High Temperature Oxidation and Corrosion of Metals, Volume 1 (Corrosion Series)

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ELSEVIER CORROSION SERIES Series Editor: Tim Burstein Department of Materials Science and Metallurgy, University of Cambridge, Cambridge, UK VOLUME 1:

High Temperature Oxidation and Corrosion of Metals – by David John Young

HIGH TEMPERATURE OXIDATION AND CORROSION OF METALS

By DAVID JOHN YOUNG

Amsterdam Boston Heidelberg London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo

Elsevier Linacre House, Jordan Hill, Oxford OX2 8DP, UK Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands First edition 2008 Copyright r 2008 Elsevier Ltd. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://www.elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-08-044587-8 ISSN: 1875-9491

For information on all Elsevier publications visit our website at books.elsevier.com

Printed and bound in Great Britain 08 09 10 11 12 10 9 8 7 6 5 4 3 2 1

PREFACE

Almost all metals and alloys of technological interest oxidize and corrode at high temperatures. However, the nature of the reaction products and the rates at which metal surfaces are degraded vary widely, and a capacity for prediction is highly desirable. This book is concerned with providing a fundamental basis for understanding the alloy–gas oxidation and corrosion reactions observed in practice and in the laboratory. Its purpose is to enable the prediction of reaction morphology, kinetics and rate as a function of temperature and the compositions of both alloy and gas. The term ‘‘oxidation’’ is used in a generic sense, for any chemical reaction which increases the metal oxidation state by forming a compound such as an oxide, sulfide, carbide, etc. Alloy oxidation reactions can be conceived of as occurring in three stages. Initially, all alloy components in contact with a hot gas are likely to react simultaneously. Subsequently, thermodynamically more stable compounds replace less stable ones, and a state of near equilibrium is locally approached. The reacting system can then be modelled as a series of spatially adjacent local equilibrium states which vary incrementally in reactant chemical potentials. During this stage, the reaction morphology and composition distribution are invariant with time. Ultimately, this ‘‘steady state’’ is lost, and all reactive alloy components are consumed in a final breakdown stage. Successful alloys are those which evidence lengthy periods of slow, steadystate reaction. For this reason, considerable emphasis is placed on analysing the underlying local equilibrium condition and testing its applicability to particular metal or alloy-oxidant systems. When an alloy–gas reaction is at steady state, the constant composition profile developed through the reaction zone can be mapped onto the system phase diagram as a ‘‘diffusion path’’. Frequent use is made of these paths in understanding reaction product distributions and in predicting, or at least rationalizing, reaction outcomes. Analysis of the alloy oxidation problem requires a multidisciplinary approach. Physical metallurgy, materials science and physical chemistry provide the tools with which to dissect alloy phase constitutions and their transformations, oxide properties and chemical kinetics. Deliberate emphasis is placed on the use of chemical thermodynamics in predicting oxidation products and describing solid solution phases. Equal attention is paid to the detailed understanding of defect-based diffusion processes in crystalline solids. The introductory Chapter 1 indicates how these various disciplines can contribute to the analysis. The lengthy Chapter 2 reviews the thermodynamic, kinetic and

ix

x

Preface

mechanical theories used in this book. It also contains tabulated data and refers to Appendices relevant to diffusion. Appendix A lists representative alloy compositions. After these preliminaries, the book is arranged in a sequence of chapters reflecting increasing complexity, which equates with greater system component multiplicity. An analysis of the reaction between pure metals and single oxidant gases is followed by a discussion of metal reactions with mixed oxidant gases and then, in Chapters 5–7, an examination of alloy reactions with a single oxidant. Much of this discussion is based on the early work of Carl Wagner, which still provides a good conceptual framework and in several cases a useful analytical basis for quantitative prediction. However, as will be shown, increasing system complexity is accompanied by a weakening in theoretical completeness. The problems arise from multicomponent effects and from microstructural complexity. Consider first the effect of increasing the number of alloy components. A steady-state reacting system consisting of a binary alloy and a single oxidant can be modelled in a two co-ordinate description of both thermodynamics and diffusion kinetics, provided that temperature and pressure are constant. Substantial thermodynamic and diffusion data are available for many such systems, and these are used in developing diffusion path descriptions. Increasing the number of alloy components leads, however, to chemical and structural interactions among them, rendering the experimental problem much less tractable. In the absence of the requisite extensive thermodynamic or diffusion data, the Wagner theory cannot be applied. Instead, higher order alloys are discussed from the point of view of dilute addition effects on the behaviour of binaries. Wagner’s theory is based on lattice diffusion. However, the transport properties of slow growing oxides are largely determined by their grain boundaries and, in some cases perhaps, microporosity. Additional alloy components can affect both the oxide grain size and the diffusion properties of the grain boundaries. Description of these phenomena is at this stage largely empirical. The latter part of the book is concerned with the effects of other corrodents and temperature variations. Chapters 8 and 9 deal with sulfur and carbonbearing gases. The very rapid diffusion rates involved in sulfidation and carburization makes them potentially threatening corrosion processes in a number of industrial technologies. Of fundamental interest are the complications arising out of the complex gas phase chemistries and the generally slow homogeneous gas phase reactions. It becomes necessary in discussing the behaviour of these gas mixtures to consider the role of catalysts, including the alloys in question and their corrosion products. It emerges that not only the gas phase, but also the gas–solid interface can be far removed from local equilibrium. In particular, analysis of the catastrophic ‘‘metal dusting’’ corrosion caused by carbon-supersaturated gases calls for the use of non-equilibrium models. The effects of water vapour on oxidation are discussed in Chapter 10. In many respects this is the least well-understood aspect of high-temperature corrosion. The reason for the difficulty is to be found in the multiple ways in which water molecules can interact with oxides. Preferential adsorption, hydrogen uptake,

Preface

xi

lattice defect changes, grain boundary transport property changes, gas generation within oxide pores, and scale and scale–alloy interface mechanical property changes need all to be considered. Finally, the effects of temperature cycling on oxide scale growth are considered in Chapter 11. A combination of diffusion modelling with a rather empirical scale spallation description is found to provide a reasonably successful way of extrapolating data for particular alloys. However, there is a need for development of more predictive descriptions of the relationship between spallation propensity, alloy properties and exposure conditions. Discussion is focused throughout on developing an understanding of the fundamentals of high-temperature oxidation. Frequent use is made of experimental information on real alloys in order to illustrate the principles involved. However, no attempt is made to survey the very extensive literature which exists for alloy oxidation. Thus most examples considered concern either iron- or nickel-based alloys, whereas cobalt-based alloys are largely ignored. Nickel aluminides are discussed, but other intermetallics are seldom mentioned. The scope of the book is further limited by the exclusion of some particular topics. Examples include ‘‘pesting’’ (disintegration by grain boundary attack) of silicides, and extensive oxygen dissolution by metals such as titanium and zirconium. No book of manageable proportions can ever be complete, or even fully up-to-date. It is remarkable that since the early, very substantial progress made by Carl Wagner and associates in understanding oxidation phenomena, the research effort has nonetheless continued to expand. The reason, of course, is the continuing need to operate equipment at ever higher temperatures to achieve greater efficiencies and reduced emissions. The need to develop suitable materials can be expected to drive even more research in years to come. Writing this book has been a large task, and its content inevitably reflects my own experience, as well as the ideas and results of others. I have tried to acknowledge important contributions to our understanding made by many researchers, and apologize for any omissions. My own research in this area has benefited from interaction with many talented students, research fellows and colleagues, all acknowledged by direct reference. It has also been sustained in large part by the Australian Research Council, a body to be commended for its willingness to support fundamental research. This book has benefited from colleagues from around the world who offered hospitality and/or generously gave expert commentary as I wrote: Brian Gleeson (University of Pittsburgh), Jack Kirkaldy (McMaster University), Daniel Monceau (CIRIMAT, Toulouse), Toshio Narita (Hokkaido University), Joe Quadakkers (Forschungzentrum, Julich), Jim Smialek (NASA, Lewis) and Peter Tortorelli (Oak Ridge National Laboratory). Finally, I acknowledge with gratitude and affection the inspiration provided by my mentors and friends at McMaster University, Walt Smeltzer and Jack Kirkaldy. D.J. Young August 2007

GLOSSARY OF SYMBOLS

Greek symbols

Explanation for symbol

a a a d d Zi Zg g g gi l l mi n ng nin np c r s y x x ec eik OX

Coefficient of thermal expansion Enrichment factor for metal in internal oxidation zone Ferrite, body-centred cubic metal phase Deviation from stoichiometry in oxide Thickness of gas phase boundary layer Electrochemical potential of component i Viscosity of gas Austenite, face-centred cubic metal phase Surface tension, free energy per unit surface area Activity coefficient of component i Interplanar distance, jump distance x=t1=2 , for parametric solutions to Fick’s equation Chemical potential of component i Stoichiometric coefficient in chemical reaction or compound Kinematic viscosity of gas Kinetic frequency term Poisson’s ratio Electrostatic potential Density Mechanical stress Fraction of surface sites Extent of reaction Mole fraction of oxide BO in solid solution A1x Bx O Critical strain for mechanical failure of scale or scale–alloy interface Wagner interaction coefficients for solute compounds i and k Mechanical strain in oxide

Symbol


A ai a0o ; a00o

Surface area of oxidizing metal Chemical activity of component i Boundary values of oxygen activity at metal–scale and scale–gas interfaces Mobility of species i Concentration of component i Boundary values of concentration at metal–scale and scale–gas interfaces Diffusion coefficient Grain boundary width Intrinsic diffusion coefficient for species A

Bi Ci C0 ; C00 D D DA

xiii

xiv

Glossary of Symbols

Symbol


DAB DA Dij

Gas phase diffusion coefficient for binary mixture A–B Tracer or self-diffusion coefficient of species A Diffusion coefficient relating flux of component i to concentration gradient in component j Chemical (or inter) diffusion coefficient Self-diffusion coefficient component B; self-diffusion coefficient for species in grain or phase boundary Self-diffusion coefficient for lattice species Diffusion coefficient for solute oxygen in alloy Diffusion coefficient for oxygen along an interface Electric field Elastic modulus of oxide Activation energy Free electron The Faraday (96,500 C) Fraction Volume fraction Total or molar Gibbs free energy Partial molar free energy Shear modulus of oxide Free energy per unit volume Volume fraction of internally precipitated oxide, BO Total or molar enthalpy Positive hole Species i adsorbed (bound) to surface site Internal oxidation zone Flux of component i Chemical equilibrium constant for reaction number n Rate constant Boltzmann’s constant Parabolic rate constant for metal consumption, corrosion rate constant Linear rate constant for scale thickening Gaseous mass transfer coefficient Surface area fraction of oxide spalled Parabolic rate constant for internal oxidation

~ D DB DL Do Do,i E EOX EA e0 F F fv G ¯ G GOX Gn gBO H h i|S ioz Ji Kn k k kc kl km ks kðiÞ p kp kw kv Kp Ksp KIC Lij L l MW mi m ; m0 n nT Ni NAV

Parabolic rate constant for scale thickening Parabolic rate constant for scaling weight gain Vaporization rate Equilibrium constant at fixed pressure Solubility product Fracture toughness, critical stress intensity factor General mobility coefficient, Onsager phenomenological coefficient Length of material over which gas flows Half thickness of alloy sheet Molecular weight Molar concentration of component i Number of charge units on lattice point defect species Number of moles Total number of moles, all species Mole fraction of component i Avogadro’s number

Glossary of Symbols

xv

Symbol


NM, i N M;min

Mole fraction of component M at scale–alloy interface Minimum mole fraction of component M required to support growth of external MO scale Mole fraction of component M originally present in alloy Mole fraction of dissolved oxygen Mole fraction of dissolved oxygen at alloy surface (solubility at ambient conditions) Pressure DA =DB , ratio of metal self-diffusion coefficients in ternary oxide Partial pressure of component i Total pressure of gas mixture Activation energy Charge General gas constant Rate constant for indicated gas–solid reaction Entropy source term (time rate of entropy production per unit volume) Total or molar entropy Spacing of periodic microstructure Surface site Species S located on crystal lattice site M, with effective charge X Temperature Time Time at temperature in cyclic exposure conditions Total or molar internal energy Building unit in crystalline compound Volume Velocity Molar volume of phase i Weight Scale thickness Position co-ordinate Metal surface recession Steady-state scale thickness when growth balanced by evaporation Depth of internal oxidation zone Position co-ordinate for scale–alloy interface relative to the original, unreacted surface location z=zs (or x/X), position within scale normalized to its thickness Effective charge, valence Position co-ordinate in reference frame with origin at scale–alloy interface

N ðoÞ M NO N ðsÞ O p p Pi PT Q q R ri s_ S S S SX M T t t U Ui V n Vi W X x XM Xss X(i) y y Z z

ABBREVIATIONS AND ACRONYMS

CTGA CVD EBSD EDAX EELS EPMA FIB IGCC ppm ppma ppmm PVD SAD SEM SIMS TBC TEM TGA TGO XRD YSZ

Continuous thermogravimetric analysis Chemical vapour deposition Electron back scattered diffraction Energy dispersive analysis of X-rays Electron energy loss spectroscopy Electron probe microanalysis Focused ion beam Integrated gasification combined cycle Parts per million (unit of relative concentration) Parts per million by atoms Parts per million by mass Physical vapour deposition Selected area diffraction Secondary electron microscope Secondary ion mass spectrometry Thermal barrier coating Transmission electron microscope Thermogravimetric analysis Thermally grown oxide X-ray diffraction Yttria-stabilized zirconia

xvii

CHAPT ER

1 The Nature of High Temperature Oxidation

Contents

1.1. 1.2. 1.3. 1.4. 1.5.

Metal Loss Due to the Scaling of Steel Heating Elements Protecting Turbine Engine Components Hydrocarbon Cracking Furnaces Prediction and Measurement 1.5.1 Oxidation rates 1.6. Rate Equations 1.6.1 Linear kinetics 1.6.2 Diffusion-controlled processes and parabolic kinetics 1.6.3 Diffusion and phase boundary processes combined 1.6.4 Volatilization 1.6.5 Thin oxide film growth 1.7. Reaction Morphology: Specimen Examination 1.8. Summary References

1 4 5 9 10 12 15 15 16 18 18 19 22 26 26

At high temperatures, most metals will inevitably oxidize over a wide range of conditions. The practical issues of material lifetimes and corrosion protection methods therefore centre around the rate of the oxidation reaction, methods of slowing it and the means for controlling its morphology. Answers to these questions turn out to be rather interesting, involving as they do the need for a fundamental understanding of several diverse aspects of solid–gas reactions. The general nature of the problem can be appreciated from a consideration of some practical examples.

1.1. METAL LOSS DUE TO THE SCALING OF STEEL Carbon steel is produced in prodigious quantities: about 1.3 109 t worldwide in 2007. Almost all of it is cast into large pieces such as slabs, which are subsequently reheated to around 1,000–1,2001C to be formed into more useful shapes (Figure 1.1). The reheating operation is carried out in direct-fired furnaces where steelworks gases, or sometimes natural gas, are combusted with excess air. 1

2

Chapter 1 The Nature of High Temperature Oxidation

Figure 1.1 Oxidized steel slab emerging from reheat furnace (Courtesy of BlueScope Steel).

The combination of high temperature, heating times of around 2 h, and oxidizing gases leads to the growth of a thick iron oxide scale on the steel. The amount of steel consumed in this way is about 1–2% of the total. Obviously, with steel losses of 13–26 Mt in 2007, plus the added cost of removing the scale and recycling it, there is considerable economic motivation to control or slow this process. However, there are difficulties. As discussed later, and as is intuitively reasonable, the steel scaling rate depends on three variables: steel chemistry, temperature and the gas atmosphere. The first cannot be changed because it is critical to the final steel properties. Temperature is determined by steel chemistry and is therefore also fixed. Changes in gas composition should, however, be possible. The reactions producing the furnace atmospheres can be described as: 3þx x CH4 þ O2 ¼ CO þ 2H2 O þ O2 (1.1) 2 2 and 1þx x CO þ O2 ¼ CO2 þ O2 (1.2) 2 2 where x represents the surplus of oxygen above stoichiometric requirements for complete combustion. In normal practice, excess air (xW0) is used to ensure complete combustion. However, it was recognized long ago [1] that for xo0, the atmosphere would be much less oxidizing and the extent of scaling might thereby be lessened. In analysing this suggestion, we recognize that it is necessary to calculate the furnace gas partial pressure of oxygen, pO2 , as a function of x and temperature, that the possible oxides of iron must be identified, and that the ranges of pO2 values at which they exist need to be established. The necessary pO2 values can be calculated from the equilibrium of reactions (1.1) and (1.2) and those of the iron oxide formation reactions, using the techniques of chemical thermodynamics described in Chapter 2. Such an analysis shows that it is not possible to lower pO2 below the value at which iron oxidizes, and still have sufficient combustion to heat the steel. Given that steel scaling cannot be prevented, it is important to know how the rate of scale growth (and steel consumption) varies with pO2 and temperature.

1.1. Metal Loss Due to the Scaling of Steel

Alloy

MO

Gas

M2+

M

O2 (g)

e-

M→M2++2e-

3

1 O2O +2e-→O 2 2

M2++O2-→MO

Figure 1.2 Reactions and transport processes involved in growth of an oxide scale.

A schematic cross-sectional view of a growing oxide scale is shown in Figure 1.2. The overall oxidation process can be subdivided into several steps. (1) Delivery of oxidant to the scale–gas interface via mass transfer in the gas phase. (2) Incorporation of oxygen into the oxide scale. (3) Delivery of reacting metal from the alloy to the alloy–scale interface. (4) Incorporation of metal into the oxide scale. (5) Transport of metal and/or oxygen through the scale. Evaluation of the rates at which these steps occur involves calculation of the gas phase mass transfer, solid-state mass transfer or diffusion in the oxide and alloy, and consideration of the interfacial redox reactions.

Fe ¼ Fenþ þ ne

(1.3)

2e þ 12O2 ¼ O2

(1.4)

where e represents an electron. The redox reactions are rapid and do not usually contribute to rate control. Other scale–gas interactions can be dealt with using the methods of surface chemistry. Gas phase mass transfer rates can be calculated from the methods of fluid dynamics, whilst mass transfer in the solid oxide and alloy is described using diffusion theory. The principal constituent of an iron oxide scale at TW5701C is wu¨stite, FeO, in which the Fe2+ species diffuses rapidly at high temperatures. At high values of pO2 , diffusion in FeO controls the rate at which this oxide accumulates [2]. However, in a combustion gas, where pO2 can be quite low, reaction with the oxidant species CO2 and/or H2O is slower than wu¨stite diffusion, and controls

4


the scaling rate [3]. Thus, it appears possible that steel scaling can be slowed by operating reheat furnaces under substoichiometric combustion conditions. Of course, the economic feasibility of this process alteration would have to be established through quantification of the actual benefit to be expected (as well as the costs). Such an exercise requires the ability to predict scaling rates as a numerical function of process variables, a principal concern of this book.

1.2. HEATING ELEMENTS The use of metals as electrical resistance heating elements is commonplace in small domestic appliances and laboratory furnaces. Of course the metals used must resist oxidation in air. Two groups of alloys are widely used for this purpose: nickel alloys containing around 20 wt% (weight percent) chromium and iron alloys containing about 20 wt% Cr and 5 wt% Al. As pure metals, each of Fe, Ni, Cr and Al oxidizes in air, but at vastly different rates. Oxidation rate measurements are discussed later in this chapter, but for the moment it is sufficient to use a comparison of different oxide scale thicknesses grown in a particular time. Data for 100 h reaction at 8001C in pure O2 at 1 atm are shown in Table 1.1. It is clear that pure iron would be quite unacceptable as a heating element, and that aluminium and chromium appear much more attractive. However, these are not practical choices: aluminium melts at 6601C and pure chromium is brittle and cannot be formed at room temperature. Nickel has neither of these deficiencies, and might have an acceptable scaling rate for some applications. However, like most metals in the pure state, nickel has quite poor high temperature strength and cannot be used. However, appropriate alloying can provide both strength and oxidation resistance. Cross-sectional views of oxidized surfaces of Ni-28Cr and Fe-20.1Cr-5.6Al0.08La alloys (all compositions in wt%) are shown in Figure 1.3. Single-phase oxides, Cr2O3 and Al2O3, respectively, grow as almost uniform scales, providing satisfactorily slow alloy consumption rates. It would be useful to be able to predict what concentrations of chromium and aluminium are required to achieve their preferential oxidation and thereby avoid reaction of the nickel or iron. To deal with this situation, it will be necessary to consider the thermodynamics of Table 1.1

a

Metal oxide scale thicknesses (t ¼ 100 h, pO2 ¼ 1 atm, T ¼ 8001C)

Metal

Scale thickness (mm)

Fe Ni Cr Ala

1.1 0.01 0.003 0.001

Measured on Ni-50Al.

1.3. Protecting Turbine Engine Components

5

Figure 1.3 Cross-sections of slow-growing protective scales (a) optical micrograph of Cr2O3 on Ni-28Cr after 24 h at 9001C; (b) bright field transmission electron microscopy view of Al2O3 on Fe-20Cr-6Al-0.08La after 400 h at 1,1501C [4]. Published with permission of Science Reviews.

competitive oxidation processes such as 2 Cr þ3NiO ¼ Cr2 O3 þ 3 Ni

(1.5)

where underlining indicates the metal is present as an alloy solute. An additional factor can be expected to complicate this prediction. Selective oxidation of a metal implies its removal from the alloy, and a lowering of its concentration at the alloy surface. Thus, it will also be necessary to consider the diffusion processes in both alloy and oxide.

1.3. PROTECTING TURBINE ENGINE COMPONENTS The gas turbine engines used to propel aircraft and to generate electric power have been developed to a remarkable extent since their invention in the midtwentieth century. As shown in Figure 1.4, fuel is combusted within a turbine to produce a large volume of hot gas. This gas impinges on angled blades in the hot (turbine) stage of the engine, causing it to rotate and drive the compressor stage, which draws in air to support combustion. Power is obtained from the engine either as rotational energy via a driveshaft, or as thrust, generated by the jet of hot exhaust gas. The efficiency of the engine, which is the proportion of the thermal energy converted to mechanical power, is related to the theoretical maximum work available, given by T To wmax ¼ q (1.6) T where q is the heat exchanged, To the ambient temperature and T the operating temperature. It is clear that the higher the turbine operation temperature, the greater is the efficiency potentially available. Since higher efficiency is the

6


Figure 1.4 Schematic diagram of gas turbine engine.

equivalent of lower cost and less greenhouse gas production per unit of output, its desirability has driven a steady increase in turbine gas temperatures. However, because this temperature is limited to whatever the materials of the first hot stage components can withstand, an increase in materials capability has also been necessary. Figure 1.5 summarizes the history of developments in turbine blade materials and the temperatures at which they have operated. In addition to alloy compositional changes, the development of these materials has seen an evolution in production technology from wrought through conventional cast and directionally solidified to single crystal production. Current hot stage materials are nickelbased superalloys, which possess excellent high temperature strength. This is necessary to withstand the enormous centrifugal forces generated by the high rotational speeds, around 10,000 rpm in the case of jet engines. The metallurgical design which provides the strength of these superalloys is such that they oxidize at unacceptably rapid rates at operating temperature. This problem has been solved by providing a coating of oxidation-resistant alloy on the component surfaces. Turbine temperatures are now exceeding the capabilities of superalloy components, and it has become necessary to cool them. This is done by pumping air or steam through cooling channels running through the component interiors, and providing thermal insulation (a thermal barrier coating or TBC) on top of the oxidation-resistant coating. The whole assembly is shown schematically in Figure 1.6. The TBC is typically a ceramic made of yttria-stabilized zirconia (YSZ); the oxidation-resistant coating, known as a bondcoat, is an aluminium-rich material (several designs are possible), and the superalloys are complex, nickelbase alloys containing chromium, aluminium and numerous other elements. Some examples of superalloy and bondcoat compositions are given in Table 1.2. Further examples of superalloy compositions can be found in Appendix A. Manufacture of these sophisticated components is complex. The superalloy itself is cast, using a directional solidification process, often as a single crystal [5]. The bondcoat can be applied in various ways [6]. Chemical vapour

1.3. Protecting Turbine Engine Components

Figure 1.5 Progressive increases in temperature capabilities of superalloys for turbine engine blades. Reproduced with permission of the National Institute of Materials (NIMS), Japan.

TBC TGO Bond coat

Superalloy

Coolant flow

Figure 1.6 Cross-sectional schematic view of TBC system for gas turbine blade.

7

8


Table 1.2

a

Some superalloy and coating nominal compositions (wt%)

Material

Ni

Cr

Al

Co

Mo

W

Ti

C

Other

IN738LC Rene´ N4 Rene´ N5 CMSX4 PWA 1480 PWA 1484 MC2 SRR99 NiCoCrAlYa b-NiAlb

bal bal bal bal bal bal bal bal bal bal

15.8 10.3 7.5 7.5 10 5 7.8 9.6 18 7

3.1 4.2 6.2 12.6 5 5.6 5.0 12.0 12.5 30

8.5 7.8 7.7 10.0 5 10 5.2 5.0 23 5

1.8 1.5 1.4 0.4

2.6 6.4 6.4 2.1 4 6 8.0 3.0

3.4 3.5

0.1

1.3 1.5

0.1

2.7

0.1

0.5Si, 0.8Ta 0.47Nb, 4.6Ta 7.1Ta, 2.8Re, 0.15Hf 2.1Ta, 1Re, 0.03Hf 12Ta 8.7Ta, 3Re, 0.1Hf 5.8Ta 0.9Ta 1Y

2 2.1 0.3

2

Overlay coating. Diffusion coating on Rene´ N4.

b

deposition (CVD) in which aluminium from a vapour phase species diffuses into the alloy surface, forms an aluminide diffusion coating. These coatings can be modified by the incorporation of platinum and the co-deposition of additional metals from the vapour phase. More complex coating chemistries can be achieved by physical co-deposition of various MCrAlY compositions in which M indicates Fe, Ni or Co or a mixture thereof. These coatings are deposited by sputtering, plasma spraying or physical vapour deposition, using a high-voltage electron beam to vaporize the source material. The outer surface of the bondcoat is oxidized to form a thermally grown oxide (TGO) that is the surface to which the TBC adheres. Application of the TBC is performed by either electron beam physical vapour deposition or plasma spraying [7]. At high temperatures, various interactions between these materials can be expected. Interdiffusion between the superalloy and its aluminium-rich coating can produce new phases as well as draining the coating of its essential aluminium. Some bondcoat constituents and metals diffusing from the superalloy through the bondcoat can dissolve in the TBC to form mixed oxides. Understanding and predicting these interactions requires knowledge of the phase equilibria relevant to each particular system. Finally, because the TBC is porous, oxygen from the hot combustion gas penetrates to the bondcoat surface, causing oxide scale growth. A high degree of resistance to this oxidation process is an essential function of the bondcoat. All of these processes are accompanied by volume changes which have the potential to mechanically disrupt the junction between the TBC and the underlying oxide scale. This in turn can lead to partial or even complete loss of the TBC, subsequent overheating of the substrate metal and component failure. To predict and thereby manage these consequences, it is necessary to understand the detailed mechanics of stress development within the superalloy substrate/bondcoat/TGO/TBC system, and the ways in which that stress is accommodated by deformation or fracture of one or more of the system components.

1.4. Hydrocarbon Cracking Furnaces

9

1.4. HYDROCARBON CRACKING FURNACES Many chemical and petrochemical processes are operated at high temperatures to achieve reasonable production rates or, as in cracking furnaces, to promote endothermic reactions. Cracking (or pyrolysis) furnaces are used to produce olefins such as ethylene and propylene, which are subsequently used to make the commodity materials polyethylene and polypropylene. The cracking reaction can be written as 2CH2 2CH2 2 ¼ 2CH ¼ CH2 þ H2

(1.7)

and is accompanied by carbon formation CH4 ¼ C þ 2H2

(1.8)

To slow the latter reaction, steam is added to the hydrocarbon feedstock. The hydrocarbon-stream mixture is heated by passing it through a tube which is suspended within a firebox. As seen in Figure 1.7, tube units (or coils) are large. The tubes are around 100 mm diameter, 10 mm wall thickness and about 10 m long. These tubes are expected to survive for 5 years or more whilst operating at wall temperatures ranging up to about 1,1001C. They must therefore possess adequate resistance to creep deformation (under their own weight), to oxidation

Figure 1.7 Pyrolysis tube unit being installed in steam cracker furnace.

10


of their external surface by combustion gas, and to attack by both carbon and oxygen on their inner surface. The materials used for pyrolysis furnace tubes are centrifugally cast heat resisting steels or nickel-base alloys, all austenitic alloys containing high chromium levels. Process economics are enhanced by higher operating temperatures, creating a demand for improved heat-resistant alloys. This demand has driven a shift in materials selection for the centrifugally cast tubes from HK grade (25% chromium, 20% nickel) to HP grade (25% chromium, 35% nickel) steel, and more recently to alloys containing 45 or 60% nickel and around 25% chromium. These higher nickel levels are intended to achieve higher creep strength. Consideration of the process gas composition reveals that the oxygen partial pressure is controlled by the equilibrium H2 O ¼ H2 þ 12O2

(1.9)

and pO2 1024 atm at 1,0001C. The carbon activity, aC, is controlled by reaction (1.8), and has the value unity. Under these conditions, the main alloy constituent which is reactive is chromium, and all of the compounds Cr2O3, Cr7C3 and Cr23C6 are possible products. The practical findings are that an external chromium-rich oxide scale grows early in the life of the tube, but that chromium carbides precipitate within the alloy, beneath its surface, later on. The results of a laboratory simulation of the process are shown in Figure 1.8. Questions arising from these observations of what happens to the alloy might include the following. Why do the alloy constituents other than chromium apparently not react? Why are the carbides formed as dispersed precipitates and not as scale layers? Why are carbides formed beneath the oxide and not vice versa? How does carbon penetrate the oxide layer to reach the alloy interiors? Why is there a layer of apparently unreacted alloy immediately beneath the scale? In addition, and as always, we wish to know the rates at which scale growth and internal carbide precipitation occur, and how these rates will vary with changes in temperature, alloy composition and gas conditions. To answer these questions, it is necessary to consider first the chemical thermodynamics governing reactions between a metal and two different oxidants. Secondly, a description of the rates of mass transfer of chromium, oxygen and carbon within the solid phases is required. Finally, knowledge of the processes whereby precipitates nucleate and grow within metals is needed, along with an ability to predict which precipitate phases can co-exist with which alloy compositions.

1.5. PREDICTION AND MEASUREMENT It is recognized from a consideration of the examples above that it is desirable to be able to predict which reaction products result from high temperature oxidation (or carburization, sulfidation, etc.), whether those products are formed as external scale layers or internal precipitates, how fast they form and what their mechanical stability will be, all as functions of alloy composition, temperature

1.5. Prediction and Measurement

11

Figure 1.8 Cross-section of cast heat-resisting steel (HP Mod grade) after laboratory exposure to steam-hydrocarbon mixture at 1,1001C for 500 cycles of 1 h each.

and gas conditions. The theoretical bases for the requisite predictive methodologies are reviewed in Chapter 2. The necessary thermodynamic, kinetic and mechanical data is not always available for complex, multi-component systems, and further experimental investigation is often necessary. Nonetheless, theoretical prediction is still useful, as it provides qualitative indications of the expected effect of experimental variables. Even if these are no more than hypotheses, they provide a rational framework for experimental design, thereby enabling efficient planning of laboratory investigations. At the same time, it is advisable to be aware of the possibilities afforded by modern experimental techniques. Useful theories provide predictions which can be tested, and the more thoroughly we can test a theory, the more confidence we are likely to have in it. Theoretical treatments should therefore be explored with the aim not only of achieving the desired performance predictions, but also of finding other implied outcomes which can be measured. The point here is that ‘‘performance’’ in terms of component lifetime might be tens or even hundreds of thousands of hours. Other predicted results, such as compositional, microstructural or phase constitutional change in alloy or reaction product, will be evident much more rapidly. Their verification therefore provides an early indication of the probability of the desired oxidation lifetime being achieved.

12


1.5.1 Oxidation rates The course of an oxidation reaction x 1 M þ 12O2 ! Mx Oy y y

(1.10)

follows a kinetic rate law dx ¼ fðtÞ dt where x is a measure of the extent of reaction at time, t. Thus, dx ¼ dnMx Oy ¼

(1.11)

dnM 2dnO2 ¼ x y

(1.12)

where ni is the number of moles of the indicated species, i. It is necessary to determine the quantitative form of the function f(t). In principle, a reaction can be followed by measuring consumption of metal or oxygen, or by observing oxide accumulation, as a function of time. If the oxide is a gas, then metal consumption can be followed continuously by attaching the metal sample to a balance of appropriate sensitivity, heating it in the reaction gas and measuring the weight loss. An apparatus suitable for this experiment is shown in Figure 1.9. In the more common case, the oxide is solid, and metal consumption cannot be directly observed in this way. Instead, a metal sample could be reacted for a time, and the amount of metal remaining after subsequent removal of the oxide measured. A series of samples reacted for different times would then yield a kinetic plot. Difficulties in removing all of the scale without damaging the underlying metal render weight change measurements of this sort

8 11

10

3

1

9 5

2 4

6

7

1. 2. 3. 4. 5. 6.

gas bottle catch bottle condenser + flask water bath for flask water pump water bath condenser 7. furnace 8. microbalance 9. specimen 10. amplifier 11. computer

Figure 1.9 Schematic view of thermogravimetric apparatus for measuring weight uptake during high temperature reaction in a controlled gas atmosphere.

1.5. Prediction and Measurement

13

inaccurate. An alternative technique is to measure the difference in metal section thickness before and after reaction. Given that the differences will be small, perhaps of order 10 mm, compared to the usual specimen thickness of some millimetre, measurement errors can be large. However, this technique has been successfully applied to the oxidation of thin foils [8]. The consumption of oxidant, dnO2 , can be followed by observing DpO2 at constant volume, or the volume change required to maintain pO2 constant. Given the vastly different densities of solids and gases, it is clear that this technique is restricted to cases of small dx, unless the oxidant can be replenished. Similar reservations apply to the use of this technique when the reaction gas is a mixture: as dx increases, the gas changes composition. By far the most common method of measuring oxidation rates is the observation of oxide accumulation with time. Gravimetric measurements can be performed continuously with a microbalance, or discontinuously by weighing a series of samples subjected to different reaction times. Continuous measurements yield a more accurate definition of Equation (1.11), but the time lapse exposure approach can be used to simultaneously react a large number of different alloys. Moreover, the multiple samples obtained for each alloy can be useful in characterizing the reaction products. When dx/dt is very small, the measurement precision provided by a high-quality microbalance is desirable, although it can be difficult to achieve. Microbalances are expensive. They must be protected against corrosion by the reaction gas by passing a counterflow of unreactive gas through the balance chamber, as shown in Figure 1.9. In the case of particularly corrosive species such as SO2 or H2S, it is advisable to use a cheap spring balance such as that shown in Figure 1.10. The elongation of a helical spring is observed as a sample suspended from it reacts and becomes heavier. The spring is usually made from silica fibre or Ni Span C wire, the latter being an alloy with an elastic modulus insensitive to temperature. The observed weight change, DW, varies with specimen surface area, A, and the measured quantity is reported as DW/A. If no metal volatilization occurs, the weight change corresponds to oxidant uptake, and it follows from Equation (1.12) that 2dnO2 1 DW ydnMx Oy ydnM ¼ ¼ ¼ 16 A A A xA

(1.13)

The loss of metal can then be expressed in terms of weight per unit surface area, DW M =A, using the atomic weight, AWM DW M dnM ¼ AWM A A

(1.14)

This loss is equivalent to a decrease in volume given by DV M 1 DW M ¼ rM A A

(1.15)

14


Figure 1.10 Schematic view of spring balance assembly for observing high temperature oxidation kinetics.

where rM is the metal density. Recognizing that uniform removal of metal from a flat surface results in a recession of the surface by a depth XM ¼ DVM =A

(1.16)

it is seen that XM ¼

AWM x DW 16rM y A

(1.17)

Similarly, the thickness X of a uniform, single-phase oxide scale grown on a flat surface can be calculated as X¼

MWOX DW 16rOX y A

(1.18)

where MWOX is the molecular weight and rOX the density of the oxide. Oxide scale thicknesses can be measured directly by examining microscopic images of cross-sections such as those shown in Figures 1.3 and 1.8. This technique, which is described below, is relatively simple and economical. For this

1.6. Rate Equations

15

reason, and also because diffusion equations are expressed in terms of position coordinates, it is preferably to rephrase the general oxidation rate Equation (1.11) as dX ¼ fðtÞ dt

(1.19)

1.6. RATE EQUATIONS 1.6.1 Linear kinetics The form of Equation (1.19) reflects the reaction mechanism in effect. As seen in Figure 1.2, the reaction steps can be classified within two groups: those occurring within the scale and those outside it. It might therefore be expected that steps in the latter group take place at rates independent of X. If they control the overall scaling rate, then dX ¼ kl dt

(1.20a)

X ¼ kl t

(1.20b)

which integrates to yield where kl is the linear rate constant, and the integration constant reflects the assumed condition X ¼ 0 at t ¼ 0. An example of such a situation is oxidation at very high temperatures in a dilute oxygen gas mixture. Under these conditions, diffusion in the oxide scale can be so fast that it does not contribute to rate control. However, transfer of oxygen from the bulk gas to the scale surface will be relatively slow, occurring at a rate controlled only by the gas properties, including pO2 and temperature. As long as these are fixed, the rate of O2(g) arrival at the scale surface is constant, and Equations (1.20) hold. Surface processes, such as molecular dissociation to produce adsorbed oxygen [9] CO2 ðgÞ ! COðgÞ þ OðadÞ

(1.21)

would, if rate controlling, lead to Equations (1.20). Linear kinetics are expected whenever a planar phase boundary process controls the overall rate. As noted in connection with Figure 1.2, scale growth requires the transfer of metal and/or oxidant through the scale. If the scale is highly porous, gas phase mass transfer takes place within the pores. If the pores are large compared to the mean free path of gas molecules, their dimensions are unimportant to the rate of gaseous diffusion, and scale thickness has no bearing on the oxidation rate. Linear kinetics then result. It was pointed out by Pilling and Bedworth [10] that if the volume of oxide is less than the volume of metal consumed in the reaction, then it is likely that a porous oxide layer will result. This condition is often stated in terms of the ‘‘Pilling–Bedworth ratio’’, and the requirement for non-porous oxides is expressed as V OX 41 (1.22) xV M

16


where Vi is the molar volume of the indicated species. However, a perusal of tabulated values [11] of this ratio reveals that only alkali and alkali earth metal oxides fail this test. By this criterion, all other metals should form compact scales. In fact, the situation is more complex, but it is nonetheless true that most metals of practical importance form more or less compact oxide scales.

1.6.2 Diffusion-controlled processes and parabolic kinetics The growth rate of compact scales is commonly controlled by diffusion of some species through the scale itself. A simplified analysis of this situation is now carried out to show that rate control by such a process leads to parabolic kinetics. dX kp ¼ dt X

(1.23a)

X2 ¼ 2kp t

(1.23b)

where kp is the rate constant and X ¼ 0 at t ¼ 0. The rate of diffusion in one dimension is described by Fick’s first law [12] as J ¼ D

@C @x

(1.24)

Here J is the flux, that is the net rate at which the moving component passes through unit area of a plane oriented at right angles to the diffusion direction, D the diffusion coefficient and C the concentration of a component. This equation, the physical origins of which are examined in Sections 2.5–2.7, expresses the empirical fact that, other things being equal, any mobile species will flow from a region of high concentration to one where the concentration is lower, until homogenization is achieved. The partial derivative in Equation (1.24) is now approximated by the difference in boundary values J ¼ D

DC DðC2 C1 Þ ¼ Dx X

(1.25)

where, as illustrated in Figure 1.11, C2 and C1 are, respectively, the diffusing component concentrations at the scale–gas and scale–metal interfaces. If diffusion is rate controlling, then the interfacial processes must be rapid and may be assumed to be locally at equilibrium. That is to say C1, C2 are time invariant, Equation (1.25) is seen to be equivalent in form to Equation (1.23a), and we may write kp ¼ ODðC1 C2 Þ

(1.26)

where O is the volume of oxide formed per unit quantity of diffusing species. This important result was first derived by Wagner [13], who thus showed that the scaling rate was determined by oxide properties: its diffusion coefficient and its composition when at equilibrium with metal and with oxidant. Wagner’s more rigorous treatment is described in Chapter 3. Parabolic oxidation kinetics were first demonstrated experimentally by Tammann [14] and, independently, by Pilling and Bedworth [10].

1.6. Rate Equations

M

MO

17

Gas

C1 C

C2

X

Figure 1.11 Simplified diffusion model for mass transport through growing metal oxide scale. C represents concentration of diffusing species, and C1, C2 its boundary values.

Metal recession is related to oxide scale thickness through Equations (1.17) and (1.18) VM XM ¼ x X (1.27) VOX where the molar volume, Vi, of each indicated substance is equal to MW/r. Thus, the rate equation for metal recession is X2M ¼ 2kc t

(1.28)

kc ¼ ðxV M =VOX Þ2 kp

(1.29)

with

the so called ‘‘corrosion rate constant’’. The rate constant measured by thermogravimetric analysis is given by DW 2 ¼ 2kw t A

(1.30)

For an oxide of stoichiometry MxOy , Equation (1.18) can be rewritten as X¼

VOX DW 16y A

(1.31)

Substitution in Equation (1.30) and comparison with Equation (1.23b) then yields V OX 2 kp ¼ kw 16y

(1.32)

18


In this book we are concerned mainly with scale thickness and metal consumption as measures of oxidation, and usually employ kp and kc.

1.6.3 Diffusion and phase boundary processes combined Because diffusion is initially rapid, but slows with increasing scale thickness, it is possible for scale growth to be controlled in the early stages by a phase boundary reaction and later by diffusion. When the scale is thin, the scaling rate predicted from Equation (1.23a) is faster than the other processes can sustain, and the rate is instead controlled by one of them, often a phase boundary reaction. As the scale thickens, the diffusion rate eventually decreases until it becomes slower than the constant phase boundary reaction rate, and then controls the overall reaction. The phase boundary process then approaches local equilibrium. The observed kinetics will be initially linear and subsequently parabolic. This behaviour has been described [15] by the rate equation X2 þ LX ¼ kt þ C

(1.33)

where L and C are constants. It is worth noting that during the initial stage, mass transfer within the scale is nonetheless occurring via diffusion, implying that the boundary values C1 or C2 in Equation (1.25) change with time.

1.6.4 Volatilization Some metals form volatile oxides. At temperatures above about 1,3001C, tungsten reacts with low-pressure oxygen to form gaseous WO3 and WO2 species, with no solid oxide formed on the surface. If the gas composition is unchanged, metal is consumed at a constant rate. This is why incandescent light filaments, which are based on tungsten, are protected by enclosure in inert gas filled light bulbs. A less severe situation arises with chromium which undergoes two oxidation reactions in dry oxygen 2CrðsÞ þ 32O2 ðgÞ ¼ Cr2 O3 ðsÞ

(1.34)

Cr2 O3 ðsÞ þ 32O2 ðgÞ ¼ 2CrO3 ðgÞ

(1.35)

the latter reaction becoming important at temperatures above about 1,0001C at pO2 ¼ 1 atm. The scaling rate law is then made up of two terms: diffusioncontrolled accumulation and a constant volatilization loss [16] dX kp ¼ kv dt X

(1.36)

This equation predicts that the scale thickness increases to a steady-state value, Xs, where dX/dt ¼ O and Xs ¼ kp/kv . Of course, metal continues to be lost at a constant rate proportional to kv .

1.6. Rate Equations

19

1.6.5 Thin oxide film growth During the early stages of reaction, X is small. At low temperatures, where diffusion and other processes are slow, the time period over which X is ‘‘thin’’ (i.e. tens of nm) can be very long. In this regime, mass transfer through the oxide film is strongly affected by electrostatic effects. These may be understood in a qualitative way from a consideration of the schematic electron energy level diagram in Figure 1.12. In the case of a very thin oxide film, electrons can be transferred from the underlying metal to surface levels at the oxide–gas interface by quantum mechanical tunnelling through the barrier represented by the film [19]. As the film thickens, this mode of electron transport is rapidly attenuated and thermionic emission becomes the most feasible mechanism. The processes of conduction and diffusion within the film itself finally control electron transport, when scattering prevents thermionically emitted elections from crossing the film in a single step. The electron transfer processes can be the oxide growth rate limiting processes, or they can be rapid, achieving a pseudo-equilibrium state with oxygen anions on the film surface. In the latter case, movement of charged ions (Mn+ or O2) through the oxide film is likely to be rate controlling. The mobile ions migrate through the oxide under the influence of an electric field, E, the magnitude of which at the surface is given by Poisson’s equation E¼

4pqs e

(1.37)

where s is the surface concentration of species with charge q, and e the dielectric constant of the oxide. Within the oxide the field is modified by any locally

Vacuum level Conduction band

Potential Energy

Øm Ø′’O Ø

Ø′’m Ø

Fermi level

O- level

Metal

Oxide |eVX| X=0

Valence band

x=X

Figure 1.12 Approximate energy level diagram for electrons in the metal-oxide-adsorbed gas system.

20


uncompensated (space) charge dE 4pr ¼ dx x

(1.38)

where r is the space charge density. The boundary conditions for this equation are supplied by the condition of overall charge neutrality Ex ¼

4pqX sX 4pqO sO ¼ þ e e

Z

X o

4pr dx e

(1.39)

where the subscripts O and X correspond to the two film interfaces. The development of an electrostatic field during oxidation has been confirmed by surface potential measurements [17]. That the oxidation rate is affected by the magnitude of the field is confirmed by experiments [18] in which an electrostatic potential difference impressed across a growing film was shown to modify the growth rate. At relatively low temperatures and high oxidant pressures, surface and gas phase processes are seldom rate controlling, and film-thickening rates will be governed by the rate of electronic or ionic transport. The transport of all charged species depends on the electric field strength, which in turn is a function of film thickness. Evaluation of the field is achieved by integrating Poisson’s Equation (1.38), for which purpose the space charge distribution must be known. This in turn can be found from a consideration of the transport equations. At the very high field strengths prevailing in thin films, of order 106 V cm1, the transport equations are non-linear, and the calculations are complex. Because this regime of behaviour is not considered in detail in this book, the reader is referred to the comprehensive treatment provided by Fromhold [19]. A more succinct account, together with a brief review of its applicability to a selection of experimental data is also available [20]. The first equation used to describe thin film growth kinetics was suggested by Tammann [14] as X ¼ k1 lnð1 þ AtÞ

(1.40)

where kl and A are constants. The various theoretical treatments reviewed in references [19, 20] do not lead to this expression (which was purely empirical) but instead yield for the case of rate control by ion transport equations of the form dX B1 ¼ A1 sinh (1.41) dt X or dX A2 B2 ¼ sinh (1.42) dt X X with Ai, Bi constants. The difficulty of distinguishing between the various models on the basis of kinetic data is illustrated in Figure 1.13, where it is seen that when the thickness

1.6. Rate Equations

21

Figure 1.13 Zinc oxidation data and regression lines found for the sinh rate Equation (1.41), the two-stage logarithmic Equation (1.43) and the parabolic rate equation. Reprinted from Ref. [20] with permission of Elsevier.

of oxide formed on zinc [21] is plotted against log (time), concave upwards curves result. This is a fairly common observation, and has led to the proposal of a two-stage logarithmic rate law [22] X ¼ k1 lnð1 þ A1 tÞ þ k2 lnð1 þ A2 tÞ þ Xo

(1.43)

on the supposition that two reaction paths operate in parallel. The data in Figure 1.13 has been curve fitted to three separate rate equations, and it is seen that their merits cannot be distinguished on this basis. It is better to test the applicability of kinetic models by seeing if the constants emerging from the fitting procedure are physically reasonable, and by verifying that the model predicts correctly the effects on reaction rate of perturbations to the system such as changes in T; pO2 and Ex. Other empirical rate laws suggested for thin film oxidation include the ‘‘inverse log law’’ 1 1 ¼ k3 logð1 þ tÞ (1.44) X X0 and the cubic rate law

X 3 ¼ k4 t

(1.45)

No physical basis exists for Equation (1.44), and only under very restricted circumstances can Equation (1.45) be justified for thin film growth [23]. However, the cubic rate equation is found to apply to alumina scale growth (Section 5.9) when oxide grain boundaries provide the means of solid-state diffusion (Section 3.9).

22


1.7. REACTION MORPHOLOGY: SPECIMEN EXAMINATION As already noted, compositional, microstructural and phase constitution information are required for both the reaction product and nearby regions of the affected alloy. As seen in Figure 1.8, these quantities can vary considerably with position in the reaction zone, and analytical methods which yield average results are inappropriate. Many features of the reaction morphology are revealed in metallographic cross-sections, such as those made use of in this chapter. Reacted samples are embedded in cold setting epoxy resin by vacuum impregnation. After the resin cures, the section is ground and polished, usually to a 1/4 mm finish. Because the corrosion products are much more brittle than metals, additional effort is required at each stage of grinding and polishing to remove the damage remaining from the preceding step. For the same reason, it can be advantageous to protect the scale outer edge by depositing a layer of nickel or copper on it prior to sectioning. The procedure is to first vacuum evaporate or plasma coat a thin metal deposit onto the reacted sample surface, making it electrically conductive. The sample can then be electroplated with the desired thickness of metal. The polished cross-section should first be examined by optical microscopy, using low and high magnifications, with a maximum resolution of about 1 mm. Digital images are then analysed, using image analysis computer software, to obtain data such as scale layer thickness, precipitate sizes and volume fraction, etc. The speed of this process permits extensive sampling and the accumulation of statistically reliable data. Higher magnification images can be acquired using scanning electron microscopy (SEM) or, for very high magnifications, transmission electron microscopy (TEM). Examples of the three image types are shown in Figure 1.14. Electron microscopy provides the opportunity to acquire compositional and crystallographic information at precisely defined locations within the reaction zone. The electron beam interacts with atoms within the sample, exciting the emission of X-rays with energies characteristic of the atomic number of the atoms involved. These X-rays are collected, analysed according to energy, and counted using the technique of Energy Dispersive Analysis of X-rays (EDAX). The spatial resolution is limited by electron scattering within the solid. Depending on electron energy and their absorption by the solid, the spatial resolution is around 1–2 mm. When appropriate standards and correction procedures are used, the technique is quantitative, yielding reliable compositional analyses for metals, but only semi-quantitative results for oxygen and, at best, qualitative results for carbon. The spatial resolution of EDAX is much better in a TEM, 1–10 nm, simply because the electrons are scattered less widely during their transit of the very thin sample. The effect is illustrated in Figure 1.15. Superior analytical precision and the capability of analysis for light elements are provided by the alternative technique of electron probe microanalysis (EPMA). In this instrument, X-rays excited by an electron beam are analysed by wavelength, using single crystals as diffraction gratings. This technique provides better analytical discrimination (e.g. between molybdenum and sulfur) and much higher count rates.

1.7. Reaction Morphology: Specimen Examination

23

Figure 1.14 Sections of internally carburized Fe-Ni-Cr alloy (a) optical metallograph, (b) SEM view of deep-etched section, (c) TEM bright field view and (d) selected area diffraction pattern from (c).

Both EDAX and EPMA are used to identify reaction product and alloy compositions as a function of position. Care is necessary when analysing multiphase regions, such as scale–alloy interfaces or precipitate-matrix assemblages, because the electron beam can be sampling two phases at the same time. The difficulty is illustrated in Figure 1.15, along with the remedy: ‘‘point counting’’. The beam, or preferably the sample, is stepped at small intervals along a line intersecting the phase boundaries, and X-rays counted at each point. Only when a constant composition is measured at several successive points, and when that composition is reproduced in several sample regions, can the analysis be accepted. The point counting techniques is also valuable for measuring composition profiles in scales and in underlying alloy regions.

24


Electron beam

(a) Electron beam

Concentration

(b)

x

(c) Figure 1.15 Spatial resolution of EDAX and EPMA limited by Compton scattering of electrons (a) bulk sample, (b) thin foil in TEM and (c) two-phase region with corresponding analysis.

The electron beam is diffracted by crystalline solids, and analysis of the resulting patterns provides information on the crystal structure and orientation of the diffraction region. The TEM is commonly used for this purpose, and an example is shown in Figure 1.14. This valuable technique is now being applied more frequently to oxidized specimens, since the introduction of the focused ion beam (FIB) milling technique for producing the required thin foil samples. The FIB provides thin foils in precisely determined locations, and is thus able to capture interfaces, grain boundaries, etc. The SEM can also be used to generate

1.7. Reaction Morphology: Specimen Examination

25

crystallographic information via the electron backscattered diffraction (EBSD) technique. This is particularly useful for identifying alloy and oxide grain orientations, to be correlated with other reaction morphological and kinetic features. Because SEM images can provide a large depth of field, the technique is suitable for examining rough surfaces such as the scale outer surface or the alloy surface after scale removal. The use of electron microscopy to identify reaction product phases by diffraction can be costly and time consuming. An alternative is provided by X-ray diffraction (XRD). The reacted sample is simply placed in the specimen holder of a diffractometer, so that the X-ray beam falls on the flat scale surface, and the intensity of diffracted beams measured. Matching the resulting diffraction pattern with tabulated standards then leads to phase identification. At the wavelengths and intensities normally used, X-rays penetrate only a short depth (1–3 mm) into the sample, so the technique provides information on only the near surface region. If a thin scale is being analysed, glancing angle XRD can be used to sample an extremely thin surface region. Because alloy oxidation frequently produces multiple reaction products disposed over a thick reaction zone (e.g. Figure 1.8), it is necessary to obtain diffraction data at a number of different depths. This is done by grinding away a thin surface region, obtaining an XRD pattern and repeating this process until the entire reaction zone has been traversed. This technique was used to identify the reaction products shown in Figure 1.8: an outer scale layer of MnCr2O4 with a thicker layer of Cr2O3 beneath it; internal Al2O3 precipitates (plus some SiO2); a singlephase austenite zone; finally an internal carburization zone of chromium carbide plus austenite. The XRD technique yields measurements of crystal lattice plane spacings. Comparison of this data with that of standards reveals any distortions in the lattice, corresponding to the existence of mechanical stress. Measurements can be carried out at a high temperature to estimate stress states under reaction conditions. Because the stress can change during reaction, it is necessary to make very rapid measurements, and this can be done using the high intensity X-rays available from a synchrotron [24, 25]. The electron beam techniques described above identify the constituent elements of a solid and define their crystallographic relationships. However, they are insensitive to atomic weight and cannot distinguish isotopes. One way of investigating the contribution of oxygen diffusion to scale growth is the use of isotopically labelled oxidants. For example, a metal can be oxidised first in 16 O2 and subsequently in 18 O2 , and the scale then analysed to determine the 16 O=18 O distribution. If they are found to be mixed, then oxygen diffusion has occurred, whereas the observation of an M18O layer on top of an M16O layer would indicate the absence of such a process. The necessary measurements are made by secondary ion mass spectrometry (SIMS). An ion beam is used to sputter away the scale surface, and the ejected ions are analysed by mass spectrometry. Sputtering is continued, removing successively deeper regions of oxide, until the underlying metal is reached. An example of the results obtained in this way is shown in Figure 1.16.

26


Figure 1.16 SIMS analysis of scale grown on an Fe-9Cr steel in N2 1%16 O2 2%H2 18 O showing different extents of oxidation by O2 and H2O in different parts of the scale. Reprinted from Ref. [26] with permission of Elsevier.

1.8. SUMMARY As seen from the oxidation cases examined here, a diversity of oxidation reaction morphologies and rates is possible. It is important, therefore, to be able to predict under which circumstances (alloy composition and environmental state) each particular form of reaction will occur, the kinetics of the process and how the rate varies with temperature and the compositions of both alloy and gas. An understanding of these fundamental principles then permits a rational approach to materials selection (or design), and the determination of suitable operating limits for temperature, gas composition, flow rate, etc. Two basic techniques are used to approach the problem. Chemical thermodynamics are employed to predict the reaction outcome, and an analysis of mass transfer processes provides an evaluation of reaction rate. The enabling theory underlying these techniques is summarized in Chapter 2, and examples relevant to high temperature oxidation are discussed. In addition, summary descriptions are provided for interfacial processes and the effects of mechanical stress in oxide scales.

REFERENCES 1. 2. 3. 4. 5. 6. 7.

J.S. Sheasby, W.E. Boggs and E.T. Turkdogan, Met. Sci., 18, 127 (1984). L. Himmel, R.F. Mehl and C.E. Birchenall, Trans. AIME, 197, 822 (1953). V.H.J. Lee, B. Gleeson and D.J. Young, Oxid. Met., 63, 15 (2005). F.H. Stott and N. Hiramatsu, Mater. High Temp., 17, 93 (2000). Superalloys II, eds. C.T. Sims, N.S. Stoloff and W.C. Hagel, Wiley-Interscience, New York (1987). G.W. Goward, in High Temperature Corrosion, ed. R.A. Rapp, NACE, Houston (1983), p. 553. B. Gleeson, J. Propulsion Power, 22, 375 (2006).

References

8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

27

B.A. Pint, J. Eng. Gas Turbines Power, 128, 370 (2006). F.S. Pettit, R. Yinger and J.B. Wagner, Acta Met., 8, 617 (1960). N.B. Pilling and R.E. Bedworth, J. Inst. Met., 29, 529 (1923). O. Kubaschewski and B.E. Hopkins, Oxidation of Metals and Alloys, Butterworth, London (1953). A.E. Fick, Pogg. Ann., 94, 59 (1855). C. Wagner, Z. Phys. Chem., 111, 78 (1920). G. Tammann, Z. Anorg.Chem., 111, 78 (1920). P. Kofstad, High Temperature Corrosion, Elsevier, London (1988). C.S. Tedmon, J. Electrochem. Soc., 113, 766 (1966). F.P. Fehlner and N.F. Mott, Oxid. Met., 2, 59 (1970). P.J. Jorgensen, J. Electrochem. Soc., 110, 461 (1963). A.T. Fromhold, Theory of Metal Oxidation, Elsevier, New York (1975). W.W. Smeltzer and D.J. Young, Prog. Solid State Chem., 10, 17 (1975). V.O. Nwoko and H.H. Uhlig, J. Electrochem. Soc., 112, 1181 (1965). I.M. Ritchie, Surface Sci., 23, 443 (1970). D.J. Young and M.J. Dignam, Oxid. Met., 5, 241 (1972). P.F. Tortorelli, K.L. More, E.E. Specht, B.A. Pint and P. Zschack, Mater. High Temp., 20, 303 (2003). P.Y. Hou, A.P. Paulikas and B.W. Veal, Mater. Sci. Forum, 522–523, 433 (2006). J. Ehlers, D.J. Young, E.J. Smaardijk, A.K. Tyagi, H.J. Penkella, L. Singheiser and W.J. Quadakkers, Corros. Sci., 48, 3428 (2006).

CHAPT ER

2 Enabling Theory

Contents

2.1.

Chemical Thermodynamics 2.1.1 Chemical potential and composition 2.1.2 Chemical equilibrium in gas mixtures 2.2. Chemical Equilibria Between Solids and Gases 2.2.1 Chemical equilibria involving multiple solids 2.2.2 Gases containing two reactants 2.3. Alloys and Solid Solutions 2.3.1 Dissolution of gases in metals 2.4. Chemical Equilibria Between Alloys and Gases 2.4.1 Equilibria between alloys and single oxide 2.4.2 Equilibria between alloys and multiple oxides 2.5. Thermodynamics of Diffusion 2.5.1 Driving forces 2.5.2 Point defects 2.6. Absolute Rate Theory Applied to Lattice Particle Diffusion 2.7. Diffusion in Alloys 2.7.1 Origins of cross effects 2.7.2 Kirkendall effect 2.8. Diffusion Couples and the Measurement of Diffusion Coefficients 2.8.1 Diffusion data for alloys 2.9. Interfacial Processes and Gas Phase Mass Transfer 2.9.1 Gas adsorption 2.9.2 Gas phase mass transfer at low pressure 2.9.3 Mass transfer in dilute gases 2.10. Mechanical Effects: Stresses in Oxide Scales 2.10.1 Stresses developed during oxidation 2.10.2 Stresses developed during temperature change References Further Reading

30 31 31 34 37 41 42 45 46 46 48 51 51 54 55 58 58 60 63 66 66 66 68 69 71 71 73 76 77

As seen in the previous chapter, we wish to predict which reaction products form when a particular alloy is exposed to a given gas, and the effects of temperature and pressure on the outcome. This requires the use of chemical thermodynamics, and in particular the use of phase equilibria. The rates at which the products form are usually governed by diffusion and interfacial processes, both involving crystallographic defects. Finally, the structural integrity of any solid is determined by its mechanical state. We now review these areas, focusing on their application to high temperature oxidation reactions. 29

30

Chapter 2 Enabling Theory

2.1. CHEMICAL THERMODYNAMICS The question of whether or not an oxide is formed is answered by determining the most stable state of the reacting system M þ 12O2 ¼ MO

(2.1)

At constant temperature and pressure the stability of a system is measured by its Gibbs free energy. The total Gibbs free energy, G, of a system is defined as G ¼ H TS ¼ U þ pV TS

(2.2)

where H is the enthalpy, S the entropy, U the internal energy and V the volume of the system, and p, T have their usual meanings. For a system in which compositional change through chemical reaction is possible, the reversible internal energy change is provided by the basic laws of thermodynamics dU ¼ TdS pdV þ

X

mi dni

(2.3)

i

where ni is the number of moles of component i, the summation is over all components in the system, and @U mi ¼ (2.4) @ni T;P;njai is the chemical potential. Combination of the differential form of Equation (2.2) with Equation (2.3) yields X dG ¼ Vdp SdT þ mi dni (2.5) i

An isothermal, isobaric system is at equilibrium where G is minimum, the location of which is determined by the differential dG ¼ 0 (2.6) Under these conditions, it is seen that the partial molar free energy of a component @G ¯ Gi ¼ ¼ mi (2.7) @ni T;P;njai is equivalent to its chemical potential. The overbar symbol will be used to denote partial molar quantities in general. Integration of Equation (2.5) leads to X mi ni (2.8) G¼ i

when dT ¼ 0 ¼ dp, and combination of Equations (2.6) and (2.8) yields the condition for chemical equilibrium X ni mi ¼ 0 (2.9) i

2.1. Chemical Thermodynamics

31

which is the Gibbs equation. To utilize this result it is necessary to evaluate the mi in terms of compositional variables.

2.1.1 Chemical potential and composition For an isothermal system of fixed composition, application of Equation (2.5) to component A yields ¯A @G ¯A ¼V (2.10) @p ¯ A is the partial molar volume. The latter is found for a perfect gas mixture to be where V ¯ A ¼ PV ¼ N A RT V pA ni

(2.11)

i

where N A ¼ nA =

P ni is the mole fraction. Rewriting Equation (2.10) as an exact i

differential and substituting from Equation (2.11), it is found that ¯ A ¼ N A RT dp dmA ¼ dG pA

(2.12)

Further substitution from Dalton’s law of partial pressures pA ¼ N A ; dpA ¼ N A dp p then leads to dm ¼ RT which upon integration yields

dpA pA

m ¼ m þ RT ln

pA pA

(2.13)

(2.14)

Here the arbitrarily chosen value pA ¼ pA is used to define the standard state at which the chemical potential has its standard (temperature-dependent) value of m . It is convenient to choose pA as unity, commonly 1 atm.

2.1.2 Chemical equilibrium in gas mixtures We consider reactions such as CO2 ¼ CO þ 12O2

(2.15)

H2 O ¼ H2 þ 12O2

(2.16)

with the intention of calculating pO2 . More generally, any reaction can be formulated as a summation over all chemical species involved X 0¼ Sni Mi (2.17) i

32


where the Mi are the symbols for the different chemical species (CO, CO2, etc.) and the ni the stoichiometric coefficients, which are negative for reactants and positive for products. Thus for reaction (2.15) n1 ¼ 1, n2 ¼ 1 and n3 ¼ 0.5. It follows that dn1 dn2 dn3 dnm ¼ ¼ ¼ ¼ dx (2.18) n1 n2 n3 nm where again x denotes the extent of reaction. Equation (2.5) may now be written as dG ¼ Vdp SdT þ mA dnA þ mm dnm ¼ Vdp SdT þ ðnA mA þ nm mm Þdx and hence

@G @x

¼ T;P

X

(2.19)

ni mi

i

The condition for chemical equilibrium is therefore given by Equation (2.9) in the specific form X ni mi ¼ 0 (2.20) i

Substituting from Equation (2.14), we find in the case of the CO2 reaction (2.15) 1=2

0 ¼ mCO þ 0:5mO2 mCO2 þ RT ln

pCO pO2 pCO2

(2.21)

Recognizing that the standard chemical potentials are, by definition, the standard free energies per unit mole, this result is recast as 1=2

mCO þ 0:5mO2 mCO2 ¼ DG ¼ RT ln

pCO pO2 pCO2

(2.22)

where DG ¼ mCO þ 0:5mO2 mCO2 is termed the standard free energy change. Since DG is a function only of temperature, the quantity Kp, called the equilibrium constant at fixed total pressure, 1=2 pCO pO2 DG Kp ¼ exp (2.23) ¼ RT pCO2 is also a function only of temperature. Tabulated values of DG are available [1–3], allowing easy calculation of Kp. A small selection of useful values is provided in Table 2.1. If the equilibrium ratio pCO =pCO2 is known, then the oxygen partial pressure is readily found from Equation (2.23). A more commonly encountered problem is that of calculating pO2 from a knowledge of the input gas composition, i.e. before equilibrium is reached. This is dealt with using the stoichiometry of the reaction, and specifying an unknown extent of reaction to be determined. If there is initially no oxygen present and the

33

2.1. Chemical Thermodynamics

Table 2.1

Standard free energiesa of reaction [1, 2] DG ¼ A þ BT ( J mol1)

Reaction

2 1 1 3AlðlÞ þ 2O2 ¼ 3Al2 O3 1 1 1 2Si þ 2O2 ¼ 2SiO2 Mn þ 12O2 ¼ MnO Zn+12O2 ¼ ZnO 2 1 1 3Cr þ 2O2 ¼ 3Cr2 O3 23 1 6 Cr þ C ¼ 6Cr23 C6 7 23 27Cr23 C6 þ C ¼ 27Cr7 C3 3 7 5Cr7 C3 þ C ¼ 5Cr3 C2 1 2Cr þ 2N2 ¼ Cr2 N Cr2 N þ 12N2 ¼ 2CrN Fe þ 12O2 ¼ FeO 3FeO þ 12O2 ¼ Fe3 O4 2Fe3 O4 þ 12O2 ¼ 3Fe2 O3 Fe þ 12S2 ¼ FeS

3Fe þ C ¼ Fe3 C Co þ 12O2 ¼ CoO 3CoO þ 12O2 ¼ Co3 O4 Ni þ 12O2 ¼ NiO H2 þ 12O2 ¼ H2 O H2 þ 12S2 ¼ H2 S O2 þ 12S2 ¼ SO2 CO þ 12O2 ¼ CO2 2CO ¼ CO2 þ C

A

B

565,900 451,040 412,304 356,190 373,420 68,533 42,049 13,389 108,575 133,890 264,890 312,210 249,450 150,247 29,037 233,886 183,260 234,345 246,440 180,580 362,420 282,420 170,700

128 86.8 72.8 107.9 86 6.45 11.9 0.84 138 174 65.4 125.1 140.7 52.6 28.0 70.7 148.1 84.3 54.8 98.8 72.4 86.8 174.5

a Referred to pure solid metals (except liquid Al), compounds and graphite. DG values for the mole numbers shown in the chemical equations.

input gas mixture consists of a moles of CO2 plus b moles of CO, we see that x/2 moles of O2 are formed with the consumption of x moles of CO2 and the generation of x moles of CO ¼ CO2 ða xÞ

CO ðb þ xÞ

þ

1 2O2 x 2

In this system Si ni ¼ a þ b þ ðx=2Þ, and the partial pressures are given by ni p (2.24) pi ¼ ða þ b þ x=2Þ

34


Thus K2p ¼

¼

p2CO pO2 p2CO2

ðb þ xÞ2 ðx=2Þ p 2 ða þ b þ x=2Þ ða xÞ

(2.25)

Although this cubic equation can be solved numerically, use can be made of the fact that x will be small, as is now seen. For reaction (2.15) DG ¼ 282; 420 86:28T J mol1 and at 1,0001C, DG1273 ¼ 172; 586 J mol1 , and therefore Kp ¼ 8:3 108 If the input gas contains nCO ¼ 0.1, nCO2 ¼ 0.9 then clearly x must be very small. Using the approximation b x a in Equation (2.25) leads to 2 0:1 x 2 15 p ¼ Kp ¼ 6:8 10 0:9 2 and if p ¼ 1 atm, we find x ¼ 1.1 1012, justifying the approximation. The value of pO2 is then given by x=2 ¼ 5 1013 atm pO2 ¼ a þ b þ ðx=2Þ It is seen that the substoichiometric combustion considered in Section 1.1 can lead to quite low oxygen partial pressures. However, to attach significance to this value, it is necessary to consider the thermodynamics of steel oxidation.

2.2. CHEMICAL EQUILIBRIA BETWEEN SOLIDS AND GASES We consider metal oxidation reactions such as (2.1), observe that they are of the general form (2.17) and note that the condition for equilibrium is therefore given by Equation (2.20). It is convenient to state this as mMO mM 12mO2 12RT ln pO2 ¼ 0

(2.26)

where the values of mMO, mM, in general depend on pressure, temperature and composition, and these dependencies have been stated explicitly for mO. However, if the metal and metal oxide are pure, immiscible solids, then their m values are independent of system composition.

2.2. Chemical Equilibria Between Solids and Gases

35

Furthermore, the chemical potentials of solids are insensitive to pressure according to Equation (2.10), because their molar volumes are small. Thus mM , mMO depend on temperature only, and Equation (2.26) can be rewritten as DG ¼ mMO mM 12mO2 ¼ 12RT ln pO2

(2.27)

DG 1 Kp ¼ exp ¼ 1=2 RT p

(2.28)

or equivalently

O2

At this precisely defined temperature-dependent value of pO2 , the metal and its oxide co-exist at equilibrium. This value is often termed the dissociation pressure, and will be denoted here as pO2 ðMOÞ. For wu¨stite formation Fe þ 12O2 ¼ FeO

(2.29a)

DG ¼ 264; 890 þ 65:4T J mol1

(2.29b)

we find from Table 2.1

and at 1,0001C, DG1273 ¼ 181; 640 J mol1 , corresponding to K2:29 ¼ 3:6 106 ;

pO2 ðFeOÞ ¼ 1:2 1015 atm

The symbol Ki will be used to denote the equilibrium constant at fixed total pressure for reaction i. For pO2 ¼ 1:2 1015 atm, iron and wu¨stite co-exist. At lower oxygen partial pressures, a clean iron surface would not oxidize and any FeO would be reduced to metal. At higher values of pO2 , iron would oxidize, a process which would continue until either the iron was consumed or, if the gas supply was limited, until enough oxygen was consumed to lower pO2 to 1.2 1015 atm. In the steel reheat furnace considered in Chapter 1, the steel can be regarded in a chemical sense as almost pure iron, and reaction (2.29a) describes its oxidation. Because fuel is continuously combusted, the ambient pO2 is maintained at a constant value. Even under the strongly substoichiometric combustion conditions leading to a 90/10 mixture of CO2/CO, this value exceeds the Fe/FeO equilibrium value. Since, moreover, the steel sections are much larger than can be consumed in the time available, reaction (2.29a) continues to form more scale. In fact, iron can form two higher oxides 3FeO þ 12O2 ¼ Fe3 O4 ;

DG ¼ 312; 210 þ 125:1T J mol1

2Fe3 O4 þ 12O2 ¼ 3Fe2 O3 ; DG ¼ 249; 450 þ 140:7T J mol1 from which we calculate for T ¼ 1,0001C pO2 ðFe3 O4 Þ ¼ 2:8 1013 atm pO2 ðFe2 O3 Þ ¼ 1:7 106 atm

(2.30) (2.31)

36


Thus the supposed gas mixture of 10% CO, 90% CO2 will form magnetite (Fe3O4), but not hematite. The question as to how wu¨stite and magnetite are disposed within the scale is dealt with in the next section. Standard free energy data for metal oxidation reactions 2x 2 M þ O2 ¼ Mx Oy (2.32) y y are conveniently summarized in Ellingham/Richardson diagrams such as the one shown in Figure 2.1. Here DG is plotted as the y-axis and temperature as the x-axis. Because all reactions are normalized to 1 mol of O2, it follows that DG ¼ RT ln pO2 and an auxiliary scale in pO2 is possible. The equilibrium value of pO2 for a particular metal oxide couple is found by drawing a straight line (the dashed line in Figure. 2.1) from the point marked ‘‘O’’ (at DG ¼ 0; T ¼ 0K) through the free energy line of interest at the desired temperature, and continuing the line to its intersection with the pO2 scale on the right-hand side of the diagram. Following this procedure for Fe/FeO at 1,0001C yields the estimate pO2 1015 atm, in agreement with the earlier calculation. The justification for this procedure is seen in the equation for the straight line y ¼ a þ bx In this case, DG ¼

DGT1 T T1

where T1 ¼ 1; 273 K, and DGT1 =T 1 ¼ R ln pO2 ðT1 Þ. The auxiliary pO2 scale is seen in the diagram to be located at T2 2; 873 K. Thus the intersection of the dashed line with the pO2 scale is at DG ¼ RT 2 ln pO2 ðT1 Þ. Additional scales are provided for the CO/CO2 and H2/H2O ratios corresponding to oxidation equilibria. We consider the example of the FeO formation reaction again. A straight line is drawn from the point marked ‘‘C’’ on the left (at DG ¼ DG reaction ð2:15Þ; T ¼ 0K) to the Fe/FeO line at 1,0001C, and continued to the CO/CO2 scale on the right-hand side, yielding an estimate pCO =pCO2 41. Thus the diagram is useful for obtaining close order of magnitude estimates. Similar diagrams are available for sulfides [4] and carbides [5]. It is seen that the free energy plots in Figure 2.1 are almost straight lines. Furthermore, most of the lines are approximately parallel, apart from changes in slope corresponding to changes of state. Rewriting Equation (2.2) as DG ¼ DH TDS

(2.33)

it is deduced from the near linearity of the plots that DH and DS are almost constant. It is also seen that the positive slopes of the lines correspond to negative entropy changes. This is a consequence of the fact that the entropy of a gas is much larger than that of a solid. Thus the largest component of DS in reaction (2.32) is associated with the removal of 1 mol from the gas phase. This also explains why the slopes of the lines are approximately equal.


