Nanocrystalline Materials

NANOCRYSTALLINE MATERIALS

i

NANOCRYSTALLINE MATERIALS A.I. Gusev, A.A. Rempel

CAMBRIDGE INTERNATIONAL SCIENCE PUBLISHING iii

Published by Cambridge International Science Publishing 7 Meadow Walk, Great Abington, Cambridge CB1 6AZ, UK http://www.cisp-publishing.com

First published 2004

© A.I. Gusev, A.A. Rempel © Cambridge International Science Publishing

Conditions of sale All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher

British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library

ISBN 1-898326-26-6 Printed by Antony Rowe Ltd, Chippenham, Great Britain

iv

CONTENTS

Preface ..................................................................................................................... vii List of Main Notations ........................................................................................ xiii

1.

INTRODUCTION ............................................................. 1 References ................................................................................................... 23

2. 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9.

SYNTHESIS OF NANOCRYSTALLINE POWDERS ............. 27 GAS PHASE SYNTHESIS ....................................................................... 27 PLASMA CHEMICAL TECHNIQUE .................................................... 32 PRECIPITATION FROM COLLOID SOLUTIONS ............................. 44 THERMAL DECOMPOSITION AND REDUCTION ......................... 49 MILLING AND MECHANICAL ALLOYING ..................................... 52 SYNTHESIS BY DETONATION AND ELECTRIC EXPLOSION .. 59 ORDERING IN NON-STOICHIOMETRIC COMPOUNDS .............. 65 SYNTHESIS OF HIGH-DISPERSED OXIDES IN LIQUID METALS ...................................................................................................... 76 SELF-PROPAGATING HIGH-TEMPERATURE SYNTHESIS ......... 78 References ................................................................................................... 79

3.

PREPARATION OF BULK NANOCRYSTALLINE MATERIALS ................................................................. 89

3.1. 3.2. 3.3 3.4. 3.5.

COMPACTION OF NANOPOWDERS .................................................. 90 FILM AND COATING DEPOSITION ................................................. 101 CRYSTALLISATION OF AMORPHOUS ALLOYS .......................... 104 SEVERE PLASTIC DEFORMATION .................................................. 108 DISORDER–ORDER TRANSFORMATIONS .................................... 114 References ................................................................................................. 124

4.

EVALUATION OF THE SIZE

4.1. 4.2. 4.3.

OF SMALL PARTICLES .... 131 ELECTRON MICROSCOPY ................................................................. 132 DIFFRACTION ........................................................................................ 137 SUPERPARAMAGNETISM, SEDIMENTATION, PHOTON CORRELATION SPECTROSCOPY AND GAS ADSORPTION .... 151 References ................................................................................................. 156 v

5.

PROPERTIES

ISOLATED NANOPARTICLES AND NANOCPOWDERS .............................................. 159

5.1 5.2 5.3 5.4 5.5

RYSTALLINE STRUCTURAL AND PHASE TRANSFORMATIONS ..................... 159 CRYSTAL LATTICE CONSTANT ....................................................... 169 PHONON SPECTRUM AND HEAT CAPACITY .............................. 177 MAGNETIC PROPERTIES .................................................................... 190 OPTICAL PROPERTIES ........................................................................ 207 References ................................................................................................. 214

OF

6.

MICROSTRUCTURE OF COMPACTED AND BULK NANOCRYSTALLINE MATERIALS ............................... 227

6.1 6.2.

INTERFACES IN COMPACTED MATERIALS ................................. 228 STUDY OF NANOCRYSTALLINE MATERIALS BY MEANS OF POSITRON ANNIHILATION TECHNIQUE ...................................... 234 STRUCTURAL FEATURES OF SUBMICROCRYSTALLINE METALS PREPARED BY SEVERE PLASTIC DEFORMATION .. 258 NANOSTRUCTURE OF DISORDERED SYSTEMS ........................ 268 References ................................................................................................. 275

6.3 6.4

7.

EFFECT OF THE GRAIN SIZE AND INTERFACES ON THE PROPERTIES OF BULK NANOMATERIALS ................... 284

7.1 7.2 7.3

MECHANICAL PROPERTIES .............................................................. 284 THERMAL AND ELECTRIC PROPERTIES ..................................... 301 MAGNETIC PROPERTIES .................................................................... 311 References ................................................................................................. 330

8.

CONCLUSIONS ........................................................... 340 References ................................................................................................. 345

SUBJECT INDEX ................................................................. 347

vi

“Tell all the Truth but tell it slant – Success in Circuit lies Too bright for our infirm Delight The Truth’s superb surprise” After Emily Dickinson

Preface In 1998 the monograph “Nanocrystalline Materials: Preparation and Properties” by A. I. Gusev was published by Ural Division of the Russian Academy of Sciences Publishing House (Yekaterinburg). The monograph was the first Russian and one of the first in the world generalisation of experimental results and theoretical considerations regarding the structure and properties of not only dispersed but also bulk solids with the nanometer size of particles, grains, crystallites and other elements of the structure. The monograph was of considerable interest to readers and became, almost immediately after publishing, a bibliographic rarity not only for readers but also for the majority of scientific and technical libraries. In more than 10 technical universities of Russia, this book is used as a basis of a course of lectures “Nanocrystalline substances and materials” for students, specialising in advanced materials science. Therefore, already in the year 2000 and subsequently in 2001, the Nauka Publishing House (Moscow) published twice a supplemented edition of the book “Nanocrystalline Materials” written by A. I. Gusev and A. A. Rempel. The English edition of the monograph by A. I. Gusev and A. A. Rempel, presented here to the reader, has been greatly refreshed, expanded and supplemented in comparison with the last Russian edition. The monograph is concerned with one of the most important current scientific problems, which is common for materials science, solid state physics and solid state chemistry, namely the nanocrystalline state of matter. It may be expected that the publication of the monograph in its new, expanded version will be available to a considerably larger number of investigators and engineers concerned with the production and application of nanocrystalline materials. vii

Until recently, the main scientific data on the nanocrystalline state of matter were published in various scientific journals, conference proceedings and compilations of articles. The authors of this monograph have taken the difficult task of presenting to the reader information on hundreds of original investigations of the nanocrystalline state, grouping these investigations in accordance with the investigated materials and properties, describing the general and special features in the results of these investigations, and focusing attention on the most interesting and practically important effects of the nanocrystalline state. The term nano, which is derived from the Greek word nanos which means dwarf, designates a milliardth (10 -9 ) fraction of a unit. Thus, the science of nanostructures and nanomaterials deals with objects in condensed matter physics on a size scale of 1 to 100 nm. The special physical properties of small particles have been utilised by peoples for a very long time, although this has been carried out unknowingly. Suitable examples are ancient Egypt glasses, colored with colloidal particles of metals, dye pigments used in different historical periods. The first scientific mention of the small particles is evidently the disordered movement of particles of flower pollen, suspended in a liquid, discovered in 1827 by the Scottish botanist R. Brown. This phenomenon is referred to as Brownian motion. The article on this microscopic observation (R. Brown, Phil. Mag. 4, 161 (1828)) laid foundations to many investigations. The theory of Brownian motion, developed independently by A. Einstein and M. Smoluchowski at the beginning of the 20 th century, is the basis of one of experimental methods of determining the size of small particles. The scattering of light by colloid solutions and glasses was studied by M. Faraday between 1850 and 1860. The starting point of examination of the nanostructured state of substance were the investigations in the area of colloid chemistry, which were already quite extensive since the middle of the 19 th century. At the beginning of the 20 th century, a significant contribution to the experimental confirmation of the theory of Brownian motion, to the development of colloid chemistry and examination of dispersed substances, and to the determination of the size of colloid particles was provided by the Swedish scientist T. Svedberg. In 1919, he developed a method of separating colloid particles from a solution using an ultracentrifuge. In 1926, he received a Nobel Prize in chemistry for his work on dispersed systems. viii

The 20 th century is characterised by extensive investigations of heterogeneous catalysis, ultrafine powders and thin films. These investigations raise question about the effect of the small size of particles (grains) on the properties of studied materials. At present, the nanostructured materials include nanopowders of metals, alloys, intermetallics, oxides, carbides, nitrides, borides and these substances in the bulk state with the grains of the nanometer size, together with nanopolymers, carbon nanostructures, nanoporous materials, nanocomposites, and biological nanomaterials. The development of nanomaterials is directly associated with the development and application of nanotechnology. The examination of nanomaterials has revealed a large number of grey areas in the fundamental knowledge of the nature of the nanocrystalline state and its stability under different conditions. On the whole, the field of nanomaterials and nanotechnology is very wide and at present has no distinctive contours. The unique structure and properties of small atomic aggregations are of considerable scientific and technical interest, because they represent an intermediate state between the structure and properties of isolated atoms and bulk solids. However, the problem at what stage of atom agglomeration the properties of bulk crystals is formed completely has not as yet been solved. It is not clear how the contributions of surface (associated with the interfaces) and bulk (associated with the size of the particles) effects to the properties of the nanocrystalline materials can be separated. The investigations in this field were carried out for a long period of time on isolated clusters, consisting from two atoms up to hundreds of atoms, small particles with a size of more than 1 nm, and ultrafine powders. The transition from the properties of isolated nanoparticles to the properties of bulk crystalline substances remained a grey area because the intermediate member, i. e. a bulk solid with the grains of the nanometer size, was not artificial created. Only after 1985, when methods of preparation of bulk nanocrystalline substances were developed, work was started to fill this gap in the knowledge of solids. The spectrum of properties of matter can be enormously enhanced if nanometer-size particles are agglomerated to a bulk material so that in addition to the crystallites with a nanometer-size they consist of a large portion of interfaces with a disordered structure and novel properties. In very small crystallites of the size of a few nanometers, this is a few millionth of a millimeter, new properties appear due to quantum size effects or scaling laws, which ix

can be controlled by the size of crystallites or particles. Indeed, the scientific interest to the nanocrystalline state of the solids in the powdered and bulk form is associated mainly with expectation of various size effects on the properties of the nanoparticles and nanocrystallites whose sizes are comparable or smaller than the characteristic correlation scale of a specific physical phenomenon or the characteristic length, which are present in the theoretical description of some property or process. These characteristic lengths are the free path of the electrons, the length of coherence in superconductors, the wavelength of elastic oscillations, the size of the exciton in semiconductors, the size of the magnetic domain in ferromagnetics, etc. The industrial interest to the nanomaterials is caused by the possibility of extensive modification and even principal changes in the properties of the existing materials at transition to the nanocrystalline state, and by new possibilities offered by nanotechnology in the creation of materials and wares from structural elements of the nanometer size. The essence of nanotechnology is the possibility to work at the atomic and molecular level, in the length scale between 1 and 100 nm, in order to produce and use materials and devices characterised by new properties and functions because of the small scale of their structure. Thus, the term “nanotechnology” relates to the sizes of structural elements in particular. Nanoproducts are already playing an important role in almost all branches of industry. The range of application of these products is huge: more efficient catalysts, films for microelectronics, new magnetic materials, protective coatings on metals, plastics and glasses. In the next couple of decades, nanostructured objects will operate in biological systems and will be used in medicine. The successes of nanotechnology may be manifested most efficiently in electronics and computer technology as a result of further miniaturisation electronic devices and the development of nanotransistors. In the content of this monograph, we have attempted to take into account both purely scientific, fundamental interest in the problem of the nanocrystalline state as a special non-equilibrium state of matter, and also some technical aspects of this problem, which are of considerable importance for materials science and the practical application of nanomaterials. The combined analysis of the structure and properties of isolated nanoparticles and nanopowders presented in the book, on the one side, and of bulk nanomaterials, on the other side, shows that the level of x

understanding and clarification of the structure and properties of isolated nanoparticles is considerably higher in comparison with bulk nanocrystalline materials. Evidently, this is a result of the considerably longer (practically from the beginning of the 20 th century) study of high-dispersed systems and nanoclusters in comparison with bulk nanomaterials which became the object of investigations only in the last 10–15 years. The monograph in the concentrated form includes a large part of the most important data on the nanocrystalline state of solids. In writing the monograph, we used a very large number of original investigations, starting from 1828 up to the year 2003, inclusive. It should be noted that more than 80 % of all references is made to studies carried out since 1988. Thus, the monograph reflects accurately the current state of investigations of the nanocrystalline state of solids. We think it will be interesting and useful to specialists in condensed state physics, solid state chemistry, physical chemistry and materials science.

Yekaterinburg, March 2004

A. I. Gusev, A. A. Rempel

xi

xii

LIST OF MAIN NOTATIONS

aB1 c Cp, Cv D, Ddiff E EF FWHM g( ω ), g( ν ) G h, k, l h, D = h/2 π Hc HV I kB K hkl m0 m* NA N(EF) p R(θ ) t T TC Tmelt T trans

lattice constant (period) of cubic unit cell with B1 structure concentration heat capacity at constant pressure and at constant volume size or mean size (diameter) of particle (grain, crystallite, domain) diffusion coefficient modulus of elasticity Fermi energy full linewidth at half maximum frequency distribution function shear modulus Miller indices Planck’s constant coercive force microhardness relative intensity Boltzmann constant Scherrer’s constant free electron mass effective electron mass Avogadro’s number density of electronic states on the Fermi level pressure angle resolution function time temperature Curie temperature melting temperature phase transformation (transition) temperature xiii

y

β βd βh βs γe η θ θD κT λ λ* µ µB µp σ σg τ τr χ

relative content of interstitial atoms X in nonstoichiometric compounds MX y broadening of diffraction reflection deformation broadening inhomogeneity broadening size broadening electronic heat capacity coefficient viscosity (liquid shear viscosity) Bragg diffraction angle characteristic Debye temperature isotermal compressibility radiation wavelength normalized positron annihilation rate chemical potential Bohr magneton magnetic permeability surface tension dispersion positron lifetime relaxation time magnetic susceptibility

xiv

Introduction

“...And freely men confess that this world’s spent, When in the planets and the firmament They seek so many new; they see that this Is crumbled out again to his atomies. ‘This all in pieces, all coherence gone, All just supply, and all relation...” After John Donne (An Anatomy of the World, 1611)

+D=FJAH Introduction The problem of production of ultrafine powders of metals, alloys and compounds and submicrocrystalline materials for different areas of technology, has been discussed in literature for many years now. In the last couple of decades, the interest in this subject has greatly increased because it was found (primarily, in metals) that a decrease in the size of crystals below some threshold value may result in a large change of the properties [1–16]. These effects form when the mean size of crystalline grains does not exceed 100 nm, and are most evident when the grain size is smaller than 10 nm. When examining the properties of superfine materials, it is necessary to take into account not only their structure and composition but also dispersion. Polycrystalline superfine materials with a mean grain size of 300 to 40 nm are referred to as submicrocrystalline, and those with a mean grain size of less than 40 nm as nanocrystalline. The conditional classification of materials on the basis of the size D of particles (grains) is shown in Fig. 1.1. Nanomaterials can also be classified on the basis of their geometrical form and the dimensionality of structural elements from which they consist. The main types of nanocrystalline materials as regards the dimensionality are cluster materials, fibrous materials, films and multilayer materials, and also polycrystalline materials whose grains have 1

Nanocrystalline Materials

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 µP

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Fig. 1.1. Classification of substances and materials on the basis of particle (grain) size D.

Clusters

(0D)

Nanotubes, filaments and rods

(1D)

Films and layers

Space network

(2D)

(3D)

Fig. 1.2. Types of nanocrystalline materials: 0D (zero-dimensional) clusters; 1D (one-dimensional) nanotubes, filaments and rods; 2D (two-dimensional) films and layers; 3D (three-dimensional) polycrystals.

comparable size in all three mutually perpendicular directions (Fig. 1.2). In this book, attention will be given mainly to the structure and properties of bulk and powdered substances and materials with a particle size of 5 to 200–300 nm, i.e. nanocrystalline and submicrocrystalline. 2

Introduction

For the investigator and the engineer used to working with traditional substances and materials in which the elements of the microstructure have the size of approximately 1 µm or more, the first encounter with nanomaterials results in at least surprise. This surprise is similar to that experienced by somebody who has seen for the first time buildings constructed in the modern style: instead of normal straight lines and angles, there are distorted planes and complicated broken contours (Fig. 1.3). However, after some time it becomes obvious that it is possible to live in such an unusual building and that it may even be interesting. This also relates to the nanocrystalline state, i.e. it is unusual but it may be interesting to work with it. The difference between the properties of small particles and the properties of bulk materials has been known for a relatively long period of time and has been utilised in different areas of technology. Suitable examples are widely used aerosols, dye pigments, glass colored by colloidal particles of metals. Suspensions of metallic nanoparticles (usually iron or its alloys) with the size from 30 nm to 1–2 µm are used as additions to engine oil during service for the restoration of worn components of automobiles and other engines. The small particles and nanosized elements are used for the production of various aviation materials. For example, radiowave-

Fig. 1.3. The first encounter with nanomaterials causes the same surprise as these buildings in the modern style, designed by F.O’Gehry in Düsseldorf (Germany).

3

Nanocrystalline Materials

absorbing ceramic materials, whose matrix is characterised by the random distribution of fine-dispersion metallic particles, are used in aviation industry. Whisker single crystals and polycrystals (fibres) are characterised by very high strength, for example, graphite whiskers have a strength of ~24.5 GPa or ten times higher than the strength of steel wire. They are used as fillers for light composite materials for aerospace applications. Carbon fibres and graphite whiskers are relatively thick (approximately 1–10 µm) and are not nanomaterials, but their production and applications were the first step on the path to the development of carbon nanomaterials. After discovery in 1984–1985 of a new allotropic modification of carbon, i.e. spherical fullerenes C 60 [17, 18], attempts were made to produce other topological forms of carbon nanoparticles. One of the proposed possible forms of carbon nanoparticles was, in particular, a quasi-one-dimensional tubular structure [19], referred to as the nanotube. The nanotubes form as a result of rotation of (0001) basal planes of the hexagonal lattice of graphite and can be single or multilayered. In fact, in 1991 and in the following years of the 20th century, researchers succeeded in detecting quasi-onedimensional tubular structures of carbon, i.e. carbon nanotubes) [20–22]. As an example, Fig. 1.4 shows a computer graphical model of a double-shell carbon nanotube, and Fig. 1.5 shows experimentally produced nanotubes [23]. Carbon nanotubes with a diameter of

Fig. 1.4. A computer graphic model for a double-shell carbon nanotube showing a helical arrangement of hexagons [23]. 4

Introduction

Fig. 1.5. Multi-shell carbon nanotubes [23].

D > 5 nm, consisting of 2 to 50 coaxial tubes, were detected for the first time by transmission electron microscopy in a condensate in an electric arc discharge between graphite electrodes [20]. The results of modelling of the structure and electronic properties of carbon nanotubes have been generalised in [24]. The carbon nanotubes have high mechanical strength and may be used for developing high-strength composites. Nanotubes have been used in different mechanical nanodevices [25], like nanoindentors for microhardness measurements. Depending on the type of helicoidal ordering of the carbon atoms in the walls of the carbon nanotubes, these nanotubes have semiconductor or metallic conductivity. Consequently, they are used as conducting elements in electronic nanotechnologies. In atomic force microscopes, the carbon nanotubes have replaced the metallic probe [26]. By joining the carbon nanotubes it is possible to produce a large number of structures with differing properties. Synthesis of these structures is very important for electronic technology. T-connected nanotubes, which may operate as a contact device, were produced in [27]. The authors of [28, 29] grew Y-shaped carbon nanotubes (Fig. 1.6); this structure is referred to as the Y-junction carbon nanotube. Synthesis was carried out by chemical deposition from the gas phase (CVD): pyrolysis of acetylene with subsequent 5

Nanocrystalline Materials

Fig. 1.6. Model of a carbon Y-nanotube.

growth of Y-nanotubes was carried out at a temperature of 920 K in branching nanochannels of the aluminium matrix. Cobalt, being a growth catalyst, was deposited on the walls and the bottom of the nanochannels. The diameter of the stem of the produced Ynanotube was approximately 60 nm, the diameter of the branches ~40 nm. The authors of [30] produce carbon Y-nanotubes by pyrolysis of organometallic precursors. As a result of a defective structure in the area joining the prongs, the Y-nanotube passes electric current only in one direction, i.e. it operates as a diode [29]. If controlling voltage is additionally applied to one of the prongs of the Y-nanotube, the nanotube operates as a current stabiliser. The possibility of controlling the current leads to the possibility of extensive application of Y-nanotubes in electronics. Recently, a group of investigators [31] from the Department of Materials Science and Engineering at the Rensselaer Polytechnic Institute (Troy, USA) proposed a method of controlled growth of carbon nanotubes on a substrate coated with a layer of SiO 2 . Pattering of SiO 2 was generated by photolithography and then subjected to combined wet and dry etching in order to produce islands of SiO 2 distributed in a specific fashion. Subsequently, bundles of nanotubes, forming a unique nanostructure (Fig. 1.7) were grown in a gas mixture of xylene/ferrocene C 8 H 10 /Fe(C 5 H 5 ) 2 on 6

Introduction

Fig. 1.7. Controlled carbon nanotube growth on a silica-coated substrate [31].

the islands of SiO 2 by the CVD method. In this process, the iron included in the composition of Fe(C 5 H 5 ) 2 plays the role of a catalyst. Each bundle includes several tens of multiwalled nanotubes with a diameter of 20–30 nm. According to the authors of [31], such nanostructures may be used in integrated systems of the next generation and in microelectromechanical devices. The first publications dealing with the production of boron nitride nanotubes appeared in 1995–1996 [32–34]. Intensive research is being carried out into the synthesis of silicon carbide nanotubes. The range of applications of these nanotubes is even wider because of higher hardness and high melting point of silicon carbide. Heterogeneous synthesis of silicon carbide fibres was described by the authors of [35], the gas phase method of production of silicon carbide nanofibres with a diameter of ~100 nm, produced from silicon and carbon powders, was described in [36], and the authors of [37] reported on hollow silicon carbide nanostructures. The nanotubes and nanofilaments of silicon carbide and also of boron carbide and SiO 2 , produced by these methods, were presented in a lecture “Elongated structures of silicon carbide: nanotubes, nanofilaments and microfibres” by A. I. Kharlamov at the NATO Advanced Study Institute “Synthesis, Functional Properties and Applications of Nanostructures” (July 26–August 4, 2002, Heraklion, Crete, Greece). At the same conference, in a lecture “The stateof-art synthesis techniques for carbon nanotubes and nanotubesbased architecture” P. Ajayan told about silicon carbide nanotubes produced on an Al 2 O 3 substrate. At the 2nd NASA Advanced 7

Nanocrystalline Materials

Materials Symposium “New Directions in Advanced Materials Systems” (May 29–31, 2002, Cleveland, Ohio), D. Larkin presented a report “High temperature nano-technology: silicon carbide nanotubes synthesis”. The results of investigations and application of various nanotubes are presented in [38]. The book starts with the playful amateur poem “Material Ethereal” by Peter Butzloff (University of North Texas, Denton, USA) on a mystery nanotube and problems of examining it. Only the first and final lines of this poem are given here: We speculate but underrate what mystery we wrangle, a Nanotube from carbon crude that nature did entangle. .............................................. Oh Nanotube of carbon crude so cumbersome we trundle through pass of phase, by time decays and character to bundle! The catalysis of chemical reactions is a very important and large area of long-term and successful application of fine particles of metals, alloys and semiconductors. Heterogeneous catalysis by means of high-efficiency catalysts produced from ultrafine powders or ceramics with grains of the nanometer size is an independent and very large section of physical chemistry. Various problems of catalysis have been discussed in hundreds of books and reviews and tens of thousands of articles. The extensive discussion of the problems of catalysis on fine particles with respect to both the content and volume is outside the framework of this book and, consequently, certain general assumptions relating to the catalytic activity of fine particles will be discussed only briefly. Catalysis on fine particles plays an exceptionally important role in industrial chemistry. Catalysed reactions usually take place at lower temperatures than non-catalysed reactions and are more selective. In most cases, the catalysts are represented by isolated fine particles of metals or alloys deposited on a carrier with a developed surface (zeolites, silicagel, silica, pumice, glass, etc.). The main task of the carrier is to support obtaining the smallest size of

8

Introduction

deposited particles and prevent their spontaneous coalescence and sintering. The high catalytic activity of fine particles is explained by electron and geometrical effects, although this division is very conventional because both effects have the same source, i.e. the small particles size. The number of atoms in an isolated metallic particle is small and, consequently, the distance between the energy levels δ ≈ E F /N (E F is Fermi energy, N is the number of atoms in the particle) is comparable with thermal energy k B T. At δ > k B T the levels are discrete and the particle loses its metallic properties. The catalytic activity of the small metallic particles starts to be evident when the value of δ is close to k B T. Consequently, it is possible to evaluate the particle size at which the catalytic properties become evident. For metals, Fermi energy E F is approximately 10 eV, at room temperature 300 K the value δ = k B T = 0.025 eV and, consequently N ≈ 400; a particle consisting of 400 atoms has a diameter of ~2 nm. In fact, the majority of data confirm that the physical and catalytic properties start to change markedly when the particles reached the size of 2–8 nm. In addition to the discussed primary electron effect, there is a secondary electron effect. This effect is caused by the fact that a large fraction of atoms is situated on the surface of small particles. The electron configuration of these atoms is different from one of the atoms distributed inside the particle. The secondary electron effect has a geometrical source and also leads to changes in the catalytic properties. The geometrical effect of catalysis depends on the number of atoms distributed on the surface (on the faces), on the edges and tops of the small particle because these atoms have a different coordination. The atoms situated on the faces have a higher coordination in comparison with the atoms on the tops and the edges. If the atoms in the small coordination are catalytically most active, the catalytic activity increases with decreasing particle size. In another case, if the atoms located on the faces are catalytically active, the rate of the catalysed reaction will be increased by larger particles. A specific role in catalysis is played by the carrier because the atoms of the catalyst which are in direct contact with the carrier may change their electronic structure because of the formation of bonds with the carrier. It is evident that as the number of atoms that are in contact with the carrier increases, the effect of the carrier on catalytic activity becomes stronger. It is clear that the 9

Nanocrystalline Materials

effect of the carrier is relatively small for large particles but increases and becomes quite strong with a decrease in the particle size. Metallic alloys (for example, alloys of catalytically inert metals of group I with metals of group VIII) are used as a catalyst because of the dilution of the metal-catalyst in the alloy increases catalytic activity. This is similar to an increase in catalytic activity with a decrease in the nanoparticle size. To a first approximation, the similarity of the effects of a decrease in the particle size and melting is caused by the fact that the valence electrons of every metal in such alloys retain their affiliation and, consequently, a catalytically inert metal (for example, copper) acts as a diluent for the particles of the catalytically active metal. Usually, the nanoparticles show catalytic activity in a very narrow size range. For example, Rh catalysts, produced by the dissociation of Rh 6 (CO) 16 clusters, fixed to the surface of disperse silica, catalyse the reaction of hydrogenation of benzene only when the particle size is 1.5–1.8 nm, i.e. only particles of Rh 12 are catalytically active in relation to this reaction. The high selectivity of the catalytic activity is also characteristic of nanoparticles of widely used catalysts such as palladium and platinum. For example, the hydrogenation of ethylene was studied at a temperature of 520 K and a hydrogen pressure of 1 atm on a platinum catalyst deposited on SiO 2 or Al 2 O 3 . A distinctive maximum of the reaction rate is observed when the size of Pt nanoparticles is about 0.6 nm. This high sensitivity of catalytic activity to the size of small particles confirms the importance of the development of selective methods of production of nanoparticles with an accuracy to 1–2 atoms. The very narrow size distribution of the nanoparticles is essential not only for catalysis but also for microelectronics. A new area of catalysis of small particles is photocatalysis using semiconductor particles and nanostructured semiconductor films. For example, this method is promising for photochemical purification of effluents to remove various organic contaminants by means of their photocatalytic oxidation and mineralisation. Detailed analysis of the effect of the size of small particles of metals and alloys, deposited on a carrier, can be found in [39] and also in reviews [40, 41] concerned with catalysis using metallic alloys and palladium. Catalysis on small metallic particles may be regarded as the chemical size effect. For example, nickel or palladium nanoparticles on a SiO 2 substrate are used as catalysists for hydrogenation of 10

Introduction

benzene. Nanoparticles are produced by decomposition of organometallic complexes. A decrease in the metallic particles size is accompanied by an increase in specific catalytic activity, i.e. the activity related to 1 surface atom of the metal. Let us consider the reaction of hydrogenation of benzene at a temperature of 373 K and at benzene C 6 H 6 and hydrogen H 2 pressures of 6700 and 46700 Pa, respectively. In this reaction, the specific catalytic activity of nickel nanoparticles is increased 3–4 times when the particle size become smaller than 1 nm and the dispersion tends to unity. (Dispersion is the ratio of the number of atoms situated on surface to the total number of the atoms in the particle.) In catalysis on palladium nanoparticles with the dispersion close to unity, the identical effect in the same reaction is observed at 300 K. A study of hydrogenolysis of ethane C 2 H 6 at a temperature of 473 K and a pressure of C 2 H 6 and H 2 equal to 6700 and 26700 Pa showed that the very rapid increase in the specific catalytic activity of the nickel nanoparticles is observed when their dispersion close to unity. The rate of the reaction of hydrogenolysis of cyclopentane and methylcyclopentane, related to 1 surface atom of the metal-catalyst, changes rapidly when the fraction of the surface atoms in the nanoparticle of the metal-catalyst (Pt, Ir, Pd, Rh deposited on glass, SiO 2 or Al 2 O 3 ) approaches unity [39]. Another chemical size effect is the shift of binding energy 3d 5/2 of the internal level of palladium in relation to the size of palladium particles [39, 41]. For palladium particles larger than 4– 5 nm, the binding energy of the 3d 5/2 level is ~335 eV, i.e. it is equal to the value characteristic of bulk palladium. A decrease in the size of palladium nanoparticles from 4 to 1 nm is accompanied (irrespective of whether the material of the substrate is a conductor (carbon) or and insulator (SiO 2 , Al 2 O 3 , zeolites)) by an increase of the binding energy of the 3d 5/2 level. The most probable reason for the positive shift is the size dependence of the electronic structure of palladium, namely the decrease of the number of valence delectrons. An identical shift of the binding energy of Pt 4f 7/2 of the internal level is recorded in the case of platinum nanoparticles [39]. The increase of the chemical activity of thin-film heterostructures is also a chemical size effect. For example, in two-layer oxide heterostructures MgO/Nb 2 O 5 the reaction of the type

MgO + Nb 2O 5 → MgNb 2O 6

(1.1)

11

Nanocrystalline Materials

takes place spontaneously at temperatures 800–1000 K lower than the temperature of the reaction between normal coarse-grained oxides. Hybrid nanocomposites of the metal–polymer type are produced by forming nanoparticles in a specially prepared polymer matrix [42, 43]. Polymer composites with metallic nanoparticles are used as electrically conducting film composite materials, and the amount of the filler in the matrix may reach 90 vol %. The introduction of metal ions into polymer fibers makes it possible to produce colored lightguides suitable for application in computer equipment. The optical properties of polymers with fillers of nanoparticles of metals, alloys or semiconductors (CdS, CdSe, InP, InAs) are interesting. Because of light machining and the possibility of producing films from these polymer nanocomposites, they can be used for the production of optical elements and light filters. The nanoparticles and nanolayers are used widely in modern technology. Multilayered nanostructures are used in the production of microelectronic devices. A suitable example are layerheterogeneous nanostructures, i.e. superlattices in which superthin layers (with a thickness from several to hundred of periods of the crystal lattice or ~1–50 nm) of two different substances, for example, oxides, alternate. The structure represents a crystal in which in addition to the conventional lattice of periodically distributed atoms, there is a superlattice of repeating layers of different composition. Owing to the fact that the thickness of the nanolayer is comparable with de Broglie wavelength of the electron, the quantum size effect is realised in superlattices in electronic properties. Utilising the effect of size quantisation in multilayered nanostructures enables the production of electronic devices with increased operating speed and information capacity. The simplest electronic device of this type is, for example, the AlAs/GaAs/AlAs two-barrier diode, consisting of a layer of gallium arsenide with a thickness of 4–6 nm, distributed between two layers of aluminium arsenide AlAs, with a thickness of 1.5–2.5 nm. Of special interest are magnetic nanostructures characterised by giant magnetoresistance. They are in the form of multilayered films of alternating layers of ferromagnetic and non-magnetic metals, for example, a ferromagnetic layer Co–Ni–Cu and a nonmagnetic copper layer alternate in the Co–Ni–Cu/Cu nanostructure. The thickness of the layers is of the order of the free path of the electron, i.e. several tens of nanometers. Changing the strength of the applied external magnetic field from 0 to some value of H it 12

Introduction

is possible to change the magnetic configuration of the multilayered nanostructure in such a manner that the electrical resistance will change in a very wide range. This makes it possible to utilise the magnetic nanostructures as detectors of the magnetic field. The highest value of giant magnetoresistance in the Co–Ni–Cu/Cu nanostructure is obtained for very thin layers of copper, thickness approximately 0.7 nm. The development of electronics over a period of several decades has also progressed along the path of miniaturisation. The first ‘jump’ in the development of electronic technology was the transition from vacuum electron valves to the transistor. The second jump is associated with the application of integrated microcircuits. The transition to integrated microcircuits became possible after understanding that all elements of the electronic circuit can be produced from the same material of the semiconductor type, instead of producing them from different materials. Silicon is such a material. The application of the material of the same type enabled the construction of all elements of the electronic circuit directly in the same specimen of this material and, connecting the elements together, produce an efficient microchip. The first necessity for decrease in the size of the electronic circuits came from military and space authorities of the USA, former USSR, European countries, and Japan, who supported appropriate research projects. If the first simplest chips (1959) consisted of tens of elements, then in 1970, microcircuits included up to 10 000 elements. Advances in electronics were accompanied by a rapid decrease of the cost of electronic devices (Fig. 1.8). In 1958, the cost of one transistor

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6DOH YROXPH PLOOLDUGV GROODUV





 

Fig. 1.8. Decrease in the minimum characteristic size of electronic components and growth of the volume of sales of electronic products [44].

