase d-hydride exhibited no macroscopic plasticity prior to fracture below 100 C while external cracks had formed at applied external stress values below the proportional limit for more ductile specimens such as those consisting of (d ? c)-hydride phases or of single phase dhydride material tested at temperatures above 100 C. This extremely brittle behavior of the material tested by Barraclough and Beevers [3, 4] may have been exacerbated by the likely presence of microscopic cracks in the interior of the specimens produced during the hydrogenation process. These micro-cracks would have affected the deformation characteristics of the specimens. To obtain a truer measure of the deformation characteristics of solid zirconium hydrides, Puls et al. [23] carried out compression tests under a confining hydrostatic pressure, thereby hoping to mitigate the effects of possible pre-existing microscopic cracks formed during hydrogenation of the specimens on the material’s deformation behavior. The testing apparatus used for this purpose was a modified Griggs machine at the Laboratoire de Métallurgie Physique of the Université de Poitiers, France [29]. This machine allowed for separate control of the uniaxial compressive stress and a surrounding pressure that was applied by means of a solid confining medium. The first set of uniaxial compression tests was carried out under a confining pressure of 1,000 MPa. Confined compression tests with this pressure were carried out mainly at room temperature with a few tests done also at 150 and 200 C. A lower confining pressure of 400 MPa was used in most of the remaining tests out of concern that the higher confining stress may have affected the accuracy of the results by creating sufficiently high levels of friction stresses to produce noticeable increases in the proportional limit. Tests carried out at 400 or 415 C were not done under confinement because of the low value of the proportional limit at that temperature and the greater ductility of the material. Initially the Griggs set up was used with the confining piston made of carbide having the same diameter as the surrounding jacket. As a result of the friction generated by this confinement, the accuracy of the stress–strain measurement results markedly decreased with this set up when the yield strength was much below 500 MPa, which turned out to be the case for specimens deformed at and above 100 C. Improvements were, therefore, made to the Griggs machine to handle lower yield strength values by using a piston made of tungsten carbide having a diameter that was less than the surrounding jacket, thus reducing the friction produced by the confinement. Finally, it became evident that above about 100 C all specimens were sufficiently ductile that they could be readily deformed into the plastic range in the Griggs machine without confinement. In addition—likely as a result of improved control in producing

32

2 Properties of Bulk Zirconium Hydrides

crack-free ZrHr specimens over the entire range of hydrogen composition of interest—it was found that unconfined compressive deformation at room temperature also gave acceptable results. Therefore, the last sets of tests consisted of standard uniaxial compression tests done with an INSTRON 4002 mechanical testing machine at AECL’s Whiteshell Laboratories in Pinawa, Manitoba, Canada. The results of all the tests at various temperatures and hydrogen composition carried out in France are listed in Table 2.2. The tests with test numbers starting with ‘I’ were unconfined tests done in an Instron machine, while all others, designated by test numbers starting with ‘G’, were done with the Griggs machine using either 1,000 or 400 MPa confining pressure as indicated. Comparing test results for specimens with closely similar hydrogen compositions, these results show that yield strength values for specimens having 1,000 MPa confinement were always somewhat higher than those obtained with specimens having a confinement of 400 MPa. Since only a few specimens tested under different confining pressure had the same hydrogen composition, it is difficult to quantify the increase in yield strength as a result of the higher confining pressure, but it appears to be in the range from 50 to 100 MPa. The exception to these results is for specimens G163 and G145 for which the reverse was true. The difference in yield strength of these two specimens of nominally identical composition was quite large, viz., 182 MPa. For the tests done under hydrostatic confinement, two types of stress–strain curves were obtained as shown in Fig. 2.17a and b. The first type shown in Fig. 2.17a, exhibited by specimens with 1.3 \ r \ 1.62, had high yield strength values showing a fairly sharp transition between the elastic and the plastic deformation regimes and little subsequent work hardening. The second type (Fig. 2.17b), with lower yield strength values and compositions r \ 1.3 and [1.62, had a more gradual transition between the elastic and the plastic regimes and strong work hardening. Deformation was generally stopped in these series of confined tests before any macroscopic fracture of the specimens. Yield strengths in all tests were determined as given by the proportional limit, similar to what was used by Barraclough and Beevers [3, 4]. The yield strength results are plotted as a function of temperature in Fig. 2.18. Included in this figure are results obtained by Barraclough and Beevers. In this plot, the yield strength data given in Table 2.2 for closely similar values of r were rounded to the nearest integer, and a corresponding average yield strength value applicable to this average value of r was determined. This figure shows that there is a sharp drop in yield strength from room temperature to 150 C, with little change up to 400 C after that. The figure shows that the yield strength values of Barraclough and Beevers are consistently and significantly lower than those of Puls et al. at all temperatures and hydrogen compositions. The results of Puls et al. show that the highest yield strength values are found just below a hydrogen composition of r  1:7. There is a large drop (minimum) in yield strength value at r  1:7 and a gradual rise beyond that. The results of Barraclough and Beevers [3, 4] at 400 C, taken from their plots of their Figs. 2.3 and 2.13, respectively, give a decrease at a composition r *1.48–1.52 that continues to a composition of r  1:78. Beyond this composition range all the nominal stress–strain curves obtained by Barraclough

2.2 Mechanical Properties of Bulk Zirconium Hydrides

33

Table 2.2 Yield strength versus hydrogen composition, r, at different temperatures from [23]; with permission from AECL Specimen No. Composition, r Yield strength (MPa) Temperature G232 G223 G170 G142 G225 G165 G140 G167 G226 G224 G166 G141 G168 G163 G145 G144 G230 G231 G169 G164 Temperature G211 G212 Temperature G235 G236 G234 Temperature I5 I6 I4 I2 I7 Temperature I15 I12 I11 I14 I10

= ambient 1.00 1.12 1.21 1.21 1.25 1.37 1.43 1.48 1.57 1.61 1.62 1.66 1.67 1.69 1.69 1.81 1.83 1.86 1.89 1.95

622# 633# 751# 794# 636# 870* 968 893* 721#* 746#* 817* 990 607 685 503 715 431# 529# 672 629

1.30 1.94

646 348

1.15 1.58 1.90

170# 122# 185#

1.21 1.43 1.61 1.72 1.96

118# 158 191 88 110

1.15 1.40 1.57 1.69 1.91

150 174 192 202 140

= 150 C

= 200 C

= 400 C

= 410 C

All samples were hydrogenated at 850 C from original Zr-2.5 Nb pressure tube material, except where otherwise indicated. (Specimens having specimen numbers starting with ‘G’ were tested under a confining pressure of 400 MPa; those bolded were tested under a confining pressure of 1,000 MPa. All other specimens were tested without confinement.) # Made from reactor grade Zr *Hydrogenated at 600 C

34

2 Properties of Bulk Zirconium Hydrides

Fig. 2.17 a One type of load–deflection curve for ZrHr hydride specimens with hydrogen composition, 1.3 \ r \ 1.62 and deformed under confinement in the Griggs machine showing a fairly sharp transition between the elastic and plastic deformation regimes and little subsequent work hardening. (Specimen G167, r = 1.48; the vertical axis has been divided by 100) (from [23]). b One type of load–deflection curve for ZrHr hydride specimens with hydrogen compositions r \ 1.3 and r [ 1.62 and deformed under confinement in the Griggs machine showing a gradual transition between the elastic and plastic deformation regimes and strong work hardening. (Specimen G168, r = 1.68; the vertical axis has been divided by 100) (from Puls et al. [23]; with permission from AECL)

and Beevers [4] exhibited load drops and, therefore, reliable yield strength values could not be obtained from these data. Puls et al. proposed that differences in the hydrogen composition dependences of the yield strength values obtained from the two sets of workers may be the result of the averaging procedure used by them in combining data obtained on material having slightly different compositions. The higher yield strength values and greater ductility, particularly at room temperature, obtained by Puls et al. are thought to be the result of their material containing fewer as-grown micro-cracks and/or the use of a confining pressure. For both sets of results, the lower yield strength values obtained at all temperatures for material containing only the e-hydride phase may be partially the result of the greater number of flaws produced in forming materials having hydrogen contents close to those of the terminal composition of ZrH2. The production of such flaws produced during hydrogenation is evident in Fig. 2.19 for a specimen of composition ZrH1.8

2.2 Mechanical Properties of Bulk Zirconium Hydrides

35

Fig. 2.17 continued

showing similar flaws both in the indentation region and outside of it. Thus, the ‘‘yielding’’ of the specimen may be the result of the (initially) stable growth of these flaws rather than being solely the result of the reduction in the onset of plastic deformation with temperature. A set of cumulative deformations starting at *400 C and decreasing in steps to room temperature were carried out for specimens of different hydrogen compositions. These results are plotted in Fig. 2.20. Such an experimental approach has the advantage that there are no differences (or uncertainties) in the hydrogen compositions of the specimens tested at different temperatures. The results are qualitatively similar to those of Fig. 2.18. However, the reduction in yield strength with temperature is greater for the non-cumulative tests. This may be connected with an annealing effect in the cumulative tests that was produced by each immediately preceding test, which was at a higher temperature. The yield strength values for the cumulative test at r  1:77 stands out, being much smaller than those with hydrogen composition values greater and less than 1.77. This dip in yield strength over the complete temperature range tested is not evident in the non-cumulative tests (Fig. 2.18), perhaps because the relevant hydrogen composition was not part of the latter set. A final set of room temperature deformation tests, but with no confinement, were carried out using an Instron machine. All specimens were deformed to failure, with the specimens broken into small pieces, indicating the extreme brittleness of this material at room temperature. The specimens with higher hydrogen compositions showed some ductility, but were still very brittle compared to non-hydrogenated, as-received material. Fig. 2.21 shows how the stress–strain

36

2 Properties of Bulk Zirconium Hydrides

Fig. 2.18 Yield strength versus temperature for ZrHr hydride specimens of different hydrogen compositions, x : r. In the legend, BB refers to results obtained by Barraclough and Beevers [3, 4]. A spline fit was used to connect the points when data for three or more temperatures at a given x : r were available (from Puls et al. [23]; with permission from AECL)

Fig. 2.19 SEM image of a micro-indentation on a specimen of ZrH1.8 (from Puls et al. [23]; with permission from AECL)

behavior varied with r. The deformation curves obtained show a similar trend with r as those obtained under confinement; i.e., the curves for compositions r = 1.4–1.6 have higher yield strength values and sharper transitions to plasticity with only a limited work hardening stage beyond yield. However, the room temperature yield strength values derived from these unconfined tests are significantly lower compared to those obtained for corresponding compositions under confinement (Table 2.2). Figure 2.22 shows the room temperature variation of the yield strength with r. An abrupt drop in the yield strength with r is observed, starting from a composition for which—on the basis of the phase diagram—the

2.2 Mechanical Properties of Bulk Zirconium Hydrides

37

Fig. 2.20 Yield strength versus temperature obtained by cumulative deformation of each sample with a given hydrogen composition, x : r, starting at the highest temperature. The solid line for Zr-2.5 Nb is the mean value for the temperature dependence of unirradiated, cold-worked Zr-2.5 Nb pressure tube material deformed in the transverse pressure tube direction (from Puls et al. [23]; with permission from AECL)

specimen is expected to consist only of the d-hydride phase and reaching a minimum level at compositions at which the specimen is expected to contain only the e-hydride phase. For both sets of results it is evident that the most brittle behavior is exhibited at compositions where d-hydride is the dominant, or only, phase (from about r *1.5–1.64). The brittleness of specimens with these compositions is further illustrated by the observation that, of five specimens tested with r ¼ 1:5, three failed prior to reaching the plastic stage and all of the tests had small load drops in the elastic stage. The deformation curves for the specimens with compositions in the single phase e-hydride range (r ’ 1:7 and higher) are interesting, showing a two-stage plastic deformation stage not evident in the corresponding tests carried out under confinement. The reason for this difference in observation is likely because the latter tests were terminated prior to reaching the second work hardening stage.

2.2.2 Fracture Toughness Simpson and Cann [25] carried out fracture toughness tests using miniature compact toughness specimens hydrogenated to hydrogen contents, ZrHr, ranging from r ¼ 0 to 1:64. Miniature specimens of 17 mm width were used to facilitate hydrogenation, but this posed problems in obtaining valid test results for specimens of low hydrogen content and/or tested at high temperature. Previous investigations had shown that tensile testing of material that had been

38

2 Properties of Bulk Zirconium Hydrides

Fig. 2.21 Examples of stress–strain curves from room temperature unconfined deformation tests of ZrHr hydride specimens. The numbers indicate the hydrogen composition, r (from Puls et al. [23]; with permission from AECL)

Fig. 2.22 Yield strength versus x : r of hydride specimens ZrHr at room temperature from unconfined compression tests. The dashed line has been drawn to indicate the trend in the results (from Puls et al. [23]; with permission from AECL)

2.2 Mechanical Properties of Bulk Zirconium Hydrides

39

Fig. 2.23 Temperature dependence of KIc for ZrHr in the composition range r = 1.5–1.64. Only the d-hydride phase was identified to exist in these specimens (from Simpson and Cann [25]; with permission from AECL)

hydrogenated to a hydrogen composition where the specimen consisted mostly or entirely of the d-hydride phase exhibited extreme brittle behavior, one reason for this behavior likely being that the hydrogenation process had resulted in the formation of many micro-cracks in the specimen. Thus, great care had to be taken to ensure the production of crack-free specimens. This turned out to be easier with reactor-grade Zr as the starting material compared to with Zr-2.5 Nb pressure tube material. Simpson and Cann [25] note that for specimens of hydrogen content, r 1:5, only four successful tests could be carried out because of the difficulty in producing crack-free specimens from the latter starting material. The results of the dependence of fracture toughness, KIc, versus temperature are reproduced in Fig. 2.23. There is a large amount of scatter in the results, as one might expect given the low room temperature toughness of the material of only *1 MPaHm. There is, however, clear indication from the upper bound values that fracture toughness increases significantly (relative to its room temperature value) within the interval from 200 to 300 C. This is in contrast to the lower-bound toughness values showing no increase at all with temperature at this hydrogen content. The variation of KIc with hydrogen content, r, is reproduced in Figs. 2.24 and 2.25. These figures show that KIc increases gradually with decrease in hydrogen content down to ZrH0.4 at which point it has a value of 10–15 MPaHm. The toughness values from the Zr-2.5 Nb pressure tube material yielded consistently lower results. Below a composition of r = 0.4 valid test results were difficult to obtain because of the increased ductility of the material combined with the small size of the compact toughness specimens. This was also true for some of the test results at 300 C at r ¼ 0:4. The KIc values calculated from the load–deflection data for these specimens are shown with a stroke through the open circle in Fig. 2.24 to indicate that they are ‘‘non-valid’’ test results.

40

2 Properties of Bulk Zirconium Hydrides

Fig. 2.24 Effect of hydrogen content on KIc of hydrogenated commercial-grade, cold-rolled zirconium starting material (from Simpson and Cann [25]; with permission from AECL)

2.2.3 Microhardness, Elastic Moduli, Internal Friction Microhardness measurements provide a measure of the elastic and plastic response of a material over a relatively small area by determining the force required with an indenter to produce an indentation of a certain depth and size in the material. The virtue of these techniques is that it is relatively easy to use and does not require large specimen sizes. From these indentations, relationships have also been developed to determine parameters such the elastic modulus, yield strength, and some measure of fracture toughness of the material. The greatest utility of the microhardness results lies, however, in the observation that abrupt changes in their values seem to correlate with changes in the phase or phases present in the specimen, thereby providing another method of phase boundary composition determinations that could, moreover, be used on a micro or nano scale on embedded hydride clusters. Among the earliest microhardness measurements on solid hydride specimens were those of Beck [7], Ambler [1] and Barraclough and Beevers [5]. Other

2.2 Mechanical Properties of Bulk Zirconium Hydrides

41

Fig. 2.25 Effect of hydrogen content on KIc of hydrogenated Zr-2.5 Nb pressure tube starting material (from Simpson and Cann [25]; with permission from AECL)

microhardness measurements were carried out by Syasin et al. [27], Yamanaka et al. [31–33], Xu and Shi [30] and Puls et al. [23]. In addition, all of these authors determined the elastic moduli of the hydrides studied, either through microhardness measurement techniques [23, 30] or using other methods, such as sound velocity [27, 31–33] and resonant oscillator technique [22]. The latter two sets of authors also measured the internal friction of the solid hydride specimens. The following account describes and assesses these results starting with the most recently published ones. Using a composite oscillator technique running at 40 or 120 kHz, Pan and Puls [22] carried out simultaneous measurements of dynamic elastic modulus, E, and

42

2 Properties of Bulk Zirconium Hydrides

Fig. 2.26 Internal friction peak, P2, a shoulder peak, P0 , and elastic modulus, E, in a ZrH1.5 hydride specimen (from Pan and Puls [22]; with permission from AECL)

internal friction, Q-1, as a function of temperature from room temperature to 300 C for hydrogenated material of a single composition, ZrH1.5. Two different source materials were used for the ZrH1.5 specimens: reactor-grade, unalloyed Zr and Zr-2.5 Nb pressure tube material. After hydrogenation at 520 C the specimens were annealed for 10 days at 400 C and slowly cooled to room temperature. As-received (i.e., non-quenched) specimens and specimens quenched into cold water after a one hour anneal at 400, 350, and 300 C, respectively, were tested. As seen in Figs. 2.26 and 2.27, the quenching treatment significantly decreased E at all temperatures from its values for as-received specimens. In addition, the different starting materials resulted in hydrogenated ZrH1.5 specimens having significantly different temperature dependencies. E in the specimens hydrogenated from reactor-grade Zr material increased in value from room temperature to a peak at *180 C, approximately independent of the quenching temperature while E obtained in the specimens prepared from Zr-2.5 Nb starting material had plateaux ranging from just above room temperature to 67 C for the two quenched specimens and to 107 C for the as-received specimen. Above these temperatures E decreased monotonically with increase in temperature. It is difficult to know the reasons for these differences. A possible cause may be the presence of *20 % volume fraction of c-hydride platelets that was predicted by Beck [7] and Barraclough and Beevers [5] to be present at room temperature in the d-hydride phase for material with this total hydrogen content. One might expect that, since the lattice structure of the c-hydride phase is not cubic that it would have anisotropic elastic properties. Hence, differences in the lattice orientations, number, size, and internal microstructures (as a result of different amounts, types and orientations of twins) of these c-hydride platelets as a result of differences in

2.2 Mechanical Properties of Bulk Zirconium Hydrides

43

Fig. 2.27 Internal friction peak, P2, and elastic modulus, E, in a Zr-2.5 Nb ? 60 at % H (equivalent to ZrH1.5) (from Pan and Puls [22]; with permission from AECL)

starting materials and heat treatments of the hydrogenated materials may then have resulted in differences in the average elastic properties. For ZrH1.5 obtained from reactor-grade zirconium starting material, the room temperature value of E ranged from 98.2 to 91.6 GPa for the as-received material and for the material quenched at 300 C, respectively, while for ZrH1.5, derived from Zr-2.5 Nb pressure tube material, E ranged from 94.3 to 90.6 GPa for the as-received material and the material quenched at 250 C, respectively. Although there were differences as a result of the quenching treatment and source material for E, there was no difference in the peak temperature of the internal friction peak, P2, and only small differences as a result of the foregoing factors. The interpretation proposed by the authors that the peak is the result of stress-induced jumps of hydrogen atom pairs in the d-hydride phase, seems, at first glance, to be strongly supported by the finding that the amplitude of the peak depends on the square of the hydrogen content of the specimen ranging from very low hydrogen content to very high hydrogen content of ZrH1.5 (60 at%). Such dependence would not be obtained if the origin of the P2 peak were actually the result of a similar stress-induced jump process occurring in the *20 % volume fraction of c-hydride phase predicted to be present at room temperature. This model would predict that the amplitude dependence of P2 depends on the amount of hydrogen contained only in the c-hydride phase. Moreover, it would predict that this dependence would be linear in this hydrogen concentration, since an internal friction peak as a result of a stress-induced jump process involving only a single hydrogen atom would be possible in the c-hydride phase because of its non-cubic lattice structure. Puls et al. [23] used a FISCHET H100 microhardness tester capable of evaluating the apparent elastic modulus E0 ¼ E=ð1  mÞ below 55 C using a diamond indenter

44

2 Properties of Bulk Zirconium Hydrides

Fig. 2.28 Apparent elastic modulus of ZrHx versus x : r at room temperature from microhardness tests using a diamond indenter (from Puls et al. [23]; with permission from AECL)

to measure the room temperature variation of E0 as a function of the hydrogen content, r, on specimens similar to those that were used in their compression tests. The results are reproduced in Fig. 2.28 showing that there is a steep reduction in E0 with increasing r starting at r *1.5–1.6. Similar measurements were also made with the same equipment on the same type of specimens at elevated temperatures by replacing the diamond indenter with a sapphire one. Since the computer program supplied by the manufacturer for data acquisition and data analysis of the microhardness tester was calibrated for the low-temperature diamond indenter material only, the values of E0 obtained are mainly useful in showing the relative variation of this parameter with temperature and composition. Fig. 2.29 gives three examples. It can be seen from this figure that E0 decreases with increase in temperature and that the effect of increased hydrogen content is to shift the lines of E0 to lower levels, while the rate of change with temperature for all of them is about the same as that of the original zirconium alloy. Puls et al. [23] also determined the elastic modulus, E, directly from load– displacement plots of compression tests such as those shown in Fig. 2.21 based on the steepest linear portion of each load–displacement curve. The mean values are plotted in Fig. 2.30. The overall appearance of the E versus r plot is qualitatively consistent with what was obtained for the apparent modulus, E0 , from the microhardness test results (Fig. 2.28). There is a significant drop in E in the range from ZrH1.6 to ZrH1.8. This drop in microhardness with composition is echoed by a similar drop in the yield strength (Fig. 2.22). Xu and Shi [30] used a nano-indentation technique for hardness measurements from which they derived values of elastic modulus and yield strength while the fracture toughness was determined using a microhardness indenter. Only bulk hydrogenated material of a single composition, ZrH1.83, consisting mostly of the single e-hydride phase, was investigated. An optical micrograph of the material

2.2 Mechanical Properties of Bulk Zirconium Hydrides

45

Fig. 2.29 Apparent elastic modulus of ZrH1.37, ZrH1.52, and ZrH1.58 at elevated temperatures from microhardness tests using a sapphire indenter. A spline fit was used to connect the data (from Puls et al. [23]; with permission from AECL)

showed that the material contained voids, particularly along the grain boundaries. The presence of such flaws is consistent with metallographic observations of such flaws in most other hydrogenated specimens consisting only of the e-hydride phase. The advantage of the nano-indentation technique, then, for such material is that the indentations are small enough that locations containing void-like defects could be avoided thus resulting in a truer picture of the intrinsic mechanical properties of the material. The authors obtained an average room temperature hardness value of 3.31 GPa for this composition, a yield strength of 478 MPa (compared to a yield strength of 780 MPa for the reactor grade zirconium starter material) and a reduced elastic modulus value of 67.66 GPa, derived from these hardness values. The relationship between the reduced modulus, Er, and the true elastic modulus, E, of the material is given by 1 1  m2 1  m2i þ ¼ Er E Ei

ð2:7Þ

With the elastic modulus and Poisson’s ratio of the indenter, Ei and mi, given by 1140 GPa and 0.07, respectively, and arbitrarily using a value of 0.3 for m since a measured value at this hydride composition is not available for this parameter, E of hydride of composition ZrH1.83 was calculated to be 65.44 GPa. (Note that Yamanaka et al. [31] obtain a value for m of 0.32, which is constant over a hydrogen composition range from r ¼ 1:5 to 1.7.) This result compares with a value of 54.95 GPa for ZrH1.83 derived from the results of E0 determined by Puls et al. using the same value of m and of 37.5 GPa obtained by the former authors directly from the load–displacement curves of their uniaxial compression tests. In comparison, E determined for the zirconium reactor-grade starting material determined by Xu and Shi, assuming a value of 0.43 for m, is 80.90 GPa compared

46

2 Properties of Bulk Zirconium Hydrides

Fig. 2.30 Elastic modulus of Zr-2.5 Nb-Hx (equivalent to ZrHx) versus x : r at room temperature from unconfined compression tests. The dashed line has been drawn to indicate the trend (from Puls et al. [23]; with permission from AECL)

to a value of 98.15 GPa obtained by Puls et al. using the same value of m. However, since the hcp lattice of the zirconium starter material is elastically anisotropic, differences could result if the indentations in the two types of tests did not sample crystal grains of the same lattice orientation. Syasin et al. [27] carried out microhardness, elastic modulus, internal friction, and micro-brittleness measurements for bulk hydrides of compositions ZrH1.66 to ZrH1.99. This composition range starts at what is thought to be approximately the (metastable) equilibrium composition of the d-hydride phase with the e-hydride phase and the phase field compositions for the (d ? e)- and e-hydride phase regions of the phase diagram (see Fig. 2.32 further on). Using a Vickers diamond indenter, their room temperature microhardness results show an oscillatory behavior as a function of hydrogen composition, giving an average value of 1.67 GPa with a maximum of 1.72 GPa at r ¼ 1:94 and minimum of 1.47 GPa at just below r  1:9. These hardness values are much lower than the average value of 3.31 GPa obtained by Xu and Shi [30] for a composition of ZrH1.83. E determined by Syasin et al. ranges from 80 to 85 GPa for r ¼ 1:66 to just below r ’ 1:9, respectively, then drops abruptly at r ¼ 1:9 to *75 GPa. Similarly, micro-brittleness exhibits a trough at just below r ¼ 1:9 followed by a peak at r = 1.94 while internal friction has a peak at just below r ¼ 1:9. No explanation was offered as to what mechanism could be responsible for the existence of these extrema which were, moreover, found to occur at the same hydrogen composition values. Yamanaka et al. [31–33] obtained room temperature microhardness and elastic modulus results for solid hydride specimens having hydrogen contents ranging from r ¼ 1:5 to 1.68. The upper range of hydrogen content of their specimens is close to the hydrogen content where other workers have observed a sharp

2.2 Mechanical Properties of Bulk Zirconium Hydrides

47

Fig. 2.31 Microhardness versus hydrogen composition (content) of Zr-at.% H (from Beck [7])

reduction in microhardness and other mechanical property values. Beck’s [7] data, reproduced in Fig. 2.31, shows the drop off in microhardness values occurring beyond r ¼ 1:61 (62.5 at.%) while Barraclough and Beevers’ data [5], reproduced in Fig. 2.32, show it to be beyond r ¼ 1:56. From the plot of effective elastic modulus, E0 obtained by Puls et al. [23], reproduced in Fig. 2.28, the drop off appears to be from r ¼ 1:51 to 1.56, although the composition intervals for these data points are very coarse. Still, the drop off would likely not be greater than r ¼ 1:6 on the basis of this plot. On the other hand, their corresponding data for the variation of E with r (Fig. 2.30) obtained from the stress–strain curves of their last set of unconfined compression tests—for which it is believed the hydrogen content was more accurately obtained—show the drop off to occur at r ¼ 1:62. This drop off with composition is close to that (r ¼ 1:65) at which a corresponding drop off in yield strength occurs (Fig. 2.22). Since the latter is derived from the same load– displacement curves, this result is, perhaps, not too surprising. Both Beck [7] and Puls et al. [23] show a slight increase in microhardness or—the equivalent—E0 , up to the drop off composition, while the microhardness results of Barraclough and Beevers remain approximately constant up to the drop off composition. These results differ from the microhardness values obtained by Yamanaka et al. (1969a, b) ranging over the composition range from r = 1.47 to 1.68 showing a slight linear reduction in microhardness over this composition range, with no indication of a drop off over this range of compositions. These authors claimed that their pellet specimens contained no microscopic cracks or pores. They give the variation of Vickers Hardness, Hv, as: Hv ðGPaÞ ¼ 7:19  2:77r

ð2:8Þ

48

2 Properties of Bulk Zirconium Hydrides

Fig. 2.32 Microhardness versus hydrogen composition (content) r = H/Zr of ZrHr (from Barraclough and Beevers [5])

The value at r ’ 1:5 is *3 GPa. The authors also determined the room temperature elastic modulus, E and shear modulus, G, from measurements of the longitudinal and shear sound velocities for both the hydrogen (protium) and deuterium forms of their solid hydride specimens. The results are reproduced in Figs. 2.33 and 2.34. The authors obtain considerably larger values of E than were obtained by Puls et al. [23] at closely similar hydrogen compositions. One reason for the difference may be because E obtained by Puls et al. is derived from low strain rate tests and would thus represent the isothermal, as opposed to the high frequency, adiabatic, elastic modulus values that are determined by Yamanaka et al. However, Pan and Puls [22], also used a dynamic technique and obtained similar values of E for specimens of composition, r ¼ 1:5 as were obtained by Puls et al. One uncertainty is that Yamanaka et al. [31, 32] do not specify the frequency of their sound waves. However, it should be noted that even the results obtained by Syasin et al. are in closer accord with those of Puls et al. [23], Puls and Pan [22] and Xu and Shi [30] than are the values obtained by Yamanaka et al., despite the fact that they are in the hydrogen composition range over which Puls et al. show the elastic modulus has substantially lower values (i.e., for material consisting only of the e-hydride phase).

2.2 Mechanical Properties of Bulk Zirconium Hydrides

49

Fig. 2.33 Change in elastic (Young’s) modulus, E, of d-hydride ZrQr (Q = D or H; r : CQ) with hydrogen composition, CQ (from Yamanaka et al. [32])

2.2.4 Summary of Mechanical Strength Results Overall, comparing the results of Barraclough and Beevers [3–5] with corresponding ones obtained by Puls et al. [23], there are two important differences. The first difference is that the room temperature yield strength values obtained by Puls et al. are considerably higher than those obtained by Barraclough and Beevers with the highest values of the former being those determined with the Griggs machine under confinement. (It should be noted that differences in yielding behavior as a result of different starting material or hydrogenation temperature were not detectable within the scatter of the results.) Puls et al. obtained room temperature yield strength values ranging from a low of 721 MPa at r ¼ 1:57 to a high of 990 MPa at r ¼ 1:66, over a hydrogen composition range of r = 1.4–1.66, with an average value of 858 MPa. This average value is close to that obtained at room temperature for unhydrogenated Zr-2.5 Nb pressure tube material, although not for the much softer reactor-grade Zr starting material. Since there is little difference in the yield strength values in the foregoing composition range depending on which starting material was used, this means that the yield strength of the hydrogenated material is mainly determined by its lattice structure and hydrogen composition and not from its starting material. The effect of confinement in increasing the measured average yield strength value over the foregoing hydrogen composition range is further evidenced by comparison with the results of the room temperature tests without confinement. The unconfined tests resulted in an average yield strength value over the composition range from r = 1.4 to 1.66 of 611 MPa compared to an average value under confinement of 858 MPa. This amounts to a reduction of *250 MPa when not using confinement. In comparison, the yield strength values obtained by Barraclough and Beevers, also without confinement, in the composition interval from r ¼ 1:27 to 1:66 ranged

50

2 Properties of Bulk Zirconium Hydrides

Fig. 2.34 Change in shear modulus, G, of d-hydride ZrQr (Q = D or H; r : CQ) with hydrogen composition, CQ (from Yamanaka et al. [32])

from *390 to 250 MPa, respectively. The reason that Barraclough and Beevers’ yield strength values are lower may be because their hydrogenation procedure resulted in the generation of a greater number of pre-existing microscopic cracks, the number of these cracks increasing with increasing hydrogen composition. These microscopic cracks could then be controlling the shape deformation of their specimens, resulting in premature fracture at room temperature and softer yielding behavior at higher temperatures compared to the specimens grown by Puls et al. containing possibly fewer, or no, as-grown microscopic cracks and/or made less susceptible to these cracks by testing under confinement. The second difference is that the plots of yield strength versus temperature obtained by Puls et al. [23] do not exhibit the small maximum in yield strength at *270 C obtained by Barraclough and Beevers [3]. A possible reason for this involves the same reasoning used in the foregoing to explain why Barraclough and Beevers’ specimens appeared more brittle at room temperature and had a softer load–displacement response at slightly higher temperatures than those obtained by Puls et al. At higher temperatures, the greater ductility of the material would result in the pre-existing microscopic cracks no longer controlling the nonlinear deformation of the material and the proportional limit obtained would more closely reflect the true plastic response of the material. The significantly higher proportional limits obtained by Puls et al. below 100 C are, then, an indication that their results more accurately reflect the true plastic response of the material, even at room temperature. This would explain why the yield strength values obtained by Puls et al. decrease continuously with temperature, compared to a peak at some intermediate temperature obtained by Barraclough and Beevers [3].

References

51

References 1. Ambler J.F.R.: The effect of temperature on the micro-indentation properties of zirconium hydride. Atomic Energy of Canada Ltd, Report AECL-2538, Chalk River, Ontario, Canada (1966) 2. Balagurov, A.M., Bashkin, I.O., Kolesnikov, A.I., et al.: Neutron diffraction determination of the kinetics of the e ? d phase transition in TiD0.74. Sov. Phys. Solid State 33, 711–714 (1991) 3. Barraclough, K.G., Beevers, C.J.: Some observations on the deformation characteristics of bulk polycrystalline zirconium hydrides. Part I: The deformation and fracture of hydrides based on the d–phase. J. Mater. Sci. 4, 518–525 (1969) 4. Barraclough, K.G., Beevers, C.J.: Some observations on the deformation characteristics of bulk polycrystalline zirconium hydrides. Part II: The deformation of e–hydrides. J. Mater. Sci. 4, 802–808 (1969) 5. Barraclough, K.G., Beevers, C.J.: Some observations of the phase transformations in zirconium hydrides. J. Nucl. Mater. 34, 125–134 (1970) 6. Bashkin, I.O., Malyshev, V.Yu., Myshlyaev, M.M.: Reversible c ? a ? d transformation in zirconium deuteride. Sov. Phys. Solid State 34, 1182–1184 (1992) 7. Beck, R.L.: Zirconium-hydrogen phase system. Trans. ASM 55, 542–555 (1962) 8. Beck, R.L., Mueller, W.M.: Mechanical properties of solid zirconium hydride. In: Nuclear Metallurgy. Metall. Soc. AIME III, 63–66 (1960) 9. Bowman, Jr. R.C., Clark, B.D.: Effects of thermal treatments on lattice properties and electronic structure of ZrH. Phys. Rev. B31, 5604–5615 (1985) 10. Cassidy, M.P., Wayman, C.M.: The crystallography of hydride formation in zirconium: I. The d ? c transformation. Metall. Trans. A 11A, 47–56 (1980) 11. Cassidy, M.P., Wayman, C.M.: The crystallography of hydride formation in zirconium: II. The d ? e transformation. Metall. Trans. A 11A, 57–67 (1980) 12. Douglass D.L.: The metallurgy of zirconium. International Atomic Energy Agency (1971) 13. Eiler E.W., Rau R.C., Tholke W.H. et al.: A metal-hydride system. GE-ANPD document, APEN 444 (1958) 14. Gulbransen, E.A., Andrew, K.F.: Crystal structure and thermodynamic studies on the Zr-H system. J. Electrochem. Soc. 101, 474–480 (1954) 15. Hon, J.F.: Nuclear magnetic resonance study of the diffusion of hydrogen in zirconium hydride. J. Chem. Phys. 36, 759–765 (1962) 16. Kolesnikov, A.I., Balagurov, A.M., Bashkin, I.O., et al.: Neutron scattering studies of ordered c-ZrD. J. Phys: Condensed Matter 6, 8977–8988 (1994) 17. Kolesnikov, A.I., Balagurov, A.M., Bashkin, I.O.: A real-time neutron diffraction study of phase transitions in the Ti-D system after high-pressure treatment. J. Phys: Condensed Matter 5, 5045–5058 (1993) 18. Libowitz, G.G.: A pressure-composition-temperature study of the zirconium-hydrogen system at high hydrogen contents. J. Nucl. Mater. 5, 228–233 (1962) 19. Mogilyanskii, D.N., Bashkin, I.O., Degtyareva, V.F., et al.: Sov. Phys. Solid State32, 1039–1041 (1990) 20. Moore, K.E., Young, W.A.: Phase studies of the Zr-H system at high hydrogen concentrations. J. Nucl. Mater. 27, 316–324 (1968) 21. Numakura, H., Koiwa, M., Asano, H., et al.: Neutron diffraction study of the metastable c titanium deuteride. Acta Metall. 36, 2267–2273 (1988) 22. Pan, Z.L., Puls, M.P.: Internal friction peaks associated with the behaviour of hydrogen in Zr and Zr-2.5 Nb. Mater. Sci. Eng. A442, 109–113 (2006)

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23. Puls, M.P., Shi, S.Q., Rabier, J.: Experimental studies of mechanical properties of solid zirconium hydrides. J. Nucl. Mater. 336, 73–80 (2005) 24. Sidhu, S.S., Murthy, N.S.S., Campos, F.P., et al.: Neutron and X-ray diffraction studies of nonstoichiometric metal hydrides. Adv. Chem. Ser. 39, 87–98 (1963) 25. Simpson, L.A., Cann, C.D.: Fracture toughness of zirconium hydride and its influence on the crack resistance of zirconium alloys. J. Nucl. Mater. 87, 303–316 (1979) 26. Singh, R.N., Ståhle, P., Massih, A.R., et al.: Temperature dependence of misfit strains of dhydrides of zirconium. J. Alloys and Comp. 436, 150–154 (2007) 27. Syasin, V.A., Boyko, E.B., Markin, V.Ya.: Investigations of hardness, internal friction and brittleness of zirconium hydride. Zeitschrift für Physikalische Chemie Neue Folge 164, 1567–1572 (1989) 28. Vaughan, D.A., Bridge, T.R.: High temperature X-ray diffraction investigation of the Zr-H system. J. Metals 85, 528–531 (1956) 29. Veyssiere, P., Rabier, J., Jaulin, J.L., et al.: Instrumental modifications for compressive testing under hydrostatic confining conditions. Rev. Phys. Appl. 20, 805–811 (1985) 30. Xu, J., Shi, S.Q.: Investigation of mechanical properties of e–zirconium hydride using microand nano-indentation techniques. J. Nucl. Mater. 327, 165–170 (2004) 31. Yamanaka, S., Yoshioka, K., Uno, M., et al.: Thermal and mechanical properties of zirconium hydride. J. Alloys Compd 293–295, 23–29 (1999) 32. Yamanaka, S., Yoshioka, K., Uno, M., et al.: Isotope effects on the physicochemical properties of zirconium hydride. J. Alloys Compd 293–295, 908–914 (1999) 33. Yamanaka, S., Kazuriho, Y., Kurosaki, K., et al.: Characteristics of zirconium hydride and deuteride. J. Alloys Compd 330–332, 99–104 (2002) 34. Zuzek, E., Abriata, J.P., San-Martin, A., et al.: H-Zr (hydrogen-zirconium): phase diagrams of binary hydrogen alloys, pp. 309–322. ASM International, Ohio (2000)

Chapter 3

Hydride Phases, Orientation Relationships, Habit Planes, and Morphologies

3.1 Introduction Most of the studies on this topic were carried out from the early 1960s to the mid 1980s. As is described in Chap. 2, hydrides can form as one of three phases, each of which has a different crystal structure and composition. A broad consensus had emerged by the 1980s that the fct c-hydride (c/a [ 1) is a metastable phase while the stable hydride phases are the face-centered-cubic (fcc) d-hydride and the face-centered-tetragonal (fct) e-hydride (c/a \ 1) phases, the stoichiometric compositions of these phases being, respectively, ZrH, ZrH1.5, and ZrH2. However, recent experimental results have reinvigorated an earlier suggestion that the c-hydride phase could be the stable phase below a peritectoid transformation temperature of *286 °C. This issue concerning which phase is the stable one when the solvus concentration is exceeded below the a/b Zr eutectoid temperature of *550 °C is not specifically addressed in this book. However, the issue has not been entirely ignored as various aspects of its role in fracture and phase relationships are raised in various places throughout this book, specifically in Chaps. 2, 6, 7, 8 and 9. Regarding the hydride habit planes, a consensus had emerged by the mid 1980s that the predominant habit of d–and c– zirconium hydrides in Zircaloy-2 and Zr-2.5Nb is on the near-basal, hexagonal-close-packed (hcp) a-Zr {1 0 1 7} planes (14° from the basal plane). The definitive results concerning hydride habit planes and orientation relationships emerged out of the studies of Carpenter et al. [12], Carpenter [13] and Perovic and Weatherly [33–35, 46].

M. P. Puls, The Effect of Hydrogen and Hydrides on the Integrity of Zirconium Alloy Components, Engineering Materials, DOI: 10.1007/978-1-4471-4195-2_3, Ó Springer-Verlag London 2012

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3 Hydride Phases, Orientation Relationships, Habit Planes, and Morphologies

Fig. 3.1 Zircaloy-2: a with g ¼ 0 1 1 0; all dislocations are in contrast. b with g ¼ 0 2 2 0; dislocations at A vanish (from Carpenter et al. [12]; with permission from AECL)

3.2 Hydride Precipitation in a-Zr Alloys: Early Analyses Derived from Observations of c-Hydride Precipitates Carpenter et al. [12], following the early work of Bailey [3] and Bradbrook et al. [6], carried out detailed examinations using transmission electron microscopy of the morphology, orientation, dislocations generated, and strain fields in and around hydride precipitates formed in high purity Zr, Zircaloy-2, Zr-1 % Al, and Zr-1 % Cr. All the hydrides observed by Carpenter et al. [12] in these low-H-containing materials were of the c-hydride phase. Figure 3.1 shows dislocations that had been generated by two needle-shaped hydrides nucleated in close proximity of an intermetallic particle. The dislocations around the hydrides were in the form of loop segments attached to the ends of the needles with Burgers vectors of type \1120 [ lying on or near the basal plane. Figure 3.2 shows schematically in a projection on the (0 0 0 1) plane the hydride orientations, dislocation loops, and their Burgers vectors observed. Large strain fields were associated with these small hydrides as seen in Fig. 3.3 which also shows how this strain field made it difficult to reveal the dislocations generated closest to the hydride’s interface. Similar strong strain contrast was observed along the basal pole (c) axis direction, but none along the needle direction. Similar results for hydride shapes and dislocations emitted were obtained for the Zr-1 % Cr and Zr-1 % Al materials except that the orientations of the c-hydrides in the latter material were in the \1 0 1 0 [ aZr directions. Less regular features for the dislocations generated by hydrides in Zr were found as shown in Figs. 3.4 and 3.5 examined in electron microscopes at 100 kV and 1 MeV, respectively. Nevertheless, these dislocations had similar Burgers vectors as those found attached to the hydrides in Zircaloy-2. Based on the orientation relationships for these hydride needles in Zr and Zircaloy-2 determined by Bradbrook et al. [6], Carpenter [11] calculated the pure lattice transformation strains for these hydrides, shown schematically in Fig. 3.6. The small magnitude of the c-hydride’s transformation strain along the \1 0 1 0 [ aZr direction is consistent with the lack

3.2 Hydride Precipitation in a-Zr Alloys

55

Fig. 3.2 Schematic showing the Burgers vectors of the dislocations generated on the basal plane by c-hydride needles in Zircaloy-2 (from Carpenter et al. [12]; with permission from AECL)

Fig. 3.3 Zircaloy-2: foil normal, z = [ 0 0 0 1 ], g ¼ 2 1 1 0; showing the effect of varying the deviation parameter: (a) Sg = 0, (b) Sg % 5 9 10-3 Å-1 (from Carpenter et al. [12]; with permission from AECL)

of any strain field in the matrix observed along this direction. An analysis by Ashby and Johnson [1] of coherent misfitting spherical precipitates had estimated the minimum magnitudes of the dilatational misfit strain and precipitate diameter of such a precipitate for which dislocations could still be nucleated at the interface. Based on this model of Ashby and Johnson [1], Carpenter et al. [12] found that for the short hydrides of *1 lm in length (and their corresponding cross-sections) the calculated transformation strain of the hydride was at the upper limit for nucleation of dislocations at the interface as a result of this misfit. This is consistent with the

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Fig. 3.4 Dislocation configurations at hydrides in zirconium viewed in a TEM at 100 kV (from Carpenter et al. [12]; with permission from AECL)

Fig. 3.5 Dislocation configurations at hydrides in zirconium viewed in a TEM at 1 MeV (from Carpenter et al. [12]; with permission from AECL) Fig. 3.6 Misfit parameters for c-hydride needles in zirconium (from Carpenter et al. [12]; with permission from AECL)

dislocations observed along the \1 0 1 0 [ directions but not with the lack of them along [0 0 0 1], for which the transformation strain is of similar magnitude. Carpenter et al. [12] rationalized this observation by noting that the energy of dislocations

3.2 Hydride Precipitation in a-Zr Alloys

57

Fig. 3.7 Schematic illustrating dislocation generation on applying the Ashby–Johnson model to c-hydride needles: a shear loop formation, b–d projection along ½1 1 2 0 showing effect of growth, e and f cross slip and g prismatic loop formation (from Carpenter et al. [12]; with permission from AECL)

having Burgers vectors c[0 0 0 1] is *2.5 larger than that of dislocations having Burgers vectors a ½1 1 2 0, assuming no dissociation of the respective dislocations. The mechanism for the generation of the observed shear dislocation loops observed by Carpenter et al. [12] was suggested by them to be similar to that proposed by Ashby and Johnson [1]. This is shown schematically in Fig. 3.7. It can be seen from Fig. 3.7 that the end result would be the production of prismatic loops, a result that had been previously postulated by Bailey [3]. However, Carpenter et al. [12] found relatively little experimental evidence for the generation of these prismatic dislocation loops, suggesting that the proposed mechanism for dislocation nucleation and growth frequently does not run to completion as shown, for instance, in Fig. 3.4a. Since the hydrides were often tapered near the end, it may be that the smaller size of the precipitate at that location results in there being reduced stress available for the first cross-slip step. On the basis of this mechanism, this cross-slip step is necessary for the formation of prismatic loops. In the foregoing discussion it was assumed that hydride formation could be achieved by requiring only the H atoms to diffuse while the Zr atoms would be moved in a shear transformation to their appropriate locations in the fct c-hydride lattice with the passage of 13 \1 1 0 0 [ Shockley partial dislocations on basal planes. In a subsequent paper, Carpenter [13] expanded on these ideas by showing

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Fig. 3.8 A possible mechanism for the homogeneous nucleation of c-hydride:  (a) generation of 0 1 1 0 partial dislocation loops by a metastable hcp nucleus, (b) cross section of resulting fct c-hydride needle (from Carpenter [13]; with permission from AECL)

that the appropriate sequence would be the passage of these partials on alternate (0 0 0 1) planes that would become (1 1 1) planes as shown in Fig. 3.8. This is the reverse of the fcc to hcp transformation postulated by Christian [14] for cobalt. Such a shear mechanism would not normally be possible homogeneously in a onecomponent system because of the large stresses required, but in a two-component system in which one of the components (H) is highly mobile, this may occur more readily by assuming that the first step in the transformation is the formation of a metastable cluster of H atoms at interstitial sites in the Zr lattice. The misfit strain produced by such a metastable cluster having a high H concentration given by rH = 1 for c-hydrides compared to the much lower rH value in the surrounding matrix might then be able to generate stresses that would be sufficient to nucleate the partial dislocations required for this shear transformation. After this, the chemical driving force would cause the glide of these dislocations into the H segregate or, alternatively, the diffusion of the H atoms into the faulted structure. Once nucleated, the misfit strains of the resultant c-hydride precipitate could result in further nucleation of the transformation partials as the hydride grows in size. Explanations as to how a shear transformation based on a dislocation multiplication mechanism could be produced had previously been developed by Cottrell and Bilby [15], based on a pole mechanism, and by Venables [45], based on a ratcheting mechanism. Carpenter examined which of these mechanisms could be operative in the formation of the c-hydrides and concluded that the absence of (c ? a) dislocations rules out the pole mechanism in these well-annealed materials, although this mechanism cannot be counted out for deformed material in which such dislocations are known to exist. However, a ratchet mechanism making use of the more common 13 \1 1 2 0 [ dislocations could work by having a screw segment dissociating on the basal plane as follows (Fig. 3.9):

3.2 Hydride Precipitation in a-Zr Alloys

59

Fig. 3.9 Creation of an fcc structure by a ratchet  mechanism from a 13 1 1 2 0 dislocation, BC: (a) dissociation and the formation of a Shockley partial loop Bb lying in the basal plane, (b) successive loops formed by alternating dissociation/recombination/ cross slip sequences (from Carpenter [13]; with permission from AECL)

1 1 1 ½1 1 2 0 ! ½1 0 1 0 þ ½0 1 1 0 3 3 3 or (using the hcp notation of Hirth and Lothe [23], Fig. 3.10) BC ! bC þ Bb There are then a number of possibilities. One is that the partials may expand to generate a loop on the basal plane by a Frank–Read mechanism before combining with the other partial. The resulting perfect dislocations could then cross slip a distance c on the primary prism plane and repeat the process (Fig. 3.9). This process could not occur if driven by mechanical stress as both partials are glissile, but it would be feasible under the influence of a chemical stress driven by the high hydrogen concentration of a critical H segregate in the hcp lattice from which the transformation initiated. An alternate mechanism could be for a mixed 13 \1 1 2 0 [ dislocation to split into short edge and screw segments, the latter lying on alternate basal planes. After that step, dissociation of the screw segments and expansion of the partials in the basal planes could occur, as before. However, for short segments—of the order of

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Fig. 3.10 The hcp notation of Hirth and Lothe (from Carpenter [13]; with permission from AECL)

Fig. 3.11 Cross section of a c-hydride needle showing how emissary of  dislocations  the type 13 1 1 2 0 may be   formed from the 13 0 1 1 0 transformation partials (from Carpenter [13]; with permission from AECL)

atomic dimensions—this would require a relatively high chemical driving stress. Nevertheless, this mechanism would still appear to be more likely than would homogeneous nucleation. The mechanism is also easier to envisage physically, as the repeated cross slip of a single screw segment is not required. The foregoing shear mechanism is closely related to that for deformation twinning in which emissary dislocations are sometimes observed [44]. The purpose of these dislocations is to reduce the high local strains associated with a large density of dislocations of the same signs that would form at the interface between the hydride and the matrix. The proposed transformation partials could give rise to perfect a-type emissary dislocations (Fig. 3.11) according to the following possible reactions:

3.2 Hydride Precipitation in a-Zr Alloys

61

The resulting partial dislocations would remain at the precipitate/matrix interface. If the partial dislocations all have the same Burgers vector, this shear transformation mechanism would have the effect of producing a cumulative shear of the lattice. Although Carpenter [13] was not able to state, on the basis of these observations, whether any of the hydrides precipitated in these annealed materials had nucleated homogeneously, the large volume and shear strain required for their formation suggest that they would likely have nucleated preferentially at dislocations. This means that a mechanism based on the dissociation of a-type dislocations, as described in the above paragraphs, probably occurred. Carpenter used the results of some further studies of hydride precipitates formed in single crystal zirconium observed by transmission electron microscopy at either 100 kV or 1 MeV to develop this model in more detail. A more extensive and detailed examination of the matrix strain field, the origin of which had previously been assumed to be the result of the dilatational misfit strain, led Carpenter [13] to the conclusion that it was actually associated with the large transformation shear that is required to form the hydride. Moreover, strain contrast changes obtained by varying the reflection vector showed that the lattice strain produced around an individual hydride was neither simply as a result of the dislocations generated during the transformation (and that are generally present in all cases) nor from the dilatational misfit strain. This conclusion was supported by the observation that there was only a relatively weak strain contrast observed around the hydride (Fig. 3.12) in a foil having a zone axis near ½1 0 1 0 in which any strain contrast at right angles to the needle’s long direction (the c-direction) could only be generated by the dilatational misfit strain because there is no shear component in this ½0 0 0 1 direction and there are also no dislocations generated from this interface as a result of this misfit. Carpenter [13], therefore, concluded that any contribution to the strain fields observed in foils with basal orientation that could be caused by the dilatational transformation strain component of the hydride in that direction must also be small. This observation in single crystal Zr is, however, not in accord with a previous finding of c-hydrides in coarse-grained Zircaloy-2, where a large strain was observed as a result of the dilatational misfit of the hydride along the c direction (Fig. 3.13). Carpenter [13] also carried out detailed measurements of the dimensions of the hydride needles observed in these latter experiments. The results of these measurements are that hydrides in pure Zr grew much faster in length than they did in their thickness and width directions. He obtained the following relationships among their length (l), width (w), and thickness (t) directions:

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Fig. 3.12 Characteristic contrast from hydride needles parallel to ð2 1 1 0 with s * 0, zone axis near [ 1 0 1 0 ] and g = 0 0 0 2 (from Carpenter [13]; with permission from AECL)

Fig. 3.13 Zircaloy-2: z ¼   1 2 1 0 ; g ¼ 1 0 1 2; Sg = 0, showing lattice strain due to the misfit along the caxis (from Carpenter et al. [12]; with permission from AECL)

w  3:6  t

ð3:1aÞ

l  270  w5=3

ð3:1bÞ

showing that these hydrides, which were generally described by Carpenter [13] as being needles, were actually sword shaped. Sword-shaped hydrides are consistent with the finding that the misfit strain in the length direction is very small and that loss of coherency in the c–direction is difficult, as seen by the lack of c–component dislocations. It is also consistent with Weatherly’s [46] description (see further on) of these hydrides having a plate-shaped morphology. On the basis of the measured hydride dimensions and the model of the creation of emissary dislocations illustrated in Fig. 3.11, Carpenter [13] estimated the theoretical number of emissary dislocations that should be emitted. He concluded that the resulting emissary dislocations would be three times greater than the

3.2 Hydride Precipitation in a-Zr Alloys

63

Fig. 3.14 Schematic cross-section of hydride needle in which shear dislocations fulfill the dual role of emissary dislocations plus providing partial relief of the dilatational misfit strain (cf. Fig. 3.11) (from Carpenter [13]; with permission from AECL)

theoretical number of shear loops that could be produced by the dilatational misfit strain. Previously, Carpenter et al. [12] had shown that the number of shear loops observed experimentally were always less than the number required to fully relieve the dilatational misfit strain. Therefore, if the observed shear loops are emissary dislocation, then the discrepancy determined previously between the number of dislocations that were observed and the number that would be necessary to relieve the misfit strain becomes even greater. This result supports the foregoing observation of a shear-type strain field around each hydride, showing that most of the partial dislocations on one side of a needle have the same sign, opposite to those on the other side. The final model for the phase transformation and resultant dislocations generated is shown in Fig. 3.14. The initial step is that shown in Fig. 3.8. After this step, shear dislocations are punched out from the interface of the hydride at regions A and B (Fig. 3.14) as a result of the dilatational misfit strain, but only the ones at A have the correct sign to also behave as emissary dislocations because the formation of these types of dislocations at B are opposed by these strains. The model illustrated satisfies the following experimental observations: (a) shear dislocations to either side of the needle are opposite in sign, (b) there is a large, shear-type strain field, (c) the dislocations have either of the two possible a–type Burgers vectors at 60° to the needles and (d) partly or completely formed prismatic loops are sometimes observed, showing that the dilatational misfit strain plays an important role in the dislocation generation process. In addition, it is possible to see why the process of prismatic punching of dislocation loops often does not go to completion since it requires the generation of dislocations at B (Fig. 3.14) having the same sign as the transformation dislocations, thereby raising the energy of the interface. It should be noted that the emissary dislocations have no effect on altering the macroscopic shape change as a result of the formation of a hydride precipitate. This is because their effect is simply to spread out the local shear over a greater volume. This shear strain, s, was estimated by Carpenter [13] to be quite large, given by s = a/2cos 30° = 0.36 over the volume of the precipitate. As precipitates grow in size it seems likely that the resultant large macroscopic shear that could be created in the matrix and transmitted to the surface of the material would be reduced by twinning within the hydride, with roughly equal numbers of twins of opposite signs being formed so that the net shear strain in the matrix would be minimized.

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Fig. 3.15 Specimen of iodide zirconium containing 200 wppm H, step quenched from 430 to 280 °C, held for 360 min and then water quenched to room temperature. The c-hydride precipitate shows distinct internal twinning (from Bradbrook et al. [6])

3.3 Hydride Precipitation in a–Zr Alloys: Determinations of Lattice Transformation Relationships of c-Hydride Precipitates 3.3.1 Introduction Motivated by the pioneering studies of Carpenter and his co-workers, Weatherly [46] examined the possibility that hydride formation could occur via an invariant plane strain transformation. This implies that the net transformation involves only a dilatation normal to the invariant habit plane (there may also be a small uniform dilatation in the plane) plus shears parallel to this plane. Since in each hydride phase the hydrogen atoms occupy tetrahedral interstices in an fcc array of Zr atoms, this means that the Zr atoms must move as well as the H atoms when hydride forms. At the hydrogen concentrations and temperatures of interest for application to pressure tubes, the diffusion rate of Zr would be too slow for the movement of Zr atoms to occur by diffusion. Their movement must, therefore, occur by a displacive or shear transformation process accompanied by the cooperative movement of H atoms by diffusion. Since the latter atoms occupy interstitial sites that are never completely filled, movement of the H atoms by diffusion is expected to be sufficiently rapid for these atoms to move in tandem with the displacive movements of the Zr atoms (see Chap. 5). Indirect evidence that hydride precipitation occurs by a displacive or martensitic transformation is the acicular morphology of the hydrides, the shape change observed at free surfaces and the internal twinning seen in large hydride platelets (Figs. 3.15–3.16, Bradbrook et al. [6]). Previous, partially successful attempts to describe hydride precipitate reactions within the framework of the general theory of martensitic transformation had been made by Bowles et al. [5] and Cassidy et al. [8] of the b-vanadium hydride transformation in vanadium as well as, sometime later, by Cassidy and Wayman [9, 10] of the d ? c and d ? e zirconium hydride transformations. The problem with these analyses concerns the question of how to account for the change in composition that is required, since this requirement is not

3.3 Hydride Precipitation in a–Zr Alloys: Determinations of Lattice Transformation

65

Fig. 3.16 Large c-hydride formed at a grain boundary showing two sets of twins (from Bradbrook et al. [6])

part of the classical martensitic phenomenology. Hydrogen must obviously diffuse to build up to the correct concentration in the hydride phase and it is not clear whether this large build-up in H concentration occurs ahead of, behind, or at the moving interface. This uncertainty concerning the H concentration build-up surrounding and during the hydride precipitation reaction translates into a corresponding uncertainty concerning the parent phase’s lattice parameter at the time of the transformation. This uncertainty affects how much the Zr atoms need to move from their locations in the a-Zr hcp lattice to the appropriate ones in the fcc or fct d- or c-hydride lattices, respectively, in order to complete the displacive transformation. This question about the correct sequence of steps for the requisite H concentration build-up in the a-Zr phase during the transformation to hydride is an important and vexing one that has so far not been resolved. Weatherly [46] based his analysis in large part on transmission electron microscope (TEM) observations that he had made of hydride habits in coarse-grained crystal bar Zr containing less than 200 wppm O and 300 wppm total metallic impurities. This material had been hydrogenated to 40 wppm H at 750 °C, furnace cooled, then annealed for 3 days at 400 °C before slowly cooling to room temperature. These conditions resulted in the formation of large c-hydride plates of length [10 lm and thickness [1 lm that were heavily twinned, while thinner plates were not. No d-hydride precipitates were observed after the foregoing heat treatment. The Zr atoms in c-hydride precipitates form a tetragonal sublattice with the H atoms fully occupying one of every second (1 1 0) plane of the tetrahedral interstices. This means that two energetically equivalent ordered configurations, called domains, are possible for this hydride phase. In his electron microscope examinations of single c-hydride precipitates, Weatherly [46] observed that the hydrides were made up of a patchwork of these domains, with each one being *20 nm in size and remaining ordered up to the solvus temperature of *280 °C (Fig. 3.17). All of the larger precipitates were internally twinned, with the twinning planes and directions given by {1 0 1} and \1 0 1 [ respectively. An internal periodic elastic strain field was also observed within the hydride plates (Fig. 3.18). Weatherly [46] surmised that these strains may be associated with dimensional differences as a result of the different ordering of the hydrogen atoms in the two

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3 Hydride Phases, Orientation Relationships, Habit Planes, and Morphologies

Fig. 3.17 The domain structure associated with ordering of the H atoms in the c-hydride, g ¼ ð1 1 0Þc . The domains (*20 nm in size) are most clearly visible at the edge of the thin foil (from Weatherly [46])

Fig. 3.18 Striations associated with a periodic elastic strain field in a chydride plate, g ¼ ð0 2 2Þ (from Weatherly [46])

types of domains. Weatherly [46] also pointed out that, since these elastic strains are visible even in c-hydride plates that are apparently readily accommodated by the matrix, it suggests that in this low H-containing Zr material the twinning observed may not be a necessary part of the shears and shuffles required to produce an invariant plane strain for this transformation. In his analyses of the observed hydride habit planes and orientation relationships, Weatherly found that of those observed, they could be divided into two groups, one consisting of plates that had habit planes of the type f1 0 1 ‘g where ‘ was typically [5 and the other of plates with f1 0 1 0g habit planes.

3.3.2 f1 0 1 ‘g Habit Planes The f1 0 1 ‘g habit planes were characterized by the orientation relationships ½1 1 0c jj½1 2 1 0aZr and ð1 1 1Þc  jjð0 0 0 1ÞaZr : These relationships were first established by Bailey [3] and are consistent with the shear mechanism for c-hydride formation suggested by Carpenter et al. [12] and Carpenter [13] as summarized in the foregoing. The habit plane is oriented approximately 14° from the (0 0 0 1) pole; i.e., it is very close to the ð1 0 1 7Þ pole, which is 15° from the ð0 0 0 1ÞaZr pole. Weatherly [46] followed Carpenter [13] in proposing that the transformation of a region of the a-Zr matrix to a c-hydride precipitate can be formally considered to occur in two steps. The first step would consist of a small dilatation to bring the c/a ratio of the Zr atom spacing from 1.593 to 1.633,

3.3 Hydride Precipitation in a–Zr Alloys: Determinations of Lattice Transformation

67

the latter being the ideal value for hcp crystals if the Zr atoms are assumed to be pffiffiffiffiffiffiffiffiffiffiffi perfect spheres. The second step would consist of a simple shear of ð2=4Þ in the ½1 0 1 0aZr direction on ð0 0 0 1ÞaZr planes. This shear would convert the hcp Zr lattice into an fcc one having lattice parameter, a0 = 4.574 Å. Finally, a dilatation of the cube axes would be required to give the correct lattice parameters of the fct c-hydride lattice, which are given by a0 = 4.596 and c0 = 4.969 Å [43]. Referred to the x1 ¼ ½1 2 1 0; x2 ¼ ½1 0 1 0and x3 ¼ ½0 0 0 1 axes in the zirconium lattice, the first step is expressed by the transformation matrix: 0 1 1 0 0 S1 ¼ @ 0 1 0:3624 A ð3:2Þ 0 0 1:0251   The foregoing x1, x2, x3 axes become x1 ¼ ½1 1 0; x2 ¼ 1 1 2 and x3 ¼ ½1 1 1 axes, respectively, in the transformed hcp a-Zr lattice, which has, at this point, become an fcc lattice. The second step in transforming this interim fcc lattice is a dilatation and associated small shears given by the transformation matrix: 0 1 1:0048 0 0 S2 ¼ @ 0 1:0592 0:0471 A ð3:3Þ 0 0:0471 1:0320 The net transformation is, then, given by the dot product S2S1 of the two matrices, with the result1. 0 1 1:0048 0 0 S2  S1 ¼ @ 0 1:0592 0:3356 A ð3:4Þ 0 0:0471 1:041 The possible a-Zr habit planes for the c-hydride plates are then found by noting first that the ½1 2 1 0aZr axis is approximately an invariant line since the pure lattice strain given by S2S1 (Eq. 3.4) for the transformation in this direction is less than 0.5 %. It turns out then that there are two possible invariant habit planes having their plane normal directions on this f1 2 1 0gaZr zone. For the first, the habit plane normal lies *12° from the ð0 0 0 1Þ pole and a rigid body rotation in the anticlockwise direction of 3° is needed after the pure lattice strain transformation, given by Eq. 3.4, to produce both an undistorted and unrotated habit plane. In the second case the habit plane normal lies 16° from a ð1 0 1 0Þ pole and a rigid body rotation of 20° is required. This second possibility appears never to have been observed experimentally, which may be because of the large rigid body rotation associated with it. Therefore, it was not analyzed further by Weatherly.

1

Weatherly [46] gives the third diagonal term as 1.0320, which seems to be in error from the dot product of S2S1.

68

3 Hydride Phases, Orientation Relationships, Habit Planes, and Morphologies

The pure lattice strain matrix, eTij , derived from the transformation matrix of Eq. 3.4 is 0 1 0:0048 0 0 eTij ¼ @ 0 0:0592 0 A ð3:5Þ 0 0 0:041 It is useful to compare this pure lattice strain matrix obtained by Weatherly [46] in the foregoing analysis for c-hydride plates having f1 0 1 7g habit planes to those calculated by Carpenter [11] for c-hydride needles having the same orientation relationships (but having ½0 0 0 1aZr habit planes), and with the transformation strains derived from the differences in the unit cell dimensions of the two lattices, viz., 0 1 0:00551 0 0 eTij ¼ @ 0 0:0564 0 A ð3:6Þ 0 0 0:0570   aZr The corresponding volume changes, Xchyd  X derived from the =XaZr Zr Zr Zr strain matrices given by Eqs. 3.5 and 3.6—taking the strains to be finite ones in the calculation of the volume change [40]—are, respectively, 10.79 [46] and 12.25 % [11]. This fairly large difference of 1.46 % in volume increase, which was not noted by Weatherly [46], stems mostly from the greater magnitude of the eT33 component of strain (which is in the ½0 0 0 1 direction of the a-Zr phase) obtained by Carpenter [11]. One reason for the difference is because of the different lattice parameter values used by the two authors. This difference, which is shown further on to be *0.52 %, can be estimated based on the following reasoning. As Carpenter [11] pointed out, the volume change between the two phases calculated in the foregoing based on the dilatational terms of the c-hydride’s transformation strain matrix can also be obtained by calculating the relative difference  in the atomic volumes of Zr in the fct c-hydride and the hcp a–Zr lattices, Xchyd  XaZr =XaZr Zr Zr Zr ; respectively. Carpenter [11] does not quote a

value for the atomic volume of Zr in the c-hydride phase, Xchyd : Using the lattice Zr parameters for a–Zr and c-hydride referenced by him, viz., a0 = 3.231 Å and c0 = 5.146 Å for a–Zr [24] and a0 = 4.617 Å and c0 = 4.888 Å for c-hydride [3],2 the corresponding atomic volumes for Zr in the a–Zr and c-hydride phases chyd 3 are, respectively, 23.262 Å3 (XaZr ). The % volume Zr ) and 26.05 Å (XZr

2

The reference for the lattice parameters of c-hydride given in Carpenter’s paper [11] is incorrectly attributed to Bradbrook et al. [6] (the typographical error in this reference is corrected here). The latter authors actually quote the lattice parameter values for c-hydride determined by Sidhu et al. [43]. Weatherly [46] also uses Sidhu et al’s lattice parameter values in his transformation strain calculations, citing an open literature version of this source [43].

3.3 Hydride Precipitation in a–Zr Alloys: Determinations of Lattice Transformation

increase,



69

 aZr Xchyd  X = 11.99 %, is 0.25 % less than the volume =XaZr Zr Zr Zr

increase obtained from Carpenter’s transformation strain matrix (Eq. 3.6). Calculating the volume change using the lattice parameters a0 = 4.596 Å and c0 = 4.969 Å for c–hydride given by Sidhu et al. [43], which were the lattice parameters used by Weatherly [46] in deriving the pure lattice strain matrix of ˚ 3 The Eq. 3.5, yields for the atomic volume of Zr in c-hydride, Xchyd ¼ 25:93 A Zr   corresponding volume increase Xchyd  XaZr ¼ 11:47 %: This volume =XaZr Zr Zr Zr increase is thus 0.69 % more compared to the volume increase of 10.79 % obtained from the strain matrix of Eq. 3.4 and *0.52 % less than the volume increase of 11.99 % based on the atomic volume difference calculated using Bailey’s [3] c-hydride lattice parameters. This reduction in the volume increase as a result of the different lattice parameter values used by the two authors—which would also be reflected in the calculated dilatational transformation strain values— cannot fully account for the total reduction of 1.46 % in the volume increase obtained from Weatherly’s transformation strain matrix compared to that obtained from Carpenter’s transformation strain matrix. The remaining *0.94 % difference likely arises from an additional dilatational transformation strain increase not included in Weatherly’s [46] pure lattice transformation strain matrix of Eq. 3.5 which is as a result of the rigid body rotation necessary—after the pure lattice strain transformation step—to achieve an undistorted habit plane. This habit plane turns out to lie close to the basal plane of the a–Zr lattice. Since the volume difference between the two lattices cannot disappear and there is no (or very little) dilatational transformation strain in the undistorted habit plane (a small amount of   transformation strain in the eT11 component of 0.44 %, which is in the 1 2 1 0 aZr direction, is retained), almost the entire volume change of the transformation would then be contained in the eT33 dilatational transformation strain component. The total value of the eT33 strain component would then be given by the sum of eT33 and eT22 transformation strain values given in Eq. 3.4 plus the additional strain of 0.94 % that is conjectured in the foregoing to arise from the rigid body rotation. A consistency check on this extra transformation strain to eT33 is obtained by calculating the difference in the volume increase between the c-hydride and the a–Zr phases based on the atomic volume differences of the Zr atoms in the respective phases in comparison to the values derived from the corresponding strain matrices. This difference is about *0.25 % less with the c-hydride lattice parameters used by Carpenter [11] and 0.69 % greater with the c-hydride lattice parameters used by Weatherly [46]. For the difference in the volume increase determined from the two different methods of calculating it to be similar, viz., +0.25 %, there would need to be an increase of 0.95 % in the eT33 component of the transformation strain matrix values determined by Weatherly, which is close to the 0.94 % increase determined from the preceding analysis. The remaining difference in volume increase of 0.52 % arises from the differences in the c-hydride lattice

70

3 Hydride Phases, Orientation Relationships, Habit Planes, and Morphologies

parameters used by the two authors. With this change in Weatherly’s [46] transformation strain matrix, the remaining slight differences in the respective volume increases obtained from the two methods of calculating these volume increases are, then, likely as a result of rounding errors. Hence, the lattice strain matrix for   c-hydride lying on the undistorted plane 1 0 1 7 is given by 0 1 0:00551 0 0 0 0 0 A ð3:7Þ eTij ¼ @ 0 0 0:117 Habit planes close to f1 0 1 7g have been observed by a number of other authors, both for pure Zr and its alloys. The normal to these planes is ½1 0 1 4, which is 15° from the [0 0 0 1] direction, in agreement with the foregoing analysis. The orientation relationships between the two phases are also consistent with these calculated habit planes. Thus, combining the effects of the pure lattice strain with the rigid body rotation leads to the result that ½1 2 1 0aZr jj½1 1 0c , while the angle   between the ½1 0 1 0aZr and 1 1 2 c directions is *1°, which means that the close packed ð0 0 0 1ÞaZr and ð1 1 1Þc planes are almost parallel, in excellent agreement with the results from Weatherly’s [46] electron microscope studies.

3.3.3 f1 0 1 0g Habit Planes The c-hydride plates that were observed to lie on f1 0 1 0g habit planes had one of two orientation relationships: ðAÞ ð0 0 1Þc jjð0 0 0 1ÞaZr ; ½1 1 0c jj½1 2 1 0aZr ðBÞ ð1 0 0Þc jjð0 0 0 1ÞaZr ; ½0 1 1 c jj½1 2 1 0aZr Relationship (A) was also observed by Akhtar [2] in crept single crystals of Zr. Weatherly observed that all of the hydrides having either type (A) or type (B) habit planes and orientation relationships contained twins in their interiors. The twins lay either parallel or perpendicular to the habit plane for those hydrides characterized by relationship (B) (Fig. 3.19) and on two sets of {1 0 1} planes obliquely inclined to the habit plane for those hydrides characterized by relationship (A) (Fig. 3.20). These latter twins were always well contained within the central region of the hydrides so that the regions of the hydride plates lying outside of the twinned regions up to the interfaces of the plates on either faces would end up being rotated relative to the twinned regions. A consequence of this is that a shape change measured over the twinned region of the hydride would be different from that measured over the un-twinned region of the plate; i.e., the misfit strains of these hydrides would not have constant values throughout their interiors.

3.3 Hydride Precipitation in a–Zr Alloys: Determinations of Lattice Transformation

71

Fig. 3.19 Twins on two sets of {1 0 1} planes in a hydride plate with a ½1 0 1 0aZr habit plane (from Weatherly [46])

The observations of hydrides having both f1 0 1 0g and f1 0 1 7g habit planes have their analog in Ti alloys for which it was found that the addition of Al to Ti changes the habit plane orientation from the former to the latter habit planes [21, 29]. This effect of alloying element on the habit plane is mirrored by similar observations in Zr where habit planes ranged from f1 0 1 0g; f1 0 1 5g and f1 0 1 7g in pure Zr [42], to only being on the f1 0 1 7g planes in Zircaloy [47]. Unfortunately, no identification of the hydride phase type (c or d) was made by either set of authors. The foregoing finding—that a realistic partial dislocation mechanism operating on the basal plane would be able to explain the observed f1 0 1 7g habits (and the associated orientation relationships) determined by Weatherly [46] for the c-hydride plates formed in slowly cooled Zr and determined by Carpenter et al. [12] and Carpenter [13] for c-hydride needles formed in rapidly quenched, coarsegrained Zr alloys—motivated Weatherly [46] to investigate whether a similar type of mechanism, based on prism plane slip, could account for hydrides with prism (f1 0 1 0g) habit planes. From a hard sphere model of the hcp lattice it is seen that slip on ð1 0 1 0Þ½1 2 1 0 can proceed by two partial steps. The first step moves the atoms from a close packed (0 0 0 1) array in the hcp lattice into the correct arrangement of a (1 0 0) plane in a (virtual) fcc lattice. Additional shuffles are then required to complete the fcc stacking on planes lying perpendicular to the atom arrangement, as shown in Fig. 3.21a. The shape change associated with this shear is depicted in Fig. 3.21b and can be described as a simple shear of 30° on the ð1 0 1 0Þ plane in a ½1 2 1 0 direction and an expansion of 16 % in the ½1 0 1 0 direction. This shear describes results of the observed orientation relationship given by relation (A). However, a further contraction of 3.5 % is required in the [0 0 0 1] direction to reduce the magnitude of the c lattice parameter of Zr (and of the virtual fcc lattice) to that of the corresponding one of the fct c-hydride’s unit cell. Hence the ð1 0 1 0Þ plane cannot be an invariant one without additional strain. Such additional strain would, first of all, need to nullify the 3.5 % contraction and, second of all, leave unaltered and unrotated a unit vector lying in the ½1 2 1 0 direction. Finally, it would need to result in a macroscopic shape change of the plate that is characteristic of an invariant plane strain with ð1 0 1 0Þ habit and the foregoing orientation relationships.

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3 Hydride Phases, Orientation Relationships, Habit Planes, and Morphologies

Fig. 3.20 a Dark field micrograph, g ¼ ð2 0 0Þc , showing a thin, acicular c-hydride plate with ½1 0 1 0a-Zr habit plane. b Two sets of twins in the central area of the plate seen in (a). Note that the twins do not extend to the hydride/matrix interface. c Same area as (b); the twins are now out of contrast but the relative rotation of the c-hydride lattice inside and outside the twinned region is indicated by the intensity variations within the plate. d Dark field micrograph, g ¼ ð111Þc ; showing a regular set of dislocations (of spacing 6 nm) at the c-hydride/Zr interface. Close examination of this figure suggests that a second set of dislocations, of * 3 nm spacing and lying almost orthogonal to the first set, is also present at the interface. set of dislocations  An equivalent  lies at the opposite interface and could be imaged using g ¼ 1 1 1 c (from Weatherly [46])

From Fig. 3.21b the simple shear as given by 0 1 1 0:5773 0 S3 ¼ @ 0 1:1603 0 A 0 0 0:965

ð3:8Þ

does not meet these requirements, whereas a pure shear—which is equivalent to a simple shear plus a rigid body rotation—as given by

3.3 Hydride Precipitation in a–Zr Alloys: Determinations of Lattice Transformation

73

Fig. 3.21 a Hard sphere model representation of glide on the ð1 0 1 0Þc ½1 2 1 0aZr system showing that slip takes place in two partial steps, b1 and b2. If each atom lying on the ð 0 0 01ÞaZr plane is displaced by b1 (in the sense indicated in the figure), the correct atom arrangement for the ð 0 01Þc plane is obtained. b The shape change corresponding to the b1 displacement illustrated in (a) (from Weatherly [46])

0

1 S4 ¼ @ 0 0

0 k 0

1 0 0 A where k ¼ 0:965

ð3:9Þ

1 k

does meet these requirements. Unfortunately, the TEM images obtained by Weatherly [46] of the hydride platelets having the type (A) orientation relationships lacked sufficient detail to confirm that the shape changes caused by internal twinning and/or by one (or two) sets of dislocations observed at the hydride’s interfaces have the directions and associated slip planes to produce the net shape change given by S4. In addition, the observed internal twinning may actually be associated with the relief of the periodic strain fields that are associated with the existence of a patchwork of the two possible domains that can exist in c-hydride rather than providing a step in the postulated invariant plane strain transformation of the Zr matrix to the c-hydride lattice structure. Although the transformation relationship given by (B) is closely similar to that of (A), it is not as easily analyzed as that of (A). This is because the ½1 2 1 0aZr direction is no longer invariant while the dilatational strain in the [0 0 0 1] direction is now 10.78 %. Therefore, a different combination of shears (after the pure lattice strain) would be required to give an undistorted ð1 0 1 0Þ plane. This difference may partly explain the different twinning orientation relationships found for this transformation variant. Weatherly suggested that the same dislocation mechanism would accomplish most of the shape change of the pure lattice strain

74

3 Hydride Phases, Orientation Relationships, Habit Planes, and Morphologies

and would differ only in the way the hydrogen atoms are ordered in the lattice, being either ahead of, or behind, the moving interface, respectively, and it would differ in the subsequent lattice shears required to produce an undistorted ð1 0 1 0Þ habit plane.

3.4 Hydride Precipitation in a/b-Zr Alloys In a/b pressure tube material, d hydrides are generally found to be the predominant hydride phase at hydrogen contents of practical interest for pressure tubes in service. However, the foregoing analyses apply to the formation of c hydride precipitates only. A similar set of analyses is needed to determine whether the observed habit planes and the associated orientation relationships of d-hydride platelets can be explained by a martensitic-type displacive transformation for which the observed habit plane is an invariant plane. Unfortunately, such analyses have not been done. One might expect the d-hydride to have a similar type of transformation as the c-hydride since Bradbrook et al. [6] and Pegoud and Guillaumin [38] had shown that d-hydride precipitates formed in Zircaloy–2 material had similar orientation relationships as those of the c-hydride plates formed in coarse-grained crystal bar Zr. Similar results for pressure tube material for both types of orientation relationships found in Zircaloy–2 were subsequently also obtained by Perovic et al. [34] while an earlier study by Northwood and Gilbert [28] had found that for the normally occurring un-reoriented as well as for   the stress reoriented d-hydrides only the orientation relationships 1 1 0 d   jj 1 2 1 0 aZr and ð1 1 1Þd  jjð0 0 0 1ÞaZr were observed. By analogy with what Weatherly [46] had shown for c-hydride plates, these orientation relationships similarly imply that the d-hydride precipitates formed in the a/b–Zr alloys have habit planes that lie close to the hcp basal planes. One might further presume that these d-hydride precipitates would have been formed by the same partial dislocation mechanism as proposed by Carpenter [13] for those c-hydride needles that had the same orientation relationships. However, because none of the principal strains of the pure lattice strains of these fcc d-hydrides are zero, further lattice strains—in addition to those proposed for the c-hydride transformation—would be required to make the f1 0 1 7g planes invariant ones. There is, however, little experimental evidence to determine what these further lattice strains should be. Nevertheless, Weatherly contended that the macroscopic shape changes observed by Bradbrook et al. [6] in Zircaloy–2 material containing large d-hydride plates provide indirect evidence that the observed f1 0 1 7g habit planes of these precipitates are likely invariant planes and therefore, by analogy with the corresponding net transformation strain matrix for the formation of c-hydrides, the resultant strain matrix for the formation of d-hydride plates in standard pressure tube material should be given by

3.4 Hydride Precipitation in a/b-Zr Alloys

0

eTij

0 ¼ @0 0

75

0 0 s=2

1 0 s=2 A 3D þ d

ð3:10Þ

where s = 0.36, D = 0.0458, and d = 0.0262. In comparison, the pure lattice transformation strain components determined by Carpenter in previous analyses [11, 13] for the same orientation relationships are given by 0 1 D 0 0 eTij ¼ @ 0 D s=2 A ð3:11Þ 0 s=2 D þ d with D and d having the same values as for the strain matrix given by Eq. 3.10. These transformation strain values are solely derived from the lattice parameter differences between the two unit cells [11] and the associated shear strain in the ½1 0 1 0 direction calculated from the a- to c-hydride transformation (Carpenter [13]). Note that in both cases the dilatational strain components have tetragonal symmetry, but the magnitude of the tetragonal misfit is much weaker for the case given by Eq. 3.11 for which the habit plane is not an invariant one. Note also that the tetragonal misfit given by Eq. 3.11 derives entirely from the tetragonal shape of the surrounding hcp lattice since the d-hydride phase has cubic symmetry, whereas for the c-hydride phase, with the same habit plane, it derives from both the latter source and the tetragonal shape of the hydride’s fct crystal structure. The tetragonal misfit for this phase is, therefore, stronger even though the magnitude of the volume misfit is less. Weatherly’s [46] analysis of the habits and orientation relationships of c-hydrides formed in large-grained Zr was subsequently applied by Perovic et al. [34] to fine-grained Zr–Nb pressure tube alloys to explain the macroscopic observations obtained from optical metallography of the morphology and orientation relationships of d-hydride precipitates in these types of alloys when hydrides are formed in the presence and absence of externally applied stress. An explanation for these observations was required because it appeared, at first glance, that the foregoing results concerning the orientation relationships and habit planes of c-hydride precipitates in coarse-grained alloys would not be applicable to the d-hydride precipitates observed by optical metallography in these fine-grained Zr–Nb pressure tube alloys. In these alloys, the long hydride stringers observed by optical metallography in slowly cooled, externally unstressed material were extended in the axial (also called longitudinal) direction of the tube, while the crystallographic texture of the material was such that if the hydrides grew on habit

planes close to 1 0 1 7 ; as found, for instance in coarse-grained Zircaloy–2 material [16, 47], most of the hydride stringers should be aligned in the radial direction. This is because there is about twice the fraction of basal planes—which

are inclined by *15° to the 1 0 1 7 habit planes—that have their normal oriented in the transverse (also referred to as the circumferential) direction in these pressure tube material than there are corresponding basal planes with their normal

76

3 Hydride Phases, Orientation Relationships, Habit Planes, and Morphologies

Table 3.1 Summary of microstructural properties of cold worked Zr-2.5Nb and XL pressure tube material Alloy

Principal alloying elements (in weight %, except as otherwise indicated)

Microstructure

a-Zr grain size (axial 9 radial) (lm2)

Zr-2.5Nb

2.4–2.8 Nb, 900–1 300 wppm O

7 9 0.4

0.32

0.62

0.06

XL

0.82 Nb 3.31 Sn 0.82 Mo 1 130 wppm O

a Grains elongated in axial-transverse directions of tube, surrounded by a thin network of b Same as for Zr-2.5Nb

7 9 0.7

0.26

0.65

0.09

Average basal pole components:

Radial Transverse Axial

oriented in the radial direction. (Table 3.1 provides typical data for the resolved basal pole fractions in the radial, transverse, and axial directions, respectively, of pressure tubes.) Related to these observations of the ‘‘natural’’ orientation of hydride stringers in pressure tubes are the observations that hydride precipitates formed under a sufficiently large applied tensile stress in the transverse direction would change their orientation and form hydride stringers extending in the radial direction. A key observation relevant to resolving this apparently different behavior of hydrides in these two types of materials (coarse-grained Zr and its alloys, versus fine-grained Zr–Nb a/b pressure tube material) was that in both types of material higher magnification observations had shown that the apparently long solid hydride stringers observed in etched samples by optical metallography were actually composed of stacks of smaller hydrides often separated by a-Zr matrix material [6, 47] (Fig. 3.22). Based on a series of studies by Perovic et al. [30–32] concerning the stability and autocatalytic nucleation of plate-shaped precipitates, Perovic et al. [33] calculated the variation of the interaction energy of hydride platelets having tetragonal dilatational misfit strain—modeled by assuming that all of the volumetric dilatational misfit strain is oriented along the plate normal direction—combined with a simple shear strain in the plane of the plate, as a function of plate separation (stacking geometry) and orientation (stacking angle) to determine which stacking was energetically favorable. The results of these calculations show that a displacive, martensitic-type transformation energetically favors an auto-catalytic nucleation process for which the tensile stress field generated in the surrounding matrix by one hydride as a result of its stress-free dilatational and shear misfit strains aids the nucleation of a second, smaller hydride at some close distance and plate orientation to the first. The angle of stacking of such a chain of hydrides is dependent upon the relative magnitudes of the net shear and dilatational strain components of the displacive transformation. These calculations predict that two types of stacking arrangements represent lowest energy

3.4 Hydride Precipitation in a/b-Zr Alloys

77

Fig. 3.22 Stacks of aligned hydrides in Zircaloy-2 viewed at high optical magnification (from Westlake [47])

configurations. These are (i) parallel arrays of slightly offset overlapping hydrides and (ii) hydride plates in close proximity to each other or in contact at their ends at angles of either 30 or 60° as shown in Figs. 3.23a–d [33]. The observed hydride stacking configurations are in agreement with these predictions (Figs. 3.24, 3.25, 3.26, 3.27). These observations provide additional qualitative confirmation that hydride precipitation has similarities to a martensitic-type, displacive transformation.3 However, it should be noted that these observations do not allow one to distinguish which of the two misfit strain matrices given by Eqs. 3.10 and 3.11 apply in the case of d-hydride precipitation on near-basal-plane habit planes, since both of them exhibit at least two of the three features of displacive, martensitictype transformations—viz., tetragonal dilatational misfit strain in the normal direction of the habit plane and shear strains parallel to the habit planes—although, of course, the tetragonal misfit of the hydride transformation strain given by Eq. 3.10 for which the habit plane is an undistorted one is much stronger. In addition, as discussed in Chaps. 2 and 8, the strain energy of a hydride formed 3

Note the observation of hydrides in the b phase shown in Fig. 3.24. The significance of this is brought out in Chap. 6, dealing with the thermodynamic relationships for the Zr–H solvus.

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3 Hydride Phases, Orientation Relationships, Habit Planes, and Morphologies

Fig. 3.23 a The interaction energy of large and small shear loops as a function of the rotation angle of the smaller loop and horizontal separation of the two loops (from Perovic et al. [33]). b As in Fig. 3.23a, but the center of the smaller loop is displaced by 0.05l along the x3 axis (from Perovic et al. [33]). c The interaction energy of large and small prismatic loops as a function of the rotation angle of the smaller loop and the horizontal separation of the two loops (from Perovic et al. [33]). d As Fig. 3.23c, but the center of the smaller loop is displaced by 0.05l along the x3 axis (from Perovic et al. [33]). e The interaction energy of large and small prismatic loops as a function of the vertical displacement of the smaller loop (along the x3 axis) for values of h = 0°, 30° and 60° (from Perovic et al. [33])

according to the invariant plane strain transformation given by Eq. 3.10 would have considerably lower elastic strain energy. The foregoing predictions of the likely stacking arrangement of the hydride precipitates assume that there are no external or internal stresses acting on the material. Perovic et al. [34] studied two types of fine-grained a/b Zr–Nb pressure tube material, viz., standard cold-worked Zr-2.5Nb pressure tube material used in CANDU reactors and tube material made from an experimental alloy, Excel (XL), which was being considered at the time for use in commercial reactors. The alloys were subjected to two sets of heat treatments, the first being designed to minimize any internal stresses. Reducing these internal stresses made it possible to isolate the effect of changes in the a-bZr interface of these two-phase alloys. The other alloy studied had a microstructure and internal stress state representative of that of pressure tubes for use in commercial reactors. The objective of the choice of these two types of material and external conditions was to investigate the effect that microstructure and internal and external stresses have on the orientation relationships during hydride nucleation and growth and, through this, to be able to

3.4 Hydride Precipitation in a/b-Zr Alloys Fig. 3.23 b continued

Fig. 3.23 c continued

79

80

3 Hydride Phases, Orientation Relationships, Habit Planes, and Morphologies

Fig. 3.23 d continued

Fig. 3.23 e continued

3.4 Hydride Precipitation in a/b-Zr Alloys

81

Fig. 3.24 Zr-2.5Nb containing 80 wppm H showing inclined stacks of parallel hydrides in the b phase (from Perovic et al. [33])

reconcile the observed macroscopic with the microscopic hydride orientation relationships. A brief description of the manufacturing process of the pressure tubes produced from the two types of alloys is given by Perovic et al. [34]. Both alloy materials were manufactured using similar thermo-mechanical treatments to produce the tubes, the most significant step of which in both tubes is the tube’s extrusion at a temperature of around 850 °C. This high temperature extrusion step is responsible for producing most of the pronounced texture of the two materials, the result of which is summarized in Table 3.1. Both tubes received a final cold work treatment with the total amount of cold work differing between the two types of tubes. The standard Zr-2.5Nb pressure tube material received the usual cold work treatment, which is specified to produce a total strain in the range from 20 to 30 % and which is achieved in two cold drawing steps, while the XL tube received a single cold draw of \5 % strain [39]. After the cold drawing, the microstructure of the two pressure tube alloys consisted of elongated a-Zr grains surrounded by an almost continuous film of metastable, retained b-Zr phase consisting of *20 wt% Nb. The volume fraction of this phase is *10 %. This metastable phase starts to decompose during service, but is generally already slightly decomposed as a result of the final 24 h/400 °C autoclaving treatment that pressure tubes receive prior to being put into service. The b-Zr phase decomposes first to the x-phase which then decomposes to the b-Nb phase, becoming progressively richer in Nb concentration with the final, equilibrium concentration of this bcc phase being 85 wt% Nb. The decomposition process is readily evident in metallographic observations of the material as the interface between the a and b phases becomes less smooth, gradually breaking up over time at elevated temperatures into discrete particles of the x and/or b-Nb phases. An example of the elongated grain structure and the thin, smooth layer of the b-Zr phase coating these a-Zr grains of Zr–2.5Nb pressure tube material—which had not yet received a cold-working and subsequent autoclaving treatment—is shown in Fig. 3.28 [34]. The figure also shows a thin d-hydride precipitate growing along the a-b interface in the a-phase. Figure 3.29 shows the same material after receiving 30 % cold work, a 24-h

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Fig. 3.25 As Fig. 3.24; nucleation of a small hydride plate of the same variant as the larger plate. Note the offset between daughter and parent phases (from Perovic et al. [33])

Fig. 3.26 Pure Zr, containing 30 wppm H, 32 % cold work, showing arrays of hydride plates. The habit plane of the two smaller plates lies at approximately 30° to that of the larger plate (from Perovic et al. [33])

Fig. 3.27 As Fig. 3.26; but with the angle between the habit planes of the plates approximately 60° (from Perovic et al. [33])

autoclaving treatment at 400 °C followed by 15 min at 500 °C. The figure shows that the decomposition of the b-Zr phase has caused the interface to become wavy

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Fig. 3.28 Elongated a-b grain structure of as-extruded Zr-2.5Nb pressure tube alloy. Also shown in this micrograph is a d-hydride plate (arrowed) which was nucleated at the a–b interface and grown in the a-phase along the interface (from Perovic et al. [34])

and a d-hydride plate-shaped precipitate has nucleated at the a-b interface (which could be a/x or a/b-Nb) and grown intra-granularly across the grain, as opposed to the previous case where it had grown inter-granularly along the interface. All of the hydrides growing along the a/b-Zr interface in the as-extruded or heat-treated alloys had one of two orientation relationships with the a-phase into which they grew. In the first, the orientation relationships were ½1 1 2d jj½1 1 2 6aZr while ð1 1 1Þd jjð2 0 2 1ÞaZr . For this relationship the closepacked ½0 1 1d and ½1 1 2 0aZr directions are almost parallel, while ð1 1 1Þd is at an angle of &8° to the ð0 0 0 1ÞaZr plane. For the second relationship, the orientation relationships were ½1 1 4d jj½1 12 6aZr and ½1 1 0d oriented 3° from the ½1 1 0 0aZr direction. For this case the ð0 0 1Þd and ð0 0 0 1ÞaZr planes were nearly parallel. These orientation relationships are similar to those found earlier by Weatherly [46] for c-hydrides in coarse-grained Zr. However, the habit planes of the hydrides growing along the a/b interface were not those found in coarsegrained, single-phase material such as Zircaloy–2, in which they were found to have f1 0 1 7gaZr habit planes. This is because, by following along the a/b interface, they must have habit planes that are perpendicular to the ð0 0 0 1ÞaZr plane as opposed to the hydrides in Zircaloy–2 having orientation relationships given by ½1 1 0d  jj½1 1 2 0aZr and ð1 1 1Þd  jjð0 0 0 1ÞaZr , which are similar to those of the foregoing first, more commonly observed, orientation relationships in the a/b alloys. However, as soon as the a/b interface starts to decompose, there is a marked change in the hydride habit planes since now the hydrides, although still nucleating at the a/b interface, grow across, not along, the a-grains and their habit planes now lie close to the basal planes as expected on the basis of their similar orientation relationships to those found in Zircaloy–2 and unalloyed Zr.

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Perovic et al. [34] speculated that the change in the growth behavior of the dhydrides with the change in the nature of the a/b interface likely reflects the different contributions that the elastic and surface energy terms make to the total energy of the hydrides. Thus in the alloys having un-decomposed a/b interfaces, the trans-granular hydride plates would have a larger surface energy compared to hydrides growing along the a/b interface. However, this would be offset by a lower strain energy provided that the conjecture of Weatherly [46] is correct and d-hydride formation within the a-Zr grains is by a displacive transformation consisting of shear parallel to an undistorted habit plane and uniaxial dilatation normal to that plane, the magnitude of this uniaxial strain accounting totally for the entire volume change that is given by the difference in unit cell volumes between the two phases. The habit planes of the similarly oriented d-hydride plates growing along a/b interfaces may, however, not be completely undistorted, which would increase the total strain energy of these plates compared to those hydrides having undistorted habit planes, but these inter-granular hydrides would have *50 % lower surface energy because one of their plate surfaces would simply be a replacement of the original a/b grain boundary interface with the hydride’s a/d interface resulting, likely, in a negligible surface energy difference. The hydrides shown in Figs. 3.28–3.29 are plates of a single orientation, but in similar material with increased hydrogen concentration, the hydrides precipitate in stacks and their macroscopic habits visible in optical micrographs as seen in Fig. 3.30a do not necessarily reflect their crystallographic habit planes as seen in Fig. 3.30b. Extant models of hydride solubility, hydride-induced overall fracture of a component containing a uniform distribution of hydrides and DHC have so far been formulated in terms of continuum models in which the macroscopically observed hydride clusters are modeled as having dimensions and morphologies that are idealizations of the actual shape of the hydride cluster assuming, additionally, that these shapes are completely made of hydride. The orientation and habit planes of these idealized hydride clusters are then obtained by determining the relationships between the orthogonal coordinates characterizing their shapes with the three orthogonal tube (or specimen) directions. The associated hydridematrix misfit (transformation) strains of these idealized, solid hydride clusters govern their stress states and their orientation relationships under external or internal stresses. To ensure that this approach is valid, these misfit strains need to be linked to the underlying microscopic ones of each individual hydride precipitate in the cluster by taking account of, or determining, their observed or expected stacking sequence. As demonstrated above, the microscopic transformation strains of an individual hydride in a cluster is derived from knowledge of its crystallographic orientation relationships, habit plane, hydrogen concentration, internal microstructure, and the mechanism of its formation. The relationship among the apparent orientations of the macroscopically observed hydride clusters in terms of the possible stacking arrangements, orientation relationships, and misfit strains of the individual hydrides in the clusters have been elucidated by Perovic et al. [34] as summarized in the following.

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Fig. 3.29 Zr-2.5Nb pressure tube alloy, extruded, cold worked 30 % and annealed for one day at 400 °C and 15 min at 500 °C. The decomposition of the retained bZr phase has caused the a=b interface to become wavy; the d-hydride plate (arrowed) nucleated at the interface and grew across the a-grain (cf. Fig. 3.28 where the hydride grows along the a=b interface) (from Perovic et al. [34])

Fig. 3.30 Precipitation of hydrides in coarse-grained Zr-2.5Nb, heat treated and doped with hydrogen (190 wppm) using the same procedure adopted for the XL alloy: a Optical micrograph showing macroscopic hydride plates. b TEM micrograph showing stacks of hydride plates. The individual plates in each stack are *100 9 10 nm in size (from Perovic et al. [34])

In pressure tube alloys the metastable, retained b-Zr phase has already begun to decompose during the final autoclave treatment that these tubes receive prior to reactor installation. Based on the results presented in the preceding paragraphs for

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annealed material in which this decomposition process of the b-Zr phase has begun, trans-granular hydrides would be expected. Since the pressure tube’s grains are elongated in the axial–transverse directions and the majority of the basal poles in these grains are oriented in that direction, one would expect the hydrides to be oriented with their platelet face normal in the transverse direction of the tube. However, the opposite is observed. Examining these macroscopic hydrides in a transmission microscope, it can be seen that they are made up of stacks of smaller hydrides running across elongated a-Zr grains. Figures 3.31, 3.32, 3.33 show examples of the types of stacking observed. The habit planes of these stacks are at an angle \30° to the a/b interface with their normal to the habit plane always close to the ½0 0 0 1aZr (c–axis) direction so that the orientation of the grains in which these stacks of hydrides are observed to grow have their c–axis oriented in the radial direction of the tube. Hydrides grown under a sufficiently high tensile stress in the transverse direction were found in optical micrographs to have their habit planes in the radial–axial plane. Figure 3.32a shows such hydrides for a radial–transverse section, while a corresponding TEM micrograph (Fig. 3.32b) shows that these stacks of hydrides now cross many grains, and appear to have nucleated at the interface of each grain boundary having similar basal pole orientation, although slightly offset from each other in the adjoining grains. These hydrides had grown in a similar fashion to those reported in single phase Zr alloys, suggesting again that nucleation of hydrides in these stacks proceeded by an autocatalytic nucleation process [31, 32]. The reoriented hydrides as well as those precipitated under zero external stress that had grown in length along the transverse direction were found to have similar orientation relationships and to lie on similar habit planes as did the hydrides observed in Zircaloy-2 material, i.e., ½1 1 0d  jj½1 1 2 0aZr and ð1 1 1Þd  jjð0 0 0 1ÞaZr with habit plane close to ð0 0 0 1ÞaZr : Therefore the stress-reoriented hydrides had grown in grains in which the c-axis was nearly parallel to the transverse direction. A similar observation had previously been made by Northwood and Gilbert [28]. Perovic et al. [34] provide a mechanistic explanation for these results by noting that in the manufacturing process of the tubes, the high temperature extrusion step produces a high level of grains with their c-axis in the transverse direction while producing large residual stresses the sign of which is such that the basal planes are either under compression in grains where the c-axis is nearly parallel to the transverse direction or are under tension in those grains where the c-axis is nearly parallel to the radial direction. The result is that for the former grains the compressive stresses would oppose hydride formation in the radial direction until there is a sufficiently large opposing tensile stress while in the latter case it aids the formation of the hydrides growing in the transverse direction (i.e., having their plate normal in the radial direction). The origin of the internal stresses could arise from grain interaction stresses that are associated with the plastic anisotropy of zirconium or it could be the result of plastic incompatibility between the a and b phases when the tube is cold drawn.

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Fig. 3.31 Longitudinal section of Zr-2.5Nb pressure tube material stress relieved for 6 h at 500 °C showing stacks of hydride plates. a In the bright field TEM micrograph the hydride appears to have a morphology similar to that seen in the optical microscope, but in dark field, b using a hydride reflection, individual plates in the stack can be seen. An earlier stage in the formation of these stacks is shown in c where two variants of the plate have nucleated (from [34])

Measurements and analyses of the former source of internal stresses—which would vary from grain to grain such that, overall, they cancel to zero—have been made by McEwen et al. [27] for cold-worked Zircaloy-2 plate material. These stresses are of the order of ±100 MPa. However, no comparable analyses are known to the author for the case of pressure tube alloys. The second mechanism proposed by Perovic et al. [34] for generating internal stresses arises from the plastic incompatibility between the a and the b phases. This mechanism comes into play when the material is cold worked. The plastic incompatibility is the result of the b phase being harder than the a phase and the latter’s plastic response, therefore, would be less than that of the a phase. During plastic deformation as a result of cold working, then, this difference in plastic response induces a positive misfit strain on the a phase that produces a compressive stress in the axial direction of the tube in this phase that is proportional to the induced misfit strain between the two phases. Assuming that the b-Nb phase has a plate-shaped morphology in between of which resides the a-Zr phase, Perovic et al. [34] used the analysis of Brown and Clark [7] who applied formulations derived from Eshelby’s [17–19] treatment of misfitting inclusions to

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Fig. 3.32 Transverse section of Zr-2.5Nb pressure tube material with hydrides reoriented under a tensile hoop stress of 400 MPa. a Optical micrograph. b TEM micrograph showing stacks of offset hydride plates (arrowed) running across a-grains. c diffraction patterns from d-hydride and a-matrix showing ð1 1 1Þd jjð0 0 0 1ÞaZr     and 1 1 0 d jj 1 1 2 0 aZr (from Perovic et al. [34])

derive a simple expression for this compressive stress. The stress is in the axial direction and given by the expression 2Gf eb =ð1  mÞ where G is the shear modulus and m Poisson’s ratio of the a phase, f is the volume fraction of the b phase, and eb is the induced misfit strain that is the result of the difference in plastic

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Fig. 3.33 Stacks of hydride plates in pure Zr, loaded with 30 wppm H, cold worked 16 %, and annealed for 1 h at 300 °C followed by furnace cooling (from Perovic et al. [34])

response in the longitudinal direction of the elongated a and b phases. An estimate of the magnitude of this stress depends on having a good estimate of the induced misfit strain, eb, in the a phase, which was not given by Perovic et al. [34]. Not obvious on the basis of this model is how much of this internal stress would be retained during the service life of the tube since the b phase will decompose and change its shape and composition during its service life. Based on the foregoing reasoning of the source, magnitude, and likely orientation of internal stresses produced by cold work in these a/b alloys, Perovic et al. [34] formulated a rationale for reconciling the observed macroscopic and microscopic orientations and habit planes of hydrides precipitated in internally and externally stressed pressure tubes. Figure 3.34 is a schematic showing hydride stacking arrangements that have been observed experimentally out of the many possible ones. All arrays shown have the normal to the hydride habit plane lying at 20° to the basal pole direction in each grain. Based on minimizing the interaction energies between hydrides in a string, as determined by Perovic et al. [34], the arrays in stacks (a), (c), and (d) are favored, since all these have interaction angles \30°, while arrays (b) and (d) are elastically repulsive because of their high interaction angles close to 90°. Array (a) can form in a single, long grain, so that even having a relatively smaller number of grains with basal poles oriented in the radial as opposed to the transverse direction—which is required to minimize their formation energies—favors their formation. In contrast, arrays (c) and (d) would require hydrides to nucleate and grow across successive adjoining grains. It may be difficult for this to occur—even when the adjoining grains have similar texture—if hydride renucleation in each grain requires the presence of an undecomposed b phase. In addition, finding successive layers of these grains, all of which have their c-axis in the radial direction–which is what would be required for array (c) to be a likely configuration—seems unlikely, since, as already noted, only approximately a third of all the grains in the material have their resolved basal pole in this direction. However, array (d) would be favored since finding clusters of parallel grains having their c-axes in the transverse direction would be expected for this texture orientation having, on average, 2/3 of the grains in the alloy with texture in this orientation. These considerations suggest that the most likely hydride stacking arrangements are

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Fig. 3.34 Possible stacking arrays of hydride plates in pressure tube alloys as viewed, a in a longitudinal section and, b in a transverse section. Note that all the arrays are shown in grains with the normal to the habit plane lying at *20° to the basal pole direction in each case (from Perovic et al. [34])

those given by arrays (a) and (d). However, in an externally unstressed pressure tube, hydride stringers forming according to array (d) would not be likely because of the compressive stresses existing in the axial/transverse directions. On the other hand, the stacking given by array (a) is favored since the component of residual compressive stress as a result of cold working in the direction of the habit plane of the hydrides in these stacks is much smaller than it is acting on the hydrides in the stacks oriented according to array (d). Only when a sufficiently large tensile stress is imposed externally in the transverse direction would array (d) be favored as it would require less work for the hydrides to precipitate in this orientation compared to when they are in the orientation given by array (a). These physical arguments, then, explain why, normally, hydride stringers are observed to lie along transverse/axial directions. Only when these stringers were formed in material that had been under a sufficiently large external tensile stress in the transverse direction during their precipitation are they observed to lie along the radial direction.