37

Figure 2.1 Ellingham/Richardson diagram showing free energies of formation for selected oxides as a function of temperature, together with corresponding equilibrium pO2 and H2/H2O and CO/CO2 ratios. Dashed line to find equilibrium pO2 for Fe/FeO and dotted line to find pCO =pCO2 for same reaction.

2.2.1 Chemical equilibria involving multiple solids In the last section we encountered the situation where the ambient pO2 value was maintained at such a value that two different oxides, FeO and Fe3O4, could form: reactions (2.29a) and (2.30) were both favoured. Of course in the fullness of time, all the iron would be converted to magnetite, which would then equilibrate with the ambient gas. Our interest is in the earlier stages, when the reaction is still in progress, and some iron still remains. In thinking about the structure of the scale,

38


it is useful to consider the metal–scale and scale–gas interfaces. At the latter, reaction (2.30) proceeds to the right at pO2 42:8 1013 atm, and we predict that the surface oxide will be magnetite. Consider now what would happen if the underlying metal was in contact with this oxide, by enquiring as to whether the reaction Fe þ Fe3 O4 ¼ 4FeO

(2.34)

will proceed. For T ¼ 1,0001C, we evaluate DG ð2:34Þ ¼ 4DG ð2:29Þ DG ð2:30Þ ¼ 57; 360 J mol1 and see that wu¨stite is more stable. It will therefore form between the iron and the magnetite, and a two-layer scale is predicted. The reasoning above is correct, but tedious. The same conclusion is reached immediately on examining the Fe–O phase diagram in Figure 2.2. This composition–temperature diagram maps the existence regions of the various possible phases. It is seen that wu¨stite is not actually ‘‘FeO’’, but is a metaldeficient oxide Fe1d O, where d varies at any given temperature. Magnetite also deviates from its nominal stoichiometry at high temperatures, but the highest oxide, hematite, is closely stoichiometric at all temperatures. The diagram also reveals that wu¨stite is unstable below 5701C. Phase diagrams summarize experimental observations of equilibrium. The Fe–O diagram informs us that for 1,370WTW5701C, iron equilibrates with

Figure 2.2 Iron–oxygen phase diagram.


39

Fe1d O. At high oxygen contents, wu¨stite can co-exist with magnetite, magnetite with hematite and hematite with O2 ðgÞ. This sequence is the one in which oxide layers are disposed within a scale grown isothermally on iron, as shown in Figure 2.3. The locus of scale composition across its width, from pure iron to oxygen gas, can be mapped onto the phase diagram, as shown in the figure. The resulting line is termed a ‘‘diffusion path’’, as it shows the concentration changes which drive diffusion within the reacting system. The significance of the single-phase regions traversed by the diffusion path is clear. However, interpretation of the two-phase regions requires consideration of the phase rule. Consider a system containing C components (chemical species) and consisting of P phases. In principle, each phase can contain all C components, and its composition is specified by C1 variables. When temperature and pressure are included, the state of each phase is completely specified by C+1 variables. For the entire system we thus find that the total number of variables is P(C+1). At equilibrium, a number of equations are in effect among the variables T1 ¼ T2 ¼ Tp

ðP 1 equalitiesÞ

p1 ¼ p2 ¼ pp

ðP 1 equalitiesÞ

m1;1 ¼ m1;2 ¼ m1;p

ðP 1 equalities for component 1Þ

and a similar set of (P1) equations for each of the other components. In all, there are (P1) (C+2) equations. It is seen that the number of variables exceeds the number of equations by a number F: F¼CPþ2

(2.35)

This result is the phase rule, and F represents the number of degrees of freedom available to the system. It tells us the number of variables which can be incrementally changed without altering the number of phases present. Confusion can sometimes arise in determining the number of components, C. The usually stated rule is that C equals the smallest number of constituents whose specification suffices to determine the composition of every phase. Evaluating this number can be a non-trivial exercise in complex chemical systems, but is straightforward for alloy oxidation: C equals the number of elements involved. For a binary oxide, C ¼ 2. The need to specify both arises from the variable composition of oxide and other solid compounds, as is now demonstrated. In an isothermal, isobaric situation, such as the oxidation of iron discussed earlier, the phase rule becomes F¼CP (2.36) For the two-component Fe–O system, a single-phase region is univariant, i.e. its composition can vary. This is self-evidently the case for Fe1dO and Fe3O4 at high temperatures. Although it cannot be seen on the diagram, Fe2O3 is also capable of very small variations in composition. It is this degree of freedom which permits the development of concentration gradients, which in turn drive the diffusion processes supporting scale growth. In binary two-phase

40


Fe

FeO

Fe3O4

Fe2O3

Figure 2.3 Cross-section of oxide scale grown on iron, with diffusion path mapped on phase diagram.


41

regions, it follows from Equation (2.36) that F ¼ 0, and compositions are fixed. In the absence of any concentration gradient, dispersed two-phase regions cannot grow and are not found in the scale. Instead, two-phase regions consist of sharp interfaces, as seen in Figure 2.3. For the same reason, wu¨stite cannot form as particles within the iron. When pure metals are oxidized isothermally, they always grow external scales rather than forming internal oxide precipitates.

2.2.2 Gases containing two reactants Gases containing two or more oxidants are commonly encountered at high temperatures. For example, most fossil fuels contain sulfur, and combustion leads to the formation of SO2 and other gaseous species. If iron is exposed to such a gas, then the possible reactions include 1 Fe þ 12S2 ¼ FeS; K2:37 ¼ (2.37) p S2 1=2

FeO þ

1 2S2

¼ FeS þ

1 2O2 ;

K2:38 ¼

pO2

1=2

p S2

Fe3 O4 þ 32S2 ¼ 3FeS þ 2O2 ;

K2:39 ¼

Fe2 O3 þ S2 ¼ 2FeS þ 32O2 ;

K2:40 ¼

p2O2 3=2

p S2

(2.38)

(2.39)

3=2

pO 2 p S2

(2.40)

as well as reactions producing sulfates, which will be ignored here for the sake of simplicity. The gas phase reactions of importance are 1=2

SO2 ¼ 12S2 þ O2 ;

K2:41 ¼

p S2 p O 2 pSO2

(2.41)

and, at high oxygen partial pressures, SO2 þ 12O2 ¼ SO3 ;

K2:42 ¼

pSO3 1=2

pSO2 pO2

(2.42)

The ternary system Fe–S–O can be analysed thermodynamically in the same way as was done for the Fe–O system, but the multiple equilibria make the process complex. As seen from the phase rule, up to three phases can co-exist at interfaces, and two-phase scale layers can grow. Interpretation of this situation is much easier using a phase diagram, such as the one drawn in Figure 2.4 on the assumption that all solids are pure and immiscible. Logarithmic scales are used for pS2 and pO2 in order to encompass the large ranges involved, and have the advantage of linearizing the equilibrium relationships of reactions (2.37)–(2.40). Thus, for example, the phase boundary

42


Figure 2.4 Thermochemical (Kellogg) diagram for Fe–S–O system at 8001C, showing three possible diffusion paths for reaction with Gas A.

between FeO and FeS is defined as a straight line by the equation log pO2 ¼ log pS2 þ 2 log K2:38

(2.43)

and has a slope equal to one. The diagram unambiguously defines the range of gas compositions in which pure iron is stable as a metal ð pO2 o1 1019 atm and pS2 o7 1010 atm at T ¼ 8001C). It also allows prediction of which of the possible reaction products can co-exist with an equilibrium gas mixture. Thus, for example, at pS2 ¼ 1 107 atm and pO2 ¼ 1 1014 atm, the surface of a scale is expected to be magnetite (point A). However, it is not possible to predict the diffusion path trajectory, from A to B, from thermodynamic information alone. Three possibilities are shown in Figure 2.4, one involving oxide, but the other two also involving sulfide. Since sulfides generally grow much faster than oxides, the question is important and is considered further in Chapters 4, 9 and 10.

2.3. ALLOYS AND SOLID SOLUTIONS Alloy phases are in general solid solutions, and the need arises to specify component activities within them. Returning to Equation (2.3) we note that the changes in composition, dni , now to be considered reflect alteration of solute concentration rather than chemical reaction. Taking the total differential of Equation (2.8) X X dG ¼ mi dni þ ni dmi (2.44) i

i

2.3. Alloys and Solid Solutions

43

and subtracting it from Equation (2.5), we obtain the Gibbs–Duhem equation X 0 ¼ Vdp SdT ni dmi (2.45) i

Again, summations are performed over all components of the system. We consider an isothermal, isobaric system in which, at equilibrium, X ni dmi ¼ 0 (2.46) i

or, dividing by the total number of component moles, nT, to obtain mole fractions X N i dmi ¼ 0 (2.47) Consistent with the approach adopted for ideal gas mixtures (Equation (2.14), the solution component activity is defined through ¯i G ¯ i mi mi ¼ RT ln ai ¼ G

(2.48)

where unit activity corresponds to the standard state in which mi ¼ mi . The choice of standard state is arbitrary, but that of pure solid is convenient. In this case, an ideal solution is defined as one in which the chemical potential of every component is related to its mole fraction by mi mi ¼ RT ln N i

(2.49)

Real solutions deviate from ideality, and are dealt with by defining an activity coefficient, gi , such that the relationship mi mi ¼ RT ln gi N i

(2.50)

holds, whatever the extent of deviation. In general, gi varies with composition, as well as with temperature and pressure. The thermodynamics of solutions can be understood from their enthalpy and entropy of mixing. At constant pressure, application of Equations (2.2) and (2.48) to a particular component in a solution of fixed composition yields ¯ i =TÞ @ðG ¯i ¼H (2.51) @ð1=TÞ and hence

@ ðmi mi Þ T ¯ i H i ¼H @ð1=TÞ

(2.52)

where Hi is the standard enthalpy per mole of unmixed component i, and overscoring indicates the partial molar quantity. Comparison with Equation (2.48) then leads to @ ln ai ¯ i H i R ¼H (2.53) @ð1=TÞ For an ideal solution, ai ¼ N i , and the partial differential in Equation (2.52) is zero, the enthalpy of the dissolved component being equal to its value in the

44


unmixed state. The enthalpy of mixing is defined for the entire solution as X X ¯i Ni H N i H i (2.54) DH m ¼ i

i

DH id m

and in the ideal case, ¼ 0. If Equation (2.48) is multiplied by N i and a sum formed for all components, we obtain X X X ¯i NiG N i Gi ¼ RT N i ln ai (2.55) i

i

i

in which the left-hand side is recognized as the free energy of mixing X DGm ¼ G N i Gi

(2.56)

i

In an ideal solution, therefore, DGid m ¼ RT

X

N i ln N i

and it follows from the equation @DG ¼ DS @T P;ni that DSid m ¼ R

X

N i ln N i

(2.57)

(2.58)

(2.59)

This expression is recognized from the Boltzmann equation S ¼ k ln o where o is a measure of randomness and k the Boltzmann’s constant, as corresponding to a random mixture. This is now illustrated for a binary mixture of A and B DSid m ¼ Smix SA SB ðnA þ nB Þ! ¼ k ln ð2:60Þ nA !nB ! where ni is the number of atoms of the indicated species and o has been evaluated as the number of distinguishable configurations of the nA þ nB atoms. Expanding Equation (2.60) with the aid of Stirling’s approximation ln n! ¼ n ln n n

(2.61)

and the relationship R ¼ kN AV (with N AV equal to Avogadro’s number) leads to DSid m ¼ RðN A ln N A þ N B ln N B Þ

(2.62)

which is merely (2.59) applied to a binary system. Thus an ideal solution is a completely random mixture of constituents which experience the same thermal interaction with all neighbouring atoms, and the entropy of mixing is purely configurational.

2.3. Alloys and Solid Solutions

45

In real solutions, interactions between dissimilar atoms give rise to non-zero ¯ i and thermal contributions to S¯ i . These are conveniently described using H ‘‘excess’’ functions of the sort Gxs ¼ G Gid

(2.63)

¯ i Gi RT ln N i RT ln gi ¼ G

(2.64)

Rewriting Equation (2.46) as and substituting for RT ln N i from Equation (2.49) we find ¯ i Gi Þ ðG ¯ id RT ln gi ¼ ðG i Gi Þ ¯ i Gid ¼G i

¯ xs ¼ H ¯ xs ¯ xs ¼G i TS i Since

¯ id H i

ð2:65Þ

¼ 0, this is equivalent to ¯ i TS¯ xs RT ln gi ¼ H

(2.66)

where the deviation from ideality of component i is seen to arise from its thermal interaction with the solution and the consequent shift in ¯ xs , H ¯ xs thermal entropy. A useful tabulation of partial molar excess quantities G xs and S¯ has been provided by Kubaschewski and Alcock [1] for binary alloy systems. An alternative approach is suited to dilute solutions where the experimental finding is that ai ¼ gi N i

(2.67)

with gi a constant. This is Henry’s law. More generally, the quantity gi varies with composition, and can be expanded, as proposed by Wagner [6], as a Taylor series which to the first order yields X ln gi ¼ ln gi þ ik N k (2.68) where the gi are the Henry’s law coefficients. It can be shown that @ ln gi @ ln gk ¼ ik ¼ ki ¼ @N k @N i

(2.69)

lessening the amount of experimentation needed. Although there are many alternative solution models available, the form (2.68) is a useful one for moderately dilute solutions.

2.3.1 Dissolution of gases in metals In studying the formation of internally precipitated oxides, carbide, etc. (see Figure 1.8), it is necessary to consider the dissolution of the oxidant in the metal, e.g. 1 2O2 ðgÞ

¼O

(2.70)

46


Table 2.2

Oxygen dissolution in metalsa

a w

Metal

¯ O (kJ mol1) DH

DS¯ O (J mol1 K1)

Referencew

Ni a Fe g Fe

182 155.6 175.1

107.6 81.0 98.8

[A6] [A9] [A9]

xs

Referred to Equation (2.72) with pO2 (atm) and N O (mole fraction). References in Appendix D.

Here, and elsewhere in this book, underscoring is used to denote a solute species in a solid. It is convenient to specify concentrations as mole fractions, Ni, and we write 1=2

N O ¼ K70 pO2

(2.71)

which is Sievert’s equation. It was the experimental demonstration of Equation (2.71) which proved that gaseous oxygen, nitrogen and sulfur dissolve in metals as dissociated atoms. The value of K70 is related to that of DG for Equation (2.70) in the usual way, but care is needed in specifying the concentration units and standard state for the solute. In much of the published work, concentration is expressed in wt%, and a standard state of 1 wt% is chosen. It is preferable to use mole fraction, N O , so that ¯ TDS¯ xs þ RT ln N O 1RT ln p (2.72) DGð2:70Þ ¼ DH 2

O2

Data for oxygen solubility in iron and nickel are summarized in Table 2.2, and corresponding data for carbon are provided in Table 9.4. The maximum value of pO2 applicable in Equation (2.71) is the equilibrium value for formation of the lowest metal oxide. Thus, for example, the maximum solubility of oxygen in austenitic iron is set by the Fe/FeO equilibrium. As seen earlier, at T ¼ 1,0001C, pO2 ðFeOÞ ¼ 1:2 1015 atm. Calculating K70 from the data in Table 2.2, this is found to correspond to a solubility limit of 3.7 106 mole fraction in the iron beneath an oxide scale. In the case of an alloy, if sufficient dissolved oxygen is present, it can react with an alloy metal solute to precipitate particles of oxide, a situation considered in the next section.

2.4. CHEMICAL EQUILIBRIA BETWEEN ALLOYS AND GASES 2.4.1 Equilibria between alloys and single oxide Consider a binary alloy A–B reacting with oxygen. In general DG ðAOÞaDG ðBOÞ and one of the metal oxides is more stable than the other. Referring to Table 1.1 we see that, e.g., the alloys Fe–Ni and Fe–Cr are of interest because the growth of NiO or Cr2O3 is much slower than that of FeO. We enquire as to the alloy concentration of nickel or chromium necessary to form the desired oxide. This situation can be formulated as a competitive oxidation reaction, e.g. Ni þFeO ¼ NiO þ Fe

(2.73)

2.4. Chemical Equilibria Between Alloys and Gases

The condition for chemical equilibrium, Equation (2.48) yields

P

47

ni mi ¼ 0, after substitution from

aNiO aFe (2.74) aFeO aNi For simplicity, we approximate the oxides as being pure, immiscible solids, so that aNiO ¼ 1 ¼ aFeO . The standard free energy change is evaluated from the equation DG ¼ mFe þ mNiO mNi mFeO ¼ RT ln

DG ð2:73Þ ¼ DG ðNiOÞ DG ðFeOÞ as +55,760 J mol1 at 1,0001C. Thus, at equilibrium, aNi ¼ 194 aFe

(2.75)

(2.76)

and the alloy needs a very high nickel content. Approximating the alloy as an ideal solution, and rewriting Equation (2.76) as N Ni ¼ 194 (2.77) 1 N Ni we find the solution N Ni ¼ 0:995. It is clear that alloying with nickel cannot be used as a method of achieving oxidation resistance for steel. Turning now to the Fe–Cr alloy, we formulate the competitive reaction 2 Cr þ3FeO ¼ Cr2 O3 þ 3 Fe

(2.78)

for which the equilibrium expression is aCr2 O3 a3Fe DG ð2:78Þ ¼ exp RT a3FeO a2Cr

(2.79)

Pure, immiscible oxides are again assumed so that their activities can be set to unity, and the standard free energy change is evaluated from the equation DG ð2:78Þ ¼ DG ðCr2 O3 Þ 3DG ðFeOÞ

(2.80)

as 244,590 J mol1 at 1,0001C, corresponding to a2Cr ¼ 9 1011 a3Fe

(2.81)

Assuming that in such a dilute solution aFe ¼ N Fe 1, it is found that aCr 1 105 . Data tabulated by Kubaschewski and Alcock [1] for ¯ Cr ¼ þ25; 100 J mol1 and ferritic Fe–Cr alloys show that for N Cr ! O, DH s 1 1 DS¯ Cr ¼ þ10:25 J mol K . Insertion of these values in Equation (2.66) yields the value gCr ¼ 3:1 at 1,0001C. Thus the required chromium activity of 1 105 is equivalent to N Cr 3 106 . Thermodynamically, at least, the use of chromium as a steel alloying addition for oxidation protection is seen to be very attractive. The question of whether the oxide forms as an external scale or as internal precipitates requires kinetic analysis. Assuming for the moment that internal oxidation occurs within a dilute alloy, it is seen that the reaction is one between

48


solute species 2 Cr þ3 O ¼ Cr2 O3

(2.82)

The value of DG (2.82) is found from the reactions 2Cr þ 32O2 ðgÞ ¼ Cr2 O3

(2.83)

Cr ¼ Cr

(2.84)

1 2O 2

(2.85)

¼O

for which we can write DG ð2:83Þ ¼ 1; 120; 270 þ 259:83T J mol1

(2.86)

¯ Cr ð2:84Þ ¼ 25; 100 10:25T þ RT ln N Cr J mol1 DG

(2.87)

¯ O ð2:85Þ ¼ 175; 100 þ 98:8T þ RT ln N O J mol1 DG

(2.88)

where Equation (2.66) has been used to find Equation (2.87), and Equation (2.88) was calculated using data for g Fe provided by Kubaschewski and Alcock [1]. From the equation ¯ Cr ð2:84Þ 3DG ¯ O ð2:85Þ DGð2:82Þ ¼ DG ð2:83Þ 2DG we find, at equilibrium, 0 ¼ DGð2:82Þ ¼ 645; 170 16:07T 2RT ln N Cr 3RT ln N O

(2.89)

and at T ¼ 1,0001C, N 2Cr N 3O ¼ Ksp ¼ 4 1028

(2.90)

The equilibrium constant, Ksp , is known as the solubility product. The maximum value of pO2 available to a dilute Fe–Cr alloy is the level set by the Fe–FeO equilibrium because a scale forms on the alloy surface. As seen earlier, this value is 1:2 1015 atm for pure iron at 1,0001C, and results in N O ¼ 3:1 106 . It follows from Equation (2.90) that the precipitation of Cr2O3 within the alloy would leave an equilibrium value N Cr ¼ 2:8 105 . It is therefore concluded that any Fe–Cr alloy containing more than 28 ppm of chromium can form internal Cr2O3 precipitates when oxidized at 1,0001C. Whether or not an external Cr2O3 scale forms cannot be predicted from thermodynamics alone. The preceding discussion of Fe–Ni and Fe–Cr alloy oxidation has been based on the simplifying assumption that the product oxides are pure, immiscible solids. This assumption is not always valid. The Fe–Ni–O system forms a solid solution spinel phase NixFe3xO4, and the Fe–Cr–O system develops several mixed oxides. These complications are best described with the help of phase diagrams.

2.4.2 Equilibria between alloys and multiple oxides A binary alloy reacting with a single oxidant constitutes a ternary system. The phase assemblages capable of co-existing at local equilibrium at a fixed

2.4. Chemical Equilibria Between Alloys and Gases

49

temperature can be represented by an isothermal section of the phase diagram. An example for Fe–Cr–O shown in Figure 2.5 is drawn as a Gibbs composition triangle. The geometry of the equilateral triangle is such that for any point within the triangle, wherever located, the sum of the perpendiculars to the three sides is always the same. This provides a convenient means of mapping compositions where N Fe þ N Cr þ N O ¼ 1, avoiding the need to calculate the third component which would arise if normal rectangular co-ordinates were used. Single-phase existence regions are marked on the diagram. The two alloy phases are shown on the Fe–Cr binary side of the triangle: austenite, containing N Cr 0:13 and ferrite, with N Cr 0:17. The three iron oxides are shown along the Fe–O binary side and the single chromium oxide on the Cr–O side. It is seen that Fe1dO dissolves a significant amount of chromium, the solubility varying with wu¨stite stoichiometry. The spinel phase Fe3O4 dissolves large amounts of chromium, up to a terminal composition of FeCr2O4. Finally, the structurally isotypic Fe2O3 and Cr2O3 form a continuous solid solution at this temperature. As the phase rule informs us, there are two degrees of freedom within a ternary single-phase region, as is illustrated by the representation on the diagram

Figure 2.5 Isothermal section (1,2001C) of Fe–Cr–O phase diagram with alloy phases omitted for clarity.

50


of single phases as areas. When two phases co-exist only one degree of freedom is available. Two-phase regions separate pairs of adjacent single phases, as shown more clearly in the enlarged schematic diagram of Figure 2.6. Each two-phase region is defined by a set of tie-lines which joins pairs of composition points along the phase boundaries. Thus, for example, compositions of wu¨stite along the line ab equilibrate with spinel compositions along the line cd. For all points on any one tie-line mFe ðWÞ ¼ mFe ðSpÞ mCr ðWÞ ¼ mCr ðSpÞ mO ðWÞ ¼ mO ðSpÞ where W denotes wu¨stite and Sp the spinel. Of course, different tie-lines correspond to different compositions of the phases, and therefore different chemical potentials. The two-phase region is univariant, and this is represented by the lines ab and cd, which define the composition of each phase in terms of a single variable. The two-phase regions bound three-phase triangles, e.g., the wu¨stite–spinel– alloy triangle bde, which represent invariants. All points within the triangle correspond to differing proportions of these three phases, always of the compositions given by the points b, d and e. Thus the relationships mFe ðWÞ ¼ mFe ðSpÞ ¼ mFe ðAlloyÞ, etc. are satisfied. As we have already seen when examining Figure 2.4, ternary phase diagrams have some utility in predicting the outcome of alloy oxidation reactions, but diffusion paths cannot be predicted without additional information. The Fe–Cr–O

d Fe3O4

c

a Fe1-δO b O

Fe e

f

Figure 2.6 Schematic enlargement of part of Figure 2.5.

Cr

2.5. Thermodynamics of Diffusion

51

diagram reveals that a necessary condition for the formation of Cr2O3 at the scale–metal interface is a surface alloy chromium concentration greater than that at the point f, i.e. N Cr 0:04. This is much higher than the value calculated from Equation (2.79) as aCr 104 at 1,2001C. The difference arises from the neglect of spinel formation in the earlier treatment. As the Fe–Cr–O diagram shows, chromium-rich spinel has a stability equal to that of Cr2O3 when N Cr 0:04 at 1,2001C. The competitive oxidation reaction is reformulated as Fe þ43Cr2 O3 ¼ FeCr2 O4 þ 23 Cr

(2.91)

for which DG ð2:91Þ ¼ 56; 690 14:0T J mol1 and 2=3

K¼

aFeCr2 O4 aCr 4=3

aCr2 O3 aFe

(2.92)

Assuming that the oxides are pure and immiscible, and approximating aFe N Fe 1, it is found that DG ¼ RT ln K ¼ 23DmCr

(2.93)

and at 1,2001C, DmCr ¼ 54; 102 J mol1. Using Equation (2.87), it is then found that the value of N Cr satisfying the equilibrium between spinel and Cr2 O3 is 0.03, in reasonable agreement with the phase diagram. Clearly the latter provides a simpler route to the answer, when available for the temperature of interest. The use of the Fe–Cr–O diagram is considered in more detail in Sections 5.3 and 7.3. However, it should be noted that the condition N Cr 40:04 is insufficient to achieve protective Cr2O3 scale formation. The main reason for this is depletion of chromium from the alloy surface by its preferential oxidation. The actual surface concentration is determined by the balance between chromium diffusion from the alloy interior and its removal into the scale.

2.5. THERMODYNAMICS OF DIFFUSION 2.5.1 Driving forces We start by considering the thermodynamic implications of matter diffusing from one part of a system to another. In an isothermal, field-free system, an amount dnA2 of component A passes from region 2 to region 1, each region being regarded as homogeneous. The changes are described using Equation (2.3) dU TdS ¼ p1 dV 1 p2 dV 2 þ ðmA1 mA2 ÞdnA2

(2.94)

and the last term reflects the fact that dnA1 ¼ dnA2 . In a slow process, the pi do not vary, and p1 dV1 p2 dV 2 ¼ dw

(2.95)

52


the amount of work done on the system. From the second law of thermodynamics we have dðU TSÞodw

(2.96)

for a spontaneous process. It follows that the necessary condition for isothermal mass transfer to occur is ðmA1 mA2 ÞdnA2 o0

(2.97)

In other words, the sign of dnA2 is the opposite of the sign of ðmA1 mA2 Þ: if dnA2 is a positive transfer of component A from region 2 to region 1, the chemical potential in region 2 must be greater than in region 1. This important result informs us that diffusion actually occurs from regions of high to low chemical potential, rather than from high to low concentration. Thus the simple description given by Fick’s law for the relationship between flux and concentration gradient in Equation (1.24) J ¼ D

@C @x

can be inaccurate to the point of predicting diffusion in the wrong direction. In developing a more accurate description of diffusion, several approaches are possible. These include geometric, random walk procedures which have been applied to crystalline solids to yield an advanced theory of correlation and isotope effects [7, 8], and the application of absolute rate theory. Before developing the latter treatment, we consider a phenomenological approach based on irreversible thermodynamics. The principal concepts were developed by Onsager [9] and extended by de Groot [10] and Prigogine [11]. Their application to solid-state diffusion has been reviewed a number of times [8, 12, 13]. An essential element of the thermodynamic treatment of diffusion is the postulate that a state of local equilibrium can be adequately approximated in each region of the solid, despite the compositional variation with position within the system. The simultaneous satisfaction of these two requirements is achieved by taking a microscopic volume element which is so small that its composition can be treated as homogeneous. Because the solid is atomically dense, the element contains a statistically meaningful number of particles. A series of such elements describes the diffusion profile within the solid (Figure 3.32). The procedures of irreversible thermodynamics enable us to calculate the rate of entropy production per unit volume, s_, in terms of the various fluxes flowing within the system. The result is a bilinear expression involving the fluxes themselves and a set of thermodynamic forces, Xi, X J i Xi T_s ¼ i

These forces are thereby identified as those responsible for the fluxes, each flux being linearly dependent on all the forces. The description is applicable only to systems that are not far removed from equilibrium, and is therefore appropriate to diffusion in a solid within which the local equilibrium state is closely approached.

2.5. Thermodynamics of Diffusion

53

For isothermal diffusion in a closed, isobaric, field-free n-component system, it is found that n X J i rZi (2.98) T_s ¼ i¼1

where r indicates gradient, i.e. partial derivative with respect to position co-ordinate, and the summation covers all components. Hence the component fluxes are given by a set of linear equations n X Ji ¼ Lij rZj (2.99) j¼1

where the Lij are the Onsager phenomenological coefficients. These each relate the flux of species i to the gradient in species j, and form a square matrix of order equal to the number of system components. The driving forces are seen to be the gradients rZi , known as electrochemical potential gradients. They are defined by Z ¼ m þ qFc

(2.100)

where q is the charge of the species, F is Faraday’s constant and c the local electrostatic potential. Gradients in potential constitute fields, but these are internal to the solid and the conditions for the validity of Equation (2.98) are maintained. In the case of a metallic alloy, the constituent atoms have no effective charge, and the driving force is the chemical potential gradient, rm. This result is intuitively satisfactory in the sense that diffusion is perceived (under the conditions specified earlier) as a process that eliminates differences in chemical potential, thereby achieving equilibration. A more profound result of the irreversible thermodynamic treatment is the recognition that the flux of any component is, in general, dependent on the chemical potential gradients of all components. The Lij ðiajÞ in Equation (2.99) are referred to as off-diagonal coefficients, and represent the ‘‘cross effects’’ between components. These cross effects can arise from thermodynamic interactions (cf 2.68) or from kinetic interactions. Aspects of the latter are outlined in Appendix B for ionic solids. For the moment, we consider the situation where cross effects are small enough to ignore. For one-dimensional diffusion in a binary alloy, the approximation L12 0 yields the simple result J 1 ¼ L11

@m1 @x

(2.101)

which, upon substitution for m1 from Equation (2.50) leads to L11 RT g1 @N 1 N 1 @g1 J1 ¼ þ g1 N 1 @x @x ¼

L11 RT @ ln g1 @N 1 1þ N1 @ ln N 1 @x

(2.102)

54


Noting that the change in molar concentration @C1 ¼ C@N 1 , with C the average molar concentration, it is found from a comparison of Equations (1.24) and (2.102) that L11 RT @ ln g1 D1 ¼ 1þ (2.103) C1 @ ln N 1 This makes clear that chemical diffusion is strongly dependent on the thermodynamic properties of the solid solution, even in the absence of kinetic cross effects. The shortcoming of the phenomenological description (2.99) is that it provides no information on the coefficients Lij relating the diffusion rate to the driving forces. For our present purposes, a more transparent description is provided by an absolute rate theory approach. Before developing this description it is necessary to consider the identity and nature of the diffusing species.

2.5.2 Point defects Solid-state diffusion involves the movement of individual particles (atoms or ions) that constitute the material. These particles are capable of movement because they vibrate around their mean positions and because the existence of defects in the solid crystal permits an occasional vibration to extend into a translation to an available lattice site nearby. Two common defects are illustrated in Figure. 2.7 for the case of a pure, single-component solid: a vacancy, or unoccupied lattice site, and an interstitial atom, i.e. one located between normal lattice sites. A lattice atom can move into an adjacent vacancy, exchanging sites with the defect. Movement via this vacancy mechanism is the most common way in

Figure 2.7 An individual vacancy and interstitial defect in a single-component crystal lattice.

2.6. Absolute Rate Theory Applied to Lattice Particle Diffusion

55

which diffusion occurs. Clearly, the concentration of vacancies present is important in determining the probability of atom translation occurring. The interstitial species can contribute to diffusion simply by moving into an adjacent interstitial site. This is improbable in pure metals, because the atoms are large, but operates for interstitial impurities such as C, H, N and O dissolved in metals. Whichever the mechanism, the concentration of defects is an important factor in the particle movement rates. The question of defect concentrations is now considered. Equilibrium concentrations of point defects in crystals are calculated by the methods of statistical thermodynamics. The application of these methods to crystals has been reviewed in detail by Schottky [14], and their use in diffusion calculations has been explored by several authors, notably Mott and Gurney [15] and Howard and Lidiard [8]. The Gibbs free energy for a monatomic crystal containing nn vacancies and n atoms is G ¼ GO þ nn gn kT ln

ðn þ nn Þ! n!nn !

(2.104)

where gn is the free energy of formation of a vacancy, and the logarithmic term is recognized as the configurational entropy resulting from the presence of defects. It is this term that makes vacancy formation inevitable at all temperatures above absolute zero. The free energy minimum representing the equilibrium state of the crystal defines the chemical potential of the vacancies as zero: @G ¼ mn ¼ 0 (2.105) @nn T;P;N Application of this to Equation (2.103) making use of Stirling’s approximation ðln N! ¼ N ln N NÞ, then yields g

nn Nn ¼ ¼ exp n (2.106) ðn þ nn Þ kT Equation (2.106) is recognized as an equilibrium expression of the same form as Equation (2.23). A more detailed discussion of point defect equilibria in ionic solids is provided in Chapter 3.

2.6. ABSOLUTE RATE THEORY APPLIED TO LATTICE PARTICLE DIFFUSION We turn now to the evaluation of individual particle jump frequencies, using absolute rate theory. The first applications to solid-state diffusion were reported by Wert and Zener [16] and Seitz [17], and subsequent extensions for various cases have been provided by others [18–20]. When a particle moves from one lattice position to another, it passes through an intermediate state that has a higher energy because adjacent particles must be perturbed from their mean lattice positions in order to accommodate the passage of the moving particle. During this lattice distortion, an activated complex involving the two interchanging species (e.g., particle plus vacancy) is formed.

56


The activity ain of the complex is described via the equilibrium constant Kin for its formation: ain DH in DSin ¼ Kin ¼ exp exp (2.107) ai an RT R where DH in is the enthalpy and DSin the entropy of complex formation. A profile of the periodic internal energy surface in a direction parallel to that of diffusion is shown in Figure 2.8. An electrostatic field can be externally imposed, or can arise through the movement of the charged species themselves, and will in this case be aligned with the diffusion direction. The height of the energy barrier to diffusion of a charged species is modified by the field, being lower for downfield movement than for upfield movement of an appropriately charged species. It will be assumed that the field does not affect DSin . We may write for the interchange of species i and a vacancy between planes (1) and (2) separated by a distance l, as shown in Figure 2.8. DSin J ¼ mlnin exp R ( " # DU in qFðcð0Þ cð1Þ Þ ð1Þ ð2Þ ai an exp RT " #) DU in qFðcð2Þ cð0Þ Þ ð1Þ ð2Þ an ai exp RT g1 in

ð2:108Þ

where m is the volume concentration of lattice sites, nin a kinetic frequency term and gin the activity coefficient for the transition-state complex. Here q is the effective charge of the vacancy, that of the cation being zero. Superscripts in

Figure 2.8 Potential energy profile in diffusion direction: upper curve, no electrostatic field; lower curve showing effect of electrostatic field.

2.6. Absolute Rate Theory Applied to Lattice Particle Diffusion

57

parentheses represent the location in Figure 2.8 at which the quantity in question is evaluated. An alternative treatment of the particle movement kinetics might be found more appealing. The rate at which ions can move from plane (1) to (2) must be proportional to the probability of finding an ion at position (1), að1Þ i , to the availability of a vacancy for it to jump into, að2Þ , to the frequency, n , with which in n the ion approaches the intervening energy barrier, and to the Boltzmann factor giving the proportion of ions possessed of sufficient energy to surmount the barrier, expðDGin =RTÞ. The overall probability of the event occurring is then given by DGin ð1Þ ð2Þ nin ai an exp RT Calculation of the corresponding flux from this probability by multiplying by the area density of sites on plane (1), ml, and expansion of DGin leads to the first term in Equation (2.108). The net flux is then calculated by subtracting the equivalent expression for the rate at which ions return from the second plane to the first, and Equation (2.108) results. Equation (2.108) is cleared of common terms and subjected to Taylor series ð2Þ ð2Þ expansion of the terms að2Þ n and ai expðqFc =RTÞ. Retention of linear terms, in the case where the field is not inordinately high, leads to J¼

ml2 nin Kin ai an rmi rmn qFrc RT

(2.109)

which, upon substituting from Equation (2.100) becomes J¼

ml2 nin Kin ai an rZi rZn RT

(2.110)

Expressions of this sort always apply to pairwise site exchanges. For diffusion of non-charged species in a metal or an alloy, q ¼ 0, and we obtain Ji ¼

ml2 nin Kin ai an rmi rmn RT

(2.111)

If the equilibrium condition of Equation (2.105) is realized and the off-diagonal terms in Equation (2.99) are ignored, this result simplifies to Equation (2.101), with 2 ml L11 ¼ nin Kin ai an RT and therefore, in the dilute (ideal) solution approximation, it is found from Equation (2.103) that Di ¼ l2 nin exp

DHin DSin exp Nn RT R

(2.112)

58


We now combine Equations (2.106) and (2.112) to determine the temperature dependence of the diffusion coefficient Q D ¼ DO exp (2.113) RT where Q ¼ DH in þ DH n with DH n the enthalpy (per mole) for vacancy formation, and the remaining constants have been collected in DO . The expected Arrhenius form is arrived at and is commonly used to interpolate or extrapolate sparse experimental data.

2.7. DIFFUSION IN ALLOYS 2.7.1 Origins of cross effects Equation (2.99) may be rewritten for atomic diffusion in an n-component system as n X Ji ¼ Lij rmj (2.114) j¼1

when it is clear that cross effects can arise through either kinetic interactions, as represented by the Onsager coefficients, Lij , or thermodynamic interactions, represented by the variation of chemical potential with composition. Experimental diffusion data is almost always collected in the form of concentration rather than chemical potential. For this reason, it is desirable to use a generalized form of Fick’s law Ji ¼

n1 X

Dij rCj

(2.115)

j¼1

where the Dij are functions of the kinetic coefficients Lij and also reflect the dependence of chemical potential on composition. A useful example is provided by the application of Wagner’s dilute solution model (Equation (2.68)). For a ternary system, it is found that D11 ¼ RT L11 ð11 þ 1=N 1 Þ þ L12 21 D12 ¼ RT L11 12 þ L12 ð22 þ 1=N 2 Þ D21 ¼ RT L22 21 þ L21 ð11 þ 1=N 1 Þ ð2:116Þ D22 ¼ RT L22 ð22 þ 1=N 2 Þ þ L21 12 In an ideal solution all ij ¼ 0, and the Dij reduce to the purely kinetic form. RTLij Dij ¼ (2.117) Nj For real solutions, if no kinetic cross effects occur, i.e. Lij ðiajÞ ¼ 0, it is clear that the diffusional cross terms Dij (i6¼j) are nevertheless non-zero. In this case the

2.7. Diffusion in Alloys

59

dilute solution limit ðN 1 ; N 2 ! 0Þ may be described by

and

D12 N 1 12 ¼ D11 1 þ 11 N 1

(2.118)

D21 N 2 21 ¼ D22 1 þ 22 N 2

(2.119)

Thus the ternary coefficients are determined uniquely by the binary ones in dilute solutions. For interstitial diffusion there are negligible correlations between crystal sublattices, so that the approximation Lij ðiajÞ ¼ 0 is valid. Practical examples are steels Fe–C–M in which M is a substitutional metal (Si, Mn, Ni, Cr, Mo, Co) and the carbon is interstitial. The variation of DCM =DCC with carbon concentration is shown in Figure 2.9, compared with the predictions of Equation (2.118) using independently measured interaction parameters [21]. Agreement is quite good. Correlations of this sort can contribute to an understanding of alloy carburization reactions. A detailed diffusion analysis employing cross terms was used by Nesbitt [22] in analysing the ability of Ni–Cr–Al alloys to supply aluminium to the surface to reheal damaged alumina scales. In the regime examined, the value of DAlCr was as high as 0:5DAlAl , leading to a significant contribution from the chromium

Figure 2.9 Variation of D12 =D11 with carbon concentration ðC1 Þ, with solid lines representing thermodynamic prediction. After Brown and Kirkaldy [21]. Published with permission from the Minerals, Metals and Materials Society.

60


concentration gradient to aluminium diffusion. Cross effects between dissolved oxygen and alloy components were considered by Whittle et al. [23, 24] in analysing alloy surface behaviour as oxygen diffused inwards. This analysis revealed that the cross effect between oxygen and a selectively oxidized component was important in driving the oxygen flux. Writing the equations (2.115) as J A ¼ DAA rCA DAO rCO J O ¼ DOA rCA DOO rCO we consider their application to diffusion in the subsurface zone of alloy AB, in which A is selectively oxidized. In the case of Ni–Cr at 1,0001C, the self-diffusion coefficients of oxygen and chromium are of order 107 and 1012 cm2 s1, respectively, and rCCr rCO . Consequently, even for small values of DOA, the off-diagonal term is important, and likely predominant, in the expression for JO.

2.7.2 Kirkendall effect We now consider another way in which diffusional interactions arise between components sharing the same lattice, but possessing different intrinsic mobilities. Their experimental manifestation is known as the Kirkendall effect, and its measurement is used to evaluate a composite alloy diffusion coefficient defined below. Consider a binary alloy AB in which one-dimensional diffusion occurs via atom-vacancy exchanges, and Equation (2.111) applies to both A and B, so that DA and DB correspond to D1 and D2 in Equation (2.112). In general the fluxes are not equal and opposite. Thus, if DA 4DB in a sample initially rich in A on the left, there will be an excess flux of A from left to right over B atoms moving to the left. Consequently, the diffusion zone as a whole drifts to the left, compensating for the accumulation of matter and hydrostatic pressure that would otherwise occur on the right. Thus the lattice planes which define the frame of reference within which Equation (2.111) applies are themselves moving. Since the diffusion zone is generally a small part of a larger sample, measurements of position that are referred to the end of the sample (the laboratory reference frame) are affected by this drift, and so, in consequence, is the estimate of diffusion rate. The problem is the same as that faced by a navigator measuring the speed of a plane using its airspeed when a wind is blowing. A knowledge of the wind speed relative to the ground resolves the difficulty. Formally, the situation is dealt with by relating the two frames of reference. In the laboratory frame of the diffusion measurement, we use n X Ji ¼ 0 (2.120) i¼1

which is equivalent to a volume-fixed frame of reference if the partial molar volumes are approximately equal. In the lattice frame, where Equation (2.111)

2.7. Diffusion in Alloys

61

applies, the expression (2.120) does not. We therefore write for the lattice frame, using J 0i to denote its fluxes J 0A þ J 0B ¼ J 0V

(2.121)

If the lattice frame moves with respect to the laboratory frame with a velocity n, then J i ¼ J 0i þ Ci n;

i ¼ A; B

(2.122)

where the non-primed fluxes refer to the laboratory frame. These equations are solved using Equation (2.120) to obtain n¼

J 0A þ J 0B CA þ CB

(2.123)

or, upon resubstitution J A ¼ J B ¼ N B J 0A N A J 0B

(2.124)

In the simple situation in which the off-diagonal Onsager coefficients are set equal to zero, and local equilibrium applies ðrmn ¼ 0Þ, Equations (2.111) and (2.112) simplify to Fick’s law (1.24). Since, moreover, for an isobaric system in ¯ are equal which partial molar volumes V ¯ A þ CB Þ ¼ 1 VðC (2.125) combination of Equations (2.123)–(2.125), and (1.24) yields ¯ A DB Þ @CA n ¼ VðD @x J A ¼ ðN B DA þ N A DB Þ

@CA @x

(2.126) (2.127)

The value of n can be measured using inert markers, as is now discussed. The first demonstration of lattice drift was performed by Smigelskas and Kirkendall [25] using the diffusion arrangement shown schematically in

Figure 2.10 Lattice drift experiment of Smigelskas and Kirkendall.

62


Figure 2.10. Molybdenum wires (the markers) were attached to a block of brass (Cu–Zn) and then an outer copper layer applied by electroplating. Annealing this couple at high temperature caused rapid outward diffusion of the more mobile zinc from the brass into the copper, slower inward diffusion of copper and inward drift of the molybdenum markers. The effect is quite general and is widely used in diffusion measurements. For an infinite diffusion couple (sample much larger than the diffusion zone), it can be shown that x CA ¼ CA ðlÞ; l ¼ 1=2 (2.128) t and hence n¼

DA DB dCA dl t1=2

(2.129)

Because the markers are located at a point of fixed composition and therefore at a fixed value of dCA =dl, Equation (2.129) integrates immediately to yield xm ¼ 2ðDA DB Þ

dCA 1=2 t dl

(2.130)

for the marker displacement. The quantities DA and DB are known as the intrinsic diffusion coefficients because they refer to diffusion with respect to the lattice planes in the presence of an activity gradient. It is necessary now to relate these to the measured tracer coefficients, DAn and DBn . These refer to the diffusional intermixing of different isotopes of the same atom or ion, and usually the enthalpy of mixing is small and the solution ideal. In this case, Equation (2.103) simplifies to L11 RT D1 ¼ (2.131) C1 However, the intrinsic diffusion coefficient refers to a non-ideal solution and Equation (2.103) must be used without approximation. As a result, d ln gA n DA ¼ DA 1 þ (2.132) d ln N A Using the Gibbs–Duhem equation for equilibrium in a solution (Equation (2.46)) we may write d ln gA d ln gB 1þ ¼1þ (2.133) d ln N A d ln N B then Equation (2.127) becomes with

~ J A ¼ DrN A

(2.134)

d ln g ~ D ¼ ðN B DAn þ N A DBn Þ 1 þ d ln N

(2.135)

~ is the chemical diffusion This is the Darken–Hartley–Crank equation [26, 27] and D ~ coefficient. The quantity D is also called the alloy diffusion coefficient, and is

63

2.8. Diffusion Couples and the Measurement of Diffusion Coefficients

obtained from a diffusion couple measurement (Section 2.8). If markers are used in the measurement, values of the self-diffusion coefficients DA and DB may also be obtained. This provides a powerful technique for exploring the compositional dependence of the Di . The above analysis has been extended to multicomponent systems (see e.g., Ref. [12]). The lack of balance among the intrinsic diffusive flows always leads to a compensating mass flow of material. That is to say, diffusional cross effects arise even in the absence of kinetic or thermodynamic correlations. Thus even a component with a negligible intrinsic mobility will move. The simple form of Fick’s law fails, and the generalized form (2.115) must be used.

2.8. DIFFUSION COUPLES AND THE MEASUREMENT OF DIFFUSION COEFFICIENTS In the most common diffusion measurements, the movement of a system towards homogeneity is observed and compared with the predictions of the diffusion equations. These equations, together with appropriate boundary conditions, yield solutions for the one-dimensional case of the general form Ci ¼ Ci ðx; t; DÞ

(2.136)

Thus D is evaluated by fitting the expressions to experimental data Ci ¼ Ci ðx; tÞ. We consider here diffusion couple experiments in which two different homogeneous mixtures are brought into contact at a planar interface and diffusion observed along a direction normal to it. Two types of diffusion couple are important. If sample dimensions and the period of diffusion are such that concentrations at the ends of the sample do not change, then the experiment is described as an infinite diffusion couple. These couples are used to measure chemical diffusion. In a tracer diffusion measurement, the couple consists of a homogeneous block of material and a thin film of isotopically labelled but compositionally identical material. The two types of diffusion couple are shown schematically in Figure 2.11. Predicted profiles of the form (2.136) are obtained from Fick’s law (Equation (1.24)), which is subject to the continuity condition @Ci @J ¼ i (2.137) @t @x leading to Fick’s second law @Ci @2 Ci ¼D 2 @t @x

(2.138)

where D has been approximated as constant. The solution of Equation (2.138) is required for the appropriate boundary and initial conditions. Methods and a number of solutions are available from Carslaw and Jaeger [28] and Crank [29]. The thin-film solution applies to the one-dimensional tracer diffusion experiment

64


t=0 A

t>0 B

CA (a)

A*

A

CA* (b)

Figure 2.11 Diffusion couples before and after diffusion. (a) Infinite couple and (b) thin-film (tracer experiment) couple.

of Figure 2.11b: Cðx; tÞ ¼

a expðx2 =4DAn tÞ 2ðpDAn tÞ1=2

(2.139)

where a is the amount of labelled material per unit area of film. After annealing, the couple is sectioned and the tracer concentration measured as a function of position. The value of DAn is then evaluated from a logarithmic plot according to Equation (2.139). For an infinite diffusion couple consisting initially of one half containing a uniform concentration C0 and the other a concentration C1 (Figure 2.11a), after diffusion time t we have Cðx; tÞ C0 1 x ¼ 1 erf pffiffiffiffiffiffi (2.140) 2 C1 C0 2 Dt where erf is the Gaussian error function Z z 2 erf ðzÞ ¼ pffiffiffi expðu2 Þdu p 0

(2.141)

Corresponding solutions are available for ternary systems [12]. Properties of the error function together with an abbreviated table of its values are shown in Appendix C. The above solutions rely on D being constant. This will apply in the tracer diffusion case, and Equation (2.139) can be used directly. However, it is improbable in the presence of a concentration gradient, the situation obtaining for the diffusion couple described by Equation (2.140) and Figure 2.11a. Either the difference C1C0 must be kept small, or the analysis of Boltzmann [30] and

2.8. Diffusion Couples and the Measurement of Diffusion Coefficients

65

pffiffi Matano [31] must be used in this case. Here the new variable l ¼ x= t is introduced. The initial conditions for the infinite diffusion couple C ¼ C0 for xo0 and C ¼ 0 for xW 0 at t ¼ 0 are independent of x, apart from the discontinuity at x ¼ 0 (Figure 2.11a). They can be described as C ¼ C0 at l ¼ 1 and C ¼ 0 at l ¼ þ1, and the Boltzmann–Matano analysis applies. Fick’s Law can then be transformed into an ordinary differential equation l dC d dC ¼ D 2 dl dl dl which integrates between zero and a value C0 such that 0oC0 oC0 , and for a fixed value of t, to yield C¼C0 0 Z 1 C¼C dC xdC ¼ Dt 2 C¼0 dx C¼0 Noting that dC=dx ¼ 0 at c ¼ 0 and c ¼ C0 , we arrive at the final solution Z C0 1 dx 0 ~ xdC (2.142) DðC Þ ¼ 2 dC 0 with

Z

C1

xdC ¼ 0

(2.143)

C0

defining the origin of co-ordinates. Graphical or numerical evaluations of the ~ 0 Þ, differential and the integral in Equation (2.142) are used to evaluate DðC as shown in Figure 2.12. Observation of marker movement in the diffusion

Figure 2.12 Concentration profile in infinite couple after diffusion, showing how the quantities required for the Boltzmann–Matano analysis (2.142) are evaluated.

66


couple then allows calculation of the self-diffusion coefficients DA and DB from Equations (2.130) to (2.135).

2.8.1 Diffusion data for alloys It is often expedient to ignore diffusional interactions, either because the necessary data are not available or because an approximate calculation is all that is required. In such cases, we rely on self-diffusion coefficients, usually measured on binary alloys. These apply to either substitutional (vacancy-exchange) diffusion of metal components or interstitial diffusion of solute oxidants. Most measurements have been carried out using tracer diffusion experiments. These are related to the intrinsic, or self-diffusion coefficients through Equation (2.132) which, in a near ideal solution approximates to DA DAn In some cases, not even tracer data are available, but a chemical diffusion ~ may have been measured. If the diffusing species of interest is both coefficient, D, dilute and highly mobile, then the expression ~ ¼ N A DB þ N B DA D can be approximated as ~ ¼ DB D A selection of self-diffusion coefficient data for binary alloys is given in Appendix D. For multicomponent systems where Equation (2.115) holds, the Matano analysis can also be applied. The origin is then defined by the condition (2.143) being simultaneously satisfied for all components. Data are available for a number of ternary alloy systems in a useful review compiled by Dayananda [32]. A rather different treatment is required for diffusion in ionic solids, where the charges on individual species must be explicitly recognized. This is dealt with in Chapters 3 and 5.

2.9. INTERFACIAL PROCESSES AND GAS PHASE MASS TRANSFER As seen earlier, linear oxidation kinetics are expected if a surface or interfacial process is rate controlling. We consider the scale–gas interface, examining first the situation where the supply of oxidizing gas is not rate determining, and gas adsorption equilibrium can be expected. The very initial reaction between gas and bare metal is not considered here. Instead, a uniform oxide scale is assumed to have already formed.

2.9.1 Gas adsorption The reaction may be written as O2 ðgÞ þ S ¼ O2 jS

(a)

2.9. Interfacial Processes and Gas Phase Mass Transfer

O2 jS þ S ¼ OjS þ OjS kc

OjS!Oxide

67

(b) (c)

where S denotes a surface site, O2 jS and OjS adsorbed molecules and atoms and kc the rate constant for the slow step (c). The pre-equilibria (a) and (b) lead to ½O2 jS ¼ ½SKa pO2 ½OjS ¼ ð½SKb ½O2 jSÞ

(2.144) 1=2

(2.145)

where square brackets indicate area concentration. Substitution of Equation (2.144) into Equation (2.145) leads to 1=2

½OjS ¼ ½SðKa Kb Þ1=2 pO2

(2.146)

Assuming now that the surface area and total concentration of sites are constant M ¼ ½S þ ½O2 jS þ ½OjS and substituting from Equations (2.144) and (2.146), one obtains M i ½S ¼ h 1=2 1 þ Ka pO2 þ ðKa Kb Þ1=2 pO2

(2.147)

(2.148)

The constant M is of order 1015 cm2. Combination of Equations (2.146), (2.147) and the rate equation for reaction (c) then leads to the result 1=2

Rate ¼

kc MðKa Kb Þ1=2 pO2 h i 1=2 1=2 1 þ ðKa Kb Þ1=2 pO2 1 þ ðKa =Kb Þ1=2 pO2

(2.149)

The rate is of course constant at fixed pO2 , but varies in a complex way with oxygen potential. Three limiting cases can be seen. At sufficiently low values of pO2 , Ka pO2 1 ðKa Kb pO2 Þ1=2 so that 1=2

Rate ¼ kc MðKa Kb Þ1=2 pO2

(2.150)

At higher pO2 values, the competitive adsorption of molecular and atomic oxygen must be considered. When atomic adsorption predominates over the molecular form 1=2 Ka 1=2 pO2 1 (2.151) Kb the term in square brackets in the denominator approximates to unity. If, furthermore, the surface is saturated, i.e. 1=2

ðKa Kb Þ1=2 pO2 1

(2.152)

then the oxidation rate is simply Rate ¼ kc M

(2.153)

68


and independent of oxygen partial pressure. However, if molecular adsorption predominates, the converse of Equation (2.151) is true and the rate equation becomes 1=2

Rate ¼

kc MðKa Kb Þ1=2 pO2 1 þ Ka pO2

(2.154)

If the surface is close to saturation with molecular oxygen, Ka pO2 1, then an inverse dependence of the rate on pO2 is predicted. Competitive adsorption treatments are particularly useful in analysing oxidation kinetics in more complex gases such as CO+CO2 mixtures [33], but have also been used for oxygen alone [34], where the competing species are O and O2. Adsorption equilibrium can only be supported if gas species arrive at the scale surface quickly enough to keep up with reaction (c). This may not be the case if the oxidant partial pressure is very low. Two such situations are of interest: pure oxidant at low pressure, and oxidant as a dilute component of an otherwise inert gas.

2.9.2 Gas phase mass transfer at low pressure This situation is described using the Hertz–Langmuir–Knudsen equation, which derives from the kinetic theory of gases [35]. In the ideal case, pi ki ¼ (2.155) ð2pmi kTÞ1=2 where ki is the rate, pi the partial pressure and mi the mass of a molecule of species i, and k is Boltzmann’s constant. This expression describes both the rate of arrival of a low-pressure gas at a flat surface and, equally, the rate of evaporation into vacuum of the same species. Using practical units of g cm2 s1 for ki and atm for pi, the rate is calculated as MWi 1=2 (2.156) ki ¼ 44:3 pi T where MWi is the species molecular weight. This equation can be used to investigate gas phase mass transfer inside porous or cracked oxide scales, as shown schematically in Figure 2.13. The question addressed is whether the values of pO2 expected from local equilibrium with the surrounding oxide can sustain significant mass transfer across the cavities to support oxide growth. If the oxide is FeO at 1,0001C, then the equilibrium pO2 is in the range 1 1015 2:8 1013 atm. Oxygen transfer rates calculated from Equation (2.156) are found to be 7 1015 2 1012 g cm2 s1 , corresponding to a thickness of wu¨stite grown on the inner side of the cavity at rates of about 1–100 nm per year. Thus closed pores and cracks are seen to be effective local barriers to continued scale growth if O2 is the only vapour species within them.

2.9. Interfacial Processes and Gas Phase Mass Transfer

M

MO

(1 − δ )MO + δ O2 = M 1−δ O

k O2

2

Figure 2.13

69

Gas

M 1−δ O = (1 − δ )MO +

δ 2

O2

Vapour phase mass transport inside pores within a growing scale.

2.9.3 Mass transfer in dilute gases The usual situation encountered in practice and in the laboratory is a gas mixture flowing past a metal surface. The Hertz–Langmuir–Knudsen equation cannot be used in this situation, because of the multiple collisions occurring between oxygen and other molecules. The rate of transfer is governed by the gas flow rate, the width of the gas boundary layer (which is retained by viscous drag), oxygen partial pressure and gas mixture properties. A readable account of how this problem is solved has been provided by Gaskell [35]. The flux of oxygen to a flat surface from a gas flowing parallel to it is given by km ðoÞ ðp pðiÞ Þ (2.157) RT where km is a mass transfer coefficient and pðoÞ and pðiÞ the oxygen partial pressures in the bulk gas and at the solid surface, respectively. The mass transfer coefficient is given by 4 1=6

DAB n 1=2 km ¼ 0:664 (2.158) L ng J¼

where DAB is the diffusion coefficient in a binary gas A-B, ng the kinematic viscosity, n the linear velocity of the gas and L the length of surface. The diffusion coefficient is found from the Chapman–Enskog formulation [36, 37] of the kinetic theory of gases. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1:858 103 T 3 ð1=MWA þ 1=MWB Þ DAB ¼ (2.159) Ps2AB OD;AB Here MWA and MWB are the molecular weights of the two gas species, sAB their average collision cross-section, OD;AB a collision integral and the numerical factor ˚ for s, Poise for Z arises from the use of non-SI units. These are cm2 s1 for DAB, A

70


and atm for P, with a dimensionless O. Tabulations of s and data permitting calculation of OD are available [35, 38]. The kinematic viscosity is defined as Zg (2.160) ng ¼ r where r is the gas density, here in g cm3, and the viscosity, Zg , is given by pffiffiffiffiffiffiffiffi 2:669 105 MT (2.161) Zg ¼ s2 O with O a different collision integral, tabulated values of which are also available. The description is valid when the dimensionless Schmidt number n Sc ¼ (2.162) D has a value between 0.6 and 50. The oxidation of steel in reheat atmospheres considered in Section 1.1 provides an example where this description can be used. Laboratory simulations of reheat furnace gases have been used [39] to investigate the effect of combustion stoichiometry on steel scaling rates at 1,1001C. The equilibrium gas composition corresponding to combustion of methane with 1% excess air is shown in Table 2.3. Reaction of a low carbon steel with this gas produced a scale consisting of single-phase FeO, which thickened according to linear kinetics. These are not the results to be expected if local equilibrium at the scale–gas interface were achieved. In that case, the surface oxide would be Fe3O4, an additional layer of FeO would grow beneath it, and diffusion-controlled, parabolic kinetics would result. The situation at the scale–gas interface therefore requires analysis. Gas phase mass transfer rates calculated from Equations (2.157) and (2.158) are shown in Table 2.3. The measured oxidation rate corresponded to J O ¼ 2 107 mol O cm2 s1 . As is clear from the comparison, molecular oxygen was not a significant reactant species, as its gas phase mass transfer rate was too slow to keep up with the scaling rate. However, gas phase transport of H2O and CO2 was more than fast enough to sustain the observed oxidation rate. Thus it is concluded on this basis that H2O and/or CO2 were the reactants, but that gas phase transport was not rate controlling, because of the relative abundance of these species. This conclusion was confirmed by the magnitude of the activation energy for the linear rate constant, measured as 135 kJ mol1. This value is much greater than the temperature effect predicted from Equations (2.157) and (2.158). Other measurements [40] of linear steel oxidation rates in dilute O2–N2 gases, where the rate is controlled by gaseous mass transfer, yielded an apparent Table 2.3 Equilibrium partial pressures and corresponding oxygen transport rates in 101% stoichiometric combustion gas at 1,1001C [39] CO

CO2

p (atm) 9.42 10 2 1 JO (mol cm s ) 5 107

2

1.3 10

N2 6

O2

H 2O 3

0.72 1.9 10 1.2 108

H2

0.188 1.3 106 6 1 10

2.10. Mechanical Effects: Stresses in Oxide Scales

71

activation energy of 17 kJ mol1. Thus it is eventually concluded that the ratecontrolling step in the linear oxidation process observed in this combustion gas at 1,1001C is a surface reaction. As seen above, quantitative gas phase mass transfer calculations can be useful in determining the feasibility of vapour transport within closed scale voids and cracks, in identifying reactant species in gas mixtures, and in distinguishing the contributions to rate control by mass transfer and interfacial reactions.

2.10. MECHANICAL EFFECTS: STRESSES IN OXIDE SCALES Oxide scales are usually subject to mechanical stress. This is of interest, because if the oxide stress is high enough, the scale will deform or even fracture. Mechanical failure of the scale will, at least temporarily, destroy the scale’s protective function. It is desirable to be able to predict such events, and preferably to design alloys which retain their scales intact. In the absence of external loading, a compressive stress in the oxide is balanced by a tensile one in the metal. Thus the mechanical state of the oxide reflects changes occurring in both phases. It is convenient to divide these into two classes: stresses developed during oxidation and those developed during temperature change. These matters have been reviewed several times, and the reader is referred in particular to Stringer [41], Taniguchi [42], Stott and Atkinson [43], Evans [44] and Schutze [45].

2.10.1 Stresses developed during oxidation Oxidation causes volume changes which, if constrained by specimen shape or constitution, are accommodated by deformation or strain in the oxide, ox . Pilling and Bedworth [46] considered scale growth occurring by inward oxygen transport, and recognized that if the ratio V ox =V m was greater than one, the resulting expansion could put the oxide into compression, whereas if the ratio was less than unity, tension and a discontinuous oxide could result. In the practically relevant case of V ox =V m 41, if the scale grows by outward metal diffusion, new oxide is formed at the free, unconstrained oxide–gas interface, and no strain results. However, if the scale grows by inward oxygen diffusion, the volume change accompanying new oxide formation has to be accommodated at the metal–oxide interface, leading to " # Vox 1=3 ox ¼ 1 (2.163) Vm if no other stress relieving mechanism is available. If the oxide behaves elastically, the corresponding growth stress would be Eox sox ¼ ox (2.164) 1 nP where Eox is the elastic modulus and nP is Poisson’s ratio for the oxide.

72


The Pilling–Bedworth description is conceptually useful, but of little quantitative use. Firstly, many oxides grow predominantly by outward metal diffusion, and the model does not apply. Even in the case of inward diffusion, the stress levels calculated from Equations (2.163) and (2.164) are found to be impossibly high [47]. The solution to this problem is proposed [45] to be mixed diffusion of both metal and oxide, leading to growth of new oxide both at the scale surface and within its interior. Mixed transport can become possible as a result of grain boundaries or microcracks facilitating oxygen access. The relative contributions of the different growth sites are expected to vary with the factors affecting individual metal and oxygen transport mechanisms (T,pO2 , oxide grain size and substrate preparation). Kofstad, in his review [48] of the extensive data available for chromium oxidation, demonstrated that the Cr2O3 scale grows by counter-current diffusion of metal and oxygen along grain boundaries. Formation of new oxide in the boundaries results in lateral stress development, deformation of the scale and its partial detachment from the metal surface. Plastic deformation increases with decreasing oxygen activity and smaller grain size. Using the assumption of elastic oxide behaviour, Srolowitz and Ramanarayanan [49] analysed the effect of new oxide growth at grain boundaries within the scale. When the grain size, d X, they find sox ¼

4Gox di 2dð1 2nP Þ

(2.165)

where Gox is the shear modulus of the oxide and di the width of new oxide grown at internal grain boundaries. Additional stresses arise when curved metal surfaces are oxidized. Consider first an infinite plane metal surface being oxidized, with V ox =VM 41. As metal is consumed, the metal–oxide interface recedes. The oxide scale, which is chemically bonded to the metal surface, remains attached and moves with the retreating metal. If scale growth is sustained wholly by metal transport, no stress results. Consider now a convex metal surface oxidizing and receding. As the oxide follows it, it is compressed tangentially into the smaller volume formerly occupied by metal. Simultaneously, a radial tensile stress develops. The differing consequences for concave and convex slopes, inward and outward diffusion and V ox =VM 4oro1 have been explored by Hancock and Hurst [50] and Christ et al. [51]. The qualitative results for V ox =VM 41 are illustrated in Figure 2.14. Oxide stresses can also be caused in other ways during oxidation. Dissolution of oxygen into metals with high solubilities (e.g., Ta, Ti) can cause large expansions [52]. Internal precipitation of oxides [53] or oxidation of internal carbides [54] in alloy subsurface regions can cause very large volume expansions and tensile stresses in external scales. Phase changes in an alloy resulting from selective oxidation also cause volume changes. In general, any deformation of the substrate metal, including that due to external loading, is transferred to an


73

Figure 2.14 Effects of metal surface curvature on growth stress development in oxide scales. Grey shading indicates newly grown oxide. Based on Refs [50, 51].

adherent scale, because ox ¼ M

(2.166)

for an intact scale.

2.10.2 Stresses developed during temperature change Metals and oxides can have significantly different coefficients of thermal expansion, a, as seen in Table 2.4. The stress produced in an intact, adherent scale by a temperature change, DT, is given by [55] sox ¼

Eox DTðaM aox Þ ½ðEox =EM ÞðXox =XM Þð1 nPðMÞ Þ þ ð1 nðoxÞ P Þ

(2.167)

where Xox and XM are the thicknesses of scale and substrate metal, nðoxÞ and nPðMÞ P are the Poisson’s ratio values for scale and metal and the values of E and a have been approximated as independent of temperature. For thin scales on thick substrates, Equation (2.167) is adequately approximated by sox ¼

Eox DTðaM aox Þ 1 nox P

(2.168)

providing that linear elastic behaviour is in effect. Clearly, the thermally induced stress is dependent on the magnitude of the temperature change and the difference between coefficients of thermal expansion. As seen in Table 2.4, values for metals are usually greater than for oxides,

74

Table 2.4


Coefficients of thermal expansion (a)

Material

106 a (K1)

T (1C)

Reference

Fe FeO FeO Fe2O3 Ni NiO Co CoO Cr Cr2O3 Cr2O3 Alloy 800 12 Cr, 1 Mo steel a Al2 O3 (single xl) Kanthal

15.3 15.0 12.2 14.9 17.6 17.1 14.0 15.0 9.5 7.3 8.5 16.2–19.2 10.8–13.3 5.1–9.8 15

0–900 400–800 100–1,000 20–900 0–1,000 20–1,000 25–350 20–900 0–1,000 100–1,000 400–800 20–1,000 20–600 28–1,165 20–1,000

[59] [60] [59] [59] [59] [59] [59] [59] [59] [59] [60] [45] [45] [61] Kanthal AB

and rapidly cooling an oxidized metal from high temperature will put the scale in compression. If the resulting stress is high enough, the scale suffers mechanical failure. The tabulated data explains why such failure is rare for oxide scales on nickel and cobalt, but common for Cr2O3 scales on austenitic chromia-forming materials such as Alloy 800. The development of stresses, both during oxidation and during temperature change, has been described here in terms of linear elastic behaviour. Thus it has been assumed that no stress relief mechanisms are in effect. This is, in fact, not the case, and a variety of outcomes can be arrived at. Stress can be relieved by plastic deformation, a process which occurs at low temperatures by dislocation movement, and at high temperature by creep. The latter is a time-dependent material flow resulting from lattice diffusion (Nabarro–Herring creep [56, 57]) or grain boundary diffusion (Coble creep [58]). Creep processes are strongly dependent on grain size and impurities, and in oxides to some extent on oxygen activity. An example where the stresses at a sample corner have been relieved by creep in the alloy is shown in Figure 2.15. If the stresses in an oxide scale develop to levels larger than can be accommodated by elastic strain, and if plastic deformation is insufficient to relieve the stresses, mechanical disruption of the system results. Depending on its stress state, properties and microstructure (which can change with temperature) the scale can fracture, form multiple microcracks, disbond from the metal (or separate along scale layer interfaces) or spall. Spallation means the separation and ejection of fragments from the scale, and is illustrated in Figure 2.16. The situation is analysed using a fracture mechanics approach, on the highly probable assumption that small defects are always present in the scale.


Figure 2.15

75

Deformation of cast, heat-resisting steel sample corner during oxidation at 1,1001C.

Figure 2.16 Partial spallation of alumina scale from a platinum-modified nickel aluminide coating system resulting from temperature cycling between 1,2001C and 801C.

76


The energy available from releasing the stress by growing a crack is compared with the energy required to form the newly created surfaces. In the linear-elastic regime, the critical stress, sc , is found in a simple calculation to be given by pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi sc pa ¼ Gc E0 (2.169) where a denotes the geometric dimensions of a pre-existing defect (length of a surface crack or half length of an internal crack), Gc the energy needed to create new surface and E0 the effective elastic modulus. The left side of Equation (2.169) represents the stress intensity factor, and the right side the fracture toughness of the material. Measured values of the latter are found [44] to be of order 1 MPa m1/2 for oxide scales. For a much more detailed discussion of the mechanical properties of oxide scales, the reader is referred to the book by Schu¨tze [45]. Mechanical failure of scales leading to their spallation, and the consequential acceleration in alloy failure rates are discussed in detail in Chapter 11. Alloy design strategies for minimizing spallation are considered in Section 7.5.

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26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61.