13

Nanocrystalline Materials

was approximately 10 US dollars, and in the year 2000 this money would purchase a microcircuit with tens of millions of transistors [44]. In currently available mass-produced microcircuits, approximately 1000 electrons are required for switching on/ switching off a transistor. At the end of the first decade of the 21st century, the required number of electrons will decrease to 10 as a result of miniaturisation [44] and work is already been carried out to develop a single-electron transistor [45]. Semiconductor heterostructures, produced from two or more different materials, are of special interest for electronics. In these heterostructures, an important role is played by the transition layer, i.e. the interface between two materials. In addition to this, according to [46], the technical device in semiconductor heterostructures is the interface itself. Semiconductor heterostructures are fabricated from materials containing such elements as Zn, Cd, Hg, Al, Ga, In, Si, Ge, P, As, Sb, S, Se, Te. These elements belong to the groups II–VI of the periodic table. Silicon occupies the most important place in the technology of electronic materials, like steel in the production of constructional materials. In addition to silicon, electronics require semiconductor compounds A III B V and their solid solutions, and also A II B VI compounds. Of the compounds of the A III B V type, gallium arsenide GaAs is used most widely, and of the solid solutions it is Al x Ga 1–x As. The application of solid solutions makes it possible to produce heterostructures with a continuous but not jump-like variation of composition. The width of the forbidden band in these heterostructures also changes continuously. In the production of the heterostructures, it is important to match the parameters of the crystal lattices of two contacting materials. If the two materials with greatly differing lattice constants grow on each other, then an increase in the thickness of the layers results in the formation of high strains at the interface and mismatch dislocations appear. Strains appear irrespective of whether the transition between the two layers is smooth or not. To reduce the strains, the lattice constants of the two materials should differ as little as possible. Therefore, special attention in the study of heterostructures is given to solid solutions of the AlAs–GaAs system because the arsenides of aluminium and gallium have almost the same lattice constants. Single crystals of GaAs are an ideal substrate for growing the heterostructures. Another natural substrate is indium phosphide InP which is used in combination with solid solutions GaAs–InAs, AlAs–AlSb, and others. 14

Introduction

A breakthrough in making thin-layer heterostructures took place with the development of a technology for the growth of thin layers by molecular beam epitaxy and liquid-phase epitaxy. It became possible to growth heterostructures with a very sharp interface. Consequently, it is possible to position the two heteroboundaries so close to each other that the size quantum effects play the controlling role in this intermediate space. The structures of this type are referred to as quantum wells. In quantum wells, the mean narrow-band layer has a thickness of several tens of nanometers which results in splitting of the electronic levels because of the size quantisation effect. This effect in the form of a characteristic stepped structure of optical spectra of absorption of the GaAs– AlGaAs semiconductor heterostructure with the superthin GaAs layer (quantum well) was detected for the first time by the authors of [47]. They also found the shift of characteristic energies with a decrease in the thickness of the quantum well (GaAs layer). In quantum wells, superlattices and other structures with very thin layers and high strains may form without the formation of dislocations and, consequently, it is not necessary to match the lattice parameters [48]. Heterostructures, especially double heterostructures, including quantum wells, quantum wires and quantum dots, enable the control of various fundamental parameters of semiconductor crystals, such as the width of the forbidden band (energy gap), the effective mass and mobility of charge carriers, the electron energy spectrum. The density of states N(E) in a three-dimensional (3D) semiconductor is a continuous function. A decrease in the dimensionality of the electron gas results in a change of the energy spectrum from continuous to discrete as a result of its splitting (Fig. 1.9). The quantum well is a two-dimensional (2D) structure in which charge carriers are restricted in the direction normal to layers, and may move freely in the plane of the layer. In quantum wires, the charge carriers are already restricted in two directions and move freely only along the wire axis. The quantum dot is a zero-dimensional (0D) structure where the charge carriers are already restricted in all three directions and are characterised by a completely discrete energy spectrum. The size of the quantum dots produced by molecular beam epitaxy and lithography ranges from 1000 to 10 nm; smaller quantum dots (1–20 nm), whose surface is protected by organic molecules preventing aggregation of the particles, may be produced by colloidal chemistry methods [49]. 15

'

G G a  





Nanocrystalline Materials





G G a   

G G a δ 



 '



'

G G a FRQVW

'





Fig. 1.9. Density of states N(E) for charge carriers as a function of the dimensionality of the semiconductor: (3D) three-dimensional semiconductor, (2D) quantum well, (1D) quantum wire, (0D) quantum dot.

The application of nanostructures in electronics will lead to further miniaturisation of electronic devices with transfer to nanosized elements for producing processors of a new generation. The i386 TM processor, produced by Intel Corporation in 1983, contained 275000 transistors and performed more than 5 million operations per second; the i486 TM processor, produced in 1989, already contained 1 200000 transistors. The most widely used processor at the end of the 20th century and the beginning of the 21st century, Pentium Pro®, contains 5.5 million transistors and carries 300 million operations per second. The size of the transistors reached the smallest value available for current technologies and, consequently, a further decrease in the size may be achieved only by the application of nanotechnology. A practical difficulty which must be overcome in making quantum dots and single-electron transistors is the time instability of structures with a small number of atoms. The stability of these quantum-electronic elements is determined by the jump (diffusion) of already a small number of atoms. Since the diffusion processes on the surface and 16

Introduction

at the interface of quantum-electronic elements are very fast, processes of failure of the elements or even their movement on the substrate as an integral unit are already detected at room temperature [50]. A problem of the stability of nanoelectronic circuits can be solved only using multicomponent materials, including oxides, carbides and nitrides of metals. These compounds have a high melting point and low mobility of atoms and, consequently, have high thermal and time stability. X-ray and ultraviolet optics uses special mirrors with multilayered coatings of alternating thin layers of elements with high and low density, for example, tungsten and carbon, molybdenum and carbon, or nickel and carbon; the thickness of a pair of layers is about 1 nm, and the layers should be atomically smooth. The possibility of producing multilayered x-ray mirrors is one of the factors determining the application in certain areas of nanotechnology, such as x-ray lithography, on the one hand, and in astronomical and astrophysical investigations, on the other hand. The formation of x-ray mirrors has been described in sufficient detail in [51], where multilayered nickel–carbon Ni/C nanostructures with a period of ~4 nm were investigated. Other optical devices with nanosized elements, intended for application in x-ray microscopy, are Frenel zone sheets with the smallest width of the zone of approximately 10 nm, and diffraction gratings with a period smaller than 100 nm. F/S heterostructures, formed by alterating thin layers of a ferromagnetic and a superconductor, are very interesting. In the F/S heterostructures, the superconducting and ferromagnetic regions are divided in space but are linked together through the interface between the layers. In most cases, ferromagnetic interlayers F are produced using Fe, Co, Gd, Ni whose Curie temperature T C is considerably higher than the superconducting transition temperature T sc of metals (Nb, Pb, V), forming the layer S. Experimental examination of these heterostructures started in [52] in which the method of rf sputtering was used to produce two-layer sandwiches F/Pb. Other methods of production of superlattices of the F/S type are molecular-beam epitaxy, electron beam evaporation, and dc magnetron sputtering [53]. Generally, superconductivity and ferromagnetism are antagonistic phenomena. Primarily, this antagonism is reflected in relation to the magnetic field. The superconductor tends to push out the magnetic field (Meisner effect), and the ferromagnetic concentrates the force lines of the magnetic field in its volume (magnetic induction effect). From the 17

Nanocrystalline Materials

viewpoint of microscopic theory, the antagonism is reduced to the following: In a superconductor, the attraction force between the electron generates Cooper pairs, whereas the volume interaction in the ferromagnetic tends to align the electronic forces paralelly. Taking this into account, the coexistence of the superconducting and ferromagnetic order in a homogeneous system is unlikely, but it can easily be achieved in artificial multilayered systems consisting of alternating ferromagnetic and superconducting layers (Fig. 1.10). The F/S type heterostructures with the layers of atomic thickness may be used in electronic devices of the next generation as logic elements and switches of superconducting current [54, 55], and superconductivity can be controlled by means of a weak external magnetic field [56]. It should be mentioned that the properties of the F/S multilayered systems, including the superconducting transition temperature, depend on the thickness of the ferromagnetic and superconducting layers. In most cases, the thickness of the ferromagnetic layer is smaller than 1 nm, the thickness of the superconducting layer is from 10 to 40–50 nm [56]. It is interesting to note that the superconducting transition temperature T sc in the F/S heterostructures may not only monotonically decreases but also oscillates with increasing thickness of the layer F. For example, in a Fe/Nb/Fe three-layer system, with ,

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$  $

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[

[

] \

] \

] \

G1 G8 [

] \ .

Fig. 1.10. Multilayer heterostructures ferromagnetic-superconductor F/S: (a) double layers, (b) triple layers, (c) superlattices

18

Introduction

an increase of the thickness of the iron layer d Fe from 0.1 to 0.8 nm, the temperature T sc initially decreases from 7 to 4.5 K and, subsequently, with an increase of d Fe to 1.0–1.2 nm, T sc increases to 5 K and with a further increase of d Fe to 3 nm, the superconducting transition temperature decreases to 3.2–3.4 K [57]. The layers of the metal and alloy, for example, Nb x Ti 1–x /Co or V/Fe x V 1–x , may alternate in the F/S heterostructures. The heterostructures of the superconductor–ferromagnetic semi-conductor type (for example, NbN/EuO/Pb or NbN/EuS/Pb) [58] with a Josephson tunnelling transition are also interesting. In these heterostructures, the thickness of the layer of the ferromagnetic semiconductor (EuO, EuS) varies from 10 to 50 nm, and the thickness of the superconducting layers is greater than 200 nm. Engineering applications have no other parts, which are working in complicated critical conditions, as blades of gas turbines in turbojet engines. The transfer to a new generation of gas-turbine engines requires constructional materials, whose strength and hardness would be 20% higher, fracture toughness 50% better, and wear resistance twice as large. Actual tests showed that the use of heat-resistant nanocrystalline alloys in gas turbines provides at least one-half of the required improvement in the properties. Ceramic nanomaterials are widely used for fabrication of parts working at elevated temperatures, under nonuniform thermal loads, and in aggressive environment. Thanks to their superplasticity, ceramic nanomaterials serve as materials of intricately shaped highly precision products used in the aerospace technology. Nanoceramics based on hydroxy-apatite possesses biocompatibility and high strength and therefore is used in orthopedy for the making of artificial joints and in stomatology for the making of dentures. The nanocrystalline ferromagnetic alloys of the Fe–Cu–M–Si–B systems (M is the transition metal of the groups IV–VI) are used as excellent transformer soft magnetic materials with a very low coercive force and high magnetic permeability. The small grain size determines a long length of the intergranular interfaces: At a grain size from 100 to 10 nm the interfaces contained 10–50% of the atoms of the nanocrystalline solid. Also, grains may have various atomic defects (vacancies or their complexes, disclinations, and dislocations), whose number and distribution differ from those in coarse grains 5 to 30 µm in size. If the dimensions of a solid in one, two or three directions are comparable with characteristic physical parameters having the length dimensionality (the size of magnetic domains, the electron free 19

Nanocrystalline Materials

path, the size of excitons, the de Broglie wavelength, etc.), size effects will be observed for the corresponding properties. Thus, size effects imply a set of phenomena connected with changes in properties of substance, which are caused by (1) a change in the particle size, (2) the contribution of interfaces to properties of the system, and (3) comparability of the particle size with some physical parameters having the length dimensionality. Because of these specific features of the structure, the properties of the nanocrystalline materials greatly differ from those of usual polycrystals. Therefore, the decrease in the grain size is viewed as an efficient method for adjustment and modification of properties of solids. In fact, there are reports on the effect of the nanocrystalline state on the magnetic properties of ferromagnetics (Curie temperature, coercive force, saturation magnetisation) and magnetic susceptibility of weak para- and diamagnetics, on the effect of shape memory on the elastic properties of metals and large changes of their heat capacity and hardness, on the variation of the optical and luminescence characteristics of semiconductors, on the appearance of the plasticity of boride, carbide, nitride and oxide materials which are relatively brittle in the normal coarsegrained state. In nanocrystalline materials, the combination of high hardness and plasticity is usually explained by difficulties in the activation of dislocation sources as a result of the small dimensions of crystals, on the one hand, and by the presence of grain boundary diffusional creep, on the other hand [13]. The nanomaterials are characterised by very high diffusibility of the atoms at the grain boundaries, which is 5–6 orders of magnitude higher than that in the normal polycrystals. However, the mechanisms of diffusion processes in nanocrystalline substances have not yet been completely explained and the literature contains contradicting explanations of this problem. The problem of the microstructure of the nanocrystals, i.e. the structure of the interfaces and their atomic density, the effect of the nanovoids and other free volumes on the properties of nanocrystals requires solution. Usually, when discussing the nonequilibrium metastable state, it is assumed that this state may be compared with some equilibrium state which exists in reality. For example, the metastable glass-like (amorphous) state corresponds to the equilibrium liquid state (melt). The specific feature of the nanocrystalline state in comparison with other well-known nonequilibrium metastable state of matter is the absence of the equilibrium state corresponding to this state in the structure and a long length of the boundaries. 20

Introduction

The nanocrystalline materials represent a special state of condensed matter, i.e. macroscopic ensembles of ultrafine particles up to several nanometers in size. The unusual properties of these materials are determined by both the specific features of separate particles (crystallites) and by their collective behaviour, which depends on the interaction between nanoparticles. The main scope of investigations of the nanocrystalline state is to answer the following questions: (1) Is there a sharp, distinctive boundary between the bulk and the nanocrystalline states?; (2) Is there some critical grain (particle) size below which the characteristic properties of nanocrystals become observable, and above which the material behaves as a bulk one? In other words, is the transition from the bulk to the nanocrystalline state the phase transformation of the first order from the thermodynamic viewpoint? The answer to this question is important for the correct procedure of experimental investigations of the nanocrystalline state and for understanding the results. At the first sight, the transition to the nanocrystalline state is not a phase transformation because the size effects increase gradually with a decrease in the size of isolated particles or grain size in bulk nanomaterials. However, all experimental investigations have been carried out on materials with a high dispersion of the particle or grain size. Obviously, one can assume that the dispersion of the dimensions leads to “bluring” of the phase transformation, if such a transformation exists. This could be confirmed by experiments with the detection of the size effect. Such experiment should be carried out on materials of the same chemical composition but different dispersion in grain size. The important condition of this experiment is equal size of the particles or grains of material investigated. Only in this experiment it will be possible to exclude completely the effect of the dispersion of the size of the particles and determine whether the size dependence of a specific property is a continuous and smooth function or whether it contains jumps, inflection points and other special features. Unfortunately, at the present time it is not possible to carry out such experiments. In solid-state mechanics, successes have been achieved in understanding of the nanocrystalline solid as an ensemble of interacting grain boundary defects. This approach is most useful in studying bulk nanomaterials. For analysis of the symmetric properties of the polycrystals as a function of the changes of the characteristics scale of structural heterogeneity, i.e. as a function 21

Nanocrystalline Materials

of the grain size, the authors of [59] used the gauge field theory [60, 61]. This theory was developed to describe the structural and physical properties of materials with defects. According to [59], a decrease in the grain size is accompanied by a topological transition from solitary waves of orientation-shear instability, which are characteristic of the polycrystalline state, to spatial-periodic structures of defects. The formation of these defects leads to the transition to the nanocrystalline state. This topological transition in an ensemble of grain boundary defects is accompanied by a large change of the characteristics of connectivity and scaling parameters. The main aim of the present monograph is to discuss the effects of the nanocrystalline state, detected in the properties of metals and compounds. The structure and dispersion (the size distribution of the grains) and, consequently, the properties of nanomaterials depend on the method of production of these materials. Therefore, in the second and third chapters of the book we discuss briefly the main methods of production of nanocrystalline powders and bulk nanocrystalline materials. It should be mentioned that significant advances in studying of the nanocrystalline state of solids were achieved after 1985 as a result of improvement of the available and development of new methods of production of both disperse and bulk nanocrystalline materials. The particle size has the strongest effect on the properties of nanocrystalline substances. Therefore, the fourth chapter considers the main methods of determination of the particle size. Special attention is given to the diffraction method of determining the particle size. This method is widely available and used. The fifth chapter is concerned with the specific features of the physical properties of isolated nanoparticles (nanoclusters) and nanopowders. These properties are determined by the small particle size. The methods of production of powdered nanomaterials have been developed sufficiently and have been known for more than 50 years. A large amount of relatively reliable experimental material has been collected for the properties of isolated particles (in most cases, metallic particles), and an efficient theoretical base has been developed for understanding their properties and structure. It should be noted that the particles of the nanopowders occupy an intermediate position between the nanoclusters and bulk solid. The chapters 6 and 7, analysing the structure and properties of bulk nanomaterials, contain the most recent experimental data. Almost all the results described in these chapters have been obtained after 1988. A great majority of the investigations of bulk 22

Introduction

nanocrystalline substances and materials have been concentrated on several problems. One of this problems is the microstructure of the bulk nanomaterials and its stability, the state and relaxation of grain boundaries. The microstructure is studied by different electron microscopy, diffraction and spectroscopic techniques. These investigations are closed to the works on the study of the structure of bulk nanomaterials by indirect techniques (investigation of phonon spectra, calorimetry, measurement of the temperature dependence of microhardness, elasticity moduli, and electrokinetic properties). It is expected that the bulk nanomaterials will be used most widely as constructional and functional materials in new technologies and also as magnetic materials. Consequently, chapter 7 pays special attention to the mechanical and magnetic properties of bulk nanomaterials. Discussion of the structure and properties of isolated nanoparticles and bulk nanomaterials should lead to united views on the current state of the investigations of this special state of matter and find common and specific features of isolated nanoparticles and bulk nanomaterials. References 1. 2.

3. 4. 5.

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H. Gleiter. Materials with ultrafine microstructure: retrospectives and perspectives. Nanostruct. Mater. 1, 1-19 (1992) R. Birringer, H. Gleiter. Nanocrystalline materials. In: Encyclopedia of Material Science and Engineering. Suppl. Vol.1. Ed. R. W. Cahn (Pergamon Press, Oxford 1988) pp.339-349 R. W. Siegel. Cluster – assembled nanophase materials. Ann. Rev. Mater. Sci. 21, 559-578 (1991) R. W. Siegel. Nanostructured materials – mind over matter. Nanostruct. Mater. 3, 1-18 (1993) H.-E. Schaefer. Interfaces and physical properties of nanostructured solids. In: Mechanical Properties and Deformation Behavior of Materials Having Ultrafine Microstructure. Eds. M. A. Nastasi, D. M. Parkin, H. Gleiter. (Kluwer Academic Press, Netherlands, Dordrecht 1993) pp.81-106 R. W. Siegel. What do we really know about the atomic-scale structures of nanophase materials? J. Phys. Chem. Solids 55, 1097-1106 (1994) I. D. Morokhov, L. I. Trusov, S. P. Chizhik. Ultra-Dispersed Metallic Substances (Atomizdat, Moscow 1977) 264 pp. (in Russian) I. D. Morokhov, V. I. Petinov, L. I. Trusov, V. F. Petrunin. Structure and properties of small metallic particles. Uspekhi Fiz. Nauk 133, 653-692 (1981) (in Russian) I. D. Morokhov, L. I. Trusov, V. N. Lapovok. Physical Phenomena in UltraDispersed Substances (Energoatomizdat, Moscow 1984) 224 pp. (in Russian) Yu. I. Petrov. Physics of Small Particles (Nauka, Moscow 1982) 360 pp. (in Russian) Yu. I. Petrov. Clusters and Small Particles (Nauka, Moscow 1986) 368 pp. (in Russian)

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Nanocrystalline Materials 12. 13. 14. 15.

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Introduction 35.

36.

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49. 50. 51.

52. 53. 54.

A. I. Kharlamov, S. V. Loichenko, N. V. Kirillova, V. V. Fomenko. Heterogeneous synthesis of silicon carbide filaments. Teoret. i Eksper. Khimiya 38, 49-53 (2002) (in Russian) A. I. Kharlamov, N. V. Kirillova. Gas-phase reactions of forming of nanothread-like silicon carbide from powdery silicon and carbon. Teoret. i Eksper. Khimiya 38, 54-58 (2002) (in Russian) A. I. Kharlamov, N. V. Kirillova, S. V. Kaverina. Hollow silicon carbide nanostructures. Teoret. i Eksper. Khimiya 38, 232-237 (2002) (in Russian) Science and Applications of Nanotubes. Eds. D. Tomanek, R. J. Enbody. (Kluwer Academic Publishers: New York – Dordrecht – Moscow 2002) 398 pp . M. Che, C. O. Bennet. The influence of particle size on the catalytic properties of supported metals. Advances in Catalysis 36, 55-172 (1989) V. Ponec. Catalysis by alloys in hydrocarbon reactions. Advances in Catalysis 32, 149-214 (1983) Z. Karpinski. Catalysis by supported, unsupported, and electron–deficient palladium. Advances in Catalysis 37, 45-100 (1990) A. D. Pomogailo. Hybrid polymer-inorganic nanocomposites. Uspekhi Khimii 69, 60-89 (2000) (in Russian). (Engl. Transl.: Russ. Chem. Reviews 69, 5380 (2000)) A. D. Pomogailo, A. S. Rosenberg, U. E. Uflyand. Nanoscale Metal Particles in Polymers (Khimiya, Moscow 2000) 672 pp. (in Russian) J. S. Kilby. Turning potential into realities: The invention of the integrated circuit (Nobel Lecture). Uspekhi Fiz. Nauk 172, 1103-1109 (2002) (in Russian) D. L. Klein, R. Roth, A. K. L. Lim, A. P. Alivisatos, P. L. McEuen. A single-electron transistor made from a cadmium selenide nanocrystal. Nature 389, 699-701 (1997) H. Kroemer. Nobel Lecture: Quasielectric fields and band offsets: teaching electrons. Rev. Modern Phys. 73, 783-793 (2001) R. Dingle, W. Wiegmann, C. H. Henry. Quantum states of confined carriers in very thin Al x Ga 1−x As - GaAs - Al xGa 1−x As heterostructures. Phys. Rev. Lett. 33, 827-830 (1974) Zh. I. Alferov. Nobel Lecture: The double heterostructure: concept and its applications in physics, electronics and technology. Rev. Modern Phys. 73, 767-782 (2001) A. P. Alivisatos. Semiconductor clusters, nanocrystals and quantum dots. Science 271, 933-937 (1996) K. Morgenstern. Dynamische Mikroskopie von Nanostrukturen. Physik Journal 1, 95-98 (2002) Yu. P. Pershin, E. N. Zubarev, V. V. Kondratenko, O. V. Pol’tseva, A. G. Ponomarenko, V. A. Sevryukova, J. Verhoeven. Features of formation of Ni/ C multilayer x-ray mirrors manufactured by electron-beam vaporization completed with an ion-beam etching. Metallofizika i Noveishie Tekhnologii 24, 795-814 (2002) (in Russian) J. J. Hauser, H. C. Theuerer, N. R. Werthamer. Proximity effects between superconducting and magnetic films. Phys. Rev. 142, 118-126 (1966) B. Y. Jin, J. B. Ketterson. Artificial metallic superlattices. Advances in Physics 38, 189-366 (1989) A. I. Buzdin, A. V. Vedyayev, N. V. Ryshanova. Spin-orientation-dependent superconductivity in F/S/F structures. Europhysics Lett. 48, 686-691 (1999)

25

Nanocrystalline Materials 55. 56.

57.

58.

59.

60.

61.

L. R. Tagirov. Low-field superconducting spin switch based on a superconductor/ferromagnet multilayer. Phys. Rev. Lett. 83, 2058-2061 (1999) Yu. A. Izyumov, Yu. N. Proshin, M. G. Khusainov. Competition between superconductivity and magnetism in ferromagnet/superconductor heterostructures. Uspekhi Fiz. Nauk 172, 113-154 (2002) (in Russian). (Engl. transl.: Physics - Uspekhi 45, 109-148 (2002)) Th. Mühge, N. N. Garif ’yanov, Yu. V. Goryunov, G. G. Khaliullin, L. R. Tagirov, K. Westerholt, I. A. Garifullin, H. Zabel. Possible origin for oscillatory superconducting transition temperature in superconductor/ ferromagnet multilayers. Phys. Rev. Lett. 77, 1857-1860 (1996) A. S. Borukhovich. Quantum tunneling multilayers and heterostructures with ferromagnetic semiconductors? Uspekhi Fiz. Nauk 169, 737-751 (1999) (in Russian). (Engl. transl.: Physics - Uspekhi 42, 653-667 (1999)) O. B. Naimark. Nanocrystalline state as a topological transition in an ensemble of grain-boundary defects. Fiz. Metall. Metalloved. 84, 5-21 (1997) (in Russian). (Engl. Transl.: Phys. Metal. Metallogr. 84, 327-337 (1997)) E. Kröner. On gauge theory in defect mechanics trends in application of pure mathematics to mechanics. In: Lecture Notes in Physics. Eds. E. Kröner and K. Kinchgassner (Springer, Heidelberg 1986) pp.281-296 A. Kadic, D. G. B. Edelen. A Gauge Theory of Dislocations and Disclinations (Springer, Berlin 1983) 186 pp.

26

Synthesis of Nanocrystalline Powders

+D=FJAH

2. Synthesis of Nanocrystalline Powders 2.1 GAS PHASE SYNTHESIS Isolated nanoparticles are usually produced by evaporation of metal, alloy or semiconductor at a controlled temperature in the atmosphere of a low-pressure inert gas with subsequent condensation of the vapour in the vicinity or on the cold surface. This is the simplest method of producing nanocrystalline powders. In contrast to vacuum evaporation, the atoms of the substance, evaporated in a rarefied inert atmosphere, lose their kinetic energy more rapidly as a result of collisions with gas atoms and form segregations (clusters). The first studies in this topic were carried out in 1912 [1,2]: examination of evaporation of Zn, Cd, Se and As in vacuum, and also in hydrogen, nitrogen and CO 2 showed that the size of the produced particles depends on the pressure and atomic weight of the gas. The authors of [3] evaporated gold from a heated tungsten filament at a nitrogen pressure of 0.3 mm Hg (40 Pa), and produced in the condensates spherical particles with a diameter of 1.5–10 nm (the mean diameter approximately 4 nm). They found that the particle size depends on the gas pressure and, to a lesser degree, on the rate of evaporation. The particle size was determined by the high-resolution electron microscopy. The condensation of vapours of aluminium in H 2, He and Ar at different gas pressures made it possible to produce particles with a size of 20–100 nm [4]. Later, the method of combined condensation of metal vapours in Ar and He was used to produce Au–Cu and Fe–Cu highly dispersed alloys, formed by spherical particles with a diameter of 16–50 nm [5, 6].