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3.5 Hydride Nucleation Studies in a/b Zr-2.5Nb Pressure Tube Material The previous studies focused on the role of prior thermo-mechanical treatment on hydride precipitation and growth in a/b Zr–Nb alloys. These studies showed the importance that the state of the b phase has on the choice of nucleation sites and how this choice influences the subsequent mode of growth of the hydrides. To understand this better in pressure tube alloys used in CANDU reactors, a follow-up study by Perovic and Weatherly [35] focused on a more detailed analysis of the state of the b phase in such pressure tube material.4 Two thermo-mechanical states were studied. In the first, the material used had been as-extruded at 850 °C, then air cooled to room temperature, while in the second it had been as-extruded at 850 °C, cold worked *40 %, then stress relieved at 400 °C for periods up to 24 h (the latter is the standard treatment given to pressure tubes) with some samples given a further heat treatment at temperatures that ranged from 500 to 850 °C to develop a more equi-axed, two-phase a/b grain structure.

3.5.1 Microstructure of Zr-2.5Nb Alloys The as-extruded material slowly cooled from 850 °C contained a two-phase microstructure. The a phase was hcp Zr with lattice parameters a0 = 3.23 Å and b0 = 5.15 Å as quoted by Carpenter [11]. The b phase in this material consisted of retained b Zr having a bcc structure and containing 20 wt% Nb. The Zr-2.5Nb alloy has a bcc b Zr crystal structure above 850 °C but on cooling below this temperature the hcp a-Zr phase is formed. The b- to a-Zr transformation follows a near Burgers vector orientation relationship, viz., ð0 0 0 1ÞaZr  jjð1 1 0ÞbZr and ½1 1 2 0aZr ffi ½1 1 1bZr [49]. The misfit at the a/b interface is accommodated by a single set, or a combination of two or three different sets, of dislocations. (The misfit at the habit plane is accommodated by a single set of ½0 1 0bZr dislocations lying along the invariant line. At the side facet planes a combination of two or three different sets of dislocations, ½1 1 1bZr ; ½1 1 1bZr and ½0 0 1bZr —again lying along the invariant line—are required to relieve the misfit [49].) Misfit dislocations also form step structures and these play a dual role in growth or dissolution as well as in providing relief from the transformation strains of the b to a phase transformation. Typical microstructures of the a/b interface containing regular arrays of misfit dislocations and steps are shown in Figs. 3.35 and 3.36, respectively. Both defects could be of importance to hydride nucleation.

4

A more detailed study in a heat-treated Zr-2.5Nb alloy of the interface formed between the retained b-Zr(bcc) and the surrounding a-Zr matrix phases has been described in a number of papers [36, 49, 50].

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Fig. 3.35 Arrays of misfit dislocations at the interface between a and retained bZr in Zr-2.5Nb, annealed for one day at 820 °C, furnace cooled to 650 °C, and helium quenched (from Perovic and Weatherly [35])

The orientation relationship between the two phases was Pitsch-Schräder (Perovic and Weatherly [35]) as follows: ð1 0 1 0ÞaZr jjð1 1 0ÞbZr ; ½1 2 1 0aZr jj½0 0 1bZr . The authors used the matrix of transformation between the hcp and bcc crystal lattices given by Kelly and Groves [20] for these orientation relationships, viz., 0 1 0:085 0 0 @ 0 D1 þ0:116 0 A ð3:12Þ þ I ¼ 0 0 þ0:03 relative to the axes ½0 1 1 0; ½2 1 1 0 and ½0 0 0 1 as the basis of the description of the interface between the two phases. This description was then used to show that there is a direct relationship between the nature of the interface and its ability to act as nucleation site for hydride precipitation. Two important defects can be seen at the interface shown in Figs. 3.35 and 3.36 that could be of importance to hydride nucleation. The first is an array of ½0 0 0 C dislocations accommodating the 3 % misfit between the ð0 0 0 2Þ planes in a–Zr and ð1 1 0Þ planes in b-Zr (Fig. 3.35). The second defect observed at the interface is a ledge structure that shows evidence of a strain field at most of the ledges (Fig. 3.36). Upon air cooling from 750 °C, or on annealing at 400 or 500 °C, the bZr phase becomes unstable and decomposes either athermally or isothermally according to bZr ! x þ br ðat temperatures below  450 CÞ or bZr ! a þ b0Nb þ br ! a þ b0Nb ðat temperatures below 450 CÞ The br phase becomes partially enriched in Nb. The lattice parameter of this phase decreases from 3.53 to 3.47 Å after annealing for 40 h at 400 °C, corresponding to an increase from 3 to 5 % in the misfit between the ð0 0 0 2ÞaZr and ð1 1 0ÞbZr planes. The x phase has hexagonal structure with its lattice parameter also dependent on heat treatment. After the same heat treatment at 400 °C, the x phase lattice parameters are a0 = 5.035 and c0 = 3.13 Å [22].

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Fig. 3.36 The same conditions as in Fig. 3.35, showing steps at the a=bZr interface with associated strain fields (arrowed) (from Perovic and Weatherly [35])

The equilibrium bNb phase containing *85 wt% Nb that is obtained after prolonged heat treatment has a lattice parameter close to that of pure Nb, viz., a0 = 3.34 Å. Two important microstructural changes accompany these phase transformations. The regular dislocation arrays observed at the a/bZr interface, as shown in Fig. 3.35, are no longer visible and the interface has changed from being faceted, with steps, to an irregular morphology with pronounced perturbations. The second change is the precipitation and growth of a high volume fraction of x particles (Fig. 3.37) within the b-Zr phase. The orientation relationship between the br and x phases found by electron diffraction is that commonly reported for this precipitation reaction, viz., ð1 1 1Þbr jjð0 0 0 1Þx ; ½1 1 0br jj½1 1 2 0x : The stress-free transformation strain describing the structural change br ! x is a nearly uniform dilatation. Taking the values of the lattice parameters quoted above, it is found that in the \ 1 1 0 [ br directions lying on the ð1 1 1Þbr plane, the expansion is 3 %, while in the orthogonal ½1 1 1br direction, it is 4 %. As a result, the x particles would be in a state of compression, while the surrounding br lattice would be in tension. It is evident from Fig. 3.37 that the decomposition of the retained bcc br phase, even though it results initially in only a small reduction in the lattice parameter, has a marked effect on the structure of the interface. The boundary develops pronounced perturbations (Fig. 3.37) and all evidence of epitaxial dislocations and steps with associated strain fields disappear. This result is rather surprising since the orientation relationship between the a and b phases remains unchanged while the lattice parameter of b decreases by only 2 % on annealing at 400 °C. Nevertheless, it is shown in Sect. 3.5.2 that this has an important effect on the sites and modes of hydride nucleation and growth. The interface appears to behave in these decomposed states as though the interfacial energy is now isotropic, and there are no longer any appreciable strain fields at the interface to attract hydrogen in sufficient numbers to result in hydride nucleation and growth.

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Fig. 3.37 Cold worked pressure tube alloy (as extruded +30 % cold work, annealed for 24 h/400 °C). Longitudinal section showing the breakup of the retained bZr phase into x þ br ; no epitaxial dislocations are seen at the interface between a and br (from Perovic and Weatherly [35])

3.5.2 Hydride Precipitation in a/bZr Microstructures Hydride nucleation in these structures is observed to occur at the a/bZr interface. At the earliest stage of the precipitation, a series of small plates or needles (10 nm in size) are found decorating the steps at the interface (Fig. 3.38). These plates are often stacked in arrays, similar to those observed in coarser, intra-granularly nucleated hydrides as shown in the previous section. The hydrides visible in Fig. 3.38 were too small to permit a positive identification of their crystal structure (c or d hydride), but larger hydrides (Fig. 3.39a) were identified as being d hydrides with an approximate ð1 1 1Þd jjð0 0 0 1ÞaZr and ½1 1 0d jj½1 1 2 0aZr orientation relationship. The hydrides that had nucleated at the a/b interface with a common habit plane and orientation relationship coalesced on continued growth into either a striated or monolithic plate as shown in Fig. 3.39b. This observation of the internal make-up of this plate is contrary to the previous findings of Perovic et al. [34], summarized in Sect. 3.4, that the habit plane (as defined by the trace of the plane along which the plate is elongated) of the inter-granularly grown hydride plate shown, for instance, in Fig. 3.28 is different from that expected if the hydride transformation matrix is representative of a martensitic, displacive type of transformation. The observations shown in Figs. 3.38 and 3.39a, revealing hydride formation at an earlier stage after nucleation then compared to that shown in Fig. 3.28, indicate clearly that the envelope of the apparently single hydride nucleated along the grain boundary actually consists of, or initially consisted of, an array of individual small hydride plates sharing similar habit planes stacked at an angle along the grain boundary. The habit planes of the hydrides in the stack were found to be consistent with the habit plane expected from a displacive lattice transformation. Thus the apparent hydride habit of the larger plate shown in Fig. 3.28 that had grown along the grain boundary represents the end product of the growth and coalescence of stacks of individual, small hydride plates that had formed at an earlier stage of the formation of such an inter-granular hydride plate. A qualitative mechanistic explanation for the observed hydride precipitation behavior given by Perovic and Weatherly [35] is that the dislocation arrays at the

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Fig. 3.38 Dark field micrograph from hydride diffraction spot showing arrays of small hydride plates decorating the steps at the a=br ; interface. (The sample was heat treated for 1.2 h/ 750 °C and furnace cooled to room temperature.) (from Perovic and Weatherly [35])

interface (i.e. the epitaxial dislocations and the virtual or DSC dislocation arrays at steps) represent sites where the H content would be higher as a result of the interaction energy term given by the product of the tensile stress at these sites with the molar volume of the H atom in solution. The hydrides can form preferentially at these sites because the tensile stress is elevated there. The behavior of the alloy in this heat-treated condition is very similar to that discussed by Banerjee and Arunachalam [4] in Ti alloys. These authors suggest that the mode of hydride precipitation in a/b Ti alloys can be explained by the partial accommodation of the strain field of the precipitate at the interface. This explanation was similarly used by Perovic and Weatherly to explain the preferential formation of hydrides at the epitaxial dislocations and at steps of the undecomposed a/b interfaces in Zr-2.5Nb alloys. However, although the sign of the energy change expected to occur is correct (i.e. there would be a reduction in energy), the physical reason for this reduction that is given in the italicized text is conceptually incorrect. The reason for the attraction of hydrogen atoms to these sites—and the consequent formation of hydrides at sufficiently high H concentration there—is the result of, to first order, the (negative) work done by the internal or external, tensile stress when there is a volume increase resulting from the positive misfit strains of the H atoms as they arrive. A more detailed explanation of this and other aspects of hydrogen/ hydride thermodynamics in stressed solids is given in Chaps. 4, 6, 8 and 10.)

3.5.3 Hydride Precipitation in a/b/x Microstructures As noted in Sects. 3.5.1 and 3.5.2, under more prolonged heat treatments that result in the formation of the x phase, the smooth, regular a/b interface begins to break up and no longer forms a preferential site for hydride nucleation. Figures 3.40 and 3.41 give examples of the arrangements of hydride precipitations for this microstructure. Figure 3.40 compares the results of an optical and a TEM examination

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Fig. 3.39 Hydride precipitation at a=bZr interfaces: a Bright field/dark field pair showing arrays of hydride plates that share a common orientation relationship with the aZr lattice, and have nucleated at the a=bZr interface. b Dark field micrograph showing a monolithic hydride plate at the a=bZr interface. (Heat treatment is as given in Fig. 3.38) (from Perovic and Weatherly [35])

of the same structure. Figure 3.40a shows a two-phase microstructure of a coarsegrained, a/b alloy containing less than 1 wppm H. With the addition of *80 wppm H by treatment at 400 °C, followed by slow cooling to room temperature, the b/x regions appear uniformly dark on examination in the optical microscope (Fig. 3.40b), but on examination in the TEM, the b/x grains can be seen to contain arrays of hydride plates (Fig. 3.40c). A clearer indication of the distribution of the three phases (br ; x and hydride) is given in Fig. 3.41. By using dark field techniques to identify the different phases, Perovic and Weatherly [35] find that the hydride plates are arranged in stacks filling the channels between the x cuboids which are produced by transformation of the retained bZr phase. A careful indexing of the associated diffraction pattern (Fig. 3.41d) indicates that there are at least three phases present (br ; x and hydride), and confirms the orientation relationship between the br ; and x phases noted in the above. The authors examined a number of diffraction patterns from different zone axes of the retained bcc bZr

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Fig. 3.40 a Optical micrograph of coarse-grained Zr-2.5Nb alloys, containing less than 1 wppm H (heat treated as given in Fig. 3.35). b Same microstructure after loading with hydrogen at 400 °C (*80 wppm H) showing that the retained b phase now etches a uniform black color. c TEM dark field micrograph from hydride diffraction spot showing arrays of hydride plates within the retained b phase (from Perovic and Weatherly [35])

phase. By noting the presence of superlattice reflections—which are not found with the d hydride phase—they deduced that the observed hydride phase is the c hydride phase with an approximate orientation relationship given by ð1 1 1Þc jjð1 1 2Þbr ; ½0 0 1c jj ½1 1 0br . It appears that the same factors that were found to control the nucleation and growth of hydride stacks in pure Zr and a=bNb Zr alloys, as summarized in the foregoing, seem to be playing a role in these a=br =x microstructures. Thus, a close examination of many of the large hydride plates running along the channels between the x cuboids shows that they are composed of many smaller plates offset in stacks (Fig. 3.42). The presence of hydride precipitates in the decomposed br phase containing positively misfitting x precipitates is of significance in understanding the possible differences in the solvi between a/b alloys and single phase Zr alloys, such as unalloyed Zr, Zircaloy-2 or -4. This aspect is discussed further in Chaps. 4, 6 and 8. Perovic and Weatherly [35] make a rough estimate of the strains produced in the br

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Fig. 3.41 The distribution of hydrides and x phase particles in a partially decomposed microstructure (pressure tube alloy, annealed for one day at 400 °C and furnace cooled). a Bright field micrograph. b Dark field using x reflection. g 1 0 1 0 x : c Dark field using hydride reflection. g  ð1 1 3Þc : d Diffraction pattern showing the orientation relationship between br and x phases discussed in the text (from Perovic and Weatherly [35])

phase as a result of the dilatational misfit of the x phase dispersed within the br phase. Based on the approach of Brown and Clark [7], they estimate that the mean tensile strain in the br matrix is roughly given by 0.01f, where f is the volume fraction of x precipitates formed. Correspondingly, the x particles would be in compression. The associated tensile stresses in the br phase could thus result in an increased H concentration in this phase compared to the normal increase as a result of equilibrium partitioning of H in solution between the a and the br phases. The increase was obviously sufficiently large for the H concentration in the latter phase to exceed the higher (compared to in the a phase) solvus concentration so that hydrides would form in that phase as well as in the a phase at some point during cool down to room temperature. The reason for the larger affinity for hydrogen in solution in the br compared to the a phase and its correspondingly higher solvus concentration (see Chaps. 4, 6 and 8) is not known, but one factor that could increase the latter is the much larger volume misfit (estimated by Perovic and

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Fig. 3.42 a Distribution of hydride plates in coarsegrained Zr-2.5Nb (produced by annealing at 750 °C, furnace cooling and heating for 3 h/400 °C). Each plate is composed of stacks of smaller plates or laths (arrowed). b Schematic diagram illustrating the distribution of hydride plates in channels between the x particles. A single variant of the hydride is observed within any one channel (from Perovic and Weatherly [35])

Weatherly to be *20 %) between the fct c-hydride and the bcc br phase lattices. This would result in a much larger hydride-matrix accommodation energy in the br compared to in the a phase, in which the total volume misfit of the c hydride is only *11 %.

3.5.4 Influence of Prior Deformation on Hydride Precipitation The final contribution to this impressive series of investigations by Perovic and Weatherly on microstructural aspects of hydride precipitation in Zr and its alloys concerns a study of the effect of prior cold work in material cut from Zircaloy-2 rod [37]. The Zircaloy-2 rod material had a mean grain size of 20 lm with a texture as shown in Fig. 3.43. The rod material was plastically deformed to 4 % tension or to 0.5 and 4 % compression. Subsequent to this, the material was doped with deuterium to a hydrogen equivalent concentration of *60 wppm. This doping procedure was done at a temperature that was well below the recovery temperature of Zircaloy-2 so that, on cooling, the hydride precipitation in these materials occurred in a cold worked matrix. The objective of this study was to obtain additional insights into the role of internal stress on hydride precipitation sites and the associated relationships between hydride habit planes and the average

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Fig. 3.43 Basal pole figure for Zircaloy-2 rod; L is the rod axis direction (from Perovic et al. [37])

orientations of macroscopic hydrides or hydride clusters. This work built on a previous study of MacEwen et al. [25, 27] using similar Zircaloy-2 rod material. In this previous study, it was found that samples annealed in the a phase at 600 °C developed an average residual tensile stress of *100 MPa on the basal plane after cooling to room temperature, with this stress being effectively independent of grain orientation. Since hydrides were generally found to precipitate with their plate normal perpendicular to the basal planes, the foregoing result was used to provide a rationale for the as-manufactured orientation of hydrides in fine-grained, cold worked Zr-2.5Nb pressure tube material. As shown in Sects. 3.4 and 3.5, in slowly cooled samples the observations were that grain boundaries often acted as preferred sites for hydride nucleation. However, it was not clear on the basis of these observations alone whether the reason for this was as a result of a favorable residual stress at some particular location along the grain boundary or the result of heterogeneous nucleation as a result of a favorable defect structure at the interface. Another uncertainty involved the role that ð1 1 0 2Þ twins might play in hydride nucleation [16, 41, 47, 48]. Neutron diffraction was used in this study to determine the lattice strains after cold work as a function of grain orientation and diffraction angle, a, ranging from 60 to 90°, relative to the longitudinal axis of the rod, (along which the tensile or compressive strains were applied) and the normal to the lattice diffracting plane of the neutrons. Since the main focus was on determining the strains on the basal planes, the foregoing angular range encompassed the majority of these grains for this material having the rod texture shown in Fig. 3.43. The results of the neutron diffraction analyses showed that after application of tensile strain of +4 %, the strains on the ð0 0 0 2Þ and ð1 0 1 0Þ planes were compressive and tensile, respectively. The magnitudes and variations with diffraction angle of the strains on the f1 0 1 7g hydride habit planes, which were estimated from the trend in the variation with a of the measured strains on the foregoing two planes, shows that it is approximately constant, with a magnitude of –15 9 10-4 (compression) over the range of diffraction angles used. Such a large

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residual compressive strain after tensile deformation beyond the elastic limit is attributed to the difficulty of pyramidal slip in Zr [25]. TEM examinations of the types, numbers, and distributions of dislocations produced also indicated that there were likely strain gradients within the grains. It should be noted that, on the basis of the measured mean strains alone, one would expect that hydride precipitation should occur on grain boundary sites—as opposed to transgranular sites—since the f1 0 1 7g habit planes of the hydrides are in compression. The mean strains (i.e., the average of the strains in many grains), after -4 % plastic external deformation, were determined to be less than 5 9 10-4. They were thus significantly smaller than those found for the same amount of deformation in tension. It was estimated that the strains on the f1 0 1 7g hydride habit planes were close to zero. The difference in the residual strains obtained after tensile and compressive plastic deformation was suggested to be the result of the presence of f1 0 1 2g twinning, which was observed only in compression. Hydride precipitation was observed at grain boundaries, twin boundaries, or as transgranular plates, the frequency of which of these occurred depending on the direction of prior plastic deformation. For 4 % tensile strain the predominant nucleation sites were at grain boundaries (with the remaining *25 % being as transgranular plates). For 0.5 % plastic compression the sites consisted of equal amounts of grain boundary or as transgranular plates, while for 4 % compression, equal amounts of grain and twin boundary sites were observed. For each of these cases the predominant site—after annealing for 3 h at 650 °C plus furnace cool— was nucleation and growth as transgranular plates with the remainder occurring at grain boundary sites. All of the hydrides observed were of the d phase and had the usual orientation relationships, viz., ð1 1 1Þd  jjð0 0 0 1ÞaZr ; ½1 1 0d jj ½1 1 2 0aZr irrespective of the nucleation site. For the hydrides nucleated at a twin boundary (Fig. 3.44) the angle between the ð1 1 1Þd and ð0 0 0 2ÞaZr planes was &1°. The hydride/twin interface contained the ½0 1 1 1aZr direction. This is the shear direction associated with the ð0 1 1 2Þ½0 1 1 1 twin system in Zr, which is nearly parallel to the ½2 0 0 direction in the d phase. Based on the above observations and in view of their previous findings, the authors concluded that the mode of formation of all macroscopically observable hydrides involved the nucleation and growth of stacked arrays of small, microscopic hydride plates—produced as a result of an autocatalytic nucleation and growth process—that had coalesced to form the macroscopically observable hydride or hydride cluster orientations and morphologies. The habit planes of the individual small hydrides in the array were always the near-basal, f1 0 1 7g planes. The macroscopically observed habit (as given by the trace in the long direction of the envelope containing the stacked array of hydrides) would, therefore, have been determined by the stacking angle of the originating stack and the growth process that lead to plate coalescence. An example of this process is shown in Fig. 3.45 of an S-shaped hydride spanning a single grain in which some of the original smaller hydride plates that make up this hydride are still visible. These hydrides have nearly flat facet planes whose habits are close to the basal plane,

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Fig. 3.44 Zircaloy-2, deformed -4 % showing twin boundary and nucleated hydride orientation (from Perovic et al. [37])

corresponding to the generally found f1 0 1 7g hydride habit planes in Zircaloy (and also in cold worked Zr-2.5Nb pressure tube material). In the center of the grain several plates had coalesced so that the apparent habit plane was closer to f1 0 1 1g than ð0 0 0 1Þ All the plates in the micrograph have an identical orientation relationship with the matrix grain, showing that the S-shape arises solely from the stacking arrangement of the original, smaller hydride plates. The existence of these S-shaped hydrides, which were fairly frequently observed in these cold worked materials, suggests that there were likely appreciable strain gradients within single grains produced by the cold work. Similar composite, curved hydride morphologies were found for hydrides nucleated at grain boundaries, an example of which is shown in Fig. 3.46. In Fig. 3.47 a large grain-boundary hydride is shown at a later stage of its growth. The apparent habit is again parallel to the grain boundary plane. However, it can be seen from the traces of the dislocation arrays, which tend to line up with the external facets of the original, smaller hydrides, that these had the usual f1 0 1 7g hydride habit planes. It was conjectured that the degree to which the apparent habit of the macroscopic hydride deviates from the f1 0 1 7g habits of the original, smaller hydride plates depend on the magnitude, sign, and type of deformation in the grain. This was based on the observation that S-shaped macroscopic hydrides were commonly observed only in the cold worked material, while in annealed material macroscopic hydrides were formed from an array of hydrides having low stacking angle, as was first shown by Westlake [47], reproduced in Fig. 3.22. In this latter case the microscopic and macroscopic habit planes are closely aligned. An example of a large hydride formed in this way for annealed Zircaloy-2 material in the present set of tests is given in Fig. 3.48. The authors point out that the mean strains provide only a rough guide on whether transgranular hydride formation is favored or not. Thus, given that a tensile strain on the basal plane favors the formation of hydride plates with nearbasal plane habit, one would expect such hydrides to form in the stress relieved and -0.5 % deformed material but not in the +4 % deformed material. However, some hydrides were found nucleated on those planes although the predominant

3.5 Hydride Nucleation Studies in a/b Zr-2.5Nb Pressure Tube Material Fig. 3.45 Array of small hydride plates forming an Sshaped large hydride. The flat facet planes of small hydrides are close to the basal plane and correspond to the expected 1 0 1 7 hydride habit plane (from Perovic et al. [37])

Fig. 3.46 Array of small hydride plates forming a ‘large’ grain boundary hydride. Note that the stacking of small hydride plates forming ‘‘large’’ grain boundary hydrides and ‘‘large’’ transgranular hydrides are identical (see Fig. 3.48) (from Perovic et al. [37])

Fig. 3.47 A later stage in the development of a grain boundary hydride. The individual small hydride plates have collapsed into a single plate and the only traces of this process are the dislocations arrays (from Perovic et al. [37])

103

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3 Hydride Phases, Orientation Relationships, Habit Planes, and Morphologies

Fig. 3.48 Zircaloy-2, well annealed showing a large hydride with the ‘apparent’ habit plane

close to the usual 1 0 1 7 hydride habit planes; b small hydride plates aligned with low stack angle and a habit close to the basal plane (from Perovic et al. [37])

mode was one of inter-granular precipitation. This shows that predictions regarding which habit plane prevails based on mean strain data ignore the roles of possible heterogeneous sites, grain-to-grain variability, and strain gradients within the grains.

3.5 Hydride Nucleation Studies in a/b Zr-2.5Nb Pressure Tube Material

105

In particular, the strain gradient in a single grain could be quite large as pointed out by MacEwen et al. [26], who suggested that the maximum deviation is expected to occur at grain boundary facets at which the difference in orientation of the ‘soft’ and ‘hard’ directions is a maximum (90°). This would produce a tensile stress at those locations and, therefore, would favor the nucleation of hydrides there. It was conjectured that the frequent observations of S-shaped hydrides also indicates the presence (and importance) of variations in the local stress state. As Perovic et al. [34] had shown, the geometry of the stacking of small hydride plates is influenced by the magnitude, sign, and type (normal or shear) of local stress. High stacking angles would be favored by tensile stress acting normal to the habit plane, while lower stack angles would be favored by shear stress acting in the same sense as the shear displacement that is required to transform a local region of the hcp a-Zr lattice to an fcc d hydride one. Assuming the correctness of these qualitative ideas, the shape of the macroscopic hydride shown in Fig. 3.45 can be rationalized by assuming that, at the center of the grain, the residual stresses acting on the habit plane were predominantly tensile, this conjecture being supported by the authors’ data for the mean strain for a C 70° in material with a compressive pre-strain of 0.5 %. On the other hand, at the periphery, the residual stress pattern must have changed to one favoring a lower stacking angle. There must also be a connection between the residual stress state in material subjected to -4 % plastic deformation (compression) and twin boundaries, since these boundaries were observed to act as nucleation sites only in the deformed material, but not in the annealed material, even though the annealing left the twin boundaries intact. The reason for this difference, however, could not be explained by the authors.

References 1. Ashby, M.F., Johnson, L.: On the generation of dislocations at misfitting particles in a ductile matrix. Philos. Mag. 20, 1009–1022 (1969) 2. Akhtar, A.: The nature of c-hydride in crept zirconium single crystals. J. Nucl. Mater. 64, 86–92 (1977) 3. Bailey, J.E.: Electron microscope observations on the precipitation of zirconium hydride in zirconium. Acta Metall. 11, 267–280 (1963) 4. Banerjee, D., Arunachalam, V.S.: On the a/b interface phase in Ti alloys. Acta Metall. 29, 1685–1694 (1981) 5. Bowles, J.S., Muddle, B.C., Wayman, C.M.: The crystallography of the precipitation of b vanadium hydride. Acta Metall. 25, 513–520 (1977) 6. Bradbrook, J.S., Lorimer, G.W., Ridley, N.: The precipitation of zirconium hydride in zirconium and Zircaloy-2. J. Nucl. Mater. 42, 142–162 (1972) 7. Brown, L.M., Clark, D.R.: The work hardening of fibrous composites with particular reference to the copper-tungsten system. Acta Metall. 25, 563–570 (1977) 8. Cassidy, M.P., Muddle, B.C., Scott, T.E., et al.: Experimental studies of the crystallography of the precipitation of b vanadium hydride. Acta Metall. 25, 829–838 (1977) 9. Cassidy, M.P., Wayman, C.M.: The crystallography of hydride formation in zirconium: I. The d ? c transformation. Metall. Trans. A 11, 47–56 (1980)

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10. Cassidy, M.P., Wayman, C.M.: The crystallography of hydride formation in zirconium: II. The d ? e transformation. Metall. Trans. A 11, 57–67 (1980) 11. Carpenter, G.J.C.: The dilatational misfit of zirconium hydrides precipitated in zirconium. J. Nucl. Mater. 48, 264–266 (1973) 12. Carpenter, G.J.C., Watters, J.F., Gilbert, R.W.: Dislocations generated by zirconium hydride precipitates in zirconium and some of its alloys. J. Nucl. Mater. 48, 267–276 (1973) 13. Carpenter, G.J.C.: The precipitation of c-zirconium hydride in zirconium. Acta Metall. 26, 1225–1235 (1978) 14. Christian, J.W.: A theory of the transformation in pure cobalt. Proc. R. Soc. London A 206, 51–64 (1951) 15. Cottrell, A.H., Bilby, B.A.: A mechanism for the growth of deformation twins in crystals. Philos. Mag. 42, 573–581 (1951) 16. Ells, C.E.: Hydride precipitates in zirconium alloys. J. Nucl. Mater. 28, 129–151 (1968) 17. Eshelby, J.D.: The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. London A 241, 376–396 (1957) 18. Eshelby, J.D.: Continuum theory of lattice defects. In: Seitz, F., Turnbull, D. (eds.) Solid State Physics 3, pp. 79–140. Academic Press, New York (1966) 19. Eshelby, J.D.: Elastic inclusions and inhomogeneities. Prog. Sol. Mech. 2, 89–140 (1961) 20. Kelly, A., Groves, G.W.: In Crystallography and Crystal Defects. Longman, London, UK (1970) 21. Hall, I.W.: Basal and near–basal hydrides in Ti-5Al-2.5Sn. Metall. Trans. A 9A, 815–820 (1978) 22. Heheman, I.F.: Transformations in zirconium-niobium alloys. Can. Metall. Q. 11, 201–211 (1972) 23. Hirth, J.P., Lothe, J.: Theory of Dislocations, p. 336. McGraw-Hill, New York (1968) 24. Lichter, B.D.: Trans. AIME 218, 1015 (1960) 25. MacEwen, S.R., Faber Jr, J., Turner, A.P.L.: The use of time-of-flight neutron diffraction to study grain interaction stresses. Acta Metall. 31, 657–676 (1983) 26. MacEwen, S.R., Faber Jr, J., Turner, A.P.L.: The influence of texture on the interpretation of diffraction data to determine residual stress. Scripta Metall. 18, 629–633 (1984) 27. MacEwen, S.R., Tome, C., Faber Jr, J.: Residual stresses in annealed Zircaloy. Acta Metall. 37, 979–989 (1989) 28. Northwood, D.O., Gilbert, R.W.: Hydrides in zirconium-2.5 wt% niobium alloy pressure tubing. J. Nucl. Mater. 78, 112–116 (1978) 29. Paton, N.E., Spurling, R.A.: Hydride habit planes in titanium-aluminum alloys. Metall. Trans. A 7A, 1769–1774 (1976) 30. Perovic, V., Purdy, G.R., Brown, L.M.: On the stability of arrays of precipitates. Acta Metall. 27, 1075–1084 (1979) 31. Perovic, V., Purdy, G.R., Brown, L.M.: Autocatalytic nucleation and elastic stabilization of linear arrays of plate-shaped precipitates. Acta Metall. 29, 889–902 (1981) 32. Perovic, V., Purdy, G.R., Brown, L.M.: The role of shear transformation strains in the formation of linear arrays of precipitates. Scripta Metall. 15, 217–221 (1981) 33. Perovic, V., Weatherly, G.C., Simpson, C.J.: The role of elastic strains in the formation of stacks of hydride precipitates in zirconium alloys. Scripta Metall. 16, 409–412 (1982) 34. Perovic, V., Weatherly, G.C., Simpson, C.J.: Hydride precipitation in a/b zirconium alloys. Acta Metall. 31, 1381–1391 (1983) 35. Perovic, V., Weatherly, G.C.: The nucleation of hydrides in Zr-2.5 wt% Nb Alloy. J. Nucl. Mater. 126, 160–169 (1984) 36. Perovic, V., Weatherly, G.C.: The b to a transformation in a Zr-2.5 wt% Nb alloy. Acta Metall. 37, 813–821 (1989) 37. Perovic, V., Weatherly, G.C., MacEwen, S.R., et al.: The influence of prior deformation on hydride precipitation in Zircaloy. Acta Metall. Mater. 40, 363–372 (1992) 38. Pegoud, J., Guillaumin, J.: Orientations crystallographiques de l’hydrure dans le Zircaloy-2 polycristallin. J. Nucl. Mater. 45, 64–72 (1972/1973)

References

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39. Puls, M.P.: The influence of hydride size and matrix strength on fracture initiation at hydrides in zirconium alloys. Metall. Trans. A 19A, 1507–1522 (1988) 40. Puls, M.P.: Review of the thermodynamic basis for models of delayed hydride cracking rate in zirconium alloys. J. Nucl. Mater. 393, 350–367 (2009) 41. Roy, C.: Hydrogen distribution in oxidized zirconium alloys by autoradiography. Atomic Energy of Canada Ltd (AECL) Report No. 2085, Chalk River Laboratories, Chalk River, Ontario, Canada (1964) 42. Roy, C., Jacques, J.G.: f1 0 1 7g hydride habit planes in single crystal zirconium. J. Nucl. Mater. 31, 233–237 (1969) 43. Sidhu, S.S., Murthy, N.S.S., Campos, F.P., et al.: Neutron and X-ray diffraction studies of nonstoichiometric metal hydrides. Adv. Chem. Ser. 39, 87–98 (1963) 44. Sleeswyk, A.W.: Emissary dislocations: theory and experiments on the propagation of deformation twins in a-iron. Acta Metall. 10, 705–725 (1962) 45. Venables, J.A.: Deformation twinning in face-centred cubic metals. Philos. Mag. 6, 379–396 (1961) 46. Weatherly, G.C.: The precipitation of c-hydride plates in zirconium. Acta Metall. 29, 501–512 (1981) 47. Westlake, D.G.: The habit planes of zirconium hydrides in zirconium and Zircaloy. J. Nucl. Mater. 26, 208–216 (1968) 48. Westlake, D.G., Fisher, E.S.: Precipitation of zirconium hydride in alpha zirconium crystals. Trans. AIME 244, 254–258 (1962) 49. Zhang, W.-Z., Purdy, G.R.: A TEM study of the crystallography and interphase boundary structure of precipitates in a Zr-2.5 wt% Nb alloy. Acta Metall. Mater. 41, 543–551 (1993) 50. Zhang, W.Z., Perovic, V., Perovic, A., et al.: The structure of hcp–bcc interfaces in a Zr-Nb alloy. Acta Mater 46, 3443–3453 (1998)

Chapter 4

Solubility of Hydrogen

4.1 Solubility in the Dilute Phase of Single Phase Metal–Hydrogen Systems For applications of zirconium alloys in nuclear applications where the metal is in contact with water at high temperatures, hydrogen enters the metal as a by-product of a slow oxidation process occurring at the water/metal surface. Since, only a small fraction of the total hydrogen that is released as a result of the oxidation process ends up in the metal, hydrogen ingress in this way tends to be a slow process. Thus, it is not a useful method for laboratory studies of the thermodynamic conditions of hydrogen solubility. For these types of studies the most convenient external source of hydrogen is in its gaseous form. The chemical driving force for absorption in this case can be readily controlled through the partial pressure of the gas. However, for these types of tests it is important to make sure that the surfaces of the metal are suitably prepared so that there are no barriers to ingress of the gas into the metal. Unfortunately, for zirconium and its alloys this method is best used at temperature above 300 °C because the partial pressure of hydrogen for hydrogen dissolved in this metal at reactor operating temperatures is very low and accurate measurements become difficult. Over this lower temperature range, then, other methods are usually used or it is assumed that extrapolation to a lower temperature range of the results obtained at higher temperature is acceptable. At any rate, the thermodynamics of the equilibrium conditions of hydrogen in gaseous form in open contact with a metal is of fundamental importance for the further development of hydrogen solubility studies. Thus, it is taken as the starting point of the exposition in this chapter. In hydrogen solubility experiments using an external, gaseous source of hydrogen, the gas molecules react at the surface of the metal, dissociating, and dissolving in it according to:

M. P. Puls, The Effect of Hydrogen and Hydrides on the Integrity of Zirconium Alloy Components, Engineering Materials, DOI: 10.1007/978-1-4471-4195-2_4, Ó Springer-Verlag London 2012

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110

4 Solubility of Hydrogen

1 H2 ðgasÞ ! HðmetalÞ 2

ð4:1Þ

Evidence for this reaction is indirectly provided when measurements of pressure– composition isotherms result in the relationship: pffiffiffiffiffiffiffi ð4:2Þ c H ¼ K S pH 2 Equation 4.2 is known as Sievert’s law. In this equation, cH is the concentration of hydrogen in the metal, pH2 the partial pressure of hydrogen gas, and KS Sievert’s constant. Equation 4.2 can be derived from thermodynamics by making use of a result from statistical mechanics for the configurational entropy of hydrogen dissolved in interstitial sites in the metal. The ideal partial configurational entropy, ScH ; is given by: " # r H ScH ¼ R ln ð4:3Þ bpha  rH where R is the gas constant, rH, the ratio of NH/NM (NH = number of hydrogen atoms, NM = number of metal atoms in the solid), and bpha is the number of interstitial sites per metal atom, M where ‘‘pha’’ stands for ‘‘phase’’). ScH is derived based on a partition function for placing hydrogen atoms in interstitial sites in the metal in a random manner with the restriction that each site can contain no more than one hydrogen atom. The statistics based on this assumption is called Fermi– Dirac. The chemical potential of hydrogen that is dissolved in the metal assuming ideal conditions (i.e., no, or negligible, H–H interactions in the solid) is then: " # " # rH rH o o ð4:4Þ lH ¼ lH þ RT ln  lH þ RT ln ðrH ! 0Þ bpha  rH bpha where loH is a standard value of the chemical potential. For dilute solutions of hydrogen in the metal, lH reduces to the second part of Eq. 4.4. Under those conditions the foregoing expression for the configurational entropy and associated chemical potential is the same as would have been obtained if Maxwell–Boltzmann, rather than Fermi–Dirac statistics had been used to derive Eq. 4.3. In the gas phase, for pressures sufficiently low that the ideal gas law applies, the chemical potential of hydrogen is given by: lo lH2 pffiffiffiffiffiffiffi ¼ H2 þ RT ln pH2 2 2

ð4:5Þ

where loH2 =2 is the chemical potential of the gas at standard pressure (0.1 MPa (1 atmosphere)) and temperature (298.15 K). At equilibrium, the chemical potentials of hydrogen per hydrogen atom in the gas and in the solid metal phases are equal, resulting in the relation:

4.1 Solubility in the Dilute Phase of Single Phase Metal-Hydrogen Systems

rH ¼ bpha exp

loH 2 2

 loH pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pH2  KS pH2 RT

111

ð4:6Þ

This relation is obtained by equating lH2 =2 of Eq. 4.5 with the second part of Eq. 4.4. Converting the hydrogen metal ratio, rH, in Eq. 4.6 to hydrogen concentration units more commonly used in experimental determinations such as atomic fractions, cH, yields, in the dilute approximation (rH, cH ? 0): cH ¼

rH  rH 1 þ rH

ðrH ! 0Þ

ð4:7Þ

Sievert’s law given by Eq. 4.2 is then obtained by inserting the dilute solution result of Eq. 4.7 into Eq. 4.6. The difference in the standard values of the chemical potentials in Eq. 4.6 is equal to the change in the Gibbs free energy of the reaction per atom of hydrogen given by Eq. 4.1. It is generally useful to express this free energy change, which can also be considered as the Gibbs free energy change of absorption (or dissolution if the direction of the reaction is reversed) in terms of the enthalpy and entropy changes of this process, given by: DGoH ¼ DHHo  TDSoH

ð4:8Þ

1 DHHo ¼ HHo  HHo 2 2

ð4:9Þ

1 DSoH ¼ SoH  SoH2 2

ð4:10Þ

where

In Eqs. 4.9 and 4.10, HHo is the formation enthalpy of hydrogen in the metal lattice at infinite dilution (rH ! 0), 12 HHo 2 is the energy of dissociation of the H2 molecule in the standard state of the hydrogen gas at 298.15 °C and 0.1 MPa, SoH is the nonconfigurational component of entropy of the hydrogen atom in the metal lattice and 12 SoH2 is the entropy of dissociation of the H2 molecule in the standard state of the gas. Thus,   loH2  loH ¼ DGoH ¼  DHHo  TDSoH 2

ð4:11Þ

From Eq. 4.8, comparing with Eq. 4.6, the Sievert’s constant is given by:  o   DSH DHHo KS ¼ bpha  exp exp  ð4:12Þ R RT The largest contribution to the entropy change of solution comes from the loss of degrees of freedom that the hydrogen atom has when brought from the gas phase into the solid phase. Hence, DSoH would be expected to be close to the standard

112

4 Solubility of Hydrogen

entropy, 12 SoH2 ; of gaseous hydrogen at room temperature of &64.9 J/(K mol H) [10]. This large, positive value of entropy contribution to the Gibbs free energy change is the reason that hydrogen can be desorbed from a metal at high temperature. The enthalpy of absorption (or dissolution) varies systematically across the periodic table, being negative for metals on the left side of the table and positive (except for Pd) for metals going to the right side of the table via the transition metals. Thus metals such as Zr, Ti, Nb, V, and Ta have negative enthalpies while metals such as Fe, Ni, Cr, and Cu have positive values. For zirconium, with a negative value, the absorption enthalpy, DHHo ; obtained experimentally is about -49 kJ/(mol H) [5]. The systematic variation in magnitude and sign of the solution enthalpy across the periodic table suggests that it is determined by the gross electronic structure of the host metal. Recent theoretical calculations for DHHo yield -58.28 kJ/(mol H) [7] for hydrogen in tetrahedral interstitial sites of the a–Zr lattice, while a value of -51.33 kJ/mol H is obtained for the octahedral sites. The lower energy, and hence, greater attraction of hydrogen for the tetrahedral sites is in accord with experimental observations [16] that find hydrogen atoms occupying tetrahedral sites in a–Zr. From Sievert’s law and the expression for Sievert’s constant given by Eq. 4.12, it is evident that the solubility of hydrogen in the metal can have two types of temperature dependencies. For metals with positive enthalpies, the solubility increases with increasing temperature for a given external pressure. These metals are called endothermic occluders. For metals with negative enthalpies, the solubility decreases with temperature and these are called exothermic occluders. Note, however, that the enthalpies obtained from the reaction given by Eq. 4.1 include the energy of dissociation of the hydrogen molecule in the gas phase, which is large and negative [-216 kJ/(mol H)] [7]. When this is subtracted from the enthalpies obtained from Sievert’s law, all metals have negative enthalpies. Nevertheless, the original classification of metals as endothermic or exothermic is useful in distinguishing the type of phase formed when the hydrogen solubility exceeds the phase boundary composition. Thus endothermic occluders form hydrogen gas—often in the form of internal bubbles in the metal—when their terminal solubility is reached, while exothermic occluders form a solid hydride phase. A consequence of this difference is that, because the solvus composition (terminal solubility for hydride formation or dissolution) increases with increase in temperature, the concentration of hydrogen in solid solution in the metal can be much higher with increasing temperature for hydride forming metals. As an example, the solubility of hydrogen in a–iron is negligible at room temperature and increases to only about 1 wppm at 500 °C, whereas zirconium has a solubility that is similarly negligible at room temperature (*0.06 wppm from extrapolated higher temperature solvus data) but increases to about *400 wppm at 500 °C. Excess thermodynamic functions for metal–H interstitial solid solutions have been defined to provide a measure of the amount of deviation of the thermodynamic parameters from their ideal state for which Sievert’s law is valid. With reference to this ideal state, given by Eq. 4.2, now identified by lid H as follows:

4.1 Solubility in the Dilute Phase of Single Phase Metal-Hydrogen Systems

"

lid H

¼

loH ðpo ; V o ; rH

rH ! 0Þ þ RT ln bpha  rH

113

# ð4:13Þ

the excess chemical potential, lEH ; is defined by: lEH ¼ lH  lid H

ð4:14Þ

from which follow the definitions of the excess enthalpy and entropy given, respectively, by: HHE ¼ HH  HHo and SEH ¼ SH  SoH

ð4:15Þ

With DSoH evaluated from the data using (rH ? 0): DSoH ¼ DSH  R ln

rH bpha

ð4:16Þ

and bpha  baZr ¼ 2 for two tetrahedral sites per Zr atom in a–Zr lattice, SEH is given by: 1 SEH ¼ DSoH þ SH2 2

ð4:17Þ

From the data of Dantzer et al. [5] DSoH ¼ 49:8 J=ðK mol HÞ; 12 SH2 ¼ 64:86 J=ðK mol HÞ; giving SEH ¼ 15:06 J=ðKmolHÞ: This is a large value for the excess entropy, a result that is supported by a similar finding in the chemically related Ti–H system [23]. Dantzer et al. [5] note that no satisfactory explanations for these large excess entropy values in the group IVB metal–H systems have been provided.

4.2 Solubility of Hydrogen in the Dilute Phase of a/b Zirconium Alloys As detailed in Chap. 3, Zr-2.5Nb used for pressure tubes in CANDU and PHW reactors is a two-phase alloy consisting of a majority a phase and a metastable minority b phase (about 10 % volume fraction). The b phase changes composition and morphology with time at service in the reactor, even during the final stages of the manufacturing process. As demonstrated, respectively, in Chaps. 5 and 8, the b phase has a higher diffusivity for hydrogen and higher solvus concentration than does the a phase. It is shown in the following that it also has a higher solubility. To determine the effect that this phase has on the overall hydrogen solubility, solvus concentration, and diffusivity of the alloy, it is necessary to know the equilibrium partitioning of hydrogen between the two phases. Sawatzky et al. [34] were the first to measure the partitioning of hydrogen between the a–Zr and the metastable Zr-20Nb b phase. For their measurements

114

4 Solubility of Hydrogen

Table 4.1 Comparison of hydrogen partitioning ratio between Zr-20Nb b phase and Zr a phase obtained by separate investigators b=a Investigators Temperature (°C) Equilibration time (h) Partitioning ratio, r ðNbÞ H

1 1 2 1 1 2 1 1 2 2

250 250 250 300 300 300 400 400 400 400

1,000 72 1,200 672 672 672 96 96 113 113

2.67 2.31 3.57 1.76 1.41 2.76 1.95 1.89 1.98 1.84

1 Sawatzky et al. [34], 2 Cann et al. [4]

they used a 1 mm thick zirconium disk (produced from sponge zirconium) that was hydrogenated to less than the solvus concentration; then sealed together with a 1 mm thick disk of Zr-20Nb in an evacuated quartz capsule. The capsule was heated at 800 °C for one hour to dissolve the oxide layers on the specimens, brought to the desired temperature, and maintained there sufficiently long to achieve the equilibrium hydrogen concentration. Subsequently, the samples were analyzed for hydrogen content. The results are summarized in Table 4.1 together with the results of later measurements by Cann et al. [4]. These authors measured the partitioning of hydrogen between the Zr-20Nb, and a–Zr using diffusion couples, since it was felt that the results obtained by Sawatzky et al. [34] may have underestimated the true equilibrium condition because the latter authors’ method required passage of hydrogen from one specimen to the other through the gas phase. This means that hydrogen had to be desorbed and absorbed, respectively, through each specimen’s surfaces, which could have slowed down the equilibration process. In addition, in Cann et al.’s experiments, different stages of decomposition of the metastable b phase were investigated. To determine the degree of break-down of the b phase, measurements were made of the b phase lattice parameter. From the results of these measurements the Nb concentration in the b phase could be deduced. The range of decomposition of the metastable Zr20Nb starting phase in these experiments largely covered the range of decomposition of the b phase stringers expected in Zr-2.5Nb pressure tube material as a result of the effects of preservice heat treatment and subsequent reactor service (see Chap. 3, Sect. 3.5.1), with some possible differences, discussed further on. As was done by Sawatzky et al. [34], the partitioning data are given in terms of the b=a partitioning ratio, rH ðNbÞ ¼ cZr20Nb =caZr ; where cZr20Nb is the hydrogen concentration in the couple having a starting composition of Zr-20Nb (b phase of varying degree of decomposition) and caZr is the hydrogen concentration in the couple consisting entirely of a–Zr. These are plotted as a function of Nb concentration in the b phase in Fig. 4.1.

4.2 Solubility of Hydrogen in the Dilute Phase of a/b Zirconium Alloys

CZr-20Nb/CZr

4

673 K

3 2 1 4

CZr-20Nb/CZr

Fig. 4.1 Measured versus fitted values of rH = CZr20Nb)/CZr ratios from Cann et al. [4]: The plotted points are those reported in Fig. 2 of Cann et al. [4] with niobium concentrations in the enriched b phase greater than 20 %. The curves are obtained using an interpolation function to the data

115

623 K

3 2

CZr-20Nb/CZr

1 4

573 K

3 2

CZr-20Nb/CZr

1 4

523 K

3 2 1 70

60

50

40

30

20

% Nb

Comparison of the partitioning results of Cann et al. [4] with those of Sawatzky et al. [34] for the starting metastable b phase of Zr-20Nb shows that the former authors obtained a larger partitioning ratio at 250 and 300 °C, but comparable results at the highest temperature tested of 400 °C. Cann et al. reasoned that the lower values obtained by Sawatzky et al. at the lower test temperatures may have been because equilibrium had not yet been achieved in the latter’s tests, since equilibration in their tests required transfer of hydrogen through the gas phase, which is a surface sensitive process that could be quite slow. From the results plotted in Fig. 4.1, two features are evident. The first is that the magnitude of the partitioning decreases as the b phase Nb concentration increases. The second is that the magnitude of the partitioning increases with decreasing temperature for the same amount of Nb concentration in the b phase. Another point to note is that the partitioning ratio is always greater than one, even when the Nb concentration in the b Zr phase has increased to 70 at %. Cann et al. estimated, however, from the ratio of the Sievert’s constants for a Zr and pure b niobium that the partitioning ratio for this case should be 0.015, i.e., much less than one.

116

4 Solubility of Hydrogen

That the partitioning ratio for 70 at % b zirconium is much larger than this and, moreover, greater than unity suggests that the partitioning is strongly influenced by the x phase. One might conjecture that since this phase has a hexagonal (although not close-packed) crystal structure, being of similar geometry, but more open than that of the a Zr phase, it may have a higher solubility for hydrogen than the latter phase. However, the studies of Perovic and Weatherly [28] have shown (see Sect. 3.5.3 of Chap. 3) that the x–phase precipitates formed within the surrounding residual bZr phase is under compression, causing this surrounding phase to be in tension. These tensile stresses induced in the residual bZr phase would enhance the hydrogen absorption affinity of this phase compared to its affinity for hydrogen in its unstressed state. The observed presence of hydrides in the br channels surrounding the x phase is strong evidence for this. Based on these partitioning results, which were obtained for temperatures and hydrogen concentrations for which hydrogen was in solution in both phases, one can derive an expression for the total concentration of hydrogen in solution in the alloy, cTH ; as a function of the concentration in solution in the a phase, caH ; as follows:     b=a cTH ¼ caH 1 þ rH ðNbÞ  1 wb ð4:18Þ b=a

where rH ðNbÞ ¼ cZr20Nb =caZr is the experimentally determined hydrogen partitioning ratio plotted in Fig. 4.1 as a function of the Nb concentration in the enriched b phase, formed when the original Zr-20Nb b phase decomposes. The weight fraction of the b phase is given by   ð4:19Þ wb ¼ ðn  na Þ= nb  na where n is the total weight percent in the alloy, nb is the Nb weight percent in the b phase, and na is the Nb weight percent in the a phase. Used with Cann’s results of b=a the experimentally determined values of rH ðNbÞ; wb is a constant given by the Nb concentrations in the a- and b-Zr phases when the Nb concentration of the b-Zr phase is 20 at %; i.e., before it has started to decompose. The reason for this is that the measured partitioning ratio is the solubility ratio of the hydrogen concentration in the a-Zr with that in either the uniform b-Zr phase at 20 wt % Nb or with that in the decomposed b-Zr phase consisting of the product phases: enriched b-Zr and x-Zr. That is, the weight fraction of this decomposed region consisting of the two product phases is the same as that of the original b-Zr phase at 20 wt % Nb.

4.3 Effect of Stress on Hydrogen Chemical Potential in the Dilute Phase An implicit assumption of the foregoing simple thermodynamic relationships has been that the metal lattice does not explicitly enter into these relationships, since its role has been assumed to be one of merely providing a lattice framework with

4.3 Effect of Stress on Hydrogen Chemical Potential in the Dilute Phase

117

reference to which the properties of hydrogen in the metal are determined. Thus, only the chemical potential of hydrogen was considered in the derivation of the equilibrium conditions. However, a more complete treatment must—at least formally—also include the chemical potential(s) of the metal atom(s) in the metal—hydrogen system. This could become important when assessing the effect of internal and external sources of stress on the equilibrium conditions in the metal—hydride systems. As is shown in Chap. 10, DHC is governed by the effect that sources of stress have on hydrogen solubility, diffusion, terminal solubility (solvus), and hydride orientation in the metal. In the following, we present a general theoretical treatment governing the solubility of hydrogen in inhomogeneously stressed metal–hydrogen systems for hydrogen concentrations less than the solvus concentration while in Chaps. 6 and 8 this treatment is extended to the case where the solvus concentration is exceeded and a hydride phase is formed. Based on Callen’s [3] approach, in which the phenomenological laws of thermodynamics are encapsulated into a series of postulates specifying the mathematical forms and properties of a set of formal thermodynamic relationships, there exists a fundamental equation of the thermodynamic system that expresses the total energy of the system, U, in terms of the entropy, S, and other independent extensive macroscopic parameters, such as the total volume, V, and component (mass) numbers, Ni. This fundamental equation is formally expressed as: U ¼ UðS; V; N1 . . . Ni Þ

ð4:20Þ

By extensive macroscopic parameters are meant parameters that have values in a composite system equal to the sum of the parameters in the subsystems; i.e., they are linearly additive parameters. This makes the energy, U, also linearly additive. The fundamental relationship given by Eq. 4.20 could be reversed by making the entropy of the system the dependent variable and the energy and other extensive variables the dependent ones. Equilibrium is obtained when the energy of the system is a minimum at constant entropy, or, conversely, the entropy is a maximum at constant energy. The equilibrium value of any unconstrained parameter of the system at equilibrium, then, is determined from the condition that the total energy of the system is a minimum or alternatively—in the entropy representation—the total entropy is a maximum. Using a Legendre transformation, one or more of the extensive independent parameters of the fundamental form given by Eq. 4.20 can be reformulated such that one or more of the extensive parameters in Eq. 4.20 are replaced by their conjugate intensive parameters, which are thermodynamic potentials of the system. We note that these potential parameters have the important property that they do not depend on the size of the system and are, at equilibrium, constant throughout the system. An important property of these alternative formulations of the thermodynamic energy of the system is that the same conditions for equilibrium apply to these energies as to the fundamental relationship given by Eq. 4.20. The various thermodynamic energies derived in this way are the enthalpy, H, the Helmholtz free

118

4 Solubility of Hydrogen

energy, F, and the Gibbs free energy, G. Their usefulness derives from each being expressed in terms of a combination of extensive and intensive (potential) variables most suitable for the experimental conditions in terms of which variables are being constrained and, thus known, and which are allowed to vary. The appropriateness of a particular choice of thermodynamic energy for a given choice of constraints derives from the property of the thermodynamic potentials which is that it is constant at equilibrium throughout the system. This allows equilibrium relationships to be derived between the thermodynamic potentials and the parameters being constrained that follow directly from the minimization of the thermodynamic energy formulated in terms of these constrained variables. In particular, the Gibbs free energy G ¼ GðT; P; N1 . . .Ni Þ ¼ U  TS þ PV

ð4:21Þ

is generally the most useful formulation of the thermodynamic energy for the determination of equilibrium conditions in multicomponent solid systems for which the temperature, T, and pressure, P, are maintained at known values while the extensive variables, Ni, which are the mole (or atom) numbers of the components of the system, are free to vary. The Gibbs free energy can also be expressed in terms of the enthalpy of the system, since H ¼ U þ PV

ð4:22Þ

G ¼ H  TS

ð4:23Þ

resulting in,

In this formulation for hydrogen dissolved in a solid, H represents the energy of solution of hydrogen in the solid under isothermal conditions as can be seen from the foregoing relations giving the equilibrium conditions of a metal-hydrogen system in contact with an external hydrogen gas reservoir. The complete differential of the Gibbs free energy is: dGðT; P; N1 . . .Ni Þ ¼ SdT þ VdP þ

i X

lk dNk

ð4:24Þ

k¼1

in which the intensive (potential) parameters, lk ; called chemical potentials, are defined by:   oGðP; V; N1 . . .Ni Þ ¼ lNk  lk ð4:25Þ oNk P;V;Niði6¼kÞ Then, at equilibrium, the condition dGðT; P; N1    Ni Þ ¼ 0

ð4:26Þ

4.3 Effect of Stress on Hydrogen Chemical Potential in the Dilute Phase

119

for given temperatures and pressures, subject to the constraint that the total numbers of moles in the system remains constant in a closed system, provides a relationship between the chemical potentials of each component.1 It was pointed out by Li et al. [22] that, in an inhomogeneously stressed crystalline solid containing only atoms on lattice sites, the assumption of Callen [3] that the chemical potential of each component is the partial derivative of that function with respect to the number of moles (or atoms) of that component does not lead to a unique value of this chemical potential at equilibrium. However, when such a system contains at least one component that could be considered to be mobile in relation to the other components forming the crystallographic framework of the solid, the classical equilibrium conditions based on constant chemical potential of the mobile species throughout the system would still be applicable, but for this species alone. More generally, it has been shown by others (for instance, in [36]) that a free energy function (and the corresponding chemical potentials) could be derived for inhomogeneously stressed solid for all species in the system by the formal introduction of lattice defects such as vacancies, di-vacancies, and self-interstitials as extensive variables in the expression of the thermodynamic potential. These defects could exist throughout the lattice and their presence would allow all species in the system to become mobile. In terms of this more general treatment, the special case considered by Li et al. [22], of a solid in which one of the components is located on interstitial sites of the lattice and considered as mobile, applies only as long as the concentration of the mobile, interstitial species does not come close to the available interstitial lattice sites. An early development in modifying the chemical potential formulation in nonhydrostatically stressed crystalline lattices was made by Herring [12] and Bardeen and Herring [2] for the purposes of analyzing diffusion creep problems. These authors considered the diffusion of lattice atoms occurring via a vacancy diffusion process. Thus by including the creation of thermal vacancies into the chemical potential formulation, these authors were able to formulate a thermodynamic driving force for diffusion that could be uniquely defined throughout a nonuniformly and nonhydrostatically stressed solid. It is informative to express the foregoing in terms of the phenomenological relations of irreversible thermodynamics linking the irreversible fluxes with the thermodynamic forces appearing in the entropy production relation. When the flux is that of atomic species in a crystalline solid and this process proceeds relatively slowly so that deviations from equilibrium are not large, one might expect that a linear relationship exists between macroscopic kinetic coefficients of the diffusing species and their thermodynamic driving forces. For such systems Onsager (see, for instance, de Groot and Mazur [6]), proposed for the case of fluids that the diffusion flux, Ji, for the ith component in a multicomponent fluid can be formally expressed by

1

Strictly speaking, this relationship determines only the extremum condition. Whether this extremum is a stable (equilibrium) one requires taking the second derivative of this function to determine whether the extremum represents a maximum, minimum or saddle point condition.

120

4 Solubility of Hydrogen

Ji ¼ 

X

Lik rlk

ð4:27Þ

k

where the Lik are mobility coefficients shown by Onsager to have the property of microscopic reversibility. These macroscopic coefficients can be related to the phenomenological diffusion coefficients by comparison of Eq. 4.27 with the usual macroscopic Fick diffusion equation given in terms of the gradients of the concentrations of the diffusing species. A more specific treatment is given in Chap. 5. The usual problem posed in diffusion of a component in a crystalline solid is the determination of the flux of this component between two locations in the crystal that are at different stresses and at which, moreover, local equilibrium prevails. The flux in this local equilibrium approximation is calculated by determining the local equilibrium values of lk at these two points in the crystal. The driving force for diffusion, then, is the gradient of chemical potential between these two points. Thus for an isothermal system at constant pressure, one must calculate the Gibbs free energy change as atoms are added or removed at these points. Subtleties arise in nonhydrostatically stressed solids concerning the appropriate formulation of the driving force for diffusion when the diffusing species are atoms located on lattice sites (referred to as substitutional species). For instance, if the diffusion mechanism of the lattice atoms is by a vacancy mechanism, then the microscopic potential driving diffusion is the difference between the chemical potential of the atom and the vacancy [32]. However, when the diffusing species are located on interstitial sites, either as self interstitials or normally located on these sites—and the concentration of these species are small in relation to the available sites—then the chemical potential for diffusion is simply the chemical potential of the diffusing species. Equilibrium in a nonhydrostatically stressed solid in this case is possible throughout the solid, not just at sources and sinks. In the case of substitutional species, equilibrium is only possible locally at sinks and sources such as at free surfaces, dislocations, and incoherent interfaces between phases. A specific development of such cases is given by Stevens et al. [36], derived from the original formulations of Herring and Herring and Bardeen [2, 12]. Since these early fairly specific treatments, Larché and Cahn [17–20] systematically set about formulating a very general thermodynamic theory capable of rigorously accounting for the constraints imposed by a crystalline lattice (which they referred to as the lattice constraint) in a multicomponent coherent system. The thermodynamic equilibrium relationships were formulated in such a way that they were applied regardless of the operative diffusion mechanism. Thus, the authors were able to demonstrate that in a system with lattice constraints (such as applies when there is coherency between phases) the chemical potentials of individual species, including vacancies, cannot be defined within the lattice (or network as the authors called it). Only at places where the lattice can be altered would this be possible and, hence, only at these locations, could local equilibrium be achieved. Thus such single species chemical potentials are not necessarily constant throughout the solid and therefore cannot be defined. The authors show that the

4.3 Effect of Stress on Hydrogen Chemical Potential in the Dilute Phase

121

most general chemical potentials for diffusion of substitutional species in a multicomponent solid—which at equilibrium is constant throughout the solid but the gradient of which leads to diffusion—are the differences in chemical potentials of the various species. For dilute concentrations of interstitial species in the general case, the chemical potential of that species remains the chemical potential for diffusion, since the latter can alter its value to adjust to the varying values of the stress throughout the solid and, for this to be possible, does not require being located at places where also the lattice can be altered. For concentrations of the interstitial species close to that of, say, a stoichiometric interstitial solid such as, for instance, ZrH2 (and possibly, ZrH, or even ZrH1.5) the chemical potential of the interstitial species (H in this case) in a nonhydrostatically stressed solid can no longer be defined on its own, since diffusion cannot proceed without some defect mechanism, such as an interstitial vacancy mechanism. In this case, similar to substitutional diffusion, only the difference in the chemical potential of the interstitial species and the corresponding site vacancy has a well defined meaning at all locations in the solid. In the analysis of Li et al. [22], the authors had already recognized the special role that the chemical potential of an interstitial component in a solid plays in terms of being able to achieve at equilibrium a constant (uniform) value throughout an inhomogeneously stressed solid. They called this component mobile, adopting for the derivation of its thermodynamic properties, the original thermodynamic development of Gibbs for a fluid component diffusing within, and causing distortion of, a porous, nondiffusible solid [11]. The authors derive, using a Moutier cycle approach, an expression for the chemical potential of any component, k, at the surface of a closed (to external species transfer) isothermal system stressed in some arbitrary way but with the surface oriented so that it is normal to one of the principal stresses. The result is:  k  Wk lk ¼ lok þ w

ð4:28Þ

where Wk ; is the work done per mole addition of component k and k  w

ow onk

ð4:29Þ

is the partial molar strain energy of component, k. The authors then show that for a mobile component, k = I, a general form of this relation, valid throughout the  k (using the Voigt notation for representing the solid, is obtained by writing for w stress tensor): I ¼ w

ri X XZ i

j

0

 

osij   VI sij þ ð1  xI ÞV ri drj oxI x0

ð4:30Þ

I is the molar volume of component I, sij are the elastic compliances of the where V  is the molar volume of the solid solution, xI is the mole fraction of solid, V

122

4 Solubility of Hydrogen

component I with the x0 outside the partial derivative indicating that the differentiation is taken with respect to xM at constant ratio of all other mole fractions and ri are the six components of the stress tensor. For Wk the expression is: X  oei   W I ¼ V ð1  xI Þ ri ð4:31Þ oxI x0 i where the ei are the six components of the elastic strains corresponding to the stresses, ri. As an example of a specific application of Eqs. 4.28–4.31, Li et al. calculated the chemical potential of hydrogen in iron which is under externally applied uniaxial stress, r, in equilibrium with an external reservoir (‘‘fluid’’) of hydrogen. The result is:   H r2 V   xFe 1 dE r2 V ol o lH ¼ lH þ  ð4:32Þ  rA E dxH onH 2E 2E H is the partial molar volume of hydrogen in the solid, E is the elastic where V modulus, ol=onH is the length change of the solid in the direction of the applied stress produced by the transfer of dnH moles of hydrogen to the solid and A is the cross-sectional area normal to the applied stress, r, with loH the chemical potential of hydrogen having the same hydrogen concentration in unstressed iron. The second and third terms in Eq. 4.32 are the self and inhomogeneity strain energies of the interstitial atom. (More general expressions for these are given in the next section.) Assuming that hydrogen expands the lattice isotropically, and neglecting the terms involving r/E, which are second order to the last term when r/E is small, Eq. 4.32 becomes: r lH ¼ loH  V H 3

ð4:33Þ

where r/3 is the hydrostatic (mean) stress component of the uniaxial stress r. This is the well-known expression for the effect of stress on the chemical potential of hydrogen in equilibrium with an external reservoir of hydrogen. This relationship was derived using the usual convention that tensile stresses have positive sign. It shows, then, that tensile stress lowers the chemical potential of hydrogen relative to that in an externally unstressed solid. Note that—as the symbol used to denote it implies—Wk in Eq. 4.28 and its specific value given in Eq. 4.33 is a work term that does not change the internal energy of the system. The physical significance of this is shown in Chaps. 6–8 giving the development of equilibrium phase relationships as they are affected by external and internal (coherency) stresses. It is straight-forward to understand the nature of this term when the source of the stress is external, as it can be seen that it represents the amount of work done in moving the external surface of the solid on which the stress is imposed as a result of the insertion of a mole of hydrogen atoms (regardless of the location in the solid where these are added). However, it is less

4.3 Effect of Stress on Hydrogen Chemical Potential in the Dilute Phase

123

straight-forward to see that this result is the same when the source of stress is internal such as at dislocations or misfitting precipitates, or even when it is applied externally but amplified near a crack tip at which value of the stress the interaction is calculated. These subtleties are discussed in Sect. 4.4.1, in which some H ; of hydrogen are important properties of the molar volume of formation, V presented.

4.4 Interaction Energy Expressions The distortion of the lattice (size effect) produced by the introduction of a misfitting mobile component—such as hydrogen in the case of Eq. 4.33—into the solid produces an amount of work done that could be interpreted as an ‘‘interaction energy’’ between the stresses in the solid and the distortions produced by the mobile component. The following sections provide a detailed treatment of this interaction energy based largely on the results derived by Eshelby [8]. Such results are of significance in understanding coherent phase equilibrium and hysteresis in the metal–hydrogen solvus, the theoretical developments of which are given in Chaps. 6, 7 and 8. The following treatment is limited to the formulation of interaction energy expressions of misfitting atoms—also referred to as point defects—within a linear elastic description of the solid. The treatment is made specific by considering the positively misfitting case of hydrogen interstitial atoms dissolved in dilute zirconium or zirconium hydride phase. It largely follows the approach provided some time ago by the author [29] where it served to provide part of the theoretical basis for dealing with the effect of hydrogen in iron.

4.4.1 Partial Molar Volume The elastic interaction of a misfitting point defect with a stress field can be developed in terms of representing the point defect as either a fixed source of stress or of strain located at the origin of the defect. In the stress representation, the defect is represented by an internal set of forces formulated such that they produce vanishing total force and torque on the solid. Following the treatment of Leibfried and Breuer [21], the displacement field caused by these forces is given by Z si ðrÞ ¼ Gik ðr  r0 Þfk ðr 0 Þdr ð4:34Þ where r and r0 are, respectively, the position vectors (bold face type is used throughout to indicate vector or matrix variables) at which the displacement and the force are calculated; Gik is the static Green’s function, fk is the kth component of the force and repeated indices imply summation.

124

4 Solubility of Hydrogen

The static Green’s function is the displacement field as a result of a unit force. The asymptotic expression for the displacements, si(r), results in si ðrÞ ¼ Pnk Gn;ik ðrÞ

ð4:35Þ

with Pnk ¼

Z

x0n fk ðr0 Þdr0 ; Pnk ¼ Pkn

ð4:36Þ

which represents the leading term in an expansion of the Green’s function in powers of r0 . The dipole tensor in Eq. 4.36 represents the (fixed) source term of the defect. It completely describes the macroscopic response of the solid to the distortions produced by the point defect. Thus, Leibfried and Breuer [21] show that the total volume change, Dv, produced by a given dipole tensor in a finite crystal depends only on the magnitudes of the forces generated by this dipole tensor and the elastic compliances of the solid. The volume change is independent of the position of the forces (provided they are not too close to the surfaces of the solid) and of the size of the solid. For a cubic crystal it is given by: Dv ¼

Pii Pii P0  ¼ C11 þ 2C12 K 3K

ð4:37Þ

where the C’s are elastic constants, K ¼ ðC11 þ 2C12 Þ=3 is the bulk modulus and P0  Pii =3, where repeated indices imply summation. Representing the source strength of the defect in terms of a dipole tensor is useful when the properties of the point defect are obtained theoretically by atomistic models, since this tensor can be more accurately and easily determined from these models than can the misfit strains. (These strains have variously also been referred to as stress-free misfit, transformation, or eigen strains. They are also sometimes referred to as initial strains. This designation is used because, unlike the defect forces that are unaltered by the relaxation of the lattice surrounding it when the defect is inserted into the crystal, the transformation strains are altered by this response.) As noted, these strains could also be used to determine the strength and symmetry of the defect. Note that, given a calculated value of the dipole tensor, the volume increase of the solid as a result of the introduction of a single misfitting atom can be calculated from Eq. 4.37. It is instructive to provide a rough estimate of the displacements generated by the defect in terms of its source strength given by Eq. 4.37 using linear continuum theory. In doing so, it is useful to separate out the displacement that is obtained for a solid of infinite size, denoted by s? and an additional displacement, sI, (where ‘I’ stands for ‘Image’) that occurs when the solid is of finite size satisfying the boundary condition of a stress-free surface. For the simplest case of a dilatation center at the origin of a solid sphere of radius, R, in an elastically isotropic material the result from Eqs. 4.35 and 4.36 is

4.4 Interaction Energy Expressions

125

P0 r^ 4pC11 r 2

ð4:38Þ

C44 P0 r P0 C44 r  pC11 3ðC11 þ 2C12 Þ R C11 3K pR

ð4:39Þ

s1 ¼ where r^ is a unit vector and sI ¼

From Eqs. 4.38 and 4.39 the respective volume changes are: Dv1 ¼

P0 P0 4C44 P0 4G  and DvI ¼ C11 C11 3K C11 3K

ð4:40Þ

where G is the shear modulus. Thus the total volume change, Dv ¼ Dv1 þ DvI ; is given by:     P0 4C44 P0 3K þ 4G Dv ¼ 1þ ð4:41Þ  3K C11 3ðC11 þ 2C12 Þ C11 With the relation C11 ¼ ð3K þ 4GÞ=3; Eq. 4.41 reduces, as required, to the same result given by Eq. 4.37. Defining a constant c¼

Dv Dv1

ð4:42Þ

then, from Eqs. 4.40 and 4.41, the expression for c for the case of an isotropic linear elastic solid is: c¼

3ðC11 þ 2C12 Þ þ 4C44 C11 3K þ 4G   3K 3ðC11 þ 2C12 Þ K

ð4:43Þ

It is evident from the ratio of sI/s? * r3/R3 that sI is negligible near the defect, but near the surface where, in some types of experiments the volume change is measured, both displacements are of equal order for a dilatation center in an isotropic solid. Moreover, for this case Eshelby [8, 9], using a misfitting sphere– in–hole model combined with linear elastic theory, demonstrated the important result that the separate contributions of Dv? and DvI are each constant for any size and shape of the crystal or the position of the forces in it. In general, however, the defect forces may not be concentrated at the center of the defect and a simple relationship such as given in the foregoing would then not exist between the two contributions to the volume increase. They would, instead, depend on the size and shape of the crystal, although their sum would still be insensitive to these details. Regarding the dilatation of the lattice as a result of the image field in a finite crystal, Eshelby [8, 9], using, again, a misfitting sphere–in–hole model inserted into the center of a finite-sized spherical solid, derived an expression for the uniform pressure, ^ ph ; created in the solid. For the case when the inserted misfitting

126

4 Solubility of Hydrogen

sphere has the same elastic constants (moduli) as that of the solid in which it is inserted the result is2 _

ph ¼ K

DvI 4G  K Dv ð1 þ mÞ Dv ¼  4G ð1  mÞ XM XM 3K þ 4G XM

ð4:44Þ

where XM is the atomic volume of the stress-free host lattice. This volume has been identified with the subscript ‘M’ to indicate that—for the applications in this text focusing on the metal-hydrogen system—it represents the atomic volume of the (defect free) metal lattice. Further on it is shown that this pressure, which is tensile for a positively misfitting defect such as hydrogen, produces an indirect interaction between such defects that is inversely proportional to the volume of the specimen and acts as an attractive force between them. Note that the total increase in Dv is actually the unconstrained misfit volume of the defect. This increase in volume, which is obtained when adding a single hydrogen atom into a finite-sized crystal, is related to the partial molar volume of H ; by the relation hydrogen, V H ¼ NA Dv V

ð4:45Þ

where NA is Avogadro’s number. The local distortions created by the interstitial hydrogen defect can be described more generally by an elastic strain dipole located at the origin of the defect.3 The elastic strain dipole is often called the k-tensor, and is obtained from the increment in strains produced in the crystal as a result of an increment of hydrogen atoms added to it by the relation kij ¼

oeij orI

ð4:46Þ

where rI is the atom ratio of interstitial to host metal atoms. These strains can be determined either by measurements of changes in the external dimensions of the solid or of the (average) lattice parameter(s) of the crystal [8]. In cubic crystals, the principal values of the k-tensor are related to the lattice constant, a0, by 1 da 1 ¼ ðk1 þ k2 þ k3 Þ a0 drH 3

2

ð4:47Þ

Note that this expression is often given in the literature with the opposite sign. This follows a sign convention where pressure is given as negative when tensile while the opposite sign convention is sometimes used for stress components. In this text, both pressure and individual stress components are taken as positive when tensile. Since the sign convention used is not always indicated in a given text, it can be indirectly inferred from the results for the interaction energy, which should yield a negative (attractive) interaction energy when the misfit strain (volume) is positive and tensile stresses are applied. 3 Hence, the components of this strain dipole are equivalent to the stress-free misfit or transformation strains of the point defect.

4.4 Interaction Energy Expressions

127

where a0 is the lattice parameter of the hydrogen free crystal. For an hexagonal metal such as Zr, with lattice parameters a0 and c0 we have 1 da 1 ¼ ð k1 þ k2 Þ a0 drH 2

ð4:48Þ

1 dc ¼ k3 c0 drH

ð4:49Þ

and

MacEwen et al. [24] experimentally determined the components of the k-tensor for deuterium atoms in Zr using time–of–flight neutron diffraction. Deuterium rather than hydrogen was used for the experiments to reduce the amount of incoherent scattering, since an isotope effect on lattice parameter changes was not expected based on results in other systems. Measurements were made at high temperatures (454 and 504 °C) to increase the maximum concentration of hydrogen (deuterium) concentration in solution, thus increasing the concentration range over which lattice parameter changes can be determined. A linear increase of lattice parameter with increase in deuterium concentration was obtained for both sets of temperatures. Although for these two temperatures, a difference was found in the dependence of the strain on deuterium concentration, this difference was not statistically significant. Assuming that the local distortion is tetragonal, the average principal k-strains derived from the neutron diffraction data at the two temperatures were then: k1 = k2 = 0.033 and k3 = 0.054. The ratio of tetragonality ¼ obtained from this result is 1.64. The relaxation volume per Zr atom, Dv=XaZr Zr 0:12 where XaZr is the atomic volume of Zr (volume of the Wigner–Seitz cell) in Zr the a–Zr phase. The increase in volume per hydrogen atom added Dv = 2.78 Å3, while the corresponding molar volume value is 1.67 9 10-6 m3/mol H. The measured dilatation of the lattice per deuterium atom is in good accord with the average lattice dilatation of 2.9 Å3 for hydrogen in metals having fcc, bcc, or hcp crystal structure [27].

4.4.2 Size–Effect Interaction As was shown in the derivation of the chemical potential of hydrogen in a stressed solid by Li et al. [22], it is sometimes necessary to calculate the change in energy of the thermodynamic system when hydrogen is brought from an unstressed solid into one that is externally stressed or brought into regions of the solid where elevated internal stresses exist such as at dislocations or cracks. The resultant additional energy that arises in this case is often called interaction energy in analogy with the interaction energy between atoms in a crystal lattice. This conceptual connection can be seen when the misfitting defect (hydrogen) is moved in a spatially varying stress field. The change in energy with distance results in a

128

4 Solubility of Hydrogen

quantity that has the properties (and units) of a force and which acts as if there were an interaction between the defect and the source of stress. In particular, this interaction provides the driving force for diffusion as a result of stress gradients for mobile defects. To derive an expression for the interaction energy we follow the approach developed in the foregoing of representing the point defect as a point dipole source of stress. The total energy for a material subjected to forces, fi(r), consisting of source forces of defects and externally applied surface and body forces is given by Z 1 fi ðrÞsi ðrÞdr E¼ ð4:50Þ 2 where s is an exact solution for a given f. We now assume that one can meaningfully separate the total force in and on the system into, at minimum, two parts, consisting of fa and fb, having corresponding displacements sa and sb, respectively. This implies that the presence of one does not affect the strength of the other, i.e., that a linear treatment applies, which also means that there are no cross terms. Then the total energy splits into three parts, with the first two parts representing the self-energies of the two source forces and the third part the interaction energy, which is the energy change that is created when one defect is moved in the presence of the other, given by: Z 1 Eint ¼  ðf a sb þ f b sa Þdr ð4:51Þ 2 The two terms in the bracket of Eq. 4.51 give identical results and, therefore, the interaction energy can be written as either twice the product of f a times the displacements at that location, sb, arising from f b ; or vice versa. Note that the interaction energy represents a work term since it has the form of force multiplied by displacement. The insertion or movement of a defect does not change the internal energy of the system when both the initial and final locations are taken as part of the thermodynamic system, since the self energies of the source forces remain the same. However, an amount of energy has nevertheless been gained or lost that arises from the insertion or movement of the defect in the stress field.4 When the defect is represented as a source of stress by means of its dipole tensor located at r0, the interaction energy is given by: Eint ¼ Pij eaij ðr0 Þ

4

ð4:52Þ

In the literature it is sometimes said that the reason that defects with positive misfit strains preferentially locate to regions of elevated tensile stress is because the enhanced lattice dilation produced by the elevated tensile stress makes it easier for the defect (which also includes hydrides) to be accommodated there. The foregoing treatment shows that this is not the reason for the attractive interaction energy. In fact, in linear elastic solids there is no interaction between the strains of the lattice produced by a stressed solid and the misfit strains of the defect.

4.4 Interaction Energy Expressions

129

where eaij ðr0 Þ is the strain imposed by internal or external stresses at the location of the defect in the absence of the defect. When the defect acts as an isotropic dilatation center, this result reduces to Eint ¼ P0 dij eaij ðr0 Þ

ð4:53Þ

where dij is the Kronecker delta function. This shows that a point defect that can be described as an isotropic dilatation center interacts only with the hydrostatic component of the applied stress. The corresponding result for the same defect represented by its volumetric misfit strain, Dv, is given by 1 Eint ¼  raii ðr0 ÞDv ¼ ph ðr0 ÞDv 3

ð4:54Þ

where ph ðr0 Þ ¼ raii ðr0 Þ=3 is the hydrostatic stress acting at the center of the point defect located at r0. For an anisotropic defect located at r0, with the defect’s lattice stress-free misfit strains given by eTij ; the interaction energy is: d raij ðr0 ÞeTij Eint ¼ V

ð4:55Þ

d is the molar volume of the defect site in a perfect, unstressed crystal lattice, where V aZr Zr or XaZr depending on which, in the case of hydrogen in zirconium, is given by V Zr whether it is expressed in terms of molar or atomic volumes, respectively.

4.4.3 Diaelastic Polarizability The foregoing treatment provides the first order term in the interaction energy, often also referred to as the size effect interaction. There are two other terms in the interaction energy the first of which is second order to the size effect interaction at low stresses, the second of which is possible when the defect is anisotropic. These terms are, respectively, the diaelastic and the paraelastic polarizabilities. The first term arises from a difference in the elastic response of the defected crystal as a result of the lattice distortions created by the defect. In the misfit strain representation of the defect this interaction is also called inhomogeneity interaction because the defect’s local elastic response is inhomogeneous with that in the rest of the (nearly perfect) crystal. The second interaction occurs only when the applied stress is nonhydrostatic. Then, if the defect is sufficiently mobile, application of a stress could cause the defect’s interaction energy to be reduced by alignment of the defect’s largest principal strain with the direction of the largest imposed principal stress.5 5

The interaction energy terms given in this section are generally of second order and not required in DHC theory. However, the existence of these interactions is sometimes exploited in experimental techniques to determine the defect’s stress-free misfit strain. For this reason, a description of their derivation is given here.

130

4 Solubility of Hydrogen

When the defect is represented as a source of stress, characterized by its dipole tensor, the macroscopic response of the crystal can be characterized by an induced dipole tensor DPdij defined as follows: DPdij ¼ Pij ðea Þ  Pij ð0Þ ¼ adij;kl eakl

ð4:56Þ

The fourth-ranked tensor, adij;kl ; having the symmetry of the elastic constant tensor, is called the diaelastic polarizability of the defect by analogy with a similar phenomenon in electrostatics for the response of a charged particle with an electric field. Expressing Eq. 4.51 in terms of the df a and df b ; the interaction energy as a result of this source is given by: Z 1 d ðdf a sb þ df b sa Þdr Eint ¼  ð4:57Þ 2 It is evident from Eq. 4.57, that in general, this interaction energy requires not only the evaluation of the work done by one source force as a result of the induced forces of the other, but also the reverse process. However, for a point defect interacting with an external surface stress, a crack, or a dislocation, the influence of the point defect on these sources can be neglected. Hence the interaction energy is given by: 1 d Eint ¼  eaij ðr0 Þadij;kl eakl ðr0 Þ 2

ð4:58Þ

where eaij ðr0 Þ is the applied strain acting at the defect site, r0. The diaelastic polarizability can be calculated from atomistic models. Examples of such calculations are given by Puls [31] for defects in ionic crystals and [30] for defects in fcc metals. Macroscopically the diaelastic polarizability is related to the change in elastic constants when adding a small concentration of, say, interstitial atoms, cI \\ 1, to a metal by: dcij;kl ¼ 

cI d a XM ij;kl

ð4:59Þ

Eq. 4.59 shows how this polarizability can be determined experimentally. Eshelby [8] produced corresponding results by representing the defect as a source of misfit strain that is assumed to have ‘‘elastic constants’’ different from those of the surrounding matrix and would, therefore, distort differently under an applied stress than the surrounding matrix material. Eshelby called this energy the inhomogeneity interaction energy.6 It is given by

6

Another formulation of this term was given by Li et al. [22] as part of Eq. 4.32 for the effect of stress on the chemical potential of hydrogen in solution under an external stress.

4.4 Interaction Energy Expressions

131

inh Eint ¼ Vd



1 dK 1 dG Wd þ Ws K dcI G dcI

 ð4:60Þ

where Wd and Ws are the interaction energy densities for a single defect as a result of, respectively, the dilatational and the deviatoric part of the induced strain. Writing A

1 dK K0  K ¼ K dcI K þ aðK 0  KÞ

ð4:61Þ

B

1 dG G0  G ¼ G dcI G þ bðG0  GÞ

ð4:62Þ

a¼

1þm 3ð1  mÞ

ð4:63Þ

b¼

2ð4  5mÞ 15ð1  mÞ

ð4:64Þ

with

where v is Poisson’s ratio, and using the notation rij = r/3 ? 0 rij with r = rii, which separates the stress tensor into dilatational and deviatoric components, the interaction energy becomes   Vd A 2 B 0 20 2 inh r þþ Eint ¼ r r ð4:65Þ 2G 2 9K where the constants A and B can be determined experimentally (using Eqs. 4.61 and 4.62) from measurements of the change in bulk and shear moduli with increase in defect concentration (which, in this case was assumed to be an interstitial atom).

4.4.4 Paraelastic Polarizability The paraelastic polarizability derives its name from corresponding phenomena in electricity and magnetism where a permanent magnetic dipole will orient itself along the lines of magnetic field. As with this phenomenon, for lattice defects having permanent anisotropic elastic dipole tensors, application of a nonhydrostatic stress causes certain orientations of these defects to have lower interaction energies. There would, in general, be an energy barrier for orientation of the defect into the various energy states, which means that the probability of preferential orientation in the lower energy state depends on temperature. Experimentally, a nonvanishing paraelastic polarizability results in an anelastic (i.e. time dependent) lattice relaxation effect called the Snoek effect.

132

4 Solubility of Hydrogen

By analogy with the diaelastic polarizability, the paraelastic change, DPpij ; in the dipole tensor is defined by DPpij ¼ Pij ðea Þ  Pij ð0Þ ¼ apij;kl eakl

ð4:66Þ

where apij;kl ¼

  1  Pij Pkl  Pij hPkl i kB T

ð4:67Þ

with the brackets \[ denoting an average over equivalent orientations. Alternatively, recognizing from Eq. 4.53 that equivalent orientations of the defect will have different energies, the probability of the defect being in orientation n is  

n =kB T proportional to exp Eint

when in thermal equilibrium. Expansion of the

deviation of the dipole tensor from its isotropic value, P0 Pn ¼ P0 þ dPn

ð4:68Þ

yields the paraelastic polarizability apij;kl

z X dPnij dPnkl ¼ zkB T n¼1

! ð4:69Þ

where z is the number of orientations. The paraelastic polarizability can also be measured as a change in the elastic constants according to Eq. 4.60. Generally, paraelastic polarizabilities are an order of magnitude greater than diaelastic ones, except at very high stresses and/or low temperatures where, in the latter case, the defect may be frozen in the orientation it had at higher temperatures.

4.4.5 Interactions Between Hydrogen Atoms in Solution The direct interaction energy between point defects when both the defect and the material are isotropic turns out to be zero because each defect produces only shear stresses in the surrounding matrix, the interaction of which with an isotropic defect is zero. However, if either the matrix material or the defect is anisotropic, which is the case for hydrogen in zirconium, there would also be an interaction between defects. This interaction energy would be quite short ranged since it can be shown [9] that both the stresses and the strains produced by either defect decrease with radial distance, r, from the defect according to 1/r3, which means that the interaction energy between them decreases as 1/r6. Thus, even when there is anisotropy, on average, the interaction between defects in a dilute interstitial solution would generally be negligible because the defects (interstitials) would be too far apart.

4.4 Interaction Energy Expressions

133

In addition to the direct interaction between defects there is, in a finite crystal, also an indirect one that arises from the hydrostatic stress produced by each defect (regardless of its symmetry) by the image term. It turns out that in the isotropic case the average energy of the crystal as a result of this indirect interaction between defects does not depend on the precise arrangement of defects in the crystals. When the defect concentration is large this image interaction energy can be of comparable magnitude, but opposite in sign to the total self strain energy of the solid that is produced by the defects. This can be shown as follows. The self energy, Es, produced by a single defect is given by Es ¼

2KG Dv2 ð1 þ mÞ Dv2  2G 3K þ 4G XM ð 1  mÞ X M

ð4:70Þ

The pressure build up as a result of the image volume increase per atom, DvI, after cI defects have been added to the crystal is, from Eq. 4.44: _

DvI 4KG Dv cI ¼ cI 3K þ 4G XM XM ð1 þ mÞ Dv ¼ 4G cI ð1  mÞ XM

ph ðcI Þ ¼ K

ð4:71Þ

The interaction energy expended per defect for adding this concentration of defects to the crystal is 1_ 2KG Dv2 Eint ðcI Þ ¼  ph ðcI ÞDv ¼  cI 2 3K þ 4G XM ð1 þ mÞ Dv2 ¼ 2G cI ¼ Es cI ð1  mÞ XM

ð4:72Þ

When this is combined with the energy of the first defect added, given by T ^ int ; is: Eq. 4.70, the average total elastic image interaction energy per defect, E T ^ int E ¼ Es þ Eint ðcI Þ ¼ Es ð1  cI Þ

ð4:73Þ

and, hence from Eq. 4.73, the total misfit strain energy is given by T ^ int EsT ¼ E cI ¼ Es cI ð1  cI Þ

ð4:74Þ

Equation 4.74 gives the heat of solution of noninteracting defects, such as for instance interstitial hydrogen atoms dissolved in the single phase region of a crystalline solid. For use in thermodynamic relationships, the energies given in the foregoing are enthalpies when calculating their magnitudes using adiabatic elastic constants and Gibbs free energies when using isothermal elastic constants. It is not immediately obvious that Eq. 4.74 would be valid across the full composition range of interstitial atoms in the phase diagram and not just for dilute solutions of this component. However, Eshelby [8] showed that in a substitutional two-component solid the simple cI ð1  cI Þ dependence for the total

134

4 Solubility of Hydrogen

nonconfigurational energy given by Eq. 4.74 obtained on the basis of the sphere-in-hole model is valid when the stress-free volumes of the defect (component 1, or solute atom) and the hole (component 2, or host (solvent) atom) for this case are equal to the crystallographic volumes of each atom in the perfect lattice in its respective pure state. This is equivalent to saying that the rule of additivity of atomic radii for solute (defect) and solvent (host) given by Vegard’s law is valid. In this case, the fractional change of lattice constant with composition is constant across the full range of composition of the phase diagram. An actual case where this is likely is when the solute and the solvent have the same crystal structure across the full range of the phase diagram while the volume difference between the two atoms is not too large, which means that the phase transformation is solely a compositional rather than a structural change. For hydrogen in metal systems this is the case for Pd–H. For the Zr–H system, measurements show that the fractional increase in lattice parameter in the a–Zr phase up to a hydrogen (deuterium)/zirconium ratio of 0.03 is constant [24], while in the d–hydride phase some data indicate there is no change (as summarized by Zuzek et al. [40]) while, based on more recent measurements, another set of authors find that there is a linear fractional increase in lattice constant with hydrogen composition [38, 39]. In the e-hydride phase, an approximately constant increase in lattice parameter with increase in hydrogen concentration has also been found (Chap. 2). The possible importance of whether Vegard’s law is obeyed or not in the Zr–H system is discussed in Chap. 8 dealing with the application of the theories of solvus relationships in coherently stressed solids. An important feature of the result of Eq. 4.74 is that the total elastic strain energy of the crystal containing cI defects (interstitial, or in our case, hydrogen atoms) is no longer linear in the number of defects added but also includes a term that depends on the square of the concentration of the added defects. This quadratic dependence arises from the image interaction term. Because of this quadratic dependence of the total strain energy on the concentration, the linear–additive property assumed in classical thermodynamics for extensive parameters such as energy, entropy, specific volume, and component number, is no longer applicable (see Chap. 7 for further discussion on this). Another property of the total elastic strain energy given by Eq. 4.74, mentioned in the foregoing, is that it is independent of the exact distribution of these defects. This means that, provided the assumptions of linear elasticity used to derive these results remain valid, the total strain energy of the solid is given by these expressions regardless of whether the hydrides are in the dilute or the highly concentrated hydride phase. It might be expected that, in reality, the validity of the assumptions used by Eshelby in deriving these strain energies might break down, particularly in the intermediate concentration range where neither the solvent nor the solute can be considered as being dilute. To determine the strain energy in the solid in this case requires accurate estimates of the nonconfigurational part of the interaction energy between defects. This, in turn, depends on the accuracy of the relations derived for the volume increase resulting from the image effect. Various methods of deriving the associated configuration independent elastic interactions were reviewed by

4.4 Interaction Energy Expressions

135

Bass et al. [1], focusing on hydrogen in Pd. Defining the coefficient c0 by the relation: 1 DvI ¼ ð1  ÞDv  c0 Dv c

ð4:75Þ

in relation to the c coefficient introduced by Eshelby [8] given by Eq. 4.42, the authors have an expression, depending on c0 , relating the volume change resulting from the image interaction to the total volume change produced in a finite solid when a misfitting defect such as a hydrogen atom is introduced into the solid. In an isotropically elastic solid, the value of c is given by Eq. 4.43, and, hence, the corresponding value for c0 is: c0 ¼

4G 3K þ 4G

ð4:76Þ

Equation 4.76 shows that in this linear elastic approximation the respective volume changes depend solely on the elastic constants of the host material. However, it is seen from Eq. 4.60 that introducing a misfitting atom into a crystal results in a change of elastic constants and therefore the value of c0 could vary if this change were nonlinear at high defect concentrations. An expression for c0 derived by Bass et al. [1] given further on provides an example of how this coefficient might vary with defect (interstitial hydrogen) concentration. Most of the results summarized by Bass et al. [1] for the values of the c and c0 coefficients defined by Eqs. 4.43 and 4.76 are based on results derived from atomistic models or from more realistic elastic calculations for the defect strain energy. In the case of the Pd–H system it is noted that Pd–H, although crystallographically isotropic, is not elastically isotropic. Nevertheless, the authors point out that in an harmonic crystal of any symmetry, because the displacements from the image term component fall off faster than r-2, only the term falling off as 1/r3, which produces the volume expansion, Dv?, has the property of creating a finite volume increase without there being any lattice dilatation in between the defect and the outer surface at which this volume change is measured. It is evident that this is a mathematical result unconnected with any specifics of the elastic behavior. In practice, however one would want to estimate the source strength of this volume increase when it is not too close to the defect where the displacements from the image dilatation contributions may be hard to distinguish from the contributions given by the 1/r3 term, and also not too close to any real surface where it matters what the shape of the surface would be. In terms of the c0 parameter defined by Bass et al. [1], the nonconfigurational average interaction energy per pair can be written NC ¼ c0 K Eint

Dv2 ðC11 þ 2C12 Þ Dv2  c0 3 XM XM 4G Dv2 K ¼ ð3K þ 4GÞ XM

ð4:77Þ

136

4 Solubility of Hydrogen

where K is the bulk modulus and the last result is with the value of c0 derived from Eshelby’s [8] solution, reproducing, as it should, the result per defect of Eq. 4.73 derived by Eshelby [8] for the elastic case. The total nonconfigurational energy ENC when there are N defects (interstitials NC independent of N is or solute atoms) in the host lattice and with Eint 1 1 NC NC ENC ¼ NðN  1ÞEint ffi N 2 Eint ðN  1Þ 2 2 2GK Dv2 ffi N 2 ð3K þ 4GÞ XM

ð4:78Þ

Assuming specifically that the defect considered is a hydrogen atom, the chemical potential for hydrogen, lNC H ð N Þ, is derived from Eq. 4.78 according to the usual definition of chemical potentials by taking the partial derivative of the Gibbs free energy (equivalent to ENC) with respect to defect number. This yields: lNC H ðNÞ ¼

o NC NC E ðNÞ ffi NEint oN

ð4:79Þ

which, converting to atomic or molar fractions, respectively, becomes from Eq. 4.77: 0 2 lNC H ¼ c KDv

h Dv2  rH Dv2  rH ¼ c0 K  c0 K M Vd bpha  XM bpha  V

ð4:80Þ

where N has been replaced by the concentration variable h ¼ rH =bpha ; which represents the fractional occupancy of the interstitial sites. The concentration, rH, has the usual definition used in this text giving the ratio of the total number of interstitial (hydrogen) atoms to the total number of metal atoms in the solid while bpha = total number of equivalent interstitial sites per metal atom in the crystal. Note that for Pd, bpha is equal to one while for a Zr it is equal to two. Defining the interstitial concentration in this way ensures that the terminal concentration— which cannot obviously be pure, solid hydrogen, but must instead be a stoichiometric interstitial compound such as the terminal stoichiometric composition of a particular hydride phase—is equal to one, similar to a binary substitutional solution. Equation 4.80 shows, as was the case for Eq. 4.78, that deriving accurate 0 expressions for lNC H depends on having accurate values for c . An alternate way of deriving Eq. 4.80—which removes it from being tied to the NC varies slowly with N. sphere-in-hole model—is to assume that generally Eint Hence, writing NC NC ðNÞ ¼ Eint ð0Þ þ Eint

then

NC dEint N dN

ð4:81Þ

4.4 Interaction Energy Expressions

137

Table 4.2 Elasticity theory results for c0 and c for Pd and Pd hydride (PdH0.66) (from Bass et al. [1]) PdH0.66 Elastic model Computational model Pd Pd PdH0.66 (0 K) (300 K) (0 K) (300 K) c: isotropic c: anisotropic c:eff. isotropic c0 : isotropic c0 :anisotropic c0 :eff. isotropic

1.25 1.52 0.20 0.34

1.20 1.34 1.43 0.17 0.25 0.30

1.18 1.33 1.43 0.15 0.25 0.30

1.08 1.37 1.47 0.07 0.27 0.32

1.21 1.34 1.43 0.17 0.25 0.30

1 1 dENC 1 ^ NC NC NC Eint ðNÞ  N 2 Eint ð0Þ þ N 3 int  N 2 E ð4:82Þ int 2 3 2 dN  NC  NC NC ^ int ¼ 13 Eint ð0Þ þ 2Eint ðNÞ is chosen as the appropriate average energy. where E d2 ENC Ignoring terms in dNint2 ; this yields for the nonconfigurational chemical potential NC 2 lNC H ðNÞ  NEint ð0Þ þ N

NC dEint NC  NEint ðNÞ dN

ð4:83Þ

This result would be identical to that given by Eq. 4.79 when using Eq. 4.78 for and assuming that c0 does not vary with N. However such variation is evident (Table 4.2) although, of the methods used to calculate c0 , only those obtained assuming isotropic elastic continuum behavior are available (see the difference in the values of c0 between pure Pd and PdH0.66). An alternative approach for deriving an expression for the nonconfigurational energy at high hydrogen concentration is to consider obtaining it from the difference between the total and the configuration-dependent interaction energies since an expression for the total interaction energy, ET, is known for an arbitrary lattice of any shape (e.g., [37]). The total energy is given by: NC Eint

1 h2 ET ¼ KDv2 2 Vd

ð4:84Þ

where Vd is the volume of the defect. Oates and Stoneham [26] have shown that the configuration-dependent energy, EC, depends on both the arrangement and number of interstitial (hydrogen) defects. However, to first order, a simple analytical form for the compositional dependence of this energy, similar to what is obtained for a substitutional binary solid, in which case the local relaxation energy varies *hð1  hÞ; might also be expected for a random distribution of interstitial atoms. Such a composition dependence of this energy seems sensible as it results in the energy becoming zero, as would be expected, at either end of the composition diagram. Thus, formally expressing the configuration-dependent energy as

138

4 Solubility of Hydrogen

~  hð1  hÞ E C ¼ A

ð4:85Þ

where à is a parameter to be determined, then the nonconfigurational energy, ENC, is the difference between the energies given by Eqs. 4.84 and 4.85, which, when compared with Eqs. 4.77 and 4.78 for a dilute solution (h ? 0), yields for Ã: ~ ¼ 1 KDv2 ð1  c0 Þh A 0 2Vd

ð4:86Þ

where c00 is the value of c0 in a very dilute solution. Thus, when hydrogen at high concentration in the solid forms a random solution such as in the hydride phase of the Pd–H system, the configurational and nonconfigurational energies are given, respectively, by:

   ENC ¼ c00 þ 1  c00 h  ET  c0 ðhÞ  ET   EC ¼ 1  c00 ð1  hÞ  ET

ð4:87Þ ð4:88Þ

Equation 4.87 shows that c0 ðhÞ equals unity when h = 1 (i.e., when all the sites are occupied), which also means that all the energy is nonconfigurational. Differentiating with respect to the total number of hydrogen atoms, N, as in Eq. 4.80 gives the various chemical potential expressions arising from the total, configurational, and nonconfigurational relaxation energies of the Pd lattice when hydrogen is introduced. Using data for hydrogen in palladium, Bass et al. [1] plot room temperature values of these chemical potentials as a function of the hydrogen concentration, rH, across the full composition range of the Pd–H phase diagram. At this point in our knowledge of the phase composition relationships of the Zr–H system it is not clear whether the foregoing expressions for the various elastic relaxation energies are good approximations for this system. There is some evidence given in Chap. 2 to suggest that the hydride phases formed appear to be closer to stoichiometric ones, and hence are not consistent with the assumption of a random interstitial solution at high hydrogen concentrations forming part of the basis of the results given by Eqs. 4.87 and 4.88. Still, it is useful to obtain some idea of the magnitudes of the various chemical potentials. Thus, the results obtained by Bass et al. [1] are reproduced in Fig. 4.2 using the data summarized in Table 4.3.