77

L.S. Darken, Trans. AIME, 184, 175 (1948). G.S. Hartley and J. Crank, Trans. Faraday Soc., 45, 801 (1949). H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, Clarendon Press, Oxford (1959). J. Crank, The Mathematics of Diffusion, Oxford University Press, Oxford (1970). L. Boltzmann, Ann. Phys., 53, 960 (1894). C. Matano, Japan Phys., 8, 109 (1933). M.A. Dayananda, in Diffusion in Metals and Alloys, Landolt and Bernstein, ed. H. Mehrer, Springer-Verlag, Berlin (1991), Ser. III, Vol. 26, p. 372. F.S. Pettit and J.B. Wagner, Acta Met., 12, 35 (1964). D.J. Young and M. Cohen, J. Electrochem. Soc., 124, 775 (1977). D.R. Gaskell, An Introduction to Transport Phenomena in Materials Engineering, Macmillan, New York (1992). D. Enskog, Arkiv. Met. Astronom. Fyz., 16, 16 (1922). S. Chapman and T.G. Cowling, The Mathematical Theory of Non-uniform Gases, Cambridge University Press (1939). R.A. Svehla, Estimated Viscosities and Thermal Conductivities of Gases at High Temperatures, NASA Technical Report R-132, NASA Lewis, Cleveland, OH (1961). V.H.J. Lee, B. Gleeson and D.J. Young, Oxid. Met., 63, 15 (2005). X.H. Abuluwefa, R.I.L. Guthrie and F. Ajersch, Oxid. Met., 46, 423 (1996). J. Stringer, Corros. Sci., 10, 513 (1970). S. Taniguchi, Trans. ISIJ, 25, 3 (1985). F.H. Stott and A. Atkinson, Mater. High Temp., 12, 195 (1994). H.E. Evans, Int. Mater. Rev., 40, 1 (1995). M. Schu¨tze, Protective Oxide Scales and their Breakdown, Institute of Corrosion and Wiley, Chichester (1997). N.B. Pilling and R.E. Bedworth, J. Inst. Met., 29, 529 (1923). D.J. Baxter and K. Natesan, Rev. High Temp. Mater., 5, 149 (1983). P. Kofstad, High Temperature Corrosion, Elsevier Applied Science, London (1988). D.A. Srolowitz and T.A. Ramanarayanan, Oxid. Met., 22, 133 (1984). P. Hancock and R.C. Hurst, in Advances in Corrosion Science and Technology, eds. M.G. Fontana and R.W. Staehle, Plenum, New York (1974), Vol. 4, p. 1. W. Christ, A. Rahmel and M. Schu¨tze, Oxid. Met., 31, 1 (1989). R.E. Pawel, J.V. Cathcart and J.J. Campbell, J. Electrochem. Soc, 110, 551 (1963). J. Litz, A. Rahmel and M. Schorr, Oxid. Met., 30, 95 (1988). N. Belen, P. Tomaszewicz and D.J. Young, Oxid. Met., 22, 227 (1984). J.K. Tien and J.M. Davidson, in Stress Effects and the Oxidation of Metals Proc. TMS-AIME Fall Meeting, ed. J.V. Cathcart, TMS-AIME, New York (1975), p. 200. F.R.N. Nabarro, in Rep. Conf. on the Strength of Solids, Physical Society, London (1948), p. 15. C. Herring, J. Appl. Phys., 21, 437 (1950). R.L. Coble, J. Appl. Phys., 34, 1679 (1963). R.F. Tylecote, J. Iron Steel Inst., 196, 135 (1960). J. Robertson and M.I. Manning, Mater. Sci. Technol., 6, 81 (1990). J.K. Tien and J.M. Davidson, Adv. Corros. Sci. Technol., 7, 1 (1980).

FURTHER READING 1. Chemical Thermodynamics and Phase Equilibria 1. K. Denbigh, The Principles of Chemical Equilibrium: With Applications in Chemistry and Chemical Engineering, 4th ed, Cambridge University Press (1981). 2. D.R. Gaskell, Introduction to the Thermodynamics of Materials, 4th ed, Taylor and Francis, Washington, DC (2003). 3. C.H.P. Lupis, Chemical Thermodynamics of Materials, North-Holland, New York (1983).

78


4. O. Kubaschewski, C.B. Alcock and P.J. Spencer, Materials Thermochemistry, Pergamon Press (1993). 5. R.A. Swalin, Thermodynamics of Solids, 2nd ed, Wiley-Interscience, New York (1972). 6. H.B. Callen, Thermodynamics and an Introduction to Thermostatics, 2nd ed, Wiley, New York (1985). 7. M. Hillert, Phase Equilibria, Phase Diagrams and Phase Transformations, Their Thermodynamic Basis, Cambridge University Press (1998). 8. M. Hansen, Constitution of Binary Alloys, McGraw-Hill, New York (1958), 2nd ed; See also First supplement, R.P. Elliott (1965); Second supplement, F.A. Shunk (1969). 9. E.M. Levin, C.R. Robbins and H.F. McMurdie, Phase Diagrams for Ceramists, American Ceramic Society, Inc., Columbia, OH (1969), 2nd ed. See also supplements (1969, 1975).

2. Diffusion in Solids 1. N.F. Mott and R.W. Gurney, Electronic Processes in Ionic Crystals, Oxford University Press (1940). 2. J.H. Holloway, Atom Movements ASM, Cleveland (1951). 3. R.M. Barrer, Diffusion in and through Solids, Cambridge University Press (1951). 4. P.G. Shewmon, Diffusion in Solids, 2nd ed, Minerals, Metals and Materials Society, Warrendale, PA (1989). 5. J. Philibert, Atom Movements: Diffusion and Mass Transport in Solids, Les Editions de Physique (1991). 6. H. Mehrer, Diffusion in Solids: Fundamentals, Methods, Materials, Diffusion Controlled Processes, Springer, Berlin (2007). 7. I. Kaur and W. Gust, Fundamentals of Grain and Interphase Boundary Diffusion, 2nd ed, Ziegler Press, Stuttgart (1989). 8. J.S. Kirkaldy and D.J. Young, Diffusion in the Condensed State, Institute of Metals, London (1987).

3. Point Defects in Solids 1. N.F. Mott and R.W. Gurney, Electronic Processes in Ionic Crystals, Clarendon Press, Oxford (1940). 2. F.A. Kro¨ger, The Chemistry of Imperfect Crystals, 2nd ed, North-Holland, Amsterdam (1973). 3. P. Kofstad, Nonstoichiometry, Diffusion and Electrical Conductivity in Binary Metal Oxides, Wiley-Interscience, New York (1972). 4. C.P. Flynn, Point Defects and Diffusion, Oxford University Press (1972). 5. H. Schmalzried, Chemical Kinetics of Solids, VCH, Weinheim (1995). 6. J. Maier, Physical Chemistry of Ionic Materials: Ions and Electrons in Solids, Wiley, Chichester (2004).

4. Mass Transfer in Fluids 1. D.R. Gaskell, An Introduction to Transport Phenomena in Materials Engineering, Macmillan, New York (1992). 2. R.B. Bird, W.E. Stewart and E.N. Lightfoot, Transport Phenomena, 2nd ed, Wiley, New York (2002). 3. R.A. Svehla, Estimated Viscosities and Thermal Conductivities of Gases at High Temperatures, NASA Technical Report R-132, NASA-Lewis, Cleveland, OH (1961).

References

79

5. Mechanical Behaviour of Scales 1. M. Schu¨tze, Protective Oxide Scales and their Breakdown, Institute of Corrosion and Wiley, Chichester (1997). 2. Mechanical Properties of Protective Oxide Scales, Special Issue of Mater. High Temp., 12 (1994).

CHAPT ER

3 Oxidation of Pure Metals

Contents

3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7.

Experimental Findings Use of Phase Diagrams Point Defects and Non-Stoichiometry in Ionic Oxides Lattice Species and Structural Units in Ionic Oxides Gibbs–Duhem Equation for Defective Solid Oxides Lattice Diffusion and Oxide Scaling: Wagner’s Model Validation of Wagner’s Model 3.7.1 Oxidation of nickel 3.7.2 Oxidation of cobalt 3.7.3 Oxidation of iron 3.7.4 Sulfidation of iron 3.7.5 Effects of oxidant partial pressure on the parabolic rate constant 3.7.6 Effect of temperature on the parabolic rate constant 3.7.7 Other systems 3.7.8 Utility of Wagner’s theory 3.8. Impurity Effects on Lattice Diffusion 3.9. Microstructural Effects 3.9.1 Grain boundary diffusion 3.9.2 Multilayer scale growth 3.9.3 Development of macroscopic defects and scale detachment 3.10. Reactions Not Controlled by Solid-State Diffusion 3.10.1 Oxidation of iron at low pO2 to form wu¨stite only 3.10.2 Oxidation of silicon 3.11. The Value of Thermodynamic and Kinetic Analysis References

82 84 85 89 91 93 96 97 98 101 104 107 109 111 112 113 115 116 122 124 127 127 131 133 135

Reaction of a pure metal with a single oxidant (oxygen, carbon, nitrogen, sulfur or a halogen) is considered. Most metals present in alloys used at high temperature form solid oxides, carbides or nitrides, but sulfides have lower melting points than the corresponding oxides, and liquid formation must sometimes be considered. We commence by surveying a selected set of experimental findings. The goal is to follow the development of a theoretical framework devised to provide an understanding of those findings, and which can be used as a predictive basis for corrosion rates under different conditions. 81

82

Chapter 3 Oxidation of Pure Metals

3.1. EXPERIMENTAL FINDINGS Cross-sections of oxide scales grown on iron, nickel and chromium are compared in Figure 3.1, and the complex sulfide scale grown on nickel is shown in Figure 3.2. All of these scales were found to thicken according to parabolic kinetics. X2 ¼ 2kp t

(3.1)

a result seen earlier to correspond to rate control by diffusion through the scale. It is to be expected then that the relative rates would correspond to the nature of the oxides. Representative values of kp are summarized in Table 3.1. Under the

Figure 3.1 Cross-sections of oxide scales on Fe reprinted from [1] with permission from La Revue de Metallurgie, Ni [3]. Reproduced by permission of The Electrochemical Society and Cr [4] with kind permission from Springer Science and Business Media.

Figure 3.2 Fracture cross-section of sulfide scale on Ni [5]. With kind permission from Springer Science and Business Media.

3.1. Experimental Findings

83

conditions examined, the scale grown on iron is approximately 95% FeO, 4% Fe3O4 and 1% Fe2O3 [1]. To a reasonable approximation, then, the kp value for iron oxidation represents the growth of the FeO layer. Noting that FeO, CoO and NiO all have the same crystal structure (isotypic with face-centred-cubic NaCl) it is seen that their growth rates, nonetheless, differ widely. The Cr2O3 phase has a hexagonal crystal structure, and is therefore not to be compared on this basis with the cubic oxides. Finally, it is seen that iron sulfidises much more rapidly than it oxidizes. The rate constant values in Table 3.1 are for specific temperatures and pressures. It is common that the temperature dependence can be described by the Arrhenius relationship Q kp ¼ kO exp (3.2) RT where Q is a constant known as the effective activation energy and kO is also a constant. As seen in Table 3.2, values of Q and kO differ widely from one metal to another. In some cases, different values apply for the same metal in different temperature regimes. An example of the oxygen pressure effect on kp is shown in Figure 3.3. The linearity of the log–log plots demonstrates the applicability of the relationship 1=n

kp ¼ kO pO2

(3.3)

where kO and n are temperature independent constants.

Table 3.1

Selected scaling parabolic rate constants, kp(cm2s1)

Metal

Gas

Fe Co Ni Cr Fe Co

Table 3.2

kp

T (1C)

Air (1 atm) O2 (1 atm) O2 (1 atm) O2 (1 atm) S2 (1 atm) S2 (1 atm)

Reference 7

1,000 1,000 1,000 1,000 900 700

2 10 3.3 109 9 1011 6 1014 7 106 2 107

[1] [2] [3] [4] [5] [6]

Arnhenius activation energy (Equation 3.2) for oxide scale growth

Metal/gas

T (1C)

Q (kJ mol1)

Reference

Fe/O2 Co/O2

700–1,000 800–950 950–1,150 600–1,100 1,100–1,400 980–1,200

164 230 150 120 240 243

[1] [2]

Ni/O2 Cr/O2

[3] [4]


log10 (kp /mg2cm-4h-1)

84

1100 °C

1000

100 950 °C

10 0.01

0.1

1 log10 (pO2 /atm)

10

100

Figure 3.3 Oxygen partial pressure effects on kp for cobalt. Data from [2].

To account properly for these observations, it is necessary to analyse more carefully the diffusion processes which support scale growth and determine their rates. Such an analysis was first carried out by Wagner [9], and we follow his treatment here, rephrasing it in terms of the Kroger–Vink description [10] of the defect solid state. A central assumption of Wagner’s theory of scale growth is that the process is supported by diffusion of crystalline lattice species through the scale. Thus the oxide is considered to be compact and free of pores and cracks. Any effect of grain boundaries and other extended defects is ignored, and attention is focused on the movement of individual lattice site, or ‘‘point’’, defects. The nature of these defects is considered first, and the relationship between defect concentration and oxide non-stoichiometry is developed. A technique of grouping point defects as ‘‘structural units’’ is used to relate micro- and macroscopic levels of thermodynamic description. Point defect diffusion is then described, and its use in the classical Wagner treatment explored. The utility of this description in accounting for experimental observations is then examined. Finally, the limitations of this treatment are identified, and their consequences for scale growth kinetics are examined.

3.2. USE OF PHASE DIAGRAMS The Wagner theory describes steady-state kinetics, controlled by diffusion within a scale under fixed boundary conditions. Thus the chemical potential of diffusing species at the metal–scale, scale–gas and any intermediate interfaces are supposed to be time invariant. In this event, local equilibrium will be in effect at those boundaries, which should therefore correspond to boundaries defined by

3.3. Point Defects and Non-Stoichiometry in Ionic Oxides

85

the metal-oxidant phase diagram. A first step in verifying that a scaling reaction is diffusion controlled is to test the validity of this proposition. We saw earlier that the three-layered oxide scale grown on iron at temperatures above 5701C was as predicted from the Fe–O phase diagram (see Figure 2.2). A more quantitative test is possible with sulfides, because EPMA can be used to measure both metal and oxidant concentrations at precisely defined (71 mm) locations within a scale. Results obtained by Bastow and Wood [7] for the nickel sulfide scale are compared with the Ni–S phase diagram in Figure 3.4, where agreement is seen to be good. In the case of reaction products with significant deviations from stoichiometry, their composition will vary with position within the scale, from the metalrich to the oxidant-rich sides of the oxide field defined by the phase diagram. This is easier to measure in sulfides than in oxides, because of the greater sensitivity of the EPMA technique to the high atomic weight sulfur. Results for an Fe1dS scale in Figure 3.4(c) show that the expected compositional gradient was indeed developed.

3.3. POINT DEFECTS AND NON-STOICHIOMETRY IN IONIC OXIDES For isothermal diffusion in the absence of external fields, there is no net flow of charge. Any physically realistic mechanism must therefore involve the movement of groups of species which conserve charge and, of course, sites. As will be demonstrated in the next section, such groups fit the definition of ‘‘relative building units’’ conceived of by Schottky [9] and Kroger et al. [10] in the development of a thermodynamic description of point defect equilibria. Since these units can be used to represent both diffusion and equilibrium, they form an appropriate link between the transport properties and local equilibrium state of a solid. In what follows, we employ the defect notation of Kroger and Vink [9] wherein the oxide lattice species are represented by the symbol SX M . Here the subscript represents the normal occupancy in a perfect crystal of the site in question, and the principal symbol represents the species actually occupying the site. The superscript represents the charge of the species relative to normal site occupancy with a prime indicating a negative, a dot positive and a cross zero charge. Thus, for example, the principal lattice species in magnesio–wu¨stite, X X (Fe,Mg)O, are the two cations FeX Fe , MgFe and the anion OO . Defect species are now introduced. Following the early work of Frenkel [11], Schottky and Wagner [12] and Jost [13], we consider first the lattice defects which can arise in a homogeneous, crystalline ionic solid. Firstly, individual lattice sites can be vacant. In a binary 0 00 oxide MO, the possibilities are V X M ,V M and V M , representing different ionization X states, on the cation sublattice, plus V O , etc. on the anion sublattice. In addition, interstitial species, e.g. Mi and O00i are possible, with the subscript i denoting an interstitial lattice site. Interstitial oxygen is unusual, because its large size makes interstitial occupancy energetically improbable in most oxides. The

86


Figure 3.4 (a) Phase diagram for Ni–S system; (b) microanalysis of compositional variation in sulfide scale on Ni at 4481C [7] (with kind permission from Springer Science and Business Media); (c) microanalysis of deviation from stoichiometry in Fe1dS scale grown on Fe at 7001C [8]. Reproduced by permission of The Electrochemical Society.

3.3. Point Defects and Non-Stoichiometry in Ionic Oxides

87

formation of charged defects always occurs in matching sets, to balance electrostatic charge. Schottky defects consist of cation and anion vacancies, e.g. V 00Ni þ V O in nickel oxide. Frenkel defects consist of matched vacancies and X interstitials, e.g. V 00Cd þ Cdi in CdO and V X O þ Oi in UO2. As seen in Section 2.2, defects of this sort always exist at temperatures above 0 K. However, their existence does not account for non-stoichiometry in oxides, for example the large deviations from stoichiometry observed in Fe1dO (see Figure 2.2). In fact, that particular system is complicated by interactions between the highly concentrated defects. We consider instead deviations from stoichiometry in a model oxide M1dO in which it is assumed fully ionized cation and anion vacancies can form. Using equilibrium expressions of the form of Equation (2.106) we write the Schottky reaction 0 ¼ V 00M þ V O

(3.4)

for which nVM nVO (3.5) N2 with KS the Schottky equilibrium constant and N ¼ n+nv , the total number of sites on each sublattice, N M and N O , which are here taken as equal for a divalent metal oxide. Deviations from stoichiometry are achieved by interchange of matter, usually oxygen, with the surrounding environment. In the metal-deficit (dW0) range KS ¼

1 2O2 ðgÞ

00 ¼ OX O þ V M þ 2h

(3.6)

and KP ¼

nVM n2h 1=2

N 3 pO2

(3.7)

while in the metal excess (do0) range 0 1 OX O ¼ V O þ 2e þ 2O2 ðgÞ

(3.8)

1=2

KN ¼

nVO n2e pO2

(3.9) N3 Here KP and KN are equilibrium constants, e0 an electron and h a positive hole: the metal excess oxide exhibits n-type semiconductivity and the metal-deficit oxide shows p-type behaviour. It is noted in formulating these equations that sites can be created or destroyed, as in Equation (3.6), but the crystalline phase is preserved by maintaining the site ratio N M =N O constant, unity in this case. Mass is conserved, effective charge is conserved and the electroneutrality of the compound is always preserved. Note also that adoption of the ‘‘effective charge’’ description means that charge is associated only with defect species. This avoids the clumsiness of counting ions, and comparing large numbers (B1022 cm3) to arrive at very small differences.

88


The relationship between intrinsic disorder, i.e. the concentration of defects when d ¼ 0, extent of non-stoichiometry and pO2 is of interest. The deviation from stoichiometry is nVM nVO (3.10) d¼ N The vacancy concentrations are found from Equations (3.7) and (3.9) using the approximations nh ¼ 2nVM

(3.11)

ne ¼ 2nVO

(3.12)

for charge balance in the relevant regimes, and their substitution into Equation (3.10) yields 1=3 1=3 KP KN 1=6 ð1=6Þ d¼ pO 2 pO 2 (3.13) 4 4 This is the desired relationship between non-stoichiometry and pO2 . We now relate it to the conditions for stoichiometry. Defining pðoÞ O2 as the equilibrium partial pressure at which the compound is stoichiometric, d ¼ 0, we find from Equation (3.13) that KN ¼ KP pðoÞ O2

(3.14)

When d ¼ 0, it follows from Equations (3.5) and (3.10) that the intrinsic disorder, D, is given by nðoÞ nðoÞ 1=2 V VM ¼ O ¼ KS N N Combination of Equations (3.5), (3.7) and (3.9) leads to D¼

KP KN ¼ KS K2e

(3.15)

(3.16)

where Ke ¼ nh ne corresponding to the electron–hole pair formation equilibrium 0 ¼ h þ e 0 Substitution from Equations (3.14) and (3.16) into Equation (3.13) yields 8 !1=6 ! 9 1=3 < ðoÞ 1=6 = p p K e 1=6 O2 O2 d ¼ KS : pðoÞ ; 4 pO2

(3.17)

(3.18)

O2

which upon substitution from Equation (3.15) leads to 8 !1=6 !1=6 9 = pðoÞ DKe 1=3 < pO2 O2 d¼ : pðoÞ ; 4 pO2 O2

(3.19)

89

3.4. Lattice Species and Structural Units in Ionic Oxides

As pointed out by Greenwood [14], this general description reveals that the greater the intrinsic disorder, D, of the stoichiometric compound, the smaller is the relative partial pressure change required to produce a given deviation from stoichiometry. Conversely, oxides which are close to stoichiometric have low D values. This is true despite the fact that compounds with the same defect type will evidence the same functional relationship between d and pO2 . The applicability of equations like (3.19) is, of course, limited to the oxide phase field, and can be of even narrower applicability if defect interactions become important. It is clear from Equation (3.19) that an oxygen potential gradient across an oxide scale gives rise to a defect gradient. It is this gradient which provides the mechanism for diffusion through an oxide scale bounded on one side by metal and on the other by oxygen gas. An example is shown in Figure 3.4(c).

3.4. LATTICE SPECIES AND STRUCTURAL UNITS IN IONIC OXIDES Consider a p-type oxide MO, containing fully ionized vacancies and positive holes, under isothermal, isobaric conditions. The lattice species are X 00 MX M ; V M ; h ; OO

and thus outnumber the single thermodynamically independent compositional variable available to the binary oxide. The removal of the dependencies among the set is accomplished by the application of the physical constraints which exist for the species. In a crystalline solid the ratio of cation to anion sites is fixed X 00 nðMX M Þ þ nðV M Þ ¼ nðOO Þ

(3.20)

and, in the absence of a field, the system is charge neutral 2nðV00M Þ þ nðh Þ ¼ 0

(3.21)

The use of these relationships has been explored by Kroger et al. [15] in arriving at their definition of building units. Building units are groups of lattice species having such a composition that the requirements Equations (3.20) and (3.21) are met when the group is added to the crystal. The obvious unit for MO is X fMX M þ OO g. A subset of building units is comprised of ‘‘relative building units’’. These are defined relative to the perfect crystal and consist of the difference between a lattice species and the lattice species corresponding to normal site occupancy. Thus relative building units represent a change in composition X resulting from the replacement of one species with another, e.g. fBX M AM g in a substitutional solid solution. Since relative building units represent compositional change they can be used X to describe diffusion. It is clear that a flux of units fBX M AM g corresponds to interdiffusion of cations A and B via a site-exchange process. The formulation of suitable relative building units emerges from the flux constraints which are analogous to the site and charge density constraints Equations (3.20) and (3.21). In the case of one-dimensional diffusion in the model system under discussion,

90


these constraints are, for a ternary oxide JA þ JB þ JV ¼ 0 ¼ J0

(3.22)

2J V ¼ J h

(3.23)

where the fluxes, J, are measured within the solvent-fixed reference frame provided by an immobile anion lattice. It follows that movement of a vacancy must be accompanied by movement of positive holes and is associated with an opposing flux of cations. Relative building units, U i , which describe these exchanges are: 00 U 1 fAX M V M 2h g

(3.24)

00 U 2 fBX M V M 2h g

(3.25)

X U 3 fBX M AM g

(3.26)

of which one unit is seen to be redundant. A further unit not contributing to diffusion but necessary to complete the structure is 00 U 4 fOX O V M 2h g

(3.27)

It is clear that combination in the appropriate proportions of units 1, 2 and 4 yields a solid (A,B)O of any desired degree of substitution and nonstoichiometry. Thermodynamic meaning is now attached to the relative building units by considering the reactions which lead to the introduction of point defects into the compound AðgÞ þ V00M þ 2h ¼ AX M

(3.28)

BðgÞ þ V00M þ 2h ¼ BX M

(3.29)

00 1 OX O þ V M þ 2h ¼ 2O2 ðgÞ

(3.30)

These equilibria are described by their corresponding Gibbs equations which, under isothermal field-free conditions, may be written in terms of molar concentrations, m, and electrochemical potentials, Z, as X Zi dmi ¼ 0 i

in each case. Since the dmi are related via the reaction stoichiometry coefficients, v, we may write X n i Zi ¼ 0 (3.31) i

whence mA ¼ ZðAM Þ ZðV 00M Þ 2Zðh Þ ¼ mðU 1 Þ

(3.32)

mB ¼ ZðBM Þ ZðV00M Þ 2Zðh Þ ¼ mðU 2 Þ

(3.33)

3.5. Gibbs–Duhem Equation for Defective Solid Oxides

1 2mO2

¼ ZðOO Þ þ ZðV00M Þ þ 2Zðh Þ ¼ mðU 3 Þ

91

(3.34)

and the potentials of U 1 ; U 2 and U 4 are seen to be the chemical potentials of the constituent elements, mi . The electrochemical potentials of individual lattice species cannot be measured. Moreover they depend on the local electrostatic potential, c, through the definition ZðSZ Þ ¼ mðSZ Þ þ ZFc where Z is the effective charge of the species and F the Faraday. The value of c is also inaccessible to measurement. It is apparent that appropriate grouping of species leads to the avoidance altogether of the need to consider directly the electrostatic potential or individual species’ chemical potentials. These quantities are indeterminate within the formalism, just as they are experimentally inaccessible. Since it is not possible to add, or remove, or diffuse lattice species other than in a way which conserves charge and lattice sites, the use of relative building units is entirely consistent with the fact that the thermodynamics and diffusion kinetics of ionic crystals can always be described in terms of elemental chemical potentials. Relative building units provide a link between the macroscopic thermodynamic/kinetic properties and the point defect structure.

3.5. GIBBS–DUHEM EQUATION FOR DEFECTIVE SOLID OXIDES For an isothermal, isobaric and chemically equilibrated system, the Gibbs– Duhem equation X ni dmi ¼ 0 (2.46) i

relates the chemical potentials of the constituent elements. The relationship applies to an open system, i.e. one which can exchange matter with its surroundings. It is therefore appropriate to the case of a solid oxide which achieves equilibrium via the transfer of oxygen to or from the ambient gas phase. As we have seen, such an oxide is generally non-stoichiometric, its composition varying continuously with oxygen activity. Such an oxide may be regarded as a solution composed of an oxide of chosen reference composition and an excess amount of one constituent. We consider here a pure binary metal-deficit oxide of composition M1dO. It is frequently convenient, if not always realistic, to adopt as a reference the stoichiometric composition MO. The formation of the metal-deficit oxide solution may then be represented as d ð1 dÞMOðsÞ þ O2 ðgÞ ¼ M1d OðsÞ 2 Although one cannot write a Gibbs equilibrium equation for this, or any other, solution formation process (because the composition of the product

92


varies with aO ), the Gibbs–Duhem equation is clearly of the form ð1 dÞdmMO þ d dmO ¼ 0

(3.35)

This result informs us that the chemical potential of the reference composition oxide varies with oxygen activity. Alternatively, one might consider the solution M1dO as being formed from its elements and write ð1 dÞdmM þ dmO ¼ 0

(3.36)

The alternative expressions given by Equations (3.35) and (3.36) are linked via the statement of equilibrium for formation of the reference oxide MðsÞ þ 12O2 ðgÞ ¼ MOðsÞ the Gibbs equation for which is dmM þ dmO ¼ dmMO

(3.37)

Since the Gibbs–Duhem equation represents the means of removing redundancy among a set of chemical potentials, it need not have a unique form. The several different, but equivalent, forms of the equation are related by the equilibria which exist among the various chemical species. Similarly, it is possible to write the Gibbs–Duhem equation in terms of lattice and defect species because the electrochemical potentials of the species are related via the appropriate building units to the chemical potentials of the elements. Thus substitution of the relationships of Equations (3.32) and (3.34) for doubly changed vacancies in a binary oxide MO into Equation (3.36) leads immediately to X 00 ð1 dÞ dZðMX M Þ þ dZðOO Þ þ ddZðV M Þ þ 2ddZðhÞ ¼ 0

(3.38)

It follows from the site and charge balances of Equations (3.20) and (3.21) that d¼

nðV 00M Þ nðOX OÞ

1d¼

nðMX MÞ X nðOO Þ

(3.39)

(3.40)

Substitution from Equations (3.39), (3.40) and (3.21) into Equation (3.38) then yields X X X 00 00 nðMX M ÞdZðMM Þ þ nðOO ÞdZðOO Þ þ nðV M ÞdZðV M Þ þ 2nðh ÞdZðh Þ ¼ 0

(3.41)

which is the form appropriate to individual species. The elemental form (3.36) and the lattice species form (3.41) of the Gibbs–Duhem equation are completely consistent. This is a necessary consequence of the imposed condition of local equilibrium expressed through Equations (3.32) and (3.34). Similar analyses can be performed for other defect types, with the same general conclusion being reached [17].

3.6. Lattice Diffusion and Oxide Scaling: Wagner’s Model

93

3.6. LATTICE DIFFUSION AND OXIDE SCALING: WAGNER’S MODEL Wagner’s original treatment [9, 16] was of critical importance in providing an understanding of the particle (atomic or ionic) processes occurring within a growing oxide scale, thereby leading to a capacity to predict the effects on oxidation rate of changes in temperature, oxide chemistry, etc. The treatment is based on the assumption that lattice diffusion of ions or the transport of free carriers (electrons or positive holes) controls scaling rates. For diffusion to be rate controlling, the scale boundaries must achieve local equilibrium. This requires that the processes occurring at the metal–scale and scale–gas interfaces are so fast that they do not contribute to rate control, and may be regarded as at equilibrium. Although this will not be the case at the very beginning of reaction, equilibrium is quickly established once a continuous scale is formed, providing that the supply of gaseous oxidant is abundant. If diffusion by lattice species is to be rate controlling, then no other diffusion process can contribute significantly to mass transfer. Thus the scale must be dense (i.e. non-porous) and adherent to the metal, so that gas phase transport within the scale is unimportant. Furthermore, the scale must contain a relatively low density of grain boundaries and dislocations so that their contribution to diffusion is unimportant, and the oxide lattice (or volume) diffusion properties dictate mass transfer rates. The Wagner model is illustrated in Figure 3.5 for the more common case of cation transport. Oxygen anion transport can sometimes occur, usually via vacancy movement. In his original model, Wagner proposed that ions and electronic species migrated independently. This is correct only to the extent that (a) charge separation can be sustained within the oxide and (b) the oxide is

Metal

Oxide Scale

Gas

ao

JV′′ Jh•

M + 2h• + VM′′ = M×M Figure 3.5

1

2

O 2 = O×O + VM′′ + 2h•

Schematic view of Wagner’s diffusion model for cation vacancy transport.

94


thermodynamically and kinetically ideal, so that the cross-terms in a complete diffusion description (see Equation 2.99) can be ignored. The latter point has been made by Wagner [18] and others [19, 20]. Wagner solved the transport problem by writing two equations, for ionic and electron species, in terms of their electrochemical potential gradients. These were of the form (2.99) without cross terms and written in terms of mobilities, Bi : J i ¼ Ci Bi rZi

(3.42)

Here the species mobility is defined as its drift velocity under an electrochemical potential gradient of unity. Comparison of Equations (3.42) and (2.99) yields L11 ¼ C1 B1

(3.43)

when cross terms are ignored. If, furthermore, the system is field free (as in, e.g. a tracer diffusion experiment) and thermodynamically ideal, we have from Equation (2.103) L11 RT (3.44) D1 ¼ C1 whence B1 RT ¼ D1

(3.45)

a form of the Nernst–Einstein relationship between diffusion and mobility. Consider the growth of a p-type (metal deficit) binary oxide scale sustained by metal vacancy diffusion. Writing Equation (3.42) explicitly, one obtains @mi þ 2FE (3.46) J V ¼ CV BV @x @mh FE (3.47) J h ¼ Ch Bh @x where the electrostatic field dc (3.48) dx and the effective charges ZV ¼ 2 and Zh ¼ 1 have been inserted. The difficulty is that the local electrostatic field cannot be measured. Recognizing that any field developed by charge within the oxide would affect the flux of other charged species, Wagner resolved the problem by invoking the condition of zero net electric current (3.23). In this way the unknown quantity E is eliminated between Equations (3.46) and (3.47) via the result 1 @m @m E¼ BV V B h h (3.49) ðBh þ 2BV ÞF @x @x E¼

Resubstitution in Equation (3.46) leads to JV ¼

CV BV Bh @mV @m þ2 h Bh þ 2BV @x @x

(3.50)

95

3.6. Lattice Diffusion and Oxide Scaling: Wagner’s Model

The expression in braces is related to thermodynamic variables via the local equilibrium Equation (3.34), rewritten for a fixed anion lattice as dmo ¼ dZV þ 2dZh

(3.51)

Since ZV ¼ 2Zh , the electrostatic potential terms on the right-hand side cancel, and Equation (3.50) becomes JV ¼

CV BV Bh dmo Bh þ 2BV dx

(3.52)

As the mobilities of free carriers are usually very much greater than those of ions, Bh BV (and Be BMi ) this result is well approximated by J V ¼ CV BV

¼

dmo dx

CV DV dmo RT dx

¼ CV DV

d ln ao dx

(3.53)

which is one form of Wagner’s original solution. The algebra leading to Equation (3.53) is tedious and, in more complex systems, quite time consuming. A simpler procedure is afforded by a description of diffusion in terms of relative buildingunits (Section 3.4). The unit of relevance 00 to a binary metal-deficit oxide is U 1 ¼ MX M V M 2h . As described earlier, the diffusion of these units necessarily satisfies site and charge balance, and is equivalent to cation diffusion through the oxide. J M ¼ JðU 1 Þ @ 00 ZðMX M Þ ZðV M Þ 2Zðh Þ @x Substitution from Equation (3.32) leads immediately to ¼ CM BM

dmM dx which is transformed via the Gibbs–Duhem Equation (3.36) to J M ¼ CM BM

(3.55)

CM BM dmo 1 d dx

(3.56)

CM DM d ln ao ð1 dÞ dx

(3.57)

JM ¼ JM ¼

(3.54)

Recognizing that because of site conservation CV DV ¼ CM DM

(3.58)

it is seen that Equations (3.53) and (3.57) are equivalent at low values of d.

96


The remaining step in this description is the relating of scale thickening rate to diffusive flux through kp dX ¼ JM VM ¼ dt X

(3.59)

where VM is the volume of oxide formed per mole of metal. It follows from Equations (3.57) and (3.59) that kp ¼

DM d ln ao 1 d dy

(3.60)

where y is the normalized position co-ordinate, y ¼ x/X. Upon integration from x ¼ 0 to x ¼ X (i.e. from one side to the other of the scale), this yields Z a00o 1 d ln ao kp ¼ DM (3.61) i 1 d ao where a0o and a00o represent the boundary values of the oxygen activity at the metal–scale and scale–gas interfaces. Use of the relationship for vacancies ZM 1 ¼ jZo j ð1 dÞ leads to the form Z

a00o

kp ¼

DM a0o

ZM d ln ao jZo j

(3.62)

which was Wagner’s original equation for metal oxidation. In the case of very small deviations from stoichiometry, d1, and ZM =jZo j is constant. In this case, Equation (3.62) can be expressed with the help of Equation (3.58) as Z 00 ZM ao kp ¼ DV CV d ln ao (3.63) jZo j a0o This useful form corresponds to Equation (1.25) as is seen below.

3.7. VALIDATION OF WAGNER’S MODEL Considerable effort has been expended in testing both the qualitative and quantitative accuracy of the Wagner description of scale growth kinetics. In the event, quantitative success was achieved in a satisfactory number of important cases: the oxidation of iron, nickel, cobalt and copper and the sulfidation of iron and silver. A review of the practically important cases of FeO, NiO, CoO and FeS scale growth is instructive.

3.7. Validation of Wagner’s Model

97

3.7.1 Oxidation of nickel Nickel forms only one oxide, NiO, which exhibits a small range of nonstoichiometry, ca. 103 at.% on the metal-deficit side. Although NiO scales are formed over a wide temperature range, it is only at temperatures above 9001C that the oxide grain size is sufficiently large for lattice diffusion to predominate over grain boundary transport. Defect concentration, electrical and diffusional properties of NiO have been interpreted in terms of non-interacting cation vacancies 1 2O2 ðgÞ

0 ¼ OX O þ V Ni þ h

V 0Ni ¼ V 00Ni þ h

Thus if V 0Ni V00Ni , the charge balance for the system is

0 VNi ¼ ½h

(3.64) (3.65)

and the equilibrium (3.64) yields 1=4

V 0Ni ¼ KpO2

Conversely, if V 00Ni V 0Ni , we have

1=6 2 V 00Ni ¼ ½h ¼ ð2K0 Þ1=3 pO2

(3.66)

(3.67)

Several investigations [21–23] have shown that the defect properties of 1=4 1=6 NiO are functions of oxygen pressure between pO2 and pO2 . For example, the self-diffusion coefficient of nickel in NiO was shown by Volpe and 1=6 1=4 Reddy [21] to be proportional to pO2 at 1,2451C and pO2 at 1,3801C, as shown in Figure 3.6. The values of DNi given in Figure 3.6 can be used in Equation (3.62) to predict scaling rate constants. The procedure is the same for any oxide for which ZM =jZO j can be approximated as constant, and the form (3.63) used. Setting 1=n

CV ¼ KpO2

(3.68)

and hence 1=n

where

DoNi

DV CV ¼ DoNi pO2

(3.69)

is the self-diffusion coefficient at pO2 ¼ 1 atom, we obtain Z a00o 1=n kp ¼ DoNi pO2 d ln pO2

(3.70)

a0o

¼

DoNi

which upon integration yields kp ¼ nDoNi

Z

a00o a0o

p00O2

1=n1

pO 2

1=n

dpO2

1=n p0O2

(3.71)

98


Figure 3.6 Self-diffusion coefficient of nickel in NiO (a) at 1 atm pressure as a function of temperature and (b) as a function of oxygen pressure. Reused with permission from Milton L. Volpe and John Reddy [21], copyright 1970, American Institute of Physics.

Rates measured at pO2 ¼ 1 atm are compared in Figure 3.7 with values predicted from Equation (3.71) using the DoNi temperature dependence provided by Volpe and Reddy [21]. Thus quantitative success was achieved with a model based on mass transport via individual point defect species. It should be noted, however, that the Volpe and Reddy diffusion description employed here could not define the relative contributions of the singly and doubly charged vacancies. More seriously, the model fails badly at temperatures below 9001C, as seen in Figure 3.7.

3.7.2 Oxidation of cobalt The monoxide CoO is also of the metal-deficit type, and shows a much larger deviation from stoichiometry than NiO, about 1 at.%. A higher oxide, Co3O4 forms at sufficiently high pO2 , but values greater than 1 atm are required at TW9001C. Growth of a single phase CoO scale occurs via cobalt diffusion, as DoB103DCo. Fisher and Tannhauser [24] and Carter and Richardson [25, 26] studied the parabolic oxidation kinetics and the self-diffusion of cobalt in CoO as a function of temperature and oxygen pressure. Diffusion data found from tracer experiments is shown in Figure 3.8. The value of D is proportional to a constant power of pO2 at each temperature, but the power changes with temperature from 0.27 to 0.35 in the range investigated. Assuming, therefore, that the ionization of


99

Figure 3.7 Parabolic rate constant for NiO scale growth: continuous lines calculated from diffusion data; individual points are measured values. Reprinted from Ref. [33] with permission from Elsevier.

cobalt vacancies varies with temperature, the authors wrote 1 2O2

m0 ¼ OX O þ V Co þ mh

If the charge balance can be approximated as

m V m0 Co ¼ h then

mþ1 1=2 mm V m0 ¼ KpO2 Co

(3.72) (3.73)

(3.74)

K being in this instance the equilibrium constant for Equation (3.72). To ease the integration of Equation (3.62) which lies ahead, it is expedient at this point to take

100


Figure 3.8 Tracer diffusion coefficient of cobalt in CoO [25, 26]. With permission of TMS.

Table 3.3 Measured and calculated parabolic oxidation rate constants for cobalt to cobaltous oxide [26, 29] Pressure: (1 atm) T (1C)

1,000 1,148 1,350

kw (g cm2 s1/2) Experimental 4

1.56 10 3.05 104 8.85 104

Calculated 4

1.65 10 3.35 104 8.26 104

kw(exptl)/kw(calcd)

0.90 0.88 1.16

the logarithmic differential of Equation (3.74) with Cv ¼ V m0 Co d ln pO2 ¼ 2ðm þ 1Þ d ln Cv Equation (3.63) then integrates immediately to yield 1=2ðmþ1Þ 1=2ðmþ1Þ o 00 p0O2 kp ¼ ðm þ 1ÞDCo pO2

(3.75)

(3.76)

where DoCo is the diffusion coefficient at pO2 ¼ 1 atm. The experimental and calculated values are in approximate agreement, as shown in Table 3.3. A more extensive examination of CoO scale growth kinetic measurements has been provided by Kofstad [27], who concluded that the Wagner model describes high temperature (TW9001C) cobalt oxidation well, with m 1.


101

A disadvantage of the integration procedure leading to Equations (3.71) and (3.76) is the treatment of n (or m) as a constant, whereas in general it varies as the relative concentrations of V0m and V 00m change. The difficulty was dealt with by Fueki and Wagner [28] by expressing Equation (3.62) in differential form DCo ¼

jZo j dkp ZCo d ln ao

(3.77)

This equation was used by Mrowec et al. [29, 30] in a careful study of cobalt oxidation kinetics. Values of DCo found from the application of Equation (3.77) to rate data were in good agreement with directly measured values [26, 31]. Gesmundo and Viani [32] considered further the variation of m with oxygen activity, and hence with position in the scale. They achieved an improved match with the experimental pO2 kp relationship by replacing the right-hand side of Equation (3.62) with the sum of two such terms, one for vacancies and one for cobalt interstitials, the latter being significant at low pO2 values near the oxide–cobalt interface.

3.7.3 Oxidation of iron At temperatures above 5701C, iron can form three oxides, wu¨stite, magnetite and hematite. The Fe–O phase diagram and Arrhenius plots for diffusion in the various phases are shown in Figures 2.2 and 3.9. As already seen (Section 2.2), the iron–oxygen diffusion couple resulting from high temperature oxidation develops a scale consisting of inner, intermediate and outer layers of wu¨stite, magnetite and hematite, respectively. The thickness of the wu¨stite layer would be predicted to be much greater than the others, because the phase field and iron diffusion coefficients for FeO are orders of magnitude larger than for the higher oxides, if the reaction is controlled by solid-state diffusion with local equilibria established at phase interfaces. Scaling kinetics determined by Paidassi [1] are shown in Figure 3.10 to be parabolic after a brief initial period of non-steady-state reaction, indicating diffusion control. The relative thicknesses of the different oxide layers quickly attain steady values, as expected for diffusion controlled oxidation. Furthermore, their values (Figure 3.11) display the expected relative magnitudes. It is clear from the Fe–O phase diagram that the approximation d 1 is inapplicable, and the simplified integral (3.63) should not be employed. Himmel et al. [35] used the radioactive tracer technique to measure DFe in wu¨stite, obtaining the results shown in Figure 3.12. As would be expected from Equation (3.58), DFe increases with departure from stoichiometry. These data were used, together with information on the variation of composition (effectively, d) with oxygen activity to carry out a graphical integration of Equation (3.57) for growth of the wu¨stite scale layer in the temperature range 800–1,0001C at pO2 ¼ 1 atm. As seen in Table 3.4, agreement with experiment is good. Similar agreement is found [36] at pO2 ¼ 3.3 106 atm. A simplified analysis has been provided by Smeltzer [40], and is perhaps more transparent. Assuming that the only defects in Fe1dO are divalent cation

102


Figure 3.9 Iron and oxygen self-diffusion coefficients in iron and iron oxides. Sources: O in Fe [34], Fe in FeO [35], Fe in Fe3O4 [37], Fe [38] and O [39] in Fe2O3. Reprinted from Ref. [33] with permission from Elsevier.

vacancies and equivalent concentrations of positive holes, and approximating Fick’s first law by a linear vacancy concentration gradient, he obtained.

DV C00V C0V J¼ (3.78) X and therefore

kp ¼ V Fe DV C00V C0V (3.79) Values for DV were obtained from the tracer diffusion data for iron in wu¨stite, using Equation (3.58). Estimates of CV ðXÞ were available from Engell [41], who coulometrically titrated the positive holes as a function of thickness in scales quenched from reaction temperature, by equating C00V ¼ 12 Ch . Rate constants calculated from Equation (3.79) are compared with experimental results in Figure 3.13, where good agreement over a wide temperature range is evident.


103

Figure 3.10 Parabolic plots for isothermal scaling of iron in air. Reprinted from [1] with permission from La Revue de Metallurgie.

Figure 3.11 Relative amounts of iron oxides in scales grown in air. Reprinted from [1] with permission from La Revue de Metallurgie.

The apparent success of Equation (3.79) and the implied validity of its assumption of diffusion via individual, doubly charged vacancies in wu¨stite are illusory. Figure 3.14 shows the measured non-stoichiometry of wu¨stite as a function of oxygen potential at a number of temperatures. If the degree of non-stoichiometry

104


Figure 3.12 Iron tracer diffusion coefficient in wu¨stite [35]. With kind permission from Springer Science and Business Media. Table 3.4 Measured and calculated parabolic rate constants for oxidation of iron to wu¨stite [35] Pressure: (1 atm) T (1C)

800 897 983

kw (g cm2 s1/2) Experimental 4

2.3 10 5.0 104 8.2 104

Calculated 4

2.3 10 4.8 104 7.7 104

kw(exptl)/kw(calcd)

1.0 1.04 1.07

were in fact equivalent to the vacancy concentration, and the defects exhibited ideal or Henrian solution behaviour, then a log–log plot such as those of Figure 3.15 would be a straight line of slope 1/6 or 1/4 for doubly or singly charged vacancies. The real plots are curved, showing that the assumed basis for Equation (3.79) is a rather crude approximation. This failure is to be expected for the large vacancy concentrations present in wu¨stite, where vacancy interactions such as cluster formation [45, 46] should be taken into account. The diffusion coefficient used in Equation (3.79) is some sort of average, representing the participating species. It must therefore be concluded that although it provides an empirically successful means of predicting the growth of wu¨stite, the model provides only limited insight into the defect nature of this oxide or its diffusion mechanism.

3.7.4 Sulfidation of iron The iron sulfidation reaction has been studied intensively as a test case for the applicability of Wagner’s theory. A review [47] of the work serves also to illustrate the considerable differences between oxidation and sulfidation reactions.


105

Figure 3.13 Calculated (curves) parabolic rate constants for wu¨stite growth on iron compared with measured values [41–43]. Reprinted from Ref. [40] with permission from Elsevier.

The Fe–S phase diagram in the Fe1dS region, is shown in Figure 3.15. As seen, the non-stoichiometry is a strong function of temperature and pS2 , and can range up to ca. 25 at.%. The material is always metal-deficit, the principal defects being metal vacancies. Usually a much larger degree of non-stoichiometry is found in sulfides than in the analogous oxides. Factors which contribute to this are the larger anion size and lower lattice energy of the sulfides. Thus point defects are more easily created and deviation from stoichiometry thereby arrived at. What is important from the point of view of metal sulfidation is that a material containing a high density of lattice defects will evidence a high diffusion rate and therefore form only a poorly protective scale. At temperatures below that of the Fe–S eutectic, pure iron sulfidises to form, in the relatively short term, a compact, tightly adherent scale. When the value of pS2 is sufficiently high (see Figure 3.15) the scale consists of a thin surface layer of FeS2 over a thick layer of Fe1dS, but at lower values of pS2 only the monosulfide phase is formed. Since the rate of formation of FeS2 is orders of magnitude less than for Fe1dS, attention is focused on the monosulfide formation reaction.

106


Figure 3.14 Non-stoichiometry of wu¨stite at several temperatures. Reprinted from Ref. [44] by permission of The Electrochemical Society, and Ref. [45], published with permission from La Revue de Metallurgie.

Figure 3.15 Phase diagram for Fe–S in the Fe1dS region with equilibrium sulfur partial pressure isobars in kPa.

The compact monosulfide scale grows according to parabolic kinetics, suggesting that the process is controlled by solid-state diffusion. Since the electron conduction characteristics of Fe1dS are metallic in nature and since the self-diffusion coefficient of sulfur, DS, is much less than that of iron, DFe,


107

then Wagner’s theory predicts that the flux of iron supports sulfide scale growth rate Z a00 S 1 d ln aS kp ¼ DFe (3.80) 0 1 d aS The variation of the tracer diffusion coefficient of iron with stoichiometry has been measured by Condit et al. [48] in single crystal Fe1dS as Q DFe ¼ Do d exp (3.81a) RT with Q ¼ 81 þ 84d kJ mol1

(3.81b)

where Do has the values 1.7 102 and 3.0 102 cm2 s1 for diffusion in the a and c directions, respectively. The way in which d varies with T and pS2 was determined by Toulmin and Barton [49], permitting the numerical integration of Equation (3.80). Fryt et al. [50, 51] found very good agreement between rates calculated in this way and measured values over wide ranges of temperature (600–9801C) and pS2 (5 1011–2 102 atm). A comparison of Fe1dO and Fe1dS scaling rates is informative. At a temperature of 8001C, a wu¨stite layer grows at 1 108 cm2 s1, whereas Fe1dS grows at 1 105 cm2 s1 when PS2 ¼ 0:01 atm. The value of d (measured by chemical analysis) at the Fe1dO/Fe3O4 interface is B0.1, and at the Fe1dS scale– gas interface B0.12. Thus the reason for the large difference in rates lies in the diffusion coefficients rather than the degree of non-stoichiometry. In wu¨stite at 8001C, DFe ¼ 107 cm2 s1, whereas in Fe1dS it is B105 cm2 s1. These differences reflect the different crystal structures (Fe1dS has the hexagonal NiAs structure rather than the cubic NaCl structure of Fe1dO) and lattice spacings of the two iron compounds.

3.7.5 Effects of oxidant partial pressure on the parabolic rate constant Wagner’s treatment of diffusion-controlled scale growth explicitly recognizes the effect of oxidant partial pressure by relating the flux of diffusing species to chemical potential gradients in the scale. Local equilibrium at the metal– scale interface for the case of negligible deviation from stoichiometry may be written M þ 12O2 ¼ MO 1 (3.82) K and fixes a0m ¼ 1 and a0o ¼ 1=K. Changing the ambient gas cannot change these values. However, at the scale–gas interface, the oxidant partial pressure can be a0m a0o ¼

108


varied, and then a00m ¼

1 1=2

KpO2

(3.83)

Thus the gradients in both metal and oxidant activity are affected by changes in the ambient atmosphere, as are the diffusive fluxes within the scale. For a metal-deficit oxide such as Fe1dO, CoO or NiO, Equation (3.63) applies if deviations from stoichiometry can be ignored. If, furthermore, DV afðao Þ, the integral is evaluated using the point-defect equilibrium (3.72) to provide the change of variable given by (3.75) resulting in (3.76). Because p00O2 is usually orders of magnitude greater than the scale–gas equilibrium value p0O2 , we can write 1=2ðmþ1Þ

kp ¼ ðm þ 1ÞDoM pO2

(3.84)

where DoM is the metal diffusion coefficient at pO2 ¼ 1 atm. Fueki and Wagner [28] tested the applicability of Equaton (3.84) to the oxidation of nickel, and found m to vary from 2 at 1,0001C to 0.75 at 1,4001C. They concluded on this basis that doubly charged vacancies, as identified in Equation (3.72), were predominant at 1,0001C, but that singly charged vacancies became more important at higher temperatures. The effect of pO2 on kp for cobalt oxidation is shown in Figure 3.3. At lower pO2 values, only CoO is formed, and Equation (3.84) describes well the variation in kp with oxidant activity, with m 1. When an outer layer of Co3O4 is formed at higher oxygen activities, it is rather thin, and the measured total weight gain corresponds essentially to CoO layer growth. As seen in Figure 3.3, the rate does not vary with pO2 in this regime. This is a consequence of the fact that the boundary value of ao at the CoO outer interface is set by the equilibrium 3CoO þ 12O2 ¼ Co3 O4

(3.85)

and is therefore unaffected by changes to the gas atmosphere. A more detailed study of the effect of pO2 on CoO scale growth was undertaken by Mrowec and Przybylski [30] who showed that 2(m+1) varied from 3.4 at 9501C to 3.96 at 1,3001C. They attributed the deviation from the value 4 expected for singly charged vacancies to a contribution from intrinsic Frenkel defects. However, when much lower pO2 values were investigated [52], the defect properties of CoO were found not to conform with the continuous power relationship of Equation (3.75). Study of the pO2 dependence of wu¨stite layer growth is difficult because the oxygen partial pressures required are so low. At 1,0001C, Fe3O4 forms on top of the wu¨stite layer at pO2 ¼ 1012 atm. As seen earlier (Figure 3.11), wu¨stite continues to constitute the majority of the scale, and measured reaction rates correspond essentially to that of Fe1dO layer growth. Since the boundary values of oxygen activity at the metal–scale and Fe1dO/Fe3O4 interfaces are fixed by the phase equilibria at these surfaces, the diffusive flux supporting wu¨stite layer growth is independent of the ambient pO2 value. The low pO2 values necessary to grow Fe1dO alone can be achieved using CO/CO2 or H2/H2O atmospheres.


109

Pettit and Wagner [53] and Turkdogan et al. [54] have oxidized iron in such atmospheres, and found the reactions to be controlled initially by surface processes involving CO2 or H2O. Eventually parabolic kinetics take over, at the rates predicted from Wagner’s theory. Growth rates of metal excess, n-type oxides show interesting oxidant pressure effects. Consider formation of interstitial cations (e.g. in Zn1+dO)

0 1 MO ¼ Mm i þ me þ 2O2 ðgÞ

(3.86)

If the charge balance can be written Ce ¼ mCMi

(3.87)

then CMi ¼

1=ðmþ1Þ K 1=2ðmþ1Þ pO2 m

(3.88)

where K is the equilibrium constant for Equation (3.86). As expected, adding more oxygen to a metal excess oxide reduces the deviation from stoichiometry. Logarithmic differentiation then yields d ln pO2 ¼ 2ðm þ 1Þ d ln CMi and integration of Wagner’s rate expression leads to 1=2ðmþ1Þ 1=2ðmþ1Þ kp ¼ ðm þ 1ÞDMi p0O2 p00O2

(3.89)

(3.90)

In the usual case where p00O2 p0O2 , the negative exponent makes the second term in the braces much less than the first, and 1=2ðmþ1Þ (3.91) kp ¼ ðm þ 1ÞDMi p0O2 Since p0O2 is established by the metal-oxide equilibrium at the base of the scale, it is independent of the gas composition. Thus the rate of growth of a metal excess oxide is usually independent of pO2 . A similar argument can be developed for metal-excess oxides in which anion vacancies are the principal defects [27]. The correctness of this prediction for the growth of Zn1+dO was demonstrated by Wagner and Grunewald [55], who obtained essentially the same oxidation rate at oxygen partial pressures of 1 and 0.02 atm, and a temperature of 3901C. The rate at which iron sulfidises varies in a complex manner with pS2 [50, 51]. This is a consequence of vacancy interactions at the high concentrations involved, and Wagner’s kinetic analysis cannot be used to provide insight into the defect properties of Fe1dS.

3.7.6 Effect of temperature on the parabolic rate constant The rate constant for growth of a metal-deficit oxide given by Wagner’s theory (3.61) is dependent on temperature in three ways. The diffusion coefficient is thermally activated, DM ¼ Do expðQ=RTÞ. The boundary value of the oxygen

110


activity, a0o , which is one of the limits of integration, is set by the temperature dependent metal-oxide equilibrium (3.82), whence, through Equation (2.28): þDH ðMOÞ DS ðMOÞ 0 ao ¼ exp exp (3.92) RT R Finally, the functional relationship between non-stoichiometry and a0o is itself temperature dependent through the temperature effect on intrinsic disorder (3.18). This last effect is significant if the degree of non-stoichiometry is large, and must be dealt with by numerical integration, as has been done for Fe1dO [35] and Fe1dS [47]. Usually, however, it is ignored. The importance or otherwise of the temperature effect on a0o depends on the nature of the oxide. For a metal-deficit oxide, we have seen that the integrated form in Equation (3.76) can be simplified on the basis a00o a0o to the form of Equation (3.84). Thus the temperature dependence of a0o is unimportant. The activation energy for the scaling rate constant is in this case the same as that of the metal diffusion coefficient. A different conclusion is reached for metal excess oxide, where the defect concentration is inversely proportional to some power of pO2 . In the usual situation where a00o 4a0o it follows that 1 1 00 (3.93) 0 ao ao and the integrated form of the rate expression is given by Equation (3.91). Rewriting this to show the temperature effect explicitly, we obtain Q DH ðMOÞ DS ðMOÞ exp exp (3.94) kp ¼ ðm þ 1ÞDo exp RT ðm þ 1ÞRT ðm þ 1ÞR thus observing that the activation energy for kp is given by ½Q þ DH ðMOÞ=ðm þ 1Þ . In the foregoing discussion of temperature effects, we have assumed that the scale was a single phase, and that its outer surface was in contact with gas at 1=2 some fixed value of a00o ¼ pO2 . However, if an additional layer develops, as in the cases of iron and cobalt (Figure 3.1), then a00o is set by the interfacial equilibrium between the two oxides, as expressed, e.g. by Equation (3.85). The temperature effect on the rate of CoO growth is then found from Equation (3.76) as Q DH DS kp ¼ ðm þ 1ÞDo exp exp exp (3.95) RT ðm þ 1ÞRT ðm þ 1ÞR where DH and DS refer to the CoO ! Co3 O4 reaction (3.85). Kofstad’s compilation [27] of cobalt oxidation rate data is reproduced in Figure 3.16. At high temperatures, where only CoO is formed, the activation energy is equal to that of DCo at 160 kJ mol1. At lower temperatures, a thin layer of Co3O4 forms on top of the CoO, but the measured overall oxidation rate corresponds closely to the growth of the majority CoO layer, and is given to a good approximation by Equation (3.95). Taking DH ð3:85Þ ¼ 183 kJ mol1, the activation energy for scaling is then predicted to be 160+183/2 ¼ 252 kJ mol1. This is in reasonable


111

Figure 3.16 Temperature effects on cobalt oxidation rates in 1 atm O2. Reprinted from Ref. [27] with permission from Elsevier.

agreement with the experimental finding of 230 kJ mol1. The rate of CoO growth is ‘‘decreased’’ at lower temperatures because a00o , as established by the CoO/Co3O4 equilibrium, is much lower than the gas phase value of 1 atm.

3.7.7 Other systems Wagner’s theory has been shown to be successful in describing the oxidation of copper to form metal-deficit Cu2O. This first demonstration is of historic interest, as it was performed by Wagner himself [55]. It is also unusual in that the transport properties of Cu2O were measured electrochemically. Later results on copper oxidation have been reviewed [27, 33] and are considered to indicate that the defect nature of Cu2O is more complex than the neutral vacancy model 1 2O2 ðgÞ

X ¼ 2V X Cu þ OO

(3.96)

deduced by Wagner. The high temperature oxidation of silicon is important in solid-state device technology, and it has accordingly been studied intensively. The reaction product is amorphous or glassy SiO2, which is highly protective. The early kinetic

112


investigations of Deal and Grove [56] led to the parabolic-linear rate equation X2 þ AX ¼ kp ðt þ tÞ

(3.97)

for reaction in dry oxygen. Here kp =A is a linear rate constant related to phase boundary reactions and t a correction to allow for the non-zero oxide film thickness at the commencement of reaction. The magnitude and activation energy of kp were shown [56, 57] to agree with those of oxygen diffusion through glassy silica. The Wagner equation for oxygen diffusion control is simply Z p00 O kp ¼ Do d ln pO2 (3.98) p0O

2

and for Do independent of oxygen activity, this integrates to yield h i kp ¼ Do p00O2 p0O2 Do p00O2

(3.99)

thus accounting for the original observation [56] that kp / pO2 , and indicating that the diffusing species are oxygen molecules. Very different results are obtained at high temperature and low pO2 values, because volatilization of SiO(g) becomes important. This situation is discussed in Section 3.10. A few other systems have been used to test the validity of the Wagner approach: silver sulfidation and bromination and CuI formation. Scaling rates were found to be in good order of magnitude agreement with predictions based on the transport properties of the relevant compounds [58–60].

3.7.8 Utility of Wagner’s theory Wagner’s equations express succinctly the parameters affecting oxidation rates: the material properties of the oxide, oxidant partial pressure and temperature. Consider the relative rates at which Fe1dO, CoO and NiO grow at 1,0001C (see Table 3.1). All three oxides have the same crystal structure and contain cation vacancies. To a first approximation, we ignore differences in atomic weights, lattice spacing and, most importantly, defect interactions, and suppose that DV has the same value in each oxide. This approximation can be tested, using DM ¼ DV CV on the assumption of uncorrelated diffusion, and measured values of D and d ¼ CV . As seen in Table 3.5, DV values calculated in this way are in fact within an order of magnitude. To this degree of approximation then, the Table 3.5

Comparative data for metal deficit oxides at 1,0001C

Oxide kp (cm2 s1)

Fe1dO CoO NiO

Calculated Dv (cm2 s1)

Measured data

7

2 10 3.3 109 9 1011

DM (cm2 s1) 7

8 10 1.2 109 1 1011

Cv (fraction)

0.13 0.01 105

6.2 106 1.2 107 1 106

3.8. Impurity Effects on Lattice Diffusion

113

differences in metal self-diffusion coefficient can be attributed directly to oxide non-stoichiometry. Recalling the earlier result for metal-deficit oxides in Equation (3.84) 1=2ðmþ1Þ

kp ¼ ðm þ 1ÞDoM pO2

then at pO2 ¼ 1 atm, kp is 1, 2 or 3 times DoM for vacancy charges of 0, 1 or 2, respectively. The rate data for cobalt and nickel at 1,0001C and pO2 ¼ 1 atm in Table 3.5 is in reasonable accord with this prediction for m ¼ 2. In the case of iron, pO2 has the value set by the FeO/Fe3O4 equilibrium and, for m ¼ 2, the value of kp predicted from Equation (3.84) is 2 108 cm2 s1, an order of magnitude lower than the measured quantity. Nonetheless, the widely different growth rates of these three oxide scales can be understood, and semi-quantitatively predicted, simply from a knowledge of their non-stoichiometry. It was the achievement of Wagner and the other early investigators in Germany to recognize that non-stoichiometry corresponded to the existence of lattice defects, and that furthermore these defects provided the mechanism of diffusion and scale growth. Wagner’s theory has been shown to be quantitatively successful in a convincing number of cases. A principal reason for the limited utility of the theory is the lack of sufficient data to permit accurate integration of rate equations like Equation (3.61). From a practical point of view, it is easier to measure a parabolic rate constant than to predict it by determining diffusion coefficients and deviations from stoichiometry as functions of oxygen activity. The real value of the theory is in providing a fundamental understanding of the oxidation mechanism. As we have seen, the thermodynamic and diffusional analysis leads to an understanding of and the ability to predict the effects of temperature and oxidant partial pressure. Despite the intellectually satisfying nature of the Wagner analysis, it is prudent to bear in mind its limitations. As we have seen, the theory works well for a moderately non-stoichiometric oxide like CoO, but fails to reveal the complexities of diffusion in highly disordered solids like Fe1dO and Fe1dS. More seriously, it cannot be used to predict the growth rates of slow growing (and therefore important) oxides like Cr2O3 and Al2O3. These oxides have immeasurably small deviations from stoichiometry, and their diffusion properties are not well understood. These difficulties result from the nature of the oxides. Firstly, the native lattice defect concentrations are so small that even low concentrations of impurities can dominate the oxide. Secondly, and for the same reason, diffusion via pathways other than the oxide lattice are usually important. We now consider these effects.

3.8. IMPURITY EFFECTS ON LATTICE DIFFUSION In reality metals are seldom anywhere near pure. Even so-called high-purity metals usually contain impurities at concentrations in the parts per million (ppm) range. One exception to this generalization might be silicon, which is routinely

114


zone-refined to very high purity levels, in order to avoid unwanted dopants which would affect semiconductor properties. The presence of impurity ions of valence different from that of the solvent species can change the defect concentration through their effect on charge balance. Consider the dissolution of chromium in NiO X X 00 2Cr þ 3ðNiX Ni þ OO Þ ¼ 2CrNi þ V Ni þ 3OO þ 3Ni

(3.100)

Cr2 O3 ðþNiOÞ ¼ 2CrNi þ V00Ni þ 3OX O ðþNiOÞ

(3.101)

or, equivalently

The different effective charge of the impurity, or dopant, is seen to be accommodated by an adjustment in the number of cation vacancies. In the first formulation, chromium metal is oxidized by NiO, displacing nickel metal, as would be predicted from the relative stabilities of Cr2O3 and NiO. In the second formulation, the NiO lattice is extended by the dissolution of some chromia. In writing these equations it is assumed that chromium is dissolved substitutionally onto the normal cation sublattice. Moreover, the cation to anion site ratio of the solvent oxide is maintained, as its crystallography is unchanged. To formulate the equations it is necessary, of course, to know the natural defect type of the solvent oxide. Consider now the cation vacancy concentration when the doped oxide is at equilibrium with a gas. In addition to Equation (3.101), we have 1 2O2 ðgÞ

00 ¼ OX O þ V Ni þ 2h

1=2

CV C2h ¼ KpO2

(3.102) (3.103)

if only doubly charged vacancies can form. The charge balance is now written 2CV ¼ CCr þ Ch

(3.104)

Substitution for Ch from Equation (3.103) then lends to 1=4

2CV ¼ CCr þ

K1=2 pO2 1=2

CV

! (3.105)

Thus doping with a higher valent cation has the effect of increasing the vacancy concentration and making it less sensitive to oxygen partial pressure. The effects on kp are predicted to be qualitatively similar. Nickel containing low concentrations of chromium has been found [61, 62] to oxidize faster than pure nickel, in agreement with this prediction. Using similar reasoning, it is found that dissolution of a lower valence cation in a metal-deficit oxide decreases the concentration of vacancies, and hence the oxidation rate. In the case of a metal excess oxide like ZnO, where the ionic defects have a positive charge, the effects of aliovalent doping are reversed. Consider the

3.9. Microstructural Effects

115

incorporation of a higher valent cation such as Cr3+ 0 1 Cr2 O3 ðþZnOÞ ¼ 2CrZn þ 2OX O þ 2e þ 2O2 ðgÞ

(3.106)

Because formation of a cation vacancy is energetically unfavourable in an n-type oxide, site balance is instead maintained via the ejection of gaseous oxygen and the formation of free electrons. If the interstitial zinc species are fully ionized, we have also ZnO ¼ Zn€ i þ 2e0 þ 12O2 1=2

(3.107)

K ¼ CZni C2e pO2

(3.108)

2CZni þ CCrZn ¼ Ce

(3.109)

and the charge balance becomes Combination of Equations (3.108) and (3.109) then yields 2CZni þ CCrZn ¼

K1=2 1=2 1=4

CZni pO2

(3.110)

and the addition of chromium is seen to decrease the concentration of zinc interstitials and would therefore be expected to decrease the zinc oxidation rate. The various combinations of higher or lower valent dopants with stoichiometric oxides (both Schottky and Frenkel type) or non-stoichiometric compounds (metal excess or deficit, lattice or interstitial species) have been considered in detail, as they are important to the study of solid-state chemistry [63]. However, the value of dopant chemistry in understanding or predicting oxidation behaviour is far from clear. Consider even the simple case of chromium doping NiO to the extent where Equation (3.105) can be approximated as CV ¼ 12CCr The rate expression Equation (3.63) then becomes Z a00o kp ¼ D V CCr d ln ao

(3.111)

(3.112)

a0o

and to proceed further we require knowledge of the chromium concentration profile within the scale. Information of this sort is not available. Moreover, the ideal or Henrian solution (Section 2.3) behaviour implicit in equilibrium concentration relationships like Equation (3.103) is highly unlikely to apply in the presence of dopants.

3.9. MICROSTRUCTURAL EFFECTS Wagner’s theory assumes the oxide scale to be continuous, coherent, perfectly bonded to the substrate metal and to be capable of diffusion only via lattice defects. As seen in the preceding sections, these assumptions can be successful, particularly at high temperatures in systems containing large concentrations of lattice defects. At lower temperatures and in oxides with less defective lattices,

116


they can fail. Nickel oxidation (Figure 3.7) provides an example of agreement between theory and experiment at high temperatures, but measured rates are much higher than predicted at low temperatures. As is clear from the measurements, the activation energy has a smaller value at lower temperatures, and a different mechanism must be in effect. That mechanism has been shown to be grain boundary diffusion.

3.9.1 Grain boundary diffusion Oxide scales are polycrystalline, and grain size can vary considerably. As seen in Figure 3.1, the NiO grain size in a scale grown at 1,1001C is tens of microns. In oxide grown at 8001C, the columnar grains seen in Figure 3.17 are about 1 mm across. At lower temperatures, the grains are even finer, and show evidence of coarsening with time (Figure 3.18). Grain boundary diffusion is often more important than lattice diffusion at low temperatures. A principal reason for this is the lower activation energy of the boundary process, corresponding to the more disordered structures in the boundaries [66]. A second reason is the usually finer grain size and hence more numerous boundaries encountered at lower temperatures, as illustrated above for NiO. The additional possibility of impurity species segregation to grain boundaries is considered in Sections 4.4, 7.5 and 10.4. A useful way of describing diffusion in a polycrystalline material was proposed by Hart [67] and adapted by Smeltzer et al. [68] to the nickel

Figure 3.17 SEM view of cross-section through NiO scale [64]. With kind permission from Springer Science and Business Media.


117

Figure 3.18 Average grain size in NiO scales as a function of oxidation time. Reprinted from Ref. [65] with permission from Elsevier.

oxidation case. An effective diffusion coefficient is defined as a weighted sum of lattice and boundary contributions Deff ¼ DL ð1 f Þ þ DB f

(3.113)

where f is the fraction of diffusion sites within the boundaries and DL and DB the self-diffusion coefficients for the bulk lattice and boundaries, respectively. Using the linear approximation to the oxidation rate equation dX DC ¼ VDeff dt X and integrating, one obtains X2 ¼ 2VDL DC

Z t DB 1þ f dt DL o

(3.114)

(3.115)

so that the effective rate constant for fixed f and predominant boundary diffusion is kp ¼ VDCDB f

(3.116)

More complex kinetics result if the oxide grains grow during the scaling reaction [64, 65, 68]. If the density of rapid diffusion sites decays according to first order kinetics [68] f ¼ f expðktÞ

(3.117)

f DB X2 ¼ 2kp t þ ð1 ekt Þ kDL

(3.118)

then

where f is the initial value of f. If, on the other hand, the decay in f is due to recrystallization and grain growth in the oxide [64] f ¼ 2d=Dt ; D2t D2o ¼ Gt

(3.119)

118


and Equation (3.118) becomes (

" #) 1=2 2 ðD GtÞ D 4D d B o o X2 ¼ 2kp t 1 þ DL G t

(3.120)

Here, the grains are modelled as cubes of edge Dt, which have grown from an original size Do with a growth constant G, and the boundaries have a width, d. Low temperature nickel single crystal oxidation rates have been successfully described [64] using Equation (3.120) with a value for kp calculated for lattice diffusion from Wagner’s theory. The success of this procedure can be seen in Figure 3.19. The reaction was controlled by boundary diffusion in the temperature range 500–8001C. Assuming d ¼ 1 nm and using measured grain sizes, the activation energy was estimated at 130–145 kJ mol1, compared with 255 kJ mol1 for lattice diffusion of nickel and for kp. Graham et al. [69] estimated the activation energy for boundary diffusion as 169 kJ mol1, using first order kinetics (3.117) for the decrease in boundary density, in approximate agreement. A review [33] of correlations of oxide microstructures with oxidation rates of several metals confirms that boundary diffusion is an important component of

Figure 3.19 Experimental results and curves calculated from Equation (3.120) for the growth of NiO scales on a (100) Ni face. Reprinted from Ref. [64] with permission from Elsevier.