27

Nanocrystalline Materials

A variant of the condensation of metal vapours in a gas atmosphere is the method of dispersion of a metal by means of an electric arc in a liquid with subsequent condensation of metallic vapours in liquid vapours, proposed as early as in the 19 th century [7]; later, this method was improved by the authors of [8–10]. The first extensive review [11], concerned with detailed discussion of the condensation method and the formation of highly dispersed metal particles by condensation of metallic vapours, was published in 1969. Several theoretical special features of condensation in supersaturated vapours, which takes place by means of the formation and growth of nuclei (clusters), were discussed in a review in [12]. The nanocrystalline particles with a size of ≤ 20 nm, produced by evaporation and condensation, are spherical, and large particles may be faceted. The size distribution of nanocrystals is logarithmico-normal and is described by the function

(

F ( D) = ( 2π ln σ g )−1 exp  − ln D − ln < Dg > 

2 ) /(2ln 2 σ g )

(2.1)

where D is the particle diameter; is the mean diameter; σ g

{

2 is dispersion; ln σ g = ∑  ni ( ln Di − ln < Dg > )  / ∑ ni  

}

1/ 2

. Analysis

shows that the majority of distributions of nanoparticles of metals, produced by the evaporation and condensation method, are described by equation (2.1) with the values σ g = 1.4 ± 0.2. Isolated nanocrystals contain no dislocations, but disclinations, which are energetically more advantageous in very small crystals than dislocations, may form [13]. The systems using the principle of evaporation and condensation, differ in the method of input of evaporated material; the method of supplying energy for evaporation; the working medium; setup of the condensation process; the system for collecting the produced powder. Evaporation of a metal may take place from a crucible or the metal may be fed into the zone of heating and evaporation in the form of wire, injected metallic powder or a liquid jet. The metal may also be dispersed with a beam of argon ions. Energy may be supplied directly by heating, the passage of electric current through a wire, electric arc discharge in plasma, induction heating with high- and superhigh frequency currents, laser radiation, and electron-beam heating. 28

Synthesis of Nanocrystalline Powders

Evaporation and condensation may take place in vacuum, in a stationary inert gas, or in a gas flow, including in a plasma jet. Condensation of the vapour–gas mixture with a temperature of 5000–10000 K may take place during its travel into the chamber with a large section and the volume filled with a cold inert gas; cooling takes place both as a result of expansion and contact with the cold inert atmosphere. There are systems in which two jets travel coaxially into the condensation chamber: the vapour–gas mixture is supplied along the axis, and a circumferential jet of a cold inert gas travels along its periphery. As a result of turbulent mixing, the temperature of metal vapours decreases, supersaturation increases and rapid condensation takes place. Favourable conditions for the condensation of metallic vapours are generated in adiabatic expansion in a Laval nozzle, when rapid expansion results in the formation of a steep temperature gradient and almost instantaneous vapour condensation takes place. An independent task is the collection of the nanocrystalline powder produced by condensation, because the individual particles of this powder are so small that they are in constant Brownian motion and remain suspended in the gas, not settling under the effect of the forces of gravity. The produced powders are collected using special filters and centrifugal deposition; in some cases, the particles are trapped by a liquid film. The main relationships of the formation of nanocrystalline particles by the method of evaporation and condensation are as follows, [11, 14]: 1. The nanoparticles form during cooling of the vapours in the condensation zone. The size of this zone increases with a decrease of the gas pressure; the internal boundary of the condensation zone is in the vicinity of the evaporator, and its external boundary may, with a decrease of gas pressure, extend outside the limits of the reaction vessel; at a pressure of several hundreds of Pa, the outer boundary of the condensation zone is situated inside the reaction chamber with a diameter of ≥ 0.1 m and convective gas flows play a significant role in the condensation process; 2. When the gas pressure is increased to several hundreds of Pa, the mean particle size initially rapidly increases and then slowly approaches the limiting value in the pressure range greater than 2500 Pa; 3. At the same gas pressure, the transition from helium to xenon, i.e. from a less dense inert gas to an inert gas with a higher density, is accompanied by a large increase in the particle size. 29

Nanocrystalline Materials

Depending on the evaporation conditions of the metal (gas pressure, the position and temperature of the substrate), its condensation may take place either in the volume or on the surface of the reaction chamber. Volume condensates are characterised by the presence of spherical particles, whereas the particles of the surface condensate are faceted. In the same evaporation and condensation conditions, metals with high melting points form smaller particles. If the gas pressure is lower than approximately 50 Pa, spherical particles of metal with a mean diameter of D < 30 nm settle on the walls of relatively large reaction chambers (diameter greater than 0.25 m). When the pressure is increased by several hundreds of Pa, the formation of highly dispersed metallic particles is completed in convective flows of the gas in the vicinity of the evaporator. Gas-phase synthesis can be used to produce particles with a size from 2 nm to several hundreds of nanometers. Smaller particles of a controlled size are produced by means of the mass distribution of clusters in a time-of-flight mass spectrometer. For example, metal vapours are passed through a cell with helium with a pressure of ~1000–1500 Pa, and are then transferred into a vacuum chamber (~10 –5 Pa), where the mass of the cluster is stabilised during the time of flight of a specific distance in the mass spectrometer. This method is used to produce clusters of antimony, bismuth and lead, containing 650, 270 and 400 atoms, respectively; the temperature of gaseous helium in the case of Sb and Bi vapours is 80 K, and in the case of Pb vapours 280 K [15]. Recently, the gas-phase synthesis of nanoparticles has been developed extensively as a result of the application of different methods of heating the evaporated substance. Highly dispersed deposits of silver and copper on glass were produced by evaporation of metals in an inert atmosphere at a pressure of 0.01–0.13 Pa [16]. The same method was used to produce clusters of Li n , containing from fifteen to two lithium atoms [17]; evaporation of lithium in high vacuum is accompanied by the formation of only separate atoms of lithium, and clusters form only in the atmosphere of a rarefied inert gas. Nanocrystalline powders of the oxides Al 2 O 3 , ZrO 2 and Y 2 O 3 were produced by evaporating oxide targets in a helium atmosphere [18], magnetron sputtering of zirconium in a mixture of argon and oxygen [19], and by the controlled evaporation of yttrium nanocrystals [20]. To produce highly dispersed powders of transition metal nitrides, the electron-beam heating of targets of appropriate metals, with evaporation carried out in the atmosphere of nitrogen or ammonia 30

Synthesis of Nanocrystalline Powders

at a pressure of 130 Pa, is used [21]. Nanocrystalline powders have also been produced by plasma, laser and arc heating methods. For example, the authors of [22, 23] produced nanoparticles of carbides, oxides and nitrides by the pulsed laser heating of metals in a rarefied atmosphere of methane (in the case of carbides), oxygen (in the case of oxides), nitrogen or ammonia (in the case of nitrides). The pulsed laser evaporation of metals in the atmosphere of an inert gas (He or Ar) or a reagent gas (O 2 , N 2 , NH 3 , CH 4 ) makes it possible to produce mixtures of nanocrystalline oxides of different metals, oxide–nitride or carbide– nitride mixtures. The composition and size of nanoparticles may be controlled by the variation of the pressure and composition of the atmosphere (inert gas or reagent gas), the power of the laser pulse, the temperature gradient between the evaporated target and the surface on which condensation takes place. The method of condensation of vapours in an inert gas is used most frequently in scientific investigations for producing small amounts of nanopowders. The powders synthesised are not agglomerated efficiently and are sintered at a relatively low temperature. The authors of [24] modified the condensation method for producing ceramic nanopowders from organometallic precursors. In the setup used in [24] (Fig. 2.1), the evaporator was a tubular reactor in which the precursor was mixed with the carrier inert gas Chamber pressure (1~50 mbar) Control valve To pump Rotated cold cylinder Carrier gas Heated tubular reactor Gas Precursor source Mass flow Needle controller valve

Particles

Scraper

Funnel

Collector

Fig. 2.1. Schematic of the chemical vapour condensation (CVC) processing apparatus for preparation of ceramic nanostructured powders from organometallic precursors [24].

31

Nanocrystalline Materials

and dissociated. The resultant continuous flow of clusters or nanoparticles travelled from the reactor in the working chamber and condensed on a cold rotating cylinder. Successful realisation of the process is ensured by a low concentration of precursor in inert gas, rapid expansion and cooling of the gas which exits from the reactor into the working medium, and a low pressure in the working chamber. The characteristics (particle size distribution, agglomeration capacity, sintering temperature) of the nanopowders produced by this method do not differ from those of the nanopowders synthesised by the standard evaporation and condensation method. The properties of isolated nanocrystalline particles are greatly determined by the contribution of the surface layer. For a spherical particle with a diameter of D and the thickness of the surface layer δ, the fraction of the surface layer in the volume of the particle 3 3 3 is ∆V / V =  π6 D − π6 ( D − 2δ)  / π6 D ≈ 6δ / D . At a thickness of the surface layer δ equal to 3–4 atomic monolayers (0.5–1.5 nm), and the mean size of the nanocrystal of 10–20 nm, the surface layer accounts for up to 50% of the substance. However, the highly developed surface of nanocrystalline particles greatly increases their reactivity and, in turn, this greatly complicates study of these particles.

2.2 PLASMA CHEMICAL TECHNIQUE One of the most widely used chemical methods of producing highly dispersed powders of nitride, carbides, borides and oxides is plasma chemical synthesis [25–31]. The main conditions of producing highly dispersed powders by this method is the occurrence of a reaction away from equilibrium and the higher rate of formation of nuclei of a new phase at a low growth rate of this phase. In the real conditions of plasma chemical synthesis, the formation of nanoparticles can be carried efficiently by increasing the cooling rate of the plasma flow in which condensation from the gas phase takes place; this decreases the size of particles and also suppresses the growth of particles by their coalescence during collisions. Plasma chemical synthesis is carried out with the use of lowtemperature (4000–8000 K) nitrogen, ammonia, hydrocarbon plasma, argon plasma of arc, glow, high frequency or microwave discharges. Starting materials are represented by elements, halides 32

Synthesis of Nanocrystalline Powders

and other compounds. The characteristics of the produced powders depend on the starting materials used, synthesis technology and the type of reactor. The particles of plasma chemical powders are single crystals and their size varies from 10 to 100–200 nm or larger. Plasma chemical synthesis ensures high rates of formation and condensation of the compound and is characterised by relatively high productivity. The main disadvantages of plasma chemical synthesis are the wide size distributions of particles and, consequently, the presence of relatively large (up to 1–5 µm) particles, i. e. the low selectivity of the process. The other disadvantage is a high content of the impurities in the powder. Up to now, the plasma chemical technique has been used to produce highly dispersed powders of nitrides of Ti, Zr, Hf, V, Nb, Ta, B, Al and Si, carbides of Ta, Nb, Ti, W, B and Si, and oxides of Mg, Y and Al [15–29, 32–37]. The plasma chemical technique is used most widely for the synthesis of groups IV and V transition metal nitrides; analysis of the structure and properties of ultrafine (with a mean particle size smaller than 50 nm) nitride powders can be found in a monograph in [38, section 1.4]. The plasma temperature, reaching up to 10000 K, determines the presence in the plasma of ions, electrons, radicals and neutral particles which are in the excited state. The presence of these particles leads to high rates of interaction and a short time of reactions (up to 10 –3 –10 –6 seconds). High temperature ensures the transfer of almost all starting substances into the gaseous state followed by their interaction and condensation of the reaction products. Plasma chemical synthesis includes several stages. The first stage is characterised by the formation of active particles in arc, high frequency and microwave plasma reactors. The highest power and coefficient of efficiency are shown by arc plasma reactors, but materials produced are contaminated with products of electrode erosion. High frequency and microwave plasma reactors do not have this shortcoming. As a result of quenching, the next stage results in the generation of reaction products. The selection of the place and the rate of quenching makes it possible to produce powders with the required composition, shape and size of the particles. The powders produced by the plasma chemical technique are characterised by regular shapes and a particle size of 10–100 nm or larger. The plasma chemical powders of carbides of metals, boron and 33

Nanocrystalline Materials

silicon are usually produced by the reaction of chlorides of the appropriate elements with hydrogen and methane or other hydrocarbons in the argon high-frequency or arc plasma; nitrides are produced by reaction of chlorides with ammonia or a mixture of nitrogen and hydrogen in low-temperature microwave plasma. Plasma chemical synthesis may also be used for producing multicomponent submicrocrystalline powders, representing mixtures of carbides and nitrides, nitrides and borides, nitrides of different elements, etc. The synthesis of oxides in the plasma of electric arc discharges is carried out by evaporation of metal with the subsequent oxidation of vapours or oxidation of particles of a metal in an oxygencontaining plasma. The authors of [39] described the plasma chemical synthesis of nanoparticles of aluminium oxide with a mean size of 10–30 nm. This study shows that the formation of nanopowders of aluminium oxide with a smallest particle size is achieved in the reaction of metal vapours with atmospheric oxygen when an air is blowed intensively. It leads to a rapid decrease of temperature. Rapid cooling not only decelerates the particle growth but also increases the rate of formation of nuclei of the condensed phase. Plasma chemical synthesis with the oxidation of aluminium particles in the flow of oxygen-containing plasma leads to the formation of larger oxide particles in comparison with the oxidation of metal vapours produced in advance. The plasma chemical synthesis is quite close to gas phase synthesis using laser heating of a reacting gas mixture [40–44]. Laser heating ensures controlled homogeneous nucleation and prevents contamination. The size of nanocrystalline particles decreases with increasing intensity (the power related to the unit area) of laser radiation as a result of an increase of temperature and the heating rate of gases-reagents. The authors [41] used this method to produce silicon nitride Si 3 N 4 with a particle size of 10– 20 nm from a gas mixture of silane SiH 4 and ammonia NH 3 . The plasma chemical method is also used to produce metal powders. For example, submicrocrystalline copper powders with a particle size smaller than 100 nm and a comparatively narrow size distribution of particles are produced by reduction of copper chloride by hydrogen in argon electric arc plasma with a temperature up to 1800 K. Gas phase synthesis using laser radiation for producing the plasma in which the chemical reaction takes place, is an efficient method of producing molecular clusters. Molecular clusters are a 34

Synthesis of Nanocrystalline Powders

new structural modification of matter. We shall discuss in more detail the achievements in the field of plasma chemical gas phase synthesis and the new possibilities of producing previously unknown polymorphous modifications of substances with the nanometer size of structural elements. The molecular clusters occupy a completely special position in the group of substances with a nanostructure. The best known of these are fullerenes [45–47], i.e. a new allotropic modification of carbon in addition to graphite and diamond. Already in November 1966, the British journal New Scientist published playful notes by D. E. H. Jones on the possibility of producing solid materials with a low density (considerably lower than the density of water). This material should consist of hollow spherical molecules whose shell is constructed from graphite sheets. A network of hexagonal rings C 6 should, for stability, also include five-member rings. At that time, nobody noticed that a similar design had already been proposed in 1951 by the well-known American architect R. B. Fuller who patented the spherical construction which was called the geodesic cupola. This construction of the cupola was used, for example, in the US pavilion at the Expo-67 World Exhibition in Montreal. The fullerenes are produced by electric arc sputtering of graphite in a helium atmosphere; gas pressure is 1.33×10 4 Pa. Combustion of the arc results in the formation of carbon black which condenses on the cold surface. The collected carbon black is processed in burning toluene or benzene. After evaporating a solution, a black condensate appears which consists of approximately 10–15% of a mixture of fullerenes C 60 and C 70 . In addition to the electric arc, fullerenes can be produced by electron beam evaporation and laser heating. The central position among the fullerenes belongs to the C 60 molecule with the highest symmetry and, consequently the highest stability. As regards the shape, the molecule of fullerene C 60 resembles a soccer-ball and has the structure of a regular truncated icosahedron (Fig. 2.2). In the molecule of fullerene C 60 the carbon atoms form a closed hollow spherical surface consisting of 5- and 6-member rings, and each atom has a coordination number equal to three and is situated in the tops of two hexagons and one pentagon. The diameter of the molecule of fullerene C 60 is 0.72-0.75 nm. Crystallisation of C 60 from a solution or gas phase is accompanied by the formation of molecular crystals with an fcc lattice; the lattice constant is 1.417 nm. Fullerene in the solid state is referred to as 35

Nanocrystalline Materials

Fig. 2.2. The structure of the most important fullerenes C 60 and C 70 . The C 60 molecule is built like a soccer-ball and its cage has a diameter of about 0.7 nm. All fullerenes exhibit hexagonal and pentagonal rings of carbon atoms.

fullerite. Higher stability is also typical of fullerene C 70 having the form of a closed spheroid. The fullerenes may be regarded as a spherical form of graphite because mechanisms of interatomic bonding in the fullerene and in bulk graphite are very similar. It is interesting to note that the high stability of fullerene C 60 was theoretically predicted already at the beginning of the 70s in calculations of potentially possible frame structures constructed from carbon atoms [48, 49]. The properties of fullerenes are very unusual. For example, crystalline fullerenes represent semiconductors and are characterised by photoconductivity in optical radiation. However, crystals C 60 , alloyed with atoms of alkaline metals, have metallic conductivity and change to the superconducting state at 30 K and higher. Transformation of crystalline fullerene to diamond takes place at room temperature at a pressure of 20 GPa. If a fullerene is heated to 1500 K, a pressure of 7 GPa is sufficient for the transition to diamond. Similar transformation of graphite to diamond requires a temperature of 900 K and a pressure of 30–50 GPa. The solutions of fullerenes are characterised by non-linear optical properties, i. e. a rapid decrease of the transparency of the solution when exceeding some critical value of the intensity of optical radiation. Recently, ferromagnetic properties are found for the polymerised form of fullerene C 60 at room temperature [50]. Investigation shows that polymerised fullerene Rh-C 60 with a rhombohedral structure has a Curie temperature of 500 K and its hysteresis curve is typical of ferromagnetics. At heating and depolymerisation, the 36

Synthesis of Nanocrystalline Powders

specimen of Rh-C 60 lost its ferromagnetic properties. At the beginning of 2001, a group of scientists [51] found a new fullerene-like form C 48 N 12 in which in comparison with usual fullerene C 60 a fifth of carbon atoms is substituted by nitrogen (Fig. 2.3). If in the fullerene crystals the molecules C 60 are connected by weak Van-der-Waals forces, then the presence of nitrogen atoms results in the formation of strong covalent bonds. For this reason, the fullerene-like crystalline material C 48 N 12 is characterised by a unique combination of strength and elasticity. The discovery of fullerenes C n (n = 60–90) and subsequent investigations show that clusters C n , containing less than 60 carbon atoms, have low stability. One of the methods of stabilising carbon fullerenes C 28 with a small number of atoms is the synthesis of endohedral complexes M@C 28 in which the atom of the doping element is introduced into the carbon sphere. A similar endohedral complex Ti@C 28 has been synthesised in particular with titanium [52]. Stabilisation of the unstable fullerene C 28 by means of intercalation into its volume of atoms of non-metallic 2p-elements (B, C, N and O) and metallic 3d-elements (Sc, Ti, V, Cr, Fe and Cu) has been examined theoretically in [53]. When evaluating the possibility of formation of endohedral complexes M@C 28 with 3dmetals, it is important to take into account the geometrical, chemical and kinetic factors. For fullerene C 28 , the limiting value of the radius of the metallic ion which can be intercalated in the internal cavity of the fullerene is 0.09–0.10 nm [54]. Therefore, all 3d-metals satisfy the geometrical criteria. The chemical factor is favourable if as a result

Fig. 2.3. The structure of fullerene C 48 N 12 [51]: the position of the symmetry axis C 6 is shown by solid line. 37

Nanocrystalline Materials

of intercalation the transfer of electronic density enhances the binding molecular orbitals. The kinetic factor takes into account the mechanism of formation of endohedral complexes. They are produced by rolling the graphen monolayer around the atom of the metal adsorbed on the surface of graphite. This process takes place in the interaction of metal when the surface of graphite strengthens the M–C bond and weakens the C–C bond (especially interlayer bonds). The results published in [53] show that intercalation of the atoms of metal M in the volume of the fullerene C 28 is accompanied (1) by the transfer of an electron charge from atom M to the atoms of the fullerene shell, (2) by the change of the population of overlapping atomic orbitals of carbon, (3) by the formation of a chemical bond of atom M with the carbon atoms and (4) by a general variation of the electron energy spectrum. Analysis of M@C 28 complexes with different metals M shows that from the viewpoint of the chemical and kinetic factors the endohedral complex Ti@C28 is most stable [53]. In this complex, the titanium atom is situated in the centre of the polyhedron C 28 . According to [55], the endohedral complex Ti@C 28 can be considered as a possible compound of the type of molecular cluster in the Ti–C system. Fullerenes, as molecular clusters, have been examined in thousands of original papers, tens of reviews and monographs. Therefore, in this book, they are only mentioned in connection with the synthesis of a new class of molecular clusters, having the composition M 8 C 12 , where M is the metal atom. After discovery of molecular clusters of carbon and initial observations of the fullerene molecule C 60 [45–47], and after intensive and greatly differing investigations of synthesis, structure and properties of fullerenes (see, for example [56–58]), special attention has been given to producing molecular clusters of other substances. By analogy with fullerenes, it was expected that these molecular clusters should have unique physical and chemical properties, differing from the properties of the available polymorphous modifications of the same substance. The search for new molecular clusters was crowned with success by the discovery in 1992 [59] of a new unusually stable charged cluster Ti 8 C 1+2 corresponding to the molecule with stoichiometric composition Ti 8C 12 in the form of a slightly distorted pentagon–dodecahedron (Fig. 2.4). In the dodecahedral molecule, all atoms are distributed on a sphere. The sphere consists of 12 pentagons including 2 atoms of titanium and 3 carbon atoms. In this molecule, all atoms of titanium and carbon have the same (as in 38

Synthesis of Nanocrystalline Powders

%K

7L& 7L

%G 7L

7L

7L

7L 7L

7L

7L 7L

&

Fig. 2.4. Dodecahedral structure of the molecular cluster Ti 8 C 12 with the symmetries T h and T d taking into account different length of Ti–C and C–C bonds.

fullerene C 60 ) co-ordination equal to 3, occupy the same positions and are distributed at the tops of the dodecahedron in such a manner that titanium is bonded only with carbon, and 6 dimers C 12 alternate with 8 atoms of titanium. The dodecahedral structure of Ti 8 C 12 can be regarded as a cube formed by 8 titanium atoms, where each face is bonded with dimer C 2 . The distance between the nearest atoms of carbon is 0.151 nm, the distance Ti–C is 0.196 nm, and the distance Ti–Ti is 0.305 nm. The point symmetry group T h of such a structure includes 24 elements of symmetry (rotations and reflections). Because of high symmetry, the ideal molecule of metallocarbohedren should be highly stable. Another possible structure of the Ti 8 C 12 cluster has the point symmetry group T d [60] (Fig. 2.4). In this configuration, the titanium atoms occupy positions of two types, and the nodes, relating to the positions of each type, form a tetrahedron. The smaller tetrahedron is rotated by 90° in relation to the larger one. The difference in the positions of the titanium atoms is due to the different position in relation to dimers C 2 . In fact, 6 dimers C 2 are parallel to the edges of the larger tetrahedron made of atoms of Ti(1) and are normal to the edges of the smaller tetrahedron formed by 4 atoms Ti(2). The atoms Ti(1) are linked with the 3 nearest carbon atoms, and Ti(2) atoms with 6 carbon atoms. The Ti(1)–C distance is 0.193 nm, Ti(2)–C is 0.219 nm, Ti(1)–Ti(2) is 0.286 nm and the distance Ti(2)–Ti(2) is 0.290 nm [56]. The linear size of the Ti 8 C 12 cluster 39

Nanocrystalline Materials

is approximately 0.5 nm. The question of which of the two structures (with symmetry T h or T d) is realised has not as yet been solved. The Ti 8 C 12 clusters were produced by the plasma chemical gasphase synthesis. The inert gas was represented by helium, reagents were hydrocarbons (methane, ethylene, acetylene, propylene, and benzene) and titanium vapours, the pressure of the gas mixture in the reactor was 93 Pa (0.7 mm Hg). A rotating titanium rod was evaporated and the ionised beam of metal vapours was produced by focused radiation of an Nd-laser with a wavelength of 532 nm. Neutral and ionised clusters were separated from the reaction products and analysed in a mass spectrometer. The mass spectra of the reaction products contained a sharp peak corresponding to the Ti 8C 12 molecule. In addition to the neutral molecules, stable ions Ti 8 C 12 form in the mixture of ionised gases. The author of [59] assumed that the Ti 8 C 12 cluster is a member of a new class of molecular clusters and referred to this cluster as metallocarbohedrene or Met-Car. In metallocarbohedrenes, the atoms of a transition metal and carbon form a cage-like structure. In fact, other clusters M 8 C 12 of transition metals such as Zr, Hf, V [61, 62], Cr, Mo and Fe [63] were soon produced. The metallocarbohedrenes and methods of producing them are described in reviews [64, 65]. According to the authors of [59], the high stability of the Ti 8 C 12 cluster is a consequence of a special geometrical and electron structure, typical of these clusters. Chemical bonds in the Ti 8 C 12 molecule are similar to those existing in carbon fullerenes. However, in contrast to fullerene C 60 , the ionised or neutral molecule of the type M 8 C 12 contains only five-member rings. As regards shape of the surface, the highly stable cluster Ti 8 C 12 corresponds to a hypothetical unstable (and, therefore, not realised in practice) fullerene C 20 . This comparison already shows that the complete identity of the chemical bonds in the M 8 C 12 clusters and in carbon fullerenes is unlikely. In fact, the calculations of the equilibrium crystall and electron structure of Ti 8C 12 [66] show that bonds of the titanium atoms with 3 adjacent carbon atoms are greatly differ from those in graphite or in fullerene C 60 ; in particular, the lengths of the Ti–C and C–C bonds in Ti 8 C 12 differ by almost a factor of 1.5 and are equal to 3.76a 0 and 2.63a 0 (a 0 = 0.052918 nm is the radius of the first Bohr orbit), respectively. According to [67], the length of the Ti–C bond is approximately 70 % greater than the length of the C–C bond. At 40

Synthesis of Nanocrystalline Powders

the same time, the atoms of carbon and titanium are situated at almost the same distances from the centre of the cluster. This means that the real dodecahedron Ti 8C 12 is greatly deformed and distorted. According to [66], the binding states of the Ti 8C 12 cluster are formed by the combination of d-orbitals of titanium and molecular orbitals C 2 , and the filled level with the highest energy is situated between the bonding and anti-bonding states of titanium. It ensures the stability of the cluster. Identical conclusions according to which the shape of the M 8C 12 is not ideal but they have the form of a distorted pentagondodecahedron, were obtained in other theoretical calculations. Slightly different results were obtained [68] in comparative examination of the electronic structure of Met-Car Ti 8 C 12 with symmetry T h and T d. According to [68], the structures of both types contain the filled level with the highest energy at a sharp peak of the density of states, formed mainly by C2p- and Ti3d-atom orbitals. The high chemical stability of the Ti 8 C 12 compound is determined by the combination of strong Ti3d–C2p-interactions between the titanium atoms and dimers C 2 , on the one hand, and C–C interactions in carbon dimers, on the other hand, in the structures of both types, Ti 8 C 12 has an open electronic shell so that it can play the role of a donor and also of the acceptor of electron density. In calculations in [68], the parameters of the structure and atomic spacing for symmetry T h were taken from [67] and for symmetry T d from [60]. The atoms in molecules of metallocarbohedrenes form strong bonds. For example, the binding energy per 1 atom of the molecule of Ti 8 C 12 is 6.1–6.7 eV/atom [66, 67, 69]. For comparison, this value in the fullerene molecule C 60 is 7.4–7.6 eV/atom [70, 71], and in titanium carbide TiC with a cubic structure B1 it is 7.2 eV/atom [66]. The considerations regarding the geometry and electron structure of the molecular clusters Ti 8 C 12 , presented in [66, 68], explain efficiently the peculiarities of reaction behaviour for these clusters in relation to polar and non-polar substances. + Investigations of the interaction between Ti 8 C 12 clusters and polar molecules of methanol CH 3OH, water H 2O and ammonia NH 3 [72] show that at room temperature the reaction between them + + Ti8C12 (P) n−1 + P → Ti8C12 (P) n

(2.2)

takes place through eight consecutive steps of annexation of the 41

Nanocrystalline Materials

polar molecule P. This means that the first solvation shell of the + Ti 8 C 12 ion is formed by eight polar molecules. In reactions with benzene and ethylene, the resultant first solvation shell includes only four hydrocarbon molecules with π-bonds. For example, at room temperature the Ti 8 C +12 cluster is fully inert in relation to nonpolar molecules of oxygen and methane. According to the authors of [72], if the clusters Ti 8 C 12 can be held together by Van-der-Waals forces and form large crystals, like fullerene C 60 , then the bulk material Ti 8C 12 will be very stable in air. It could also + be mentioned that, regardless of reactivity of Ti 8 C 12 in relation to many substances, interaction between them takes place only as association of ligands, without rupture of any chemical bonds in a cluster. This confirms the high stability of metallocarbohedrenes. It is very interesting to note that in plasma chemical gas phase synthesis [59, 61, 62] there was preferential formation of cluster particles of M 8 C 12 and M mC n (M – Ti, Zr, Hf, V) with the ratio M:C 1.5–2.0, and not of nanoparticles of carbides TiC, ZrC, HfC, VC with the fcc crystal structure. In identical synthesis, systems TaC and Nb–C were characterised by the formation, in addition to clusters Ta mC n and Nb mC n with the compositions similar to M 8 C 12 , of small amounts of nanocrystalline particles M m C n with m ≈ n, with a cubic structure. At the same time, conventional plasma chemical synthesis (without laser heating of plasma) makes it possible to produce only carbide nanoparticles. Thus, in gas phase synthesis, two structures – cubic and the metallocarbohedrene type – can form in the ‘transition metal–carbon’ systems. Because each M–C system usually should initially contain clusters (or particles) of only one structural type, it could be assumed that the selective formation of a specific structure is determined by its thermodynamic stability. However, the author of [73] reported that as a result of synthesis in the experimental conditions used in the study, the Ti– C and V–C systems are characterised by the simultaneous formation of cubic MC (M 14 C 13 ) and dodecahedral M 8 C 12 structures. The more extensive formation of cubic nanoparticles in comparison with M 8C 12 took place at a relatively low laser radiation power. In [73] it is assumed that the formation of metallocarbohedrenes may take place by photo-dissociation of cubic nanoparticles, indicating high stability of M 8C 12 clusters. Taking into account the results in [59, 61–63, 73], it is natural to assume that possible reasons for the preferential formation of carbide fcc nanoparticles or molecular clusters M 8 C 12 may also be 42

Synthesis of Nanocrystalline Powders

of a kinetic nature. The correct answer to the question of the reasons for the preferential formation of a specific structure is important for practice because it makes it possible to produce the crystalline modification of the nanostructural material which is required, i.e. carry out in practice the directional synthesis of the nanomaterial. To solve this problem, investigations were carried out into the formation of Nb m C n nanoclusters in the Nb–C system [74] in relation to the synthesis conditions (the concentration of a carboncontaining reagent in a gas atmosphere, laser radiation power). To evaporate a rod of metallic niobium, heat and sustain the plasma, the authors used the radiation of an Nd-laser with a wavelength of 532 nm. The buffer gas was helium, the total pressure of the gas mixture was 0.4–0.65 MPa. The mass spectra of ionised clusters Nb m C +n were recorded directly from the plasma using a quadrupole mass spectrometer. Analysis of the mass spectra shows that the nanoparticles with a cubic structure and with the ratio Nb:C ≈1:1 (Nb 14 C 13 ) form at a relatively low (4%) concentration of methane CH 4 in helium and a radiation power of 10–15 mJ pulse –1 . At a methane concentration from 8 to 20% and the radiation power not lower than 15 mJ pulse –1 , dodecahedral particles with the Nb:C ratio close to 1:2 (for example, Nb 11 C 21 , Nb 13 C 22 ) form preferentially. It is interesting that an increase in the concentration of hydrocarbons is accompanied by the growth of clusters: for example, at a methane concentration of 8%, the largest clusters are Nb 8 C 12 , and a concentration of 12% the clusters Nb 11 C 21 , Nb 12 C 22 , Nb 13 C 22 and Nb 14 C 25 appear. The identical growth of clusters is detected in laser gas phase synthesis in the Zr–C system, where clusters Zr 13 C 22 , Zr 14 C 21 and Zr 13 C 23 appear [75]. It indicates the formation of structure representing doubled dodecahedrals. The authors of [74] concluded that the metallo-carbohedrenes (in particular, large clusters consisting of two or more connected dodecahedrals) are formed at a high concentration of hydrocarbon and a high laser radiation power, causing dehydrogenation of carbon. Thus, the metallocarbohedrenes are formed at a higher carbon content in the plasma. A decrease in the hydrocarbon concentration or in radiation power decreases the carbon content in the plasma. As a result, a relative decrease in carbon content leads to the formation of carbide nanoparticles MC with the cubic structure B1, in which the carbon content is lower than in molecular clusters M m C n . This really shows that in the gas-phase synthesis conditions, the formation of cubic or 43

Nanocrystalline Materials

dodecahedral structures in the M–C systems is determined to a greater extent by kinetic factors instead of thermodynamic ones. On the whole, plasma chemical synthesis with different methods of formation of plasma is one of the most promising methods of producing different nanostructured materials. 2.3 PRECIPITATION FROM COLLOID SOLUTIONS Precipitation from colloid solutions is evidently the first method of producing nanoparticles. Visitors to the Royal Institution’s Faraday Museum in London, opened in 1973 by Queen Elizabeth II, may see two glass vessels with a colloid solution of gold (Fig. 2.5) produced by M. Faraday in the first half of the 19 th century. These solutions have retained their stability for almost 200 years. The production and optical properties of colloid solutions of gold were described by Faraday in 1857 [76]. The conventional method of producing nanoparticles from colloid solutions is based on a chemical reaction between the components of the solution and interrupting the reaction at a specific moment in time [77–81]. Subsequently, the dispersed system is transferred from the liquid colloidal state to the nanocrystalline solid state. For example, nanocrystalline powders of sulfides are produced by reaction of hydrosulfuric acid H 2 S or sulfide Na 2 S with the water-

Fig. 2.5. Colloid solutions of gold, produced by M. Faraday (the Museum of the Royal Institute in Great Britain, London).