4.5 Interaction of Hydrogen Atoms in Solution with Internal Defects Before treating the case of hydrogen atoms interacting with internal defects such as impurity atoms and dislocations, it is useful to describe a mathematical construct for the occupation of hydrogen atoms in lattice sites that is based on the

4.5 Interaction of Hydrogen Atoms in Solution with Internal Defects

139

Fig. 4.2 Contributions to the chemical potential in Pd over a range of hydrogen concentrations. The total chemical potential in elasticity theory is lT with configurational part lC (Eqs. 4.84 and 4.88) and nonconfigurational part lNC (Eq. 4.80). The experimental excess chemical potential, lE, is the sum of the elastic part, lT, and a chemical part lCH. The combination of terms appropriate for statistical calculations is lCH ? lC (from Bass et al. [1])

widely used construct for the filling of energy levels of electrons in metals using Fermi–Dirac statistics. This construct is the Density Of Site Energies (DOSE).

4.5.1 Density of Site Energies (DOSE) and Fermi–Dirac Statistics DOSE is defined as the normalized number of sites, nðEÞ; in a given interval of energy between E and E ? dE such that Zþ1

nðEÞdE ¼ 1

ð4:89Þ

1

The DOSE is equivalent to the density of states for electrons and other particles in the quantum mechanical description of these particles in metals, although in that case the distribution of energy states is most usefully formulated in reciprocal space whereas for DOSE the distribution over sites in real space is most appropriate [13]. It is to be noted that the DOSE does not contain any information about

140

4 Solubility of Hydrogen

Table 4.3 Physical parameter values for Pd and Pd hydride (PdH0.66) used in Fig. 4.2 (from Bass et al. [1]) K(Pa) dK/dh (Pa) c00 , PdH0.66 Molar volume (m3/mol) (300 K) 8.87 9 10-6

1.20 9 1011

1.18 9 1011

0.25

the localization of the sites nor are the spatial correlations between sites of different energies accessible. The energy scale used is that based on the standard state of gaseous hydrogen at 0.1 MPa and 298.15 K. For a perfect crystal where only one of a possible number of interstitial sites is occupied (because it has the lowest energy) all the sites have equal energy and the system is totally degenerate. The DOSE is then given by the Dirac delta function. Following the treatment of Kirchheim [13], for a crystal containing dislocations, there is a range of sites of different energies around the dislocation center that can be occupied. The Gibbs free energy of the system is reduced when the lowest energy (most negative or most attractive) sites are filled first while filling sites of the same energy that are the most numerous increases the configurational entropy. Since each site can contain only a single hydrogen atom or none at all, Fermi–Dirac statistics apply to this case. The minimum Gibbs free energy for all energy intervals results in the following thermal occupancy distribution of energy levels, Ei, for the corresponding site: oðEi Þ ¼

1 1 þ exp½ðEi  lÞ=kB T

ð4:90Þ

In Eq. 4.90, l is the derivative of the Gibbs free energy with respect to particle concentration. Thus, when the particle in Eq. 4.90 is a hydrogen atom, l : lH is the chemical potential of hydrogen, although it is usually called the Fermi energy in the context of this type of distribution. Assuming a continuous distribution of sites, integration of the DOSE with the thermal occupancy distribution given by Eq. 4.90 gives the following relation for the ratio of dissolved hydrogen atoms, N, to the total number of available sites, N0: N ¼ N0

Zþ1 1

nðEÞdE 1 þ exp½ðE  lH Þ=kB T

ð4:91Þ

To illustrate the use of this relation in the simplest case of a metal assumed to be in equilibrium with an external reservoir of hydrogen gas, the concentration of hydrogen atoms in the metal is expressed in terms of the ratio, rH = NH/NM and the interstitial site number bpha. Then, using Eq. 4.5, the hydrogen solubility is given by rH NH  ¼ bpha bpha NM

Zþ1 1

nðEÞdE Eðlo =2Þ H2 1 ffi 1 þ pffiffiffiffiffi exp kB T pH 2

ð4:92Þ

4.5 Interaction of Hydrogen Atoms in Solution with Internal Defects

141

For a perfect crystal with nðEÞ ¼ dðE  Eo Þ and dilute solution (rH \\ 1) the result  o  lH2  2Eo pffiffiffiffiffiffiffi rH ¼ pH2 bpha exp ð4:93Þ 2kB T of Eq. 4.6 is reproduced with loH  2Eo and the trivial difference that Eq. 4.6 expresses energies per mole rather than per atom as in Eq. 4.93. In general, Eq. 4.92 is amenable to a closed solution for simple cases only. One such case—which illustrates also the competition between energy decrease and entropy increase—is based on the step (or T = 0 K) approximation of the Fermi– Dirac function. In this case oðEÞ  1 for E\lH and oðEÞ  exp½ðlH  EÞ=kB T for E [ lH : Writing this in terms of cH to be consistent with the notation used in the literature, where cH ¼ rH =bpha is the fraction of interstitial sites occupied by hydrogen, yields cH ¼

ZlH 1

nðEÞdE þ

Zþ1 lH

nðEÞdE  cH1 þ cH2 exp½ðE  lH Þ=kB T

ð4:94Þ

where cH1 is the concentration of particles below the Fermi level and cH2 is the concentration of particles above the Fermi level. It is evident that occupation of sites below the Fermi level decreases the energy of the system, whereas occupation of sites above the Fermi level increases the entropy of the system. Hence, for cH1 [ cH2, energy plays a bigger role than does entropy. In many cases, a sample contains a small fraction of defects, f, only. In this case the majority of sites are perfect crystal sites and the DOSE can be expressed by nðEÞ ¼ ð1  f ÞdðE  Eo Þ þ f  nf ðEÞ with f 1

ð4:95Þ

If the average energy of nf ðEÞ (its first moment) is greater than Eo, the defects have a negligible effect and the hydrogen atoms are almost all in perfect lattice sites. On the other hand, when the average energy of nf ðEÞ is less than Eo then integration over the first part of Eq. 4.95 gives the concentration contained in perfect lattice sites given by:   1f l  Eo  o   exp H cfrH ¼ kB T H 1 þ exp E kl BT ð4:96Þ  o  2E  loH2 pffiffiffiffiffiffiffi  pH2 exp 2kB T where cfrH refers to the concentration of hydrogen atoms not bound to lattice defects, which means that the activity of hydrogen, or its partial pressure, is solely determined by the concentration of free (mobile) hydrogen.

142

4 Solubility of Hydrogen

4.5.2 Interaction of Hydrogen with Dislocations: DOSE Method The treatment in Sect. 4.4.2 shows that hydrogen has negative (binding) interaction energy with any defect that produces tensile stresses in the material. There could also be interactions with isotropically misfitting impurity atoms with positive misfit strain since hydrogen in zirconium has anisotropic misfit strain while, in addition, the metal is elastically anisotropic. Overall, one would expect that the practical impact of any interaction of hydrogen with positively misfitting impurity atoms would be small, both because the interaction energy is expected to be small based on results in other metals and because the number of impurity atoms is small. By far, the greatest impact could be the attraction of hydrogen to dislocations since the number of sites around a dislocation for which the magnitude of this interaction may be substantial could be fairly large. Moreover, the region around dislocations over which hydrogen would be attracted and bound could be sufficiently large that hydride precipitates could form there even at temperatures of practical concern for DHC. (The case for hydrogen attraction to cracks is dealt with in Chap. 10, dealing with the theory of DHC.) It would appear at first glance that determination of the interaction energy of a point defect with a dislocation would involve—at some distance from the origin of the dislocation—all of the interaction energy elements presented in Sects. 4.4.2 to 4.4.5. However, usually only the size effect interaction is considered since the core region of the dislocation where the induced interaction energies become important is excluded from calculations of the elastic interaction energy because the linear elastic model is not expected to be reliable there. The hydrostatic part, ph, of the stress field of an edge dislocation is given by [13] ph ¼

rii Gbð1 þ mÞ sin H ¼ 3pð1  mÞ r 3

ð4:97Þ

where7 b is the magnitude of the Burger’s vector, r and H are cylindrical coordinates defined in Fig. 4.3, and the z-axis is along the dislocation line. The interaction energy of hydrogen with the dislocation on a circle of constant pressure (excluding the core region) is H H ¼ V Eint ðdisloc; rÞ ¼ ph V H A sin H V r

7

Gbð1 þ mÞ sin H 3pð1  mÞ  r

ð4:98Þ

When dealing with the stress field produced by dislocations most authors have used the sign convention in which compressive stresses have positive signs and tensile stresses negative signs. The opposite sign convention has mostly been used in describing the stress state in front of cracks or for externally applied stresses. In the following, to be consistent with the predominant usage in the literature, we use the sign convention usually employed for describing the stress field of dislocations.

4.5 Interaction of Hydrogen Atoms in Solution with Internal Defects

143

Fig. 4.3 Cylindrical coordinates with the origin at the dislocation core, glide plane (dashed line), and circle (cylinder in three dimensions) of constant hydrostatic stress, p (ph in text) of diameter 2R (2Rhyd in text) (from Maxelon et al. [25])

where A is defined by the last relation in Eq. 4.98. Note that the energy decreases only slowly with distance from the center of the dislocation. Attractive interactions of hydrogen with dislocations are in regions where the hydrostatic stress is negative (tensile), based on this sign convention for stress. Figure 4.3 shows that the loci of constant (isobaric) hydrostatic stress are circles tangent to the slip plane at the dislocation center. When such circles are located below the slip plane they represent circles of tensile isobaric stress. Kirchheim [13, 15] used the foregoing equation to calculate the number of hydrogen atoms surrounding dislocations in palladium. Kirchheim [13, 15] assumed that the total hydrogen content of the solid was sufficient to saturate at least some part of the core region of the dislocation, thus resulting in the formation of a hydride. Hydride formation is the result of attractive (negative) hydrogen pair interactions. This means that such interaction energies should be included with the dislocation interaction energy for hydrogen atoms located in the core region of the dislocation. It also seems reasonable to assume that this attractive interaction energy would be of similar magnitude to the formation energy of b–hydride formed when the solvus concentration in the a–Pd phase is exceeded, which, at room temperature is *18 kJ/ mol H. Thus, calling this energy, Ehyd, i.e., the hydride formation enthalpy, Kirchheim [15] further assumed that the composition of this ‘‘hydride’’ would not vary much with temperature and hence its energy of formation could be assumed to be constant. Thus, with these assumptions, the total energy, E, of hydrogen located along circular loci of constant hydrostatic stress is given by: E ¼ Eint ðdisloc; rÞ þ Ehyd

ð4:99Þ

From Eq. 4.98, at H = -p/2 an outer cut-off radius, r = 2Rhyd, of the isobaric stress locus which forms the boundary inside of which hydride is expected to be formed is then obtained, giving

144

4 Solubility of Hydrogen

H AV  Rhyd ¼  2 E  Ehyd

ð4:100Þ

Then the total number of hydrogen atoms per unit volume, NHhyd ; inside a cylinder of unit height and radius, Rhyd ; is given by sðhydÞ

NHhyd ¼ NH

hyd aqd pR2hyd ¼ NPd aqd pR2hyd

ð4:101Þ

where the number of interstitial (hydrogen) sites per unit volume in this hydride, sðhydÞ hyd NH ¼ baPd NPd which, since ba-Pd = 1 for hydrogen occupying the octahesðhydÞ

hyd hyd ¼ NPd where NPd is dral interstitial sites in fcc a Pd or b hydride, yields NH the number of Pd sites per unit volume in the hydride; a is the ratio of hydrogen to Pd atoms in the hydride and qd is the dislocation density. It is expected that the hydride radius would increase with increase in hydrogen content in the material as the lowest energy levels start to fill up with hydrogen. From Eq. 4.101 the average hydrogen to Pd concentration ratio, rHhyd ðdislocÞ; locked up in hydrides at the dislocation cores is given by

rHhyd ðdislocÞ ¼

NHhyd hyd NPd

¼ aqd pR2hyd

ð4:102Þ

Using Eq. 4.102, a rough estimate of the number of hydrogen atoms that could be bound up in hydrides formed in the core regions of dislocations can be calculated. Assuming that the isobaric core cylinders of the dislocations where all the sites are occupied by hydrogen have radii of Rhyd = 0.5 nm and each of these cylinders consists of a hydride of composition PdH0.5, then in a material containing a dislocation density of 1 9 1014 m-2, an average hydrogen concentration of *40 appm would be locked up in hydrides in the dislocation cores. Thus to determine, experimentally, the chemical potential of hydrogen interacting with dislocations would require preparing samples having starting total hydrogen content values that are about an order of magnitude lower. Doing this calculation for Zr assuming that d-hydrides of composition rHhyd ðdislocÞ ¼ 1:5 would be formed in the dislocation cores and using the same core (hydride) radii as for Pd, yields an average total hydrogen concentration of 59 appm (*0.65 wppm) that could be locked up in these dislocation cores. The total concentration of hydrogen locked in hydrides at dislocations in Zr is 1.5 times greater than that in Pd because hydrogen occupies tetrahedral sites in any of the three hydride phases in Zr and consequently there are now two interstitial sites per metal (Zr) atom for hydrogen (bd–hyd = 2), but only 1.5 of these are occupied in the case of d hydride. Thus, the occupation fraction—changing from a to b—is now 0.75 compared to 0.5 as was assumed for ba-Pd. The DOSE must meet the constraint that the total number of hydrogen atoms, N0, in the sample is related to it by the relation

4.5 Interaction of Hydrogen Atoms in Solution with Internal Defects

N0 ¼

Z

þ1

1

nðEÞdE   H 1 þ exp El RT

145

ð4:103Þ

The density of states in the region containing the hydride, nhyd(E), is obtained from Eqs. 4.100 and 4.101, giving: dNHhyd dNHhyd dRhyd ¼ dE dRhyd dE H Þ2 ð AV hyd aqd p  ¼ NPd 3 2 E  Ehyd

nhyd ðEÞ ¼

ð4:104Þ

For the dilute regions outside the hydride cutoff radius, Rhyd, the restriction on a applied to the inner region is removed since no hydride is assumed to form which also means that Ehyd = 0; hence, E = Eint. Then the density of states becomes aPd ndilute ðEÞ ¼ NPd qd p

H Þ ð AV 2E3

2

ð4:105Þ

where the positive sign applies to E [ 0 and vice versa (both negative and positive interaction energies are possible depending on the location of the isobaric circle as shown in Fig. 4.4). An inner cutoff radius is also needed since the density of states diverges at E = 0. This is obtained from the condition aPd NPd

¼

ZE0 1

nðEÞ dE þ

Zþ1

aPd nðEÞdE ¼ NPd qd p

 H Þ2 ð AV 2E

ð4:106Þ

E0

yielding:  1 H qd p 2 E 0 ¼ AV 2

ð4:107Þ

However, this energy cannot be calculated directly from Eq. 4.98 as shown in Fig. 4. 4 showing the spatial relationship between dislocations spaced such that their inner energy cutoff circles touch each other. The integral given by Eq. 4.103, is obtained by combining the results from Eqs. 4.104 to 4.107, giving

146

4 Solubility of Hydrogen

Fig. 4.4 Pattern of dislocations with touching circles of constant energy corresponding to the inner cut-off energy e0 defined by Eq. 4.108 (E0 in the text), which is not related to Eq. 4.99 to the outer cut-off radius R0 (from Kirchheim [15])

rH ¼

N0 aPd NPd

" Z  3 E1 þEhyd  H Þ2 a E  Ehyd dE qd pð AV  ¼ 1 þ exp½ðE  lH Þ=ðRT Þ 2 1 Z E0 E3 dE  1 þ exp½ðE  lH Þ=ðRT Þ E1

Z þ1 E3 dE þ 1 þ exp½ðE  lH Þ=ðRT Þ E0

ð4:108Þ

where rH is the mole fraction of hydrogen in the material. Assuming rH \\ 1 then E1 is the elastic energy at the hydride boundary given by E1 ¼

H AV Rhyd

ð4:109Þ

The integrals given by Eq. 4.108 represent the fraction of the average hydrogen content in the sample located, respectively, as hydrogen in hydrides at the dislocation core, and as solvent in the a–Pd matrix in the dilated and compressed regions below and above the slip planes. Closed solutions for these integrals are, in general, not possible but for low hydrogen content the last two integrals can be neglected and a simple approximate solution can be obtained as follows. For the case that jlH j  RT the Fermi–Dirac function in Eq. 4.108 can be replaced by a step function and the solution of Eq. 4.108 takes the form:

4.5 Interaction of Hydrogen Atoms in Solution with Internal Defects

 H Þ2 Z l  3 qd pðAV rH ¼  a E  Ehyd dE 2 1  H Þ2 aqd pðAV ¼  2 4 E  Ehyd

147

ð4:110Þ

which, upon rearrangement, yields the following expression for the chemical potential:   H aqd p 12 AV ð4:111Þ lH ¼ Ehyd  rH 2 The approximate solution given by Eq. 4.111 is valid provided lH [ 20 kJ/mol H. It shows that the chemical potential of hydrogen in the material depends on the inverse square root of the total hydrogen content in the sample, rH. Comparing this prediction with electrochemical measurements of lH in cold rolled Pd, Kirchheim [14, 15] found good agreement with the experimental results except at low and high values of rH. The deviation from the predictions of Eq. 4.111 at high rH was judged to be the result of increasing contributions to lH from hydrogen–hydrogen interactions forming progressively larger regions of hydrides spreading out from the dislocation cores (over which the average interaction energy with the dislocation would become small compared to the H–H binding energy), while the large constant value obtained at low values of rH was thought to be the result of the largely constant value of the total interaction energy in the core of the dislocation consisting of hydride formation (H–H interaction) and dislocation interaction energies. In this region, the dislocation interaction energy given by Eq. 4.98 would give an overestimate of this energy with the actual energy being lower and assumed to be approximately constant with position around the origin of the dislocation. The reference (or standard) state for these electrochemical measurements was assumed to be given by lH ¼ kB T ln cfrH

ð4:112Þ

where the concentration of free interstitials, cfrH ; is assumed to be equal to the total hydrogen content of the sample. That is, interactions with defects are assumed to be negligible for this reference crystal and hence, according to Eq. 4.96, Eo ¼ 0: Besides measurements of the electrochemical potential in samples containing dislocations at various concentrations in comparison to well-annealed ones in which the material is a nearly perfect crystal, the concentration of free interstitials can also be determined from measurements of electrical resistivity. A study of H in Pd has shown [33] that the electrical resistivity of hydrogen trapped at dislocations in the form of hydrides contributes negligibly to the overall electrical resistivity. This result made it possible to determine the amount of free hydrogen in the material. From these measurements, it is concluded that the fraction of free hydrogen sites at low temperatures is negligible, which implies that they are all

148

4 Solubility of Hydrogen

tied up in hydrides at dislocations. With increasing temperature, hydrogen becomes partitioned between sites far from the dislocation cores and close to them. However, sites close to the dislocation cores never become saturated because of both long-range elastic interaction with the dislocations and attractive H–H interactions that result in the formation of a hydride phase (assumed in Kirchheim’s calculations [15] to have a room temperature composition ratio H/Pd of *0.5). The indirect indications of hydride formation at dislocation cores obtained from the foregoing measurement techniques were found to be consistent with the results of small angle neutron scattering (SANS) measurements [25]. The SANS measurements show for a scattering vector of 0.02–0.2 Å-1 that loading palladium samples with hydrogen containing dislocations to a density of 1015 m-2 resulted in an additional intensity compared to samples that were relatively free of dislocations. The corresponding net cross-section was found to be inversely proportional to the scattering vector as expected for line-type scattering objects with a superimposed exponential decrease stemming from scattering within the Guinier regime, suggesting the presence of small precipitate particles. The dislocations were introduced into the samples by doping them with hydrogen to H/Pd concentration ratio of 0.73 which transformed the entire sample to the b–hydride phase. The total amount of hydrogen absorbed by the samples was subsequently removed to bring the sample back to a composition containing undetectable amounts of hydrogen. In this way the samples were cycled through the two-phase a–Pd/b–hydride phase field twice. The dislocations produced by this process are the result of the large volumetric misfit strain of ±12 % between hydride and matrix when hydrides are formed and subsequently dissolved. Introducing dislocations into the crystal in this way had been found from previous experiments to result in dislocations having favorable configurations for attracting hydrogen to their cores. In comparison, dislocations produced in Pd by heavy cold working segregated into walled structures in which they seemed to have diminished attractiveness for hydrogen.

4.5.3 Formation of Hydrides at Dislocations: Thermodynamic Method The DOSE method for treating hydrogen segregation to defects such as dislocations having high interaction energies near their origins and slowly decreasing weak ones further from their cores is useful in showing the details of how hydrogen partitions between sites of varying interaction energies in the crystal. However, to determine solely the conditions for hydride formation in the core regions of dislocations, a simpler, thermodynamic treatment is more useful. This is particularly the case for segregation of hydrogen to cracks loaded in tension. Although this approach is treated in detail in Chap. 10, it is useful to document here a thermodynamic treatment used by Maxelon et al. [25] to interpret the results of the SANS measurements on Pd.

4.5 Interaction of Hydrogen Atoms in Solution with Internal Defects

149

In a system without any sources of internal and external stress, the equilibrium condition between hydrogen in solution and in the hydride is expressed by the equality of the chemical potentials for hydrogen dissolved in the solvent (a phase) with that dissolved in the hydride (b phase). This equality gives lsH ðrii ¼ 0Þ ¼ loH þ RT ln csH

ð4:113Þ

where csH is the solvus composition. Since csH ; expressed as H/Pd, is %0.01 at room temperature, the ideal solution approach can be used for the configurational energy expression (the logarithmic term in Eq. 4.113). Along the circle forming the boundary between the cylindrical hydride and the solid solution, where the hydrostatic stress ph ¼ rii =3 is constant and given by Eq. 4.97 at H = -p/2, the hydrogen chemical potential in the dilute phase at the solvus concentration is given by H lsH ðph Þ ¼ loH þ RT ln csH þ ph V

ð4:114Þ

Far from the dislocation, where there are no hydrides and hydrogen in solution is free, the chemical potential given by Eq. 4.113 applies. Defining this chemical potential relative to a reference state loH ; which is the same as that chosen for the chemical potentials defined by Eqs. 4.113 and 4.114, and equating these three chemical potentials, yields the following expression for the free hydrogen concentration in terms of the solvus concentration:   H 1 AV fr s cH ðhydrideÞ ¼ cH exp  ð4:115Þ 2RT Rhyd where A is defined by Eq. 4.98. Equation 4.115 gives the minimum value of the free hydrogen concentration far from the dislocation at which hydride would form within a circle of radius Rhyd tangent to the origin of the dislocation below the slip plane. For an edge dislocation with Burgers vector, b = 0.275 nm in Pd at room H Þ=ð2RT Þ ¼ 1 nm: Thus, for Rhyd ¼ 0:5 nm, the free hydrogen temperature, ðAV concentration for which hydrides with this radius are predicted to form at dislocations is csH 0:135 (i.e. at a hydrogen concentration in the dilute phase that is about 14 % lower than the solvus concentration). From Eq. 4.115, the relationship between the total hydrogen content, ctotal H ; the hydrogen located in the cylindrical hydrides of radii, Rhyd at dislocations and those hydrogen atoms that are free is given by ctotal ¼ aqd pR2hyd þ cfrH H ¼

aqd pR2hyd

þ

csH

  H 1 AV exp  2RT Rhyd

ð4:116Þ

Solving this implicit relation for the hydride cylinder radii as a function of the  total shows that the predicted dependence of this total hydrogen content, Rhyd cH

150

4 Solubility of Hydrogen

Fig. 4.5 Radii obtained from small angle neutron scattering for a deformed Pd sample at various hydrogen concentrations. The line is calculated by assuming the formation of a hydride of cylindrical shape in the strain field of dislocations. The dislocation density q = 2 9 1015 m-2 was used as a fitting parameter to obtain agreement with experimental data (from Maxelon et al. [25])

parameter on ctotal compares well with the results obtained from SANS, assuming H a = 0.6 and qd = 2 9 1015 m-2 as shown in Fig. 4.5. A similar result would be obtained for Zr at room temperature. However, in zirconium the extrapolated room temperature solvus concentration is *6 9 10-6 H/Zr (*0.06 wppm), which is much lower than in Pd at that temperature. Since in most practical cases, zirconium and its alloys contain at least that amount of hydrogen in their as-received condition, the solvus concentration would be exceeded at all locations in the solid and not just at dislocations. The total hydrogen content that could be locked up in hydrides formed at dislocations is calculated in the foregoing for a dislocation density of 1 9 1014 m-2 and hydride radii of 0.5 nm to be equal to 59 appm. An estimate of the solvus temperature in zirconium at which the solvus concentration equals this concentration of hydrogen locked up at dislocations is *80 °C. Thus, if Rhyd = 0.5 nm represents the limit at which hydrides would be formed within isobaric circles of tensile hydrostatic stress at dislocations and not elsewhere in the solid, then below this temperature there would be very few ‘‘free’’ hydrogen atoms remaining in the lattice since they would either be contained in hydrides located throughout the bulk of the material or as small hydride precipitates formed in the dislocation core regions. With increasing solvus concentration at higher temperatures, a decreasing fraction of the total hydrogen content would, however, be contained within the bulk hydrides. This extra hydrogen would then be free to migrate to dislocations, forming hydrides there, first in the core region, and then possibly increasing in size, increasing the total hydrogen content locked up at dislocations. Note that, even at 150 °C the solvus concentration is still only 429 appm. The solvus concentration represents the hydrogen concentration that is potentially free for diffusion. However, even if the amount locked up in hydrides at dislocations is limited to a radius of 0.5 nm there would still be a substantial fraction of this free hydrogen content locked up in the dislocation cores and not available for diffusion.

4.5 Interaction of Hydrogen Atoms in Solution with Internal Defects

151

Note that, the amount of hydrogen that could be bound to dislocations is linearly proportional to the dislocation density of those dislocations assumed to be suitable for attracting hydrogen. Changes in factors of two or more of the density of these dislocations would, therefore, have a significant effect on the amount of hydrogen that could be bound to dislocations. The effect that hydrogen trapped in hydrides at dislocation cores can have on DHC growth rate requires including the results of the foregoing derivations into the flux equations for hydrogen diffusion to a crack tip loaded in tension. To make this possible, however, it is necessary to derive formal theoretical relationships and summarize experimental results for the diffusivity of hydrogen in Zr. This is done in the following chapter (Chap. 5).

References 1. Bass, R., Oates, W.A., Schober, H.R., et al.: Configuration-independent elastic interactions in metal-hydrogen solutions. J. Phys. F: Met. Phys. 14, 2869–2880 (1984) 2. Bardeen, J., Herring, C.: Diffusion in alloys and the Kirkendall effect. In: Schockley, W. (ed.) Imperfections in nearly perfect crystals, Wiley, NY, pp. 261–288 (1952) 3. Callen, H.B.: Thermodynamics. Wiley, New York (1960) 4. Cann, C.D., Sexton, E.E., Duclos, A.M., et al.: The effect of decomposition of beta-phase Zr20 at % Nb on hydrogen partitioning with alpha-zirconium. J. Nucl. Mater. 210, 6–10 (1994) 5. Dantzer, P., Lou, W., Flanagan, T.B., et al.: Calorimetrically measured enthalpies fort the reaction of H2 (g) with Zr and Zr alloys. Metall. Trans. A. 24A, 1471–1479 (1993) 6. De Groot, S., Mazur, P.: Non-equilibrium Thermodynamics. North-Holland Publishing Co., Amsterdam (1961) 7. Domain, C., Besson, R., Legris, A.: Atomic-scale ab initio study of the Zr-H system: I. Bulk properties. Acta. Mater. 50, 3513–3526 (2002) 8. Eshelby JD: The continuum theory of lattice defects. In: Seitz, F., Turnbull, D. (eds.) Advances in Solid State Physics 3, pp. 79–144 (1956) 9. Eshelby, J.D.: Interaction and diffusion of point defects. In: Smallman RE (ed.) Vacancies 76, London, The Metals Society, pp. 3–10 (1976) 10. Fukai, Y.: The metal-hydrogen system: basic bulk properties. Springer, Berlin (2005) 11. Gibbs, J.W.: The Scientific Papers of J.W. Gibbs. vol. 1, Dover, New York, USA, (1961) 12. Herring, C: Surface tension as a motivation for sintering. In: Kingston, W.E. (ed.) Physics of Powder Metallurgy, McGraw-Hill, NY, pp. 143–179 (1951) 13. Kirchheim, R.: Solid solutions of hydrogen in complex materials. Solid State Phys. 59, 203–305 (2004) 14. Kirchheim, R.: Interaction of hydrogen with dislocations in palladium—I. Activity and diffusivity and their phenomenological interpretation. Acta. Metall. 29, 835–843 (1981) 15. Kirchheim, R.: Interaction of hydrogen with dislocations in palladium—II. Interpretation of activity results by Fermi-Dirac distribution. Acta. Metall. 29, 845–853 (1981) 16. Khoda-Bakhsh, R., Ross, D.K.: Determination of the hydrogen site occupation in the a phase of zirconium hydride and in the a and b phases of titanium hydride by inelastic neutron scattering. J. Phys. F: Met. Phys. 12, 15–24 (1982) 17. Larché, F.C., Cahn, J.W.: A linear theory of thermochemical equilibrium of solids under stress. Acta. Metall. 21, 1051–1063 (1973) 18. Larché, F.C., Cahn, J.W.: A nonlinear theory of thermochemical equilibrium of solids under stress. Acta. Metall. 26, 53–60 (1978)

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19. Larché, F.C., Cahn, J.W.: Thermochemical equilibrium of multiphase solids under stress. Acta. Metall. 26, 1579–1589 (1978) 20. Larché, F.C., Cahn, J.W.: The interactions of composition and stress in crystalline solids. Acta. Metall. 33, 331–357 (1985) 21. Leibfried, G., Breuer, N.: Point Defects in Metals I. Springer, Berlin (1978) 22. Li, J.C.M., Oriani, R.A., Darken, L.S.: The thermodynamics of stressed solids. Zeitschrift für Physikalische Chemie Neue Folge 49, 271–290 (1966) 23. Luo, W., Clewley, J.D., Flanagan, T.B.: Calorimetrically measured enthalpies for the reaction of H2(D2) (g) with Ti and Ti-Ni alloys at 323 K. Metall. Trans. B 24B, 867–873 (1993) 24. MacEwen, S.R., Coleman, C.E., Ells, C.E., et al.: Dilation of h.c.p. zirconium by interstitial deuterium. Acta. Metall. 33, 753–757 (1985) 25. Maxelon, M., Pundt, A., Pyckhout-Hintzen, W., et al.: Interaction of hydrogen and deuterium with dislocations in palladium as observed by small angle neutron scattering. Acta. Mater. 49, 2625–2634 (2001) 26. Oates, W.A., Stoneham, A.M.: Strain-induced interaction energies between hydrogen atoms in palladium. J. Phys. F: Met. Phys. 13, 2427 (1983) 27. Peisl, H.: Lattice strains due to hydrogen in metals. In: Alefeld, G., Völkl. (eds.) Hydrogen in Metals, vol. 1, Springer, Berlin, pp. 53–74 (1978) 28. Perovic, V., Weatherly, G.C.: The nucleation of hydrides in Zr-2.5 wt% Nb Alloy. J. Nucl. Mater. 126, 160–169 (1984) 29. Puls, M.P.: Elastic interactions of hydrogen in the lattice of iron alloys. In: Oriani, R.A., Hirth, J.P., Smialowski, M. (eds.) Hydrogen Degradation of Ferrous Alloys. Noyes Publications, Park Ridge, NJ, USA, pp. 114–130 (1985) 30. Puls, M.P., Woo, C.H.: Diaelastic polarizabilities as a result of vacancies and interstitials in metals. J. Nucl. Mater. 139, 48–59 (1986) 31. Puls, M.P.: Dipole tensors and changes in elastic constants produced by defects in ionic crystals. Phil. Mag. A 51, 893–911 (1985) 32. Puls, M.P., Dutton, R., Stevens, R.N.: The chemical stress applied to creep and fracture theories—II. Application to the growth of sub-critical Griffith cracks. Acta. Metall. 22, 639– 647 (1974) 33. Rodrigues, J.A., Kirchheim, R.: More evidence for the formation of a dense Cottrell cloud of hydrogen (hydride) at dislocations in niobium and palladium. Scripta. Metall. 17, 159–164 (1983) 34. Sawatzky, A., Ledoux, G.A., Tough, R.L. et al. Hydrogen diffusion in zirconium-niobium alloys. In: Nejat, V. Metal-Hydrogen Systems, Pergamon, Oxford, UK, pp. 109–120 (1982) 35. Skinner, B.C., Dutton, R.: Hydrogen diffusivity in a-b zirconium alloys and its role in delayed hydride cracking. In: Moody, M.R., Thompson, A.W. (eds.) Hydrogen Effects on Material Behavior, The Minerals, Metals and Materials Society, Warrendale, PA, USA, pp. 73–83 (1990) 36. Stevens, R.N., Dutton, R., Puls, M.P.: The chemical stress applied to creep and fracture theories—I. A general approach. Acta. Metall. 22, 629–638 (1974) 37. Wagner, H., Horner, H.: Elastic interactions and the phase transition in coherent metalhydrogen systems. Adv. Phys. 23, 587–637 (1974) 38. Yamanaka, S., Yoshioka, K., Uno, M., et al.: Thermal and mechanical properties of zirconium hydride. J. Alloys Compd. 293–295, 23–29 (1999) 39. Yamanaka, S., Yoshioka, K., Uno, M., et al.: Isotope effects on the physicochemical properties of zirconium hydride. J. Alloys Compd. 293–295, 908–914 (1999) 40. Zuzek, E., Abriata, J.P, San-Martin, A et al.: H-Zr (Hydrogen-Zirconium) In: Phase Diagrams of Binary Hydrogen Alloys. ASM International, Materials Park, Ohio, USA, pp. 309–322 (2000)

Chapter 5

Diffusion of Hydrogen

5.1 Phenomenological Flux Equations In describing the diffusion of hydrogen in metals in a thermodynamic system not too far from equilibrium, it is useful to formulate this in terms of the theory of irreversible thermodynamics (De Groot and Mazur [3]). This theory provides a general framework for the macroscopic description of irreversible processes. It is formulated as a continuum theory in which the state parameters are field variables (continuous functions of time and space) that are locally defined. The latter requirement is generally not needed in equilibrium thermodynamics since the state variables are usually independent of space coordinates. In the theory of irreversible thermodynamics the balance equation for entropy plays a central role. This balance equation states that the entropy of a volume element changes with time for two reasons. One is because of entropy flow into and out of the volume (which can be positive or negative) while the other is because of irreversible processes occurring inside the volume. The latter processes always result in non-negative entropy production since entropy can only be created, never destroyed. This entropy increase vanishes for reversible processes. The theory was developed to provide expressions that relate the source of entropy production explicitly to various possible irreversible processes that can occur in a system. To accomplish this, macroscopic conservation laws of mass, momentum, and energy are required in local (differentiable) form. From these relations, the entropy increase is calculated making use of equilibrium thermodynamic relations connecting the rate of entropy production of each mass element to the rate of change of energy and of compositions. The relations developed for these purposes turn out to have very simple forms consisting of the product of the flux characterizing an irreversible process with a quantity that is related to the nonuniformity of the system (gradient of temperature or composition, for instance) and that can be characterized as a generalized force. To be solvable, the entropy production rate needs some physical description of the fluxes contained in it. Hence the second sets of relations making up the basic M. P. Puls, The Effect of Hydrogen and Hydrides on the Integrity of Zirconium Alloy Components, Engineering Materials, DOI: 10.1007/978-1-4471-4195-2_5,  Springer-Verlag London 2012

153

154

5 Diffusion of Hydrogen

equations of irreversible thermodynamics are phenomenological expressions linking the irreversible fluxes with the thermodynamic forces appearing in the entropy production relation. As a first approximation the relationship between the fluxes and forces should be linear. Examples of such linear relations developed outside the framework of irreversible theory are Fick’s law of diffusion, Fourier’s law of heat conduction, and Ohms law of electrical conduction. The linear proportionality between the forces and the fluxes is given by matrices of phenomenological coefficients. An important macroscopic observation concerning these coefficients is encapsulated in the Onsager-Casimir reciprocity theorem which is based on the time reversal invariance of the microscopic equations of motion. In fact, the development of the phenomenological relations of irreversible processes was motivated by the desire to explore the physical consequences of the existence of such reciprocal relationships. A specific application given in the following is the development of the diffusion equation for the transport of hydrogen in a temperature and stress gradient arising from the reciprocal relationship between mass (particle) and heat flow. We start with the most general form of the linear phenomenological flux equations, viz., X Ji ¼ Lij Xj ð5:1Þ j

where the Ji and Xi are any of the Cartesian components of the independent fluxes and thermodynamic forces appearing in the entropy production expression given by X DS_ ¼ Ji Xi ð5:2Þ i

and Lij are phenomenological mobility coefficients of the diffusing quantities such as mass (atomic species), heat, electricity, etc., multiplied by the associated gradients causing the diffusion represented by the generalized forces, Xj. For application to particle and thermal diffusion, Eq. 5.2 is reformulated in terms of the specific gradients of chemical potential and temperature that are conjugate to the fluxes causing the entropy production. The result is: DS_ ¼ 

n  l   1 1X J q rT  Jk Tr k  Fk 2 T T k¼1 T

ð5:3Þ

where the index q refers to the heat flux and k to the particle flux. Fk is a force derived from the gradient of a potential. It turns out to be useful to divide the thermodynamic force multiplying the Jk diffusion flux term into temperatureindependent and temperature-dependent parts using. Td

l   hk k ¼ ðdlk ÞT  dT T T

ð5:4Þ

5.1 Phenomenological Flux Equations

155

where the index T indicates that this differential is taken at constant temperature and with  hk the partial specific enthalpy of component k. Using Eq. 5.4 we can define a new heat flux given by J 0q ¼ J q 

n X

 hk J k

ð5:5Þ

k¼1

Now, from Eq. 5.5, plus the condition n X

Jk ¼ 0

ð5:6Þ

k¼1

the entropy production given by Eq. 5.3 becomes DS_ ¼ 

n1   1 0 1X J rT  J k rðlk  ln ÞT q 2 T T k¼1

ð5:7Þ

This relation thus defines a constrained force (Xk)T that is conjugal to Jk according to ðX k ÞT  rðlk  ln ÞT

ð5:8Þ

Based on Eq. 5.7 for the entropy production rate the corresponding flux equations—which must contain the same gradients of forces conjugate to the fluxes that are given in the entropy production rate relation—become J0q ¼ 

n1   Lqq 0 1X J rT  Lqk rðlk  ln ÞT q 2 T k¼1 T

n1   Liq 1X Lik rðlk  ln ÞT Ji ¼  2 J0q rT  T k¼1 T

ð5:9Þ

At this point it is useful to introduce two quantities of transfer; these are the heat of transfer (Qk ) and reduced heat of transfer (Q0 k ). These quantities are, respectively, defined by Jq ¼

n1 X

Qk Jk ; rT ¼ 0

and

k¼1

Jq0

¼

n1 X

ð5:10Þ Q0 k Jk

; rT ¼ 0

k¼1

Then, from Eq. 5.9: Q0 i ¼

n1 X k¼1

Lqk L1 ki

ð5:11Þ

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5 Diffusion of Hydrogen

and it can be shown that in the stationary state of a closed system containing an applied rT, the resultant expression for the corresponding concentration gradient gives an expression proportional to the term given by Eq. 5.11. Introducing the quantities Xk ¼ ðX k ÞT  Q0 k J q ¼ J 0q 

n1 X

rT T

ð5:12Þ

Q0 k Jm

m¼1

which are in a form that leaves the entropy production rate of Eq. 5.7 invariant, a ‘‘diagnonal’’ form of the phenomenological equations is obtained given by ! n1 X  0 0 rT Qm Lmk Qk ð5:13Þ J q ¼  Lqq  T2 m;k¼1 for the heat flux and Ji ¼

n1 X

Lik Xk

ð5:14Þ

k¼1

for the flux of diffusing species. These relations correspond to the following relationship between the heat of transfer and the reduced heat of transfer:   Qk ¼ Q0 k þ hk  hn

ð5:15Þ

For the present case of a two-component metal containing one mobile component (hydrogen) and one immobile metal atom component that fixes the lattice framework with respect to which the diffusion of hydrogen takes place, we obtain from Eq. 5.12 and 5.14:   rT J H ¼ LHH rðlH ÞT þ Q0 ð5:16Þ H T It can be seen from Eq. 5.15 that Q0 k represents the heat content transferred by the hydrogen as a result of the cross-term between the heat and matter (hydrogen) flow. Now with lH given by lH ¼ loH þ RT lnðaH Þ

ð5:17Þ

where aH is the activity of hydrogen, the hydrogen flux given by Eq. 5.16 becomes   Q0 ð5:18Þ JH ¼ LHH RT ðr lnðaH ÞÞT þ H rT T where all the terms having a dependence on rT have been grouped in the 0 Q0 H rT=T term. From the definition of Qk given by Eq. 5.15 it is seen that this

5.1 Phenomenological Flux Equations

157

parameter represents the energy transported that is in excess of the heat, CPdT = HH, (CP being the specific heat) that must be supplied by the surroundings when a quantity of hydrogen diffuses in a temperature gradient. A value of zero for this parameter means that there would be no thermal diffusion. Converting the mobility coefficient, LHH, into the scalar diffusivity coefficient, Dchem, defined by Fick’s first law through the ratio -JH/rcH, we obtain the relation LHH ¼

Dchem cH RT

ð5:19Þ

for an isotropic dilute solution for which aH % cH. Substituting this result into Eq. 5.18, the relation for the diffusion flux becomes:   Q0 chem H JH ¼ D cH ðr lnðaH ÞÞT þ rT ð5:20Þ RT 2 In the general anisotropic case, in which the diffusivity is a second rank tensor it is assumed that Q0 eff remains a scalar, Eq. 5.20 becomes   X o lnðaH Þ Q0 H oT ðJH Þi ¼  Dchem c þ ð5:21Þ H ij 2 oxj T RT oxj j¼1;2;3 where i, j refer to the three orthogonal axes. With the effect of stress contained in the ln (aH) term, Eq. 5.21 now contains all the terms that affect diffusion of hydrogen in an immobile metal lattice. Expanding the gradient terms o lnðaH Þ=oxj by the chain rule for each orthogonal direction and assuming that the activity depends linearly on the concentration, cH, according to aH = cHcH, where cH is the activity coefficient, then:   d lnðaH Þ o lnðaH Þ dcH o lnðaH Þ drm ¼ cH þ cH dxj ocH dxj orm dxj   H drm o lnðcH Þ dcH o lnðcH Þ dcH V ¼ cH þ  ð5:22Þ ocH dxj ocH dxj RT dxj   H cH drm o lnðcH Þ dcH V ¼ 1þ  o lnðcH Þ dxj RT dxj To evaluate the second term in the square bracket of Eq. 5.22 we have used, for H given by H  rm V brevity, the isotropic relation: RT lnðaH Þ ¼ ðr=3ÞV Eq. 4.33 in Chap. 4 for the effect of the hydrostatic (mean) stress component, rm, on the chemical potential. The implicit assumption in the use of this relation in Eq. 5.22 is that the diffusion process is slow enough compared to the random movement of atoms that local thermodynamic equilibrium conditions apply at each local value of stress. Substituting the results of Eq. 5.22 into Eq. 5.21, the expression for the flux of hydrogen now becomes:

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5 Diffusion of Hydrogen

2 3 H cH drm o lnðcH Þ dcH V 1 þ  6 X o lnðcH Þ dxj RT dxj 7 6 7 ðJ H Þi ¼  Dchem 6 7 ij 0 4 5 Q c oT H j¼1;2;3 þ H2 RT oxj

ð5:23Þ

The expression for the flux of hydrogen in a concentration, stress, and temperature gradient is now complete. Solutions for this equation require experimental values for the diffusion coefficient. However, if diffusivity is obtained using methods where the hydrogen concentration gradient is the source of the diffusion, then, as can be seen from the first term in Eq. 5.23, unless the concentration of hydrogen is very small, the contribution to the diffusivity as a result of the gradient in chemical potential needs to be extracted from the measured diffusivity to obtain the diffusivity that determines total diffusion flux. In addition, it is useful to derive expressions that show how the diffusivity is affected by microstructure as a result of, for instance, trapping at various types of defects or the presence of fast diffusion paths. These objectives are developed in the following sections with the theoretical development based, for the most part, on methods provided by Kirchheim [9] using the DOSE approach documented in Chap. 4.

5.2 Diffusivity—Theory The diffusion of hydrogen in metals occurs by means of a direct interstitial mechanism with a tracer diffusion coefficient, D , given by ! ‘2 C rH  f 1 D ¼ ð5:24Þ 2d bpha where C is the jump frequency, ‘ the jump distance, and d the dimensionality of the lattice, which is three when considering diffusion in the bulk of the material, and f is a correlation factor that goes to unity when rH ! 0. This factor becomes less than unity when rH =bpha ! 1 at which point the interstitial sites become filled (such as in hydrides) and blocking of sites comes into play. This is accounted for by the bracketed term which means that a vacancy mechanism becomes applicable. It is implicit in Eq. 5.24 that, for the case of thermally activated hopping, all the sites visited by the H-atoms have the same average jump frequency. In the following we reproduce various derivations given by Kirchheim [9] for the calculation of the tracer diffusion coefficient from atomic models. In these models the density of site energy (DOSE) of interstitial hydrogen atoms at various locations in the lattice is derived using Fermi–Dirac statistics (Kirchheim [9]). Unlike in Kirchheim’s approach, we assume that the formulation is in terms of molar, not atomic, quantities of thermodynamic parameters, such as the site energies and the Fermi energy (the latter being the equivalent to the hydrogen chemical potential).

5.2 Diffusivity—Theory

159

This simply means that the thermal energy in the various expressions derived by Kirchheim [9] is given by RT rather than kBT. In a tracer diffusion experiment the concentration in the sample is constant. The diffusion of the tagged particle is then obtained from the average of the jump frequencies that the atoms make. Assuming, as in Eq. 5.24, a jump distance ‘, the tracer diffusion coefficient, D , is given by  * Qo þ Eo  E1 + 2 2 C0 exp   ‘ ‘ RT D ¼ hCi ¼ ð5:25Þ 6 6 nðE1 ÞoðE1 Þ½1  nðE2 ÞoðE2 Þ Equation 5.25 shows that the jump rate for particles hopping from sites of energy, E1, to sites of energy, E2, is a product of the following: • A constant factor, which is equal for all sites (C0) which means that an uncorrelated random walk has been assumed (constant saddle point energy), • A Boltzman factor exp½ðQo þ Eo  E1 Þ=RT , where Qo is the activation energy of diffusion in the reference material, • The partial concentration in sites of energy E1 ½¼ nðE1 ÞoðE1 Þ; • The availability of empty sites of energy E2 ½¼ nðE2 Þf1  oðE2 Þg. The result of integration of the last expression in Eq. 5.25 gives     C0 Qo lH  Eo exp  ð1  c H Þ2 exp hC i ¼ cH RT RT

ð5:26Þ

which, when re-inserted into Eq. 5.25, yields for the tracer diffusion coefficient:   ‘2 Qo D ¼ C0 exp  ð1  cH Þ2 c0 ¼ Do ð1  cH Þ2 c0 ð5:27Þ 6 RT Comparison between Eqs. 5.26 and 5.27 shows that   ‘2 Qo Do  C0 exp  6 RT

ð5:28Þ

with the activity coefficient, 

l  Eo a0 ¼ c0 cH  exp H RT

 ð5:29Þ

From these definitions it is seen that Do is the tracer diffusion coefficient in a reference material with sites of energy Eo and the activity coefficient represents the activity with respect to this lattice. Note that from Eq. 5.27 the factor ð1  cH Þ occurs twice, once to account for blocking of sites already occupied, the second time to account for the fact that c0 ! ð1  cH Þ1 when cH ! 1. From Eqs. 5.26 and 5.27 it is evident that the temperature dependence for self-diffusion is determined solely by the exponential dependence of the average

160

5 Diffusion of Hydrogen

jump frequency, which has an effective activation energy given by ðQo þ Eo  lH Þ. This activation energy expresses the energy difference between the saddle point and Fermi energy. An Arrhenius law for self-diffusion is expected because the Fermi energy does not change very much with temperature for a broad DOSE. Hence, the tracer diffusion coefficient obeys an Arrhenius law despite having a distribution of site energies and, therefore, of jump frequencies in the solid. The concentration dependence of the activation energy is very pronounced with increasing cH because the second derivative of the Gibbs energy is positive at equilibrium. Thus, an increasing hydrogen concentration results in an increased diffusivity. Physically this is explainable by the fact that the Fermi energy rises with increasing cH and, therefore, moves closer to the saddle point energy, decreasing the activation barrier. The foregoing treatment is now applied to the case when there is a concentration gradient. The object is to provide a relationship between the tracer diffusion coefficient and the chemical (intrinsic) diffusion coefficient. Often in experiments it is the latter that is measured, based on solutions of the Fick equations. The following treatment also provides a relationship between the mobility coefficient and the coefficient, LHH that occurs in the phenomenological flux equations derived from the theory of irreversible thermodynamics given by Eq. 5.16 in Sect. 5.1. Consider a one-dimensional flow of atoms between two adjacent parallel lattice planes separated by a distance, ‘, with average concentrations, cH1 \cH2 within these planes. The flux of atoms in a direction parallel to the normal of these planes is then given by Fick’s first law; viz., JH ¼ 

bpha chem ocH cH2  cH1 bpha ¼ Dchem D XM ox ‘ XM

ð5:30Þ

which defines the chemical diffusion coefficient Dchem. In Eq. 5.30 bpha has the same meaning as before and XM is the atomic volume of the metal. The ratio of these two quantities are included here because the concentration units in Fick’s equation need to be given in terms of atoms per unit volume of the diffusing species while the units of cH used in the other thermodynamic relationships are more conveniently given as dimensionless, either in units of atomic ratio or atomic fraction. To derive an expression for the flux of hydrogen atoms caused by a hydrogen atom concentration gradient a similar approach is used as was done in deriving the relation for the tracer diffusion coefficient. Thus, the flux is given by formulating the difference in the average jump frequencies of atoms located in plane 1 and plane 2. This approach results in 3 2 C1 ðxÞnðE1 Þ   7 H ðxÞ ZZ 6 7 6 1 þ exp E1 l bpha RT 7 6 JH ¼ ð5:31Þ 7dE1 dE2 6 6 6XM C2 ðx þ ‘ÞnðE2 Þ 7 4  5 H ðxþ‘Þ 1 þ exp E2 lRT

5.2 Diffusivity—Theory

161

With the jump frequencies given as before as thermally activated jumps over a barrier of constant energy, Qo þ Eo , that is, weighted to account for blocking of available sites according to nðEi Þ½1  oðEi ; lH Þ, the jump frequencies are given by   ðQo þ Eo  E1 Þ C1 ¼ C0 nðE2 Þ½1  oðE2 ; lH ðx þ ‘ÞÞ exp  ð5:32Þ RT   ðQo þ Eo  E2 Þ ð5:33Þ C2 ¼ C0 nðE1 Þ½1  oðE1 ; lH ðxÞÞ exp  RT The combinations of Eqs. 5.31–5.33, when combined with expanding lH ðx þ ‘Þ in a Taylor series, result in:   bpha o lH  Eo 1 olH 2 JH ¼  ð5:34Þ D ð1  cH Þ exp RT ox XM RT Equation 5.34 is now in the same form as Eq. 5.16 derived from the phenomenological theory of irreversible thermodynamics (in the linear approximation) where the gradient of the chemical potential is the driving force for diffusion. Equation 5.34 can be converted to one involving the gradient of the activity coefficient instead of the gradient of the chemical potential by using the relation between chemical potential and activity coefficient given by Eq. 5.17 and using the chain rule for partial differentiation as before. This gives olH RT oaH ¼ c0 cH ox ox

ð5:35Þ

Using Eqs. 5.35 and 5.29 in Eq. 5.34 yields   bpha o lH  Eo 1 oaH 2 JH ¼  D ð1  cH Þ exp c0 cH ox XM RT bpha o oa H D ð1  cH Þ2 ¼ XM ox bpha o o lnðaH Þ ¼ D ð1  cH Þ2 c0 cH ox XM o lnðaH Þ ¼ Do ð1  cH Þ2 c0 cH ox

ð5:36Þ

where cH ¼ rH =XM is the hydrogen concentration (hydrogen to metal ratio) per unit volume. Using the relations for the tracer diffusion coefficient given by Eqs. 5.27–5.29, we also have the following relationship from Eq. 5.34:

162

5 Diffusion of Hydrogen

D CH olH RT ox o lnðaH Þ ocH ¼ D o lnðcH Þ ox  o ln cH ocH  ¼ D 1 þ o ln cH ox

JH ¼ 

ð5:37Þ

where cH is the hydrogen concentration per unit volume. Comparison with Eq. 5.23 shows that Dchem in the Fick equation is equivalent to D . Therefore, since D is predicted to be concentration dependent, so is the chemical diffusion coefficient. The result given by Eq. 5.23 was obtained by replacing the LHH coefficient in the phenomenological equation with the chemical diffusion coefficient as defined by Fick’s first law in the dilute solution limit, leading to the correspondence expressed by Eq. 5.19. In that approximation it is expected that Dchem would be concentration independent. The foregoing derivation shows, however, that such a restriction on the solute concentration can be avoided and a more general result can be obtained. It also shows how the state of the microstructure of the solid—as characterized by the number and types of defect sites—each producing interaction energy contributions to the hydrogen perfect lattice site energies—affects both the solubility and the diffusivity of hydrogen. To apply the foregoing treatment to experimental determinations of hydrogen diffusion and permeation, the material and its defect have to be first characterized and a description in terms of the DOSE given. Then Eq. 4.91 of Chap. 4 would provide—either numerically or in terms of a closed solution—the chemical potential as a function of hydrogen concentration, lH ðcH Þ. Based on the solution to the density of state expression given by Eq. 4.91, it is then possible to calculate Dchem using Eqs. 5.27 and 5.37. Then with the appropriate boundary conditions, Fick’s first law can be solved, but with a concentration-dependent diffusivity, Dchem. Of course, if measured values of lH ðcH Þ are available, these can be inserted into Eqs. 5.27 and 5.37 as well.

5.3 Diffusivity in Dilute Phase—Results There have been relatively few measurements of diffusivity in zirconium and its alloys and none that have tried to elucidate—either experimentally or theoretically—the more fundamental aspects of diffusion in this material. Summaries of published results were provided by Greger et al. [5] and Watanabe et al. [20]. The former authors used a novel tritium implantation method to determine the diffusivity in Zircaloy-2. Table 5.1 summarizes all of the presently available results including those discussed further on while Fig. 5.1 shows the temperature dependences of the diffusion coefficients in terms of an Arrhenius plot. In this figure, taken from Greger et al. [5], results in the literature that were obtained using hydrogen as the

5.3 Diffusivity in Dilute Phase—Results

163

Table 5.1 Diffusion coefficient data in Zr and its alloys for hydrogen isotopes, where the diffusion coefficient, DH ¼ DoH exp½Q=RT  Reference (number in [] refers to Isotope Material D0H  107 Q (kJ/ (m2s-1) (mol H) reference indication in Fig. 5.1) H H H T H H T H T H T T T T

T T T T T

Zr 1.09 Zr 0.714 Zircaloy-2 2.17 Zr 1.53 b-Zr 5.32 Zr, Zircaloys 7.0 Zircaloy-2 0.21 Zr 40.0 Zircaloy-2 1.04 Zr-2.5Nb, Zr-20Nb 1.17 Zr-2.5Nb PT: 0.18 as extruded Zr-2.5Nb PT: Pickering 2.4 NGS, axial Pickering NGS PT: 3.8 transverse Darlington NGS, as 4.9 received, tube 1, Sect. 1 Ibid, tube 1, 4.0 Sect. 15 Ibid, tube 2 1.4 Ibid, tube 342 Zr-2.5Nb PT: 400 C 3.5 heat treatment Zr-2.5Nb PT: 565 C 5.2 heat treatment

47.69 29.53 35.05 37.94 34.80 44.34 35.56 48.13 42.1 33.60 16.40

Gulbransen and Andrew [6] [1] Mallet and Albrecht [11] [2] Sawatzky [15] [3] Cupp and Flubacher [2] [4] Gelezunas et al. [4] Kearns [8] [5] Austen et al. [1] [6] Mazzolai and Ryll-Nardzewski [13] [7] Greger et al. [5] Sawatzky et al. [16] Skinner and Dutton [18]

34.75

Skinner and Dutton [18]

38.62

Skinner and Dutton [18]

41.51

Skinner and Dutton [18]

40.55

Skinner and Dutton [18]

34.75 39.58

Skinner and Dutton [18] Skinner and Dutton [18] Skinner and Dutton [18]

42.48

Skinner and Dutton [18]

diffusing species have been normalized to that for tritium by multiplying the data by 1/H3 to account for the isotopic mass difference. The justification for the use of this simple relationship in scaling the diffusion coefficients to account for the effect that isotopic mass has on the diffusion rate was based on the experimental results of Gulbransen and Andrew [6]. These authors found that the measured diffusion coefficient for deuterium was a factor of 1.49 lower compared to that for hydrogen. This result compares well with the usual isotopic mass difference factor of pffiffiffi pffiffiffiffiffiffiffiffipffiffiffiffiffiffi mp = md ¼ 1= 2 ffi 1:41 between protium and deuterium, respectively, derived from the random walk model of atomic diffusion. However, there is a paucity of data indicating whether this rule is applicable over the full range of temperatures over which diffusion coefficient measurements have been made in zirconium. It can be seen from Fig. 5.1 that the magnitudes and temperature dependencies of the diffusion coefficients obtained deviate from each other by more than an order of magnitude. These differences are much greater than the generally stated

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Fig. 5.1 Arrhenius plot of tritium diffusion coefficients. Dashed lines are hydrogen diffusion coeffcients divided by H3 in order to account for expected isotopic effect. The number of each curve corresponds to the reference numbers in Table 5.1 with the data obtained by Greger et al. [5] given by the thick solid line (from Greger et al. [5])

experimental uncertainties of *20 % and the scatter in the individual data points (except for the very much larger upper bound uncertainty stated without explicit explanation by Austen et al. [1]). Some of the reasons for these large systematic differences in the results obtained may be: • In experiments in which a diffusion profile was measured for a given diffusion time there were uncertainties in the thickness of the hydride layer formed on the specimen’s surface. An estimate of the thickness of the hydride layer is required for an accurate comparison between the theoretically and experimentally determined hydrogen concentration profiles. • There are differences in the thermal history and production processes of the materials used that could affect the concentration of hydrogen in the material that is free to migrate. • For the case of some experiments using tritium, the annealing times may have been too short to generate a sufficiently spread out diffusion profile, limiting the accuracy of the experimental determination of the shape of the tritium concentration profile. • The role of the state of the surface can lead to highly variable hydrogen uptake rates in experiments where the source of hydrogen is an external source of the gas. Greger et al. [5] attributed this as the reason for the much lower values of the diffusion coefficient obtained by Gulbransen and Andrews [6]. This concern is supported in general in the literature, where a large scatter in the results of diffusion coefficients obtained in other metals using permeation or gassing/ de-gassing methods is found Völkl and Alefeld [19]. The purpose of the experiments of Greger et al. [5] was to determine the effect of increased levels of oxygen on the diffusivity. To avoid the effects on the measurements

5.3 Diffusivity in Dilute Phase—Results

165

Fig. 5.2 Arrhenius plot of measured diffusion coefficients. Open circles, untreated samples (1,350 wppm O; 15 wppm H). Open squares, oxidized samples with an average oxygen concentration of: (1) 3,470 wppm, (2) 2,950 wppm, (3) 9,880 wppm, (4) 11,320 wppm (uniform oxygen distribution. Full circles, hydrogen loaded samples: (5) 50 wppm H, (6) 200 wppm H, (7) 500 wppm H, (8) 1,000 wppm H (from Greger et al. [5])

of the diffusivity that could arise from the aforementioned difficulties, a new method for the determination of tritium diffusion was developed. The oxygen concentration of their as-manufactured alloy was about 1,350 wppm, which is similar to that in Zr-2.5Nb pressure tube material. Their material was doped with increasing amounts of oxygen, up to 11,300 wppm. The authors found, as shown in an Arrhenius plot of the results of their tritium diffusion measurements (Fig. 5.2), that within experimental error, no effect of oxygen concentration on the tritium diffusion coefficient could be detected over the range of concentration of oxygen dissolved in the lattice. Watanabe et al. [20] examined the literature data for values of the pre-exponentials and activation energies of hydrogen diffusion coefficients determined experimentally for a large number of metals. For the elements C, Al, a–Fe, Ni, Cu, and Pd they found that there is a linear relationship between the pre-exponential factors and the corresponding activation energies. They point out that this is similar to the phenomena known as the compensation effect in heterogeneous catalysis. To explain these results they proposed models based, variously, on the assumptions that there exist • energetically heterogeneous sites; • energetically homogeneous sites for which there exists a linear relationship between the entropies and heats of adsorption; • two distinct types of diffusion paths having different diffusion rates. In contrast to the results for the foregoing metals that could be explained on the bases of these models, there was no such clear correlation observed for V, Nb, Zr, and Ta belonging to the IVB and VB groups of elements in the periodic table. Note that the third explanation based on the material having a microstructure consisting of two distinct diffusion paths could be a possible candidate for explaining the results of diffusion coefficient measurements in Zr-2.5Nb pressure tube material. However, the authors were not aware of two sets of result for this alloy, one set by

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5 Diffusion of Hydrogen

Léger [10] and Sawatzky et al. [16]; the other set by Skinner and Dutton [18] published in the same year as the paper by Watanabe et al. [20]. (The results of the former authors are discussed in more detail in the following.) The models proposed by Watanabe et al. [20] are such that, in an Arrhenius plot of the log of the diffusion coefficient versus 1/T there would be a bend in its variation with temperature at a certain temperature. That this bend is often not observed in Arrhenius plots of experimental data is suggested by the authors to be because of the narrow range of temperatures over which the measurements are usually made for which any possible small curvature disappears within the scatter of the data. As noted in the foregoing, the results summarized by Greger et al. [5] and Watanabe et al. [20] did not include diffusion coefficient measurements on material from Zr-2.5Nb pressure tubes used in CANDU and PHW reactors. In Chap. 3 it is pointed out that this alloy has a metastable two-phase (a/b) structure. This structure changes its morphology and phase composition continuously with time starting during the tube’s final autoclaving treatment of 48 h at 400 C, which is the final step in its manufacturing process before it is ready for reactor installation. In Chap 4 it is shown that the equilibrium partitioning of hydrogen between the a- and the b-phases results in an increased hydrogen concentration in the latter phase. The magnitude of this increase diminishes as the b-phase decomposes. In the following, we present results showing that there is also increased hydrogen diffusivity in the metastable b-phase that decreases as this phase decomposes over time in service. Measurements of the diffusion coefficient in Zr-2.5Nb pressure tube material based on concentration gradient methods have been carried out by Léger [10] and Sawatzky et al. [16]. These results show that above 450 C the diffusivity has a magnitude that is similar to that obtained by Kearns [8] for unalloyed Zr and for the single phase Zircaloys. However, at lower temperatures the diffusivity is greater for Zr-2.5Nb pressure tube material with the divergence increasing with decreasing temperature. The reason for this divergence was attributed by Sawatzky et al. [16] to the higher rate of hydrogen diffusion in the b phase. Results of measurements of the hydrogen diffusion coefficient in a solid of composition Zr-20Nb—the starting composition of the metastable b phase after pressure tube high temperature extrusion and subsequent cold working—showed that it is about two orders of magnitude greater in this phase than in the a phase. The b phase starts out as a thin, nearly continuous intergranular layer between the a phase grains elongated in the pressure tube’s axial direction. For hydrogen diffusion in the axial direction of the pressure tube, then, the b phase can act as a fast diffusion path. Sawatzky et al. [16], using an idealized representation of the geometry and thickness relationship between the two phases, developed the following expression for the overall diffusion coefficient of hydrogen in Zr-2.5Nb pressure tube material in the axial direction: b=a

a=b

DH ¼

DaH þ rH d b=a DbH b=a

1 þ rH d b=a

;

ð5:38Þ

5.3 Diffusivity in Dilute Phase—Results

167

where db=a is the ratio of the thickness of the b- grains to that of the a-grains and b=a rH is the equilibrium partitioning ratio of the concentration of hydrogen in the b- to that in the a-grains. Based on their results from independent measurements of b=a rH , DaH , and DbH and microstructural data for d b=a , Eq. 5.38 predicts a factor of a=b

five greater effective diffusion coefficient, DH , than was measured. The authors attributed the cause for this difference to a reduction in db=a caused as a result of b-phase decomposition. However, as is evident from results subsequent to their study, which is documented in Chap. 4, there would also be corresponding b=a reductions in rH . Clearly there was a need for improving the theoretical and experimental understanding of hydrogen diffusion in these a/b pressure tube alloys. This was the motivation of the work of Skinner and Dutton [18] described in the following. Concentration gradient techniques for measuring diffusion coefficients require relatively thick specimen slices for adequate accuracy of the hydrogen content determination in each slice. At low temperatures this requirement also leads to impractically long diffusion times needed to ensure that a sufficiently spread out concentration gradient is formed that could be accurately measured. However, with radioactive tracer techniques these deficiencies can be minimized since thinner slices can be used for sufficiently accurate determinations of the concentration of the tracer in the slice. Skinner and Dutton used a tritium implantation technique to determine the hydrogen diffusion coefficient for a range of materials having different Nb composition and morphologies of the b phase. Implantation of the anodically oxidized specimens was done at room temperature using ionized hydrogen-tritium molecules. Implantation was done with an acceleration voltage of 35 keV for 25 min. This resulted in an implantation of about 8 9 1014 hydrogen (protium) and an equal amount of tritium atoms in a thin layer just below the oxide surface. After implantation, the specimens were stored at liquid nitrogen temperature until needed for the diffusion anneal experiments. The authors did not determine whether the concentration of implanted protium and tritium atoms in this initial layer exceeded the solvus concentration. The diffusion profile, which was determined by cutting successive 10 lm slices, was analyzed by assuming that the concentration profile spreads out from its original, as-implanted distribution according to the relation  2 x cT ðx; tÞ ¼ cT ð0; tÞ exp ð5:39Þ 4DT t where cT ðx; tÞ is the tritium concentration at a distance, x and time, t, and DT is the tritium tracer diffusion coefficient, found from the slope of the data plotted according to ln½cT ðx; tÞ vs x2. It was assumed, as was also done by Greger et al. [5], that the corresponding hydrogen diffusion coefficient could be derived from this plot according to the simple, square root isotopic mass ratio relationship, pffiffiffi giving DH ¼ 3DT . The temperature dependence of the diffusion coefficient was obtained by assuming that the results followed an Arrhenius relationship and thus

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5 Diffusion of Hydrogen

Fig. 5.3 Diffusion coefficient measurements in as-extruded and heat-treated Zr-2.5Nb pressure tube material. References: Austen et al. [1] Sawatzky et al. [16]; Watanabe et al. [20] Kearns [8] (from Skinner and Dutton [18]; with permission from AECL)

a regression of DH plotted versus 1/T gave the values of the pre-exponential D0H and activation energy, Q, of DH ¼ DoH exp½Q=RT . Measurements were carried out on the following Zr-2.5Nb pressure tube materials with the associated b phase microstructure indicated in brackets: • as-extruded (continuous b phase of composition Zr-20Nb); • autoclaved (a continuous inter-boundary layer of b phase containing some x–Zr precipitates with the b phase now enriched with Nb, distinguishing between those produced for the Canadian Pickering and Darlington CANDU Nuclear Generating Stations (NGS)); • autoclaved two times (b phase further decomposed); • autoclaved plus a heat treatment for 24 h at 565 C (b-phase stringers completely decomposed into the terminal b-Nb and a-Zr phases). In addition, measurements were carried out on specimens consisting entirely of the Zr-20Nb b phase in its as-manufactured state and after it had been heat treated at 375 C for 45 h and 42 d, respectively, to produce two levels of decomposition into the enriched beta (benr) and x-Zr phases. Results of measurements on the foregoing materials in the approximate fuel channel inlet-to-outlet temperature range from 267 to 313 C are reproduced in Fig. 5.3. All measurements were carried out with diffusion occurring in the tube’s axial direction except for three results of diffusion in the transverse direction. The latter results show a consistent anisotropy, as expected, for the diffusion rate

5.3 Diffusivity in Dilute Phase—Results

169

Fig. 5.4 Tube-to-tube variation of diffusion coefficient (from Skinner and Dutton [18]; with permission from AECL)

between these two directions. The plot also contains the results from the other two sets of authors who made diffusion coefficient measurements on Zr-2.5Nb pressure tube material. It can be seen from the results obtained from pressure tube material manufactured for the Pickering and Darlington reactors that there is a variability in the diffusion coefficient, presumably arising from differences in the manufacturing route. Additional indications of the degree of variability of the data plotted in Fig. 5.4 is seen by the results from three additional tubes manufactured for the Darlington reactors, as well as for different locations along one particular tube. The figure also plots the results for the diffusion coefficient obtained for the Zr-20Nb b phase material obtained by Sawatzky et al. [16]. Figure 5.5 contains results for the Zr-20Nb material for various stages of decomposition using the tritium tracer technique in comparison with those obtained by Sawatzky et al. [16] using the diffusion couple approach. The greater diffusion rate obtained by the tritium versus the diffusion couple approaches in the undecomposed Zr-20Nb (100 % b phase) material is thought by Skinner and Dutton to be because the diffusion rate in this material is so high that tritium had already diffused a substantial distance during the time it took to implant the tritium. This explanation is supported by the closeness of their results with those of Sawatzky et al. [16] when they apply the same diffusion couple approach to measure the hydrogen diffusion rate in the material used in the tritium diffusion experiments. All of Skinner and Dutton’s results for the diffusion coefficient constants, DoH and Q are summarized in Table 5.1.

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5 Diffusion of Hydrogen

Fig. 5.5 Tritium diffusion coefficient measurements in Zr-20Nb in various stages of decomposition. References: Austen et al. [1] Sawatzky et al. [16]; Watanabe et al. [20] Kearns [8] (from Skinner and Dutton [18]; with permission from AECL)

Skinner [17] developed a theory of hydrogen diffusion that could account for the presence of hydrides in the diffusion path. Hydride precipitates located throughout the diffusion path could act to slow the overall hydrogen diffusion rate because the rate of diffusion of hydrogen in this phase is slower. The theory is, however, quite general and could be applied to derive an effective diffusion coefficient in any material containing microstructural features or phases where the diffusion rate is different from that in the bulk of the material. When this model is applied to the case of diffusion in a/b alloys such as Zr-2.5Nb pressure tube material, the following a=b expression for the effective diffusion coefficient DH is obtained: b=a

a=b

DH ¼

t

DaH staa þ rH db=a sbb DbH b=a

1 þ rH db=a

ð5:40Þ

where sa and sb are the total times a hydrogen atom spends in the respective a and b phases and ta and tb the corresponding times in these phases resulting in net diffusion. All other symbols in Eq. 5.40 have the same meaning as in Eq. 5.39. It is evident from the definitions of these resident times that their ratios, ta/sa and tb/sb—which must be B1—are factors that determine the effective rate of

5.3 Diffusivity in Dilute Phase—Results

171

diffusion of hydrogen in each phase and would depend implicitly on the morphology, volume fraction, and orientation of the minority phase. Limiting values suggest themselves for these factors for various idealized cases. Thus, in the axial direction, the grains can be approximated as being infinite in length with very few b phase cross-links in the radial direction. Then for this geometry the two ratios are both approximately equal to unity and the expression for the effective diffusion coefficient reduces to that given by Eq. 5.39 derived by Sawatzky et al. [16]. These authors had attributed the difference between the predictions derived from Eq. 5.39 and the experimental results as caused by the decomposition of the b phase resulting in a reduction in its volume fraction whilst assuming that the higher rate of diffusion would nevertheless remain the same. The results given by Skinner and Dutton show, however, that there is also a reduction in the diffusion rate in the b phase as it decomposes. Therefore, these authors concluded that the most important reason for the reduction of the effective diffusion coefficient in the autoclaved as well as additionally annealed pressure tube materials is the result of the reduction in the diffusion rate in the b phase as it decomposes. There is, however, as documented in Chap. 4, also a greater reduction in the hydrogen partitioning ratio as the b phase decomposes than was assumed by Skinner and Dutton. Skinner and Dutton also make an attempt to explain the reduction obtained for the effective diffusion coefficient in the transverse direction of the pressure tube. Based on the average of the three available data points, they find that the diffusion rate is a factor 0.78 lower than in the axial direction. Assuming that tb/sb is reduced according to the volume fraction ratio  tb tt ð5:41Þ ¼ sb t ð tt þ tr Þ where tt and tr are the volume fractions of the b phase aligned in the transverse and radial directions, respectively, Eq. 5.40 can be rewritten as h i tt   DaH þ rHb=a db=a ðt þt DbH t rÞ a=b DH ¼ ð5:42Þ b=a t 1 þ rH db=a On the basis of the measured anisotropy for the effective diffusion coefficient, agreement is obtained with the predictions of Eq. 5.42 for tt = 1.22 tr, which appears to be a reasonable result based on the observed microstructure. Equation 5.41 could also be used to estimate the reduction in the diffusion coefficient in the radial direction using tr instead of tt in the denominator of Eq. 5.41. Inserting this into Eq. 5.50, the expression for the effective diffusion coefficient in the radial direction becomes: h i tr   DaH þ rHb=a db=a ðt þt DbH t rÞ a=b DH ¼ ð5:43Þ b=a r 1 þ rH db=a

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5 Diffusion of Hydrogen

Table 5.2 Summary of heat of transfer for hydrogen and deuterium obtained by different investigators Reference Diffusing species Material Q (kJ/mol H) H D H

Zircaloy-2 rod Zircaloy-2 rod Zr-2.5Nb pressure tube material

19.68 and 25.54 26.0 and 28.47 17.17–21.35

[14] [14] [7]

Using the value of 1.22 for the ratio of tt over tr obtained from Eq. 5.42 for the transverse data, Eq. 5.43 predicts a reduction in the effective diffusion coefficient of 0.73 in the radial compared to in the axial direction. Unfortunately there were no experimental results for the diffusion coefficient of hydrogen in pressure tube material in the radial direction to check whether this approach has merit—a situation that has not changed since then. However, there are many data of the DHC growth rate available for the radial and axial directions from which the ratio of the two diffusion coefficients could be inferred. Such an analysis has, to the writer’s knowledge, not yet been done.

5.4 Thermal Diffusion—Results Only a few results of experiments to measure the thermal diffusion in zirconium and its alloys have been published in the open literature. The earliest appears to be that of Sawatzky [14] who measured the heat of transport for both hydrogen and deuterium in Zircaloy-2. The simplest solution is obtained for the steady-state condition (JH = 0) in a temperature gradient with the hydrogen concentration in the solid everywhere less than the solvus composition and no hydrogen pickup at external surfaces. From Eq. 5.23, but without the stress gradient term, the steady-state solution is:   Q 0 cH ¼ cH exp ð5:44Þ RT where the heat of transport, Q , is equivalent to the phenomenological constant, Q0 H , defined by Eq. 5.15. The experiments involved charging a rod of Zircaloy-2 with hydrogen after which it was placed into an apparatus for generating a constant temperature gradient along its length. The sample was maintained with this temperature gradient for a sufficient time to achieve the steady-state hydrogen distribution as a function of temperature given by Eq. 5.44. After that it was removed from the apparatus, cooled, and the specimen sectioned into successive 1-mm discs along its length each of which was analyzed for hydrogen content. Two runs were made of 44 days each, with the hot and cold ends of the rod held at 500 and 300 C, respectively, and a total hydrogen content in the sample of 60 wppm. The latter ensured that the hydrogen concentration in the rod was nowhere above the solvus composition. Based on Eq. 5.44, assuming a linear temperature gradient between the hot and cold parts of the rod, a plot of the natural

5.4 Thermal Diffusion—Results

173

log of the hydrogen concentration versus the inverse of the corresponding temperature at that location resulted in two straight lines, the slopes of which yielded Q =R. Two similar runs were made using deuterium as the diffusing species. The results are summarized in Table 5.2. Note that the heat of transfer is positive, which means that hydrogen diffuses from hot to cold parts of the material. When the solvus is not exceeded along the temperature gradient, then no hydride is formed and a concentration gradient is built up that, at steady state, exactly opposes the flow as a result of the thermal transport effect. At the time of the thermal diffusion study of Sawatzky [14], thermal transport was mostly of interest for fuel cladding, where contact of the fuel cladding with the fuel would result in a hot spot, and the resultant thermal gradient of which would generate a heavily hydrogenated region away from this location that could result in failure of the cladding. However, in 1983 multiple contacts between pressure tubes and their surrounding calandria tubes along their lengths in a Pickering NGS resulted in the formation and failure of hydride ‘‘blisters’’ at cold spots generated by the contacts. This failure rekindled interest in thermal diffusion and led Jovanovic et al. [7] to carry out measurements of Q in Zr-2.5Nb pressure tube material. The results of these measurements are also listed in Table 5.2. It is seen that the results obtained are on the low side of those obtained by Sawatzky [14].

References 1. Austen, J.H., Elleman, T.S., Verghese, K.: Tritium diffusion in Zircaloy-2 in the temperature range—78–204 C. J. Nucl. Mater. 51, 321–329 (1974) 2. Cupp, C.R., Flubacher, P.: An autoradioactive technique for the study of tritium in metals and its application to diffusion in zirconium at 149–240 C. J. Nucl. Mater. 6, 213–228 (1962) 3. De Groot, S., Mazur, P.: Non-equilibrium Thermodynamics. North-Holland Publishing Co, Amsterdam (1961) 4. Gelezunas, V.I., Conn, P.K., Price, J.: The diffusion coefficient for hydrogen in b-Zr. J. Electrochem. Soc. 110, 799–805 (1963) 5. Greger, G.U., Münzel, H., Kunz, W., et al.: Diffusion of tritium in Zircaloy-2. J. Nucl. Mater. 88, 15–22 (1980) 6. Gulbransen, E.A., Andrew, J.: Diffusion of hydrogen and deuterium in high purity zirconium. J. Electrochem. Soc. 101, 560–566 (1954) 7. Jovanovic, M., Stern, A., Kneis, H., et al.: Thermal diffusion of hydrogen and hydride precipitation in Zr-Nb pressure tube alloys. Can. Metall. Q. 27, 323–330 (1988) 8. Kearns, J.J.: Diffusion coefficient of hydrogen in alpha zirconium, Zircaloy-2 and Zircaloy-4. J. Nucl. Mater. 43, 330–338 (1972) 9. Kirchheim, R.: Solid solutions of hydrogen in complex materials. Solid State Phys 59, 203–305 (2004) 10. Léger, M.: Hydrogen diffusion in the axial direction of Zr-2.5 wt % Nb pressure tubes. Ontario Hydro Research Division Report 80–233-K (1980) 11. Mallet, M.W., Albrecht, W.M.: Low-pressure solubility and diffusion of hydrogen in zirconium. J. Electrochem. Soc. 104, 142–146 (1957) 12. Markowitz, J.M.: The thermal diffusion of H in alpha-delta Zircaloy-2. Trans. Metal Soc. AIME 221, 819 (1961)

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13. Mazzolai, F.M., Ryll-Nardzewsi, J.: An anelastic study of the diffusion coefficient of hydrogen in a-zirconium. J. Less-Common Met. 49, 323–327 (1976) 14. Sawatzky, A.: Hydrogen in Zircaloy-2: Its distribution and heat of transport. J. Nucl. Mater. 2, 321–328 (1960) 15. Sawatzky, A.: The diffusion and solubility of hydrogen in the alpha-phase of Zircaloy-2. J. Nucl. Mater. 2, 62–68 (1960) 16. Sawatzky, A., Ledoux, G.A., Tough, R.L., et al.: Hydrogen diffusion in zirconium-niobium alloys. In: Veziroglu, N. (eds.) Metal-hydrogen Systems, pp. 109–120, Pergamon, Oxford (1982) 17. Skinner, B.: Effect of precipitated hydrides on the mobility of hydrogen in zirconium. Unpublished research, Atomic Energy of Canada Ltd., Chalk River, Ontario, Canada (1986) 18. Skinner, B.C., Dutton, R.: Hydrogen diffusivity in a-b zirconium alloys and its role in delayed hydride cracking. In: Moody, M.R., Thompson, A.W. (eds) Hydrogen Effects on Material Behavior, pp. 73–83, The Minerals, Metals and Materials Society (TMS), Warrendale (1990) 19. Völkl, J., Alefeld, G.: Diffusion of hydrogen in metals. In: Völkl, J., Alefeld, G. (eds.) Hydrogen in Metals I, pp. 321–348, Springer, Germany (1978) 20. Watanabe, K., Ashida, K., Sonobe, M.: The compensation effect on diffusion constants of hydrogen in metals. J. Nucl. Mater. 173, 294–306 (1990)

Chapter 6

Characteristics of the Solvus

6.1 Introduction The solvus is defined as the hydrogen concentration in the dilute phase that is in equilibrium with the hydride phase at a given temperature. It is sometimes also referred to as the terminal solubility of hydrogen or, particularly in the zirconium literature, as the Terminal Solid Solubility (TSS). In a closed system with constant hydrogen content, it represents the temperature during a temperature scan at which the last remaining hydrides have dissolved or have just started to precipitate for a given value of total hydrogen content. The most striking and important characteristic of the solvus in metal-hydrogen systems is the large and reproducible hysteresis between the dissolution and formation stages of the hydride phase. Depending on the direction of the transformation this hysteresis manifests itself in closed systems of constant hydrogen content as a difference in temperature at which the transformation occurs during a temperature scan and in systems open to an external source of hydrogen gas as a difference in the plateau pressures of the gas. Although all first-order phase transformations exhibit hysteresis to some extent because of the finite differences in extensive physical properties between the transforming phases, the greater magnitude of these differences in metal-hydrogen systems appears to be a associated with the disparity between the high mobility of hydrogen relative to that of the metal component. This allows the phase transformation to rapidly proceed at temperatures at which the lack of mobility of the underlying metal lattice could force the crystal into a metastable state. In early experimental determinations of the solvus relationships in hydride forming metals and, particularly in zirconium and its alloys, it was at first not appreciated that there were in fact at least two reproducible, apparently stable, phase relationships, and that neither of these represented a true equilibrium state of the thermodynamic system even though each was highly reproducible. To some extent this lack of understanding (or appreciation of its importance, particularly for DHC) has persisted to this day. As an example, a widely quoted source of the literature M. P. Puls, The Effect of Hydrogen and Hydrides on the Integrity of Zirconium Alloy Components, Engineering Materials, DOI: 10.1007/978-1-4471-4195-2_6, Ó Springer-Verlag London 2012

175

176

6 Characteristics of the Solvus

data on the temperature-composition phase diagram of the zirconium-hydrogen system by Zuzek et al. [43] gives no indication that the data obtained do not represent equilibrium states of the system. It was thought initially by some early experimenters, including this writer, that the lower solvus temperature obtained during cooling relative to that observed during heating reflected, at least in part, the fact that a barrier to hydride nucleation had to be overcome that required an increase in thermodynamic driving force over that which exists when macroscopic quantities of each phase are in equilibrium with each other. The latter conditions were presumed to be equivalent to those prevailing at the termination of the dissolution phase of the phase transformation. However, as understanding of the physical origins of hysteresis increased it became evident that the observed ‘‘supersaturation’’ was much larger than if it were simply the result of a nucleation barrier. A useful comprehensive description of the characteristics and interpretation of first-order phase transformations exhibiting hysteresis, particularly as it applies to metal-hydrogen systems, has been given by Flanagan et al. [11]. These authors have emphasized the importance of comparing the characteristics of this phenomenon in a broad range of systems where it occurs and under a variety of experimental conditions. It was felt that in this way key features of hysteresis common to all of them could be identified that would provide clues as to the thermodynamic or mechanistic origin of this phenomenon. As is shown in the next section, the authors have concluded that the feature common to all hysteresis phenomena is the necessity to impose a finite driving force on the system before the phase transformation can proceed in either direction. The hysteresis arises from the fact that this finite threshold driving force is associated with a unidirectional, finite increment of phase boundary movement that is dissipated through internal entropy production. This dissipation of internal work does not show up as a transfer of heat to the surroundings during the irreversible part of a hysteresis cycle. It can be quantified through the thermodynamic link relating the chemical driving force needed to initiate the transformation with the internal entropy production associated with this. The latter can be derived from experimental results of the independent parameters that must be changed to affect a change in the direction of the transformation. In this chapter we summarize these important conclusions of Flanagan et al. [11] followed by a summary of some early, key experimental results and their thermodynamic interpretations. The focus of these studies is restricted to the solvus side of the phase diagram. In Chap. 7 we review the more recent theoretical studies of coherent binary phase relationship as these are relevant to understanding hysteresis on both the low hydrogen (solvus) and high hydrogen (hydride) side of the metal-hydrogen phase diagram. In Chap. 8 the theoretical results presented in Chaps. 6 and 7 are used to provide a new theoretical basis for interpreting extant solvus data in the Zr–H system. Physical understanding of experimentally observed solvus relationships is a key component for underpinning our understanding of the DHC process in zirconium alloys. It is for this reason that three chapters have been devoted to this topic.

6.2 General Considerations Concerning Hysteresis in Phase Transformations

177

6.2 General Considerations Concerning Hysteresis in Phase Transformations Hysteresis manifests itself in phase transformations when there is a barrier to the phase transformation requiring that a threshold driving force of macroscopic extent be applied before the transformation can proceed. An important physical attribute of this barrier in systems exhibiting hysteresis is that it cannot be overcome by thermal fluctuations alone. This means that the transformation cannot proceed, regardless of the time allowed, until a sufficiently large, finite threshold driving force has been applied. This situation is different from fully reversible transformations where infinitesimal changes produced solely by thermal fluctuations can lead to correspondingly infinitesimal amounts of transformation in either direction. It is also not the same as the supersaturation that is observed on first formation of a new phase in first-order phase transitions arising from the need to overcome a nucleation barrier. This, latter, type of supersaturation cannot be sustained once nucleation has been completed whereas in a system exhibiting hysteresis, a state of supersaturation continues beyond this stage until the threshold driving force for the transformation is reached. An example of a metal hydrogen system that illustrates the differing roles of nucleation and metastability on the pressure hysteresis in metal/hydrogen gas systems is shown in Fig. 6.1 for Pd-H. The figure shows a reduction in the absorption pressure needed to maintain the continued formation of hydride in the plateau region once the nucleation stage has passed, but this plateau pressure remains constant throughout the hydride formation stage and is greater than the plateau pressure at which hydride dissolution occurs. An analogous example where it is thought the data show a similar phenomenon is shown in Fig. 6.2 from the results of Pan et al. [24] for a closed Zr–H system with constant hydrogen content. A consequence of the need for a finite driving force is that a finite amount of entropy is produced in the complete hysteresis loop. This finite amount of entropy is a measure of, and reflects, the inherent irreversibility of the process. Formally, the entropy production can be expressed as: di S ¼ dS  de S

ð6:1Þ

where dS and deS are the entropy changes of the system and of an associated quantity of heat transferred to or from a surrounding heat bath, respectively. Then, for reversible transformations diS = 0 while for irreversible ones, diS [ 0. These relations are equivalent to those forming the foundations of the thermodynamics of irreversible processes leading to the macroscopic diffusion equations derived in Chap. 5 except that in that case, dealing with the thermodynamic formulation for diffusion flux, entropy production is more properly expressed as a temporal entropy production rate. For a closed system, deS = 0 and, thus, the fundamental equation is given by dU ¼ TdS  Tdi S  PdV ¼ TdS  dQ0  PdV

ð6:2Þ

178

6 Characteristics of the Solvus

Fig. 6.1 Hydrogen absorption data for annealed Pd-H at 333 K where the aphase datum point at the maximum above the hydride formation plateau is metastable due to the difficulty of nucleation of the hydride phase in the well annealed metal (from Flanagan et al. [11])

100

Zr-2.5Nb + 86 wppm H Cooling from Tmax1 (450oC)

Hydrogen Solubility, wppm

Fig. 6.2 Solvus boundaries derived from the analysis of Young’s modulus as a function of temperature measured on a single specimen containing 0.77 at % hydrogen. The solid lines are the fit to the data shown in Fig. 7 of Pan et al. [24] (from Pan et al. [24])

Cooling from Tmax2 (370oC)

30

TSSP1

20

TSSD 10

1.6

1.8

2

2.2

TSSP2 2.4 2.6

2.8

3

Reciprocal TSS Temperature; 1000/K

where the prime on dQ0 indicates that it is not a true heat increase in the system that can be detected by changes to the surroundings. Rather, it represents what has been called an ‘‘uncompensated heat’’ as introduced by Clausius (as quoted in [33]). since it is the result of irreversible structural changes generated internally, the extent of which does not result in any detectable changes to the usual external thermodynamic variables that control the conditions under which the phase transformation proceeds. For a reversible process TdS = TdeS, dQ0 ¼ 0 and dU = TdeS - PdV. Writing Eq. 6.2 in terms of the Gibbs potential one has: dG ¼ SdT þ VdP  Tdi S ¼ SdT þ VdP  dQ0

ð6:3Þ

6.2 General Considerations Concerning Hysteresis in Phase Transformations

179

This relation reduces at constant T and P to: dG ¼ Tdi S ¼ dQ0 [ 0

ð6:4Þ

In Eq. 6.4 dQ0  Wirrev  Wrev ¼ Wloss can also be viewed as a quantity that represents the loss of useful work produced either in a single irreversible step or over the full hysteresis cycle. The former cannot, however, be applied unambiguously without knowledge of the equilibrium pathway, although knowledge of this is not necessary for the full cycle, in which case one has I Tdi S ¼ Q0 ðcycleÞ ¼ Wloss ðcycleÞ ð6:5Þ In the case of thermal hysteresis, where temperature excursions are the driving forces for evaluation of the loss of work, it is useful to imagine a Carnot cycle connected to the system in which hysteresis is occurring [38]. In thermal hysteresis (i.e., for closed systems at constant hydrogen content) no work is transferred between the system and the surroundings. Therefore, in contrast to pressure hysteresis in open systems, the degradation of energy is not obvious. The degradation occurs because the heat of transition is transferred from the system to the surroundings at Tf and then returned from the surroundings to the system at Td with Td [ Tf. The net effect is the degradation of energy because a given amount of heat has been transferred from a hot (Td) to a cold (Tf) reservoir without any work being done. The corresponding work, which could have been obtained with a notional Carnot heat engine, WCarnot : Wloss operating between these two heat baths at Tf and Td has been evaluated by Torra and Tachoire [38] for martensitic transformations. These authors call the work measured in this way a fictitious frictional work, because it is of the same type and magnitude as the driving force that is needed to overcome a certain amount of internal friction occurring during the transformation, leading to dissipated heat transferred to the surroundings. This fictitious work corresponds to the uncompensated heat accompanying the entropy production. In the case of a martensitic transformation this work is associated with the driving forces     oDG DGðat Tf Þ  ð6:6Þ DT ¼ Tf  Teq DSp!m oT and DGðat Td Þ 

    oDG DT ¼ Td  Teq DSm!p oT

ð6:7Þ

that would be required to perform the fictitious frictional work (‘‘p’’ stands for parent and ‘‘m’’ for martensite in these equations). However, since the work driving the interface during the transformation is really associated with an entropy production occurring only within the solid phase, it should not be interpreted literally as an actual work required to drive the interface, since no work is done in the sense that its source could be considered to be a weight attached to the system

180

6 Characteristics of the Solvus

that is being raised or lowered. Moreover, there is no accompanying heat released to the surroundings during the transformation. The heat and work, analogous to that found in observations of pressure hysteresis for open systems, are virtual quantities arising from the degradation caused by heat absorption of the sample coming from a high temperature reservoir and its subsequent reduction (and, hence, loss) coming from a low temperature reservoir. Similarly, as shown further on, in a corresponding open-system pressure-hysteresis case the net work done on the sample occurs during the isothermal compression and expansion of the external gas source. These processes can, usually, be considered to behave ideally and therefore without losses when done quasi statically. Now, during these processes of reversible compression and expansion of such an ideal gas, an equivalent heat is released to the surroundings. However, because these steps are done reversibly, there is no entropy production. In the irreversible steps, which are the steps during which hydride is formed or dissolved, there is entropy production, but this is not sensed externally since no heat is actually exchanged with the surroundings. This seemingly paradoxical situation arises because the irreversibility in this case is, again, as in the thermal hysteresis case, the result of an uncompensated heat, socalled because it is not sensed by the surroundings during these two irreversible steps of formation and dissolution of the hydride phase. As noted in the foregoing, the physical manifestation of this irreversibility is the production of internal microstructural changes of the material resulting, for instance, in the creation and annihilation of dislocations during the hydride formation and dissolution processes. These changes are not sensed by external thermodynamic parameters. Experimentally, phase diagram determinations are affected by hysteresis occurring during the phase transformation in two ways. One is that the transition temperature is not uniquely defined, occurring over a range of values, while the other is that phase boundary compositions determined during heating and cooling (or, more generally, during dissolution and formation) differ. The magnitude of the free energy dissipated, or the uncompensated heat for solvus hysteresis, can be derived from the results of, for instance, thermal hysteresis scans of a martensitic transformation, giving: I DT Q0 ¼ Tdi S ¼ DHsolv ð6:8Þ Teq In Eq. 6.8, DHsolv is the enthalpy for the phase transformation that is unaffected by hysteresis (i.e., it consists only of the chemical energy of bonding when hydride is formed from hydrogen within the solid (see Eq. 6.10 further on), Teq  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðTheat Tcool Þ with Theat and Tcool the temperatures at which the phase transformation terminates or starts during heating and cooling, respectively, at a given constant total hydrogen content and DT ¼ ðTheat  Tcool Þ: The free energy dissipated can also at Tcool and for heating be determined from the solvus composition for cooling cs;cool H s;heat cH at Theat using

6.2 General Considerations Concerning Hysteresis in Phase Transformations

181

Fig. 6.3 Solvus hysteresis for zirconium hydrogen where ‘a’ is the terminal solid solubility (after [5] (from Flanagan et al. [11])

0

Q ¼

I

" di S ¼ RTav ln

cH s;cool cs;heat H

# ð6:9Þ

where ‘‘s’’ stands for ‘‘solvus.’’ Equations 6.8 and 6.9 are applicable, respectively, for small values of csH ; or high temperatures, where the metal hydrogen system behaves close to ideally. In applications of Eq. 6.9 an average temperature, Tav ; should be chosen that is close to the value of Teq defined through Eq. 6.8. Flanagan et al. [11] illustrate the use of Eqs. 6.8 and 6.9 on the basis of early data by Erickson and Hardy [5] for the Zr–H system, reproduced in Fig. 6.3, and the results of a calorimetric study of this system by Dantzer et al. [4]. Note that calorimetric studies yield solvus enthalpies that are unaffected by hysteresis. The solvus enthalpy is derived from the results of the calorimetric study using the relation:     DHsolv ¼DHplat   DHðcsH Þ ð6:10Þ ¼87  49:3 ¼ 37:7 kJ/(mol H)   In Eq. 6.10, DHplat is the plateau enthalpy and DH csH is the enthalpy for solution of hydrogen in a-Zr. From the results of Erickson and Hardy [5], Flanagan et al. [11] obtain, using Eq. 6.9 at Tav ¼ 667 K; that Q0 ¼ 3:46 kJ=ðmol HÞ while using Eq. 6.8, Q0 ¼ 3:76 kJ=ðmol HÞ: The agreement between these two results is gratifying considering the different origins of the two equations. Further support for the foregoing good correspondence for the Zr–H system obtained from the relations given by Eqs. 6.9 and 6.10 is derived from the observations by Dantzer et al. [4] that ideal solution behavior in this system—and also the Ti–H system [11]—extends to quite high values of solvus concentration. Many of the insights regarding the nature of the hysteresis in metal hydrogen systems have been obtained from the results of experimental and theoretical studies of hydride formation in the Pd-H system. Hysteresis in this system is

182

6 Characteristics of the Solvus

Fig. 6.4 Hydrogen isotherm data at 393 K for Pd-H (from Flanagan et al. [11])

usually characterized isothermally through a pressure hysteresis scan in which the hydrogen content of the palladium phase is increased, driven by an increasing external partial pressure of hydrogen until a ‘‘plateau’’ pressure is reached at which the hydrogen content of the sample increases at constant external hydrogen pressure, signaling the start of the formation or decomposition of the hydride phase. Hysteresis is exhibited because the reverse process results in a plateau pressure that is lower than the corresponding pressure for hydride formation. An example of this is shown in Fig. 6.4. The hydrogen pressure where the transition would be able to take place reversibly cannot be realized experimentally. Nevertheless, it must lie somewhere between the plateau pressures for hydride formation, pf, and dissolution, pd. Its exact location is unknown, but could, in principle, for a miscibility gap transition such as Pd-H, be obtained fairly accurately from an extrapolation of an analytical representation of the experimental results in the single phase regions above and below the critical point. One could also measure the thermal hysteresis for this system at constant hydrogen content [9]. The equations for the magnitudes of the dissipated energy for each of these experimental methods of determining the hysteresis are given in Eqs. 6.11 and 6.12 in the following. These apply to both miscibility gap systems such as Pd-H and structural phase transformation systems such as Zr–H. In both cases, the experimental situation is that the hydride phase forms at Tf and dissolves at Td by adding or removing hydrogen at constant pressure or else isothermally adding hydrogen at pressure, pf, and removing it at pressure, pd. For pressure hysteresis the magnitude of the free energy dissipated is obtained from the formation and decomposition plateau pressures by:   1 pf 0 Q ¼ RT ln ð6:11Þ 2 pd

6.2 General Considerations Concerning Hysteresis in Phase Transformations

183

while for thermal hysteresis (such as for a martensitic transformation) the free energy dissipated is obtained from the temperature difference between hydride formation and decomposition (DT = Td – Tf):   DT Q0 ¼ DHeq  Teq

ð6:12Þ

In Eq. 6.12 DHeq–equivalent to DHsol given in Eq. 6.8–is the enthalpy change for the transition unaffected by hysteresis, i.e., it is a calorimetric value, while qffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi Teq  Tf Td . The two aspects of hysteresis simply reflect experimental situations for which different variables in the pH2  rH  T space are held constant. Experimentally it has been shown, as expected, that the values of the free energy dissipations for the two types of experimental conditions are approximately the same [8]. In the derivation of the foregoing equations, the plateau properties are assumed to be constant and the phase boundary compositions are assumed to be uninfluenced by hysteresis. However, in reality, the boundary compositions are always affected to some extent by hysteresis, as shown in Fig. 6.4. If these shifts are significant, or if the plateau slopes are appreciable, then the following integral must be used to evaluate the isothermal extent of hysteresis: I pffiffiffiffiffiffiffiffiffiffiffiffiffiffi RT ln pd ðH2 ÞdrH ð6:13Þ It is unnecessary to carry out the full hysteresis cycle if the constant pressure plateaux are horizontal (i.e., invariant in the dependent variable) in order to determine the magnitude of the hysteresis. It can be evaluated using a smaller hysteresis loop within a complete cycle. Such small loops extend only to a small fraction of conversion from one phase to the other. A pressure scanning loop is indicated in Fig. 6.4 and a temperature scanning loop during thermal hysteresis for the Pd-H system is illustrated in Fig. 6.5. It might seem that the extent of hysteresis should be related to the volume expansion or contraction that takes place upon hydrogenation or dehydrogenation. Based on this assumption, it would then seem that hysteresis would be proportional to cs;incoh ð½a þ d=dÞ  cs;incoh ða=½a þ dÞ: This relation gives the difference between H H the equilibrium compositions of the coexisting phases. Since the actual phase boundary compositions are affected by hysteresis, their difference was approximated by their average values, the latter being assumed to be equivalent to the incoherent and, hence, equilibrium phase boundary compositions. The reason for this suppoð½a þ d=dÞ  cs;incoh ða=½a þ dÞhas been sition is that the difference given by cs;incoh H H found to be proportional to the volume change upon hydride formation or decomposition while in miscibility gap systems such as Pd-H there is usually also a direct proportionality between lattice parameter and hydrogen content. However, there are examples where this is not the case. Thus, for the b to d transformation in the Nb-H system the extent of hysteresis is very large [15] but for a comparable difference in

184

6 Characteristics of the Solvus

Fig. 6.5 A temperature scan during thermal hysteresis for the Pd-H system at pH2 ¼ 1:0 kPa (from Flanagan et al. [11])

hydrogen content in the V–H system corresponding to the b0 to c transition, the hysteresis is quite small for vanadium in bulk form [42], while it is surprisingly large for the same metal in the form of thin films [25]. This result is opposite to that found for the corresponding physical difference of the Pd-H system [12, 35]. One reason for this may be because, although the misfit volume is always the same, the elastic strain (accommodation) energies of the hydrides in these systems might be different as a result of different morphologies and corresponding misfit strain anisotropies of the hydrides. This is elaborated on further in Chap. 8 in applications of the present concepts to the Zr–H system. The extent of hysteresis in metal-hydrogen systems generally decreases with increasing temperature but not always in a linear manner. For instance, hysteresis often seems to vary little with temperature at lower temperatures. However, this changes to a temperature dependence at higher temperatures (e.g., for the system U-H [20, 41]. In the Pd-H system, of course, the hysteresis must vanish at the critical point where the two branches of the coexistence compositions meet [12]. * ( From the van’t Hoff plots (plots of ln p or ln p (or equivalent   parameter, such as ln csH versus 1/T) it can also be seen that jDHðpd Þj [ DHðpf Þ with the difference between them appearing to be approximately constant at low temperature. An important point to note is that repeatability of the hysteresis cycle, which is a characteristic of true hysteresis, only occurs if the system is internally returned to the same state. Generally, this requires that the system is brought through a hysteresis cycle a few times so that any internal changes to the material resulting from hydride formation and dissolution have reached a constant state. Practically, this means that in measurements of the solvus temperature (or composition) the results of the first or more cycles should usually be discarded. In addition, it is important to keep other conditions of the scan, such as the upper and lower temperatures achieved and the rates of heating and cooling, the same. A severe

6.2 General Considerations Concerning Hysteresis in Phase Transformations

185

example of changes brought about by repeated hysteresis scans is obtained in the case of complete conversion of the metal from its hydrogen free to its solid hydride state and back again. In this case, too rapid a conversion rate can result in the complete disintegration of the material into small particles during the first few hysteresis scans. Examples of similar effects on hysteresis of embedded hydrides in structural metal-hydrogen systems are given by Birnbaum et al. [2] for Nb-H and Pan et al. [24] for Zr–H (see Fig. 6.2). Finally, an example of an analysis due to Flanagan et al. [11] of a simplified pressure hysteresis cycle is given since an understanding of the significance of the results of such extensively used methods to study hysteresis can provide useful perspectives for assessing theories of solvus hysteresis based on other approaches such as the accommodation energy models given in Sect. 6.3 and the total coherency energy models given in Chap. 4, Sect. 4.4.5, and Chaps. 7 and 8. Consider a system shown schematically in Fig. 6.6 (consisting of one mole of H distributed between the gas and solid phases and one mole of metal in a heat bath at constant temperature). To simplify the illustration of the pressure hysteresis cycle shown in Fig. 6.7, it is assumed that the phase boundary compositions are unchanged by hysteresis (in reality they would be affected) and that the compositions of the phase boundaries are at the extremes of the phase diagram, at a = 0 and b = 1. However, the final result will, nevertheless, be corrected to account for the more realistic condition for which a [ 0 and b \ 1. Each step in the hysteresis cycle is assumed to be carried out quasi statically to ensure that steps that are inherently irreversible are carried out as reversibly as possible. The steps involving * irreversibility are given in terms of the limiting (plateau) pressures, p (with for( ward arrow for Steps 1 to 2) and p (with backward arrow for Steps 3 to 4). The arrows in the steps indicate the direction in which it is thermodynamically possible to proceed. Only a step in which the arrows point in both directions can be carried out reversibly. Thus Steps 5–6 can be carried out in either direction and the value of the pressure for this step is assumed to be the pressure for the equilibrium transformation. This transformation cannot be achieved experimentally but its physical state can, nevertheless, be thermodynamically defined so that it can act as a meaningful reference state, the use of which provides an equilibrium path along which the otherwise irreversible step of hydride formation or dissolution can proceed, thus making possible the use of a thermodynamic analysis. Referring to Fig. 6.7, thermodynamic expressions for the irreversible path, Steps 1 to 2, can be obtained by carrying this out along a reversible path as follows: DHð1 ! 2Þ ¼ DHð1 ! 5Þ þ DHð5 ! 6Þ þ DHð6 ! 2Þ

ð6:14Þ

which reduces to *

*

DHð1 ! 2Þ ¼ 0 þ DHðpeq Þ þ 0 ¼ DHðpeq Þ

ð6:15Þ

186

6 Characteristics of the Solvus

Fig. 6.6 Schematic system for measuring pressure hysteresis in a metalhydrogen system. The system consists of one mole H distributed between the gas and solid phases and one mole of metal in a heat bath at constant temperature (from Flanagan et al. [11])

Fig. 6.7 A simplified pressure hysteresis cycle for a metal hydride system (from Flanagan et al. [11])

Equation 6.15 follows from Eq. 6.14 because the enthalpy changes along the plateau branches of the cycle vanish (provided the pressure of the gas is such that * ideal behavior can be assumed) while DHðpeq Þrepresents the enthalpy for the equilibrium transformation with the arrow indicating the direction of the change and, hence, the sign of the enthalpy change. Note that this enthalpy corresponds to the enthalpy change measured calorimetrically, not the change determined from * van’t Hoff plots. From Eq. 6.15 it can also be seen that DHðpeq Þ would be the true equilibrium value if the phase boundary compositions were unaffected by hysteresis. The change in entropy for the formation of hydride is given by: DSð1 ! 2Þ ¼ DSð1 ! 5Þ þ DSð5 ! 6Þ þ DSð6 ! 2Þ

ð6:16Þ

6.2 General Considerations Concerning Hysteresis in Phase Transformations

187

which becomes *

1 p DSð1 ! 2Þ ¼ DSðpeq Þ þ R ln 2 p~

!