119


Table 3.6

Activation energies for parabolic oxidation kinetics and for oxide lattice diffusion [33]

Metal/oxide

T (1C)

Oxidation activation energy EA (kJ mol1)

Diffusion activation energy Q (kJ mol1)

EA/Q

Cr/Cr2O3 Ni/NiO Cu/Cu2O Zn/ZnO Ti/TiO2 Zr/ZrO2

700–1,100 400–800 300–550 440–700 350–700 400–860

157 159 84 104 122 134

330 254 151 305 251 234

0.48 0.62 0.56 0.34 0.49 0.57

scale growth. This is evident in Table 3.6 where activation energies for oxidation at intermediate temperatures are compared with the corresponding quantity for cation lattice diffusion. The former are around half the magnitude of the latter, as is typical of grain boundary diffusion. Isotope diffusion measurements in growing NiO scales [70] have demonstrated that boundary diffusion is dominant at 5001C and 8001C. The success of the grain growth model (3.119) was demonstrated by Atkinson et al. [71, 72], who used independently measured values for lattice, dislocation and grain boundary diffusion to predict low temperature nickel oxidation rates in agreement with experimental results. The intensively studied nickel oxidation reaction has been shown conclusively to be dominated by grain boundary diffusion at temperatures below about 9001C. It seems likely that the same will be true for all oxides, in an appropriate temperature regime, and that the lower the value of DL , then the higher the temperature range in which boundary diffusion will be the predominant mechanism of mass transport. An example of practical importance is Cr2O3. Lattice diffusion has been measured in single crystal Cr2O3, and found to be extremely slow. Several investigators [73–75] found that DCr was independent of 3=4 pO2 over a wide range, but increased with pO2 at high oxygen potentials, ð3=4Þ and perhaps [27] with pO2 at low potentials. The oxygen potential effects are not well understood, although Kofstad [27] has suggested that Cr2O3 shows metal-deficit behaviour at high pO2 and metal excess behaviour at low pO2, and that the intermediate range where DCr afðpO2 Þ may reflect intrinsic behaviour. The most important finding, however, is the very low magnitude of DCr , 1016 cm2 s1 at pO2 ¼ 1 atm and 1,1001C. The activation energy for lattice diffusion of chromium is about 515 kJ mol1. Chromium oxide scaling studies are restricted to relatively low temperatures, to avoid volatilization of both metal and oxide CrðsÞ ¼ CrðgÞ; Cr2 O3 þ 12O2 ¼ 2CrO3 ;

DG ¼ 396; 000 224 T J mol1

(3.121)

DG ¼ 561; 730 357 T J mol1

(3.122)

Metal evaporation becomes important at temperatures above about 1,0001C and CrO3 formation at pO2 ¼ 1 atm is significant above 9001C. Even when

120


consideration is restricted to low temperatures, the thermogravimetric kinetic data shows a remarkable degree of scatter (Figure 3.20). Caplan and Sproule [76] showed that much of the observed variation is due to the diversity of scale microstructures developed. These authors were able to use rather high temperatures by surrounding their samples with Cr2O3, to saturate the gas with CrO3. As seen in Figure 3.21, the scale grown on an etched polycrystalline chromium surface varied considerable, with thin oxide growing on some grains and thick oxide on others (and at grain boundaries). Figure 3.21 compares SEM views of scale fracture sections taken from oxidized samples of etched and electropolished chromium. The latter is made up of multiple layers of detached, convoluted finely polycrystalline Cr2O3, whereas the former appears to be a single crystal of Cr2O3, still attached to the metal surface. The authors attributed these different outcomes to the very thin oxide left by the electropolishing procedure being finely polycrystalline, and nucleating a scale of

Figure 3.20 Comparison of reported rate constants for chromium oxidation [76]. With kind permission from Springer Science and Business Media.


121

Figure 3.21 Fracture cross-sections of chromia scales grown at 1,0901C on (a) etched Cr and (b) electropolished Cr [76]. With kind permission from Springer Science and Business Media.

similar microstructure [84]. The different morphological evolutionary paths of the two structures shown in Figure 3.21 was accounted for in terms of their different diffusion mechanisms. The polycrystalline oxide grew rapidly, and developed compressive stresses, leading to convolution and eventual detachment form the metal. The compressive stresses were attributed to new oxide formation within the scale resulting from simultaneous metal and oxygen diffusion along grain boundaries. The single crystal oxide scale grew slowly and developed no significant compressive stress because, in the absence of grain boundaries, only lattice diffusion of chromium occurred. In this case, new oxide would form at the free outer surface, generating no stress. The difference in observed weight change kinetics is clearly related to the different scale morphologies. However, the rate of single crystal scale growth was not quantitatively correlated with the lattice value of DCr , there being no single crystal diffusion data available at that time. Because the diffusion coefficient is so small, data is still scant. If the value measured [75] at pO2 ¼ 1 atm and 1,1001C of DCr ¼ 1016 cm2 s1 is used in Equation (3.84) with pO2 ¼ 1 atm, a value of kp ¼ (m+1) 1016 cm2 s1 is predicted. The value measured by Caplan and Sproule at 1,0901C corresponds to kp ¼ 2 1013 cm2 s1, three orders of magnitude faster. Moreover, the measured activation energy for oxidation was 240 kJ mol1, compared with 515 kJ mol1 for diffusion. It would be concluded on this basis that lattice diffusion via cation vacancies cannot support the observed rate of chromia scale growth, even when no grain boundaries are present, and presumably lattice diffusion is important. Consider now the possibility of scale growth supported by interstitial cation diffusion, in which case Equations (3.90) and (3.94) should apply. We formulate

122


the interstitial defect equilibrium m 0 3 X 3 CrX Cr þ 2OO ¼ Cri þ me þ 4O2 ðgÞ

(3.123)

along with the charge balance Ce ¼ mCCri obtaining ð3=4ðmþ1ÞÞ

CCri ¼ Constant pO2

(3.124)

The value of pO2 is that given by the Cr/Cr2O3 equilibrium. Using the value (see Table 2.1) DH ðCr2 O3 Þ ¼ 746 kJ mol1 of O2, and Kofstad’s [27] value of 515 kJ mol1 for the diffusion activation energy, we find from Equation (3.94) that the activation oxidation energy for oxidation is [550+746 3/4(m+1)] kJ mol1. If the interstitial species is singly charged, then the predicted activation energy is 236 kJ mol1, in close agreement with the 240 kJ mol1 measured by Caplan and Sproule. Thus the data is consistent with lattice diffusion via chromium interstitials. However, in the complete absence of diffusion data for the relevant regime of T and pO2 , it would be unwise to view this agreement as conclusive. The very low oxygen pressures needed to explore the behaviour of chromia near the Cr/Cr2O3 equilibrium can only be controlled by using H2/H2O or CO/ CO2 mixtures. Unfortunately, these molecular species have their own interactions with Cr2O3 [85–89], and these may obscure the oxygen effects which are relevant to chromia scales grown in pure oxygen. Data obtained [88] in H2/H2O atmospheres at 9001C corresponded to growth of Cr2O3 as the only reaction product under conditions where volatilization would be slow. In this gas, pO2 ¼ 1 1019 atm and the rate constant was 8.6 1011 g2 cm4 s1. Reference to Figure 3.20 shows that this value is of the same order as other measurements made at pO2 ¼ 1 atm, and much faster than the single crystal rate measured by Caplan and Sproule [76]. The fast rate is consistent with grain boundary diffusion, and the lack of pO2 dependence indicates that chromium interstitials are the mobile species.

3.9.2 Multilayer scale growth As we have seen in Sections 2.2 and 3.2, multilayered scales can form during metal oxidation, and the scale structure is qualitatively predictable from the relevant phase diagram. Because local equilibrium is in effect at each of the interfaces, the values of ao are fixed at these boundaries. Accordingly, we expect that the diffusion flux in each layer is inversely proportional to its thickness. However, we cannot evaluate layer-thickening rates directly from these fluxes, because there is an additional mass transfer process at each interface. This problem has been treated by a number of authors [27, 90–95]. Consider the growth of a duplex scale (Figure 3.22) made up of MOa and MOaþb . Under steady-state conditions, the thickness of each layer increases


M

Gas

Moa+b

MOa

CM

123

a/b

CM

b/a

CM

x=0

Figure 3.22

x =X1

x=X2

Schematic view of two-layered scale growth.

parabolically with time X2i ¼ ki t

(3.125)

where ki is a rate constant (which is not equal to kp ) and the subscript indicates the layer identity. The values of ki depend on the diffusional fluxes in the oxide, and on the interface reaction b b (3.126) MOaþb þ M ¼ 1 þ MOa a a This situation can, in principle, be dealt with from a mass balance point of view. If metal is the only diffusing species

dX1 b=a a=b J1 J2 ¼ CM CM (3.127) dt b=a a=b where J i is the metal flux in the indicated layer and CM and CM the concentrations of metal in the outer and inner oxide, respectively, both at the MOa =MOaþb interface. Evaluation of the J i is difficult, and it is useful instead to relate the ki values to other, simpler reactions. If a single layer of MOa is grown at pO2 ða; bÞ, the equilibrium value for Equation (3.126), the rate constant can be evaluated from Wagner’s theory, assuming only metal diffuses, as Z pa=b ð1Þ O2 Z M Dð1Þ ka ¼ (3.128) M d ln pO2 0 j j Z o pO 2

The rate constant kb is defined in terms of growth of a higher oxide layer on the surface of lower oxide, in the absence of any base metal and therefore of diffusion from it, but exposed at its outer surface to pO2 ðgÞ4pO2 ða=bÞ. This rate is related to k2, the rate of outer layer thickening when both layers grow simultaneously on

124


the metal, via the volume change accompanying the phase transformation (3.126). Thus X2 V MOaþb (3.129) kb ¼ 1 þ k2 X1 VMOa Z kb ¼

ðgÞ

pO

2

ða=bÞ

pO

2

Zð2Þ M Dð2Þ d ln pO2 jZo j M

(3.130)

Recognizing that in the duplex scale relative layer thicknesses reflect the fractions of the metal flux consumed in growing each of them, it can be shown that ka ðX1 Þ2 1 þ aðða=ða þ bÞÞðX2 =X1 ÞÞ ¼ 1 þ ð1=aÞðX1 =X2 Þ kb ðX2 Þ2

(3.131)

where a ¼ V MOa =V MOaþb . Thus the ratio of the layer thicknesses in a duplex scale is related to the ratio of the rate constants for the exclusive growth of the individual layers. Finally, the overall rate constant kov which describes the rate of total scale thickness increase is related to the layer growth rates by

2 1=2 1=2 kov ¼ k1 þ k2 (3.132) The applicability of this description to the growth of FeO+Fe3O4 scales on iron has been demonstrated by Garnaud and Rapp [96], who used independently measured values of ka and kb to predict XFe3 O4 =XFeO ¼ 0.041 at 1,1001C. This value is in good agreement with that of 0.053 derived from Paidassi’s metallographic observations [1].

3.9.3 Development of macroscopic defects and scale detachment As discussed in Section 2.10, oxide scale growth by outward metal transport leads to new oxide formation at the free outer surface, and no growth stresses result. However, as the metal surface recedes, the scale can maintain contact with it only if it is free to move. In the case of a flat sample of limited size, a rigid scale would be constrained at specimen edges and therefore unable to maintain contact. Even when the oxide has limited plasticity, growth of a sufficiently thick scale will eventually lead to detachment from the metal, starting at the edges. An example of the effect [97] is shown for a silver sufide scale in Figure 3.23. The cross-section reveals that the sulfide had remained in contact with the flat sides of the specimen, forming a thick, compact scale as metal was consumed. Much less silver was reacted at the specimen edges, where geometrical constraints had prevented the sulfide from maintaining good contact. As seen in the figure, a porous reaction product had developed, rather than an empty gap. This interpretation was confirmed by the ‘‘pellet’’ experiment [98, 99] shown schematically in Figure 3.24. A pellet of the same material as the scale was placed in contact with the metal specimen. A tube containing the oxidant (liquid sulfur in this case) was placed on top of the pellet and held there under load.


125

Figure 3.23 Cross-section of initially flat silver sample after sulfidation at 4441C for 9 min [97].

Load

S(l)

S(l) tube

Ag2S

Ag2S

Ag2S

Ag2S

Ag

Porous Ag2S Ag Rigid support

Figure 3.24

Schematic illustration of Rickert’s [98, 99] pellet experiment. Description in text.

Heating the whole assembly was found to cause growth of more scale up into the tube (confirming outward diffusion of silver in Ag2S). As the metal was consumed, the pellet and loaded tube moved downward, maintaining contact with the metal. No region of porous sulfide developed. However, if at some time after commencement of the reaction, the metal and tube of sulfur were each independently clamped in position, porous sulfide formed at the pellet-metal interface. The development of porous material was described by Mrowec and co-workers [97] using their dissociation mechanism. Once local scale–metal separation occurs, the metal activity at the underside of the scale can no longer be maintained. Metal continues to diffuse outward, driven by the oxidant chemical potential gradient, and aM decreases. Consequently, as seen from Equation (3.82),

126


as increases, and gas phase transport commences from the underside of the scale to the underlying metal surface. A porous product grows under gaseous mass transfer control, until it bridges the gap between metal and separated scale, whereupon outwards metal transport resumes. As calculated in Section 2.9 (and as pointed out by Gibbs [100]) the oxidant partial pressures prevailing at metal–scale interfaces are usually too low to support any significant transport. However, as diffusion through the scale takes place, the ao value at the scale underside will rise and gaseous mass transport could thereby be enabled. Birks and Rickert [101] showed that in the case of NiO growth, the likely pO2 values were sufficient to account for the observations. Furthermore, most metals contain carbon impurities which will oxidize. As shown by Graham and Caplan [102], the resulting CO/CO2 mixture occupies the voids. In this case, the gas can act as an oxygen carrier via the reactions CO þ MO ¼ CO2 þ M taking place in different directions on opposite sides of the cavity (Figure 3.25). Finally, it will be recalled that real oxide scales are polycrystalline, and inward oxygen diffusion via grain boundaries can occur. Atkinson et al. [70] used 18 O tracer studies to show that oxygen did not penetrate NiO scales during their initial growth, but that long-term penetration occurred when an inner, porous NiO scale sublayer developed. This transport of oxidant molecules is suggested to take place through microchannels or pores developed in the outer layer. Mrowec and co-workers [97] have proposed that the underside of a separated scale will dissociate preferentially at oxide grain boundaries, where outward diffusion of metal is fastest. This process could then create microchannels along favourably oriented boundaries, allowing subsequent inward transport of molecular oxidant. The possibility of molecular species penetrating scales is discussed in Chapters 4, 9 and 10.

Gas

JM

MO

CO + MO →CO2 + M

CO

CO2

CO2+M →CO+MO

Figure 3.25 Action of CO/CO2 couple within a void accelerating oxygen transport.

3.10. Reactions Not Controlled by Solid-State Diffusion

127

3.10. REACTIONS NOT CONTROLLED BY SOLID-STATE DIFFUSION As observed in Section 1.6, parabolic scaling kinetics are not invariably observed at high temperatures, and processes other than solid-state diffusion can control the reaction rate. For pure metals, this will be the case if either an interfacial process or gas phase mass transfer is slower than diffusion in the scale. The principles involved are discussed here with reference to the oxidation of iron and silicon at low oxygen potentials.

3.10.1 Oxidation of iron at low pO2 to form wu¨stite only Linear scaling kinetics have long been reported [103] for the oxidation of iron at low oxygen potentials where only Fe1dO forms. In order to obtain the low pO2 values required, gas mixtures of CO–CO2 or H2–H2O are used. Because the pO2 values are so low (1015–1013 atm at 1,0001C), molecular oxygen is far less abundant than the CO2 or H2O species. Given that the homogeneous gas phase dissociation reactions of both CO2 and H2O are rather slow at these temperatures, it is clear that the relevant species of importance are CO2 and H2O. In the case of CO2, the linear rate was found to depend on pCO2 and the total pressure of CO+CO2 mixtures [53, 104–110]. It was concluded that the rate was controlled by the reaction CO2 ðgÞ þ S!COðgÞ þ OjS

(3.133)

where, as before, S represents a surface adsorption site, and the net rate can be written Rate ¼ kf pCO2 yv kr pCO ð1 yv Þ

(3.134)

with kf and kr denoting the forward and reverse rate constants for reaction (3.133), and yv the fraction of surface sites empty. At the Fe/FeO equilibrium oxygen activity, ano , the net rate is zero. Substituting from the CO/CO2 equilibrium expression (2.15) p ano ¼ Kc CO2 (3.135) pCO into Equation (3.134), we obtain from the zero rate condition the general result kr ð1 yv Þ ¼ kf yv

ano Kc

enabling us to rewrite Equation (3.134) as ano Rate ¼ kf yv pCO2 pCO Kc

(3.136)

(3.137)

In gas mixtures containing only CO and CO2, the total pressure is PT ¼ pCO þ pCO2

(3.138)

128

and


an an Rate ¼ kf yv PT N CO2 1 þ o o Kc Kc

(3.139)

where N CO2 ¼ pCO2 =PT . As seen in Figure 3.26, the data of Pettit et al. [104] confirms the dependence on both total pressure and CO2 mol fraction, providing that yv is constant. A similar expression has been shown to apply for the linear kinetics of wu¨stite scaling in H2/H2O atmospheres [53]. Grabke [111] showed that the linear rate constant values in CO/CO2 atmospheres agreed with those obtained for surface exchange of oxygen on wu¨stite equilibrated with iron. As the wu¨stite scale thickens, diffusion through it slows until a thickness is reached at which diffusion becomes rate-controlling and the kinetics parabolic [53]. It has been noted [106, 108] that reaction at high pCO2 values produces scales of wu¨stite only, although the equilibrium pO2 values calculated from Equation (2.15) exceed the value for Fe3O4 formation. Clearly the supposed gas phase equilibrium is not in effect, and instead the local CO/CO2 ratio is set at the gas–scale boundary. As noted by Kofstad [27], parabolic scaling in H2/H2O gases is faster than in CO/CO2 gases of the same calculated equilibrium oxygen potential. Again this indicates that the scale–gas boundary conditions cannot be calculated from the CO–CO2 equilibrium. Part of the reason for this is the rapid rate at which oxygen is incorporated into the fast growing scale. As shown in Section 2.9, the oxidation of low-carbon steels in substoichiometric combustion gases leads to wu¨stite scale formation according to linear kinetics. Mass transfer calculations showed that gas phase mass transfer did not control the rate, but a surface reaction process did. A regime of behaviour was

Figure 3.26 Dependence of initial linear iron oxidation rate on composition and total pressure in CO/CO2 mixtures. Reprinted from Ref. [104] with permission from Elsevier.

129


0.24

k1 (mg cm-2 min-1)

0.22 0.2 0.18 0.16 0.14 0.12 0.1 8.8

8.9

9.1

9

9.2

9.3

9.4

9.5

102 pCO2 (atm)

Figure 3.27 Linear scaling rates for a low-carbon steel in simulated reheat furnace gas, T ¼ 1,1001C [112]. With kind permission from Springer Science and Business Media.

found for low carbon, low silicon steel [112] in which a small fractional change in oxidant partial pressure led to a relatively large change in rate, as shown in Figure 3.27. The expression in Equation (3.139) cannot be used because ðpCO2 þ pCO Þaconstant. Even Equation (3.138) is unreliable, because yv can vary, and a different treatment of the surface processes is to be preferred. The surface reactions are reformulated as CO2 ðgÞ þ S ¼ CO2 jS OjS þ COðgÞ

CO2 jS k2

OjS!OX o þS

(a) (b) (c)

in order to track vacant surface sites. Assuming a fixed concentration of surface sites M ¼ ½S þ ½OjS þ ½CO2 jS

(3.140)

defining Ka as the adsorption equilibrium constant for reaction (a), and using the rate constants ki specified in (b) and (c), we formulate the steady-state approximation for the surface concentration ½OjS : d½OjS ¼ 0 ¼ k1 ½CO2 jS k1 ½OjS pCO k2 ½OjS (3.141) dt It is found by substituting Ka pCO2 ½S for ½CO2 jS in Equations (3.140) and (3.141), followed by elimination of [S], that ½OjS ¼

k1 Ka MpCO2 Ka pCO2 ðk1 þ k1 pCO þ k2 Þ þ k1 pCO þ k2

A similar scheme can be proposed for reaction with H2O.

(3.142)

130


The rate of the oxygen uptake reaction (c) is proportional to [O|S], given by Equation (3.142). It is concluded then that k2 is not the dominant term in the numerator (because the reaction rate is not proportional to pCO2 ) and the reverse of reaction (b) must therefore be significant. Similarly, it can be concluded that the surface is not saturated with adsorbed CO2, as the rate does change with changing gas compositions, and therefore Ka pCO2 cannot be large. Proceeding on the assumption that, in fact, Ka pCO2 is small, Equation (3.142) is approximated as ½OjS ¼

k2 Ka MpCO2 k2 k1 pCO

and the oxidation rate expression becomes pCO2 kl ¼ a þ b pCO

(3.143)

(3.144)

where a and b are constants. This expression was found to fit the data well [112] with a ¼ 0.375 mg1 cm2 s atm and b ¼ 27.3 mg1 cm2 s. The large change in pCO had a much greater effect than did the very small one in pCO2 . Yet another regime of behaviour is found for iron oxidation in the case of exposure to dilute oxygen-bearing gases. Abuluwefa et al. [113] oxidized a lowcarbon, low-silicon steel in N2–O2 mixtures containing 1–16% O2, at temperatures of 1,000–1,2501C. They found initially linear rates, followed by steady-state parabolic kinetics. The linear rate constant was directly proportional to pO2 , and displayed a very small activation energy, 17 kJ mol1. The observed scaling rates were in good agreement at low pO2 values with predictions made for gas phase diffusion control using Equation (2.157), as shown in Figure 3.28. The small

Figure 3.28 Comparison of measured rates for carbon steel oxidation at 1,2001C with values calculated from Equation (2.157) [113]. With kind permission from Springer Science and Business Media.


131

activation energy is also consistent with such a mechanism. The difference between this situation and the combustion gas oxidation discussed above was the larger total oxidant partial pressures of the latter, leading to higher gaseous transfer rates.

3.10.2 Oxidation of silicon As seen earlier, scales of amorphous SiO2 are extremely slow growing and provide excellent protection. However, volatile species can form at elevated temperatures, causing wastage of silicon. Partial pressures of the various possible gas species are shown in Figure 3.29, where pSiO is seen to reach a maximum near the Si/SiO2 equilibrium oxygen partial pressure. At lower values of pO2 , SiO(g) forms and, in the absence of a protective silica scale, silicon is lost through this volatilization process. Wagner [58] analysed this phenomenon, which he called ‘‘active’’ oxidation, in terms of gas phase mass transfer. Because oxygen is consumed at the silicon surface, a diffusion gradient is established in the gas mixture near the surface (Figure 3.30). Thus the value at the surface, pnO2 , could be below the minimum necessary for solid SiO2 formation, even with a pO2 (gas) value above it. It is recognized that the initial value of pO2 (gas) necessary to passivate the silicon surface is therefore higher than the equilibrium Si/SiO2 value. The critical value can be calculated from a consideration of gas phase mass transfer. Most situations of practical interest involve the viscous flow regime, and Equations (2.157) and (2.158) apply. To use them, we need boundary values for both pO2 and pSiO , which are related via local equilibrium at the silicon–gas interface SiðsÞ þ 12O2 ðgÞ ¼ SiOðgÞ

(3.145)

pnSiO ¼ KðpnO2 Þ1=2

(3.146)

4 SiO2(s)

Si (s)

0 -4

Si (g)

log pSixOy

-8

SiO (g) SiO2(g)

-12 -16 -20 -24 -28 -44

Figure 3.29

-40

-36

-32

-28

-24 -20 log pO2

-16

-12

-8

-4

Equilibrium vapour pressures in the Si–O system at T ¼ 1,1271C.

0

132


SiO2 Gas

Si

) pO(gas 2

pO∗ 2 ∗ pSiO ∗ pSiO ≈0

δ

Figure 3.30 Filamentary SiO2 growth on silicon at high temperatures, showing gas phase partial pressure gradients.

From Equation (2.157) J O2

kO2 pO2 DO2 pO2 ¼ RT dO2 RT

(3.147a)

DSiO pSiO dSiO RT

(3.147b)

J SiO ¼

where d is the thickness of the boundary layer (Figure 3.30). The steady-state condition for SiO volatilization is J SiO ¼ 2J O2 (3.148) and therefore pO2 ¼ 12

dO2 DSiO p dSiO DO2 SiO

It can be shown [114] that for a laminar boundary layer dSiO DSiO 1=2

dO2 DO2 and Equation (3.149) becomes

DSiO 1=2 pO2 ¼ 12 pSiO DO2

(3.149)

(3.150)

(3.151)

Consider now the critical condition for protective SiO2 formation 1 1 2SiðsÞ þ 2SiO2 ðsÞ

¼ SiOðgÞ

(3.152)

3.11. The Value of Thermodynamic and Kinetic Analysis

133

Figure 3.31 Silica nanofibres formed by oxidation of silicon at 1,1301C in CO/CO2 [118]. With kind permission from Springer Science and Business Media. Eq

which defines an equilibrium value, pSiO . The critical value of pO2 is therefore given by DSiO 1=2 Eq pO2 ðcritÞ ¼ 12 pSiO (3.153) DO2 If the gas phase pO2 is higher, then protective SiO2 forms. If it is lower, pnSiO adjusts through Equation (3.146), and volatilization or active oxidation results. A similar analysis can be made for the molecular flow regime, using Equation (2.155). Its effectiveness in predicting the transition between active and protective oxidation has been verified experimentally [115–117]. Behaviour in the viscous flow regime is more complex, however, because when solid SiO2 does form, it can be in the form of a non-protective, fast growing deposit. Hinze and Graham [118] observed three regimes of behaviour in Ar–O2 mixtures at 1,2271C: linear weight loss at low pO2, fast linear weight gain at somewhat higher pO2, and protective oxidation at pO2 4 103 atm. The explanation for the intermediate regime was suggested to be formation of SiO2 whiskers growing from the silicon surface. The outer tips of these whiskers acted as growth sites, redefining the diffusion distance dSiO (see Figure 3.30) and explaining the rapid reaction through an accelerated SiO flux (3.147b). Improved imaging capabilities which have become available since that work have allowed the production of Figure 3.31. A highly ordered structure of SiO2 is seen to develop [119] in confirmation of the Hinze and Graham proposal.

3.11. THE VALUE OF THERMODYNAMIC AND KINETIC ANALYSIS In this chapter we have explored the application of thermodynamic and kinetic analysis techniques to the simplest high temperature oxidation situation: reaction

134


of a pure metal with a single oxidant. It is clear that the usual hypothesis of local equilibrium at interfaces between contacting phases is commonly correct. Thus the oxide (or sulfide, etc) predicted to be at equilibrium with the gas is usually found at the scale surface; the oxide shown by the phase diagram to equilibrate with the metal is found to grow in contact with the metal. When this is not so, it can be concluded that solid-state diffusion does not control the reaction rate, and that instead either a gas phase process or a surface reaction is rate controlling. Calculation of gas phase mass transfer rates has been found to be quantitatively successful in determining when these processes are capable of controlling the rate. More importantly, the model of local equilibrium within the growing oxide scale is also successful. The state of a scale interior is well described as a series of microscopic local equilibrium regions, each incrementally different (in the growth direction) from adjacent regions, as shown schematically in Figure 3.32. This allows the use of the diffusion path description and justifies the application of irreversible thermodynamics to the diffusion problem. In a very large number of cases, scale growth is controlled by solid-state diffusion. The Wagner treatment of this situation is found to succeed when adequate information on oxide defect properties is available. This success provides proof that lattice diffusion of point defect species can support scale growth in a number of cases. It leads to very useful predictions as to how scaling rate will vary with oxide type (degree of non-stoichiometry and lattice species mobility) and with oxidant activity and temperature. It also succeeds in predicting qualitatively the effect of dilute oxide solute impurities on scaling rates. The value of the Wagner treatment is less obvious in the case of slow-growing oxides such as Cr2O3 and Al2O3. Our knowledge of the defect properties of these oxides, and the effect of oxygen potential upon them is very limited, and testing the Wagner model is in this sense difficult. Morever, as will be discussed in M

MO

O2(g)

aO″ aO aO′

Figure 3.32 The local equilibrium description: a series of very small regions each of which can be approximated as homogeneous.

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Chapter 7, the diffusion properties of these oxides are often dominated by grain boundary transport. As seen in Section 3.9, grain boundary diffusion can lead to oxidation rates very different from those predicted by Wagner, and even to different kinetics when microstructural change occurs with time. Nonetheless, the mechanism is still one of diffusion, and the basic concepts underlying Wagner’s theory still provide insight and a basis for experimental design. To obtain value from the theory, however, it is essential to add to it a detailed description of microstructural phenomena. Arriving at a definitive version of such a description is a continuing pre-occupation in high temperature corrosion research. For this reason, microstructural evolution receives considerable attention in the remainder of this book.

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CHAPT ER

4 Mixed Gas Corrosion of Pure Metals

Contents

4.1. Introduction 4.2. Selected Experimental Findings 4.3. Phase Diagrams and Diffusion Paths 4.3.1 Scaling of chromium in oxidizing–nitriding and oxidizing–carburizing gases 4.3.2 Scaling of chromium in oxidizing–sulfidizing–carburizing gases 4.3.3 Scaling of iron in oxidizing–sulfidizing gases 4.3.4 Scaling of nickel in oxidizing–sulfidizing gases 4.4. Scale–Gas Interactions 4.4.1 Identity of reactant species 4.4.2 Rate determining processes in SO2 reactions 4.4.3 Production of metastable sulfide 4.4.4 Independent oxide and sulfide growth in SO2 4.5. Transport Processes in Mixed Scales 4.5.1 Effect of pre-oxidation on reaction with sulfidizing– oxidizing gases 4.5.2 Solid-state diffusion of sulfur 4.5.3 Gas diffusion through scales 4.5.4 Scale penetration by multiple gas species 4.5.5 Metal transport processes 4.6. Predicting the Outcome of Mixed Gas Reactions References

139 140 147 147 150 150 151 154 154 156 159 163 168 169 172 172 174 175 175 181

4.1. INTRODUCTION Atmospheres encountered in practice are very rarely constituted of a single oxidant. Even in the case of air, both oxygen and nitrogen can react with a base metal such as chromium. Examples of more complex gases are frequently encountered. A common example is provided by combustion gases which invariably contain carbonaceous species, usually water vapour and commonly sulfur species deriving from the impurities present in most fossil fuels. Another example is the production of synthesis gas. The two processes used to produce hydrogen on a large scale are steam reforming CH4 þ H2 O ¼ CO þ 3H2

(4.1) 139

140

Chapter 4 Mixed Gas Corrosion of Pure Metals

and coal gasification C þ H2 O ¼ CO þ H2

(4.2)

Clearly both processes involve handling gases which, at the necessarily low oxygen potentials, are likely to be carburizing as well as oxidizing. In general, it is necessary to consider the possibility of more than one oxidant reacting with the metal. After a brief review of selected experimental findings, the use of phase diagrams and diffusion paths to arrive at an understanding of scale constitutions is examined, and surface processes are analysed. The mechanisms of mass transport are then considered in a discussion of scaling rates. Much of the literature in the area of mixed gas corrosion is of an applied nature, involving complex engineering alloys and simulated, multicomponent process gases. Although of obvious practical utility, this literature provides little in the way of fundamental understanding. Fortunately, a substantial number of model studies involving pure metals are also available, particularly for sulfidizing–oxidizing gases [1–7]. The behaviour of a number of metals in gases containing both oxygen and sulfur was studied rather intensively in the 1970s and 1980s, in the aftermath of sudden oil price increases, when alternative routes to liquid fuels were being sought. The matter is becoming of renewed interest, as oil prices rise again, and the combustion of high sulfur content coals for power generation increases. Attention is focused here on the behaviour of chromium, iron and nickel in mixed gases.

4.2. SELECTED EXPERIMENTAL FINDINGS Key questions in the case of mixed gas corrosion concern whether or not reaction products other than oxide form, and to what extent they are harmful. Iron exposed to SO2 or SO2–Ar can form a lamellar mixture of sulfide plus oxide [9, 10] or a two-phase mixture overlaid by oxide alone [8, 9, 11–14] as shown in Figure 4.1. Layered structures as shown in Figure 4.2 can be formed on nickel [8, 15–17] and sometimes on cobalt [18, 19] although results reported for cobalt are not all in agreement. More complex gas mixtures of CO/CO2/SO2/N2 have been used to simulate aspects of combustion gas corrosion, and to permit independent control of pS2 and pO2 . The earlier literature concerns reaction in pure SO2 or in diluted SO2/Ar mixtures. In these gases the equilibrium SO2 ¼ 12S2 þ O2

(4.3)

requires that pS2 ð1=2ÞpO2 , if SO3 formation can be neglected. Values of DG (4.3) are given in Table 2.1. In the CO/CO2/SO2/N2 mixtures, Equation (4.3) still holds, but the equilibrium CO2 ¼ CO þ 12O2

(4.4)

4.2. Selected Experimental Findings

141

Figure 4.1 Oxide–sulfide scales grown on iron in different SO2/CO2/CO mixtures at 8001C. Grey phase is oxide, light phase sulfide.

Figure 4.2 Layered sulfide–oxide scale grown on nickel in SO2 at 6001C [33]. With kind permission from Springer Science and Business Media.

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can be used to control pO2 , and the value of pS2 is then given by 1=2

pS2 ¼ K4:3 pSO2 =pO2

(4.5)

Additional species such as COS and CS2 can be important under some conditions. In these cases equilibrium calculations are best carried out using numerical free energy minimization procedures, for example those in software packages such as ThermoCalc or FACTSage. A practical difficulty arises from the rather slow rates of the homogeneous gas phase reactions involved, and it is essential that laboratory gas mixtures be brought to equilibrium by passing them through a heated catalyst bed (such as alumina-supported platinum) before contacting the experimental specimen. The effect is illustrated in Figure 4.3 for manganese exposed at 8001C to a gas mixture of inlet composition 23-CO2, 45-CO, 22-SO2, 10-N2 vol.%. The calculated equilibrium composition contained pS2 ¼ 8:6 106 atm and pO2 ¼ 5:7 1016 atm. As seen in the figure, the catalysed gas produced a scale of MnO plus MnS, but the non-catalysed gas led to a scale which evolved with time from a two-phase mixture to almost single-phase MnO. The formation and behaviour of Cr2O3 in mixed gases have been the subject of many research programs because of the protective nature of the Cr2O3 scale, upon which many technologically important alloys depend. In the presence of secondary oxidants, chromia scales have been found to behave in a diversity of ways. For example, a sublayer product of Cr2N has often been found growing underneath an outer Cr2O3 scale on pure chromium after heating in air [21–26]. Pre-oxidation for 2.5 h in oxygen (pO2 ¼ 40 kPa) was found not to stop the

Figure 4.3 Scales grown on manganese in CO2/CO/SO2/N2 mixture at 8001C (light phase sulfide, grey phase oxide) (a) gas passed over Pt catalyst, (b) gas uncatalysed [20]. At equilibrium, MnS is stable with respect to MnO. (With kind permission from Springer Science and Business Media).

143


Table 4.1 Gas no. 1 2 3 4 5

Carburizing–oxidizing–nitriding gas mixtures reacted with chromium [27] Starting gas composition (vol.%) CO 96.6 62.2 49.7 12.4

CO2

H2

H 2O

Reaction potentials (9001C) N2

P(O2) (atm)

P(N2) (atm)

19

3.40 2.20 17.6 44.0 56.5

2.30

35.6 32.7 43.6 41.2

1 10 1 1019 1 1017 1 1015 1 1019

0.36 0.33 0.44 0.41

ac 0.5 0.5 0.04 0.001

nitridation of chromium during subsequent exposure to nitrogen (pN2 ¼ 40 kPa), indicating that the previously established oxide film does not constitute an effective barrier to nitrogen ingress. Obviously the oxide scale formed under these conditions was not impermeable to gas penetration, and nitrogen from the air had reached the chromium. When exposed at 9001C to a CO/CO2 mixture (Table 4.1), chromium is found [27] to develop a two-layered scale (Figure 4.4) consisting of a Cr2O3 outer layer and an inner layer of Cr7C3 containing finely distributed oxide particles. Adding N2 to CO–CO2 results in a three-layered scale (Figure 4.4). The outer layer is again pure Cr2O3. The intermediate layer, now thicker than the chromia, is a mixture of Cr7C3, oxide and a small amount of Cr2N. The innermost layer is pure, compact Cr2N. In a gas mixture of H2/H2O/N2 corresponding to the same equilibrium pO2 value as the CO/CO2/N2 gas, and a closely similar value of pN2 , chromium grows a single layer of pure Cr2O3, and no nitride develops. Addition of SO2 to the gas (Table 4.2) leads to sulfide formation and suppresses nitridation [28]. The resulting scale is shown in Figure 4.4 to be multilayered. The outermost layer is principally Cr5S6 with a Cr2O3 content varying from 1 wt% at the scale–gas interface to 12 wt% near its inner boundary. This two-phase mixture consists of a fibrous structure aligned approximately normal to the metal substrate surface. The underlying scale region is highly porous. Its outer region is largely oxide with a small sulfur content, but its inner region is principally Cr5S6 with minor amounts of Cr2O3 and Cr7C3. The innermost layer is mainly Cr7C3 and Cr2O3 with very little sulfide. Lower pSO2 values have less effect [28]. The gas composition represented by point B in Figure 4.10 produces a thick compact scale of Cr2O3 containing about 24 wt% Cr5S6 as finely dispersed particles. Gas C (Figure 4.10) produces an outer, buckled layer of Cr2O3 containing 0.5 wt% S. A thin sublayer made up of Cr7C3 with oxide dispersions also forms adjacent to the metal. Rather different observations have been reported for reaction of chromium in pure SO2. An early investigation [29] reported the simultaneous growth of CrS and Cr2O3, whereas later work [30] using the same gas led to the finding that only oxide formed, containing 1 wt% S. However, scaling rates in SO2 were found to be 2–3 orders of magnitude faster than in oxygen at 800–1,0001C. The reaction of chromium with H2/H2O/H2S gases at 9001C has also been studied [31]. These gases were such that Cr2O3 was stable with respect to sulfides, but two-phase

144


Figure 4.4 Scales grown on chromium exposed at 9001C to (a) CO/CO2 and (b) CO/CO2/N2 [27] (with kind permission from Springer Science and Business Media); (c) SO2/CO2/CO/N2 (reprinted from Ref. [28] with permission from Elsevier).

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Table 4.2 Gas

Sulfur-bearing gas mixtures reacted with chromium [28] Input gas composition (vol.%) CO

A B C

74.7 62.2 74.6

CO2

2.56 2.18 2.63

N2

22.7 35.6 22.8

SO2

0.039 0.010 0.0039

Reaction potentials (atm) at 9001C P(O2)

P(S2) 19

1 10 1 1019 1 1019

7

3 10 3 108 3 109

P(N2)

ac

0.23 0.36 0.23

0.6 0.5 0.6

Figure 4.5 Initial linear and subsequent parabolic kinetics: Fe reacted with dilute Ar2SO2 at T ¼ 8001C [9]. Published with permission of Taylor and Francis Ltd., http:// www.informaworld.com.

oxide-plus-sulfide scales formed at low H2O/H2S ratios. At intermediate ratios, the two-phase product was overgrown by oxide, and at high ratios only oxide was formed. Scaling kinetics and rates can vary considerably as the nature of the reaction product changes with gas composition. When iron is reacted with dilute Ar–SO2 mixtures, an initial period of linear reaction is followed by parabolic kinetics as shown in Figure 4.5. Flatley and Birks [9] demonstrated that the linear rate constant was proportional to both pSO2 and the gas flow rate, and concluded that gas phase diffusion of SO2 was rate controlling in this regime. The reaction product was a lamellar oxide–sulfide mixture, like that shown in Figure 1a. The subsequent parabolic stage of reaction reflected the onset of solid-state diffusion control in the thicker scale. At low pSO2 values this scale consisted of a coarse FeO+FeS outer

146


layer on top of the first formed lamellar structure. Reaction rates were reported to be faster than those for the oxidation of iron in pure oxygen. At high pSO2 values, the initially formed FeO+FeS structure was overgrown by pure oxide, and the parabolic rate constant was equal to that for the oxidation of iron in pure O2 [10]. The same result was found for reaction in CO/CO2/SO2/N2 gases [20]. Chromium scaling kinetics in the gas mixtures of Tables 4.1 and 4.2 are shown in Figures 4.6 and 4.7. Formation of additional carbide and nitride layers augments the rate, and sulfide formation increases the rate by up to an order of magnitude, depending on the sulfur partial pressure.

Figure 4.6 Chromium scaling kinetics at 9001C in gases of Table 4.1 [27]. With kind permission from Springer Science and Business Media.

Figure 4.7 Chromium scaling kinetics at 9001C in gases of Table 4.2 [32]. Published with permission from Trans Tech Publications Ltd.

4.3. Phase Diagrams and Diffusion Paths

147

4.3. PHASE DIAGRAMS AND DIFFUSION PATHS Thermochemical diagrams of the sort described in Section 2.2 provide a useful basis for analysing and rationalizing the morphologies of scales grown in dual oxidant gases. However, as we now discuss, they seldom provide a means of predicting the outcome of a particular reaction. The essence of this approach is simple: calculate the partial pressures of the two oxidants, locate the co-ordinates on the thermochemical diagram (Section 2.2) and thereby define the reaction product. Even if this calculation is successful, it provides no information as to which compounds will exist within the scale interior, where the oxidant activities are not the same as in the gas. More seriously, the prediction often fails even at the scale–gas interface, where one might hope to predict the equilibrium phase, as is done in the case of a single oxidant (Section 3.2). One reason for such a failure was illustrated in Figure 4.3. In the absence of a catalyst, the gas phase was far removed from equilibrium and a completely different reaction product resulted. Even in pure SO2 this can be a problem because the additional reaction SO2 þ 12O2 ¼ SO3

(4.6)

can, depending on temperature, affect the value of pO2 by orders of magnitude [30]. Unfortunately, much of the early work on reaction with pure or diluted SO2 failed to employ a catalyst for the SO3 reaction. If the scaling reaction is rapid and the reactant species is dilute or at low pressure, then it will be depleted from the gas at the scale surface. In the absence of a catalyst at this surface, the gas composition will be different from that of the bulk gas. Furthermore, the kinetics of the solid–gas reactions can lead to changes in the relative oxidant activities, a point which is discussed in Section 4.2 with reference to oxidation–sulfidation of nickel, cobalt and iron. We consider first the reactions of chromium with oxygen–carbon and oxygen– nitrogen gases, where scaling rates are slow and the complications described earlier should be avoided.

4.3.1 Scaling of chromium in oxidizing–nitriding and oxidizing–carburizing gases The Cr–O–C and Cr–O–N phase diagrams are shown in Figure 4.8, with the equilibrium oxidant potentials for the gases in Table 4.1 marked. The carbon activity, aC, is defined through Equation (2.48), with pure solid graphite as reference state. The oxide is much more stable than both carbide and nitride, and is predicted to form in contact with these gases. As seen in Figure 4.4, the prediction is borne out. Unfortunately, however, the protection expected of a chromia scale is not realized in the CO/CO2-based gases, or even in air, as carbides and/or nitrides form beneath the oxide. As will be shown subsequently, the inner carbide and nitride layers continue to grow as the chromia layer thickens, showing that carbon and nitrogen are diffusing through the oxide.

148


Figure 4.8 Thermochemical diagrams for Cr–O–C and Cr–O–N at 9001C. Points correspond to equilibrium for gases in Table 4.1. Dashed lines show diffusion paths for oxide–carbide and oxide–nitride scales. Dotted line represents equilibrium C þ 21 O2 ¼ CO at fixed pCO.

A schematic diffusion path for the oxide over nitride layered structure grown in air [21–25] is shown in Figure 4.8, where the Cr2O3/Cr2N phase boundary is seen to correspond to the interface between the two scale layers. A diffusion path for the oxide over carbide+oxide layered structure developed in CO/CO2/Ar is shown in Figure 4.8. The transit of the path through the inner, two-phase layer is represented by the line along the carbide– oxide phase boundary. Although the activity ratio, ao/ac, is defined at this boundary (assuming pure, stoichiometric compounds), the individual values are


149

not. Put another way, the three-component system has a degree of freedom in the two-phase region, and the activity gradients necessary for mass transfer and scale growth can and do develop. The compositions of all the above gases were such that chromium oxide was stable with respect to chromium carbide and nitride. However, the gas phase carbon and nitrogen activities were high enough to react with chromium in the absence of oxygen. The observed sequence of reaction products in the scales is in accord with thermodynamic prediction. Thus, at the scale surface where the chromium activity is lowest, the most stable product, oxide, is formed. At the scale base where the chromium activity is highest and oxygen activity at a minimum, the least stable product, nitride, is located (when it forms). The intermediate stability phase Cr7C3 is found in the middle regions of the scale. The formation of the lower stability phases implies an ability of the secondary oxidants to penetrate the Cr2O3. Schematic activity profiles for these cases are shown in Figure 4.9. The thermodynamic analysis leaves many questions unanswered. Most obviously, the reason for development of an inner Cr2N layer in air and CO/CO2/N2 but not in H2/H2O/N2 gas is not revealed. The ao/aN values of the

aC O CrCr 7O73+3 +Cr O Cr 2O23 3

Cr

Cr2O3

aO

aCr (a)

aC

aN

aO Cr

Cr2O3 Cr2N Cr7C3+Cr2O3 +Cr2N

aCr

(b)

Figure 4.9 Schematic activity profiles representing the penetration of (a) carbon and (b) nitrogen and carbon through a Cr2O3 layer.

150


two are almost identical and the thermodynamic driving forces for oxide and nitride formation are the same in each gas. The difference is one of reaction kinetics. This raises the more general questions as to how the secondary oxidants can penetrate the oxide layer, and what the mass transfer processes are in the inner layers. These questions are considered in Section 4.4.

4.3.2 Scaling of chromium in oxidizing–sulfidizing–carburizing gases The Cr–O–S phase diagram is shown in Figure 4.10. In all the gas mixtures shown, the oxide is stable with respect to sulfide. The appearance of chromium sulfide at the scale–gas interface (Figure 4.4c) thus demonstrates that the scale surface was not at equilibrium with the bulk gas composition. Carbide grew beneath the oxide developed in the gas, just as in the sulfur-free gases. However, no nitride ever formed in the SO2-containing gases, although it did in sulfur-free CO/CO2/N2. Clearly this complex pattern of behaviour cannot be predicted from the thermochemical diagrams.

4.3.3 Scaling of iron in oxidizing–sulfidizing gases The Fe–S–O phase diagram is shown in Figure 4.11, with a number of different gas compositions marked on it. These compositions were controlled using CO/CO2/SO2/N2 mixtures. In the more commonly reported experiments, a gas of pure SO2 or SO2 diluted with N2 or Ar is used. In this case, the sulfur and oxygen pressures are given by the equilibrium (4.3) plus the stoichiometric

Figure 4.10 Thermochemical diagram for Cr2O2S at 9001C. Dotted line shows diffusion path for sulfide forming under oxide.


151

Figure 4.11 Thermochemical diagram for Fe2O2S at 8001C [48]. Numbered points represent equilibrium compositions for reaction gases. Dashed line represents pSO2 ¼ 7:9 102 atm. With kind permission from Springer Science and Business Media.

requirement pS2 ¼ ð1=2ÞpO2 . Values corresponding to pSO2 ¼ 7:9 102 atm are marked in Figure 4.11. The Fe–S–O diagram reveals that scales in equilibrium with pure SO2 at 1 atm should consist of oxide only at the scale–gas interface. This prediction is in fact borne out [9, 10], at least in the long term, when the reaction products had the appearance of the scale in Figure 4.1b. However, scales grown in diluted SO2 varied in their phase constitution with pSO2 . At pSO2 ¼ 7:9 102 atm, the scale had the same appearance as at pSO2 ¼ 1 atm. At lower pSO2 values, scales were two-phase lamellar mixtures of oxide and sulfide [9, 14]. Gases corresponding to points 7 and 8 (pSO2 ¼ 2 104 atm) corroded iron to produce the scale shown in Figure 4.1a. Evidently local equilibrium at the scale–gas interface might be achieved at high pSO2 values, but not at low values, where sulfide apparently can exist despite the fact it is in contact with a gas in which the reaction FeS þ 12O2 ¼ FeO þ 12S2

(4.7)

is thermodynamically favoured. Furthermore, sulfide has been found to form in gases 7 and 9, which contain equilibrium pS2 values below the minimum necessary for FeS formation in the absence of oxygen. Similar difficulties have been found in the much-studied Ni–S–O system, which is now briefly reviewed.

4.3.4 Scaling of nickel in oxidizing–sulfidizing gases The Ni–O–S phase diagram is shown in Figure 4.12 for T ¼ 6001C. The point labelled X represents the equilibrium oxygen and sulfur potentials in pure SO2 at

152


Figure 4.12 Thermochemical diagram for Ni2O2S at 6001C. Point X represents equilibrium in pure SO2 at 1 atm. Diffusion path for oxide+sulfide layer over Ni3S2 layer.

1 atm. It is clear that the only nickel reaction product stable in contact with this gas is the oxide. However, the experimental findings do not conform with this prediction. The reaction of nickel with pure SO2 in the temperature range 500–1,1001C almost always produces a scale consisting of an inner layer of singlephase sulfide surmounted by a thick layer of duplex NiO+nickel sulfide mixture [8, 15–17, 33–36]. The inner sulfide is the one stable in equilibrium with nickel: Ni3S2 for To5331C, Ni37dS2 for 533oTo6371C and Ni–S liquid at higher temperatures. The phase Ni37dS2 ranges in stoichiometry from metal deficit to metal excess (Figure 3.4a). The sulfide in the duplex layer formed at about 6001C has been identified as Ni37dS2 [17, 34], but that found in scales grown at higher temperatures has not been directly identified. Scales grown in SO2/argon mixtures [17, 34, 35] had the same morphologies. The only exceptions to this pattern are the observations of a scale of NiO only at 1,0001C and pSO2 ¼ 0:01 atm [16], and at the same temperature in an SO2-50% Ar mixture [38]. The detailed morphology of the duplex layer varies with temperature. The concentration of sulfur in the inner part of this layer is very low at To5251C [37],


153

while at around 6001C it is lower than in the outer part of the layer [17, 33]. At these lower temperatures, the sulfide precipitates in the outer part of the duplex layer are large and irregular [15, 37] and because of their shape (Figure 4.2) are described as ‘‘flames’’. As the temperature increases, the flames are replaced by a finer distribution. A duplex layer grown at 6031C was found to have a high electrical conductivity at room temperature, suggesting the presence of a continuous path made up of the metallic conductor Ni37dS2 within the oxide matrix [17]. At higher temperatures, 700–8001C, the duplex scale morphology is quite complex, reflecting a tendency for the liquid sulfide to be extruded from the inner region to form protrusions at the scale–gas interface where they are subsequently oxidized [16, 33]. At still higher temperatures, the overall sulfur content of the duplex layer is much reduced and the sulfide particles are coarser and more isolated [15, 33, 37]. The kinetics of nickel reaction with pure and diluted SO2 are correspondingly complex, as shown in Figure 4.13. Rapid rates correspond to the existence of a continuous sulfide network in the two-phase layer, and slow kinetics are observed when the sulfide content of the layer becomes small. The high diffusion coefficient of the sulfide explains these observations [17]: an interconnected sulfide network provides a continuous rapid mass transfer medium, whereas a discontinuous distribution contributes much less to mass transfer. Reaction kinetics in the temperature range 600–8001C, where sulfide-rich layers grow, have been described as linear [16, 37], protective [15] or irregular [36, 39]. Linear rate constants were reported [16, 34] to be approximately proportional to pSO2 . In SO2/Ar mixtures, the kinetics at 6031C showed an initial linear stage, a second stage of increasing rate, and were finally parabolic [17]. Complex kinetics in SO2/Ar were also reported by Wootton and Birks [37].

Figure 4.13 Scaling kinetics for Ni exposed to SO2 [33]. With kind permission from Springer Science and Business Media.

154


Reaction of nickel in SO2/O2 gas mixtures conducted at To6371C to avoid liquid sulfide formation, led to closely similar structures: an inner layer of singlephase sulfide, and an outer oxide-plus-sulfide mixed layer [8, 39–45] except when the gas was strongly oxidizing, and only a small amount of sulfide formed at the scale–metal interface [8, 44]. After extended periods of reaction, the scale layers started to separate, and the outer layer was converted to NiO. When separation became extensive, and nickel mass transfer was interrupted, the reaction essentially stopped and a thin outermost layer of NiSO4 was formed [43]. The SO2/O2 reaction with nickel follows kinetics which are initially linear and then parabolic until the onset of scale separation [43]. Other investigators [37, 40, 41] have described the kinetics as approximately parabolic. For so long as the duplex scale is produced, its growth rate is independent of pO2 and pSO2 , but decreases as pSO3 increases. A schematic diffusion path is shown in Figure 4.12 for a duplex oxide-plussulfide outer scale layer and Ni37dS2 inner layer. Although showing clearly that the phase diagram provides no predictive capacity in dealing with the Ni+SO2 reaction, it serves to identify the problems confronting us in understanding the scale morphology. Firstly, what are the scale–gas interaction processes which apparently permit sulfide to exist in contact with a gas which is oxidizing to sulfide? Secondly, what are the processes within the scale which constrain the diffusion path to lie along the oxide–sulfide phase boundary? Thirdly, how does the inner sulfide layer form? We consider the scale–gas interactions first.

4.4. SCALE–GAS INTERACTIONS It can usually be assumed that the bulk gas has been catalysed and brought to equilibrium with respect to the otherwise slow reactions (4.3) and (4.6). However, this does not mean that the gas has its equilibrium composition at the scale surface. If a minority species such as SO3 or O2 is a reactant, then it will be consumed at the sample surface. In the absence of a catalyst at this location, the SO3 or O2 cannot be replenished from the gas phase, and its activity is consequently lower than the equilibrium value. The question of just what are the reactant species is seen to be important.

4.4.1 Identity of reactant species The idea of slow transport within the gas coupled with rapid selective removal of some gas species into the scale leading to a different gas composition at the interface was proposed by Birks [9, 46] to explain the formation of both oxide and sulfide at the scale surface under conditions where the sulfide is not stable. It was proposed that reaction of metal at a high activity reduced the local oxygen activity in the gas to the equilibrium value with respect to oxide formation, a very low value. As a result, the sulfur activity would rise through readjustment to maintain the SO2 dissociation equilibrium (4.3), thereby stabilizing the sulfide. Although qualitatively appealing, the mechanism fails quantitatively.

4.4. Scale–Gas Interactions

155

The low pO2 values proposed (about 1019 atm for the Fe–FeO equilibrium at 8001C) are simply too small to support a measurable oxidation rate. The concept of gas phase depletion is nonetheless correct, and likely to succeed when applied to less dilute species. Consider the situation in pure SO2 and dilute Ar–SO2. Equilibrium values of pS2 and pO2 at 8001C, calculated from Equation (4.5), neglecting SO3 formation, are shown in Table 4.3. Fluxes of the various gas species to a sample surface can be calculated for the viscous flow regime (Section 2.9). Taking representative values for laboratory experiments of sample length 1 cm and gas flow rate 0.5 m/min, we calculate the molecular fluxes shown in Table 4.3. A comparison of these values with measured weight uptake rates during corrosion in SO2 gases is revealing. The weight uptake rates in Table 4.4 correspond to two-phase oxide and sulfide growth on nickel (Figure 4.2), chromium (Figure 4.4) and cobalt, and to single-phase oxide outer layer growth on iron (Figure 4.1b). In all two-phase cases, the measured rates far exceed the calculated fluxes of molecular oxygen and/or sulfur. It could be concluded on this basis that the reactant species was SO2 and not oxygen or sulfur. However, it might be argued that catalysis of reaction (4.3) by the scale surface itself, if it occurred, could rapidly replenish gaseous sulfur and oxygen which could, therefore, act as the real reactants. The question has been resolved experimentally in the cases of iron and manganese. Gas mixtures of CO/CO2/SO2/N2 have been used [20, 48, 49] to independently control two of the three variables of interact (pS2 , pO2 and pSO2 ), the third being set by the equilibrium (4.5). Referring to Figure 4.11, it is seen that gases 4–6, 9 and 10 all fall very close to the dashed line corresponding to pSO2 ¼ 0:079 atm. Reaction of iron with all of these catalysed gases led to the same Table 4.3

Partial pressures and mass transfer fluxes in SO2 gases (8001C)

Gas

pSO2

pO2

pS2

SO2 1 7.3 1010 3.6 1010 Ar-7.9%SO2 0.079 1.3 1010 6.5 1011

Table 4.4

a

JSO2

JO2

JS2

3.2 102 2.5 103

6 1012 5 1012 1 1012 1 1013

Scaling rates in SO2 at 8001C

Metal

pSO2 (atm)

Scale surface

Ni Co Fe

1 1 0.2 0.01 0.04

Oxide+sulfide (l) Linear Oxide+sulfide Parabolic Oxide Parabolic Oxide+sulfide Linear Oxide+sulfide Parabolic

Cra

Fluxes (g cm2 min1)

Equilibrium composition (atm)

In CO/CO2/SO2.

Kinetics

Weight uptake rate after 1 h (mg cm2 min1)

Reference

1.0 0.2 0.015 1.5 0.6

[31] [47] [9] [9] [28]

156


results: an initially two-phase oxide and sulfide reaction product which was overgrown with oxide at extended times. This evolution of structure is illustrated in Figure 4.14. Reaction kinetics were in all cases parabolic after an initial period of more rapid reaction. Typical data are shown in Figure 4.15. At this temperature and pSO2 value, the kinetics of reaction with catalysed gas became parabolic after about 36 min, when the outer oxide layer was established. The parabolic rate constants corresponding to oxide growth were 3.270.7 mg2 cm4 min1 for all gases. Thus all gases produced the same reaction products at the same rate, despite the fact that pS2 varied from 1010 to 103 atm and pO2 from 1013 to 1011 atm. It is therefore concluded that SO2 was the reactant. Further support for this conclusion is provided by the results of reaction with gases 1, 5 and 8. All these involve essentially the same value of pS2 ¼ 106 atm, but vastly different levels of pSO2 and pO2 . Experiment 1 led to a complex, fourlayered sulfide and oxide scale which grew according to slow parabolic kinetics. Experiment 8 led to a lamellar two-phase scale (Figure 4.1a) which grew according to rapid parabolic kinetics with kp ¼ 7.8 mg2 cm4 min1. This variation in product morphology and growth rate demonstrates that the gaseous sulfur activity was not the controlling factor. Similarly, experiments 6 and 7 were carried out at the same pO2 ¼ 1011 atm, but different pSO2 and pS2 levels. Whereas experiment 6 produced the oxide overgrowth shown in Figure 4.1b, experiment 7 led to the lamellar structure of Figure 4.1a and linear, rather than parabolic kinetics. Clearly the gas phase oxygen activity was not the important factor. In only one case, experiment 3, was SO2 not the reactant species. In this case the equilibrium value of pS2 ¼ 8:6 102 atm was even higher than the equilibrium pSO2 ¼ 1:2 102 atm. The reaction product was almost pure FeS, which grew according to rapid parabolic kinetics, with kp ¼ 5 103 g2 cm4 min1, a rate which could be sustained by the high partial pressure of molecular sulfur. Thus the SO2 species ceases to be the important reactant only when its partial pressure is significantly exceeded by that of another chemically reactive species.

4.4.2 Rate determining processes in SO2 reactions The possibility of gas phase mass transport being the rate controlling process was considered briefly earlier. It was recognized that the high stability of SO2 (g) with respect to O2 (g) and S2 (g) means that the latter species are necessarily in the minority. As seen in Table 4.3, the rates at which they reach a reacting sample surface are negligibly small, and cannot support observed reaction rates. However, in dilute SO2 gas mixtures, the rate at which the SO2 species diffuses to the surface is, at 8001C, in order of magnitude similar to the scaling weight uptake rates reported in Table 4.4 for 1 h of reaction. In the case of parabolic kinetics, corrosion rates would be faster at earlier times and, at some point, too high for the gas phase process to keep up. At short times then, linear rates controlled by gaseous diffusion are predicted. Such a situation was reported by Birks [9, 46] and Rahmel [11] and later confirmed by Kurokawa et al. [10] in the case of iron reaction with SO2 gases. It was shown that the linear rate constant for mixed sulfide-plus-oxide scale growth on iron was proportional to both the flow


157

Figure 4.14 Evolution of sulfide–oxide scales on Fe in catalysed gas 5 (Figure 4.11). (a) 4 min; (b) 36 min; (c) 144 min and (d) 400 min. [48]. With kind permission from Springer Science and Business Media.

158


Figure 4.15 Kinetics of Fe reaction with gas 5 of Figure 4.11, illustrating the effect of gas phase catalysis [48]. With kind permission from Springer Science and Business Media.

rate of an SO2/Ar gas mixture and its pSO2 value. As seen in the equations for viscous flow mass transfer (2.157 and 2.158), these are almost the dependencies expected for a rate controlled by gas phase transfer. The gas mixtures used in these experiments were such that, at equilibrium, oxide was stable with respect to sulfide. Mixed oxide–sulfide scales grown in gas mixtures where, at equilibrium, sulfide was stable with respect to oxide, have also been shown [50] to thicken according to linear kinetics, attributed to gas phase mass transfer control. In these latter experiments, the gas mixtures were based on CO/CO2/COS, containing rather high pCOS values, and COS was the reactant species. The reaction of manganese with SO2 gas mixtures is very similar in morphological evolution to that of iron [3, 20] when the gas compositions fall in the oxide stability field: an initial period of dual-phase oxide-plus-sulfide scale growth is succeeded by development and thickening of an oxide outer layer. However, the kinetics of both reaction stages are parabolic, with SO2 the dominant reactant species in both cases. Obviously, neither gaseous mass transfer nor an interfacial reaction, each of which leads to linear kinetics, could be involved. Instead, it must be concluded that both stages are diffusion-controlled. The way in which a solidstate diffusion-controlled process can occur under non-equilibrium conditions (in which a metastable sulfide phase grows) is discussed in the next section. Reactions of nickel with SO2 gas mixtures are difficult to study because of the changes of reaction product stabilities with both temperature and gas composition. Nonetheless, it is clear that the growth rates of two-phase (oxideplus-sulfide) scales are strongly dependent on pSO2 . Linear kinetics are attributed [16, 34] to surface reaction rate control. A similar conclusion has been reached


159

for the growth of two-phase oxide-plus-sulfide scales on cobalt, based on the dependence of scaling rate on pSO2 and flow rate [51, 52]. To summarize, then, when SO2 is the reactant gas species, the reaction kinetics can be controlled by (a) solid-state diffusion, leading to parabolic kinetics, (b) surface reactions leading to linear kinetics or (c) gas phase mass transfer, also leading to linear kinetics. Parabolic kinetics can be associated with either twophase scale growth or oxide layer overgrowth. Linear kinetics appear always to be associated with a two-phase reaction product. All these findings apply to gas compositions such that oxide is stable with respect to sulfide (although they can in some cases apply to other regimes). To dissect these reaction mechanisms, it is clearly necessary to understand first how reaction with SO2 can produce a thermodynamically unstable product.

4.4.3 Production of metastable sulfide As we have seen, pure SO2, SO2+O2 or SO2/CO/CO2 gases very commonly provide environments in which the stable reaction product is oxide. The observation of parabolic kinetics is indicative of steady-state diffusion control in the scale, and might therefore be expected to correspond to the achievement of local equilibrium at its surfaces, and the formation of oxide in contact with the gas. However, both oxide and sulfide are commonly found at the scale surface after reaction of nickel and cobalt with oxidizing–sulfidizing gases, and also for iron at short times or low pSO2 values. Clearly, in these cases the scale surface is not at equilibrium with the gas. In a gas in which SO2 is the only reactant species, possible reactions at the scale surface include 2M þ SO2 ¼ 2MO þ 12S2

(4.8)

M þ SO2 ¼ MS þ O2

(4.9)

3M þ SO2 ¼ 2MO þ MS

(4.10)

If local equilibrium for reactions (4.8) and (4.9) is reached, their combination is thermodynamically equivalent to the formation of oxide and sulfide from the elements at their equilibrium partial pressures. In this event, the two reactions can occur simultaneously only if the gas composition falls exactly on the oxide– sulfide equilibrium line in the stability diagram. The possibility of this occurring is remote, and, as suggested long ago [9, 11, 50], the direct reaction (4.10) must be considered. Reaction (4.10) can produce a mixture of oxide and sulfide, even if one of them is not in equilibrium with the bulk gas, provided that the metal activity at the scale–gas interface, asM , is larger than the equilibrium value for reaction (4.10), að10Þ M . The latter is seen from the reaction stoichiometry to be given by ð1=3Þ

að10Þ M ¼ K 10

ðpSO2 Þð1=3Þ

(4.11)

160


where K10 is the equilibrium constant. Under these circumstances (which amount to a steady, although non-equilibrium, state), the unstable sulfide can form, even though not at equilibrium with the gas phase sulfur potential. However, even though molecular oxygen and sulfur are kinetically irrelevant, equilibrium could nonetheless be achieved via destruction of the sulfide through reaction with SO2: 2MS þ SO2 ¼ 2MO þ 32S2

(4.12)

As always, the reason for failure to achieve equilibrium lies in the kinetics of the situation. The sulfide will grow and be perpetuated if the rate of reaction (4.10) exceeds that of reaction (4.12). If it does not, then sulfide will be eliminated, and true equilibrium is established between an oxide scale and the gas. The metastable equilibrium (4.10) is sustained by surface activities of sulfur and oxygen which lie on the sulfide–oxide equilibrium line of the stability diagram. If the SO2 dissociation reaction is also at equilibrium on the surface, then the surface state is defined. Consider the example of iron reacting with pSO2 ¼ 0:079 atm depicted in Figure 4.11. The intersection of the dashed line 1=2 (representing pO2 pS2 ¼ 0:079K3 ) with the oxide–sulfide phase boundary represents the supposed metastable equilibrium. It also defines the minimum value of að10Þ Fe necessary at the scale surface for this equilibrium to be sustained. If, as a result of depletion, the effective value of pSO2 at the interface is less than in the surrounding gas, a steady state can nonetheless be maintained, providing that a higher asFe value is available. There remains the significant question as to just how this remarkable metastable state is arrived at. The surface state corresponds to a higher sulfur activity, but lower oxygen activity than in the gas. This has been explained [3, 53] on the basis of selective removal of oxygen from the adsorbed layer gas into the scale. At first sight, this is difficult to accept because it is the simultaneous reaction of both oxygen and sulfur we are trying to explain. We turn aside for a moment to question whether the stoichiometry of reactions such as (4.10) is actually achieved. A convincing demonstration is available in the case of iron reacting with dilute CO/CO2/SO2 gases of particular compositions [49]. The scale shown in Figure 4.16 is a lamellar mixture of FeO+FeS, with a sublayer of Fe3O4+FeS near the surface. It contains a sulfide volume fraction, fS, measured as 0.4870.06. The reaction appropriate to the equilibrium gas composition is 5Fe þ 2SO2 ¼ Fe3 O4 þ 2FeS

(4.13)

which would form sulfide and oxide in a molar ratio of 2:1. Subsequent conversion of magnetite to wu¨stite via the reaction within the scale Fe3 O4 þ Fe ¼ 4FeO

(4.14)

leads to a molar ratio of 1:2 for FeS to FeO. Using standard density data, it is calculated that the resulting value of fS would be 0.42. The good agreement of the measured value shows that, at least under the parabolic scaling conditions of this experiment, SO2 is reacted with the stoichiometry shown in reaction (4.13). It should be noted for later reference that at lower pSO2 values, where two-phase


161

Figure 4.16 Scale grown on iron in CO/CO2/SO2/N2 (gas 7 of Table 4.5) at 8001C.

scales grow according to linear kinetics, the observed sulfide volume fractions are significantly lower, indicating a different mechanism. Returning to the question of sulfur enrichment on the scale surface, we recognize that preferential sulfur adsorption will account for the observations, providing that surface concentrations are insensitive functions of activity. When kinetics are parabolic, as in the case of the scale in Figure 4.16, the boundary conditions are fixed, and it can be assumed that SO2 exchange between the surface and the surrounding gas is faster than SO2 incorporation into the scale. In other words, gas adsorption is expected to approach equilibrium with respect to SO2 (g), but not with the minority species S2(g) and O2(g). Given that two solid phases are present at the scale surface, adsorption of SO2 can be represented formally as taking place on the oxide (4.15) SO2 ðgÞ þ X ¼ S X þ 2OX þ 2V X o

and on the sulfide

M

X SO2 ðgÞ þ 2Y ¼ 2O Y þ SX s þ VM

(4.16)

together with the surface exchange processes SjX þ Y ¼ SjY þ X

(4.17)

OjY þ X ¼ OjX þ Y

(4.18)

Here X and Y represent surface adsorption sites on the oxide and sulfide, respectively, and cation vacancies have been assumed neutral for the sake of simplicity. If oxygen incorporation via Equation (4.15) is favoured over sulfur incorporation, then the adsorbed phase becomes enriched in sulfur. Such a

162


situation, coupled with a low probability of sulfur desorption, can lead to the non-equilibrium surface activities required for simultaneous oxide and sulfide formation. These non-equilibrium activities can exist in a situation where both oxide and sulfide grow together. The growth of each phase involves consumption of vacancies, V X M , and the incorporation of additional sulfur or oxygen. If these growth processes proceed in parallel, then the balance between adsorbed sulfur and oxygen activities is preserved. The situation is thus seen to be self-sustaining for so long as several conditions are met: (a) The value of pSO2 is much greater than those of pS2 and pO2 . (b) The metal activity at the scale surface is no less than the minimum required for reactions such as (4.10), given by Equation (4.11), or the equivalent for other stoichiometries. (c) The rate of reaction (4.10) producing the two-phase scale is faster than the sulfide oxidation reaction (4.12). (d) Solid-state diffusion within the scale is fast enough to satisfy requirements (b) and (c). As already seen, simultaneous oxide and sulfide formation can be maintained for lengthy times, and large extents of reaction, in the case of nickel and cobalt. In the case of iron, the two-phase product is quickly overgrown by oxide at high pSO2 values, but continues for long times at low pSO2. In the case of chromium, the two-phase oxide and sulfide product grows for long times at relatively high pSO2 and not at all at low pSO2. A special situation arises when the gas composition lies in the sulfate stability field. In this case, the formation of oxide and sulfide at or close to the scale surface can be explained either by the mechanism described above, or by assuming the formation of an outer layer of metal sulfate: M þ SO3 þ 12O2 ¼ MSO4

(4.19)

A two-phase scale could then form beneath the sulfate layer through the reaction 4M þ MSO4 ¼ MS þ 4MO

(4.20)

This mechanism was originally suggested by Alcock et al. [39] and subsequently adopted by Kofstad and co-workers [44, 45] in describing the nickel reaction. In that reaction, the two-phase product was found to be Ni3S2+NiO at about 6001C. Reference to Figure 4.12 reveals a difficulty in that the sulfate phase field is seen not to contact the observed Ni3S2 area. A detailed consideration of possible metastable diffusion paths has been provided by Gesmundo et al. [14]. These are based on the supposition that kinetic hindrances exist for the formation of, e.g., a single-phase NiO layer, which the phase diagram predicts would develop between NiSO4 and Ni3S2 if no other sulfide formed. Of necessity, these possibilities remain speculative. Considerable effort has been expended [32, 33, 43, 54] in seeking to determine whether the sulfate mechanism actually operates. At 6031C, sulfate formed only when the two-phase layer separated from the metal, so that the nickel flux was


163

greatly reduced, and scale–gas equilibrium perhaps more closely approached. At higher temperatures, however, small amounts of sulfate were found in the absence of major scale separation. The sulfate was present as thin (o1 mm), scattered islands [38, 44, 45] on the surface. At 6031C, the rate decreased as pSO3 increased (and pSO2 decreased), indicating that the SO2 was the reactant species. In view of the apparently marginal kinetic stability of the sulfate phase, it is uncertain whether reaction (4.19) together with 9Ni þ 2NiSO4 ¼ Ni3 S2 þ 8NiO

(4.21)

or the direct reaction 9 2Ni

þ SO3 ¼ 12Ni3 S2 þ 3NiO

(4.22)

is the more important. Both would increase in rate with pSO3 , as observed experimentally. Slightly different volume fractions of sulfide and oxide are predicted from the two stoichiometries fS ¼ 0.38 for reaction (4.22) and 0.31 for reaction (4.21). The direct reaction with SO2 7Ni þ 2SO2 ¼ Ni3 S2 þ 4NiO

(4.23)

would produce fS ¼ 0.48, a distinctly higher value. The question of relative amounts of sulfide and oxide appears not to have been investigated.

4.4.4 Independent oxide and sulfide growth in SO2 Most discussions of SO2 corrosion are based on the occurrence of reactions such as (4.10) providing the explanation for the simultaneous formation of both oxide and sulfide. As we have seen, this is equivalent to the development of a metastable surface state which is supersaturated with respect to sulfur. If such a state develops, there seems no reason why sulfide and oxide cannot form independently via reactions such as (4.8) and (4.9). This does in fact occur during reaction of iron in CO/CO2/SO2/N2 gas mixtures at 8001C, if the SO2 partial pressure is low [49]. Catalysed gas mixtures with the equilibrium compositions shown in Table 4.5 produced scales with the morphologies shown in Figure 4.17. It is clear that both the sulfide volume fraction and lamellar spacing, l, varied with gas compositions. Measured values are listed in Table 4.6, along with scaling rates and phase constitutions of the scale surfaces. At pSO2 ¼ 2:2 102 atm, the two-phase product was overgrown with oxide. Two-stage parabolic kinetics were observed (Figure 4.18) with the second stage kw ¼ 1.6 mg cm2 min(1/2). This rate is in reasonable agreement with the value of 1.8 mg cm2 min(1/2) reported [55] for iron oxidation in pure oxygen at 8001C. In both experiments the bulk of the oxide is FeO, under thin external layers of Fe3O4 and Fe2O3 and the growth kinetics will therefore reflect largely the accumulation of wu¨stite. The diffusional flux responsible for the growth of this layer is determined by the boundary conditions at its inner and outer solid–solid interfaces, and is therefore independent of gas composition. Thus at high SO2 partial pressures, the reaction is ultimately one of oxidation only [9, 10, 50, 57].