44

Synthesis of Nanocrystalline Powders

soluble salt of a metal: Nanocrystalline cadmium sulfide CdS is produced by precipitation from a mixture of solutions of cadmium perchlorate Cd(ClO 4) 2 and sodium sulfide Na 2 S

Cd (ClO 4 ) 2 + Na 2S = CdS ↓ +2NaClO 4

(2.3)

The growth of nanoparticles of CdS is interrupted by a sudden increase of the pH of the solution. The colloid particles of metal oxides are produced by hydrolysis of salts. For example, TiO 2 particles form easily in hydrolysis of titanium tetrachloride

TiCl4 +2H 2 O = TiO 2 ↓ +4HCl

(2.4)

The formation of metallic or semiconductor clusters with a very low dispersion of the dimensions (or even monodispersed clusters) is possible inside pores of a molecular sieve (zeolite). Isolation of the clusters inside the pores is maintained at heating to very high temperatures. For example, semiconductor clusters (CdS) 4 are synthesised inside cavities of zeolites [82]. Analysis of the properties of clusters, produced in ultrafine channels and, in particular, in pores of zeolites, has been the subject of a review in [83]. Large semiconductor nanoparticles are synthesised by annexation of additional molecules to the initial small cluster which is stabilised in advance in a colloid solution by organic ligands. This synthesis of large nanoparticles can be regarded as polymerization of inorganic compounds. Nanoparticles can also be produced by means of ultrasound treatment of colloid solutions, containing large particles. Precipitation from colloid solutions makes it possible to synthesise nanoparticles of a mixed composition, i.e. nanocrystalline heterostructures. In this case, the core and the shell of the mixed nanoparticle are produced from semiconductor substances with different electron structures. Formation of heterostructures, for example, CdSe/ZnS or ZnS/CdSe, HgS/CdS, ZnS/ZnO, TiO 2 /SnO 2 takes place as a result of controlled precipitation of molecules of a semiconductor of one type on pre-synthesised nanoparticles of a semiconductor of another type [84–87]. These heteroparticles may be coated with a layer of another semiconductor. Nanocrystalline heterostructures are used in photocatalysis.

45

Nanocrystalline Materials

In the group of all the methods of producing isolated nanoparticles and other powders, the method of precipitation from colloid solutions is characterised by high selectivity and makes it possible to produce stabilised nanoclusters with a narrow size distribution which is very important for the application of nanoparticles as catalysts or in nanoelectronic devices. The main problem of precipitation from colloid solutions is how to avoid coalescence of the produced nanoparticles. The chemical synthesis of large metallic clusters using colloid solutions has been discussed in detail in [88]. There are different chemical methods of producing nanoparticles in colloid solutions but in any case it is necessary to protect particles in order to prevent their coalescence. Stabilisation of the colloid particles and clusters is possible using ligand molecules. Various polymers are used as ligands. The schematic reaction of production of a ligand-stabilised metallic cluster M n has the following form: mL nM + + ne − → M n  → M n L m ,

(2.5)

where L is a ligand molecule. The metallic clusters of gold, platinum and palladium, produced by this method, may contain from 300 to 2000 atoms. The metallic clusters have a cubic or hexagonal close-packed structure. In these clusters, the central atom is surrounded by several shells in which the number of atoms is 10k 2 + 2 (k is the number of the shell), i.e. the first shell contains 12, the second 42, the third 92 atoms, etc. The total number of atoms in a cluster is n =  2 N + 1 + 10

N

Σk k =1

2

, where N is the number

of atomic shells (layers). In the clusters, stabilised with ligands, there is a metallic core where the nearest neighbours of the metal atom are only metallic atoms, and the external shell of the metallic atoms, partially bonded with the ligand molecules. Protection of the clusters by a means of an outer shell is shown in Fig. 2.6: The surface of a dark colloid nanoparticle of gold with a size of approximately 12 nm is coated with a light shell of trisulfonated triphenylphosphine P(m-C 6 H 4 SO 3 Na) 3 ligand molecules. The metallic clusters, consisting of 55 atoms, distributed in two shells, are evidently particles with a smaller size still retaining some of the properties of the metal; however, examination by scanning tunneling spectroscopy at room temperature already indicates the 46

Synthesis of Nanocrystalline Powders

Fig. 2.6. High resolution microscopic image of a single gold colloid of about 11×13 nm, covered by a shell of P(m-C 6 H 4 SO 3 Na) 3 ligands (image obtained by J.-O. Bovin and A. Carlsson, University of Lund) [88].

splitting of electron levels in these particles. The hydrolysis of metal salts is carried out to produce colloid oxide particles [89–91]. For example, the nanocrystalline oxides of titanium, zirconium, aluminium and yttrium may be produced by hydrolysis of appropriate chlorides or hypochlorides. Ultrafine titanium oxide is also produced by the hydrolysis of titanyl-sulfate with subsequent heating of the amorphous deposit at 1000–1300 K. Colloid solutions are stabilised to prevent coalescence of the nanoparticles using polyphosphates, amines, and hydroxyl ions. The colloid solutions of semiconductor oxides and sulfide nanoparticles are used directly (without precipitation) in photocatalytic processes of synthesis and destruction of organic compounds, and dissociation of water. To produce highly dispersed powders, the deposits of colloid solutions, consisting of agglomerated nanoparticles, are heated at 1200–1500 K. For example, a highly dispersed powder of silicon carbide (D ~ 40 nm) is produced by hydrolysis of organic salts of silicon with subsequent heating in argon at 1800 K [92]. Highly dispersed powders of oxides of titanium and zirconium are often produced by precipitation by means of oxalates. Cryogenic drying is also used to produce highly dispersed powders from colloid solutions. A solution is pulverised in a chamber with 47

Nanocrystalline Materials

a cryogenic medium and, consequently, freezes up in the form of fine particles. Subsequently, the pressure of the gas medium is reduced in such a manner that it becomes lower than the equilibrium pressure above the frozen solvent and the material is heated with continuous pumping for sublimation of the solvent. This leads to the formation of the finest porous granules of the same composition, and heating these granules gives powders. The precipitation methods also include the production of nanocrystalline composites from tungsten carbide and cobalt used for producing hard alloys [93, 94]. Colloid solutions of tungsten and cobalt salts are dried by spraying. The produced powder is subjected to low-temperature carbothermal reduction in a suspended layer thus retaining high dispersion. To decelerate the growth of grains and decrease the solubility of tungsten carbide in cobalt, nonstoichiometric vanadium carbide in the amount of up to 1 wt.% is added to the mixture. The hard alloy, produced from this nanocrystalline composite, is characterised by the optimum combination of high hardness and strength [93–95]. It is shown [96] that every nanocomposite particle of WC–Co, with a size of ~75 µm, consists of several millions of nanocrystalline WC grains smaller than 50 nm, distributed in the cobalt matrix. Sintering of the nanocomposite mixture of tungsten carbide with 6.8 wt.% Co and 1 wt.% VC leads to the formation of alloys in which 60% of WC grains were smaller than 250 nm and 20% smaller than 170 nm. An even finer grain structure was found in the alloy containing, in addition to tungsten carbide, 9.4 wt.% Co, 0.8 wt.% Cr 3 C 2 and 0.4 wt.% VC. After sintering at 1670 K 60% of tungsten carbide grains in this alloy are smaller than 140 nm and 20% smaller than 80 nm. Comparison of the nanocrystalline and the conventional polycrystalline alloy, having the same hardness, shows that the fracture toughness of the nanocrystalline alloy is 1.2–1.4 times higher than the fracture toughness of the conventional coarsegrained alloy [96]. According to [97], the hard WC–Co alloy, produced from the nanopowder of tungsten carbide with a particle size of 30–50 nm, has a more uniform fine-grain structure and high hardness and strength than the conventional hard alloy of the same composition. In [97] it is also reported that the addition to the coarse-grained charge of the standard WC–Co alloy of 3–5 wt.% of tungsten carbide nanopowder decreases the scatter of the hardness and strength values, i. e. stabilises the properties of the hard alloy.

48

Synthesis of Nanocrystalline Powders

2.4 THERMAL DECOMPOSITION AND REDUCTION Thermal decomposition is usually carried out using complex element-organic and organometallic compounds, hydroxides, carbonyls, formiates, nitrates, oxalates, amides and imides of metals which at a specific temperature decompose with the formation of a synthesised substance and generation of the gas phase. The production of highly dispersed metallic powders by thermal decomposition of different salts has been described in detail in [98]. For example, the pyrolysis of formiates or iron, cobalt, nickel, and copper in vacuum or an inert gas at 470–530 K produces the metallic powders with a mean particle size of 100–300 nm. A variant of pyrolysis is the decomposition of organometallic compounds in a shock tube. Subsequently, free metal atoms are condensed from the supersaturated vapours [14]. A long steel tube, closed on both ends, is sectioned into two different parts with a thin diaphragm made of mylar film or aluminium foil. The longer part of the tube is filled with argon under a pressure of 1000–2500 Pa with an addition of 0.1–2.0 mol.% of a organometallic compound. The other part of the tube is filled with helium or a mixture of helium with nitrogen until the membrane is ruptured. Rupture of the membrane leads to the formation of a shock wave. The temperature at the front of this wave may reach 1000–2000 K. Impact heating of the gas results in the decomposition of the organometallic compound several microseconds after the passage of the wave front, and free metal atoms form a strongly supersaturated vapour capable of rapid condensation. This method is used to produce ultrafine powders of iron, bismuth and lead. A combination of thermal decomposition and condensation is the supersonic discharge of gases from a chamber, in which increased constant pressure and temperature is maintained, through a nozzle into vacuum [14]. In this case, the thermal energy of gas molecules is transformed into the kinetic energy of the supersonic flow. During expulsion the gas cools down and transforms into supersaturated vapour in which clusters, containing from 2 to a million of atoms may form. An increase of the initial pressure in the chamber at a constant temperature increases supersaturation. The authors of [99] describe the production of submicrocrystalline powder (Si 3 N 4 + SiC) by pyrolysis of liquid polysilazane [CH 3 SiHNH] n , discharged in the form of an aerosol through an ultrasound nozzle. The aerosol was heated by the radiation of a continuous-wave CO 2 laser. 49

Nanocrystalline Materials

The highly dispersed powders of silicon carbide and nitride are produced by pyrolysis of polycarbosilanes, polycarbosiloxanes and polysilazanes [100–102]. Initial heating is carried out by a means of low temperature plasma or laser radiation and the products of pyrolysis are subsequently annealed at a temperature of ~1600 K to stabilise the structure and composition. Poly[2-(vinyl)pentaborane] is used to produce nanocrystalline boron carbide. Boron-containing polymers of the type of polyborazine, polyborazole and poly(B-vinylborazine) are proposed for the production of highly dispersed powders of boron nitride and also as additions to the titanium powder for synthesis of nanocrystalline compositions TiN + TiB 2 [102]. The nanocrystalline AlN nitride powder with a mean particle size of 8 nm is produced by decomposition of ammonium polyamide imide [Al(NH 2) 3NH] n in ammonia at 900 K [103]. The submicrocrystalline titanium nitride powder with a mean particle size of 100–300 nm is prepared by decomposition of polytitanimide Ti[N(CH 3 ) 2 ] 4 [100]. The transition metal borides may be produced by pyrolysis of borohydrides at 600–700 K, i. e. at a temperature which is considerably lower than the conventional temperatures of solid phase synthesis. For example, high-dispersion powders of zirconium boride with a specific surface of 40–125 m 3 g –1 are produced by thermal decomposition of zirconium tetraborohydride Zr(BH 4) 4 under the effect of pulsed laser radiation [104]. The powders, produced by thermal dissociation of monomer and polymer compounds, must be additionally annealed to stabilise the composition and structure; the annealing temperature of nitrides and borides is 900–1300 K, and for oxides and carbides it is 1200– 1800 K. Section 2.1 describes the method of producing nanocrystalline powders in which the thermal decomposition of the organometallic precursor is combined with the condensation of nanoparticles on the cold surface in the atmosphere of a rarefied inert gas. The main disadvantage of the thermal decomposition method is the relatively low selectivity of the process, because the reaction product is usually a mixture of the target product and other compounds. A widely used method of producing highly dispersed metallic powders is the reduction of compounds of metals (hydroxides, chlorides, nitrates, carbonates) in a hydrogen flow at a temperature of 10 4 K. The capability of electrically exploded conductors to rapidly change properties and efficiently transform the primary electric or magnetic energy of accumulators to other types of energy (thermal energy, the energy of radiation of the plasma, energy of the shockwaves) is used in particular for producing ultrafine powders. In the initial stage of electric explosion, the Joule heating of a conductor is accompanied by its linear expansion at a relatively low 63

Nanocrystalline Materials

rate of 1–3 m s –1 . In the stage of an explosion the passage of a current pulse superheats the metal above the melting point, the expansion of the material of the exploded conductor takes place at a rate of up to 5×10 3 m s –1 and the superheated metal is dispersed in an explosion-like manner [136]. The pressure and temperature at the front of the shockwave reach several hundreds of MPa (thousands of atmospheres) and ~10 4 K, respectively. Particles of very small sizes form as a result of condensation in the flow of the rapidly expanding vapour. Regulating the explosion conditions, it is possible produce powders with a particle size from 100 µm to 50 nm. The mean size of the particles monotonically decreases with increasing current density and shortening of the pulse time. Electric explosion in an inert atmosphere makes it possible to produce powders of metals and alloys. Adding of additional reagents into the reactor (air, mixture of oxygen and inert gas, nitrogen, distilled water, decane C 10 H 22 , paraffin, commercial oil) allows to produce ultrafine powders of oxides, nitrides, carbides or their mixtures (Table 2.1). The authors of [134] described copper powders produced by electric explosion in an inert gas at a pressure of 200 Pa with the maximum size distribution corresponding to ~20 nm, and aluminium powders with a mean particle size of approximately 50 nm. According to the experimental data [137], the submicrocrystalline powders, produced by electric explosion of wires, have a very large excess energy. For example, aluminium powders with a mean particle size of 500–800 nm have an excess energy of 100–200 kJ mol –1 , and silver powders with a mean particle size of ~120 nm Table 2.1 Some nanopowders produced by electrical explosion in a vacuum and various media Metal

Vacuum 100° may be the result of large static displacements of vanadium atoms.

larger (by 0.00047 nm) than the lattice constant of the disordered carbide VC 0.875 . According to [140, 141, 148, 152] this large difference of the lattice constants of the ordered and disordered carbides VC 0.875 may be observed only at maximum or nearly maximum degree of ordering. The ratio of the intensities of the structural and superstructure reflections confirms that the degree of long-range order is nearly at a maximum in nanostructured vanadium carbide. In addition, this relationships shows that the ordered phase occupies the entire volume 70

Synthesis of Nanocrystalline Powders

of the substance, i.e. the nanopowder consists of a single ordered phase. Analysis of the ratio of the intensities yields another important conclusion regarding the almost complete absence of oxygen in the carbon sublattice; this is in agreement with the chemical analysis data according to which the lattice contains only 0.1 wt.% of oxygen. Attention should be given to the fact that the intensity of superstructure reflections I super of the coarse-grained ordered carbide VC 0.875 decreases with increasing diffraction angle 2θ (Fig. 2.17b), whereas the intensity of superstructure reflections of the VC 0.875 nanopowder in the range 2θ > 100° not only does not decrease but even increases (Fig. 2.17c). A possible reason for this is the presence of large relaxation static displacements of vanadium atoms in the vicinity of carbon vacancies. Earlier, these displacements were actually detected in the ordered carbide V 8 C 7 [153]. Regardless of the nanometer thickness of the nanocrystals, analysis of the width of diffraction reflections did not reveal any large deviations from the instrumental width. Since all atoms inside the crystal are scattered coherently, the absence of broadening of the diffraction reflections is in agreement with the presence of a relatively large number of atoms in the nanocrystallites because of their large size in two other dimensions. The pycnometric density of nanostructured VC 0.875 powder is considerably lower than the theoretical density and is 5.15 g cm –3 . This could have been caused by the presence in the specimen of a substance with a low density (for example, adsorbed water) or by the high imperfection of the metal sublattice determined by the presence of vacant metal sites (metal structural vacancies). To explain this, the pycnometric density was measured after heating the carbide in vacuum. The removal of water at a temperature of 470 K resulted in a pycnometric density of 5.48 g cm –3 which is also slightly lower than the theoretical value and is associated with the presence of chemisorbed oxygen in the specimen. Only after vacuum annealing at 900 K it is possible to achieve a pycnometric density of 5.62 g cm –3 coinciding with the theoretical density. The establishment of theoretical density as a result of annealing indicates that the metal sublattice did not contain structural vacancies and the reduced initial density is determined by adsorbed impurities of water and oxygen. Although after annealing at 900 K 1–2 atomic layers of oxides remained on the surface of nanocrystallites, the difference between the theoretical and pycnometric densities cannot be detected because the content of the 71

Nanocrystalline Materials

Fig. 2.18. Position of vanadium V and carbon C atoms in the unit cell of the ordered cubic (space group P4 3 32) phase V 8 C 7 : vacant (unfilled by carbon atoms) octahedral interstitials of the metallic sublattice are indicated).

surface oxide phase is less than 0.1 wt.%, and the densities of the oxides and vanadium carbide are similar. The technique of magnetic susceptibility has been used successfully for analysis of the structural state of weak magnetics [160, 161]. This technique is highly sensitive to phase transformations in non-stoichiometric carbides [140, 141, 144–146] and, consequently, it was used in the study of VC 0.875 nanopowder. The magnetic susceptibility γ of the nanopowder of vanadium carbide VC 0.875 was measured in the temperature range from 300 to 1200 K in a vacuum of 1×10 –3 Pa after treatment in hydrochloric acid for removal of vanadium oxide [157, 159]. The temperature dependence of the magnetic susceptibility of the vanadium carbide (Fig. 2.19) is in good agreement with the data [162, 163] for coarse-grained specimens. This confirms the slight effect of the nanostructure of vanadium carbide on its electronic properties. The most effective and sensitive method of study of defects at the interfaces and surfaces of nanoparticles is electron-positron annihilation. Trapping of positrons by defects such as vacancies or nanovoids leads to an increase in the positron lifetime in 72

Synthesis of Nanocrystalline Powders









χ  FP J 

QDQR9&











%





.

Fig. 2.19. Temperature dependence of magnetic susceptibility χ of VC 0.875 nanopowder [153, 154].

comparison with the lifetime for a defect-free material [164]. The value of the lifetime may be used to determine the type of defect. In [150, 157–159] the lifetime of positrons was measured on a powder of carbide VC 0.875 subjected to preliminary heating at 400 K to remove water. For comparison, measurements were taken of the positron lifetime in a coarse-grained sintered specimen of vanadium carbide VC 0.875 . The measured positron lifetime spectra are shown in Fig. 2.20. The spectra show that the mean positron lifetime in the nanopowder is considerably longer than that in the polycrystal. The spectrum of the coarse-grained specimen of the vanadium carbide contains only a short component 157 ± 2 ps which corresponds to the annihilation of positrons in the structural vacancies of the carbon sublattice [156, 157]. The quantitative analysis of the spectrum of the nanocrystalline specimen shows that in addition to the short component, equal to 157 ± 2 ps, it contains a long component 500 ps with a relative intensity of I 2 = 7%. According to [164], the long component is determined by the annihilation of positrons in defects on the surface of the particles. Trapping of the positron by a structural vacancy indicates the absence of diffusion of the positron over large distances; in this case, the intensities of the components are proportional to the volume fractions of the phases containing defects of different type. Thus, the relative intensity of the long component, I 2, coincides with 73

Nanocrystalline Materials







&RXQWV DUELWUDU\ XQLWV







QDQRFU\VWDOOLQH 9 & 







FRDUVHJUDLQHG 9 & 











 7LPH QV





Fig. 2.20. Positron lifetime spectra of nanocrystalline and coarse-grained vanadium carbide V 8C 7 (VC 0.875) [146, 155].

the volume fraction of the surface ∆V surf = δS/V in the vanadium carbide nanopowder. Estimates show that the thickness of the surface layer is δ = 0.5–0.7 nm and corresponds to 3–4 atomic monolayers. Thus, it follows from the measurements of the positron lifetime that the internal part of the nanocrystallites contains only non-metal structural vacancies, and the surface layer of the nanocrystallites of the vanadium carbide contains defects of the type of vacancy agglomerates. Authors [150, 157–159] proposed the following model of the structure of vanadium carbide nanocrystallites. The nanocrystallites represent strongly bent plates-sheets 400–600 nm in diameter and 1520 nm thick. The internal part of the nanocrystallites consists of ordered carbide V 8C 7 with a high degree of the long-range order and a negligible small content of dissolved oxygen. The surface layer of the nanocrystallites contains 3.1 wt.% of chemisorbed oxygen and a large number of vacancy agglomerates indicating to a loose structure of a layer. The thickness of the surface phase does not exceed 0.7 nm or 4 atomic monolayers. This morphology of the nanocrystalline powder of the nonstoichiometric vanadium carbide may be a consequence of cracking of 74

Synthesis of Nanocrystalline Powders

the particles at the interfaces between the disordered and ordered phases. In fact, high temperature X-ray measurements [149] show that at a temperature of 1413±20 K the order–disorder VC 0.875 → V 8 C 7 phase transition results in a sudden large increase of the lattice constant of basic B1 structure by 0.0004 nm (Fig. 2.21a); the size of the domains of the ordered phase is ~20 nm. According to [151, 152], the ordering VC 0.875 → V 8C 7 takes place via the mechanism of phase transition of the first kind at a temperature of 1368 ± 12 K; at 300 K constant a B1 of the basic crystal lattice of the quenched disordered

, 



,

QP

9&  9&



 





%





.



E

7

 .

, QP





 9&

9&







\







&9 DWRP UDWLR

Fig. 2.21. Variation of the basic lattice constant a B1 (B1 cubic structure, space group Fm3m ) of non-stoichiometric vanadium carbide as a result of ordering: a) sudden change in the basic lattice constant a B1 at the order → disorder V 8C 7 → VC0.875 transition at a temperature of T trans = 1413 ± 10 K [149]; b) the dependence of the basic lattice constant a B1 on the composition of vanadium monocarbide VC y in the disordered (full circle) and ordered (open circle) states at a temperature of 300 K [151, 152]. 75

Nanocrystalline Materials

carbide VC 0.875 is 0.0002 nm lower than that of the ordered carbide with the same carbon content (Fig. 2.21b). The difference in the volumes of the disordered and ordered phases leads to the formation of stresses and subsequent cracking at the phase boundaries. Another mechanism of formation of nanostructures is also possible. The first kind of the disorder–order phase transition results in the formation of domains of the ordered phase between which stresses form as a result of mismatch of the atomic structure of antiphase domains. With time, the stresses lead to cracking of the grains of the initial disordered phase at the boundaries of the antiphase domains of the ordered phase. Special investigations are required for reliable explanation of the mechanism of refining of the grains due to order-disorder transformation in vanadium carbide. Thus, the formation of the nanostructure of the powder of nonstoichiometric vanadium carbide VC 0.875 is determined by the disorder–order VC 0.875 → V 8 C 7 phase transformation taking place in this material. The nanocrystallites of the ordered nonstoichiometric vanadium carbide in the form of distorted discs with a diameter not exceeding 600 nm and a thickness of 15–20 nm form as a result of cracking of particles up to 1 µm in size. The surface layer of the nanocrystallites has thickness 0.5–0.7 nm and contains chemisorbed oxygen and a large number of vacancy agglomerates. This indicates its loose structure. It may be assumed that the disorder–order transformations, taking place with spasmodic volume changes, may be used for the formation of the nanostructured state of other materials, including strongly non-stoichiometric compounds and substitutional solid solution . 2.8 SYNTHESIS OF HIGH-DISPERSED OXIDES IN LIQUID METALS This new method of synthesis of highly dispersed oxides is proposed by the authors of [168]. The working medium is represented by melts of gallium at a temperature of 323–423 K, melts of lead at 653–873 K or of a lead–bismuth alloy at 453–873 K. Synthesis takes place in two stages. In the first stage, metal M is dissolved in the melt. The chemical affinity of this metal for oxygen is higher than that of the metal forming the melt. The solubility of metal M in the melt should not be lower than 0.1 wt.%. In the second stage, the dissolved metal M is oxidised by bubbling 76

Synthesis of Nanocrystalline Powders

the melt with water vapour or a gas mixture (H 2 O + Ar). The content of the water vapours in the oxidation gas mixture is 15– 30 vol.%. Below, as an example, we present the reactions taking place in the process of oxidation of metal M in a gallium melt: 2Ga (liquid) + 3H 2 O (gas) = Ga 2 O 3 (solid) + 3H 2↑,

(2.6)

Ga 2 O 3 (solid) ⇒ Ga 2 O 3 (dissolved) ,

(2.7)

x M (dissolved) + y Ga 2 O 3 (dissolved) = M x O 3y (amorphous) + 2y Ga (liquid) . (2.8) Selective oxidation leads to the formation of amorphous highly dispersed metal oxides. In particular, the selective oxidation of aluminium, dissolved in gallium in an amount of 1 wt.%, results in formation of flakes of an amorphous high-porosity substance with chemical composition Al 2O 3·H 2O. Examination of the microstructure shows that the substance consists of fibres with a diameter of 5100 nm, oriented in one direction. The distance between the residual fibres is 5–400 nm. The produced nanomaterial has a porosity of 96.5–99.5 vol.% and a specific surface from 30 to 800 m 2 g –1 may be used for thermal insulation. The stability of the composition, structure and properties of the nanomaterial is retained during longterm annealing in the temperature range up to 1000 K. Selective oxidation of indium, dissolved in lead, has made it possible to produce indium oxide In 2 O 3 in the flaky form with a macroparticle size of up to 5 mm. Examination by scanning electron microscopy shows that flakes of In 2 O 3 consists of needles with a diameter of 50–100 nm spaced at 80–200 nm. The number of metals, whose solubility in melts of gallium, lead or lead–bismuth alloys is 0.1 wt.% or higher, is relatively large. By means of selective oxidation of the system Ga (liquid) – M (dissolved) at a temperature of up to 423 K it is possible to produce highdispersion oxides Na 2 O, Al 2 O 3 , MgO and CaO. Selective oxidation in the systems Pb (liquid) – M (dissolved) and Pb/Bi (liquid) – M (dissolved) at temperatures of up to 873 K makes it possible to synthesise nanosized oxides SbO 2 , TeO, NiO, GeO 2 , SnO 2 , In 2 O 3 , K 2 O, ZnO, Ga 2 O 3 , Na 2 O, MnO, Li 2 O, Al 2 O 3 , BaO, SrO, MgO and CaO. According to [168], the method developed is suitable for producing highly dispersed nitrides, sulfides and halides in melts. In this case, the melt with a dissolved metal should be subjected to the treatment by an inert gas with nitrogen N 2 , hydrogen sulfide H 2 S or gaseous chlorides of gallium or lead. 77

Nanocrystalline Materials

2.9. SELF-PROPAGATING HIGH-TEMPERATURE SYNTHESIS The self-propagating high-temperature synthesis (SHS) represents a rapid process of solid combustion of reagents (a metal and carbon for carbides or a metal in nitrogen for nitrides) at a temperature from 2500 to 3000 K [169]. Usually carbides are synthesized in a vacuum or an inert atmosphere (argon). The mean size of grains in carbides produced by the SHS method is 10–20 mm, while the size of nitride grains usually is smaller and equals 5–10 mm. SHSsynthesized carbides and nitrides of group IV and V transition metals have, as a rule, an inhomogeneous composition and require additional grinding and annealing for homogenisation. To decrease the grain size in synthesized carbides or nitrides, the starting mixture is diluted with the final product (for example, up to 20 mass % TiC carbide is added to the Ti + C mixture). For the same purpose, some carbon in the mixture is replaced by polymers (polystyrene, polyvinylchloride) during synthesis of carbides. As a result, carbides and nitrides with grains 1–5 mm in size on the mean can be synthesized. It is proposed [169] to use an inert dilutant in SHS of nanosized powders of titanium carbide. The inert dilutant is NaCl, which is chemically inert to starting powders of metallic titanium Ti and carbon C, as well as to the final titanium carbide TiC. During combustion of titanium and carbon, NaCl, whose melting point T melt = 1074 K, forms a melt, which prevents the growth of the synthesized carbide particles. Moreover, NaCl readily dissolves in water and can be easily separated from the synthesized carbide. Titanium carbide is synthesized using a Ti + C + mNaCl mixture, where m was 0.2 to 0.7 mole of sodium chloride per Ti or C atom. The size of particles in the starting powders of Ti, C and NaCl did not exceed 50, 0.1 and 150 mm respectively. Samples 30 mm across and 40 mm long are pressed from the powder mixture. The synthesis is realized in a standard reactor under an argon atmosphere at a pressure of 0.5 MPa. Combustion is initiated by a current pulse (the voltage U = 15–20 V and the pulse time of 1.0–1.5 s), which is fed to the sample via tungsten wire. When the quantity of sodium chloride, m, increased from 0.2 to 0.7, the combustion temperature dropped from 2500 to 1950 K. The starting mixture is homogenised spontaneously during combustion thanks to melting of Ti and NaCl. Microscopic examination shows that the titanium carbide particles are distributed in the NaCl melt and are isolated one from another by a thin layer 78

Synthesis of Nanocrystalline Powders

of the melt which prevents their growth. No intermediate phases form. The size of the titanium carbide particles decreases with growing concentration of NaCl in the starting mixture. After NaCl is washed off, the synthesized powder of titanium carbide is analyzed by X-ray diffraction and electron microscopic methods. The synthesized titanium carbide has a B1 cubic structure with the lattice spacing of 0.433 nm. The powder includes irregularly shaped particles 20 to 300 nm in size with an mean size of about 100 nm. According to [169], the optimal concentration of NaCl for the synthesis of titanium carbide nanoparticles is m = 0.4 or about 30 mass %. The chemical composition of the synthesized titanium carbide is not determined in [169]. If the synthesized titanium carbide is free of impurities, the comparison of the lattice constant measured in [169] and an exact experimental dependence a B1 (y) for TiC y [55, 140, 141, 158, 170] suggests that the carbide has a nearly stoichiometric composition TiC 1.0 .