*

ð6:17Þ

From Eqs. 6.15 and 6.17, and the definition of the Gibbs free energy, the free energy change for hydride formation is given by: DGð1 ! 2Þ ¼DHð1 ! 2Þ  TDSð1 ! 2Þ *

1 p ¼DHðpeq Þ  TDSðpeq Þ  RT ln ~p 2 ! * 1 p ¼  RT ln \0 ~ 2 p *

!

*

ð6:18Þ

*

In Eq. 6.18, the last equality follows from the condition, DGðpeq Þ ¼ 0. To develop insight into the thermodynamic connection between irreversibility and hysteresis, various entropy relationships need to be derived. From Eqs. 6.1 and 6.17 we have: Di Sð1 ! 2Þ ¼DSð1 ! 2Þ  De Sð1 ! 2Þ DHð1 ! 2Þ ¼DSð1 ! 2Þ  T! * * DHðpeq Þ 1 p * ¼DSðpeq Þ þ R ln  ~ 2 T p ! * 1 p ¼ RT ln [0 ~ 2 p

ð6:19Þ

where Di Sð1 ! 2Þ is given by: Di Sð1 ! 2Þ ¼ 

DGð1 ! 2Þ T

ð6:20Þ

Equations 6.19 and 6.20, thus, show that DGð1 ! 2Þ is the free energy dissipated internally in the system and is, therefore, the uncompensated heat during hydride formation. The process is assumed to have taken place quasi statically which ensures that there are no other sources of dissipation. A similar set of equations as the foregoing holds for hydride decomposition. It is evident from the resulting equations that absolute values cannot be evaluated from them without knowledge of ~ p which is, however, an experimentally unrealizable state, although it can be theoretically defined. This means that only values relative to this reference state can be obtained. Note that the steps over the plateau regions (Steps 4 to 1 and 2 to 3) correspond to reversible changes since the only change done along these paths is that the gas is reversibly compressed or

188

6 Characteristics of the Solvus

expanded, respectively. Hence, entropy is not produced along these paths. Thus, the entropy production for the complete hysteresis cycle can be calculated from Eq. 6.19. The corresponding result for hydride dissolution is given by:   ~ 1 p Di S ¼ RT ln * [ 0 ð6:21Þ 2 p This results in the relation: 

I

dG ¼ T

I

*

1 p di S ¼ Q ¼ RT ln ( 2 p

!

0

The work during this cycle is obtained as follows: I dW ¼Wð1 ! 2Þ þ Wð2 ! 3Þ þ Wð3 ! 4Þ þ Wð4 ! 1Þ ! ! * * 1 p 1 p ¼RT þ 0  RT þ RT ln ( ¼ RT ln ( 2 2 p p

ð6:22Þ

ð6:23Þ

As this cycle is carried out under isothermal conditions, the sample is returned to its initial state at the end of the cycle and the H total internal energy change over the complete cycle is zero. Therefore, with dU ¼ 0 and, from the first law, we have: ! I I I * 1 p dW ¼  dQ ¼ RT ln ( ¼  dG ¼ Q0 ð6:24Þ 2 p The contributions to the net work in Steps 1 to 2 and 3 to 4 cancel but appear in the two reversible steps of gas compression (4 to 1 and 2 to 3). Heat must be dissipated to the surroundings during these two reversible, isothermal paths and not in the internally irreversible paths during which hydride is being formed and dissolved. It is interesting that the mechanical work done on the gas phase is degraded to heat in the two reversible rather than the irreversible paths. Note also that the only heat that is transferred between the system and the surroundings during the irreversible steps is the heat of the chemical reaction which is contained in DHsol(calorimetric). Thus, there is an ‘‘uncompensated heat’’ appearing in the irreversible steps that is equal to the dissipated free energy. This free energy change does not appear in the surroundings and is therefore not heat in the usual thermodynamic sense. The calorimetrically measured heats for hydride formation or decomposition (Steps 1 to 2 or 3 to 4) are enthalpy changes because the pressures are constant during these changes and there is no other work. For the simple hysteresis cycle shown in Fig. 6.7 where there are no changes in the lower and upper boundaries for the phase transformations (although in a real system there would be such changes), the enthalpy changes must have the same magnitudes for the two sets of steps because the changes of state functions must be of opposite signs for the two processes where

6.2 General Considerations Concerning Hysteresis in Phase Transformations

189

the initial and final states are interchangeable for the two paths. The enthalpy change for the equilibrium path must, correspondingly, also be the same. A study of the Pd-H and Pd-D systems by Flanagan et al. [10] was carried out that demonstrates the foregoing points. The  authors  found the following heats evolved and absorbed at 298 K. For jQj ¼ DHplat they obtained 19.09 ± 0.1 kJ/ mol H for hydride formation  and 19.28 kJ/mol H for hydride dissolution. *

(

In addition, for Q0 ¼ 12RT ln p=p they obtained 1.09 kJ/mol H. If the ‘‘frictional’’

work to drive the interface during the formation and dissolution of Pd-H subsequently appeared as heat in the surroundings, then the heat measured by calorimetry for hydride formation would be |–19.09 – 1.09 | = 20.18 kJ/mol H and for hydride dissolution it would be |19.28 – 1.09 | = 18.19 kJ/mol H. Flanagan et al. [10] point out that such a difference in measured heats would have been readily detected by calorimetry if it had been present. Hence, they concluded–similar to the conclusions of Torra and Tachiore [38] for a martensitic transformation–that this so-called ‘‘frictional’’ work to drive the interface is not converted into heat that is detectable in the surroundings. Therefore it is, in fact, the ‘‘uncompensated heat.’’ Compared to martensitic transformations though, this ‘‘uncompensated heat’’ is generally found to be much larger for metal hydride phase transitions. In summary, the foregoing development on thermodynamics has shown that for pressure hysteresis in solid/gas systems the net work done on the system is done during the reversible steps. Since the work corresponds to an isothermal compression and expansion of an ideal gas, an equivalent amount of heat is released to the surroundings during these reversible steps. The irreversibility is reflected by an uncompensated heat that is not sensed by the surroundings during the irreversible parts of the phase transformation. In contrast, in thermal hysteresis for a closed solid phase transformation, such as, for instance, in the case of martensitic transformation, work is not done on the solid and there is no corresponding heat evolution. Heat and work in thermal hysteresis that would be analogous to those found in pressure hysteresis experiments for an open system are virtual quantities arising from the energy dissipation caused by heat absorption in the solid from a high-temperature reservoir and subsequent loss of this heat to a low-temperature reservoir. Despite these apparently quite different forms of hysteresis with regard to energy dissipation, the crucial existence of the uncompensated heat or entropy production during the irreversible paths of the transformations is a common feature.

6.3 Theories of Solvus Hysteresis Based on Accommodation Energy Paton et al. [26] were the first to propose a quantitative model of hysteresis for the hydrogen solvus in titanium and its alloys. Ti and its alloys have structural hydride phase transformations like zirconium and its alloys. By structural (polymorphic) phase transformation is meant that the crystal lattice structure of the hydride phase

190

6 Characteristics of the Solvus

is different from that of the parent phase from which it transforms. It turns out that the thermodynamic relationships for coherent equilibrium for metal-hydrogen systems with such changes in crystal structure are fundamentally different from those for isomorphic systems such as Pd-H where the two phases have the same crystal lattice structure. This is dealt with in detail in Chap. 7. Solvus hysteresis in the model of Paton et al. [26] was assumed to arise from the irreversible work done in relaxing a part of the accommodation energy by plastic deformation. By accommodation energy is meant the sum of the total molar strain energy and plastic work of matrix plus precipitate that is produced when a misfitting hydride precipitate forms in an infinite matrix of the parent phase. The model of Paton et al. [26] was based on qualitative ideas first proposed by Westlake [39, 40] and is similar in approach to that proposed by Schultus and Hall [36] for hysteresis in Pd-H. The former authors assumed that the solvus for hydride dissolution, which is determined during heating, represented the equilibrium solvus. The model of internal energy dissipation derived by these authors provided an expression for the shift in the solvus determined during cooling from its value determined during heating. In the model of Paton et al. [26] the shift consists of both elastic and plastic contributions to the accommodation energy. The plastic work contribution is a consequence of the very high stresses–exceeding the yield strength–produced in the matrix (and possibly also in the hydride) as a result of the large stress-free transformation strains that the hydride imposes on its parent phase on formation or dissolution. Hysteresis in the solvus in this model arises because the plastic work component of the accommodation energy cannot be recovered during reversal of the transformation. However, seen from the present perspective, the model of Paton et al. [26] cannot be quite correct because of the assumption that plastic deformation occurs only in the cooling part of the hysteresis cycle. Moreover, these authors underestimated the amount of plastic work that is done by implicitly assuming that plastic work was generated only over the volume of the hydride. Birnbaum et al. [2] applied the foregoing accommodation energy approach to interpret the solvus data obtained in the Nb-H system. Their data were derived from thermal hysteresis scans in which the phase transformation is sensed by changes in plots of electrical resistivity with temperature. The authors found that the solvus temperature obtained on cooling was lowest for the first cycle and increased slightly until a constant value was achieved after the nth cycle. In contrast, the solvus temperature determined on heating remained unchanged with cycle number. The authors fitted their data assuming, as is usually done, that a plot of the hydrogen solvus concentration versus temperature has an Arrhenius-type temperature dependence. Additionally, the authors assumed that the fitted nonconfigurational entropy term would be the same for the two sets of solvus data obtained after the nth cooling cycle and for the heating cycles while, for reasons not stated, a separately fitted value was used for the solvus data obtained on first cooling. It was not made clear what criterion was used to determine a common value for this parameter since, generally, each set of solvus data, when fitted independently, would yield a different value.

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In interpreting the difference in the enthalpy of hydride formation and dissolution obtained from their data in going through the complete hysteresis scan, the authors assumed that the difference in solvus enthalpy obtained from the separate fits to the p first cool down and heat up data represented the unrecoverable plastic work, DHtotal , which is lost during the formation and dissolution stages of the hysteresis cycle. In addition, they assumed that the plastic work done during the formation and dissolution stages of the scan are equal. Hence, the total plastic work over the complete cycle was given by p

p

p

 total ¼ DH  a!b þ DH  b!a DH p  a!b  2DH  pb!a ¼ 2 600 J/mol ¼ 2DH

ð6:25Þ

In Eq. 6.25, a is the dilute and b is the hydride phase. The energy, 2 600 J/mol, is the difference between the experimentally determined solvus enthalpies for hydride formation during the first cool down cycle and the subsequent hydride dissolution cycle. The authors then assumed that the first cool down cycle could be taken as an approximation of what they called the ‘‘constrained’’ solvus since it resulted in the largest difference in hydride formation enthalpy between it and the heat up solvus. The enthalpy of hydride formation for this ‘‘constrained’’ solvus was assumed to consist of the sum of a chemical (H–H interaction) enthalpy term and an accommodation energy term accounting for the misfit strain energy between hydride and matrix. Because of the large misfit strain it was further assumed, as was done by Paton et al. [26], that only a part of this accommodation energy would be retained as elastic strain energy while the remainder would be dissipated as plastic work. The driving force for this conversion of elastic strain energy to plastic work was assumed to derive from the pure elastic strain energy, D wel inc , arising from the 12 % volumetric misfit strain between hydride and matrix. Thus, the following relationship between the elastic–plastic and pure elastic accommodation energies, respectively, was obtained: e

p

 a!b ¼ D  a!b þ DH wel DH inc ¼ 3 100 J/mol

ð6:26Þ

In Eq. 6.26 the magnitude of the pure elastic strain energy of 3 100 J/mol was obtained using an expression derived by Eshelby [6] assuming that the shape of the hydride is spherical and the elastic constants in hydride and matrix are the same. Then, from Eqs. 6.25 and 6.26, the actual elastic strain energy contribution to the  ea!b ; was calculated to be 1 800 J/mol which means that enthalpy of formation, DH the elastic contribution represented more than 50 % of the total accommodation energy given by Eq. 6.26 for the first cool down solvus. The total enthalpy for the equilibrium hydride formation/dissolution solvus was then determined by subtracting the accommodation energy value given by Eq. 6.26 from the total (negative) enthalpy for hydride formation obtained experimentally for the first cooling cycle. The authors also derived an expression for the effect of uniform external stress on the solvus, a more detailed derivation of which was given separately by

192

6 Characteristics of the Solvus

Grossbeck and Birnbaum [13]. From this analysis, the authors concluded that tensile stress at the crack tip could result in a substantial increase in the temperature at which hydride formation and dissolution could occur. However, as was later demonstrated by Flanagan et al. [7] and noted by Oates and Flanagan [23], the ‘‘thermodynamic arguments offered by Birnbaum and Grossbeck [13] in support of the mechanism of stress induced hydride precipitation are difficult to understand because of their failure to clearly differentiate between the effects of uniform and non-uniform stress on the solvus behaviour.’’ Using the expression for the effect of stress on chemical potential of hydrogen in solution given by Li et al. [19], Birnbaum and Grossbeck [13] concluded that the increase with tensile stress of the hydrogen concentration in solution would not be large. In addition, they concluded that an increase in hydrogen concentration in solution at the crack tip cannot, in any case, result in hydride precipitation ‘‘since at equilibrium the chemical potential of hydrogen at the crack tip is equal to that in the solid solution.’’ As discussed in detail in Chap. 10 dealing with DHC, and also previously pointed out by Flanagan et al. [7] and, recently, by Puls [32], this statement is only true when the increase in hydrogen concentration at the crack tip is insufficient to reach the solvus concentration for hydride formation. Birnbaum et al. [2] also claimed that there is a further substantial reduction in the solvus under external stress because this stress could ‘‘assist the plastic accommodation process.’’ This latter claim was based on interpretations of experimental observations of hydrides that apparently only formed after a tensile stress was applied in regions that had been cooled to slightly more than 100 K below the experimentally obtained constrained solvus. From a theoretical point of view their explanation of these results is problematic. If external stress were to facilitate the formation of hydride by facilitating its formation plastically rather than elastically, one would expect a decrease in the accommodation energy compared to that if all of the misfit strain energy were accommodated elastically. Such a reduction in accommodation energy would result in an increase in the temperature at which hydride formation first starts during a temperature cool down scan opposite to what they observed. Moreover, results of recent calculations of the plastic work of hydride formation subjected to a uniform, triaxial stress field of magnitude similar to that which would exist at a crack tip for spherically shaped hydrides by Lufrano et al. [21, 22] and for plate-shaped hydrides by Puls et al. [27] do not support such large reductions in accommodation energy. These authors show that the effect is not very large for plate-shaped hydrides, becoming possibly observable only for spherically shaped hydrides at very large deviatoric tensile stresses. Puls [28] used a similar approach to that employed by Birnbaum et al. [2] in his first attempt to interpret the solvus data for Zr–H systems. As was done by the latter authors, Puls assumed that even though some of the hydride’s misfit strains would be accommodated by plastic deformation, the total magnitude of the combined elastic strain energy and plastic work would be given by the pure elastic strain energy D wel inc inc  t for this energy). Aside from this (in the original papers Puls used the notation w similarity, the analysis differed from that of Birnbaum et al. [2] in two ways.

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One was that Puls assumed the results of solvus measurements obtained by Kearns [14] on Zircaloy-2 and -4 materials using a diffusion couple approach represented the equilibrium solvus data for that material. In a diffusion couple approach two specimens containing initially different hydrogen contents are joined together and equilibrated at a given temperature. One side of the couple has hydrogen content sufficient for a finite volume fraction of hydrides to be present at each chosen equilibration temperature, while the other always starts out approximately hydrogen free. Upon a sufficiently long equilibration period such an approach should result in a uniform hydrogen concentration in the initially hydrogen free side corresponding to the solvus concentration existing in the hydride-containing part of the diffusion couple. Puls [28] assumed that the solvus concentration obtained in this way represented the equilibrium solvus. Lending credibility to this assumption at the time was the result that the solvus data for hydride dissolution obtained by another investigator [37] for nominally the same alloy material, and using dilatometry methods with a thermal scan approach to detect the presence or absence of hydrides, were slightly shifted to higher temperatures compared to those obtained by Kearns [14]. On further reflection, however, as Flanagan subsequently pointed out (private communication to the author), the solvus obtained by Kearns using such a diffusion couple approach is actually a dissolution solvus since equilibration between the two sides of the diffusion couple requires some hydride dissolution to occur in the hydrogen-rich side of the couple. Another difference in the analysis of the solvus data by Puls [28] of the Zr–H system compared to that for the Nb-H system by Birnbaum et al. [2] was that Puls assumed that the solvus data for hydride formation obtained by Erickson and Hardy [5] and Slattery [37] under cooling conditions contained a contribution resulting from the additional (surface) energy required for hydride nucleation. Birnbaum et al. [2] had implicitly assumed that this initial barrier to hydride formation would be smaller than the scatter in their solvus temperature data. Puls [28] also provided a derivation for the effect of stress on the solvus based on an adaptation of the Moutier cycle approach used by Li et al. [19] to the Zr–H system under a uniform external stress. From the results of this derivation it could be concluded that such a stress imposed on a system having a constant average hydrogen composition would not result in any discernable effect on the solvus if the partial molar volumes of hydride (per mole of H) and of hydrogen in solution had similar magnitudes. The data available at the time, however, were that for Zr–H systems the molar volume of hydrogen in solution was half that of the hydride molar volume (per mole H). This led Puls [28] to conclude that an important contributor to preferential hydride formation at a crack tip would be a reduction in the solvus concentration directly associated with an elevated tensile stress at that location. Later results, as given in Chaps. 2 and 4, showed, however, that there is little difference in the partial molar volumes of hydrogen in the two phases and hence the effect of uniform external stress on the solvus is expected to be negligible. A revision of the initial analysis of the hysteresis in the solvus data in Zr–H systems was given by Puls [29] after the author became aware of the results of analytical elastic–plastic accommodation energy calculations by Lee et al. [16] for

194

6 Characteristics of the Solvus

spherical misfitting precipitates in a solid having isotropic-elastic, perfectly-plastic constitutive behavior. Applying the results to those Zr–H systems for which solvus data had been obtained, it became evident that for yield strength values applicable to these materials the combined elastic strain energy and plastic work term contributions to the total accommodation energy were always smaller than the corresponding pure elastic strain energy value, D wel inc : As an example, for a material with a yield strength of 500 MPa and an isotropic transformation (misfit) strain of 0.057 (for d hydride, assuming the volume misfit is isotropic), Eq. 6.26 with the left hand side divided by D wel inc becomes  e þ DH  pa!d DH a!d  0:3 D wel inc

ð6:27Þ

as compared to unity assumed by Birnbaum et al. [2] and Puls [28]. In addition,  pa!d  2DH  ea!d for this case while previously it had been assumed that DH e  a!d [ DH  pa!d : Lower yield strength values would decrease the result given by DH Eq. 6.27 and increase the difference between the plastic and elastic enthalpy values. Using a particle-size dependent model developed by Ashby and Johnson [1], Lee et al. [16] estimated that the foregoing elastic–plastic result should apply to precipitates having dimensions of the order of 1 lm and larger, which means that hydrides with dimensions just above those of critical nuclei, would not be able to reduce their strain energies by plastic deformation until they had grown to somewhat larger sizes. Based on these predictions and the elastic–plastic accommodation results obtained from the analytic model of Lee et al. [16], Puls [29–31]) proposed a revised interpretation of solvus hysteresis for metal-hydrogen systems. In the revised model, Puls [29–31] proposed that there are three reproducible solvus temperatures that could, in principle, be detected in a thermal hysteresis scan, corresponding to three distinct stages of hydride formation and dissolution. The first solvus was called the nucleation solvus because it was thought to represent the hydride nucleation stage of hydride formation. Experimentally, it is the solvus having the lowest reproducible formation temperature during cool down. An example would be the cool down solvus obtained by Birnbaum et al. [2] during the first cycle for Nb-H and the solvus boundary identified as TSSP1 by Pan et al. [24] for Zr–H (see Fig. 6.2). Because of the small precipitate sizes during this stage, the hydrides are expected to exist as fully coherent precipitates with the hydride-matrix accommodation energy not reduced by any plastic deformation. For this reason this solvus was also referred to as the coherent solvus. The additional drop in temperature beyond the coherent phase boundary needed for hydride nucleation was assumed to be within the scatter of the data. A second reproducible solvus, detected under cool down conditions after hydrides of substantial size had already formed in the material, would result in solvus temperatures higher than those for hydride nucleation. It was assumed, in this case, that a substantial amount of the pure elastic accommodation energy governing the nucleation solvus would be reduced since the hydride’s misfit strains for these larger hydrides could be accommodated locally by plastic deformation. This solvus was called the hydride

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195

growth solvus and corresponds to the experimentally determined solvus identified by Pan et al. [24] as TSSP2 for Zr–H. The third solvus, detected during heat up or in a diffusion couple approach, was assumed to require the production of the same amount of plastic work during hydride dissolution as was expended during hydride growth. This solvus, referred to as TSSD by Pan et al. [24] for Zr–H, was significantly less affected by cycle numbers and conditions compared to the other two solvi. On the basis of the foregoing interpretations, none of the experimentally determined solvi could be considered to be equilibrium (isothermally reversible) solvi, although numerically, from theoretical considerations on the basis of the accommodation energy model of hysteresis (see further on) the dissolution solvus might be a close approximation of the latter, particularly for a soft material. Both experimental observations and theoretical considerations have shown that cool down (hydride formation) solvi are significantly affected by the prior thermomechanical history of the material while this is not observed for the hydride dissolution solvus. Because of this dependence of the formation solvi on prior thermo-mechanical history of the material and on the experimental conditions, there is uncertainty in knowing what stage of hydride formation is reflected in a particular set of experimental results. The solvi temperatures for nucleation and growth defined in the foregoing are, therefore, considered as providing lower and upper bound solvi temperatures, respectively, within which an experimentally determined hydride formation solvus might fall under cool down (or hydride formation) conditions. Theoretical expressions for the foregoing solvi were derived by Puls [28–30] using an approach developed by Li et al. [19]. These theoretical solvi relations were formulated with reference to the equilibrium solvus. As noted in Sect. 6.2, the equilibrium solvus is not an experimentally determinable quantity, although suitable choices for this state can, nevertheless, be thermodynamically defined. As a result, however, such theoretical solvi relations cannot be unambiguously specified although the differences in their experimental values can be. One of the methods used by the latter authors to derive relationships for coherent equilibrium was the Moutier cycle approach. This cycle was based on Moutier’s theorem, which says that the work done in proceeding isothermally and reversibly from one thermodynamic state to another is independent of the path. Thus by choosing two different paths, one of which involves the transfer of hydrogen from the hydride to the solution phase, the other of which involves returning the system to its original state by transferring the same amount of hydrogen from the solution phase back to the hydride, the sum of the work done over each path must equal zero. This approach is, however, strictly applicable only to coherent phase equilibrium in which the full cycle can be carried out reversibly and there is no change in the coherency energy of the two phases during the phase transformation. Experimentally, it is evident that a thermal cycle scan—i.e., a scan that starts and ends with the system in the same state—cannot be done isothermally since, for the same hydrogen content, the solvus temperatures for hydride formation and dissolution are different. Nevertheless, Puls [28, 29] used the Moutier cycle approach in deriving theoretical expressions for the solvus in hydride forming materials by modifying

196

6 Characteristics of the Solvus

Fig. 6.8 A schematic thermal hysteresis cycle for a martensitic transition. The same schematic is also used to illustrate a thermal hysteresis cycle for zirconium-hydride formation and dissolution (from Flanagan et al. [11]). Solid lines with arrows going in only one direction represent irreversible steps while the dashed line with arrows going in both directions represents the reversible steps

the cycle to account for irreversible energy losses arising from plastic deformation when hydrides are formed and dissolved. In the following, we present an updated version of this derivation based on the hysteresis scan approaches summarized in Sect. 6.2. Furthermore, improvements to the theory that have evolved out of taking account of phase relationships derived from the thermodynamic theory of coherent equilibrium are then detailed in Chap. 7 while the applicability of all of these theories to solvus data in Zr–H systems is addressed in Chap. 8. Consider a system under external stress, ra. Only the concentration of the hydrogen in the binary (or pseudo-binary) hydrogen-metal system needs to be considered, since it can be reasonably assumed that diffusion of the metal atoms can be ignored over the temperature range of practical interest up to 400 °C. Hence, the metal lattice provides the reference lattice frame from which changes in stresses and strains can be calculated. It is further useful to define the hydrogen concentration variable in such a way that its terminal concentration for the hydride phase is equal to one, similar to the bounding mole or atom fraction concentrations in a binary substitutional solid. We then assume a hysteresis cycle, shown schematically in Fig. 6.8, similar to that given in Sect. 6.2, used by Salzbrenner and Cohen [34] to interpret the hysteresis generated during martensitic transformation in a single crystal in which a single interface travels across the crystal during the phase transformation. We have modified the hysteresis cycle in Fig. 6.8 in two ways. First, the cycle is compressed over the path from Steps 1 to 2 by assuming that it ranges over only a small incremental amount of conversion of hydrogen in solution to hydride. Second, we assume that—as was done in the earlier Moutier cycle derivations—all free energy losses would be as a result of irreversible plastic work contributions to the accommodation energy. A key assumption of the

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197

following derivation based on accommodation energy is that any (elastic) coherency energy that remains stored in the solid after completion of the phase transformation is available to aid the transformation upon phase reversal. Regarding the first point, rather than going directly from Point 1 to 2 we follow the sequence that takes us through the equilibrium state in Steps 5 to 6 by going from Point 1 to 5 (Step 1), Point 5 to 6 (Step 2) and arriving at Point 2 from Point 6 (Step 3). We start at Point 1 with a system consisting of zb phase fractions of b hydride that had been formed at temperature, Tf2, from the a phase, containing a dilute concentration of hydrogen in solution. (The reason for labeling this temperature as ‘f2’ is explained further on.) Now the following steps are carried out in going from Points 1 to 2 which involves increasing the zb phase fractions of b hydride to zb ? dzb phase fractions: Step 1:

The temperature is increased from Tf2 to Teq assuming that any in energy associated with the specific heat of the system neglected. Since there is a difference in composition between temperatures, the work, W1 ; in this step is:  b a W1 ¼ lH s;f 2 ðTf 2 ; ra Þ  ls;eq H ðTeq ; r Þ z

changes can be the two ð6:28Þ

a where ls;eq H ðTeq ; r Þ is the equilibrium chemical potential and 2 a ls;f H ðTf 2 ; r Þ is the chemical potential for hydride formation.

Step 2a: At temperature Teq, the zb phase fraction of hydride is excised and removed from the externally stressed solid, while maintaining the external stresses on the remaining solid solution (a) phase. The work required, W2a, is the reduction in the residual, plastically relaxed coherency energy in matrix, and hydride and increase in external work (when external stresses are tensile and misfit strains positive) as a result of the removal of the hydride phase from the external stresses. int W2a ¼ Dwee inc  Dwinc

Dwee inc Dwint inc

ð6:29Þ

= elastic hydride-matrix coherency energy difference1 during hydride formation (expansion) for the case where some of the misfit strain has been plastically relaxed = interaction energy of hydride arising from application of external stress

Only the interaction energy associated with the external stress on the hydride is removed here because the external stress is still acting on hydrogen in solution, although the matrix no longer has the coherency stresses of the hydride acting on it.

1

The coherency energy change is defined as the difference in energy between the constrained and stress-free states and is, therefore, always a positive quantity whist the work done is defined as the difference in energy between the final and initial states and can, therefore be either positive or negative.

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6 Characteristics of the Solvus

Step 2b: The external stresses are reduced to zero. The work required, W2b, is given by: W2b ¼ WL  WE WL WE

ð6:30Þ

= energy of loading system = elastic energy of system

In addition, since there is no change in H concentration in the solution phase when a the external stress is removed, the chemical potential ls;eq H ðTeq ; r Þ becomes s;eq s;eq a H relative to its externally lH ðTeq ; r ¼ 0Þ  lH ðTeq Þ since it is increased by pV s;eq a  stressed state where it is given by ls;eq ðT ; r Þ ¼ l eq H H ðTeq Þ  ph VH : The stress, ph ¼ raii =3 is the hydrostatic (mean) component of the external stresses, raij : The a chemical potential reverts back to ls;eq H ðTeq ; r Þ in Step 2d and, therefore, has not been included in the present and later steps, since the work done as a result of this change cancels. Step 2c: At Teq the excised hydride phase is increase by dzb phase fraction to a total phase fraction of zb ? dzb of hydride phase. The work, W2c, required to do this is zero since the transfer occurs when the system is in equilibrium and at zero external stress: W2c ¼ 0

ð6:31Þ

s;eq a  The chemical potential remains at ls;eq H ðTeq Þ ¼ lH ðTeq ; r Þ þ ph VH

Step 2d: The external stress is replaced. The work for this step is: W2d ¼ WL þ WE

ð6:32Þ

Step 2e: (zb ? dzb) phase fraction of hydride is put back into the previously relaxed cavity created when zb phase fraction of the hydride phase was cut out (excised) in Step 2a. The work, W2e ; is: int W2e ¼ Dwee inc þ Dwinc

þ Dwpe inc ¼

oDwee oDwpe oDwint b b inc inc inc  dz þ  dz þ  dzb ozb ozb ozb

ð6:33Þ

plastic work done during hydride formation (expansion) stage.

The first two terms in Eq. 6.33 refer to the elastic work in replacing the zb phase fraction of hydride excised in Step 2a and the second three terms refer to the elastic and plastic work plus the interaction energy for the additional dzb phase fraction that is added in this step. Using the chain rule for differentiation with the mass balance equation:

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199

  zb cbH þ 1  zb cbH ¼ ^cH

ð6:34Þ

we obtain: int W2e ¼Dwee inc þDwinc

þ

b b oDwee oDwpe inc  dz inc  dz þ o^cH cb  ca o^cH cb  ca H

oDwint inc

H

H

H

b

dz  b cH  caH  ee   inc þ Dw  pe  int Dw inc inc þ Dw ee int  Dwinc þDwinc þ dzb cbH  caH þ

ð6:35Þ

o^cH

The partial molar interaction energy, D wint inc ; is given by: D wint inc ¼ 

Zr a T V  rij  eij rH

ð6:36Þ

where the bar over an energy or volume change means that it refers to a partial molar energy or volume. The concentration variables are defined by: caH ¼ NHa =N a ; cbH ¼ NHb =N b ; zb ¼ NHb =N; za ¼ NHa =N; NHa and NHb the total numbers of hydrogen atoms in the a and b phases, respectively, N a and N b the total numbers of equivalent interstitial lattice sites in the a and b phases, respectively, and N ¼ N a þ N b the total number of equivalent interstitial sites in the solid, which for this case of an interstitial solution, is the total number of interstitial sites that could, for a given phase, be occupied by hydrogen atoms. These concentrations are identical to those expressed previously in Chap. 4 by the variable, h  rH =bpha ; where bpha in this case stands for the total number of equivalent interstitial sites per metal atom sites in the phase (‘‘pha’’) considered and rH is, as used elsewhere in this text, the ratio of interstitial (hydrogen) to metal atoms. It is important to note that, with these composition variables, the molar volume for zirconium (which is also required implicitly in the accommodation energy calculations) must be divided by rH, as shown in Eq. 6.36 for these energies to properly correspond to partial molar energies per mole of hydrogen. Step 3: The temperature of the solid consisting of (zb ? dzb) phase fraction of hydride is decreased from Teq to Tf2. This results in chemical work, W3 ; as follows:      2 a b b ð6:37Þ Teq ; ra  ls;f W3 ¼ ls;eq H H ðTf 2 ; r Þ z þ dz

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6 Characteristics of the Solvus

The chemical work arises because the solid has a solvus hydrogen concentration and temperature corresponding to the chemical potential 2 a ls;f H ðTf 2 ; r Þ whereas the chemical potential for the reversible transfer a between the solution and hydride phases is given by ls;eq H ðTeq ; r Þ ¼ s;eq s;eq H corresponding to a solvus concentration, cH ; and sollH ðTeq Þ þ ph V vus temperature, Teq. Summing the work contributions given by Steps 1 to 3 to zero yields (after canceling identical terms of opposite sign and dividing the remaining terms by dn):    ee  pe  int Dw w 2 a incþDw inc þD  inc ¼ 0 Teq  ls;f ls;eq H H ðTf 2 ; r Þ þ ph VH þ cbH  caH

ð6:38Þ

Transposing terms and changing signs yields:   ee  pe  int Dw w s;eq  2 a incþDw inc inc þD  þ p V ðT ; r Þ ¼ l T þ ls;f f 2 eq h H H H cbH  caH

ð6:39Þ

Assuming that the hydrogen composition, cbH ; of the hydride phase is fixed, the chemical potential expressions in Eq. 6.39 are required only for the dilute hydrogen solution side of the phase diagram. These can be approximated by: 2 s;f 2 ls;f H ðTf 2 Þ ffi RTf 2 ln cH

ð6:40Þ

  ls;eq Teq ffi RTeq ln cs;eq H H

ð6:41Þ

and

s;eq 2 Note that the chemical potentials for the concentrations cs;f H and cH are at two different temperatures. This represents one of the differences between the earlier Moutier cycle derivations by Puls [28, 29, 32] and the present hysteresis cycle derivation. Flanagan et al. [11] point out that, for the difference in these chemical potentials to be consistent with the temperature at which the work loss term arising from entropy production is determined, an average temperature, Tav % Teq should be used for Tf2. Making this substitution in Eq. 6.39, then, the concentration of the partially coherent, externally stressed hydride growth solvus in the dilute phase, 2 cs;f H ; is given in terms of the solvus concentration in the incoherent, externally unstressed dilute phase, cs;eq H by: " #

 H  pe  int D wee ph V s;eq s;f 2 inc þDw inc inc þDw cH ffi cH exp exp ð6:42Þ RTeq cbH RTeq

 where cbH  caH has been replaced by cbH since caH  cbH :

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201

The second difference is that the hydrogen to metal ratio, rH, has been replaced by cbH ¼ rH =bpha ; where bpha refers to the total number of equivalent interstitial sites per metal atom in the phase. As discussed further in Chap. 8, this factor could range from bhyd = 1 to 1.5 to 2 if the hydride formed is, respectively, the c, d or e phase. This correction accounts for the fact that the number of interstitial sites has a fixed relationship to the number of normal metal lattice sites and is, therefore, not independent of the latter as assumed in the previous derivations. This factor, rH, has not disappeared, however, as noted in the discussion after Eq. 6.36, since it must divide the molar volume of zirconium when given per mole Zr to account properly for the number of hydrogen atoms per zirconium atoms in the hydride phase. The foregoing result is for the case of hydride formation. Implicit in this derivation is that hydrides of phase fraction zb are present when the temperature, Tf2, for further precipitation of the hydride phase is reached from above and that these hydrides have their misfit-induced elastic accommodation energies partially reduced by plastic relaxation. Because of this, it had been assumed [24, 29] that further increases in the phase fraction of these hydrides would result in similar partial molar changes in elastic and plastic accommodation energies as had led to the reduced elastic accommodation energies of the hydrides already present. In total, three solvus relationships were derived by Puls [29–31] based on the foregoing type of analysis. The first of these is given by Eq. 6.42. If hydride formation occurs with no prior hydrides present (zb = 0), then it had been assumed that the experimentally determined solvus [24, 29] likely reflects the early post-nucleation stage of hydride formation. Thus, in this case it was assumed by Puls [29, 30] that all of the accommodation energy would be elastic since the hydrides would be too small for plastic relaxation to be possible. In this case Eq. 6.42 becomes:

el 

 Dwinc þ Dwint Ph V H s;eq s;f 1 inc cH ffi cH exp exp ð6:43Þ RTeq cdH RTeq  el Dw inc = pure elastic partial molar hydride-matrix coherency energy change during hydride formation (no reduction due to plastic deformation) For this solvus, the shift in composition or temperature from its equilibrium value is governed by the pure elastic accommodation energy. There would, consequently, be no hysteresis for the full loop if hydride dissolution were to occur prior to the hydrides growing to a size where plastic relaxation would be able to reduce the elastic accommodation energy. The corresponding experimentally determined solvus was previously referred to as TSSP1 by Pan et al. [24], which is why the formation composition in Eq. 6.43, has been labeled ‘‘f1.’’ This distinguishes it from the one given by Eq. 6.42, labeled ‘‘f2’’ indicating that it is the theoretical equivalent of the experimental solvus referred to as TSSP2 by Pan et al. [24]. Finally, a theoretical expression for the hydride dissolution solvus can be obtained by carrying out the hysteresis cycle steps over the dissolution side of the full cycle. In the dissolution half of the cycle, the starting point, Step 1, is assumed to be just above

202

6 Characteristics of the Solvus

Point 4 where the amount of hydride phase is given by zb and the starting temperature is at Td which is then decreased to Teq. In Step 2, with all the coherency stresses removed by excising the hydride phase, the hydride phase fraction is then decreased from zb to zb - dzb. This differs from the formation cycle in which the hydride phase fraction was increased from zb to zb ? dzb. The result of this hysteresis cycle is: " #

 H  int D wes wps ph V s;eq s;d inc  D inc inc þ Dw cH ffi cH exp ð6:44Þ exp RTeq cbH RTeq  es Dw inc ¼  ps Dw inc ¼

elastic partial molar hydride-matrix accommodation energy change during hydride dissolution (shrinkage) partial molar plastic work done during hydride dissolution (shrinkage)

In this thermodynamic model, it is assumed that a small phase fraction, dzb, of pre-existing, partially relaxed hydride is converted to the a-phase state. However, in the earliest finite element calculations to simulate this process [17], for simplicity, an entire hydride precipitate was transformed to the a phase. In later calculations [18, 27] this approach was refined slightly by starting from the prior plastically relaxed state of the hydride when calculating the energy change of the hydride for converting it back to the a phase. It was found that the complete conversion of the entire plastically relaxed hydride to the a phase resulted in the latter phase being in a state of tension over the region where the hydride had been. The residual elastic strain energy associated with this prior-hydride a-phase state was, nevertheless, less than the initial elastic strain energy of the hydride and, therefore, the net elastic energy change,D wes inc turned out to be numerically positive for the cases considered. It should be noted that, strictly speaking, the energy changes for hydride formation (expansion) should also have been started from this prior hydride dissolution stage to take account of the changes in the a phase after the first cycle of hydride formation and dissolution had occurred. In retrospect, and in view of the results and interpretation of hysteresis provided by Flanagan et al. [11], one can identify a number of difficulties with the accommodation energy model in being able to provide a universal explanation for hysteresis for the entire range of experimental conditions, measurement techniques and systems in which it has been observed. One of the difficulties with the model is that it requires making a choice concerning hydride shape and transformation (stress-free misfit) strains that are conjectural because they may not be experimentally available. Thus, for the solvus relationships given by Eqs. 6.42–6.44, the magnitude of thermal hysteresis is largely determined by the magnitude of the pure elastic accommodation energy of the hydride. However, calculations for Zr–H show that for plate-shaped precipitates this energy would be very small if–consistent with the observation of these precipitates lying on basal planes (see Chap. 3)–all of the volumetric transformation strains were oriented in the plate’s normal direction. A much larger value, in close agreement with the observed thermal hysteresis, is obtained assuming that the

6.3 Theories of Solvus Hysteresis based on Accommodation Energy

203

transformation strains are largely volumetric, given by the pure lattice strain values calculated by Carpenter [3]. This was the main justification used by Puls [29] for favoring the latter transformation strains as being the appropriate ones to use in accommodation energy calculations. Another difficulty with the accommodation energy model was pointed out by Flanagan et al. [11]. These authors noted that, because of the generally observed constancy of the hysteresis for average compositions ranging over the two-phase state in many systems exhibiting hysteresis, the composition dependence of the derivative of the partial accommodation energy with respect to an increment of phase conversion must be constant over this composition range, which means that the total accommodation energy must be a linear function of phase conversion while also having a different slope between hydride formation and dissolution. Such behavior is not obtained with any of the accommodation energy models described here or elsewhere in the literature (see Flanagan et al. [11] for a list of these). In Chap. 7, dealing with theories of coherent phase equilibrium, it is shown that the total coherency energy of a coherent two-phase mixture is more correctly given by the sum of the self strain energy of each precipitate combined with the nonconfigurational interaction energy of each precipitate with the average dilatational field produced by all the other misfitting precipitates in the finite solid. This latter contribution to the total strain energy of the solid was neglected in the accommodation energy model. Its inclusion fundamentally alters the nature of the energy barrier that needs to be overcome during phase transformation, resulting in a new source of energy dissipation (hysteresis) connected with it. The energy dissipation arises because the component of the total coherency energy of the solid derived from the interactions of the precipitates with the dilatational field of all of the other precipitates in the solid is of macroscopic extent. Such a macroscopic barrier cannot be solely overcome by the small incremental changes produced isothermally by thermal fluctuations. Hence, a finite driving force–sufficient to overcome such a macroscopic energy barrier–needs to be applied before an increment of phase transformation can occur in any direction. Since this increment of phase boundary movement is of finite extent, it ends up being dissipated through the entropy production that is generated by the finite movement of the phase boundary. This entropy production manifests itself in the hysteresis in some independent external parameter such as the phase transformation temperature at constant average composition, or vice versa, the phase transformation composition at constant temperature at which the phase transformation proceeds in the two directions.

References 1. Ashby, M.F., Johnson, L.: On the generation of dislocations at misfitting particles in a ductile matrix. Philos. Mag. 20, 1009–1022 (1969) 2. Birnbaum, H.K., Grossbeck, M.L., Amano, M.: Hydride precipitation in Nb and some properties of NbH. J. Less-Common Met. 49, 357–370 (1976)

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3. Carpenter, G.J.C.: The dilatational misfit of zirconium hydrides precipitated in zirconium. J. Nucl. Mat. 48, 264–266 (1973) 4. Dantzer, D., Luo, W., Flanagan, T.D., et al.: Calorimetrically measured enthalpies for the reaction of H2 (g) with Zr and Zr alloys. Metall. Trans. A 24A, 1471–1479 (1993) 5. Erickson, W.H., Hardy, D.: The influence of alloying elements on the terminal solubility of hydrogen in a-zirconium. J. Nucl. Mater. 13, 254–262 (1964) 6. Eshelby, J.S.: Continuum theory of defects. Prog. Solid State Phys. 3, 79–144 (1956) 7. Flanagan, T.B., Mason, N.B., Birnbaum, H.K.: The effect of stress on hydride precipitation. Scripta Metall. 15, 109–112 (1981) 8. Flanagan, T.B., Clewley, T., Kuji, T. et al.: Isobaric and isothermal hysteresis in metal hydrides and oxides. J. Chem. Soc., Faraday Trans. I. 82, 2589–2604 (1986) 9. Flanagan, T.B., Kuji, T.: Hysteresis in metal hydrides evaluated from data at constant hydrogen content: Application to palladium-hydrogen. J. Less-Common Met. 152, 213–226 (1989) 10. Flanagan, T.B., Luo, W., Clewley, J.D.: Calorimetric enthalpies of absorption and desorption of protium and deuterium by palladium. J. Less-Common Met. 172–174, 42–55 (1991) 11. Flanagan, T.B., Park, C.-N., Oates, W.A.: Hysteresis in solid state reactions. Prog. Solid St. Chem. 23, 291–363 (1995) 12. Frieske, H., Wicke, E.: Magnetic susceptibility and equilibrium diagram of PdHn. Ber. Bunsenges. physik. Chem. 77, 48–52 (1973) 13. Grossbeck, M.L., Birnbaum, H.K.: Low temperature hydrogen embrittlement of niobium II: Microscopic observations. Acta Metall. 25, 135–147 (1977) 14. Kearns, J.J.: Terminal solubility and partitioning of hydrogen in the alpha phase of zirconium, Zircaloy-2 and Zircaloy-4. J. Nucl. Mater. 22, 292–303 (1967) 15. Kuji, T., Oates, W.A.: Thermodynamic properties of Nb-H alloys II: The b and d phases. J. Less-Common Met. 102, 261–271 (1984) 16. Lee, J.K., Earmme, Y.Y., Aaronson, H.I., et al.: Plastic relaxation of the transformation strain energy of a misfitting spherical precipitate: Ideal plastic behavior. Metall. Trans. A 11A, 1837–1847 (1980) 17. Leitch, B.W., Puls, M.P.: Finite element calculations of the accommodation energy of a misfitting precipitate in an elastic-plastic matrix. Metall. Trans. A 23A, 797–806 (1992) 18. Leitch, B.W., Shi, S.-Q.: Accommodation energy of formation and dissolution for a misfitting precipitate in an elastic-plastic matrix. Modelling Simul. Mater. Sci. Eng. 4, 281–292 (1996) 19. Li, J.C.M., Oriani, R.A., Darken, L.S.: The thermodynamics of stressed solids. Z. Physik. Chem. Neue Folge 49, 271–290 (1966) 20. Libowitz, G.G., Gibb, T.: High pressure dissociation studies of the uranium-hydrogen system. J. Phys. Chem. 61, 369–381 (1957) 21. Lufrano, J., Sofronis, P., Birnbaum, H.K.: Elastoplastically accommodated hydride formation and embrittlement. J. Mech. Phys. Solids 46, 1497–1520 (1998) 22. Lufrano, J., Sofronis, P.: Micromechanics of hydride formation and cracking in zirconium alloys. Comput. Modell. Eng. Sci. (CMES) 1, 119–131 (2000) 23. Oates, W.A., Flanagan, T.B.: The solubility of hydrogen in transition metals and their alloys. Prog. Solid St. Chem. 13, 193–283 (1981) 24. Pan, Z.L., Ritchie, I.G., Puls, M.P.: The terminal solid solubility of hydrogen and deuterium in Zr-2.5Nb alloys. J. Nucl. Mater. 228, 227–237 (1996) 25. Papaconstantopolis, K., Wenzl, H.: Pressure-composition isotherms of hydrogen and deuterium in vanadium films measured with a vibrating quartz microbalance. J. Phys. F. 12, 341–360 (1982) 26. Paton, N.E., Hickman, B.S., Leslie, D.H.: Behavior of hydrogen in a-phase Ti-Al alloys. Metall. Trans. 2, 2791–2796 (1971) 27. Puls, M.P., Leitch, B.W., Shi, S.Q.: The effect of applied stress on the accommodation energy and solvi for the formation and dissolution of zirconium hydride. In: Moody, N.F., Thompson, A.W., Richer, R.E. et al. (eds.) Hydrogen Effects on Material Behaviour and Corrosion

References

28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43.

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Deformation Interactions, pp. 233–248. TMS (The Minerals, Metals and Materials Society) (2003) Puls, M.P.: The effects of misfit and external stresses on terminal solid solubility in hydrideforming metals. Acta Metall. 29, 1961–1968 (1981) Puls, M.P.: Elastic and plastic accommodation effects on metal-hydride solubility. Acta Metall. 32, 1259–1269 (1984) Puls, M.P.: On the consequences of hydrogen supersaturation effects in Zr alloys to hydrogen ingress and delayed hydride cracking. J. Nucl. Mater. 165, 128–141 (1989) Puls, M.P.: Effects of crack tip stress states and hydride-matrix interaction stresses on delayed hydride cracking. Metall. Trans. A 21A, 2905–2917 (1990) Puls, M.P.: Review of the thermodynamic basis for models of delayed hydride cracking rate in zirconium alloys. J. Nucl. Mater. 393, 350–367 (2009) Prigogene, I, Defay, R.: Chemical Thermodynamics. Longmans, Green and Co., London, UK (1954) Salzbrenner, R., Cohen, M.: On the thermodynamics of thermoelastic martensitic transformations. Acta Metall. 27, 739–748 (1979) Salomons, E., Feenstra, R., Degroot, D., et al.: Pressure-composition isotherms of thin PdHc films. J. Less-Common Met. 130, 415–420 (1987) Schultus, N., Hall, W.: Hysteresis in the palladium-hydrogen system. J. Chem. Phys. 39, 868–870 (1963) Slattery, G.F.: The terminal solubility of hydrogen in zirconium alloys between 30°C and 400°C. J. Inst. Metals 95, 43–47 (1967) Torra, V., Tachoire, H.: Martensitic transformations in shape-memory alloys. Successes and failures of thermal analysis and calorimetry. Thermochim. Acta 203, 419–444 (1992) Westlake, D.G.: A generalized model for hydrogen embrittlement. Trans. ASM 62, 1000–1006 (1969) Westlake, D.G.: Mechanical behavior of niobium (columbium)—hydrogen alloys. Trans. TMS-AIME 245, 287–292 (1969) Wicke, E., Otto, K.: Über das System Uran-Wasserstoff und die Kinetik der Uranhydrid bildung. Z. Physik. Chem. Neue Folge 31, 222–248 (1962) Wiswall, R.H.: Hydrogen storage in metals. In: Alefeld, G., Völkl, J. (eds.) Hydrogen in Metals I. Application-Oriented Properties, Springer, Berlin (1978) Zuzek, E., Abriata, J.P., San-Martin, A. et al.: H-Zr (Hydrogen-Zirconium) In: Phase Diagrams of Binary Hydrogen Alloys, pp. 309–322. ASM International, Materials Park, Ohio (2000)

Chapter 7

Theories of Coherent Phase Equilibrium

7.1 General Features As pointed out in Chap. 6, the theoretical models for the solvus derived in that chapter, based on accommodation energy concepts, turn out to be incapable of providing an explanation for hysteresis for the entire range of experimental conditions, experimental methods, and systems in which such hysteresis has been observed. Moreover, application of the accommodation energy models was narrowly focused on the effect of hysteresis to the solvus relations in closed systems at low volume fraction of the hydride phase. The models did not specifically address whether and how the corresponding compositions on the hydride side of the phase diagram would be affected. The critical examination of these types of models carried out in Chap. 6, therefore, led to the conclusion that there was a need to improve upon them. It turns out that in parallel with the development of accommodation energy models for the solvus in hydride forming metals, a series of comprehensive studies were carried out independently by others to develop a general theoretical framework of cohesively stressed solids with particular application to the thermodynamic equilibrium conditions applicable in such solids. Specific simplified models for the determination of thermodynamic relationships for coherent equilibrium in binary substitutional solids were derived. These were based on models for the total coherency energy between the misfitting phases in substitutional binary solids taking account of the additional strain energy produced by indirect (non-configurational) interactions between the misfitting phases in a finite solid. Two bounding cases of the application of these coherency energy models are reviewed in this chapter with a view toward determining their overall viability in providing a universal description of hysteresis in open and closed metal-hydrogen systems in which hydrides are formed. Application to the Zr–H system is addressed in the following chapter (Chap. 8). We commence by noting, as previously shown in Chap. 4, that the addition of misfitting defects such as interstitial hydrogen atoms into a crystalline material M. P. Puls, The Effect of Hydrogen and Hydrides on the Integrity of Zirconium Alloy Components, Engineering Materials, DOI: 10.1007/978-1-4471-4195-2_7, Ó Springer-Verlag London 2012

207

208

7 Theories of Coherent Phase Equilibrium

results in an increase in the energy of mixing of the material that, in a continuum linear-elastic model of the solid, is made up of at least three linearly additive contributions. The first contribution comes from the defect’s self strain energy that is produced by its transformation strain. This was actually the only energy that was considered in the accommodation energy models. The second contribution is a configuration-dependent work, or interaction energy term that occurs when one defect is moved in close proximity to the other. This configuration-dependent interaction energy arises from the strain field produced by one defect interacting with the stress field of the other (or vice versa), the result of which is that a certain amount of (reversible) work is produced when their positions are changed. This work is configuration-dependent because it depends sensitively on the relative locations of the defects. The resulting interaction energy, however, generally drops off rapidly with distance between the defects and can be assumed to be negligible for dilute concentrations of defects because the average separation between them would be large. The third contribution to the energy of mixing is an indirect interaction energy—called an image or non-configurational energy—that occurs in solids of finite size and is the result of an approximately uniform dilatation produced throughout the solid by the dilatational component of each misfitting defect. Like the second term, it also acts like a (reversible) work term that is produced when there is a transfer of defects (misfitting atoms or precipitates) to or from the solid or between phases. It differs from the other strain or interaction energy terms by being independent of the distance between the defects, depending only on the number of defects already present and on their transformation strains. Hence, it is independent of the specific distribution of these defects in the solid. For this latter reason it is called a non-configurational energy. This non-configurational energy is also applicable when the sole source of misfitting defects is a distribution of coherently misfitting precipitates. For the simple linear-elastic case dealt with in Chap. 4, the total coherency energy arising from the contributions of the foregoing three terms (assuming zero configuration-dependent interaction energy) is proportional to the square of the average defect concentration. As a result, the partial molar coherency energy derived from this depends linearly on the number of defects (misfitting atoms) in the solid. Because it results from the average dilatational strains produced by all the defects in the finite solid, it constitutes a macroscopic contribution to the total coherency energy. It is shown in this chapter that these physical characteristics of the total non-configurational coherency energy result in thermodynamic relationships for coherent mixtures of misfitting phases that are fundamentally different and novel compared to those derived on the basis of the non-interacting strain (accommodation) energy models considered in Chap. 6. The present chapter consists of a general treatment of the possible effects that these concentration dependent coherency energy contributions have on phase relationships under conditions of average hydrogen content and temperatures for which the volume fraction of the hydride particles becomes important. It is shown that, as a result of the non-linear dependence of the total coherency energy on the defect concentration in the solid, the equilibrium phase relationships for such

7.1 General Features

209

coherently (non-hydrostatically) stressed solids deviate significantly from those described by classical, Gibbsian thermodynamics such as liquid and hydrostatically stressed (incoherent) solid binary mixtures. Beginning in the 1970s, a series of comprehensive studies were carried out by Larché and Cahn [14–18] to determine the general conditions for thermo-chemical equilibrium and diffusion in multi-component crystalline solids self-stressed by coherency strains. The important contribution of these authors was the development of a rigorous thermodynamic framework dealing with the linkage between composition and stress in coherently stressed solids. However, the very generality of this treatment, involving lengthy and complex relationships and requiring the introduction of many new physical parameters, seems to have obscured recognition of some unusual and potentially important physical characteristics of phase equilbria in such solids. These features were recognized in two important papers by Williams [26, 27], who based his thermodynamic analysis of the effect of coherency strains on a simpler, more intuitive approach. Williams’ idea was that the total Gibbs free energy of a thermodynamic system stressed by coherency strains can simply be expressed by the weighted sum of the total chemical (incoherent) Gibbs free energies of the coexisting phases in the mixture plus an additional term that accounts for the total coherency energy of the solid produced by the misfitting precipitates. The condition that the stresses generated by the coherency strains must sum to zero for a finite solid in static equilibrium produces an expression for the total coherency energy of the system dependent on the product of the phase fractions of the two phases. The dependence of the total coherency energy on the product of the phase fractions of the two phases results in phase equilibrium compositions being affected by the average composition or the phase fractions of the phases present (in a closed system these two parameters are linked), in contrast to incoherent equilibrium where this is not the case. The reason for this effect is that the coexisting coherently misfitting phases are each under an average stress that is opposite to that of the other phase. In a finite solid under static equilibrium these stresses must balance in order for the overall average internal stress in the solid to equal zero. Hence, the overall composition of the solid, which governs the proportion of phases present, now becomes an additional parameter affecting the equilibrium conditions. It is this dependence on phase (volume) fraction of the coexisting phases and, hence, average composition that leads to new equilibrium behavior, radically different from those found in Gibbsian thermodynamics for fluids and incoherently misfitting phases in solids. It is significant that the dependence of the total coherency energy of a mixture of misfitting coherent precipitates in a binary solid on the product of the phase fractions of the two phases is similar to that obtained by Eshelby [6] for the strain energy of a finite solid consisting of a random distribution of misfitting atoms in which Vegard’s law is obeyed (see Chap. 4). Prompted by Williams’ findings [26, 27], Cahn and Larché [4] used their previously developed methodology to derive phase relationships for coherent equilibrium based on a very simple model of a binary substitutional solid in which the only source of misfit stress derives from a difference in lattice parameter

210

7 Theories of Coherent Phase Equilibrium

between the two coexisting phases. In addition, it was assumed that the elastic constants of each phase are independent of composition and, therefore, the same in each phase. In addition, each phase was assumed to be homogeneous in composition and stress—conditions that would obviously restrict the shapes and distributions of second phase particles to which the results of this analysis would strictly apply. More complex conditions were subsequently evaluated by others by including the effect of varying composition on the lattice parameter of each phase and on the elastic constants. In the latter cases it is evident that, for a fully rigorous treatment, one must also include the effects on the phase relationships of the shapes of the coexisting phases and their locations in the solid since the stresses in each phase would, in general, be affected by these factors resulting, moreover, in the possibility of non-uniform distributions of stresses and compositions. Numerous other studies followed these initial ones, each one addressing slightly different aspects of the problem and using different assumptions regarding the physical features of the thermodynamic system [5, 12, 13, 19, 20, 22, 24, 25]. In the following sections (Sects. 7.2 and 7.3), results of coherent phase equilibrium relationships are presented for two types of phase transformations treated extensively in the literature. These two phase transformations are distinguished by whether the crystal structures of the two phases are different (polymorphic) or the same (isomorphic). They could be considered to represent the bounding cases between which the coherent phase relationships of a real system such as Zr–H can be found since the Zr–H system appears to be a mixture of the two (see Chap. 8). In Sect. 7.4, the stability conditions for the extremum solutions presented in Sects. 7.2 and 7.3 for the two types of phase transitions are explored and examples given showing that there could be abrupt breaks (‘‘forbidden’’ regions) in the composition ranges over which such a two-phase mixture is stable. The relevance of the foregoing derivations to the system Zr–H is then explored in Chap. 8 in which key experimental results of solvus relations are also summarized. Although most of the treatments of coherent equilibrium given in the foregoing literature have been carried out specifically for binary substitutional solid solutions, application of these results to metal-hydrogen systems can be carried over straight-forwardly when the appropriate choice of composition unit is made for the latter. Throughout this chapter when the treatment is generic, the a phase is designated as the phase containing a dilute concentration of interstitial hydrogen, while the phase containing a high concentration of hydrogen interstitials (the hydride phase) is referred to as the b phase.

7.2 Polymorphic Phase Transformation In this section a simple example of a binary, polymorphic coherent phase transformation is analyzed. By polymorphic is meant a transformation where the transformed phase has a different crystal structure from that of the parent one. This is also frequently referred to as a structural phase transition. In this simple

7.2 Polymorphic Phase Transformation

211

example, it is further assumed that coherency strains result only from the misfits between the lattice parameters when the two phases form a coherent mixture. This simplifies the thermodynamic treatment since there is no coupling between composition and strain energy. In general, the equilibrium state of a thermodynamic system at constant temperature and external stress and constant overall composition, i.e., a system closed to external sources of atoms, is given by minimization of the Gibbs free energy. In the case considered here, as noted, there is no coupling between the composition of each phase and the coherency stresses generated by the misfit strain produced between the two phases. As a result, the total Gibbs free energy of the coherent state can be given by simply adding the elastic energy change DGel arising from the coherency stresses generated by the lattice parameter misfit between the two phases to the total Gibbs free energies of the corresponding unstressed system. Since, this elastic coherency energy must obviously be zero when either phase fraction equals unity (i.e. when the system is in one of its singlephase states), the simplest dependence of DGel on phase fraction is obtained by assuming that it is the product of the two phase fractions, similar to the dependence on concentration found by Eshelby (6, 7) for point defects. Hence, the coherency energy of the system is assumed to have the following dependence on phase fractions, za and zb:   DGel ¼ za zb W  zb 1  zb W ð7:1Þ where W is the partial molar coherency energy equivalent to D wel inc given in Chap. 6. The second form for the coherency energy in Eq. 7.1 follows from the condition given by Eq. 7.3, further on. It is assumed that the compositions in each phase of the two phase system are homogeneous. To simplify the derivation it is further assumed that the elastic constants are independent of composition and are the same in each phase. As previously noted, these conditions then allow a decoupling between the chemical and mechanical contributions to the free energy, simplifying considerably the derivation. The total Gibbs free energy of the coherent solid then is given by:   Gcoh ¼ za gachem þ zb gbchem þ DGel zb ð7:2Þ where gachem and gbchem are the molar chemical Gibbs free energies of the a and b phases in their single-phase states, respectively. Note that the phase fractions are related by the condition za þ zb ¼ 1

ð7:3Þ

Any changes in surface energy as a result of the phase transformation are neglected in the present treatment. (The neglect of surface energy implies that the relation given by Eq. 7.1 applies to macroscopic equilibrium conditions such as those represented in phase diagrams.)

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7 Theories of Coherent Phase Equilibrium

For a closed system with total hydrogen content (average composition), ^cH , the total number of interstitial atoms is conserved, which means that the following mass (atom) balance equation must hold: zb cbH þ za caH ¼ ^cH

ð7:4Þ

where, as in Chap. 6, the concentration variables are defined by: caH ¼ NHa =N a ; cbH ¼ NHb =N b ; zb ¼ NHb =N; za ¼ NHa =N; ^cH ¼ NH =N with NH the total number of hydrogen atoms in the solid, NHa and NHb the total numbers of hydrogen atoms in the a and b phases, respectively, N a and N b the total numbers of equivalent interstitial lattice sites in the a and b phases, respectively, and N ¼ N a þ N b , which is the total number of equivalent interstitial sites in the solid, which for this case of an interstitial solution, is the total number of interstitial sites that could, for a given phase, be occupied by hydrogen atoms. These concentrations are identical to those expressed previously in Chap. 4 by h  rH =bpha , where bpha , stands, as before, for the total number of equivalent interstitial sites per metal (M) atom sites in phase ‘‘pha’’, and rH is, as used elsewhere in this text, the ratio of interstitial (hydrogen) to metal atoms. With this definition of the hydrogen concentration variable, the concentration of hydrogen in a metal-hydrogen solid solution takes on the same range of values as that of atom B in a substitutional binary solid solution of A and B atoms for which the concentration of each atom in the latter case is expressed in terms of atom or mole fractions. Hence, by replacing the concentration variable cB (or similar such variable) found in the literature for substitutional solid solutions with the hydrogen concentration variable cH, defined as in the foregoing, all of the results for the equilibrium concentrations obtained for binary substitutional solid solutions can be straightforwardly applied to the equilibrium conditions relevant in metal-interstitial (hydrogen) solid solutions. To obtain the state of equilibrium for the coherent Gibbs free energy given by Eq. 7.2, this free energy needs to be minimized with respect to the independent variables ^cH ; caH and cbH ; where the phase fractions are dependent variables through Eqs. 7.3 and 7.4. Liu and Ågren [20] have shown, however, that it is simplest to keep the phase fraction as a variable in the minimization by using the Lagrange multiplier method for the minimization and using Eq. 7.4 as an auxiliary (constraint) condition. Hence, using the Lagrange method, an extra unknown, the Lagrange multiplier that we will call, lH, is introduced for each constraint condition and a new function, the Lagrangian, L, is defined, viz.,   L ¼ Gcoh þ lH ^cH  zb cbH  za caH ð7:5Þ The extrema of Gcoh and L are now found by taking the partial derivatives of Eq. 7.5 and equating the result to zero. This yields three equations. The constraint, given by Eq. 7.4 yields one more equation, which means that there are now sufficient equations to solve for all four unknowns.

7.2 Polymorphic Phase Transformation

213

The simplest case to solve is one in which DGel depends only on the relative phase amounts as given by Eq. 7.1. This results in the following three equations plus the balance equation given by Eq. 7.4:   oL oDGel b b a a ¼ g  g þ  l c  c ¼0 ð7:6Þ H H chem H chem ozb ozb oL oga ¼ za chem  lH z a ¼ 0 a ocH ocaH oL ocbH

¼ zb

ogbchem ocbH

 lH z b ¼ 0

ð7:7Þ

ð7:8Þ

The Lagrange multiplier is obtained from Eqs. 7.7 and 7.8, with the result that: ogachem ogbchem ¼ l ¼ H ocaH ocbH

ð7:9Þ

From Eq. 7.9, it is now evident why the Lagrange multiplier has been given the same notation as that of the chemical potential of hydrogen since the result given by Eq. 7.9 shows that it is identical to it. Equation 7.9, then, shows that one of the extremum conditions is the well-known one stating that the chemical potentials of hydrogen in each phase are equal to each other. Now, substituting the result for lH given by Eq. 7.9 into Eq. 7.6 gives:   oga oDGel chem ¼ ð7:10Þ gachem  gbchem þ cbH  caH ocaH ozb Using the mass balance equation given by Eq. 7.4, Eq. 7.10 takes the following form:   oga   b b a oDGel chem  g ¼ c  c gachem þ cbH  caH H H chem ocaH o^cH

ð7:11aÞ

Equation 7.11a, on rearranging, gives the following expression for the chemical potential of the two-phase mixture:   gbchem ðcbH Þ  gachem ðcaH Þ oDGel   ð7:11bÞ lH ð^cH Þ ¼ þ o^cH cb  ca H

H

Equations 7.10 and 7.11a have a very straight-forward physical interpretation. The left hand sides of these equations represent the chemical driving force needed to overcome the coherency energy change that is given on the right hand side for an incremental increase in the b-phase fraction from a large reservoir of a-phase having composition, caH . Liu and Ågren [20] note that the effect represented by Eq. 7.10 is physically the same as if the b-phase were formed in the presence of an

214

7 Theories of Coherent Phase Equilibrium _

imposed average uniform pressure ph such that the work involved in converting a M of the a-phase to the b-phase is given by: molar volume V M ¼ ph V

_

   el dDGel  ¼ 1  2zb W ¼ 1  2zb D winc b dz

ð7:12Þ

M is the molar volume of the metal and the second expression on the right Where V hand side follows from Eq. 7.1. Note the similarity of this result with that derived by Eshelby [6, 7] given in Chap. 4 for point defects (Eqs. 4.70–4.74). In fact, when the phase transformation results in the formation of isotropically misfitting precipitates having elastic constants equal to those of the surrounding untransformed phase, the coherency energy expression is identical to that obtained by Eshelby [6, 7] for the sphere-in-hole model. This is evident from the characteristics of the solution since there is no explicit dependence on size in such continuum linearelastic models. Hence, such models apply equally well to point defects as to larger defects such as precipitates provided the same assumptions concerning their misfit strains and elastic constants apply. Note that, depending on which is the majority phase in the system, this work term can be either positive or negative, ranging from b þD wel wel inc to D inc between z = 0 and 1, respectively, because the ‘‘pressure’’ in the majority phase changes from being tensile to compressive, respectively while also having its maximum absolute value at these limits. Figure 7.1 illustrates graphically the conditions given by Eqs. 7.9 and 7.11a. This graphical construction also shows the essential difference between coherent and incoherent equilibrium. To simplify the illustration, the frame of reference has been chosen so as to make the usual common tangent construction for incoherent equilibrium horizontal (i.e., gachem and gbchem have equal minimum values) while the composition dependencies of the chemical Gibbs free energies of the single-phase states have been chosen to be the same. This means that the first and second derivatives of these free energies with respect to the corresponding minimum energy compositions are also equal. It is evident that the common tangent construction for incoherent equilibrium has been replaced in the case of coherent equilibrium by a weaker, parallel tangent, construction (see also Pfeifer and Voorhees [22]). The parallel tangent construction requirement means that equilibrium can only be achieved simultaneously when the tangents to the slopes at compositions, caH ; cbH and ^cH ; of the respective single-phase Gibbs free energies and coherency energy curves are equal to each other. The dependence of the coherency energy on the product of the phase fractions of the two coexisting phases results in a concave down parabola of positive magnitude that is zero at the compositions caH and cbH giving the limit of stability of the respective single-phase Gibbs free energies. It has a maximum positive value at an average composition that is, in this example, halfway between the coherent equilibrium compositions, caH and cbH : The graphical construction illustrated in Fig. 7.1 should also be a good approximation when the chemical Gibbs free energy curves of the incoherent (chemical) single-phase states have different compositional dependencies.

7.2 Polymorphic Phase Transformation

215

bo Fig. 7.1 Conditions for coherent a ? b equilibrium where xao (cao (cbo H ) are the H ) and x incohererent equilibrium compositions (symbols in parentheses are those used in the text) and xa (caH ) and xb (cbH ) represent possible coherent equilibrium compositions. The parabola between xa and xb represents the variation of the elastic coherency energy. The tangents through xa and xb are parallel with the tangent through the average composition, xo (^cH ), at least approximately parallel to the other two (from Hillert [11])

To illustrate some of the characteristics of this type of phase equilibrium, a simple parabolic dependence for the variations of the chemical free energies of each phase with composition, similar to that used by Cahn and Larché [4], was assumed by Hillert [11]. With the exception of the case that one of the terminal phases would be a pure compound, this is expected to be a good approximation for small deviations of the coherent equilibrium compositions from their incoherent equilibrium values. Thus, Eq. 7.2 becomes:  2   bc bo ao 2 b b Gcoh ¼ za K a cac  c þz K c  c þza zb W ð7:13Þ H H H H bo where cao H and cH are the composition values for incoherent equilibrium of the a and b phases and Gcoh is derived relative to these incoherent equilibrium states, using the common tangent line as reference. Again, for simplicity, this common tangent line has been assumed to be horizontal, as illustrated in Fig. 7.1. With the total coherent Gibbs free energy given by Eq. 7.13, the equilibrium conditions given by Eqs. 7.9 and 7.11a now result in the following relationships:     bo ao b K a cac cbc ð7:14Þ H  cH H  cH ¼ K

and       bo bc bo ao 2 ao b  c þ c  c c  c K a cac  2K H H H H H H  2   bc bo K b cH  cH ¼ 1  2zb W

ð7:15Þ

Hillert [11] used Eqs. 7.14 and 7.15 with the mass balance equation given by ao cH and zb for a series of cac Eq. 7.4 to calculate cbc H; ^ H values at a given set of cH , bo a b cH ; K ¼ K ¼ K and W values. The results are graphically illustrated in Fig. 7.2. The plot shows that the coherent phase boundary compositions fall within those of

216

7 Theories of Coherent Phase Equilibrium

Fig. 7.2 Linearized binary phase diagram with coherent phase boundaries is shown. The points where the two coherent phase boundaries and the T0 line meet is a Williams point. The boundary lines of the coherent a ? b phase field lines fall inside the incoherent (equilibrium) phase field lines. The tie-lines representing the coherent equilibrium compositions of the two-phase mixture fall within the respective single-phase fields. In this figure xo (^cH in the text) is the average composition of the solid and W and K are defined in the text (from Hillert [11])

the incoherent phase boundary ones, which is as expected from the results in Chap. 6. Note that phase boundary compositions define the limits of stability of the twophase field; i.e., they represent the compositions at which a second phase first appears or is completely dissolved. In the case of incoherent equilibrium, since the equilibrium compositions of the two phases are independent of their phase fractions, phase boundary compositions also define tie-line compositions. The latter are the compositions of the two phases in mutual equilibrium with each other. This is not the case for coherent equilibrium as is evident from Fig. 7.2. For instance, at an average composition equal to that of the coherent solvus composition, which is the limit of stability of the a single-phase state (the top dashed tie-line in the figure), the composition of the b-phase in equilibrium with the a-phase is symmetrically (for this case of equal K’s) to the right of the incoherent b-phase composition compared to its coherent phase field composition. The latter is located the same distance to the left of the incoherent phase field composition. The effect on the tie-line compositions for values of phase fractions other than the limiting values of zero and one is shown graphically in Fig. 7.3 relating the tie-line bc compositions cac cH , for different relative H and cH to the average composition, ^ coherency energy strengths, W/K. This figure shows that as the average composition increases to values greater than those given by the coherent phase field boundary compositions on the a/(a ? b) side of the phase field, the tie-line compositions of both the a- and b-phases decrease. The reason for this is

7.2 Polymorphic Phase Transformation

217

Fig. 7.3 The changes in compositions of two coherent phases as the average composition, xo (^cH ), is varied. The effect of different strengths of the coherency effect is illustrated. The bc b compositions, xa (cac H in the text) and x (cH in the text), are those for coherent equilibrium. The arrows for fb (zb in the text) = 0 and fa(za in the text) = 0 point to the ends of the tie-lines at which the respective phase fractions equal zero. The symbols in brackets refer to the notation used in the text (from Hillert [11])

graphically evident from the plot of Fig. 7.1. This plot shows that the slope of the tangent to the DGel curve is greatest when the average composition is equal to the respective coherent phase field compositions (i.e., at the limits of zero phase fraction of the respective phases). (Note that the value of DGel at these compositions is zero but not its derivative with respect to change in phase fraction.) Equilibrium is achieved at average compositions at which the slopes of the tangents of the two compositional derivatives of the chemical free energies are each equal to the slope of the compositional derivative of the coherency energy at that composition. From this construction, it can be seen that the tangent points giving these parallel slopes would be located at phase composition values furthest from their corresponding incoherent equilibrium values when the average composition in the material is equal to one of the single-phase composition limits. At average compositions in-between these single-phase composition limits the tangent point to the coherency energy parabola produces a slope that decreases monotonically with increase in average composition until, at an average composition of 0.5 (in this case of assumed symmetry in coherency energy variation between the single-phase chemical Gibbs free energy states), the coherent equilibrium compositions happen to be the same as those for incoherent equilibrium. Further increases in average composition beyond this mid-point value now result in the coherent equilibrium compositions of both phases continuing to decrease, but now the compositions of both phase are to the left of (i.e., less than) the corresponding

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7 Theories of Coherent Phase Equilibrium

incoherent equilibrium compositions. From Fig. 7.3, it can also be seen that the difference of the coherent phase field compositions between the two phase boundaries decreases with increasing W/K until the apex point is reached at which it disappears altogether. At this apex point the coherency energy has the same value as the chemical driving force and, therefore formation of a coherent two-phase mixture ceases to be the lowest Gibbs free energy solution over the entire composition range. At and above this point, the stable phase is either the coherent a single-phase state starting at average compositions B0.5 (for the symmetric case plotted in Fig. 7.3) or the b single-phase state starting at average compositions C0.5. There is thus a path dependence on which phase would be present in the system. For incoherent phase changes such a point does not represent equilibrium except at congruent points where there are no composition changes on phase transition. However, this T0 line found in incoherent phase diagrams becomes a field boundary for a coherent equilibrium phase diagram. Even though composition differences between phases are permissible in this case, the equilibrium coherent phase change will now occur without composition changes at the T0 line. This phase transition is of first order, having a finite volume change and a latent heat. It has no counterpart in incoherent phase diagrams. To distinguish it from higher order phase transition points such as critical or tri-critical points, Cahn and Larché [4] proposed to call it a Williams point after the author [26, 27] who first identified this point. This terminology has been adopted by subsequent authors. It should be noted, however, as Hillert [11] points out, that such a first-order, diffusionless phase transformation might be difficult to observe in practice since the system could also achieve this phase transformation gradually by diffusion at the high temperatures where this point is expected to occur. It should be noted, as Johnson and Voorhees [12] point out, that for values of W/K below the Williams point there is no path dependence, and hence no hysteresis for the foregoing symmetrical case and no loss of coherency as the phase fractions of each phase are increased from zero. By symmetrical case is meant that the two phases have equal elastic constants and their chemical free energy curves as function of composition are the same. However, when there are deviations from these symmetry conditions, there is the possibility that some of the extremum solutions represent energy maxima rather than minima. The presence of these maxima within the composition range for two-phase mixture would then create forbidden zones, resulting in processing path dependencies for the stable phase relationships. The conditions that determine the composition ranges over which the extremum solutions result in energy maximum values need to be determined in general to determine whether such forbidden regions and path dependencies could occur in the Zr–H system over the temperature and average composition ranges of practical interest. The results of the stability analyses are provided in this chapter in Sect. 7.4, while their applications to the Zr–H system are given in Chap 8.

7.3 Isomorphic Phase Transformation

219

7.3 Isomorphic Phase Transformation In this section, phase equilibrium conditions are derived for a two-component cubic crystalline solid in which the two solid solution phases have the same crystal structure and the lattice parameter varies linearly with concentration across the full composition range. This was the case first considered by Cahn [3] in his seminal studies on coherent fluctuations and spinodal decomposition of isotropic, twocomponent substitutional solids in systems exhibiting miscibility-gap phase boundaries [1–3]. It would appear, at first glance, that the thermodynamic behavior of systems having miscibility-gap type phase fields would not apply to Zr–H. However, as discussed in Chap. 8, despite the a-phase in Zr–H having an hcp lattice structure and the b-(d-hydride)phase an fcc structure, so that the phase transformation is polymorphic (structural), in the a-(and, possibly the b-)phase the system exhibits trends in the variation of physical properties with composition that resemble those expected to exist in systems having miscibility-gap phase boundaries. Hence, it seems sensible to also examine the thermodynamic equilibrium conditions for such non-structural (isomorphic) phase transitions. In addition to the initial study by Cahn [3], Roıtburd [23] examined the equilibrium conditions relevant to coherent isomorphic and polymorphic phase transitions. The latter author arrived at similar conclusions as, independently, did Williams [26, 27] regarding differences in phase equilibrium conditions between such coherent and incoherent systems. Prompted by the insights provided by Williams’ work [26, 27], a series of studies were subsequently published in quick succession by Chiang and Johnson [5], Pfeifer and Voorhees [22], and Lee and Tao [19] focusing on coherent equilibrium in binary substitutional solids. A recent application to metal–hydrogen systems with miscibility-gap phase boundaries, such as Pd–H, was carried out by Schwarz and Khachaturyan [24, 25]. The following summary makes use of the latter authors’ methodology. The isomorphic phase transition model they considered is referred to in subsequent discussions as an ideal isomorphic model. As was shown in Chap. 4, solute atoms in a binary solid solution, either of substitutional or interstitial type, can be considered as acting like point defects generating a fixed amount of internal strain as a result of their lattice misfit when replacing a solvent atom in its crystal lattice. It is shown in that chapter that the interaction energy between defects produced by the average image dilatation strain of the solute atoms makes a substantial contribution to the overall energy of the solid solution. From Eqs. 4.70 and 4.74, the total strain energy per atom is given by: EsT ¼ 2G

ð1 þ mÞ Dv2 ^cH ð1  ^cH Þ ð 1  mÞ X M

ð7:16Þ

where cI in Eq. 4.74 has been replaced by ^cH , the average composition of the solid solution while all other symbols are as defined in Chap. 4.

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7 Theories of Coherent Phase Equilibrium

As shown in Chap. 4, the total strain energy is the sum of an inhomogeneous local strain field centered on each solute atom and a homogeneous part produced by the interaction of the stress field of an inserted solute atom with the image dilatation generated by all the other solute atoms in a finite crystal. When it can be assumed that direct pair interactions between solute atoms are negligible, then Eq. 7.16 gives the total strain energy of the solid solution. Since this energy is independent of the exact location of each solute atom, it is commonly referred to as non-configurational energy. Moreover, the ^cH ð1  ^cH Þ concentration dependence of this energy given by Eq. 7.16—derived from a sphere-in-hole model— was shown by Eshelby [6] to be valid across the full composition range of the solid solution when the rule of additivity of atomic radii for solute (defect) and solvent (host) given by Vegard’s law is valid. Vegard’s law refers to the case when the fractional change of lattice constant with composition is constant across the full range of composition of a two-component system. Thus Eq. 7.16 can be expressed in terms of Vegard’s linear relation: a ¼ ð1 þ ko^cH Þao

ð7:17Þ

where a and ao are the lattice parameters of the solid solution and solvent crystal lattices, respectively, and ko is the homogeneous, isotropic strain per solute atom given by Eq. 4.77 for the isotropic (cubic) case for which k1 = k2 = k3 and rH  ^cH . This gives: EsT ¼ 2XM G

ð 1 þ mÞ 2 k ^cH ð1  ^cH Þ ð 1  mÞ 0

ð7:18Þ

If the solid solution decomposes into two compositionally homogeneous phases, a and b, of compositions caH and cbH ; then Eq. 7.17 for each phase becomes: aa ¼ ð1 þ k0^cH Þaa0

and

ab ¼ ð1 þ k0^cH Þab0

ð7:19Þ

Neglecting terms quadratic in k0, the crystal lattice misfit between the twophases in their unconstrained states is:   ab  a b a  k c  c 0 H H aa a

k¼

ð7:20Þ

The total strain energy of each phase is then given from Eq. 7.18 by: Esa ¼ 2XM G

ð 1 þ mÞ 2 a k c ð1  caH Þ ð 1  mÞ 0 H

ð7:21Þ

Esb ¼ 2XM G

ð 1 þ mÞ 2 b k c ð1  cbH Þ ð 1  mÞ 0 H

ð7:22Þ

Thus the total strain energy of the incoherent (i.e. elastically decoupled, unconstrained) mixture is given by:

7.3 Isomorphic Phase Transformation

  T Es;incoh ¼ 1  zb Esa þ zb Esb

221

ð7:23Þ

where zb is the phase fraction of the b phase, as defined in Eqs. 7.3 and 7.4. The strain energy caused by the elastic constraint of the b phase when the twophases are coherently in contact with each other is given by: 2 b b Esa=b ¼ 2XM G ð1þmÞ ð1mÞ k z ð1  z Þ h  i2 b a zb ð1  zb Þ ¼ 2XM G ð1þmÞ ð1mÞ k0 cH  cH

ð7:24Þ

where the second relation is obtained when the condition given by Eq. 7.20 holds. The total strain energy of the coherent mixture is given by the sum of the strain energies of the incoherent phases given by Eqs. 7.21 and 7.22 plus the coherency energy given by Eq. 7.24, viz., T Es;coh ¼ Esa þ Esb þ Esa=b Þ 2 ¼ 2XM G ðð1þm cH ð1  ^cH Þ 1mÞ k0^

ð7:25Þ

In contrast to the total incoherent strain energy relation given by Eq. 7.23, the strain energy terms Esa and Esb in the expression for the total coherency energy of Eq. 7.25 are not proportioned according to the phase fractions of the two-phases since this proportioning is contained in the elastic constraint energy term, Esa=b , which accounts for the coherency energy arising from the lattice mismatch between the a- and b-phases. The second equality term of Eq. 7.25 is obtained by combining the balance relation given by Eq. 7.4 with the relations for Esa and Esb T given by Eqs. 7.21–7.22 and eliminating terms. Note that the net Es;coh then reduces to the same relation as that given by Eq. 7.16 for point defects. This result is not surprising since, for this case of an isotropic solid solution obeying Vegard’s law, Eshelby [6] has shown that the strain energy of a distribution of point defects is independent of the location of the defects in the solid. Thus, unlike the case of a polymorphous transition with concentration independent lattice parameters of each phase for which the sole source of misfit strain in the solid arises from a difference in lattice parameter between the a- and b-phases as a result of their different crystal lattice structures, the coherency energy in a solid obeying Vegard’s law depends only on the average composition of the solid solution and not on the phase fractions of the two-phases. It is implicit in the validity of the strain energy expressions of Eqs. 7.16 and 7.25 that the distribution of hydrogen interstitials in the crystal is a disordered one since an ordered structure would be one in which all of the interstitial sites would be occupied. Such ordering would result in the elimination of coherency strains within an individual phase. In an ordered arrangement of interstitials, when all of the available interstitial sites are occupied, the distortions produced by the individual interstitials would be uniformly spread throughout the crystal. Such an ordered (stoichiometric) state is one of the reference states of the strain energy expression given by Eq. 7.16 or 7.25. The other reference state is the

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7 Theories of Coherent Phase Equilibrium

interstitial-free state of the crystal. The concentrations corresponding to these reference states are ^cH ¼ 1 and 0, respectively resulting, as required, in the strain energy being zero when the material is in either of these two states. The Gibbs free energies per host atom of the a and b single-phase states, then, consist of two parts: the ‘‘chemical’’ free energy of the undeformed (or unrelaxed) phase which is made up of energy and entropy contributions arising from atomic bonds and configurational plus lattice vibration terms, respectively, plus the total strain energy generated by the sum of the self and interaction energies of the disordered (randomly) distributed mixture of misfitting interstitial (point) defects, viz., Gaincoh ¼ gachem ðP; T; cH Þ þ Esa  gachem ðP; T; cH Þ þ AcH ð1  cH Þ

ð7:26Þ

Gbincoh ¼ gbchem ðP; T; cH Þ þ Esb  gbchem ðP; T; cH Þ þ AcH ð1  cH Þ

ð7:27Þ

and

where A  2XM G

ð 1 þ mÞ 2 k ð 1  mÞ 0

ð7:28Þ

The constant, A, is thus a material parameter that is assumed, in this example, to have the same value in both of the single-phase states (i.e., it is assumed that the elastic properties are independent of composition). (Note that for an isotropically misfitting precipitate having the same elastic constants as the matrix, A  D wel inc , as defined in Chap. 6). The chemical Gibbs free energies are defined at constant pressure, P, and temperature, T, of the solid. The Gibbs free energies of the singlephase states, a and b, have been denoted as incoherent to indicate that coherency— or the lack of it—does not play a role when the solid is in one or the other of the two single-phase states or when it is in its incoherent two-phase state. The Gibbs free energy of the coherent two-phase mixture of compositionally homogeneous a- and b-phases, then, consists of the linear sum of the chemical free energies of the individual single-phase states, proportioned according to the phase fraction of each phase given by:     gchem P; T; caH ; cbH ; zb ¼ 1  zb gachem ðP; T; caH Þ ð7:29Þ þzb gbchem ðP; T; cbH Þ T to which is added the coherent energy Es;coh of this phase mixture given by Eq. 7.25. This results in the following expression for the coherent Gibbs free energy:

7.3 Isomorphic Phase Transformation

    T Gcoh P; T; caH ; cbH ; zb ¼ gchem P; T; caH ; cbH ; zb þ Es;coh   ¼ 1  zb gachem ðP; T; caH Þ þzb gbchem ðP; T; cbH Þ þ A^cH ð1  ^cH Þ

223

ð7:30Þ

It is evident from Eq. 7.30 that in a closed system where the average composition is fixed, the strain energy contribution to the Gibbs free energy given by the last term in Eq. 7.30 is a constant and, therefore, reduces to zero in the free energy minimization. This means that the two-phase equilibrium conditions are determined entirely from the chemical Gibbs free energies. This result also means that, unlike the previous case of a polymorphic phase transformation, the usual common tangent rule first established by Gibbs for fluid phases and later for incoherent solid solution phases by others, establishes the equilibrium conditions. This useful result was first recognized by Cahn [3]. It is worth noting at this point that the total coherent Gibbs free energy given by Eq. 7.30 is formally very similar to the corresponding expression given in Sect. 7.2 for the polymorphic case. In both cases, the coherency energy is assumed to be proportional to the product of the phase fractions in the two-phase composition range. However, in the isomorphic case there exist similar strain energy contributions also in the single-phase states, while they are absent in the polymorphic one. The presence of these strain energy contributions in the single-phase states in the isomorphic case results in the total coherent Gibbs free energy depending only on the average composition for this case. Minimization of the Gibbs free energy given by Eq. 7.30 under the constraint of conservation of total number of interstitial atoms results in the following relationships:      a  a b b b b a a og c g H chem chem cH  gchem cH ogchem cH ¼ ¼ ð7:31Þ ocaH ocbH cbH  caH These relationships give the compositions of the two phases when they are in coherent equilibrium with each other. Note that, unlike the equilibrium conditions for polymorphic transformations in a closed system given by Eqs. 7.9–7.11a, the relation given by Eq. 7.31 does not explicitly contain a coherency strain energy term. Thus the geometrical solution of these equations is simply determined by the usual Gibbs common tangent construction applied to the chemical Gibbs free energy curves for the a and b single-phase states. Physically, the compositions bc cac H and cH obtained by this procedure from Eq. 7.31 are the terminal compositions of the coexisting a- and b-phases when in coherent equilibrium with each other. This means that they delineate the coherent phase field compositions. In addition, since these equilibrium compositions do not depend on the average composition, ^cH , over the full range of composition of the two-phase mixture of the solid, they also represent the tie-line compositions. Hence, tie-line and phase field boundary compositions coincide in this special case of isomorphic coherent equilibrium. Given the average composition of the solid, ^cH , and the coherent equilibrium

224

7 Theories of Coherent Phase Equilibrium

bc compositions, cac H and cH , then from the mass (atom) balance relationship given by Eq. 7.4, the following lever law relationship for the phase fraction of the coherent b phase is obtained:

zbc ¼

^cH  cac H ac cbc H  cH

ð7:32Þ

It is useful to see how the foregoing result differs from the case when the phase fractions formed in the two-phase state are no longer coherently related, corresponding to incoherent equilibrium. In this case, the total Gibbs free energy relation for incoherent equilibrium is simply the sum of the respective phase fractions of the Gibbs free energies of the elastically relaxed a and b phases. These are the corresponding single-phase states given by Eqs. 7.21 and 7.22. This results in the following expression for the incoherent total Gibbs free energy:   GincohðcH Þ ¼ 1  zb Gaincoh þ zb Gbincoh  a a  1 hzb gachem þ Ac ð7:33Þ  H 1 ciH b b b b þz gchem þ AcH 1  cH To better illustrate the essential difference between the total coherent and incoherent Gibbs free energy expressions, Eq. 7.33 is reformulated to have the same structure as that given by Eq. 7.30, viz.,   T Gincoh ðcH Þ ¼ gchem P; T; caH ; cbH ; zb þ Es;incoh ð7:34Þ   T by Eq. 7.23. This where gchem P; T; caH ; cbH ; zb is given by Eq. 7.29 and Es;incoh shows that in the incoherent case the cross-term strain energy for the misfit strains between the two phases given by Esa=b is absent. Similar to the case for coherent equilibrium, to determine the equilibrium compositions in a closed system, the usual common tangent rule applies to the relationship given by Eq. 7.33 or 7.34, but now the equilibrium compositions are determined from the composition derivatives of the single-phase Gibbs free energies, Gaincoh and Gbincoh given by Eqs. 7.26 and 7.27. These energies include, for each phase, a total strain energy contribution produced by the randomly distributed point defects (interstitial hydrogen atoms). This differs from the result for the coherent bc equilibrium case where the equilibrium compositions cac H and cH are determined   from the composition derivatives of the chemical free energies gbchem cbH   and gachem caH : It seems counter-intuitive at first glance that the coherent compositions are obtained from the derivatives of only the chemical Gibbs free energies and not from the total Gibbs free energies—the latter of which contain also the strain energy contributions of the point defects. An explanation for this is given further on, when comparing the present results with those obtained for the polymorphic transition case given in Sect. 7.2.

7.3 Isomorphic Phase Transformation

225

Based on Eq. 7.33, the following conditions are obtained for incoherent equilibrium:    a b b a oG incoh cH oGincoh cH ¼ ocaH ocb h    H i  ð7:35Þ     gbchem cbH þ A 1  2cbH  gachem caH þ A 1  2caH ¼ cbH  caH The geometrical solution of Eq. 7.35 can be obtained, similar to the coherent case, by the Gibbs common tangent construction. This yields the terminal (and hence bo phase field) compositions, cao H and cH ; of the a- and b-phases coexisting in incoherent equilibrium. Since the last term in the equalities of Eq. 7.35 does not depend on the average composition, the chemical potentials of hydrogen in the two phases, given by the equality of the first two terms in Eq. 7.35, respectively, are constants throughout the composition range for values of average composition equal to or bo within the incoherent two phase field boundary compositions given by cao H and cH . Note that the phase field compositions for coherent two-phase equilibrium, bc cac H and cH ; are inside these field boundaries for incoherent two-phase equilibrium. This result is qualitatively similar to that obtained for polymorphic phase transitions. In both cases, the shift between coherent and incoherent equilibrium compositions arises from the strain (coherency) energy contribution to the total Gibbs free energy. There is, however, a physical difference between the two cases. In the isomorphic case with linear dependence of the lattice parameter of the two phases on bo composition, the incoherent equilibrium compositions, cao H and cH ; can be considered to have been shifted to lower and higher values, respectively, compared to bc the corresponding coherent equilibrium compositions, cac H and cH ; because there are no additional coherency energy contributions to the overall Gibbs free energies when the compositions of these phases reach their single-phase stability limits and a two-phase mixture is produced. The reason for this is because the two-phase state is distinguished from the two single-phase states only by a change in the distributions of the point defects (hydrogen in our case) producing macroscopic regions containing either a dilute or a concentrated collection of solute atoms. For this special case of a two-component, cubic, isotropic crystalline solid in which Vegard’s law is assumed to hold across the full composition range, such clustering does not, however, result in any additional coherency energy contributions to the Gibbs free energy. Thus one can think of the incoherent two-phase compositions shifting outward from the coherent two-phase compositions as a result of a negative and positive strain energy contribution, respectively, to the latter. For the polymorphic case, on the other hand, there are no coherency energy contributions in the single-phase states. Thus the incoherent equilibrium compositions contain no implicit coherency energy contributions as in the isomorphic case. Only within the composition range of coherent two-phase equilibrium is there a

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7 Theories of Coherent Phase Equilibrium

coherency energy contribution to the Gibbs free energy. This coherency energy is proportional to the product of the respective phase fractions of the two phases. The coherent two-phase equilibrium compositions obtained in this case can thus be visualized as being the result of a shift inward from their corresponding incoherent two phase composition boundary values as a result of a positive, misfit-induced coherency energy contribution to these incoherent equilibrium compositions. As noted in the foregoing discussion, an important difference between the polymorphic and isomorphic cases is that phase field and tie-line (equilibrium) compositions are the same in the isomorphic, but not in the polymorphic case. This difference exists, even though in both cases it is assumed that there are no differences in the curvatures (second derivative with respect to composition) of the chemical Gibbs free energies and in the elastic constants of the two single-phase states. However, this correspondence between the tie-line and phase field compositions in this special isomorphic case can be easily disturbed, as shown by Chiang and Johnson [5]. Thus when the elastic constants depend on composition— and, hence, are different in each phase—qualitatively similar differences between phase field and tie-line compositions as in the polymorphic case are also obtained in the isomorphic case. In addition, in the systems that have consolute critical points (miscibility-gaps) analyzed by Chiang and Johnson [5], even when the elastic constants for such systems are assumed to be the same in each phase, there are still predicted differences between tie-line and phase field compositions for temperatures close to the critical temperature. At such temperatures, the differences between coherent and incoherent phase field compositions become quite large. These large differences between the coherent and incoherent equilibrium compositions are not captured in analyses in which the general equations of equilibrium are linearized functions with respect to these differences. To complete this section, expressions are derived in the following for hydrogen chemical potentials. These derivations serve to illustrate some important characteristics of coherent phase transformations illustrating the distinguishing features making them different from incoherent ones in closed and open thermodynamic systems. By definition, the chemical potential of a component (hydrogen) is given by the partial derivative of the Gibbs free energy with respect to that component. For the a- and b-phases in their incoherent (unconstrained) states corresponding to, respectively, Gaincoh and Gbincoh ; given by Eqs. 7.26 and 7.27 we thus obtain for the corresponding chemical potentials: laH ¼

oGaincoh ogachem ¼ þ Að1  2cH Þ ocH ocH

ð7:36Þ

lbH ¼

oGbincoh ogbchem ¼ þ Að1  2cH Þ ocH ocH

ð7:37Þ

In Eqs. 7.36 and 7.37, cH  ^cH since these relations represent single-phase states. However, they also give the chemical potentials for the individual phases

7.3 Isomorphic Phase Transformation

227

when they are in incoherent two-phase equilibrium. With the condition for incoherent equilibrium given by Eq. 7.35, and the relations given by Eqs. 7.36 and 7.37, it is seen that within the two-phase composition range, cao cH  cbo H , the H ^ b bo Þ ¼ l chemical potentials of the two phases must be equal. Thus, laH ðcao H ðcH Þ, H which also shows that they are constant within this composition range since they bo depend only on the phase field compositions cao H and cH : It is shown in the following that this is not the case for coherent equilibrium. First of all, for coherent equilibrium, the range of the phase field boundary is smaller, given by cac cH  cbc H . At these terminal compositions the corresponding H ^ single-phase states are:  a  a oGincoh ð^cH Þ ogchem ð^cH Þ ¼ þAð1  2cac ð7:38Þ laH ð^cH Þ ¼ HÞ o^cH o^ c ac ac H ^cH ¼ c ^cH ¼ c H

lbH ð^cH Þ ¼

oGbincoh ð^cH Þ o^cH

H

! ¼ ^cH ¼ cbc H

ogbchem ð^cH Þ o^cH

! þAð1  2cbc HÞ

ð7:39Þ

^cH ¼ cbc H

Replacing the derivatives of the chemical Gibbs free energies in Eqs. 7.38 and 7.39 by their common value given in Eq. 7.31 at the phase field compositions for coherent equilibrium gives the chemical potentials of hydrogen in the two phases in their single-phase states at these limits of stability, viz.,     gbchem cbc  gachem cac H H laH ð^cH Þ ¼ þ Að1  2cac ð7:40Þ HÞ bc ac cH  cH     gbchem cbc  gachem cac H H b lH ð^cH Þ ¼ þ Að1  2cbc ð7:41Þ HÞ bc ac cH  cH Now, the chemical potential of hydrogen in the two-phase state is obtained from the derivative of the Gibbs free energy given by Eq. 7.30 with respect to the average composition. Making this substitution results in   h  ac i dzb a lH ð^cH Þ ¼ gbchem cbc þ Að1  2^cH Þ  g H chem cH d^cH

ð7:42Þ

which, by use of the relation for the derivative dzb =d^cH obtained from Eq. 7.32, gives     gbchem cbc  gachem cac H H þ Að1  2^cH Þ ð7:43Þ lH ð^cH Þ ¼ bc ac cH  cH There are two differences in the expression for the chemical potential of hydrogen between the coherent and incoherent two-phase states. One is that in the

228

7 Theories of Coherent Phase Equilibrium

      Fig. 7.4 Schematic of the hydrogen chemical potentials l ca (lH cac in the text) and l cb H   in the text) for the coherent two-phase solid. The symbols in parentheses are the (lH cbc H corresponding ones used in the text. The arrows indicate the transformation paths when the solid is in equilibrium with a source of hydrogen of constant chemical potential. No equilibrium coexistence of phases is possible along these arrows (from Schwarz and Khachaturyan [25])

coherent case there is now a coherency energy term depending on the average composition. The second is that the phase field composition limits, which govern the magnitude of the first term in Eq. 7.42, cover a narrower range. The presence in the chemical potential of an explicit coherency energy term depending on the average composition means that, unlike incoherent equilibrium, the chemical potential is not constant in the two-phase composition range but, as shown in Fig. 7.4, decreases linearly as the average composition at a given temperature is increased from its lowest value when ^cH ¼ cac H to its highest value when bc ^cH ¼ cH (although the tie-line compositions are still given by their phase field values). Beyond this composition the system is in the single-phase state of the b phase in which the chemical potential increases again with average composition in a similar fashion as it does when it is in the single-phase state of the a phase. Mathematically the variation of the chemical potential with overall composition in the two-phase region is a consequence of the difference in the sign of the derivative of the coherency energy at the compositions of coherent equilibrium on opposite sides of the phase diagram. For instance, assuming that the locations of the minima of the chemical Gibbs free energies of the single-phase states are symmetrically located with respect to the end point compositions of zero to one on the phase diagram, then the chemical potential of hydrogen is lower by the constant, A, for values of the average composition located at the a/(a ? b) and (a ? b)/b sides of the phase field boundaries, respectively. Physically this is because the coherency energy is a positive quantity increasing from zero for the pure a and hydride phases and has a maximum value in-between these limits

7.3 Isomorphic Phase Transformation

229

(in this case of assumed symmetry at ^cH ¼ 0:5). This follows from the result that—in this sphere-in-hole model—there can be no coherency energy produced by these interstitial defects (hydrogen atoms) when the crystal either contains none of them (at ^cH ¼ 0) or they form an ordered lattice (^cH ¼ 1).