164

Table 4.5


Equilibrium gas compositions (p/atm) used in FeS volume fraction study at 8001C [49]

Gas

pN2

pCO2

pCO

pSO2

pS2

pO2

1 2 3 4 5 6 7 8 9 10 11

0.12 0.12 0.12 0.12 0.24 0.26 0.24 0.24 0.12 0.12 0.12

0.88 0.88 0.88 0.88 0.76 0.73 0.76 0.76 0.88 0.88 0.88

5.3 104 1.7 104 5.3 104 1.7 104 4.6 105 4.4 105 4.6 105 1.5 104 2.3 104 3.0 104 9.5 105

2.2 105 2.2 105 2.2 104 2.2 104 2.2 104 2.2 102 2.2 103 2.2 103 1.2 104 3.5 104 6.9 105

1.2 1013 1.2 1015 1.2 1011 1.3 1013 1.2 1015 1.2 1011 1.2 1013 1.2 1011 1.2 1013 3.2 1012 1.2 1015

1.0 1012 1.0 1011 1.0 1012 1.0 1011 1.0 1010 1.0 1010 1.0 1010 1.0 1011 5.5 1012 3.2 1012 3.2 1012

At pSO2 ¼ 2:2 103 atm, rapid parabolic kinetics (Figure 4.18) accompanied the formation of a lamellar oxide and sulfide scale (Figure 4.16). The sulfide volume fractions (Table 4.6) are in good agreement with the value fS ¼ 0.42 calculated earlier for stoichiometric uptake of SO2 via reactions (4.13) and (4.14). Thus direct reaction with SO2 produced a metastable surface state, providing a fixed boundary condition, so that diffusion-controlled parabolic kinetics resulted. The reason for the more rapid rate is discussed in the next section. At low pSO2 values in the range 2.2 105–3.5 104 atm, two-phase oxide and sulfide scales grew according to linear kinetics. Although the value of pSO2 in gases 3, 4 and 5 was 10 times higher than in gas 1, the rate was increased less than two-fold. Accordingly, it can be concluded that gas phase mass transfer, which is proportional to pSO2 , was not in effect. The reaction must therefore have been controlled by a surface process, but this was not reaction (4.10) or (4.13), as shown by the diverse values of the sulfide volume fraction fS (Table 4.6). Assuming that iron was delivered to the scale–gas interface by rapid diffusion in a process which did not control the rate, but instead led to a steady-state surface iron activity, the accumulation rates of FeO and FeS may be written from reactions (4.8) and (4.9) as dnFeS ¼ k8 pSO2 dt

(4.24)

dnFeO 1=2 ¼ k9 pSO2 dt

(4.25)

1 k9 1 ¼1þ N FeS k8 p1=2

(4.26)

and for time-independent rates

SO2

Figure 4.17 Scales grown on iron in CO/CO2/SO2/N2 gas mixtures 1, 5, 9 and 10 of Table 4.5 [49]. With kind permission from Springer Science and Business Media.


165

166


Figure 4.18 Parabolic scaling kinetics for iron exposed to CO/CO2/SO2/N2 gases (Table 4.5) at 8001C [49]. With kind permission from Springer Science and Business Media.

Table 4.6

Iron oxide+sulfide scale constitutions and growth rates at 8001C [49]

Gas kw (mg cm2 min1/2)

k1 (mg cm2 min1)

Phases at surface fS

l (mm)

1 2 3 4 5 6 7 8 9 10 11

0.035

FeO/FeS Fe3O4 FeO/FeS FeO/FeS Fe3O4/FeS Fe2O3/Fe3O4 Fe3O4/FeS Fe2O3/FeS FeO/FeS Fe3O4/FeS FeO/FeS

8

1.0 0.063 0.064 0.064 5.2, 1.6 4.15 4.51 0.054 0.102 0.046

0.0870.02 0.0370.02 0.1370.05 0.1770.04 0.4070.05

2.7 2.4 2.5

0.4870.06 0.3670.09 0.2170.06 0.3270.11 0.1870.06

o1 o1 3.6 o1 3.4

Here n denotes mole number and N mole fraction. This is expressed in terms of volume fraction, using the molar volume ratio, as 1 k9 1 ¼ 1:68 þ 0:68 fS k8 p1=2

(4.27)

SO2

Data for FeO/FeS scales plotted in Figure 4.19 are seen to be in only rough agreement with this prediction. The slope implies that k8 2k9 .


167

Figure 4.19 Variation in scale sulfide volume fraction with pSO2 for (FeO+FeS) scales [49]. With kind permission from Springer Science and Business Media.

For the reaction to continue, iron must be available at a sufficient activity at the surface of both phases. If diffusion of iron through FeS and perhaps along phase boundaries predominates over diffusion through FeO, then lateral diffusion of iron must occur at the scale–gas interface in order to sustain the two-phase morphology. Treating the growth of a lamellar sulfide–oxide scale as a cellular (co-operative) phase transformation, it is recognized that the lamellar spacing, l, would therefore be inversely dependent on scaling rate [56]. Basically, a rapid scale growth rate allows time for only a closely spaced microstructure to develop because of the need for lateral mass transfer on the scale surface. Conversely, when scale growth rates are relatively slow, and both gas and surface diffusion are fast, a widely spaced microstructure which lowers the overall surface energy will develop. For a unidirectional co-operative or cellular phase transformation propagating at a velocity, v, kD (4.28) l where k is a constant which includes the driving force for reaction and D the diffusion coefficient for lateral mass transfer on the scale surface. If k is approximately independent of pSO2 , then v varies inversely with l at fixed temperature. v¼

168


Figure 4.20 Variation of lamellar spacing with scale surface growth velocity for (FeO+FeS) scales [49]. With kind permission from Springer Science and Business Media.

Values of v (calculated from kl, fS and molar volumes) are shown plotted according to Equation (4.28) in Figure 4.20 for FeO/FeS scales. Agreement is good in the range examined, but the relationship fails at very large lamellar spacings. In the latter case the principal phase is wu¨stite, the sulfide is discontinuous and the basis for Equation (4.28) no longer exists. It is therefore concluded that the linear kinetic regime of the Fe–SO2 reaction is supported by two independent scale–gas reactions (4.8) and (4.9), which control the relative amounts of sulfide and oxide formed. However, the spacing of the resulting lamellar mixture of phases is controlled by delivery of the other reactant, iron, which diffuses to the surface via FeS or phase boundaries, and then laterally to the reacting oxide. These results provide experimental support for the notion of a co-operative or cellular reaction hypothesized in the past [3, 59].

4.5. TRANSPORT PROCESSES IN MIXED SCALES Two very different patterns of morphological evolution have been observed: on the one hand, continued growth of multiple scale layers and on the other, accretion of different layers, reflecting successive stages of reaction. The first is illustrated by the growth of multiple layers on chromium (Figure 4.4), the kinetics of which are seen in Figure 4.21 to be parabolic, reflecting diffusion control. An example of the second is provided by the reaction of iron with gases containing high levels of SO2 (Figure 4.14). In this case, a two-phase reaction product grows in the initial reaction, but is then overgrown by single-phase oxide, and no further growth of the buried oxide-plus-sulfide layer occurs.

4.5. Transport Processes in Mixed Scales

169

Figure 4.21 Kinetics of layer growth for scale shown in Figure 4.4 [27]. With kind permission from Springer Science and Business Media.

It is clear that carbon and nitrogen must penetrate the outer Cr2O3 scale layer in the first case, whereas this is evidently not so for sulfur in the case of iron oxide. Further information on this matter is available from pre-oxidation studies.

4.5.1 Effect of pre-oxidation on reaction with sulfidizing–oxidizing gases Many investigations into the effect of a preformed pure oxide scale on subsequent reaction with sulfur-containing gases have been reported. The practical aim of this work was to slow the damaging sulfidation process by using a compact, adherent oxide layer as a barrier to reaction. The results are relevant to an understanding of the transport processes involved. Unfortunately, data for iron is limited to that of Rahmel and Gonzalez [58] who studied the reaction of pre-oxidized iron with CO/CO2/COS gases of high sulfur and low oxygen potential at 8001C and 9001C. These gases were in the sulfide phase field and had pO2 values less than the Fe/FeO equilibrium value. A preformed FeO scale did inhibit subsequent reaction, to an extent which increased with oxide thickness. An outer layer of FeS developed during reaction with CO/CO2/COS. Depending on gas composition, the FeS layer was in direct contact with FeO, or an intermediate layer of Fe3O4 developed for a short time, and then disappeared. The outer part of the FeO layer and the intermediate Fe3O4 layer developed stringers of FeS. Thus sulfur penetration of the oxide took place to only a limited extent.

170


Considerably more information is available for nickel. Alcock et al. [39] showed that NiO layers of submicron thickness slowed the subsequent reaction with SO2+O2 gas, but did not prevent sulfide formation. Pope and Birks [59] pre-oxidized nickel at pO2 ¼ 1 atm and 1,0001C, exposing it to CO/CO2 mixtures and finally to CO/CO2/SO2. Sulfide formed beneath the preformed oxide after an incubation period. Subsequent work [60] in which pSO2 was varied led to formation of sulfide beneath the oxide, except when pSO2 was too low for reaction (4.10) to be thermodynamically possible. Similar results were obtained by Kofstad and Akesson [61], who showed that the induction period was longer for thicker NiO scales. Worrell and Rao [32] showed that pre-oxidation in air at 8001C for a day to produce NiO layers of 5–10 mm provided protection against attack by SO2 for up to 14 days. A detailed study [62] of the effect of porosity in the initial NiO layer on subsequent corrosion in SO2 or SO2/O2 was revealing. Samples with high porosity reacted with SO2 according to rapid linear kinetics, quickly forming an inner sulfide, then filling the pores with nickel sulfide and finally developing a two-phase product on top of the preformed oxide. Samples with low porosity reacted significantly more slowly, and those with ‘‘extremely low’’ porosity reacted according to parabolic kinetics with a rate constant lower than that observed in pure oxygen. Subsequent reaction with SO2/O2 was different. The initial porosity was less important because the initial NiO reacted with this gas to form a surface layer of NiSO4 and then oxide and sulfide product on top of the sulfate together with sulfide channels through the underlying oxide. Information is also available on the practically important question of chromium pre-oxidation. Labranche et al. [63] pre-oxidized chromium in H2/H2O gases at 9001C and exposed it immediately to H2/H2O/H2S atmospheres with a composition in the Cr2O3 stability field. No sulfide was detected after 20 h, although about 1 wt% of sulfur was found in the chromia. However, chromium sulfide was found mixed with oxide through the entire scale thickness after 111 h of reaction. Similar results were found [64] for reaction in CO/CO2/N2-based gases. Preoxidation at 9001C produced an adherent Cr2O3 layer of 3 mm thickness, with underlying sublayers containing carbide and nitride, similar to the scale shown in Figure 4.4b. Additions of SO2 to the gas were designed to yield compositions in the Cr2O3 phase field, with sulfur potentials higher than the Cr/CrS equilibrium value. Reaction in these SO2-bearing gases led to formation of a layer of Cr3S4+Cr2O3 on top of the oxide (Figure 4.22) at pS2 ¼ 3 107 atm, and a dispersed Cr3S4 phase at the surface of porous Cr2O3 which had grown over the preformed chromia at pS2 ¼ 3 108 atm, while no surface sulfide formed at pS2 ¼ 3 109 atm. Addition of sulfur to the gas led to the disappearance of the Cr2N layer, although the Cr7C3+Cr2O3 sublayer continued to grow, and incorporated particles of Cr2N. No sulfur was found beneath the preformed oxide when pS2 ¼ 3 107 atm, a significant level (ca. 8 atm%) accumulated in this region when pS2 ¼ 3 108 atm, and less than 1% was found after reaction at pS2 ¼ 3 109 atm.


171

Figure 4.22 Scales produced on chromium by exposure at 9001C to CO/CO2/SO2/N2 after pre-oxidation in CO/CO2/N2 (a) pS2 ¼ 3 107 atm, (b) pS2 ¼ 3 108 atm and (c) pS2 ¼ 3 109 atm. Reprinted from Ref. [64] with permission from Elsevier.

172


It is clear that sulfur can penetrate NiO and Cr2O3 under at least some conditions where the oxides should be thermodynamically stable in contact with the gas in question. This problem has been discussed many times in the literature, usually in connection with sulfur transport in scales formed without pre-oxidation [46, 51, 59, 60, 65, 66].

4.5.2 Solid-state diffusion of sulfur An obvious possible method of sulfur penetration is via dissolution in and diffusion through the oxide lattice. Little information is available on either of these processes. Analytical measurements of sulfur solubility in NiO and CoO have been reported by Pope and Birks [59], who measured maximum values at 1,0001C of 0.026 wt % in NiO and 0.050 wt % in CoO. Sulfur has been found to diffuse more rapidly than oxygen in NiO, but much slower than nickel [67–69]. In NiO single crystals, it is reported [70] that Ds ¼ 5.4 1013 cm2 s1, Do ¼ 8 1014 cm2 s1 and DNi ¼ 1011 cm2 s1 at 1,0001C. However, the combined values of sulfur solubility and diffusivity in NiO are too small to account for the sulfidation rate of nickel in mixed atmospheres [71]. In summary, there appears to be no evidence that solid-state diffusion of sulfur through metal oxide can ever account for the penetration by sulfur of preoxidized scales. However, sulfur can diffuse at significant rates in the sulfides of nickel and cobalt [16, 34, 39–41, 72, 73] thereby accounting for the inward growth of the sulfide layer which develops at the interface between these metals and their scales.

4.5.3 Gas diffusion through scales A second mode of sulfide formation beneath single-phase oxide could be via gas diffusion of sulfur or its compounds. As seen earlier (Tables 4.3 and 4.4), however, sulfur pressures in SO2 atmospheres are generally too low to support the observed reaction rates, and the question of molecular S2(g) diffusion is therefore probably irrelevant. However, if transport occurs via the much more abundant SO2 (or SO3 in oxygenated gas), a viable mechanism is available. Inward transport of sulfur by this means has been considered in detail by Birks [46, 51, 59, 60, 65], who proposed that gas compositions within scale cracks could change by reaction with the scale. Lowering the oxygen potential to values at equilibrium with the scale interior would then lead to an increase in sulfur potential through the equilibrium relationship (4.5). In this way, sulfide formation could become thermodynamically possible within the scale. The interface between a crack surface and the gas phase is equivalent to that between the scale exterior and the gas, so the discussion provided in Section 4.4.3 is equally applicable. In particular, the condition that the thermodynamically favoured reaction (4.12) be kinetically hindered applies in this situation. Diffusion of SO2 molecules provides a satisfactory explanation of the observations reported for reaction of pre-oxidized nickel with SO2-bearing gases.


173

The improved resistance to sulfur penetration of a NiO scale with increased thickness and decreased porosity is an obvious consequence. Direct information on iron oxides is very limited. However, the observation during reaction without pre-oxidation that the first formed oxide and sulfide layer ceases to grow once an oxide layer forms on top indicates that FeO and perhaps Fe3O4 are more resistant to SO2 diffusion than NiO. The much larger grain size of FeO, and the consequently reduced availability of grain boundaries, could be a factor. The greater plasticity of FeO and hence its lower frequency of cracking during growth could also be important in limiting the availability of pathways for molecular diffusion. The behaviour of Cr2O3 scales when exposed to sulfurous gases is interestingly complex. At a high sulfur potential and high pSO2 , chromium sulfide nucleated at the oxide–gas interface (Figure 4.22). Sulfide formed at both the surface and beneath the oxide at an intermediate sulfur potential (and pSO2 value), whereas at a low sulfur potential and low pSO2, some enrichment of sulfur developed beneath the oxide, but not on top. These observations are not readily understood on the basis of inward SO2 diffusion and the diffusion path shown in Figure 4.10. The diffusion path illustrates the ideas first articulated by Stringer and Whittle [74] and by Stringer [75] in connection with mixed gas corrosion. In essence, it was proposed that the reduced oxygen activity within and beneath the oxide allowed an increase in as through the SO2 dissociation equilibrium. The line A-X in Figure 4.10 follows such a path. This effect, together with the relatively high chromium activity can then stabilize the sulfide with respect to the oxide. The activity gradients for sulfur, oxygen and chromium within the two-phase layer (along the path XY) are all appropriate for its growth. A disadvantage of the description is the strong sulfur gradient within the single-phase oxide layer, along the path X-A in a direction appropriate to outward sulfur diffusion. This can be rationalized on the basis that such a process would be kinetically hindered by the low solubility of sulfur in the oxide phase. However, a more fundamental difficulty exists with the inability of the description to cope with the reaction morphology of Figure 4.22. This is overcome by applying again the description of a sulfur-enriched surface-adsorbed layer given earlier (Equations (4.15)–(4.18) in Section 4.4.3). It must be recognized that the lowering of oxygen activity relative to that of sulfur can occur both within and beneath an oxide as well as at its surface, by preferential adsorption. The line segment AX of the diffusion path in Figure 4.10 represents both cases, providing no information on the spatial location. It thus provides no ability to predict whether sulfur enrichment occurs between an (oxide) scale surface and its interior, or between the bulk gas and an adsorbed layer. It is recognized that the diffusion path description lacks utility in this situation. The case of pre-oxidized chromium reacting with SO2 illustrates very clearly how surface processes can displace the location at which oxidant activity changes occur. The oxide shown in Figure 4.22a was obviously gas permeable, as evidenced by the continued growth of the carbide subscale. The failure of sulfur

174


to penetrate this material was due to its immobilization in an outer Cr3S4 layer. At a lower pSO2 value (Figure 4.22b), sulfur permeated the preformed oxide more freely because the surface sulfide formed in this gas was discontinuous. At a still lower pSO2 value (Figure 4.22c), no surface sulfide at all was formed, and sulfur slowly permeated the oxide, enriching beneath it. Assuming that a separate sulfide phase was formed in the third gas, we see that all three cases can be described, at least in part by the diffusion path of Figure 4.10. The differences in the path segment X-Y are due to (surface) processes other than diffusion. We see that a difference arises in the case of the high pSO2 gas (Figure 4.22a) where a single-phase oxide sublayer still exists. Given the failure of equilibrium thermodynamic phase diagrams to predict scale surface constitutions in the case of SO2 reaction, and the inability of diffusion paths to cope with surface reactions, it is reasonable to ask how predictive capacity could be arrived at. Unfortunately, only qualitative statements can be made, as will be seen in Section 4.6.

4.5.4 Scale penetration by multiple gas species As has been experimentally demonstrated, and discussed in some detail earlier, SO2 molecules can both react at scale surfaces and penetrate oxides to react in the scale interior. Scales which are permeable to one gas species might be expected to transmit others, and this is indeed the case. As seen earlier (Figure 4.4) nitride and carbide can form beneath a Cr2O3 scale during exposure to mixed gases. As is clear from Figure 4.21, carbon and nitrogen continue to penetrate the oxide layer, supporting diffusion-controlled growth of the underlying carbide and nitride layers throughout the observed reaction. Gas phase ac and aN values were high enough to stabilize the carbide and nitride at high aCr values, and the schematic activity profiles of Figure 4.9 illustrate the diffusional steady-state. The exception to this pattern was the single-phase oxide scale grown in H2/H2O/N2 gas, which was obviously not permeable to nitrogen. It is known [76] that carbon solubility in Cr2O3 is negligible, and it seems likely that the same would be true of nitrogen. Neither of these species could penetrate the oxide by lattice diffusion, and molecular transport via scale imperfections is indicated. Diffusion along these imperfections must be rather slow to produce the activity gradients corresponding to the layered scales which result. Accordingly, it is proposed that the transport mechanism is one of diffusion of adsorbed gas molecules along grain boundaries or similar internal surfaces. Competitive adsorption processes give rise to interactions among the diffusing species. Thus the non-polar N2 molecules are displaced by relatively strongly adsorbing H2O, and the absence of any nitride layer underneath Cr2O3 grown in H2/H2O/N2, despite its thermodynamic stability, is thereby explained. However, nitrogen is able to diffuse along these internal surfaces if the oxidant is CO2 and the corresponding adsorbed species is CO. As seen earlier (Section 4.3.2), addition of SO2 to the CO/CO2/N2 gas also suppressed the expected nitride formation, even when the pSO2 value was too low to form external surface sulfide. Under these circumstances, the rate of inner

4.6. Predicting the Outcome of Mixed Gas Reactions

175

carbide growth was slowed, but not stopped. Obviously, these effects would not be possible if the mechanism of penetration by secondary oxidants was one of gaseous diffusion through large defects. Again it is concluded that mass transport involved much smaller defects, such as internal surfaces. Preferential adsorption of sulfur on oxide boundaries would be expected. It seems that the more reactive CO molecule can adsorb to some extent on the sulfur-poisoned grain boundaries, whereas the unreactive N2 cannot.

4.5.5 Metal transport processes In a two-phase scale such as the examples shown in Figures 4.1a, 4.2 and 4.4c, metals can diffuse as cations in the lattices of both phases and along the boundaries between them. If these phases are continuous, in the sense of providing an unbroken diffusion pathway between metal and the scale–gas interface, then the flux of metal in each phase is described by Equation (3.62). However, as has become abundantly clear, the boundary conditions at the scale–gas interface are far from equilibrium, and unknown. This makes the application of Equation (3.62) impossible. Moreover, the diffusional properties of phase boundaries such as those between oxide and sulfide, which can be so abundant, are unknown. In the absence of basic data, it is appropriate to assess the contributions of the different diffusional processes by comparing scaling rates. Table 4.7 lists rate constants for oxidation, sulfidation and reaction with SO2 for several metals at particular temperatures. Most values refer to reaction with the relevant gas at a pressure of 1 atm. The comparison is not ideal because the pS2 and pO2 values in effect at a scale–SO2 gas interface will be much less than 1 atm. However, as the effect of oxidant partial pressure on reactions involving only one species (Equations (3.76) and (3.90)) is small, the values shown are sufficient for our purposes. Corrosion in SO2 is seen to be faster than oxidation in pure O2 whenever a two-phase product is formed. The difference for iron and manganese is only moderate, at about an order of magnitude, because both FeO and MnO have large concentrations of defects, and oxidation is in any case rapid. Conversely, the difference for nickel and chromium is very large. The sulfidation rates of both metals are many orders of magnitude faster than the oxidation rates because the sulfides possess much more defective structures. The formation of a continuous sulfide phase in the scales developed by these metals therefore provides an alternative, much more rapid diffusion pathway for cations.

4.6. PREDICTING THE OUTCOME OF MIXED GAS REACTIONS The corrosion reactions examined in this chapter exhibit a diversity of outcomes, and an attempt to arrive at a unified view is worthwhile. As we have seen, the use of solid–gas thermodynamic equilibrium in predicting scale surface constitutions is successful in some cases (oxidation–carburization–nitridation of

800

800

600 900

Fe

Mn

Ni Cr

[30] [79] [81]

1.6 109

1.0 1013 2 1013

5.5 10

[55]

Reference

8

kw

O2

6

9 107 8 107

3.2 109

8.1 10

kw

Notes: (a) Single-phase oxide scale surface and (b) two-phase oxide+sulfide scale surface.

T (1C)

Metal S2

[80] [80]

[78]

[77]

Reference 8

(a) 4 10 (b) 3 107 (a) 3 109 (b) 1.1 108 (b) 2.5 106 (b) 1.1 108

kw

Table 4.7 Values of kw (g2 cm4 s1) for reaction in O2 (1 atm), S2 (1 atm) and SO2 (p indicated)

2

2.2 10 2.2 103 7.3 102 2.5 104 1.0 3.9 102

p (atm)

SO2

[49] [49] [20] [20] [17] [28]

Reference

176 Chapter 4 Mixed Gas Corrosion of Pure Metals

177


chromium, sulfidation–oxidation of iron and manganese under certain condition) and is without value in other situations, notably the sulfidation–oxidation of nickel and chromium. Among the factors leading to these different outcomes are the differing stabilities of the reaction products, the existence of heteronuclear molecules appropriate to biphase solid production and the relative rates at which secondary, metastable reaction products can be incorporated into the scale or destroyed by interaction with gas species. Selected DG values for reactions of interest are shown in Table 4.8. We consider first the question of why chromium carbide and nitride develop beneath Cr2O3 scale layers, but never at the surface, whereas chromium sulfide and oxide can develop at both locations. Appropriate gas molecules for simultaneous formation of two products exist in all cases: CO, NO and SO2. However, the feasibility of the various reactions differs greatly. Firstly, the species NO is present at only very small concentrations, as can be seen from the equilibrium 1 2N2

þ 12O2 ¼ NO; DG ¼ 90; 136 12:4 T ðJ mol1 Þ

(4.29) 5

At 9001C, for example, a gas mixture containing pN2 ¼ 1 atm and pO2 ¼ 10 atm has an equilibrium value pNO ¼ 2 107 atm. This is far too low to support a surface reaction, and the process (e) in Table 4.8 is kinetically irrelevant. The same evaluation is arrived at for other nitrogen oxides, and we conclude that no reaction is available for the formation of metastable nitride at a Cr2O3 surface. Chromium nitride can only develop by inward diffusion of nitrogen to a zone where the oxygen activity is low and chromium activity is high. The question of nitride formation on iron or nickel at high temperatures does not arise, as no stable compounds exist. A similar situation exists with carbides. No nickel carbide is stable, and the iron carbide, Fe3C, is of marginal stability. In fact it is metastable below 7481C, where values of acW1 are required for its formation. Although Fe3C is formed on iron in strongly carburizing gases (see Chapter 9), the simultaneous formation of both carbide and oxide on pure iron has not yet been reported. However, elemental carbon can be deposited in the inner parts of an iron oxide scale grown in CO2 at low temperatures, about 400–5001C [84, 85]. Gas molecules diffuse inwards through the oxide and the CO/CO2 ratio alters according to CO2 ¼ CO þ 12O2 Table 4.8

(4.30)

Selected mixed gas reaction free energies [82, 83]

Reaction

T (1C)

DG1 (kJ mol1)

(a) 3Fe+SO2 ¼ FeO+2FeS (b) 72Ni þ SO2 ¼ 12Ni3 S2 þ 2NiO (c) 3Cr þ CO ¼ 13Cr7 C3 þ 13Cr2 O3 (d) 73Cr þ SO2 ¼ CrS þ 23Cr2 O3 (e) 83Cr þ NO ¼ Cr2 N þ 13Cr2 O3

800 600 900 900 900

195 92 124 536 379

178


as the oxygen activity of the scale decreases towards the metal–scale interface. As pCO rises, the Boudouard reaction 2CO ¼ CO2 þ C

(4.31)

becomes favoured (providing the temperature is low) and carbon deposition results. This is analogous to the formation of chromium carbide beneath a Cr2O3 scale, the difference being that no iron carbide can form and instead graphite precipitation results. A significant volume expansion accompanies reaction (4.31), and the resulting compressive stresses lead to oxide ‘‘bursting’’. Chromium carbides are significantly more stable (Table 2.1) and the possibility of simultaneous Cr7C3+Cr2O3 formation is now considered. As seen in Table 4.8 their formation by reaction (c) with CO is thermodynamically favourable, but the driving force is rather small, at only 42 kJ mol1 of chromium. The reaction 2Cr þ 3CO2 ¼ Cr2 O3 þ 3CO (4.32) with DG ¼ 274 kJ mol1 at 9001C is much more favourable at gas component activities near unity. The possibility of the energy barrier to surface carbide formation being overcome by oxygen depletion and consequent carbon enrichment via CO ¼ C þ 12O2

(4.33)

is remote. If this reaction dominated the surface (and CO2 processes could be neglected), the situation could be represented by the line AB in Figure 4.8. It is seen that oxygen depletion at constant pCO would lead to carbon precipitation, but not to carbide formation. We therefore conclude that formation of carbide at a Cr2O3 scale surface is not thermodynamically favoured. Instead, carbide can form beneath the oxide, where the oxygen activity is low and chromium activity high. The situation is quite different when SO2 is the reactant species, as seen in Table 4.8. The simultaneous formation of Cr2O3 and CrS in contact with the gas via reaction (d) is strongly favoured, with DG ¼ 230 kJ mol1 of metal at T ¼ 9001C and pSO2 ¼ 1 atm. The alternative reaction 2Cr þ 32SO2 ¼ Cr2 O3 þ 32S2

1

(4.34) 1

has DG ¼ 399 kJ mol at 9001C, corresponding to 200 kJ mol of chromium at unit activity of gas components. Although the reaction is favoured even at low pS2 values, a mechanism for sulfur enrichment on the surface (represented by the shift from A to B in Figure 4.10) is available. The combination of a strong driving force plus a mechanism for selective adsorption makes two-phase product formation on chromium much more favoured thermodynamically in SO2 than in CO. As is now clear, the greater stability of metal sulfides compared to that of carbides and nitrides is the fundamental reason that metastable surface sulfide formation is sometimes possible in sulfidizing–oxidizing gases. The question of interest then is why in some cases the formation of two-phase scale continues for long reaction times, whereas in others it ceases after a short time. The situation


179

was described earlier in terms of competing reactions for two-phase product formation (4.10) and for sulfide oxidation (4.12). Whereas the latter is purely a surface reaction, the former is controlled at least partially by the rate of metal transport through the scale. Thus, the faster the rate of scale growth, the less likely it is that sulfide formed at the surface will be oxidized. On this basis (Table 4.7) it would be predicted that sulfide formation would continue for longer times in the case of nickel, but for shorter times in the case of iron, manganese and chromium. The analysis is qualitatively successful for manganese, iron and nickel, but not for chromium, and a further factor must be involved. It has been suggested [3] that an important factor is the metal activity at the ð2-phaseÞ surface required for two-phase scale formation through reaction (4.10), aM , relative to the activity required for true local equilibrium of reaction (4.8), aðoxideÞ . m The ratio of metal activities required for these competing processes is 1=6

ð2phaseÞ

aM

aðoxideÞ M

¼

1=2 K8 pSO2 1=3

1=4

K10 pS2

(4.35)

Thus lower values of pSO2 and higher values of pS2 lower the metal activity level required to form a two-phase scale rather than the oxide. The metal activity at the scale–gas interface is high during the initial stages of reaction and presumably decreases progressively as the scale thickens, until its value is below the minimum of Equation (4.35). This accounts satisfactorily for the observed behaviour of iron, which forms a two-phase product for long times at low pSO2 values, but quickly converts to an oxide-only outer layer at high pSO2 . It is also broadly consistent with observations on the chromium reaction. When reacted at pSO2 ¼ 1 atm, chromium formed only oxide [30] but at pSO2 ¼ 0:04 atm it developed a two-phase product at the scale–gas interface [28]. However, the latter gas when fully equilibrated would have contained pSO2 ¼ 1010 atm, a value too low to be kinetically significant, but pCOS ¼ 4 104 atm. In short, a different surface mechanism was in effect, and the utility of Equation (4.35) cannot be tested. The value of aM at the scale–gas interface is certainly an important parameter. As noted by Gesmundo et al. [3], any pre-oxidation treatment carried out at a high pO2 would result in a low surface aM value, and an inability of the scale to form any oxide-plus-sulfide product when subsequently exposed to SO2. As seen earlier, pre-oxidation of nickel at high pO2 (air or pure oxygen) produced scales which resisted sulfide formation at the surface, and eventually formed sulfide beneath the oxide when exposed to SO2. This is to be contrasted with the continued growth of oxide-plus-sulfide on nickel exposed to SO2 without preoxidation, where a higher surface aNi value must have been available. An analogous effect has been observed [64] in the reaction of chromium. As discussed earlier, chromium reacts with CO/CO2/N2/SO2 gas mixtures to form an outer layer of Cr2O3+Cr5S6. If, however, the metal was first pre-oxidized and then exposed to the same oxidizing–sulfidizing gas mixture, the sulfide formed at the scale surface was identified as Cr3S4. The Cr5S6 phase is not stable at the reaction temperature of 9001C, being formed during cooling from the high

180


temperature Cr1dS phase. The growth of the lower sulfide indicates that a higher surface aCr value was available. Pre-oxidation depressed asCr , allowing the higher sulfide to form. The experimental data for both nickel and chromium provide qualitative support for the proposal that surface sulfide formation is favoured by high surface aM values, and that these can be lowered by suitable pre-oxidation treatments. High surface aM values are supported by rapid diffusion of metal through the sulfide phase, or along interphase boundaries. However, despite these successes, Equation (4.35) is not universally applicable. Evaluation of the relevant equilibrium constants using the data in Table 4.8 leads to the prediction that formation of a two-phase product requires a higher activity of nickel than of iron. Despite this, a sulfide-plus-oxide scale persists on nickel whereas on iron it is soon overgrown by oxide. Here the kinetic factors outweigh the thermodynamics because diffusion in the nickel sulfides is fast enough to maintain high asNi values. As will be discussed in Chapter 7, further complexities arise in the mixed gas corrosion of alloys. It is sufficient therefore to conclude that the formation of additional reaction products (carbides, nitrides and sulfides) during oxidation of metals is governed by both thermodynamic and kinetic factors as well as gas adsorption processes. In general, thermochemical diagrams are successful in predicting the phases formed in contact with carburizing–oxidizing and nitriding–oxidizing gases. This success is attributable to the low stability of carbides and nitrides compared to that of oxides. However, the diagrams frequently fail to predict scale constitutions formed in contact with sulfidizing–oxidizing gases. One reason for this failure is the higher stability of metal sulfides which can be sufficient to enable a metastable oxide-plus-sulfide mixture to form. Another reason is the ability of sulfur to adsorb preferentially on the scale surface, resulting in a surface richer in sulfur than the gas phase. The differences between the scales grown on different metals exposed to oxidizing–sulfidizing gases cannot be rationalized in terms of phase stabilities alone. The ability of the scale to support rapid metal diffusion, thereby maintaining high surface metal activities is an important factor in promoting two-phase product formation in direct reaction with SO2. In this respect, the behaviour of nickel is unique, as a result of its formation of high diffusivity Ni37dS2 (DNiE105 cm2 s1 at 6001C) at temperatures of 533–6351C, and liquid sulfide at higher temperatures. For this reason, the formation of mixed oxide-plus-sulfide on nickel continues for very long times, and metal destruction is extensive. The use of diffusion paths to describe phase distributions within reaction product scales is only sometimes of value. Inward diffusion of gaseous species adsorbed on internal surfaces can be represented by diffusion paths in the cases of oxidation–carburization and oxidation–nitridation of chromium. However, interaction between diffusing species (e.g. CO, N2, SO2) via competitive adsorption can prevent the diffusion of nitrogen and slow the transport of carbon, changing the diffusion paths. The diffusion path concept is of even less value in describing sulfidation–oxidation reaction. In the case of nickel and chromium,

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SO2 usually reacts at the scale surface rather than penetrating the oxide. In the case of iron, SO2 reacts at the surface initially, but cannot penetrate the iron oxide outer layer once it is formed. A very large research effort into mixed gas corrosion reactions has yielded a substantial body of descriptive knowledge, an appreciation of the multitude of factors involved and a capacity to interpret and understand the results. However, our predictive capacity is at best qualitative. Nonetheless, the understanding which has been developed does provide a rational basis for experimental design to use in any future research. This may prove to be of considerable value as new technologies for fossil fuel processing are developed to improve efficiencies and reduce greenhouse gas emissions.

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33. F. Gesmundo, C. DeAsmundis and P. Nanni, Oxid. Met., 20, 217 (1983). 34. M. Seiersten and P. Kofstad, Corros. Sci., 22, 497 (1982). 35. W.L. Worrell and B.K. Rao, in Proc. Int. Conf. High Temperature Corrosion, ed. R.A. Rapp, NACE, Houston, TX (1983), p. 295. 36. N. Jacobson and W.L. Worrell, in Proc. 3rd Int. Conf. Transport in Nonstoichiometric Compounds, eds. G. Simkovich and V.S. Stubican, Plenum Press, New York (1984), p. 451. 37. M.R. Wootton and N. Birks, Corros. Sci., 12, 829 (1972). 38. H. Nakai, K. Okada and Y. Kato, in Proc. 3rd JIM Int. Symp. High Temperature Corrosion of Metals and Alloys, Japan Institute of Metals, Sendai (1983), p. 427. 39. C.B. Alcock, M.G. Hocking and S. Zador, Corros. Sci., 9, 111 (1969). 40. V. Vasantasree and M.G. Hocking, Corros. Sci., 16, 261 (1976). 41. M.G. Hocking and V. Vasantasree, Corros. Sci., 16, 279 (1976). 42. K.L. Luthra and W. Worrell, in Proc. Symp. Properties of High Temperature Alloys, eds. Z.A. Foroulis and F.S. Pettit, Electrochem. Soc., New York (1976), Vol. 1, p. 318. 43. K.L. Luthra and W.L. Worrell, Met. Trans. A, 10A, 621 (1979). 44. B. Haflan and P. Kofstad, Corros. Sci., 23, 1333 (1983). 45. P.K. Lillerud, B. Haflan and P. Kofstad, Oxid. Met., 21, 119 (1984). 46. N. Birks, in Proc. Symp. High Temp. Gas-Metal Reactions in Mixed Environments, eds. S.A. Jansson and Z.A. Foroulis, Met. Soc. AIME, New York (1973), p. 322. 47. K. Holthe and P. Kofstad, Corros. Sci., 20, 919 (1980). 48. G. McAdam and D.J. Young, Oxid. Met., 37, 281 (1992). 49. J. Unsworth and D.J. Young, Oxid. Met., 60, 447 (2003). 50. A. Rahmel and J.A. Gonzalez, Werkst. Korros., 22, 283 (1971). 51. P. Singh and N. Birks, Oxid. Met., 12, 23 (1978). 52. N.S. Jacobson and W.L. Worrell, J. Electrochem. Soc., 131, 1182 (1984). 53. P. Kofstad, High Temperature Corrosion, Elsevier, London (1988). 54. V. Guerra-Brady and W.L. Worrell, in Proc. 10th Int. Symp. Reactivity of Solids, eds. P. Barret and L.C. Dufour, Materials Science Monographs, 28A, Elsevier, Amsterdam (1985), p. 61. 55. M.H. Davies, M.T. Simnad and C.E. Birchenall, Trans. AIME, 191, 889 (1951). 56. M. Hillert, ed., ‘‘The Mechanisms of Phase Transformations in Crystalline Solids, Institute of Metals, London (1969). 57. N. Birks and G. Meier, Introduction to High Temperature Oxidation of Metals, Edward Arnold, London (1983). 58. A. Rahmel and J.A. Gonzalez, Corros. Sci., 13, 433 (1973). 59. M.C. Pope and N. Birks, Oxid. Met., 12, 173 (1978). 60. P. Singh and N. Birks, Werkst. Korros., 31, 682 (1980). 61. P. Kofstad and G. Akesson, Oxid. Met., 13, 57 (1979). 62. W.L. Worrell and B. Ghosal, in Proc. 3rd JIM Int. Symp. High Temperature Corrosion of Metals and Alloys, Japan Institute of Metals, Sendai (1983), p. 419. 63. M.H. La Branche, A. Garrat-Reed and G.J. Yurek, J. Electrochem. Soc., 130, 2405 (1983). 64. X.G. Zheng and D.J. Young, Corros. Sci., 38, 1877 (1996). 65. N. Birks, in Proc. Symp. Properties of High Temperature Alloys, eds. Z.A. Foroulis and F.S. Pettit, Electrochemical Society (1976), Vol. 1, p. 215. 66. K.N. Strafford and P.J. Hunt, in Proc. Int. Conf. High Temperature Corrosion, ed. R.A. Rapp, NACE, Houston, TX (1983), p. 380. 67. D.R. Chang, R. Nemoto and J.B. Wagner, Met. Trans. A, 78A, 803 (1976). 68. W.Y. Howng and J.B. Wagner, J. Phys. Chem. Solids, 39, 1019 (1978). 69. P. Dumes, A. Fauvre and J.C. Colson, Ann. Chim. Fr., 4, 269 (1979). 70. J.S. Choi and W.J. Moore, J. Phys. Chem., 66, 1308 (1962). 71. J.B. Wagner, Oxid. Met., 23, 251 (1985). 72. B. Gillot and D. Garnier, Ann. Chim. Fr., 5, 483 (1989). 73. J. Gillewicz-Wolter and K. Kowalska, Oxid. Met., 22, 101 (1984). 74. J. Stringer and D.P. Whittle, Rev. Int. Htes Temp. et Refract., 14, 6 (1977).

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CHAPT ER

5 Oxidation of Alloys I: Single Phase Scales

Contents

5.1. 5.2. 5.3. 5.4. 5.5.

Introduction Selected Experimental Results Phase Diagrams and Diffusion Paths Selective Oxidation of One Alloy Component Selective Oxidation of One Alloy Component Under Non-Steady-State Conditions 5.6. Solid Solution Oxide Scales 5.6.1 Modelling diffusion in solid solution scales 5.7. Transient Oxidation 5.7.1 Transient behaviour associated with alumina phase transformations 5.8. Microstructural Changes in Subsurface Alloy Regions 5.8.1 Subsurface void formation 5.8.2 Scale–alloy interface stability 5.8.3 Phase dissolution 5.8.4 New phase formation 5.8.5 Other transformations 5.9. Breakdown of Steady-State Scale 5.10. Other Factors Affecting Scale Growth References

185 187 193 196 202 206 209 216 219 226 226 229 230 235 236 237 241 243

5.1. INTRODUCTION Most metallic materials of practical importance are alloys. Even mild steel invariably contains some level of carbon, and usually small amounts of other elements. The alloys and coatings employed for heat resisting applications are usually based on iron, nickel or cobalt and contain chromium and/or aluminium to provide protective oxide scales. The choice of chromium and aluminium is based on the slow rate at which their oxides grow (see Table 1.1) and the fact that these oxides are considerably more stable than those of iron, nickel or cobalt (see Table 2.1). Thus Cr2O3 and Al2O3 are thermodynamically favoured and will form preferentially if this is kinetically possible. Alloy additions of silicon would seem to meet the same

185

186

Chapter 5 Oxidation of Alloys I: Single Phase Scales

criteria, and to possess the added advantage of being readily available in the economic form of ferrosilicon. However, the use of silicon as an alloying additive is limited by metallurgical constraints. Since silicon decreases the weldability and toughness (or impact resistance) of steels and nickel-base alloys, its concentration is limited to such low levels that formation of a silica layer alone cannot be achieved. However, silica scales play a protective role in the high temperature performance of a number of non-oxide ceramics, e.g. MoSi2, SiC and Si3N4. A very large research and development effort has led to the production of present-day heat-resisting materials, and is continuing in the search for better ones. The literature in this field is consequently vast, and no attempt will be made to review it here. Instead, the focus will be on developing an understanding of the different modes of alloy oxidation, finding ways of predicting the circumstances under which each of these modes operates, and calculating the rate of alloy consumption in each case. This chapter is concerned principally with the factors which determine whether an alloy forms a protective scale of the desired low diffusivity oxide (usually Cr2O3 or Al2O3) or some other reaction products. The number of possible outcomes is large and increases with alloy complexity, each additional alloy component providing an additional degree of thermodynamic freedom, as seen in Equation (2.35). Even a simple binary alloy can develop two-phase regions within the scale and/or form non-planar interfaces between adjacent phases. It can also precipitate oxide of its more reactive component inside the alloy in the process of internal oxidation. Alloy components will form oxides of different stabilities. Moreover, the component metals will generally have differing mobilities in each relevant phase, alloy and oxides, as well as varying intersolubilities. The situation can be further complicated by the existence of multiple phases within the alloy (e.g. ferrite+Fe3C in steel; g þ g0 in superalloys; etc.), the formation of ternary product phases, and occasionally the development of low-melting eutectics such as FeO–Fe2SiO4 on silicon steels. In addition, the distribution of reaction product phases, that is to say the reaction morphology, will usually change with time. In an initial, transient stage, all reactive alloy components oxidize, yielding a product with essentially the same relative proportions of metallic constituents as the parent alloy. Subsequently, local equilibrium is achieved at alloy–scale and scale–gas interfaces, and steady-state reaction follows. During this stage, the scale morphology and composition are invariant with time, and the phase diagram ‘‘diffusion path’’ description applies. Heat resisting alloy design is directed towards achieving a slow growing oxide scale during the steady state. Ultimately, the alloy component responsible for protective oxide formation is exhausted or ceases to be available, and a final breakdown stage ensues. Eventually, of course, the final oxidation product contains reactive alloy components in the same proportions as did the original alloy. We consider first some experimental results for alloy oxidation selected to illustrate the diversity of reaction product morphologies. Attention is then concentrated on the situation in which a single-phase oxide is the only reaction product. Its possible morphologies, growth rates and protective value are then

5.2. Selected Experimental Results

187

investigated, drawing heavily on the extensive theoretical treatments developed by Wagner and added to by others. Finally, some consideration is given to predicting the lifetime of an alloy component forming a protective scale in the steady-state regime. For detailed descriptions of the oxidation behaviour of individual alloys, the reader is referred to the proceedings of the many conferences addressing the topic. Some have been established as regular international conference proceedings [1–3]. In addition, a useful summary of commercial alloy performance in various high temperature environments has been provided by Lai [4].

5.2. SELECTED EXPERIMENTAL RESULTS Many high temperatures alloys are designed to form protective chromia (Cr2O3) scales. Examples include stainless and heat-resisting steels, and nickel-based alloys such as Incoloys and some Inconels. Compositions of some representative alloys are given in Table 5.1. A larger collection is found in Appendix A. The range of possible steady-state behaviour of chromia forming alloys is shown in Figure 5.1. A model binary alloy of Fe–28Cr (all compositions in wt%) is seen in Figure 5.1a to form an external scale of Cr2O3 alone when reacted with oxygen at 9001C. The scale grows slowly, with kp ¼ 1.4 109 cm2 s1, Table 5.1

a

Nominal compositions of some heat resisting alloys (wt%)

Alloy

Fe

Ni

Cr

Si

Mn

P91

bal

0.3

9

0.4

304L 310 347 253MA 353MA 800 HP (cast) 601 617

bal bal bal bal bal bal bal 14 1.5

8–12 20 11 11 35 31 35 bal bal

18–20 25 18 21 25 21 25 23 22

1 1.5 1.0 1.7 1.6 1.8

0.2

0.5

0.5

214 Kanthal A Kanthal AF MA956 PM2000

3

bal bal

0.5 0.5

bal bal bal

16 0.2 20.5–23.5 0.7 21 20 19

JA13

bal

16

0.1

Reactive element metals.

Al

C

Other

0.4

0.10

1Mo, 0.2V

2 2.0 2.0 0.8 1 r2.0

0.03 0.25 0.08 0.08 0.06 0.07 0.44 0.06 0.07

1.4

0.3

4.5 5.3 5.1 4.5 5.8

0.05 0.08 0.01 0.01

5.0

0.03

Nb (8 %C) 0.08Ce REMa 0.25Al, 0.35Ti 0.8Nb 0.5Ti 12.5Co, 9Mo, 0.3Ti, 0.2Cu 0.01Y 0.08Ti, 0.06Zr 0.5Ti, 0.5Y2O3 0.5Ti, 0.2Ni, 0.5 Y2O3 0.3Y

188


10m

(a)

(b)

MCr2O4 Cr2O3

20μm (c)

(d)

Figure 5.1 (a) External Cr2O3 scale grown on Fe–28Cr at T ¼ 1,0001C, (b) iron-rich scale grown on Fe–7.5Cr at T ¼ 8501C in pure O2, (c) simultaneous external Cr2O3 scale growth and internal Al2O3 precipitation, alloy IN 601 at T ¼ 1,0001C and (d) two-layered scale of spinel and Cr2O3 together with internal attack on HP35 cast, heat resisting steel oxidized in steam at 1,0001C.

representing the desired outcome of single-phase protective layer growth. However, when the alloy chromium content is too low, this protective layer does not form. A model alloy of Fe–7.5Cr reacted with oxygen at 8501C to form a fastgrowing iron-rich scale, as shown in Figure 5.1b. Dilute alloys can fail to form a protective scale even if the oxygen partial pressure is too low to oxidize any component except chromium. As will be seen in Figure 6.1, Cr2O3 is internally precipitated within an Fe–5Cr alloy reacted at 1,0001C, in a low pO2 atmosphere. Oxygen had dissolved in the alloy, diffusing inwards to react with solute chromium and precipitate its oxide. It is obviously desirable to be able to predict the minimum alloy chromium level required to form external rather than internal Cr2O3, and thereby protect the alloy iron base from oxidation. Commercial chromia-forming alloys can be even more complex in their reaction morphologies. Oxidation of Inconel 601, which contains a low level of


189

aluminium in addition to 23% Cr (Table 5.1), is seen in Figure 5.1c to form an external Cr2O3 scale and internal Al2O3 precipitates. A cast heat-resisting steel (HP grade, 25Cr–35Ni) oxidized in steam at 1,0001C to form a two-layered scale (Figure 5.1d). A continuous chromia inner layer formed, but a manganese-rich spinel (MCr2O4 with M a mixture of Mn, Fe and Ni) layer grew on top. Clearly it is necessary to establish the conditions under which the desired oxide scale can prevent the oxidation of other alloy components. In the example of Figure 5.1d, the chromia layer had allowed outward diffusion of manganese to form a surface layer of spinel. The chromia scale shown in Figure 5.1c had allowed oxygen to diffuse into the alloy, precipitating alumina. In these particular cases, the chromia scale nonetheless protected the alloy iron and nickel from reaction. Alloy oxidation can also lead to changes within the alloy itself. We have already encountered the example of internal oxidation, which reflects the inward diffusion of oxidant. Microstructural changes in the alloy can also result from outward diffusion of alloy components. A common example is decarburization. Figure 5.2 shows a cross-section of a tube wall from a failed boiler superheater unit. Overheating of this tube led to rapid oxidation, tube wall thinning and subsequent mechanical failure as the steam pressure inside the tube ruptured its wall. The tube metal was a 1¼Cr–1Mo steel which has a microstructure of ferrite plus pearlite. The pearlite (a mixture of lamellar Fe3C and ferrite) is seen in the lower part of the wall cross-section to have coarsened and spheroidized. In the upper part of the cross-section, near the rapidly oxidizing external surface,

Figure 5.2 Cross-section of tube wall (1¼Cr+1Mo steel) from failed boiler superheater unit, showing decarburization beneath the outer (upper) surface.

190


the carbide has almost completely disappeared. Here the solute carbon from within the ferrite was oxidized at the steel–scale interface via the reaction: C þFeO ¼ Fe þ COðgÞ

(5.1)

DG1 ¼ 147; 760 150:1T J mol1

(5.2)

for which

The resulting CO(g) escaped through the porous scale. As a result, the carbon activity in the steel at the surface was lowered, causing the cementite to dissolve: Fe3 C ¼ 3Fe þ C

(5.3)

Solute carbon diffused from the interior carbide dissolution front to the steel– scale interface, there to be oxidized and removed. Selective oxidation of an alloy constituent lowers its concentration within the alloy. If alloy diffusion is rapid compared with the scaling rate, then the change in alloy concentration is averaged over a large region, and the concentration change at the alloy–scale interface will be small. However, if alloy diffusion is relatively slow, replenishment of the selectively removed metal is hindered, and the concentration of that element is depleted in the subsurface zone, as illustrated schematically in Figure 5.3. When the concentration of one component is decreased, the concentrations of others are increased. The extent to which they are enriched is governed by the rates at which they diffuse away from the surface O2 (g)

BO

Alloy AB

NA NB

Figure 5.3 Depletion of selectively oxidized alloy component and enrichment of non-oxidized component in binary alloy.

191


into the alloy interior. These changes in subsurface composition can cause alterations in the phase constitution of this region. A simple example of practical importance is the precipitation of a copper-rich phase at the surface of steel during hot working [5]. A significant quantity of steel is produced by remelting scrap. Most scrap impurities are removed during the steelmaking process, but some, such as copper and tin, remain. Several successive cycles of steelmaking and recycling as scrap lead to an increase in the concentration of ‘‘residuals’’ such as copper to levels far above those found in steel produced from iron ore. Oxidation of steel is inevitable during hot working (reheating, rolling, etc.), producing a scale of iron oxide, but leaving the copper unreacted. The resulting increase in copper concentration in the steel beneath the scale can exceed the solubility limit, precipitating a copper-rich Cu–Fe phase. An example is shown in Figure 5.4. If the temperature is above about 1,1001C, this phase is liquid, and penetrates the steel grain boundaries. Mechanical working of the steel in this state causes cracking, a phenomenon known as ‘‘hot shortness’’. The selective removal from an alloy of a metal by its preferential oxidation can drive other phase changes within the subsurface zone. An example is shown in Figure 5.5, where selective oxidation of aluminium from a two-phase g-Ni plus gu-Ni3Al alloy led to dissolution of the aluminium-rich gu-phase, as aluminium diffused out of the subsurface zone. Selective oxidation of aluminium from b-NiAl does not at first cause a phase change, but it does lead to the development of cavities at the alloy–scale interface, as shown in Figure 5.6. Taking as a measure of success the ability of an alloy to form a single-phase scale of the desired oxide, it is desirable to be able to predict the conditions (alloy composition, pO2 and temperature) under which this will be the result. As seen in this preliminary examination, it is necessary to predict not only when the desired

100

FeO

Steel

Cu

Concentration (at%)

90

Fe

80 70 60

O

50 40 30

Cu

20 10

Sn

0 0

2

4

6 8 10 Distance (μm)

12

Figure 5.4 Copper enrichment beneath iron oxide scale grown on a 0.47Cu steel at T ¼ 1,1001C. Left: SEM view of cross-section and right: EPMA scan.

14

16

192


γ

γ + γ’

100µm Figure 5.5 Dissolution of gu-Ni3Al in subsurface region of g+gu model alloy (Ni–23Al) due to selective aluminium oxidation at 1,2001C.

Al2O3

Figure 5.6 FIB image showing cavity formation at b-NiAl surface due to selective aluminium oxidation at 1,2001C.

193


oxide will form preferentially, but also when it forms as an external layer rather than an internal precipitate. Further, it is necessary to predict the effect of the external layer on the oxidation of other alloy components and on microstructural changes in the alloy subsurface region. We consider first the utility of phase diagrams in predicting diffusion paths and thereby reaction morphologies.

5.3. PHASE DIAGRAMS AND DIFFUSION PATHS During the steady-state period of alloy oxidation, the scale morphology, i.e. the identity and spatial arrangement of phases in the reacting system, is timeinvariant. This situation is conveniently represented by a diffusion path mapped onto the relevant phase diagram on the assumption that local equilibrium is in effect. The general nature of the problem is examined using the Ni–Cr–O isothermal section [6] shown in Figure 5.7. In this particular system, two simple oxides, NiO and Cr2O3, and a ternary spinel, NiCr2O4, can exist, and the degree of miscibility or intersolubility of the

O

NiCr2O4(S) Cr2O3 NiO NiO+S +Alloy

O3 Cr 2 + S

+

loy

Al

Alloy+ Cr2O3

Ni

Figure 5.7 Isothermal section of Ni–Cr–O phase diagram at T ¼ 1,0001C [6]. With kind permission from Springer Science and Business Media.

Cr

194


oxides is very limited. As seen from the diagram, all three oxides can co-exist at equilibrium with pO2 ¼ 1 atm, as can also two-phase mixtures of NiO+NiCr2O4 and NiCr2O4+Cr2O3. Thus specifying the ambient conditions is insufficient to determine the oxide which will be stable at the surface of a growing scale. Obviously, at least the alloy composition is required as well. Local equilibrium at the alloy–scale interface is specified by the tie-lines joining alloy composition points to the corresponding oxides. According to Figure 5.7 then, all Ni–Cr alloys containing N Cr 0:03 should form the desired oxide Cr2O3 at 1,0001C. However, experimental investigations of alloy reactions with pure oxygen and other oxidizing gases [7–10] have shown that a minimum chromium level of 10–20% is required to ensure the selective formation of Cr2O3. The major reason for the discrepancy is chromium depletion in the alloy subsurface zone. The chromium concentration at the alloy–scale interface is reduced (by its selective oxidation) to a value significantly lower than that of the bulk alloy. To relate the alloy chromium mole fraction value, NCr, to the interfacial value, NCr,i, it is necessary to analyse the alloy diffusion process. The same difficulty arises in the case of Fe–Cr alloy oxidation. An isothermal section of the Fe–Cr–O phase diagram is shown in Figure 2.5. As discussed earlier, the minimum value of NCr necessary to thermodynamically stabilize Cr2O3 formation is about 0.04. Experimental observations [7, 11], however, put the critical alloy concentration at about 14% at 1,0001C. Again, a principal reason for the difference is a lowering of the value NCr,i as a result of relatively slow alloy diffusion. The Co–Cr system exhibits an even greater extent of depletion, requiring up to 30% chromium to provide the critical interfacial value, estimated as N Crit 0:01, required for Cr2O3 formation. The question of how to deal with the alloy depletion problem is addressed in the next section. Returning now to the different reaction morphologies shown in Figure 5.1, we see that they are consistent with the phase diagram, once allowance is made for surface alloy depletion. Thus the Fe–28Cr has enough chromium to sustain N Cr; i 4N Cr; Crit , and a Cr2O3 scale results. A small degree of depletion in the Fe–7.5Cr alloy is sufficient to lower NCr,i below the critical value and the diffusion path shown in Figure 5.8 results. It is noted that the Fe–28Cr alloy develops a convoluted alloy–scale interface, at which voids are nucleated. Neither effect is predictable from the phase diagram without detailed knowledge of the local diffusion processes. Reaction morphologies depend on oxygen partial pressure as well as alloy composition. The internal precipitate morphology of Figure 6.1 was produced in a gas with pO2 too low to oxidize iron, so that Cr2O3 was the only stable reaction product. The formation of internally precipitated Cr2O3 corresponds to the development of a two-phase region, as shown by the diffusion path in Figure 5.8. For comparison, a diffusion path is shown for external (single-phase) Cr2O3 formation at a higher alloy NCr value. Both paths terminate at the same oxygen potential, but at different mCr and mFe values. It is obviously not possible to predict from the phase diagram alone which of the two morphologies will result for a particular alloy composition. Again, a diffusional analysis will be required.


195

Figure 5.8 Diffusion paths on Fe–Cr–O phase diagram corresponding to (a) depletion and iron-rich oxide growth, (b) internal, (c) combined internal and external and (d) external chromium oxidation.

As already mentioned, even when an external single-phase Cr2O3 layer forms in contact with the alloy surface, there remains the possibility of additional oxide formation. Formation of a spinel layer on top of the chromia, as illustrated in Figure 5.1d, is a common result for heat resisting steels. A schematic diffusion path for this type of scale morphology is shown for the Fe–Cr–O system in Figure 5.8. Clearly the outer layer can develop only if the second metal is soluble in Cr2O3 and can diffuse through it at a sufficient rate. We return to this question in Chapter 7. The complex reaction morphologies shown here are all consistent with steady-state local equilibrium having been established within the reacting systems. This was shown experimentally in each case by the observations that the morphologies were time invariant as the extent of reaction increased. It is also evident from the fact that the sequence of phase assemblages making up the morphology can in each case be represented by a diffusion path on the relevant phase diagram. It is clear that phase diagrams of the type A–B–O can be used to describe the oxidation morphologies of binary alloys, AB. However, diffusion within the alloy

196


in general leads to surface concentrations which differ from those of the bulk alloy. To predict alloy oxidation behaviour it is necessary to be able to calculate these concentration changes. We now consider the diffusion processes supporting growth of a single-phase scale on a binary alloy. In general, such an oxide can contain both alloy components, depending on the intersolubility of AO and BO, and their relative stabilities. We consider first the simplest case, where one alloy component remains completely unoxidized, and a pure binary oxide results from selective oxidation of the other.

5.4. SELECTIVE OXIDATION OF ONE ALLOY COMPONENT Selective oxidation will occur if only one oxide is stable. This is the case for alloys consisting of a noble metal such as Pt, Ag or Au, which does not form an oxide under normal conditions, and a reactive metal which does. An example of this alloy class is Pt–Ni, which was analysed by Wagner [12]. It is also the case for more practically relevant alloy systems such as Fe–Cr and Ni–Cr if the ambient oxygen potential is below the minimum necessary to form any iron- or nickelbearing oxide, but still above the value required for Cr2O3 formation. In the case of exclusive scale formation, it is of interest to know whether the scaling rate is controlled by diffusion in the scale or in the alloy. In a formal sense this question lacks meaning, as the rates at which B diffuses in the alloy and the scale must be in balance for a steady state to exist. However, it is reasonable to classify the process as being controlled by scale diffusion if the scale grows at the same rate as it does on pure B metal. Conversely, scaling is described as being controlled by alloy diffusion if the rate at which BO grows on the alloy is significantly less than on pure B metal. We therefore compare scaling rates on the two materials. The growth rate of a NiO scale on a Pt–Ni alloy is proportional to the nickel cation flux in the oxide, given by Equation (3.71), rewritten here as ðiÞ 1=6 ½ðp00O2 Þ1=6 ðpO Þ 2 (5.4) ¼ Constant X ðiÞ where, as before, p00O2 is the ambient oxygen partial pressure and pO the value at 2 the alloy–scale interface. Doubly charged cation vacancies have been assumed, but a different charge can be accommodated by changing the exponent of pO2 . Scale growth on pure nickel is related to the corresponding flux expression

J NiO Alloy

ðeqÞ

½ðp00O2 Þ1=6 ðpO2 Þ1=6 (5.5) X ðeqÞ Here pO2 represents the partial pressure for equilibrium between pure nickel and its oxide. The ratio of the two fluxes at a given scale thickness is therefore J NiO Metal ¼ Constant

a¼

J NiO Alloy J NiO Metal

¼

1=6 ðp00O2 Þ1=6 ðpðiÞ O2 Þ ðeqÞ

ðp00O2 Þ1=6 ðpO2 Þ1=6

(5.6)

The boundary value oxygen pressures are next related to nickel concentrations.

5.4. Selective Oxidation of One Alloy Component

197

The equilibrium condition for the reaction 2 Ni þO2 ¼ 2NiO

(5.7)

is written ðeqÞ

a2Ni pO2 ¼ K1 7 ¼ pO2

(5.8) ðeqÞ pO 2

has the where pure nickel is the reference state, for which aNi ¼ 1 and standard equilibrium value. If the alloy is assumed to be ideal, we can write for the alloy–scale interface ðeqÞ

ðiÞ N 2Ni;i pO ¼ pO2 2

(5.9)

where N Ni;i denotes the nickel mole fraction at the alloy–scale interface. Since platinum is unreactive, it is possible to identify an alloy nickel level, N Ni;e , which will equilibrate with NiO and an oxygen partial pressure equal to the ambient value, p00O2 ðeqÞ

N 2Ni;e p00O2 ¼ pO2

(5.10)

00 Substitution in Equation (5.6) for pðiÞ O2 and pO2 from Equations (5.9) and (5.10) leads to

a¼

1 ðN Ni;e =N Ni;i Þ1=3 1 ðN Ni;e Þ1=3

(5.11)

Values of N Ni;e are very low in the case of Pt–Ni alloys. Wagner [12] calculated values of 6 107 and 6 105 at 8501C and 1,1001C, for p00O2 ¼ 0:21 atm. According to Equation (5.11), then, a 1 and oxidation is controlled essentially by diffusion in NiO, if N Ni;i 0:01. To make use of this finding, it is necessary to relate the interfacial mole fraction N Ni;i to the original alloy level N ðoÞ Ni . The concentration at the interface is established by the diffusion of nickel towards the interface from the alloy, and away from it into the oxide. Assuming ~ is independent of composition, Fick’s that the alloy diffusion coefficient, D, second law applies 2 @N Ni ~ @ N Ni ¼D @t @x2

(5.12)

where x is the distance from the alloy surface and the initial condition N Ni ¼ N ðoÞ Ni

for t ¼ 0;

x40

(5.13)

applies. The problem is simplified considerably if movement of the alloy surface can be ignored. This will be a reasonable approximation if only a very thin oxide scale is formed, and alloy surface recession is consequently small. Diffusion within the alloy is then treated as the semi-infinite case (a limiting case of Equation (2.140) in which Ci ¼ (C0+C1)/2) with a fixed boundary, leading to the steady-state solution ! x ðoÞ pffiffiffiffiffiffi N Ni ¼ N Ni; i þ N Ni N Ni; i erf (5.14) ~ 2 Dt

198


The analysis is continued by enquiring as to what interfacial nickel concentration corresponds to an alloy flux sufficient to sustain NiO scale growth. The flux of nickel towards the alloy surface is given by ~ @N Ni D J AB ¼ (5.15) V AB @x x¼0 where V AB is the alloy molar volume. The differential is evaluated from Equation (5.14), recalling the error function definition (see Appendix C) Z z 2 erf ðzÞ ¼ pffiffiffi expðZ2 ÞdZ (5.16) p o and obtaining

!2 1 @N Ni 2 x ðoÞ ¼ N Ni N Ni; i pffiffiffiffiffiffi pffiffiffi exp pffiffiffiffiffiffi @x p ~ ~ 2 Dt 2 Dt

(5.17)

Evaluation at x ¼ 0, followed by substitution into Equation (5.15) then yields ~ 1=2 N ðoÞ N Ni;i Ni D (5.18) J AB ¼ VAB pt To sustain external scale growth, this flux must equal the rate at which nickel is incorporated into the growing scale, dðnNi =AÞ (5.19) dt where nNi =A represents the number of moles of nickel in the scale per unit surface area. This rate is found from scale thickening n X Ni d (5.20) ¼d VNiO A J AB ¼

and since X2 ¼ 2kp t,

2 kp dðnNi =AÞ 1 ¼ dt VNiO t

Combining Equations (5.18), (5.19) and (5.21), we obtain 2 V AB pkp ðoÞ N Ni N Ni;i ¼ ~ V NiO D

(5.21)

(5.22)

Wagner pointed out that the maximum flux available from the alloy was delivered if N Ni;i 0, and this enables the calculation from Equation (5.22) of a minimum level of N ðoÞ Ni necessary to sustain external scale growth. This point is discussed further below. The approximation used above of a zero rate of scale–alloy interface movement can be avoided and the moving boundary incorporated into the description. The interface movement is related to the scaling rate by Equation (1.29), using V AB in place of V M . The position of the alloy–scale interface relative


199

to its original location, DxM , is given by DxM ¼ ð2akc tÞ1=2 kc

(5.23) kc =kc .

is the corrosion rate constant for pure nickel and a ¼ The mass where balance of Equations (5.21) and (5.22) is then replaced by a balance for platinum, which is rejected from the oxide and diffuses from the scale–alloy interface into the alloy. The amount of platinum (per unit surface area) made available when the interface advances by an increment dy is equal to ð1 N Ni; i Þdy. The rate at which that occurs is equated to the diffusion rate, which is given by Fick’s first law, evaluated at the interface. Thus dy @ð1 N Ni Þ ~ ¼ D ð1 N Ni;i Þ (5.24) dt @x x¼y assuming that the alloy molar volume is independent of composition. Wagner found the result 1=2 N ðoÞ kc Ni N Ni;i ¼F ~ 1 N Ni;i 2D

(5.25)

where the function F(u) is defined by FðuÞ ¼ p1=2 uð1 erf uÞ exp ðu2 Þ

(5.26)

Thus N Ni;i and a can be found from the simultaneous solution of Equations (5.11) and (5.25). It is useful to observe that when uc1, F(u)E1 and when u{1, FðuÞ p1=2 u. If kc is small enough, then u{1. In this case of negligible interface recession, the solution (5.25) reduces to the form Equation (5.22), obtained on the basis that the interface movement can be ignored. In any event, the analysis predicts that scale growth is controlled by oxide diffusion ðkc ¼ kc Þ if N Ni 4N Ni;min , and by alloy diffusion if N Ni oN Ni;min . In the former case, a ¼ kc =kc is independent of ðoÞ ðoÞ N Ni and in the latter case a decreases as N Ni is lowered. ðoÞ Comparison of experimental results [13] for the dependence of a on N Ni with Wagner’s predictions for the Pt–Ni system showed that the latter were reasonably successful in the regime where N ðoÞ Ni o0:5 and alloy diffusion controlled the scaling rate (ao1). However, at higher nickel levels, the measured rates were significantly slower than predicted. As a result, the predicted critical values of N oNi at which rate control should transfer to scale diffusion (0.7 at 8501C and 0.6 at 1,1001C) were ~ lead to incorrect. However, the significant errors in measured values of kc and D ~ and the accurate calculation of Equation (5.25) large compounded errors in kc =D, is therefore difficult. Moreover, the assumption that NiO scale growth is controlled by lattice diffusion is not applicable at temperatures lower than about 9001C, where grain boundary diffusion is more important. ~ 1=2 1, i.e when scale–alloy interface moveIn the case where u ¼ ðkc =2DÞ ment is slow compared to alloy diffusion, then FðuÞ p1=2 u

(5.27)

200


Substitution into Equation (5.25) then yields 1=2 pkc ðoÞ N Ni ~ 2D N Ni;i ¼ 1=2 pkc 1 ~ 2D

(5.28)

This approximation is not applicable to the Pt–Ni alloy situation, because the NiO growth rate leads to relatively high kc values. It might be appropriate, however, for slower growing oxides such as Cr2O3 and Al2O3. Data on several alloys collected by Whittle et al. [14] is reproduced in Table 5.2, along with data for Ni–Al [15, 16]. It is seen that the condition u{1 is met in these cases. Interface concentrations calculated from Equation (5.28) are also shown in the Table. The extent of depletion predicted from Equation (5.28) is, in fact, unrealistically small. Chromium concentration profiles measured in an Fe–32Cr alloy after selective formation of a Cr2O3 scale are shown in Figure 5.9. These confirm that depletion occurs, but show that the effect is much greater than predicted. Given the sensitivity ~ this is perhaps not surprising. We note in of the calculation to errors in kc and D, ~ particular that D will normally be a function of alloy composition (see Section 2.7). Bastow et al. [18] showed that an interfacial value N Cr;i ¼ 0:19, in approximate agreement with their EPMA measurement, was consistent with a rate constant kc ¼ 3.9 1012 cm2 s1, a value three times faster than the rate they actually observed. This discrepancy has led to a more detailed examination of the relationship between alloy and scale diffusion, which we discuss below. First, however, an examination of qualitative trends revealed by the data in Table 5.2 is useful. Comparing iron- and nickel-based chromia forming alloys, it is seen that the ~ is somewhat greater for the latter. This reflects mainly the fact that ratio kc =D diffusion in austenite is slower than in ferrite. Consequently, the chromium concentration at the alloy–oxide interface will be depleted to a lower value in a Ni–Cr alloy than in an equivalent Fe–Cr alloy under the same conditions. ~ is much smaller Comparing Fe–Cr and Fe–Al alloys, it is seen that the ratio kc =D Table 5.2 Alloy

Kinetic parameters for alloy diffusion and selective scale growth [14] NðoÞ A

Ni–28Cr 0.30

T (1C)

1,000 1,200 Fe–28Cr 0.29 1,000 1,200 Fe–4.4Al 0.087 1,000 1,200 Fe–12Al 0.22 1,000 1,200 Ni–10Al 0.19 1,200

D~ AB (cm2 s1)

kc (cm2 s1)

u

NA;i (Equation 5.28)

4.1 1011 3.9 1010 4.1 1010 3.9 109 8.4 109 2.1 106 8.4 109 2.1 106 1 109 [16]

1.2 1013 7.2 1013 1.2 1012 3.9 1012 2.6 1016 1.4 1013 2.0 1018 6.4 1016 4.0 1013 [15]

3.8 102 3.0 102 3.8 102 2.2 102 1.8 104 1.8 104 1.2 105 1.2 105 1.4 102

0.24 0.26 0.24 0.26 0.087 0.087 0.22 0.22 0.17


201

Figure 5.9 Chromium depletion in Fe–32Cr measured by electron probe microanalysis after selective oxidation of chromium at T ¼ 9771C. Reprinted from Ref. [17] with permission from Elsevier.

in the alumina forming alloys, because the oxidation rate is much slower and alloy diffusion is faster. As a result, Fe–Al alloys are predicted to maintain rather flat aluminium concentration profiles, with N Al;i N ðoÞ Al . This has been verified [19] for Fe–Cr–Al alloys under circumstances where a scale of alumina only forms. Microprobe analysis, with a spatial resolution of 1–2 mm, showed no detectable variation in the alloy aluminium level from the alloy interior to the alloy–scale interface. Thus any depletion zone was of thickness less than 1–2 mm. ~ values lower than either Ni–Cr or Fe–Cr, leading to a The Ni–Al alloys have kc =D reduced extent of depletion. In comparison to Fe–Al alloys, however, Ni–Al is subject to significantly more depletion. Values of the minimum concentration, N B;min , of scale-forming element necessary to support external scale growth were calculated from Equation (5.22) and are listed in Table 5.3. Comparison with experimental observations of N B;min , however, reveals that these predictions are not useful. As already noted, one reason for this lack of success is the sensitivity of the calculation to error in the ~ An example is shown in Figure 5.10, where calculations basic data used, kc and D. ~ which differ by a factor of 2 are seen to [18] for Fe–Cr assuming two values for D result in values of N Cr;i which differ by 0.05, i.e. 40%. A further reason for its lack of quantitative success in predicting values of N B;min , is that Wagner’s treatment was designed to assess the minimum alloy concentration necessary to supply a flux to the surface sufficient to sustain the growth of a single-phase scale presumed to have formed already. Thus the theory does not provide guidance on

202

Table 5.3

a


Values of NB,min to support selective oxide scale growth

Alloy

Scale

T (1C)

Predicted (Equation 5.25)

Observed

Reference

Ni–Cr Fe–Cr Ni–Al Fe–Ala

Cr2O3 Cr2O3 Al2O3 Al2O3

1,000 1,000 1,200 1,300

0.07 0.07 0.02 104

0.15 0.14 0.12–0.24 0.02–0.04

[7] [7] [15, 20, 21] [22]

Observed on Fe–Cr–Al alloys growing scales of Al2O3 only.