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73 (1986)) Eds. C. J. Brinker, D. E. Clark and D. R. Ulrich (MRS, Pittsburg, 1986) pp.71-78. G. Franz, G. Schwier. Starting materials for advanced ceramics – production and properties. In: Raw Materials for New Technologies. Ed. M. Kursten (Nagele and Obermuller, Stuttgart 1990) pp.139-158. V. M. Chertov, V. I. Litvin, I. F. Mironyuk, V. V. Tsyirina. Synthesis and texture of xerogels based on ultradispersed powders of alumina and aluminium monohydroxide. Neorgan. Materialy 29, 1019-1020 (1993) (in Russian). F. Hatakeyama, Sh. Kanzaki. Synthesis of monodispersed spherical b-silicon carbide powder by a sol-gel process. J. Amer. Ceram. Soc. 73, 2107-2110 (1990). L. E. McCandlish, B. H. Kear, B. K. Kim. Processing and properties of nanostructured WC-Co. Nanostruct. Mater. 1, 119-124 (1992). L. Wu, J. Lin, B. K. Kim, B. H. Kear, L. E. McCandlish. Grain growth inhibition of nanostructured WC–Co alloys. In: Proc. of the 13 th Intern. Plansee Seminar. V.3. Eds. H. Bildstein and R. Eck. (Metallwerk Plansee, Reutte 1993) pp.667-675. Z. Fang, J. W. Eason. Study of nanostructured WC-Co composites. In: Proc. of the 13th Intern. Plansee Seminar. V.3. Eds. H. Bildstein and R. Eck. (Metallwerk Plansee, Reutte 1993) pp.625-638 P. Seegopaul, L. E. McCandlish, F. M. Shinneman. Production capability and powder processing methods for nanostructured WC-Co powder. Intern. J. Refr. Met. Hard Mater. 15, 133-138 (1997). N. I. Borisenko, V. A. Moldaver, A. V. Lebedev, N. V. Kobzarev. Application of nanosize powders in technology of hard alloys. In: Physics and Chemistry of Ultra-Disperse (Nano-) Systems / Proc. of VI th All-Russian Conference,August 19-23, 2002, Tomsk, Russia. (Moscow Physico-Technical Institute, Moscow 2002) pp.339-340. I. D. Morokhov, L. I. Trusov, S. P. Chizhik. Ultra-Dispersed Metallic Substances (Atomizdat, Moscow 1977) 264 pp. (in Russian). K. E. Gonsalves, P. R. Strutt, T. D. Xiao, P. G. Klemens. Synthesis of silicon (carbide, nitride) nanoparticles by rapid laser polycondensation/ crosslinking reactions of an organosilazane precursor. J. Mater. Sci. 27, 32313238 (1992) K. E. Gonsalves, K. T. Kembaiyan. Synthesis of advanced ceramics and intermetallics from organometallic/polymeric precursors. Solid State Ionics 32/33, 661-668 (1989). M. Peuckert, T. Vaahs, M. Bruck. Ceramics from organometallic polymers. Advanc. Mater. 2, 398-404 (1990). M. G. Mirabelli, A. T. Lynch, L. G. Sheddon. Molecular polymeric precursors to boron-based ceramics. Solid State Ionics 32/33, 655-660 (1989). T. Wade, J. Park, E. G. Garza, C. B. Ross, D. M. Smith, R. M. Crooks. Electrochemical synthesis of ceramic materials. 2. Synthesis of aluminum nitride (AlN) and an AlN polymer precursor: chemistry and materials characterization. J. Amer. Chem. Soc. 114, 9457-9464 (1992). G. W. Rice, R. L. Woodin. Zirconium boronitride as a zirconium boride precursor. J. Amer. Ceram. Soc. 71, C181-C183 (1988). E. G. Avvakumov. Mechanical Methods of Activation of Chemical Processes (Nauka, Novosibirsk 1988) 305 pp. (in Russian). Mechanical Alloying in Inorganic Chemistry. Ed. E. G. Avvakumov. (Nauka, Novosibirsk 1991) 264 pp. (in Russian).

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Mechanical Alloying (Proc. Intern. Symp. On Mechanical Alloying, Kyoto, Japan, May 7-10, 1991) Ed. P. H. Shingu. Materials Science Forum 8990 (1991) (Trans Tech Publications, Switzerland 1992) 816 pp. A. R. Yavari, P. J. Desre, T. Benameur. Mechanically driven alloying of immiscible elements. Phys. Rev. Lett. 68, 2235-2238 (1992). H.-J. Fecht. Nanostructure formation by mechanical attrition. Nanostruct. Mater. 6, 33-42 (1995). J. Balogh, L. Bujdoso, G. Faigel, L. Granasy, T. Kemeny, I. Vincze, S. Szabo, H. Bakker. Nucleation controlled transformation in ball milled FeB. Nanostruct. Mater. 2, 11-18 (1993). A. A. Popovich, V. N. Vasilenko. Mechanical alloying of high-melting compounds. In: Mechanical Alloying in Inorganic Chemistry. Ed. E. G. Avvakumov. (Nauka, Novosibirsk 1991) pp.168-176 (in Russian). A. A. Popovich, V. P. Reva, V. N. Vasilenko. Mechanochemical synthesis kinetics and structure formation of refractory compounds. Neorgan. Materialy 28, 1871-1876 (1992) (in Russian). V. Yu. Davydkin, L. I. Trusov, P. Yu. Butyagib, V. V. Moskvin, I. V. Kolbanev, V. I. Novikov, S. S. Plotkin. Structure of high-melting carbides synthesized by mechanical alloying. In: Mechanical Alloying in Inorganic Chemistry. Ed. E. G. Avvakumov. (Nauka, Novosibirsk 1991) pp.183-185 (in Russian). N. Atsumi, K. Yoshioka, T. Yamasaki, Y. Ogino. Nitriding of transition metals by mechanical alloying in nitrogen gas. Funtai oyobi Funmatsu Yakin (J. Japan. Soc. Powd. and Powd. Metall.). 40, 261-264 (1993) (in Japanese). A. Teresiak, H. Kubsch. X-ray investigations of high energy ball milled transition metal carbides. Nanostruct. Mater. 6, 671-674 (1995). D. Oleszak, H. Matyja. Nanocrystalline Fe-based alloys obtained by mechanical alloying. Nanostruct. Mater. 6, 425-428 (1995. M. A. Xueming, J. I. Gang. Nanostructured WC-Co alloy prepared by mechanical alloying. J. Alloys and Comp. 245, L30-L32 (1996). M. S. El-Eskandarany, M. Omori, T. Kamiyama, T. J. Konno, K. Sumiyama, T. Hirai, K. Suzuki. Mechanically induced carbonization for formation of nanocrystalline TiC alloy. Sci. Reports of Res. Inst. Tohoku Univ. (Sendai, Japan) 43, 181-193 (1997). P. S. de Carli, J. C. Jamieson. Formation of diamond by explosive shock. Science 133, 1821-1822 (1961). L. F. Trueb. Microstructural study of diamonds synthesized under conditions of high temperature and moderate explosive shock pressure. J. Appl. Phys. 42, 503-510 (1971). S. A. Gubin, V. V. Odintsov, V. I. Pepekin. Diagramm of carbon phase states and its account in calculation of detonation parameters. Khimich. Fizika 5, 111-120 (1986) (in Russian). S. A. Gubin, V. V. Odintsov, V. I. Pepekin. Thermodynamic calculation of ideal and nonideal detonation. Fizika Goreniya i Vzryva 23, 75-84 (1987). (in Russian) A. M. Staver, N. V. Gubareva, A. I. Lyamkin, E. A. Petrov. Ultrafine grained diamond powders prepared with using of explosion energy. Fizika Goreniya i Vzryva 20, 100-103 (1984) (in Russian). A. I. Lyamkin, E. A. Petrov, A. P. Ershov, G. V. Sakovich, A. M. Staver, V. M. Titov. Production of diamonds from explosives. Doklady AN SSSR 302, 611-613 (1988) (in Russian). I. Yu. Mal’kov, L. I. Filatov, V. M. Titov, B. V. Litvinov, A. L. Chuvilin, T. S. Teslenko. Diamond formation from liquid carbon phase. Fizika Goreniya 85

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i Vzryva 29, 131-134 (1993) (in Russian). A. E. Aleksenskii, M. V. Baidakova, A. Ya. Vul’, V. Yu. Davydov, and Yu. A. Pevtsova. Diamond-graphite phase transition in ultradisperse-diamond clusters. Fizika Tverd. Tela 39, 1125-1134 (1997) (in Russian). (Engl. Transl.: Physics of the Solid State 39, 1007-1015 (1997)). V. L. Kuznetsov, M. N. Aleksandrov, I. V. Zagoruiko, A. L. Chuvilin, E. M. Moroz, V. N. Kolomiichuk, V. A. Likholobov, P. M. Brylyakov, G. V. Sakovich. Study of ultradispersed diamond powders obtained using explosive energy. Carbon 29, 665-668 (1991). M. Yoshikawa, Y. Mori, H. Obata, M. Maegawa, G. Katagiri, H. Ishida, Ishitani. Raman scattering from nanometer-sized diamond. Appl. Phys. Lett. 67, 694-696 (1995). M. Gamarnik. Energetical preference of diamond nanoparticles. Phys. Rev. B54, 2150-2156 (1996). A. L. Vereshchagin. Detonation Nano-Diamonds (Biisk Technological Institute of Altai State Technical Unversity, Barnaul 2001) 177 pp. (in Russian). A. G. Beloshapko, A. A. Bukaemskii, A. M. Staver. Formation of ultrafine grained compounds at impact wave loading of porous aluminium. Investigation of obtained particles. Fizika Goreniya i Vzryva 26, 93-98 (1990) (in Russian). A. G. Beloshapko, A. A. Bukaemskii, I. G. Kuz’min, A. M. Staver. Ultrafine grained powder of stabilized zirconium dioxide. Synthesis by dynamic method. Fizika Goreniya i Vzryva 29, 111-112 (1993) (in Russian). M. M. Martynyuk. Importance of evaporation and boiling of liquid metal in process of electric explosion of conductors. Zh. Tekhnich. Fiziki 44, 12621276 (1974). Yu. A. Kotov, N. A. Yavorski. Study of particles forming at electric explosion of conductors. Fizika i Khimiya Obrabotki Mater. No 4, 24-29 (1978) (in Russian). V. A. Burtsev, N. V. Kalinin, A. V. Luchinskii. Electric Explosion of Conductors and its Application in Electrophysical Devices (Energoatomizdat, Moscow 1990) 289 pp. (in Russian). A. P. Il’in, O. B. Nazarenko, D. V. Tikhonov. Physical Principles of Preparing Dispersed Metals by Impulses of Electric Current. In: Ultrafine Grained Powders, Nanostructures, Materials. Proc. of 2 nd Interdistrict Conf., Krasnoyarsk, October 5-7, 1999. (Krasnoyarsk State Technical University, Krasnoyarsk 1999) pp.31-33 (in Russian). A. P. Il’in. On excess energy of ultrafine grained powders prepared by electric explosion of wires. Fizika i Khimiya Obrab. Mater. No 3, 94-97 (1994) (in Russian). A. P. Il’in. Features of energy saturation of a structure of small metal particles prepared in strongly non-equilibrium conditions. Fizika i Khimiya Obrab. Mater. No 4, 93-97 (1997) (in Russian). Properties, Production and Application of Refractory Compounds / Handbook. Ed. T. Ya. Kosolapova (Metallurgiya, Moscow 1986) 928 pp. (in Russian). A. I. Gusev, A. A. Rempel. Nonstoichiometry, Disorder and Order in Solids (Ural Division of the Russ. Acad. Sci., Yekaterinburg 2001) 580 pp. (in Russian). A. I. Gusev, A. A. Rempel, A. J. Magerl. Disorder and Order in Strongly Nonstoichiometric Compounds: Transition Metal Carbides, Nitrides and Oxides

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(Springer, Berlin - Heidelberg - New-York, 2001) 607 pp. A. I. Gusev, A. A. Rempel. Phase diagrams of metal–carbon and metal– nitrogen systems and ordering in strongly nonstoichiometric carbides and nitrides. Phys. Stat. Sol. (a) 163, 273-304 (1997) A. I. Gusev. Order–disorder transformations and phase equilibria in strongly nonstoichiometric compounds. Uspekhi Fiz. Nauk 170, 3-40 (2000) (in Russian). (Engl. transl.: Physics - Uspekhi 43, 1-37 (2000)). A. I. Gusev, A. A. Rempel. Structural Phase Transitions in Nonstoichiometric Compounds (Nauka, Moscow 1988) 308 pp. (in Russian). A. A. Rempel. Effects of Ordering in Nonstoichiometric Interstitial Compounds (Nauka, Yekaterinburg 1992) 232 pp. (in Russian). A. A. Rempel. Atomic and vacancy ordering in nonstoichiometric carbides. Uspekhi Fiz. Nauk 166, 33-62 (1996) (in Russian). (Engl. transl.: Physics Uspekhi 39, 31-56 (1996)). A. I. Gusev, A. A. Rempel. Superstructures of non-stoichiometric interstitial compounds and the distribution functions of interstitial atoms. Phys. Stat. Sol. (a) 135, 15-58 (1993). V. N. Lipatnikov, A. I. Gusev. Ordering of Titanium and Vanadium Carbides (Ural Division of the Russ. Acad. Sci., Yekaterinburg 2000) 265 pp. (in Russian). T. Athanassiadis, N. Lorenzelli, C. H. de Novion. Diffraction studies of the order-disorder transformation in V8C7. Ann. Chim. France 12, 129-142 (1987). A. A. Rempel, A. I. Gusev. Nanostructure and atomic ordering in vanadium carbide. Pis’ma v ZhETF 69, 436-442 (1999) (in Russian). (Engl. transl.: JETP Letters 69, 472-478 (1999)). V. N. Lipatnikov, W. Lengauer, P. Ettmayer, E. Keil, G. Groboth, E. J. Kny. Effects of vacancy ordering on structure and properties of vanadium carbide. J. Alloys Comp. 261, 192-197 (1997). V. N. Lipatnikov, A. I. Gusev, P. Ettmayer, W. Lengauer. Phase transformations in non-stoichiometric vanadium carbide. J. Phys.: Condens. Matter 11, 163-184 (1999) D. Rafaja, W. Lengauer, P. Ettmayer, V. N. Lipatnikov. Rietveld analysis of the ordering in V8C7. J. Alloys Comp. 269, 60-62 (1998). A. I. Gusev. Nanocrystalline Materials:Preparation and Properties (Ural Division of the Russ. Acad. Sci., Yekaterinburg 1998) 200 pp. (in Russian) A. I. Gusev. Effects of the nanocrystalline state in solids. Uspekhi Fiz. Nauk 168, 55-83 (1998) (in Russian). (Engl. transl.: Physics - Uspekhi 41, 49-76 (1998)). A. I. Gusev, A. A. Rempel. Nanocrystalline Materials (Nauka-Fizmatlit, Moscow 2000) 224 pp. (in Russian) A. A. Rempel, A. I. Gusev, O. V. Makarova, S. Z. Nazarova. Nanocrystalline nonstoichiometric vanadium carbide with high microhardness. In: Ultrafine Grained Powders, Nanostructures, Materials. Proc. of 2nd Interdistrict Conf., Krasnoyarsk, October 5-7, 1999. (Krasnoyarsk State Technical University, Krasnoyarsk 1999) pp.168-174 (in Russian). A. A. Rempel, A. I. Gusev, O. V. Makarova, S. Z. Nazarova. Physical and chemical properties of nanostructured vanadium carbide. Perspektivnye Materialy No 9, 9-15 (1999) (in Russian). A. I. Gusev, A. A. Tulin, V. N. Lipatnikov, A. A. Rempel. Nanostructure of dispersed and bulk nonstoichiometric vanadium carbide. Zh. Obsh. Khimii

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72, 1067-1076 (2002) (in Russian). (English transl.: Russ. J. General Chem. 72, 997-1006 (2002)). A. A. Rempel, A. I. Gusev, R. R. Mulyukov, N. M. Amirkhanov. Microstructure, microhardness and magnetic susceptibility of submicrocrystalline palladium. Nanostruct. Mater. 7, 667-674 (1996). A. A. Rempel, A. I. Gusev, S.Z. Nazarova, R. R. Mulyukov. Imputity superparamagnetism in plastically deformed copper. Doklady Akad. Nauk 347, 750-754 (1996) (in Russian). (Engl. transl.: Physics - Doklady 41, 152-156 (1996)). R. Caudron, J. Castaing, P. Costa. Electronic structure of face centred cubic titanium and vanadium carbide alloys. Solid State Commun. 8, 621-625 (1970). A. S. Borukhovich, N. M. Volkova. On conduction band of nonstoichiometric vanadium monocarbide. Izv. AN SSSR. Neorgan. Materrialy 7, 1529-1532 (1971) (in Russian). R. Wurschum, H.-E. Schaefer. Interfacial free volumes and atomic diffusion in nanostructured solids. In: Nanomaterials: Synthesis, Properties and Applications. Eds. A. S. Edelstein and R. C. Cammarata. (Institute of Physics Publishing, Bristol 1996) pp.277-301. A. A. Rempel, M. Forster, H.-E. Schaefer. Positron lifetime in carbides with B1 structure. Doklady Akad. Nauk SSSR 326, 91-97 (1992) (in Russian). (Engl. transl.: Sov. Physics Doklady 37, 484-487 (1992)). A. A. Rempel, M. Forster, H.-E. Schaefer. Positron lifetime in non-stoichiometric carbides with a B1 (NaCl) structure. J. Phys.: Condens. Matter 5, 261-266 (1993). A. A. Rempel, L. V. Zueva, V. N. Lipatnikov, H.-E. Schaefer. Positron lifetime in the atomic vacancies of nonstoichiometric titanium and vanadium carbides. Phys. Stat. Sol. (a) 169, R9-R10 (1998). R. Sh. Askhadullin, P. N. Martynov. Synthesis of ultrafine oxides in nonalkaline liquid metals (Ga, Pb, Pb-Bi): Properties and application possibilities of substances obtained. In: Physics and Chemistry of Ultra-Disperse (Nano-) Systems / Proc. of VI th All-Russian Conference. (Moscow PhysicoTechnical Institute, Moscow 2003) pp.451-455 (in Russian) H. H. Nersisyan, J. H. Lee, C. W. Won. Self-propagating high-temperature synthesis of nano-sized titanium carbide powder. J. Mater. Res. 17, 28592864 (2002). L. V. Zueva, A. I. Gusev. Effect of nonstoichiometry and ordering on the period of the basis structure of cubic titanium carbide. Fizika Tverd. Tela 41, 1134-1141 (1999) (in Russian). (Engl. Transl.: Physics of the Solid State 41, 1032-1038 (1999))

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+D=FJAH! 3. Preparation of Bulk Nanocrystalline Materials Regardless of the large variety and development of the methods of production of nanocrystalline materials (this relates in particular to the most likely known methods of gas-phase evaporation and precipitation from colloidal solutions), the investigations of the structure and properties of nanoparticles are very complicated. This is associated with, in particular, the high reactivity of the particles resulting from their highly developed surface. Therefore, bulk nanocrystalline materials are the subject of fundamental and applied science. In many cases, these materials are more suitable for investigation and application. The main methods of production of bulk nanomaterials were described in a review [1]. None of these methods is universal, because each of them is suitable for a limited number of substances. The conventional methods of powder technology are used most widely, i. e. different types of pressing and sintering (cold pressing and sintering, hot axial and isostatic pressing, magnetic pulsed pressing). Powder technology also includes the method of vacuum compacting of nanoparticles, produced by condensation from the gas phase, proposed by H. Gleiter [2–5]. The main difficulty when using powder technologies for production of pore-free or with least porosity ware from nanopowders is associated with intensive recrystallisation and the residual porosity. By shortening the duration of the high temperature treatment, it is possible to decrease the degree of recrystallisation and grain growth during sintering. The application of high static or dynamic pressure for the pressing of nanopowders at room or high temperatures reduces the residual porosity and increases the relative density of produced materials. Powder technology is suitable for chemical elements, compounds and alloys. The deposition of films and coatings makes it possible to produce 89

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pore-free material. Films with a thickness lower than 100−150 nm are considered as nanocrystalline materials. Films are universal in composition, and the size of crystallites in them can changes in a wide range. There are amorphous, crystalline and multilayered films. This offers great possibilities for using films in instrument industry and electronics. In fact, regardless of the small thickness of the coating, they greatly increase the mechanical properties of ware. For example, coatings of titanium nitride TiN or titanium carbonitride TiC xN y greatly increase the wear resistance and cutting properties of metal-working tools, the corrosion resistance of metals and alloys. Films of different composition are used widely in electronic microcircuits. Films and coatings are produced by chemical (CVD) and physical (PVD) vapour deposition from the gas phase, electrodeposition and by means of sol– gel technology. Pore-free nanostructured materials may also be produced by crystallisation from the amorphous state, but this method is suitable only for alloys, which can be quenched from the melt to the amorphous state. Crystallisation of amorphous alloys is carried out at normal and high pressure, and is combined with deformation treatment. Severe plastic deformation makes it possible to produce porefree metals and alloys with a grain size of approximately 100 nm and is suitable mainly for plastically deforming materials. The formation of the nanostructure in non-stoichiometric compounds such as carbides, nitrides and oxides MX y of transition metals (M = Ti, Zr, Hf, V, Nb, Ta; X = C, N, O) and in substitutional solid solutions A x B 1–x is possible by means of atomic ordering. This method is applicable if the disorder–order transformation is the phase transition of the first kind and is accompanied by an abrupt change of volume. 3.1. COMPACTION OF NANOPOWDERS The method for production of bulk nanocrystalline materials, proposed by the authors of [2–6] in 1982–1986, is well known and used widely. The technology described in [2–6] uses the method of evaporation and condensation for the production of nanocrystalline particles, which are deposited on the cold surface of a rotating cylinder. Evaporation and condensation are carried out in the atmosphere of a rarefied inert gas, mostly helium He. At the same gas pressure, the transition from helium to xenon, i. e. from a less dense inert gas to an inert gas with a higher density, is 90

Preparation of Bulk Nanocrystalline Materials

accompanied by a large increase of the size of the particles. Usually, the particles of the surface condensate are faceted. In the same evaporation and condensation conditions, the metals with higher melting points form smaller particles. The deposited condensate is removed from the cylinder surface using a special scraper and is collected in a collector. After removing the inert gas, the nanocrystalline powder is compacted in a vacuum first at a pressure of ~1 GPa and finally at a pressure up to 10 GPa (Fig. 3.1). The setup for producing bulk nanocrystalline substances by Gleiter’s methods [2, 5] is shown in Fig. 3.2. This setup is used to produce sheets with a diameter of 5–15 mm, a thickness of 0.2– 3.0 mm and a density equal to 70–90% of the theoretical density of the appropriate material. For example, the density of nanocrystalline metals reaches 97%, and that of nanoceramic materials up to 85% [7]. The bulk nanomaterials, produced by this 5RWDWLQJ FROG ILQJHU OLTXLG QLWURJHQ 6FUDSHU

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Fig. 3.1. Schematic representation of a gas-condensation chamber for synthesis of nanocrystalline substances [5]. The material, which is evaporated or sputtered from one or several sources, condenses to crystallites in a noble-gas atmosphere and is transported via convection to a finger cooled down by liquid-nitrogen. The clusters are scraped off, collected and compacted in two compaction units in situ after evacuation. 91

Nanocrystalline Materials

Fig. 3.2. External view of setup for producing nanocrystalline substances by the method of evaporation, condensation and compacting (Institut für Theoretische und Angewandte Physik, Universität Stuttgart).

method, consists of particles with a mean diameter from 1–2 nm to 80–100 nm, depending on the evaporation and condensation conditions. One of the initial investigations, concerned with the production of bulk nanomaterials, was carried out in 1983 by a group of Russian authors using a powder of nanocrystalline nickel [8]. The nickel powder with a mean particle size of 60 nm was produced by the evaporation and condensation method. To produce bulk specimens, the powder was pressed for 30 seconds at temperatures from 673 to 1173 K and at a pressure up to 5 GPa. As a result of short term heating, it was possible to retain the nanostructure in the produced bulk specimens. It was reported in [8] that the hardness of the bulk nanostructured specimens of nickel is considerable higher than that of the coarse-grained nickel specimen. The exclusion of contact with the environment in the production of the nanopowder and during its pressing makes it possible to prevent contamination of the bulk nanocrystalline specimens. This is very important when investigating the nanocrystalline state of metals and alloys. The setup described in [2–6] may be used for producing bulk nanocrystalline oxides and nitrides. In this case, the metal evaporates into an oxygen- or nitride-containing atmosphere. As an example, Fig. 3.3 shows a bulk specimen of nanocrystalline oxide ZrO 2 , 92

Preparation of Bulk Nanocrystalline Materials

Fig. 3.3. Bulk specimen of nanocrystalline oxide ZrO 2 prepared by evaporation, condensation and compacting, i. e. by the method proposed by H. Gleiter [2, 5]. Specimen diameter is 5 mm, thickness is approximately 1 mm. The mean grain size of the specimen is 20 nm. The specimen was prepared by Gregor Knöner, Ralf Röwe and Alexej Rempel (Institut für Theoretische und Angewandte Physik, Universität Stuttgart, 2001).

produced by this method. The mean grain size in the specimen is 20 nm. The diameter of the specimen is 5 mm, the thickness approximately 1 mm. The porosity of nanoceramics, produced by compacting of powders, is associated with triple points of the crystallites. A decrease of the particle size of the powders is accompanied by a large decrease of their compacting capacity in pressing using the same pressure [9]. The decrease and the more uniform distribution of porosity are achieved by pressing at higher temperature, which do not yet result in intensive recrystallisation. For example, conventional sintering of the highly dispersed powder of zirconium oxide with a particle size of 40–60 nm at 1370 K for 10 seconds makes it possible to produce a relative density of 72% at a mean grain size in the sintered specimen of 120 nm. Hot pressing at the same temperature and a pressure of 1.6 GPa results in a sintered material with a relative density of 87% and a mean grain size of 130 nm [10]. A decrease in sintering temperature to 1320 K and an increase in sintering time to 5 hours enable production of bulk zirconium oxide ZrO 2 with a relative density higher than 99% and a mean grain size of 85 nm [11]. The authors of [12] used hot pressing of titanium nitride powder (D ~ 80 nm) at 1470 K and a 93

Nanocrystalline Materials

pressure of 4 GPa to produce bulk specimens with a density of 98 % of theoretical density but (according to diffraction data), the mean grain size after hot pressing was not smaller than 0.3 µm, because of intensive recrystallisation. Investigation [13] showed that the densest (relative density 98%) specimens of titanium nitride are produced by sintering specimens pressed from the finest nanopowders (D ~ 8–25 nm) with a narrow size distribution of the grains. On the whole, to produce bulk nanocrystalline materials, especially ceramic ones, it is promising to use pressing of nanopowders with subsequent high temperature sintering. When using this method, it is important to avoid growth of grains when sintering the pressed specimens. This is possible at a high density of pressed specimens (not lower than 0.7 of theoretical density), when the sintering processes are relatively rapid and take place at a relatively low temperature T ≤ 0.5T melt (T melt is the melting point). The production of dense pressed specimens is an important task because the nanocrystalline powders are difficult to press and the conventional methods of static pressing do not result in sufficiently high density. The physical reasons for the low pressing capacity of the nanopowders is the existence of interparticle adhesion forces whose relative value rapidly increases with decreasing particle size. The application of the dynamic methods of compaction of the nanopowders makes it possible to overcome the adhesion forces of the particles and obtain, at the same pressure, more dense bulk specimens in comparison with the static pressing conditions. The magnetic pulsed method, proposed by the authors of [14– 17] is used efficiently for compacting of nanocrystalline powders. This method represents dry high-intensity pressing of the powders. The magnetic pulsed pressing makes it possible to generate pulsed compression waves with an amplitude of up to 5 GPa and a duration of up to several microseconds. The method is based on concentration of the force effect of the magnetic field of powerful pulsed currents and allows to control quite easily the parameters of the compression wave. The magnetic pulsed pressing does not led to pollution of environment and is far safer than the dynamic methods using explosive substances. The principal scheme of uniaxial magnetic pulsed pressing is shown in Fig. 3.4 [14]. The inductor (1) generates a pulsed magnetic field B. The mechanical pulse of force F, compressing the powder is generated as a result of the interaction of the pulse magnetic field with the conducting surface of the concentrator (2). 94

Preparation of Bulk Nanocrystalline Materials ,

-











 





 





Fig. 3.4. Diagram of uniaxial magnetic pulsed pressing [17]: (a) compression stage, (b) stage of ejection of the finished specimen, (1) inductor, (2) concentrator, (3) upper and lower plungers, (4) powder, (5) die, (6) device for removing the specimen

The concentrator activates the upper plunger (3), which compresses the powder. The displacement of the concentrator is based on using of diamagnetic effect of pushing out of the conductor from the region of the pulsed magnetic field. The die and the specimen are placed in the vacuum chamber and all operations with the powder are carried out in vacuum. In contrast to the stationary pressing methods, the pulsed compression waves are accompanied by rapid heating of the powder as a result of rapid generation of energy during friction of particles in the course of packing. If the size of the particles is relatively small (D ≤ 0.3 micrometer), the duration of heating of the particles by diffusion of heat from the particle surface is considerably shorter than the characteristic duration of the pulsed compression wave (1– 10 µs). Under specific conditions, by optimising the parameters of the compression wave, it is possible to carry out dynamic hot compacting of the ultrafine powder as a result of the high powder surface energy. At the same pressing pressure, the magnetic pulsed method makes it possible to produce more dense specimens than static pressing (Fig. 3.5). As an example, Fig. 3.6 shows the variation of the pressing pressure, shrinkage rate and density of 95

Nanocrystalline Materials 

QDQR$O2 



J FP







ρ







 









5 *3D

Fig. 3.5. Dependence of the density ρ of the nanocrystalline oxide n-Al 2 O 3 on the pressure in stationary and magnetic pulsed pressing [16]: (1), (2), (3) stationary pressing at 300, 620 and 720 K, respectively; (4) magnetic pulsed pressing. 