7.4 Stability Conditions and Path Dependences for Coherent Phase Transformations The analyses given in Sects. 7.2 and 7.3 for closed systems have provided phase equilibrium conditions that are, strictly speaking, only extremum conditions. Hence, they cannot tell us whether the solutions obtained represent stable thermodynamic states or not. Two bounding cases were analyzed: (1) coherent polymorphic phase transitions where coherency energy is generated only over the two-phase composition range and, (2) coherent isomorphic phase transitions in a cubic lattice with the lattice parameter depending on composition according to Vegard’s law where coherency energy is generated in both the single- and twophase states, but no additional coherency energy is generated when the system passes from one of the single-phase states to the two-phase state. The extremum solutions for these two cases have shown that there are important differences in the phase relationships between these two types of coherent systems relative to those for fluid or incoherent solid systems. In both coherent cases, it is found that the coherent phase field compositions lie within those of the incoherent phase field compositions. However, for polymorphic phase transitions the tie-line compositions, which give the compositions of the phases in two-phase equilibrium with each other, vary with phase fraction, and hence also with average composition in a closed system. This means that the tie-line compositions depend on the phase fraction and differ from the corresponding coherent phase field compositions. On the other hand, for the ideal isomorphic coherent phase transformation, tie-line compositions coincide with phase boundary compositions. Thus for such closed isomorphic systems the stability conditions would be similar to those of fluid or incoherent solid solution systems. In these cases, the extremum solutions obtained for the phase relationships represent stable thermodynamic states, regardless of the average composition. It should be noted, however, that this latter result applies only to closed, ideal isomorphic systems which are systems in which the solid has cubic lattice symmetry and the variation of the lattice parameter with composition follows Vegard’s law. They would not necessarily apply to more realistic systems, for which there is generally some deviation from ideality, or when the restriction to a closed system is removed, a case that is treated in Sect. 7.4.2. In the following section (Sect. 7.4.1), stability criteria are presented to determine under what conditions the extremum solutions obtained in Sect. 7.2 for closed, polymorphic phase transformations result in stable thermodynamic states. In the subsequent section (Sect. 7.4.2), a similar stability analysis is presented for

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7 Theories of Coherent Phase Equilibrium

the case of isomorphic phase transitions but, in this case, only for open systems, since the extremum solutions given in Sect. 7.3 for closed ideal isomorphic systems represent stable thermodynamic states.

7.4.1 Stability Conditions for Closed Polymorphic Systems Various methods have been adopted by different authors to determine the composition ranges over which the extremum solutions in polymorphic systems represent stable thermodynamic states. One approach, used by Liu and Ågren [20, 21], is based on the usual approach in thermodynamics in which the stability conditions are determined from the signs of the first and second derivatives of the total Gibbs free energy function of the system. In general, the Gibbs free energy expressions for coherent systems are complex and differentiation of these expressions can be cumbersome and/or require numerical methods. Johnson and Voorhees [12], therefore, avoided using this approach, circumventing it by assuming a somewhat artificial but suitably simple geometric arrangement for the spatial equilibrium arrangement of a coherent mixture of a- and b-phases. This choice of geometry of the equilibrium arrangement of the two phases made it possible for the authors to derive—based on the general equilibrium solutions derived for coherent phase equilibrium by Larché and Cahn [18]—tractable equilibrium relationships permitting all conditions of thermal, mechanical, and chemical equilibrium to be satisfied simultaneously. The geometrical arrangement chosen by Johnson and Voorhees [12] is of a spherical region of b-phase located at the center of an all-inclosing a-phase sphere with which it can interchange material in both directions. The ratio of the respective radii of the embedded b-phase sphere versus the surrounding a-phase spherical shell represents, in their model, the phase fraction of each phase. The model allows for differences in the elastic moduli of the two phases and in the dependencies of the chemical free energies of each phase on composition. A set of three algebraic equations for the three unknowns—the equilibrium compositions of the a- and b-phases and their phase fractions—were derived. The unknowns were formulated in terms of reduced compositions Ya and Yb of the a- and b-phases and of the average composition, W, viz.,     bo ao ao Y a ¼ 1 þ 2 cac ð7:44Þ H  cH = cH  cH     bo b0 ao Y b ¼ 1 þ 2 cbc H  cH = cH  cH

ð7:45Þ

    bo ao W ¼ 1 þ 2 ^cH  cbo H = cH  cH

ð7:46Þ

To make the solutions as general as possible the original equations were reformulated in terms of dimensionless parameters, K, f, and d. These parameters,

7.4 Stability Conditions and Path Dependences for Coherent Phase Transformations

231

which give, respectively, the ratio of the coherency energy to the chemical driving force, the ratio of the curvatures of the chemical free energies versus composition and the ratio of the elastic constants of the two single-phase states, a and b, are defined as follows: 9e2 K b G K a C

ð7:47Þ

d¼

Kb  Ka K a C

ð7:48Þ

f¼

va  vb vb

ð7:49Þ

K¼

with Ka

G¼

 ao va cbo  c H H

C ¼

3K b þ 4Ga 4Ga

ð7:50Þ

ð7:51Þ

and

 q00 kT o ln cH 1 o2 G v¼ o 1þ ¼ cH ð1  coH Þ NA XM oc2H co o ln cH

ð7:52Þ

H

where q00 is the number of lattice points per unit volume in the reference state, XM is the atomic volume of the metal in the reference state, NA is Avogadro’s number, cH is the activity coefficient of component, H, and the first and second derivatives of the Gibbs free energy, G, are evaluated at the hydrogen compositions, bo coH ¼ cao H or cH in the reference state, as appropriate. The misfit strain, e, is assumed to be isotropic and defined by:  b   V a V T eij ¼ edij ¼ ð7:53Þ a 3V  are the partial molar volumes of the unstressed phases. Since the where the V’s molar volumes are assumed to be unaffected by hydrogen composition, they are equivalent to the molar volumes of the stoichiometric hydride (b) and pure metal (a) phases. The other parameters in Eqs. 7.47–7.52 are the elastic moduli for the indicated phases. The notations for these are the same as those given in Chap. 4 except that the shear moduli are given here by Ga or Gb for the a- and b-phases, respectively, to distinguish these constants from the Gibbs free energy, which is simply given by the symbol, G.

232

7 Theories of Coherent Phase Equilibrium

In terms of the foregoing reduced compositions and dimensionless material parameters, the following three equations for the reduced compositions and the phase fraction, zb of the two-phase mixture are obtained:    2 K  d  zb 2zb þ 1 ð7:54Þ Y a ¼ 1 þ 2 ð1 þ dzb Þ    2   K  d  zb 2 zb þ 1 ð7:55Þ Y b ¼ 1 þ ð 1 þ fÞ 2 ð1 þ dzb Þ and 8      9 < 2d2  Kdf zb 3 þ d2 ð1 þ W Þ þ 4d  K  ðd þ 2fÞ zb 2 = :

þ½2d  ð1 þ W Þ þ 2 þ K  ðf  2Þzb þ K  W  1 ;  2 = 1 þ dzb ¼ 0

ð7:56Þ

Equation 7.56 for zb can be considered as the equation of state for the extremum condition. However, it does not equal zero for just any physical combination of material and elastic constants. Rather, it is a cubic equation for the volume fraction of the b-phase. It can, in principle, be solved exactly. To determine the stability of the thermodynamic state obtained from the solutions to these ‘‘equilibrium’’ equations for particular cases, Johnson and Voorhees [12] examined the nature of these solutions in terms of bifurcation analysis. The results of these bifurcation analyses were graphically displayed using what the authors called phase stability diagrams. These are plots of phase fractions versus average alloy compositions showing the ranges over which single- or two-phase states represent conditions for which the total Gibbs free energy is a minimum or not. Hence, by this method the authors were able to establish the composition and phase fraction ranges over which the results obtained represent stable (or metastable1) equilibrium conditions. Johnson and Voorhees [12] give a series of examples of possible behavior for various choices of differences in the properties of the two phases as expressed by the dimensionless parameters f and d. The case f and d = 0 is the simplest one giving a stable solution over the full range of phase fraction zb and a linear dependence of the phase fraction on average composition similar to that obtained for fluid and incoherent solid solution systems. The results for this case are discussed in more detail further on. There are two main types of effects as a result of non-zero values for the parameters f and d. One is that there are excluded regions resulting in abrupt changes in the relationship between phase fraction and average composition; this

1

It is not possible with this method to determine whether the state obtained represents an absolute minimum or not.

7.4 Stability Conditions and Path Dependences for Coherent Phase Transformations

233

also results in differences in the equilibrium compositions between the phases (the lengths of the tie-lines) depending on the phase fraction. The second is that the phase fraction is no longer a linear function of the average composition. Depending on the values of f and d, the results can be very sensitive to the value of K. It should be noted that such excluded composition regions do not occur for systems having isomorphic phase transitions since the coherency energy curve in this case is always tangent to the chemical free energy curves at the compositions marking the limits of the coherent phase field. From the point of view of application to the Zr–H system, it is useful to examine in detail the solutions for two cases: (1) the fully symmetric case, for which f = 0 and d = 0, and (2) the partially symmetric case for which f = 0 and d = 0. The results for case (1), when f = 0 and d = 0, are: zb ¼

1 W þ 2 2ð1  KÞ

ð7:57Þ

WK ð1  K Þ

ð7:58Þ

Y a ¼ 1  Yb ¼ 1 

WK ð1  K Þ

ð7:59Þ

These results are identical to those obtained by Cahn and Larché [4] derived from a simpler model. Because of the simplicity of the latter authors’ model their results appear, at first glance, to be less general than those of Johnson and Voorhees [12]. Cahn and Larché [4] approximated the dependences of the Gibbs free energies on composition by a Taylor series expansion about the equilibrium compositions of the respective incoherent phases to second order in the compositions, thus obtaining parabolic functions of the differences between the coherent and incoherent phase compositions for the chemical Gibbs free energies. (This approach is also identical to that used by Hillert [11], the results of which are graphically illustrated in Fig. 7.2). The results of Johnson and Voorhees [12], on the other hand, depend only on the second derivatives of the Gibbs free energies evaluated at the equilibrium compositions of the incoherent phases. Identical results were, nevertheless, obtained because the analytic solutions of Johnson and Voorhees [12] are linearized approximations assuming that differences between the coherent and incoherent equilibrium compositions of each phase are small. The results show that the phase fraction is a linear function of the average composition, and the difference in the reduced equilibrium composition of the two phases, Yb Ya = 2. Both of these results are identical to those for the incoherent case for which the dimensionless coherency strain energy factor, K = 0. However, in the coherent case it is evident from Eqs. 7.58 and 7.59 that the tie-line compositions now depend on the average composition and, hence, through Eq. 7.57 on the phase fraction, zb. This dependence on zb results in each of the dimensionless composition variables Yb and Ya to be shifted by the same magnitude of the factor WK/

234

7 Theories of Coherent Phase Equilibrium

(1-K) in the same direction from their respective incoherent values. Since, for bo zb = 0, W = -1/(1-K), the shift is to higher values of cao H and cH : The results for case (2) f = 0 and d = 0 are: 8 " #12 9 < = ½ 2 þ D  ð f  2 Þ  8Kf  ð K  1  W Þ 1 1þ ð7:60Þ zb ¼ : 4Kf ½2 þ K  ðf  2Þ2 ;   Y a ¼ 1 þ K  1  2zb 

Y b ¼ 1 þ ð1 þ fÞ  K  1  2z

ð7:61Þ  b

ð7:62Þ

Note that the difference in the reduced equilibrium compositions of the coexisting a and b phases, Yb - Ya, now depends on the phase fraction, and hence average composition. The real roots of Eq. 7.60 that are physically meaningful are those in the range 0 B zb B 1. Alloy compositions that result in real roots for given f and K are bounded by the condition that the quantity under the radical sign in Eq. 7.60 is greater than or equal to zero. This gives the following conditions: Wc ¼ K  1 þ

½ 2 þ K  ð f  2Þ  2 8Kf

ð7:63Þ

W  Wc

when

f[0

ð7:64Þ

W Wc

when

f\0

ð7:65Þ

For alloy compositions corresponding to Eq. 7.64, the solution is a parabola that opens to the left while for the case corresponding to Eq. 7.65 it is a parabola that opens to the right, toward larger values of W. The magnitude of the curvature is governed by the magnitude of K, increasing as K increases. Of special significance are solutions for volume fractions corresponding to (Wc, zbc ) that are within the range 0 B zb B 1. When this occurs, two real solutions exist in this range, with one of the solutions being a maximum, the other a minimum. The point (Wc, zbc ) at which the two solutions intersect on the phase stability diagram is a turning point and the slopes of each of the two solutions at this point are infinite. The critical phase fraction, zbc , at which this occurs is given by: zbc ¼

2 þ K  ðf  2ÞÞ 4Kf

ð7:66Þ

Physically, the turning point given by Eq. 7.66 represents the phase fraction at which the energy minimum solution abruptly ceases to exist. Therefore, the phase fraction of the system must jump abruptly to an end-of-range extremum value (zb = 0 or 1). It turns out that only for certain combinations of material

7.4 Stability Conditions and Path Dependences for Coherent Phase Transformations

235

parameters would the turning point have a value that is within the physical limits of the stability diagram. The values of K for which these are realized are: 2  f 1 3f þ 2 \ \ 2 K 2 2f 1 3f þ 2 [ [ 2 K 2

ðf [ 0Þ

ð7:67Þ

ðf\0Þ

ð7:68Þ

These conditions show that K must be bounded from above and below, which also means that for low enough values of K the turning point will disappear. To evaluate the magnitudes of the effects described in the foregoing for the Zr–H system over the temperature and average composition ranges of interest in practical applications to operating conditions of pressure tubes and fuel cladding materials in nuclear reactors, it is necessary to determine what might be sensible magnitudes of the parameters K, f, and d. This is done in Chap. 8.

7.4.2 Stability Conditions for Open Isomorphic Systems In Sect. 7.3 thermodynamic equilibrium conditions are given that are applicable to closed ideal isomorphic systems. These are systems having cubic crystal structure, a linear dependence of the lattice parameter of each phase on composition (Vegard’s law), and a constitutive response of the material that is isotropic elastic. The results show that the solutions obtained represent stable thermodynamic states over the full range of equilibrium compositions because the tie-line compositions do not depend on the phase fractions of each phase. This result is similar to that obtained for liquid mixtures and incoherent solid phase mixtures. Moreover, provided there is no reduction of the coherency energy through plastic relaxation when the system is in its two-phase state, then the theory also predicts that there would be no hysteresis in a closed isomorphic system. This prediction for the isomorphic system applies for all values of the dimensionless coherency factor less than the Williams point. The result is identical to that for a polymorphic system when both phases have similar properties and there is no reduction in the coherency energy with increasing phase fraction. However, when the system is open to an external source of hydrogen one might expect that the lack of dependence on phase fraction of the tie-line compositions in the isomorphic system would not apply. To demonstrate this—and the consequences of this result—stability relationships are derived in this section for the ideal isomorphic system treated in Sect. 7.3, but now evaluated when open to a very large reservoir of hydrogen so that the external hydrogen chemical potential in open contact with the system can be assumed to remain at a constant value. Experimentally such a situation would apply, for instance, when the material is in contact with hydrogen gas provided by an effectively inexhaustible supply.

236

7 Theories of Coherent Phase Equilibrium

The thermodynamics of such a system are given by the grand thermodynamic potential, viz., Xð^cH ; lrH Þ ¼ Gcoh  lrH ^cH

ð7:69Þ

where Gcoh is given by Eq. 7.30 and lrH is the (external) chemical potential of the reservoir. For this system, only the external chemical potential of the reservoir is fixed. The average composition of the system at equilibrium is obtained by minimizing Eq. 7.69 with respect to this composition variable, viz., oXð^cH ; lrH Þ dGcoh ¼  lrH ¼ 0 o^cH d^cH

ð7:70Þ

From the definition of the chemical potential lH ð^cH ; P; TÞ of hydrogen in the two-phase composition range given by the derivative of the coherent Gibbs free energy of Eq. 7.30 with respect to the average composition, the equilibrium condition of Eq. 7.70 can be expressed as: lH ð^cH ; P; TÞ ¼ lrH

ð7:71Þ

Writing Xð^cH ; lrH Þ in Eq. 7.69 in terms of phase fraction, zb ; rather than average composition, Schwarz and Khachaturyan [24, 25] show that the variation of Xðzb ; lrH Þ with zb has four topologically different types of behavior as graphically illustrated in Fig. 7.5. Curves ‘a’ and ‘d’ describe the dependence of r a ac Xðzb ; lrH Þ on zb for lrH \lbH ðcbc H Þ and lH [ lH ðcH Þ; respectively. In these cases b r there is a monotonic increase or decrease in Xðz ; lH Þ with increasing zb with only one minimum located at zb ¼ 1 or 0, which correspond to the single-phase state for the a phase for curve a, and the single-phase state for the b phase for curve d. Between these two extremes there are two curves, b and c, for which there exists a maximum in-between two minima located at zb ¼ 1 and 0. These maxima occur r a ac when lrH is in the range: lbH ðcbc H Þ\lH \lH ðcH Þ. The minimum for curve a at b zb ¼ 0 corresponds to ^cH \cac H while the minimum for curve d at z ¼ 1 correbc bc ac sponds to ^cH [ cH , with cH and cH defining the limits of the composition ranges of the single-phase states of the a and b phases, respectively. The barrier inbetween these two states is produced by a contribution from the coherency energy  2 that—as in the polymorphic case—is proportional to zb : The presence of this term results in there being a maximum in DXa!b proportional to the macroscopic volume of the b phase. As in the case for the polymorphic transformation, this barrier cannot be overcome by thermal fluctuations alone and needs a corresponding increase in thermodynamic driving force. Without this increase, the state on the other side of the barrier cannot be reached. This means, as pointed out by Schwarz and Khachaturyan [25] that the barrier completely locks the transformation from one phase to the other in this direction. Thus, although both phases may be present during the charging process, they are not in thermodynamic equilibrium and even infinitesimal changes in phase fraction from one to the other

7.4 Stability Conditions and Path Dependences for Coherent Phase Transformations

237

Fig. 7.5 Free  energy,  Xcoh ðxÞ (X ^cH ; lrH in the text) of an open, coherent two-phase a/b solid as a function of the volume fraction x (zb in text) of the solute-rich b phase. The solid is in equilibrium with a source of interstitials of constant chemical potential. Four curves are shown for increasing values of lr (lrH in text) (from Schwarz and Khachaturyan [25])

would require an increase in thermodynamic driving force, lrH ; in excess of that existing for the case shown by curve b. The presence of a macroscopic thermodynamic barrier for this type of phase transformation arising from the dependence of the coherency energy on the total volume of the system differs from thermodynamic barriers found in classical firstorder phase transformations; i.e., in systems in which any stresses existing within the phases in contact with each other are decoupled (incoherent equilibrium). Such first-order phase transformations also have barriers, but they are microscopic nucleation barriers that can be overcome by thermal fluctuations. Such fluctuations can be slowed down, reducing the rate of the transformation, but they cannot block the transformation entirely. In addition, in a classical first-order transformation, the forward and reverse transformations occur when the chemical potentials of the parent and product phases are equal, whereas in the present open system, in which the thermodynamic driving force is kept constant, the transformation continues until it is complete. Furthermore, ergodicity, a main requirement of Gibbsian thermodynamics, is also no longer applicable to this coherent system since the

238

7 Theories of Coherent Phase Equilibrium

state of the system becomes history dependent (i.e., it exhibits hysteresis). Note that, the stability conditions of the open isomorphic system share similarities with those of a system having a polymorphic phase transition when the physical conditions are such that a turning point is reached. That the open isomorphic system exhibits hysteresis can be illustrated through the phase transformation loop shown in Fig. 7.4. In this loop the effect on phase stability and associated hydrogen composition of an increasing and then decreasing external chemical potential is shown. Starting with a material containing only residual amounts of hydrogen (i.e. in the a-phase) the external chemical potential is increased monotonically causing the system to absorb hydrogen atoms. As long as the external chemical potential increase is such that lrH \laH ðcac H Þ the material remains in the single-phase state of the a-phase. For the r a ac composition range lbH ðcbc H Þ  lH  lH ðcH Þ the solid can exist in either the a- or bphase, but since it started out in the a-phase it will not transform into the b-phase until lrH ¼ laH ðcac H Þ: At this point the a-phase is no longer stable and transformation to the b-phase starts. This transformation continues, driven by the fixed external chemical potential, until the solid converts entirely to the b-phase, of composition cb;end : From Fig. 7.4 it is evident that cb;end [ cbc H H H : (In fact, it can be b;end bo shown that cH is bigger even than cH ; which is the composition for incoherent equilibrium of the b-phase.) The composition cb;end can be deduced from the H intersection of the horizontal dashed line in Fig. 7.4 with the limit of the singlephase state of the b-phase line; i.e., when the following equality between chemical b b;end potentials applies: lr;a!b ¼ laH ðcac Þ: H H Þ ¼ lH ðcH It is instructive to note that this increase in the composition of the b-phase is similar to that prevailing for the case of polymorphic phase transformations in a closed system. This is because in the closed polymorphic case the tie-line compositions are dependent on the phase fraction. However, in that case equilibrium is maintained between the two phases. Note, though, that in the closed polymorphic case these equilibrium tie-line compositions are derived with a parallel tangent construction whereas for the closed ideal isomorphic system the usual common tangent construction gives the equilibrium compositions. The fact that establishment of equilibrium compositions in the former case requires a parallel tangent construction means that, in the polymorphic case, hysteresis is possible when there is a reduction in coherency energy prior to reversal of the phase conversion process. On the face of it, this differs from the case of the closed isomorphic system. However, the common tangent construction for establishing equilibrium compositions in that case would only be valid when there is no dependence of the  2 coherency energy on zb ; which applies when full coherency is maintained throughout the phase transformation cycle, but is not the case when this energy is plastically relaxed. In contrast, in an open ideal isomorphic system with an external source of hydrogen, equilibrium can only be achieved when the transformation to the b single-phase state of composition cb;end is complete. A similar H

7.4 Stability Conditions and Path Dependences for Coherent Phase Transformations

239

argument applies to the reverse transformation starting from, say, composition cb;end of the b single-phase state, a result which is also shown in Fig. 7.4. H The lack of equilibrium between the two phases in the open isomorphic case during the phase transformation process results in hysteresis between the forward and reverse transformations. This can be seen from the foregoing analyses and the graphical constructions of Fig. 7.4 which shows that the complete hysteresis cycle from a ? b?a produces a hysteresis gap, expressed in terms of chemical potential differences, given by:   b bc bc ac laH ðcac ð7:72Þ H Þ  lH ðcH Þ ¼ 2A cH  cH Note that, the hysteresis expressed by Eq. 7.72 derives solely from the rate of change with composition of the coherency energy at the start of the phase transformation when minute amounts of a- or b-phase are first formed within the respective single-phase starting phases. A part of the rate of change of coherency energy arises from the rate of change of an elastic work contribution (i.e., a nonconserved quantity) produced by the interaction of the inserted or removed misfitting hydrogen atoms with the uniform image expansion field produced by all the other misfitting hydrogen atoms or precipitates already present in the material. Since the compositions at which the instability limits for the single-phase states are reached are always the same, it does not matter whether the coherency is maintained or lifted during the subsequent increase in size and phase fraction of the newly formed phase. That the magnitude of the hysteresis for this open, ideal isomorphic system is proportional to the coherency energy barrier appears to be special to this case in which any other possible sources of dissipation are contained within the bounds of this coherency energy barrier. The reason for this appears to be because, in going from one single-phase state to the other, the system starts and finishes in states for which the coherency energy is the same. However, as Flanagan et al. [10] have argued (Sect. 6.2), in general the macroscopic increase in chemical Gibbs free energy that is needed to drive the reaction over the cohesive energy barrier to completion in each direction results in entropy production associated with these movements. These authors also present experimental evidence to suggest that this must be the dominant source of energy dissipation in all metal–hydrogen systems in which an energy barrier with these properties must initially be overcome, including under closed conditions or for hysteresis scans in which the phase transformation is only partially completed in each direction. This latter possibility was not addressed by Schwarz and Khachaturyan [25]. A contribution to hysteresis produced by finite phase boundary movement during phase transformation is expected, however, when the transformation during the hysteresis cycle is reversed before full phase transformation to a single-phase state has occurred. In addition to this source of hysteresis, there might also be a contribution from any enthalpy change produced by unrecoverable plastic work as a result of plastic relaxation of the coherency energy as the new phase forms, if this is large enough to be measurable. Similar arguments would apply to a closed

240

7 Theories of Coherent Phase Equilibrium

system cycled through a thermal phase transformation loop regardless of whether the transformation occurs in a polymorphic or isomorphic system. The global thermodynamic argument used by Schwarz and Khachaturyan [25] for the open isomorphic system cannot be used to determine the path that the system might take to get from one single-phase state to the other during absorption/desorption since the system is not in equilibrium during this process. The theory can be definitive only in establishing the final compositions of the singlephase states after complete conversion from the starting single-phase states. Thus, it cannot address what the state of the system would be if the absorption process were reversed at some point prior to full conversion. In addition, the theory predicts that in a cubic crystalline solid with a miscibility-gap phase boundary such as Pd–H, there would be no hysteresis in a closed system during a thermal cycle. However, experimental evidence for the Pd–H system (Sect. 6.2) is that for opensystem pressure–hysteresis scans the hysteresis gap has in some cases been found to be the same whether reversal of the process occurred prior to, or after, the system had reached its opposite single-phase state [8, 10]. In addition, Flanagan et al. [9] have found approximate equality between the hysteresis gaps measured in Pd–H systems for both closed as well as open systems. Both of these results are contrary to the theoretical predictions of Schwarz and Khachaturyan [25] that the closed system should exhibit no hysteresis and suggest that these authors’ argument may apply only to the ideal case considered. Such a system, however, may not exist in practice. Thus, theoretical considerations have been advanced in Chap. 4 in the case of the Pd–H system that this system cannot be considered to behave as an ideal isomorphic one. Further support for this comes from Chiang and Johnson [5] who point out (Sect. 7.3) that various deviations from the ideal case can result in the isomorphic phase transition to have thermodynamic properties similar to those of a polymorphous phase transition. Based on these considerations, one must conclude that the role of internal entropy production needs to be factored into these thermodynamic arguments for a complete description of the total irrecoverable free energy dissipation during the phase transformation. Combining the results of the analyses of the equilibrium properties of coherent polymorphic and isomorphic phase transformations given in this chapter with the thermodynamic models of hysteresis of first-order phase transformations given at the beginning of Chap. 6, one arrives at the following physical picture. When the phase transformation involves overcoming a macroscopic (i.e., a finite) energy barrier for the phase transformation to proceed in each direction, then there is a finite phase boundary movement associated with overcoming such a barrier. This movement of the phase boundary results in internal entropy production, which is the main source of hysteresis. In a closed system during a thermal phase transformation scan, the transformation of a small phase fraction of a-phase to the b-phase then starts during temperature reduction when a temperature is reached where the difference in the original and present value of chemical potential is sufficient to overcome the macroscopic coherency energy barrier created as a result of the formation of the b-(hydride)phase. The results given in Sects. 7.2–7.4 show that when there is no reduction in coherency energy by plastic deformation during

7.4 Stability Conditions and Path Dependences for Coherent Phase Transformations

241

this process, there would be no hysteresis attributable to this contribution to the Gibbs free energy change upon temperature reversal. We show in the next chapter that, in the limit of the volume fraction of the second phase approaching zero, the results obtained in this chapter are the same as those derived with the accommodation energy model. In this model, it was assumed that there is a reduction in the coherency energy by plastic deformation and that this is the sole source of hysteresis. The accommodation energy model requires, however, that this energy, and its subsequent reduction by plastic deformation, be quite large for the calculated hysteresis gap to be in accord with experimentally observed gaps. In systems where the calculated coherency energy from this accommodation energy term is not large, then the bulk of the hysteresis must come from another source. We suggest that this is the internal entropy production proposed by Flanagan et al. [10]. Missing in this thermodynamic approach are mechanistic models that would provide links between the magnitude of the energy barrier, the phase boundary movement, and the resultant entropy production. These considerations are further examined in the next chapter.

References 1. Cahn, J.W.: On spinodal decomposition. Acta Metall. 9, 795–801 (1961) 2. Cahn, J.W.: On spinodal decomposition in cubic crystals. Acta Metall. 10, 179–183 (1962) 3. Cahn, J.W.: Coherent fluctuations and nucleation in isotropic solids. Acta Metall. 10, 907–913 (1962) 4. Cahn, J.W., Larché, F.C.: A simple model for coherent equilibrium. Acta Metall. 32, 1915–1923 (1984) 5. Chiang, C.S., Johnson, W.C.: Coherent phase equilibria in systems possessing a consolute critical point. J. Mater. Res. 4, 678–687 (1989) 6. Eshelby, J.S.: Continuum theory of defects. Prog. Solid State Phys. 3, 79–144 (1956) 7. Eshelby, J.D.: Interaction and diffusion of point defects. In: Smallman, R.E.(ed.) Vacancies 76, London, The Metals Society, pp. 3–10 (1976) 8. Everett, D.H., Norton, P.: Hysteresis in the ‘‘Palladium-Hydrogen System’’. Proc. Roy. Soc. A 259, 341–360 (1960) 9. Flanagan, T.B., Clewley, T., Kuji, T et al.: Isobaric and isothermal hysteresis in metal hydrides and oxides. J. Chem. Soc., Faraday Trans. 1. 82, 2589–2604 (1986) 10. Flanagan, T.B., Park, C.-N., Oates, W.A.: Hysteresis in solid state reactions. Prog. Solid St. Chem. 23, 291–363 (1995) 11. Hillert, M.: In: Phase Equilibrium, Phase Diagrams and Phase Transformations: Their Thermodynamic Basis, 2nd edn. Cambridge University Press, Cambridge, UK (2008) 12. Johnson, W.C., Voorhees, P.W.: Phase equilibrium in two-phase coherent solids. Metall. Trans. A 18A, 1213–1228 (1987) 13. Johnson, W.C., Müller, W.H.: Characteristics of phase equilibria in coherent solids. Acta Metall. Mater. 39, 89–103 (1991) 14. Larché, F.C., Cahn, J.W.: A linear theory of thermochemical equilibrium of solids under stress. Acta Metall. 21, 1051–1063 (1973) 15. Larché, F.C., Cahn, J.W.: A nonlinear theory of thermochemical equilibrium of solids under stress. Acta Metall. 26, 53–60 (1978)

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16. Larché, F.C., Cahn, J.W.: Thermochemical equilibrium of multiphase solids under stress. Acta Metall. 26, 1579–1589 (1978) 17. Larché, F.C., Cahn, J.W.: The effect of self-stress on diffusion in solids. Acta Metall. 30, 1835–1845 (1982) 18. Larché, F.C., Cahn, J.W.: The interactions of composition and stress in crystalline solids. Acta Metall. 33, 331–357 (1985) 19. Lee, J.K., Tao, W.: Coherent phase equilibria: effect of composition-dependent elastic strain. Acta Metall. Mater. 42, 569–577 (1994) 20. Liu, Z.-K., Ågren, J.: On two-phase coherent equilibrium in binary alloys. Acta Metall. Mater. 38, 561–572 (1990) 21. Liu, Z.-K., Ågren, J.: Two-phase coherent equilibrium in multicomponent alloys. J. Phase Equil. 12, 266–274 (1991) 22. Pfeifer, M.J., Voorhees, P.W.: A graphical method for constructing coherent phase diagrams. Acta Metall. Mater. 39, 2001–2012 (1991) 23. Roıtburd, A.L.: Equilibrium and phase diagrams of coherent phases in solids. Sov. Phys.— Solid State 26, 1229–1233 (1984) 24. Schwarz, R.B., Khachaturyan, A.G.: Thermodynamics of open two-phase systems with coherent interfaces. Phys. Rev. Letters 74, 2523–2526 (1995) 25. Schwarz, R.B., Khachaturyan, A.G.: Thermodynamics of open two-phase systems with coherent interfaces: application to metal-hydrogen systems. Acta Mater. 54, 313–323 (2006) 26. Williams, R.O.: Long-period superlattices in the copper-gold system as two-phase mixtures. Metall. Trans. A 11A, 247–253 (1980) 27. Williams, R.O.: The calculation of coherent phase equilibria. CALPHAD 8, 1–14 (1984)

Chapter 8

Experimental Results and Theoretical Interpretations of Solvus Relationships in the Zr–H System

8.1 Introduction In Chap. 7 derivations were presented for the phase relationships of coherent binary systems for two bounding cases. In the first case the only source of coherency strain energy arises from the misfit strains between the two phases (polymorphic case) while for the other (ideal isomorphic case) the only source of misfit is that produced by the atoms themselves, each treated as a point defect that is misfitting in relation to the crystal lattice made up of a random mixture of the two components. In this case there is no additional misfit strain produced when these atoms segregate into two separate phases. These two cases lead to equilibrium phase relationships that are not only different in important ways from the usual Gibbsian equilibrium conditions for incoherent solid or fluid systems but also from each other. The equilibrium relationships obtained for both of these cases were described because it is not entirely clear which of these two bounding cases correspond to the Zr–H system, as discussed further in the following. Correspondence of the Zr–H system with the ideal polymorphic case requires, at a minimum, that precipitate and matrix have different crystal structures, thereby generating misfit strains between the two phases. Certainly there is a difference in crystal structure between the two phases for the Zr–H system. Despite this difference, as shown in Chaps. 2 and 4, the molar (misfit) volumes of hydrogen per mole of Zr in the a-Zr and c- and d-hydride phases are approximately the same in each phase. However, there is a volume difference between the molar volumes of Zr per mole of Zr in each of these phases, thus producing volumetric strain between them, in addition to possible shear misfit strains (although these could be internally relaxed). Correspondence of the Zr–H system to the ideal isomorphic case requires, at a minimum, that there is a linear dependence between the lattice parameter and the concentration of each component. The experimental determinations by MacEwen et al. [22] of the molar volume of hydrogen in a-Zr from measurements of the M. P. Puls, The Effect of Hydrogen and Hydrides on the Integrity of Zirconium Alloy Components, Engineering Materials, DOI: 10.1007/978-1-4471-4195-2_8,  Springer-Verlag London 2012

243

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8 Experimental Results and Theoretical Interpretations

variation of the lattice parameter of the a-Zr phase with hydrogen concentration (described in Chap. 4) shows that this dependence is linear (albeit over a fairly narrow concentration range). The same was found from the data of Yamanaka et al. [53, 54] for the dependence of the lattice parameter of the d-hydride phase with hydrogen composition, although earlier studies, as summarized by these authors and Zuzek et al. [55], found no such dependence. From all of these results one could conclude that the conditions of coherent equilibrium may be somewhere in between those of coherent polymorphic and ideal isomorphic systems. In the following, we have assumed that the Zr–H system has the equilibrium characteristics of a closed, coherent, polymorphic phase transformation system. This is because there could be conditions in this system where the minima of the Gibbs free energy do not correspond to stable equilibrium states and, therefore, the conditions for the existence of these unstable states in the Zr–H system needs to be determined. In the closed, ideal isomorphic system, on the other hand, the minima in the Gibbs free energy always correspond to stable equilibrium states.

8.2 Application of Coherent Phase Stability Analysis to the Zr–H System As discussed in Sect. 7.4, stability analyses of coherent equilibrium (extremum) solutions show that there could be physical conditions resulting in excluded ranges of composition throughout which stable two-phase equilibrium is not possible. These ranges would be accompanied by abrupt jumps in the phase fraction relationships because the equilibrium phase(s) on the other side of the excluded zone could only be reachable when approached from the opposite composition side upon changing the average composition of the solid. This results in a path dependence of the equilibrium phase relationships. The examples given by Johnson and Voorhees [11] are categorized in relation to the magnitudes of the dimensionless parameters, K, f, and d of Eqs. 7.47–7.49 giving, respectively, the ratio of the coherency energy to the chemical driving force, the degree of asymmetries in the variation with composition of the elastic constants and of the chemical free energies of the two single-phase states, a and b. We start by considering the case f = 0 and d = 0. In this case the free energy curves for both phases are the same as well as the elastic constants. For this case, K reduces to the following expression: K¼

4Gð1 þ mÞe2  ð 1  mÞ

1 ao cbo H  cH

2

ð8:1Þ va

From Eq. 7.52, assuming ideal solution behavior, va reduces to:

8.2 Application of Coherent Phase Stability Analysis to the Zr–H System

va ¼

q00 kB T 1 RT  ao ao ao VM ð1  cao NA ð1  cH ÞcH H ÞcH

Inserting the result of Eq. 8.2 into Eq. 8.1 yields:   ao M 1  cao 4Gð1 þ mÞe2 V H cH K¼  2 ð1  mÞ RT bo cH  cao H

245

ð8:2Þ

ð8:3Þ

Using the notation of Chap. 6, it is seen that Eq. 8.3 can be written as:   ao ao 2D wel inc 1  cH cH K¼ ð8:4Þ  2 RT ao cbo  c H H where, for the spherical geometry considered by Johnson and Voorhees [11], wel D wel inc  D inc ðsphereÞ ¼

2Gð1 þ mÞe2  VM : ð 1  mÞ

ð8:5Þ

Note that the expression for D wel inc ðsphereÞ is identical to that for the self strain energy of an isotropically misfitting point defect, as given by Eq. 4.70. This is as expected for this continuum elastic model of misfit induced strain energy since the result does not depend on the size of the precipitate—only the full coherency energy has a dependence on the aggregate size of the phase through its dependence on the volume fraction of each phase. To evaluate K for the Zr–H system we recall that in the foregoing derivations cH  rH =bpha : This choice of composition variable ensures that the terminal composition of the hydride phase is equal to one, similar to what is obtained for a binary substitutional solution. Assuming, for the purposes of the following analysis that d-hydride is the phase that is in equilibrium with the a-Zr phase throughout the temperature range from ambient to the eutectoid temperature, we note from phase diagram determinations that the d-hydride phase has a composition close, but not exactly equal, to the stoichiometric value of rH = 1.5. That is, the phase diagram determinations indicate that pure d-hydride exists over a range of composition that is fairly narrow and straddles the stoichiometric value. The fact that the composition range of the fcc d-hydride phase is so close to the stoichiometric value— above which it transforms to the fct e-hydride phase—suggests that the choice of the available equivalent interstitial sites, bd-hyd, applicable for this crystal structure in determining the terminal composition of the Zr d-hydride system is 1.5 and not 2 as would be inferred from geometry of the available quadrahedral interstitial sites of the fcc lattice. Hence, this is the value of bd-hyd that is used in the following numerical evaluations. For the incoherent solvus concentration we assume that the average of the formation and dissolution solvi, which is denoted by cao H ; provides a sensible estimate of this composition. Using the linear regression fit to the solvi data of Pan

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8 Experimental Results and Theoretical Interpretations

et al. [31] given in units of atomic fraction (which is approximately numerically the same as rH for the low concentration considered), we obtain at 260 C that cao H ¼ 0:00485=1:5 ¼ 0:00323: We assume the corresponding value for 1 From Leitch and Puls [21] we have for cbo H ¼ 1:45=1:5 ¼ 0:967: el Zr ¼ 6; 176 J/(mol Zr) while RT = 4,432 (J/mol Zr) at D winc ðsphereÞ ¼ 441:1  V 260 C. Inserting these data into Eq. 8.4 we obtain KZrH ¼ 0:00694  D wel inc ðsphereÞ=RT  0:00967: This shows that KZrH  1; and is, therefore, well below the Williams point given by KZrH ¼ 1. Hence, the Gibbs free energy is a minimum throughout the full range of phase fractions from zb ¼ 0 to 1; with no excluded zones. The extrema values for the tie-line compositions given by Eqs. 7.57–7.59 then represent composition ranges over which the system is in stable equilibrium. It is also evident that the magnitude and temperature dependence of KZrH is dominated by the value of the solvus concentration since the combined value of the other terms is approximately equal to unity. It is of interest now to compare the result obtained from Eqs. 7.54–7.56 with those obtained from the accommodation energy approach given in Chap. 6. We assume that the results given in Chap. 6 correspond to those of Eqs. 7.57–7.59 only at the start of the transformation to the two-phase mixture when zb ¼ 0: From Eq. 7.57, taking account of the foregoing result for Zr–H system showing that KZrH  1; we then have that W ffi 1. Substituting this value into Eq. 7.58 for the dimensionless solvus concentration, Ya, this relation simplifies to: Y a ¼ 1 þ K From the definition of Ya given by Eq. 7.44, we have:  K  bo ao ao c  c ¼  c cac H H H 2 H

ð8:6Þ

ð8:7Þ

ao which, from Eqs. 8.4 and 8.5, and utilizing the condition that cbo H  cH and   ao 1  cH ffi 1 (for the case of Zr–H), reduces to:

cac D wel H inc ðsphereÞ ¼1þ ao cH cbo H RT

ð8:8Þ

This equation can be viewed—in this linearized solution—as being equivalent to the first term in a Maclaurin series expansion of f ð xÞ; where x is equal to the second term on the right-hand side of Eq. 8.8 and f ð xÞ ¼ expð xÞ: Making this correspondence, then, shows that the result given by Eq. 8.8 is equivalent to that obtained by Puls [33]—reproduced in Chap. 6—given by:

1

In the following specific application of the theory the b phase refers to the d-hydride phase in the Zr–H system.

8.2 Application of Coherent Phase Stability Analysis to the Zr–H System

" # cac D wel H inc ðsphereÞ ¼ exp cao cbo H H RT

247

ð8:9Þ

This comparison shows that the result obtained by Puls [33] is rigorously applicable only to the case zb ffi 0: Moreover, in applications of this result, Puls assumed that the hydride precipitates could have any shapes and could also be anisotropically misfitting. From the analysis of Johnson and Voorhees [11] and others it is clear, however, that application of Eq. 8.9 to non-spherically shaped precipitates having, additionally, anisotropic misfit strains, can only be approximately correct. In addition, from the full solution of Johnson and Voorhees [11], for average composition values ranging over the full phase diagram, it is clear that the relationship between coherent and incoherent solvi compositions given by Eq. 8.9 applies only to the first appearance or the last remaining dissolution of hydride precipitates (corresponding to the condition zb ffi 0) in which case the average hydrogen composition of the solid would correspond to that of the coherent solvus concentration (the a single-phase composition limit in the coherent case). Evaluation of Eq. 8.7 yields: ao cac H  cH ¼

0:00967 ð0:964Þ ffi 0:00466 2

ð8:10Þ

As a result of the symmetry assumed in this case, there is an identical, positive shift in magnitude between the coherent and incoherent tie-line compositions of bo the b phase, given by cbc H and cH ; respectively. Therefore, in relation to the much larger composition values prevailing on the pure hydride side of the phase diagram, the magnitude of this composition shift is predicted to be below the level of measurability for this phase. It is of interest now to determine whether, over the range of phase fractions from 0 B zb B 1, there are predicted to be any free energy maxima and, hence, excluded compositions with abrupt changes in phase fractions when either or both of f and d = 0. We consider the case f = 0 and d = 0. For this case the shifts in reduced compositions as a result of coherency effects are given by Eqs. 7.61 and 7.62. It is seen that the composition shifts are no longer the same for each phase since the shift for Yb is greater than for Ya by the factor (1 ? f). We next determine whether there are any values of average composition in the range Ya and Yb for which the free energy extremum represents an energy maximum. The turning point at which such a change from minimum to maximum would occur is given by zbc (Eq. 7.66). We need to determine whether this critical phase fraction value falls within the physically possible range of 0 B zb B 1 for the calculated values of f and K. Assuming ideal solution behavior for vb as well as for va we have: q0 kB T 1 RT 1 RT      ¼  ð8:11Þ vb ¼ 0  bo bo V NA 1  cbo cbo V M 1c Zr 1  cbo cbo c H H H H H H

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8 Experimental Results and Theoretical Interpretations

ao Using the data for cbo H ; cH and T assumed in the foregoing examples yields b 9 2 v ¼ 9:920  10 J=m while va ¼ 9:801  109 J=m2 : From these values of v we get f ¼ 8:88. Inserting this value into Eq. 7.66 with the present value of KZrH ¼ 0:00967 we obtain zbc ¼ 8:238; which is well outside the range of possible physical values of zb from zero to one. Thus the stability analysis predicts that the extremum values given by Eqs. 7.61 and 7.62 represent energy minima over their entire range for these values of f and K. In fact, from the limit Eq. 7.67 for KZrH it is seen that KZrH must be [0.0967 for there to be a turning point (where the energy minimum changes to an energy maximum) within the physically possible range of zb values. For this to be possible with the present value of KZrH , f would have to be approximately a factor of ten larger. However, having such large positive values for f seems physically unlikely, assuming the correctness of the extant phase diagram data for the Zr–H system. The general expressions for the shifts in the reduced coherent tie-line compositions are given by Eqs. 7.61 and 7.62. Using Eqs. 7.44 and 7.45, which relate the actual compositions to the reduced compositions, with the numerical results obtained for the reduced compositions, we obtain:    ac   bo  ao K 1  2zb ¼ 0:00330 cH  cao ð8:12Þ H ¼ cH  cH 2       bo ao K 1  2zb ffi 0:0338 cbc ¼ ð1 þ fÞ cbo ð8:13Þ H  cH H  cH 2

where the numerical results are those obtained with the foregoing data and for as zb ffi 0: In this example va has the same magnitude   in the fully symmetric case. ao is the same as that in the Therefore, the shift in composition for cac  c H   H bc bo symmetric case while that for cH  cH is increased by the factor ð1 þ fÞ since f [ 0. A positive value for f was obtained because it was assumed that the variation with composition of the free energy of the a phase from its equilibrium value is steeper than that of the b phase. When assuming ideal solution behavior ao for both phases, the larger value of cbo H compared to cH produces a less rapid change with composition in the variation of the chemical free energy of the b, compared to the a phase.   bo when f is \ 0. Such a value could A smaller shift is obtained for cbc H  cH

arise if the composition of the d-hydride phase is very close to being exactly stoichiometric. Only negative values less than one have physical significance since va =vb must be positive. Assuming for this case that f = -1/2, for which va =vb ¼ 1=2 and zb ffi 0 the result is:     bo ao K ffi 0:00171 ð8:14Þ cbc ¼ cbo H  cH H  cH 4

8.2 Application of Coherent Phase Stability Analysis to the Zr–H System

249

Again, because the value of va has been chosen to be the same in all three cases only the value of vb has been varied, there is no change in the value of and ac cH  cao H : We have given two possible values for vb because one might expect less accuracy in the experimental determination of the (a ? d)/d phase boundary compared to the determination of the composition dependence with temperature of the solvus. We come to this conclusion because the accuracy with which the composition of the solid hydride phase can be determined is expected to be less as a result of the much larger amount of hydrogen that this phase contains, which generally means that their compositions were determined by the usually less accurate weight gain measurement methods. In addition, considerably more effort has been devoted to obtaining accurate experimental values of the solvus than of the corresponding pure hydride phase composition. Finally, insertion of the value of f into Eq. 7.68 shows that, with f = -1/2, the value of KZrH is well below the threshold above which there would be excluded zones over the range of tie-line compositions from 0 B zb B 1. The only differences for the case f \ 0 are that the ranges of tie-line compositions are now less than compared to the symmetric case while the dependence of the phase fraction of the b-phase on average composition results in a parabola that opens to the right in a zb versus W (reduced average composition) phase stability diagram. This contrasts with the case f [ 0 for which it opens to the left. In both cases, however, the curvature of the parabola is very small since it depends on the magnitude of KZrH ; which is small in the ranges of average composition and temperatures of practical interest to the Zr–H system. It should be noted that because the dependence of the coherency energy is generally nonlinear in the phase fraction zb, resulting in the generation of internal stresses in the solid depending, at the very least, on the phase fractions of the coexisting phases, there is now an apparent violation of the Gibbs phase rule applicable for liquid and incoherent solid phase mixtures at constant temperature and pressure. This is because there is now an additional degree of freedom available to the system [11] on top of the usual ones of pressure, temperature, and the number of components minus one (the latter imposed by the mass balance constraint). This may be one reason why the c-hydride phase, which has lower hydrogen composition and transformation strains compared to the d-hydride phase, can coexist with the d-hydride phase, as found under certain conditions.

8.3 Evaluation of Models of Hysteresis for the Zr–H System In the next section (Sect. 8.4) experimental results are summarized for the solvus in the Zr–H system. These results show that there is considerable hysteresis between the formation and dissolution solvi. Almost all of these results were obtained from temperature scans in closed systems at constant average hydrogen composition. These scans give the temperatures at which the first and last appearances of the hydride (b) phase first occurred. These results and similar ones

250

8 Experimental Results and Theoretical Interpretations

determined in other structural phase transformation systems, such as Nb–H, Ti–H, and V–H, were rationalized in terms of the accommodation energy model. However, lack of data for the phase compositions in open systems as well as in closed systems for increasing values of average compositions masked the inadequacies of these accommodation energy models, as pointed out by Flanagan et al. [8], in providing an explanation for hysteresis across the entire two-phase composition range. These authors argued that the accommodation energy model cannot provide a universal explanation for the source of hysteresis when considering the results in other systems and over broader ranges of experimental conditions. It also became apparent from parallel developments of phase equilibrium relationships in coherently stressed solids that the accommodation energy model represents an incomplete accounting of the total coherency energy in solids containing misfitting atoms and two-phase mixtures. Two examples of phase relationships predicted with the use of these more complete coherency energy models are given in Chap. 7. However, as Johnson and Voorhees [11] have pointed out for the case of closed, symmetric, polymorphic systems, theory predicts that below the Williams point the equilibrium phase fractions over the two-phase region are independent of the path chosen to reach that state. That is, regardless of whether a targeted average composition in the solid is achieved by a processing path in which this composition is reached starting from either lower or higher values, a unique value for the equilibrium compositions of the two coexisting phases and, hence, for the phase fractions, is obtained at a given temperature. This independence of the volume fractions of the phases on the path taken is equivalent to that of a two-phase fluid system. Hence these models predict that there would be no hysteresis between hydride formation and dissolution over the entire range of tie-line (phase equilibrium) composition values with average composition provided there is no reduction in the coherency strain energy such as could be caused by plastic relaxation and/or diffusion of the metal atoms in these systems during or after an increment of phase transformation has occurred. There could, however, in principle, be path dependencies in the phase fractions of these systems when there are deviations from the fully symmetric system. General stability analyzes of the extremum solutions showed that under some conditions these solutions represent energy maxima rather than minima, resulting in ‘‘forbidden zones’’ and path dependencies giving the appearance of hysteresis in the compositions of the two phases. In the preceding section it is shown, however, that for temperatures and average hydrogen compositions of practical interest this is not likely in the Zr–H system. Nevertheless, the phase relationships in binary solids subjected to coherency strains derived in Chap. 7 exhibit characteristics that are similar to those proposed in the past to explain the origin of hysteresis in such systems. For one, in a closed system it is found that the presence of coherency strains results in tie-line compositions being displaced from their corresponding phase field compositions with these shifts changing with average composition. Pfeifer and Voorhees [32] show graphically, from plots of the chemical free energies with composition, that these changes in equilibrium compositions with average composition in coherent solids

8.3 Evaluation of Models of Hysteresis for the Zr–H System

251

are a consequence of the requirement of a parallel tangent construction to establish the equilibrium compositions. This construction is required to take account of the dependence of the total coherency energy of the solid on the square of the average composition or phase fraction. From the derivative of the coherency energy with average composition it is seen that the greatest difference in the coherent equilibrium compositions of the two co-existing phases occurs between the average composition at which the first increment of b phase transforms from the a singlephase state and the average composition at which the first increment of a phase forms from the b single-phase state. These diametrically opposed sets of equilibrium compositions would, in this model, show up only upon complete conversion from one single-phase state to the other. Yet the experimental results for Zr–H and other metal-hydrogen systems show that there is also hysteresis between the solvus compositions for hydride dissolution and formation when both are determined at close to the same (zero) phase fraction values. The hysteresis between forward and reverse transformation in the case of complete conversion from one single-phase state to the other in an open system was analyzed by Schwarz and Khachaturyan [51] as summarized in Sect. 7.4.2. It is relatively easy to understand the reason for the hysteresis in this case, since the phase conversion can only proceed in each direction when the external chemical potential exceeds a threshold value dictated by the compositional derivative of the coherency strain energy at that phase fraction or average composition value. When this occurs, however, equilibrium with the external source of hydrogen cannot be reached until complete conversion of one phase to the other has occurred. That there exists a similar hysteresis gap of the same magnitude in both open and closed systems for thermal cycles extending over only a small fraction of the entire composition range between the two single-phase states is, at first glance, not so easy to understand, particularly if one discounts the role of plastic relaxation of the coherency strain energy as playing a role in this. In the following we propose a tentative explanation that accounts for both sets of results. This is based, in part, on insights provided by Flanagan et al. [8] concerning the thermodynamic origin of hysteresis, as summarized in Sect. 6.2. Flanagan et al. [8] postulate that hysteresis occurs between the forward and reverse phase transformations when a finite (macroscopic) energy barrier must be overcome in each direction before phase conversion is possible. Because the threshold value imposed by this barrier is of finite extent, it results in a finite phase boundary movement when it is exceeded. From the theory of irreversible thermodynamics, such finite phase boundary movements always lead to internal entropy production. The authors further suggest that any unrecoverable expenditure of enthalpy from plastic work generated during or after the phase transformation makes only negligible contributions to the overall free energy losses giving rise to the hysteresis gap.2

2

On theoretical grounds, there is the possibility that this could be the case for the Zr–H system since it is observed (see Chap. 3) that hydride precipitates form long, thin platelets and these

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Based on these considerations, Flanagan et al. [8] then derived thermodynamic relationships showing the link between the finite Gibbs free energy driving force needed for the phase transformation to proceed in either direction and the corresponding internal entropy increases produced by the finite boundary movements. This interpretation of the source of hysteresis through internal entropy production is opposite to that assumed in the accommodation energy models in which the hysteresis derives from irrecoverable accommodation energy changes produced, with no other assumed sources of irreversibility. Flanagan et al. [8] do not provide any definitive answers as to the source of the finite energy barrier that is the cause of the internal entropy production, pointing out only that the source of hysteresis is unlikely to be from irreversible changes in the accommodation energy during phase transformation because these changes are too small to explain the experimentally observed hysteresis gap. The theory of coherent phase equilibrium given in Chap. 7 and in Sect. 8.2 shows, however, that the accommodation energy is actually only the self strain energy component of the total coherency energy of a misfitting two-phase mixture. The other component consists of the image interaction term. This term differs from the accommodation (self strain) energy in two ways. One is that it is of macroscopic (finite) extent. The other is that it cannot be reduced by plastic deformation because it depends only on the dilatational components of the transformation strains. In contrast, such reduction is possible for the accommodation energy (for a finite-sized defect such as a precipitate) because large shear stresses that can be plastically relaxed can be generated by this term depending on the shape of the precipitate. The image interaction component of the total coherency energy thus represents an energy contribution to the total energy that cannot be dissipated by plastic deformation during phase conversion,3 while its sign and magnitude would vary across the twophase average composition range. It is proposed, therefore, that this component of the total coherency energy is the sought-for source of hysteresis since it requires a finite increase in chemical free energy to overcome it from which follows a finite phase boundary movement and, hence, internal entropy production. Qualitatively, the effect that the coherency energy has on the two-phase coexistence compositions as a result of the foregoing two characteristics of the image interaction energy component of the coherency energy is then twofold.

(Footnote 2 continued) shapes are the results of hydride formation by an invariant plane strain transformation for which almost all of the volumetric transformation strain would be directed in the platelet normal direction. For this extreme anisotropy of the transformation strain, using an expression given by Puls [35] for an aspect ratio of 0.1, the elastic accommodation energy is reduced by a factor of *6.5 compared to the value obtained for a hydride precipitate of the same shape but with transformation strains assuming hydride formation occurred by a pure lattice strain transformation. 3 It could, however, be reduced to zero if all the stresses resulting from the misfit strains of the precipitates are relieved through diffusion of the atoms of the underlying lattice structure with respect to which the misfit strains are imposed.

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The first effect is that the two-phase coexistence compositions vary with phase fraction (or average composition). This then results in a difference in phase compositions between the start of the forward and reverse transformations if these occur at different phase fractions. Note, however, that this difference in forward and reverse compositions—if it can be detected experimentally—is not a hysteresis effect. If, in addition, the self strain energy component of the coherency energy is plastically reduced upon reversal of the phase transformation, then this term also provides a contribution to a difference in the two-phase coexistence compositions between the forward and the start of the reverse transformations. On the other hand, there would be no change (approximately) in the compositions for forward and reverse isothermal phase transformations when these occur at the same phase fraction value and the self strain energy components of the total coherency energy for these two reactions are either negligible or remain elastically constrained. An important point, however, is that the change in the self strain energy has the same magnitude across the entire two-phase field while that of the image interaction energy varies linearly with phase fraction. Thus, at phase field compositions at which the phase field fractions reach their bounding values, the self strain energy is the most important term. There would then be hysteresis in these phase field compositions if, as assumed in the accommodation energy model, the self strain energy is plastically relaxed during or before the reverse transformation. This source of contribution to the hysteresis in the (tie-line) compositions becomes less important for intermediate phase fraction values. At these phase fraction values an increasing amount of hysteresis is created through internal entropy production because of the increasing importance of the image interaction energy term to the total coherency energy. The macroscopic extent of the energy barrier due to this term results in hysteresis in both the forward and reverse (tieline) compositions even at identical phase fraction values because overcoming this barrier results in internal entropy production even though there is now no, or little, reduction in the coherency energy by plastic deformation between forward and reverse transformations. Qualitatively, it would seem possible on the basis of the foregoing considerations that the contributions of these two sources of hysteresis vary in magnitude in such a way across the two-phase average composition field that their combined effects on overall hysteresis of the phase field or tie-line compositions produces an approximately constant difference, in accord with results of experimental observations.

8.4 Summary of Results of Experimental Solvus Determinations From the start of interest in the effects of hydrogen and hydrides on zirconium alloys there have been numerous experimental studies to determine the solubility limit of hydrogen in these materials. In Chap. 6 general features of the solvus are

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described focusing on theoretical models of the hysteresis. Here some key experimental results are summarized with emphasis on those results that have found use in practical applications or have provided data and insights into the effect on the measured solvus compositions of differences in material properties and measurement techniques.

8.4.1 Effect of Experimental Methods There have been many methods to determine the composition/temperature relationship of the solvus boundary. In this section the influence these methods have on the results obtained is explored. The following techniques have been employed: • • • • • • • • •

Differential scanning calorimetry (DSC) Dynamic elastic modulus (DEM) Internal friction (IF) Electrical resistivity Dilatometry Diffusion equilibration Equilibrium vapor pressure Neutron diffraction (ND) Metallography

The earliest determinations of the solvus in Zr and its alloys used equilibrium vapor pressure (see references in Kearns [12]), diffusion equilibration [12], dilatometry [7, 52], and electrical resistivity [25]. Kearns [12] adopted a diffusion equilibration method to avoid what he thought were supersaturation effects in the methods for solvus determinations during heating or cooling. These supersaturation effects were assumed at that time to be the reason for the large differences in the values of the solvus temperatures obtained between the cooling and heating segments of the temperature cycle. Subsequently, it became evident that most of the differences between the solvus temperatures determined during cooling and heating, corresponding to hydride dissolution and formation, respectively, was the result of hysteresis. Although there are various possible mechanisms contributing to hysteresis-like effects in the solvus, as discussed in Sect. 6.2 and the previous sections in this chapter, true hysteresis is a reproducible effect, originating from an irreducible free energy loss mechanism that occurs no matter how slowly the phase transformation proceeds. Thus the solvus obtained from the diffusion equilibration method of Kearns [12] is also affected by hysteresis, and actually is equivalent to the Terminal Solid Solubility of Dissolution (TSSD) obtained during a heating cycle since hydrides must partially dissolve in the high hydrogen containing part of the diffusion couple to supply hydrogen to the part of the couple initially containing low hydrogen content.

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Kearns’ data—combined with those of others available at the time—extend down to only about 260 C. To provide data for TSSD to lower temperatures, Cann and Atrens [3] used a metallographic technique. With this technique, using unalloyed Zr samples doped with total hydrogen contents ranging from 230 to 4 wppm, the authors obtained TSSD data down to *156 C. They combined these observations with determinations of the hydride phase (d or c) present in the solid as a function of hydrogen content. The authors obtained good agreement with Kearns’ TSSD equation at high temperature and also, as explained further on, at lower temperatures (\280 C), where they identified the c-phase as the only hydride phase present. c-hydrides are calculated to have smaller elastic accommodation energies than do d-hydrides, mainly because of their smaller transformation strains [4, 35]. Therefore, according to the accommodation energy model of TSS, the c-hydride phase might be expected to have a slightly different solvus relationship if it were actually the stable phase at those low temperatures as suggested by Mishra et al. [26] if also the chemical binding energies of the two phases are similar (which is likely: see Domain et al. [6]). Cann and Atrens [3], therefore, carried out a separate linear regression fit of the data for the temperature range over which the c-hydride phase was the only one observed. The results provide a straight line on a van’t Hoff plot having a slightly different slope than that given by Kearns’ regression fit to all the data from unalloyed Zr available at the time. Cann and Atrens [3] other three data points at higher hydrogen content— where both c- and d-hydrides were observed—fall on Kearns’ regression fit of the combined data for unalloyed Zr. Extrapolation of this line to the temperature range containing the four data points where only c-hydride precipitates were found, show that these points fall on Kearns’ fit as least as well as do the higher temperature data points. Hence, in this soft material, the solvus line obtained is not inconsistent with predictions from the accommodation energy model assuming that the chemical enthalpy of hydride formation would be the same for the two hydride phases. This result implies that there is little difference in the solvi for the two hydride phases, assuming that the c-hydride phase actually represents the stable phase at low temperature. Although the data obtained by Kearns [12] for unalloyed Zr, Zircaloy-2, and Zircaloy-4 with the diffusion equilibration technique and those of others using similar or different techniques all have slightly different regression lines when fitted separately, the overall impression from these data is that, within experimental uncertainty, they represent a statistically similar population of points and could just as well be combined into a single linear regression fit, as was done by Kearns [12]. Pan et al. [31] used the DEM technique, employing a range of heating and cooling conditions on unirradiated Zr–2.5Nb pressure tube material to determine the sensitivities of the measured solvus temperatures to variations in these conditions. For TSSD, changes in heating rate and maximum temperature reached during a thermal cycle did not result in noticeable differences as long as the heating rate was the same as the immediately preceding cooling rate. For TSSP, a large number of factors were shown to affect it. The factors investigated were maximum temperature reached during a thermal cycle; hold time at maximum

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Fig. 8.1 Constant strain amplitude drive signal (proportional to IF) as a function of temperature for heating and cooling of a sample of pure zirconium containing 9 wppm hydrogen. Torsion pendulum measurements at f = 2.2 Hz and  = 1.92910-5. The arrow indicates that the onset of the precipitation peak in cooling is taken as the position of the TSSP boundary. (from Ritchie and Sprungmann [38]; with permission from AECL)

temperature and cooling rate. The range of solvus compositions obtained turned out to be bounded between limits denoted TSSP1 and TSSP2 with TSSP1 referring to the solvus having the highest solvus composition. The difference between these two bounds can be substantial. For example at 250 C, the difference between TSSP1 and TSSP2 was *15 wppm. Some limited studies on the effect of microstructure on TSSP and IF peaks associated with the presence of hydrides were also carried out by Ritchie and Pan [39, 40]. Using an inverted torsion pendulum and a counterbalanced flexure pendulum at low-frequency, Ritchie and Sprungmann [38] carried out a series of rising and falling temperature cycles passing through the solvus temperature on two sets of unalloyed zirconium, one referred to as ‘‘pure’’ having extremely low oxygen content (17–23 wppm) while the other set—of Marz-grade zirconium—had somewhat higher oxygen content (120 wppm). Hydrogen contents ranged from 9 to 680 wppm. In addition, unalloyed zirconium of high purity with low oxygen content was also tested, plus a single crystal sample containing 34 wppm hydrogen and 200 wppm oxygen. Torsion pendulum tests on pure Zr for a range of hydrogen content varying from 9 to 34 wppm always yielded a reproducible precipitation peak (after the second and greater thermal cycles) on both heating and cooling, with the peak on cooling sharper and more distinct than the one on heating, as shown in Fig. 8.1. The position of the solvus was taken by the authors as being located at the onset of the sharp increase in the precipitation peak obtained during cooling (*184 C at 9 wppm H). No attempt was made to locate a similar solvus temperature on heating from the corresponding dissolution peak. However, from Fig. 8.1, using the same criterion as for the precipitation peak, the heating solvus is located at *230 C, showing that there is a hysteresis gap of *46 C at that hydrogen content between the two solvi based on this criterion. The results of TSSP temperature determinations carried out in this way for the complete set of hydrogen contents tested are summarized in the authors’ Table 1 and reproduced here in Fig. 8.2.

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Table 8.1 List of TSSD, TSSP, and DHCi equations used in Fig. 8.9. Unit of concentration is in wppm while the unit of the solvus enthalpy is in J/(mol H) TSS/TSSP equations (measurement method) Reference TSSD (DEM): cH = 8.080 9 104 exp[-34 520/(RT)] TSSP1 (DEM): cH = 2.473 9 104 exp[-25 840/(RT)] TSSP2 (DEM): cH = 3.15 9 104 exp[-27 990/(RT)] TSSPI (DEM): cH = 1.249 9 105 exp[-34 362/(RT)] DHCi (DHC init.): cH = 1.337 9 105 exp[-35 142/(RT)]

Pan et al. [31] Pan et al. [31] Pan et al. [31] Pan [27] Shi et al. [42]

Fig. 8.2 Hydrogen solubility as a function of reciprocal temperature for the TSSP results obtained by [Ritchie and Sprungmann [38] compared with the TSSD solvus relation of the regression fit by Kearns [12] of the compilation of data for unalloyed zirconium. (from Ritchie and Sprungmann [38]; with permission from AECL)

Remarkable is the close agreement of these TSSP data to the regression fit derived from a compilation of TSSD data of unalloyed Zr material by Kearns [12]. Note, however, that these TSSP temperatures correspond to the point of earliest detectable precipitation and not to the maximum slope location of the precipitation peak, which would be *11 C lower at 9 wppm H (from Fig. 8.1). In addition, as discussed further on, the TSSD regression line obtained by Kearns stands out from those obtained in other studies for unalloyed zirconium and a-phase zirconium alloy material in having significantly lower dissolution temperatures. For the single

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Fig. 8.3 Auto-twisting strain as a function of temperature during a thermal cycle of a Marzgrade zirconium sample containing 209 wppm of hydrogen. The position of the knee on the cooling branch was taken by Ritchie and Sprungmann [38] to be the position of the TSS boundary (cooling rate of 0.8 C/min). (from Ritchie and Sprungmann [38]; with permission from AECL)

crystal specimen with a hydrogen content of 34 wppm—tested in the low-frequency flexure pendulum—the precipitation peaks on heating and on cooling were much sharper than for the pure, polycrystalline zirconium material tested in torsion. Moreover, there was only an approximately 5 C difference between the peak temperatures of these two peaks. The authors surmised that the precipitation peak in the single crystal material was analogous to that observed in polycrystalline material of similar purity. One may speculate that the latter peak is broader because it corresponds to the precipitation of hydrides in different orientations, different grains, and at locations having different levels of internal strains, all of these factors contributing to a variation in signal strength with temperature of the IF peak for the polycrystalline material. The authors also carried out auto-twisting tests in a lowfrequency torsion pendulum. Auto-twisting is spontaneous twisting that arises when a wire sample of a hydride forming material in a torsion pendulum at rest is cycled through the hydride solvus temperature during cooling and heating. Although there are breaks in the auto-twisting curve in both the heating and cooling directions, the sharpest is during cooling. An example is shown in Fig. 8.3 for a wire of Marz-grade zirconium containing 209 wppm H. It is thought that the sharp break in the autotwisting curve on cooling indicated on the figure represents the TSSP temperature for this specimen. The physical origin of the auto-twisting effect—although evidently associated with hydride precipitation—is not clear, but the observed shear strain may be the result of the movements of dislocations punched into the lattice around precipitating hydrides. A small increase in TSSP of *5 C was also observed in the auto-twisting curves upon applying a small biasing shear stress in the range from 0.5 to 5 MPa. This increase in solvus temperature cannot be attributed to a shift resulting from an interaction energy effect proportional to the product of the transformation shear strain of the hydride with the applied shear stress, since the applied shear stress is too low for such an effect to be detectable. The authors conjectured that the shift to higher temperatures may be a secondary effect resulting

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Fig. 8.4 Example of DSC data from an unalloyed a-Zr specimen containing 63 wppm hydrogen, obtained during heat up at a heating rate of 30 C/min. Shown are the basic heat flow response and its temperature derivative. (from Khatamian and Ling [13]; with permission from AECL)

from the increased applied shear stress making it possible for the shorter dislocation loops formed at an earlier (higher temperature) stage of hydride growth to contribute to auto-twisting. It should be noted that methods based on dilatometry [7, 43, 52], electrical resistivity [25], DSC [13–15, 24, 41, 44–50], DEM, and IF [31, 38–40] all rely on there being a distinct, reproducible feature in the change of the particular property being monitored as function of temperature that can be identified—depending on the direction of the temperature change—with either the initial stages of formation or final stages of dissolution of hydride precipitates in the solid. The solvus compositions obtained in this way are thus terminal solvus compositions, equivalent to what in Chap. 7 are referred to as phase field compositions. Of the foregoing techniques, the DSC technique was the last to be used for solvi determinations but this technique has since found wide application to solvus determinations in zirconium and its alloys, one reason being that it requires much smaller sample sizes for comparable signal strengths than do the other methods. The differential heat flow data produced in this technique is the result of a strong exothermic or endothermic production of heat flow when hydride transformation occurs. It generally gives a strong and easily recognizable change in the heat flow signal identifiable with the terminal or initial stages of hydride phase transformation. A problem with this method—intrinsic not only to this but to all other such thermal cycling methods—is determining which feature of the heat flow signal can be identified with the solvus temperature. There are at least three identifiable features near the terminal stages of the phase transformation as shown in Fig. 8.4 that could be taken to represent the terminal solvus temperature. These features are labeled ‘‘peak’’, ‘‘maximum slope’’, and ‘‘completion’’ temperatures. The range in predicted phase transformation temperatures between these three choices can be quite large; for instance it is *30 C in the example given in Fig. 8.4. The maximum slope temperature has been the choice of many investigators for locating the solvus temperature because it provides: (1) the most sharply delineated feature (through its derivative), (2) a solvus temperature that is intermediate between the other two choices, and, therefore, (3) is the part of the signal expected

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to be least affected by variations in the response function of the calorimeter. However, complete agreement in the literature concerning adopting the foregoing criterion for determining the solvus temperature does not currently exist and either of the other two criteria have variously been used to establish the solvus relationships. In some cases these other choices were prompted, not by physical considerations, but by the quality of the signal. This was the case for the irradiated material studied by Vizcaíno et al. [49], discussed in Sect. 8.4.3 further on, in which only the peak temperature could be relied upon to give a well-defined indication of the location of the solvus temperature. Of the researchers not in agreement with the maximum slope temperature as being the best choice for solvus temperature determination, Khatamian has been the most persistent in proposing an alternate choice. Khatamian has argued that the peak temperature criterion is to be preferred, not only on the basis of practical considerations imposed by limitations resulting from indistinct signal variations with temperature for the other points, but also, more importantly, on physical and experimental grounds [13–15, 17]. The physical justification given by Khatamian and Ling [13] for favoring the peak temperature criterion over the other two was that for unalloyed zirconium this criterion gives a solvus relationship that is in better agreement with that obtained for similar material by Kearns [12] derived from the diffusion equilibration technique. Similar agreement using the DSC technique can be seen in the solvus relations obtained by McMinn et al. [24] for Zircaloy-2 and -4 materials. These latter authors found that the solvus temperatures derived with the peak temperature criterion correlated (although not perfectly) with that obtained by Kearns for similar materials obtained with the diffusion equilibration technique. However, unlike Khatamian [15], McMinn et al. obtain almost exact correlation between the diffusion anneal temperatures (during charging of their specimens with hydrogen from a surface hydride layer) and the maximum slope temperature (Fig. 2 in McMinn et al. [24]). For this reason, the latter authors chose the maximum slope temperature as being the appropriate criterion for determining the solvus temperature in their tests. Khatamian argued, however, that, because the diffusion equilibration method is a quasi-static isothermal technique, it provides true ‘‘equilibrium’’ solvus relationships and thus can serve as a benchmark technique for comparison with the results from other TSSD determination techniques. This reasoning cannot be supported—as noted some time ago by Puls [35] and, again, here at the beginning of this section—because it is clear that the diffusion equilibration technique must also be subject to hysteresis effects. Therefore the results obtained with this method are not actually representative of an ‘‘equilibrium’’ solvus any more than are the solvi obtained from thermal cycling techniques. As pointed out in Chap. 6, true equilibrium in these types of systems can never be realized experimentally. Another reason advanced by Khatamian [13] for the choice of peak temperature in DSC measurements was the correspondence subsequently provided by Khatamian and Root [17] for this temperature from solvus temperature determinations using ND techniques. The authors showed that there was good agreement between the temperature at which—during long isothermal holds approached on heating—

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the neutron scattering results indicate all hydride precipitates had disappeared at the DSC peak temperature. Since the ND measurements provide a direct indication of the presence or not of hydrides and since they are carried out isothermally with the temperature approached slowly from a lower value, Khatamian felt that these features also made the ND technique a suitable benchmark technique for determining the ‘‘equilibrium’’ solvus temperature. Recently, Giroldi et al. [9] cast doubt on the accuracy of the ND method for low hydride phase fraction values and thus on it being sufficiently accurate as a benchmark method for dissolution solvus temperature determinations. In addition, the technique requires that all hydride precipitates be aligned in the same orientation for all of them to be detected by the selected diffraction peak. This is generally not the case and particularly during the final stages of dissolution there may still be hydrides present not detectable by this technique in most materials used in practical applications. In fact, these authors pointed out that convincing support for the maximum slope criterion was given by the experimental results of Pan and Puls [29]. These latter authors monitored changes in the DEM over time by using very slow, essentially quasi-static temperature increases and decreases (0.1 C/min) in successive ramps of 5 C each, followed by 3 h hold times after each temperature increase. The results of these experiments showed that hydride phase transformation could not keep up with even the very slow rate of rise or fall of temperature when increasing or decreasing the temperature in 5 C steps close to the termination or initiation stages of the phase transformation. This manifested itself in an asymmetric peak in the amount of hydride transformed during identical 3 h holding periods after 5 C quasi-static temperature changes. The maximum slope points on the high temperature side of these peaks—which were also the sides of the peaks for which the transformation rate changed most slowly—closely agree with those given by the inflection points on plots of variations of DEM with temperature.4 In the standard DEM method, using heating and cooling rates typically around 2 C/min, these inflection points were chosen as representing the TSS temperatures (Figs. 8.5, 8.6). In addition, in tests on unalloyed Zr, where, unlike in the alloy material, there exists a distinct IF peak associated with the start or completion of the phase transformation, the maximum slope temperature on the high temperature side of this peak (Fig. 8.7) coincides with the point of inflection of the variation of DEM with temperature. Comparing the results from the DEM and IF techniques in Figs. 8.5, 8.6, and 8.7 to the DSC results reproduced in Fig. 8.4 one can also see the close similarity in the temperature dependences of the three sets of signals. Thus, the DSC ‘‘peak’’ temperature can be seen to be equivalent to the location of the hydride transformation rate peak derived from the quasi-static DEM measurements as well as to the location of the IF peak for unalloyed Zr material, while the DSC ‘‘maximum slope’’ temperature is equivalent to the location of the maximum slope temperatures of the DEM and IF peaks on

4

Similarly dilatometry or electrical resistivity techniques use the inflection point as representing the TSS temperature.

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Fig. 8.5 Increments of hydrogen concentration in solution for identical isothermal holds of 3 h as a function of temperature; a curve has been fitted to the heights of hydrogen increments. (from Pan and Puls [29]; with permission from AECL)

Fig. 8.6 Increments of hydrogen concentration in solution for identical isothermal holds of 3 h as a function of temperature; a curve has been fitted to the heights of hydrogen increments. (from Pan and Puls [29]; with permission from AECL)

their high temperature sides. The sharp IF peaks on heating and cooling obtained by Ritchie and Sprungmann [38] from a specimen consisting entirely of a single crystal of pure Zr, when compared to the much broader peaks obtained by these authors in a similar polycrystalline material, are particularly relevant to this discussion, since they show that dissolution of hydrides in material having a heterogeneous microstructure such as would be expected in a polycrystalline material, results in a range of detected hydride solubility. The maximum slope

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263

Fig. 8.7 Internal friction, Q-1, and frequency squared (F2) as a function of temperature determined during a thermal cycle at a rate of 2 C/min for a pure Zr sample containing 141 wppm hydrogen, using a flexure pendulum at F = 2.2 Hz. (from Pan and Puls [29]; with permission from AECL)

criterion, then, is a compromise choice producing solvus values closest to complete hydride transformation while also being subject to the least amount of judgment in locating the temperature at which it occurs. The quasi-static DEM measurements of Pan and Puls [29] also show that there is still a considerable range of *±20 C on either side of the maximum slope temperature location over which there is an observable rate of hydride transformation during a 3 h hold. A somewhat smaller difference between these two temperatures is obtained for TSSP. Since this spread is approximately the same in these very slow (quasi-static) heating and cooling tests as in the other three tests where the heating and cooling rates are greater, it suggests that this difference represents a true physical variation in the solubility of the individual hydride precipitates and is not just an effect of heating rate resulting from, for instance, thermal lag.5 It likely arises as a consequence of the heterogeneity of the hydride precipitation sites, producing hydride precipitates having a range of retained coherency strain energies, particularly as they become smaller and their shapes change (which eventually also results in their surface energies becoming a factor in the total energy changes driving their dissolution). Such a heterogeneous twophase mixture could not truly be in coherent or even semi-coherent (metastable) equilibrium and would transform over time. One might, thus, expect the variation in the transformation rate close to the terminal temperature to be a reflection of this variation in ‘‘equilibrium’’ conditions among the hydride precipitates. In addition, associated with the foregoing considerations, metallographic observations have shown that hydride precipitates formed in a-Zr vary considerable in sizes, shapes, and spatial distributions (see Chap. 3). Further support along

5

This effect can also be seen in plots of the derivative of Young’s modulus versus temperature [31].

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these lines has recently also been provided by Vizcaíno et al. [50], as further discussed in Sect. 8.4.3. Moreover, as pointed out in Chap. 4, hydride precipitates are likely to form at dislocation cores and close to the high tensile stress fields of other hydrides (as shown in Chap. 3) since hydrogen is attracted to these locations. Many of these hydrides might grow to sizes where only a small part of their total size would be located at the high tensile stress region in the dislocation core or close to another hydride. Hydrides would precipitate and dissolve at these cores and in the proximity of other hydrides at different temperatures than would hydrides formed in the bulk or grown outside the dislocation core region. In addition, larger hydrides,—whether located at high tensile stress regions or not— would take longer to dissolve than smaller ones. The degree to which all of these factors control the sizes, shapes, and individual solubilities of hydride precipitates would depend on many factors that affect hydride sizes and locations, such as the material’s microstructure, its average hydrogen composition, and the cooling and heating rate. Thus—particularly for hydride dissolution—one would expect there to be a variation in the rate of dissolution of hydrides as the temperature is increased and the hydride phase fraction approaches zero since not all the hydrides would completely disappear at the same time. One could conclude, then, from all of the foregoing considerations that the spreads in the signals with temperature shown in Figs. 8.4, 8.5, 8.6, and 8.7 obtained with the DSC and DEM techniques, reflect true variations in local TSSD values of individual hydride precipitates at the start and completion of their transformation processes, respectively. In comparing the solvus relationships obtained by Vizcaíno et al. [49] using either of the two choices of dissolution temperatures, one might expect that the fits to the data would have produced solvus lines that are merely shifted from each other, having approximately the same slopes. This turned out, however, not to be the case. The difference in solvus enthalpies obtained using either the peak or the completion temperature criterion is *2.5 kJ/mol H, which is of the magnitude expected for the elastic–plastic accommodation energy assuming the components of the hydride transformation strains are approximately equal. This difference may, however, not be physically meaningful since, as mentioned further on in Sect. 8.4.3, the fit to the data is a two parameter one where the pre-exponential is also a fitting parameter. As an aside, the difference in solvus enthalpies obtained by Khatamian and Ling [13] in unalloyed Zr from regression fits using either the peak or the maximum slope criterion is *1.13 kJ/mol H. This result may be physically meaningful since the pre-exponential turned out, in this case, to be the same for each fit. It is tempting to conclude, then, that the difference in solvus enthalpies of a factor of two between the latter and former cases reflects the difference in solvus temperatures obtained with the two choices of solvus temperature criteria, which is also about a factor of two. On the other hand, Giroldi et al. [9] recorded the results of solvus temperature values obtained using each of the three possible criteria and found that the regression fits to these three sets of solvus data all gave identical solvus enthalpies and nearly identical, monotonically decreasing pre-exponential values, in accord with this writer’s initial expectation.

8.4 Summary of Results of Experimental Solvus Determinations

265

Finally, it should be noted that the terminal solvus relationships obtained from thermal cycling techniques give phase field solvus relationships only. These relationships are therefore, strictly speaking, not applicable in models for calculating hydride growth at flaws or cold spots because, as shown in Chap. 7, as long as some coherency strain is present in the system the equilibrium tie lines do not fall on the phase field composition lines except at limiting values of phase fractions. This means that the solvus compositions would vary depending on average composition or phase fraction and account must be made of this in diffusion calculations. Germane to this, the solvi compositions in these cases are actually approached under slow, isothermal hydrogen ingress conditions, and not by temperature increases or decreases at constant average hydrogen content, as in thermal cycling methods. To investigate whether there is a difference in the precipitation solvus detected during slow hydrogen ingress conditions with those obtained with thermal cycling techniques, Pan et al. [31] carried out experiments in which they continually monitored changes in the DEM of samples exposed to an external source of hydrogen at constant temperature. The surfaces of these samples were prepared in such a way that the rate of hydrogen ingress was slow enough that no hydride layer was formed on the outside of the specimen (This was verified subsequent to the tests.) The first such tests were carried out at 250 C on an experimental pressure tube material, Excel. It was found that TSSP obtained during isothermal hydrogen ingress (called TSSPI) occurred at a lower concentration (correspondingly, higher temperature) than did TSSP2, obtained during cooling in thermal cycling tests. The difference was about 13 wppm in this alloy, which, it should be noted, has considerably higher TSS values than all of the other zirconium materials tested to date. A similar, more extensive study was subsequently carried out on Zr–2.5Nb by Pan [27]. Figure 8.8 plots the change in hydrogen content in solution as a function of time, showing similar behavior of the effect of thermal cycles as that given in Fig. 3 of Khatamian et al. [16] for Excel. In the tests on Zr–2.5Nb, thermal cycles between room temperature (*30 C) and the test temperature of 250 C, were inserted periodically, after each 4 or 5 days of hydrogen charging at constant temperature. As shown in Fig. 8.8, the positions of these thermal cycles superimposed on the curve of cH in solution versus charging time are indicated by the letters A to K. At each thermal cycle a corresponding TSSP2 temperature was determined during the cooling run. When the total hydrogen concentration cH B 30 wppm, e.g., at the points A, B, C, or D in Fig. 8.8, the maximum temperature of the thermal cycles was higher than the TSSD temperature of the total hydrogen content at that time. Hence, the measured effective TSSP2 from these cooling runs was used to estimate the total hydrogen concentration in the sample for that time. This is indicated on Fig. 8.8 by the starred symbols. Therefore, the linear section, A to D, of the plot of cH versus time represents the total hydrogen content in the sample versus time. From the linear extrapolation of this linear section from A to D to the end of charging time, which is given in Fig. 8.8 by the dashed line, a total hydrogen content in the sample was derived and found to be equal to 70 wppm, in good accord with the total content of 71 wppm that was subsequently measured using hot vacuum extraction mass spectrometry (HVEMS). From these types of ingress experiments done at six

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8 Experimental Results and Theoretical Interpretations

Fig. 8.8 Hydrogen concentration in solution as a function of charging time in Zr-2.5Nb at a constant temperature with a charging rate of *0.05 wppm. At the times A to K, a double thermal cycle between 30 and 250 C was inserted. During the cooling run TSSP2 or equivalent TSSP2 was determined and shown by the star symbol. (from Pan [27]; with permission from AECL (copyright 2012, AECL))

different temperatures, a solvus expression for TSSPI was obtained (listed in Table 8.1). Figure 8.9 shows that around 280 C TSSPI is close to the TSSP2 relationship obtained by Pan et al. [31] but becomes progressively lower than TSSP2 below this temperature. In addition, as expected, TSSPI is well above TSSD. An interesting result is that there is a break in the linear rate of ingress with time that occurs at a total hydrogen content around TSSD where the ingress rate becomes nonlinear, gradually decreasing its slope with time. A similar such deviation was observed in Excel ([28], Fig. 3) but in that material it occurred at hydrogen concentrations even lower than at the TSSD composition. This change in linear ingress rate may be associated with the filling up of dislocation cores with sufficient hydrogen for hydride precipitation to form there, as discussed in Chap. 4. That this starts when the total hydrogen content is at or slightly below TSSD, depending on the material, may be coincidental since it would depend on the magnitude of the interaction energy between the positively misfitting hydrogen atoms and the tensile stresses existing at dislocation cores, which bear no relationship with TSSD corresponding to the solubility of hydrides located in the bulk of the material (i.e., not located in dislocation cores). Table 8.1 lists the solvus equations shown in Fig. 8.9, including their sources. Note that the isothermal charging experiments for TSSPI have, in a sense, their parallel in the TSSD relationships of Kearns [12] obtained by the diffusion equilibration technique. In analogy with TSSPI, one could label the solvus relationship obtained by Kearns, TSSDI6 to express the fact that it is obtained through an isothermal ingress experiment. One can see from the results of Khatamian and

6

Strictly speaking, TSSDI is not a terminal solvus and should more properly be labeled SSDI.

8.4 Summary of Results of Experimental Solvus Determinations

267

Fig. 8.9 Isothermal TSSP (TSSPI) of hydrogen in Zr2.5Nb determined using DEM during isothermal hydrogen ingress. The solvus relation obtained is compared with those relations obtained using thermal cycling approaches. Two ingress rates were used in two sets of tests, one of 1–2 wppm/h, the other of 0.01–0.1 wppm/h. The plotted line gives the regression fit to both sets of data. The solvus relations displayed are listed in Table 8.1. (from Pan [27]; with permission from AECL (copyright 2012, AECL))

Ling [13] for zirconium and McMinn et al. [24] for Zircaloy-2 and -4 as shown in Fig. 8.10 that, similarly as for hydride precipitation during isothermal ingress, there are also differences with the TSSD relationships obtained with the DSC technique for hydride dissolution during isothermal ingress, but only when using the maximum slope criterion. On the basis of the coherent equilibrium relationships given in Chap. 7, one might, in fact, expect a difference between TSSDI and TSSD because of the dependence of the tie-line compositions on phase fraction. Thus TSSD, being a terminal solvus, gives the phase field, zero phase-fraction compositions, while TSSDI—obtained by equilibration with a hydride phase fraction mixture that is always greater than zero—represents solvus compositions given by tie-lines that would progressively shift to lower values with increasing phase fraction from those given by the phase field compositions. In fact, the opposite shift in compositions is observed, which, on the face of it, would seem to give credence to Khatamian’s assertion that the difference in the solvus compositions is the result of the wrong choice of solvus temperature criterion for determining the TSSD solvus. Finally it should be noted that the closeness of TSSPI to TSSP2 rather than to TSSP1—the latter of which has been deemed to represent the solvus for hydride nucleation [31]—is consistent with the view expressed in Sect. 6.2 that TSSP1 does not represent a true hysteresis state (i.e., a state that is reproducible during successive thermal cycles), but is a time- and condition-dependent state reflecting the initial supersaturation needed to provide the additional driving force to overcome the hydride nucleation barrier (mainly as a result of the surface energy). Once nucleation is achieved and the hydrides grow in size, the surface energy term ceases to be important and the need for supersaturation to provide this extra energy disappears and the path followed during further temperature reduction would be

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8 Experimental Results and Theoretical Interpretations

Fig. 8.10 TSSD relationships for different Zr materials determined using different methods

that corresponding to the true hysteresis state, which is presumed to be given by TSSP2, as shown in Fig. 8.9. The amount of supersaturation observed for overcoming the nucleation barrier would depend—among other conditions—on the hold time at a given temperature, increasing with increasing cooling rate. Thus, under the isothermal, slow hydrogen ingress conditions prevailing in the experiments of Pan and Puls [29], one would expect that the saturation needed to overcome the nucleation barrier would be undetectably small. The results derived from these tests, then, indicate that TSSP2 is the precipitation solvus most relevant in models of the formation and growth under isothermal conditions of hydrides at cold spots and flaws. Considering all of the foregoing difficulties in determining the true solvus composition, it is important to establish—for practical application of these results to assessments of the susceptibility of a component to DHC—how the solvus relationships obtained with a particular method and solvus temperature selection criterion relate to the hydrogen composition limits for DHC initiation at cracks (loaded such that the applied stress intensity factor is well above its threshold for DHC initiation). Details for determining this limit and the results obtained for unirradiated Zr–2.5Nb pressure tube material have been given by Shi et al. [42]. A regression fit to these data, labeled DHCi, is included in Table 8.1 and plotted in Fig. 8.9. Comparison of the DHCi relationship with TSSD derived for the same material allows one to determine the level of conservatism in the methodology between the solvus and the hydrogen composition limit for DHC initiation. Figure 8.9 shows that the TSSD relationship obtained using a combination of DEM and DSC techniques, with the latter based on the maximum slope criterion, does provide a conservative—but not overly so—limit for DHC initiation for

8.4 Summary of Results of Experimental Solvus Determinations

269

unirradiated material. In Sect. 8.4.4 a similar comparison is provided for irradiated Zr–2.5Nb pressure tube material.

8.4.2 Differences Between Solvi in a and a/b Zr Materials: Effect of b-Zr Phase The effect of the composition and morphology of the b-Zr phase on TSSD was evaluated by Cann et al. [5] on the basis of their experimental results of hydrogen partitioning between the a- and b-Zr phases (The accommodation energy theory predicts that both TSSD and TSSP are affected in the same direction (but not necessarily the same magnitude) by the effects described in this and the following sections; however, the emphasis in these sections is on the effect on TSSD.) Relative to the TSS in a-Zr and its alloys such as Zircaloy-2 and -4 containing no b-phase, the presence of the b-phase will increase the apparent TSS concentration, the increase depending on the volume fraction and niobium concentration in this phase. Note that the generally greater concentration of hydrogen in the b-Zr phase in equilibrium with that in the a-Zr phase is not expected to lead to hydride formation in the b-phase because the TSS concentration is much higher than in the a-Zr phase [14]. (The exception could be at very high cooling rates when there could be insufficient time to equilibrate the hydrogen concentration in the b-phase in relation to that in the a phase.) For cold worked Zr–2.5Nb pressure tube material, the largest shift relative to the TSSD for unalloyed Zr and Zircaloy-2 and -4 is for tube material that has not been autoclaved. The standard pressure tube autoclaving treatment of 24 h at 400 C and the subsequent thermal aging in the reactor will each, in turn, decrease the TSSD concentration, all other factors being equal. A TSSD relationships for cold worked Zr–2.5Nb pressure tube material was derived in the 1990s by Canadian researchers for use in a CSA Nuclear Standard [1]. The data was obtained from measurements carried out in two separate laboratories on material from the same pressure tube, with one laboratory using DEM, the other DSC. Added to these data were TSSD measurement results obtained using DSC from specimens taken (by a punching method) from locations just in-board of the burnish mark of the inlet-end rolled joint region of five ex-service, cold worked Zr–2.5Nb pressure tubes from a CANDU reactor. Although the maximum fluence experienced by these ex-service pressure tube materials—which varied with location—was mostly less than the fluence for saturation of the irradiation-induced increase in yield strength, it was subsequently found that material removed from pre-irradiated pressure tubes by punching had solvus temperatures, as determined by DSC, that were similar in magnitude to those of unirradiated material. It is known that the rate of breakdown of the b-phase can be reversed by the effect of fast neutron irradiation. The combined effects of temperature and neutron flux varying along the axis of the pressure tube result in a broad minimum state of b phase decomposition located over a broad range approximately mid-way

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8 Experimental Results and Theoretical Interpretations

between the inlet and outlet ends of the pressure tube [10]. Hence, little or no decrease in TSSD concentration as a result of b-phase transformation is predicted around that location. At the rolled joints, where there is minimal irradiation effect (however, see comments in next paragraph), the b phase will gradually break up, with the outlet end of the pressure tube having a greater break up rate than the inlet end, since the former is at a higher temperature. Therefore the TSSD composition would be expected to be depressed least at the inlet half of the pressure tube and most at the outlet end relative to the TSS concentration for unirradiated pressure tube material. Figure 8.10 shows a comparison of the TSSD relationship obtained from the foregoing Zr–2.5Nb pressure tube material [1] with the TSSD relationships obtained for unirradiated Zr-2 and Zr-4 material by Kearns [12] using the diffusion equilibration technique and by McMinn et al. [24] using DSC. As noted in Sect. 8.4.1, these solvus relationships, obtained with these two different methods, differ noticeably from each other. Also plotted are the TSSD relationships for unalloyed Zr material. The solvus relationship given by Khatamian and Ling [13] was determined using the peak temperature criterion while the one given in this text and plotted in Fig. 8.10—provided privately by Khatamian—was obtained using the maximum slope criterion. Kearns’ relationship for unalloyed Zr follows the expected trend of having a lower solvus composition than in the alloys, but only below about 200 C. The result for unalloyed Zr obtained by Khatamian and Ling [13] using the maximum slope criterion does follow the expected trend of being below those for the Zircaloy-2 and -4 materials, but the agreement with Kearns’ [12] TSSD relationship is poor. As noted in Sect. 8.4.1, this poor agreement led Khatamian to suggest that the peak temperature criterion is the more appropriate choice for solvus determinations since it brings the two solvus relationships into alignment. It is seen that, except for the relationships obtained by Kearns for unalloyed Zr and Zircaloy-2/-4, which were obtained by the diffusion equilibration technique, the Zr–2.5Nb a/b phase pressure tube alloy material has a higher solvus composition compared to those determined in material consisting of a-phase alloys. Table 8.2 lists the solvus equations used in Fig. 8.10, including their sources. Khatamian and Ling [13] and Khatamian [14] carried out a systematic study of the effect of the b-phase on TSS. Using DSC and the maximum slope temperature to determine the TSS temperature these authors obtained the following TSSD relationships: For annealed Zr–2.5Nb:   32 900 cTH ¼ 6:97  104 exp ð8:15Þ RT For a-Zr: caH ¼ 1:47  105 exp

  38 430 RT

ð8:16Þ

8.4 Summary of Results of Experimental Solvus Determinations

271

Table 8.2 List of TSSD equations plotted in Fig. 8.10. Unit of concentration is in wppm while the unit of the solvus enthalpy is in J/(mol H) Reference TSSD equationsa: Material, Measurement Method Zrb, DSC: cH = 1.47 9 105 exp[-38 430/(RT)] Zr-2, -4: Diff. Equil.: cH = 9.87 9 104 exp[-34 541/(RT)] Zircaloy-2, DSC: cH = 1.064 9 105 exp[-35 990/(RT)] Zr-2.5Nb PT, FFSG, DSC and DEM: cH = 8.19 9 104 exp[-34 500/(RT)] Zr, Diff. Equil.: cH = 1.306 9 105 exp[-36 467/(RT)]

Khatamian and Ling [13] Kearns [12] McMinn et al. [24] CSA [1] Kearns [12]

a

All TSS relationships determined by DSC listed in this and the following tables and plotted in the associated figures unless otherwise indicated have been obtained with the solvus temperature determined using the maximum slope criterion b The TSSD relationships listed in the references by Khatamian and Ling [13] and Khatamian [14] are those derived from the peak temperature criterion. The relationships given in this text were derived with the maximum slope criterion, the results of which were made available to this writer by the author

Table 8.3 List of TSSD equations plotted in Fig. 8.11. Unit of concentration is in wppm while the unit of the solvus enthalpy is in J/(mol H) TSSD Equations: Material, Measurement Method Reference Zircaloy-2, -4, Diff. Equil.: cH = 9.87 9 104 exp[-34 541/(RT)] Zircaloy-2, -4, DSC: cH = 1.064 9 105 exp[-35 990/(RT)] Zr-2.5Nb PT, FFSG: DSC and DEM: cH = 8.19 9 104 exp[-34 500/(RT)] Irrad. Zircaloy-2, -4, DSC: cH = 2.424 9 104 exp[-28 195/(RT)]

Kearns [12] McMinn et al. [24] CSA [1] McMinn et al. [24]

From these solvus relationships the amount of hydrogen in solution at a given b=a temperature can be calculated and, hence, the partitioning ratio, rH ðNbÞ using Eqs. 4.18 and 4.19. Assuming that n has an uncertainty range from 2.4 wt% Nb to 2.8 wt% Nb, while na has an uncertainty range from 0.5 to 1.5 wt% Nb, yields wb b=a values ranging from 0.1179 to 0.0486. The partitioning ratio, rH ðNbÞ at 300 C, then, ranges from 4.38 to 7.50, assuming that the a-phase TSSD given by Eq. 4.18 is increased by an accommodation energy factor of 1.15 to account for the assumed strength difference between the a-phase contained in the a/b-phase material and in a pure a-phase material, which is what is given by Eq. 8.15. At the same temperature, for an undecomposed b-phase (Zr–20Nb), Cann, in an internal b=a report, obtained a partitioning ratio, rH ðNbÞ ¼ 3:8;7 which is close to, but less than the estimated lower bound value derived from Khatamian’s TSSD measurements. Clearly, further work is required to determine what the cause of the discrepancy is between the two sets of results.

7

b=a

In the paper by Cann et al. [5] rH ðNbÞ is given as 2.76; it is not clear which of the two values is the correct one.