Figure 5.10 Chromium depletion profiles calculated for (1) DFeCr ¼ 1 1012 and (2) 2 1012 cm2 s1 [18]. With kind permission from Springer Science and Business Media.

how much of the alloy component is required to form this scale in the first place. Before returning to this question we consider again the depletion profiles in Figure 5.9.

5.5. SELECTIVE OXIDATION OF ONE ALLOY COMPONENT UNDER NON-STEADY-STATE CONDITIONS Although concentration measurements near a phase boundary are subject to error, it seems from Figure 5.9, that N Cr;i first decreases then increases with time.

5.5. Selective Oxidation of One Alloy Component Under Non-Steady-State Conditions

203

Although N Cr;i ultimately reaches a constant value, the steady-state assumption of fixed boundary conditions is apparently inapplicable for a significant period at the commencement of reaction. The steady-state assumption is the basis for Wagner’s analytical solution (5.25), which could for this reason be inapplicable. The non-steady-state situation has been analysed for Fe–Cr oxidation, by Whittle et al. [14], Wulf et al. [17] and Bastow et al. [18], using a finite difference method. In this numerical approach, it is possible to allow for a compositiondependent alloy diffusion coefficient, but this has been shown to have little effect on the interfacial concentration in Fe–Cr alloys. The possible variation in N Cr;i with time is reflected in the mass balance for chromium at the alloy–scale interface. The situation is shown schematically for a binary alloy AB in Figure 5.11, where C represents concentration (moles/ volume), and an average value, CB;OX , is specified for B in the scale of BOn. The general statement of mass balance at a moving interface is written 0 J AB J OX B ¼ nðCB;OX CB;i Þ

J OX B

(5.29)

where is the flux of B away from the interface into the oxide, n the velocity of the interface and C0B;OX the boundary value of CB in the oxide at the interface. All of J AB , J OX B and n must be defined in the same frame of reference (see Section 2.7). The choice is arbitrary, but solution of the diffusion profile in the alloy is

Figure 5.11

Mass transfer at moving scale–alloy interface.

204


facilitated by using a reference frame with its origin at the original alloy surface, marked by a dashed line in Figure. 5.11. The displacement of the scale–alloy interface from its origin is specified as xc , and hence n ¼ dxc =dt. The flux of B from the alloy towards the interface is given by @CB (5.30) J AB ¼ DAB @x x¼xc

Component B also diffuses away from the interface through the scale, allowing it to grow. This is normally expressed with respect to a reference frame with its origin at the metal–scale interface. Defining the scale thickness as zs , then 0 V BOn ð J OX B Þ ¼

dzs dt

(5.31)

or dzs (5.32) dt where the prime is used to denote the different frame of reference, z. This is transformed to the desired reference frame, x, using the relationship 0 ð J OX B Þ ¼ CB;OX

OX 0 J OX B ¼ ð J B Þ þ CB;OX v1;2

(5.33)

where n1;2 is the velocity of the oxide frame with respect to the original alloy surface n1;2 ¼

dxc dt

Combination of the Equations (5.29) to (5.34) leads to @cB dzs dxc dxc ðCB;OX CB;i Þ CB;OX D ¼ @x x¼xc dt dt dt where the approximation C0B;OX CB;OX has been used. Noting that VBOn xc zs ¼ V AB it is found from Equation (5.35) that @cA dxc V MOn ¼ CB;OX CB;i DAB @x x¼xc dt V AB

(5.34)

(5.35)

(5.36)

(5.37)

Numerical solution of Equations (5.15) and (5.37) together with an expression for the alloy recession rate dxc V AB dzs ¼ dt V BOn dt

(5.38)

coupled with a rate law zs ¼ fðtÞ then reveals the alloy depletion profiles. Whittle et al. [14] proposed that an appropriate formulation of the rate law was z2s ¼ 2kp t þ k

(5.39)

205

5.5. Selective Oxidation of One Alloy Component Under Non-Steady-State Conditions

which differs from that of Wagner. Their evaluation of the change with time of the interfacial concentration relative to the bulk alloy value, N B;i y ¼ ðoÞ (5.40) NB is shown in Figure 5.12 for a model alloy. It is seen that a rapid decrease in N B;i occurs in the initial stages of reaction, when the oxide growth flux, (and rate at which B is withdrawn) is maximal. This initial decrease is followed by an increase, until a steady-state value is reached. This theoretical prediction is in agreement with experimental observation for the Fe–Cr system. Thus we conclude that the steady state assumed by Wagner is in fact arrived at, but that during an initial period this is not the case. The existence of a steady state is actually a pre-requisite for parabolic kinetics to be in effect. As seen in Section 3.7, diffusion controlled scale growth leads to parabolic kinetics only if the boundary conditions are fixed with time. The boundary values in a scale are related to the alloy interfacial composition through a local equilibrium condition such as Equation (5.10). More generally 2 2 B þO2 ¼ BOn ; DG41 (5.41) n n DG41 2=n aB pO2 ¼ K41 ¼ exp (5.42) RT If the activity coefficient for B is denoted by g, then K41 pðiÞ O2 ¼ ðgN B;i Þ2=n

(5.43)

and, in general, rate expressions such as Equation (5.4) lead to parabolic kinetics only if N B;i afðtÞ. Conversely, the observation of parabolic kinetics is an indication

1.0

θ

0.75

0.5

0.25

0

2.5

5 7.5 10 Dimensionless time

12.5

×10-3

Figure 5.12 Calculated variation of interfacial concentration with time during non-steadystate oxidation. Reprinted from Ref. [14] with permission from Elsevier.

206


00 that N B;i is constant. However, if pðiÞ O2 pO2 , the effect on oxidation rate of transient variation in N B;i could be small, as seen from Equation (5.4). The initial time dependence of N B;i predicted by Whittle et al. [14] was a consequence of their use of Equation (5.39) to describe scaling kinetics. As seen from the differential form

kp dzs ¼ dt ð2kp t þ kÞ1=2

(5.44)

the deviation from parabolic kinetics is greatest in the early stages of reaction, when 2kp tok. A non-zero value for k is realistic, reflecting as it does the existence of an oxide film on the metal surface before commencement of the high temperature reaction. Even in the absence of such a pre-formed oxide, strictly parabolic kinetics cannot obtain at extremely short times. If k ¼ 0, then dzs/dt is predicted to approach infinity as t approaches zero, an impossibility, as diffusion from the alloy is limited. It is recognized that the very initial kinetics cannot be parabolic, just as the exclusive oxidation of only one alloy component when the ambient pO2 is sufficient to oxidize others is initially impossible. This initial period of reaction, referred to as ‘‘transient’’ because it precedes the establishment of steady-state conditions, is discussed further in Section 5.7. Other oxidation morphologies result if selective oxidation of one component to form an external scale does not occur. Their natures vary with the reactivity of other alloy components. If no other alloy metals are reactive at the oxidant activity and temperature in question, and the one reactive component cannot reach the surface quickly enough to develop a scale, then internal oxidation results. This situation is considered in Chapter 6. Another reactive alloy component will oxidize simultaneously. Oxides which have limited intersolubility develop as separate phases, a situation described in Chapter 7. However, if the degree of intersolubility is large, it is still possible that a single-phase external scale of solid solution oxide can result. The questions of interest then concern the nature of this scale, its growth rate and how these properties vary with alloy composition.

5.6. SOLID SOLUTION OXIDE SCALES Pairs of binary oxides, AOn1 þ BOn2 , will dissolve in one another to an extent which is greater if (a) n1 ¼ n2 , (b) the oxides are crystallographically isotypic, (c) the cations A2nþ and B2nþ are similar in size and polarisability and (d) the stabilities of the oxides are not too different. The oxides MnO, FeO, CoO and NiO, all of which have the face-centred cubic NaCl structure, form ternary solid solutions A1x Bx O in which O x 1. Similarly, a-Fe2O3 and Cr2O3, both of which have a hexagonal crystal structure, are completely miscible at high temperatures. In the same way, FeS and NiS are fully intersoluble, as are FeS and CoS, all three monosulfides having the hexagonal NiAs structure. In general, the ratio N A =N B ¼ ð1 xÞ=x in the oxide differs from the corresponding alloy ratio

5.6. Solid Solution Oxide Scales

207

because the more reactive metal enters the scale preferentially. Furthermore, as the cation self-diffusion coefficients in the oxide, DA and DB , will differ, the cation ratio will vary with position in the scale. To calculate the scale growth rate as a function of alloy composition, it is necessary to know the distributions of the two metals within the scale. This problem has been analysed by Wagner [23] and the results extended by Coates and Dalvi [24]. The co-ordinate systems shown in Figure 5.11 are again employed. The z frame, attached to the alloy–scale interface, is used to describe transport in the oxide; the x frame, with its origin at the original alloy surface describes transport in the alloy. Of course it is understood that the oxide concentration profile will in general not be flat. The molar flux of each cation species in the oxide, J i , is given by @ ln ai (5.45) @z where i is A or B, and kinetic cross-effects are ignored. The Gibbs equation (2.9) relates the chemical potentials of the binary oxides and their constituents zA moA þ RT ln aA þ ðmoO þ RT ln ao Þ ¼ moAOn þ RT ln aAOn (5.46) zO J i ¼ Di Ci

with a similar equation for BOn. Assuming for the sake of simplicity that the valences, zi , are related by (5.47) zA ¼ zB ¼ j zO j ¼ 2 so that n ¼ 1, one obtains from Equations (5.45) and (5.46) DA ð1 xÞ @ ln aAO @x @ ln ao JA ¼ V OX @x @z @z DB x @ ln aBO @x @ ln ao JB ¼ V OX @x @z @z

(5.48)

(5.49)

It is supposed that V OX does not vary with x and that @ ln aAO =@x and @ ln aBO =@x are known from the solution thermodynamics of the mixed oxide. The scale-thickening rate is then found from ð J A þ J B ÞVOX ¼

dzs kp ¼ dt zs

(5.50)

where zs is the instantaneous scale thickness. Because diffusion control is in effect, the system is in a steady state, and both x and ao can be expressed in terms of a normalized position parameter z y¼ (5.51) zs Substitution from Equations (5.48), (5.49) and (5.51) into Equation (5.50) yields @ ln aAO @x @ ln ao @ ln aBO @x @ ln ao þ þ DA ð1 xÞ þ DB x ¼k (5.52) @x @y @y @x @y @y

208


Application of the continuity condition for the diffusion of one component on the basis again that x and ao are functions of y only leads to @x d @ ln aBO @x @ ln ao DB x yk ¼ (5.53) @y dy @x @y @y Each of Equations (5.52) and (5.53) apply only within the scale. Within the alloy phase, the distribution of component B is found by solving Fick’s second law @N B @2 N B ¼ DAB @t @x2 assuming DAB to be a constant. Using the Boltzmann transformation x l ¼ 1=2 t we obtain the ordinary differential equation

(5.54)

(5.55)

d2 N B l dN B ¼0 (5.56) þ 2 dl dl2 Solution of the three simultaneous Equations (5.52), (5.53) and (5.54) requires appropriate boundary conditions. These are provided by the initial conditions DAB

N B ¼ N ðoÞ B

for

x40;

t ¼ 0;

x ¼ 1;

t40

(5.57)

and ao ðy ¼ 1Þ ¼ a00o

(5.58)

together with the thermodynamic relationships xðy ¼ 1Þ ¼ f 1 ða00o Þ

(5.59)

xðy ¼ 0Þ ¼ f 2 ðN B;i Þ

(5.60)

a0o ¼ f 3 ðx0 Þ

(5.61)

and the mass balances which apply at the scale interfaces. The mass balance for B at the scale–gas interface is J B ðy ¼ 1Þ ¼

x00 dzs V OX dt

(5.62)

which, upon substitution from Equations (5.51), (5.52) and (5.53), becomes @ ln aBO dx d ln ao ¼ x00 k (5.63) DB x @x dy dy y¼1 Similarly, a mass balance for B at the scale–alloy interface (y ¼ 0) is used to evaluate x0 . Wagner treated this by relating the average mole fraction of B in the scale, xAV to the amount consumed from the alloy.


209

Using the valences of Equation (5.47), his result can be written 23=2 DAB V OX dN B N B;i þ dl l¼xs =t1=2 k1=2 VAB DB x @ ln aBO dx d ln ao ¼ k @x dy dy y¼0

(5.64)

Coates and Dalvi extended the range of applicability of this treatment by including dissolution of oxygen in the alloy and its diffusion into that phase. Even without that complication, it will be appreciated that solution of the simultaneous Equations (5.52), (5.53) and (5.54) together with the mass balances of Equation (5.63) and (5.64) represents a substantial undertaking. Since, moreover, the diffusional properties of the oxide can be expected to vary with both x and ao , a solution is essentially impossible without a relatively simple diffusion model.

5.6.1 Modelling diffusion in solid solution scales A fruitful approach was proposed by Dalvi and Coates [25] using the data [26] shown in Figure 5.13 for the distribution of nickel and cobalt in a (Ni,Co)O scale grown on a binary alloy. The mixed oxide is a nearly ideal solution d ln aCoO ¼1 d ln x

(5.65)

and the Gibbs–Duhem equation can be written as d ln aNiO ¼

x d ln aCoO 1x

(5.66)

Substitution from these thermodynamic equations into Equation (5.52) leads to ðDNi DCo Þ

dx d ln ao ½ð1 xÞDNi þ xDCo ¼ k þ dy dy

(5.67)

Investigations into the NiO–CoO solid solution by Zintl [27, 28] revealed that the vacancy mole fraction, NV, decreased almost exponentially with additions of NiO. Based on this finding and the vacancy model for each of CoO and NiO 1 2 O2 ðgÞ

¼ OXO þ V 00M þ 2h

(5.68)

Wagner [29] suggested that NV in the solid solution oxide could be modelled as 1=6

x N V ¼ N NiO V b pO 2

(5.69)

reflecting a law of mixtures for the free energy of vacancy formation via Equation (5.68), i.e. DGV ¼ ð1 xÞðDGV ÞNiO þ xðDGV ÞCoO

(5.70)

210


Figure 5.13 Distribution of CoO in CoO–NiO solid solution scale grown on Ni–10.9Co at 1,0001C. Experimental data [26] is compared with model curve calculated [25] from Equations (5.77) and (5.78) [25]. With kind permission from Springer Science and Business Media.

Here N CoO V (5.71) N NiO V is the vacancy mole fraction in the indicated binary oxide at and N MO V pO2 ¼ 1 atm. Recalling that for substitutional diffusion with NME1, DM ¼ DVNV, it was further suggested [29] that diffusion in the ternary oxide could be described by ! NV o DCo ¼ DCo (5.72) N CoO V b¼

and DNi ¼

DoNi

NV N NiO V

! (5.73)


211

where DoM denotes the diffusion coefficient of the indicated metal in its pure binary oxide at pO2 ¼ 1 atm. Combination of Equations (5.69), (5.72) and (5.73) with the definition p¼

DNi DCo

(5.74)

then leads to 1=6

(5.75)

1=6

(5.76)

DCo ¼ DoCo bx1 pO2

DNi ¼ pDoCo bx1 pO2

Substitution from Equations (5.75) and (5.76) into Equation (5.67) yields ( ) k0 dx ðp 1Þ 1=6 dy bx1 pO2 d ln ao ¼ ½p ðp 1Þx dy

(5.77)

where k0 ¼ k=DoCo . Application of this diffusion model to the scale–gas interface mass balance (5.63) yields, after some algebra, a differential equation describing the variation within the scale of x with normalized position y

2 d2 x 1 dx y 2þ 1 ðp 1Þ þ y ln b dy 6 dy

þb

x00 x

)

00 1=6 ( p 1 ð1=6Þ yy2 ao dx dx ¼0 dy y¼1 dy ao x00 ð1 x00 Þðp 1Þ

(5.78)

where y ¼ p ðp 1Þx. Simultaneous solution of Equations (5.77) and (5.78) using measured values of p and b then yields x and ao as functions of y. It was found expedient to treat the exponent of oxygen activity appearing in the defect equilibrium (5.69) as a variable. 1=5

When pO2 was used, the calculated composition profiles shown in Figure 5.13 were found to fit the experimental data very well. The index 1=5 was interpreted as corresponding to a mixture of singly and doubly charged vacancies. The cobalt enrichment at the scale surface resulted from the fact that p ¼ DNi =DCo 0:5. Single-phase (Fe,Mn)O scales grow according to parabolic kinetics on Fe–Mn alloys oxidized in CO2–CO atmospheres [30]. Microprobe concentration profiles showed that the scale compositions were rather uniform, and approximately the same as the alloy compositions. This reflects the fact that diffusion in the oxide was about 104 times faster than in the alloy. The relatively flat, linear gradients in the scale could be approximated by dx ¼b dy

(5.79)

212


and the ideality of the FeO–MnO solution [31] allowed use of Equation (5.65). In this case, Equations (5.53) and (5.63) yield the simple result kx00 ðp 1Þð1 x00 Þ (5.80) b after elimination of @ ln ao =@y. This yielded a value of p ¼ 0.99, consistent with the lack of segregation of the metals within the scale. The diffusion coefficient of iron in wu¨stite is proportional to the oxide non-stoichiometry [32] and an equation analogous to Equation (5.72) applies to the (Fe,Mn)O scale. Values of DFe deduced from the alloy scaling rates were used (in Equation (5.72)) to calculate the non-stoichiometry of the mixed oxide. Figure 5.14 compares the calculated results with those measured for powdered oxide after equilibrating with the gas. The good agreement provides additional support for the validity of the diffusion model. Similar analyses have been carried out for solid-solution oxide scales developed on Co–Fe [33, 34] and Ni–Fe [35] alloys, and for monosulfide scales on Fe–Ni [36, 37] and Fe–Co [38, 39]. An unusual pattern of component segregation was found in the (Co,Fe)O scales, where at low ambient pO2 values, the more mobile iron was enriched towards the scale surface, as seen in Figure 5.15a. However, at high pO2 values, a maximum in iron concentration developed in the scale interior (Figure 5.15b). The explanation for this is the DMn ¼ pDFe ¼

Figure 5.14 Non-stoichiometry of (Fe,Mn)1dO: points deduced from alloy scaling rates and Equation (5.72); dashed curves measured by equilibrating powdered oxides with gas [30] Reproduced by permission of The Electrochemical Society.


213

ξ ξ

(a)

(b)

(c)

Figure 5.15 (a), (b) Compositional profiles in (Co,Fe)O scales at low and high pO2 values [36, 37, 41] and (c) DFe/DCo as a function of pO2 [40]. Reproduced by permission of The Electrochemical Society.

214


curious variation in p ¼ DFe =DCo with ao , as measured by Crow [40] and shown in Figure 5.15c. Incorporating this information into the numerical solution procedure for Equations (5.77) and (5.78) allowed Narita et al. [41] to calculate the scale concentration profile successfully (Figure 5.15a and b). The reason for the change in p with ao is not apparent. It has been suggested by Whittle and coworkers [42] that correlation effects can lead to variation of p with vacancy concentration, and hence with ao . However, even this model cannot account for the reversal in relative mobilities of iron and cobalt evident in Figure 5.15. Sulfide scales provide the investigator with the advantage of being able to measure accurate concentration profiles for the oxidant species using an electron microprobe. Results for an (Fe,Ni)1dS scale are shown in Figure 5.16, where the sulfur concentration varies with position in a non-monotonic fashion. The mixed sulfide grew at a faster rate than Fe1dS scales grew at the same pS2 on pure iron. The indicated enhancement in DFe could arise either through a decreased activation energy for diffusion, or from an increase in defect concentration above that predicted for an ideal solution. The former possibility may be rejected on the basis of self-diffusion data [43–46] for Fe1dS and Ni1dS. The latter possibility is supported by the concentration profiles in Figure 5.16 as is now discussed. Recognizing that the deviation from stoichiometry is given by NM d¼1 (5.81) NS

0.150

53

0.125

52

0.100

51

0.075

50

0.005

0

100 200 Distance from scale-alloy interface/μm

atomic % S

NNi /NNi + NFe

it is clear that d varies with position in an unusual fashion in the (Fe,Ni)1dS scale. The values of d calculated in this way range up to 0.04, much greater than the value of 0.02 reported [47] for Fe1dS under these conditions. Since Ni1dS has a smaller deviation from stoichiometry than Fe1dS, it is obvious that the solution is

49

Figure 5.16 Compositional profiles in (Fe,Ni)1dS scale grown on Fe–41Ni at T ¼ 6651C [37]. Reproduced by permission of The Electrochemical Society.

215


not ideal with respect to the defect species nor, equivalently, to sulfur. The conclusion that a ternary solid solution may be close to ideal with respect to its component binary compounds but deviate strongly from ideality for the electronegative species is a common one. If it is assumed that the psuedobinary solution FeS–NiS is ideal, that deviations from stoichiometry can be ignored and that p ¼ DFe =DNi is constant, independent of composition and as , then Equation (5.67) can be rewritten as ð1 pÞ

dx d ln as kp þ ð1 x þ pxÞ ¼ dy DFe dy

(5.82)

If it is further assumed that the relationship between DFe and N V (or d) in the (Fe,Ni)1dS scale is the same as that given by Condit et al. [43] for Fe1dS ð81 þ 84dÞ kJ mol1 DFe ¼ DO d exp (5.83) RT then Equation (5.82) can be applied to the data in Figure 5.16 for x and d as functions of y. This procedure permits the evaluation of the gradient d ln as =dy, and p can then be varied to match the sulfur activity profile to the boundary values. The results of this calculation are shown in Figure 5.17, where it is seen

Figure 5.17 Sulfur activity profile in (Fe,Ni)1dS scale (y ¼ x/X) calculated from Equations (5.82) and (5.83) [37]. Reproduced by permission of The Electrochemical Society.

216


that d ln as =dy is constant throughout the scale, despite the unusual behaviour of N S . The value found for p was 0.4, consistent with the observed enrichment of nickel towards the scale surface. The methods of calculating solid solution scale compositions and growth rates are complex, and require a great deal of information on the thermodynamic and kinetic properties of the oxide. It is therefore much easier to measure scaling rates than it is to model them. Nonetheless, the experimental validation of the scaling theory has led to useful conclusions. The growth of single-phase, solid solution scale layers is controlled by diffusion, and parabolic kinetics result. Scale compositions vary with position within the scale, but are time invariant during steady-state reaction. The average scale composition is related to the ability of an alloy to deliver metal by diffusion to the scale–alloy surface. A useful form of this relationship has been provided by Bastow et al. [42] Z 1 N ðoÞ N B;i xAV ¼ þ N B;i x dy ¼ B (5.84) FðuÞ 0 where FðuÞ is as defined in Equation (5.29) and, as before, 1=2 kc u¼ 2DAB

(5.85)

If scaling is much faster than alloy diffusion, the situation for the MO and MS scales examined so far, then N B;i N ðoÞ B and the scale has the same average metal ratio as the alloy. If the reacting system is not at steady state, then N B;i changes with time, as must therefore xAV . If an alloy becomes depleted in one component, then the other component will become enriched in the scale. If that component is the faster diffusing one, then its further enrichment at the scale surface may lead to the formation of a new oxide phase. The subsequently changed oxide constitution and morphology can be associated with loss of protective behaviour, as is discussed in Section 5.9 and Chapter 7.

5.7. TRANSIENT OXIDATION Discussion so far has been focused on the growth of an external scale under steady-state conditions. However, the time taken to achieve this steady state could be lengthy, in which case considerable scale would accumulate. The situation where only one oxide is stable was considered in Section 5.4, where we concluded that the scale–alloy boundary conditions (and therefore the scaling rate) changed with time only if the kinetics were non-parabolic. Gesmundo et al. [48, 49] have investigated this situation further, noting that a more realistic description of the early stage transient kinetics should involve a contribution to rate control by the scale–gas interaction processes. Thus scaling kinetics are expected to show a transition from an initial linear form to subsequent parabolic behaviour as the scale thickens and eventually diffusion becomes slower than the scale–gas interfacial process. It was shown that under these conditions the value

5.7. Transient Oxidation

217

of N B;i decreased monotonically from N ðoÞ B to the steady-state value, with no minimum of the sort suggested by Whittle et al. [14]. The different conclusions were consequences of the different kinetic models used for the transient stage. The consequences of the transient oxidation stage are potentially more significant in the case where more than one oxide can form, and the oxides have limited intersolubility. An example is provided by the oxidation of binary Cu–Zn alloys, studied long ago by Dunn [50] and subsequently by others. Relative oxidation rates of these alloys are indicated by the data in Figure 5.18. Alloys containing up to 10% Zn react at 8001C according to parabolic kinetics at essentially the same rate as pure copper, producing a Cu2O scale with inclusions of ZnO [51]. If the alloy zinc level is 20%, the oxidation rate is orders of magnitude less, independent of N ðoÞ Zn , and corresponds to the growth mainly of the more stable ZnO. Wagner [12] calculated the value of N Zn;min from Equation (5.25), modified to take into account the variation of DZn with composition [52]. The resulting values for N Zn;min of 0.14, 0.15 and 0.16 at 7251C, 8001C and 8001C are in reasonably good agreement with experimental observation (Figure 5.18). Alloys containing intermediate zinc levels of 10–20% showed wide deviations from parabolic kinetics [53], as seen in Figure 5.19. The rate was initially similar to that of pure copper, but subsequently decreased significantly as a ZnO layer developed at the base of the scale. If the reaction was interrupted by a 1-h anneal under argon, the ZnO layer developed during this time. When oxidation was resumed, slow parabolic kinetics were observed, and the rate was characteristic of high zinc content alloys. This pattern of behaviour can be understood in terms of an initial, transient reaction period during which both Cu2O and ZnO nucleate on the surface [54, 55]. The faster growing Cu2O overgrows the ZnO, which remains as slow-growing

Figure 5.18 Oxidation of Cu–Zn alloys: weight uptake after 5 h reaction at pO2 ¼ 1 atm [51]. With permission of TMS.

218


3.0 (a)

2.0

(ΔW/A/mgcm-2)2

1.0

0 (b) 2.0

1.0

0

0

40

80

120 Time, min

160

200

Figure 5.19 Oxidation kinetics observed for Cu–15Zn at T ¼ 7001C and pO2 ¼ 1 atm (a) continuous (b) interrupted by 1 h anneal in Ar [53]. Reproduced by permission of The Electrochemical Society.

particles at the scale–alloy interface (Figure 5.20). During this stage, the overall scaling kinetics are similar to those of single-phase Cu2O layer growth, since this phase constitutes the majority of the scale. This initial stage of preferential copper oxidation leads to zinc enrichment at the alloy–scale interface, and the reaction Zn þCu2 O ¼ ZnO þ 2Cu

(5.86)

commences. This process is thermodynamically favoured, with DG ¼ 164 kJ mol1 at 8001C, corresponding to a2Cu ¼ 9:6 107 aZn

(5.87)

Thus, at local equilibrium, aZn 108 , and the transient formation of Cu2O is a consequence of the reaction kinetics. Whereas Cu2O growth is rapid, the displacement reaction (5.86) is slow. Eventually, however, the displacement


Cu+

Cu2O ZnO Cu-Zn

Zn2+ Cu-Zn

219

Cu2O ZnO Cu-Zn

Figure 5.20 Schematic view of transient Cu2O overgrowing ZnO and eventually being isolated from Cu–Zn alloy as the ZnO layer becomes complete.

reaction becomes kinetically favoured, and the alloy surface area fraction covered with ZnO increases to unity, as shown schematically in Figure 5.20. Once coverage with ZnO is complete, further Cu2O growth ceases because copper is essentially insoluble in the zinc oxide. Further scale growth then consists of ZnO layer thickening under steady-state diffusion control. Wagner [12] carried out a similar analysis for the oxidation behaviour of Cu–Ni alloys. Using Equation (5.25), he calculated that for exclusive NiO formation a value of N Ni;min ¼ 0:75 was required at 9501C. This was in satisfactory agreement with the change in alloy oxidation rate observed by Pilling and Bedworth [56] at a value of about 0.7. However, scaling rates in the range 0:7oN ðoÞ Ni o1 were greater than for pure nickel, increasing with the level of copper. As with the Cu–Zn system, the high diffusion coefficient of Cu2O meant that regions of this oxide remaining from the initial transient stage of oxidation continued to grow fast. Evidently the displacement reaction Ni þCu2 O ¼ NiO þ 2 Cu

(5.88)

is slow and regions of Cu2O persist at the scale–alloy interface for long times [53, 57]. It may be that nucleation of new NiO regions at the Cu2O/alloy interface is energetically unfavourable, and that the lateral spreading of original surface NiO nuclei is also slow. Transient oxidation processes occurring before the establishment of steadystate protective scales of Cr2O3 or Al2O3 are rather different from the Cu–Zn and Cu–Ni systems described earlier. The much greater stability of chromia and alumina makes internal precipitation of these oxides more likely. Discussion is therefore postponed until internal oxidation processes are considered in Chapter 6. Even when only one metal is oxidized, non-steady-state oxidation can take place in an initial transient period associated with phase transformations in the oxide. The technologically important example of alumina scale formation is now considered.

5.7.1 Transient behaviour associated with alumina phase transformations Alumina exists in a number of crystalline forms only one of which, the hexagonal a-phase, is thermodynamically stable [58]. However, the other phases retain their crystalline forms indefinitely below certain temperature limits [58] as shown

220


γ

(a)

θ

γ

(b)

500

δ 700

900

α

θ+α

θ

α

1100

Figure 5.21 Approximate Al2O3 transformation temperatures observed [58] on bulk material used for catalyst supports (a) g-Al2O3+3% Pt (b) g-Al2O3. Reproduced with the permission of The American Ceramic Society.

approximately in Figure 5.21. The long-term existence of these metastable phases arises from the difficulty of achieving the transformations through which the material must pass to reach the stable a-phase. Activation barriers may, of course, be overcome thermally, but the magnitude of the barriers may also be altered by the presence of foreign phases, either gaseous or solid [59, 60], and by dissolved impurity species [61]. As seen in Figure 5.21, the presence of platinum in contact (as a dispersed catalyst) with g-Al2O3 alters the sequence of its phase transformations, and generally lowers the temperatures at which they occur. Nickel has also been shown [62] to accelerate transformation to a-Al2O3 at temperatures of 8501C and 9501C. As discussed later in this section, chromium and iron also affect the transformation. Oxidation of alumina forming alloys at temperatures below about 1,2001C often leads initially to formation of transient, metastable alumina scales [63–88]. This is significant, because the metastable aluminas grow much more rapidly than a-Al2O3 [63–88]. A comparison of scaling rates for y and a-Al2O3 in Figure 5.22 illustrates this point. An example of the transition from fast transient oxidation to slow, steady-state a-Al2O3 growth observed by Rybicki and Smialek [64] for the intermetallic b-NiAl containing 0.05 at. % Zr is shown in Figure 5.23. The metastable aluminas have lower densities than a-Al2O3 and transformation is accompanied by a 13% reduction in volume. The higher growth rates of the metastable oxides are related to their different crystal structures (g-Al2O3 has a cubic spinel type structure [66], the structure of d-Al2O3 is the subject of some disagreement [67] and y-Al2O3 is monoclinic) and looser packing than the a-Al2O3 structure of hexagonal close packed oxygen with aluminium occupying octahedral interstitial sites. The different morphologies developed by the alumina phases also contribute to their differing growth rates: whereas a-Al2O3 is a dense layer, the metastable forms tend to develop as blades and whiskers. Considerable information is available for the oxidation of the intermetallic bNiAl. This material has good oxidation resistance due to its ability to form scales which are exclusively Al2O3 [15]. It has been studied intensively because it is the principal constituent of diffusion coatings grown on nickel base superalloys to provide protection against oxidation. The transient oxide grown on b-NiAl+Zr at 9001C was found to have a blade or platelet structure. The oxide was identified by XRD as y-Al2O3. At 8001C and 9001C, the first formed oxide was g-Al2O3 but was replaced by y-Al2O3 after


221

Figure 5.22 Rates of y-, g- and a-Al2O3 scale growth on b-NiAl+Zr [65]. Reprinted from Ref. [65] with permission from Elsevier.

Figure 5.23 Transition from fast transient oxidation to steady-state a-Al2O3 growth on b-NiAl+Zr [64]. With kind permission from Springer Science and Business Media.

about 1 h [68]. At these temperatures, the y-phase persisted for at least 100 h. At 1,0001C and 1,1001C, however, the y-phase was replaced by a-Al2O3, which nucleated in the prior y-Al2O3 scale. These nuclei grew laterally, until they impinged to form grain boundaries [68, 69], the transformation to a-Al2O3 then being complete. Shrinkage cracks within the grains resulted from the volume

222


change accompanying the y–a transformation. The grain boundaries formed where the a-Al2O3 islands met provided pathways for rapid diffusion, leading to the development of oxide ridges, as proposed by Hindam and Smeltzer [20]. Plan and cross-sectional views of the ridge structure are shown in Figure 5.24. The ridges remain on the surface, but do not continue to grow in proportion to the underlying scale thickness. The nucleation sites for a-Al2O3 formation are of interest. On the basis of their TEM observations, Doychak et al. [69] suggested that nucleation commenced preferentially at the oxide-gas surface. Smialek and Gibala [71] concluded that the transient oxidation of Ni–Cr–Al alloys was ended by nucleation of a-Al2O3 at the scale–alloy interface. Both of these investigations relied upon TEM examination in which the electron beam was transmitted through the scale thickness, and the location of the a-nuclei was therefore ambiguous. Subsequent observations [72] of fracture sections of scales grown on b-NiAl, reproduced in Figure 5.25, show that the a-phase grew at the metal–scale interface. Minor alloy additions to the b-NiAl can affect the rate at which steady-state a-Al2O3 growth is achieved. Both zirconium and ion-implanted yttrium slow the transformation from y- to a-Al2O3 [73, 74]. Fine oxide dispersions in the alloy can also affect the transformation. Pint et al. [75] showed that dispersed Y2O3, ZrO2, La2O3 and HfO2 all delayed slightly the y- to a-Al2O3 transformation during initial oxidation of b-NiAl at 1,0001C. However, dispersions of a-Al2O3 and TiO2 both accelerated the transformation. The delays caused by Y, Zr, La and Hf oxides were attributed to the effect of dissolution into the transient oxide. According to Burtin et al. [61] larger ions inhibit the y–a transformation. It was suggested that such dopants could interfere with both the surface area reduction and the diffusionless transformations required to convert y-Al2O3 blades to dense a-Al2O3. The accelerating effect of a-Al2O3 inclusions was presumably simply one of nucleation. Alloy additions of chromium can also accelerate the transformation through initial formation of Cr2O3 which, being isotypic with a-Al2O3, promotes its nucleation [76]. Ferritic FeCrAl alloys such as Kanthal (Table 5.1) are also alumina formers. At temperatures of 1,0001C and higher, the a-phase is quickly formed, providing good protection. This is thought to be due to transient formation of Fe2O3 which is also structurally isotypic with a-Al2O3, and promotes its nucleation. Confirmation of this has been provided by N’Dah et al. [77], who oxidized commercial FeCrAl alloys in Ar–H2–H2O atmospheres at 1,1001C and 1,2001C. If the H2O(g) level was high enough to yield a pO2 value above the Fe2O3/FeAl2O4/ Al2O3 equilibrium value, a scale of 100% a-Al2O3 was obtained. However if the water vapour level was lower, a mixture of a- and y-Al2O3 resulted. At lower temperatures, the scales formed on FeCrAl alloys can contain metastable aluminas, and consequently provide poor protection [78–80]. Figure 5.26 shows a TEM cross-sectional view [81] of the scale grown on Kanthal AF (Table 5.1) at 9001C in an atmosphere of O2+40%H2O. An EDAX line scan across the scale revealed a narrow central region rich in chromium. This was a residue of the initial stage of transient oxidation in which Fe, Cr and Al all oxidized. The iron had subsequently diffused into the outer scale region, where oxygen


223

30µm

whiskers

ridge

Al2O3

β cavities

2µm

Figure 5.24 Ridges of a-Al2O3 developed on b-NiAl where islands of a-Al2O3 had met: upper: SEM plan and lower: FIB cross-section views. Localized spallation visible in plan view [70]. Reproduced by permission of the Electrochemical Society.

224


Figure 5.25 SEM view of fracture section of alumina scale grown at 1,1001C on b-NiAl, showing a-grains at the scale–alloy interface [72]. Published with permission from Science Reviews.

Figure 5.26 TEM cross-sectional view of scale grown on Kanthal AF at 9001C [81]. Bright material in the middle of scale is chromium-rich remnant from transient oxidation. Published with permission from Science Reviews.

activities were higher. The outer layer was g-Al2O3 whereas the inner layer was a-Al2O3. The latter had nucleated at the chromium-rich region and grown inwards and laterally to form a protective layer. Before that layer was complete, the outer g-Al2O3 layer developed. Its stability was thought to be enhanced by the presence of water vapour. An increase in the amount of transient oxidation of a variety of alumina forming alloys when exposed to humid air has also been reported by Maris-Sida et al. [82]. The more rapid growth of transient metastable aluminas can cause more severe depletion of the alloy aluminium. Pragnell et al. [83] studied the oxidation of commercial FeCrAl foils of nominal thickness 50 mm at 9001C. They observed rapid initial growth of transient y-Al2O3 which was transformed only slowly to


225

a-Al2O3. The total weight uptake after 72 h was B0.4 mg cm2, much more than that corresponding to protective a-Al2O3 scale growth. Measurements of alloy aluminium concentrations (Figure 5.27) show that the depletion levels were consequently significant. A strongly beneficial effect of titanium in promoting a-Al2O3 formation has been reported. As noted earlier, dispersed TiO2 in b-NiAl accelerated transformation of transient alumina to the a-phase. Comparisons [84] of the oxidation kinetics of different FeCrAl grades at 850–9251C have shown that Kanthal AF reached steady-state a-Al2O3 growth the fastest. This grade contains nominally 0.1% Ti. Prasanna et al. [85] showed that titanium from the alloy was incorporated into the oxide scale, possibly accelerating the y–a transformation. The application of a slurry of TiO2 to the FeCrAl surface before oxidation has also been shown [81, 86] to accelerate a-Al2O3 formation. Since TiO2 was used by one set of investigators [81] in the form of rutile and by the other [86] as anatase, it seems that the chemical rather than the structural nature of TiO2 was important. Finally, oxidation of g-TiAl alloys produces a-Al2O3 along with TiO2 at temperatures where other alumina formers develop transient oxides [87]. Pint et al. [75] have suggested that the accelerating effect of titanium is consistent with the findings of Burtin et al. [61] in that the Ti4+ ion is of similar size to Mg2+, which has been found also to be a phase change accelerator. It seems that the transient behaviour of alumina scales is affected by a large number of variables, and that information is still being collected. Nonetheless, it also seems that ways of accelerating the phase transformations, and thereby lessening the amount of transient oxidation, are being developed. Quantification of alumina transformation kinetics under various circumstances is highly desirable. Temperature–time–transformation plots, such as those in Figure 5.28 due to Andoh et al. [88] provide a useful representation of such data. 6

Al concentration / wt%

5 4 3 2 Measured Predicted

1 0

0

20

40

60

80 100 X distance / μm

120

140

160

Figure 5.27 Aluminium depletion caused by rapid transient oxidation of FeCrAl at 9001C [83]. Published with permission from Science Reviews.

226


Figure 5.28 Temperature–time–transformation plots for alumina formed on Fe–20Cr–5Al [88]. Published with permission from Trans. Tech. Publications.

5.8. MICROSTRUCTURAL CHANGES IN SUBSURFACE ALLOY REGIONS As is by now clear, scale growth almost always leads to the development of compositional changes in the alloy subsurface as the result of the different rates at which alloy components are oxidized. The diffusion processes involved can lead to volume changes, breakdown in the morphological stability of the scale–alloy interface, depletion and dissolution of minority phases, precipitation of new phases and other transformations resulting from the compositional changes, as discussed later. The additional possibility of inward oxygen diffusion leading to internal oxide precipitation will be dealt within Chapter 6.

5.8.1 Subsurface void formation An example of void formation within b-NiAl beneath an alumina scale was shown in Figure 5.6. The alloy surface revealed by scale removal shows the voids to be faceted, and of varying aspect ratios. The cross-sectional view shows that the Al2O3 undersurface is flat, the void having developed in the metallic phase. There are several possible ways in which such voids could form. Growth of an external scale by outward metal transport means that new oxide is formed at the scale–gas interface, and cannot in any direct sense fill the space vacated by the reacted metal. However, plastic deformation of the scale can allow it to retain contact with the retreating metal surface, if scale movement is unconstrained. To the extent that plastic deformation is not available, void space develops somewhere within or beneath the scale. In the case of a completely rigid scale, the total void volume would equal the volume of metal consumed by

5.8. Microstructural Changes in Subsurface Alloy Regions

227

oxidation. The location of the voids depends on the detailed transport mechanisms in effect. In solid solution alloys, mass transport occurs via vacancy diffusion, and the origins and sinks for these defects must be considered. It is assumed [89–91] that vacancies are injected at the scale–metal interface, as metal atoms move into the scale. If these are annihilated at dislocations, they cannot cause void formation within the metal, but nonetheless the reacting metal shrinks. If, as is being supposed, the oxide scale is unable to conform with the shrinking metal core, void space must be generated elsewhere by the creation of new vacancies. These can be emitted from dislocations in the reverse of the annihilation process. Thus dislocations serve as very rapid pathways for the transmission of vacancies and thereby of void space. Voids develop where vacancies can aggregate, i.e ‘‘coalesce’’ or ‘‘condense’’, in what must be a nucleation process. Preferred sites for void nucleation will therefore be phase interfaces and alloy grain boundaries. Moreover, the development of a vacancy concentration gradient from a maximum at the scale–alloy interface to a minimum in the alloy interior will lead to a greater number of voids nucleating immediately beneath the scale than deeper into the alloy. This was the experimental finding of Hales and Hill [89] in the case of pure nickel. Of course, the vacancy injection, transport and condensation model is applicable to both metals and alloys. Alloys are subject to an additional effect, arising from the different mobilities of the constituent metals. Consider the case of b-NiAl forming an external scale of pure Al2O3, and voids at the alloy–scale interface [92–95]. Brumm and Grabke [96] have investigated void formation on a series of b-NiAl compositions within the homogeneity range of this phase (20% at 1,2001C). They found that void formation decreased with increasing alloy N Al =N Ni ratio. This was explained using the diffusion data [97] shown in Figure 5.29. As seen from the figure, DNi =DAl 3 for N Al 0:5. The selective oxidation of aluminium from b-NiAl necessarily depletes aluminium from the alloy surface, and enriches nickel, as shown schematically in Figure 5.3. In the case of nickel-rich alloys, the high value of DNi =DAl means that the inward flux of nickel exceeds the outward flux of aluminium. Such a situation of unbalanced material flows is known as the Kirkendall effect, and was analysed in Section 2.7. In that discussion it was assumed that the lattice was free to move, and its resulting drift rate, n, reflected the different metal self-diffusion coefficients n ¼ V AB ðDA DB ÞrCA

(5.89)

In the case of b-NiAl oxidation, however, the alloy surface is anchored to an almost rigid alumina scale, and is not free to move. The vacancy flux induced by the imbalance between J Ni and J Al therefore leads to void formation rather than lattice drift. Evidently void nucleation at the alloy–surface interface is energetically favoured over other sites within the bulk alloy. In aluminium-rich NiAl, however, diffusion of aluminium is the dominant process (Figure 5.29), and the Kirkendall effect ceases to drive vacancies towards the alloy surface [98, 99]. The voids continue to enlarge with time as NiAl oxidation proceeds, despite the gaps developed between alloy and oxide. At 1,2001C, the vapour pressure of aluminium above the depleted alloy is sufficient to transport Al(g) across the

228


Figure 5.29 Interdiffusion and self-diffusion coefficients in b-NiAl. Reprinted from Ref. [97] with permission from Elsevier.

cavity to the scale, sustaining its continued growth [22, 96]. At temperatures below 1,0001C, the value of pAl is too low, according to Equation (2.155), to maintain the observed oxide scaling rate. Some other transport mechanism, perhaps surface diffusion, must be involved [96]. The development of the interfacial voids obviously weakens scale adhesion, making scale loss more likely. Platinum is added to NiAl to improve its scale adherence [100]. The improvement is associated with a reduction in cavity formation [101], an effect thought to result from interactions within the alloy increasing DAl and/or decreasing DNi . Gleeson et al. [102] have confirmed that platinum increases DAl in b-NiAl. A completely different mechanism of void formation is available in cases where the alloy contains carbon. Inward diffusing oxygen can react with solute


229

carbon to form bubbles of CO2, as has been shown experimentally [103–105]. Fracture of oxidized specimens in a vacuum chamber attached to a mass spectrometer revealed the presence of CO2. The extent of void formation was shown to increase with carbon content, and could be suppressed by decarburization before oxidation. This mechanism can operate in both alloys and single metals.

5.8.2 Scale–alloy interface stability The additional degree of freedom available in a binary alloy plus oxygen system permits the development of a two-phase region in a diffusion zone, unlike the case of pure metal oxidation, where such zones are thermodynamically impossible in the absence of capillarity effects. For this reason, pure metal–scale interfaces are stable. However, no such thermodynamic constraint applies to alloy–scale interfaces, the shapes of which are kinetically controlled. An example of an unstable interface is shown in Figure 5.1a. The general nature of the problem is rather simple, as shown in Figure 5.30, where a perturbation in an otherwise flat alloy–scale interface is represented. If such a perturbation grows, the interface is unstable; if it decays, the interface is stable.

Figure 5.30 Schematic view of growth or decay of perturbation at alloy–scale interface, according to which phase is the slower diffusing.

230


The effect of the perturbation is locally to decrease the scale thickness from X to Xu and increase the alloy depletion depth from XD to X0D . Clearly, if scale growth is controlled by scale diffusion, i.e. dX=dt ¼ kp =X, then growth is faster at the site of the perturbation. Accordingly metal is consumed faster at this site than on the flat surface, a process which continues until a uniform scale thickness is restored. It is seen that the interface is stable when scale diffusion is rate controlling. Consider now the situation where alloy diffusion is rate controlling, ~ AB =XD . Clearly this and to a first approximation, the flux of B is proportional to D flux is slowest at the site of the perturbation shown in Figure 5.30, because X0D 4XD . Thus oxidation of the flat part of the interface is faster than at the perturbation, causing the flat surface to recede faster than the locally perturbed region. In this case, the irregularity grows and the interface is unstable. The conditions under which diffusion in the alloy controls scaling rates were examined by Wagner [12], as discussed in Section 5.3. Wagner [106] extended that analysis to consider the possibility of morphological breakdown, assuming that no oxygen dissolved in the alloy and that surface capillarity effects can be neglected. He found from a two dimensional analysis of diffusion at a sinusoidal scale–alloy interface that the condition for interface stability is N B;i DAB VOX 41 (5.90) 1 N B;i DB V AB where now N B;i represents alloy composition at the average interface location. When, however, the interface is unstable, it is likely that particles of the more noble metal will be occluded into the oxide. Whittle et al. [107] have examined the effect of relaxing the assumptions of negligible oxygen solubility in the alloy and of the more noble metal in the oxide. They found that internal precipitation of BO behind the alloy–scale interface was a likely outcome under the supposed conditions.

5.8.3 Phase dissolution The situation considered is that of a two-phase alloy in which a precipitate phase rich in the more reactive solute element acts as a reservoir for the continued exclusive growth of the solute metal oxide scale. A schematic representation is shown in Figure 5.31, using the example of an Ni–Si alloy. The concentration profile of reactive metal B is defined by the original alloy mole fraction N ðoÞ B , the solubility limit in the matrix phase, N aB , and the alloy–scale boundary value. It is assumed that precipitate dissolution is fast enough to maintain local precipitatematrix equilibrium. If diffusion of B through the solute-depleted subsurface alloy region is also fast enough, then N B;i will be approximately constant, and steadystate parabolic kinetics result. Diffusion analysis [108] yields the concentration profile of B in the single-phase dissolution zone. Approximating the scale–alloy interface as immobile, one finds from a mass balance for B that u N ðoÞ þ p1=2 u erf ðgÞ (5.91) B N B;i ¼ g exp ðg2 Þ


231

Figure 5.31 Selective oxidation of two-phase alloy AB causing dissolution of B-rich phase and diffusion through depletion zone.

where u¼

kc ~ 4DAB

1=2

g¼

Xd 1=2 ~ AB t 4D

(5.92)

(5.93)

and Xd represents the precipitate dissolution depth. A slightly more accurate description is obtained by taking scale–metal interface movement into account [108]. Application of (5.91) to the kinetics of precipitate dissolution zone widening during oxidation of Ni–Si alloys consisting of a g-matrix and b-Ni3Si precipitates, and of Co–Si alloys containing a-Co2Si precipitate led to successful prediction [109] of depletion depths (Figure 5.32). Two questions arise when considering the selective oxidation of protective scale-forming metals from two-phase alloys. Firstly, will the precipitates dissolve fast enough to maintain the solute level at its equilibrium value N aB ? Secondly, will diffusion through the depleted zone be fast enough to maintain N B;i at a high enough level to sustain selective scale growth? The second problem is similar to

232


300

250

Xd [μm]

200

150

100

50

0 0

50

100

150

200

250

300

350

t1/2 [s1/2]

Figure 5.32 Depletion zone deepening kinetics for Ni–16Si (’) and Co–19Si (~). Continuous lines predicted from Equation (5.91). Reprinted from Ref. [109] with permission from Elsevier.

the situation of diffusion from a single-phase alloy considered by Wagner [12], and discussed here in Section 5.3. In both cases, diffusion through a single-phase, ~ AB subsurface alloy zone delivers B to the scale–alloy interface, and the ratio kc =D is a key factor. This has been demonstrated [110] by comparing austenitic and ferritic modifications of a series of cast iron–chromium carbide alloys. The software package THERMO-CALC [111] was used to predict how alloying additions would affect the phase constitution and to calculate alloy and precipitate compositions and weight fractions. The base alloy chosen for investigation was Fe–15Cr–0.5C at 8501C, where it is austenitic. Alloy compositions are listed in Table 5.4 along with their predicted phase constitutions. Matrix chromium contents were around 11 wt.% and the coarse interdendritic carbides varied in volume fraction from 6% to 10%. Adding silicon to the iron-based alloy altered its phase constitution to a+M7C3. To study the chemical effect of silicon in isolation from its effect on the matrix crystal structure, another alloy was designed to retain the austenite structure by using nickel as an austenite stabilizer. To complete the iron-based alloy set, an a+carbide steel was produced to investigate the effect of changing matrix to ferrite without simultaneously introducing silicon. Molybdenum was chosen as the ferrite stabilizer. To verify that the molybdenum had no major effect other than producing a ferrite matrix, a molybdenum-bearing austenitic alloy was designed, again using nickel as the austenite stabilizer. Measured oxidation rates and observed reaction morphologies (Table 5.4) fell into two classes. Either a protective chromium-rich oxide scale developed in association


Table 5.4

233

Oxidation of cast ferrous alloys in oxygen at 8501C Phase constitution

Reaction morphology

kp ðg2 cm 4 s1 Þ

Fe–15Cr–0.5C

g þ M23 C6

2.5 108

Fe–15Cr–0.5C–3Mo

a þ M23 C6

Fe–15Cr–0.5C–3Mo–3Ni

g þ M23 C6

Fe–15Cr–0.5C–1Si

a þ M7 C3

Fe–15Cr–0.4C–1Si–1Ni

g þ M23 C6

Iron oxide scale Carbides engulfed Cr2O3 scale Carbide dissolution zone Iron oxide scale Carbides engulfed Cr2O3 scale Carbide dissolution zone Cr2O3 scale Carbide dissolution zone

Composition (wt%)

1.4 1011

9.3 109 1.3 1012

1.4 1012

with subsurface alloy carbide dissolution, or a fast-growing iron oxide scale engulfed the carbide phase. Whereas the ferritic materials always formed protective chromia scales, the austenitic alloys formed non-protective, rapidly growing iron oxide scales except in the case of the austenite containing silicon. The discussion will return to this last observation after consideration of the alloy diffusion processes. Carbide dissolution depths were measured metallographically and chromium concentrations by electron probe microanalysis, leading to the results shown in Table 5.5. Values for kc were calculated from the weight gain kinetics using measured scale compositions. Values for DCr were then calculated from Equation (5.91) leading to the results shown in Table 5.5. Examination of these values reveals that DCr calculated for ferritic alloys are of the order of those reported in the literature. Chromium diffusion in austenitic alloys was slower, as expected, but not as slow as independently measured diffusion coefficients would suggest. Subsequent use of high temperature XRD to identify the surface phase constitution of reacted alloys confirmed that decarburization of this region had transformed it from austenite to ferrite. Whether the alloy was ferritic to begin with, or was converted to ferrite in its subsurface zone, the relatively rapid lattice diffusion of chromium to the alloy surface sustained a protective Cr2O3 scale. The effect of silicon on the oxidation behaviour of cast Fe–Cr–C was very strong. Adding 1% to the base alloy made it ferritic and led to growth of a protective Cr2O3 scale. Even with an austenitic matrix, which resulted from the further addition of nickel, the silicon-bearing alloy developed a Cr2O3 scale. It was concluded that the effect of silicon on oxidation was related not to the change it produced in alloy diffusion, but rather its ability to alter the scale–alloy interface. Variation in carbide size was important to the reaction morphology [110, 112]. Whereas the base alloy Fe–15Cr–0.5C developed a thick iron oxide scale when reacted in its cast and annealed form, the same alloy formed a protective Cr2O3

234


Table 5.5

Calculated DCr values for Cr2O3-forming alloys at 8501C

Alloy

Matrix

Xd (mm)

NiCr

DCr (cm2 s1)

Fe–15Cr–0.5C (forged) Fe–15Cr–0.5C–1.0Si Fe–15Cr–0.4C–1.0Si–Ni Fe–15Cr–0.5C–3.0Mo

g a g a

35 22 25 45

0.06 0.10 0.10 0.07

6 1012 4 1011 1 1011 4 1011

Cr23C6 particles b

Figure 5.33 Oxide scales grown at 8501C on g-Fe–15Cr–0.5C (a) as cast (b) forged, demonstrating effect of carbide size on chromium release [110]. With kind permission from Springer Science and Business Media.

scale after hot forging (Figure 5.33). The value of kp in this latter case was 6.8 1012 g2 cm4 s1. The volume fraction of chromium-rich carbide was the same in both alloy forms, but the precipitates were much smaller (around 1 mm) in the forged material than the 3–5-mm interdendritic carbides typical of the cast alloys. Thus precipitate size as well as volume fraction is important in achieving delivery of scale-forming metal to the alloy surface. In the literature on multiphase oxidation, frequent reference is made to the 1974 study performed by El Dahshan et al. [113] on Co–Cr–C alloys. This work was the basis of the subsequent suggestion [114] of using a minority alloy phase as a ‘‘reservoir’’ of scale-forming metal. Additions of up to 2 wt% carbon to Co–25Cr caused precipitation of large quantities of chromium-rich carbide and consequently lower chromium content in the metal matrix of these alloys. Nonetheless, the alloys oxidized protectively at 1,0001C in pure oxygen. Formation of a protective chromium-rich oxide scale was accompanied by dissolution of the chromium-rich carbides within a shallow alloy subsurface region. It was therefore concluded that localization of much of the alloy chromium content into precipitates had no effect on oxidation performance, as rapid carbide dissolution yielded the chromium required to form the protective scale. Viewed in the light of the findings for Fe–Cr–C alloys discussed earlier, the conclusions reached by El Dahshan et al. are surprising. Their cast, cobalt-based

235


alloys had coarse carbides which would be expected to dissolve slowly. Furthermore, the austenitic alloys might not provide the rapid diffusion required to sustain Cr2O3 growth on a Co–Cr alloy. A re-examination [115] of Co–25Cr–C oxidation at 1,0001C has demonstrated that their protective behaviour was in fact due to the presence of silicon contamination, as suggested by Jones and Stringer [116]. Silicon was incorporated into the alloys during annealing in evacuated SiO2 ampoules. The silicon was thought to promote rapid chromia formation through a surface nucleation effect.

5.8.4 New phase formation The example of copper hot shortness was described in Section 5.1. Accumulation of a layer of copper-rich phase results from noble metal rejection at the scale–alloy interface, just as in the Pt–Ni case investigated by Wagner [12], coupled in this case with a limited solubility for copper in iron. The concentration profile for copper in the reacting system is represented schematically in Figure 5.34, where the steel is represented as a binary Fe–Cu alloy, and the solubility of copper in FeO is set at zero. At low temperatures, diffusion in the alloy can be neglected, and the thickness of the copper-rich layer can be estimated from a simple mass balance ðoÞ ¯ yCðoÞ Cu ¼ xðCCu CCu Þ

(5.94)

where C¯ Cu is the average copper concentration in the copper-rich layer and the distances x and y are defined in Figure 5.34. Combination with Equation (5.36) Fe(Cu)

Cu(Fe)

FeO

CCu

y

(0)

CCu x

z

Figure 5.34 Schematic concentration profile for copper in oxidized copper-bearing steel, neglecting diffusion into substrate. Dashed line shows location of original alloy reference.

236


then leads to x¼

CðoÞ Cu

VFeCu z ðoÞ V ¯ ðCCu CCu Þ FeO

(5.95)

where the scale is approximated as being entirely wu¨stite. Under steady-state conditions of parabolic scale growth, the copper layer also thickens according to ¯ Cu estimated from the parabolic kinetics. If V FeCu is approximated as VFe and C Fe–Cu phase diagram, then for a 0.47 wt% copper steel, we calculate x ¼ 2:83 103 z. Measured [117] rates of copper layer accumulation were found to be in agreement with values predicted from scaling rates at 1,1501C, using the above mass balance. However, measured copper layer thicknesses were less than predicted at 1,2501C, particularly in the early stages. This occurred because diffusion of copper into the substrate steel cannot be neglected at high temperatures, as seen in the measured concentration profile in Figure 5.4. Another example of new phase formation is provided by the technically important alloys based on g-TiAl. These have an attractive combination of high temperature strength and low density, but their high temperature oxidation performance needs improvement. Initial selective oxidation of aluminium leads to formation of the z-phase (approximately Ti50Al30O20) as a layer at the alloy surface [118, 119]. Examination of the diffusion path in Figure 5.35 shows that little titanium diffusion is involved, but inward oxygen diffusion through the z-phase matches the outward aluminium diffusion. This steady state is not maintained with continued oxidation. Instead, slow aluminium diffusion in the parent g-phase towards the Z/g interface leads to its local depletion, morphological breakdown of the interface and ultimately precipitation of oxygen-rich a2, as shown in Figure 5.35. The accompanying volume change leads to cracking of both the z-layer and Al2O3 scale, followed by TiO2 formation and loss of protective behaviour.

5.8.5 Other transformations Alloys of three or more components are obviously capable of a greater diversity of phase changes than the relatively straightforward binaries considered so far. An example of practical importance is the Ni–Cr–Al systems, which forms the basis of a number of heat resisting alloys and coatings. An isothermal section at 1,1501C of the phase diagram for this system is shown in Figure 5.36 [120]. Isothermal oxidation of three-phase (a-Cr+b-NiAl+g-Ni) alloys led to selective aluminium removal from the alloy, and development of a transformed subsurface region [121], as shown in Figure 5.36. The phases present were identified by electron probe microanalysis: the bright white phase is a-Cr, the mid-grey phase b-NiAl the light grey one g-Ni. As seen from the schematic diffusion path in Figure 5.36, depletion of aluminium from the three-phase alloys must lead eventually to single g-phase formation. Dissolution of the b-phase is immediately understandable in terms of the large gradient in aluminium activity developed by the selective oxidation process.

5.9. Breakdown of Steady-State Scale

237

(a)

(c)

(b)

(d)

Figure 5.35 Oxidation of g-TiAl at T ¼ 1,0001C (a, b) initial protective behaviour and (c, d) after a2 precipitation at g–Z interface. Reprinted from Ref. [119] with permission from Elsevier.

Dissolution of a-Cr, however, was driven by smaller, local gradients in aCr resulting from the increased solubility for chromium in g-Ni developed as the aluminium concentration decreased. For this reason an alloy with a large N ðoÞ Cr value formed a subsurface g+a region whereas a chromium alloy formed single-phase g-Ni.

5.9. BREAKDOWN OF STEADY-STATE SCALE When a protective scale of slow growing oxide can no longer be maintained, other alloy components start to oxidize and alloy consumption is accelerated.

238


30μm Figure 5.36 Isothermal section at 1,1501C of Ni–Cr–Al phase diagram [121] and metallographic section of oxidized alloy, showing diffusion path for selective Al2O3 formation on three-phase alloy.

This phenomenon of breakdown or breakaway oxidation becomes inevitable when the interfacial concentration N B;i decreases to a value lower than the minimum necessary to maintain the exclusive growth of the desired BOn scale. It may even become possible at higher values of N B;i , which are adequate to maintain growth, but insufficient to reform a new scale if the existing one is damaged or removed. Although there is no satisfactory way of predicting the latter value, it can be measured experimentally. The problem then becomes one of predicting when the capacity of the alloy to supply B to the interface is exhausted. Similar considerations apply in the case of a scale with some solubility for a second alloy component. Taking the example of an Fe–Cr alloy, it is clear

5.9. Breakdown of Steady-State Scale

239

that as N Cr;i decreases, N Fe;i increases and the iron content of the scale also rises. If in the oxide DFe =DCr 41, iron is increasingly enriched towards the scale–gas interface until an iron-rich oxide precipitates. The ability of an alloy to supply the desired metal to its surface obviously ~ varies with N ðoÞ B , DAB and t, along with the total amount of B in the alloy specimen. Assuming the specimen to be a large, thin sheet so that edge effects can be neglected, the problem is one of diffusion in a single dimension, normal to the oxidizing surfaces. We consider first the situation where scale growth is very slow, but alloy diffusion rapid, as will be the case with ferritic alumina formers. In this event, the N Al profile in the alloy will be almost flat, and the value of N Al;i is equal to the average value N Al remaining after aluminium is withdrawn from the alloy into the scale. Clearly, the change in N Al with time is significant only if the sheet is extremely thin. This is, in fact, a situation of practical interest because thin sheet material is used in some heat exchangers and as catalyst supports. This problem has been treated by Quadakkers and Bongartz [122] on the basis that the small movement of the scale–alloy interface can be ignored. The materials examined were Fe–20Cr–5Al and oxide dispersion strengthened (ODS) versions of this and similar alloys. Their oxidation weight gain kinetics are not strictly parabolic [123], obeying instead a power law DW ¼ k t1=n (5.96) A where n 3. The approximately cubic rate law results from the fact that mass transfer in the scale is predominantly via grain boundary diffusion, and the density of grain boundaries changes with time [124] (see Section 3.9). The corresponding amount of aluminium withdrawn from each side of the sheet is DW Al ¼ 1:125kt1=n (5.97) A where the dimensionless numerical factor is the Al/O weight ratio in Al2O3. Setting the alloy sheet thickness at 2l, we find for the reduction in alloy aluminium content, DCAl (mole/volume) 1:125kt1=n (5.98) 27l with 27 the atomic weight of aluminium. If the critical value for breakaway is CCrit , the time taken to reach it, tB is therefore !n ðoÞ 27l CAl CCrit tB ¼ (5.99) 1:125 k ðoÞ CAl CAl ¼ DCAl ¼

Quadakkers and Bongartz [122] examined sheets of several ferritic alumina formers oxidized at 1,2001C, and verified that the concentration profiles of aluminium were essentially flat. Using a critical aluminium level of 1.3 wt% for breakdown of the alloy MA956, they predicted from Equation (5.99) the times for breakdown of different sheet thicknesses, shown in Figure 5.37 as the line

240


Figure 5.37 Lifetime limits for breakdown of Al2O3 scales on MA956 sheet. Straight lines predicted from Equations (5.99) and (5.101), and points observed experimentally [122]. Published with permission from Wiley-VCH.

labelled ‘‘no spalling’’. Agreement is seen to be good, as would be expected of a simple mass balance. At greater sheet thicknesses, and longer lifetime, the times to failure are seen to be shorter than predicted. This was attributed to repeated scale cracking and spallation, which occurred at regular intervals. After each of these events, alumina grew again, according to the same kinetics until the next scale spalled. Assuming equal amounts of aluminium are lost in each spallation event, DW n ¼ 1:125kðtn Þ1=n (5.100) A where t is the time between spallation events and DW n the corresponding aluminium loss, then ðoÞ 27l ðCAl CCrit Þ DW n n1 (5.101) tB ¼ 1:125 A kn The dashed line in Figure 5.37 shows behaviour times calculated from Equation (5.101) on the basis of the observed average DW n =A ¼ 2 mg cm2 . Again the simple mass balance prediction is seen to be successful. The more difficult question of predicting when scale spallation will occur is deferred to Chapter 11. Diffusion in austenitic alloys is significantly slower, and the above description does not apply. Instead, the diffusion profile inside the alloy must be found by solving the general diffusion equation (5.12). Because the interface concentration N B;i becomes a function of time as breakdown is approached, no analytical solution is available. However, a simpler approach is to assume that the surface

5.10. Other Factors Affecting Scale Growth

241

concentration remains constant until the depleted zones on the two sides of the sheet meet in the middle. At that stage, the surface concentration starts to decrease and breakdown follows. For diffusion out of a thin, plane sheet –loxol in which the concentration is initially CðoÞ B and the interfacial concentration is fixed at CB;i the solution is quoted by Crank [125] as 1 CB CB;i 4X ð1Þn Dð2n þ 1Þ2 p2 t ð2n þ 1Þpx (5.102) exp ¼ cos 2 ðoÞ p 2l 2n þ 1 4l CB CB;i n¼0 for fixed interfaces. As shown by Carslaw and Jaeger [126], diffusion depletion reaches the middle of the sheet when Dt 0:05 (5.103) l2 The subsequent decrease in CB;i with time has been treated approximately by Whittle [127] on the assumption that N B;i 1. Whittles’ solution was " ! !# 1 pkc 1=2 X 2nl þ x 2ðn þ 1Þl x ðoÞ erfc N B;i ¼ N B þ erfc (5.104) 2DAB 2ðDAB tÞ1=2 2ðDAB tÞ1=2 n¼0 For the specific example of Ni–20Cr oxidized at 1,2001C, with ~ AB ¼ 2 1010 cm2 s1 , kc ¼ 2 108 cm2 s1 and 2l ¼ 0.25 mm, he set x ¼ 0, D and calculated N Cr;i ¼ fðtÞ. If the critical interface concentration necessary to prevent spinel formation is N Cr ¼ 0:03, then a breakdown time of 6 105 s would be predicted. This compares with a value of 2 105 s predicted from Equation (5.103) for the time at which depletion reaches the sample centre. Consistent with these predictions, Douglass and Armijo [128] showed that NiCr2O4 had started to form beneath the chromia scale on this alloy in less than 444 h at 1,2001C. Evans and Donaldson [129] have demonstrated that the approximate solution (5.104) for diffusion out of a thin plane sheet describes the remnant chromium profile reasonably well. The above analyses are of at least indicative value for thin alloy sections, where consumption of the scale-forming metal can occur in a reasonable time. For larger sections, the predictions are optimistic. At 1,1001C, a 5-mm section of MA956 (Table 5.1) is predicted from Equation (5.101) to last for more than 105 h. At 1,0001C (a realistic maximum for a chromia former), a 5-mm section of Ni–20Cr is predicted on the conservative basis of Equation (5.103) to last for 8 107 h. However, the latter estimate is based on the benign assumption that the Cr2O3 scale never cracks or spalls. Moreover, as will be described in later chapters, other modes of failure become likely before the alloy is exhausted of chromium.

5.10. OTHER FACTORS AFFECTING SCALE GROWTH When alloys scale under steady-state conditions, the identity of the oxide in contact with the alloy is determined by the metal composition at this interface.

242


This composition is related to the original alloy composition and can be calculated from Wagner’s analysis of diffusion in the alloy and scale, assuming the latter to be a single phase, continuous layer. The ratio kc =DAB is found to be critical in determining interfacial concentrations and, therefore, the minimum original alloy concentration of a component necessary to sustain the exclusive growth of its oxide scale. Quantitative application of the theory yields limited success, because of its ~ AB . Although the sensitivity to error in experimental measurements of kc and D theory has been extended to cover solid solution scales, the complexity of their solution thermodynamics and diffusion behaviour means that an even larger body of experimental information is required to permit predictions of scale composition and growth rate. Nonetheless, the theory has been verified in a number of cases, and can clearly be relied upon in a qualitative sense. In describing N B;i in terms of kc and DAB , the theory successfully accounts for differences between ferritic and austenitic alloys, and between chromia and alumina scales. It also succeeds in relating the spatial distribution of components within solid solution scales to the relative oxide stabilities and mobilities. These successes are of use in interpreting and to some extent predicting scale breakdown. The values used for the alloy diffusion coefficient have been assumed in this chapter to be those characteristics of bulk or lattice diffusion. Whereas this is reasonable at very high temperatures, it will often be an underestimate at low and intermediate temperatures, where other diffusion pathways such as grain boundaries and dislocations can be more important. The surface finish given to an alloy component before placing it into service can affect the density of grain boundaries and dislocations in the subsurface region. Any low-temperature mechanical working of the surface, such as machining, grinding, blast cleaning, shot peening, etc deforms the subsurface metal, introducing large numbers of dislocations. As the alloy is heated, the deformed metal recrystallizes, forming a generally finer grain and subgrain structure. These subsurface defects will be present during the transient stage of oxidation, and will persist for long times at low temperatures. The consequently higher effective alloy diffusion coefficient is obviously of benefit in rapidly achieving and maintaining protective steadystate growth of chromia or alumina. Several experimental studies [8, 130–132] have demonstrated the more rapid formation of Cr2O3 on cold-worked alloy surfaces. Diffusion theory allows calculation of the minimum concentration of an alloy component necessary to sustain the exclusive growth of its oxide. However, this concentration may not be sufficient to achieve such a steady state. In the initial, transient oxidation stage of reaction, essentially all alloy components capable of forming oxides do so. The subsequent development of scale morphology then depends on the competition between continued growth of fast diffusing oxides and replacement of less stable oxides by more stable, but slow growing ones at the oxide–alloy interface. Because this morphological evolution is controlled in part by nucleation and solid–solid interfacial processes, it cannot be described by diffusion alone.

References

243

The presence of minority components in the alloy can be critical in their effect on the transient reaction. As discussed in Section 5.6, the phase transformations leading to a-Al2O3 formation can be accelerated by many alloy additions, either by their chemical doping of alumina or by providing isostructural oxides which act as ‘‘templating’’ sites for a-phase nucleation. Similarly, the addition of cerium to Fe–Cr alloys has been shown [133, 134] to promote Cr2O3 nucleation. Several metals which form very stable oxides (e.g. Ce, La, Y, Hf, etc.) are for this reason known as ‘‘reactive elements’’. Their addition to chromia and alumina forming alloys often affects the strength of the scale–alloy interface, the scale microstructure and the mass transfer mechanisms governing scale growth. These effects are discussed in Chapter 7.