ρ

'

!



!



ρ

'



*3D

P V





J FP



 





 





 9



µV

Fig. 3.6. Dynamic parameters of magnetic pulsed pressing of nanocrystalline oxide n-Al 2O 3 [16]: variation of pressing pressure P, shrinkage rate V and density ρ during the passage of a pulsed compression wave.

nanocrystalline oxide nano-Al 2 O 3 during the passage of a pulsed compression wave [16]. Aluminium nitride AlN powder, produced by means of electric explosion, is pressed by the magnetic pulsed method under a pressure of 2 GPa to a density of 95% of the 96

Preparation of Bulk Nanocrystalline Materials

theoretical value, and Al 2 O 3 to a relative density of 86%. The magnetic pulsed pressing is used for producing ware of different shape, and in the majority of cases these ware do not require any additional mechanical treatment. In particular, in operation with superconducting oxide ceramics [15] it was possible to produce ware with a density higher than 95% of the theoretical value. In a general case, the application of pulsed pressure results in a higher density of the specimens in comparison with static pressing as a result of the effective overcoming forces of interparticle interaction during rapid movement of the particles in powder. The short heating time of the nanopowder allows reducing the recrystallisation at high temperature and retaining the small particle size. The magnetic pulsed method has been used for pressing nanocrystalline Al 2 O 3 [18, 19] and TiN [20]. The results published in [20] show that an increase of pressing temperature to ~900 K is more efficient than an increase of pressure in cold pressing. At a pulsed pressure of 4.1 GPa and a temperature of 870 K it is possible to produce bulk specimens of nanocrystalline titanium nitride with a grain size of ~80 nm and a density of approximately 83% of the theoretical value. A decrease in pressing temperature to 720 K is accompanied by a decrease in density to 81%. To produce gasdense ceramic pipes with an external diameter of up to 15 mm and up to 100 mm long from nanopowders, the authors of [21] used radial magnetic pulsed pressing. The powder is placed in a cylindrical gap between a hard metallic bar and an external cylindrical copper shell. Pressing is carried out by the radial compression of the outer shell by pulsed current. The developed pulsed pressure may reach 2 GPa. The starting material is represented by nanopowders of Al 2 O 3 and Y 2 O 3 –ZrO 2 with a mean particle size of 10–30 nm. As a result of radial magnetic pulsed pressing of these nanopowders, it was possible to produce pipes with a relative density of ceramics of more than 95%. Dry cold ultrasound pressing [22, 23] is a promising and efficient method for compacting ceramic nanopowders without the application of plasticising agents. The treatment of the powder by powerful ultrasound in the pressing process decreases the friction amongst the particles and the friction of the powder with the walls of the pressing mould, breaks up the agglomerate and large particles, increases the surface activity of the powder particles and the uniformity of distribution of these particles in the volume. This increases the density of the pressed ware, accelerates diffusion processes, restricts the grain growth in subsequent sintering and 97

Nanocrystalline Materials

Fig. 3.7. Ceramic ware produced by ultrasound pressing of nanopowders [24].

results in the retention of the nanostructure. For example, ultrasound pressing of the nanopowder of ZrO 2 , stabilised with Y 2 O 3 oxide with subsequent sintering of the specimens in air at a temperature of 1923 K was used to produce ceramics with a relative density ~90%. The mean size of the particles in the starting powder was ~50 nm. The mean grain size in sintered ceramics depends on the power of ultrasound oscillations during pressing: Increase in the power of ultrasound from 0 to 2 kW decreases the mean grain size from 440 to 200 nm. Ultrasound pressing of the nanopowders is especially efficient for producing ware of complicated shapes: sleeves, conical gears, spirals, etc. (Fig.3.7) [24]. The produced ceramic ware are characterised by a uniform microstructure and density. Thus, there are several methods for compacting nanocrystalline powders. These methods can be used to produce ceramic specimens with high relative density and homogeneity. Subsequent sintering of these ceramic specimens allows to retain their higher density and, to a lesser degree, the nanostructure. In fact, recrystallisation in nanomaterials takes place at a relatively high rate and the crystallites and grains grow even at room temperature [25– 27] (see also Section 6.1). Taking this into account, it is clear that the role of sintering in producing nanostructured ceramics is very 98

Preparation of Bulk Nanocrystalline Materials

important. Conventional methods of sintering and their effect on the evolution of the microstructure of the powder materials have been discussed in [28, 29]. A new method of sintering ceramic materials by means of superhigh frequency radiation is quite interesting [30–35]. This method is based on superhigh frequency heating of sintered specimens. Heating takes place by radiation in the millimeter range (frequency range from 24 to 84 GHz). The volume absorption of superhigh frequency energy leads to uniform simultaneous heating of the entire specimen because the heating rate is not limited by heat conductivity, as in conventional sintering methods. Consequently, sintered ceramics with a homogeneous microstructure can be produced. A processing gyrotron system for high temperature superhigh frequency treatment of materials [30], developed at the Institute of Applied Physics of the Russian Academy of Sciences in Nizhny Novgorod, is shown in Fig. 3.8. The microwave energy source is a continuous wave gyrotron (1) with a power of 10 kW at a

Fig. 3.8. Gyrotron system for microwave processing of materials [30, 33]: (1) continuous wave gyrotron with a power of 10 kW at a frequency of 30 GHz, (2) a microwave power transmission line transforming the gyrotron operating mode into a Gaussian wave beam, (3) superhigh frequency radiation furnace as a supermultimode cylindrical cavity (Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia). 99

Nanocrystalline Materials

frequency of 30 GHz. Electromagnetic radiation from the gyrotron is shaped as a Gaussian beam and transferred by means of an open quasi-optical line (2) into a superhigh frequency furnace (3). The furnace is a supermultimode cylindrical cavity with a diameter of 50 cm, height 60 cm. Inside the cavity the radiation is uniformly distributed by means of a spherical scatterer. The unifotm volume distribution of microwave energy ensures uniform heating of the material. The sintering temperature is 1300–2300 K and is regulated with an accuracy of 0.2 K. The application of the gyrotron system for sintering ceramic materials greatly reduces the risk of overheating. Microwave sintering of bulk specimens with a relative density 70–80% and compacted from TiO 2 nanopowders with a mean particle size of 20–30 nm, makes it possible to produce sintered specimens with a relative density of 97–99% [31, 33, 34]. The mean grain size of the sintered specimen is 200–220 nm. The conventional sintering methods do not always make it possible to ensure strong joining of different ceramic materials. For example, the conventional methods cannot be used to produce mechanically strong joining between ZrO 2 and Al 2 O 3 , which is required for developing devices of the thermal barrier type. The application of nanocrystalline materials and of microwave sintering enables this problem to be solved. The joining of ZrO 2 and Al 2 O 3 is ensured by means of sintered interlayer of nanosized composite ceramic 60 vol.% ZrO 2 + 40 vol.% Al 2 O 3 with a mean grain size of 100 nm. The relative density of the interlayer is 96–98% of theoretical density. Short term microwave heating of the ZrO 2 / interlayer/Al 2O 3 assembly to 1700 K results in strong joining ZrO 2 and Al 2 O 3 oxides. On the whole, the currently available methods of compacting nanocrystalline powders and sintering compacted nanomaterials already make it possible to produce high-density ceramic ware of complicated shapes. However, the small grain size characteristic of the starting nanopowders cannot be retained in sintered nanomaterials. In the majority of sintered nanomaterials, the grain size is 200–300 nm, i. e. approximately 5–10 times larger than in the starting nanopowders. To retain the small grain size, it is necessary to decrease the sintering temperature, to shorten sintering time, and carry out sintering at high dynamic or static pressure.

100

Preparation of Bulk Nanocrystalline Materials

3.2. FILM AND COATING DEPOSITION Deposition on a cold or preheated substrate is used to produce films or coatings, i. e. continuous pore-free layers of a nanocrystalline material. In this method, in contrast to gas-phase synthesis, nanoparticles form directly on the surface of the substrate and not in the volume of the inert gas in the vicinity of the cooled wall. As a result of the formation of the dense layer of the nanocrystalline materials, it is not necessary to carry out pressing. Deposition takes place from vapours, plasma or a colloid solution. In vapour deposition, the metal is evaporated in vacuum, in oxygenor nitrogen-containing atmosphere, and vapours of metal or compound (oxide, nitride) condense on the substrate. Varying the evaporation rate and substrate temperature regulates the size of the crystallites in the film. A film of zirconium oxide, alloyed with yttrium oxide, with a mean crystallite size of 10–30 nm was produced by means of pulsed laser evaporation of metal in a beam of oxygen ions and subsequent deposition of oxides on a substrate heated to 350–700 K [38]. In deposition from plasma, the electric discharge is sustained using an inert gas. Varying the gas pressure and discharge parameters regulates the continuity and thickness of the film and the size of crystallites in the film. In deposition from plasma, a metallic cathode with a high degree of ionisation (from 30–100%) is a source of metallic ions. The kinetic energy of ions is 10–200 eV, deposition rate up to 3 µm min –1 . The authors of [39, 40], acting on chromium with plasma, produced by arc discharge in low-pressure argon, deposited a chromium film with a mean crystallite size of ~20 nm on a copper substrate; a film with a thickness less than 500 nm had an amorphous structure, and more thick films were in the crystalline state. High hardness (up to 20 GPa) of the film is determined by the formation of supersaturated solid solutions of interstitial impurities (C, N) in chromium. Ion-plasma coatings of titanium nitride and carbonitride are used widely. Heating the substrate to 500–800 K makes it possible to retain the nanocrystalline structure of the coating. The methods of production and properties of the coatings and films of refractory compounds are discussed in detail in a review [41]. Deposition from plasma is carried out using mainly reactive working media (mixtures of argon with nitrogen or hydrocarbons at a pressure of ~0.1 Pa) and metallic cathodes. The main 101

Nanocrystalline Materials

disadvantage of ion-plasma arc sputtering is the formation of fine metal droplets due to partial melting of the cathode and a possibility of penetration of metallic droplets into deposited films. Deposition from plasma is used to produce not only simple films of nanometer thickness but also films with a nanostructure. Recently, Fujimori et al [42] reported that Co–Al–O granular thin films exhibit very large magnetoresistance in spite of their high electrical resistivity. This unique property is related to the metal– oxide granular microstructure, which contains metallic nanoparticles, embedded into the matrix of a non-metallic insulating oxide. Giant magnetoresistance forms in the presence of superparamagnetism and, consequently, the magnetic particles in the film must be in the nanoscale range. To explain this, Ohnuma et al [43] investigated a microstructure of films by means of high resolution transmission electron microscopy and small angle X-ray scattering. Co-Al–O thin granular films were prepared on a glass substrate by Ar + O 2 reactive sputtering using a Co 72 Al 28 alloy target. Oxygen concentration in the films was varied from 0 to 47 at.% by controlling the partial pressure of oxygen in the gas mixture for reactive sputtering. The results show that giant magnetoresistance in the film appears when Co particles are fully surrounded by the amorphous aluminium oxide. The microstructure of granular films Co 61Al 26 O 13 and Co 52Al 20 O 28 is shown in Fig. 3.9. Light regions are the amorphous aluminium oxide, and dark regions correspond to metallic particles with a size of 2–3 nm. In the Co 52 Al 20 O 28 films, the metallic particles consist of pure cobalt with an hcp or fcc structure. In the Co 61 Al 26 O 13 films, containing a larger amount of aluminium, metallic particles represent the CoAl phase with a CsCl

Fig. 3.9. HREM image of Co 61 Al 26 O 13 (a) and Co 52 Al 30 O 28 (b) films [43]. 102

Preparation of Bulk Nanocrystalline Materials

structure. The value of the giant magnetoresistance changes quite markedly in relation to the oxygen content in the film and is maximum when the mean distance between the metallic nanoparticles is minimum. Thus, regulating the deposition conditions and the oxygen content in the Ar + O 2 gas mixture it is possible to change the microstructure and properties of the Co–Al–O films. A variant of deposition from plasma is magnetron sputtering which uses not only cathodes of metals and alloys but also cathodes produced from different compounds. In this case, the substrate temperature can be reduced to 100–200 K or lower. This widens the possibilities of producing amorphous and nanocrystalline films. However, the degree of ionisation, the kinetic energy of ions and the deposition rate in magnetron sputtering are lower than when using electric arc discharge plasma. In [44] the method of magnetron sputtering of a Ni 0.75Al 0.25 target and of deposition of metallic vapours on an amorphous substrate was used to produce Ni 3Al intermetallic films with a mean crystallite size of ~20 nm. Oxide semiconductor films are produced by deposition from colloid solutions on a substrate. This method includes the preparation of the solution, deposition on the substrate, drying and annealing. Deposition of nanoparticles of oxides was used to produce semiconductor films of ZnO, SnO 2 , TiO 2 , WO 3 [45–49]. Nanostructured films, containing nanoparticles of different semiconductors, are produced by the co-precipitation method. The preparation of nanocrystalline ZrO 2 films is described in [50]. Chemical and physical vapour deposition from the gas phase (CVD and PVD) are conventional methods of depositing films. These methods have been used for a very long time for producing films and coatings for different applications. Usually, the crystallites in these films are quite large but in multilayered or multiphase CVD films it is also possible to produce nanostructures [41, 51]. Deposition from the vapour phase is usually associated with hightemperature gas reactions of metal chlorides in the atmosphere of hydrogen and nitrogen or hydrogen and hydrocarbons. The temperature range of deposition of CVD films is 1200–1400 K, the deposition rate is 0.03–0.2 µm min –1 , whereas laser radiation makes it possible to reduce to 600–900 K the temperature developed in deposition from the gas phase. This leads to the formation of nanocrystalline films. Recently, organometallic precursors of the type of tetradimethyl(ethyl)amids M[N(CH 3) 2]4 and M[N(C 2H 5) 2] 4 which have 103

Nanocrystalline Materials

a high vapour pressure are used often in deposition from the gas phase. In this case, the dissociation of the precursor and activation of the reagent gas (N 2 , NH 3 ) are carried out using electron cyclotron resonance. Films of transition metal nitrides, produced by different deposition methods, are characterised by a superstoichiometric (in comparison with phases MN 1.0 with a B1 basic structure) content of nitrogen, for example Zr 3 N 4 , Hf 3 N 4 , Nb 4 N 5 , Ta 3 N 5 , etc. 3.3 CRYSTALLISATION OF AMORPHOUS ALLOYS In this method, the nanocrystalline structure is produced in an amorphous alloy by its crystallisation. Melt spinning, i.e. production of thin ribbons of amorphous metallic alloys by means of rapid cooling (at a rate ≥10 6 K s –1 ) of the melt on the surface of a spinning disc or drum has been developed quite sufficiently. The amorphous ribbon is annealed at a controlled temperature for crystallisation. To develop a nanocrystalline structure, annealing is carried out in such manner as to ensure the formation of a large number of crystallisation centres and a low growth rate of the crystals. The first stage of crystallisation may be the precipitation of fine crystals of intermediate metastable phases. For example, the authors of [52] in investigating an amorphous alloy of the Ni–P system found that the initial stage is characterised by the formation of small crystals of a metastable highly supersaturated solid solution of phosphorous in nickel Ni(P) and crystals of nickel phosphides appear only then. It is assumed that the amorphous phase is a barrier for the growth of crystals. Nanocrystalline ribbons can also be produced directly during melt spinning. In [53], the spinning method is used to produce a ribbon of Ni 65 Al 35 alloy. The ribbon consists of crystals of NiAl intermetallic with a mean grain size of ~2µm; in turn, these crystals are characterised by a highly uniform microtwin substructure with characteristic size of several tens of nanometers. This substructure prevents the propagation of microcracks and increases the ductility and toughness of the brittle intermetallic NiAl. Figure 3.10 shows changes in the X-ray diffraction pattern of a hard magnetic Fe 90 Zr 7 B 3 alloy in transition from the amorphous to nanocrystalline state. The amorphous alloy is produced by melt spinning and additionally subjected to relaxation annealing at a temperature of 673 K for two hours. The nanocrystalline state is produced by annealing at 873 K in a vacuum of 10 –5 Pa for 1 hour. 104

&RXQWV DUELWUDU\ XQLWV

Preparation of Bulk Nanocrystalline Materials

QDQRFU\VWDOOLQH )H=U% DOOR\ DPRUSKRXV )H=U% DOOR\ 













θ GHJUHHV

Fig. 3.10. Changes in the X-ray diffraction pattern of amorphous Fe 90 Zr 7 B 3 alloy after annealing and transforming to nanocrystalline state (the grain size of the precipitated bcc phase α-Fe(Zr) is approximately 10 nm).

The grain size of the crystalline bcc phase α-Fe(Zr), formed in the amorphous phase, was determined by high resolution transmission electron microscopy (HRTEM) and is equal to ~10 nm. Crystallisation of amorphous alloys has been studied actively because of the possibility of producing nanocrystalline ferromagnetic alloys of Fe–Cu–M–Si–B (M = Nb, Ta, W, Mo, Zr) systems with a low coercive force and high magnetic permeability, i. e. soft magnetic materials. Examination of thin films of a Ni–Fe alloy is shown that soft magnetic properties improve with a decrease in effective magnetocrystalline anisotropy [54]. This may be achieved by increasing the number of grains taking part in exchange interaction in thin magnetic films. In other worse, a decrease in the grain size increases the exchange interaction, decreases magnetocrystalline anisotropy and improves the soft magnetic properties. This concept was later realised by experiments in directional crystallisation of multicomponent amorphous alloys. Soft magnetic materials are Si-containing steels and, consequently, initial attempts to improve the soft magnetic properties by means of crystallisation of amorphous alloys were carried on alloys of the Fe–Si–B system with copper additions. However, it was not possible to produce alloys with a nanocrystalline structure in this system. Only the addition to the Fe–Si–B amorphous alloys of group IV–VII transition metals and copper and subsequent 105

Nanocrystalline Materials

Fig. 3.11. Nanocrystalline microstructure evolution of Fe 73.5 Cu 1 Nb 3 Si 13.5 B 9 alloy by primary crystallization [56].

crystallisation enabled a nanocrystalline structure to be produced [55]. An alloy with a homogeneous nanocrystalline structure was produced by crystallisation of Fe–Cu–Nb–Si–B amorphous alloys at 700–900 K. In this alloy, grains of the bcc phase α-Fe(Si) with a size of ~10 nm and copper clusters smaller than 1 nm are uniformly distributed in the amorphous phase. Crystallisation of Fe 73.5 Cu 1 Nb 3 Si1 3.5 B 9 amorphous alloys is investigated by high-resolution electron microscopy in [56]. The consecutive change of the microstructure of the alloys during crystallisation, based on these results, is shown schematically in Fig. 3.11. Preliminary (prior to crystallisation annealing) deformation by rolling of Fe–Cu–Nb–Si–B amorphous alloys or preliminary lowtemperature annealing enable a further decrease in the grain size to ~5 nm [57, 58]. For example, cold rolling of Fe 73.5 Cu 1 Nb 3 Si1 3.5B 9 amorphous alloy up to a strain of ~6% and subsequent annealing in vacuum at 813 K for 1 hour leads to the precipitation of nanocrystalline grains of the bcc phase α-Fe(Si) in the amorphous phase; the mean grain size is ~6–8 nm. The mean grain size of the nanocrystalline alloy subjected to only annealing at 813 K for 1 h is 8–10 nm. Low-temperature annealing of Fe 73.5 Cu 1 Nb 3 Si1 3.5 B 9 amorphous alloy at a temperature 723 K for 1 hour combined with subsequent short-term (for 10 seconds) high-temperature annealing at 923 K results in a mean grain size of the bcc phase of 4–5 nm. The decrease in the grain size of Fe–Cu–Nb–Si–B alloy after stepped annealing approximates the structure of this alloy to the structure of a pure bulk nanocrystalline metal with a grain size of 2–5 nm, produced by compacting [2–6]. 106

Preparation of Bulk Nanocrystalline Materials

Additional deformation or heat treatment, decreasing the grain size, does not result in any changes in the phase composition of the alloy. According to [58], this means that the phase composition of Fe 73.5 Cu 1 Nb 3 Si1 3.5 B 9 alloy is finally formed in the last hightemperature stage of treatment. A decrease in the grain size of the nanocrystalline phase as a result of preliminary deformation or heat treatment is caused by the formation of additional crystallisation centres in the amorphous phase. Crystallisation of rapidly solidified amorphous aluminium alloys Al–Cr–Ce–M (M = Fe, Co, Ni, Cu), which contained more than 92 at.% Al, led to the formation of a structure containing an amorphous phase and Al-rich icosahedral nanoparticles (D ~ 5–12 nm) precipitated in the amorphous phase [59]. As an example, Fig. 3.12 shows an HRTEM image of a rapidly solidified Al 94.5 Cr 3Ce 1 Co 1.5 alloy with dispersed precipitates of the icosahedral phase; electron diffraction patterns are taken from several regions with a diameter of 1 nm. It is interesting to note that the type of electron diffraction pattern of the dispersed phase depends on the size of the region in which diffraction of the accurately focused electron beam takes place. For example, the electron diffraction pattern obtained from a region of a diameter 1 nm, belonging to region B, has an fcc structure (Fig. 3.12b), whereas the electron diffraction patterns, obtained from a region with a diameter of 3

Fig. 3.12. High resolution TEM image (a) of a rapidly solidified Al 94.5 Cr 3 Ce 1 Co 1.5 alloy [59, 60]: icosahedral nanoparticles B, D and others are distributed in amorphous matrix C; (b),(c) and (d) are nanobeam diffraction patterns taken from the regions with a diameter of 1 nm marked B,C and D, respectively. 107

Nanocrystalline Materials

nm, show reflections corresponding to the symmetry axis of the 5 th order. This means that the precipitated nanoparticles at distances approximately 1 nm have a disordered structure (without the symmetry of the 5 th order), and at distance of approximately 2 nm or greater they have an icosahedral structure with long-range order [60]. Al 94.5Cr 3 Ce 1 Co 1.5 alloys have extremely high tensile strength (up to 1340 MPa), close to or higher than the strength of special steels. The main reason for the high tensile strength is the formation of spherical nanoparticles of the icosahedral phase and the presence of a thin aluminium layer around these particles. At present, the production of nanocrystalline alloys by crystallisation from the quenched amorphous state is developing and the number of alloys with the nanocrystalline structure produced by this method rapidly increases. 3.4. SEVERE PLASTIC DEFORMATION Severe plastic deformation is a very attractive method of producing submicrocrystalline (or, which is the same, superfine grain) materials with a mean grain size of ≤100 nm [61–65]. This method is based on the formation of a highly fragmented and misoriented structure retaining the residual features of the recrystallised amorphous state. The formation of this structure in metals and alloys is a result of multiple severe shear plastic deformation with in a true logarithmic degree of deformation of e = 4–7. Various methods are used to achieve high deformation of the material: rotating under quasi-hydrostatic pressure, equal channel angular pressing, rolling, uniform forging. In addition to increasing the mean grain size, the application of severe plastic deformation makes it possible to produce bulk specimens without damage and residual porosity. This cannot be achieved by compacting highly dispersed powders. Plastic deformation is an efficient means of the formation of the structure of metals, alloys and some other materials. During deformation, the dislocation density increases, the grains are refined and the concentration of point defects and static faults increases. The combination of these changes leads to the formation of a specific microstructure. The main relationships governing the formation of the structure during plastic deformation are determined by the combination of parameters of the starting structural state of the material and specific deformation conditions, and also by the mechanics of the deformation process. With other conditions being 108

Preparation of Bulk Nanocrystalline Materials

Fig. 3.13. Schematic representation of severe plastic deformation setup [62]: (a) high pressure torsion, (b) equal-channel angular pressing

equal, the main role in the formation of the structure and properties of the material is played by the mechanics of the deformation process: If the process ensures the uniform stress and strain states throughout the entire volume of the material, the deformation process is most efficient. The main methods for creating high strains, which lead to appreciable grain refining without failure of the specimens, are high pressure torsion and equal channel angular pressing (Fig. 3.13.) In high-pressure torsion, shear deformation is developed in discshaped specimens with a radius R and thickness l. The geometrical form of the specimens is such that the main volume of the material is deformed in the conditions of quasi-hydrostatic compression and the specimen does not fail inspite of the high strain. The true logarithmic degree of deformation e, obtained by high-pressure torsion, is calculated from the equation [61]: e = ln (θ R/l),

(3.1)

where θ is the rotation angle in radians. The following equation is 109

Nanocrystalline Materials

used to calculate the degree of shear strain ε s at some point x situated at distance R x from the axis of the specimen ε s = 2πN(R x /l),

(3.2)

where N is the number of rotations. To compare the degree of shear strain in torsion with the strain in other deformation methods, the value of ε s is usually converted to equivalent strain ε equiv = ε s /√3. Equation (3.2) shows that the strain should change in a linear manner from 0 in the centre of the specimen to the maximum value at the perimeter of the specimen. However, the results of a large number of investigations show that after several revolutions, the structure in the central part of the specimen is refined and, on the whole, the structure is uniform throughout the specimen bulk [62]. The conventional methods of plastic deformation based on shear (rolling, drawing, pressing, forging, torsion, etc.) make it possible to obtain relatively high degrees of strain as a result of multiple treatment but do not provide the uniform distribution of the parameters of the stress and strain state. The formation of a uniform structure is most efficient when using the stationary deformation process based on a simple shear. The process is based on pushing a workpiece through two channels with an equal cross section intersecting under an angle of 2Φ = 90–150° (Fig. 3.14). On the z P

x

P0

y

Fig. 3.14. Diagram of plastic deformation by the equal-channel angular pressing [68]: Φ is half of the angle of intersection of the channel, P is pressing pressure, and P 0 the counter pressure from the side of the outlet channel 110

Preparation of Bulk Nanocrystalline Materials

plane of intersection of the channels, the uniform localised simple shear deformation is concentrated with an intensity of ∆Γ = 2cotΦ.

(3.3)

Multiple cyclic treatment of the material by this method results in superhigh strain intensity Γ = N∆Γ = 2NcotΦ,

(3.4)

where N is the number of cycles. The produced material is in a uniform stress–strain state, but the size of the workpiece does not change. The true logarithmic degree of deformation is determined from the equation e = sinh –1(Γ/2) = ln{(Γ/2) + [(Γ/2) 2 + 1] /2 }.