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8 Experimental Results and Theoretical Interpretations

8.4.3 Effect of Cold Work and Irradiation The results presented in the previous section indicate that, for the most part, zirconium alloy materials have slightly higher solvus compositions compared to unalloyed zirconium materials. The reason for this could be attributed to the larger strengths of the alloys resulting in the hydride precipitates in the alloy materials having greater elastic–plastic accommodation energy values compared to in unalloyed materials. On the same basis, one should expect that increased cold work should also increase the solvus compositions if it results in an increase in yield strength over that in the same alloys not subjected to cold work. One would similarly expect that materials subjected to fast neutron irradiation would have increased solvus compositions relative to the same materials in their unirradiated states since fast neutron irradiation increases their yield strengths. However, the microstructural changes responsible for the respective increases in yield strength eventually saturate at some level of cold work and/or fast neutron fluence [30]. On the basis of the accommodation energy model, then, there should also be a corresponding saturation in the increase of this energy and, hence, of the solvus composition. One would expect, however, that the level of cold work and fast neutron fluence at which this saturation occurs would be different for different materials with the softer materials having proportionally greater increases in yield strengths and accommodation energies before saturation occurs compared to the stronger materials. In addition, changes in yield strengths as a result of the forgoing factors, since they are generally obtained based on tests using macroscopic tensile specimens are often geometry and texture dependent and the results obtained may not necessarily reflect the actual changes in yielding strength thresholds occurring at the microscopic level. In the following, results of changes in the solvus compositions produced as a result of increases in both level of cold work and fast neutron fluence are summarized for the simpler a-phase alloys while the next section deals with the effects that these factors have in the more complex a/b-phase alloys. Regarding the effect of cold work, McMinn et al. [24] carried out DSC measurements on b-quenched Zircaloy-2 forging material both in its as-received and 30 % cross-rolled (cold) conditions. The cross rolling changed the nearly random texture of the starting material such that the resolved basal pole fraction perpendicular to the rolling directions increased from 0.365 to 0.724 with a corresponding decrease in the other two orthogonal directions. Subsequent to the DSC measurements the cross-rolled material was annealed at 550 C for 1 h and a final set of DSC measurements carried out. The results for all four conditions of the material (0, 30 % cold work, before and after anneal at 550 C) were within the scatter of results from the other Zircaloy-2 and -4 materials for both TSSD and TSSP despite the fact that there was an increase in average yield strength ranging from 326 to 556 MPa, respectively, between the zero and 30 % cold worked materials. In contrast, the oxygen strengthened TIG welded Zircaloy-2 plate had increases in solvus compositions relative to those obtained from the combined data

8.4 Summary of Results of Experimental Solvus Determinations

273

Fig. 8.11 Comparison of TSSD relationships for unirradiated and irradiated Zircaloy-2 and -4 materials and for unirradiated Zr-2.5Nb pressure tube material. See Table 8.3 for list of equations and references

for all other materials, even though the average yield strength (and corresponding difference compared to the non-strengthened material) was only slightly greater in the TIG welded plate compared to in the b-quenched material. Regarding the effect of irradiation on solvus composition, McMinn et al. [24] found large increases in the irradiated Zircaloy-2 and -4 materials compared to the solvi compositions in similar unirradiated material. However, the slope of these authors’ solvus relationship is much less than that for the unirradiated material, crossing the solvus line for the corresponding unirradiated material at about 330 C. The materials had received fast neutron fluences in the range from 5.5 9 1024 to 1.0 9 1026 n/m2 at temperatures in the range from 250 to 300 C, which means that these solvus results are meaningful only up to the maximum irradiation temperature of 300 C. The large increase in the solvus shown in Fig. 8.11 at low temperatures is only qualitatively in agreement with theoretical predictions based on the accommodation energy model when taking account of the irradiation-induced increase in the macroscopic yield strength of this material. As summarized in Sect. 8.4.4 further on, the increases in the solvus compositions between irradiated and unirradiated Zircaloy-2 and -4 materials are also larger than corresponding increases for irradiated Zr–2.5Nb pressure tube material. The possible reason for this is discussed in Sect. 8.4.4. McMinn et al. [24] proposed that the large increases in the solvi compositions produced by fast neutron irradiation are brought about by hydrogen trapping at irradiation-induced dislocations rather than by increases in the accommodation energy. The authors assumed that such trapping would make the hydrogen atoms unavailable for hydride formation in the same way that hydrogen in the b-phase in a/b-phase alloys would be unavailable for hydride formation in the a-phase. This

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8 Experimental Results and Theoretical Interpretations

trapping of hydrogen atoms at dislocations would thus increase the average hydrogen composition at which hydride dissolution or formation occurs at a given temperature. No attempt to quantify these ideas was made, however. Although the authors’ hydrogen trapping proposal seems, at first glance, reasonable, it is likely not the full story. First of all, some component of the increase must come from an increase in the accommodation energy. Second of all, the treatment in Chap. 4 shows that trapping of hydrogen at dislocations would likely also lead to hydride formation in the core region of these dislocations, the formation or dissolution of which may be detectable during DSC measurements. Kirchheim [18, 19] and Maxelon et al. [23] have shown, theoretically and experimentally, that in the case of the Pd–H system the accumulation of hydrogen atoms at the many closely equivalent, high binding energy sites in the core of dislocations also likely results in the formation of small hydride precipitates (a detailed account of this is given in Chap. 4). In a closed system at constant average hydrogen content, these small dislocation-core hydrides would not start to dissolve and contribute to the DSC signal until temperatures are reached that are much higher than those required for the larger bulk hydrides located at crystal sites of lower binding energies to dissolve. The overall slope of the solvus, then, reflects the fact that at lower temperatures the trapped hydrogen contained in the hydrides locked at dislocation cores is not involved in the dissolution process of the bulk hydrides, which means that the actual free hydrogen concentration (hydrogen in solution) in equilibrium with the bulk hydrides is lower than what is assumed in a van’t Hoff plot in which the inverse of the dissolution temperature is plotted versus total hydrogen content. Unlike trapping of hydrogen in the b-phase, trapping of hydrogen in hydrides at dislocation cores is strongly temperature-dependent since a temperature is reached at which these hydrides will also start to completely dissolve. This effect of the variation of the solvus with temperature—which is most evident at lower temperatures—is consistent with the findings by several investigators [24, 46] in both irradiated and unirradiated material that a single linear regression fit to the solvus data for precipitation could not be made when including the data below about 200 C. The reason for observing this only in the solvus data for precipitation may be because these data extend to lower temperatures than do the data for dissolution. It is expected that an increase in the number of dislocations through cold work or irradiation would also have the effect of making hydride formation and dissolution signals less distinct compared to in unirradiated material with a lower dislocation density. As shown in Fig. 8.12, this type of behavior was, in fact, observed by Vizcaíno et al. [49, 50]. These authors used the DSC technique to determine solvus relationships on neutron irradiated Zircaloy-4 coolant channel material removed from the Atucha 1 PHW reactor which had very high levels of hydrogen content ranging from *150 to 250 wppm and neutron fluences up to 1026 n/m2. It was argued by Vizcaíno et al. [50] that—based on the results of observations from optical and transmission electron microscopy—the less distinct hydride transformation signals in these irradiated materials are a consequence of the existence of a wide spread in hydride sizes ranging over four orders of magnitude from 10 nm to 100 lm. These observations by these authors are in accord

8.4 Summary of Results of Experimental Solvus Determinations

275

Fig. 8.12 DSC heating curve sequence (sample 2 R2CC). a The peak shifts to higher temperature after each annealing. b Baseline corrected curves. A recovery peak superimposed to the initial curve is observed that may be related to damage recovery. (from Vizcaíno et al. [50])

with those proposed by this writer when account is taken of the preferential formation of hydride precipitates in the cores of irradiation-produced dislocations. Other evidence for the presence of numerous small hydride precipitates at freshly formed dislocations can be inferred from metallographic observations showing the presence of rows of numerous small black dots arranged along lines emanating from flaws in unirradiated Zr–2.5Nb pressure tube material deformed in tensile or torsion specimens. These rows of black dots are arrayed in lines along which one would expect freshly formed dislocations to be created ([36] Fig. 8, [37] Fig. 1). Implicit in this trapping model is that the trapping of hydrogen/ hydride at freshly produced dislocations is additional to the likely pre-existing trapped hydrogen/hydrides at the numerous dislocations already present in the material produced during the manufacturing process. The spreading out of the DSC signal in irradiated zirconium alloys obviously poses a challenge in accurate determinations of the solvus. McMinn et al. [24] do not mention whether the DSC signals from their irradiated materials were more difficult to interpret than those from corresponding unirradiated materials. Nevertheless, except for TSSP at temperatures \170 C, the authors obtain absolute solvus enthalpy changes between unirradiated and irradiated material from DSC determinations that are about a factor of two greater than those obtained between unirradiated and irradiated Zr–2.5Nb pressure tube material by other workers (see Sect. 8.4.4 further on). (It should be noted that, as discussed further on, differences in solvus enthalpies are not an accurate indicator of the true enthalpy differences when fits to the data also produce different pre-exponential values.)

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8 Experimental Results and Theoretical Interpretations

Considerably greater differences between unirradiated and irradiated Zircaloy-4 coolant tube material were recently documented by Vizcaíno et al. [47–50]. The authors point out that the more irregular features of the DSC signals in irradiated materials made it more difficult to identify the exact location of the terminal stage of hydride dissolution based on the maximum slope and completion temperature locations of the DSC signal. Therefore these authors based all their solvus temperature determinations for these irradiated materials on the peak temperature criterion. In addition, instead of the usual method of obtaining the solvus enthalpy from a regression fit of the solvus data in a van’t Hoff plot, the authors employed a more direct approach that proved to be useful for better interpreting the solvus enthalpy changes obtained in their irradiated material. In this approach the solvus dissolution heat, Qd?a, is obtained directly from the DSC signal through numeric integration of the area enclosed between the calculated dqðtÞ=dt versus time curve and the x-axis. Given the sample mass and total hydrogen content in each sample after completion of the tests, the total number of moles of hydrogen in the sample, mH, can then be calculated and the solvus enthalpy determined using:  d!a ¼ DH

Qd!a mH

ð8:17Þ

A plot of Qd?a versus mH for the unirradiated material shows that the two parameters are linearly proportional to each other over the entire composition range. The slope of the linear regression fit to these data gave a dissolution  d!a ¼ 39:3 kJ/mol H. In comparison, the dissolution enthalpies enthalpy, DH obtained directly from regression fits to the data from van’t Hoff plots (ln X versus 1/Td where X = mH/mZr and Td is the dissolution temperature) were 35.6 and 38.1 kJ/mol H when Td is obtained from the peak (Td1) and completion (Td2) temperatures, respectively. Comparing the solvus enthalpies obtained by these three different criteria, it is seen that the results are not far apart. However, the difference between them is of the magnitude expected for the accommodation energy. One might also have expected the solvus enthalpy obtained using the approach of Eq. 8.17 to give an average value in-between those obtained using Td1 and Td2, which turned out, however, not to be the case. Because of the demonstrated proportionality between Qd?a and mH it is possible to determine the solution enthalpy without knowing mH by normalizing the van’t Hoff solvus relation given by: ln X ¼

 d!a DSd!a DH  R RT

ð8:18Þ

by combining Eq. 8.18 with Eq. 8.17. This yields: ln Xmodif ¼

 d!a DSd!a DH  R RT

ð8:19Þ

8.4 Summary of Results of Experimental Solvus Determinations

277

 ðAÞ Þ; with DH  ðAÞ a normalization factor for Qd?a where Xmodif ¼ Qd!a =ðmZr DH d!a d!a obtained in the usual way from a van’t Hoff plot of the same data. Then, plotting the data in a van’t Hoff plot according to Eq. 8.19 (which is essentially a plot of ln Qd?a versus 1/Td) and fitting a linear regression line to these data provides yet  d!a but without the need, when using this approach, of another value for DH specifying the hydrogen concentration, mH. This approach was adopted for the irradiated material where it turned out to be useful in interpreting the results assuming that trapping of hydrogen at dislocations is the reason for the variation of the solvus enthalpies with thermal cycle number and annealing temperature and time. Of course, when applied to the data for the same material this approach provides no extra benefit, since it still does require the hydrogen concentration in the material to be known in order to calculate the reference solvus enthalpy,  ðAÞ ; through a regression fit of the data plotted in the usual van’t Hoff relaDH d!a tionship of ln X versus 1/Td. However, when this direct approach is used to  d!a in irradiated material, it is useful because DH  ðAÞ acts as a refcalculate DH d!a erence enthalpy that allows one to quantify the additional dislocation trapping  d!a in these irradiated material. effect on the measured DH Vizcaíno et al. [47–50] did not discard the results of their first heating runs, as is the usual custom, because these authors felt that some insight concerning the effect of irradiation on the dissolution solvus could be gleaned from the results of this run. The solvus enthalpies calculated from Eq. 8.17 for different specimens taken from different locations (varying in irradiation temperatures and fluences) were always much lower for the first cycle than for those obtained on subsequent cycles. Also, as shown in Fig. 8.13, unlike the results obtained by McMinn et al. [24] for presumably closely similar material, the solvus enthalpies obtained from the irradiated material by Vizcaíno et al. [49] gave increasingly larger (absolute) solvi enthalpy values with thermal cycles up to about the third cycle after which there was little further change. Except for material taken from coolant channels in the outer layer of channels in the reactor, the absolute values of the solvus enthalpies at saturation were, however, considerably lower than those obtained by McMinn et al. [24]. In addition, after annealing the irradiated material for 2 h at either 508 or 611 C, the solvus enthalpies only increased by a small amount, remaining well below the value obtained for the unirradiated material. For the measurements carried out on the unirradiated material the authors recorded the results obtained from both the peak and the completion temperatures. The solvus enthalpies are similar to those obtained by McMinn et al. [24]. There are two important differences between the starting conditions of the irradiated materials investigated by McMinn et al. [24] and Vizcaíno et al. [47, 49]. The first difference is the higher hydrogen content in the material taken from the coolant channel component. The hydrogen content in these materials ranged from 156 to 260 wppm, compared to the irradiated EB welded Zr-2 and -4 materials used by McMinn et al. [24] in which the hydrogen content ranged from 15 to 75 wppm and in which, moreover, additional hydrogen was added to some of the specimens after the material had been irradiated. Adding hydrogen after

278

8 Experimental Results and Theoretical Interpretations

Fig. 8.13 Evolution of  irrad with successive runs DH da of three samples from Region 3 CC (containing ffi157 wppm H). (from Vizcaíno et al. [50])

irradiation limits the maximum amount of hydrogen that can be added without annealing out some of the irradiation effects. The differences in available diffusible hydrogen resulting from these non-overlapping hydrogen content levels may be the reason why the effect of irradiation and the recovery of these effects on the solvus are so different in these two sets of materials. The second difference is that the higher hydrogen content levels in the materials from the Atucha reactor were reached by a slow absorption of hydrogen in the material as a by-product of the surface corrosion reaction. This slow hydrogen ingress occurred in conjunction with a simultaneous build up of fast neutron fluence, resulting in a gradual and simultaneous increase in irradiation-induced dislocation loops. The foregoing two differences between the two materials likely resulted in the locations and size distributions of hydride precipitates being different—at least during the initial thermal cycle—and may be the reasons for the different rates of recovery of the solvus composition to that in unirradiated material.

8.4.4 Combined Effect of Thermal Aging and Irradiation in Zr–2.5Nb The foregoing interpretations of the changes in solvus composition as a function of fluence, irradiation temperature, and post-irradiation annealing in a-phase zirconium alloy material exposed to fast neutron irradiation has indicated that the precipitation of hydrides in the core of dislocations may play an important role. However, in a/b zirconium alloys, the effect of irradiation on the solvus is further confounded by associated effects of fast neutron irradiation and thermal aging on the composition and morphology of the b phase. The resulting changes of the b phase

8.4 Summary of Results of Experimental Solvus Determinations

279

Fig. 8.14 TSSD determined using DSC in samples saw-cut from ex-service pressure tubes BB and AA at inlet, outlet, and middle of the pressure tube locations. See Table 8.4 for list of equations and references. (from Pan [27]; with permission from AECL (copyright 2012, AECL))

affect the partitioning of hydrogen between the two phases and, hence, the solvus composition. This complication is not present in the a-phase zirconium alloys such as Zircaloy-2 or -4 fuel cladding and cold worked Zircaloy-2 pressure tube materials. Based on results and considerations given in Sects. 8.4.2 and 8.4.3, the solvus compositions in pressure tubes in operating reactors relative to the corresponding compositions in pressure tubes at the time of installation would be expected to decrease over time in the rolled joint regions because of the greater break up of the b phase (particularly at the outlet end) in these regions. In terms of the accommodation energy model, any offsetting effect from increases in solvus composition brought about by an irradiation-induced increase in yield strength and, hence, accommodation energy, is expected to be less than in the body of the tube, the net effect being a reduction in the solvus composition in these regions. The variation in solvus composition as a function of axial position in the body of the tube would depend on the offsetting contributions from irradiation-induced strength (and, hence, accommodation energy) increases versus changes in beta-phase composition brought about by aging and fast neutron fluence, both of which are functions of time and location. The following summarizes some recent experimental results illustrating these possibilities. For the outlet and inlet rolled joints from ex-service pressure tubes BB and AA, Pan [27] determined the TSSD relationships from van’t Hoff plots of data derived from DSC measurements of the TSSD temperature combined with corresponding HVEMS measurements of the hydrogen isotope concentration. The results (Fig. 8.14) show that there is a small difference between the TSSD relationships for the outlet and inlet rolled joint pressure tube regions, with the outlet having a lower TSSD composition. Cann’s [2] prediction was that for the degree of b-phase

280

8 Experimental Results and Theoretical Interpretations

Table 8.4 List of TSSD equations plotted in Fig. 8.14. Unit of concentration is in wppm while the unit of the solvus enthalpy is in J/(mol H) TSSD Equations: Tube, Location along Tube Reference Ex-service tubes AA and BB, middle: cH = 6.472 9 104 exp[-33 016/(RT)] Ex-service tubes AA and BB, inlet: cH = 1.202 9 105 exp[-35 907/(RT)] Ex-service tubes AA and BB, outlet: cH = 2.951 9 105 exp[-40 116/(RT)] Ex-service tubes AA and BB, all samples: cH = 1.048 9 105 exp[-35 320/(RT)] FFSG: cH = 8.19 9 104 exp[-34 500/(RT)]

Pan [27] Pan [27] Pan [27] Pan [27] CSA [1]

Table 8.5 List of TSSD and DHCi equations plotted in Fig. 8.15. Unit of concentration is in wppm while the unit of the solvus enthalpy is in J/(mol H) TSSD/DHCi equations: Tube, Solvus Type, Location along Tube Reference Ex-service tube BB, TSSD, all: cH = 1.102 9 105 exp[-35 593/(RT)] Ex-service tube BB, DHCi, all: cH = 2.296 9 105 exp[-38 171/(RT)] Ex-service tube AA, TSSD, all: cH = 5.976 9 104 exp[-32 637/(RT)] Ex-service tube AA, DHCi, all: cH = 1.247 9 105 exp[-34 953/(RT)] Unirradiated, DHCi: cH = 1.337 9 105 exp[-35 142/(RT)] FFSG: cH = 8.19 9 104 exp[-34 500/(RT)]

Pan [27] Pan [27] Pan [27] Pan [27] Pan [27] CSA [1]

decomposition expected in the inlet and outlet rolled joint regions of the pressure tube, the difference in TSSD in these regions and in autoclaved pressure tubes having had no other thermal treatment should be less than 10 %.8 Cann’s b-phase partitioning results also lead to the prediction that TSSD compositions in the outlet rolled joint region should be slightly lower because the outlet end of the pressure tube would have a slightly more decomposed b-phase. Similar small differences would be expected to prevail for TSSD values obtained at axial locations along the pressure tube between the locations where the decomposition of the b-phase would be least compared to where it would be greatest. Table 8.4 lists the solvus equations used in Fig. 8.14, including their sources. The effect of an increase in TSSD composition with fluence as predicted from the accommodation energy model is confirmed by the results of Pan on material taken from the middle of three ex-service pressure tubes BB, AA, and AB (Fig. 8.14). For this material, there should be minimal b-phase decomposition. Hence the results should reflect just the effect of increase in irradiation-induced strength of the material. Based on the very simple accommodation energy model, assuming that it is the macroscopic yield strength that is affecting its magnitude and hence the solvus, there is also predicted to be little fast neutron fluence dependence of the solvus in pressure tube material that has been in service for at least a year or two, since the

8

The review of Cann’s results in Chap. 4 suggests that the change in TSSD should be somewhat less than what Cann estimates since he assumed that the partitioning of the hydrogen to the b phase would be reduced both by the decrease in the measured partitioning ratio and (erroneously, it appears) the reduction in the volume fraction of the beta phase.

8.4 Summary of Results of Experimental Solvus Determinations

281

Fig. 8.15 Comparison between the solubility for DHC initiation and TSSD in ex-service pressure tubes BB and AA with that from an unirradiated pressure tube. See Table 8.5 for list of equations and references. (from Pan [27]; with permission from AECL (copyright 2012, AECL))

increase in yield strength with fluence has saturated at a fluence of *1 9 1025 n/m2. Table 8.5 lists the solvi used in Fig. 8.15, including their sources.

8.4.5 Effect of Manufacturing Variables, Microstructure, and/or Composition The foregoing summary shows that differences in manufacturing variables, microstructure, and/or alloying or impurity compositions can, in some cases, affect the solvus composition. The most recent measurements of terminal solvus relationships were, therefore, done to determine whether this is the case for some of the more recently developed fuel cladding materials or for the pressure tube material manufactured for the Indian PHWRs. All of these evaluations were done on unirradiated material. Using dilatometry Singh et al. [43] carried out solvus determinations on cold worked Zircaloy-2 and Zr–2.5Nb pressure tube material manufactured for the Indian PHW reactors. Three methods of processing the data for the variation of strain versus temperature were used. These were: (1) direct monitoring of strain, (2) instantaneous derivative of the strain, and (3) differential of the strain between the hydrogenated and unhydrogenated reference samples. The results obtained are summarized in Table 8.6. The temperature cycles used in these tests were such that the TSSP values obtained can be assumed to be equivalent to those labeled TSSP1 by Pan et al. [31]. They have, therefore been identified as such in Table 8.6. The TSSD relationships obtained by Singh et al. [43] for the Indian-

282

8 Experimental Results and Theoretical Interpretations

Table 8.6 List of TSSD and TSSP equations for different materials (cw PT stands for cold worked pressure tube). Unit of concentration is in wppm while the unit of the solvus enthalpy is in J/(mol H) TSS Equation: Material Reference TSSD, Zircaloy-2 cw PT: cH = 3.818 9 104 exp[-30 004/(RT)] TSSP, Zircaloy-2 cw PT: cH = 2.854 9 104 exp[-25 930/(RT)] TSSD, Zr-2.5Nb cw PT: cH = 6.358 9 104 exp[-35 440/(RT)] TSSP, Zr-2.5Nb cw PT: cH = 3.235 9 103 exp[-17 210/(RT)] TSSD, Zr-2.5Nb cw PT: c = 5.26 9 104 exp[-31 819/(RT)] TSSP; Zr-2.5Nb cw PT: cH = 2.22 9 104 exp[-24 618/(RT)] TSSD, Zircaloy-2 ? high Fe Zircaloy: cH = 1.28 9 105 exp[-36 540/ (RT)] TSSP, Zircaloy-2 ? high Fe Zircaloy: cH = 5.26 9 104 exp[-28 068/ (RT)]a TSSP, Zircaloy-2 ? high Fe Zircaloy: cH = 1.07 9 104 exp[-21 026/ (RT)]b TSSD, Zr: cH = 1.41 9 105 exp[-38 104/(RT)] TSSP, Zr: cH = 3.39 9 104 exp[-27 291/(RT)] TSSD, N18: cH = 5.364 9 104 exp[-31 809/(RT)] TSSP, N18: cH = 2.973 9 104 exp[-25 690/(RT)] TSSD, Zircaloy-4: cH = 5.258 9 104 exp[-33 117/(RT)] TSSP Zircaloy-4: cH = 4.014 9 104 exp[-27 336/(RT)] TSSD, M5: cH = 8.497 9 104 exp[-34 187/(RT)] TSSP, M5: cH = 3.064 9 104 exp[-26 180/(RT)] a

For T [ 533 K;

b

Singh et al. [43] Singh et al. [43] Singh et al. [43] Singh et al. [43] Giroldi et al. [9] Giroldi et al. [9] Une and Ishimoto [45] Une and Ishimoto [45] Une and Ishimoto [45] Une and Ishimoto [46] Une and Ishimoto [46] Tang and Yang [44] Tang and Yang [44] Tang and Yang [44] Tang and Yang [44] Tang and Yang [44] Tang and Yang [44]

For T B 533 K

sourced Zr–2.5Nb pressure tube material are well outside the tight band of results obtained by others for this and for Zircaloy-2 materials. This is likely because of poor signal quality, particularly for the Zr–2.5Nb material, for which the authors note that results obtained with the second and third criteria for solvus temperature determinations were unusable. Giroldi et al. [9] also recently carried out a study of the solvus in cold worked Zr–2.5Nb CANDU type pressure tube material. There were two main objectives of this work. One was to explore the effect of the increasingly greater rate of b phase decomposition for hydrogen contents where the solvus temperature is greater than 400 C. The other was to explore the viabilities of the dilatometry and DSC techniques for solvus temperature determination in relation to the different criteria, as discussed in Sect. 8.4.1, for establishing these temperatures with each technique. Regarding the differences in the results between the dilatometry and DSC techniques, Giroldi et al. [9] conclude that—for Zr–2.5Nb pressure tube material, at least—the dilatometry technique is inherently more uncertain. This is consistent with the less distinct solvus transition point obtained by Singh et al. [43] for this material using this technique. The solvus relationships obtained with the DSC technique using the maximum slope criterion are listed in Table 8.6.

8.4 Summary of Results of Experimental Solvus Determinations

283

Une and Ishimoto [45] used DSC with the maximum slope criterion to obtain solvus relationships for high Fe-containing Zircaloy-2 fuel cladding material. As a comparison, similar measurements were also carried out for regular Zircaloy-2 fuel cladding material. Using a cooldown/heat up rate of 10 C, the authors found no differences in the TSSD relationships for the two Zircaloy-2 materials. An interesting result during heat up is shown for a sample containing 106 wppm H. Relative to a baseline drawn to join the low and high temperature sides of the DSC signal, there is a small peak at 201 C, in addition to the one at 360 C, the latter corresponding to the TSSD temperature. The authors speculate that the former peak is the result of the conversion of c- to d-hydrides during heat up. Preexponential values and enthalpies obtained from linear regressions of van’t Hoff plots for TSSD and TSSP are summarized in Table 8.6. The authors do not state what the maximum temperature was to which each specimen was taken before cool down nor whether the maximum temperature was the same regardless of the total hydrogen content in the specimen.9 (The hydrogen content in their specimens ranged from 40 to 542 wppm.) The results obtained show that TSSP data below 260 C lie above the extrapolated TSSP line fitted to the higher temperature data. McMinn et al. [24] obtain roughly similar results for Zircaloy-2 (starting at 170 C in their case), but only for the irradiated material. However, below hydrogen contents of 10 wppm these authors’ data also show that there is less of a reduction in the solvus composition versus temperature compared to the low-temperature extrapolation of the high temperature regression fit. This is not the case for the data of Pan et al. [31] for Zr–2.5Nb pressure tube material where even the data at 10 wppm and lower follow closely the overall linear regression fit to all of the data. As discussed in Chap. 4 and Sect. 8.4.2, one reason for these varying results of TSSP data at low temperatures may be because of small hydride precipitates locked-up in dislocation cores, the binding energy at that location being so high that these hydrides do not dissolve on heating until a considerably higher temperature is reached than for the hydrides in the bulk of the material and, conversely, precipitate on cooling at much higher temperatures than hydrides forming in the bulk, thus effectively reducing the amount of free hydrogen available for dissolution or precipitation in relation to the total hydrogen content of the material. This effect would likely only become noticeable at low temperatures when the hydrogen locked up in dislocation cores becomes comparable to the amount retained in the a phase at the solvus composition corresponding to the hydride precipitates located in the bulk of the material. Une and Ishimoto [45] compared the hysteresis obtained experimentally for Zircaloy-2 with that obtained from the accommodation energy model, using pure elastic and elastic–plastic accommodation energy values calculated by Puls [34]. These accommodation energy values were calculated using analytical expressions 9

In a subsequent paper they state that the maximum temperature used was 500 C with a hold time of 5 min. However, this maximum temperature could not have been used by the authors for specimens with hydrogen content close to and up to the maximum value of 542 wppm, since the dissolution temperature for the maximum hydrogen content is above 500 C.

284

8 Experimental Results and Theoretical Interpretations

for isotropically misfitting spheres derived by Lee et al. [20]. However, the hysteresis determined experimentally by Une and Ishimoto [45] was obtained only from the enthalpy difference of the linear regression fits of the solvus values for dissolution and precipitation (Table 8.6). In such fits, both the pre-exponential and the solvus enthalpy terms are fitting parameters. Thus they typically turn out to be different for each solvus. As a result, the solvus enthalpy difference obtained from such fits does not have any well-defined physical meaning in the sense that it represents, for instance, the accommodation energy difference between a dissolving and a precipitating hydride, because some of the contribution to hysteresis is now contained also in the pre-exponential term. To obtain corresponding physically meaningful values from such fits to the data, the pre-exponentials— representing all entropy contributions to the solvus other than entropy of mixing— need to be determined either from separate thermodynamic data or given a fixed value by choosing, for instance, a suitable average value of the pre-exponential terms obtained from the standard regression fits of the two solvi data sets. This was done by Shi et al. [42] for the data obtained by Pan et al. [31]. The enthalpy difference of 3 296 kJ/(mol K) calculated from the data of TSSP1 and TSSD for Zr–2.5Nb turned out to be much smaller than the value of 8 680 kJ/(mol K) obtained directly from the original regression fit by Pan et al. [31]. Une and Ishimoto [45] obtained a similarly large difference of 8 472 kJ/(mol K) from their regression fits to their TSSD and TSSP1 Zircaloy-2 data in which the pre-exponential is a fitting parameter. Comparing this experimentally obtained value with 4  627 J/(mol H) (D wel inc ðsphereÞ  Qheat ¼ 4 178  ð449Þ J/(mol H) obtained from the accommodation energy model assuming a yield strength value of *500 MPa led Une and Ishimoto [45] to conclude that there was poor agreement between the hysteresis gap predictions obtained from the accommodation energy model compared to the value obtained experimentally. However, based on the revised experimentally determined enthalpy difference given by Shi et al. [42], for the solvus data in Zr–2.5Nb, agreement between the predictions of the accommodation energy model and experimental results is now much closer. One would expect similar improved agreement to prevail for the Zircaloy-2 data obtained by Une and Ishimoto [45] when re-analyzed according to the approach of Shi et al. [42]. In a follow-up study Une and Ishimoto [46], using the same DSC method and solvus temperature selection criterion, compared their previous solvus results with those in unalloyed Zr from two different sources, viz., Zr liner, and NBS Zr, having, respectively, oxygen contents of 320 and 850 wppm. The objective was to remove any uncertainty arising from the use of different measurement techniques and solvus temperature selection criteria from a comparison of the solvus relationships obtained between Zircaloy-2 and unalloyed Zr materials. Only two data points were obtained for the NBS Zr material and these did not show any meaningful differences in the TSSD and TSSP data obtained for this and the Zr liner material. The preexponential and solvus enthalpy values obtained from regression fits of the data are

8.4 Summary of Results of Experimental Solvus Determinations

285

listed in Table 8.6. The van’t Hoff plots of these data—combined with the data from the authors’ previous study on Zircaloy-2—show that the unalloyed Zr results have lower solubility for both TSSD and TSSP, with a larger difference obtained for the latter. Only Kearns [12] has carried out a similar study using the same method (diffusion equilibration) on these two types of materials. Kearns found a similar difference in TSSD between unalloyed Zr with different oxygen contents (\100 wppm and 1180 wppm) and the Zircaloy-2 and -4 materials. Une and Ishimoto speculate that the difference between the alloy materials and the unalloyed ones is an effect arising from the difference in alloy addition and not an effect of oxygen content since both the latter authors and Kearns found no significant difference in solvus relations between the two unalloyed Zr materials containing different levels of oxygen. However, the difference in solvus enthalpies between unalloyed Zr and Zircaloy-2 and -4 materials is of the right sign and magnitude to be consistent with a difference in elastic–plastic accommodation energy between the two materials arising from the latter having lower yield strength. Une and Ishimoto [45] were the first to carry out a systematic study—using the same measurement technique and solvus temperature selection criterion—of the difference in TSSP solvi between unalloyed Zr and Zircaloy-2 materials. From the accommodation energy model, one would expect the difference between the two sets of materials to be greatest for the solvus for precipitation, but only for TSSP2, not TSSP1. In terms of the maximum temperature during the cooling cycle to which the specimens were taken, the precipitation solvus obtained by Une and Ishimoto may have been closer to TSSP1 than to TSSP2. This conjecture notwithstanding, there may be indirect effects coming into play for the softer material in terms of the nature and amount of dislocation debris left behind when hydrides fully dissolve that could result in the detected precipitation solvus ending up being closer to TSSP2 than to TSSP1. A study using the DSC technique was carried out by Setoyama et al. [41] on the solvus for unalloyed Zr, Zircaloy-2, and Zr–M (M = Fe, Sn, Cr, Ni) materials for use in integrity assessments of these as choices for fuel cladding material. Only TSSD was measured because of its better reproducibility. No mention was made of the cooling and heating rates used nor was the criterion for selecting the solvus temperature stated. However, from the example given of a typical DSC curve, it is inferred that the peak temperature criterion was used. The authors found that increasing levels of Sn and Cr increased TSSD whereas no effect on this solvus was found with increasing levels of Fe and Ni. These results have not been reproduced in Table 8.6, because of the different solvus temperature criterion used, which is not consistent with the maximum slope criterion used to obtain the solvus relationships of the other results in the table. A study of an experimental fuel cladding material for use in PW reactors at high burn up was carried out by Tang and Yang [44]. The material studied was N18, one of a number of Zr–Sn–Nb fuel cladding alloys being developed in China. For

286

8 Experimental Results and Theoretical Interpretations

comparison, the materials Zircaloy-4 and M5—the latter being a Zr–Nb fuel cladding alloy—were also studied. The DSC method, combined with the maximum slope criterion, was used to identify the solvus temperatures for precipitation and dissolution. The hydrogen content in the test samples ranged from 20 to 240 wppm. A heat up/cooldown rate of 10 C was chosen, but, except for the first conditioning cycle (the results of which were discarded), the upper and lower temperatures and the hold times at these temperatures for the cycles employed was not given. From the example plot given, however, it can be inferred that the maximum temperature was at least 400 C (but the time at temperature was not indicated) from which it can be inferred that TSSP1 might have been the precipitation solvus measured for most specimens in these studies. The exception would be those specimens at the highest hydrogen content and slightly below for which the TSSD temperatures would have been at, to above, 400 C, respectively. The results obtained, summarized in Table 8.6, show that there was close agreement between the TSSD relationships for the three materials studied by the authors as well as those for similar materials studied by other investigators. Similarly good agreement was obtained for the TSSP relationships except for those obtained by Singh et al. [43] and McMinn et al. [24]. Of these latter two, the TSSP solvus relationship obtained by McMinn et al. [24] in unirradiated Zircaloy-2 and -4 materials stands out for being much lower than those obtained by all other investigators. It is seen that the hysteresis gap between TSSP and TSSD obtained by these authors was much smaller than what was found by all other investigators. The reason for the smaller hysteresis gap obtained by McMinn et al. [24] may be because of the greater amount of hydride trapping at dislocations in these materials. Consistent with this speculation is the trend in the data below 7 wppm hydrogen, where there was essentially little variation in solvus composition with temperature. In fact, the authors point out that these data points were not included in the regression fit to the remainder of the data. As noted in Sect. 8.4.3, this leveling off becomes even more pronounced and moves to higher temperatures for TSSP for the irradiated material, which is also consistent with the increased numbers of dislocations available that can harbour hydrides in their cores.

References 1. CSA: Technical Requirements for the In-service Evaluation of Zirconium Alloy Pressure Tubes in CANDU Reactors. Canadian Standards Association, Mississauga, Ontario, Canada, Nuclear Standard N285.8-10 (2010) 2. Cann, C.D.: Unpublished data. AECL, Chalk River Laboratories, Chalk River, Ontario, Canada (1994) 3. Cann, C.D., Atrens, A.: A metallographic study of the terminal solubility of hydrogen in zirconium at low hydrogen concentrations. J. Nucl. Mater. 88, 42–50 (1980) 4. Cann, C.D., Puls, M.P., Sexton, E.E., et al.: The effect of metallurgical factors on hydride phases in zirconium. J. Nucl. Mater. 126, 197–205 (1984)

References

287

5. Cann, C.D., Sexton, E.E., Duclos, A.M., et al.: The effect of decomposition of beta-phase Zr-20 at% Nb on hydrogen partitioning with alpha-zirconium. J. Nucl. Mater. 210, 6–10 (1994) 6. Domain, C., Besson, R., Legris, A.: Atomic-scale ab initio study of the Zr–H system: I. Bulk properties. Acta Mater. 50, 3513–3526 (2002) 7. Erickson, W.H., Hardy, D.: The influence of alloying elements on the terminal solubility of hydrogen in a-zirconium. J. Nucl. Mater. 13, 254–262 (1964) 8. Flanagan, T.B., Park, C.-N., Oates, W.A.: Hysteresis in solid state reactions. Prog. Solid State Chem. 23, 291–363 (1995) 9. Giroldi, J.P., Vizcaíno, P., Flores, A.V., et al.: Hydrogen terminal solid solubility determinations in Zr–2.5Nb pressure tube microstructure in an extended concentration range. J. Alloys Comp. 474, 140–146 (2009) 10. Griffiths, M., Davies, P.H., Davies, W.G., et al.: Predicting the in-reactor mechanical behavior of Zr–2.5Nb pressure tubes from postirradiation microstructural examination data. In: Moan, G.D., Rudling, P (eds.) Zirconium in the Nuclear Industry: Thirteenth International Symposium. ASTM STP, vol. 1423, pp. 507–523 (2002) 11. Johnson, W.C., Voorhees, P.W.: Phase equilibrium in two-phase coherent solids. Metall. Trans. A 18A, 1213–1228 (1987) 12. Kearns, J.J.: Terminal solubility and partitioning of hydrogen in the alpha phase of zirconium, Zircaloy-2 and Zircaloy-4. J. Nucl. Mater. 22, 292–303 (1967) 13. Khatamian, D., Ling, V.C.: Hydrogen solubility limits in a- and b-zirconium. J. Alloys Comp. 253, 162–166 (1997) 14. Khatamian, D.: Solubility and partitioning of hydrogen in meta-stable Zr-based alloys used in the nuclear industry. J. Alloys Comp. 293–295, 893–899 (1999) 15. Khatamian, D.: DSC ‘‘peak temperature’’ versus ‘‘maximum slope temperature’’ in determining TSSD temperature. J. Nucl. Mater. 205, 171–176 (2010) 16. Khatamian, D., Pan, Z.L., Puls, M.P.: Hydrogen solubility limits in Excel, an experimental zirconium-based alloy. J. Alloys Comp. 231, 488–493 (1995) 17. Khatamian, D., Root, J.H.: Comparison of TSSD results obtained by differential scanning calorimetry and neutron diffraction. J. Nucl. Mater. 372, 106–113 (2008) 18. Kirchheim, R.: Interaction of hydrogen with dislocations in palladium—I. Activity and diffusivity and their phenomenological interpretation. Acta Metall. 29, 835–843 (1981) 19. Kirchheim, R.: Interaction of hydrogen with dislocations in palladium—II. Interpretation of activity results by Fermi-Dirac distribution. Acta Metall. 29, 845–853 (1981) 20. Lee, J.K., Earmme, Y.Y., Aaronson, H.I., et al.: Plastic relaxation of the transformation strain energy of a misfitting spherical precipitate: ideal plastic behavior. Metall. Trans. A 11A, 1837–1847 (1980) 21. Leitch, B.W., Puls, M.P.: Finite element calculations of the accommodation energy of a misfitting precipitate in an elastic-plastic matrix. Metall. Trans. A 23A, 797–806 (1992) 22. MacEwen, S.R., Coleman, C.E., Ells, C.E., et al.: Dilation of h.c.p. zirconium by interstitial deuterium. Acta Metall. 33, 753–757 (1985) 23. Maxelon, M., Pundt, A., Pyckhout-Hintzen, W., et al.: Interaction of hydrogen and deuterium with dislocations in palladium as observed by small angle neutron scattering. Acta Mater. 49, 2625–2634 (2001) 24. McMinn, A., Darby, E.C., Schofield, J.S.: The terminal solid solubility of hydrogen in zirconium alloys. In: Sabol, G.P., Moan, G.D. (eds.) Zirconium in the Nuclear Industry: Twelfth International Symposium. ASTM STP, vol. 1354, pp. 173–195 (2000) 25. Mishima, Y., Ishino, S., Nakajima, S.: A resistometry study of the solution and precipitation of hydrides in unalloyed zirconium. J. Nucl. Mater. 27, 335–344 (1968) 26. Mishra, S., Sivaramakrishnan, K.S., Asundi, M.K.: Formation of the gamma phase by a peritectoid reaction in the zirconium-hydrogen system. J. Nucl. Mater. 45, 235–244 (1972/73) 27. Pan, Z.L.: Unpublished data. AECL-Chalk River Laboratories, Chalk River, Ontario, Canada (2000/2001)

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28. Pan, Z.L., Puls, M.P., Ritchie, I.G.: Measurement of hydrogen solubility during isothermal charging in a Zr alloy using an internal friction technique. J. Alloys Comp. 211–212, 245–248 (1994) 29. Pan, Z.L., Puls, M.P.: Precipitation and dissolution peaks of hydride in Zr–2.5Nb during quasistatic thermal cycles. J. Alloys Comp. 310, 214–218 (2000) 30. Pan, Z.L., St Lawrence, S., Davies, P.H., et al.: Effect of irradiation on the fracture properties of Zr–2.5Nb pressure tubes at the end of design life. J. ASTM Int. 2/9, 1–22 (2005) 31. Pan, Z.L., Ritchie, I.G., Puls, M.P.: The terminal solid solubility of hydrogen and deuterium in Zr–2.5Nb alloys. J. Nucl. Mater. 228, 227–237 (1996) 32. Pfeifer, M.J., Voorhees, P.W.: A graphical method for constructing coherent phase diagrams. Acta Metall. Mater. 39, 2001–2012 (1991) 33. Puls, M.P.: The effects of misfit and external stresses on terminal solid solubility in hydrideforming metals. Acta Metall. 29, 1961–1968 (1981) 34. Puls, M.P.: On the consequences of hydrogen supersaturation effects in Zr alloys to hydrogen ingress and delayed hydride cracking. J. Nucl. Mater. 165, 128–141 (1989) 35. Puls, M.P.: Elastic and plastic accommodation effects on metal-hydride solubility. Acta Metall. 32, 1259–1269 (1984) 36. Puls, M.P.: Determination of fracture initiation in hydride blisters using acoustic emission. Metall. Trans. A 19A, 2247–2257 (1988) 37. Puls, M.P., Rogowski, A.J.: Hydride formation and redistribution in Zr–2.5wt% Nb stressed in torsion. In: Latanision, R.M., Pickens, J.R. (eds.) Atomistics of Fracture. Plenum Publishing Corporation, pp. 789–794 (1983) 38. Ritchie, I.G., Sprungmann, K.: Hydride precipitation in zirconium studied by pendulum techniques. Atomic Energy of Canada Report AECL-7806 (1983) 39. Ritchie, I.G., Pan, Z.L.: An internal friction study of Zr–2.5wt% Nb–H alloys. Phil. Mag. A 63, 1105–1113 (1991) 40. Ritchie, I.G., Pan, Z.L.: Internal friction and Young’s modulus measurements in Zr–2.5Nb alloy doped with hydrogen. In: Kinra, V.K., Wolfenden, A. (eds.) M3D: Mechanics and Mechanism of Material Damping. pp. 385–395. ASTM, Phildadelphia (1992) 41. Setoyama, D., Matsunaga, J., Ito, M., et al.: Influence of additive elements on the terminal solid solubility of hydrogen for zirconium alloy. J. Nucl. Mater. 344, 291–294 (2005) 42. Shi, S.Q., Shek, G.K., Puls, M.P.: Hydrogen concentration limit and critical temperatures for delayed hydride cracking in zirconium alloys. J. Nucl. Mater. 218, 189–201 (1995) 43. Singh, R.N., Mukherjee, S., Gupta, A., et al.: Terminal solid solubility of hydrogen in Zr-alloy pressure tube material. J. Alloys Comp. 389, 102–112 (2005) 44. Tang, R., Yang, X.: Dissolution and precipitation behaviors of hydrides in N18, Zry-4 and M5 alloys. Int. J. Hydrogen Energy 34, 7269–7274 (2009) 45. Une, K., Ishimoto, S.: Dissolution and precipitation behavior of hydrides in Zircaloy-2 and high Fe Zircaloy. J. Nucl. Mater. 322, 66–72 (2003) 46. Une, K., Ishimoto, S.: Terminal solid solubility of hydrogen in unalloyed zirconium by differential scanning calorimetry. J. Nucl. Sci. Technol. 41, 949–952 (2004) 47. Vizcaíno, P., Banchik, A.D., Abriata, J.P.: Solubility of hydrogen in Zircaloy-4: irradiation induced increase and thermal recovery. J. Nucl. Mater. 304, 96–106 (2002) 48. Vizcaíno, P., Banchik, A.D., Abriata, J.P.: Calorimetric determination of the d hydride dissolution enthalpy in ZIRCALOY-4. Metall. Mater. Trans. A 35A, 2343–2349 (2004) 49. Vizcaíno, P., Banchik, A.D., Abriata, J.P.: Hydride phase dissolution enthalpy in neutron irradiated Zircaloy-4. J. Nucl. Mater. 336, 54–64 (2005) 50. Vizcaíno, P., Flores, A.V., Bozzano, P.B., et al.: Hydrogen solubility and microstructural changes in Zircaloy-4 due to neutron irradiation. J. ASTM Int. 8: Paper ID JAI102949 (2011) 51. Schwarz, R.B., Khachaturyan, A.G.: Thermodynamics of open two-phase systems with coherent interfaces: application to metal-hydrogen systems. Acta Mater. 54, 313–323 (2006) 52. Slattery, G.F.: The terminal solubility of hydrogen in zirconium alloys between 30 and 400 C. J. Inst. Metals 95, 43–47 (1967)

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53. Yamanaka, S., Yoshioka, K., Uno, M., et al.: Thermal and mechanical properties of zirconium hydride. J. Alloys Compd. 293–295, 23–29 (1999) 54. Yamanaka, S., Yoshioka, K., Uno, M., et al.: Isotope effects on the physicochemical properties of zirconium hydride. J. Alloys Compd. 293–295, 908–914 (1999) 55. Zuzek, E., Abriata, J.P., San-Martin, A., et al.: H–Zr (Hydrogen-Zirconium). In: Phase Diagrams of Binary Hydrogen Alloys, pp. 309–322. ASM International, Materials Park, Ohio, USA (2000)

Chapter 9

Fracture Strength of Embedded Hydride Precipitates in Zirconium and its Alloys

9.1 Introduction Failure of a component can occur in two major ways. One is a rapid process that depends on the overall fracture toughness of the material. The other is a slow process driven by diffusion of an atomic constituent in the material to a tensile stress concentrator, the increase in concentration of this component causing a decrease in local fracture toughness. This latter process generally consists of two stages often having different time dependencies: (1) an initiation stage and (2) a subsequent repeated propagation stage during which the subcritical crack produced in the initiation stage slowly increases in length until it reaches its critical length for unstable propagation. At this point the process becomes a rapid one, the crack length at which this occurs depending on the overall fracture toughness of the material. Hydrogen in a metal can lead to both types of failure or only one of them, depending on whether the occlusion (retention) of hydrogen in the metal is endothermic or exothermic. When it is endothermic, such as in iron, chromium, nickel, etc., generally only the second failure mode is of concern, while when it is exothermic, such as in zirconium, titanium, niobium, etc., in which hydrides are formed when the solubility limit of hydrogen is exceeded, both types of failure modes can be of concern. In each case the enabling step in the embrittlement process is the fracture of individual hydride precipitates embedded in the zirconium matrix. Thus the fracture strength of such hydride precipitates is a key material parameter that is required in understanding and modeling the conditions for loss of fracture toughness as a result of the presence, or the possibility, of locally formed hydride precipitates. Determination of this fracture strength at the microscale level in terms of continuum models is the topic of this chapter. Experimental evidence has shown that hydride precipitates of concern to degradation of fracture toughness of the material—either locally or throughout the bulk—roughly speaking must be at least as large as to be visible in standard optical micrographs. On the usual size scale used in optical micrographs the hydride M. P. Puls, The Effect of Hydrogen and Hydrides on the Integrity of Zirconium Alloy Components, Engineering Materials, DOI: 10.1007/978-1-4471-4195-2_9, Ó Springer-Verlag London 2012

291

292

9 Fracture Strength of Embedded Hydride Precipitates

Fig. 9.1 Comparison of a optical and b SEM micrograph of the same hydrided region formed at a sharp (15 lm radius) flaw tip in Zr–2.5Nb pressure tube material under an effective KI = 9 MPaHm (from Cui et al. [19])

precipitates appear as discrete, irregularly shaped linear arrays having a darker appearance than the surrounding zirconium matrix. As brought out in Chap. 3, these precipitates are generally not single solid hydrides but consist of a dense agglomeration of smaller hydride precipitates. This can be seen in comparing the appearance of the same hydride clusters using both optical and scanning electron microscope (SEM) microscopy as shown in Figs. 9.1 and 9.2. In Chap. 3 it is also shown that each of these smaller hydride platelets invariably has the same orientation relationship with the surrounding zirconium matrix material. We will sometimes refer to such a collection of hydride precipitates as hydride clusters, or particularly when located at flaws, as a hydrided region. For the purpose of calculating stress states in and around such experimentally observed hydride arrays, their shapes need to be simplified, usually by representing their overall boundaries as plates or oblate spheroids compressed in such a way that the interior consists totally of hydride. The thickness and widths of these idealized precipitates are derived from their appearance in optical micrographs by enveloping their shapes by rectangles or ellipsoids of dimensions corresponding to the average thickness and average length of such arrays. This is obviously a somewhat arbitrary and inexact method and parameters derived from such idealized hydrided regions such as, for instance, their stress states should be considered as being more useful in a comparative rather than an absolute sense when comparing hydrided regions of different shapes and dimensions.

9.2 Early Work In the early years of the nuclear industry, since there had been no failures attributable to the slow fracture process, it was not recognized that this could be a possibility. Therefore, all effort was devoted to characterizing and understanding

9.2 Early Work

293

Fig. 9.2 Comparison of a optical and b SEM micrograph of the same hydrided region formed at a blunt (100 lm radius) flaw tip in Zr–2.5Nb pressure tube material under an effective K = 14 MPaHm (from Cui et al. [19])

the possibility of the rapid mode of fracture which could be caused by the reduction in overall fracture toughness of zirconium alloys as a result of an increase in volume fraction of hydride precipitates. Studies were carried out on the overall effect that various uniform distributions of hydride precipitates might have on reduction in fracture toughness of the material [15, 22, 26, 27, 33, 34, 52]. Emphasis in these experiments was on overall embrittlement caused by an increasing volume fraction of hydride precipitates as a function of their morphologies, orientations, and sizes. Observations were limited to specimens that had been taken to rupture. However, with the discovery of DHC and its relevance to pressure tubes [16, 21] there emerged also a need to understand what controlled the fracture of hydride clusters observed to form at flaws during DHC. In Zr–2.5Nb pressure tube material DHC propagation occurs preferentially in the axial-radial directions of the pressure tube, driven by the largest principal stress, which is the hoop (or transverse) stress. The hydride clusters formed at flaws are roughly plateshaped, embedded in the surrounding a-Zr matrix and mostly oriented with their plate normal directions parallel to the externally applied tensile stress direction, which is the transverse direction. These hydride clusters are variously also referred to as radial and/or reoriented hydrides, where radial refers to one of the principal pressure tube directions. The hydrides are called radial because in optical micrographs of the radial/transverse plane the traces of their plate edges are aligned in this direction. These same hydride clusters are also called reoriented because in Zr–2.5Nb pressure tube material their normal orientation is with their plate edges aligned along the transverse-axial plane (Chap. 3). It was thought that understanding the characteristics of fracture of these reoriented radial hydrides would provide important input for models of the threshold stress intensity factor, KIH, which determines the loading conditions at which DHC initiation at cracks would occur. A program with this in mind was, therefore, initiated by Simpson [45]. This author used Acoustic Emission (AE) to detect cracking in hydrides during continuous tensile loading of smooth and notched tensile specimens containing a

294

9 Fracture Strength of Embedded Hydride Precipitates

uniform distribution of hydrides. AE had been shown to be a reliable indicator of hydride cracking in DHC initiation and growth tests. These types of tests were subsequently continued by Puls and co-workers [14, 36, 37, 44]. In a parallel approach, fracture tests on macroscopic samples of bulk solid hydride material consisting of a range of hydrogen compositions were carried out by Simpson and Cann [46] and Puls et al. [38], repeating some of the initial studies carried out by Barraclough and Beevers [7]. Results of these studies are summarized in Chap. 2. However, as recognized also by others in recent years [1, 2, 50], the latter type of tests involve the production of macroscopic hydride specimens having microstructures different from those of the hydride clusters embedded in a zirconium matrix. In addition, their macroscopic sizes likely make them more susceptible to fracture. There have been many other studies on the fracture properties of zirconium alloy specimens containing embedded hydrides, but their main focus has generally been on understanding hydride embrittlement of fuel cladding material (see, in particular, the earlier work of Bai and co-workers and the references cited therein [3–6]). None of these studies were specifically focused on the fracture properties of ‘‘radial’’ hydrides that would be of interest for input into models of DHC initiation and growth, which is the main emphasis of this chapter. The following summarizes key findings from these latter types of studies that were initiated by Simpson [45].

9.3 Fracture Strength of Radial Hydrides: Rising Load Tensile Tests Using both smooth and notched tensile specimens, with the latter having semicircular notches, unirradiated pressure tube materials containing uniform distributions of reoriented radial hydride clusters were tested at ambient temperature as a function of average length of the radial hydride clusters. It was found that as the average length of these hydride clusters increased, the magnitude of the applied stress at which first fracture of these hydrides was detected by AE decreased [36]. For cold-worked Zircaloy-2 pressure tube material—for which deformation past the yield point of the specimen resulted in the formation of a diffuse neck—an applied stress just below the 0.2 % offset uniaxial yield strength was sufficient to initiate fracture at hydrides in specimens containing hydride cluster distributions of average length in the 90 lm range, whereas distributions of radial hydride clusters with shorter average lengths required the application of a finite amount of macroscopic plastic strain before the onset of first hydride fracture. It was found that with decreasing hydride cluster length a corresponding increase in the amount of macroscopic plastic strain was required before the onset of a steady increase in AE events indicating that fracture of hydride precipitates had commenced. This trend could not be unequivocally confirmed in the other two pressure tube

9.3 Fracture Strength of Radial Hydrides: Rising Load Tensile Tests

295

materials tested (Zr–2.5Nb and the experimental alloy, Excel1) because of the different necking behaviors of these materials. There were, however, indirect indications from the deformation and fracture characteristics of these specimens that indicated a similar trend in hydride cracking behavior in these materials. The difference in necking behavior was the result of differences in the preferred textures of Zr–2.5Nb and Excel materials compared to that in the cold-worked Zircaloy-2 pressure tube material. The latter material had equal magnitudes of resolved basal pole fractions in the transverse and radial tube directions, which allowed equal deformation in these two directions, whereas the greater resolved basal pole texture in the radial versus the transverse directions in the former made it impossible for these materials to sustain the formation of a diffuse neck. In these materials a localized neck oriented *60° to the tensile axis started to form soon after the yield point was reached. Fracture of the specimens ultimately occurred along this localized neck in specimens containing radial hydride distributions of the shortest average lengths studied. Necking in this way means that most of the uniformly dispersed hydride precipitates experienced only negligible plastic strains above the yield strain and therefore did not fracture. With failure being initiated only in the localized plastic neck, the plastic strain to fracture the radial hydrides could not be determined from the engineering strain. This was exacerbated by the fact that there were now also fewer hydrides involved in the rupture process along the localized neck and, hence, fewer AE fracture events associated with the fracture of hydrides. This made it difficult to distinguish AE generated by hydride fracture with background AE produced by other sources. In specimens containing intermediate-sized hydride clusters (lengths ranging from *50 to 100 lm) the formation of a localized neck was disrupted and failure would only partially produce a localized neck with the remainder of the fracture path following the trace of the radial hydrides running at right angles to the tensile stress direction. Finally, in specimens containing the longest average radial hydride lengths (ranging from *150 to 450 lm), a localized neck would start, but failure of the specimen at right angles to the loading direction was always initiated from one of the edges of the specimen at the location where the localized neck intersected the edge of the specimen. Important findings from this study were that, up to hydride platelet lengths from 50 to 100 lm, a measurable amount of plastic strain was required for first indication of fracture initiation in the hydrides. Within the large scatter of the data, this result was independent of the matrix strength, which varied from an average value of 627 MPa for Zircaloy-2 to 780 MPa for Zr–2.5Nb and 930 MPa for Excel pressure tube material. The amount of plastic strain required for hydride fracture initiation decreased with increase in hydride cluster platelet length and axiality of the stress state. No detectable plastic strain was observed at first hydride fracture when the average hydride cluster lengths were longer than *50–100 lm in Zircaloy-2.

1

An experimental cold worked Zr–Nb pressure tube material that was never used in practice.

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9 Fracture Strength of Embedded Hydride Precipitates

9.4 Mechanism for Fracture of Embedded Radial Hydride Clusters in Rising Load Tests To explain the foregoing results, Puls [36] proceeded on the premise that the criterion for fracture of an embedded hydride cluster is governed by the attainment of a critical normal stress. This was the approach used by Bourcier and Koss [10] to rationalize their results of deformation tests of titanium tensile specimens containing hydride clusters having their longest dimensions oriented along the tensile stress axis. These authors discounted the use of fiber loading models to explain their results because of observations that transverse cracks occurred randomly along the lengths of the hydrides, a result which appears at variance with the predictions of such a model (however, see further on for a more detailed assessment on the basis of such models). Bourcier and Koss [10], therefore, adopted an approach based on the inhomogeneity of strain between the plastically deforming and the (presumed) elastically deforming hydride clusters. This approach had previously been applied by Beremin [8] for estimating the breaking stress of parallel and perpendicular MnS inclusions in steels. The method is based on a formulation by Berveiller and Zaoui [9] who used Eshelby’s [23–25] theory of inclusions to calculate the stresses and strains inside and at the boundary of inhomogeneous, elastically deforming misfitting ellipsoidal inclusions coherently embedded in a plastically deforming matrix. Fracture of the precipitates is assumed to occur when the maximum normal tensile stress inside the inclusions reaches a critical level. It is assumed that this fracture threshold value is a material parameter of the embedded hydride precipitates, independent of the morphologies and sizes of the precipitates. The morphologies and orientations of the precipitates simply determine the stresses generated inside them. It was assumed that the sole misfit strains imposed on the hydrides are induced ones, occurring when the matrix starts to deform plastically and the deformation of the hydride either remains elastic or deforms plastically, but to a lesser degree. This difference in strain rate in the two materials produces additional induced strains on the hydride when coherency is maintained between matrix and hydride precipitate. Implicit in the application of this model is that a plastic flow-induced misfit strain is required to increase the net normal tensile stress inside the precipitate to a magnitude sufficient to fracture it. To simplify the calculation, the strains resulting from the transformation strains of hydride were neglected. For mathematical convenience the hydride platelet shape was approximated as a flat oblate (disc-shaped) spheroid with k = c/a  1, where a and c are the semi-major and semi-minor axes of the spheroid, respectively. Oblate platelets with the a-axis oriented along the tensile stress direction were called parallel hydrides while platelets with the a-axis oriented perpendicular to the tensile stress direction were called perpendicular (radial) hydrides. Assuming isotropic plasticity, the following relation was obtained for the critical normal stress, rhf at fracture (i.e., the stress induced in the c-direction of the perpendicular (radial) hydride):

9.4 Mechanism for Fracture of Embedded Radial Hydride Clusters ðnÞ

rhf ¼ rN þ Epl



297

 8  6pk  ehyd N 9pk

ð9:1Þ

where rN and ehyd N are the applied net section stress and plastic strain in the tensile stress direction in the matrix at the location of those hydrides first detected to crack ðnÞ in the specimen and Epl is the ‘‘plastic equivalent’’ Young’s modulus where n refers to the axiality of the stress state (n = 1–3). The induced strain in the precipitate, ehyd N , is the difference between the applied strain in the matrix and in ðnÞ

ð3Þ

the precipitate. An estimate of Epl for Zircaloy-2 showed that Epl is about twice ð1Þ

as large as Epl : This means that plastic strains in the region between notches in notched specimens need only be half as large as those in smooth tensile specimens to generate the same increase in induced internal stress in the hydride. This prediction accounts for why fracture of hydrides in the notched specimens occurred at lower plastic strains. The inverse dependence on k of the geometric factor given by the expression in the square brackets shows that as the hydrides increase in length (at constant maximum thickness), there is a decrease in the net-section plastic strains required to fracture them. For parallel hydrides, the geometric factor is given by (20-6pk)/9pk, showing that the stress increase induced inside parallel hydrides is about 2.5 times higher than that in perpendicular hydrides. That is, this model predicts that parallel hydrides are more likely to fracture than perpendicular ones if the fracture strength of plate-shaped hydrides is independent of direction. This prediction is contrary to observations. The reason for the error in the prediction is likely the neglect of the effect of the transformation strains on the internal stresses in the hydride. When the internal stresses generated by the transformation strains alone are considered, it is seen that very much larger compressive stresses are produced in the a-direction compared to in the c-direction, with the difference increasing with decrease in k [32]. This fact was not recognized at the time of publication of these results. However, the calculations by Leitch and Puls [32] show that for isotropic—or nearly isotropic— misfit strains, the reduction in the stress in the c-direction is not sufficient to compensate for the increase in the induced stresses between parallel and perpendicular hydrides. It should be noted, first of all, that the values of the internal stresses given in the elastic–plastic case by Leitch and Puls [32] are average values. As stated in that paper, the actual stresses inside coherent hydrides embedded in an elastic–plastic matrix vary with distance along the a-direction with the lowest absolute values at the mid-point of the spheroid. In addition, it is likely that the misfit strains are highly anisotropic with almost all of the irreducible volumetric transformation strain oriented along the c-direction. This would result in almost negligible stresses in the c-direction at the center of thin hydrides shaped like oblate spheroids compared to in the other two orthogonal directions. No attempt was made to quantify the fracture threshold strength of the embedded hydride precipitates in this study, but not accounting for the stresses produced by the transformation strains would have made any estimate inaccurate at any rate.

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9 Fracture Strength of Embedded Hydride Precipitates

Comparing the predicted fracture strength values for the data obtained with the Zircaloy-2 material assuming k = 0.025 under uniaxial and triaxial loading, the foregoing theory predicts the following hydride fracture strengths for the two types of loading: rhf ¼ 1:078rys þ 0:759rys

ðuniaxialÞ

ð9:2Þ

rhf ¼ 1:20rys þ 0:367rys

ðtriaxialÞ

ð9:3Þ

where rys is the uniaxial yield strength of the matrix. In Eqs. 9.2 and 9.3, the first term is the stress applied directly by the matrix at the location of the fracturing hydrides and the second is the induced stress. From this it is seen that the fracture strength under uniaxial loading is predicted to be somewhat larger than under triaxial loading. Since this prediction runs counter to the observations, an alteration to the model, or a change in the assumption concerning the lack of orientation dependence of the fracture strength, would seem to be required. From metallographic observations of the ‘‘flow-reorientation’’ of the perpendicular (radial) hydrides in aligning themselves along a direction parallel to the direction of the uniaxial stress near the fracture face under uniaxial, but not triaxial loading it seems that hydride precipitates can plastically deform but only under uniaxial loading because the greater constraint under triaxial loading would cause the hydrides to fracture before they would start to yield and/or ‘‘flowreorient’’. Therefore, assuming some plastic flow occurs within the hydrides under uniaxial, but not under triaxial loading, Puls [36] concluded on the basis of the foregoing data for Zircaloy-2 that a plastic strain of 0.008 inside hydrides under uniaxial and zero under triaxial loading would result in equal fracture strength values for the two stress states. This result was obtained from the difference between the experimentally observed plastic strain at fracture of 0.04 and the calculated, corrected plastic strain at fracture of 0.032 that is needed to achieve equality of fracture strength values between the two loading conditions. Thus, with this assumption the main reason that hydrides fracture at lower applied matrix plastic strains in the notched compared to in the smooth specimens is that the plastic equivalent Young’s modulus is much larger in the former case. Bourcier and Koss [10], in analyzing their results of embedded—mostly parallel—hydrides in deformed titanium, did not come to this conclusion. Based on their experimental results showing the linear dependence of the initiation strain as a function of the stress state between uniaxial and biaxial tension and the limited variation of the geometric factor with stress state, they concluded that a common value of Epl applies to all stress states. The ‘‘experimental’’ value of Epl was then compared to the calculated one using the foregoing theory. From this comparison matrix the authors obtained, ehyd which led them to conclude that a conpl ffi 0:75epl siderable amount of plastic strain is possible in parallel hydrides during plastic deformation of the matrix under uniaxial loading. The level of plastic strain calculated by Bourcier and Koss [10] is considerably larger than that derived from the same theory for the plastic deformation of perpendicular hydrides under uniaxial

9.4 Mechanism for Fracture of Embedded Radial Hydride Clusters

299

loading determined by Puls [36] from the results of similar types of experiments. In a subsequent study on hydride precipitates in Zircaloy-2 sheets, Yunchang and Koss [53] found that there was only a small variation with stress state in the plastic strain to fracture hydrides. They attributed this result to the greater hydride aspect ratio observed after deformation in the less elongated hydrides in the equibiaxial tension case. This reduced the induced (inhomogeneity) contribution of the total applied strain in these hydrides compared to in the hydrides deformed under uniaxial tension. As shown in the following, this result is consistent with this writer’s own findings of considerable plasticity exhibited by parallel hydrides near the fracture surfaces of smooth tensile specimens loaded to failure at higher temperatures or in softer materials. To obtain additional information concerning the interplay between matrix strength and plastic deformation on the threshold for hydride fracture, a study was carried out by Puls [37] using tensile specimens cut from both the rolling and transverse directions of unalloyed zirconium plate material. The specimens were hydrogenated to hydrogen contents of 20 and 100 wppm. Different hydride size distributions were obtained by either air or furnace cooling the specimens from a temperature at which all the hydrides had been dissolved. No attempt was made to produce radial hydrides in these specimens. This was because hydride reorientation requires application of a sufficiently large and constant biasing stress during hydride nucleation. The much lower values of matrix strengths in these specimens (\200 MPa at room temperature for the specimens cut along the rolling direction of the plate) would have made it impossible to achieve a sufficiently large constant stress at the elevated temperature at which hydride nucleation first initiates. Only the tests on the specimens oriented along the plate’s rolling direction gave useful results. The hydrides in these specimens were randomly oriented and—depending on cooling rate and hydrogen content—varied in average length with little variation in their thickness. Plastic deformation of these tensile specimens resulted in diffuse necks. Fracture of the hydride platelets in the region of diffuse necking always required some matrix plastic deformation, but the (engineering) strain required was only *0.2 % under uniaxial loading and slightly less than that under triaxial loading. This was particularly evident in the arrested tests, an example of which is shown in Fig. 9.3. In these types of tests the total plastic strain was limited to 3 % after which the load was removed and reapplied. All of the specimens in these arrested deformation tests gave similar initiation stress values, slightly above the yield strength, with an average value of 217 MPa and an average strain of 0.29 %. The second deformation to 3 % strain yielded an average initiation stress of 313 MPa and an average plastic strain of 0.18 % while the third and final 3 % deformation gave an average initiation stress of 329 MPa and an average plastic strain of 0.23 %. It was also found that there was little variation with hydride platelet length in the plastic strain at the onset of hydride fracture initiation. This is consistent with metallographic observations showing that many of the hydride precipitates fractured through-thickness at multiple locations along their lengths, this occurring either simultaneously or at different times as evidenced by either identical (Fig. 9.4) or differing (Fig. 9.5) sizes of voids that

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Fig. 9.3 Typical room temperature AE event rate generation curves for the three stages of arrested deformation (to *3 % plastic strain each) for a smooth tensile specimen oriented in the rolling direction of unalloyed zirconium plate (from Puls [37]; with permission from AECL)

Fig. 9.4 A hydride precipitate in unalloyed zirconium plate material, fractured throughthickness at numerous locations along its length (from Puls [37]; with permission from AECL)

had been created along the lengths of the hydrides, respectively. Only parallel hydrides were observed in the region of the diffuse neck, indicating that the previously randomly oriented hydrides had ‘‘flow-reoriented’’. This is another

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Fig. 9.5 Micrographs showing through-thickness fractures along lengths of hydride precipitates in unalloyed zirconium plate material: a small rows of voids of receding sizes along a thin hydride precipitate, b hydride precipitate containing a void across its thickness at A, plus additional fractures at B and C not yet grown into voids (from Puls [37]; with permission from AECL)

Fig. 9.6 Graphical method of estimating the load for onset of crack initiation in hydrides for a tensile specimen of Zr–2.5Nb pressure tube material tested at 100 °C, where ri is the stress at onset of hydride crack initiation and ry is the yield strength of the material (from Choubey and Puls [14]; with permission from AECL)

indication of the ductility of hydrides under uniaxial deformation, even at ambient temperature. Because of the lower strength of the matrix—even though this limits the maximum matrix stress that can be applied on the hydrides—the hydrides were able to fracture at relatively low applied stress, with only a small subsequent increase in matrix plastic strain. This can be rationalized by the finding [32] that in softer matrix material, because there is greater relaxation of the transformation strains locally at the hydride/matrix boundary, both the mean and the plate-normal compressive stresses in the hydride are reduced compared to in harder material. A subsequent follow-up study by Choubey and Puls [14] of the earlier work of Puls [36] on pressure tube material concentrated on the effect of elevated temperature on the fracture strength of long radial hydride clusters while a final study by Shi and Puls [44] examined the effect on hydride fracture strength of test

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Fig. 9.7 Smooth tensile specimen of Zr–2.5Nb pressure tube material tested to failure at 200 °C showing plastic flow-induced reorientation of long radial hydrides in the necked region and microvoids formed close to the fracture region (from Choubey and Puls [14]; with permission from AECL)

temperature and radial hydride distributions with different average lengths. In both studies only the fracture of radial hydrides in smooth tensile specimens oriented along the transverse direction of unirradiated Zr–2.5Nb pressure tube material was investigated. In the study by Choubey and Puls [14] it was observed that fracture initiation and propagation was always along the lengths of such long, radial hydride clusters (50–100 lm long), and occurred when the applied stress ranged from being just below, to just above, the material’s yield strength at temperatures ranging from ambient to about 120 °C. The threshold value for hydride fracture up to 120 °C was calculated from the correspondence between the stress/strain plot and the AE signal, as shown in Fig. 9.6, combined with finite element results by Leitch (unpublished results from the work of Leitch and Puls [32]) for the stresses inside hydride platelets with aspect ratios of 0.02 and 0.04 produced solely by their stress-free transformation strains. The previous results obtained by Puls [36] for the same material were similarly re-evaluated. The model by Beremin [8] for calculating any additional induced stresses produced by differential plastic deformation between hydride and matrix did not need to be applied for these results because there was little macroscopic plastic strain observed at first fracture initiation for these long, radial hydride clusters. Combining these finite element results with the externally measured threshold stress, the fracture strength of hydrides in unirradiated Zr–2.5Nb pressure tube material was calculated to range from 575 MPa (from the data obtained by Puls [36] at ambient temperature) to 520 MPa (from the data of Choubey and Puls [14] at 100 °C). This study also showed the effect that the decrease in the magnitude of the yield strength has on hydride fracture mode. In a matrix with high enough matrix strength, perpendicular (radial) hydrides fractured along their lengths either before or after the applied

9.4 Mechanism for Fracture of Embedded Radial Hydride Clusters Fig. 9.8 Dependence of the nominal fracture strength of hydrides as a function of average hydride lengths in Zr–2.5Nb pressure tube material (tensile axis is along transverse direction of the pressure tube). The tensile stress at crack initiation is the engineering stress of the specimen (from Shi and Puls [44]; with permission from AECL)

Fig. 9.9 The nominal lower bound fracture strengths of radial hydrides in Zr–2.5Nb tensile specimens oriented with their tensile axes along the tube’s transverse direction. All specimens tested in this series had hydride distributions with average lengths of *85 lm and maximum lengths of *230 lm (from Shi and Puls [44]; with permission from AECL)

303

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stress equaled that of the yield strength, while in a softer matrix or at higher temperatures—below some threshold matrix yield strength value—hydrides flowreoriented during plastic deformation, becoming longer, thinner, parallel hydrides that ultimately fractured through-thickness. This was evidenced, as shown in Fig. 9.7, in the region near the specimen’s fracture surface. Another factor favoring hydride flow-reorientation with increase in temperature may come from the much greater softening over the temperature range from ambient to 150 °C of the d-hydride phase compared to that of the Zr–2.5Nb pressure tube matrix material, as illustrated in the plots of Figs. 2.18 and 2.20. In the final study along the foregoing lines, Shi and Puls [44] determined the hydride fracture initiation threshold of hydride distributions having different average lengths (and correspondingly different average thickness values) in Zr–2.5Nb pressure tube material as a function of temperature. Measurements were made of representative hydride length and thickness values for hydride distributions having different average lengths. Figure 9.8 shows the variation of the applied (nominal) stress at which hydride fracture was first observed versus average hydride length. It is evident that at hydride lengths, Lave \ 25 lm, the nominal hydride fracture strength increased with decrease in average hydride length. Note that the nominal fracture strength for these hydride lengths is less than the yield strength of the material. For average hydride lengths, Lave [ 25 lm, the fracture strength remained the same with increase in average length. The likely reason is that the internal stresses produced by the hydride’s transformation strains remained the same because the aspect ratios of the hydrides remained the same with increase in hydride length. Only hydrides in size distributions with the lowest average hydride length had greater aspect ratios. From the data determined from the hydride size distribution with the longest average length, the lower bound fracture strength of radial hydrides as function of temperature dependence up to *150 °C was obtained. As with the data for hydride fracture strength obtained by Choubey and Puls [14], no correction for possible induced stresses in the hydrides resulting from differential plastic deformation between matrix and hydride was needed for these long, radial hydrides since they fractured with negligible matrix plastic strain. The results are limited to 150 °C because, as in the Choubey and Puls [14] study, above that temperature the radial hydrides started to flow reorient into parallel hydrides that eventually fractured in their through-thickness directions. This is graphically illustrated in Fig. 9.9. Note that the dependence of the hydride fracture strength on temperature listed in Fig. 9.9 was derived directly from the values of the externally applied load at which hydride cracking was first detected. That is, no correction was made to these engineering fracture stress values to account for the compressive stresses produced by the hydride’s transformation strains. From this equation, the lower bound fracture strength at 100 °C is *640 MPa. At that temperature the aspect ratios of these long, radial hydrides were estimated from optical micrographs to be in the 0.04–0.02 range, from which a normal compressive stress of -175 MPa was calculated based on unpublished results by Leitch (as documented in [14]) of elastic–plastic finite element calculations of hydride precipitates having oblate spheroidal shapes. Adding this stress to

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the applied engineering stress brings the lower-bound net hydride fracture strength for length-wise fracture of radial hydrides to 465 MPa. This value is in close agreement with the lower bound value estimated for the fracture strength of long, radial hydrides grown under constant load from a nominally smooth surface under cantilever beam loading in pre-irradiated, ex-service pressure tube material [39]. This and similar results of such tests are described in Sect. 9.6

9.5 Hydride Stress State Determinations in Tensile Tests Observed Under Synchrotron X-ray Irradiation The limitations of the foregoing tests are that they do not provide a direct measure of the stress in the hydride clusters when cracking is detected. The stress in the hydride at which cracking by AE is detected must be inferred from additional information concerning the deformation properties of bulk matrix and hydride material, supplemented by assumptions or information on the states of coherency of the embedded hydrides with the matrix and their associated transformation strains, shapes, and orientations with respect to the applied stress. In recent years, sources of very high energy X-rays have become available from the latest generation of synchrotron accelerators. The availability of these high energy sources of X-rays has made it possible to obtain information on the internal state of stress of both matrix and precipitate in components having dimensions close to those used in the usual mechanical tests. In addition, the high intensity of these sources of synchrotron X-ray irradiation has made it practical to carry out such mechanical testing in situ over an adequate time period for sufficient data to be collected with very high spatial resolution. Results of such tests on zirconium alloys with embedded hydride precipitates have recently appeared in the literature and are described in the following. Kerr et al. [28] were one of the first to apply synchrotron X-ray diffraction methods to the mechanical deformation of zirconium alloy specimens containing hydride precipitates. For these tests, a slice of dimensions 2.25 9 30 9 60 mm3 was machined from a warm-rolled Zicraloy-2 slab having 20 lm grain size. The texture and mechanical properties of this plate material had been previously wellcharacterized by Xu et al. [54, 55]. The slice was doped with hydrogen to a composition of *90 wppm, giving a similar volume fraction of hydrides as in the tensile tests of Zr–2.5Nb pressure tube material carried out by Puls and co-workers (see previous sections). The strong orientation in the plate normal direction of the basal poles of the Zircaloy-2 grains resulted in precipitation of hydride clusters of average dimensions 2 9 20 9 20 lm3 with their long dimensions oriented along the rolling/transverse directions of the plate. Tensile and compressive tests were carried out in all three principal directions of the plate, viz., the rolling, transverse, and normal directions (RD, TD and ND, respectively). For the tensile tests, only TD and ND orientations of specimen dimensions 7 9 1 9 2.5 mm3 could be cut parallel to the loading directions while specimens for compression testing of

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Fig. 9.10 Applied stress versus lattice strain evolution from the movement of the { 1 1 1 } hydride peak for specimens loaded axially along TD in a tension and b compression. RD is perpendicular to the applied load in both cases. The strain components both parallel (TD) and perpendicular (RD) to loading are plotted (from Kerr et al. [28])

dimensions 4 9 3 9 2.25 mm3 were cut with loading axes in all three principal plate directions. Thus, given the texture and preferred orientation of the hydride clusters, these platelet-shaped clusters were oriented parallel to the loading direction in the TD tension tests while in the ND tension and compression tests they were oriented in the perpendicular orientation, with the latter orientation corresponding to that of the radial hydrides in the tensile tests carried out by Puls and co-workers. Diffraction data was obtained with the scattering vector both parallel and perpendicular to the loading direction. For the d-hydride precipitates, only the peak location of the d-spacing distance between the {1 1 1} planes was observed. Changes in the d-spacing of this peak with deformation of the specimen were used to obtain the strain components, exx and eyy, in the tensile and transverse tensile directions, respectively. Figure 9.10 shows plots of results obtained for the load applied parallel to TD. Qualitatively similar results were obtained for all other specimen orientations. The most important feature of the variation of lattice strain with applied engineering tensile stress is that there are three stages in the hydride deformation, labeled I to III. From the slope of the long linear Stage I an average

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Fig. 9.11 Comparison of results from FE calculations and experimental data for both phases under applied tension. The specimen is loaded along the TD direction, with the RD direction perpendicular to the applied load (from Kerr et al. [28])

elastic modulus value for the d-hydride phase of *100 GPa is obtained, consistent with results obtained by Puls et al. [38] from bulk d-hydride deformation and indentation experiments. The location of the inflection point between Stages I and II gives the value of the yield strength of the composite material. Load transfer occurs from the plastically deforming matrix to the harder hydride precipitates during the subsequent Stage II. This stage has a decreased slope from that of Stage I culminating in Stage III where there is no further increase in stress in the hydride clusters with increase in strain. Concerning the latter stage, in both the tensile loading specimens (only the TD tensile loading direction results are shown in Fig. 9.10) the strain evolution in this stage could not be accurately tracked because during tensile loading the movements of the {1 1 1} d-hydride and f1 0 1 0g a-Zr peaks causes them to overlap at a stress starting somewhat above the Stage I/II transition. However, the characteristics of Stage III, including the saturation strain values for the different specimen orientations, could be observed in the strain variations with stress in the perpendicular directions. It should be noted that the strains in the hydride clusters plotted in Fig. 9.10 are relative to the initial, externally unstressed states of the hydride clusters. In these states, hydride clusters, if coherently misfitting with the matrix, would be in compression as a result of their positive transformation strains with respect to the surrounding zirconium lattice matrix. These compressive strains produced by the transformation strains would have to be added to the plotted strains to obtain their true values, defined relative to the hydride’s stress-free state such as when it is in bulk form. To interpret their data, the authors carried out a finite element analysis using a model containing a parallel hydride of 20 lm length with an aspect ratio of 0.1. The hydride dimensions used in the model were chosen to be consistent with the measured average hydride cluster dimensions observed using optical microscopy. Unit cell boundary conditions were applied to mimic a non-interacting, regular infinite array of this configuration. It was assumed that both hydride and zirconium matrix have the same elastic modulus and only the zirconium matrix deforms plastically above the yield strength. The stress–strain behavior of the zirconium

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matrix in the model was obtained from experimental results of tensile tests of non-hydrogenated Zircaloy-2 plate material. Comparison of the model predictions with the experimental results, which also include the average lattice strain data for the a-Zr matrix in addition to those for the d-hydrides, is given in Fig. 9.11. The combined data show that, in Stage I, since both hydride and matrix have the same elastic modulus, the increase in strain with applied stress is the same in both materials. In Stage II, at applied stresses from the yield point of the matrix up to the inflection point to Stage III (*340 MPa), there is load transfer from the matrix to the hydride, which now extends elastically more than does the matrix in which the deformation has become plastic. Under the assumption that the deformation in the hydride remains elastic and equal to the elastic–plastic strain in the matrix, the corresponding stress in the hydride would continue to increase linearly as in Stage I. However, with the experimentally determined strain increase in the hydride plotted as a function of the external stress, it appears as if the hydride has a softer response than the calculated value because the external stress increase in this plot reflects the elastic–plastic response of the matrix and not that of the hydride precipitates in which the stress continues to increase linearly with strain throughout this short stage. This softer response of the matrix is given by the solid line, labeled FE (Zr), which has the same dependence on external stress as the average elastic strain in the hydrides obtained from diffraction. At the Stage III inflection point, assuming, as discussed in the following, that fracture has occurred, then the stress in the hydrides at fracture can be calculated if deformation in the hydride remains elastic. The strain in the hydrides at the inflection point is *8 9 10-3 and the applied stress is 340 MPa. Since the elastic modulus in the hydride is *100 GPa, this means the stress in the hydride at fracture generated by the applied load and imposed through the strain produced by the elastic–plastic deformation of the matrix is *800 MPa. However, to obtain the actual stress in the hydride would still require a calculation of the compressive stresses produced in the hydrides by their transformation strains in the same way that this was required to interpret the results of the tensile tests that were monitored by AE as given in Sect. 9.3. Regarding the interpretation of the saturation in Stage III, the authors consider a number of possible explanations, viz.: (1) hydride fracture, (2) interface failure, (3) plastic deformation of the hydrides and, (4) work hardening of the hydrides leading to load transfer back to the matrix. The authors concluded that only the first explanation is likely based on the results of tensile tests in the literature, summarized in Sect. 9.3, where cracking of hydrides is detected by AE. In these experiments AE indications of hydride cracking during specimen load increase were found to occur throughout the plastic deformation stage, peaking between matrix plastic strains from 1 to 4 % depending on alloy composition and stress state. The authors note that such behavior has also been observed in conventional composite materials where it has been rationalized in terms of shear lag models, the original version of which was developed by Cox [18]. In a shear lag model applied to parallel hydride platelets, the stress in the hydride in the lengthwise direction below some critical length (the stress transfer

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length, ‘t) reaches its peak value halfway along its length. This peak value increases with increase in length of the hydride provided its length is less than ‘t. Above the hydride’s stress transfer length, however, this peak stress reaches a limiting plateau value that spreads outward with increasing length of the hydride from the mid-point of the hydride toward its ends, the plateau stress depending only on the elastic properties of the two phases and the externally applied stress. If this plateau stress value exceeds the through-thickness fracture strength of the hydride as the external stress is increased, through-thickness fracture of the hydride can occur along multiple locations along its length. The external stress at which this occurs is initially relatively insensitive to the length of the parallel hydrides. Because of this initial lack of sensitivity of hydride fracture on hydride length, the authors propose that in their diffraction experiments, the onset of fracture in parallel hydrides detected by AE would not be observable through changes in hydride strain with applied load if the average hydride lengths after initial fracture remained significantly longer than ‘t so that the average load transferred to the hydrides by the matrix remained relatively unchanged, limited to its plateau value. As multiple fractures occur along the lengths of the hydrides their effective lengths would eventually decrease to their stress transfer lengths. Further fracture of these segments with increasing load would limit the peak stress (and corresponding strain) in the hydrides that can be imposed on them and there would be no further increase in strain in the hydrides with increase in applied stress. It should be noted that—of the experimental support for this interpretation obtained from results of tensile tests where hydride fracture is monitored with AE—only the results from tests of the fracture of parallel hydrides in unalloyed zirconium by Puls [37] would appear to be relevant to rationalizing the results in terms of a shear lag model. In all the other tests, the zirconium alloy material contained mostly radial (perpendicular) hydrides for which the length (thickness, actually) extending in the applied stress direction is very short. In these perpendicular hydride clusters the cracks—running lengthwise in the clusters—almost always extended to the ends of these clusters. Thus each fractured hydride cluster would generate only one AE fracture event during load increase, unlike in the case of parallel hydrides where each hydride could be the source of multiple fractures detected by AE. Yet the trend in the AE signal with increase in applied strain was similar in both types of cases. A similar set of tests as those done by Kerr et al [28] were also carried out by Steuwer et al. [48], using slightly different zirconium material, specimen geometries, and experimental conditions. The specimens used by Steuwer et al. [48] were prepared from material cut from Zircaloy-2 and -4 fuel cladding tubes. In both cases, the tubes from which the samples were prepared had 20 mm outer diameter and 1.7 mm wall thickness. Test specimens were obtained by cutting out strips oriented along the long axes of these tubes. The strips were flattened and cold rolled in their long direction to achieve a 30 % thickness reduction from 1.7 to 1.2 mm. From these strips matchstick tensile specimens were cut, 3 mm (Zircaloy-2) or 1.4 mm (Zircaloy-4) wide by 1.2 mm thick and 40 mm long. The specimens were annealed at 580 °C for 2.5 h and gaseously hydrogenated to

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Fig. 9.12 Macroscopic stress–strain curve (left graph) compared with the lattice strain evolution measured using EDXRD (right graph) for the Zircaloy-2 specimen. The evolution of the d-311/c113 peak shows an elastic (lattice) strain of *2.25 %. Error bars of the order of the symbol size or less (\10-4) have been omitted (from Steuwer et al. [48])

*400 wppm hydrogen at 400 °C for 1 h. The plate-shaped hydride clusters in these tensile specimens were oriented with their long dimensions in the tensile stress direction (i.e., these are parallel hydrides as described in Sect. 9.3) and were identified from their diffraction data as being d hydrides. Unlike the single d-hydride peak observed from only the reflections of {1 1 1} planes by Kerr et al. [28], Steuwer et al. [48] were able to observe three d-hydride peaks corresponding to reflections from the {3 3 1}, {1 1 3}, and {2 2 0} d-hydride planes, but no peak was observed from the {1 1 1} d-hydride planes as found by Kerr et al. [28]. The reason for the different observations likely is because the range of d-spacing over which diffraction data were obtained differed between the two studies. A further contributing factor may be the significantly higher hydrogen content of the specimens tested by Steuwer et al. [48, 49] Figure 9.12 shows the strain evolution in the hydrides derived from the d-311 peak and in the zirconium matrix phase derived from selected peaks for this material. Only those a-Zr peaks that exhibited relatively small amounts of elastic/ plastic anisotropy during deformation were selected to plot the strain evolution in this phase. As can be seen from the accompanying engineering stress–strain plot, the variation of elastic strain in the matrix determined by diffraction versus macroscopic strain follows the same dependence as the macroscopic engineering stress versus strain data, exhibiting the characteristics expected for this ductile material. However, in step with this increase in strain in the a-Zr phase up to 4 % macroscopic strain was a very large linear strain increase in the d-hydride phase that shows up for all three peaks, but most clearly for the d-311 peak. At 4 %

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macroscopic strain, where the large, linear increase in the strain of the hydride phase stops abruptly, the relative increase in strain in the hydride as determined by diffraction is equal to *2.25 %. Above this value of strain, further strain increases in the hydrides follow a similar trend to those in the a-Zr matrix. Comparison of the shifts to higher d-spacing of the various a-Zr and d-hydride peaks with increase in applied stress shows that the hydride peaks shift much more, particularly the d-311 peak. Superimposing this shifted spectrum on the calculated spectrum over the d-spacing range of the diffraction measurements shows that at 4 % macroscopic strain the d-311 peak has shifted to where the {1 1 3} c-hydride reflection would be if c hydrides were present in the material. The authors note that the difference in the lattice d-spacing of the d-311 and c-113 peaks (1.43761 Å versus 1.47011 Å, respectively) is 2.26 %, suggesting that the applied stress has produced a change in the crystal structure from the d- to the c-hydride structure. To rationalize the foregoing result, the authors propose that the application of an external tensile stress produces a gradual, stress-induced ordering of hydrogen atoms in the subset of those d hydrides whose crystallographic a-axes are most closely aligned with the loading direction. The ordering would be most pronounced in the d-311 and c-113 reflection pairs since, of the three observed d-hydride peaks, the d-311 peak is the closest to the [0 0 1] axis. This axis naturally aligns itself along the loading direction as the applied load elongates the cubic d-hydride phase in the direction of loading to form the tetragonal cell of the c-hydride phase. The authors supposed that hydrogen atoms that, before loading, were randomly occupying some of the available eight tetrahedral sites were induced by the strain produced in the direction of the applied load to occupy those ordered tetrahedral sites that elongate the crystal structure toward that of the c-hydride phase, which has positive c/a tetragonal structure. This preferential pairing mechanism is akin to the Snoek-like relaxation of the hydrogen atoms in fcc d hydride when hydrogen atom pairs preferentially align along the applied stress direction. This mechanism was proposed by Pan and Puls [35] as being responsible for the P2 peak observed in internal friction experiments. The P2 peak is observed in both hydride-containing zirconium specimens—increasing in magnitude with increase in hydrogen content—as well as in bulk d-hydride specimens. The continuous nature of the strain increase with increase in load is support for this ordering mechanism as is its linear dependence on applied stress. If, instead, the applied stress were to result in the shrinkage of d hydrides in favor of the nucleation and growth of c hydrides, this would have manifested itself over the observed spectrum by the simultaneous disappearance and appearance, side by side, of the d- and c-hydride peaks. A problem with the foregoing interpretation is that for the d-hydride phase to truly convert to the c-hydride phase there must also be a reduction in H/Zr atom ratio from *1.5 to 1 corresponding to the compositions of the two phases during this conversion. These excess hydrogen atoms cannot remain in solution in the a-Zr phase if—as seems likely—the two hydride phases have closely similar solvus compositions, even though the latter is likely a metastable phase. One way to eliminate this excess hydrogen in the d-hydride phase would be for the extra

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hydrogen to precipitate as c-hydride phase material within the remaining d-hydride matrix during the stress-induced transformation. This assumption is rather loosely based on similar conclusions drawn by Beck—as summarized in Chap. 2—who argued that, during cooling, c hydride is a metastable decomposition phase of d hydride rather than this phase decomposing directly to the equilibrium a-Zr phase in order to reduce its H/Zr ratio with decrease in temperature. Thus one would expect to also observe the simultaneous formation of a c-hydride peak during the shifting of the d-hydride peak with increasing load. As this is not observed, perhaps the rejection of the excess hydrogen occurs only when the shift to the c-hydride peak is complete. There should still be an observable effect since the greater volume fraction of c-hydride precipitates formed as a result of this conversion would result in an increase in the c-hydride peak height from that calculated based on the assumption of equal volume fractions of the two phases. A point to consider, however, is that, unlike in internal friction experiments where the shift in hydrogen atom pairs is continually reversed, this large-scale phase conversion process involves hysteresis since there is a difference in hydride-matrix misfit volume of *4 % between the two phases resulting in a difference in accommodation energy. A consequence of hysteresis is that removal of the stress alone would not cause a reversion of the phase transformation. This is, in fact, what was observed in a follow up study by Steuwer et al. [49]. In this study, the internal stresses were measured that were generated in parallel hydrides located at a fatigue crack in a single edge notched (SEN) specimen loaded at room temperature in the same direction as the parallel hydrides in the smooth tensile specimens. These hydrides showed the same large increase in strain under load near the fatigue crack as in the tensile tests, but these tensile strains did not disappear when the load was removed. This result is consistent with what one would expect when there is hysteresis and there is no temperature increase upon stress removal to dissolve the transformed hydrides. An interesting feature of the results of Steuwer et al. [48] is that the maximum linear strain increase of *2.25 % in the hydrides coincides with a macroscopic strain value of 4 %. The authors suggest that only beyond this value of macroscopic strain do the hydrides start to bear some of the overall load applied on the specimen. The value of 4 % for the macroscopic strain is interesting because it is claimed by the authors to provide further support that the applied stress has induced a d- to c-hydride transformation. The authors note that, from the lattice parameter differences of the stress-free phases,2 the d-hydride has a *4 % greater unit volume than does the c-hydride phase. A similar difference of *4 % linear strain is obtained between c- and d-hydrides when the former hydrides form as acicular precipitates in bulk d-hydride material. This strain difference was the Bain shape strain estimated by Cassidy and Wayman [11] who studied the d- to 2

It should be noted that lattice parameter measurements of the c-hydride phase can only approximately be of its stress-free state since bulk specimens consisting only of the c phase—and therefore unconstrained by a misfitting surrounding phase—have never been produced, a point that was also recognized by Steuwer et al. [48].

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c-hydride transformation showing that it has the characteristics of a martensitic (i.e. invariant plane strain) transformation with a 4 % strain increase normal to the invariant plane. (This result can also be obtained by simply taking the difference in lattice parameters in the c-direction of the tetragonal c-hydride phase with that in the a-direction of the cubic d-hydride phase.) However, these two results cannot both be used to explain the foregoing correspondence of the macroscopic strain with the hydride strain peak at which it has shifted to the position of the c-hydride peak. The difference in strain of 4 % from the stress-free volume differences of the two phases is negative and is close to the magnitude of the volumetric strain difference between the two precipitates when on their observed habit planes in the a-Zr phase. However, the respective linear strain differences in the three orthogonal directions in this case are less than 4 %. The positive difference of 4 %, on the other hand, applies only to the case where the c-hydride phase is precipitated in the d-hydride matrix. It may be that this is actually what happens initially, but since the latter are embedded in the a-Zr matrix, the net strain should still end up being negative. Not accounted for in these considerations are the original transformation-strain-induced compressive strains in the hydrides. Their magnitudes inside the hydrides would depend on the matrix phase in which the hydride precipitates are precipitated, as well as their shapes and other factors, such as the degree of prior relaxation between matrix and hydride of these strains. Only if one could argue that these starting strains are the same for each phase—before and after transformation—could one extract a meaningful connection between the macroscopic and internal hydride strains. The latter point is germane to all of the elastic strain determinations made so far with these diffraction methods. That is, the evolution of the elastic strains in matrix and hydride precipitates with applied stress determined in these diffraction experiments are relative to the strains in both materials at the start of application of the external load. Therefore these strains do not account for the local internal stresses of matrix and hydrides prevailing at the start—and presumably throughout—the application of a rising external load. Thus similar supplemental information or assumptions as in the interpretations of the tensile tests of hydride fracture determinations by AE are still required to derive from the measured strains the absolute stress/strain values of hydride fracture that would be of use in model calculations. Even the calculated relative increase in stress in the hydrides during Stage II shown in Fig. 9.11, which is derived from the measured strain increase determined by diffraction, is only correct provided the assumptions are correct that (1) hydrides continue to deform elastically up to fracture, and (2) the model for load transfer between matrix and hydride is correct. Account must also, of course, be taken of the considerable compressive stresses generated inside hydride precipitates by their positive transformation strains. The foregoing studies were extended by Kerr et al. [29, 30] to examine the stress state of radial hydrides formed at stress concentrators. A specimen manufactured from Zr–2.5Nb pressure tube material containing a blunt notch with 15 lm root radius and containing 60 wppm hydrogen was taken to a test temperature of 250 °C from a maximum temperature chosen so that all hydrogen

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would be dissolved at that temperature. At 250 °C, TSSP is *60 wppm so the hold at that temperature should have retained little or no hydride precipitates in the bulk of the specimen and hydride precipitates would only have formed at the notch root. The specimen was loaded to an applied KEFF * 8 MPaHm and diffraction measurements were made continuously for *6.5 h. The applied KEFF in this case was chosen to be below the estimated threshold value for crack initiation of the hydrided region at the flaw tip. The observations from the diffraction data reproduced in Fig. 9.13a and b show that hydrides had precipitated during a very short time at the test temperature, creating a hydrided region of *100 lm after only 2 h that stayed at this length for the remainder of the constant load hold. After this time, overloads of 110 and 125 % of the original KEFF value were applied in succession, the results of which are shown in Fig. 9.13c and d. From these figures it can be seen that the presence of hydride precipitates relaxes the strain field in the surrounding zirconium matrix. However, the strain in the hydride precipitates had increased relative to its value in hydrides away from the influence of the notch-tip stress field. What is significant in this result is that it shows that although the relaxation of the matrix strain by the transformation strains of the hydrides reduces the strain value of the former to an approximately constant plateau over the hydrided region, the strain in the hydrides relative to their initial, externally unstressed value is increased by the applied stress, reaching a maximum at the flaw tip. Results of coupled finite element calculations have shown (see Chap. 10 for a description of the results of these) in a continuum sense the effect of the transformation strains of the hydrides on the overall strains. These calculations give the same trend in reduction of matrix strain, but the interpretation by the authors of these finite element calculations has been that it also represents the strains in the hydrided region. As discussed in Chap. 10 where some of the results of these calculations are reviewed, this interpretation does not make physical sense. Kerr et al. [30] make a similar point, supported by their experimental findings. From Fig. 9.13c and d it is seen that application of overload on the specimen to the values indicated after growth of the hydrided region had ceased did not substantially increase the elastic strains in the hydrides. This is in contrast to the room temperature measurements, discussed in the following, where increasing overload produced an increase in the observed differential elastic strain between matrix and hydride phases. These findings are consistent with the results of overload experiments by Shek et al. [43] showing that the percentage overload required at 250 °C is considerably greater than at room temperature. A possible reason for this difference may lie in the significant reduction of the yield strength of the hydride phase between room temperature at 250 °C [38] resulting in the hydride phase being able to deform plastically in unison with the plastic deformation of the matrix. It is significant that results of fracture toughness tests of compact toughness specimens of unirradiated and irradiated material containing radial hydrides reproduced in Figs. 9.14 and 9.15, similarly show that 250 °C is approximately the temperature where these hydrides no longer act as fracture initiation sites, and, therefore, no longer primarily contribute as nucleation sources for the overall ductile failure of the specimen [47, 51].

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Fig. 9.13 Strain profiles along x-axis with nominal notch tip at zero (KI in this figure is denoted KEFF in the text). a 100 % load: initial hydride precipitation. b 100 % load: after 5–6.5 h of hydride growth and zirconium matrix relaxation. c 110 % load: strain increases as a result of overload, and there is not much strain relaxation at notch tip produced by creep. d 125 % load: strain increases in zirconium matrix, but strain in notch-tip hydride is essentially unchanged when compared to 110 % loading (from Kerr et al. [30])

In a previous set of tests by Kerr et al. [29], tensile tests were carried out on a specimen containing a previously grown flaw tip hydrided region of *100 lm. This hydrided region had been grown at 250 °C under a KEFF * 8 MPaHm, which was insufficient to fracture it. This specimen was monitored with synchrotron X-rays during a subsequent rising load tensile stress test at room temperature. The load in this test was increased in increments of 50, 100, 110, 120, and 130 % of the initially applied KEFF value. The applied load was held at each of these values to record the X-ray diffraction data. Data for the hydrided region came from the d-111 peak and, because of the strong texture of this material, only strain increments in the hydrides parallel to the loading direction could be measured. All strains measured are relative to the reference lattice spacing obtained for the material far from the influence of the elevated strains at the notch which means that the original level of strain in the hydride phase produced by its transformation strains alone is not known. Figure 9.16 shows the evolution of the strain along the x-axis, away from the root of the notch in a direction perpendicular to the direction of the applied load. When the load reached 100 % of the previously applied value at which the hydrided region had been formed one would expect that the original

316 Fig. 9.14 Fracture toughness at maximum load versus temperature for pressure tube alloys containing radial hydrides. The hydrogen content for the Zircaloy-2 data was 90 wppm, whilst that for the Zr–2.5Nb data was 200 wppm (from Simpson and Chow [47]; with permission from AECL)

Fig. 9.15 Fracture toughness versus temperature for Zr– 2.5Nb with various hydride morphologies. HCC refers to the hydride continuity coefficient, ranging from 0 to 1. HCC is a measure of the degree to which hydrides are oriented in the radial-axial plane of the pressure tube wall. A low value of HCC corresponds to a situation where only a few hydrides, predominantly oriented in the axial-tangential plane, are present (from Wallace et al. [51])

9 Fracture Strength of Embedded Hydride Precipitates

9.5 Hydride Stress State Determinations in Tensile Tests Observed

317

Fig. 9.16 Strain profiles along the x-axis with nominal notch tip at zero (KI in this figure is denoted KEFF in the text). a Zero load: residual strain profile. Strain of hydride tracks that of matrix. b 100 % load: load transferred to notch tip hydrides, consistent with plasticity in zirconium matrix. c 120 % load: continued load transfer. Intensity map below indicates location of notch tip hydrides and how strain was averaged. d 130 % load: peak strain is now in the zirconium matrix, in front of the notch tip hydride, consistent with hydride fracture (from Kerr et al. [29])

strain state of hydride and a-Zr matrix formed during growth would approximately be recovered. It is then evident that the larger strain in the hydrided region relative to that in the a-Zr matrix shows that load had been transferred from the plastically deforming matrix to the hydrided region as the hydrided region grew in size. This ex-situ result, then, is similar to that obtained from the in situ hydride growth tests. At the first overload applied stress of 120 % the figure shows a further increase in strain in the hydride phase relative to the matrix strain with a peak close to the root of the notch. The next applied load of 130 % overload turned out to be sufficient to fracture a part of the hydrided region. This is evidenced by the peak strain now being located in the matrix material. This result shows, as mentioned in connection with the discussion on the results of the in situ tests, that the overload required to fracture such a hydrided region of similar length and initial KEFF is much less at

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room temperature than when applied at 250 °C. From Fig. 9.16a it can also be seen that the residual strains at room temperature in both phases are compressive, as expected from a specimen previously loaded in tension at elevated temperature during which the hydrided region had formed. Note that the strains in the hydrides are opposite to those observed by Steuwer et al. [49] in which the strains in the hydrides remained in tension after load removal because of a permanent lattice expansion in the direction of applied tensile stress. No obvious explanation for these differences in the results of these two sets of tests can be advanced on the basis of these limited data.

9.6 Fracture Strength of Radial Hydrides: Constant Load Tests After the discovery of DHC initiation at the inner pressure tube surfaces near the inlet rolled joints of over rolled pressure tubes in the earliest generations of CANDU reactors, long-term tests were set up to determine the threshold stress for DHC initiation on the inside surfaces of pressure tube materials with nominally smooth surfaces. By nominally smooth surfaces are meant surfaces containing flaws of depth B0.1 mm. Cheadle and Ells [12] reported the first set of results obtained with these types of tests. Transverse CB specimens, 38 9 3.1 9 4.1 mm3, were machined from unirradiated Zr–2.5Nb pressure tube material. Some of the specimens were gaseously hydrogenated at *400 °C to a total hydrogen content ranging from 40 to 120 wppm. The loading of the CB specimens was such that the maximum outer fiber tensile stress in the transverse direction was imposed on the inside surface of the specimens. After loading, the specimens were heated to *300 °C and then cooled to the test temperature under load so that some fraction of the hydrogen content dissolved at the maximum temperature would precipitate as radial hydrides and be able to grow in length under load at the test temperature. Specimens with as received hydrogen content (typically in the range from 10 to 15 wppm) were tested only at 77 °C, while those that had been hydrogenated to a total hydrogen content ranging from 40 to 120 wppm were also tested at higher temperatures of 152 and 252 °C. It should be noted that for some specimens the loading resulted in imposed outer fiber stresses that exceeded the yield strength of the material during the initial cycle to 300 °C. This meant that the actual outer fiber stress was plastically reduced somewhat from its elastically calculated value. From the note in the table given by the authors it appears, however, that a correction had been made for this reduction. At the time of publication of these results by Cheadle and Ells [12], most of the tests had only been under load for about a year. The results listed in the following provide an update of the results of these tests after considerably longer test times, amounting, in some cases, to an additional 5 years of testing [42]. The results of such tests fall into two groups depending on their hydrogen content.