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CHAPT ER

6 Oxidation of Alloys II: Internal Oxidation

Contents

6.1. Introduction 6.2. Selected Experimental Results 6.3. Internal Oxidation Kinetics in the Absence of External Scaling 6.4. Experimental Verification of Diffusion Model 6.5. Surface Diffusion Effects in the Precipitation Zone 6.6. Internal Precipitates of Low Stability 6.7. Precipitate Nucleation and Growth 6.8. Cellular Precipitation Morphologies 6.9. Multiple Internal Precipitates 6.10. Solute Interactions in the Precipitation Zone 6.11. Transition from Internal to External Oxidation 6.12. Internal Oxidation Beneath a Corroding Alloy Surface 6.13. Volume Expansion in the Internal Precipitation Zone 6.14. Success of Internal Oxidation Theory References

247 248 255 260 267 273 278 284 290 299 301 305 306 311 312

6.1. INTRODUCTION As recognized in Chapter 5, when an alloy component is selectively oxidized but cannot reach the surface quickly enough to develop a scale, then internal oxidation results. Furthermore, an alloy which initially contains sufficient of the reacting metal to form a scale can become depleted in that component to the extent that internal oxidation commences. Under some circumstances, internal oxidation and external scaling can occur simultaneously. It is emphasized that ‘‘oxidation’’ means forming a compound (oxide, carbide, nitride, etc.) of the reactive alloy solute metal, and the description given here applies to internal oxidation, carburization, nitridation, etc. Internal oxidation is the process in which a gas phase oxidant dissolves in an alloy and diffuses inward, reacting with a dilute solute metal to form dispersed precipitates of metal oxide or metal carbide, etc. This class of precipitation reactions is distinguished by its dependence on gas–solid interaction, and the formation of a reaction product zone adjacent to the alloy surface. It does not

247

248

Chapter 6 Oxidation of Alloys II: Internal Oxidation

include homogeneously distributed phase changes, such as carbide precipitation occurring during alloy aging. The practical reality [1] is that a large percentage of high-temperature corrosion failures involve internal oxidation. The internally precipitated reaction products cause embrittlement and dilation of the alloy subsurface region, which can cause the affected zone to flake off. Because the process is supported by diffusion of interstitial species (dissolved oxygen, carbon, nitrogen or sulfur), it is rapid. Internal oxidation is a very destructive process. Both chromia and alumina formers can be attacked by internal oxidation, even when the alloys contain sufficient chromium or aluminium to sustain external scale growth according to Wagner’s criterion (5.22). Just what leads to this outcome is obviously of interest. It is important to establish not only the conditions under which this mode of attack occurs, but also the rate of the process and how it varies with alloy composition and ambient conditions. The general features of internal oxidation reactions were first established by Smith [2, 3], Rhines et al. [4, 5] and Meijering and Druyvesten [6, 7]. Many subsequent investigations have added to our descriptive knowledge of the process. We consider first some experimental results, with the aim of relating reaction morphologies to the phase diagrams which describe the phase assemblages encountered. The conditions under which these morphologies develop are then established, and the kinetics described using Wagner’s diffusion analysis [8] and its explication by Rapp [9]. These descriptions are then extended to other, more complex situations, where the simplifying assumptions adopted by Wagner are no longer applicable. As always, our purpose is to understand the mechanisms of the processes, develop means of calculating their rates and ultimately arrive at means for their mitigation.

6.2. SELECTED EXPERIMENTAL RESULTS Typical reaction morphologies of internally oxidized alloys are shown in Figure 6.1, where chromium-rich oxide has precipitated inside an Fe–5Cr alloy reacted in Ar/H2/H2O atmospheres where the ambient pO2 value was too low for FeO to form. Clearly oxygen had dissolved in the alloy and diffused inwards to react with alloy solute chromium, precipitating its oxide. The depth of the precipitation zone, XðiÞ , is seen in Figure 6.1 to increase according to parabolic kinetics X2ðiÞ ¼ 2kðiÞ p t

(6.1)

where kðiÞ p is the internal oxidation rate constant. This is an almost universal observation [11, 12] and indicates that the process is diffusion-controlled. The effect of alloy chromium content is shown in Figure 6.2. Dilute alloys form only internal oxide, Fe–10Cr forms both external and internal oxide and Fe–17Cr forms only an external scale. A schematic phase diagram in Figure 6.3 illustrates diffusion paths corresponding to the steady-state morphologies of


249

Figure 6.1 Internal oxidation of Fe–5Cr at pO2 ¼ 8:7 1017 atm in Ar–H2–H2O; kinetics at 1,0001C. Reproduced from Ref. [10] by permission of The Electrochemical Society.

Figure 6.2a–d. The diagram has been constructed on the basis that pO2 is too low for any iron-bearing oxide, such as FeCr2O4, to form. Thus pure iron equilibrates directly with oxygen. Paths (a) and (b) show variation in oxygen content at a ~ AB , where oxygen fixed NCr/NFe ratio, and correspond to the situation DO D diffuses into the alloy so fast that chromium diffusion can be neglected. These paths represent local equilibrium situations, and do not encompass the supersaturation zones necessary to drive precipitate nucleation (Section 6.7). Path (c) represents simultaneous internal and external oxidation, and path (d) shows external scaling only. The chromium oxide precipitates shown in Figures 6.1 and 6.2 are dispersed and generally spheroidal in shape, although non-uniform in size. Moreover, the volume fraction of precipitate appears to vary somewhat with depth at higher temperatures, although it is approximately constant at 9001C. A very different

250


Figure 6.2 Change in oxide morphology with composition of Fe–Cr alloys exposed to pO2 ¼ 8:7 1017 atm in Ar–H2–H2O at 1,0001C. Reproduced from Ref. [10] by permission of The Electrochemical Society.

precipitate shape is obtained by internal nitridation, as seen in Figure 6.4. Lamellar precipitates of Cr2N have grown into the alloy, aligned approximately normal to the sample surface, i.e. in a direction parallel to that in which the reaction is proceeding. Clearly the competition between precipitate nucleation and growth has led to very different outcomes in the oxidation and nitridation reactions. It was observed in Chapter 5 that cold working an alloy surface by grinding introduced subsurface defects which accelerated alloy diffusion, making external scale formation by the selectively oxidized component more likely at moderate temperatures. As seen in Figure 6.5, Incoloy 617 (Table 2.1) forms a protective Cr2O3 scale when surface ground before reaction. However, when the deformed region is removed by chemical polishing, both internal and external oxidation develop during subsequent reaction. Of further interest is the finding that internal oxidation occurs preferentially at grain boundaries, rather than within the grains. Penetration along the grain boundaries involved oxidation of alloy


251

Figure 6.3 Schematic phase diagram for Fe–Cr–O with Cr2O3 as the only stable oxide. Diffusion paths (a)–(d) correspond to the reaction morphologies in Figure 6.2a–d.

Figure 6.4 Optical micrograph of lamellar Cr2N precipitates formed in Fe–20Ni–25Cr reacted at 1,0001C in N2–10%H2.

carbides, and was remarkably fast. It turns out that internal oxidation at grain boundaries is common in austenitic alloys. In a number of alloys, the selectively reacted component can form more than one product phase. A frequently encountered example is the precipitation of

252


(a)

(b)

Figure 6.5 Oxidation of IN 617 at T ¼ 7001C, pO2 ¼ 1 1023 atm in Ar–CO–CO2. (a) Grain boundary precipitation of Cr2O3 in material prepared by chemical polishing and (b) external Cr2O3 scale on material prepared by surface grinding.

chromium-rich carbides during carburization of heat-resisting alloys. Figure 6.6 shows a cross-section of carburized Fe–37.5Ni–25Cr where two precipitation zones have been revealed by their different response to stain etching. The carbides in the near-surface zone are chromium-rich M7C3, and those in the deeper zone are chromium-rich M23C6. Carburization reactions are discussed in detail in Chapter 9. As already indicated, diverse precipitate morphologies are possible. Further examples are shown in Figure 6.7. Strongly directional growth of alumina precipitates in the diffusion direction has occurred, Widmansta¨tten plates of


253

Figure 6.6 Internal carburization of Fe–37.5Ni–25Cr at 1,0001C in gas with ac ¼ 1, in H2–C3H6. Near-surface zone contains Cr7C3 precipitates and deeper zone contains Cr23C6. Etched with Murakami’s reagent.

Cr2N have developed and apparently lamellar, chromium-rich M23C6 has grown into an Fe–25Cr alloy. Questions of interest concern the factors controlling the predominance of precipitate growth over nucleation, what controls the orientation of the precipitate with respect to the metal matrix and the diffusion direction, and whether or not the aligned precipitate–matrix interfaces can provide preferred diffusion pathways for the oxidant, thereby accelerating the corrosion rate. Alloys can contain more than one component capable of internally precipitating an oxide. Oxidation of a model alloy Ni–3.5Cr–2.5Al led to the internal precipitation of both chromium- and aluminium-rich oxides, as shown in Figure 6.8, a cross-sectional image obtained by SEM. The image brightness is related to the average atomic number of the material being imaged. Thus the metal matrix, which is mainly nickel, is bright, the chromium-rich oxide is grey and the aluminium-rich oxide appears dark. Clearly, the more stable aluminiumrich oxide is precipitated to greater depth than the chromium-rich oxide. This reflects the gradient in oxygen activity from its maximum at the alloy surface to a minimum in the alloy interior. The conditions under which internal oxidation is possible can be specified in a general way, and are formulated here for a binary alloy AB. Internal precipitation of BOn can occur if this oxide is more stable than that of metal A, per mole of oxygen. Precipitation will occur if oxygen can dissolve in the alloy and diffuse inward so as to achieve an activity high enough to stabilize BOn , but not AO. The precipitates will be distributed internally rather than aggregating to form a scale if NB is sufficiently low. It is desirable to be able to specify the critical value of NB

254


Figure 6.7 Diverse precipitate morphologies resulting from internal oxidation reactions. (a) Oxidation of Ni–2.5Al at T ¼ 1,0001C, pO2 ¼ 4:6 1011 atm, in Ar–H2–H2O; (b) nitridation of Ni–15Fe–25Cr in N2–10%H2 at 1,0001C and (c) carburization of Fe–25Cr at 1,0001C in H2–C3H6 gas with ac ¼ 1, showing internal reaction front.

6.3. Internal Oxidation Kinetics in the Absence of External Scaling

255

Figure 6.8 Simultaneous internal oxidation of chromium and aluminium in Ni–3.5Cr–2.5Al at T ¼ 1,0001C, pO2 ¼ 9:8 1013 atm, in Ar–H2–H2O. Grey oxide is chromium-rich and dark oxide aluminium-rich. A pure nickel layer is present at the surface.

separating these two regimes of oxidation. As always, we wish to predict the rate of the process, and how it varies with material properties and environmental factors. As seen from the brief survey of experimental results, a full description of the process also involves predicting precipitate size, shape, orientation and distribution. The kinetics of internal oxidation are considered first, and a number of simple limiting cases are identified. The factors affecting precipitate nucleation, growth, morphology and distributions are then considered. Predictions for the transition between internal and external oxidation are then compared with experimental data. Finally, the effects of the volume expansion accompanying internal oxide precipitation are discussed.

6.3. INTERNAL OXIDATION KINETICS IN THE ABSENCE OF EXTERNAL SCALING We consider an alloy AB exposed to an oxygen potential high enough to react with B, but not with A, and suppose that alloy diffusion is negligible compared with inward oxygen movement. Internal oxidation will result if the oxygen solubility in the B-depleted alloy, expressed as a mole fraction, N ðsÞ O , and its diffusion coefficient, DO , are high enough. If, furthermore, the precipitate BOn is extremely stable, then the reaction zone is assumed to consist of precipitates embedded in a matrix of almost pure A. This assumption is based on the thermodynamics of the reaction B þn O ¼ BOn ;

DGP

(6.2)

256


discussed in Section 2.4. The solubility product for local equilibrium between precipitate and matrix DGP N B N nO ¼ Ksp ¼ exp (6.3) RT with DGP ¼ DGf ðBOn Þ DH B nDH O

(6.4)

is very small for a high-stability precipitate. Although it is not necessarily so, it was originally assumed [8, 9], and is often the case, that both N B and N O are very low throughout the precipitation zone, as represented in Figure 6.9. Thus oxygen diffuses through a metal matrix of almost pure A, between the BOn precipitates which have already formed, to reach the reaction front at a depth X(i), where more B is available for reaction. An approximate estimate of the internal penetration rate can be made from a mass balance at the reaction front. Reformulating the standard expression (5.29) for mass balance at a moving boundary in terms appropriate to the development of a two-phase zone, we can write ioz all all J ioz O J O ¼ nðCO CO Þ

(6.5)

where the superscripts ioz and all refer to the internal oxidation zone and the base alloy, and Cioz O the overall oxygen concentration in the oxide-plus-matrix two-phase region. Given the assumption that the oxygen concentration at the reaction front is zero, it follows that all Call O ¼ 0 ¼ JO

(6.6)

Approximating further that the oxygen flux J ioz O ¼ DO Gas

Alloy

NBOυ

CðsÞ CðXÞ @CO O DO O @x XðiÞ Gas

NB(0)

(6.7) Alloy

NBOυ (0 NBB(0)

N0(s) N0

N0(s) N0

(a)

Xi

(b)

Figure 6.9 Schematic representation of internal precipitation of a very stable oxide and the reactant concentration profiles: (a) component B immobile and (b) both oxygen and B diffuse.


257

where the superscripts (s) and (X) denote the boundary values of the oxygen concentration at the surface and internal limit of the precipitation zone. Setting ðXÞ CO ¼ 0 (Figure 6.9) and substituting Equation (6.7) in Equation (6.5) yields DO CðsÞ dXðiÞ iox O C ¼ XðiÞ dt O

(6.8)

Integration of Equation (6.8) and substitution from the stoichiometric relationship ðoÞ Ciox O ¼ nCB

where with

CðoÞ B

(6.9)

is the original alloy concentration of B, then leads to Equation (6.1), kðiÞ p ¼

DO CðsÞ O

(6.10)

nCðoÞ B

It is usually assumed that the molar volumes of the alloy and the matrix A are the same, and hence kðiÞ p ¼

DO N ðsÞ O

(6.11)

nN ðoÞ B

This simple result is intuitively reasonable in that it reflects the fact that the penetration rate is proportional to oxygen permeability, N ðsÞ O DO , and inversely proportional to the concentration of reactant metal. It should be noted that it has been assumed that the oxide precipitates do not interfere with inward oxygen diffusion. A more rigorous and less restrictive analysis has been provided by Wagner [8, 9], allowing for the possibility that component B also diffuses. The diffusion model is shown schematically in Figure 6.9b. Again it is assumed that Ksp is extremely small, and that both N O and N B are vanishingly small at the reaction front. The problem then is to solve the diffusion equations for both B and O: @N i @2 N i ¼D 2 @t @x

(6.12)

for the boundary conditions N O ¼ N ðsÞ O NO ¼ 0

t40

(6.13)

x XðiÞ ;

t40

(6.14)

for x40;

t¼0

(6.15)

x XðiÞ ;

t40

(6.16)

for

N B ¼ N ðoÞ B NB ¼ 0

x ¼ 0;

for

for

The solutions are NO ¼

N ðsÞ O

pffiffiffiffiffiffiffiffiffi erf x=2 DO t 1 erf g

(6.17)

258


( NB ¼

N ðoÞ B

1

pffiffiffiffiffiffiffiffi) erfc x=2 DB t erfcðgf1=2 Þ

for parabolic kinetics, where Equation (6.1) applies, with !1=2 kðiÞ p g¼ 2DO

(6.18)

(6.19)

and f¼

DO DB

(6.20)

Wagner dealt with the mass balance at the reaction front ðx ¼ XðiÞ Þ by supposing that all precipitation took place at this location, and therefore the fluxes of O and B towards the interface were equivalent: @N O @N B DO ¼ nDB (6.21) @x x¼XðiÞ @x x¼XðiÞ þ Here e is a very small increment in x, used to indicate that the fluxes are evaluated very close to, but on opposite sides of the reaction front. Substitution from Equations (6.17) and (6.18) into Equation (6.21) leads, after differentiation, to N ðsÞ O N BðoÞ

¼

expðg2 Þerf g f1=2 expðg2 fÞerfcðgf1=2 Þ

(6.22)

The quantity g, and hence kp , can be evaluated numerically from this equation. In the special case where N ðsÞ DB O 1 DO NB

(6.23)

then g 1 and gf1=2 1, and Equation (6.22) can be accurately approximated by !1=2 N ðsÞ O (6.24) g 2nN ðoÞ B Substitution of this result into Equation (6.19) then yields the simple result (6.11). Inspection of Equation (6.23) reveals that the required condition amounts to a high oxygen permeability relative to any B diffusion, which was the basis for the derivation of Equation (6.10), and is represented by Figure 6.9a. If, however, diffusion of B is important, another special case can arise if N ðsÞ O N ðoÞ B

DB 1 DO

(6.25)

In this case, g 1 and gf1=2 1, and Equation (6.22) can be approximated by g

p1=2 f1=2 N ðsÞ O 2nN ðoÞ B

(6.26)


which, when combined with Equations (6.19) and (6.20), yields ! ðsÞ 2 p DO N O ðiÞ kp ¼ DB 2nN ðoÞ B

259

(6.27)

This is the situation represented by Figure 6.9b, and corresponds to enrichment of B within the precipitation zone as a result of its rapid diffusion from within the alloy towards the surface. In distinguishing the two limiting cases represented by Equations (6.11) and (6.27) it is necessary to evaluate the oxidant permeability N ðsÞ O DO and the corresponding alloy quantity, N BðoÞ DB . The oxidant solubility is related to the surface oxygen activity via Sievert’s Equation (2.71). The maximum value of pO2 available to a bare alloy surface is that at which component A forms an external scale. Thus, for example, internal oxidation of Fe–Cr is limited to a maximum N ðsÞ O value given by 1=2 N ðsÞ O ¼ K½pO2 ðFeOÞ

(6.28)

where K is the Sievert’s law constant for O in iron. To avoid the complications of scale formation (see Section 6.12), it is common to study internal oxidation by controlling pO2 at the level set by the A/AO equilibrium. This is conveniently done using a ‘‘Rhines pack’’ [4]: a sealed, evacuated capsule containing a large quantity of powdered metal A mixed with its lowest oxide, along with the AB alloy sample. Alloy solubility data for oxygen shown in Table 6.1 are calculated from Table 2.2. Their use is based on the supposition that all of the reactive alloy components are precipitated near the surface, and oxygen solubility in the remaining, almost pure iron or nickel is set by the Rhines pack condition. Data for both ferritic and austenitic iron are provided, for reasons which are now discussed. Table 6.1 Alloy

Fe–Cr

a

Permeability data for internal oxidation in Rhines packsa T (1C)

1,000

Fe–Al

1,000

Fe–Si

1,150

Ni–Cr Ni–Al Ni–Si

1,000 1,200 1,000

In alloy ABb

In matrix A NðsÞ O

DO (cm2 s1)

4.5 106(a) 3.3 106(g) 4.5 106(a) 3.3 106(g) 1.5 105(a) 9.0 106(g) 4.8 104 9.4 104 4.8 104

3.5 106(a) 7.3 107(g) 3.5 106(a) 7.3 107(g) 9.3 106(a) 3.9 107(g) 9.1 109 7.5 108 9.1 109

DB (cm2 s1)

NðoÞ B

0.054 1.5 1011(g) 4.2 1010(a)

0.020

3.0 1.9

6.3 109(a)

0.016

1.4

7.2 1012 1.0 109 3.9 1011

0.056 0.043 0.016

10.8 1.6 7

Described in Section 6.3. Alloy compositions chosen to match examples studied experimentally.

b

ðoÞ NðsÞ O DO =NB DB

260


Diffusion in ferritic alloys is complicated at certain temperatures by the appearance of a gðfccÞ-phase. Reference to the phase diagrams for Fe–Cr, Fe–Al and Fe–Si in Figure 6.10 shows that all alloys, when sufficiently dilute, are austenitic at temperatures of about 900–1,4001C. Consider, for example, an alloy of original composition Fe–15Cr, which at T ¼ 1,0001C is ferritic. Internal oxidation removes most of the chromium from the metal phase, shifting its composition into the g-region. If the small concentration of dissolved oxygen can be ignored, the diffusion path in the metal region of the reacting alloy is as shown in Figure 6.10a. For this reason, it is appropriate to consider oxygen dissolution and diffusion through austenite. The a ! g transformation can be suppressed [13] by the addition of an unreactive ferrite stabilizer such as tin, and data for ferrite is also provided in Table 6.1. Similarly, data for DB (calculated from data in Appendix D) in both a- and g-Fe are provided, where available. ðoÞ It is seen in Table 6.1 that for the conditions chosen, DO N ðsÞ O 4DCr N Cr and the conditions for Equations (6.24) and (6.11) are met. Even in the case of much more mobile silicon and aluminium, the conditions are close to being realized, and (6.11) is expected to provide a reasonable approximation. In this situation, the internal oxidation process is controlled by inward oxygen diffusion, and counter diffusion of the alloy solute metal can be ignored. If, however, counter diffusion of the reacting metal is important, then it will enrich in the internal oxidation zone as additional oxide precipitates. Such a situation can be expected during oxidation at very low pO2 values, when the oxygen permeability is consequently lowered. Wagner [8] also calculated the degree of solute enrichment in the precipitation zone. Defining f BO as the mole fraction of BOn precipitate in the internal oxide zone, an enrichment factor a¼

f BO N ðoÞ B

is identified, and was evaluated by Wagner as h i1 a ¼ p1=2 u expðu2 Þerfc u

(6.29)

(6.30)

with u ¼ gf1=2 . Under the limiting conditions of Equation (6.25), this result can be approximated as a ¼ p1=2 u ¼

2nN ðoÞ B DB pN ðsÞ O DO

(6.31)

6.4. EXPERIMENTAL VERIFICATION OF DIFFUSION MODEL As already mentioned, internal oxidation almost invariably follows parabolic kinetics. The applicability of the simple form (6.11) is first investigated. One obvious and useful prediction from this equation is that for a given solvent A, the rate of internal oxidation is independent of the chemical identity of B, and is

6.4. Experimental Verification of Diffusion Model

Figure 6.10 Phase diagrams for (a) Fe–Cr, (b) Fe–Al and (c) Fe–Si showing g-phase regions, and diffusion path in metallic part of internally oxidized Fe–Cr (see text).

261

262

Figure 6.10


(Continued ).

determined solely by the permeability of oxygen in A, together with the oxide stoichiometry. If correct, this provides a method for measuring oxygen permeability. Alloys based on silver provide a good test of this possibility, because Ag2O is unstable at high temperatures, and reliable, independent measurements of N ðsÞ O and DO are available [14]. Values of DO N ðsÞ O derived from measurements of X i as a function of time (Equations (6.1) and (6.11)) have been collected by Meijering [11] and are compared in Figure 6.11 with independent permeability measurements [14] which yielded 107:2 kJ mol1 ðsÞ 4 N O DO ¼ 2:4 10 exp (6.32) cm2 s1 RT Agreement is seen to be good. It may be concluded that, at least for the dilute alloys involved here, the assumption that oxide precipitates do not interfere with oxygen diffusion is reasonable. The internal oxidation of silver alloys is of more than academic interest: the process is used to provide hardness in silver-based electrical contact materials. Good-quality data for oxygen permeability in nickel have been provided by Park and Altstetter [22], using solid-state electrochemical techniques to measure independently 164 kJ mol1 DO ¼ 4:9 102 exp (6.33) cm2 s1 RT


263

Figure 6.11 Permeability of oxygen in silver deduced from internal oxidation kinetics in 1 atm O2 of: ’ Ag–1.3Zn [6], & Ag–1.0Mg [6], K Ag–1.75Mg [15], Ag–1.8Al [15], E Ag–1.0Cd [15], B Ag–0.95Cd [16], Ag–4.8Cd [17], X Ag–1.7Li [18], + Ag–0.3Pb [19] and J Ag–In alloys [20]. Continuous line represents Equation (6.32). Published from Ref. [11] with permission from Wiley.

4

2 N ðsÞ exp O ¼ 8:3 10

7

55 kJ mol1 RT

(6.34)

for pO2 set by the Ni/NiO equilibrium. Internal oxidation kinetics for various nickel-based alloys have been used to deduce the oxygen permeability values shown in Figure 6.12. Agreement with Equations (6.33) and (6.34) is seen to be reasonable. It should be noted that permeabilities deduced from internal ðoÞ oxidation of Ni–Al alloys were in fact a function of N Al , as will be discussed later. The values shown in Figure 6.12 were obtained [21] by extrapolating to ðoÞ N Al ¼ 0. Another prediction available from Equation (6.10) is that for a given matrix A, and fixed T and pO2 , the rate constant for internal oxidation is inversely proportional to N ðoÞ B . Internal oxidation rates for a series of Fe–Cr alloys [23] are seen in Figure 6.13 to vary with 1=N ðoÞ Cr as predicted. Internal oxidation depths observed in Ni–Cr [21] and Cu–Si [24] alloys are seen in Figure 6.14 to vary as predicted from Equations (6.1) and (6.11), i.e. X2ðiÞ / 1=N ðoÞ B . As we have seen, the Wagner diffusion theory achieves considerable success in quantitatively accounting for internal oxidation rates. The theory also applies

264


log10(N0(s)D0 /cm2s-1)

-10

-11

-12

-13

-14

6

7

8 104K/T

9

10

Figure 6.12 Permeability of oxygen in nickel deduced from internal oxidation kinetics under Rhines pack conditions: ’ Ni–Cr [21], Ni–Al [21], J Ni–0.12Al [11]. Continuous line represents NðsÞ O DO according to Equations (6.33) and (6.44). Published from Ref. [11] with permission from Wiley.

3

Figure 6.13 Internal oxidation rates for Fe–Cr alloys at pO2 ¼ 8:7 1017 atm (E) and 2.6 1026 atm (’) values [23]. Published with permission from Trans Tech Publications.

to internal attack by other oxidants, although reaction rates can be very different because of the different permeabilities. Some comparative data in Table 6.2 illustrate this point. The corresponding internal precipitation reaction rates are shown in Table 6.3. The data are plotted according to Equation (6.10) in Figure 6.15, using logarithmic scales to encompass the large ranges of values. The slope is close to unity, confirming that Equation (6.10) provides a very useful predictive tool.


265

24 20

10−3(Xi / m)2

16 12 8 4 0 0

20

40

80

60 (0) 1/ NCr (a)

100

100

10-10kp (cm2s-1)

80

60

40

20

0 0

200

400

600

800

1000

1200

1/Nsi(0) (b)

Figure 6.14 Internal oxidation depth as a function of alloy solute content. (a) Ni–Cr alloys in Rhines pack at 1,0001C for 10 h [21] (Published with permission from Taylor & Francis Ltd., http://www.tandf.co.uk/journals). (b) Cu–Si alloys in Rhines pack at 7501C for 100 h [24] (Published with permission from The Minerals, Metals & Materials Society).

Table 6.2 Comparative permeabilities (cm2 s1) for different oxidants at 1,0001C: oxygen in Rhines packs, carbon at ac ¼ 1 and nitrogen at pN2 ¼ 1 atm

a

Solvent metal

NðsÞ O DO

Ni g-Fe

4.3 1012 2.4 1012

a

Oxygen permeability data from Chapter 2.

NðsÞ N DN

NðsÞ C DC

1.5 1011 [25, 26] 1.6 1011 [25, 26]

3.1 109 [27, 28] 1.4 108 [27, 28]

266


Table 6.3 Comparative internal oxidation, nitridation and carburization rate constants 2 1 at 1,0001C kðiÞ p /cm s Oxidationa

Nitridationb

Carburizationc

a-Fe–5Cr 5 1010 (8.7 1017 atm) [23] 2.4 107 [23] 10 11 (4.7 10 atm) [29] g-Ni–5Cr 1 10 g-Fe–20Ni–25Cr 6.6 109 [30] 1.1 107 [30] a

At indicated pO2 values. At pN2 ¼ 0:9 atm. c At ac ¼ 1. b

-6

C

Ferrite

log 10(kp(i))

-7

Austenite

N

-8

-9

C

O O

-10

-11 -11

-10

-9

-8

-7

-6

log 10(Ni(s)Di/υNCr (0))

Figure 6.15 Internal precipitation reaction rates for different oxidants in ferritic and austenitic alloys under reaction conditions specified in Table 6.3.

Despite the considerable successes of the Wagner diffusion model in describing internal precipitation reactions in the absence of any external scale, its applicability is limited by the assumptions on which it is based. The assumptions which may prove incorrect for some reacting systems are as follows: (a) The precipitate is extremely stable, and both N O and N B are vanishingly small within the precipitation zone. (b) As a consequence of (a), f BO is constant throughout the precipitation zone, and changes discontinuously to zero at the reaction front. (c) Precipitate nucleation and growth have no effect on overall reaction kinetics. (d) Mass transfer within the internal oxidation zone occurs solely via lattice (bulk) diffusion, is unaffected by the presence of precipitates and is not subject to cross-effects resulting from kinetic or thermodynamic interactions with other solutes.

6.5. Surface Diffusion Effects in the Precipitation Zone

267

We consider first the effect of precipitates, and microstructure in general, on oxidant diffusion, while retaining the assumptions of a highly stable precipitate, and a matrix which is strongly depleted in reactive solute B.

6.5. SURFACE DIFFUSION EFFECTS IN THE PRECIPITATION ZONE As seen in Figures 6.1 and 6.5, internal oxidation can be favoured at alloy grain boundaries. The situations in the two cases depicted are quite different. Although the precipitates formed on grain boundaries in Fe–Cr are larger, the penetration depth is the same as within the grains themselves, and the overall reaction kinetics are not affected. The austenitic alloy IN 617, however, has undergone rapid, preferential intergranular attack, forming a continuous internal oxide network along the grain boundaries. Preferential intergranular penetrations of internal oxide have been observed for Ni–Al [31–34] and Ni–Cr alloys [35, 36], to an extent which becomes more marked at lower temperatures and higher N ðoÞ B values. Intergranular morphologies of internal oxidation were reported earlier for Fe–Al [37], tin-based alloys [5] and copper-based alloys [4]. A related phenomenon is the in situ oxidation of prior interdendritic carbides in cast materials [38] shown in Figure 6.16. Intergranular oxidation can be much faster than the rate at which the intragranular precipitation front advances. The parabolic rate constant for intergranular oxidation in Ni–5Cr at 1,0001C is found [21] to be about 103 times the value of kðiÞ p . Similarly, the rate of in situ carbide oxidation in cast heatresisting steels (Figure 6.16) is observed [38, 39] to be much faster than intragranular precipitation. Clearly, these rapid rates cannot be sustained by volume (lattice) diffusion of oxygen, and a faster transport process must be involved. A model based on diffusion along the oxide–metal grain boundary is shown schematically in Figure 6.17 for the case of in situ carbide oxidation. A very similar situation arises when intragranular precipitates form with elongated plate or rod shapes, aligned in the growth direction (Figure 6.7). The example of elongated Al2O3 precipitate growth in dilute Ni–Al alloys has been studied intensively [21, 29, 40–46], leading to an understanding of the diffusion processes involved in the growth of these cellular morphologies. The kinetics of internal oxidation are parabolic, reflecting diffusion control, but the rate ðoÞ constant is independent of N Al . The behaviour of these alloys is compared with that of Ni–Cr in Figure 6.18. Clearly, the data for Ni–Cr alloys conform with Equation (6.11), but that for Ni–Al does not. If, nonetheless, effective oxygen permeability values are deduced from Equation (6.11), they are found [29] apparently to increase with aluminium levels ðoÞ N ðsÞ O DO ðeffÞ ¼ a þ b N Al

(6.35)

where a and b are constants. This is interpreted to mean that oxygen diffuses both through the metal matrix and along precipitate–matrix interfaces, the concentration of the latter being proportional to the original alloy aluminium content. On this basis, the effective flux of oxygen through a precipitation zone containing

268


Figure 6.16 Rapid penetration of internal oxide along prior carbide network in cast Fe–35Ni–27Cr alloy at T ¼ 1,0001C [39]. With kind permission from Springer Science and Business Media.

lath-shaped oxides oriented as shown in Figure 6.19 can be written as J eff ¼ J O AO þ J i Ai þ J OX AOX

(6.36)

where AO , Ai and AOX are the cross-sectional area fractions of alloy, alloy-oxide interface and oxide, normal to the diffusion direction. Because diffusion in Al2O3

269


Scale

Alloy

2 ro

Cr2O3

Cr23C6

2 rc

X

NO(s)

No (e)

NO

Figure 6.17 Schematic model for enhanced internal boundary oxidation of prior carbide in situ [38]. With kind permission from Springer Science and Business Media.

200 Ni-Al Ni-A 200

Depth (μm)

Depth (μm)

Ni-Cr

100

100

pack pack

0

0 1

2

3 4 wt % Cr (a)

5

1

2

3

4

wt % Al (b)

Figure 6.18 Extent of internal oxidation of Ni–Cr and Ni–Al alloys in Rhines packs for 20 h at 1,0001C [21]. Published with permission from Taylor & Francis Ltd., http://www.tandf.co.uk/ journals.

270


d W

XB

ξ Surface

Front of internal oxidation Allo Alloy

Figure 6.19 Schematic view of oriented Al2O3 laths in internal oxidation zone [21]. Published with permission from Taylor & Francis Ltd., http://www.tandf.co.uk/journals.

is so slow, the third term is set at zero. The effective oxygen diffusion coefficient is then defined as DO;eff ¼ DO;O AO þ DO;i Ai

(6.37)

where DO;O is the usual diffusion coefficient of oxygen in nickel and DO;i the interfacial coefficient. The area fractions and diffusion coefficients are assumed to be independent of position within the internal oxidation zone. The mole fraction of oxide, N BO , is related to the precipitate dimensions and their number density, FN. Using the dimensions w and d specified in Figure 6.19, and assuming that the precipitates are continuous across the full width of the internal oxidation zone, we write for small values of FN V All N BO ¼ FN wd (6.38) VOX and Ai ¼ 2ðw þ dÞFN di 2wFN di

(6.39)

where di is the width of the interface diffusion zone and the approximation is based on w d. The cross-section of matrix metal remaining after oxide precipitation is AO ¼ 1 Ai AOX ¼ 1 2wFN di FN wd

(6.40)

which, upon substitution along with Equations (6.38) and (6.39) into Equation (6.37) yields

DO;eff DO;i di 2 V OX 1 ¼1þ N BO (6.41) DO;O DO;O d VAll A similar result is obtained for rod-shaped precipitates [21, 29] and indeed will be found for any prismatic precipitate morphology. If no aluminium enrichment occurs, the amount of oxide corresponds to the original alloy concentration, N BO ¼ N ðoÞ Al , then the form of Equation (6.41) is seen to correspond with the experimental result (6.35). Comparison of experimentally determined values for b with the corresponding term in Equation (6.35) yields the results shown in Table 6.4.

271


Table 6.4

a

Interfacial and matrix oxygen diffusion in internally oxidized Ni–Al [21]a

T (1C)

DO,idi/DO,Od

DO,i/DO,O

1,100 1,000 900 800

39 85 85 173

3.9–39 102 8.5–85 102 8.5–85 102 1.7–17 103

DO,i/DO,O calculated for d ¼ 10–100 nm and di assumed to be 1 nm.

The ratios between interfacial and lattice diffusion coefficients of oxygen seem reasonable, and increase with decreasing temperature as would be expected. If the interfaces concerned are incoherent, as was assumed [21], then the chemical identity of the oxide will be of secondary importance, and a similar enhancement in oxygen diffusion can be anticipated for any oxide–austenite interface. The example of in situ oxidation of interdendritic chromium carbide (Figure 6.16) is now analysed on this basis. As is clear in the micrograph, oxygen penetration at the interdendritic locations was much faster than within the austenite grains, where only a shallow internal oxidation zone had formed. Oxidation of a rodshaped carbide is shown schematically in Figure 6.17. The chemical reaction 23 Cr23 C6 þ 69 2 O ¼ 2 Cr2 O3 þ 6 C

(6.42)

is accompanied by a volume expansion. If accumulation of chromium from the surrounding metal matrix can be ignored, the rod radii are related by rO ¼ krC where the subscripts denote oxide or carbide, and k is the ratio 11:5V OX k¼ VC

(6.43)

(6.44)

with V i the molar volume of the indicated substance. In the figure, N O denotes the local concentration of oxygen, and the zone of rapid inward interfacial diffusion is defined as an annular region, of width d, around the oxide rod. Boundary values of the oxygen concentration are set at the ðeÞ alloy–scale interface, N ðsÞ O , and by local carbide–oxide equilibrium, N O . The molar flux of oxygen, J O , per unit cross-sectional area of carbide rod is given by the linear approximation to Fick’s Law as JO ¼

ðsÞ ðeÞ dð2rO þ dÞ DO;i ðN O N O Þ VA XB r2C

(6.45)

where r is the rod radius, DB the boundary or interfacial diffusion coefficient, V A the alloy matrix molar volume and XB the boundary oxidation depth. If this oxygen is entirely consumed in reaction with the carbide rod, then the resulting oxide rod lengthens at a rate given by dXB J ¼ O VC dt 69=2

(6.46)

272


Combination of Equations (6.43)–(6.46) leads to ðsÞ ðeÞ dXB 2 dDO;i V OX N O N O ¼ 3 rC V A dt XB

(6.47)

for the case d rC . Integration of Equation (6.47) leads to X2B ¼ 2kðiÞ B t

(6.48)

where the parabolic rate constant for internal oxidation kðiÞ B ¼

2 dDO;i V OX ðsÞ ðN O N ðeÞ O Þ 3 rC V A

(6.49)

is independent of primary carbide volume fraction, but inversely proportional to carbide diameter. A similar conclusion is reached if other prismatic carbide shapes, such as uniform sheets, are chosen. Inward oxygen diffusion along phase boundaries according to Equation (6.49) explains the observation [38] that several heat-resisting alloys all had approximately the same internal oxide penetration rates, despite their considerable variations in composition. Comparing the interdendritic oxidation rate constant of Equation (6.49) with the normal bulk material value of Equation (6.10), we obtain kðiÞ B kðiÞ p

¼

N ðoÞ Cr V OX di DO;i rC V A DO;O

(6.50)

ðsÞ ðiÞ if N ðeÞ O N O . The value of kB measured for an austenitic Fe–35Ni–27Cr cast steel 11 2 1 18 at 1,0001C was 3 10 cm s compared with the value kðiÞ cm2 s1 p ¼ 6 10 expected for lattice diffusion under the same conditions. Substitution of these values in Equation (6.50) together with rC ¼ 2 mm and N ðoÞ Cr ¼ 0:29 leads to the estimate di DO;i =DO;O ¼ 800. This is similar to estimates of boundary diffusion along Al2O3–austenite interfaces (Table 6.4). As seen in Table 6.4, enhancement of oxygen diffusion at boundaries is of decreased importance at higher temperatures. Hindam and Whittle [42] showed that at 1,2001C, lath or rod-shaped precipitates (depending on N ðoÞ Al ) grew into dilute Ni–Al alloys according to parabolic kinetics, but at rates which were controlled by oxygen diffusion through the matrix. Thus it can be concluded that boundary diffusion of oxidant is not a necessary condition for the development of a cellular precipitation morphology. Because the elongated precipitate–matrix interfaces have the effect of accelerating internal attack at lower temperatures, the question of how to predict their formation is an important one to which we return in Section 6.8. Finally, it should be noted that the precipitates formed during internal oxidation of dilute chromium and aluminium alloys are in fact more complex than has been implied. In both cases a spinel phase, MCr2O4 or MAl2O4, is formed near the surface if the oxygen activity is high enough. The binary oxide, M2O3, forms in a second, deeper precipitation zone. The general question of multiple precipitation zones is discussed in Section 6.9. When the internally formed precipitates are small and disperse, their surfaces cannot provide any significant contribution to diffusion. However, their presence

6.6. Internal Precipitates of Low Stability

273

reduces the metal matrix cross-section available for diffusion, as stated in Equation (6.40). For this reason, it is common to rewrite Equation (6.10) as kðiÞ p ¼

DO N ðsÞ O nN ðoÞ B

(6.51)

where e is an empirical constant designed to take into account the diffusional blocking effect of the precipitates. The quantity e would be expected to be related to f BO , but no information is available on this point, possibly because precipitate fractions are often small, and 1.

6.6. INTERNAL PRECIPITATES OF LOW STABILITY The Wagner diffusion model assumes Ksp to be vanishingly small and both N O and N B extremely dilute within the precipitation zone, which is therefore essentially oxide embedded in pure solvent metal A. However, this is not a realistic description for many cases. Consider precipitation of chromium compounds within an alloy a Cr þna X ¼ Cra Xna

(6.52)

Ksp ¼ N aCr N na X

(6.53)

where X is a generic oxidant and Ksp the equilibrium solubility product. If Ksp is not extremely small, then the necessarily low values of N X mean that N Cr will not always be small, and the assumption of complete precipitation fails. Even for rather stable precipitates such as Cr2O3, this condition can be difficult to meet at low oxidant activities. Values for Ksp are calculated for Equation (6.52) using the free energies of compound formation and alloy dissolution Cr þ nX ¼ CrXn

(6.54)

Cr ¼ Cr

(6.55)

1 2X2

(6.56)

¼X

using tabulated values [47] for oxide and carbide formation, together with Rosenqvist’s data [48] for DGf ðCr2 NÞ. Measured carbon [28] and oxygen (Table 2.2) solubility data are available, but the situation for nitrogen is less clear. Although the expression DGN2 ðg FeÞ ¼ 5690 þ 118:6T þ 2RT ln N N J mol1

(6.57)

is available [47], it is recognized that no accurate data are available for Ni–N solutions. Following Savva et al. [49] in conjecturing a temperature insensitive solubility of 1 ppma, we find DGN2 ðNiÞ ¼ 292; 460 þ 2RT ln N N J mol1

(6.58)

274


These estimates together with partial molar free energies of solution of chromium in iron [47] lead to the precipitate stability data shown in Tables 6.5 and 6.6. The quantity N Cr;min in the tables is the minimum concentration (mole fraction) of chromium required in the matrix metal to stabilize the designated precipitate at the alloy surface where N X has its maximum value of N ðsÞ X . Clearly the assumption of complete precipitation is in considerable error for the chromium carbides and nitride. Even the rather stable oxide precipitates leaving a significant concentration of chromium in the alloy. The temperature effect is significant, both through the decreased oxide stability at higher temperature and the retrograde oxygen solubility. The calculated results of Table 6.6 correspond to greatly decreased extents of chromium precipitation at higher temperatures. This effect is apparent in the internal oxidation of Fe–Cr alloys (Figure 6.1), where the volume fraction of oxide decreases substantially at higher temperatures. It is recognized that the calculated N Cr;min values apply at the alloy surface. As depth within the precipitation zone increases, N O must decrease, and therefore the concentration of chromium in the matrix, N Cr , must increase, in order to stabilize the precipitate, according to Equation (6.53). The amount of oxide precipitated, N BO , must therefore be a function of position, decreasing from a maximum at the alloy surface to a minimum at the reaction front. In view of this, it is necessary to investigate the effect of incomplete precipitation on the practically important quantity: the rate at which the internal oxidation front advances. Qualitatively, the consequence is clear. A lower value of fBO reflects, in

Table 6.5

Chromium compound precipitate solubilities at 1,0001C in g-Fe Cr2O3

N ðsÞ X Ksp NCr,min

Table 6.6 T (1C)

Carbides (ac ¼ 1)

pO2 ¼ 8:7 1017 atm

pO2 ¼ 2:6 1020 atm

Cr7C3

3.5 106

6 108

0.066

25

1.4 10 6 105

1.4 10 0.02

25

3.8 10 0.03

Nitride

Cr23C6

Cr2N at pN2 ¼ 1 atm

1 103

0.066 15

3.6 10 0.14

27

3 105 0.17

Cr2O3 solubilities in g-Fe at low pO2 values (atm) 900

1,000

1,100

pO2 ðatmÞ

8.7 1017

2.6 1020

8.7 1017

2.6 1020

8.7 1017

2.6 1020

N ðsÞ O Ksp NCr,min

6.8 106

1.2 107

3.5 106

6 108

1.6 106

2.8 108

1.1 1027 2 106 8 104

1.4 1025 6 105 0.02

8.6 1024 1 103 0.62


275

effect, a reduced availability of chromium, i.e. an effectively lower value of N BO in Equation (6.10), and hence larger values of kðiÞ p . The precipitate volume fraction varies with position, reflecting the changing values of N O and N Cr . Figure 6.20 shows schematic concentration profiles and a

M - Cr

M + Cr2N

NCr(0)

NN(s)

%N

(a)

Cr 2N

M %Cr (b)

Figure 6.20 Formation of low-stability Cr2N precipitates. (a) Concentration profiles and (b) diffusion path for DN DCr O.

276


diffusion path for the case of Cr2N precipitation. The equilibrium fraction of precipitate can be related to composition via the lever rule: N BO ¼

N ðoÞ B NB p NB NB

(6.59)

p

where N B and N B refer to local values in the precipitate and matrix, respectively, and negligible diffusion of B has been assumed. Defining a precipitate fraction r, normalized to its value at the alloy surface r¼

N ðoÞ B NB ðsÞ N ðoÞ B NB

(6.60)

and recognizing that the local equilibrium is described by combining Equations (6.53) and (6.60), we obtain NB ¼ 1 ar (6.61) N ðoÞ B with the solubility parameter a¼1

K1=a sp ðsÞ 1=v N ðoÞ B ðN X Þ

(6.62)

For Wagner’s Equation (6.10) to apply, precipitation must be uniform and complete, i.e. r ! 1 and N B ! 0. From Equation (6.61) it is seen that this requires ðsÞ an a a ! 1, a condition met when Ksp ðN ðoÞ B Þ ðN X Þ , but which will not be met for chromium carbide or nitride. The diffusional kinetics of this situation were analysed by Kirkaldy [50] and independently by Ohriner and Morral [51], and have been applied to the specific case of Cr2 N in Fe–Cr [52]. Assuming still that metal diffusion is unimportant and that KSPN 3N , one obtains

@r 4Ksp DN @ 1 @r ¼ (6.63) 3 @x @t ð1 arÞ2 @t ðN ðoÞ Cr Þ This equation can pffiffibe converted via the Boltzmann transformation (Section 2.7), l ¼ x= t, to an ordinary differential equation which upon integration yields Z 0 N ðoÞ ð1 aÞ2 dx r Cr ¼ x dr (6.64) dr o ð1 arÞ2 8DN tN ðsÞ N where r0 is the value in the interval [0, 1] chosen for evaluation. The variation in Cr2N volume fraction f n , with depth in an internally nitrided alloy is shown in Figure 6.21. The value of f n decreases approximately linearly with depth, and is everywhere much lower than the stoichiometric equivalent of N ðoÞ Cr . Also shown in the figure is the value of f n calculated from TEM-EDAX measurements of N Cr as a function of depth in the matrix of the internal nitridation zone. This calculation is based on the assumption that chromium diffusion is negligible, and the difference ðN ðoÞ Cr N Cr Þ is therefore equivalent to


277

Figure 6.21 Nitride volume fractions in internally nitrided Fe–20Ni–25Cr at 1,0001C compared with stoichiometric equivalent of NðoÞ Cr , and as calculated from mass balance assuming no Cr diffusion.

the amount of nitride precipitated. Agreement is seen to be excellent, confirming that chromium diffusion can be neglected. Application of Equation (6.64) requires knowledge of several parameters. Unfortunately, the assumption of ideal solution behaviour, i.e. N ðsÞ N afðN Cr Þ is incorrect, as is discussed in Section 6.10. For the moment, however, it is 10 sufficient to use the effective permeability N ðsÞ cm2 s1 at N DN ¼ 8:8 10 1,0001C and pN2 ¼ 0:9 atm, as deduced from internal nitridation kinetics [53]. Solution of Equation (6.64) using this permeability value and the measured r ¼ rðxÞ in Figure 6.21 yields a ¼ 0.82. The corresponding value of Ksp ðCr2 NÞ is then calculated from Equation (6.62) using the nitrogen solubility N ðsÞ N . If the effect of residual chromium on nitrogen solubility is ignored, then a value of Ksp ¼ 6 107 results. The values calculated thermodynamically from the method of Equations (6.54)–(6.56) are 3 105 in g Fe and 2 108 in nickel. The agreement between the value deduced from the precipitate distribution in Equation (6.64) and the expected range for thermodynamic equilibrium is good. The semi-quantitative success of the diffusion model implies that local equilibrium in the metal matrix (as expressed by Equation (6.53)) is maintained by steady-state diffusion of dissolved nitrogen, and the local extent of precipitation is therefore controlled by the precipitate–matrix equilibrium (Equation (6.60) and Figure 6.20). In short, the precipitate distribution is controlled by the diffusion path, i.e. the diffusion coefficients and the phase diagram, and not by nucleation phenomena. The extent to which internal nitrogen penetration exceeds the predictions of Equation (6.11) depends on the deviation of r from the ideal value of 1, i.e. on a. Ohriner and Morral [51] have calculated that for a ¼ 0.8 the quantity X=t1=2 exceeds the model prediction by a factor of approximately 1.7. This corresponds

278


to an increase in kp by a factor of about 3. Experimentally measured [53] values of kp are in fact up to 5 times faster than predicted from Equation (6.11). The additional acceleration is due to higher N ðsÞ N values enhanced by a thermodynamic interaction with solute chromium. As we have seen, the Kirkaldy/Morral theory succeeds in describing the distribution of low-stability precipitates. To gain an understanding of why precipitate sizes and number densities vary with position within the internal oxidation zone, it is necessary to examine the process of precipitate nucleation and growth.

6.7. PRECIPITATE NUCLEATION AND GROWTH It has been assumed so far that the internal oxidation front corresponds to the position where the equilibrium (6.2) is just satisfied. However, new precipitates cannot form if Equation (6.3) is precisely obeyed. To nucleate a new precipitate, an excess of oxidant is required 1=n Ksp NO4 (6.65) NB to drive the nucleation event. The need for this supersaturation was recognized by Wagner [54], but it was not incorporated into his description. The need for supersaturation can be understood from a consideration of the energetics of oxide nucleus formation. For simplicity, we consider first the formation of a spherical nucleus within a homogeneous, isotropic alloy matrix and assume for the moment that the molar volume of B is the same in both oxide and alloy. The overall free energy change is DG ¼ VDGV þ Ag

(6.66)

where V is the volume of the precipitate, DGV the free energy per unit volume accompanying the chemical reaction (6.2), A the precipitate surface area and g the precipitate–matrix interfacial tension. For a spherical precipitate of radius r DG ¼ 43pr3 DGV þ 4pr2 g

(6.67)

which is represented schematically in Figure 6.22. At small values of r, the second term is more important than the first, but at larger values the reverse is true. The shape of the curve in Figure 6.22 reflects a negative value for DGV and a positive one for g. As is seen, for rorn, a nucleus will spontaneously decay, whereas for r4rn , free energy is reduced by precipitate growth. For this reason, rn is known as the critical nucleus size, and sufficient supersaturation must be present to provide DGV large enough to overcome the surface energy barrier, DGn , to nucleus formation. The assumptions underlying Equation (6.67) are unrealistic. Recognizing that precipitates may not be spherical, that their volume will generally be larger than that of the metal they replace and that nucleation sites are usually local defects

6.7. Precipitate Nucleation and Growth

279

ΔG

ΔG ∗ r r∗ ΔGr

Figure 6.22 Free energy of spherical nucleus formation according to Equation (6.67).

we write instead DG ¼ VðDGV þ DGS Þ þ

X

Ai gi DGd

(6.68)

i

Where DGS is the strain energy resulting from the volume change, Ai and gi the areas and surface tensions of the precipitate-matrix interfaces and DGd the energy associated with defect site annihilation. At equilibrium, DGV just balances the strain and surface energy barriers to nucleation. In the case of spherical precipitates, Equation (6.68) can be rewritten as DG ¼ 43pr3 ðDGV þ DGS Þ þ 4pr2 g DGd n

(6.69)

n

The critical values r and DG are found by differentiating Equation (6.69) to locate the maximum in the curve 2g (6.70) rn ¼ ðDGV DGS Þ DGn ¼

16pg3 DGd 3ðDGV DGS Þ

(6.71)

where, for clarity, the fact that DGV is of opposite sign to both DGS and g has been explicitly recognized. Thus the more stable the precipitate, the lower the barrier to nucleation. The nature of the defect at which nucleation occurs is important, as the magnitude of DGd can vary considerably. In order of increasing DGd , i.e. decreasing DGn , the sequence would be approximately: homogeneous sites, vacancies, dislocations, stacking faults, grain and interphase boundaries and free surfaces. This is evident in the frequently observed preferential precipitation of internal oxides at alloy grain boundaries, e.g. Figures 6.1 and 6.5.

280


Figure 6.23 Schematic concentration profiles for internal precipitation of BOn showing a supersaturated region ahead of the internal oxidation zone.

The effects of the supersaturation requirement on precipitate size distributions and penetration kinetics were examined by Kahlweit et al. [16, 55, 56], and their treatment has since been extended by Gesmundo et al. [57]. The diffusion model is shown schematically in Figure 6.23. A key difference between this description and that used by Wagner (Figure 6.9) is that the precipitation front is not precisely defined, but instead is spread over a small region wherein N O and N B change with time as particles nucleate and grow. This can be appreciated from a consideration of mass transfer in the region of a newly formed precipitate (Figure 6.24). For the precipitate to develop, both O and B must be delivered to its surface. In the usual case of N BðoÞ DB N ðsÞ O DO , precipitate growth is limited by the availability of the metal, which becomes depleted ahead of the precipitate. A point is reached at which the oxide particle can grow no further and the inwardly diffusing oxygen sweeps past it, deeper into the alloy. To form the next precipitate, sufficient supersaturation must be achieved to overcome the nucleation barrier. At the position, X, where this is achieved, the reactant concentrations are denoted as N nO and N nB , and N nB ðN nO Þn ¼ Sn 4Ksp

(6.72)

In contrast, the last-formed precipitate relieved the local supersaturation when it nucleated, and at that location, X0 , N 0B ðN 0O Þn ¼ Ksp

(6.73) 0

a relationship which is observed throughout the region 0 x X , i.e. most of the internal oxidation zone. For parabolic internal oxidation kinetics, the diffusion equation solutions [16] for the reactant concentrations are n pffiffiffiffiffiffiffiffiffi N ðsÞ O NO erf x=2 DO t N O ¼ N ðsÞ for xoXn (6.74) O erfðgÞ

281


Figure 6.24 Mass transfer near a growing precipitate at the internal oxidation front: continuous concentration profiles at time of nucleation, dotted profiles after precipitate growth.

and N B ¼ N ðoÞ B

n N ðoÞ B NB

1=2

erfcðgf

Þ

pffiffiffiffiffiffiffiffi erfc x=2 DB t

for

x4Xn

(6.75)

with g and f as defined in Equations (6.19) and (6.20), respectively. In general [57], n N ðsÞ O NO n N ðoÞ B NB

¼n

GðgÞ Fðgf1=2 Þ

þ

N nB N 0B 0 N ðoÞ B NB

GðgÞ

(6.76)

where GðuÞ ¼ p1=2 u expðu2 ÞerfðuÞ

(6.77)

and FðuÞ ¼ p1=2 u expðu2 ÞerfcðuÞ 1=2

2

(6.78) n

with u ¼ gf . Under the limiting conditions g 1 and N O Gðg2 Þ 2g2 , and then kðiÞ p ¼

DO N ðsÞ O 0 nðN ðoÞ B NBÞ

N ðsÞ O,

then

(6.79)

replaces Equation (6.11), as found by Kahlweit et al. [16, 56]. Thus the penetration rate is greater than predicted by Wagner’s model, to the extent necessary to reach a higher solute metal concentration N 0B , where sufficient supersaturation for precipitate nucleation can be achieved.

282


The distance DX ¼ Xni X0i

(6.80)

represents the spacing between successive nucleation events, and is therefore representative of the local precipitate number density, f N 3 1 Xi 1 fN ¼ 3 (6.81) 3 DX Xi ðDXÞ Kahlweit et al. [16, 56] derived the relationship ðoÞ ðoÞ DX N nO N B N 0B N 0O N nO N oB N 0B nðN B N 0B ÞðN nB N 0B Þ ¼ ðsÞ ¼ ¼ ðoÞ n X nN nB N nB N 0B fN ðsÞ NO N ðsÞ O O ðN B N B Þ

(6.82)

ðsÞ , from which it follows that for DO ; DB and N nO ðBnB Þn independent of X and N O ðsÞ ðsÞ then ðN O DX=Xi Þ is also independent of Xi and N O . Equation (6.81) can therefore be rewritten as

fN ¼

3 kðN ðsÞ OÞ X3i

(6.83)

n n n where the constant k is a function of DO =DB ; N ðoÞ B ; K sp and N O ðN B Þ . The last is unknown, but assumed to be constant. It is then predicted from Equation (6.83) 3 that under fixed reaction conditions, f N is proportional to ðN ðsÞ O Þ , i.e.

3=2

f N ðXi Þ ¼ constant pO2

(6.84)

If, furthermore, Ksp is very small, solute enrichment is negligible ða 1Þ and the precipitates are spherical, their radius, r, is given by 4pr3 ¼ VOX N ðoÞ B 3 which upon substitution from Equation (6.83) yields !1=3 V OX N ðoÞ Xi B r¼ 4pk N ðsÞ fN

(6.85)

(6.86)

O

Bo¨hm and Kahlweit [16] tested these predictions using internal oxidation of a dilute Ag–Cd alloy at 8501C and confirmed that f N ðCdOÞ decreased with X3i and 3 increased with ðN ðsÞ O Þ . However, the assumption that K sp is very small and hence f BO afðxÞ is frequently incorrect. The predictions of Equations (6.83) and (6.86) will not be obeyed in such cases. An example of this situation is shown in Figure 6.25, where fBO decreases sharply with increasing depth. Particle size also increases with depth, but not in accord with Equation (6.86). Numerical evaluations by Gesmundo et al. [57] have shown that kðiÞ p is quite sensitive to the critical degree of supersaturation required for nucleation when Ksp is large. However, in the case of low Ksp values considered by Wagner, the predicted values of kðiÞ p are essentially unaffected. Rhines [4] pointed out that the more stable the oxide, the easier is nucleation (see Equation (6.71)) and the greater the number of precipitates which result.

283


0.16 Fe-5%Cr

Oxide-Volume Fraction

Fe-7.5%Cr Fe-10%Cr

0.12

0.08

0.04

0 0

10

20

30

40

50

Depth (μm)

(a)

(b) Figure 6.25 Internal oxide. (a) Volume fraction and (b) particle radius in Fe–5Cr at 1,0001C [10]. Published with permission from The Electrochemical Society.

While this may be correct for very dilute alloys, where metal diffusion is slow, it is clearly not a useful generalization in the cases shown in Figures 6.4 and 6.7. These cellular growth morphologies are found for both a low Ksp precipitate, alumina, and high Ksp compounds, carbides and nitrides. Indeed, the Kahlweit theory of repeated supersaturation and new precipitate nucleation is clearly inapplicable to these cases where the growth of existing needle or platelet-shaped

284


precipitates is the dominant process, and nucleation is no longer important. As already noted, these morphologies can lead to faster alloy penetration by facilitating oxidant diffusion. The reasons for their development are therefore of interest.

6.8. CELLULAR PRECIPITATION MORPHOLOGIES The application of classical nucleation theory to internal oxidation developed by Kahlweit et al. [16, 55, 56] assumes that the extent of precipitate growth is limited by the local supply of reacting solute metal. Since this is usually a relatively slow process, it would seem to be a relatively good assumption. Nonetheless, the growth of rods or laths of Al2O3, Cr2N and Cr23C6 has been found to continue across virtually the complete internal precipitation zone. Other examples of these morphologies include MoS2 precipitates in internally sulfidized Ni–Mo alloys [58], Al2O3 in ferritic iron [13], In2O3 in Ag–In alloys [59], TiO2 in Co–Ti alloys [60], Cr2N in binary Ni–Cr alloys [61] and various commercial heat-resisting alloys [62, 63] and Cr23C6 in a variety of ferritic and austenitic alloys [10, 23, 64–66]. The example of Cr2N lamellar precipitate growth in austenite is now investigated. A low-magnification image of an internal nitridation zone is shown in Figure 6.4, and a high-magnification image of the precipitation front in Figure 6.26 reveals that a grain boundary had developed at the reaction front. Analysis by selected area diffraction (SAD) showed that the precipitates were Cr2N and the matrix austenite. Their orientation relationship (Figure 6.26b) was found to be

½112 :ð0002Þ

ð111Þ ½1210

(6.87) Cr2 N

g

Cr2 N

g

The same orientation relationship was found throughout the precipitation zone, and at all reaction times. The parallel orientations of the Cr2N lamellae are clear in the dark field images of Figure 6.27. The parent austenite grain ahead of the precipitation front and the product austenite grain behind the front are of different orientations. Neither the Cr2N nor the reacted austenite grain has rational orientation relationships with the parent grain. Chemical microanalysis performed by energy dispersive spectrometry in the TEM yielded results for the reaction front. A scan across the unreacted–reacted austenite grain boundary (Figure 6.28) shows a small step function decrease in N Cr at the boundary. A scan in the orthogonal direction, parallel to the reaction front, reveals a completely flat profile between precipitates. The morphology, structure and compositional variations observed at the nitridation front are characteristic of the cellular ‘‘discontinuous precipitation’’ reaction. Such a reaction is characterized by lamellar or rod-shaped precipitates growing with an orientation relationship to their matrix, a high angle boundary at the precipitate growth front where the unreacted alloy and reacted matrix are the same phase but are differently oriented, and a step function or ‘‘discontinuous’’ change in composition at the precipitation front [67]. In a closed system,

6.8. Cellular Precipitation Morphologies

285

Figure 6.26 Internal nitridation reaction front in Fe–20Ni–25Cr at 1,0001C. (a) Bright field TEM and (b) SAD pattern near precipitation front: large, bright spots show [112] zone axis of austenite, small spots show ½1120 zone axis of Cr2N.

it would be written as g ¼ gD þ P

(6.88)

with g indicating the parent austenite, gD the chromium-depleted matrix and P the precipitate. For the internal nitridation reaction we write g þ N ¼ gD þ Cr2 N

(6.89)

and show the mass transport processes schematically in Figure 6.29. Nitrogen is transported from the alloy surface to the discontinuous precipitation front by a mixture of lattice diffusion through the matrix phases and diffusion along the interfaces between the lamellae and matrix. The g=gD high angle boundary provides a mechanism for rapid lateral transport of chromium, allowing it to segregate to the advancing Cr2N lamellae tips, and for the rejection of iron and nickel from the nitride. In this situation, the rate of precipitate penetration into the alloy is controlled by the nitrogen diffusion rate, but the spacing of the precipitate lamellae is controlled by chromium diffusion at the precipitation front. If the latter process is

286


Figure 6.27 Dark field images of Cr2N precipitates 5 mm below the alloy surface and at the precipitation front (Xi ¼ 60 mm) in Fe–20Ni–25Cr after 90 min nitridation at 1,0001C.

one of grain boundary diffusion, then [68, 69] dXi kDCr;i ¼ dt S2

(6.90)

where k is a constant, DCr;i the diffusion coefficient at the g=gD boundary and S the cellular dimension defined in Figure 6.29. Equation (6.90) reflects the fact that the rate at which the precipitates advance must balance the rate at which chromium is delivered to their tips ð DCr;i =SÞ together with the requirement that the total precipitate–matrix surface area created (which is proportional to 1/S) is minimized. Precipitate lamellae spacings developed during internal nitridation of austenite are shown in Figure 6.30. The spacing at a given depth remains constant with time, and the spacing at the moving reaction front increases with depth. Calculating the reaction front speed from dXi =dt ¼ kðiÞ p =X i and the measured [70] value of kpðiÞ , the spacing data of Figure 6.30 is plotted according to Equation (6.90) in Figure 6.31. Agreement with the discontinuous precipitation theory is seen to be good.


287

Figure 6.28 EDS analysis across the unreacted–reacted austenite grain boundary at the internal nitridation front of Figure 6.26. Gas

γ D+Cr2N

γ

S

JCr

JN

x= 0

x=Xi

Figure 6.29 Mass transport processes involved in discontinuous precipitation of Cr2N during internal nitridation of austenite.

There remain the questions as to why and how the discontinuous precipitation morphology is adopted by the reacting system. Two key factors are involved: the existence of a precipitate–matrix orientation relationship which can reduce surface energy, and the low stability of Cr2N with the consequently small driving force for precipitate nucleation at low N ðsÞ N levels.

288


3

precipitate spacing, µm

2.5 2 1.5

22.5 min 40 min 90 min

1 0.5 0 0

10

20

30 depth, µm

40

50

60

Figure 6.30 Variation of nitride spacing with position within precipitation zone at different reaction times for Fe–25Cr–20Ni at 1,0001C.

Figure 6.31 Test of discontinuous precipitation Equation (6.90) for internal nitridation of Fe–25Cr–Ni at 1,0001C.

The precipitates are constrained in their growth direction by the availability of chromium and nitrogen. The average direction normal to the alloy surface minimizes the nitrogen diffusion distance to where immobile chromium remains as yet unreacted. Thus a lamellar morphology is kinetically favourable, but creates a large internal surface area. The Cr2N–austenite system is able to reduce the surface energy by adopting a largely coherent precipitate–matrix orientation relationship. However, to accommodate both the preferred precipitate growth direction and the energetically favourable orientation relationship, the austenite matrix needs to adopt the appropriate orientation, which will in general be different from that of the parent grain. For this reason, the austenite undergoes reorientation at the reaction front, a process facilitated by the diffusion occurring along this high angle boundary.


289

Clearly the cellular precipitation process is self-sustaining. However, it is of enduring stability only because new precipitates do not nucleate ahead of the reaction front. The primary reason for this is the low stability of Cr2N and its high solubility product (Table 6.5). The free energy barrier to nucleation (6.71) is consequently high, and the nitrogen supersaturation necessary to overcome it is not achievable when growth of the existing lamellae is supported by accelerated chromium diffusion to their tips. A similar situation arises during lamellar Cr23C6 growth, as discussed in Chapter 9. However, the formation of alumina rod and lath-shaped precipitates is apparently different. As seen in Figure 6.7, and even more clearly in Figure 6.32, the precipitates extend across the width of the internal oxidation zone. However, no grain boundary develops in the metal phase at the reaction front, no orientation relationship is established between the Al2O3 and the metal and it must be concluded that the discontinuous precipitation mechanism is not in effect. The reasons for the formation of elongated Al2O3 precipitates have not been clearly established, although it is reasonable to speculate [60] that rapid diffusion of precipitating metal to the growing particles prevents nucleation of

Figure 6.32 Rod-shaped oxide precipitates formed during internal oxidation of Ni–4Al [42], as revealed by SEM examination of deep etched samples. With kind permission from Springer Science and Business Media.

290


new ones. In that case, it would be expected that varying temperature and pO2 so as to alter DO =DAl would affect the precipitate morphology. This question appears not to have been adequately investigated.

6.9. MULTIPLE INTERNAL PRECIPITATES We consider first the case where the solute metal forms two different precipitates corresponding to different oxidation states. The example of internal carburization forming zones of Cr7C3 and Cr23C6 precipitates is illustrated in Figure 6.6. Another common example is the formation of a spinel phase together with either Cr2O3 or Al2O3. In all such cases, the existence of different precipitate zones is a consequence of the gradient in oxidant potential between its maximum at the alloy surface and minimum in the alloy interior. A diffusion path for the internal oxidation of an Ni–Al alloy is shown in Figure 6.33, drawn on the basis that DO DAl . The two reaction fronts within the internal precipitation zone correspond to the reactions 2 Al þ3 O ¼ Al2 O3

(6.91)

Al2 O3 þ Ni þ O ¼ NiAl2 O4

(6.92)

In the aluminium alloy case, precipitate growth predominates over nucleation, and the amount of oxygen supersaturation required at each interface is presumably small. The general situation of two precipitate zones was described by Meijering [11] on the assumptions that B is immobile, both precipitates have very low Ksp values and that the interface between the two zones is sharp, i.e. reaction (6.92) or its equivalent instantaneously achieve equilibrium. The diffusion model is shown schematically in Figure 6.34. The intermediate reaction front where BOn1 is further oxidized to BOn2 is located at x ¼ XI . The oxidant concentration at this point is denoted by N IO. Using the linear concentration gradient approximation in Fick’s law (as in Equation (6.7)) and utilizing the mass balances at the two reaction fronts (as in Equation (6.8)), Meijering wrote DO N IO dXi ¼ n1 N ðoÞ B Xi XI dt

(6.93)

I DO ðN ðsÞ dXi dXI O NOÞ þ ðn2 n1 ÞN oB ¼ n1 N ðoÞ B dt dt XI

(6.94)

and

It is seen that both zones widen according to parabolic kinetics. The analysis leads to the expression for total penetration X2i ¼

2DO N ðsÞ O t neff N B

(6.95)

6.9. Multiple Internal Precipitates

291

Figure 6.33 Diffusion path for internal oxidation of dilute Ni–Al at 1,0001C [42]. With kind permission from Springer Science and Business Media.

which is seen to be equivalent to Equations (6.1) and (6.11). The effective stoichiometric coefficient is given by " !# n1 ð1 þ 4mð1 mÞðn2 n1 Þ=n1 Þ1=2 1 1 neff ¼ (6.96) 2mðn2 n1 Þ=n1 m where m ¼ N IO =N ðsÞ O . While this analysis can in principle predict the ratio X I =X i , this requires a knowledge of N IO , which is indeterminate within the formalism. The practical utility of Equation (6.95) lies in its application when XI =Xi ; n1 and n2 are all known, and penetration kinetics are used to deduce the permeability N ðsÞ O DO .

292


NB

N0(s) N0

X1

Xi

Figure 6.34 Reactant concentration profiles when two precipitate zones form from a single oxidant and one solute metal.

NC(0)

NB(0)

(s)

N0

N0

Figure 6.35 Simultaneous internal oxidation of two solute metals in ternary alloy.

A somewhat similar situation can arise in the internal oxidation of ternary alloys, if two components are reactive as shown schematically in Figure 6.35. Studies of this sort date back to the pioneering work of Rhines [4] on copper alloys containing tin or zinc as well as one of the metals: aluminium, beryllium or silicon. He produced two internal oxidation zones, with the inner one containing the more stable oxide. Figure 6.8 illustrates the example of simultaneous internal oxidation of chromium and aluminium in a nickel-based alloy. A more sophisticated treatment of the simultaneous internal oxidation of two solute metals has been provided by Niu and Gesmundo [71]. However, it too fails to provide a prediction of where the reaction front for the less stable precipitate will be located. As a result, the theory cannot predict the degree of enrichment in


293

the near-surface zone, because this is supported by diffusion of the less reactive solute metal through the inner precipitation zone. The theory was applied with qualitative success to the internal oxidation of aluminium and silicon in Ni–Si–Al, using the approximation that the intermediate SiO2 precipitation front coincided with that of Al2O3, thereby removing the uncertainty. However, Yi et al. [72] showed clearly that Al2O3 was precipitated at greater depth than SiO2. Another example of multiple internal precipitation zones arises when a dilute alloy is simultaneously attacked by two or more different oxidants. This situation also was first analysed by Meijering [11], and is shown schematically in Figure 6.36. The oxidant forming the less stable precipitate under reaction

NB N2(S)

N1(S) N2 N1

X1

X2

(a)

X1 X2

Ln ac

BC

Surface

BO B

Ln a0 (b)

Figure 6.36 (a) Concentration profiles for simultaneous internal attack on alloy AB by two oxidants, assuming essentially immobile B. (b) Corresponding diffusion path for the oxidizing–carburizing case.

294


conditions will be found in the deeper reaction zone, if it is formed. The reason for this is that the more stable precipitate forms at or near the surface if the reaction, for example, BN þ 12O2 ¼ BO þ 12N2

(6.97)

is favoured by the gas composition. Beneath the surface, N O decreases with depth, until reaching a low value at the oxide precipitation front. If diffusion of nitrogen through the near-surface oxide precipitation zone is rapid, then N N does not decrease very much, and a position is reached where N N =N O exceeds the value necessary for the reaction BO þ N ¼ BN þ O

(6.98)

where nitride precipitation commences. The Meijering analysis assumes (a) neither precipitate significantly affects the inward oxidant diffusion rates; (b) the less stable precipitate is converted to the more stable one via a displacement reaction involving dissolved oxidant (1); (c) the displacement reaction goes rapidly to completion at precisely defined oxidant activity values, i.e. no intersolubility exists between the two precipitate compounds; (d) no thermodynamic or kinetic interaction of importance takes place in the solution phase and (e) precipitates are extremely stable, and N B ffi 0 throughout the two precipitate zones. Under these conditions, the two zones grow each according to parabolic kinetics. Meijering further assumed that the alloy solute B is essentially immobile, and no solute enrichment occurs in the precipitation zone. The approximate Meijering treatment leads to the results X21 ¼ 2

X22

¼2

D1 N 1ðsÞ n1 N ðoÞ B

D1 N ðsÞ 1 n1 N ðoÞ B

þ

t

D2 N ðsÞ 2 n2 N ðoÞ B

(6.99) ! t

(6.100)

These simple forms result from the way in which the intermediate precipitation front at X1 is treated. The Meijering treatment assumes that the oxidant (2) released at this position by the displacement reaction, which is the reverse of reaction (6.98), all diffuses inward to extend the inner precipitation zone. Thus if an alloy was first reacted to internally precipitate the less stable compound, e.g. BN, and then exposed to oxygen alone, the advancing oxidation front would displace the internally nitrided zone inwards, but the thickness of the nitride zone would remain constant. In essence, therefore, the innermost precipitation zone is predicted to widen at the same rate, whether or not another precipitation zone develops near the surface.