(3.5)

It is most efficient to use angle 2Φ close to 90° which leads to the highest levels of deformation intensity at a small increase of contact friction. A lubricant is used to minimise contact friction. This deformation method, proposed by V. M. Segal [66] and developed in [67, 68], is referred to as equal-channel angular pressing. In comparison with other methods of plastic deformation, equal-channel angular pressing has produced the most uniform submicrocrystalline structure of material. Materials with such uniform structure possess stable and reproducible physical properties. In a general case, the structure of the material, produced by equal-channel angular pressing, depends not only on the nature of material and the magnitude of applied strain, but also on a number of technical parameters such as the size and shape of the cross section of the channels (the diagonal of the square section or the diameter of circular channels), and the direction of passage of the workpiece through the channels. If a material is difficult to deform, equalchannel angular pressing is carried at elevated temperatures. Analysis of the results of investigation of the structure and properties of submicrocrystalline materials has been carried out in reviews [62, 64, 69]. The main feature of the structure of submicrocrystalline materials, produced by the deformation methods, is the presence of non-equilibrium grain boundaries which act as sources of high elastic stresses. Triple points of the grains are another source of stresses. The diffusion contrast of the boundaries and intragranular flexural 111

Nanocrystalline Materials

extinction contours, which are observed on TEM images of submicrocrystalline materials, indicate that the grain boundaries are non-equilibrium boundaries. According to different estimates, the width of the intergranular boundaries in submicrocrystalline materials varies from 2 to 10 nm. The non-equilibrium grain boundaries contain large numbers of dislocations. Additionally, noncompensated disclinations are found in the grains boundaries. The dislocation density in the submicrocrystalline materials, produced by severe plastic deformation, is ~3×10 15 m –2 , and the power of disclinations is 1–2°. It should be noted that the density of the dislocations inside the grains is considerably lower in comparison with that at the boundaries. The dislocations and disclinations are the reason for the excess energy of the grain boundaries and generate long-range stress fields, which concentrate in the vicinity of the grain boundaries and triple points. For example, the excess energy of intergranular boundaries of submicrocrystalline copper with a mean grain size of ~20 nm is almost 0.5 J m –2 . Annealing of submicrocrystalline materials leads to the evolution of their microstructure. Evolution can be conventionally divided into two stages. In the first stage, as a result of annealing at temperature about 1/3 rd of the melting point, stress relaxation takes place, the grain boundaries transfer from the non-equilibrium to more equilibrium state, and grains slowly grow. A further increase in annealing temperature or an increase in annealing time results in collective recrystallisation, i. e. grain growth. The method of severe plastic deformation is used for producing submicrocrystalline structures of such metals as Cu [70–72], Pd [73–76], Fe [77–79], Ni [70, 72, 80–82], Co [83], alloys based on aluminium [63], magnesium [84] and titanium [85, 86]. The authors of [72] noted different microstructures of Ni and Cu produced by severe plastic deformation with the same magnitude of deformation: in submicrocrystalline nickel, the size of the majority of grains was approximately 100 nm, whereas the grain size in submicrocrystalline copper was 5–100 nm, and the copper grains contained a larger number of defects (dislocations and twins) than the grains of submicrocrystalline Ni. This means that in submicrocrystalline nickel, the redistribution of dislocations into energy favourable configurations (for example, rows of dislocations) already takes place in the process of severe plastic deformation, whereas in submicrocrystalline copper such redistribution does not even start. The results of [72] show that the microstructure of any material, produced by severe plastic deformations, should greatly 112

Preparation of Bulk Nanocrystalline Materials

differ in different stages of deformation. In addition to this, the microstructure depends greatly on the type of deformation (pressure, shear or torsion) and deformation parameters (temperature, magnitude, rate and duration of deformation). In fact, in order to understand the structure and properties of submicrocrystalline materials, it is very important to take into account the phase and structural transformations which take place in these materials during heating and cooling, i.e. such phenomena as recrystallisation, dissolving, and precipitation of a second phase, etc. The threshold of the temperature stability of the submicrocrystalline structure depends on the state of intergranular boundaries which, in turn, depends on the conditions of production of this structure. The structure and recrystallisation of submicrocrystalline materials also depend on the composition of the alloy and the type of crystal lattice, but these problems have not been discussed in sufficient detail. Severe plastic deformation has been used for producing the submicrocrystalline structure of not only metals, alloys and intermetallic compounds with relatively high plasticity, but also certain compounds with high brittleness. It is interesting that after plastic deformation of a comparable magnitude, the grain size of brittle compounds is smaller than that of metals. For example, in [87, 88] torsion under quasi-hydrostatic pressure was used to produce bulk nanocrystalline titanium carbide specimens (Fig. 3.15) with a grain size of ~30–40 nm from the coarse-grained (D ~ 2– 5 µm) powder of non-stoichiometric titanium carbide TiC 0.62 . The formation of the submicrocrystalline structure by deformation methods is accompanied by significant changes in the

Fig. 3.15. Compacted specimen of nanocrystalline titanium carbide TiC 0.62 with a grain size of ~30–40 nm, produced by severe plastic deformation (torsion under quasi-hydrostatic pressure) of coarse-grained titanium carbide [87].

113

Nanocrystalline Materials

physical properties of metals, alloys and compounds. Metals with a submicrocrystalline structure are suitable objects for experimental investigation of the intercrystallite boundaries because they can be examined by certified methods of metal physics and solid state physics [77, 78]. 3.5. DISORDER–ORDER TRANSFORMATIONS In the conditions of thermodynamic equilibrium, the strongly nonstoichiometric carbides MC y and nitrides MN y may be in the disordered or ordered state. The disordered state is thermodynamically equilibrium at a temperature above 1300–1500 K and at a temperature below 1000 K only the ordered state [89– 96] is thermodynamically equilibrium. However, the disordered state of non-stoichiometric carbides and nitrides is easily retained at low temperatures by quenching from high temperature. In this case, the disordered state is metastable. The disorder–order phase transformations in strongly non-stoichiometric compounds are usually transitions of the first kind and are accompanied by an abrupt change of the lattice constant. Consequently, disorder–order transformations, i.e. ordering, can be used to produce a nanostructure in non-stoichiometric compounds. The application of ordering for producing a nanocrystalline powder of nonstoichiometric vanadium carbide VC 0.875 is described in Section 2.7. However, using ordering, the nanostructure can also be produced in bulk specimens of non-stoichiometric compounds. Let us consider producing a nanostructure in bulk nonstoichiometric vanadium carbide VC 0.875 . The disordered carbide VC 0.875 has a cubic B1 basic crystal structure. Bulk specimens of VC 0.875 carbide were produced by hot pressing a powder of disordered vanadium carbide VC 0.875 at a temperature of 2000 K in a high-purity argon flow [97]. Examination of the surface of sintered disordered carbide VC 0.875 by the EDX method (Fig. 3.16a) showed the presence of vanadium V, carbon C and impurities of metallic Ti, Nb and Ta. The presence of Ti, Nb and Ta is the result of surface segregation of small impurities during high-temperature sintering of the specimen. Earlier, similar phenomena of surface segregation of a small impurity of zirconium carbide ZrC were detected on sintered NbC specimens containing approximately 1 mol.% ZrC [98, 99]. After polishing sintered specimens of VC 0.875 with the removal of a surface layer approximately 50µm thick, the impurity disappeared (Fig. 3.16b). 114

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Fig. 3.16. EDX spectra of the surface of the sintered specimen of vanadium carbide VC 0.875 : (a) immediately after sintering the disordered carbide there are impurities Ti, Nb, Ta and W on the surface; (b) grinding with the removal of the surface layer approximately 50 µm thick leads to the disappearance of impurities Ti, Nb, Ta and W, the formation of traces of Al and Si associated with the use of Al–Sicontaining abrasive for grinding, (c) there are no impurities on the surface of the annealed ordered carbide V 8 C 7 (VC 0.875), EDX spectra of quenched specimens also have the same form and do not contain lines of impurity elements.

Sintered bulk specimens were treated under three thermal routes: 1) annealing at a temperature of 1370 K for two hours followed by slow (at a rate of 100 K h –1 ) cooling to 300 K, 2) quenching from 1420 to 300 K, and 3) quenching from 1500 K to 300 K [97]. For quenching the bulk specimens are put in quartz ampoules evacuated to 10 –3 Pa and are annealed at a maximum temperature (1420 or 1500 K) for 15 min. The ampoules with specimens are then dropped in water; quenching rate was 100 K s –1 .Figure 3.17 shows the surface of a bulk carbide VC 0.875 specimen after quenching from 1500 K. The grain boundaries of the basic cubic phase are clearly visible. The grain size is 10–60 µm. The EDX spectra of annealed (Fig. 3.16c) and quenched bulk specimens of VC 0.875 contain only lines of main elements: vanadium V and carbon C. The structure of annealed and quenched bulk specimens of the VC 0.875 carbide was examined by X-ray diffraction. Both after annealing and quenching the X-ray diffraction patterns contain 115

Nanocrystalline Materials

Fig. 3.17. Microstructure of the bulk specimen of carbide VC 0.875 after quenching from 1500 K [97]. Sharp grain boundaries are due to their high susceptibility to oxidation. The image obtained in a ISI-DS 130 electron microscope at a magnification of 100

additional weak reflections, in addition to structural reflections (Fig. 3.18). Judging by their position, additional reflections are superstructure reflections and correspond to an ordered cubic V 8 C 7 phase with the P4 332 space group. Superstructure reflections of the annealed and quenched specimens have nearly equal integral intensity, but greatly differ in the width. The widest superstructure reflections are observed in the X-ray diffraction pattern of the specimen quenched from 1500 K. According to the equilibrium phase diagram of the V–C systems [92, 94, 100] the formation of an ordered phase V 8 C 7 takes place as a result of the disorder–order transformation at a temperature of T trans = 1380 K; the experimental temperature of phase transformation is 1413 ± 20 K [101]. These data show that rapid cooling from 1420 or from 1500 K must lead to quenching of the disordered non-stoichiometric vanadium carbide VC 0.875 and saving of the disordered VC 0.875 carbide as a metastable phase. However, even if a specimen is quenched from 1500 K, an ordered V 8 C 7 phase appears and the relative intensity of superstructure reflections is approximately equal to that for the specimens after quenching from 1420 K or after annealing at 1370 K (see Fig. 3.18). As a result of ordering, every grain of the disordered basic phase breaks down into domains of the ordered phase. The degree of 116

Preparation of Bulk Nanocrystalline Materials



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Fig. 3.18. X-ray diffraction patterns of bulk VC 0.875 carbide specimens produced by hot pressing and subjected to additional heat treatment (radiation CuKα1,2, structural reflections of B1 phase and superstructure reflections of phase V 8 C 7 are shown) [97]: (a) annealing at 1370 K for 2 hours followed by slow cooling for 300 K; (b) quenching from 1420 to 300 K at a rate of 100 K s –1 ; (c) quenching for 1500 to 300 K at a rate of 100 K s –1 . All X-ray diffraction patterns show superstructure reflections of an ordered cubic (space group P4 332) phase V 8C 7; the largest broadening of the superstructure reflections is seen on the X-ray diffraction pattern (c) of specimen VC 0.875 quenched from 1500 K

ordering in a domain is high, while the mutual spatial distribution of the domains is chaotic and depends on the ratio between the structures of the ordered phase and the disordered matrix. Optical microscopic examination at a magnification of 200 times shows that the formation of the ordered phase starts at the grain boundaries of the disordered phase. In the disordered carbide, the grains of basic phase in the disordered carbide have sharp straight boundaries (Fig. 3.19a), and after annealing these boundaries became broken 117

Nanocrystalline Materials

Fig. 3.19. Changes in the microstructure of a bulk non-stoichiometric vanadium carbide VC 0.875 as a result of ordering: (a) in disordered carbide VC 0.875 the grains have sharp straight boundaries, (b) in the ordered V 8 C 7 (VC 0.875 ) carbide the grain boundaries of the basic phase appear to be “eaten” as a result of the formation of domains of the ordered phase (image is obtained by V.N. Lipatnikov, Vienna University of Technology, 1995).

as a result of ordering (Fig. 3.19b). It means that the domains of ordered phase grow through the grains of the disordered basic phase in the direction from the boundaries to the centre of the grain. The small size of the domains and the same cubic symmetry of the disordered and ordered phases do not allow the domain boundaries to be examined by optical microscopy. X-ray examination shows that the width of structural reflections is independent of the thermal treatment conditions of the bulk VC 0.875 specimens. Therefore, it may be assumed that the size of grains of the disordered basic phase remains unchanged during ordering. On the contrary, the superstructure reflections greatly broaden. This broadening may be due to the small size of the domains of the ordered phase which forms under different thermal treatment conditions. Does a nanostructure appear in bulk specimens 118

Preparation of Bulk Nanocrystalline Materials

of VC 0.875 carbide subjected to thermal treatment and containing an ordered V 8 C 7 phase? How small are the domains of the ordered phase? These questions may be answered if one measures the width of experimental diffraction reflections and compares it with the instrumental width determined from the resolution function of the diffractometer. The resolution function of a Siemens D-500 X-ray diffractometer is determined in a special diffraction experiment using an annealed specimen of a stoichiometric tungsten carbide with a grain size of 10–20 µm. The tungsten carbide has no homogeneity range and there is no reflection broadening caused by inhomogeneity. The size broadening of the diffraction reflections does not take place with these large grains either. As a result of annealing of the specimen there is no strain broadening. Thus, the width of some diffraction reflections of the tungsten carbide completely coincides with the resolution function θ R of the diffractometer for the given diffraction angle θ. The dependence of the second moment θ R of the resolution function of the Siemens D500 X-ray diffractometer on diffraction angle θ is shown in Fig. 3.20.

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Fig. 3.20. Broadening of diffraction reflections of bulk vanadium carbide VC 0.875 specimens after heat treatment and formation of the ordered phase V8C7 [97] (comparison of the second moments θ exp of the experimental diffraction reflections with the angular dependence of second moment θR(θ) of the resolution function of diffractometer): ( l ) annealing at 1370 K followed by slow cooling, ( ¡ ) quenching from 1420 K, ( ™ ) quenching from 1500 K. Angular dependence of the resolution function of diffractometer, θ R (θ), is shown by the solid line. 119

Nanocrystalline Materials

Comparison of the width of superstructure diffraction reflections of bulk vanadium carbide specimens with the resolution function θ R shows that experimental reflections broaden (Fig. 3.20). Since the width of these reflections depends on the domain size and the instrumental width, the size of domains can be determined from the 2 − θ2R . To a first approximation, measured broadening β = 2.235 θexp

it will be assumed that the strain broadening is absent and the observed broadening β is caused only by a small size of domains, therefore β = β s . In turn, the size broadening β s (2θ) = 2β s (θ), measured in radians, is associated with the mean domain size by the equation

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where K hkl ≈ 1 is Scherrer’s constant whose value depends on the shape of the domain (crystallite, particle) and on Miller indices (hkl) of diffraction reflection; λ is the radiation wavelength. Figures 3.18 and 3.20 show that the broadening of the superstructure reflections is the largest for the specimen quenched from 1500 K. The smallest broadening of the superstructure reflections is observed for the vanadium carbide specimen annealed at 1370 K. This means that domains of the ordered phase are the smallest in the specimen quenched from 1500 K. In the annealed vanadium carbide specimens, the size of the domains is the largest because annealing and subsequent slow cooling create favourable conditions for domain growth. The determination of broadening β s and subsequent evaluation of the mean domain size of the ordered phase using equation (3.8) gave the following results. In the annealed specimens (Fig. 3.18a), the size of domains is 127 ± 10 nm with a probability of 95%; in the specimens quenched from 1420 K (Fig. 3.18b) and 1500 K (Fig. 3.18c) the size of domains is equal to 60 ± 9 nm and 18 ± 12 nm with a probability of 95%. Thus, annealing of bulk specimens of the non-stoichiometric vanadium carbide VC 0.875 and quenching of these specimens from a temperature of T trans ± 100 K cause the appearence of a nanostructure consisting of domains of the ordered phase. The size of domains increases with a decrease in annealing or quenching temperature and with a decrease in the cooling rate. The authors of [97, 102] also produced a bulk specimen of 120

Preparation of Bulk Nanocrystalline Materials

vanadium carbide from a nanopowder of non-stoichiometric ordered carbide V 8 C 7 (VC 0.875 ). The method of production and the microstructure of the vanadium carbide nanopowder were descrided in Section 2.7. The nanostructured vanadium carbide VC 0.875 powder was compacted by the cold method at room temperature and a pressure of 10 MPa. The density of compacts was 68% of the theoretical density of vanadium carbide; this value is much larger than the density of the nanopowder which is equal to 36%. Stepped sintering of compacted specimen was carried out in a vacuum of 1×10 –3 Pa at temperature from 400 to 2000 K with a step of 100 K; the holding time at each temperature was 2 hours. No large change was detected in the density of the sintered specimen in comparison with the density of the compact. The relative variation of the mass ∆m/m of the sintered specimen as a function of sintering temperature is shown in Fig. 3.21. There are three main stages of mass loss. The first stage (from room temperature to 500 K) is associated with the loss of water during heating. The second stage corresponds to the temperature interval from 900 to 1500 K and is caused by the dissociation and removal of the surface oxide phase thanks to the presence of free carbon in the specimen. After sintering in this temperature range, the colour of the specimen changed from black, determined by the surface oxide phase, to grey corresponding to the pure vanadium carbide. The third stage of the mass loss starts at a temperature of 1900 K and is associated with vacuum congruent evaporation of vanadium carbide VC 0.875 which takes place without any change in the composition of the specimen [90]. 

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Nanocrystalline Materials

In spite of the large porosity (approximately 30%) of the sintered specimen, sintering allowed Vickers microhardness to be measured. The sintered specimen was ground and microhardness measurements were taken in automatic regime at a load of 200 g and 500 g and loading time 10 s. Microhardness does not exhibit any dependence on load within the measurement error. Microhardness H V was equal to 60–80 GPa. For comparison, the measured microhardness of a coarse-grained carbide VC 0.875 specimen, produced by hot pressing, is 21 GPa at a load of 0.1 kg. According to the literature data [103], the microhardness of the coarse-grained vanadium carbide is 29 GPa at a load of 0.1 and 0.2 kg. Thus, the microhardness of the specimen produced by sintering of vanadium carbide nanopowder is 2 to 3 times higher than the microhardness of the coarse-grained disordered vanadium carbide and approaches that of diamond. Usually, at 300 K the microhardness of nanomaterials is 2–7 times higher than H V of the conventional polycrystalline materials [1, 104]. The high microhardness of the carbide VC 0.875 specimen, produced by sintering of the nanopowder, may be explained by the Hall–Petch law H V ≈ H 0 + kD –1/2 , i.e. H V is proportional to D –1/2 . Analysis of the experimental data on the microhardness of bulk nanocrystalline materials, published in [104], shows that the Hall–Petch law is fulfilled when the grain size varies in the range 500 to 20 nm. The examined carbide corresponds to this range of the nanocrystallite sizes. It should be noted that no nanostructure is found in the sintered specimen of VC 0.875 . The study of sintered specimens in an ISIDS130 scanning electron microscope shows that its microstructure represents a set of well sintered agglomerates with free spaces between them (Fig. 3.22). This is consistent with the large porosity of the sintered specimen. In order to determine whether the sintered agglomerates have a nanostructure, additional high-resolution transmission electron microscopic studies are necessary. Diffraction investigations showed no superstructure reflections in the X-ray diffraction patterns of the sintered specimen. This is obvious because the maximum sintering temperature of 2000 K is considerably higher than the temperature of the order–disorder phase transformation T trans . To obtain the ordered state in the specimen, it is necessary to carry out thermal treatment in the vicinity of temperature T trans . The formation of the nanostructure in the bulk non-stoichiometric vanadium carbide VC 0.875 is determined by the disorder–order 122

Preparation of Bulk Nanocrystalline Materials

Fig. 3.22. Scanning electron micrograph of the microstructure of a sintered vanadium carbide nanopowder. Highly dense sintered conglomerates and the free space between them with the size of up to tens of micrometers are clearly visible [97].

VC 0.875 –V 8 C 7 phase transformation and by the appearance of domains of the ordered phase. The size of the domains decreases with an increase in the temperature of the start of thermal treatment (quenching or annealing) and with an increase in the cooling rate. The discussed results together with the data presented in Section 2.7 indicate that the ordering is a new efficient method of producing a nanostructure in bulk and powdered non-stoichiometric compounds. The disorder–order transformations, which are accompanied by an abrupt change of the crystal volume, may be used for producing a nanostructured state not only of strongly nonstoichiometric compounds but also of substitutional solid solutions, including some alloys. The formation of a nanostructure in solid solutions also takes place as a result of solid phase decomposition. For example, it has been established that the decomposition of carbide solid solutions (ZrC) 1–x (NbC) x [105] leads to the formation of a nanostructure with a grain size of approximately 70 nm. The formation of a nanostructure in these solid solutions will be considered in detail in Section 4.2 when discussing the determination of the size of small particles from the broadening of diffraction reflections. 123

Nanocrystalline Materials

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by Fine Particles Technology. Eds. M.-I. Baraton, I. V. Uvarova. (Kluwer Academic Publishers, Netherlands, Dordrecht 2000) pp.135-142 Yu. V. Bykov, S. V. Egorov, A. G. Eremeev, A. A. Sorokin, V. V. Holoptsev. Microwave sintering of nanostructured ceramic materials. In: Physics and Chemistry of Ultra-Disperse (Nano-) Systems / Proc. of V th All-Russian Conference. V.2. (Ural Division of the Russ. Acad. Sci., Yekaterinburg 2001) pp.14-19 (in Russian) Yu. Bykov, V. Holoptsev, Y. Makino, S. Miayke, I. Plotnikov, T. Ueno. Kinetics of densification and phase transformation at microwave sintering of silicon nitride with alumina and yttria or ytterbia as additives. J. Japan. Soc. Powd. Powder Metallurgy 48, 558-564 (2001) H. J. Höfler, H. Hahn, R. S. Averback. Diffusion in nanocrystalline materials. Defect and Diffusion Forum 75, 195-210 (1991) S. Okada, F. Tany, H. Tanumoto, Y. Iwamoto. Anelasticity of ultrafine-grained polycrystalline gold. J. Alloys Comp. 211-212, 494-497 (1994) Yu. A. Bykovskii, V. P. Kozlenkov, Yu. B. Krasil’nikov, I. N. Nikolaev. Preparation of ZrO 2-Y 2 O 3 films by laser evaporation of metals in an oxygen ion beam. Poverkhnost No 12, 69-73 (1992) (in Russian) D. A. Dudko, V. G. Aleshin, A. E. Barg, N. V. Dubovitskaya, L. N. Larionov, A. A. Smekhnov. On the nature of high hardness of vacuum-plated chromium. Doklady AN SSSR 285, 106-109 (1985) (in Russian) A. E. Barg, V. N. Dubovitskaya, D. A. Dudko, L. N. Larikov. Formation of amorphous phase based on chromium by ion-plasma deposition. Metallofizika 9, 118-119 (1987) (in Russian) R. A. Andrievski. The synthesis and properties of interstitial phase films. Uspekhi Khimii 66, 57-77 (1996) (in Russian). (Engl. Transl.: Russ. Chem. Reviews 66, 53-72 (1996)) H. Fujimori, S. Mitani, S. Ohnuma. Tunnel-type GMR in metal-nonmetal granular alloy thin films. Mater. Sci. Eng. B 31, 219-223 (1995) M. Ohnuma, K. Hono, H. Onodera, J. S. Pedersen, S. Mitani, H. Fujimori H. Distribution of Co particles in Co-Al-O granular thin films. In: Advances in Nanocrystallization. Proceedings of the Euroconference on Nanocrystallization and Workshop on Bulk Metallic Glasses (Grenoble, France, April 21-24, 1998). Ed. A. R.Yavari. (Trans Tech Publications, Switzerland 1999). Materials Science Forum 307, 171-176 (1999) / J. Metastable Nanocryst. Mater. 1, 171-176 (1999) H. van Swygenhoven, P. Böni, F. Paschoud, M. Victoria, M. Knauss M. Nanostructured Ni 3Al produced by magnetron sputtering. Nanostruct. Mater. 6, 739-742 (1995) S. Hotchandani, P. V. Kamat. Charge-transfer processes in coupled semiconductor systems. Photochemistry and photoelectrochemistry of the colloidal CdS - ZnO system. J. Phys. Chem. 92, 6834-6839 (1992) I. Bedjia, S. Hotchandani, P. V. Kamat. Photoelectrochemistry of quantized WO3 colloids. Electron Storage electrochromic and photoelectrochromic effects. J. Phys. Chem. 97, 11064-11070 (1993); Preparation and photoelectrochemical characterization of thin SnO 2 nanocrystalline semiconductor films and their sensitization. J. Phys. Chem. 98, 4133-4140 (1994) B. O’Regan, M. Grätzel, D. Fitzmaurice. Optical electrochemistry. 1. Steadystate spectroscopy of conduction-band electrons in a metal oxide semiconductor electrode. Chem. Phys. Letters 183, 89-93 (1991); Optical electrochemistry. 2. Real-time spectroscopy of conduction-band electrons in a metal oxide

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semiconductor electrode. J. Phys. Chem. 95, 10525-10528 (1991) H. Yoshiki, K. Hashimoto, A. Fujishima. Reaction mechanism of electroless metal deposition using ZnO thin film (I): process of catalys formation. J. Electrochem. Soc. 142, 428-432 (1995) L. Kavan, T. Stoto, M. Grätzel, D. Fitzmaurice, V. Shklover. Quantum size effects in nanocrystalline semiconducting TiO 2 layers prepared by anodic oxidative hydrolysis of TiCl 3 . J. Phys. Chem. 97, 9493-9498 (1993) K. Yamada, T. Y. Chow, T. Horihata, M. Nagata. A low-temperature synthesis of zirconium oxide coating using chelating agents. J. Non-Cryst. Solids 100, 316-320 (1988) R. A. Andrievski. The synthesis and properties of nanocrystalline refractory compounds. Uspekhi Khimii 63, 431-448 (1994) (in Russan). (Engl. transl.: Russ. Chem. Reviews 63, 411-428 (1994)) K. Lu, J. T. Wang, W. D. Wei. Thermal expansion and specific heat capacity of nanocrystalline Ni-P alloy. Scripta Metal. Mater. 25, 619-623 (1991) T. Cheng. Nanometer substructures and its effects on ductility and toughness at room temperature in nickel-rich NiAl. Nanostruct. Mater. 2, 19-28 (1993) H. Hoffmann. Magnetic properties of thin ferromagnetic films in relation to their structure. Thin Solid Films 58, 223-233 (1979) Y. Yoshizawa, S. Oguma, K. Yamauchi. New iron-based soft-magnetic alloys composed of ultrafine grain structure. J. Appl. Phys. 64, 6044-6046 (1988) K. Hono, D. H. Ping. APFIM studies of nanocomposite soft and hard magnetic materials. In: Advances in Nanocrystallization. Proceedings of the Euroconference on Nanocrystallization and Workshop on Bulk Metallic Glasses (Grenoble, France, April 21-24, 1998). Ed. A. R.Yavari. (Trans Tech Publications, Switzerland 1999). Materials Science Forum 307, 69-74 (1999) / J. Metastable Nanocryst. Mater. 1, 69-74 (1999) N. I. Noskova, N. F. Vil’danova, A. P. Potapov, A. A. Glazer. Influence of deformation and annealing on structure and properties of amorphous alloys. Fiz. Metall. Metalloved. 73, No 2, 102-110 (1992) (in Russian) N. I. Noskova, E. G. Ponomareva, A. A. Glazer, V. A. Lukshina, A. P. Potapov. Influence of preliminary deformation and low temperature annealing on a size of nanocrystallites of Fe 73.5 Cu 1 Nb 3 Si 13.5 B 9 alloy obtained at crystallization of amorphous ribbon. Fiz. Metall. Metalloved. 76, No 5, 171-173 (1993) (in Russian) A. Inoue, H. M. Kimura, K. Sasamori, T. Masumoto. Ultrahigh strength of rapidly solidified Al 68-x Cr 3 Ce 1 Co x (x = 1, 1.5 and 2 %) alloys containing an icosahedral phase as a main component. Mater. Trans. Japan. Inst. Met. 35, 85-94 (1994) A. Inoue. Preparation and novel properties of nanocrystalline and nanoquasicrystalline alloys. Nanostruct. Mater. 6, 53-64 (1995) N. A. Smirnova, V. I. Levit, V. P. Pilugin, R. I. Kuznetsov, M. V. Degtyarev. Peculiarities of low temperature recrystallization of nickel and copper. Fiz. Metall. Metalloved. 62, 566-570 (1986) (in Russian) R. Z. Valiev, I. V. Aleksandrov. Nanostructured Materials Obtained by Severe Plastic Deformation (Logos, Moscow 2000) 272 pp. (in Russian) R. Z. Valiev, N. A. Krasilnikov, N. K. Tsenev. Plastic deformation of alloys with submicron-grain structure. Mater. Sci. Engineer. A137, 35-40 (1991) R. Z. Valiev, A. V. Korznikov, R. R. Mulyukov. Structure and properties

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of metallic materials with submicrocrystalline structure. Fiz. Metall. Metalloved. 73, No 4, 70-86 (1992) (in Russian) V. M. Segal. Materials processing by simple shear. Mater. Sci. Engineer. A 197, 157-164 (1995) V. M. Segal. Methods of investigation of strain-deformed state in processes of plastic change of shape. Ph. D. Thesis (Physico-Technical Institute of Acad. Sci. of Belorus. SSR, Minsk 1974) 30 pp. (in Russian) V. M. Segal, V. I. Reznikov, F. E. Drobyshevskii, V. I. Kopylov. Plastic treatment of metals by simple shear. Izv. AN SSSR. Metally No 1, 115123 (1981) (in Russian) V. M. Segal, V. I. Reznikov, V. I. Kopylov, D. A. Pavlik, V. F. Malyshev. Processes of Plastic Formation of Structure of Metals (Nauka i Tekhnika, Minsk 1994) 232 pp. (in Russian) R. Z. Valiev, A. V. Korznikov, R. R. Mulyukov. Structure and properties of ultrafine-grained materials produced by severe plastic deformation. Mater. Sci. Engineer. A168, 141-148 (1993) N. A. Akhmadeev, R. Z. Valiev, V. I. Kopylov, R. R. Mulyukov. Formation of submicron-grained structure in copper and nickel with the use of severe shearing strain. Metally No 5, 96-101 (1992) (in Russian) A. A. Rempel, A. I. Gusev, S.Z. Nazarova, R. R. Mulyukov. Imputity superparamagnetism in plastically deformed copper. Doklady Akad. Nauk 347, 750-754 (1996) (in Russian). (Engl. transl.: Physics - Doklady 41, 152-156 (1996)) A. Cziraki, I. Geracs, E. Toth-Kadar, I. Bakonyi. TEM and XRD study of the microstructure of nanocrystalline Ni and Cu prepared by severe plastic deformation and electrodeposition. Nanostruct. Mater. 6, 647-550 (1995) A. A. Rempel, A. I. Gusev, R. R. Mulyukov, N. M. Amirkhanov. Microstructure and properties of palladium subjected to severe plastic deformation. Metallofizika i Noveishie Tekhnologii 18, 14-22 (1996) (in Russian) A. A. Rempel, A. I. Gusev. Magnetic susceptibility of palladium subjected to severe plastic deformation. Phys. Stat. Sol. (b) 196, 251-260 (1996) A. A. Rempel, A. I. Gusev, R. R. Mulyukov, N. M. Amirkhanov. Microstructure, microhardness and magnetic susceptibility of submicrocrystalline palladium. Nanostruct. Mater. 7, 667-674 (1996) A. A. Rempel, A. I. Gusev, S. Z. Nazarova, R. R. Mulyukov. Magnetic susceptibility of palladium subjected to plastic deformation. Doklady Akad. Nauk 345, 330-333 (1995) (in Russian). (Engl. transl.: Physics - Doklady 40, 570-573 (1995)) R. Z. Valiev, R. R. Mulyukov, V. V. Ovchinnikov, V. A. Shabashov, A. Yu. Arkhipenko, I. M. Safarov. On physical width of intercrystallite boundaries. Metallofizika 12, 124-126 (1990) (in Russian) R. Z. Valiev, R. R. Mulyukov, V. V. Ovchinnikov. Direction of a grain-boundary phase in submicrometer grained iron. Phil. Mag. Letters 62, 253-256 (1990) R. Z. Valiev, R. R. Mulyukov, V. V. Ovchinnikov, V. A. Shabashov. Mossbauer analysis of submicrometer grained iron. Scripta Metal. Mater. 25, 27172722 (1991) R. Z. Valiev, R. R. Mulyukov, Kh. Ya. Mulyukov, L. I. Trusov, V. I. Novikov. Curie temperature and saruration magnetization of nickel with submicrongrained structure. Pis’ma v ZhTF 15, 78-81 (1989) (in Russian) R. Z. Valiev, Ya. D. Vishnyakov, R. R. Mulyukov, G. S. Fainstein. On the decrease of Curie temperature in submicron-grained nickel. Phys. Stat. Sol.