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In the first group are the results from specimens containing as received hydrogen contents. Because of the low hydrogen content, testing for fracture at hydrides was limited to the lowest of the three test temperatures, which was 77 °C. At outer fiber stress of 620 MPa, 64 % of specimens failed, 20 % failed at 585 MPa while no failures were obtained at the next lowest stress of 550 MPa and lower. Taking the average of the last two failure stresses gives an average hydride fracture strength value of 568 MPa at 77 °C. In the second group are the results for specimens that had been hydrogenated up to *100 wppm. Overall, the failure stresses for this group were lower than those for the first group of specimens containing as received hydrogen content. At a test temperature of 77 °C, 100 % of the specimens failed at 620 MPa, 80 % failed at 585 MPa, 67 % at 550 MPa while no failures occurred at the next lowest stress of 414 (nor at the next lowest of 276 MPa). Taking the average of the last two failure stresses gives an average hydride fracture strength value of 482 MPa at 77 °C. At 152 °C, 75 % of specimens failed at 620 MPa, 55 % at 585 MPa, 42 % at 550 MPa, 40 % at 414 MPa, and no failures occurred at the next lowest stress of 276 MPa. Taking the average of the last two failure stresses gives an average hydride fracture strength value of 345 MPa at 152 °C. Finally, at 252 °C, 62 % of specimens failed at 620 MPa, 43 % at 585 MPa, 11 % at 550 MPa, 33 % at 472 MPa, 0 % at 414 MPa, 13 % at 330 MPa, and no failures occurred at the next lowest stress of 275 MPa. Taking the average of the last two failure stresses gives an average hydride fracture strength value of 303 MPa at 252 °C. At the time of publication by Cheadle and Ells [12] there had been no failures for stresses less than 414 MPa for all specimens under the loads given in the foregoing. Additional results, also not given by Cheadle and Ells [12], were later obtained from thermally cycled specimens with hydrogen contents C40 wppm that were continually cycled from room temperature to 252 °C. In these tests, 100 % of the specimens tested failed at 350 MPa while 40 % failed at 250 MPa. This result—and the foregoing result from the tests without thermal cycles in which the lowest stress for failure at 252 °C was 330 MPa—led Cheadle et al. [13] to conclude (see their Table 1) that the lower bound failure stress at 252 °C for nominally smooth surfaces (containing flaws B0.1 mm) with hydrogen contents ranging from 40 to 120 wppm is 250 MPa while at room temperature it is 550 MPa. These results give a rather large drop in failure strength at 250 °C compared to the trend in the data at all lower temperatures. This suggests that the low failure strength values of 330 and 250 MPa at 250 °C may not be correct and should be discarded because such a large decrease in failure strength over such a small temperature interval is difficult to rationalize on physical grounds. These low failure strength values are also inconsistent with the results obtained by Shi and Puls [44] for fully embedded hydride clusters. It should be noted that, although Cheadle et al. [13] indicated that the thermally cycled specimens had nominally smooth surfaces, the internally published summary of the results by Scully [42] gives the notch depth of these specimens as ranging from 0.08 to 0.13 and 0.19 mm, respectively, for the specimens that failed at 350 and 250 MPa. In addition, for the non-thermally cycled results at 250 °C,

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for which the lowest failure stress was at 330 MPa, there are two unusual aspects associated with this result. One is that there were no failures at the next highest stress of 414 MPa. The other is that the single failure out of a total of nine specimens on test at 330 MPa occurred after an unusually long time (27,880 h). This time at failure was approximately double the maximum failure times for specimens at higher applied stress values and suggests that failure might have been caused by an unknown external disturbance to the specimen. Assuming that this was the case, then revised values of 472 and 414 MPa, respectively, are obtained for the stress at failure and the next lowest stress at which there were no failures. Taking the average of these stresses gives a corrected average hydride fracture strength value of 443 MPa at 250 °C. Post test optical microscopy of the results reported by Cheadle and Ells [12] show that failures in all these tests were initiated by the fracture of a radial hydride that had grown from the top (original inside pressure tube) surface of the specimen. The crack formed from the surface at such a hydride cluster then continued to extend by DHC in the thickness direction of the specimen. Failure of a specimen occurred through ductile rupture of the remaining ligament when the specimen could no longer support the load. For the as received specimens, it was observed that all failures occurred in hydrides at some small stress raiser such as at a surface scratch or a sand particle. The embedded sand particles likely were leftovers from the sand blasting operation used for cleaning purposes during manufacture of these pressure tubes (this practice was subsequently discontinued). Failure of the specimens containing hydrogen contents C40 wppm was not associated with any surface flaws or embedded particles. The fracture surfaces in these cases were characterized by flat areas similar in depth to the lengths of the unfractured, reoriented hydride clusters that had formed and grown from the surface. Before proceeding, it should be noted that even if one accepts the hydride fracture strength of embedded hydride clusters formed at the surface of unirradiated Zr–2.5Nb pressure tube material to be as low as 250 MPa at 250 °C at hydrogen contents of *100 wppm as reported by Cheadle et al. [13], this does not mean that failure of pressure tubes with nominally smooth inner surfaces is possible by DHC initiation for these hydrogen concentrations at these hoop stress values. Fracture initiation at surface hydrides at such a low externally applied stress value would only be possible if radial hydrides formed at the pressure tube’s surface would be able to grow to very long lengths. However, the reason for the growth of such long, radial hydrided regions from small surface imperfections in CB specimens is because of the linear, through-thickness stress gradient in these specimens that is created by the bending stress. This macroscopic stress gradient favors the extension in the through-thickness direction of radial hydride clusters formed at the top surface of the CB specimen since hydrogen is attracted to this region by stress-driven diffusion. Such a macroscopic stress gradient in the through-wall direction does not exist in operating pressure tubes containing only small surface imperfections. New impetus for the measurement of the fracture strength of long hydride clusters grown at smooth surfaces under constant load conditions arose from the

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development of the process zone model. This model was developed (see Scarth and Smith [40, 41] and Chap. 11 for a description of the model and list of references) for assessments of volumetric flaws detected during periodic inspections in pressure tubes of fuel channels of operating Canadian CANDU reactors [17]. In this methodology there are two experimentally determined, lower bound material parameters, KIH and pc, that need to be specified. In the theory, pc is defined as the fracture strength of an infinitely long hydrided region grown from a nominally smooth, planar surface. In three dimensions such a hydrided region would correspond, geometrically, to a long, thin plate having finite extent in the third orthogonal direction. For such a shape, it can be analytically shown that the plate normal compressive stress produced by the hydride precipitate’s positive transformation strains approaches zero in the limit of the hydrided region’s length going to infinity. Thus, for this case, the only stress acting on the hydride cluster at the specimen’s surface would be the externally applied outer fiber stress and the value of this stress when the hydrided region fractures and, hence, the specimen ultimately fails, is then equivalent to the hydrided region’s fracture strength, pc. In practical determinations of this parameter, since the length of the hydrided region at fracture is of necessity finite, the outer fiber stress at specimen failure is always an overestimate of the actual failure strength of the embedded hydride cluster. For a more conservative estimate, it is necessary to account for the compressive stress in the thickness direction of the hydrided region at the surface of the specimen produced by the hydride’s transformation strains. There are, however, a lot of uncertainties associated with calculating these stresses theoretically. If one wishes to avoid doing this, a rough estimate of the aspect ratio that needs to be achieved by the hydride cluster is given in the following. From the results of these calculations it is concluded that hydride clusters with aspect ratios B0.01 could be considered as having close to negligible normal compressive stresses produced by their transformation strains. Hence, for hydrided regions with these aspect ratios the externally applied outer fiber stress at specimen failure can be considered to be a good estimate of pc and, correspondingly, of their fracture strengths. From the finite element results of Leitch cited by Choubey and Puls [14] compressive stresses in the disc-normal direction at the center of an oblate ellipsoidallyshaped hydride precipitate were calculated to be -399 and -173 MPa for aspect ratios k = 0.04 and 0.02, respectively. In these calculations it was assumed that the hydride’s transformation strains are isotropic, equal to 5.4 % in each direction, and hydride and matrix have equal yield strength values of 700 MPa, this being approximately the yield strength values of Zr–2.5Nb pressure tube and solid d– hydride materials at room temperature. These compressive stresses would be expected to be approximately equal to those at the planar surface of a hydrided region of identical shape and aspect ratio grown from this surface. Note that the finite element results show that for the two aspect ratios considered, reducing the aspect ratio by half results in a similar reduction in absolute value of the normal compressive stress. These compressive stresses would also decrease somewhat (in absolute value) with decrease in matrix yield strength. At 250 °C, the yield strength in unirradiated Zr–2.5Nb pressure tube material is *500 MPa. Leitch and Puls [32]

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show that for an ellipsoidally-shaped hydride with positive isotropic transformation strains there is 18.7 % reduction in the absolute value of the average compressive stress in the plate-normal direction when the matrix and hydride yield strengths are educed from 800 to 500 MPa. Assuming a similar reduction for this normal stress at the planar surface of a surface-grown hydrided region of identical aspect ratio—note that such a hydrided region would have the same thickness at the surface, but half the length of a fully embedded hydride cluster—the compressive stress at the planar surface would be reduced to *-140 MPa. It is evident from the calculations of Leitch and Puls [32] that the magnitude of the normal stress produced by the hydride cluster’s transformation strain depends strongly on the aspect ratio of the hydride cluster. From the results of Shi and Puls [44], fully embedded hydrides have k values in the range from 0.04 to 0.02 once they grow to average lengths [*50 lm. It should be noted, however, that measurements of the thickness values of hydride clusters have high uncertainties associated with them because the irregularities of the shapes of such hydride clusters in their thickness directions are large in relation to their enveloping, overall thickness values. The observed thickness taken from optical micrographs is also sensitive to the method and intensity of etching used to reveal the hydrides. Regarding the test results provided by Cheadle and Ells [12] and Cheadle et al. [13], no information was provided of the thickness and lengths that hydride clusters grown from planar surfaces had attained when a specimen failed or for specimens in which no failure had occurred under similar loading conditions. Fortunately such information does exist from the results of similar types of tests for the determination of pc. An example is the unfractured hydrided region loaded under an outer fiber stress of 432 MPa at 250 °C in pre-irradiated material shown in Fig. 9.17. The longest hydrided region has a length of *570 lm with a slight taper in thickness from the surface to its tip. Another example is the unfractured hydride cluster shown in Fig. 9.18 grown in unirradiated pressure tube material at 230 °C. This hydrided region has a thickness at the surface of 12.5 lm and a length from the root of the small notch of 450 lm, giving an aspect ratio of 0.0156. (Note that the hydrided region has a tapered shape.) Keeping in mind the different yield strength values of the two materials at the testing temperatures, both hydride clusters have aspect ratios that are close to achieving the limit derived from finite element calculations at which the normal compressive stress produced by the transformation strains becomes less than 100 MPa so that the fracture strength of the hydride cluster approximates that given by the applied outer fiber stress at specimen failure. Similar to the original constant load DHC initiation tests at nominally smooth surfaces, all experiments to determine pc (including the ones described in the foregoing) have been carried out using CB specimens loaded in bending. The specimens were cut from pressure tubes and oriented in the bending rig with the inner surface of the tube as the top surface of the specimen such that the bending load generates its maximum, outer-fiber tensile stress on this face. To ensure that hydride precipitation and subsequent growth occurs preferentially from the surface, shallow scratches (if not already present) were applied to the surfaces of

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Fig. 9.17 Metallographic appearance of long (maximum *570 lm), radial hydrided regions grown at 250 °C from the inside pressure tube surface of Zr– 2.5Nb pressure tube material removed from an ex-service pressure tube. The CB specimen was loaded at an outer fiber stress of 432 MPa and the specimen section shown is the radial-transverse plane (from Rodgers [39]; with permission from COG)

Fig. 9.18 Optical micrograph showing a 450 lm long radial hydride cluster grown under constant load (outer fiber stress of 650 MPa) at 230 °C from a small surface notch (radius 5 lm and depth 15 lm) in unirradiated Zr–2.5Nb pressure tube material (from Lee and Vesely [31]; with permission from COG)

the CB specimens. The stress gradient generated across the wall of the specimens by the CB loading ensures that hydrogen is attracted to the top surface of the specimen where the scratches are located and the tensile stress (in the transverse direction of the pressure tube specimen) has its maximum value. A thermal cycle was applied such that the amount of hydrogen taken into solution was approximately equal to the precipitation solvus when the temperature was reduced to the test temperature. The slightly greater hydrogen concentration at the small surface flaws then would bias the formation of radial hydrides at these locations over those

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in the bulk. Once precipitated, these hydride clusters would continue to grow in length as a result of the stress gradient in the through-thickness direction of the specimen. The most difficult part of these tests is choosing an appropriate thermal cycle to ensure that long, radial hydride clusters would be formed during cooling at the tensile surface in preference to a homogeneous distribution of radial hydride precipitates distributed throughout the specimen. A series of specimens, each loaded to progressively higher loads, were tested with the maximum load such that the outer fiber tensile stress would be less than the yield strength of the material at the test temperature. The average between the calculated outer fiber tensile stress at the lowest load at which no DHC initiation had occurred and the next highest load at which it had occurred was taken as pc. Post test metallographic examinations of the latter two sets of specimens were carried out to ensure that long hydride clusters had indeed been successfully grown from the surfaces in each of these sets of specimens and that DHC initiation had occurred from one of these long clusters and not from some other physical feature. There are only limited data of successful results from these types of tests, none of which are available in the open literature. The lower bound value, pc = 450 MPa, used in current fitness assessments of in-service pressure tubes in fuel channels of Canadian CANDU reactors [17] was derived from the results of experiments by Rodgers [39] on pre-irradiated material taken from an ex-service pressure tube. (These tests were actually a continuation of the initial sets of tests reported by Cheadle and Ells [12] to determine whether small surface imperfections, such as fuel bundle scratches, could act as DHC initiation sites.) There was no need to add additional hydrogen to these pre-irradiated specimens since—at the location of the ex-service pressure tube from which the CB specimens were taken—sufficient hydrogen isotope content had accumulated in the tube for hydrides to be present at 250 °C. Long hydride clusters were successfully grown, appearing to initiate from shallow fuel bundle scratches located on the inside surface of this tube. Such scratches are a common feature in pressure tubes that have been in service for some years. These types of scratches were (and are) not thought to represent an increased risk of DHC initiation under normal operating conditions. The purpose of these tests was to demonstrate that this was, in fact, the case in irradiated material. Very long radial hydride clusters—starting from the top (inside pressure tube) surface of the CB specimens—were produced in these tests. An example of such a long hydride cluster is shown in Fig. 9.17. Post-test examination of the CB specimens in which DHC initiation had occurred showed that the origin of DHC initiation was at the root of a long hydrided region that had grown from a fueling scratch at the surface. This hydrided region was of similar length as those not fractured at lower applied outer fiber tensile stresses. Tests on unirradiated Zr–2.5Nb pressure tube material for the purpose of pc determination were subsequently carried out by Lee and Vesely [31]. Material from three different pressure tubes was used. Five specimens per material were tested at each applied stress level. Three surface flaws were machined, 2 mm apart, in the transverse direction of the tube at the middle ID surface of each CB

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specimen. The root radii of the machined flaws were 5 lm, with depths ranging from 10 to 15 lm. The 4-day-long thermal cycle used included a 1 h hold at the peak temperature of 275 °C. The test temperature during the remainder of the cycle was 230 °C. At the end of the desired number of cycles the specimens were air quenched and unloaded at the same time to prevent further radial hydride precipitation. Post-test examination of the specimens showed that long radial hydride clusters had formed at the surface flaws with lengths ranging from 100 to 450 lm. Figure 9.18 shows an unfractured radial hydride cluster grown to a length of *450 lm from a surface flaw. Four different outer fiber stress values were applied: 450, 500, 600, and 650 MPa. Failures occurred at radial hydrides only in the specimens loaded to 600 and 650 MPa. Post-test examination confirmed that initiation in each case had always occurred at one of a number of long radial hydride clusters grown from the various surface flaws, but there were also many of these clusters that had not fractured, one of which is shown in Fig. 9.18. The outer fiber stresses of 600 and 650 MPa at which DHC initiation occurred were likely close to or above the yield strength of the material. Therefore some plastic flow of the matrix near the surface at the root of each long radial hydride cluster, where fracture had initiated, had likely decreased the net stress on these radial hydride clusters at that location below the externally applied value. The lengths of the radial hydride clusters at which DHC initiation had occurred were not given by the authors, but one example micrograph indicates that the cluster was at least as long as 150 lm, while an adjoining unfractured one had a length of 175 lm and a thickness at the surface of 9.6 lm giving an aspect ratio of 0.027. From the foregoing finite element calculations such a hydride would have an induced compressive stress of -140 MPa produced by its transformation strains alone, giving net pc, or hydride fracture strength values of 450 and 500 MPa. An accurate comparison of the results of the fracture strength of embedded hydride precipitates obtained by Lee and Vesely [31] in unirradiated material with those of Rodgers [39] is not possible because there is a large gap in test data between 500 MPa at which no failures were observed and 600 MPa at which they were. Taking the average between these two values and crudely correcting for internal compressive stress of -140 MPa in the unirradiated material, yields a fracture strength value of 410 MPa. In the experiments by Rodgers [39] tests were carried out over smaller applied stress intervals of 410, 432, and 475 MPa. Failure occurred at 475 MPa but not at the two lower applied stress values. The value given in the CSA standard was chosen to be 450 MPa, which is approximately the average between the two bounds of 432 and 475 MPa and close to the value crudely calculated from the data of Lee and Vesely [31] for the unirradiated material. The difference in fracture strength obtained between the results from the tests on unirradiated and irradiated Zr–2.5Nb pressure tube materials is less than 50 MPa, keeping in mind the much greater gap between the loads at which failure occurred and at which it did not occur. The main difference in the two materials is that the material with the lower fracture strength had been irradiated to a dose that had increased the yield strength value of this material to its saturation level while

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the other was for unirradiated material having lower yield strength values. The average yield strength value in the transverse direction for pressure tube material irradiated to above the saturation fluence value (listed as ‘‘fully irradiated’’ in [17] at 250 °C is about 699.5 MPa [17] while for unirradiated material at 230 °C it is 508.2 MPa [17]). Keeping in mind that these are average values and the actual ones in the specimens tested by Lee and Vesely [31] could differ, it is likely that much of the total lengths of the radial hydride clusters grown in the unirradiated materials were subjected to some matrix plastic flow along their entire length for the tests at 600 and 650 MPa while the applied stresses for the irradiated materials were well below their yield strengths. In the unirradiated material, therefore there is more uncertainty concerning the effects on hydride fracture strength of applied outer fiber stresses above the yield strength. The resulting plastic flow at some distance below the surface would lower the applied outer fiber stress there. But it could also impose an additional internal tensile stress on the growing hydrided region through the imposition of differential, plastic-flow-induced misfit strains that would increase the net tensile stress in the plate-normal direction of this hydrided region. Therefore it is not possible to determine from the present data whether the fracture strength of embedded hydride clusters is different between pre-irradiated and unirradiated materials.

9.7 Comparison of Rising Load and Constant Load Hydride Fracture Strength Results The results from the foregoing rising and constant load tests have resulted in lower bound fracture strength values for embedded hydride precipitates that are approximately the same in both cases. The physical basis for determination of the fracture strength of embedded hydride precipitates used in both cases was the attainment of a critical normal tensile stress in the hydride. Thus, given that similarly shaped hydride clusters with low k values were grown in both cases, the attainment of the critical state for fracture of embedded hydride clusters was shown to be relatively unaffected by the differences in the physical paths used in arriving at this state. The foregoing lower bound hydride fracture strength value of 450 MPa at 250 °C obtained from the constant load tests and its temperature dependence at lower temperature obtained from the rising load tests can thus be applied to both (1) models of gross embrittlement of components containing a uniform distribution of hydride precipitates in which hydride fracture occurs under a rising load and (2) in models of DHC initiation and propagation in which hydride fracture occurs under constant (externally applied) load with a crack tip hydrided region growing by diffusion to its critical configuration. The results of these types of tests also contain an implicit size scale over which the hydride fracture strength values derived from the present sets of experiments are applicable in model calculations. Considering the range of hydride lengths (aspect ratios) over which

9.7 Comparison of Rising Load and Constant Load Hydride Fracture Strength Results

327

determinations of the hydride fracture strength of embedded hydride clusters have been made, this should be in the size scale from 10 to 1,000 lm. Overload tests were carried out by Cui [20] on the specimens containing long radial hydride clusters that had not fractured during the constant load tests of Lee and Vesely [31]. Overload refers to the higher load applied to a specimen in relation to the constant load under which the radial hydride clusters had been grown but had not fractured. The results were surprising in that the load to fracture of those previously unfractured hydrides was about 50 % greater than the lower bound constant load for failure during hydride growth. One possibility for this significant difference could be that it simply reflects the results of variability in hydride structure and internal stress state produced by local microstructural variability. Thus, the hydride clusters that did not fracture simply had microstructures more resistant to fracture. The fact that in the original constant load tests only two out of five specimens fractured at an outer fiber stress of 600 MPa and two out of four specimens at 650 MPa, with both sets grown under nominally the same conditions, is indirect evidence for this variability [31]. Such variability is also evident in the AE results supported by post rupture examinations of the tensile tests carried out by Puls and co-workers [14, 36, 37, 44]. In these specimens— containing a distribution of similarly-sized hydrides—there is a wide range of combinations of applied stresses and plastic strains at which hydride clusters, or different parts of these clusters, fracture. Alternatively, for the specimens with hydrided regions previously grown under constant load, a different net stress might be imposed on the hydrided regions on removal and subsequent reapplication of the same load under rising load conditions, particularly if the latter were done at different temperatures (i.e., room temperature) from those used during the growth of these long, radial hydrides. As noted in the foregoing, the outer fiber stresses under which the long radial hydride clusters had been grown by Lee and Vesely [31] likely were close to, or exceeded, the yield strength value over some depth from the outer surface of the specimens at the temperature of hydride growth. Removal of the load would have produced a compressive back stress near the surface acting against the externally applied stress upon reapplication of the original external stress during lower temperature overload tests. In modeling the stress state of radial hydrides under various types of applied stresses, knowledge of the constitutive relationship of the hydride precipitates and particularly whether the hydride is capable of sustaining—prior to fracture—any amount of tensile plastic deformation is of great importance, as this would influence the stresses imposed on the hydride during rising load tests when the load is increased beyond the yield strength of the matrix. Indirect experimental evidence [14, 26, 44] indicates that hydrides embedded in a zirconium matrix are capable of what appears to be considerable plastic deformation. This can happen either when the temperature is high enough or when the matrix material is soft enough so that the yield strength of the matrix is less than the fracture strength of the hydride. It can also happen when the hydrides are small enough and, as a result, have sufficiently high aspect ratios that their compressive stresses produced by their transformation strains alone are high. In these cases it is observed that radial

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9 Fracture Strength of Embedded Hydride Precipitates

hydrides are ‘‘flow-reoriented’’ with their long axes oriented in the tensile flow direction. They also appear longer and thinner than do the hydrides that were already aligned with their long axes in the tensile flow direction (Fig. 9.7). Moreover, only the hydrides in the localized neck region in Zr–2.5Nb pressure tube material [14] were fractured through their thickness, suggesting that very high plastic strains were required to fracture these hydrides in this way. In the unalloyed (commercially pure) Zr tensile specimens, on the other hand, the yield strength was sufficiently low—and the texture such—that the specimen was capable of generating large plastic strains within the diffuse neck region of the specimen, resulting in the appearance of many flow-reoriented hydrides with multiple fractures across their thickness at various locations along their lengths [37] as shown in Figs. 9.4 and 9.5. The mechanism for the ‘‘flow-reorientation’’ of these hydrides under the aforementioned conditions, however, is not clear. Is this an indication that some plastic deformation of the embedded hydrides is possible? One possibility for how the hydrides could have been ‘‘flow-reoriented’’ is that these longer, thinner hydrides are really the result of a rearrangement of smaller hydrides as shown in Figs. 9.1 and 9.2 making up what appears, at magnifications typically used in micrographs, to be a solid hydride precipitate. These smaller hydrides could have been carried with the flowing matrix located within and around the cluster into a linear chain of hydrides forming what look under optical examination of etched surfaces to be solid, but much thinner and longer hydride precipitates. In flowing with the matrix, these small hydrides might then act like hard particles in a flowing, viscous fluid and might be little deformed themselves, but simply ‘‘carried’’ by the flowing matrix. If the foregoing situation were applicable, it is not clear whether coherency is maintained between matrix and hydride. If not, very little strain would be transferred to the hydrides from the flowing matrix and the stress increase in the through-thickness direction of these tangential hydrides would be very small. If, however, coherency is maintained, then large stresses could be induced normal to the through-thickness direction.

References 1. Arsène, S., Bai, J.B., Bombard, P.: Hydride embrittlement and irradiation effects on the hoop mechanical properties of pressurized water reactor (PWR) and boiling-water reactor (BWR) ZIRCALOY cladding tubes: Part I. Hydride embrittlement in stress-relieved, annealed, and recystallized ZIRCALOYs at 20 °C and 300 °C. Metall. Trans. A 34A, 553–566 (2003) 2. Arsène, S., Bai, J.B., Bombard, P.: Hydride embrittlement and irradiation effects on the hoop mechanical properties of pressurized water reactor (PWR) and boiling-water reactor (BWR) ZIRCALOY cladding tubes: Part III. Mechanical behavior of hydride in stress-relieved annealed and recrystallized ZIRCALOYs at 20 °C and 300 °C. Metall. Trans. A 34A, 579–588 (2003) 3. Bai, J.B., Prioul, C., François, D.: Effect of microstructure factors and cold-work on the hydride precipitation in Zircaloy-4. J. Adv. Sci. 3, 188–200 (1991)

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4. Bai, J.B., Prioul, C., François, D.: Hydride embrittlement in ZIRCALOY-4 plate: Part I. Influence of microstructure on the hydride embrittlement in ZIRCALOY-4 at 20 °C and 350 °C. Metall. Trans. 25A, 1185–1197 (1994) 5. Bai, J.B., Ni, J., Prioul, C., et al.: Hydride embrittlement in ZIRCALOY-4 plate: Part III. Interaction between tensile stress and the hydride morphology. Metall. Trans. A 25A, 1199–1208 (1994) 6. Bai, J.B.: Effect of hydriding temperature and strain rate on the ductile-brittle transition in b treated Zircaloy-4. J. Nucl. Sci. Technol. 33, 141–146 (1996) 7. Barraclough, K.G., Beevers, C.J.: Some observations on the deformation characteristics of bulk polycrystalline zirconium hydrides—part I. The deformation and fracture of hydrides based on the d-phase. J. Mater. Sci. 4, 518–525 (1969) 8. Beremin, F.M.: Cavity formation from inclusions in ductile fracture of A508 steel. Metall. Trans. A 12A, 723–731 (1981) 9. Berveiller, M., Zaoui, A.: An extension of the self-consistent scheme to plastically-flowing polycrystals. J. Mech. Phys. Solids 26, 325–344 (1979) 10. Bourcier, R.J., Koss, D.A.: Hydrogen embrittlement of titanium sheet under multiaxial states of stress. Acta Metall. 32, 2091–2099 (1984) 11. Cassidy, M.P., Wayman, C.M.: The crystallography of hydride formation in zirconium: I. The d ? c transformation. Metall. Trans. A 11A, 47–56 (1980) 12. Cheadle, B.A., Ells, C.E.: Crack initiation in cold-worked Zr-2.5 wt percent Nb by delayed hydrogen cracking. In: Proceedings of 2nd International Congress on Hydrogen in Metals. Pergamon, Oxford, Paper C38 (1977) 13. Cheadle, B.A., Coleman, C.E., Ells, C.E.: Prevention of delayed hydride cracking in zirconium alloys. In: Adamson, R.B., Van Swam, L.F.P. (eds.) Zirconium in the Nuclear Industry: Seventh International Symposium. ASTM STP, vol. 939, pp. 224–240 (1987) 14. Choubey, R., Puls, M.P.: Crack initiation at long radial hydrides in Zr–2.5Nb pressure tube material at elevated temperatures. Metall. Trans. A 25A, 993–1004 (1994) 15. Coleman, C.E., Hardie, D.: The hydrogen embrittlement of a-zirconium—A review. J. LessCommon Met. 2, 168–185 (1966) 16. Coleman, C.E., Ambler, J.F.R.: Delayed hydrogen cracking in Zr–2.5Nb alloy. Rev. Coat. Corros. III (2 and 3), 105–157 (1979) 17. CSA: Technical Requirements of the In-service Evaluation of Zirconium Alloy Pressure Tubes in CANDU Reactors. Canadian Standards Association, Mississauga, Ontario, Canada, Nuclear Standard N285.8-10 (2010) 18. Cox, H.L.: The elasticity and strength of paper and other fibrous materials. Br. J. Appl. Phys. 3, 72–79 (1952) 19. Cui, J., Shek, G.K., Scarth, D.A., et al.: Delayed hydride cracking initiation at notches in Zr–2.5Nb pressure tube material. J. Press. Vess. Technol. 131, 041407 (2009) 20. Cui, J.: Unpublished. Kinectrics Inc., Toronto, Ontario, Canada (2003) 21. Dutton, R., Nuttall, K., Puls, M.P., et al.: Mechanisms of hydrogen induced delayed cracking in hydride forming materials. Metall. Trans. A 8A, 1553–1562 (1977) 22. Ells, C.E.: Hydride precipitates in zirconium alloys. J. Nucl. Mater. 28, 129–151 (1968) 23. Eshelby, J.D.: The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. London A 241, 376–396 (1957) 24. Eshelby, J.D.: Elastic inclusions and inhomogeneities. Prog. Sol. Mech. 2, 89–140 (1961) 25. Eshelby, J.D.: Continuum theory of lattice defects. In: Seitz, F., Turnbull, D. (eds.) Solid State Physics, vol. 3, pp. 79–140. Academic, New York (1966) 26. Evans, W., Parry, G.W.: The deformation behaviour of Zircaloy-2 containing directionally oriented zirconium hydride precipitates. Electrochem. Technol. 4, 225–231 (1966) 27. Hardie, D.: The influence of the matrix on the hydrogen embrittlement of zirconium in bend tests. J. Nucl. Mater. 42, 317–324 (1972) 28. Kerr, M., Daymond, M.R., Holt, R.A., et al.: Strain evolution of zirconium hydride embedded in a Zircaloy-2 matrix. J. Nucl. Mater. 380, 70–75 (2008)

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29. Kerr, M., Daymond, M.R., Holt, R.A., et al.: Fracture of a minority phase at a stress concentration observed with synchrotron X-ray diffraction. Scripta Mater. 61, 939–942 (2009) 30. Kerr, M., Daymond, M.R., Holt, R.A., et al.: Observation of growth of a precipitate at a stress concentration by synchrotron X-ray diffraction. Scripta Mater. 62, 341–344 (2010) 31. Lee W.K., Vesely, P.J.: Unpublished. Kinectrics Inc., Toronto. Ontario, Canada (2001) 32. Leitch, B.W., Puls, M.P.: Finite element calculations of the accommodation energy of a misfitting precipitate in an elastic-plastic matrix. Metall. Trans. A 23A, 797–806 (1992) 33. Louthan Jr., M.R.: Cleavage in hydrided Zircaloy-2. Trans. A ASM 57, 1004–1008 (1964) 34. Marshall, R.P., Louthan Jr., M.R.: Tensile properties of Zircaloy with oriented hydrides. Trans. ASM 56, 693–700 (1963) 35. Pan, Z.L., Puls, M.P.: Internal friction peaks associated with the behaviour of hydrogen in Zr and Zr–2.5Nb. Mater. Sci. Eng. A 442, 109–113 (2006) 36. Puls, M.P.: The Influence of hydride size and matrix strength on fracture initiation at hydrides in zirconium alloys. Metall. Trans. A 19A, 1507–1522 (1988) 37. Puls, M.P.: Fracture initiation at hydrides in zirconium. Metall. Trans. A 22A, 2327–2337 (1991) 38. Puls, M.P., Shi, S.Q., Rabier, J.: Experimental studies of mechanical properties of solid zirconium hydrides. J. Nucl. Mater. 336, 73–80 (2005) 39. Rodgers, D.K.: Unpublished. Atomic Energy of Canada Ltd., Chalk River Laboratories, Chalk River, Ontario, Canada (1990) 40. Scarth, D.A., Smith, E.: Developments in flaw evaluation for CANDU reactor Zr–Nb pressure tubes. J. Press. Vess. Technol. 123, 41–48 (2001) 41. Scarth, D.A., Smith, E.: The effect of plasticity on process-zone predictions of DHC initiation at a flaw in CANDU reactor Zr–Nb pressure tubes. J. Press. Vessels Pip. 437, 19–30 (2002) 42. Scully, C.J.: Unpublished. AECL, Chalk River Laboratories, Chalk River, Ontario, Canada (1984) 43. Shek, G.K., Cui, J., Perovic, V.: Overload fracture of flaw tip hydrides in Zr–2.5Nb pressure tubes. J. ASTM International 2: Paper ID JAI12435 (2005) 44. Shi, S.Q., Puls, M.P.: Fracture of hydride precipitates in Zr–2.5Nb alloys. J. Nucl. Mater. 275, 312–317 (1999) 45. Simpson, L.A.: Criteria for fracture initiation at hydrides in zirconium-2.5 pct niobium. Metall. Trans. A 12A, 2113–2124 (1981) 46. Simpson, L.A., Cann, C.D.: Fracture toughness of zirconium hydride and its influence on the crack resistance of zirconium alloys. J. Nucl. Mater. 87, 303–316 (1979) 47. Simpson, L.A., Chow, C.K.: Effect of metallurgical variables and temperature on the fracture toughness of zirconium alloy pressure tubes. In: Adamson, R.B., Van Swam, L.F.P. (eds.) Zirconium in the Nuclear Industry: Seventh International Symposium. ASTM STP, vol. 939, pp. 579–595 (1987) 48. Steuwer, A., Santisteban, J.R., Preuss, M., et al.: Evidence of stress-induced hydrogen ordering in zirconium hydrides. Acta Mater. 57, 145–152 (2009) 49. Steuwer, A., Daniels, J.E., Peel, M.J.: In situ crack growth studies of hydrided Zircaloy-2 on a single-edge notched tensile specimen. Scripta Mater. 61, 431–433 (2009) 50. Veleva, M., Arsène, S., Record, M.-C., et al.: Hydride embrittlement and irradiation effects on the hoop mechanical properties of pressurized water reactor (PWR) and boiling-water reactor (BWR) ZIRCALOY cladding tubes: Part II. Morphology of hydrides investigated at different magnifications and their interaction with the processes of plastic deformation. Metall. Trans. A 34A, 567–578 (2003) 51. Wallace, A.C., Shek, G.K., Lepik, O.E.: Effects of hydride morphology on Zr–2.5Nb fracture toughness. In: Van Swam, L.F.P., Eucken, C.M. (eds.) Zirconium in the Nuclear Industry: Eighth International Symposium. ASTM STP, vol. 1023, pp. 66–88 (1989) 52. Westlake, D.G.: Initiation and propagation of microcracks in crystals of zirconium-hydrogen alloys. Trans. ASM 56, 1–10 (1963)

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53. Yunchang, F., Koss, D.A.: The influence of multiaxial states of stress on the hydrogen embrittlement of zirconium alloy sheet. Metall. Trans. A 16A, 675–681 (1985) 54. Xu, F., Holt, R.A., Daymond, M.R., et al.: Development of internal strains in textured Zircaloy-2 during uni-axial deformation. Mat. Sci. Eng. A 488, 172–185 (2008) 55. Xu, F., Holt, R.A., Daymond, M.R.: Evidence for basal \a[-slip in Zircaloy-2 at room temperature from polycrystalline modeling. J. Nucl. Mater. 373, 217–225 (2008)

Chapter 10

Delayed Hydride Cracking: Theory and Experiment

10.1 Introduction As noted in Chaps. 1 and 9, zirconium alloy components can fail by a time–dependent mechanism of cracking if hydrides can preferentially form and subsequently crack at locations of elevated tensile stress. This process of time-dependent hydride cracking, called delayed hydride cracking (DHC), is based on the mechanism of diffusion of hydrogen to a region of elevated tensile stress followed by nucleation, growth, and fracture of the hydrided region. By repeating these steps, a crack can propagate in a component at a rate that, above a threshold stress intensity factor, KIH ; is mainly dependent on temperature. In this chapter we present the status of the present state of understanding of DHC in zirconium alloys, emphasizing the connections between the models developed and the experimental data. The intent of this chapter is not to provide an exhaustive review of the literature but to focus on results that are deemed to have collectively advanced our understanding of this phenomenon. This chapter is divided into two main parts, the first part dealing with DHC propagation, the other with DHC initiation. Although it may appear backwards to start with DHC propagation rather than initiation, this approach is consistent with the historical development of the field and thus also helps somewhat in simplifying the exposition. Readers wishing to obtain a succinct recent overview of DHC and other aspects of the effects of hydrogen and hydrides on the integrity of zirconium alloys are referred to a recent review by Coleman [15]. In addition, an earlier review by Northwood and Kosasih [47] provides a comprehensive summary account of work published in this field up to the date of that publication.

M. P. Puls, The Effect of Hydrogen and Hydrides on the Integrity of Zirconium Alloy Components, Engineering Materials, DOI: 10.1007/978-1-4471-4195-2_10,  Springer-Verlag London 2012

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10.2 Historical Overview Although a few laboratory experiments in the 1960s hinted that zirconium alloys may fracture by a time-dependent mechanism involving hydrogen, the first confirmation of such a mechanism was the cracking in 1972 reported by Simpson and Ells [80] of experimental fuel cladding made from Zr-2.5Nb material. After several months of storage at room temperature, cracks associated with hydrides had been found in the heat affected zone in the weld between the cladding and its end-cap at the root of an internal notch created between the fuel cladding and the ‘‘upset’’ produced by the magnetic force resistance weld. The authors concluded that, despite the low temperature at which the material had been held, hydrides had accumulated at this notch as a result of a diffusion-driven build up of hydrogen to a concentration above its solvus composition. The hydrides formed at the notch had eventually grown to some critical size at which they had fractured. The crack initiated in this way had then continued to propagate by this mechanism, starting a process that the authors called delayed hydrogen embrittlement. Although many different names were subsequently used to describe this phenomenon, it is now universally described as DHC. To study this process in more detail, several weld sections were prehydrogenated by Simpson and Ells [80] to hydrogen contents of 100 and 300 wppm. These specimens were held at room temperature and periodically examined. This study showed that there was a gradual increase in the lengths of the hydrided regions formed at the roots of the upset notches,verifying the diffusive nature of this process. An attempt was made by the authors to quantify the rate of growth of the hydrides formed at the ‘‘upset’’ notch. They based their analysis on a recent treatment by Waisman et al. [92] who attributed the failure of a titanium vessel near its weld fusion line to a similar mechanism of delayed hydrogen embrittlement. To demonstrate that the hydride accumulation at the weld was driven by diffusion of hydrogen in a tensile stress gradient, Waisman et al. [92] formulated a hydrogen flux equation derived from the theory of irreversible thermodynamics. This equation is similar in form to that given by Eq. 5.23 in Chap. 5. Simpson and Ells [80] used a simplified version of this equation (without the thermal gradient term) to calculate the rate of hydrogen arrival at the upset notch. The authors compared this arrival rate with the experimentally observed growth in length of the hydrided region at the notch. Assuming that the hydrogen concentration in the material is everywhere given by its solvus composition, the one-dimensional hydrogen flux equation was reduced to an equation depending only on the tensile stress gradient as follows: JH ¼ DH csH

H dr V 3RT dx

ð10:1Þ

where DH is the chemical diffusion coefficient of hydrogen, csH is the solvus composition and r is (uniaxial) stress. An approximate solution to this equation was obtained by simply assuming a numerical value for the stress gradient. A much lower diffusion flux was predicted

10.2

Historical Overview

335

Fig. 10.1 Through-wall crack in a pressure tube of a CANDU reactor, showing oxidized crack growth bands. The crack initiated at the inside surface just inboard of the rolled joint (from IAEA [26])

than the observed rate of hydride growth at the notch. The discrepancy was attributed to uncertainties in extrapolating to room temperature the relations for hydrogen diffusion and solvus composition obtained from fits of the data of these parameters at much higher temperatures. Soon after the observation of DHC in Zr-2.5Nb fuel cladding material described in the foregoing, leakage in 19 out of 780 Zr-2.5Nb pressure tubes in Units 3 and 4 of the Pickering Nuclear Generating Station was discovered [53]. All the leakages originated from cracks grown in the axial and radial directions of the tubes, extending in a series of bands, Fig. 10.1, starting from the inside walls of these tubes at a location just inboard from the stainless steel end fitting. The end fitting connects the pressure tube by a rolled joint to the remaining, outboard part of the fuel channel. The interpretation was that the cracks had initiated and grown at low temperatures by DHC, but once the reactor was at power and the pressure tubes were at a higher temperature, [250 C, cracking had stopped because the initial, low hydrogen concentration in these tubes—typically \15 wppm—had all been in solution. The surface of the arrested crack oxidized during this period of reactor operation, but cracking continued during subsequent reactor shutdowns with the stopped crack continuing to oxidize during each period of power production. Each band on the fracture surface corresponded with a reactor shutdown and a period of operation, with cracking at the expected rates for DHC at the temperature of the shutdown [8]. It was found that the tubes in which this cracking was observed had been over rolled, generating high residual tensile stresses in the region just inboard from the point where the pressure tube is no longer surrounded and supported by the stainless steel end fitting into which it is rolled. To prevent further occurrences of such cracking, depending on the reactor, the residual stresses were either minimized by stress relief or redesign of the rolled joint [17], the latter to reduce further the residual stresses in properly rolled joints. The occurrence of DHC in these pressure tubes led to the inception of a large research program in the Canadian nuclear industry that ultimately resulted in the development and subsequent continued updating of the technical requirements for the in-service evaluation of zirconium alloy pressure tubes in CANDU reactors. These technical requirements have been published as part of a series of documents provided by the industry through the Canadian Standards Association [5]. In addition, the information has been utilized in improving the designs of new CANDU reactors.

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Fig. 10.2 Micrograph showing tapered hydride cluster along crack path with uncracked hydride in front of crack at 3 (‘H’ refers to hydride cluster) (from Shek et al. [69])

10.3 General Features of DHC It is useful at this stage to provide a brief overview of the broad characteristics of DHC encapsulating our current understanding of the phenomenon. From the earliest experimental observations, it was clear that DHC involved the accumulation of hydrides to some critical size at a region of elevated tensile stress, the fracture of this hydrided region up to its leading edge and the repetition of this process. In pressure tube material, DHC is largely limited to the radial-axial plane of the tube. Optical micrographs of arrested DHC cracks show that hydrided regions formed at a crack can be idealized as plate shaped, extending in the crack growth directions with thickness much smaller than their in-plane lengths (Fig. 10.2). Lying on the pressure tube’s radial-axial plane these plate shaped (often tapered) hydrided regions are also referred to as radial hydrides. Observations of fracture surfaces at low magnifications often showed periodic rows of ridges (striations) extending in rows parallel to the crack front (Fig. 10.3). The fracture surfaces between these striations were of brittle appearance, being flat with cleavage-like river patterns (Fig. 10.4). The lengths between these striations were strongly temperature dependent (Figs. 10.3 and 10.20). The interpretation of the foregoing observations is that the hydrided region grows to some critical length from the crack tip, fracturing along its length up to its leading edge. The fracture of the hydrided region causes an abrupt increase in length of the macroscopic crack, which is arrested at the leading edge of the hydrided region by the ductile matrix. The striations are the physical evidence of this arrest and the distance between each row of striations, called the striation spacing, represents the critical fracture length of the hydrided region while the striation length itself is associated with the plastic zone of the macrocrack at its point of arrest. Direct confirmation of the stepwise sequence of events characterizing DHC was obtained in a limited number of TEM in situ experiments in

10.3

General Features of DHC

337

Fig. 10.3 DHC fracture surfaces of Zr-2.5Nb pressure tube material showing striations at (a) 400 wppm H tested at 325 C, (b) 400 wppm H tested at 250 C, (c) 100 wppm H tested at 200 C, (d) 400 wppm H tested at 150 C. The crack growth direction is vertical with average striation spacing indicated by accompanying arrows (from Dutton et al. [19]; with permission from AECL)

different hydride forming metals [6, 36, 41, 73], while indirect confirmation was obtained from metallographic and fractographic observations of hydrides at crack tips and from striation spacing length measurements, respectively [79]. In some favorable cases these observations could be correlated with corresponding stepwise intervals of acoustic emission and electrical resistivity signals [27, 69, 94] corroborating the metallographic and fractographic evidence of stepwise growth and fracture of the hydrided region. The rate of growth of the DHC crack suggests that the process is diffusion driven. The presence of a ductile stretch zone (the striation) after each growth event suggests that the first step for continuation of the process is the nucleation of new radial hydrides. Although there may be, by chance, transverse hydrides located at the new location of the crack tip, these hydrides are not in favorable orientations to cause embrittlement of the crack front. The requirement for the nucleation of new, reoriented (radial) hydrides after each microgrowth step then means that the concentration of hydrogen at the flaw must be at—or be able to increase to—a concentration that is at least as great as the solvus composition for nucleation of new radial hydrides.

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Fig. 10.4 TEM micrograph replica (92,812) of DHC fracture surface of Zr-2.5Nb CT specimen tested at 200 C containing 130 wppm H. Crack growth direction is vertical, with striation stretch marks along line indicated by two sets of X–X (from Dutton et al. [19]; with permission from AECL)

Another set of experimental observations concerns the stress dependence of DHC. Although the earliest experimental studies of DHC involved tests in which smooth and notched specimens were held under a constant load and the conditions and times to failure were determined, all subsequent testing involved choosing specimen geometries in which the dependence of the crack growth rate as a function of the applied stress intensity factor starting from a crack or notch could be readily measured. In order for the results to correspond as closely as possible to what would occur in an operating pressure tube, the specimens needed to be cut from sections of such tubes. The radial-axial crack plane on which DHC occurs in this material and the wall thickness of only *4 mm put some constraints on the types, dimensions, and crack orientations of the test specimens. The specimen types chosen were compact toughness (CT), or curved compact toughness (CCT) and cantilever beam (CB) specimens. Figure 10.5 shows a schematic of these types of specimens and their relationships to the pressure tube wall from which they were cut. It is evident from Fig. 10.5 that determination of the DHC growth rate (referred to as DHC velocity in much of the past literature) in the axial direction required the use of CCT specimens while the curved CB specimen could only be used for DHC growth rate measurements in the radial pressure tube direction. It took a few years to arrive at testing procedures that were relatively free of experimental artifacts [83]. Once this was achieved it was found, as shown schematically in Fig. 10.6, that the DHC growth rate has a three stage dependence on Mode I (crack opening) stress intensity factor, KI : This type of dependence on stress intensity factor is common to many other environmentally assisted subcritical crack growth mechanisms. The figure shows that below a threshold

10.3

General Features of DHC

339

Fig. 10.5 CCT (a) and CB (b) specimens used in DHC tests

stress intensity factor, KIH ; crack growth stops completely, while from this threshold on there is a short interval of KI (Stage I) over which there is an abrupt increase in rate, culminating in a plateau region (Stage II) over which the growth rate is approximately constant until an applied KI is reached (Stage III) where it increases abruptly again spanning a short interval up to the fracture toughness limit, KIc, of the material. Measurements of KIH have shown that it is approximately independent of temperature (Fig. 10.7) or increases weakly with temperature until a threshold temperature is approached at which there is a steeper increase (Fig. 10.8). A theoretical explanation for this is given further on in Sect. 10.5.2. KIH also depends on texture and strength of the material (Fig. 10.9). As the fraction of resolved basal poles orientated in the direction of the crack plane normal increases, KIH decreases since the increase in resolved basal pole fractions favors an increasing density of hydride platelets precipitated on the crack plane. In addition, there is an increase in yield strength with this increase in texture. By itself, however, the effect of increasing yield strength on KIH is relatively weak. This is evident from the fairly small decrease of KIH in fast neutron irradiated material compared to its value for material in its unirradiated state (Fig. 10.10). These trends in KIH variations are, however, hard to quantify accurately since there is also always a fairly large scatter in KIH brought about by variations from place to

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Fig. 10.6 Schematic of the dependence of DHC crack growth rate (velocity) versus stress intensity factor (KI)

Fig. 10.7 Temperature dependence of KIH from unirradiated Zr-2.5Nb pressure tube material (from Sagat et al. [65]; with permission from AECL)

place in the microstructure of the material, even when all of the material has been taken from the same pressure tube. Over the plateau range of KI , where there is no dependence on applied stress, the DHC growth rate depends exponentially on the inverse of the first power of the temperature. Thus it acts like a thermally activated process. An early example of that is given in Fig. 10.11 by Simpson and Puls [83]. There is also a difference in the activation energy of growth rate between cracks growing entirely in the axial

10.3

General Features of DHC

341

17 16 15

13

1/2

KIH (MPa m )

14

12 11 10 9 8 7 6 5 100

150

200

250

300

350

o

Temperature ( C)

Fig. 10.8 Dependence of KIH versus temperature of pre-irradiated Zr-2.5Nb pressure tube material removed from an ex-service pressure tube from a CANDU reactor, doped with 153 wppm [Heq] (from Resta Levi and Puls [60]; with permission from AECL) Fig. 10.9 The texture dependence of KIH in various Zr alloys obtained from CCT and CB specimens. Data sources are: [3] [30]; [4] [14]; [5] [13]; [6] [24]; [7] [46]; [10] [31]; [11] [32]; [this study] [34] (from Kim [34])

compared to the radial directions. (The reason for the difference in the average DHC rate for the as-received and hydrided material and for the two pressure tube directions is connected with the state of the b phase, as discussed in Chaps. 4 and 5). Simpson

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Fig. 10.10 Effect of fluence on KIH . Material from CANDU reactor outlet rolled joint region irradiated at about 290 C and tested at temperatures between 140 and 270 C (from Rodgers et al. [63]; with permission from AECL)

and Puls [83] also noted that the DHC growth rates are different depending on the direction of approach to the test temperature such that, above a test temperature of *200 C, DHC growth rates become negligibly small when the test temperature is approached from below compared to when it is approached a sufficiently large temperature interval from above. These authors were the first to propose that this direction-of-approach effect arises from a difference in the solvus temperature for hydride formation and dissolution. As shown in Chaps. 6 and 8, this difference in solvus temperature or composition is now understood to be the result of hysteresis. Hysteresis has been found to be a common occurrence in metal hydrogen binary, or pseudo-binary systems and, for this reason, three chapters (Chaps. 6–8) have been devoted to this and related aspects of phase thermodynamics. Contrary to what is often stated, because of the hysteresis in the phase relationships, the phase boundary compositions at both ends of the two-phase coexistence range do not represent equilibrium states. Their high reproducible gives them the appearance of being equilibrium states, but the fact that there is hysteresis in the compositions or temperatures between the forward and reverse phase transformations means that these transformations are in each direction irreversible and a macroscopic change in the state of the system (such as a change in temperature) is required to reverse the transformation. Moreover, as discussed in Chaps. 6 and 8, the experimentally determined forward and reverse phase relationships are only truly reproducible when it can be established that the difference between them represents true hysteresis; that is, it represents a time-independent difference in physical states. Figure 10.12 shows schematically how the concentration of hydrogen in solution at the same temperature depends on how this temperature is approached. Further on, it is shown that DHC rate is proportional to the magnitude of the diffusible hydrogen. As discussed in Chaps. 4 and 5, diffusible hydrogen is the hydrogen in solid solution in the a-Zr phase that is not bound to nondiffusing species or defects and is, therefore, as the name implies, free to diffuse. The first systematic examination of this direction-of-approach effect on DHC growth rate was made by Ambler [1]. Subsequent to this, a more extensive series of studies on Zr-2.5Nb pressure tube material were carried out over a period of 10 years by Shek and co-workers [70]. Figure 10.13 gives an example of the large difference in DHC rate at 250 C that is obtained depending on the peak temperature reached before cooling to the test temperature. A summary of most of the results of

10.3

General Features of DHC

343

Fig. 10.11 DHC growth rate (velocity) versus temperature relationship for unirradiated Zr-2.5Nb pressure tube material (from Simpson and Puls [83]; with permission from AECL)

these studies is given in Shek’s PhD dissertation [70].1 Experiments to determine this effect for pre-irradiated Zircaloy material were carried out by Schofield et al. [67] who used the Dutton-Puls DHC growth rate model as applied by Shi et al. [77] (described in the next section) to analyze these results, finding good agreement with the predictions of the model and the experimental results. A schematic representation of the dependence of the DHC rate on test temperature and direction of approach to test temperature, reproduced in Fig. 10.14, was first given by Cheadle et al. [9] and has since been used by many authors to illustrate the possible temperature dependencies of DHC.

1

Unfortunately, very few of the results of these very interesting series of studies have appeared elsewhere in the readily accessible open literature.

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Fig. 10.12 Schematic of TSS boundary curves showing how hydrogen in solution varies during a temperature cycle along the path ABCD (from IAEA [26])

Fig. 10.13 DHC growth rate (velocity) at 250 C in Zr2.5Nb pressure tube material containing 60 wppm H as a function of peak temperature reached prior to cooling to test temperature (from [68])

From Fig. 10.12, it can be seen that the DHC growth rate will have its highest possible value when the thermal history for the given hydrogen content is such that the concentration in solution (the hydrogen available for diffusion) has been maximized. To choose the thermal cycle for the specimen that will achieve this, it is necessary to recognize, as seen from Fig. 10.12, that the hydrogen concentration in solution in the bulk is the lesser of the following three concentration values: 1. The total hydrogen content; 2. The composition of the solvus for hydride dissolution at the peak temperature; 3. The composition of the solvus for hydride precipitation at the test temperature. The maximum possible DHC rate for a material—when not dependent on the total hydrogen content—occurs when the thermal history of the sample is such that there is sufficient hydrogen in solution in the bulk at the peak temperature of the thermal cycle such that, at the test temperature, the solvus composition for precipitation has been reached. Figure 10.14 illustrates schematically the effects of solubility hysteresis on measurements of DHC crack rates (velocities). (Note that this figure is not meant to imply that the DHC rates at low temperatures are the same whether the test temperatures have been approached from above or below.) For a specimen with a specific hydrogen concentration, DHC growth rate can be

10.3

General Features of DHC

345

Fig. 10.14 A schematic of the DHC crack growth (velocity) showing the effect of temperature history

different depending on whether the test temperature is achieved by heating or cooling. If the specimen is heated to progressively higher test temperatures, the velocities follow a profile similar to that described by the sequence: T1 ? T2 ? T3 ? T4. The maximum growth rate occurs at T2. When the propagation rates are measured after cooling from temperature, T4, they follow the profile given by the sequence: T4 ? T5 ? T6 ? T2. The rate at T6 is the maximum for the specimen and this rate is only achieved by cooling to T6 from a sufficiently high temperature, T4, where all hydrogen is in solution. The decrease in rate on heating above T2 reflects a decreasing driving force since the total hydrogen available for transport continues to increase with increasing temperature. On cooling from T4, the crack can grow at T5 because there is now sufficient driving force to precipitate and grow hydrides at the crack tip up to a critical hydrided region length at which it cracks. Upon further cooling to T6, the rate increases as a result of an increased driving force as the concentration of hydrogen required to precipitate at the crack tip decreases with decrease in temperature while the concentration in solution in the bulk material remains constant (all hydrides in the bulk dissolved) corresponding to the solvus composition at T4. In addition to the effect on DHC rate of direction-of-approach to the test temperature, there exists a similar effect involving KIH : Shek [70]—see also early data by this author reproduced in Shi and Puls (Fig. 5 in [75])—showed that KIH increases with decrease in hydrogen supersaturation. Hydrogen supersaturation was defined by this author as the concentration of hydrogen in solution in the bulk of the

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specimen that is in excess of the hydrogen concentration at the same locations and temperature given by the solvus for hydride dissolution. A theoretical explanation for this effect, discussed in more detail in the next section (Sect. 10.4), was given by Shi and Puls [75]. It should be noted, as pointed out in Chap. 6, that denoting this concentration difference as supersaturation is a misnomer since it implies that it is time dependent when it is actually a time-independent concentration difference produced by the hysteresis between the two solvi. However, since a concise, physically more accurate description of this difference does not currently exist, the terminology introduced by Shek [70] is retained here.

10.4 Theory of DHC Growth Rate Soon after the discovery of the leaking pressure tubes in the Pickering 3 and 4 NGS that was caused by DHC, a simple model for the growth rate by this mechanism was developed by Dutton and Puls [18]. The model was developed at a time when only few experimental data concerning DHC initiation and propagation existed. The model had its origin in the authors’ previous studies concerning the thermodynamic driving force for diffusion of substitutional and/or interstitial species in a crystalline solid. These studies resulted in the formulation of a model to quantify the growth of a Griffith-type crack by diffusion of vacancies [54]. The diffusion of vacancies in a stress gradient (akin to that of hydrogen in a stress gradient) forms a key component of this model and represents a particular application in crystalline substitutional or interstitial solids of the general diffusion flux relations derived from the linear theory of irreversible thermodynamics as presented in Chap. 5. For the case of hydrogen migration, such equations had previously been used by Waisman et al. [92] and Simpson and Ells [80] to explain the preferential formation and growth of hydrides at regions of elevated tensile stress. These flux equations can be formulated in terms of different thermodynamic variables, but when a tensile stress gradient is the source of hydride accumulation, it is most useful to formulate the driving force and boundary values in terms of chemical potential differences. Since zirconium alloys are fairly ductile metals, account must also be taken of the presence of a plastic zone at the crack tip. As shown schematically in Fig. 10.15, it was assumed that the DHC growth rate—which in actuality, is given by the repetitive process of hydrogen diffusion to the crack tip, followed by hydride nucleation, growth and fracture of these hydrides once they have reached their critical dimensions—could be approximated by the diffusional growth of a preexisting hydride platelet. In this model, hydride nucleation is assured when it can be shown that the hydrogen concentration in the bulk of the material that is free to diffuse to the crack tip under a tensile stress gradient is of sufficient magnitude that it would be able to increase to at least its solvus composition at the crack. Determining whether this is possible, therefore, is an implicit prerequisite for the applicability of this model. Another assumption in the model is that the average crack growth rate can be approximated by the

10.4

Theory of DHC Growth Rate

347

Fig. 10.15 Schematic of crack and hydride geometries in Dutton-Puls DHC growth rate model (from Dutton and Puls [18]; with permission from AECL)

growth rate of a plate-shaped hydride at a single location in the plastic zone. That is, there is no explicit failure criterion built into the model and any stress variations in the plastic zone affecting the growth rate of the hydrided region at its leading edge are ignored. Implicitly, this assumption means that there are no impediments for the hydrided region to grow in size within the plastic zone to its critical dimensions for fracture. In this connection, it was further assumed in the calculation of the average macrocrack growth rate that the time interval over which fracture of the hydrided region occurs after it reaches its critical dimensions can be neglected in relation to the time taken for hydride to grow to this length by diffusion. Finally, it was assumed—consistent with the tenants of the theory of linear irreversible thermodynamics—that local equilibrium conditions prevail at the crack tip and bulk hydrides. This ensures that the hydrogen concentrations in the bulk and at the crack tip remain at constant values given by their stress affected solvus compositions. Regarding this approximation for the bulk, it is not a necessary feature of the model, but this approximation makes possible an exact, closed form, steady-state solution to the diffusion flux equation. Removing this constraint and assuming that another—approximately fixed—concentration exists in the bulk causes the accuracy of the predictions of the steady-state solution to be diminished. The reduction in accuracy is expected to be small; however, since the region from which hydrogen is drawn in the bulk is large compared to the number of hydrogen atoms required to grow the hydrided region at the crack tip to its critical dimensions.

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Referring to the cylindrical geometry used to model the diffusion process shown in Fig. 10.15 and ignoring any h-dependence of the stress field at the crack, the diffusion equation is given by2: a cD a H ðr; r Þ  DH rlD H ðr; r Þ  RT cD ðr; ra Þ  DH a rlD  H H ðr; r Þ XZr  RT

JH ¼ 

ð10:2Þ

where number of hydrogen atoms/unit volume cD H atom fraction of hydrogen concentration in a-Zr cD H DH chemical diffusion coefficient of hydrogen in a-Zr XZr atomic volume of Zr in a-Zr hydrogen chemical potential for diffusion lD H applied stress ra It is important to note that the hydrogen chemical potential gradient in Eq. 10.2 refers only to the hydrogen that is free to diffuse. In the literature, this is generally referred to as the hydrogen concentration in solution. Only for this fraction of the total hydrogen content does there exist a chemical potential definable everywhere throughout a nonhydrostatically stressed crystalline solid (i.e., its value is uniquely given everywhere). It differs from the chemical potential of hydrogen at a sink or source. Such a chemical potential is constrained by the prescribed local conditions, such as the condition of local phase equilibrium, or of equality with the chemical potential of an external source of hydrogen. As discussed in Chaps. 4 and 5, the hydrogen chemical potential can depend on many variables. However, for DHC in solids containing no temperature gradients, the most important variables are the hydrogen concentration in solution and the stress. In addition, it was found that within the overall accuracy of the constants in the model, it is sufficient to include only the first-order linear stress interaction energy contribution to the chemical potential for diffusion. In addition, for simplicity in the following formulation, it is assumed that the partial molar volumes of hydrogen in the various phases are isotropic and the h-dependence of the crack tip stresses can be ignored. With these restrictions, the chemical potential for diffusion is equivalent to that given by Eq. 4.33 in Chap. 4, written here as:  D  o  lD ð10:3Þ H ðr; ph Þ ¼ lH þ RT ln cH ðrÞ  ph ðrÞVH H is the molar where loH is the chemical potential of an arbitrary reference level, V volume of hydrogen in a-Zr and ph ¼ raii ðr Þ=3 is the hydrostatic (mean) stress at

2

The present formulation uses the usual sign convention for stress where tensile stress has a positive sign. In the original derivation [18], [19] the reverse sign convention was used.

10.4

Theory of DHC Growth Rate

349

the location of the hydrogen atom. Inserting this expression into Eq. 10.2 yields the following hydrogen diffusion flux equation:    H cD V DH dcD H ðr; ph Þ H ðr; ph Þ dph ðrÞ  JH ¼  ð10:4Þ dr RT dr XZr This diffusion flux equation is equivalent to that given by Eq. 5.23 in Chap. 5. Here we have assumed that the activity coefficient for hydrogen in solution in the a-Zr phase is unity. To solve Eq. 10.4, the following assumptions were made: (1) Hydride growth occurs under steady-state conditions, (2) Any hydrogen entering into the crack tip region across the circle at r = ‘ converges onto the leading edge of the preexisting, plate-shaped hydride lying with plate normal parallel to the crack plane normal (approximating the hydride cluster shapes and orientations at the crack tip observed experimentally) and (3) The nearest hydrides away from the crack tip are at r = L, where L is the average inter-hydride spacing. Figure 10.15 has been reproduced from the original references to clarify a number of points omitted in these references concerning the foregoing assumptions and the representation of the DHC process in the figure. The first is that it was implicit in the original formulation that DHC crack growth could be either in the axial or radial direction of pressure tubes used in CANDU or PHW reactors. This means that the crack plane is the radial-axial plane. The preexisting (bulk) hydride platelet clusters in these tubes are observed in optical micrographs to have their platelet edges predominantly oriented in the tube’s transverse direction and are, therefore, referred to as transverse (or circumferential) hydrides. When hydrides can nucleate and grow at the crack tip, they are reoriented with their platelet normal oriented along the transverse direction and, hence, lie on the radial-axial planes. These are referred to as radial or axial hydrides. Both sets of hydride clusters–despite being differently oriented—have their habit planes on the near-basal planes of the hcp lattice of the a-Zr phase. Support for this comes from TEM examinations, as detailed in Chap. 3, and theoretical calculations showing that the lowest strain or accommodation energy for hydride platelets is when their habit planes are the basal a-Zr planes [55]. Since both macroscopically observed hydride cluster orientations have the same crystallographic orientation they would have similar accommodation energies and, hence, would have the same solvus composition in the externally unstressed state. For Zr-2.5Nb pressure tube material, hydrides are able to precipitate on their preferred habit plane in both pressure tube orientations, because the fraction of resolved basal poles in both directions is sufficient for this to be possible. In Fig. 10.15, the bulk hydrides at L are shown to have the same orientation as the radial hydride cluster at the crack tip. The schematic was drawn this way for simplicity of representation. The actual transverse hydrides would either be oriented with their plate normal directions in the x-direction or out of the plane of the paper (in the implicit z-direction), depending on whether the crack growth direction depicted is the radial or axial one, respectively. The steady-state condition is satisfied when r  JH ¼ 0

ð10:5Þ

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For cylindrical coordinates, with JH given by Eq. 10.4, Eq. 10.5 reduces to the following differential equation for the steady state hydrogen flux (note that there are misprints in Eqs. 10 and 12 of Dutton and Puls [18]):   1 d dcH cH  dph r r VH ¼0 ð10:6Þ r dr dr RT dr A detailed derivation of the solution of this equation is given in Puls [59]. Here we repeat only the derivation of the boundary conditions. From the expression for the chemical potential for diffusion, given by Eq. 10.3 the variation of the hydrogen concentration with stress can be obtained. At zero stress we have: o D lD H ðr; 0Þ ¼ lH þ RT ln cH ðr; 0Þ

ð10:7Þ

Equilibrium between a region at zero and a region at nonzero stress is obtained when the chemical potentials for diffusion given by Eqs. 10.3 and 10.7 are equal. The result shows that the concentration in the stressed part of the solid is increased over that in the unstressed part according to   H ph ð r Þ V D D cH ðr; ph Þ ¼ cH ðr; 0Þ exp ð10:8Þ RT where the superscript, D, indicates that this refers only to the diffusible hydrogen in solution. When hydrides are present in the unstressed region, we obtain from Eq. 10.8:   H p h ðr ÞV Do s cH ðr; ph Þ ¼ cH ðr; 0Þ exp ð10:9Þ RT in which csH ðr; 0Þ is the hydrogen concentration at the solvus composition for a/(a ? b)3 phase equilibrium under zero external stress at location, r. This concentration is convenient to use as the reference concentration when solving for the boundary concentrations of hydrogen in solution in local equilibrium with hydrides at the crack tip and in the bulk. It is, therefore, identified with the superscript ‘o’ in Eq. 10.9. When the hydrogen concentration in the a-Zr phase has reached its terminal (solvus) concentration, which could be either in a uniformly stressed solid, or locally at a hydride under stress, ph(r), the chemical potential of hydrogen at the hydride is given by:    H þ p h ðr ÞV Hh lBH ðr; ph Þ ¼ loH þ RT ln cBH ðr; ph Þ  ph ðr ÞV ð10:10Þ Hh is the partial molar volume of hydrogen in the hydride phase (defined in where V Chap. 2).

3

b refers to the hydride phase, which could be either d or c.

10.4

Theory of DHC Growth Rate

351

Local equilibrium is achieved when the chemical potential for diffusion, given by Eq. 10.3, is equal to the boundary chemical potential given by Eq. 10.10. Equating these chemical potentials and combining this with the reference concentration for cD H ðr; ph Þ given by Eq. 10.9, yields—after some algebra—the following boundary (solvus) concentration at location, r:     Hh H ph ðr ÞV ph ð r Þ V B s ðr; p Þ ¼ c ðr; p Þ ¼ c ðr; 0Þ exp exp ð10:11Þ cD h h H H H RT RT where csH ðr; 0Þ is the solvus concentration at zero external stress at r. Equation 10.11 is an important relation as it shows how external stress affects the solvus, csH ðr; 0Þ, obtained at zero external stress. It is equivalent to the relations given by Eqs. 6.42–6.44 in Chap. 6 that account for the effect of differences in accommodation H , Eq. 10.11 predicts that external stress Hh ¼ V energies on the solvus. Note that if V has no effect on the hydrogen solvus composition. From Eq. 10.11 we then obtain the following boundary conditions for the solution of the steady-state flux given by Eq. 10.6: At r = ‘:     H Hh ph ð‘ÞV ph ð‘ÞV D s cH ð‘; ph Þ ¼ cH ð‘; 0Þ exp exp ð10:12Þ RT RT while at r = L cD H ðL; ph Þ

¼

csH ðL; 0Þ exp



   H Hh ph ðLÞV ph ðLÞV exp RT RT

ð10:13Þ

As noted at the start of this section, these two boundary conditions assume that hydrides are able to form at the crack tip and are also present in the bulk, respectively. These equations are used to solve for the steady-state diffusion flux, JH ðsteady stateÞ subject to the boundary conditions given by Eqs. 10.12 and 10.13. The result is DH csH ½EL  E‘ ; rUXZr   ZL H dr ph ðrÞV exp U¼ r RT

JH ðsteady stateÞ ¼ 

ð10:14Þ

‘

where EL and E‘ are defined by Eqs. 10.15 and 10.16 in the following:   Hh ph ðLÞV EL ¼ csH ðL; 0Þ exp ð10:15Þ RT

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E‘ ¼ csH ð‘; 0Þ exp



Hh ph ð‘ÞV RT

 ð10:16Þ

It can be seen from Eqs. 10.14–10.16 that the model predicts that hydrogen diffusion to the crack tip hydride occurs when the hydrogen concentration difference, EL - E‘, is positive. Moreover, it appears from this result as if the flux of hydrogen to the crack tip hydride is driven by an apparent concentration gradient given by the difference in EL and E‘ and not at all by a stress gradient, the latter having the effect of increasing the hydrogen concentration at regions of elevated tensile stress. The reason is that the difference in concentrations obtained from Eqs. 10.15 and 10.16 expresses the net effect of tensile stress on the diffusion flux through: (1) its effect on the concentration of hydrogen in solution, and (2) its effect on the local equilibrium concentration of the solvus given by Eqs. 10.12 and 10.13. Note that there is no effect of stress on the solvus if the volume difference between hydrogen in solution and in the hydride is zero. A common factor in both (1) and (2) is the effect of stress on the concentration of hydrogen in solution, H Þ=ðRT Þ term, which cancels out the same given by the exponential of the ðph ðrÞV term in the boundary concentrations of Eqs. 10.12 and 10.13, leaving only the effect of stress on hydrogen in hydride in the EL and E‘ terms. These two terms are, therefore, not the solvus compositions at the sink (r = ‘) and source (r = L) hydride locations. This can be most easily seen when transfer of hydrogen between the a-Zr and hydride phases does not result in a net volume change. In that case, the solvus composition is unaffected by applied stress and is the same throughout the solid and, therefore, also at the bulk and crack tip hydrides. With the source and sink hydride compositions unaffected by stress, there is no concentration difference between them, but flow of hydrogen to the crack tip hydrides would still occur because the stress gradient drives the diffusible hydrogen to regions of elevated tensile stress. However, any attempt by the system to increase the hydrogen at the crack tip hydride by this stress gradient is continually prevented, because the local equilibrium condition forces the concentration there to remain fixed at the appropriate local solvus composition. Hence, all hydrogen arriving there—driven by the stress gradient to levels equal to the solvus composition (which in this thought experiment is assumed to be uniform throughout the system)—must be absorbed by the hydrides located there, causing them to grow. Now this tensile stress-driven mechanism for hydride growth at regions of elevated tensile stress is always present even if there were a net volume difference upon transfer of hydrogen between the hydride and the a-Zr phases. It is, therefore, the fundamental mechanism driving diffusion of hydrogen to flaws and, hence, DHC. If there is also an effect of stress on the solvus this simply provides an additional, concentration-gradient-driven flux produced by a difference in solvus compositions between the sink and source hydrides that would, from Eqs. 10.15 and 10.16, either aid or impede the stress-gradient-driven flux depending on the sign of the volume difference. An alternative interpretation of the foregoing result—which implicitly supports the foregoing explanation—has been provided by Flanagan et al. [23].

10.4

Theory of DHC Growth Rate

353

In recent reviews of the theory [58, 59], a summary is given of the modifications of this model as experimental results became available and the importance of hysteresis in the solvus became evident. In the final version of the Dutton-Puls model, use was made of the experimentally determined solvi for hydride dissolution and precipitation and the interaction energy of hydrogen in hydride was replaced by the hydride’s interaction energy, D wint inc ; given by Eq. 6.36 in Chap. 6. With these changes the following relations apply for EL and E‘: " # int D w ð L Þ inc EL ¼ cs;H heat exp ð10:17Þ cbH RT and " E‘ ¼

cs;H cool

exp

D wint inc ð‘Þ

#

cbH RT

ð10:18Þ

and cs;cool are the experimentally determined solvi for hydride where cs;heat H H dissolution and precipitation under zero applied stress, respectively, and D wint inc ðLÞ int and D winc ð‘Þ are the hydride interaction energies at the dissolving and precipitating hydrides at L and ‘, respectively with cbH defined as given after Eq. 6.39 in Chap. 6. In converting the hydrogen arrival rate at the crack tip given by Eq. 10.14 to a hydride growth rate (and, hence, given the foregoing assumption, a DHC growth rate) various assumptions were made over the years. For definiteness in discussing the predictions of this model, the last version produced by the writer [56] is reproduced here. In this model it was assumed that the crack-tip hydrides grow with a constant thickness, thyd ; with this thickness taken from experimental observations. The DHC crack growth rate, da/dt, in Stage II, is then given by: da 2pDH ¼ ½EL  E‘  dt XZr thyd qhyd rH UðL; ‘Þ

ð10:19Þ

where a is the length of the macrocrack, qhyd the hydride density, rH is the H/Zr ratio in the hydride and the concentration difference, ½EL  E‘ ; must be positive for DHC to be possible. It has been found, experimentally, that when the applied KI ’s are large enough that Stage II DHC propagation rates apply, the critical length for fracture of the hydrided region at the crack tip is always less than the plastic zone length [96]. In these cases, the stresses on the hydrided region at the crack tip as it grows in length are independent of KI and hence the dominant E‘ term is independent of KI : In the earliest applications of the model it was assumed that the distance, L, was short enough that it would still be within the KI -dependent range of the crack tip stresses. The model then predicts a small, negative dependence on KI stemming from this KI -dependence of the EL term. As discussed in Puls [56], the source location for the bulk hydrogen is somewhat arbitrary and it is not realistic to assume that hydrogen—even when its

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source is from the bulk hydrides—comes only from the hydrides nearest to the crack-tip hydride and that, therefore, L should be a fixed distance given by the inter-hydride spacing. It was also assumed that the model could be used to a good approximation to account for the experimentally determined differences in DHC rate depending on whether the test temperature is approach from below or above. In the latter case, the bulk hydrogen concentration is greater than given by the composition of the solvus for dissolution of bulk hydrides. In that case, the bulk hydrides cannot act as sources of hydrogen in solution to replenish the hydrogen in the bulk that has migrated to the crack tip. It cannot, therefore, remain at a fixed value, but would gradually decrease as the length of the DHC crack increases. However, considering the large region from which hydrogen can flow to the crack tip, it was judged that the decrease in bulk hydrogen concentration would be small. This was also the assumption made recently in numerical evaluations of the model by Cirimello et al. [12] and McRae et al. [43]. Taking account of all of the foregoing considerations and rearranging Eq. 10.19 to more clearly indicate the source of the temperature dependence of the DHC propagation rate, the expressions given in the following Sects. 10.4.1 and 10.4.2 are obtained for the two cases of test temperature approached from below and above, respectively.

10.4.1 Test Temperature Approached from Below It is assumed that the immediately prior temperature history is such that the bulk hydrogen concentration is given by the hydride dissolution solvus:   D wint s; cool " # # inc ð‘Þ " 2pDH cH exp b s; heat cH RT da D wint inc ð‘Þ cH ¼ exp 1 ð10:20Þ dt XZr thyd qhyd rH UðL; ‘Þ cs;H cool cbH RT

10.4.2 Test Temperature Approached from Above It is assumed that the immediately prior temperature history and hydrogen content is such that the bulk hydrogen concentration is given by the hydride precipitation solvus: " # D wint s; cool inc ð‘Þ " " # # 2pDH cH exp cbH RT da D wint inc ð‘Þ ¼ exp 1 ð10:21Þ dt XZr thyd qhyd rH UðL; ‘Þ cbH RT From inspection of Eqs. 10.20 and 10.21, the conclusions given in the following Sects. 10.4.3 to 10.4.6 can be made for the predictions of the model.

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Theory of DHC Growth Rate

355

10.4.3 Dependence on Direction of Approach to Temperature The model predicts that there is a difference in DHC growth rate depending on whether the test temperature is approached from above or below. This difference becomes most obvious as the test temperature approaches TA (equivalent to T3 in Fig. 10.14) which we define as the temperature at which the DHC growth rate has decreased to zero. TA is given by the condition " # s;heat D wint inc ð‘Þ cH ¼1 ð10:22Þ exp cs;cool cbH RTA H This temperature depends on the yield strength, which is also temperature dependent. Equation 10.22 is, therefore, an implicit relation for TA that must be solved numerically or graphically. Shi et al. [77] provided such a solution, formulating the model by assuming from the start that the solvi relations are unaffected by external stress. The earliest systematic experimental studies of this effect were carried out by Ambler [1], although the theoretical interpretation of the results provided at the time—based on one of the earliest version of TSS hysteresis derived by Puls [55]—is not supportable on the basis of the present understanding of the solvus. Ambler’s experimental results also did not show a difference in DHC growth rate depending on direction of approach of test temperature at very low temperature, which is an implicit prediction of the model and which was subsequently confirmed experimentally by Shek [70].

10.4.4 Dependence on KI and Activation Energy In Stage II, the DHC growth rate is independent of KI and the temperature dependence is approximately given by the temperature dependence of the product of the chemical diffusion coefficient and the solvus for hydride precipitation or dissolution. In a recent international study of DHC propagation rate in the axial pressure tube direction of Zr-2.5Nb pressure tube material from various sources [16, 26] the average activation energy for DHC rate was found to be 47.9 kJ/mol H. This is lower than the predicted value obtained from the sum of the solvus  D ; when using, for instance,  sol ; and activation energy for diffusion, Q enthalpy, DH  D by Sawatzky  sol ðTSSP2Þ by Pan et al. [52]—see Table 8.1 Chap. 8—and Q DH et al. [66]—see Table 5.1 Chap. 5—giving 61.6 kJ/mol H. Obviously, from the data in the two tables, other combinations of energies could be used that would give either lower or higher values. Nevertheless, the comparison shows that the predicted value from the model is mostly larger than the measured one. This suggests that there are dependencies on temperature affecting DHC rate that are not captured by assuming that the temperature dependence of the model comes solely from the temperature dependence of the product csH DH : One of these is the

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temperature dependence of the yield strength, discussed in Sect. 10.4.5, while a further possibility could be found in the simplification of assuming steady-state diffusion conditions. This assumption was made to obtain an analytic closed form solution of the hydrogen diffusion rate to the crack tip hydride. A good discussion of some of the many factors affecting the temperature dependence of DHC is given by Jovanovic et al. [28] as well as in the two publications summarizing the results of the IAEA study [16, 26]. Concerning the assumption of steady-state diffusion, Shmakov et al. [84] were able to obtain excellent agreement between the activation energy of the DHC growth rate obtained from experiments and the predictions of their model. These authors’ model is formulated on the same physical principles and conditions as the Dutton-Puls model except that the authors used a piecewise, numerical approach for calculating the flux of hydrogen atoms arriving at the crack-tip hydride rather than imposing steady-state conditions as was done by Dutton and Puls [18]. Using this numerical method, Shmakov et al. [84] were able to reproduce the experimentally observed KI independence of the DHC growth rate as well as the steep increase in the DHC growth rate with KI in Stage I just above KIH . In addition, they were able to mimic the temperature dependence of the drop-off in DHC growth rate at high temperature resulting from the limit imposed by the total hydrogen content in the material, which ultimately makes it impossible for hydride formation anywhere in the material including at the crack tip. The critical temperature, Tc (equivalent to T5 in Fig. 10.14), at which this occurs compares well with that obtained from the Dutton-Puls model by Shi et al. [77] for the same total hydrogen content. (It is important to note that predictions from the Dutton-Puls model of Tc (and also TA) do not depend on the methodology used to solve the diffusion equation.)

10.4.5 Dependence on Yield Strength The DHC growth rate, da/dt, depends on the yield strength. There is a weak dependence through the term, UðL; ‘Þ; and a stronger dependence through the hydride interaction energy term, D wint inc ð‘Þ: Comparison between the predictions of the model and the experimentally determined values for the variation of da/dt along the length of a pressure tube in which there is a corresponding variation in the yield strength shows reasonably good agreement [65]. Later comparisons with other data also show good agreement [12, 43]. The results obtained in the recent IAEA Coordinated Research Program on DHC growth rate measurements [16, 26] summarizes the results obtained by different investigators participating in this research program. A plot containing all of these data is given in Fig. 10.16. This shows that the DHC growth rate increases with increase in yield strength. Hence, since irradiation (up to some saturation fluence) increases the yield strength, the DHC growth rate is faster in irradiated material than in the same material in its unirradiated state [65]).

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Theory of DHC Growth Rate

357

Fig. 10.16 Dependence on yield strength (stress) of normalized DHC growth rate (velocity) of all of the various Zr-2.5Nb materials tested over a wide range of temperatures (from IAEA [26])

10.4.6 Dependence on Total Hydrogen Content Provided the test temperature is chosen so that hydrides are present in the bulk, the model implicitly predicts that the DHC growth rate is independent of total hydrogen content. This is because the rate is dependent, to first order, only on the product of the hydrogen available for diffusion (given by the solvus composition) times its diffusion rate (given by the chemical diffusion coefficient). Hence, increasing the volume fraction of hydride precipitates in the bulk has little effect. (There could be a secondary effect at high hydrogen content produced by fracture of the un-reoriented hydrides, which results in the delamination of the material in the crack growth direction. This delamination has the effect of changing the stress state over a short distance in front of the crack from plane strain to plane stress, lowering the growth rate.) Note, though, that when testing for the effect of hydrogen content on DHC growth rate in Zr-2.5Nb pressure tube material, account must be taken of the changes in microstructure affecting the diffusion rate and yield strength in a/b zirconium alloy material if, during any hydrogenation process that increases the hydrogen content above 100 wppm, this requires a substantial increase in time and temperature compared to the usual values for these parameters when hydrogenating to lower total hydrogen content. From the foregoing results it is remarkable—considering its simplicity—how well the Dutton-Puls model, both qualitatively and quantitatively, captures many of the observed features of DHC growth rate. However, since this model contains no physical conditions that provide for a prediction of the shape and size of the hydrided region as it grows, nor any conditions as to what might be the critical dimensions of this hydrided region for fracture, it cannot be used to predict the experimentally determined threshold stress intensity factor, KIH ; that must be exceeded for DHC initiation to occur. The model does predict that there is a sharp drop-off in growth rate in Stage I with decrease in KI ; but this is a rate dependent, not a crack-initiation-dependent, limit for DHC propagation.

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10.5 General Theory of KIH 10.5.1 Introduction It took some time before the myriad observations of the factors and characteristics of the fracture of the hydrided region in front of a DHC crack and the separate results from tests of fracture strength and toughness of both bulk and embedded hydrides (Chap. 9) could be combined into a viable theoretical model. Thus it was not until the 1990s that the first KIH models were developed by Smith [87, 88] and Shi and Puls [74]. The model provided by Smith is not as complete as that of Shi and Puls and was not further developed by that author. Hence, the following section focuses on the development and predictions of the Shi and Puls KIH model [74]. This expression provides a fracture-based theoretical lower limit for the 1 stress intensity factor for DHC, denoted KIH ; applicable to conditions where there would be no limits to the length that a hydrided region extending from the crack tip could grow.

10.5.2 Fracture Condition of Crack Tip Hydrided Region: KI Versus Lc Relationship The basis of the KIH model is the establishment of a relationship between KI and the critical length, Lc, for fracture of the crack tip hydrided region. Referring to the coordinate system and hydride cluster configuration shown in Fig. 10.17, it was assumed that the crack plane normal stress, rh, in the hydride cluster produced by its transformation strains alone is linearly additive to the crack tip effective stress normal to the crack plane, raeff (subsequently expressed as r\ in this and the original derivation), acting on the region around the crack tip in the absence of any hydrides. For ease of computation it was also assumed that the hydride cluster consists entirely of hydride. The net normal stress in the hydride, then, is the sum of rh and raeff ð r? Þ: This ignores nonlinear coupling effects arising from any possible elastic–plastic responses of matrix and hydride cluster when it forms at the crack tip. The elastic–plastic constitutive behavior of the matrix in the absence of the hydride cluster was, however, accounted for in the sense that it was assumed that there would be a plastic zone in front of the crack tip in which r? arising from the influence of the crack alone has an approximately constant value up to the plastic zone boundary on the crack plane. Fracture of the hydride cluster as it grows in front of the crack tip was assumed to occur when the net normal stress in the hydride cluster, rlocal, becomes equal to or greater than the hydride’s fracture strength, rhf ; as follows: rlocal ¼ r? þ rh  rhf

ð10:23Þ

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General Theory of KIH

359

Fig. 10.17 Coordinate system, schematic of blunted crack (blunting not shown) and location of hydrided (hatched) region at tip of blunted crack; rh is the normal stress (in the Y-direction) in the hydrided region due only to the transformation strain, eTyy ; raeff is the normal stress produced by the crack in the absence of hydride, ry (rys in the text) is the yield strength, 2d is twice the crack tip opening displacement; rPZ is the plastic zone boundary on the crack plane (LPZ in this text) and rlocal is the net stress in the hydride (from Shi and Puls [74]; with permission from AECL)

A further simplification was made by assuming that the thickness, thyd, of the hydride cluster as it grows remains constant and has the shape of a plate. For ease of analysis, it was assumed that this plate is of infinite extent in the direction along the crack front (i.e., plane strain conditions apply for hydride cluster and crack). It was recognized that actual crack tip hydride clusters are often slightly wedge shaped, but it was judged that the results predicted by assuming that the hydride cluster is plate shaped with constant thickness would not qualitatively be too different than if a wedge-shaped hydride cluster had been assumed. The elastic solution derived by Chiu [10] for a plate having the same elastic constants as the matrix and in which only the transformation strain in the plate normal direction is nonzero was used for an analytic expression of the plate normal stress in the hydride and in the matrix at its leading and trailing edges. The solution for the normal stress in the hydride platelet is quite lengthy involving, additionally, trigonometric terms. Hence, a closed-form solution of Eq. 10.23 for Lc as a function of KI with this algebraically complicated expression for rh cannot be obtained except in the limit of an infinitely long hydrided region [74]. In the general case of a hydrided region of finite length, Eq. 10.23 must be solved numerically using an iterative approach since the location at which the net stress is equal to the hydride fracture strength is not known a priori and needs to be assumed. The KI versus Lc relationship shown in Fig. 10.18 was obtained in this way using the same input data as were used to produce this relationship in Shi and Puls (Fig. 4 in [75]). Also plotted in Fig. 10.18 is the variation of the plastic zone length on the crack plane, LPZ, as a function of KI and the variation in 1 the hydride’s aspect ratio, thyd/Lc, also as a function of KI . KIH —calculated using Eq. 10.26 further on—is 4.07 MPaHm in the example given in Fig. 10.18. This compares to a value of *5.4 MPaHm extracted from the plots of Figs. 4 and 5 of Shi and Puls [75], while a direct calculation using Eq. 13 of Shi and Puls [74]

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Fig. 10.18 Plot of KI versus Lc, LPZ and t/Lc using the same input data as in Shi and Puls [75] and with hydride thickness, thyd = 2 lm (thyd is denoted by t in graph) and T = 250 C

yields 5.75 MPaHm. A possible reason for the differences between these values 1 of KIH is given further on. There are two important physical characteristics of the transformation-straininduced normal compressive stress (taken as negative) in this plate-shaped hydride cluster growing at constant thickness (i.e., growing at decreasing aspect ratio). One is that the plate normal stress is nonuniform along its length, having sharply higher negative values inside the hydride at both its leading and trailing edges and being least negative at its midpoint. The second feature is that the overall stress along the length of the hydride, and therefore also its midpoint value, becomes less negative as the hydride cluster grows in length at constant thickness (decreasing aspect ratio). Thus, with the hydride growing in length away from the crack tip starting at a fixed distance from the tip (which was taken as zero by Shi and Puls [74]) the location of the misfit induced minimum normal stress in the hydride cluster moves away from the crack tip and becomes less negative as the hydride cluster grows in length. It was not noted explicitly by Shi and Puls [74] that it is these two characteristic of the hydride cluster as it extends in length that result in there being: • A KI dependence of the critical hydride length as the hydride cluster grows in length outside of the plastic zone along the crack plane, and • A sharp bend in the KI versus Lc relationship, the asymptotic values at each end defining threshold values for Lc and KIH corresponding to the limiting cases, respectively, when either the corresponding applied KI or the critical hydride cluster’s length increases to an effectively infinite value. Theoretical support for the foregoing results obtained with this linear elastic model comes from the results of elastic–plastic finite element calculations by Ellyin and Wu [22] who determined the normal stress in a plate shaped, misfitting solid hydride precipitate located at the crack tip. The misfit strain in these

10.5

General Theory of KIH

361

Fig. 10.19 Distribution of stress normal to the crack plane from (a) hydride expansion (length of hydride platelet is 10 lm and X1 = 0 denotes the crack tip, and (b) variation of hydride length, L, at constant expansion of 0.1 lm. The results are normalized to the yield strength, ro (from Ellyin and Wu [22])

calculations was added to the plate as a fixed displacement and maintained at this level during the elastic–plastic relaxation, whereas the actual situation would be one where the insertion of the hydride plate would impose an initial transformation strain that could then decrease locally at the boundary between hydride and matrix under the influence of the surrounding matrix to a constrained value determined either elastically or elastic–plastically. (How the normal stress inside the hydride platelet differs between the two methods of imposing the transformation strain is shown in Fig. 10.23, further on.) As shown in Fig. 10.19, the magnitude of the stress in front of the crack tip decreases with increased hydride expansion and length and the steep stress gradient at the crack tip either vanishes or changes.

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A compressive stress develops near the leading edge of the hydride plate and for the higher values of hydride expansion plastic deformation with compressive stresses takes place at that location. The maximum tensile stress shifts to the front end of the hydride platelet where fracture of the hydride is predicted to initiate given sufficient magnitude of applied KI . These finite element calculations support the results from the noncoupled elastic calculations of the Shi-Puls KIH model in which it was found that the peak stress in the hydride platelet moves towards the plastic zone boundary. From the foregoing, it can be seen that the threshold values become material parameters at their limiting values. The simple, closed form expressions for these material parameters derived by Shi and Puls [74, 75] are reproduced in Sect. 10.5.3. Numerical solutions are required to obtain KI versus Lc relationships in-between these two limiting conditions since, away from their limiting values, these parameters depend on loading as well as material conditions. An important characteristic of the KI versus Lc dependence given in Fig. 10.18 is its relationship to the variation of the plastic zone length, LPZ, as a function of KI : The plot shows that Lc C LPZ only when KI C 10 MPa while Lc does not markedly increase from its limiting value until KI has decreased to about 6 MPaHm. From this result one can see that the DHC growth rate associated with this KI versus Lc relationship does not reach its KI -independent stage (Stage II) 1 until KI is about 6 MPaHm greater than KIH . Up to that value of applied KI , the DHC growth rate is less than its maximum possible, KI -independent value because the leading edge of the hydride cluster must extend outside the plastic zone to achieve its critical length. Outside the plastic zone the stress gradient, and hence the diffusion flux, drops rapidly with distance from the plastic zone boundary. As discussed in more detail further on, it is this aspect of the relationship between the variation of LPZ and Lc with KI that determines the nature of the various DHC limiting conditions.

10.5.3 Limiting Values of Lc and KIH The limiting value of Lc, which applies to all cases for which Lc B LPZ, was determined by Shi and Puls [75] to have the following relationship: Lc ¼

aEeTyy thyd h i 1 rys  rhf ð1  m2 Þ 12m

ð10:24Þ

where a is a constant and the identity and numerical values for the parameters in Eqs. 10.24–10.26 are given in Table 10.1. This result for Lc was obtained based on the relationship for the plastic zone length along the crack plane given by Banks and Garlick [4]:

10.5

363

General Theory of KIH

Table 10.1 Parameters used in theoretical calculations Parameter Value Molar volume of zirconium Poisson’s ratio Elastic modulus Yield strength Yield strength Hydride fracture strength Hydride cluster thickness rH = H/Zr

Reference

V Zr ¼ 14:00  m3 =mol Zr

Chap. 2

m ¼ 0:436  4:8  104  ½TðKÞ  300 E ¼ 95 900  57:4½ TðKÞ  273 MPa rys ðunirradiated) ¼ 961  1:02  TðKÞ MPa rys ðpre-irradiated) ¼ 1 199:1  1:0957  TðKÞ MPa rhf ¼ 675:7  0:09096  TðKÞ MPa

Shi and Puls [74] Shi and Puls [74] Shi et al. [77] Resta Levi and Puls [60] Shi and Puls [78]

thyd = (3.1/130 9 T (C) – 0.8) 9 10-6 lm

Chap. 10

1.5 for d zirconium hydride

Chap. 8

LPZ ¼

 ð1  2mÞ2 KI 2 2p rys

ð10:25Þ

The parameter, a & 0.45 in Eq. 10.24, is a constant that accounts for the morphology of the hydrided region. It was estimated from the limiting condition, KI ? ? of the full numerical solution, such as the one shown graphically given in Fig. 10.18. Note that the factor 1=ð1  2mÞ in Eq. 10.25 reappears as the factor multiplying rys in Eq. 10.24. The value of the hydride cluster’s transformation strain, eTyy ; in the plate normal direction used in the calculations given in Fig. 10.18 is 0.072, the same as was used by Shi and Puls [75]. Equation 10.24 for Lc predicts that the critical hydride fracture length is linearly proportional to the elastic modulus, the thickness of the hydrided region, the hydrided region’s transformation strain, and the shape of the hydrided region. In general, one would expect the critical length to depend on the hydrided region’s aspect ratio, but the assumption of constant thickness of the hydrided region in the derivation means that this parameter is replaced by just its thickness. Lc also depends inversely on the difference in the yield strength of the matrix (multiplied by a stress state constraint factor) and the hydrided region’s fracture strength. The temperature dependence of Lc thus depends on the temperature dependence of all of the foregoing parameters and it is not obvious from inspection of this relation which of the temperature dependencies of the various parameters dominates. As noted in Sect. 10.3, experimentally it has been found that the striation spacing is a good measure of Lc [69]. A plot of the results of measurements of these striation lengths at different temperatures obtained in the recent IAEA tests [26] combined with those from earlier tests are reproduced in Fig. 10.20. An accompanying plot (Fig. 10.21) of the dependence of the striation spacing versus yield strength shows an approximately inverse linear relationship with this parameter, consistent with the theoretical prediction given by Eq. 10.24 keeping in mind that the temperature dependence of the hydrided region’s fracture strength is not as

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Fig. 10.20 Temperature dependence of striation spacing for pressure tube material for CANDU and RBMK reactors (from IAEA [26])

strong as that of the yield strength. In an application of the Shi-Puls KIH model given in Sect. 10.6.3 further on, plots of the theoretical predictions of the temperature dependences of Lc and Lpz (for KI = 23 MPaHm) given by Eqs. 10.24 and 10.25, respectively, are compared (Fig. 10.34). Interestingly, at that value of KI , Lc has the same magnitude and temperature dependence as Lpz even though from Eq. 10.25, its dependence on yield strength is different. In the plots of Fig. 10.34 both the hydrided region’s thickness and its ‘‘effective’’ transformation strain were given specific temperature dependencies as explained in Sect. 10.6.3. It is evident from Eq. 10.24 that there is a temperature at which Lc ? ? but from the experimental data given in Table 10.1 and the plot of Lc versus temperature in Fig. 10.34, it is seen that this occurs at a temperature higher than the maximum temperature used in the experiments and evaluated in the plots. To obtain a closed-form expression for the theoretically lowest minimum value 1 of KIH (denoted KIH ) Shi and Puls [74] assumed that the crack-tip hydride cluster could grow to effectively infinite length. In this case, the location at which the net tensile stress in the hydride cluster has its maximum value is at the plastic zone boundary on the crack plane. (This had previously also been noted by Smith [87].) Beyond that location there is a rapid decrease in the KI -dependent elastic normal stress produced by the crack alone. In the plastic zone, the stress produced by the crack alone is approximately constant from r = LPZ to 2d for Zr-Nb pressure tube material in the transverse tube direction, where d is the crack tip opening displacement. At r B 2d (see below) the plastic zone normal stress decreases until it becomes equal to the yield strength at the tip of the blunted crack. On the other hand, the stress in the hydride in the direction normal to the hydride platelet produced by its transformation strain, rh, becomes increasingly more compressive (negative) as r ? 0 (i.e., as it approaches the crack tip).. For a blunting crack tip, the distance at which the crack tip normal stress is a maximum is equal to 2d. In the case where the stress–strain relationship is close to being elastic, perfectly plastic, the Hutchinson-Rice-Rosengren (HRR) singular solution [25, 61, 72], valid along the crack plane over a distance between 2 and 10d, gives a normal stress that decreases only slowly with distance towards the plastic zone boundary from this maximum stress location. This is the case for

10.5

General Theory of KIH

365

Fig. 10.21 Dependence of striation spacing on yield strength for various pressure tube materials (from IAEA [26])

unirradiated Zr-2.5Nb pressure tube material for tensile stress in the tube’s transverse direction. Although the HRR expressions give probably the most realistic representation of the stresses between 2 and 10d in front of the crack tip, there does not appear to be a general expression available for the plastic zone length that is consistent with the variation of the plastic zone stresses given by the HRR solution. Rice and Johnson [62] derived an approximate expression for materials having an elastic, perfectly plastic, uniaxial stress–strain response. This plastic zone length expression has been widely used in the literature, but it is limited in its general applicability since it is given only for a Poisson’s ratio of 1/3. More importantly, the HRR stress values at the plastic zone boundary do not connect smoothly to the asymptotic values of the elastic KI -dependent singular solution when the plastic zone boundary given by the Rice-Johnson expression is well within the validity range of the HRR stresses. This is presumably because the elastic KI -dependent singular solution [93] becomes valid only at a distance of r * 100d from the crack tip, while another solution prevails between these two limits. Various results, derived from finite element calculations, exist in the literature for the stress variation over this intermediate distance from the crack tip [50, 90], but these are not available in simple analytic form for arbitrary values of Poisson’s ratio as required in the present calculations. In view of the foregoing considerations, rather than using the HRR expression for the plastic zone normal stress in the present calculations, it was assumed that the crack tip normal stress is constant from a distance r = 2d away from the crack tip to the plastic zone boundary at r = LPZ and given by its value at the plastic zone boundary. Since the plastic zone length given by Rice and Johnson [62] was derived only for a single value of Poisson’s ratio of one-third, the plastic zone expression given by Banks and Garlick [4], which has an explicit dependence on m, was used in Shi and Puls [74, 75] and in the calculations given here. No analytic solution exists for the variation of the crack tip normal stress from the blunted crack tip to r = 2d, except that it is equal to rys at the tip. Having a relationship for this variation is, however, only important at high KI values and at temperatures less than *250 C where the critical hydrided region becomes very short. For these cases a simple linear reduction from its peak value at r = 2d of rys =ð1  2mÞ to its value at r = 0 of rys was assumed.

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With the maximum net tensile stress in the hydride cluster assumed to occur at r = LPZ as Lc becomes effectively infinite in length, a closed-form expression for KIH is obtained. As stated, this approximate expression for KIH —which implicitly assumes that there is no impediment to the hydride cluster’s growth outside of the 1 plastic zone to effectively infinite length—is denoted by KIH ; indicating that it is the theoretically lowest bound value for KIH : Combining the crack tip tensile stress given by Eq. 5 in Shi and Puls [74] with that given by Eq. 10 for rh in Shi and Puls [74], both evaluated at r = LPZ, and equating the resultant sum to rhf ; therefore, 1 yields the following expression for KIH : 1 KIH

8
T T T > = < e þ e þ 2me  xx yy zz 1 KI VZr ð10:32Þ Lmax ¼

s;fi b c > 2p > ; :RT rH cH ln Ho cH

Equation 10.32 is the equivalent of Eqs. 15 or 17 in Shi and Puls [75]6, but accounting formally for the effect of stress on the solvus in the present equation. As is the case for Eq. 10.28, numerically there is little difference in the results for the two equations. From Eqs. 10.28–10.30, it can be seen that TA depends on the respective values of the solvi for precipitation and dissolution, the stress elevation at the crack tip and the magnitude (and, hence, temperature dependence) of the yield strength. In Shi et al. [77] and the present treatment, it also depends on the magnitude (and, hence, temperature dependence) of Poisson’s ratio, since the value of this ratio determines the stress elevation in the plastic zone at the crack tip. Relatively, small changes in the magnitudes of these parameters can change the DHC limit temperature substantially. For instance, in Shi et al. [77] the solvus for precipitation given by TSSP2 was used. With the yield stress for unirradiated pressure tube material given in that paper and the same Poisson’s ratio, TA * 153 C, which increases to *180 C when applying the larger yield strength value used by Ambler [1] (an increase in yield strength of 127 MPa). In fact, there is a large scatter in the yield strength values for Zr-2.5Nb pressure tube material at a given temperature [5, 11]. Thus increasing the yield strength value by a further 70 MPa, which brings it closer to its two-standard-deviation upper bound value, yields TA * 320 C. From this one can see that one possible explanation for the small, but detectable DHC growth rate observed by Simpson and Puls [83] in one specimen at 300 C when this temperature was approached from below may be because this particular specimen had a yield strength value close to the upper bound for this parameter. It should be noted that if one uses TSSP1 as the solvus for precipitation for crack tip hydrides, then TA * 77 C. This is a much lower limit temperature than has been experimentally observed. Previously, it had been thought that TSSP1 is the solvus concentration that needs to be reached to ensure that hydride nucleation

6

There is a misprint in Eq. 15 in Shi and Puls [75]; the expression in the curly brackets should have been squared.