295

The formation of two separate precipitation zones in sequence according to thermodynamic stability has been verified [73, 74], but kinetic data have become available only recently. When heat-resisting alloys were exposed [75] to twocomponent (CO/CO2) or three-component (CO/CO2/SO2) gases, they developed discrete internal precipitation zones which each grew according to parabolic kinetics, as shown in Figure 6.37. However, the assumption of a single precipitate species in each zone, while appropriate for binary alloys, was found to be inapplicable to these multicomponent materials. The observation of chromiumrich oxide and sulfide precipitates in the same zone was common. The sulfide also contained iron. Approximating this compound as a thiospinel, one can write FeCr2 S4 þ 3 O ¼ Fe þCr2 O3 þ 4 S

(6.101)

for precipitate co-existence. Since the iron activity, aFe , can vary within the matrix of a multicomponent alloy, as and ao are independent, and the two-precipitate zone can be both thermodynamically and kinetically stable. Unfortunately, the experiments behind the data of Figure 6.37 also produced external scales, boundary conditions were uncertain and further analysis is not possible. Gesmundo and Niu [76] have relaxed the requirement that DB O, and have avoided other approximations in the Meijering treatment. However, they retained the assumption that Ksp in both precipitate zones is very small, and consequently N B O. Enrichment of precipitated element B in the internal reaction zones was found to affect the rates at which the oxidants penetrated deeper into the alloy. However, predictions made for the simultaneous internal carburization and oxidation of Ni–3.9Cr in CO/CO2 at 8211C were in poor agreement with the experimental data of Copson et al. [77, 78]. While the basic ðsÞ data used in the calculation ðN ðsÞ O ; DO ; N C ; DC ; DCr Þ were of high quality, it had

Figure 6.37 Internal precipitation zone growth kinetics for 310 stainless steel exposed to CO–CO2–SO2–N2 at 1,0001C [75]. With kind permission from Springer Science and Business Media.

296


been measured at high temperatures. Extrapolation to low temperatures, such as the 8211C used by Hopkinson and Copson, is always somewhat questionable for diffusion coefficients in view of the increasing importance of boundary and dislocation mechanisms. The simultaneous carburization and oxidation of chromium-bearing alloys is an important technical problem, leading to a form of failure known as ‘‘green rot’’ [79]. The name comes from the green colouration of fracture surfaces in the embrittled material resulting from this form of internal attack. The general reaction morphology is shown in Figure 6.38 for a Type 304 stainless steel (Fe–18Cr–8Ni base) exposed at 7001C to a CO/CO2 mixture corresponding to pO2 ¼ 1023 atm and a supersaturated carbon activity of 7. An external scale formed, but was repeatedly disrupted and spalled by regular temperature cycling. As seen in the micrograph, two internal precipitation zones were formed: oxides beneath the surface and carbides at greater depths. The oxide zone actually consisted of two regions: spinel nearest the surface, and Cr2O3 further in. The practical effect of the carburization is profound. In the absence of carbon, this alloy would form an external oxide scale rather than undergoing internal oxidation (see Section 6.11). Because carbon permeability in the alloy is so high 11 ðDC N ðsÞ cm2 s1 at 7001C) internal carburization is rapid, removing C 6 10 much of the chromium from solution. The precipitated chromium is immobilized, and is therefore unavailable to form an external oxide scale. Instead, the carbides are oxidized in situ, just as proposed by Meijering. Only in this way could the enormous oxide volume fractions seen in Figure 6.38 be formed internally. The Cr2O3 is responsible for the green colour and the expressive term, green rot.

Figure 6.38 Internal carburization and oxidation of a 304 stainless steel exposed to CO–CO2–Ar at 7001C. Reprinted from Ref. [80] with permission from Elsevier.


297

The carburization front was found to have penetrated 520 mm in 396 h at 7001C, corresponding to ! ðsÞ DO N ðsÞ D N C O C ¼ 1:9 109 cm2 s1 2 þ ðoÞ 1:5N ðoÞ 0:344N Cr Cr according to Equation (6.100). Permeability data are not available for such a low temperature, so high-temperature data are extrapolated, yielding ðoÞ ðoÞ 18 10 DO N ðsÞ cm2 s1 and DC N ðsÞ cm2 s1 . O =1:5N Cr ¼ 2 10 C =0:344N Cr ¼ 9 10 Clearly the oxygen permeability data are not applicable to the observed internal oxidation rates, but the carbon permeability are roughly in accord with the experimental carburization depth. Alloys containing two or more reactive solute metals exposed to mixed oxidant gases can form very complex internal precipitation zones. Oxidation in air under thermal cyclic conditions (which remove scale and allow internal attack) leads to internal formation of nitrides and oxides of both aluminium and chromium [81] in Ni–Al–Cr alloys (Figure 6.39). As expected, the nitrides are located deeper within the reaction zone, reflecting their lower stability and the high nitrogen permeability. The sequence of chromium and aluminium nitrides is, however, unexpected. It can be understood in terms of thermodynamic interactions within the matrix phase, as is discussed in Section 6.10.

Figure 6.39 1,1001C.

Simultaneous internal nitridation and oxidation of Ni–Cr–Al exposed to air at

298


10 Internal penetration rates are rapid, in the range kðiÞ cm2 s1 p ¼ 1:5 5:1 10 at 1,1001C for a range of alloy compositions. The applicability of Equation (6.100) was tested using oxygen and nitrogen permeability data. For N Al ¼ 0:2, ðoÞ ðoÞ 10 10 N ðsÞ cm2 s1 and N ðsÞ cm2 s1 . The N DN =N Al ¼ 5 10 O DO =1:5N Al ¼ 3 10 overall rate constant predicted from Equation (6.100) to be 5.3 1010 cm2 s1 is in satisfactory agreement with the experimental results. The kinetic models for internal attack by multiple oxidants have a critical shortcoming. Both the approximate Meijering model [11] and the more accurate Gesmundo and Niu [76] description treat the reactant metal B as being at a negligible matrix concentration throughout the multiple precipitation zones. While this might be a reasonable approximation for oxide formation, it is nowhere near correct for carbides or nitrides (Table 6.5). A large concentration of chromium remains in the matrix of the inner carbide or nitride zone, and will react with inwardly diffusing oxygen when it arrives. Thus the description of the mass balance at the intermediate interface X1 (Figure 6.36) solely in terms of a reaction such as Equation (6.98) is in considerable error. This is illustrated in Figure 6.40 for the case of internal nitridation followed by carburization. The arrival of carbon has converted the large lamellar prior nitrides in situ to carbide,

Figure 6.40 TEM bright field view of morphology produced by sequential internal nitridation followed by carburization of a Ni–20Ni–25Cr alloy at 1,0001C.

6.10. Solute Interactions in the Precipitation Zone

299

as expected. In addition, however, it has reacted with matrix chromium to form additional fine carbides. A further complication is the appearance in this zone of CrN, a phase which is unstable at the ambient nitrogen pressure employed. The reason [82] is likely the release of nitrogen via the reaction C þ76Cr2 N ¼ 13Cr7 C3 þ 76 N

(6.102)

The saturation level N ðsÞ N corresponds in this reaction to an equilibrium value of N C ¼ 1:4 103 . However, much higher levels of N C are available from carbon dissolution, up to about 0.06, leading to higher (supersaturated) N N values according to reaction (6.102). Under these circumstances CrN is stabilized. Given all these complexities, the kinetic theory cannot be expected to provide any better than order of magnitude predictions. The further approximation in the Meijering theory that N IO ¼ 0 is an additional source of error in the case of carburization– oxidation reactions.

6.10. SOLUTE INTERACTIONS IN THE PRECIPITATION ZONE We have assumed so far that the various alloy solutes, oxidants and reacting metals, behave in an ideal fashion, each having no effect on the thermodynamic activity or diffusion of the others. Given that the oxidant and solute species interact chemically to the extent of forming a precipitate, the supposition is seen to be improbable. Nonetheless, as seen earlier in this chapter, the diffusion theories of Rhines, Wagner and Meijering have proven remarkably successful in providing at least semi-quantitative descriptions of internal penetration rates in many cases. The questions of interest therefore concern how large the solute interactions are, and when they become important. Gesmundo and Niu [83] have considered the general quaternary system A–B–C–O, in which the only oxides possible are the pure binaries. It is supposed that the stability of the oxides increases in the order AO, BO, COn, and that the oxygen potential is sufficient to oxidize only C. Assuming that Ksp ðCOn Þ 1, the situation is one of oxygen dissolving in and diffusing through a single-phase A–B matrix. The effects of B on oxygen permeability and hence internal oxidation can therefore be investigated. Ternary diffusion interactions were ignored in this analysis, and attention was focused on oxygen solubility. The model originally proposed by Alcock and Richardson [84] for oxygen solubility in liquid binary alloys was employed ln Ks ðABÞ ¼ N A ln Ks ðAÞ þ N B ln Ks ðBÞ þ N A ln gA ðABÞ þ N B ln gB ðABÞ

ð6:103Þ

where Ks ðiÞ is the Sievert’s constant (Equation (2.71)) for oxygen in the indicated solvent, and gA ðABÞ and gB ðABÞ are the metal activity coefficients in the binary alloys. Approximate ideality was assumed for the substitutional alloy solution, yielding the simplified result ln Ks ðABÞ ¼ N A ln Ks ðAÞ þ N B ln Ks ðBÞ

(6.104)

300


and the oxygen solubility in the alloy is given by 1=2

N ðsÞ O ðABÞ ¼ K s ðABÞpO2

(6.105)

The oxygen diffusion coefficient also varies with AB composition. The original model of Park and Altstetter [85] for oxygen dissolution in binary alloys DO ðABÞ ¼ DO ðAÞ

gO ðABÞ gO ðAÞ

(6.106)

was examined. However, because Ks ¼ 1=gO , this description leads to the unacceptable result DO ðBÞ ¼ DO ðAÞ

Ks ðAÞ Ks ðBÞ

To avoid this difficulty, the empirical description DO ðBÞ NB DO ðABÞ ¼ DO ðAÞ DO ðAÞ

(6.107)

(6.108)

in which the diffusion coefficient ratio is raised to the power N B, was adopted. The solutions to the diffusion equation for N O within the precipitation zone and N C in the alloy ahead of the precipitation front are the same as Equations (6.17)–(6.27). Application of this model for alloy interaction effects on oxygen permeabilities to the systems Cu–Al, Ni–Al and Cu–Ni–Al leads to the results shown in Table 6.7. While the model successfully predicts that adding nickel to Cu–Al will reduce greatly the extent of internal oxidation, it overestimates the size of the effect and is unsuccessful in relating rates to nickel concentrations. A much more detailed analysis was undertaken by Guan and Smeltzer [86] who examined the Ni–Cr–Al system. Their approach was based on the use of Wagner’s formalism (2.68) for solute interactions to evaluate N ðsÞ O (Ni–Cr), and a full solution of the diffusion equations, including cross-effects. Results for the variation of oxygen solubility with N ðoÞ Cr are shown in Figure 6.41. Such large changes in N ðsÞ would be expected to affect the rate of internal aluminium O oxidation, and perhaps limit the possibility of it occurring at all. The results of this calculation are examined in Section 7.4.

2 1 Table 6.7 Estimates [83] of kðiÞ from Equation (6.108) for internal oxidation at 8001C in p /cm s a Rhines packs

Cu–0.72Al Ni–0.54Al Cu–10.16Ni–0.76Al Cu–20.11Ni–0.79Al Cu–30.07Ni–0.80Al a

Measured

Calculated

2.0 109 1.4 1010 2.9 1011 2.5 1011 1.5 1011

1.3 108 1.4 1011 4.8 1012 6.3 1012 8.1 1012

Oxygen partial pressure controlled by Cu/Cu2O equilibrium.

6.11. Transition from Internal to External Oxidation

301

Figure 6.41 Oxygen solubility in Ni–Cr–Al as a function of NðoÞ Cr at 1,2001C [86]. With kind permission from Springer Science and Business Media.

A final example of the importance of solute interactions is provided by the internal nitridation of Ni–Cr–Ti alloys [87]. Reaction rates at nitrogen potentials high enough to react with titanium but not chromium were found to increase with N ðoÞ Cr . The effect was shown to be due to the Cr–N thermodynamic interaction which increased N ðsÞ N.

6.11. TRANSITION FROM INTERNAL TO EXTERNAL OXIDATION As is discussed in Section 5.4, if an alloy contains a sufficient concentration of its most reactive component, then the metal can form a protective external scale. Conversely, if the component is dilute, and no other alloy component is oxidized, then internal oxidation results, destroying the alloy. We now consider what concentration of solute metal is necessary to ensure external rather than internal oxide formation. Darken [88] recognized that the volume fraction of internally precipitated oxide would affect the reaction, and that internal oxidation could only occur up to a maximum value of f BO , and hence N ðoÞ B . Wagner [8] proposed that a transition from internal to exclusive external oxidation would occur when N ðoÞ B is increased to a critical value at which the internally precipitated particles reduced the oxygen flux to a sufficient extent. Since the oxide is essentially impermeable to oxygen, diffusion is restricted to the metal channels between precipitate, so that the average flux is lowered. This slows the rate at which the supersaturation needed for new precipitate nucleation can be achieved, and the outward flux of

302


component B is then of greater relative importance. If N ðoÞ B is high enough to sustain a sufficient flux for continued growth of precipitates, their enlargement leads to continuous oxide layer formation. The mole fraction of internal oxide is found by definition (6.29) to be N BO ¼ aN ðoÞ (6.109) B where a is the enrichment factor calculated from Equation (6.30). Under the limiting conditions (6.25), where metal solute diffusion is important, the limiting form (6.31) applies. Recognizing that the volume fraction of BO; gBO ; is given by VBO gBO ¼ N BO (6.110) VA we combine Equations (6.31), (6.109) and (6.110) to obtain !1=2 ðsÞ p V A N O DO ðoÞ N B ¼ gBO 2n V OX DB

(6.111)

If a critical value can be specified for gBO , then the minimum value of N ðoÞ B for external scale formation can be calculated from Equation (6.111). Rapp [89] studied the internal oxidation of Ag–In alloys at 5501C, where Ag2O is unstable over a wide range of pO2 , the conditions (6.23) were met and a ¼ 1. ðoÞ Systematic variation of N In established that the critical value for scale formation rather than internal oxidation was N InO1:5 ¼ N ðoÞ In ¼ 0:15. This corresponds to an oxide volume fraction, gBO ¼ 0:30. At low pO2 values, where N ðsÞ O is reduced, the conditions (6.25) are met, and Equation (6.111) applies. As is seen from ðsÞ this equation, the critical value of N ðoÞ B varies with N O , and hence with pO2 . Rapp determined metallographically the levels of N ðoÞ In required for scale formation at different oxygen pressures. These results are compared with theoretical predication in Figure 6.42, where agreement is seen to be quite good. We conclude that formation of a critical volume fraction of internal oxide constitutes a correct criterion for the transition to external scale formation. We also observe that oxidation at low pO2 provides a suitable way of inducing protective scale formation on dilute alloys. Providing these scales maintain their mechanical integrity, a low-pressure pre-oxidation treatment can be used to provide protection against subsequent exposure to high oxygen potential gases. We now use Equation (6.111) with gBO set at 0.3, to calculate critical alloy compositions necessary for external, rather than internal oxide formation. Results for chromia and alumina formers are calculated using oxygen solubility data from Table 2.2, and diffusion coefficients taken from Table D2 (Appendix D). Critical ðoÞ values of N ðoÞ Cr and N Al calculated on this basis are compared in Table 6.8 with minimum values estimated from the kinetic criterion (5.25) for the concentration necessary to sustain external scale growth. Similar results were obtained by Nesbitt [90], using somewhat different permeability data. It is seen that the concentrations necessary to avoid internal oxidation are greater than those required merely to support scale growth, and should therefore be preferred. As is also seen, fairly good agreement between prediction and experimental reality is achieved. Although the precision is much less than would be required

6.11. Transition from Internal to External Oxidation

303

Nln(O)*

External Oxidation

Internal Oxidation

log pO2* (atm)

Figure 6.42 Transition from internal to external oxidation of Ag–In alloys at 5501C: continuous line calculated from Equation (6.111), points measured experimentally. Reprinted from Ref. [89] with permission from Elsevier.

Table 6.8 Calculated minimum solute concentrations (mole fraction) for exclusive Cr2O3 or Al2O3 scale formation under Rhines pack conditions

a

Alloy

Scale

T (1C)

Support scaling kinetics (5.22)

Prevent internal oxidation (6.111)

Experimental

Ni–Cr g-Fe–Cr Ni–Ala Fe–Alb

Cr2O3 Cr2O3 Al2O3 Al2O3

1,000 1,000 1,200 1,200

0.07 0.07 0.02 104

0.29 0.16 0.11 0.15

0.15 0.14 0.06–0.13 0.10–0.18

gBO set at 0.2 [90]. Data for a-Fe.

b

for alloy design, it is concluded that the form of Equation (6.111) may be relied upon for semi-quantitative prediction. Of particular importance is the prediction that the critical alloying content required to avoid internal oxidation increases with N ðsÞ O and hence with ambient oxygen potential. As is clear from Equation (6.111), the competition between internal and external reaction is critically dependent on the oxidant permeability. Using the

304


representative values of Table 6.2, it is found that the minimum value of N ðoÞ B necessary to prevent internal nitridation is two to three times higher than the value required to avoid internal oxidation in austenite. Internal carburization is even more difficult to prevent, with the necessary values of N ðoÞ B 25–70 times higher than those required to form an oxide scale. This prediction is realistic only in the sense that chromia forming alloys are almost always found to carburize internally. In the absence of a protective oxide scale, internal carburization of heat-resisting steels and many alloys is unavoidable. Because, moreover, the process is also very rapid, it constitutes a serious practical problem. Carburization and related corrosion phenomena will be discussed in detail in Chapter 9. Another reaction morphology can develop during the preferential oxidation of a single alloy component: simultaneous external scale growth and internal precipitation. Wagner [91] analysed the conditions under which this could occur, by comparing the concentration product within the alloy, N B N nO , with the solubility product of the oxide, Ksp . For parabolic scale growth, Equations (5.23)–(5.26) apply. At the alloy–scale interface, the reaction B þn O ¼ BOn

(6.112)

n N B;i ðN ðsÞ O Þ ¼ K sp

(6.113)

is at equilibrium, and

The solutions to Fick’s second law for oxygen diffusion from the interface into the alloy and for diffusion of B from the alloy to the interface are Ksp 1=2 erfc½x=2ðDO tÞ1=2

(6.114) NO ¼ N B;i erfc½ðkc =2DO Þ1=2

and ðoÞ N B ¼ N ðoÞ B ðN B N B;i Þ

~ AB tÞ1=2

erfc½x=2ðD ~ AB Þ1=2

erfc½ðkc =2D

(6.115)

These solutions are then used to evaluate the gradient in the logarithm of the concentration product at the interface ðx ¼ xc Þ ~ AB Þ N ðoÞ N B;i @ ln N B N nO 1 expðkc =2D ¼ B 1=2 ~ @x N B;i ðpDAB tÞ erfc½ðkc =2DAB Þ1=2

x¼xc n expðkc =2DO Þ ð6:116Þ ðpDO tÞ1=2 erfc½ðkc =2DO Þ1=2

~ AB , we approximate the second term to Noting that DO DAB and kc D n=ðpDO tÞ1=2 . It is then found that 2 3 ðoÞ 1=2 v ð1 N B Þðkc DO Þ @ ln N B N O n 6 p 1=2 n o 17 ¼ 4 5 (6.117) 1=2 1=2 2 @x ~ ~ ðpDO tÞ x¼y N B;i nDAB 1 F½ðkc =2DAB Þ

Here the auxiliary function F(u) is as defined in Equation (6.77).

6.12. Internal Oxidation Beneath a Corroding Alloy Surface

305

If the right-hand side of Equation (6.117) is negative, the concentration product decreases in the alloy from its saturation value at the interface, and oxide precipitation is impossible. However, if it is positive, the alloy beneath the scale becomes supersaturated and internal oxidation results. Thus the condition for internal oxidation beneath a scale of the same oxide is p 1 N ðoÞ ðkc DO Þ1=2 B n o 41 (6.118) 2 N B;i nDAB 1 F½ðkc =2D ~ AB Þ1=2

and the interfacial concentration is found from N B;i ¼

1=2 ~ N ðoÞ

B F½ðkc =2DAB Þ 1=2 ~ 1 F½ðkc =2DAB Þ

(6.119)

Alternatively, the condition for avoiding internal oxidation beneath the scale may be expressed as ~ AB Þ1=2

R þ F½ðkc =2D (6.120) Rþ1 ~ AB . If kc is small enough, then kc =2D ~ AB 1 and where R ¼ ðpkc DO =2Þ1=2 =nD 1=2 ~ ~ ~ ~ 2 the F kc =2DAB ðpkc =2DAB Þ . If, furthermore, DO DAB and kc DO D AB condition (6.120) may be approximated by pkc DO 1=2 1 4 (6.121) N ðoÞ B ~ AB 2 nD N ðoÞ B 4

Simultaneous internal and external oxidation of B is predicted to occur when N BðoÞ is less than the level predicted from Equation (6.120) and greater than the value set by Equation (6.111) for external scale formation, providing that N BðoÞ is sufficient to support external scale growth (see Equation (5.25)). The range of conditions permitting both internal and external oxidation of the solute metal can be rather restricted, as demonstrated by Atkinson [92] for Fe–Si alloys.

6.12. INTERNAL OXIDATION BENEATH A CORRODING ALLOY SURFACE In many practical situations, the oxidant activity will be high and an external scale will grow. Alloys such as Ni–Cr, Ni–Al, Fe–Cr and Fe–Al will, if sufficiently dilute, form external scales of iron or nickel-rich oxides together with internally precipitated chromium or aluminium-rich oxides. A schematic view of this reaction morphology is shown in Figure 6.43. The interactions between the internal precipitates and iron or nickel oxides when they come into contact are considered in the next chapter. For the moment, our interest is in the effect of the receding alloy surface on the internal oxidation kinetics. Diffusional analyses of internal oxidation in conjunction with scale growth according to parabolic kinetics have been provided by Rhines et al. [24] and ðsÞ Maak [93]. In this situation, N O denotes the dissolved oxygen concentration at the

306


NiO

Ni+Cr2O3

Ni-Cr

NCr=NCr(0)

N0(s)

N0

Xc

Figure 6.43

Xi

Oxidation of a dilute Ni–Cr alloy at high pO2 .

alloy–scale interface, Xi the distance of the internal precipitation front from the original alloy surface and Xc the position of the scale–alloy interface with respect 1=2 to the original surface. In the common case ðkðiÞ 1 and Xc oXi , then p =2DO Þ ðsÞ pffiffiffiffiffiffiffiffi N DO (6.122) Xi ðXi Xc Þ ¼ 2 O ðoÞ F Xi =2 DB t t nN B where the function F(u) is as defined in Equation (6.78). When both Xc and DB are small, Equation (6.122) yields Equation (6.11). Experimental verification of Equation (6.122) has not been completely successful [9]. Permeability values deduced from internal oxidation kinetics under an external scale were apparently smaller than those determined from exclusively internal reactions. In view of the microstructural complexity of the scale–alloy interface (Figure 6.42), it seems quite likely that local scale separation could occur from time to time, as a result of reduced oxide plasticity. In this case, the boundary value oxygen activity, and hence N ðsÞ O , would vary with time, and (6.122) would no longer apply.

6.13. VOLUME EXPANSION IN THE INTERNAL PRECIPITATION ZONE The precipitation of internal oxides is almost always accompanied by a volume expansion. As is seen from the molar volumes listed in Table 6.9, the expansions are large. The effect of internal oxide precipitation on the molar volume of the internal oxidation zone can be calculated for a binary alloy Ni–B as ðoÞ V T ¼ V Ni ð1 N ðoÞ B Þ þ N B V BOn

(6.123)

where it is assumed that no solute element enrichment or depletion occurs. The volume increase ratio DV=V Alloy ¼ ðV T V Alloy Þ=V Alloy is then calculated for

6.13. Volume Expansion in the Internal Precipitation Zone

307

Table 6.9 Molar volumes of internal oxides; alloy expansion on internal oxidation of Ni-based alloys VOX (cm3)

SiO2a a-Al2O3 Cr2O3 a

25.8 25.6 29.2

Alloy DV/V NðoÞ B ¼ 0:01

0.05

0.10

0.026 0.008 0.011

0.13 0.04 0.05

0.26 0.08 0.11

b-Cristobalite.

various solute concentrations, leading to the results shown in Table 6.9. Similar calculations for nitridation and carburization of chromium show that resulting expansions are less, principally because of the higher densities of Cr2N and the chromium carbides, all of which are interstitial compounds. In the case of internal oxidation, the enormous volume increase generates stresses which must be relieved. Shida et al. [33, 43] suggested stress relief mechanisms of grain boundary sliding and extrusion of internal oxide-free metal adjacent to grain boundaries, in the case of intergranular oxidation at low temperature. However, internally oxidized Ni–Cr alloys were thought [36] to be able to accommodate stress by metal flow within the grains. In fact, outward transport of the more noble metal was first reported by Darken [94] in a study of Ag–Al alloy oxidation. Mackert et al. [95] found nodules of palladium and silver on the external surface of internally oxidized Pd–Ag–Sn–In alloys, and proposed that Pd and Ag diffused via the Nabarro–Herring mechanism. Guruswamy et al. [96] observed silver nodules on the surface of internally oxidized Ag–In alloys, and concluded that dislocation pipe diffusion was the mechanism of silver transport. An example of outward metal displacement during internal oxidation of Ni–Cr–Al is shown in Figure 6.8. Yi et al. [34] demonstrated that in the case of Ni–Al–Si internal oxidation, the volume of metal accumulated outside the precipitation zone was close to the equivalent of the volume increase calculated for silicon and aluminium oxidation. This result is shown in Figure 6.44, and confirms that the driving force for outward nickel displacement is the volume increase within the precipitation zone. The mechanism whereby the nickel moves is obviously of interest. Yi et al. [34] proposed that the mechanism was one of Nabarro–Herring creep [97]. In the case of internal oxidation, the volume expansion at the internal oxidation front causes compressive stress and a reduction in vacancy concentration. Thus a vacancy gradient is established between the free alloy surface, where the equilibrium concentration N v prevails, and a much reduced concentration at the reaction front. Assuming a linear gradient, we can write J Ni ¼ J V ¼

DV DN V Xi VNi

(6.124)

308


Figure 6.44 Comparison of volume of Ni transported outward with volume increase calculated for internal oxidation of Al and Si in Ni–Al–Si. Reprinted from Ref. [34] with permission from Elsevier.

where DV is the vacancy diffusion coefficient. To conserve mass, this flux must equal the rate at which nickel is displaced by newly precipitated oxide J Ni ¼

N ðoÞ 1 dXi B ðV BO V B Þ V Ni dt V Ni

(6.125)

where the amount of nickel displaced has been calculated from the volume of new materials divided by the nickel molar volume. Equating Equations (6.124) and (6.125) and integrating, we find X2i ¼ 2kðiÞ p t with kðiÞ p ¼

DV DN V VNi N BðoÞ ðV BO V B Þ

(6.126)

If the further approximation N V ðx ¼ Xi Þ O is made, then DN V ¼ N V . Recalling from Equation (3.58) that DNi DV N V , we obtain DNi V Ni kðiÞ (6.127) p ¼ ðoÞ N B ðV BO V B Þ Values of kðiÞ p predicted from Equation (6.127) are compared with experimental measurements for an Ni–4Al alloy in Figure 6.45, where agreement is seen to be good. Thus internal oxidation is, in this case, controlled by outward diffusion of nickel, although driven by inward oxygen diffusion. It is likely that the rapid

6.13. Volume Expansion in the Internal Precipitation Zone

309

Figure 6.45 Comparison of experimentally measured internal oxidation rates with predictions of Nabarro–Herring creep mechanism. Reprinted from Ref. [34] with permission from Elsevier.

oxygen diffusion associated with the fine precipitate platelets was a factor contributing to this result. The Nabarro–Herring model was found by Yi et al. [34] to be inapplicable to ðoÞ Ni–4Al–xSi alloys with x ¼ 1 or 5 wt%. Instead of decreasing with increasing N Si as predicted by Equation (6.127), the rate increased. Although the magnitude of the rate constant was satisfactorily accounted for by dislocation pipe diffusion ðoÞ of nickel, the variation of kðiÞ p with N Si was not. It is possible that increasing silicon levels led to a greater multiplicity of precipitate–matrix interfaces and consequently higher effective DO values. If dislocation pipe diffusion is sufficiently rapid, the process does not contribute to rate control, and internal oxidation kinetics are described using an expression like Equation (6.35) for the ternary alloy: ðoÞ ðoÞ N ðsÞ O DO ¼ a þ bðN Al þ N Si Þ

(6.128)

More detailed study of precipitate morphologies is required. The swelling effect caused by internal oxidation is not universally observed to cause metal ejection. For example, internal precipitation of Cr2O3 and MCr2O4 in Fe–Cr (Figures 6.1 and 6.2) or Ni–Cr [36] does not lead to external iron or nickel accumulation. The volume changes are nonetheless large (Table 6.9) and significant deformation must occur. It is possible that outward movement of nickel simply carries with it the embedded chromium-rich particles. The latter are large and spheroidal and drift with the moving nickel lattice. Alumina, however, precipitates as rods and platelets normal to the alloy surface, i.e. parallel to the direction of nickel movement. In such a configuration, it seems possible that nickel can transport past the fixed alumina precipitates, ‘‘extruding’’ to the outer surface. Even when this favourable morphology develops, nickel displacement has been found to be suppressed if an external NiO scale grows during internal Al2O3 precipitation within a binary Ni–Al alloy oxidized at pO2 ¼ 1 atm [72].

310


The suggested reason [33, 39, 72] is that growth of an external scale by outward cation diffusion leads to metal vacancies being injected into the alloy at the scale– alloy interface. These then diffuse inwards, permitting more rapid outward nickel movement. Put more simply, consumption of metal at the scale–alloy interface provides the space needed to accommodate the internally precipitated oxide. A similar situation arises during in situ oxidation of primary carbides (Figure 6.16) beneath a growing Cr2O3 scale [39]. If the carbide oxidation is described by reaction (6.42), then the weight change corresponding to oxygen consumption is given by !2 kðiÞ ðDW i =AÞ2 p f Cr23 C6 ðiÞ ¼k ¼ 576 (6.129) 2t V CrC0:261 where the mass conversion number 576 is computed on the assumption that no carbon is lost from the alloy. The volume increase due to carbide oxidation, normalized to the alloy surface area is then given by ðDV i =AÞ2 ðiÞ ðV CrO1:5 V CrC0:261 Þ (6.130) ¼ kðiÞ n ¼k 24 2t 11 The rate of the process was measured as kðiÞ cm2 s1 , corresponding p ¼ 2:9 10 ðiÞ 15 to a volume expansion accumulation rate of kn ¼ 3 10 cm2 s1 . This is to be compared with the volume consumption rate corresponding to external scale growth, DV ex . Approximating the scale as pure Cr2O3 and assuming a fixed alloy–scale interface, one finds ðDV ex =AÞ2 V Cr 2 ¼ kðexÞ ¼ kw (6.131) n 2t 24 where kw is the parabolic weight gain rate constant for scale growth. The measured value of kw ¼ 8 1012 g2 cm4 s1 leads to an estimate of kðexÞ ¼ 6:5 1013 cm2 s1 . The volume made available, if external scaling causes n injection of vacancies at the alloy surface, is very much larger that the volume required to accommodate the expansion due to interdendritic carbide oxidation. Intragranular precipitation of Al2O3 rods within Al-bearing heat-resistant steels takes place according to parabolic kinetics [39, 98] together with external chromium-rich oxide scale growth at high pO2 values. No oxide-free surface region of metal develops (Figure 6.46). This is quite unlike the extensive metal ejection observed in the absence of scaling on a dilute alloy (Figure 6.8). Measurement of internal penetration and scale thickening rates allowed calculation of the volume 14 expansion rate kðiÞ cm2 s1 and the rate of free volume generation n ¼ 6:3 10 ðexÞ by vacancy injection kn ¼ 1 1012 cm2 s1 at 1,0001C and pO2 ¼ 0:2 atm. In this instance the internal expansion corresponds to the reaction Al þ32 O ¼ Al2 O3

(6.132)

Again, the volume potentially made available by the scaling process is substantially larger than that needed to accommodate the expansion due to internal oxidation, and the absence of visible metal displacement is thereby explained.

6.14. Success of Internal Oxidation Theory

311

6.14. SUCCESS OF INTERNAL OXIDATION THEORY Internal precipitation in alloys resulting from reaction with external oxidants is a highly destructive process, frequently leading to alloy failure when it occurs. As we have seen in this chapter, these reactions can develop a diversity of morphologies, at rates which vary over orders of magnitude with oxidant identity and the alloy composition and phase constitution. However, the reactions all involve simple, solid-state precipitation processes: B þn X ¼ BXn where X is a generic oxidizing solute. Consequently, local equilibrium is closely approached at intermediate and high temperatures, and solubility product calculations work well. For this reason, the diffusion path description applies and diffusion-controlled parabolic kinetics result. In the case of the fast diffusing oxidants carbon and nitrogen, the diffusion path for the system A–B–X can usually be defined on the basis DX DB simply as a straight line from the X-corner of the ternary to the AB alloy composition. This approach correctly describes the sequence of precipitate phase constitutions, the variation in composition of mixed carbides such as (Cr,Fe)7C3, and the change in volume fraction with depth of these low-stability compounds. Even for the slower diffusing oxygen, this approach is useful. Prediction from these simplified diffusion paths is successful for multiple oxidants and multicomponent alloys, but inaccurate when fast diffusing alloy solutes such as silicon and aluminium are involved (Figure 6.46).

Figure 6.46 Internal aluminium oxidation and external chromia scale growth on 60HT heat-resisting steel (T ¼ 1,2501C, pO2 ¼ 4 1014 atm).

312


Rather simple diffusion theory usually succeeds in predicting parabolic rate constants very well for binary alloys, providing that both DX and N ðsÞ X are known. The measured permeabilities of carbon and oxygen in austenite and ferrite provide good order of magnitude predictions of the relative rates of the various internal precipitation reactions, internal carburization being almost three orders of magnitude faster than oxidation. Unfortunately, data for nitridation are scant. In the absence of such data, internal oxidation kinetic measurements can be used to evaluate permeabilities. The kinetic theory is particularly valuable in predicting the increase in rate with oxidant solubility and diffusivity, and hence with aX and temperature. It also successfully predicts the decrease in rate with increasing N ðoÞ B for cases of dispersed precipitates. However, the theory has mixed success in describing the growth of multiple precipitation zones. The total depth of attack is reasonably well predicted, but quantitative calculation of the individual precipitate zones is not yet possible. From a practical point of view, this may be unimportant to the prediction of alloy failure. However, if the reaction is to be used as a method of fabricating nanostructures, this deficiency needs to be addressed. The classical theories of internal oxidation all assume uniform distributions of precipitates. This insistence upon a strict chemical stoichiometry ignores the effects of low precipitate Ksp values, microstructure and alloy phase transformations and can lead to error. As we have seen, the competition between precipitate nucleation and growth can have important effects. It alters size distributions, and therefore penetration depths. In the extreme, it can produce cellular precipitation morphologies which are associated with rapid boundary diffusion and accelerated reaction. The alloy phase transformations or crystallographic reorientations accompanying this process have been well characterized in a number of cases, but a satisfactory description of fibrous alumina precipitate growth has not yet been arrived at. From an alloy design (or selection) point of view, the most important thing about internal oxidation is avoiding it. The diffusion-based theory provides a method for predicting how much alloy solute metal is required to ensure external scale growth rather than internal precipitation of the preferentially formed oxide. Its predictions are approximately correct, and a sound basis for alloy design is potentially available. However, for this method to be a useful design tool, we require greater accuracy. To achieve this, a much better knowledge of the solute interactions which determine thermodynamic activities and diffusivities of oxidant and alloy components is required.

REFERENCES 1. 2. 3. 4. 5.

R.C. John, in Corrosion 96, NACE, Houston, TX (1996), p. 171. C.S. Smith, Min. Met., 11, 213 (1930). C.S. Smith, J. Inst. Met., 46, 49 (1931). F.N. Rhines, Trans. AIME, 137, 246 (1940). F.N. Rhines and A.H. Grobe, Trans. AIME, 147, 318 (1942).

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CHAPT ER

7 Alloy Oxidation III: Multiphase Scales

Contents

7.1. Introduction 7.2. Binary Alumina Formers 7.2.1 The Ni–Al system 7.2.2 The Fe–Al system 7.3. Binary Chromia Formers 7.3.1 The Ni–Cr and Fe–Cr systems 7.3.2 Transport processes in chromia scales 7.4. Ternary Alloy Oxidation 7.4.1 Fe–Ni–Cr alloys 7.4.2 Ni–Pt–Al alloys 7.4.3 Ni–Cr–Al alloys 7.4.4 Fe–Cr–Al alloys 7.4.5 Third-element effect 7.5. Scale Spallation 7.5.1 The sulfur effect 7.5.2 Interfacial voids and scale detachment 7.5.3 Reactive element effects 7.6. Effects of Minor Alloying Additions 7.6.1 Silicon effects 7.6.2 Manganese effects 7.6.3 Titanium effects 7.6.4 Other effects 7.7. Effects of Secondary Oxidants 7.8. Status of Multiphase Scale Growth Theory References

315 316 316 322 326 326 328 330 330 331 334 336 338 341 342 343 344 347 347 350 350 351 352 355 356

7.1. INTRODUCTION Practical heat-resisting alloys have multiple components (Tables 5.1 and 9.1), nearly all of which are susceptible to oxidation in a wide range of environments. When these alloys are exposed at high temperatures, an initial, transient period of reaction in which all alloy components oxidize, is followed by a steady-state reaction. The rapid development of a corrosion-resistant, steady-state scale

315

316

Chapter 7 Alloy Oxidation III: Multiphase Scales

morphology is the basis for alloy (or coating) design and selection, and is the central concern of this chapter. We wish to predict the nature of the steady-state reaction morphology as a function of alloy composition and environmental variables. Of particular interest are the conditions leading to the development of a protective, slow-growing oxide scale on the alloy surface. The ability of this scale to resist penetration by gaseous impurities such as sulfur and carbon is of obvious interest, as is also its ability to block outward diffusion of other alloy components. It turns out that diffusion through Cr2O3 scales of Fe, Ni and Mn is difficult to avoid, and simultaneous growth of two or more oxides has to be considered. As always, we wish to predict reaction rates and, ultimately, component lifetimes. The prediction of steady-state reaction morphologies is a realistic goal for single-oxidant environments, because the oxidant activity must decrease monotonically from the scale–gas interface to the alloy interior. The activity gradient provides the driving force for diffusion and interfacial mass transfer. Recognition of its existence permits the construction on phase diagrams of diffusion paths, if the alloy diffusion properties are known and concentration changes at the alloy–scale interface can be predicted. Discussions of oxidation morphologies commenced with Wagner’s analyses [1–4] of binary alloys. These distinguished alloy classes on the basis of the relative affinities for oxygen of the constituent metals. Subsequent reviews [5–7] have established a taxa of reaction morphologies for binary alloys based on the thermodynamic stabilities and transport properties of the oxidation products. Unfortunately, this systematic approach is not easily extended to multicomponent alloys. Instead, we focus here on chromia and alumina scale formation and the processes that can accompany them. A brief review of binary alloy oxidation is followed by an examination of the effect of ternary alloy additions. Minority component effects are then considered, with particular attention directed to reactive element additions. Finally, the behaviour of alloys reacted with multiple oxidants is discussed. Consideration is restricted throughout to isothermal reaction conditions.

7.2. BINARY ALUMINA FORMERS 7.2.1 The Ni–Al system Nickel-base alloys can be described using the phase diagram of Figure 7.1. The g-phase is the basis of the Inconel alloys (e.g. 601 and 617 in Table 5.1), nickel-base superalloys have the g þ g0 phase constitution and b-NiAl is a principal constituent of aluminide coatings, so this system is of considerable practical interest. The classic study of its oxidation behaviour was carried out by Pettit [8], using pure oxygen at 0.1 atm. His results are reproduced in the oxidation map of Figure 7.2, which defines regions I, II and III, corresponding to different reaction morphologies and mechanisms. The dilute alloys of region I develop external scales of NiO and internal precipitates of Al2O3 and NiAl2O4 at all temperatures

7.2. Binary Alumina Formers

Figure 7.1 Ni–Al phase diagram.

Figure 7.2 Oxidation map for Ni–Al alloys [8]. Reaction morphologies I, II and III are described in the text. Dotted line shows temperature dependence of NAl,min according to Equation (7.6). Dashed line shows variation of NAl,min in CO/CO2 gas with Al2O3 the only stable oxide.

317

318


investigated. In region II, a protective a-Al2O3 scale develops initially according to slow parabolic kinetics. Subsequently, however, a thick scale containing both NiO and spinel grows more rapidly, while a discontinuous alumina layer grows at and beneath the scale–alloy interface. Increasing either temperature or N ðoÞ Al changes the behaviour to type III, in which a protective a-Al2O3 scale is the only reaction product. The broad bands separating the three regions arise through irregular behaviour which varies with alloy surface preparation. Subsequent investigations have broadly confirmed these results. Hindam et al. [9–11] also found internal precipitation of Al2O3 and NiAl2O4 beneath a scale of NiO on dilute alloys, and irregular, non-reproducible kinetics for a Ni–6Al alloy followed by the development of a three-layered scale. The innermost layer was Al2O3, the intermediate layer NiAl2O4 and the outermost layer NiO. A scale of this type is shown in Figure 7.3. Wood and Stott [12] identified the critical aluminium content necessary to form a stable Al2O3 scale at 1,0001C as being in the range 7–12.5 wt%. More recently, Niu et al. [13] determined this critical level to be N Al;min ¼ 0:1020:15 (5–7.5 wt%) at 1,0001C. At the still lower temperature of 8001C, alloys containing up to 10 wt% Al undergo internal oxidation [14]. Thus there is disagreement as to the critical level at lower temperatures. The different reaction morphologies are readily understood in terms of diffusion paths mapped onto Ni–Al–O phases diagrams, such as that of Figure 7.4. The dilute alloy situation is shown in Figure 6.33. In essence, precipitation of Al2O3 and NiAl2O4 within the alloy depletes it so severely in

Figure 7.3 Three-layered scale grown on Ni–22Al shown in FIB-milled section.


319

Figure 7.4 Ni–Al–O phase diagram section at 1,0001C [15]. Reproduced by permission of The Electrochemical Society. Diffusion path for scale of Figure 7.3 mapped as dotted line.

aluminium that NiO is stable in contact with the metal. The situation for highaluminium content alloys is shown in Figure 5.24. If N ðoÞ Al is high enough, alumina forms in contact with the alloy, yielding Pettit’s Type III reaction morphology. At lower N Al (and higher N Ni ) values, the alumina scale is overlaid by spinel and NiO. This sequence reflects the relative stabilities of the oxides, as we now establish. Reactions at the interfaces shown in the schematic diagram of Figure 7.5 can be written as (a)

2 Al þ32O2 ¼ Al2 O3

(7.1)

(b)

Ni þ Al2 O3 þ 12O2 ¼ NiAl2 O4

(7.2)

(c)

Ni þ 12O2 ¼ NiO

(7.3)

on the basis that nickel diffuses through Al2O3 to form the outer layers. The oxygen activity at the scale–alloy interface clearly depends on aAl . The minimum value of aAl required to form Al2O3, rather than nickel-rich oxides, can be estimated by the methods of Section 2.4. The requisite value of aAl corresponds

320


Gas

NiAl2O4

NiO

Al2O3

Alloy

Ambient NAl pO2 /atm

10-10 10-12

10-30 (d)

(c)

(b)

(a)

Figure 7.5 Schematic view of multiple layer scale grown on Ni–Al in Type II reaction. pO2 values calculated for 1,0001C, assuming aNi ¼ 1.

Table 7.1 Spinel-free energies of formation [187] DGf ¼ A þ BT ðJ mole1 Þ

Spinel

FeCr2 O4 NiCr2 O4 MnCr2 O4 FeAl2 O4 NiAl2 O4 MnAl2 O4

A

B

1,450,670 1,376,880 1,583,600 1,988,442 1,933,667 2,119,897

324 332 331 406 408 414

[8] to less than 1 ppm by weight, reflecting the very high stability of Al2O3 relative to NiO (Figure 7.4). The actual value will depend on alloy diffusion. For high N ðoÞ Al values, depletion is minimal, and pO2 values calculated from Equation (7.1) are of order 1030 atm at 1,0001C. Turning next to reaction (7.2), the local equilibrium at interface (b) can be written as h i 1=2 (7.4) aNi pO2 ¼ exp ðDGf ðNiAl2 O4 Þ DGf ðAl2 O3 ÞÞ=RT where unit activity oxides have been assumed. Again the metal component activity will be controlled by diffusion. If it is low enough, as a result of the alumina layer blocking nickel diffusions, then spinel will not form at all. However, if nickel diffuses easily through the inner layer, its activity will be close to that of the alloy, i.e. approximately unity. In this event, pO2 is calculated from Equation (7.4), using DGf ðNiAl2 O4 Þ from Table 7.1, to be of order 1012 atm. For reaction at interface (c), a value of pO2 1010 atm is calculated for aNi 1 using


321

DGf ðNiOÞ from Table 2.1. Thus the oxygen activity decreases monotonically from the outside to the inside of the scale, as it must for the scale to grow. Conversely, the oxide layers can be predicted to form in this sequence on the basis of their relative stabilities. The corresponding diffusion path is shown in Figure 7.4. Whilst the oxidation morphologies can be understood on the basis of Ni–Al–O thermodynamics, the conditions under which the regimes I, II and III develop cannot. These conditions are determined largely by kinetic factors, principally diffusion in the various phases. We consider first the boundary between internal and external oxidation, i.e. between regions I and II. Wagner’s criterion [4] stated in Equation (6.111) yields the minimum aluminium level, N Al;min , necessary to form external scale rather than internal precipitate, if the critical precipitate volume fraction, g, for formation of a continuous layer is known. Nesbitt [16] set g ¼ 0:2 and found for 1,2001C, that N Al;min ¼ 0:0720:09 (4.4 wt%) at high pO2 , where NiO can form. Using the more conventional value g ¼ 0:3, and taking data for DO ; N ðsÞ O and DAl from Chapter 2 and Appendix D, we calculate the value N Al;min ¼ 0:14 at 1,2001C, equivalent to 7 wt% for these conditions. These two estimates lie within the experimental transition band between internal and external alumina formation (Figure 7.2). It is of interest to explore the effect of temperature on N Al;min , with the aim of testing the utility of Wagner’s expression in predicting the measured effect shown in Figure 7.2. Combining Wagner’s condition (Equation (6.11)) with Sievert’s Equation (2.71) for oxygen solubility, we obtain !1=2 1=2 p VA KpO2 Do n N B;min ¼ gBOn (7.5) 2n V BOn DB The temperature dependence of N B;min can therefore be expressed as R @ ln

N B;min DH O DHðO2 Þ QO þ QB ¼ @ð1=TÞ 2

(7.6)

where DH O is the partial molar heat of dissolution of oxygen in solvent metal, DHðO2 Þ the enthalpy of the interface reaction producing ½O2(g), and Qi the activation energy for diffusion of the indicated species. The transition between regimes I and II is subject to the pO2 value characteristic of the reverse of reaction (7.3), for which DHðO2 Þ ¼ 234; 000J mol1 . Taking DHO from Table 2.2 and the Qi from Appendix D, the right-hand side of Equation (7.6) is calculated as 21,828 J mol1. The predicted dependence of N Al;min on temperature is shown as a dotted line in Figure 7.2. Agreement with experiment is reasonably good for these high oxygen activity conditions. Figure 7.2 also shows results for the transition from internal to external alumina under low oxygen potentials, where only Al2O3 can form. These experiments were carried out in a CO/CO2 gas mixture of fixed composition pCO =pCO2 ¼ 0:2, so that pO2 was controlled by the reaction CO2 ¼ CO þ 12O2 , for which DHðO2 Þ ¼ 282; 420 Jmol1 . The enthalpy term in Equation (7.6) is then evaluated as 4,420 J mol1, and the value of N Al;min is predicted to increase with temperature. This is contrary to the experimental observations in Figure 7.2.

322


The calculated value of N Al;min at 1,2001C is 0.05 (2.4 wt%), slightly less than the observed value of 3.0 wt%. It is possible that the slow gas phase reaction led to a failure to achieve equilibrium, and the calculation for pO2 is therefore inapplicable. Despite the success of diffusion theory in accounting for the variation with temperature and N ðoÞ Al of the initial oxidation morphologies of Ni–Al alloys, it is evident from Figure 7.2 that the formation of an initial alumina scale in region II did not correspond to long-term protection. Pettit [8] attributed this loss of protection to a lowering of the interfacial aluminium content, of N Al;i , resulting from diffusion being slower than the rate of aluminium consumption by alumina scale growth. According to the Wagner description, if this was the case, no alumina scale could form in the first place. The two views are reconciled by recognizing that behaviour in regime II is not steady-state, and Wagner’s analysis therefore cannot apply. The non-steady-state behaviour is explicit in the observed transition to approximately linear kinetics when protection is lost. This could result from a change in mass transfer mechanism within the scale, any such change in the alloy being improbable. Scale diffusion mechanisms can change in response to microstructural alterations or the precipitation of new phases. The slow diffusion of nickel into the alumina scale followed by formation of spinel and even NiO appears to be the reason for this behaviour. As pointed out by Pettit, it is prevented by ðoÞ increasing either N Al or the temperature, thereby maintaining a higher value of N Al;i (and a lower N Ni;i ). The effect of N ðoÞ Al is obvious, but the temperature effect implies that the activation energy for alloy diffusion (188 kJ mol1 [17]) is greater than that of the alumina diffusion process. Tracer diffusion studies have led to activation energy estimates of 477 kJ mol1 for aluminium [18] and 460 kJ mol1 for oxygen [19] in polycrystalline Al2O3 at high oxygen pressures and temperatures above 1,4501C. However, extrapolation of these diffusion coefficients to the temperatures of oxidation experiments leads to values much lower than those implied by alumina scale growth rates. Hindam and Whittle [20] compared directly measured diffusion coefficient values with those deduced from alumina scaling rates. The results (Figure 7.6) yielded approximate agreement for scale growth controlled by grain boundary diffusion of oxygen through a fine-grained (0.1–5 mm) structure (Equation (3.113)). The effective activation energy is 130 kJ mol1, less than that of alloy diffusion, as suggested by Pettit. Before leaving the Ni–Al system, it is appropriate to note that even when a protective scale is formed in regime III, the scale is not of practical use. The problem is that the scale cracks and spalls profusely on cooling from reaction temperature. Alloy developments aimed at preventing this problem are discussed in Section 7.5.

7.2.2 The Fe–Al system An isothermal section of the Fe–Al–O phase diagram is shown in Figure 7.7, and the Fe–Al diagram is shown in Figure 6.10. The Fe–Al–O diagram is similar to


323

Figure 7.6 Comparison of diffusion coefficients deduced from alumina kp values with diffusion data [20]. With kind permission from Springer Science and Business Media.

that of Ni–Al–O in that Al2O3 is by far the most stable oxide in both systems, with the consequence that all alloy compositions down to extremely low levels equilibrate with this phase. Important differences exist with respect to oxide intersolubilities. The spinel and Fe3O4 form a continuous solid solution, and Al2O3 and Fe2O3 have limited mutual solubility, the extent of which increases at higher temperature. On the contrary, nickel has very little solubility in Al2O3, and the NiAl2O4 spinel is a true ternary compound of closely stoichiometric composition. Dilute Fe–Al alloys oxidize under Rhines pack conditions (in which pO2 is controlled by the Fe/Fe1dO equilibrium) to produce internally precipitated aluminium-rich oxides [23, 24]. Early work aimed at establishing aluminium levels necessary to reduce alloy scaling rates have been reviewed by Tomaszewicz and Wallwork [25]. Boggs [26] found that at To5701C and pO2 1 atm, aluminium levels of about 2.4 wt% were sufficient to form an inner scale layer of FeAl2O4 spinel. This layer acted as a partial barrier to iron diffusion, reducing the thickness of the outer Fe3O4 layer by 75%. At higher temperatures, Al2O3 appeared in the scale in increasing amounts as the temperature increased

324


O

FeAl2O4 Fe2O3

Al2O3

Fe3O4 FeO

Alloy+Spinel+Al2O3

Fe

Al

Figure 7.7 Phase diagram for Fe–Al–O. Data from Refs [21, 22].

to 800–9001C. The alumina was g-phase at low temperatures, but increasingly a-phase at higher temperatures. At 8001C and 9001C, an essentially pure Al2O3 film developed after the transient stage of reaction, and oxidation rates were very low. However, protection was lost after some time, and iron-rich nodules grew through the alumina whilst aluminium was internally oxidized beneath the nodules. As seen in Figure 7.8, a transparently thin Al2O3 layer covered most of the surface, but the sporadic nodules grew quickly, causing rapid attack on the alloys. This general pattern of reaction morphologies has been confirmed by others [27–31]. The minimum value of N ðoÞ Al necessary to prevent internal oxidation is 0.048 at 5001C [26] in the range 0.038–0.048 at 8001C [27] and 0.05 at 9001C [29]. The value required to form a protective alumina scale has been estimated as more than 0.15 at 6001C [26], 0.13 at 8001C [27] and 0.14 in the range 800–1,0001C [32]. A more recent investigation [33] into the oxidation of an Fe–Al alloy with N ðoÞ Al ¼ 0:10 at 1,0001C confirmed that this was sufficient to prevent internal oxidation, but not enough to stop nodule formation after an alumina scale was established. The Fe–Al system is seen to be qualitatively similar to Ni–Al in possessing the same three regimes of behaviour. The same competition between oxygen diffusion into the alloy and aluminium diffusion to its surface determines the reaction morphology. Zhang et al. [33] have analysed the system at 1,0001C in this way, making use of Wagner’s criteria for scale formation. As noted in Section 6.3, depletion of aluminium from iron by either scale formation or internal precipitation causes the alloy a ! g transformation. Unfortunately, data for DAl in g-Fe is unavailable. Using an estimate for this quantity, they calculated an


325

Figure 7.8 Iron-rich nodules growing out of thin alumina film on Fe–4.9Al at 8001C, pO2 ¼ 0:92 atm [26]. Reproduced by permission of The Electrochemical Society.

N Al;min of 0.04 to be required to prevent internal oxidation. Wagner’s criterion for the N Al value required to sustain a continuous Al2O3 scale (Equation (5.22)) was found to yield 4.6 103 for a-Fe and 0.04 for g-Fe. The experimental results for ðoÞ N Al ¼ 0:10 showed that internal oxidation did not occur, as predicted, but that iron-rich nodules or a mixed scale developed, and no continuous Al2O3 scale was maintained. The same problem arises for Fe–Al as was noted for Ni–Al: Wagner’s steadystate analyses do not succeed. The same reason is in effect: neither system achieves a long lasting steady-state. In the case of Fe–Al, there is agreement that cracking of the alumina scale allows gas access to the underlying alloy. If this is depleted in aluminium, as might be the case if a subsurface g-iron layer is present, then scale rehealing would be impossible, and iron-rich nodule formation thereby explained [26, 27, 31, 34]. An alternative explanation suggested by Zhang et al. [33] is that iron oxides remaining from the initial period of transient oxidation react with the alumina, forming spinel. This decreases the Al2O3 layer thickness, balancing the growth process. If as a result the alumina layer thickness is approximately constant, aluminium metal is consumed according to linear kinetics, and depletion could be even more severe. Whether this could destabilize the alumina scale with respect to other oxides in the time scale required has not been established.

326


7.3. BINARY CHROMIA FORMERS 7.3.1 The Ni–Cr and Fe–Cr systems Isothermal sections of the Fe–Cr–O and Ni–Cr–O systems are shown in Figures 2.5 and 5.7. The obvious difference between the two is the much greater intersolubility of oxides in the iron-based system. Thus a single-phase field extends between the isotypic Fe3O4 and FeCr2O4 compositions, whereas the nickel spinel is a true ternary phase. This reflects the fact that an Fe3+ cation exists but no such nickel species is formed. Similarly, a continuous Fe2O3–Cr2O3 solid solution can form, whereas the nickel solubility in Cr2O3 is extremely small. Oxidation morphologies for Fe–Cr and Ni–Cr, together with their associated diffusion paths, were discussed in Section 5.3. In both cases, the behaviour of dilute alloys is controlled by monoxide (MO) scale layer growth. Depending on temperature, internal precipitation of Cr2O3 is also observed. As the scale–metal interface advances, the Cr2O3 precipitates are incorporated into the scale and transformed into spinel. This reaction morphology is shown schematically in Figure 7.9. The volume fraction of spinel increases with N ðoÞ Cr until the Cr2O3 phase appears. The extensive compositional range of the Fe–Cr spinel allows the formation of an almost continuous spinel layer on low chromium alloys, as illustrated by the 9Cr steel in Figure 7.10. These changes in morphology are reflected in oxidation rates. A compilation by Wood et al. [36] of oxidation rate data for model alloys is reproduced in Figure 7.11. Very small additions of chromium increase the rate of nickel

Figure 7.9 Schematic view of M–Cr alloy oxidation at subcritical NðoÞ Cr levels. If M ¼ Fe, outer layers of Fe3 O4 and Fe2 O3 form at high pO2 values.

7.3. Binary Chromia Formers

327

Figure 7.10 Spinel formation in inner scale layer grown on P91 steel at 6501C. Reprinted from Ref. [35] with permission from Elsevier.

oxidation, but not that of iron. This is generally thought to be a dopant effect, due to an increase in V 00M concentration to compensate for dissolved CrM . It is not observed for Fe–Cr alloys, because Fe1dO is already highly defective. The decrease in rate observed as N ðoÞ Cr is further increased is due to a growing volume fraction of spinel particles within the MO layer. Because diffusion in the spinel phase is relatively slow, the particles effectively reduce the diffusional crosssection of the MO layer, slowing its growth. In addition, porosity develops within the MO+spinel scale layer, because the two-phase oxide is unable to deform plastically to accommodate the volume loss caused by outward diffusion of iron or nickel. Gas phase transport of oxygen within the pores is slow (Section 2.9) if O2 is the only gas species available, and pore formation also slows scale growth. The reduction in rate as alloy chromium levels are increased to about 10 wt% is much greater for Fe–Cr than for Ni–Cr. This difference is partly due to the fact that diffusion in NiO and NiFe2O4 is much slower than in iron oxides, and the basis for comparison therefore differs. It also reflects the more ready formation of a continuous spinel layer on Fe–Cr alloys. The limited intersolubility of NiO and NiCr2O4 means that the latter phase remains as dispersed particles, providing much less diffusional blocking. At higher chromium levels, continuous scales of Cr2O3 develop, and the rate constant drops sharply (Figure 7.11). The chromium levels predicted from Equation (7.5) to be necessary for chromia scale formation are shown in Table 6.8. They are in only approximate agreement with the experimental results of Figure 7.11. The slower rate of chromia scaling on nickel-base alloys is attributed to more severe chromium depletion resulting from its slower diffusion in these

328


Figure 7.11 Oxidation rates of M–Cr alloys in pure O2 at 1,0001C [36]. Reproduced with permission from Wiley-VCH.

alloys. Under these circumstances, alloy diffusion contributes to oxidation rate control [37], as discussed in Section 5.4. Iron-base alloys with chromium levels near the critical value N Cr;min do not achieve long-term oxidation resistance. The high solubility in Cr2O3 of iron permits its outward diffusion and the formation of iron-rich oxides at the scale surface. Chromium levels of about 25 wt% are required to prevent this. Nickelbase alloys are superior in this respect, partly as a consequence of the much lower solubility of nickel in Cr2O3, and perhaps reflecting also differences in diffusion coefficients, as is discussed below. To understand the difference between Fe–Cr and Ni–Cr oxidation in detail, and also to analyse the effects of additional alloy components it is necessary to consider diffusion in the scale.

7.3.2 Transport processes in chromia scales Much of the early data on Cr2O3 scale growth rates and mechanisms have been reviewed by Kofstad [39], who concluded that chromia scales grow by outward diffusion of chromium. Although the defect properties of Cr2O3 are not fully understood (see Section 3.9), subsequent work has shown that grain boundary

329

7.3. Binary Chromia Formers

diffusion is much faster than lattice diffusion for both chromium [40, 41] and oxygen [41–43]. These data indicate that chromia growth on simple binary alloys is supported mainly by chromium diffusion, but oxygen diffusion also contributes to overall mass transport. For spinel MCr2O4 to grow on top of the Cr2O3 scale layer, the metal M must also diffuse outwards. Lobnig et al. [44] studied the diffusion of vacuum deposited Fe, Ni, Mn and Cr into thin (1–2 mm) Cr2O3 scales, which had been grown on Fe–20Cr or Fe–20Cr–12Ni alloys. By analysing the penetration profile shapes, they determined the diffusion coefficient values shown in Table 7.2 for short diffusion times. Assuming a value for the boundary width d ¼ 1 nm, the DB values for Fe, Cr and Ni were found to be several orders of magnitude greater than the corresponding lattice diffusion coefficients. Surface roughness led to inaccuracies in the estimates of both DL and DB , but the errors were small compared with the orders of magnitude differences in the data of Table 7.2. Using Equation (3.113) to calculate effective values Deff , and for simplicity assuming cubic oxide grains, we see that Deff ðCrÞ has closely similar values in the two scales: 1015–1014 cm2 s1. Furthermore Deff ðFeÞ Deff ðCrÞ for Fe–20Cr, thereby explaining the rapid growth of an outer FeCr2O4 layer on high iron activity alloys. The values of Deff ðNiÞ are, in the Fe–20Cr scale, an order of magnitude lower that that of chromium, but in the Fe–20Cr–12Ni scale about half that of chromium. In the absence of a value for a Ni–Cr scale, the data for DðNiÞ seems inconclusive. The relatively large value of Deff ðFeÞ in Cr2O3 is relevant to the technique of ‘‘pre-oxidation’’. This is the method of first oxidizing an alloy at low pO2 so that FeO is unstable, and the selective oxidation of chromium assured. After a protective Cr2O3 scale has formed, the alloy is placed into service at what will usually be a higher pO2 value. Unfortunately, the high oxygen pressure provides a gradient in aO which constitutes a driving force for iron diffusion through the chromia scale to form iron oxide. Pre-oxidation of Fe–9Cr and Fe–7.5Cr in H2/H2O (pO2 ¼ 6 1020 atm) at 8501C produced chromia scales of about 1 mm thickness [45]. Subsequent exposure, without change in temperature, to pure Table 7.2 Values of lattice diffusion coefficient DL, grain boundary diffusion coefficient DB times boundary width d, and grain size, Dt in Cr2O3 scales at 9001C [44] Base alloy

Diffusant

DL (cm2 s1)

dDB (cm3 s1)

Dt (mm)

Fe–20Cr

Fe Ni Mn Cr Fe Ni Mn Cr

2 1014 3 1015 2 1014 1 1014 4 1015 5 1015 2 1013 7 1015

1 1016 2 1019 2 1017 1 1016 1 107

0.1 0.2 0.1 0.1 0.1 0.2 0.4 0.1

Fe–20Cr–12Ni

5 1019 – 2 1017

330


oxygen at 1 atm led to continued slow growth of these scales until the rates accelerated with the precipitation of iron-rich oxide at the scale–gas interface after 2–3 weeks. Taking Deff ðFeÞ ¼ 1 1015 cm2 s1 , and estimating the iron diffusion penetration distance as X2 4Deff ðFeÞt

(7.7)

a penetration time of 29 days is calculated for a 1 mm scale. Agreement with the experimentally observed times for iron to reach the chromia scale surface can be regarded as satisfactory, given the approximate basis of Equation (7.7) and the uncertainty in the value of Deff ðFeÞ. We therefore conclude that pre-oxidation of marginal Fe–Cr alloys needs careful investigation before use. The high temperature growth of relatively thick Cr2O3 scales before service at substantially lower temperatures could nonetheless prove successful.

7.4. TERNARY ALLOY OXIDATION Our interest is in alloys for which selective oxidation of one component leads to the development of a slow-growing, protective scale. We therefore consider firstly alloys in which one component is much more reactive to oxygen than the other two, and secondly alloys in which two components are each much more reactive than the third. The first case is exemplified by Fe–Ni–Cr, the basis of heatresisting steels, and Ni–Pt–Al, the basis of a number of high temperature coatings. Examples of the second are Fe–Cr–Al (Kanthal) and Ni–Cr–Al (superalloys and Inconels). In many cases the reactive metals can be regarded as solutes in iron and/or nickel, although they may also partition to minority phases.

7.4.1 Fe–Ni–Cr alloys Single-phase Fe–Ni–Cr alloys should in principle be easily understood. However, accurate prediction of N Cr;min even for binary alloys was found to be difficult (Table 6.8). At this time it cannot even be attempted for the ternary alloys, because data for N ðsÞ O and DO in Fe–Ni binaries are not available. In the case of attack by carbon, the necessary data is available and provides a quantitative description of Fe–Ni–Cr carburization (Chapter 9). In the absence of such data for oxidation, discussion is necessarily qualitative. As seen in Figure 7.12, differences between the Fe–Cr and Ni–Cr systems are reflected in ternary alloy oxidation rates. For a given chromium level, oxidation rates decrease with increasing Ni/Fe ratio. At chromium levels less than about 10%, the differences reflect changing volume fractions of Fe1dO and the slower diffusing NixFe3xO4, and at high nickel levels, NiFe2O4 and NiO [38]. Scales formed on alloys with more than about 20% chromium consist of an inner Cr2O3 layer, overlaid by spinel. Increases in Ni/Fe ratio lead to decreases in alloy iron activity and its consequently smaller solubility in Cr2O3. This in turn affects the extent of spinel formation. The behaviour shown in Figure 7.12 is relevant to the performance of heat-resisting steels, which typically contain about

7.4. Ternary Alloy Oxidation

331

Figure 7.12 Oxidation weight gains of Fe–Ni–Cr alloys reacted in pure O2 at 1,0001C for 100 h [38]. With kind permission from Springer Science and Business Media.

10–20 wt% nickel, and are austenitic. At the higher N Ni levels, long-term protection against iron spinel formation can be achieved. Consequently, austenitic stainless and heat-resisting steels based on formulations in the range Fe–(10–20)Ni–(20–25)Cr are widely used at temperatures up to 900–1,1001C, depending on the atmosphere. Examples of alloy compositions are shown in Appendix A.

7.4.2 Ni–Pt–Al alloys It has been known for some time [46–48] that the addition of platinum to nickel aluminide intermetallics improves their oxidation resistance. Platinum-modified b-NiAl is used as a bondcoat on superalloy components in turbines [49] (Section 1.3) and new coatings based on g=g0 constitutions have recently been investigated [50–52]. An isothermal section of the Ni–Pt–Al phase diagram in Figure 7.13 shows that the solubility of platinum in each of the g; g0 and b phases is large. Copland [54, 55] has shown that substitution of platinum for nickel in these phases at constant N Al has the effect of reducing the aAl value.

332


Figure 7.13 Isothermal section (T ¼ 1,1501C) of Ni–Pt–Al phase diagram. Reprinted from Ref. [53] with permission from Elsevier.

Platinum is not completely inert to oxygen at high temperatures, instead forming a volatile oxide Pt þ O2 ¼ PtO2 ; DG ¼ 164; 300 3:89TðJ mol1 Þ

(7.8)

If pO2 ¼ 1 atm, then pPtO2 values of 2 106–4 105 atm are predicted for 1,100–1,2001C. However, exposure of platinum-bearing nickel aluminides to oxygen or air leads to the growth of external scales which protect the platinum from oxidation. Oxidation of b-NiAl produces a scale of pure Al2O3. Although this behaviour is in regime III of Pettit’s classification (Figure 7.2), the reaction rate is determined by which alumina phase grows (Section 5.7) and the frequency of scale spallation. The extent of spallation, which can occur under isothermal conditions, is determined by cavity formation at the scale–alloy interface (Section 5.8) and the amount of impurity sulfur in the system [56–61]. The nature of the sulfur effect is discussed in Section 7.5. For present purposes, the important finding is that the addition of platinum to b-NiAl suppresses spallation. The cavities developed at a b-NiAl/Al2O3 interface (Figure 5.12) are observed even in the very early stages of reaction [62, 63]. The addition of platinum to the intermetallic decreases both their size and number density, whether or not sulfur is present in the alloy [64–67]. This decrease in void volume fraction is not due to


333

any decrease in the amount of aluminium oxidation. In fact, alumina scaling rates are accelerated by the presence of platinum [68, 69] as shown in Figure 7.14. As is discussed in Section 5.8, the cavities are Kirkendall voids, and their accumulation represents the mismatch between aluminium and nickel alloy fluxes. These fluxes are driven by the chemical potential gradients arising from the concentration profiles in the alloy (Figure 5.3): J i ¼ Ci Bi

@ ln ai @x

(7.9)

Figure 7.14 Effect of Pt on NiAl oxidation at 1,1001C for low and high sulfur alloys [69]. With kind permission from Springer Science and Business Media.

334


The presence of platinum depresses aAl , increasing the chemical potential gradient and hence the flux of aluminium. This accounts for the reduction of Kirkendall porosity. It also supports the more rapid alumina scale growth observed for b-Ni(Pt)Al. The increased DAl in Ni(Pt)Al has been observed directly in diffusion couple studies [51]. The decrease in cavity formation decreases the amount of bare metal surface beneath the oxide where sulfur may segregate. Somewhat similar effects are observed for g0 and g=g0 alloys when platinum is added. In the absence of platinum, these alloys fall in regime II of Figure 7.2, and are marginal alumina formers, growing NiO and NiAl2O4 on top of their alumina scales. When platinum is added to the alloys, NiO formation is decreased, and suppressed completely at high platinum levels [70]. Gleeson et al. [51, 70] ascribe this to two effects: the chemical potential gradient effect described above, and the decrease in oxygen permeability in the gphase observed with increasing platinum levels. As seen in Equation (7.5), such a decrease reduces the initial value of N ðoÞ Al required to form external rather than internal Al2O3. The addition of platinum to Ni–Al alloys can profoundly affect their oxidation behaviour, despite the fact that platinum does not participate directly in the oxidation process. The effects arise out of the strong interactions within the alloy between platinum and the other constituents. These change aAl values, and hence aluminium diffusion rates, and appear also to lower oxygen permeabilities, at least in the gphase. Faster aluminium diffusion not only helps stabilize the alumina scale, but decreases the amount of Kirkendall voidage. Attention is now directed to the technologically important M–Cr–Al alloys, where M is Fe or Ni. Whilst also important, Co–Cr–Al alloys are less commonly used, and will not be discussed here. It is important to enquire into the circumstances under which these alloys act as chromia or alumina formers. A very large and complex literature has accumulated in this area, extensive reviews of which have been provided by Wood and Stott [71] and Stott et al. [72].

7.4.3 Ni–Cr–Al alloys The phase constitutions of these alloys [73] can be seen in Figure 5.36. The g-Ni phase has extensive solubilities for both aluminium and chromium, and the Ni–Al intermetallics have smaller, but significant solubilities for chromium. The construction of a three-dimensional quaternary Ni–Cr–Al–O diagram for each temperature is unrewarding. Instead, a concise way of describing the oxidation behaviour of such a broad range of alloy compositions and correspondingly diverse set of phase assemblages is provided by oxide mapping [74–77]. Compositional regions in which particular oxides predominate in the steadystate scale are plotted onto the Gibbs triangle. The map proposed by Wallwork and Hed [75] for Ni–Cr–Al at 1,0001C is shown in Figure 7.15. Although empirical, these maps can be very useful, for example in indicating the relationship between the Ni/Cr ratio and the ability to form a highly protective alumina scale.

335


Figure 7.15 Oxide map for Ni–Cr–Al ternaries at 1,0001C [75]. With kind permission from Springer Science and Business Media.

The thermodynamics of the oxide system are useful in understanding the development of different reaction morphologies. We enquire as to the location where chromium-rich oxides would be stable in the oxide layer sequence shown in Figure 7.5, by altering the alloy to Ni–Cr–Al, and considering the reaction 2 Al þCr2 O3 ¼ 2 Cr þAl2 O3 for which the equilibrium condition is i9 8 h

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