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is possible at the crack tip and would, therefore, be the controlling solvus determining the various limit temperatures. However, it has already been seen in the foregoing sections that this assumption would lead to high temperature threshold values that are not in accord with experimental results. The interpretation proposed by Pan et al. [52], that TSSP1 represents the solvus concentration for hydride nucleation, whereas TSSP2 represents that for hydride growth, was speculative and there is no independent, or direct, proof that this interpretation is correct. In fact, isothermal ingress experiments carried out by Pan et al. (see Chap. 8), show that under isothermal, slow hydrogen-charging conditions hydride formation occurs at approximately the solvus concentration given by TSSP2, which is the solvus for hydride precipitation determined during continuous cooling. The hydride formation solvus determined during isothermal ingress has been designated TSSPI (Table 10.2). TSSPI would seem to correspond more closely to the physical situation in DHC where hydrogen flows up the tensile stress gradient towards the crack tip and increases in concentration at constant temperature until it exceeds the solvus for hydride precipitation in the plastic zone of the crack. One reason that hydride nucleation may be quite easy to achieve, requiring little supersaturation, is because small hydride precipitates are always present at dislocation cores throughout the material (see Chap. 4). In particular, since new, initially undecorated dislocations would be generated in the plastic zone of a crack during first loading and subsequent DHC steps, these dislocation cores could serve as hydrogen traps, ultimately leading to hydride formation. Filling these newly created dislocation cores with sufficient hydrogen atoms so that their concentration (and extent) would exceed the threshold for hydride nucleation, would, however, take time and would likely require a similar time interval as is needed to increase the stress-gradient-driven hydrogen concentration at a crack tip to its solvus composition. This is not a problem during DHC growth because this process is driven by the same diffusion mechanism, but hydride nucleation by this mechanism might not be as effective in aiding hydride nucleation when TSSP is determined during continuous cooling and there are no freshly nucleated sources of dislocations. In this case some under-cooling would be required, thus accounting for the larger value of TSSP (TSSP1). An explanation for the increase in TA of the ice-water quenched specimens is based on the observation that the higher cooling rate results in the hydride precipitates having increased aspect ratios associated with their decreased sizes. This increase in aspect ratios results in these hydrides having increased accommodation energies. Associated with this are correspondingly higher TSSD compositions. In addition, the metallographic evidence is that the hydrides in these specimens are of submicrometer size and therefore less likely to have their elastic accommodation energies reduced by plastic deformation. As a result, the solvus for hydride dissolution for these submicrometer-sized hydrides might be similar in magnitude to that of the solvus for hydride precipitation given by TSSP2. This means that the bulk hydrogen concentration would be similar in magnitude to that given by the solvus for hydride precipitation and the limiting bulk concentration would be determined by the total hydrogen content in the specimen, which also

10.6

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399

means that the DHC limit temperature, TA * TC. The experimental DHC velocity results of Ambler [1] are consistent with this explanation, the most salient being that TDAT for the specimens that were ice-water quenched increases with total hydrogen content in the sample and is always just below the solvus temperature for hydride dissolution. Calculating TA (*TC) using Eq. 10.28 for the ice-water quenched specimens containing 54 and 110 wppm total hydrogen content yields TA * 290.2 and 349.4 C (563.2 and 622.4 K), respectively. These predicted arrest temperatures are close to those that can be inferred from the extrapolated results of Ambler [1] for the case when the test temperature in the furnace cooled specimen is approached from 400 C. In this case, the concentration in the bulk is either given by TSSP2 or by the total hydrogen content, whichever is smaller. The corresponding experimentally determined TDAT temperatures obtained in the icewater quenched specimens are *13–21 C lower than the equivalent experimentally determined TDAT temperatures obtained for furnace cooled specimens cooled to the test temperature from 400 C. These results imply that the solvus for hydride dissolution for the ice-water quenched specimens is significantly higher than that for specimens containing furnace cooled hydrides and is close to, but possibly slightly less than, that given by the magnitude (but not the direction) of TSSP2. This writer is not aware of any TSSD measurements in pressure tube material containing ice-water quenched hydrides to verify this inference from the DHC growth rate results of Ambler [1]. Finally, the higher value obtained for TDAT for the furnace cooled specimens containing 110 wppm H in the specimens that were initially furnace cooled from 300 C (573 K) compared to those cooled from 250 C (523 K) is likely the result of the greater remaining volume fraction of the original, un-reoriented, hydride platelets lying on their natural transverse-axial planes. Simpson and Nuttall [81] first observed that these hydrides had fractured along their lengths during DHC growth, creating long slits in the crack plane along the crack growth direction. These slits have the effect of locally, near the crack plane, breaking up the specimen in its through-thickness direction into thin sheets (i.e., delaminating it), thus reducing the stress elevation in front of the crack to values closer to those that would prevail under plane stress, compared to plane strain conditions. This delamination would occur immediately after each DHC step, since these un-reoriented hydrides are present everywhere in the specimen and do not need to be formed by a process of diffusion. It would result in a lower TDAT because of a reduction in the driving force for hydrogen diffusion in those specimens in which, prior to heating up to the test temperature, they were cooled from a maximum temperature of 250 C (523 K). This maximum temperature is well below the solvus for hydride dissolution (TSSD) for these specimens containing 110 wppm H. Thus, as Ambler [1] also pointed out, only 35 wppm would be precipitated as reoriented hydrides in this case compared to the specimens cooled from 250 C in which approximately 72 wppm would be precipitated as reoriented hydrides, while in the specimens containing 42 wppm the same difference in maximum temperature would result in all of the total H content to be available for reorientation and only slightly less so for the specimens cooled from 250 C. Since there would still

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Fig. 10.36 Plot of KI versus Lc, LPZ, Lmax and thyd/Lc for unirradiated material at T = 225 C with thyd = 3 lm, eTyy = 0.072 and eTxx ¼ eTzz = 0. The line given by the ‘‘bullet’’ markers is a plot of the experimental data of Amouzouvi and Clegg [3]

be a significant volume fraction of un-reoriented hydrides in the specimens containing 110 wppm H that were cooled from 300 C (573 K), TDAT obtained in these specimens would also be subject to some lowering of its stress amplification by the delamination process as a result of the presence of residual, un-reoriented hydrides. This would explain why TDAT in the specimens having an H content of 50 wppm tested by Amouzouvi and Clegg [2] and others was at least as high as 200 C (572 K), which is *20 C higher than the highest TDAT of 182 C (542 K) obtained in Ambler’s [1] experiments. The final aspect that needs explaining of the results from the ice-water quenched specimens is their KI dependence. Figure 10.36 shows the results of the calculated KI versus Lc dependence compared to the measured variation of striation spacing versus KI using values for the crack tip hydrided region’s thickness and transformation strain that brings the calculated, lowest bound Lc value given by Eq. 10.24 close to the experimentally determined one [3]. Also plotted on this graph are the dependences of LPZ and Lmax versus KI (given by Eqs. 10.25 and 10.32, respectively). Lmax is calculated using the precipitation solvus TSSP2 determined by Pan et al. [52] (Table 10.2) and a total hydrogen content in the specimens of 50 wppm. The corresponding TA is 284.4 C. As discussed in the foregoing paragraphs, for these ice-water quenched specimens TA is calculated assuming that the hydrogen concentration in the bulk is determined by the solvus for hydride dissolution, but with a magnitude equal to that given by TSSP2.

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It is seen that the calculated values of Lc increase much more steeply with KI than do the measured ones, which could be because, in actuality, there is a decrease in thickness of the hydrided region at the crack tip with increase in KI ; which has not been included in this model because of the lack of input data. Moreover, the calculated Lc values increase slightly with KI above *15 MPaHm. The dependence of Lc with KI above this value (assuming no change in the hydrided region’s thickness with increase in KI ) depends sensitively on the spatial dependence of the reduction in the normal stress produced by crack tip blunting. This has been approximated in this model simply as being a linear decrease between the value of the normal stress at r = 2d given by rys =ð1  2mÞ and its value at the blunted crack tip (r = 0) given by 1  rys : The reason for this sensitivity to the spatial variation of the crack tip stress between r = 0 and r = 2d is because, at 225 C, the distance 2d is only 2.47 9 2d away from the plastic zone boundary along the crack plane and therefore, at a sufficiently high KI value, the location at which the hydride cluster’s fracture strength is first reached as a function of the hydride cluster’s length is between r = 0 and 2d. This sensitivity to the spatial variation of the normal stress from the tip of the blunted crack to 2d increases with decreasing temperature because the plastic zone length decreases with temperature more than does the location at which r = 2d. With the values for the parameters used, its length becomes equal to 2d at T = 115 C. This suggests that for the purposes of obtaining the relationship between Lc and KI , the present model is not capable of reproducing reality very well below, say, 200 C. Comparison of the plastic zone length with the measured striation spacing shows that below *14 MPaHm the critical hydrided region’s length becomes longer than the plastic zone length. At KI magnitudes smaller than this value some of the growth of the hydrided region before it fractures must occur in the rapidly declining KI -dependent elastic tensile stress field of the crack, resulting in the average growth rate to decrease with decrease in KI from its maximum possible (Stage II) value. This is consistent with the experimental results in Fig. 3 in [2] where the last two points at and above KI * 14 MPaHm show a flattening of the DHC growth rate versus KI curve. The calculated dependence of Lmax versus KI crosses the corresponding calculated Lc curve at KI * 8 MPaHm while it crosses the corresponding experimentally determined curve at *10 MPaHm. The crossing point determines KIH . From Fig. 3 in Amouzouvi and Clegg [3] the experimentally determined value of KIH is *7 MPaHm.

10.6.5 DHC Limiting Conditions: Summary Discussion and Conclusions The DHC limit temperature obtained experimentally for the unirradiated material with the test temperature approached from above and the total hydrogen concentration in the material not a limiting factor for DHC can be rationalized on

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the basis that the hydrided region’s critical length at that temperature is much longer than the plastic zone length at the given applied KI (*20 MPaHm). Using the steady-state Dutton-Puls DHC hydride growth rate model to obtain an approximate value for the crack-tip hydride cluster’s growth rate, it is concluded that the time for the crack tip hydrided region to reach its critical length would be significantly longer than the waiting times used experimentally to determine whether DHC growth had ceased. This suggests also that, although DHC growth rate had not actually stopped completely at that temperature, the slow rate at which it was proceeding was an indication that only a small further temperature increase would have been needed to produce a negligible growth rate. The analyses of the full Shi-Puls DHC fracture model show that increasing the applied KI values to above 30 MPaHm would significantly increase the DHC growth rate since almost all of the hydrided region’s growth would then proceed within the higher stress field of the plastic zone. This is in accord with the early observations by Simpson and Nuttall [81] of DHC growth at high temperatures when using such high KI values. The DHC threshold temperature of 365 C observed in pre-irradiated material containing only 153 wppm hydrogen is concluded to be the result of a combination of two factors. First, to reach the temperature dependent (i.e., high temperature fracture-stress-dependent) threshold means that KI must be sufficiently large to exceed the hydrogen-concentration-limited value of KIH at that temperature. (KIH is increased in these specimens containing only 153 wppm hydrogen because the maximum possible hydrogen supersaturation in the bulk has not been achieved at 365 C.) Second, since it was shown that sufficiently large KI values had been applied, the observed threshold temperature was the result, not of insufficient matrix yield strength compared to the hydrided region’s fracture strength as previously conjectured [60], but of a rapid decline in the DHC growth rate because the critical hydrided region at that temperature extended well outside the plastic zone length. An explanation for the results of the DHC arrest temperature limit, TA, when the test temperature is approached from below, based on the findings of the foregoing analyses, is as follows. First, best agreement with experimentally observed threshold temperatures derived from the Dutton-Puls DHC crack growth rate model is obtained when TSSP2 rather than TSSP1 is used as the solvus concentration that needs to be achieved at the crack tip for DHC to be possible. Support for this conclusion is based on the following considerations. One is that the newly formed dislocations in the plastic zone at the crack tip could provide the initial large local tensile stress values at their cores that facilitate the nucleation and growth of hydrides to sizes just above their critical nucleus values. These hydrides then are the seeds for subsequent auto-catalytic growth of these dispersed hydride clusters into a much larger and denser hydrided region as hydrogen is transported to the crack tip plastic zone. Another is that the isothermal hydrogen ingress experiments have shown that hydride precipitation occurs at values close to TSSP2 rather than TSSP1. These

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403

experiments mimic more realistically the actual process occurring during DHC than do the usual experiments that measure TSS under slow cooling conditions. Given the foregoing, it was then shown that large scatter in TA values is possible and this scatter was a consequence of equally large scatter in yield strength values in pressure tube material. Thus, with the yield strength for unirradiated pressure tube material and Poisson’s ratio given in Shi et al. [77], these authors predicted that TA * 153 C. This increases to *180 C when applying the larger yield strength value used by Ambler [1] (an increase in yield strength of 127 MPa). Increasing the yield strength value by a further 70 MPa, which brings it close to its twostandard-deviaton upper bound value, yields TA * 320 C. One may then conclude from this analysis of DHC data in unirradiated pressure tube material that DHC should be possible in pre-irradiated material up to at least 300 C when approaching the test temperature from below since the yield strength in pre-irradiated material is *250 MPa greater. (A recent study of the DHC arrest temperature in irradiated Zircaloy-2 material by Schofield et al. [67] provides partial support for this prediction.) Of course, the magnitude of the DHC growth rate is expected to be significantly less when the test temperature is approached from below compared to when it is approached from above. The increase in TA of the ice-water quenched specimens is the result of the decreased size of the precipitated hydrides. Associated with this decreased size is an increased aspect ratio, which increases the accommodation energy. The smaller sizes of these hydrides may also have resulted in these accommodation energies being less reduced by plastic deformation. As a result, the solvus for hydride dissolution for these submicrometer-sized hydrides is expected to be similar in magnitude (but not direction of phase transformation) to that of the solvus for hydride precipitation, given by TSSP2. Because of this significant increase in the solvus for hydride dissolution to a magnitude given by TSSP2, the concentration of hydrogen in solution in the bulk has the same magnitude in ice-water quenched specimens as required for hydrides to precipitate and grow at the crack tip (both bulk and crack tip solvus composition being controlled by TSSP2). This means that the DHC arrest temperature, TA, in this case, is the same as the DHC starting temperature, TC, because the bulk hydrogen concentration would be the same, regardless of whether the test temperature is approached from above (giving TA) or below (giving TC). These conclusions are in accord with Ambler’s [1] experimental results.

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3. Amouzouvi, K.F., Clegg, L.J.: Peak amplitude distribution of acoustic emission events during delayed hydride cracking of Zr-2.5% Nb alloys. In: Tyson, W.R., Mukherjee B. (eds.) Proceedings of International Symposium on Fracture Mechanics, pp. 107–110. Pergamon Press, Toronto, Canada (1988) 4. Banks, T.M., Garlick, A.: The form of crack tip plastic zones. Eng. Fract. Mech. 19, 571–581 (1984) 5. CSA (2010) Technical Requirements for the In-service Evaluation of Zirconium Alloy Pressure Tubes in CANDU Reactors. Canadian Standards Association, Mississauga, Ontario, Canada, Nuclear Standard N285.8-10 6. Cann, C.D., Sexton, E.E.: An electron optical study of hydride precipitation and growth at crack tips in zirconium. Acta Metall. 28, 1215–1222 (1980) 7. Carpenter, G.J.C.: The dilatational misfit of zirconium hydrides precipitated in zirconium. J. Nucl. Mater. 48, 264–266 (1973) 8. Causey, A.R., Urbanic, V.F., Coleman, C.E.: In-reactor oxidation of crevices and cracks in cold-worked Zr-2.5 wt.% Nb. J. Nucl. Mater. 71, 25–35 (1977) 9. Cheadle, B.A., Coleman, C.E., Ambler, J.F.R.: Prevention of delayed hydride cracking in zirconium alloys. In: Franklin, D.G., Adamson, R.G. (eds.) Zirconium in the Nuclear Industry—Seventh International Symposium. ASTM STP, vol. 939, pp. 224–240 (1987) 10. Chiu, Y.P.: On the stress field due to initial strains in a cuboid surrounded by an infinite space. J. Appl. Mech. 44, 587–590 (1977) 11. Christodoulou, N., Turner, P.A., Ho, E.T.C.: Anisotropy of yielding in a Zr-2.5Nb pressure tube material. Metall. Trans. A 31A, 409–420 (2000) 12. Cirimello, P., Domizzi, G., Haddad, R.: Influence of metallurgical variables on delayed hydride cracking in Zr-Nb pressure tubes. J. Nucl. Mater. 350, 135–146 (2006) 13. Coleman, C.E.: Effect of texture on hydride reorientation and delayed hydrogen cracking in cold-worked Zr-2.5Nb. In: Franklin, D.G. (ed.) Zirconium in the Nuclear Industry: 5th International Symposium. ASTM STP, vol. 754, pp. 393–411 (1982) 14. Coleman, C.E., Sagat, S., Amouzouvi, K.F.: Control of microstructure to increase the tolerance of zirconium alloys to hydride cracking. Atomic Energy of Canada Report, Report No. AECL-9524. Atomic Energy of Canada Ltd, Chalk River Laboratories, Chalk River, Ontario, Canada (1987) 15. Coleman, C.E.: Cracking of hydride-forming metals and alloys. In: Milne, I., Ritchie, R.O., Karihaloo, B. (eds.) Comprehensive Structural Integrity, vol. 6, Chapter 6.03, pp. 103–161. Elsevier, Oxford (2003) 16. Coleman, C.E., Inozemtsev, V.V.: Measurement of rates of delayed hydride cracking (DHC) in Zr-25Nb alloys: An IAEA coordinated research project. J. ASTM Int. 5, Paper ID JAI101091 (2008) 17. Dunn, J.T., Kakaria, B.K., Graham, J., et al.: CANDU-PHW Fuel Channel Replacement Experience, AECL. Report No. AECL-7538. Atomic Energy of Canada Ltd, Chalk River Laboratories, Chalk River, Ontario, Canada (1982) 18. Dutton, R., Puls, M.P.: A theoretical model for hydrogen-induced sub-critical crack growth. In: Effect of Hydrogen on Behavior of Materials, pp. 512–525. TMS-AIME, New York (1976) 19. Dutton, R., Nuttall, K., Puls, M.P., et al.: Mechanisms of hydrogen induced delayed cracking in hydride forming materials. Metall. Trans. A 8A, 1553–1562 (1977) 20. Domizzi, G., Haddad, R.: Reply to Young S. Kim: Comment on ‘‘Influence of metallurgical variables on delayed hydride cracking in Zr-Nb pressure tubes: by Cirimello et al. J. Nucl. Mater. 350 (2006) 135–146. J. Nucl. Mater. 374, 331–333 (2008) 21. Edsinger, K.: A review of fuel degradation in BWRs. In: Proceedings of International Topical Meeting on Light Water Reactor Fuel Performance. ANS, Park City, UT, USA, pp. 62–179 (2000) 22. Ellyin, F., Wu, J.: Effect of hydride precipitation on the elastoplastic stress field near a crack tip. Acta Metall. Mater. 42, 2709–2717 (1994)

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69. Shek, G.K., Jovanovic, M.T., Seahra, H., et al.: Hydride morphology and striation formation during delayed hydride cracking in Zr-2.5% Nb. J. Nucl. Mater. 231, 221–230 (1996) 70. Shek, G.K.: The effect of material properties, thermal and loading history on delayed hydride cracking in Zr-2.5 Nb alloys. PhD Thesis, University of Manchester, Manchester, UK (1998) 71. Shek, G.K., Metzger, D.R.: Effect of hydrogen concentration on the threshold stress intensity factor for delayed hydride cracking in Zr-2.5Nb pressure tubes. In: Proceedings of PVP2011, 2011 ASME Pressure Vessels and Piping Division Conference, Paper PVP2011-57624 (2011) 72. Shih, C.F.: Tables of Hutchinson-Rice-Rosengren singular field quantities. Materials Research Laboratory, Brown University Report MRL E-147, Providence, RI, USA (1983) 73. Shih, D.S., Robertson, J.M., Birnbaum, H.K.: Hydrogen embrittlement of a-titanium: in-situ TEM studies. Acta Metall. 36, 111–124 (1988) 74. Shi, S.Q., Puls, M.P.: Criteria for fracture initiation at hydrides in zirconium alloys I. Sharp crack tip. J. Nucl. Mater. 208, 232–242 (1994) 75. Shi, S.Q., Puls, M.P.: Dependence of the threshold stress intensity factor on hydrogen concentration during delayed hydride cracking in zirconium alloys. J. Nucl. Mater. 218, 30–36 (1994) 76. Shi, S.Q., Liao, M., Puls, M.P.: Modelling of time-dependent hydride growth at crack tips in zirconium alloys. Mod. Simul. Mater. Sci. Eng. 2, 1065–1078 (1994) 77. Shi, S.Q., Shek, G.K., Puls, M.P.: Hydrogen concentration limit and critical temperatures for delayed hydride cracking in zirconium alloys. J. Nucl. Mater. 218, 189–201 (1995) 78. Shi, S.Q., Puls, M.P.: Fracture strength of hydride precipitates in Zr-2.5Nb alloys. J. Nucl. Mater. 275, 312–317 (1999) 79. Simpson, L.A.: The critical propagation event for hydrogen-induced slow crack growth in Zr2.5%Nb. In: Miller, K.J., Smith, R.E. (eds.) Mechanical Behaviour of Materials, vol. 2, ICM3, pp. 445–455. Pergamon Press, Oxford, UK (1979) 80. Simpson, C.J., Ells, C.E.: Delayed hydrogen embrittlement of Zr-2.5wt.%Nb. J. Nucl. Mater. 52, 289–295 (1974) 81. Simpson, L.A., Nuttall, K.: Factors controlling hydrogen assisted subcritical crack growth in Zr-2.5Nb alloys. In: Lowe, A.L., Parry, G.W. (eds.) Zirconium in the Nuclear Industry, ASTM STP vol. 633, pp. 608–629. ASTM, Philadelphia, PA (1977) 82. Simpson, L.A., Cann, C.D.: Fracture toughness of zirconium hydride and its influence on the crack resistance of zirconium alloys. J. Nucl. Mater. 87, 303–316 (1979) 83. Simpson, L.A., Puls, M.P.: The effects of stress, temperature and hydrogen content on hydride-induced crack growth in Zr-2.5 pct Nb. Metall. Trans. A 10A, 1093–1105 (1979) 84. Shmakov, A.A., Kalin, B.A., Ioltukhovskii, A.G.: A theoretical study of the kinetics of hydride cracking in zirconium alloys. Met. Sci. Heat Treat. 45, 315–320 (2003) 85. Shrire, D., Grapengiesser, B., Hallstadius, L. et al.: Secondary defect behaviour in ABB BWR fuel. In: Proceedings of International Topical Meeting on Light Water Reactor Fuel Performance, ANS West Palm Beach, pp. 398–409 (1994) 86. Shmakov, A.A., Singh, R.N., Yan, D., et al.: A combined SIF and temperature model of delayed hydride cracking in zirconium materials. Comp. Mater. Sci. 39, 237–241 (2007) 87. Smith, E.: The fracture of hydrided material during delayed hydride cracking (DHC) crack growth. Int. J. Press. Ves. Pip. 61, 1–7 (1995) 88. Smith, E.: Near threshold delayed hydride crack growth in zirconium alloys. J. Mater. Sci. 30, 5910–5914 (1995) 89. Smith, E.: The stress in a zirconium alloy due to the hydride precipitation misfit strains: I Hydride region in an infinite solid or a free surface. Manchester University—UMIST Materials Science Centre, Manchester, UK. Unpublished report prepared for Ontario Power Corporation, Toronto, Ontario, Canada (1995) 90. Tracey, D.M.: Finite element solutions for crack-tip behavior in small-scale yielding. Trans. ASME J. Eng. Mater. Tech. 98, 146–151 (1976) 91. Wäppling, D., Massih, A.R., Ståhle, P.: Model for hydride-induced embrittlement in zirconium-based alloys. J. Nucl. Mater. 249, 231–238 (1997)

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92. Waisman, J.L., Sines, G., Robinson, L.B.: Diffusion of hydrogen in titanium alloys due to composition, temperature and stress gradients. Metall. Trans. 4, 291–302 (1973) 93. Williams, M.L.: On the stress distribution at the base of a stationary crack. ASME Trans. J. Appl. Mech. 24, 109–114 (1957) 94. Yan, D., Eadie, R.L.: The near threshold behaviour of delayed hydride cracking in Zr-2.5 wt% Nb. Int. Press. Vess. Pip. 77, 167–177 (2000) 95. Yan, D., Eadie, R.L.: The critical length of the hydride cluster in delayed hydride cracking of Zr-2.5wt% Nb. J. Mater. Sci. 35, 5667–5672 (2000) 96. Yan, D., Eadie, R.L.: An approach to explain the Stage I/II behaviour of the delayed hydride cracking velocity vs. KI curve for Zr-2.5 Nb. Scripta Mater. 43, 89–94 (2000) 97. Yan, D., Eadie, R.L.: Calculation of the delayed hydride cracking velocity vs. KI curve for Zr-2.5 Nb by critical hydride cluster length. J. Mater. Sci. 37, 5299–5303 (2002) 98. Zhenk, X.J., Luo, L., Metzger, D.R., et al.: A unified model of hydride cracking based on elasto-plastic energy release rate over a finite crack extension. J. Nucl. Mater. 218, 174–188 (1995)

Chapter 11

DHC Initiation at Volumetric Flaws

11.1 Introduction Volumetric flaws in pressure tubes are distinguished from cracks in that they occupy, as the name implies, a volume in the material and generally have a blunt flaw tip profile even in the absence of applied tensile stresses. Examples of such flaws in pressure tubes are bearing pad fretting and debris flaws. Bearing pad fretting flaws can occur if a fuel bundle in the pressure tube experiences excessive vibrations or is improperly supported resulting in the fuel bundle spacers wearing indentations over time into the pressure tube. Debris flaws are produced when debris accidentally enters the fuel channel and gets wedged between a fuel bundle and the pressure tube wall, creating a gouge or gouges on the inside of the pressure tube. When such flaws are detected during periodic inspections they need to be assessed for their potential to initiate DHC. The simplest initial approach is to assume, conservatively, that such a flaw is actually a crack having the same length as the maximum depth of the flaw and to calculate the stress intensity factor of this flaw subjected to the pressure tube wall stresses experienced during normal operation. If the calculated stress intensity factor is less than KIH up to the next inspection (the amount of fatigue crack extension must also be considered when the flaw is assessed as a crack), a pressure tube containing such a flaw is considered fit for service. However, often this approach places too onerous a limit on pressure tube life and a less conservative approach is required. Experimental programs were thus started to determine the conditions for DHC initiation from volumetric flaws with different depths and root radii, hydrogen content, thermal cycles, and material conditions, etc. An early example of results from such a program on unirradiated Zr–2.5Nb CANDU reactor pressure tube material is given by Sagat et al. [9].

M. P. Puls, The Effect of Hydrogen and Hydrides on the Integrity of Zirconium Alloy Components, Engineering Materials, DOI: 10.1007/978-1-4471-4195-2_11,  Springer-Verlag London 2012

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11.2 Early Models of Blunt Flaw Assessment The DHC initiation models developed by Shi and Puls [15] and Smith [17, 18, 20] for the determination of the threshold for DHC initiation from blunt flaws were formulated on the basis that fracture initiation occurs once the net normal tensile stress in the hydrided region in front of the flaw becomes equal to or greater than the (micro-) fracture strength of this region. This is the same approach used by these authors in deriving their KIH models [16, 18, 19]. For the blunt flaw analysis, however, the problem of determining the net stress in the hydrided region was simplified in the model of Shi and Puls [15] by using Eshelby’s elastic solutions for misfitting ellipsoidal inclusions [7, 8] to determine the compressive stresses inside the inclusion produced by the hydride’s transformation strains. For these hydride morphologies the compressive stresses are uniform when the transformation strains are also uniform throughout the inclusion. It is shown in Sect. 10.5 that if this result were to pertain for a crack with a hydrided region growing within the plastic zone of the crack, it would not lead to a KIH threshold for fracture, only to a threshold critical length for the hydrided region for a given value of applied KI . However, for blunt flaws in an elastic solid the peak normal stress produced by the flaw alone increases with increase in far-field applied stress, unlike the case of the peak normal stress in the plastic zone in front of a crack in which it remains constant, with only its position shifting with applied external load. Hence for a given blunt flaw geometry and external load there is a unique relationship between the peak normal applied stress produced by a given flaw geometry and the critical length of the hydrided region at which it fractures. Therefore, in this model, the maximum length (more accurately, minimum aspect ratio) to which a hydrided region could grow with time and thermal cycles is the key parameter that determines whether DHC initiation will occur or not. However, it was not necessary in this model to deal with the added complication of a non-uniform transformation-strain-induced stress that is produced in an elastic–plastic material for hydrided regions growing at a crack. As with all of the analytical relationships derived to describe DHC characteristics in the previous chapter, the foregoing models are too approximate to be directly useful in flaw assessments for reactor application. Instead, empirical relationships for the peak flaw tip threshold stress were derived as a function of thermal cycles and time at test temperature from the results of experiments performed on a large number of cantilever beam specimens having different notch depths and root radii. These specimens were subjected to different outer fiber tensile stresses and had sufficient total hydrogen content that hydride formation at the machined flaws under the testing conditions would be possible. Figure 11.1 shows results of such experiments. It is seen that the calculated peak threshold stress ahead of the notch for DHC initiation, given by the dividing line separating data points for which DHC initiation occurred from those for which it did not occur, decreases from a maximum at no thermal cycles to a constant saturation value above a threshold number of cycles [9]. Data such as this was used to develop a safe operating envelope for peak threshold stress as a function of reactor cycles, a schematic of which is shown in Fig. 11.2.

11.2

Early Models of Blunt Flaw Assessment

411

Fig. 11.1 Minimum number of temperature cycles to failure as a function of peak stress (from Sagat et al. [9]; with permission from AECL)

1000

Peak Stress (MPa)

800

600

400

200

0 1

10

100

1000

Allowable Number of Heatup/Cooldown Cycles

Fig. 11.2 Blunt flaw DHC initiation acceptance criteria in trial version of Fitness-For-Service Guidelines (from Scarth [10])

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DHC Initiation at Volumetric Flaws

This enveloping approach was used in the first trial-use version of a fitness for service document [1] that was developed for the assessment of flaws in pressure tubes of CANDU nuclear reactors when such flaws are detected as a result of periodic inspections. However, the reactor cycle dependence of this approach proved to be too restrictive on reactor operation. In addition the peak stress approach had two other weaknesses. One was that, although calculation of the peak normal stress imposed by the external load at blunt flaws required knowledge of the flaw geometry such as, for instance, its root radius, this information was not explicitly contained in the peak threshold stress for DHC initiation. In addition, it was not clear how to deal with the presence of small—but possibly very sharp—secondary flaws at the root of blunt flaws in this type of approach. For this reason another model was developed in which the effect of root radius and the presence of small, sharp, secondary flaws could be explicitly treated. This is the hydride process zone model developed by Scarth and Smith [12–14].

11.3 Hydride Process Zone Approach to Volumetric Flaw Assessment: General Considerations The strategy employed by Scarth and Smith [13] for this approach was to parallel, as much as possible, the methodologies developed to account for nonlinearity in the fracture of ductile solids. Another was that the model should be anchored by experimental input at the two extremes of blunt flaw root radii, ranging from sharp (crack) to smooth (planar) surfaces. Implicit in this approach is that the methodology no longer requires, as do the models for KIH and the early blunt flaw assessment models, knowledge of the microstructural features of the hydrided zone, such as its effective transformation strains, its hydride density, and its morphology and morphological evolution. In the earliest developments of elastic–plastic fracture of ductile metal alloys, accounting for nonlinearity of the fracture process produced by plasticity was done by the introduction of a process zone that was taken to be an infinitesimally thin strip of material emanating from the crack tip in which plastic yielding had occurred while the remainder of the material was assumed to deform elastically. The genesis of this approach is attributed to the combined derivations of Dugdale [5] and Bilby-Cottrell-Swinden [2], referred to by Smith [20] as the DBCS representation. Smith, as described further on, translated this model to the prediction of DHC crack initiation at blunt flaws by assuming that the nonlinearity associated with the formation and fracture of a hydrided region in front of a flaw in an elastic–plastic solid could be similarly represented by a process zone. In the DBCS representation only the process zone strip represents a region of nonlinear response—uniform along the strip—while the remaining part of the solid behaves elastically. These restrictions made it possible to obtain a closed-form analytic solution for the failure threshold criterion of a crack in a ductile solid.

11.3

Hydride Process Zone Approach to Volumetric Flaw Assessment

413

Referring to a schematic of a process zone at a blunt flaw shown in Fig. 11.3 [10], the process zone emanating from the flaw tip along the eventual fracture plane has length, s, on which is imposed a crack tip opening displacement, vT , that depends on the flaw configuration and external loading. The process zone strip is assumed to be held together by cohesive forces, pH ¼ rys , assumed to be uniform along the length of the strip. Failure occurs when vT achieves a critical value, vc . By analogy with the original Mode III results, the following Mode I relationship between crack tip opening, vT , and applied stress intensity factor, KI , is then obtained1 vT ¼

KI2 E0 rys

ð11:1Þ

where E0 ¼ E=ð1  m2 Þ in plane strain fracture mode and rys is the uniaxial yield strength of the material. The critical condition,   vT  vc , for unstable fracture occurs when KI ¼ KIc , and, hence, vc ¼ KIc = E0 rys . This result applies to a long crack where the process zone length, s, at failure is small compared with the crack length and any other geometrical dimensions such as the remaining ligament width. It can be seen from Eq. 11.1 that this model represents one of the earliest and simplest attempts to quantify the plastic blunting of the crack tip as a result of the plastic deformation imposed by the elevated stresses on the material at the tip of the crack. It provided a means of calculating the applied crack tip opening displacement at which unstable failure occurs, assuming that all plastic deformation is localized in an infinitesimally thin strip along the crack plane. The model was, therefore, in this sense, easy to understand intuitively because one can readily see how the length of this strip must increase with increasing value of vT until KIc is reached. However, when this model is recast to apply to the conditions for fracture of a hydrided region in front of a blunt flaw its physical meaning is not so obvious. Reformulating Eq. 11.1 to be applicable to the failure of a hydrided region at the flaw of root radius, q, as illustrated in Fig. 11.3, the following correspondences were made: pH is now the cohesive strength of the hydrided region and KIc  KIH where KIH is the stress intensity facture for DHC initiation at a crack. Strictly speaking the correspondence should have been KIc  KIchyd where KIchyd is the fracture toughness of solid hydride material having the same composition as that of the hydrided region. As the data for KIchyd are, however, limited and the materials from which these are determined do not generally have the same integrity and microstructure as do the embedded hydrided regions formed at flaws, Scarth and Smith [12] replaced it with the experimental value of KIH . The following relationship then holds for a hydrided region process zone at a blunt flaw: vT ¼

1

KI2 E 0 pH

ð11:2Þ

This expression was not actually given in the original derivation but follows from standard fracture mechanics theory for a strip-yield model.

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DHC Initiation at Volumetric Flaws

Leading Edge of Process Zone

Trailing Edge of Process Zone Fig. 11.3 Hydrided region process zone emanating from a flaw tip (from Scarth, Scarth and Smith [10, 12])

This relationship leads to the following two threshold conditions that must both be met for fracture of the hydrided region: vT  vc ¼

2 KIH E 0 pc

ð11:3Þ

where vc is given by Eq. 11.2 when KI ¼ KIH and pH  pc

ð11:4Þ

Note that in this model vT is the relative flaw root displacement produced by the hydrided region alone. The material elsewhere is assumed to deform elastically as in the original model. Hence vT is not, as in the original DBCS formulation, the relative crack tip opening displacement produced by plasticity along the process zone strip. Note also that Eq. 11.3 meets one of the conditions set by Smith for the requirements of the model, which is that it is anchored through experimental input at the two extremes of blunt flaw root radius ranging from zero (crack, KIH ) to infinite (planar surface, pc ). It is seen that there are now two fracture parameters, vc (through KIH ) and pc , but one of them depends on the threshold value of the other through Eq. 11.3. These fracture parameters are also material parameters when they are independent of loading conditions (or length of process zone) and flaw geometry (or root radius). (The other conditions appropriate to the original model must also apply.) The threshold for fracture of the hydrided region and, hence, DHC initiation, is achieved when both conditions are met. This is illustrated schematically in Fig. 11.4.

11.3

Hydride Process Zone Approach to Volumetric Flaw Assessment

415

Fig. 11.4 Schematic of various pH versus vT behavior as a hydrided region develops at a flaw tip (from Scarth, Scarth and Smith [10, 12])

As seen schematically in Fig. 11.4, the hydride process zone model is more complex and, therefore, more difficult to understand in physical terms in comparison to the fracture toughness process zone model because the former model encapsulates an additional fracture threshold condition not contained in the latter one. In addition, it is less straight forward to visualize the physical process as the system approaches the threshold conditions because in the hydride process zone case the material conditions are changing over time while the loading remains constant, whereas for the latter, it is the opposite. The difficulty, then, in visualizing the exact process of arriving at the fracture threshold for the hydride process zone model is that the hydride process zone forms as a result of changes in internal variables that are neither under direct experimental control nor known during the test compared to the loading conditions in the original fracture toughness model. Figure 11.4 is thus an attempt to illustrate this internal process schematically for different possible paths of microstructural changes (sequence of hydrided region formation at the flaw). The figure illustrates the expectation that over time under an external load, pH , decreases continuously as a hydrided region forms and grows in size and length from the root of the flaw. The value of pH ¼ rTH at vT ¼ 0 imposed by the external nominal load, rn , is then the value of the normal tensile stress at the root of the flaw in the absence of any hydrided region (process zone). Guided by the results of coupled finite element calculations such as those illustrated in Chap. 10 for the case of a hydrided region forming at a crack, it is assumed that the gradual formation of a hydrided region results in an average decrease in the net normal tensile stress on the flaw root plane. The reduction in net tensile stress is because of an opposing normal compressive stress produced by the hydrided region alone as a result of its positive transformation strains. (This is an idealized result and may not always be achievable for reasons discussed in Chap. 10.) All

416

11

DHC Initiation at Volumetric Flaws

loads giving pH values larger than that cause DHC initiation while all lower loads result in no initiation even though a substantial hydrided region—indicated by a continuing increase in vT —could have accumulated at the flaw. As noted at the start, the threshold conditions of the process zone model are given in terms of limiting values of the parameters pH (through pc ) and vT (through KIH and pc ) that are experimentally determined material parameters. Thus the model formulated in this way can make no predictions concerning the values of pH for stages of hydrided region development that are not at these limiting values. Formulation of the model in this way is both its strength and its weakness. The strength derives from the fact that no microstructural information concerning the hydrided region is needed to establish failure threshold predictions. The weakness arises when wishing to address situations where these failure threshold values would be too conservative (such as for the formation of hydrided regions under non-ratcheting conditions) or are explicitly required (such as to determine overload failure threshold conditions). In both of these latter cases, additional information needs to be built into the value of pH , usually involving input from microstructural information concerning the hydrided region. Particularly for threshold determinations of overload failure it is currently not clear how best to modify the hydride process zone model to account for the starting state of the hydrided region grown at the flaw prior to overload application (see further on for a brief description of these cases). Support for the viability of the foregoing approach was provided by Smith in two of his earliest publications on this topic. These studies were based on analytical solutions of idealized flaw/hydrided region configurations. In the first of these papers, Smith [17]—employing the same fracture criterion (Eq. 10.23) as used by Shi and Puls [16] in deriving the latter authors’ KIH model—derived analytical expressions for the net normal stress that is developed in a hydrided region growing from (1) a free surface, (2) a sharp crack, (3) a circular hole, and (4) an elliptically-shaped hole. These calculations show that when the flaw geometry, external (far field) loading, and yield strength of the material are such that the stresses produced at the tip of the flaw remain elastic, then, once the applied stress produced by the flaw alone reaches values where the net stress in the hydrided region is everywhere positive, the highest value of this stress along the hydrided region is at the root of the flaw. This justified the peak flaw threshold criterion approach used by Shi and Puls [15] and the simplification in that study of assuming a hydrided region shape for which the normal compressive stress induced by the transformation strain is uniform along the length of the hydrided region. For the flaw, restriction to the linear elastic case makes the calculation for the crack somewhat artificial, while for all other examples this approximation is not too restrictive, particularly in irradiated material. Assuming a hydrided region of constant thickness equal to 2 lm, elastic modulus of 80 GPa (it was assumed that hydrided region and matrix have the same elastic properties), and a normal transformation strain of 0.17, Smith [17] calculated the normal compressive stress in the hydrided region at the flaw root for three different cases: (1) smooth surface, (2) semi-circular flaw of depth 0.8 mm, and (3) flaw of depth 0.8 mm and root

11.3

Hydride Process Zone Approach to Volumetric Flaw Assessment

417

radius of 0.2 mm. The results show that for a hydrided region of length 20 lm, the normal compressive stress at the root does not vary much with type of surface, the stresses being 216 MPa, 219 MPa, and 229 MPa for cases (1) to (3), respectively. This result also shows that the compressive stress in the hydride at the flaw root is essentially independent of flaw profile for a wide range of flaw shapes. In addition, the calculations give normal compressive stresses for a hydrided region length of 200 lm for the three cases (1) to (3) of 21 MPa, 23 MPa and 31 MPa, respectively. Thus for this highly idealized model, the results show that a hydrided region grown to this length from any of these types of locations has effectively negligible compressive normal stress that would oppose the externally imposed tensile stress. In the second paper of this series, Smith [20] derived relationships for the predicted threshold values for two extreme cases; one, when the nonlinearity of the fracture process occurring in the hydrided region—assumed to be driven by plastic deformation processes that are, however, not an essential element of this case—is taken into account; the other, when it is not. The objective of these analyses seems to have been two-fold. In the first instance, it was meant to provide a theoretical explanation for why experimental data for DHC initiation at notches showed that there is a decrease in peak threshold stress with increase in thermal cycles. As noted, these results were incorporated into the first attempt at a blunt flaw DHC initiation acceptance criterion methodology for fitness for service assessments of detected volumetric flaws in pressure tubes as shown schematically in Fig. 10.2. Physically the reason for the effect of thermal cycles is as follows. In material in which flaw geometry, loading conditions, and total hydrogen content are such that a hydrided region forms at a flaw only during the cool down part of a thermal cycle, there is a range of total hydrogen content for which this hydrided region does not completely dissolve during the heat up part of the cycle when returning to the original test (or reactor operating) temperature. As a result, each thermal cycle produces a slightly longer hydrided region. This process of thermal-cycle-induced hydrided-region growth at flaws is called ratcheting [6]. Since the number of such thermal cycles has the effect of increasing the length of the hydrided region in such a way that it decreases its aspect ratio, it decreases the compressive stresses generated inside of it that are produced by its transformation strains alone. The decrease in these compressive stresses means that a smaller applied tensile stress generated by the flaw in the absence of the hydrided region is required before the hydrided region’s facture strength is reached. In the case where a hydrided region is capable of growing during isothermal hold at the test temperature, thermal cycles could also contribute to this ratcheting effect by accelerating the growth rate of the hydrided region compared to its isothermal growth rate. In the second instance, Smith’s [20] analyses served to introduce the concept of the hydrided region’s threshold stress, pc , as the equivalent of the yield strength in the DBCS process zone model. This parameter can be physically interpreted in going from the original DBCS process zone model to the hydrided region process zone model as follows.

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DHC Initiation at Volumetric Flaws

In a process zone model, the simplest way to express the nonlinearity of fracture processes, such as would be occurring in the hydrided region at the flaw tip, is to assume that this region can be replaced by an infinitesimally thin strip behaving nonlinearly up to fracture and failing (loosing its cohesiveness) when the stress in this strip achieves its fracture stress value, pc . At this point the relative displacement across the strip attains a critical value, vc , corresponding to a fracture strain ec ¼ vc =thyd where thyd is the initial thickness of the hydrided region at the position where loss of cohesion occurs. When failure occurs in this strip the stress in this strip would, in general, progressively decrease. However, for mathematical simplicity, it is assumed that pH in the strip would retain its constant value, pc , as the relative displacement, vT , ranges from zero to vc , at which point pc drops abruptly to zero. Smith proposed that in such a model the actual hydrided region at the flaw could also be longer than the strip over which its stress is given by pc and that, therefore, only that length of the hydrided region that is fracturing is assumed to be at stress, pc . A later and final version of the model assumed the process zone length, s, over which the stress is equal to pc to be the length of the hydrided region. However, this process zone length does not generally correspond to the critical hydrided region’s length determined experimentally. Smith [20] shows in these analyses2 that when failure is assumed to occur by plastic straining of the hydrided region, the value of the global threshold stress, pTH , cannot be unique. Smith quantifies this non-uniqueness in terms of the ratio pTH =pc where pc is the threshold stress appropriate to the baseline case of a hydrided region forming at a planar surface of a semi-infinite solid subject to a uniform tensile stress with no allowance being made for the compressive stress induced by hydride formation. Smith finds that the extent to which the ratio, pTH =pc , is greater than unity depends on the ductility of the hydrided region and the stress gradient in the vicinity of the notch (which is increased as the notch root radius decreases). It is also greater when the hydrided region length decreases. Smith’s simple analytical results, therefore, show physically why there is a thermal cycle dependence (i.e., a non-uniqueness of the ratio, pH =pc ) when using the peak threshold stress approach.

11.4 Hydride Process Zone Model: Closed Form Solution To illustrate in a very simple way the application of the process zone methodology to DHC initiation, the situation shown in Fig. 11.5 of a 2D, ‘‘infinitely long’’ blunt surface flaw with a semi-elliptical cross-section in a semi-infinite solid is analyzed. The semi-axes are a and b, and the solid is subjected to an applied nominal tensile

2

The notation for the hydrided region’s stress and relative displacement at the flaw tip (p and v , respectively) used in Smith’s paper is different from what is used here. The notation has been changed to be consistent with Smith’s later notation for these parameters.

11.4

Hydride Process Zone Model: Closed Form Solution

419

Fig. 11.5 Model of a blunt, semi-elliptical flaw with associated process zone in the surface of a semi-infinite solid (from Scarth, Scarth and Smith [10, 12])

stress, rn . There is a uniform stress, pc , in the process zone that represents the hydrided region emanating from the flaw tip. Elastic behavior of the matrix material outside the process zone is assumed. In order to clearly see the interaction between material and geometrical parameters, the analytical Mode III results by Smith [21] are used to simulate the Mode I situation with the shear modulus being replaced by E0 =2, with E0 defined as in Eq. 11.1. These results give the applied nominal stress, rnTH , required for DHC initiation at threshold conditions. The relative displacement, vT , across the process zone at the flaw tip attains the critical value, vc , at threshold conditions, giving:   0  rnTH b=a 2 p E vc þ sec1 exp ¼ ð11:5Þ pc 1 þ ðb=aÞ p½1 þ ðb=aÞ 8 pc a The peak flaw tip stress, rTH , at threshold conditions is given by

420

11

DHC Initiation at Volumetric Flaws

rTH ¼ kt rnTH ¼ ½1 þ ða=bÞ rnTH

ð11:6Þ

where kt is the flaw tip elastic stress concentration factor. With the flaw root radius given by q ¼ b2 =a, substitution of Eqs. 11.3 and 11.6 into Eq. 11.5 gives the threshold peak stress, rTH , as     2  rTH 2 a 1=2 1 p KIH ¼1þ sec exp pc p q 8 p2c a

ð11:7Þ

This equation shows that the threshold peak stress is geometry dependent. Defining the nondimensional parameter, w, by w¼

2 p KIH ; 8 p2c a

ð11:8Þ

then, if w is less than 0.5, Eq. 11.7 can be simplified to give rTH to within 4 % by the expression rTH KIH ¼1þ pc pc ðp qÞ1=2

ð11:9Þ

For a value of pc ¼ 450 MPa and KIH ¼ 4.5 MPaHm, w ¼ 0.5 corresponds to a flaw depth, a, equal to 0.08 mm. For these values of pc and KIH , Eq. 11.9 can therefore be used for flaws with depths not less than 0.08 mm. This simple expression, which does not involve the flaw depth, a, shows that the threshold peak flaw tip stress for DHC initiation increases as the flaw root radius decreases. Based on the analytical Mode III results of Smith [21] the length, s, of the process zone at threshold conditions is given by: hq h i i1=2 s ¼ exp½w  1 þ ðexp½wÞ2  1 a a

ð11:10Þ

where w is given by Eq. 11.8. The foregoing expressions are applicable only when DHC initiation occurs under elastic loading conditions in the surrounding zirconium alloy matrix material. In the context of this idealized model, elastic conditions exist when the peak stress, rTH , is less than the yield stress, rys . From Eq. 11.9, elastic conditions exist when the flaw root radius q is sufficiently large that q[

KIH 2  2 p rys  pc

ð11:11Þ

For smaller flaw root radii, DHC initiation cannot occur without plastic deformation at the flaw tip. In this case, the closed-form process zone approach is extended by simulating the plastic deformation by a plastic process zone. This is shown in the next section (Sect. 11.5).

11.4

Hydride Process Zone Model: Closed Form Solution

For a semi-circular flaw, where a ¼ b, Eqs. 11.5 and 11.7 give   2  rTH 2rnTH 2 1 p KIH ¼ ¼ 1 þ sec exp pc pc p 8 p2c a

421

ð11:12Þ

This expression shows that at threshold conditions, the stresses, rTH and rnTH , increase as the flaw depth decreases. Thus the process zone methodology predicts that for a given peak stress, DHC initiation becomes less likely as the flaw depth decreases, which is physically realistic.

11.5 Hydride Process Zone Model: Effect of Flaw Tip Plasticity The threshold conditions given by Eqs. 11.2–11.4 do not explicitly account for plasticity at the root of the flaw in the matrix prior to hydride formation since the process zone displacement is interpreted as a measure of the expansion of the thickness of the hydrided region. The relative displacement, vT , subsequent to plastic deformation is used as contributing to DHC initiation in terms of comparing with vc. The plasticity effects do, however, factor into the calculation of vT when this parameter is calculated using, for instance, the Engineering Process Zone model or the weight function method for determining the elastic–plastic stress field in front of the flaw that is produced as a result of the flaw alone. As this contribution to vT is not captured in the threshold value, vc , the model would predict that reaching the hydride crack initiation threshold requires the application of higher applied loads than would actually be the case. To eliminate this underestimate, account was taken of the effects of this plasticity on the threshold condition. The effect of prior plasticity was demonstrated by Scarth and Smith [14] using an approximate analytical treatment based on a dual process zone formulation. The purpose of this derivation was to show by a simple analytical model the advantages to be gained by incorporating plasticity into the process zone methodology. Before presenting this derivation we record here some analytical results obtained with the hydride process zone model from an earlier derivation by Scarth and Smith [13] that can also be derived from the developments in Sect. 11.4. In this analysis both the flaw root radius, q, and process zone size, s, are assumed small compared to the flaw depth, a. Under these conditions, the following relation for the process zone displacement, vT , at the flaw tip when the tensile stress within the hydrided region is equal to pH is obtained: " #2 1 pH ðpqÞ1=2 KEFF  ð11:13Þ vT ¼ 0 E pH 2 The parameter KEFF is the effective stress intensity factor for the blunt flaw assuming the crack to have the same planar dimensions as that of the flaw. It is

422

11

DHC Initiation at Volumetric Flaws

evident that this expression satisfies the threshold condition given by Eq. 11.3 when pH and vT are equal to their critical values and q ? 0 and that the elastic peak flaw tip stress rp ¼ pH when vT ! 0. It is also evident from this result that the threshold (critical) value for vT when vT ¼ vc has an explicit dependence on flaw root radius. The lack of such dependence, as noted at the end of Sect. 11.2, was one of the deficiencies of the initial blunt flaw assessment methodology. Equation 11.13 was derived assuming that the material ahead of the flaw tip deforms elastically prior to formation of a hydrided region. With the relation given by Eq. 11.2, it follows from Eq. 11.13 that DHC initiation occurs when KEFF ¼ KTH where KTH is the stress intensity threshold equivalent of rnTH and rTH given by Eqs. 11.5 and 11.6, respectively, of a flaw with small root radius given by 1=2

KTH ¼ ½E0 pc vc 

þ

pc ðpqÞ1=2 pc ðpqÞ1=2  KIH þ 2 2

ð11:14Þ

In Eq. 11.14, the second expression is obtained by using Eq. 11.3 to replace the square bracketed expression in the first relation for KIH . The simplest way to account for stress relaxation by plastic deformation is to represent the relaxation through a uniform process zone within which there exists a tensile stress, ryc, that is limited by plasticity but has a value that is both greater than pc and the uniaxial yield strength, rys, the latter condition accounting for constraint effects at the flaw tip. Thus, superimposing on the hydride process zone another process zone of uniform stress, ryc, that is the result of plastic deformation at the crack tip by the crack alone, the following expression for the plastic zone displacement is obtained: " #2 1 ryc ðpqÞ1=2 KEFF  ð11:15Þ vy ¼ 0 E ryc 2 With the applied elastic displacement, vT ðeÞ, given by Eq. 11.2 with pH ¼ pc , then the elastic–plastic displacement, vT ðepÞ, is completely defined, given by the relation: vT ðepÞ ¼ vT ðeÞ  vy

ð11:16Þ

The ratio of elastic–plastic to elastic flaw root displacement is then given by h i2 ryc ðpqÞ1=2 2 vT ðepÞ pc KEFF  ¼1 h i 1=2 2 vT ðeÞ ryc KEFF  pc ðpq2 Þ

ð11:17Þ

Equation 11.17 shows that when using the following data: q = 10 lm, ryc = 980 MPa, KIH ¼ 4.5 MPaHm, pc ¼ 450 MPa3 and KEFF ¼ 6.0 MPaHm,

3

These two values are presently considered as the nominal lower bound values for KIH and pc for use in assessment methodologies.

11.5

Hydride Process Zone Model: Effect of Flaw Tip Plasticity

423

then vT ðepÞ=vT ðeÞ ¼0.78. Now, for this same set of parameters, Eq. 11.13, with pH ¼ pc together with Eq. 11.3 gives vT ðeÞ=vc ¼ 1.11, while vT ðepÞ=vc ¼ 0.87. This shows then that when flaw tip elastic behavior is assumed, DHC initiation is predicted to occur, whereas when plasticity is taken into account no DHC initiation is predicted. Accounting similarly for crack tip plasticity in the process zone expression for KTH results in 2

312 2 pqp r K c ys 5

IH  þ KTH ðepÞ ¼ 4 4 1  rpysc

ð11:18Þ

Evaluating Eqs. 11.18 and 11.14 with the foregoing parameters we obtain KTH ðeÞ ¼ 5.76 MPaHm whilst KTH ðepÞ ¼ 6.40 MPaHm. Taking the ratio of these two results shows that the threshold stress intensity factor with plasticity taken into account is 11 % larger than the same factor when it is not. In looking at the basis of the process zone threshold condition, it seems, however, that the foregoing relations represent a double accounting of the effect of plasticity. The inconsistency is particularly evident as the flaw root radius, q, in Eq. 11.18 goes to zero. In this limit, KTH would be the stress intensity factor of a crack given by: 2

KTH

312 2 K

IH 5 ¼4 1  rpysc

ð11:19Þ

The relation given by Eq. 11.19 for KTH makes sense only when replacing KIH with KIchyd . Values for KIchyd have been experimentally determined by Simpson and Cann [22] using CT specimens of material consisting of solid ZrHr having a range of composition, r prepared from either a-Zr or Zr–2.5Nb pressure tube material. Appropriate values for KIchyd for use in Eq. 11.19 could be obtained from these results by choosing data for bulk specimen compositions similar to those of the hydride distributions obtained at cracks. Using KIchyd in Eqs. 11.18 and 11.19, instead of KIH , would then correctly account for the effect on KTH of the elastic– plastic deformation of the surrounding zirconium matrix material, because these plasticity effects are then not also implicitly contained in KIchyd as they are in KIH . On balance, however, the use of KIH rather than KIchyd in vc seems the appropriate choice to make, since there exists a large and fairly reliable database of KIH values obtained using material from a large number of different ex-service pressure tubes under a range of service conditions. However, it would seem that the additional implicit inclusion of elastic–plastic deformation in the matrix surrounding the hydrided region at the flaw tip contained in the experimental values of KIH would also need to be accounted for when modifying the threshold conditions for the effect of prior plastic relaxation and creep on the applied value of vT .

424

11

DHC Initiation at Volumetric Flaws

The correction provided by Eq. 11.18 was derived only to get an approximate idea of the magnitude of this effect. It is too approximate and would not be practical if one were also to replace KIH with KIchyd in the threshold condition of Eq. 11.3. For use in the elastic–plastic Engineering Process Zone model, therefore, an empirical correction factor, CV ¼ 1:15 dividing vc given by Eq. 11.3 was introduced instead by Scarth [11]. This factor was, however, initially introduced to account for what was believed to be inaccuracies in the fit of the cubic polynomial equation in representing the flaw tip stress distribution. Later studies by Xu and Scarth [23] using weight functions, however, determined that the cubic polynomial equation did, in fact, provide a reasonable estimate of vT . It was then determined that the reductions in the calculated value of vT as a result of plasticity are not adequately captured when calculating vc from KIH by assuming elastic material behavior in the matrix material outside of the process zone. As a result the foregoing factor of CV ¼ 1:15 was retained, but the reason for its retention has now been realized to be different. Also, since the factor corrects for the plasticity implicit in the use of KIH in the threshold criterion (Eq. 11.3), more accurate methods of estimating the flaw tip stresses, such as those based on weight functions, have not changed the need for its use in correcting the threshold condition.

11.6 Engineering Process Zone Model Various features of the foregoing process zone methodology have been examined by Smith [21] using a variety of simple idealized models that allow for analytical results. These studies were done in order to derive readily visualized relationships between material and geometrical parameters controlling DHC initiation. Unfortunately these analyses have not been externally published. The insights gained from these studies were incorporated by Scarth and Smith [12–14] into a numerical procedure for engineering flaw assessment calculations called the Engineering Process Zone Model. This procedure has evolved over time as data emerged and improvements were made in the calculation schemes of the various steps. From the start, the procedure has involved the following steps: 1. As shown in Fig. 11.6, with rð xÞ given by a generic, elastic flaw tip stress distribution, a cubic polynomial equation is fitted to rð xÞ using the least squares method. 2. Generalized, closed form equations for the stress intensity factor and crack mouth opening displacements, vT , for cracks emanating from the roots of blunt notches covering a wide range of notch geometries are developed. 3. The generalized equations are incorporated into a closed form process zone model based on a polynomial stress distribution to calculate the process zone displacement, vT , at the root of the notch. This step involves the following two procedures:

11.6

Engineering Process Zone Model

425

Fig. 11.6 Illustration of the method of superposition used in the process zone model (from Scarth, Scarth and Smith [10, 12])

a. A fictitious crack of length, s, is subjected to the applied cubic polynomial stress distribution produced by the flaw. b. A fictitious crack of length, s, is subjected to a uniform restraining cohesive stress, pc . c. The end of the process zone away from the flaw tip is modeled as the tip of a fictitious crack at which the sum of the two stress intensity factors, KI , from the two loadings of 3(a) and 3(b) must be zero. This allows evaluation of the length, s, of the process zone from which the opening displacement, vT , at the mouth of the process zone (at the root of the flaw) is evaluated. 4. The equations solved in steps 3 are rearranged to calculate the effective peak stress for DHC initiation. As illustrated graphically in Fig. 11.6, the method makes use of the principle of superposition to calculate the net stress distribution and, hence, length, of the process zone model. This linear superposition of stress fields along the process zone is justified because, outside of the infinitesimal strip of process zone on the crack plane, the material in this model is assumed to behave elastically. Before continuing, it is worth pointing out that from a practical engineering viewpoint one of three possible measures of applied load level can be used to predict threshold conditions for DHC initiation at notches. The first measure of applied load is the peak notch-tip stress, which is generally used for blunt notches with root radii greater than about 50 lm. For notches with smaller root radii, the peak stress level becomes very large, and is not a practical measure of threshold conditions for DHC initiation. The second measure is the effective stress intensity factor, KEFF , as calculated for a crack with the same planar dimensions as the notch. This measure is generally used for notches with root radii less than 50 lm. As the notch root radius approaches zero, which is the situation of an elastic crack, it is intuitive that it is more relevant to use KEFF rather than peak stress to

426

DHC Initiation at Volumetric Flaws

Predictions Based on Lower-Bound KIH Predictions Based on Mean KIH 1.25

/ σnTH(e)

1.00

nTH (ep)

Fig. 11.7 Ratio of predicted threshold nominal stress for DHC initiation with plasticity divided by predicted threshold nominal stress without plasticity for DHC initiation experiments on notched specimens (from Scarth and Smith [14])

11

0.75

0.50

0.25

0.00 0.00

0.10

0.20

0.30

0.40

0.50

0.60

Notch Root Radius, ρ (mm)

characterize threshold conditions for DHC. The third measure of applied load is the nominal applied stress, rn . This measure is generally used when it is not practical to use either peak notch-tip stress or KEFF , such as is the case with a small secondary flaw. For a given notch geometry and material condition, the three measures of applied load are, in principle, interchangeable. The effect on the process zone threshold nominal stresses, rnTH ðepÞ and rnTH ðeÞ with and without plasticity, respectively, derived analytically in Sect. 11.5 was evaluated using Version 2.1 of the Engineering Process Zone model [14]. Figure 11.7 shows how the ratio of these threshold stresses varies with notch root radius. As one would expect, the ratio increases as the notch root radius decreases. Comparisons of the predicted threshold peak stress for DHC initiation, rp  rTH , are given in Figs. 11.8 and 11.9. In these figures, each datum point corresponds to nominally ten specimens at the same notch root radius and peak stress. Predictions of threshold peak stress values based on the lower bound of 7.3 MPaHm for that material and mean value of 9.0 MPaHm are shown. In Fig. 11.8 the applied peak stress values at the root of the notch are calculated assuming elastic response of the material while in Fig. 11.9 they are calculated using an elastic–plastic model. It can be seen from the figures, first of all, that in both cases the process zone model predicts an increasing peak failure threshold stress with decrease in root radius. This trend is also evident in the experimental results, which are obtained from tests using unirradiated Zr-2.5Nb CANDU reactor pressure tube material. The failed data points are all above the predicted lower bound curve. The model predictions demonstrate a conservative trend of under-predicting the experimental results in

Engineering Process Zone Model

427 Experiments: Failed Experiments: Did Not Fail Prediction Based on Lower-Bound KIH Prediction Based on Mean KIH

2500

2000

Peak Notch-Tip Stress,

p

Fig. 11.8 Comparison of predicted threshold peak stress for DHC initiation without plasticity with experimental results in terms of elastic peak stress (from Scarth and Smith [14])

(MPa)

11.6

1500

1000

500

0 0.00

0.10

0.20

0.30

Notch Root Radius,

0.50

0.60

(mm)

Experiments: Failed Experiments: Did Not Fail Prediction Based on Lower-Bound KIH Prediction Based on Mean KIH

(MPa)

1200

1000

Peak Notch-Tip Stress,

p

Fig. 11.9 Comparison of predicted threshold peak stress for DHC initiation with plasticity with experimental results in terms of elastic– plastic peak stress (from Scarth and Smith [14])

0.40

800

600

400

200

0 0.00

0.10

0.20

0.30

Notch Root Radius,

0.40

0.50

0.60

(mm)

both cases but with reduced under-predictions for the elastic–plastic case (Fig. 11.9). There is also a slightly better fit in the elastic–plastic case between data where no failures have occurred with the two prediction curves.

428

11

DHC Initiation at Volumetric Flaws

11.7 Validation of the Engineering Process Zone Model In the foregoing section the hydride engineering process zone model is described and the predictions of the model compared to a limited set of DHC initiation data from blunt flaws. Over the last decade extensive test programs have been carried out to validate the engineering process zone model for a wide range of factors. A summary of these and their status up to 2002 is given by Scarth [10]. The factors considered and their results are summarized in the following subsections.

11.7.1 Flaw Shape The effect of flaw shape for cases where there are large differences such as between bearing-pad fretting flaws and debris fretting flaws was addressed by using separate engineering process zone model parameters for each type of flaw. The need for this arises because the bearing-pad fretting flaw geometry has a lower stiffness in response to a process zone, and therefore has a larger process zone displacement and a lower threshold stress. This has produced different threshold peak stress levels for bearing pad fretting flaws and debris fretting flaws.

11.7.2 Flaw Root Radius As shown in Fig. 11.10, experimental results from the DHC initiation test programs indicate that the threshold peak stress for DHC initiation increases as the flaw root radius decreases. The same trend is obtained with the engineering process zone model, which explicitly takes flaw root radius into account. A possible initial concern in the application of the engineering process zone model to evaluate flaws arose with root radii that are in the range .10 lm. The model initially over-predicted the threshold stress results on specimens with very small root radius notches but this over-prediction disappeared when using the plasticity correction factor, CV ¼ 1:15. Therefore, it was shown that the model can properly deal with blunt flaws having very small root radii.

11.7.3 Flaws with Depth Greater than 1 mm Nearly all of the initial experimental data on DHC initiation from blunt flaws used to validate the engineering process zone model were based on specimens having notch depths B1.0 mm. To determine whether the predictive capabilities of the

Validation of the Engineering Process Zone Model

429

Experiment: Failed Experiment: Did Not Fail Prediction Based on Lower-Bound KIH Prediction Based on Mean KIH

1400

(MPa)

1200

1000

p

Fig. 11.10 Comparison of predicted threshold peak stress for DHC initiation from elastic engineering processzone model analysis with test results in terms of elastic– plastic peak stress from a preirradiated ex-service pressure tube under constant loading (from Scarth, Scarth and Smith [10, 12])

Peak Notch-Tip Stress,

11.7

800

600

400

200

0 0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Notch Root Radius, (mm)

model is only limited to flaws having depths B1.0 mm, data from unirradiated material with depths ranging over flaw depths of 1.0, 1.5, and 2.0 mm with identical root radii of 0.10 mm were produced. These data showed that the effect of notch depths greater than 1.0 mm on the threshold peak stress for DHC initiation is not significant. This demonstrates that the process zone-based acceptance criteria and evaluation procedures are applicable to flaws with depths greater than 1.0 mm.

11.7.4 Flaw Surface Roughness and Secondary Flaw Significance The profiling of machined flaws in the various sets of tests showed that secondary flaws existed in some of the cantilever beam DHC initiation test specimens from pre-irradiated and unirradiated material the data from which formed the technical basis for the blunt flaw, process zone-based DHC initiation acceptance criteria in the CSA Standard [3]. The engineering process zone model explicitly takes into account secondary flaws, and was used to determine the significance of secondary flaws as described by Cui et al. [4]. In this study constant load DHC initiation tests from simulated bearing pad or V-notched debris flaws with or without simulated secondary flaws machined into CB test specimens of material taken from two unirradiated CANDU reactor pressure tubes were carried out. The samples were

430

11

100 µm

DHC Initiation at Volumetric Flaws

100 µm

Fig. 11.11 a Optical micrograph of Sample M-164 in Test Group SF1-1 after 9 cycles, no secondary flaw, applied bending stress 250 MPa (from [4]). b. Optical micrograph of sample M14 in Test Group SF2-2 after 9 cycles, secondary flaw depth 15 lm, applied bending stress 250 MPa (from Cui et al. [4])

subjected to flaw-tip hydride ratcheting thermal cycles. The differing effects of small secondary flaws of different depths on DHC initiation from a primary flaw were demonstrated from the experimental results. Threshold conditions for DHC initiation were established for the primary flaw and secondary flaw geometries and the thermal cycling conditions used in the program and the results compared with predictions from the engineering process zone model. Important conclusions are as follows: 1. The presence of small secondary flaws had a significant impact on the distribution of re-oriented hydrides as shown in Figs. 11.11a, b. The secondary flaw resulted in a local area with high peak stress providing a larger stress gradient as driving force for hydrogen diffusion to the secondary flaw-tip. 2. Secondary flaws 15 lm deep (root radii of all secondary flaws were approximately 8 lm) at either the corner or at the flat bottom of a blunt primary bearing pad flaw did not significantly reduce the threshold nominal stress for DHC initiation from the primary flaw. However, secondary flaws with depths greater than 25 lm could be significant. The results showed that the resistance to DHC initiation decreases as the secondary flaw depth increases. 3. Secondary flaws 15 lm deep at the bottom of V-notched primary flaws reduced the threshold nominal stress for DHC initiation from the primary flaw. This indicates that the reduction in the threshold nominal stress for DHC initiation as a result of a small secondary flaw is higher at V-notch as opposed to at bearing pad primary flaws. 4. The overall agreement between the experimental results and the engineering process zone model predictions was reasonable as shown in Fig. 11.12.

11.7

Validation of the Engineering Process Zone Model

431

350

Applied Bending Stress (MPa)

325 300

SF2-4 (no failure)

275

SF2-3 (no failure)

SF3-4 (6/10 failed)

SF2-2 (no failure)

SF3-3 (9/11 failed)

SF2-1 (no failure)

SF3-2 (no failure)

SF4-1 (11/11 failed)

250

SF1-1 (no failure)

225 200

SF3-1 (no failure)

175 150 0

5

10

15

20

25

30

35

40

45

50

Secondary Flaw Depth (µm) Prediction for Mean KIH

Prediction for lower-Bound KIH

Fig. 11.12 Comparison of predicted threshold nominal stresses for DHC initiation from the engineering process zone model with experimental results from samples that contained a bearing pad primary flaw with or without a secondary flaw at the corner (from Cui et al. [4])

11.7.5 Use of Flaw Tip Plasticity and Creep in the Engineering Process Zone Model Application When using elastic–plastic stress distribution from finite element analysis in the engineering process zone model, it is assumed that the plastic strain distribution is unaltered as the process zone forms (last part of Sect. 11.4). As this is likely not true, because of additional plastic strains as the hydrided region/process zone forms, there will be a redistribution of stress beyond the usual redistribution that would occur in an elastic material. Use of the engineering process zone model in this context is therefore approximate. It is not clear, in the context of the process zone model, whether this change as a result of the elastic–plastic nature of the stress distribution is significant. This is because the process zone model tends to average the stress distribution in the notch-tip region.

11.7.6 Accuracy of the Cubic Polynomial Expression in the Engineering Process Zone Model Application The cubic polynomial stress distribution that is fitted using least squares to the finite element stress distribution provides an expedient but not necessarily optimum fit. Although it was originally believed that this was a major factor in the

432

11

DHC Initiation at Volumetric Flaws

over-prediction of the results of the threshold stress levels for the results of DHC initiation experiments with very small root radii flaws, it was subsequently found, as noted at the end of Sect. 11.5, that this was not the case.

11.7.7 Scatter and Material Variability All pressure tubes tested have exhibited tube-to-tube variability in the DHC initiation properties, including the threshold peak stress. This material variability is taken into account in the process zone model by using the lower-bound value of KIH for axial flaws of 4.5 MPaHm given in the property data section of the present assessment methodology [3]. This lower bound value of KIH was determined from a statistical analysis of a relatively large database of results of tests of material taken from pre-irradiated, ex-service pressure tubes over a range of temperature from a variety of pressure tubes. For pc , there is less statistical reliability because the experiments to measure this parameter are more difficult to perform, as explained in Chap. 9, and therefore there are very few data for this parameter. The main justification for using 450 MPa for pc —obtained from limited data of material from only one ex-service CANDU reactor pressure tube (see Chap. 9) —is that use of this value has resulted in reasonable comparisons of predictions with results from DHC initiation experiments. In addition, this lower bound value is in accord with data obtained from other types of tests, as discussed in Chap. 9. Overall, scatter in the peak stress levels for DHC initiation arises from variability in: • Tube-to-tube differences in DHC initiation properties. • Differences in hydrided region microstructural differences which are likely produced by differences in the original underlying a–Zr phase; such differences were evident even when specimens were taken from the same pressure tube. • Differences in notch geometry produced in the machining of the cantilever beam specimens. These effects of scatter were addressed by testing several specimens at nominally the same conditions.

11.7.8 Hydrogen Isotope Content and Number of Reactor Cooldown/Heatup Cycles The effects of hydrogen isotope content are addressed by determining whether hydride ratcheting conditions exist at the flaw tip, which then determines the appropriate level of the process zone restraining stress, pH . When the hydrogen

11.7

Validation of the Engineering Process Zone Model

433

isotope content is such that hydride ratcheting conditions prevail, then pH ¼ pc is used while when it is not, pH [ pc is used. The reason for the higher critical value of pH in this case is because the length of the fracturing hydrided region would be shorter, resulting in there being some tensile-stress-shielding compressive stresses produced by the hydrided region’s effective transformation strains. The process zone-based acceptance criteria establish conditions for prevention of DHC initiation for any number of reactor heatup/cooldown cycles when the value of the process zone threshold criterion, pc under ratcheting conditions is used in the acceptance procedure with this model. This is because, for these conditions, essentially the lowest possible value for pc is obtained since it represents the fracture strength of an effectively infinitely long hydrided region at the flaw tip. The fracture strength of such a hydrided region could not be any lower no matter what the total number of reactor cooldown/heatup cycles would be.

11.7.9 Above Threshold Conditions Most of the focus on threshold conditions for DHC initiation has been for conditions for which the hydrided region stress, pH , is equal to the threshold stress, pc , for failure of an infinitely long hydrided region growing from a planar surface. These conditions are appropriate to flaw-tip ratcheting situation where a hydrided region is able to grow in size with each reactor heatup/cooldown cycle. However, the process zone methodology can be used for flaw-tip hydride non-ratcheting conditions which correspond to the case when pH [ pc , as shown schematically in the top curve of Fig. 11.4. However, in this case additional information is required to determine the appropriate critical value of pH . This information would need to come either from experimental observations of the hydrided region’s microstructural characteristics, or from theoretical models, to account for the compressive stresses produced by the effective transformation strains of a hydrided region of finite length

11.7.10 Effect of Irradiation The effects of irradiation are taken into account in the process zone model by validation against experimental results of DHC initiation from ex-service, pre-irradiated pressure tube material, as well as through use of the lower bound value of KIH for axial flaws of 4.5 MPaHm in the process zone-based acceptance criteria. This value is consistent with data obtained from measurements on both unirradiated and pre-irradiated material.

434

11

DHC Initiation at Volumetric Flaws

11.7.11 Cyclic Loading and Overload Conditions The basis for the engineering process zone model described in this text is that hydride precipitation and fracture is under a constant applied load. However, cyclic loading occurs during normal reactor operation. As well, a hydrided region overload can occur during reactor heatup subsequent to a cooldown at reduced pressure. A hydrided region overload can also occur during some type of upset pressure transient. A hydrided region overload condition is defined as the loading condition when an applied stress level that is acting on a flaw with an existing hydrided region exceeds the stress level at which this existing hydrided region had precipitated. Current limited experimental results support the position that it is justified to perform an engineering process zone-based flaw evaluation by assuming that flawtip hydride formation occurs at the maximum pressure value during normal operating conditions during which a flaw-tip hydrided region is predicted to exist. Nevertheless the effect of overload needs quantification. The effect of cyclic loading on fatigue damage and hydrided region overloads needs to be considered in combination. This is currently an active area of investigation in the Canadian nuclear industry. Except for some results presented in Chap. 9, it is not dealt with in this text, awaiting possible future updates.

References 1. Anon: Technical Basis for the Fitness-for-service Guidelines for Zirconium Alloy Pressure Tubes in Operating CANDU Reactors. Internal report of CANDU Owners Group (COG) COG Report No. COG-96–651, Revision 0 (1996) 2. Bilby, B.A., Cottrell, A.H., Swinden, K.H.: The spread of plastic yield from a notch. Proc. Roy. Soc. London A 272, 304–314 (1963) 3. CSA: Technical Requirements for the In-service Evaluation of Zirconium Alloy Pressure Tubes in CANDU Reactors. Canadian Standards Association, Mississauga, Ontario, Canada, Nuclear Standard N285.8–10 (2010) 4. Cui, J., Scarth, D.A., Shek, D.K., et al.: Delayed hydride cracking initiation at simulated secondary flaws in Zr–2.5 Nb pressure tube material. Int. J. Pres. Ves. Piping 474, 53–65 (2004) 5. Dugdale, D.S.: Yielding of steel sheets containing slits. J. Mech. Phys. Solids 8, 100–104 (1960) 6. Eadie, R.L., Metzger, D.R., Léger, M.: The thermal ratcheting of hydrogen in zirconiumniobium: An illustration using finite element modeling. Scripta Metall. 29, 335–340 (1993) 7. Eshelby, J.D.: The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. Roy. Soc. London A 241, 376–396 (1957) 8. Eshelby, J.D.: Elastic inclusions and inhomogeneities. Prog. Sol. Mech. 2, 89–140 (1961) 9. Sagat, S., Shi, S.Q., Puls, M.P.: Crack initiation criterion at notches in Zr–2.5Nb alloys. Mater. Sci. Eng. A176, 237–247 (1994) 10. Scarth, D.A.: A new procedure for evaluating flaws in CANDU nuclear reactor pressure tubes for the initiation of delayed hydride cracking. PhD Thesis, University of Manchester, Manchester (2002) 11. Scarth, D.A.: Unpublished. Kinectrics Inc., Toronto, Ontario, Canada (2004)

References

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12. Scarth, D.A., Smith, E.: Developments in flaw evaluation for CANDU reactor Zr–Nb pressure tubes. Int. J. Pres. Ves. Piping 123, 41–48 (2001) 13. Scarth, D.A., Smith, E.: The use of failure assessment diagrams to describe DHC initiation at a blunt flaw. Int. J. Pres. Ves. Piping 412, 63–73 (2002) 14. Scarth, D.A., Smith, E.: The effect of plasticity on process zone predictions of DHC initiation at a flaw in CANDU reactor Zr–Nb pressure tubes. Int. J. Pres. Ves. Piping 437, 19–30 (2003) 15. Shi, S.Q., Puls, M.P.: Criteria for fracture initiation at hydrides in zirconium alloys II. Shallow notch. J. Nucl. Mater 208, 243–250 (1994) 16. Shi, S.Q., Puls, M.P.: Criteria for fracture initiation at hydrides in zirconium alloys I. Sharp crack tip. J. Nucl. Mater. 208, 232–242 (1994) 17. Smith, E.: The initiation of delayed hydride cracking at a blunt flaw. Int. J. Pres. Ves. Piping 62, 9–17 (1995) 18. Smith, E.: The fracture of hydrided material during delayed hydride cracking (DHC) crack growth. Int. J. Pres. Ves. Piping 61, 1–7 (1995) 19. Smith, E.: Near threshold delayed hydride crack growth in zirconium alloys. J. Mater. Sci. 30, 5910–5914 (1995) 20. Smith, E.: Threshold stress criterion for delayed hydride crack initiation at a blunt notch in zirconium alloys. Int. J. Pres. Ves. Piping 68, 53–61 (1996) 21. Smith, E: Unpublished. Kinectrics Inc., Toronto, Ontario, Canada (1996–2006) 22. Simpson, L.A., Cann, C.D.: Fracture toughness of zirconium hydride and its influence on the crack resistance of zirconium alloys. J. Nucl. Mater. 87, 303–316 (1979) 23. Xu, S., Scarth, D.A.: Unpublished. Kinectrics Inc., Toronto, Ontario, Canada (2009)

Chapter 12

Applications to CANDU Reactors

12.1 Introduction We have seen in this book that the earliest studies on the effect of hydrogen and hydrides on the mechanical properties of zirconium alloys were principally concerned with how much overall reduction in fracture toughness the material would experience as a result of an increase in the bulk distribution of zirconium hydride precipitates. Emphasis was placed on the effect that hydride precipitates would have on fuel cladding, because these components are subjected to the highest temperatures and fluences in the reactor and are therefore expected to absorb the most amount of hydrogen as by-product of a corrosion reaction. A similar, but lesser, concern involved other components in the reactor core made of zirconium alloys, the most important of these being the pressure tubes in CANDU and other PHW reactors. In the early 1970s, the focus shifted in the Canadian nuclear industry towards the study of DHC in zirconium alloys. This was prompted by the discovery at that time of some through-wall cracks formed in a small number of cold worked Zr-2.5Nb pressure tubes that had been incorrectly installed (over rolled) in two of the earliest CANDU reactors. It was found that these cracks had formed and propagated as a result of a localized hydride embrittlement process, now universally referred to as DHC. The discovery of these cracks formed the start of an entire new field of study ongoing for approximately 40 years now. That the discovery of DHC in the over rolled pressure tubes would be the start of an ongoing and extensive field of study was not anticipated at the time, since the focus was on understanding the cause of that particular incidence and prevention of its future occurrence in pressure tube reactors. Over time, other sources of potential DHC initiation were discovered in reactors such as the formation of bearing pad fretting flaws and, initially, these discoveries were similarly addressed. However, it gradually became apparent that pressure tubes could (and sometimes did) contain flaws produced during operation of the reactor and that these flaws had the potential to be the source of DHC initiation and growth as the total hydrogen M. P. Puls, The Effect of Hydrogen and Hydrides on the Integrity of Zirconium Alloy Components, Engineering Materials, DOI: 10.1007/978-1-4471-4195-2_12,  Springer-Verlag London 2012

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content in the tubes increases with service time. The recognition of this resulted in the need for a more systematic, structured approach to address these potential concerns. The focus of studies on hydrogen and hydrides then shifted to contributing to the development of an industry-wide methodology for the assessment of fitness for service of pressure tubes. The first official version of such a document was issued in 1991 on a trial-use basis within the Canadian nuclear industry. Throughout the studies on the effects of DHC and bulk hydrides on the integrity of reactor components, two different approaches to generating data and understanding were employed. One was a mechanistically oriented approach focusing on the underlying processes occurring at size scales ranging from the micro to the atomic scale making up the behavior at the macroscopic level, the other an engineering approach focusing primarily on generating data for direct use in applications to pressure tubes while covering ranges of variables relevant to their operating conditions. The struggle, throughout, has been to find the appropriate balance between these two approaches. The early work was very broadly based, as is to be expected when a field of study is first discovered, and both types of research orientations were equally pursued. However, as the knowledge base matured and the direct needs of the operating reactors started to dominate the requirements, a greater emphasis on the engineering approach emerged, a not too surprising outcome in a field of study that was becoming more mature, in an industry in a similar state of development. One of the objectives of this text is to show the importance of these early, more fundamentally based studies in underpinning the technical basis for the data derived from the engineering-oriented approaches. The evidence for success of this underpinning in the Canadian nuclear industry can be found in the generation of a working document for pressure tube fitness for service assessments with official versions now in their second edition [1]. This document meets the practical needs of reactor operators, but also owes its existence and viability to the many underlying, largely self-directed research efforts of individual teams of researchers who, in the early years of the studies in this field, provided the results that now underpin the technical bases of the practical material contained in this standard. The objectives of this chapter then is to provide present workers in the field with an overall description of the engineering contents of this document in as much as it addresses practical concerns associated with DHC and hydride-related fracture toughness issues and show how these are supported by both engineering and mechanistically oriented studies described in this book. Although, in this book, the topic of hydrogen and hydrides in zirconium alloys has almost entirely focused on the potential role that DHC plays in compromising the integrity of pressure tubes, the knowledge and insight gained overall through this intensive study of DHC has also benefited the understanding of the effects of hydrogen and hydrides in other zirconium alloy components and materials. An example is the current effort of an IAEA Coordinated Research Programme on the measurement of DHC growth rate in fuel cladding material [2] that was preceded, as an initial step, by a similar study on pressure tube material [11].

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12.2 Overview of Assessment Approach The general approach for the assessment of DHC initiation at flaws specified in the CSA Standard N285.8 ([1], referred to hereinafter as the Standard) involves a series of steps and decision points, the principles of which have been organized here under the headings related to the main DHC parameters that are required to address these issues.

12.2.1 DHC Initiation at Planar Flaws: KIH When a flaw has been detected in a pressure tube, it can be treated either as a planar flaw (i.e., a crack) or a volumetric (i.e., blunt) flaw. For treatment as the latter, there must be appropriate information regarding the geometry of the flaw that justifies this choice. The requirements as concerns DHC properties of the latter type of flaws are discussed in a subsequent Sect. 12.2.2. When the flaw is treated as planar, it is evaluated to determine its stress intensity factor KI under normal reactor operating conditions. In addition, the hydrogen content of the pressure tube in the bulk near the flaw but away from the influence of its stress field is evaluated. The planar flaw is unacceptable for service if both of the following two conditions apply. • The calculated applied stress intensity factor KI exceeds the lower bound value of KIH . • The bulk hydrogen concentration near the crack is equal to or greater than the solvus composition given by TSSD at reactor operating temperature. It is shown in Chap. 10 from the results of experiments on small test specimens of material taken from both unirradiated and preirradiated ex-service pressure tubes as well as from theoretical predictions that, because of the hysteresis between TSSD and TSSP, the higher temperature limit given by TSSD is a conservative (but not overly so) choice for the threshold temperature above which there would be no DHC initiation for a given hydrogen content. If the bulk hydrogen concentration is less than the TSSD composition during sustained operating conditions, but KI  KIH , a calculation is also required to determine the amount of DHC growth during reactor cooldown. This is required for assessments concerning pressure tube failure as a result of unstable crack propagation and/or plastic collapse. This is discussed further in Sect. 12.2.3. If KI \KIH , the possible extension of the flaw by fatigue over the next service interval must be evaluated and the new KI value of the flaw at the end of the assessment period compared to the lower bound value of KIH . The lower bound, isothermally determined value of KIH in the current Standard is 4.5 MPaHm. A value of this magnitude had already been suggested to apply to unirradiated pressure tube material from very early data published by Coleman and Ambler [3]

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Fig. 12.1 Effect of on time to failure of round notched bar specimens containing 100 wppm hydrogen at 147 C showing lack of cracking below 4.5 MPaHm (from Coleman and Ambler [4]; with permission from AECL)

reproduced by these authors in Coleman and Ambler [4], with some additional data added. The authors suggested that the threshold is at 4.5 MPaHm based on extrapolation of the data for times to failure by DHC of round notch bar specimens of unirradiated Zr-2.5Nb pressure tube material loaded in tension at 147 C, the results of which are shown in Fig. 12.1. Note the rapid increase in times to failure by DHC for these samples as the applied KI approaches the lower bound KIH value. According to analyses using the theoretical models of DHC growth rate and KIH in Chap. 10, this reflects the reduction in the rate of growth of the flaw tip hydrided region as it approaches its critical dimensions (length). Note, though, that the times to failure in these tests for the specimens loaded to low applied KI values are dominated by the time for the first DHC initiation step to occur, which is expected to require the growth of a longer critical hydrided region than in subsequent steps, because initiation is from machined notches that have not been sharpened by fatigue. The KIH theory suggests that there would be little difference in the lower bound KIH values between unirradiated and irradiated material at these test temperatures. This has been substantiated over the years by a statistical analysis of data obtained from a selection of materials taken from preirradiated, ex-service pressure tubes (Gutkin [5]). The analysis by Gutkin shows that there is no statistically significant variation with test temperature for the KIH data, which

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were obtained at test temperatures in the interval from 130 to 250 C. This result is consistent with theoretical predictions over this small temperature interval. Hence, a single value for KIH suffices for application to CANDU reactor pressure tube assessments. There is a large scatter in the data, even from material from the same pressure tube. This is not surprising on theoretical grounds since KIH would be affected by local variations in microstructure that can sometimes be considerable over a 6 m long pressure tube. Unfortunately, very little of the KIH data from these preirradiated ex-service pressure tubes have yet been externally published.

12.2.2 DHC Initiation at Volumetric Flaws: TSSD When a detected flaw is treated as volumetric, it must be evaluated to determine whether the tensile stresses externally applied to it are sufficient to exceed the threshold conditions of this flaw under normal operating conditions as calculated from the process zone model. The Standard allows this to be done on the basis of the calculated value of the applied peak stress at the root of the flaw and comparison of this value with either: • A table of results of calculated threshold peak stress values as a function of flaw root radius obtained from a generic evaluation with the process zone model, or • The threshold peak stress calculated using an explicit process zone evaluation procedure. In either case, the first step is the determination of whether ratcheting conditions apply at the flaw tip. The presence or not of this condition determines whether credit can be taken for the shorter hydrided regions that are observed to form at the crack tip under nonratcheting conditions. Establishing whether ratcheting is possible at a flaw requires three inputs. The first is the total hydrogen concentration in solution in the bulk near the flaw. The second—a material parameter—is the solvus composition for hydride dissolution. The third is a theoretical relation for the increase in hydrogen concentration in solution at the flaw tip produced by the elevated peak hydrostatic stress existing there. Numerical evaluation of this relation also requires an accurate value for the molar volume of hydrogen in solution, described in Chaps. 2, 4 and 6. Comparing the calculated increase in hydrogen in solution with the solvus composition for hydride dissolution determines whether ratcheting conditions prevail at the flaw tip. When the total hydrogen content and stress elevation at the flaw tip is such that not all the hydrides formed during cooldown at the flaw will be dissolved at the flaw tip, then ratcheting conditions must be assumed to apply and the standard process zone threshold conditions must be used. When ratcheting conditions do not apply, application of a constant factor specified in the Standard is permissible that reduces the applied load and, thus, effectively results in an increase in the threshold conditions for DHC initiation.

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It is assumed in the procedure in the Standard for calculating whether hydrides at the flaw tip can dissolve or not that the effect of local applied stress on the solvus composition can be ignored. Justification for this comes from the following considerations. There are two possible effects on the solvus as a result of the presence of stress at the flaw. The first is a direct effect through the difference in interaction energies of hydride and hydrogen in solution at that location. It can be seen from the results in Chaps. 2 and 4 of the molar volumes of hydride (or, approximately, of hydrogen in hydride) and hydrogen in solution that these are approximately equal to each other when all molar volumes are evaluated per mole of hydrogen. Therefore, there is no direct effect of applied local stress on the solvus. An indirect effect might be possible if the local stress affects the elastic–plastic accommodation energy of the hydride, since this would affect the solvus composition, data for which have been obtained only under no applied stress. Only theoretical estimates are available for this effect. Puls et al. [17] used finite element methods to calculate the elastic–plastic accommodation energy of a penny-shaped hydride having isotropic transformation strains and aspect ratio (thickness to length) of 0.1 under externally applied biaxial stress approaching, but slightly less than, the yield strength of the material. The authors found that a net uniaxial external stress changed the accommodation energy by an amount that is not significant in terms of its effect on the solvus composition, considering the usual experimental uncertainties in measurements of this composition. The effect of stress on the accommodation energy when assuming that all of the volumetric misfit strain of the hydride is directed normal to the plane of the disk was not calculated, but would be even less, since the elastic accommodation energy for this choice of transformation strain is about a factor of 6.5 less (see Chap. 8). Lufrano et al. [12] and Lufrano and Sofronis [13] carried out similar calculations as Puls et al. [17] but for spherical hydride particles. For such particle shapes the effect of external stress at high deviatoric applied stress was slightly larger and in the opposite direction (a reduction in accommodation energy versus increase in deviatoric stress) as obtained by Puls et al. [17] for disk-shaped hydrides. This is as expected since a spherical particle with isotropic transformation strain generates only deviatoric stresses in the matrix, which results in plastic deformation when the yield strength is exceeded. It is clear from this discussion that the application of these calculations and the validity of their results to the threshold conditions derived with the process zone model are strongly dependent on the correctness of the thermodynamic relationships for hydrogen in the a-Zr and hydride phases, specifically as it concerns the effect of hysteresis and tensile stress on the solvus and the effect of the tensile stress gradients on hydrogen redistribution. The current state of understanding in this regard is summarized in Chaps. 6–8. It can be concluded from the information provided in these chapters that our understanding of solvus relations remains somewhat incomplete and there are still uncertainties in understanding that will require further studies.

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12.2.3 Planar Flaw Growth to the End of an Assessment Period: DHC Growth Rate For detected planar flaws that meet the criterion KI  KIH but for which the bulk hydrogen concentration near them is less than the TSSD concentration at temperatures corresponding to sustained reactor operating conditions, the increase produced by DHC growth in the flaw dimensions up to the end of the evaluation period needs to be determined. The purpose of this calculation is to determine whether the detected flaw continues to be acceptable against (unstable) crack initiation and plastic collapse. In general, the planar flaw could have a profile and flaw depth that allows it to extend in both the radial and axial pressure tube directions. Hence, in this case, calculation of DHC growth in both directions is required. Requirements for DHC crack growth calculations that involve DHC properties are: 1. Bulk hydrogen isotope concentration near the flaw. 2. TSSD temperature for this bulk hydrogen isotope concentration. 3. DHC growth rate as a function of temperature in both the axial and radial pressure tube directions. 4. Under-cooling temperature below the TSSD temperature during reactor cooldown at which DHC crack growth starts, with the TSSD temperature calculated according to item (2). There are two reactor operating conditions that need to be considered for this calculation: • Sustained hold. • Reactor cooldown transient. For the case of sustained hold, calculations of DHC crack growth in either direction at the temperature and over the time of the hold are only required when the pressure tube temperature during the hold is less than or equal to the TSSD temperature given by item (2) in the foregoing. For the case of a reactor cooldown transient, DHC crack growth is evaluated by integrating the isothermal crack growth rate equation according to the temperature versus time history of the reactor’s cooldown transient. In the Standard, relationships are given, based on the results of laboratory testing, to take account of reactor pressure reductions during cooldown and of the effect of cooling rate on the under-cooling temperature. Also, at some temperature during reactor cooldown the pressure will have been reduced sufficiently so that DHC is no longer possible (KI \KIH ). The temperature at which this occurs is determined and the maximum temperature interval during which the material is susceptible to DHC crack growth during continuous cooling can then be calculated from this. From the foregoing, it can be seen that reliable data concerning DHC growth rate for preirradiated material is required. The studies summarized in Chap. 10 have shown how the DHC growth rate becomes independent of KI over a

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considerable range of KI values between KIH and KIc . The data used in the Standard was generated with applied KI being in this KI -independent range. Theory and experimental data also show that the DHC growth rate is an exponential function of temperature since the exponent is proportional to the product of activation energy and the inverse of the absolute temperature. The activation energy is obtained by a regression fit of the log of the DHC growth rate data versus the inverse absolute test temperature. Separate databases have been generated for DHC growth rates in the radial and axial pressure tube directions, because these two growth rates have different activation enthalpies and preexponentials. The databases for the two directions consist of DHC growth rates measured on preirradiated, ex-service pressure tubes covering a range of irradiation (service) temperatures. For the DHC growth rate in the radial direction, separate fits have been made for measurements at temperatures above and below 100 C. The reason for the separate fits to the data at low and high temperature is because the trend in the data indicates that the decrease in growth rate with decrease in temperature below 100 C is not as great as above this temperature. There is currently no definitive physical explanation for why the decrease in growth rate with temperature becomes less below 100 C (although speculations concerning this are made in Chaps. 4, 5 and 8 involving trapping of hydrogen atoms as hydrides formed in the dislocation cores). However, by making a separate fit to the lower temperature data, a higher growth rate is obtained than would otherwise be given by use of the fit to the higher temperature data. Therefore, use of these two separate fits produces more conservative results for low temperatures. The full range of temperatures over which DHC growth rate measurements were made is from 270 to 40 C. Values of growth rate at higher and lower temperatures than these limits must be obtained by extrapolation of the fits to the respective data sets covering the higher and lower temperature ranges. For the axial DHC growth rate data, there are insufficient data at temperatures below 100 C. Therefore, only a single fit to the data was made and a recommendation made that for calculations at temperatures less than 100 C, conservatively, the growth rate at 100 C should be used. In deterministic assessments, upper bound values are used, whilst the full distribution functions for the fits to the data are used for probabilistic assessments. All of the DHC growth rate data are conservative in the sense that they were obtained by approaching the test temperature from above, with the reduction in temperature and the total hydrogen isotope content in the specimens being such that the maximum concentration of hydrogen in solution at the test temperature is obtained in relation to the solvus temperature differences between hydride dissolution and precipitation. However, no distinction in the present analysis of the data sets has been made to account for different material irradiated at different temperatures (i.e., distinguishing the results from material taken from different locations along the length of the pressure tube1). It is known (see, for instance [18]) that the DHC

1

In a CANDU reactor the temperature along the pressure tube typically ranges from approximately 250 to 300 C.

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growth rate at a given temperature varies with location along the pressure tube. Work is ongoing to account for these differences in fits to the database.

12.2.4 Reactor Core Assessment: Leak Before Break Analysis Leak before break assessments are required in cases where the hydrogen equivalent concentration is greater than the terminal solid solubility for hydrogen dissolution (TSSD) at sustained hot conditions. These leak before break assessments can be done using either a deterministic or a probabilistic approach. An example of a probabilistic approach for leak before break assessment is given in Puls et al. [15]. The parameters required for these assessments are the axial DHC growth rate and fracture toughness (from which the critical crack length in the axial direction can be derived). It should be noted that meeting leak before break conditions up to the design life of the reactor is also a fuel channel design requirement for CANDU reactors. In the context of this requirement, another design requirement generally is that the total hydrogen content must not exceed the TSSD composition along the body of the tube at normal operating conditions to the design end-of-life of the pressure tubes. Nevertheless, leak before break assessments carried out for design assume, conservatively, that DHC growth is possible during normal reactor operation. In a reactor core assessment pressure tube aging effects also need to be taken into account. Aside from the foregoing parameters for leak before break assessments, another parameter of interest is KIH . Understanding the trends in the aging of these parameters also requires aging data for underlying parameters shown to have an effect on the foregoing parameters. The parameters known to be of importance are the strength of the material (yield strength, ultimate tensile strength (UTS), and ductility) and those parameters characterizing microstructural changes such as the evolution with fluence and time at temperature of dislocations (their densities, types, structures, and distributions) and the morphology and composition of the b phase. These microstructural changes affect the mechanical properties, and hydrogen solubility and diffusion rate which, in turn, affect DHC and fracture toughness properties. Theoretical models for hydrogen solvus and DHC properties described in the previous chapters have shown the expected dependences of these parameters on the underlying ones. Therefore, how aging in these underlying parameters affect the DHC parameters plays a role in reactor core assessments. Related to the foregoing property changes, the dimensional changes of the pressure tube as a result of irradiation creep and growth and corrosion must also be monitored since the pressure tube dimensions, particularly its wall thickness, affect the stresses imposed on the pressure tube wall by the pressurized coolant. As discussed in Chap. 10, increasing hydrogen content in excess of the minimum amount needed for DHC to be possible at normal reactor operating conditions, which would result in increasing volume fractions of hydrides in the bulk, are not expected to affect most DHC material parameters except possibly the solvus

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composition, as discussed in Chap. 8. The latter is a new theoretical prediction, however, and needs experimental confirmation. A discussion of aging effects of KIH , axial DHC growth rate, yield strength, and fracture toughness was given by Puls [16] using data derived mostly from studies by Sagat et al. [18]. The results indicate that with only a small increase of fluence from zero (over about one year of reactor operation in the body of the tube) there is a sharp change in value in all of these parameters from their unirradiated values, but little change beyond that. The smallest change with increase of fluence from zero was found for KIH , resulting in a relatively small reduction, ranging from 1 to 2 MPaHm, in the average value of KIH . As with the other parameters, further increases in fluence result in no apparent further changes of the average KIH within the scatter of the data (see Fig. 10.10). The more recent results discussed in Sect. 12.2.1 show, additionally, that there may be little difference in the lower bound KIH values between unirradiated and irradiated material. At any rate, since KIH in unirradiated material is not expected to have lower values than in irradiated material, data for irradiated material are always used for assessment purposes. A more recent summary [14] confirms that the previously determined trends of imperceptible change in the material parameters with fluence after an initial steep change continues to levels of fluence that are expected at the design end-of-life of the pressure tubes. The changes in the fracture properties and DHC growth rate show that these are closely linked with the irradiation-induced microstructural evolution [9, 7], which includes: 1. Formation of dislocation loops and increase in dislocation density, and 2. Decrease in Nb concentration in solution of the metastable b-Zr phase and precipitation of b-Nb in the a-Zr phase. The observed variation in prism plane line broadening is consistent with the observed changes in mechanical properties, specifically the yield strength, UTS, and total elongation [10, 6]. At very low fluence, both the strength and density of a-type dislocation loops increases rapidly until these reach approximately constant values. As the fluence continues to rise, there is a small but steady increase in both. This is most likely the main cause of the long-term variation in mechanical properties of the UTS, as shown in Fig. 12.2. At fluence levels [20 9 1025 n/m2, there is a clear trend towards saturation in the c-component dislocation structure, and there is a corresponding saturation in mechanical properties. The results of the b-phase analysis show a strong tendency to saturation in Nb concentration after a fluence of about 5 9 1025 n/m2 and therefore one can also conclude that any effect of the state of the b-phase on DHC properties will have saturated. The state of the b-phase is strongly dependent on flux [8], and there is a possibility that the dislocation loop density is also dependent on flux. However, for the range of fluxes applicable to the data set reported by Pan et al. [14], fluence and not flux seems to be the most dominant factor controlling the irradiation damage densities. Accordingly, there is no evidence showing the presence of accelerated change or breakaway in microstructure with irradiation up to the design end-of-life fluence.

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Fig. 12.2 Transverse ultimate tensile strength (UTS) and total elongation (TE) at 240 C versus fast neutron fluence (E [ 1 MeV) for Erable I and II specimens in the OSIRIS test reactor (from Pan et al. [14]; with permission from AECL)

The factors governing the fracture mechanism, such as the increase in the number of void nucleation sites with irradiation and the increase in a-type dislocation density, occur at a lower fluence \0.4 9 1025 n/m2 and thus the reduction in fracture toughness also occurs in the early lifetime of a pressure tube as shown in Fig. 12.3. After the initial transient, the fracture toughness remains approximately constant up to the maximum fluence of 26.1 9 1025 n/m2 (E [ 1 MeV). The small but steady increase in the c-component dislocation density does not affect the fracture toughness, but there is an accompanying small gradual increase in tensile strength. The dependence of the DHC growth rate with fluence is plotted in Fig. 12.4. The DHC growth rate in irradiated pressure tubes is affected by (Hosbons et al. [10] and Sagat et al. [18]): 1. Irradiation-induced hardening, causing the DHC growth rate to increase with increasing transverse yield strength; and 2. State of the decomposition of the b phase. Concerning item (2), during neutron irradiation the b-Zr phase, which is partially decomposed as a result of the stress relief treatment that is the last step in the manufacture of the pressure tube, reconstitutes and approaches a steady-state condition representing a balance between thermal decomposition and radiationinduced mixing. For the samples in the study by Pan et al. [14], the neutron flux and temperature were such that the thermally induced decomposition was reversed during irradiation. This effect of irradiation can be inferred from the results of the

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Fig. 12.3 Initial crack growth toughness, dJ/da, and maximum load toughness, Jml, at 240 C versus fast neutron fluence (E [ 1 MeV) for Erable I and II specimens irradiated in the OSIRIS test reactor (from Pan et al. [14]; with permission from AECL)

Fig. 12.4 Variation in radial DHC velocity at 240 C versus fast neutron fluence (E [ 1 MeV) for Erable I and II specimens irradiated in the OSIRIS test reactor (from Pan et al. [14]; with permission from AECL)

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Fig. 12.5 Nb concentration in solution of the b phase (determined from bcc lattice parameters) as a function of fast neutron fluence (E [ 1 MeV) for Erable I and II specimens irradiated in the OSIRIS test reactor at about 250 C (from Pan et al. [14]; with permission from AECL)

measurements of the decrease in Nb concentration in the b phase, consistent with an increase in volume fraction of the b-Zr phase, as shown in Fig. 12.5. As discussed in Chap. 5, hydrogen diffuses faster in the bcc b-, than in the hcp a-Zr phase. After extrusion, subsequent heat treatment of the pressure tube results in decomposition of the b phase. One might then expect that the diffusivity of hydrogen is decreased from its value in the as-extruded state. The results of the study by Pan et al. [14] show that the reduction in the concentration of Nb in the b phase with increasing fluence can be attributed to the increase in the volume of the bcc component of the b phase filament [6]. This increase, in turn, increases the cross section for fast diffusion of hydrogen in the b phase. Since, as shown in Chap. 10, the DHC growth rate is proportional to the rate of hydrogen diffusion, it first decreases after heat treatment (autoclaving), but then increases as it is being irradiated. The increased strengthening of the a phase as a result of irradiation damage also contributes to the increased partial reconstitution of the b phase. Thus, the variation of the DHC growth rate with increasing fluence is complicated by the partial reconstitution of the b phase. Given that there is a sharp transient of transverse strength at very low fluence of \0.4 9 1025 n/m2, as shown in Fig 12.2, but the DHC growth rate remains relatively unchanged, one can conclude that DHC growth rate is not only controlled by irradiation-induced hardening occurring at low fluences but also by the reconstitution of the b phase that occurs over a larger fluence range. The sharp

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Applications to CANDU Reactors

increase in the hardening of the matrix material affects the DHC growth rate through changes in the crack tip hydrostatic stress (which is proportional to the yield strength) and changes in the solvus composition, while the rate of hydrogen diffusion is affected through the more gradual change in the state of the b-phase decomposition. The net effect is that the DHC growth rate gradually approaches a saturation level at fluences [5 9 1025 n/m2 as shown in Fig. 12.4, rather than the sharp change at very low fluence experienced by tensile strength. The result is consistent with the similarity of the fluence dependencies of the curves of DHC growth rate and Nb concentration. These results by Pan et al. [14] confirm that after an initial transient the fracture toughness remains constant and the DHC growth rate gradually increases with fluence up to the design end-of-life fluence. There is no evidence of an accelerated decrease in the former or an accelerated increase in the latter.

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