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(a) 117, 549-553 (1990) Kh. Ya. Mulyukov, G. F. Korznikova, R. Z. Abdulov, R. Z. Valiev. Magnetic hysteresis properties of submicron grained nickel amd their variation upon annealing. J. Magn. and Magn. Mater. 89, 207-213 (1990) Kh. Ya. Mulyukov, G. F. Korznikova, R. Z. Valiev. Microstructure and magnetic properties of submicron grained cobalt after large plastic deformation and their variation during annealing. Phys. Stat. Sol. (a) 125, 609-614 (1991) R. Z. Abdulov, R. Z. Valiev, N. A. Krasilnikov. Formation of submicrometergrained structure in magnesium alloy due to high plastic strains. Mater. Sci. Letters 9, 1445-1447 (1990) R. M. Galeyev, O. R. Valiakhmetov, G. A. Salishchev. Dynamic recrystallization of coarse grained titanium-base VT30 alloy in (a + b) field. Metally No 4, 97-103 (1990) (in Russian) O. R. Valiakhmetov, R. M. Galeyev, G. A. Salishchev. Mechanical properties of titanium VT8 alloy with submicrocrystalline structure. Fiz. Metall. Metalloved. 70, No 10, 204-206 (1990) (in Russian) A. A. Rempel, A. I. Gusev, R. R. Mulyukov. Preparation of nanocrystalline titanium carbide. In: Solid State Chemistry and New Materials. V.1. (Institute of Solid State Chemistry, Yekaterinburg 1996) pp.244-245 (in Russian) A. A. Rempel, A. I. Gusev, R. R. Mulyukov. Preparation of nanocrystalline titanium carbide by severe plastic deformation. In: Ultrafine grained powders, materials and nanostructures. Proc. of Interdistrict Conf., Krasnoyarsk, December 17-19, 1996. Ed. V. E. Red’kin. (Krasnoyarsk State Technical University, Krasnoyarsk 1996) pp.131-132 (in Russian) A. I. Gusev, A. A. Rempel. Structural Phase Transitions in Nonstoichiometric Compounds (Nauka, Moscow 1988) 308 pp. (in Russian) A. I. Gusev. Physical Chemistry of Nonstoichiometric Refractory Compounds (Nauka, Moscow 1991) 286 pp. (in Russian) A. I. Gusev, A. A. Rempel. Nonstoichiometry, Disorder and Order in Solids (Ural Division of the Russ. Acad. Sci., Yekaterinburg 2001) 580 pp. (in Russian) A. I. Gusev, A. A. Rempel, A. J. Magerl. Disorder and Order in Strongly Nonstoichiometric Compounds: Transition Metal Carbides, Nitrides and Oxides (Springer, Berlin - Heidelberg - New-York 2001) 607 pp. A. I. Gusev, A. A. Rempel. Phase diagrams of metal-carbon and metal-nitrogen systems and ordering in strongly nonstoichiometric carbides and nitrides. Phys. Stat. Sol. (a) 163, 273-304 (1997) A. I. Gusev. Order-disorder transformations and phase equilibria in strongly nonstoichiometric compounds. Uspekhi Fiz. Nauk 170, 3-40 (2000) (in Russian). (Engl. transl.: Physics - Uspekhi 43, 1-37 (2000)) A. A. Rempel. Effects of Ordering in Nonstoichiometric Interstitial Compounds (Nauka, Yekaterinburg 1992) 232 pp. (in Russian) A. A. Rempel. Atomic and vacancy ordering in nonstoichiometric carbides. Uspekhi Fiz. Nauk 166, 33-62 (1996) (in Russian). (Engl. transl.: Physics - Uspekhi 39, 31-56 (1996)) A. I. Gusev, A. A. Tulin, V. N. Lipatnikov, A. A. Rempel. Nanostructure of dispersed and bulk nonstoichiometric vanadium carbide. Zh. Obsh. Khimii 72, 1067-1076 (2002) (in Russian). (English transl.: Russ. J. General Chem. 72, 985-993 (2002)) S. V. Rempel’, A. I. Gusev. ZrC segregation to the surface of dilute solid solutions of zirconium carbide in niobium carbide. Neorgan. Materialy 37,

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1205-1211 (2001) (in Russian). (Engl. transl.: Inorganic Materials 37, 10241029 (2001)) S. V. Rempel, A. I. Gusev. Surface segregation of ZrC from a carbide solid solution. Fiz. Tverd. Tela 44, 66-71 (2002) (in Russian). (Engl. transl.: Physics of the Solid State 44, 68-74 (2002)) A. I. Gusev. A phase diagram of the vanadium–carbon system taking into account ordering in nonstoichiometric vanadium carbide. Zh. Fiz. Khimii 74, 600-606 (2000) (in Russian). (Engl. transl.: Russ. J. Phys. Chem. 74, 510-516 (2000)) T. Athanassiadis, N. Lorenzelli, C. H. de Novion. Diffraction studies of the order-disorder transformation in V 8 C 7 . Ann. Chim. France 12, 129-142 (1987) A. A. Rempel, A. I. Gusev, O. V. Makarova, S. Z. Nazarova. Physical and chemical properties of nanostructured vanadium carbide. Perspektivnye Materialy No 9, 9-15 (1999) (in Russian) L. Ramqvist. Variation of hardness, resistivity and lattice parameter with carbon content of group 5b metal carbides. Jernkontors Annaler. 152, 467475 (1968) A. I. Gusev. Nanocrystalline Materials:Preparation and Properties (Ural Division of the Russ. Acad. Sci., Yekaterinburg 1998) 200 pp. (in Russian) A. I. Gusev, S. V. Rempel. X-ray diffraction study of the nanostructure resulting from decomposition of (ZrC) 1–x (NbC) x solid solutions. Neorgan. Materialy 39, 49-53 (2003) (in Russian). (Engl. trans.: Inorganic Materials 39, 43-46 (2003)).

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+D=FJAH" 4. Evaluation of the Size of Small Particles Investigations of materials with superfine grains show that a decrease in the crystallite size below some threshold value results in large changes of the properties. Size effects appear if the mean size of crystalline grains does not exceed 100 nm, and manifest themselves most vividly when the grain size is smaller than 10 nm. The particle size is one of the most important parameters determining the peculiarities of the properties and the application field of the nanomaterial. Particles, grains or crystallites of what sizes can be called nanoparticles? Where is the limit behind which we achieve the nanocrystalline state? As a first step, one can accept the conventional division of any substance on the basis of the grain size. On this level of understanding, materials with a mean grain (particle) size greater than 1 µm are referred to as coarse-grained materials. Polycrystalline materials with a mean grain size of 100–150 to 40 nm are usually referred to as submicrocrystalline materials and those with a mean grain size smaller than 40 nm are called nanocrystalline materials. However, this conventional division is not scientifically justified. The physically justified determination of the nanocrystalline state requires deep understanding of the problem. From the physical viewpoint, the transition to the nanocrystalline state is associated with the appearance of size effects on the properties. While there are no size effects, there is no nanocrystalline state. Actually, if the length of the solid in one, two or three dimensions is comparable with or smaller than the characteristic correlation length of a specific physical phenomenon or physical parameter, used in the theoretical description of some property or process, size effects will be detected on the appropriate properties. Such physical parameters as the size of magnetic domains in ferromagnetics, the free path 131

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of the electron, the de Broglie wavelength, the coherence length in superconductors, the wavelength of elastic oscillations, and the size of the exciton in semiconductors may be used for evaluating the transition to the nanocrystalline state. Thus, the size effects refer to a set of phenomena associated with the change in the properties of a substance as a result of a decrease in the particle size and a simultaneous increase in the fraction of the surface contribution to the general properties of the system. The substance or material is referred to as nanocrystalline substance when its particle size is equal to or smaller than some characteristic physical parameters having the dimension of length. Only in this case is it possible to detect the real and not far-fetched changes of the properties, which are determined by the size effects. Now we understand what the nanocrystalline state is from the physical viewpoint. Let us consider how the particle size in a substance can be determined. It should be mentioned that in many cases the size and shape of particles are determined to a large extent by the method of preparing a nanomaterial. 4.1. ELECTRON MICROSCOPY The most widely used method of determining the size of small particles and objects is microscopy. In fact, the invention of the optical microscope already resulted in a large number of inventions in crystallography, medicine, biology and made microscopy very popular and well known. The development of electron microscopy made it possible to examine objects which are considerably smaller than 1 µm and resulted in its extensive application in solid state physics, crystallography, solid state chemistry, materials science and mineralogy. At present, high-resolution electron microscopy [1–3] may be used to examine the distribution of individual atom columns, the distribution of atom columns in crystalline and colloid particles, various crystal lattice defects, the mutual location of the molecules in biological objects, for example, in a DNA helix. Electron microscopy is used for the direct determination of the size of nanoparticles and nanocrystallites. It is used widely in investigations of nanomaterials and in nanotechnologies for advanced electronics. Electron microscopy is the only technique for determining the constitution of the nanoparticles produced from colloid solutions. Figure 4.1 shows a bimetallic colloid particle produced by reduction of HAuCl 4 with tri-sodium citrate. Full-shell clusters are formed, for instance, if palladium acetate is reduced in an acetic acid 132

Evaluation of the Size of Small Particles

Fig. 4.1. A bimetallic colloid, consisting of a gold core (dark area) and a shell of palladium (light area) [4].

Fig. 4.2. HRTEM micrograph of a Ni–TiN nanocomposite particle [5].

solution by gaseous hydrogen in the presence of ligand molecules. The dark core of the particle consists of gold and has a characteristic hexagonal shape, the light shell is formed by palladium atoms; parallel planes of Pd atoms are seen clearly. The particle size is approximately 20 nm. In turn, the palladium shell is protected by ligand molecules but on this image the shell is not visible. Electron microscopy makes it possible to examine the interfaces in nanocomposite materials. In particular, Fig. 4.2 shows a composite nanoparticle Ni–TiN produced by active plasma–metal reaction methods [5], i.e. by evaporation and condensation of nickel 133

Nanocrystalline Materials

and titanium in nitrogen using DC arc plasma. The interface between the particles of nickel and titanium nitride TiN is clearly visible; the size of the Ni particle is 30–50 nm. Figures 4.3 and 4.4 were obtained at a magnification of 1 000 000 times in a JEOL electron microscope. They show clearly bcc nanocrystallites of molybdenum produced by gas phase condensation. The spherical particle of molybdenum is located between eight smaller particles of molybdenum carbide, which form the vertices of the cube (Fig. 4.3). The mean size of the crystallite is 12–15 nm, and the quality of high-resolution electron microscopic image makes it possible to determine interplanar distances. For example, the distance between (110) atomic planes is about 0.22 nm. The application of electron microscopy makes it possible to examine an unusual nanostructure produced as a result of ordering of non-stoichiometric vanadium carbide [6, 7]. At a magnification of 100 times the vanadium carbide powder is made up of large (up to 20 µm) separate irregular-shaped agglomerates which comprise particles of about 1 µm in size. However, at a larger magnification it becomes clear that each 1 µm particle has a complicated structure and actually represents a set of a large number of

Fig. 4.3. BCC nanoparticles of molybdenum Mo (central part) with smaller particles of molybdenum carbide MoC grown on it (image was obtained by W.M. Straub, F. Phillipp and H.-E. Schaefer, Institut für Theoretische und Angewandte Physik, Universität Stuttgart, 1994). 134

Evaluation of the Size of Small Particles

Fig. 4.4. BCC Mo nanoparticles produced by gas condensation method (image was obtained by W.M. Straub, F. Phillipp and H.-E. Schaefer, Institut für Theoretische und Angewandte Physik, Universität Stuttgart, 1994).

nanocrystallites. The SEM image of vanadium carbide nanopowder (Fig. 4.5) was obtained at a magnification of up to 30000 times in a DSM 982 Gemini high-resolution scanning electron microscope. It is seen that the vanadium carbide nanocrystallites have the shape of distorted lobes, which grow into each other and form a nanostructure resembling corals. Figure 4.6 shows a nanoparticle of titanium carbide Ti 44 C 56 powder produced by ball milling metallic titanium and carbon, which was taken at a 44:56 ratio [8]. The size of the nanoparticle is 12– 18 nm, the distance between the atomic planes is 0.25 nm and corresponds to the (111) B1 interplanar spacing of the cubic titanium carbides close to the stoichiometric composition TiC 0.9–1.0 . With all advantages of electron microscopy as a method of determination of the particle size, it must be taken into account that it is a local method and provides information on the size of the object only in the field of observation. However, the observed area may not be representative, i. e. not characteristic of the entire volume of the substance. Let us consider a simple example. Electron microscopy is used 135

Nanocrystalline Materials

Fig. 4.5. Nanocrystallites of the powder of ordered vanadium carbide V 8C 7 (VC 0.875), magnification up to 30 000 [6].

Fig. 4.6. HRTEM image of mechanically alloyed Ti 44 C 56 powder after 720 ks of ball-milling [8].

to examine a section with a size of 10×10 µm (i.e. 10 –10 m 2 ) of a specimen with a surface area of 1 cm 2 . In this case, the examined area contains 0.000001 part of the entire surface of the specimen. Is this much or little? The surface area of our planet is equal to approximately 510 million km 2, and a millionth part is 510 km 2. This 136

Evaluation of the Size of Small Particles

small area of the earth surface may be covered with water (ocean, sea or large lakes) or ice (Antarctic, Greenland, and large mountain glaciers), may be a desert or marsh, may be overgrown with forests, etc. It is clear that examination of only one millionth part of the earth surface is insufficient for making an accurate conclusion on the structure of the entire planet surface. An adequate representation may be obtained only by examining several sections. Therefore, electron microscopy examination must be carried out on several areas in order to obtain statistically averaged out information for the entire substance. Electron microscopy is the only direct method of determining the size of very small particles. All other methods are indirect, because the information on the mean particle size is extracted from the data on the variation of some property of the substance or process parameter. Indirect methods include diffraction, magnetic, sedimentation and gas-adsorption methods. 4.2. DIFFRACTION The diffraction method occupies the main position amongst indirect methods for determining the particle size. At the same time, this method is simplest and accessible because the X-ray examination of the structure is used widely and efficient equipment is available. In diffraction experiments on powder or polycrystalline specimens, the defects of a structure are studied on the basis of the broadening of diffraction reflections. However, in the application of this method in practice, one often compares the width of the diffraction reflections from a coarse-grained substance and from the same substance in the nanocrystalline state. This determination of the broadening and the subsequent evaluation of the mean particle size sometimes are not accurate and could be characterised by a very large error (up to several hundreds of percent). The point is that the broadening should be determined in relation to diffraction reflections from an infinitely large crystal. It means that the measured width of the diffraction reflection should be compared with the instrumental width, i.e. with the width of the resolution function of the diffractometer. This function should be determined in a special diffraction experiment. In addition to this, the accurate determination of the width of diffraction reflections is possible only by means of computational description of the shape of the experimental reflection. This is very important because there may also be other (in addition to the small crystallite size) physical 137

Nanocrystalline Materials

reasons for the broadening of diffraction reflections. Therefore, it is important to determine not only the broadening but also to separate the different contributions to the broadening, i. e. to find the values of these contributions. Since the diffraction method of determination of the particle size is most widely used and is accessible, special features of this application will be discussed in detail here. The diffraction method is based on the effect of broadening of diffraction reflections associated with the size of the particles (crystallites). All types of defects cause displacement of the atoms from the lattice sites. The author of [9] derived an equation for the intensity of Bragg reflections from a crystal defect, which enabled all the defects to be divided conventionally into two groups. The defects in the first group only lower the intensity of diffraction reflections but do not cause reflection broadening. The broadening of reflections is caused by defects of the second group. These defects are microdeformations, inhomogeneity (non-uniform composition of the substance over their volume) and the small particle size. The broadening, caused by microdeformations and randomly distributed dislocations, depends on the order of reflection and is proportional to tan θ , where θ is the scattering angle in the diffraction experiment. In substitutional solid solutions A 1–x B x and in non-stoichiometric interstitial compounds MX y there is another reason for broadening, which is called inhomogeneity. Inhomogeneity causes a variation of the composition over the volume of the specimen [10, 11]. The broadening, caused by inhomogeneity ∆x, is proportional to (sin 2θ )/cos θ [10–12]. In the case of nanocrystalline substances, the broadening associated with a small size D of crystallites (D < 200 nm) is most interesting, and in this case the magnitude of broadening is proportional to sec θ . The broadening, caused by the inhomogeneity or small particle size, is independent of the order of reflection. Let us consider the derivation of the equation, which takes into account the diffraction reflection broadening, caused by the finite size of the particles of a polycrystalline substance. Let V be the height of the column of planes of coherence scattering averaged out with respect to volume, while is the particle diameter averaged out with respect to volume. For spherical particles, integration leads to the equation: = 4 V /3 .

(4.1)

138

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Let us consider the scattering vector s = 2sin θ / λ , where λ is the radiation wavelength. Mathematically, its differential (or indeterminacy from the physical viewpoint, because in a finite crystal the wave vector becomes a poor quantum number) is equal ds = 2cos θ d θ / λ = cos θ d(2 θ )/ λ .

(4.2)

In this equation, quantity d(2 θ ) is the integral width of the diffraction reflection expressed in angle 2 θ and measured in radians. The integral width is determined as the integral intensity of the reflection divided by reflection height and does not depend on the shape of the diffraction reflection. Therefore, it is possible to use the integral width for analysis of X-ray, synchrotron or neutron diffraction experiments, carried out by different techniques with differing resolution functions of the diffractometer and in different angle ranges. The indeterminacy of the scattering vector, ds, is inversely proportional to the V . Therefore, the product of these quantities is equal to unity V × ds = 1. It is clear from this relationship that the indeterminacy ds is equal to zero at the infinite height of the column (i.e. the infinitely large size of the crystals). If the height of the column is small and tends to zero, indeterminacy ds of the wave vector and, consequently, width d(2 θ ) of the diffraction reflection become very large. We shall assume that all grains are spherical. Considering equations V = 1/ds, (4.1) and (4.2), the mean grain size for the diffraction reflection of an arbitrary shape may be determined as =1.33 λ /(cos θ d(2 θ ))

(4.3)

where d(2 θ ) is the integral width of the diffraction reflection. This width is often replaced by the full width at half maximum (FWHM). The relationship between the integral width of reflection and FWHM depends on the shape of the experimental diffraction reflection and must be determined in every specific case. For the reflections in the form of a rectangle and a triangle, the integral width of the reflection is equal to FWHM exactly. For the Lorentz and Gauss functions, this relationship has the form d(2 θ ) L ≈ 1.6×FWHM L (2 θ ) and d(2 θ ) G ≈ 1.1×FWHM G (2 θ ), respectively. For the pseudo-Voigt function, which will be examined later, this relationship is more complicated and depends on the ratio of the Gauss and Lorentz contributions. For diffraction reflections in small 139

Nanocrystalline Materials

angles, the relationship between the integral broadening and FWHM may be assumed to equal to d(2 θ ) ≈ 1.47×FWHM(2 θ ). Substituting this relationship into (4.3), we have the Debye formula [13]: = 0.9 λ /[(cos θ ) × FWHM(2 θ )] .

(4.4)

In a general case, when the particles of the substance have an arbitrary shape, the mean particle size can be determined by the Debye–Scherrer formula: = K hkl λ /[(cos θ ) × FWHM(2 θ )] ≡ K hkl λ /[(2cos θ )×FWHM( θ )] (4.5) where K hkl is the Scherrer’s constant whose value depends on the shape of the particle (crystallite, domain) and on diffraction reflection indices (hkl). In a experiment as a result of the finite resolution of the diffractometer, the reflection broadens and cannot be smaller than the instrumental width of reflection. In other words, in equation (4.5) the width FWHM(2 θ ) of the reflection should be replaced by broadening of the reflection, β , in relation to the instrumental width. Therefore, in the diffraction experiment, the mean particle size is determined using Warren’s method [14–16]: = K hkl λ /[(cos θ )× β (2 θ )] ≡ K hkl λ /[(2cos θ )× β ( θ )]

(4.6)

where β = (FWHM exp )2 − (FWHM R ) 2 is the broadening of the diffraction reflection. It should be noted that β (2 θ ) ≡ 2 β ( θ ). The full width at half maximum FWHM R or the instrument width of reflection may be measured on an efficiently annealed and completely homogeneous substance (powder) with a particle size of about 1–10 µm. In other words, the reference should be represented by a reflection without any additional broadening, with the exception of instrument broadening. If the resolution function of the diffractometer is described by the Gauss function, and θ R is its second momentum, then FWHM R = 2.235 θ R . The diffraction reflections are described by the Gauss function g( θ ) and Lorentz function l ( θ )

140

Evaluation of the Size of Small Particles

g (θ ) = A exp[−(θ − θ 0 ) 2 /(2θ G2 )] ,

(4.7)

l(θ ) = A[1 + (θ − θ 0 ) 2 / θ L2 ] −1

(4.8)

or by their superposition V( θ ) = cl( θ ) + (1 – c)g( θ ), i.e. by the pseudo-Voigt function

V (θ ) = cA[1 + (θ − θ 0 ) 2 / θ L2 ] −1 + (1 − c) A exp[−(θ − θ 0 ) 2 /(2θ G2 )] , (4.9) where c is the relative contribution of the Lorentz function to the total reflection intensity; θ L and θ G are the parameters of the Lorentz and Gauss distributions, respectively; A is the normalising factor. Let us consider special features of the Gauss and Lorentz distributions required for further analysis. Parameter θ G is the second momentum of the Gauss function. The second momentum θ G , expressed in angles θ , is associated with the full width at half maximum, measured in angles 2 θ , by the well-known relationship θ G ( θ ) = [FWHM(2 θ )]/(2×2.355). This relationship can easily be derived directly from the Gauss distribution. Figure 4.7a shows the Gauss distribution described by the function

g (θ ) = hG exp[−(θ − θ 0 ) 2 /(2θ G2 )] ,

(4.10)

where θ G is the second momentum of the Gauss function, i.e. the value of the argument corresponding to the inflection point of the function where ∂ 2 g( θ )/∂ θ 2 = 0. If the value of function (4.10) is equal to to half its height, i.e., h G exp [–( θ – θ 0 ) 2 /(2 θ G2 )] = h G /2, then exp [–( θ – θ 0 ) 2 /(2 θ G2 )] = 1/2. From this it follows that ( θ – θ 0 ) = θ G 2 ln 2 . As can be seen from Fig. 4.7a, the Gauss function full width at half height is FWHM G = 2( θ – θ 0 ) = 2θG 2 ln 2 = 2.355 θ G . Parameter θ L of the Lorentz distribution coincides with the half width of this function at half height. Let it be that the Lorentz function

141

Nanocrystalline Materials a

b

g(θ ) = hG exp[ -(θ - θ 0)2/ 2θ G2 ]


(θ ) = hL[1 + (θ - θ0)2/θ L22] -1

FWHML = 2θ L θ L /√3

(θ)

g(θ )

___ FWHMG = 2θ G√2ln2

θG

FWHM L

FWHMG

hL /2

hG /2

θ

θ0

θ0

θ

Fig. 4.7. Gauss g( θ ) and Lorentz l( θ ) distributions, used for model description of the shape of the diffraction lines: (a) relationship between the second momentum θ G of the Gauss function and the full width of the function g( θ ) at half of its maximum, FWHM G ; (b) for the Lorentz function parameter θ L = FWHM L /2, and the second momentum of the function l( θ ) is θ L / 3.

 (θ − θ 0 ) 2  l (θ ) = hL 1 +  θ L2  

−1

(4.11)

has the value, which is equal to half of the height, i.e. l( θ ) = h L /2 (Fig.4.7b). The value of the argument, which corresponds to this value of the Lorentz function, is determined from the  (θ − θ 0 )2   equation 1 + θ L2  

−1

=

(θ − θ 0 ) 2 1 = 1 and . It then follows that θ L2 2

θ = θ 0 + θ L . Thus, for the Lorentz function FWHM L = 2× θ L . The second momentum of the Lorentz function, i. e. the value of the argument, corresponding to the inflection point of the function, can be determined from the condition ∂ 2l( θ )/∂ θ 2 = 0. Calculation shows that the second momentum of the Lorentz function is equal to θ L /√3. The pseudo-Voigt function (4.9) ensures the best description of experimental diffraction reflection in comparison with the Gauss and Lorentz functions. Taking this into account, the resolution function of the diffractometer R( θ ) is represented as a pseudo-Voigt function; it will be assumed for simplicity that A = 1 in equation (4.9). Then we have

142

Evaluation of the Size of Small Particles

R (θ ) = c[1 + (θ − θ 0 ) 2 / θ L2 ] −1 + (1 − c) exp[−(θ − θ 0 ) 2 /( 2θ G2 )] . (4.12) The resolution function is the superposition of the Lorentz and Gauss functions. As the zero approximation, the width of the resolution function may be approximated as FWHM RpV = cFWHM L + (1 – c)FWHM G = 2c θ L + 2.235(1 – c) θ G . (4.13)

If θ L = θ G ≡ θ R , then FWHM RpV = 2(1.1775 – 0.1775c) θ R . Let some effective Gauss function G eff ( θ ), the area under which coincides with that of the pseudo-Voigt function, have the width FWHM, equal to FWHM RpV . Then the second momentum of this function is θ eff G = (1 – 0.15c) θ R . Thus, the pseudo-Voigt resolution function R( θ ) and the effective Gauss function G eff ( θ ) are equivalent with respect to half width. To a zero approximation, this makes it possible to replace function (4.12) by the function 2 2 2 Reff (θ ) = exp[−(θ − θ 0 ) 2 /( 2θ eff G )] ≡ exp[−(θ − θ 0 ) /( 2θ R * )] , (4.14)

where

θ R* ≡ θ eff G = (1 – 0.15c) θ R

(4.15)

on the condition that θ L = θ G ≡ θ R . The experimental function I( θ ), describing the shape of an arbitrary diffraction reflection, is a convolution of the distribution function g( θ ) and of the resolution function R eff ( θ ) (4.14), i. e.

I (θ ) =

+∞

∫ Reff (θ '−θ ) g (θ ' )dθ ' =

−∞

+∞

= I ∫ exp[−(θ '−θ + θ 0 ) 2 /( 2θ R2 * )] exp[−(θ '−θ + θ 0 ) 2 /( 2θ x2 )]dθ '= −∞

{

}

2 2 = [I / 2π (θ R* + θ x2 ) ] exp − (θ − θ 0 ) 2 / [2(θ R* + θ x2 )] .  

143

(4.16)

Nanocrystalline Materials

Equation (4.16) shows that the second momentum θ exp of the 2 = θ R2 * + θ x2 or, with allowance for experimental function is θ exp (4.15), 2 θ exp = [(1 − 0.15c)θ R ] 2 + θ x2 .

(4.17)

Broadening β of the diffraction reflection is expressed by means of the full width of reflection at half height FWHM exp as β = (FWHM exp ) 2 − (FWHM R* ) 2 . If the second momenta and the full

width are expressed in the same units (in angles θ or in angles 2 θ ), then FWHM exp, R = 2.235 θ exp,R and the broadening of reflection (hkl) is 2 2 2 β = 2.235 θ exp − θ R* = 2.235 θ exp − [(1 − 0.15c )θ R ]2 .

(4.18)

As already mentioned, the values of the broadening, caused by the small grain size, deformations and inhomogeneity are proportional to sec θ , tan θ and (sin 2 θ )/cos θ , respectively. Three different types of broadening can be separated because of the different angular dependence. It is worth noting that the size of regions of coherent scattering, determined from size broadening, may correspond to the size of the individual particles (crystallites), but may also reflect the subdomain structure and characterise the mean distance between the stacking faults or effective size of mosaic blocks, etc. In addition to this, it is important to point out that the shape of diffraction reflection depends not only on the size but also on the shape of nanoparticles [17, 18]. The separation of the broadening, caused by several different factors, will be examined on the example of nanostructured carbide solid solutions of the ZrC–NbC systems. In X-ray investigation of these solid solutions it was established that diffraction reflections on the X-ray diffraction pattern of a (ZrC) 0.46 (NbC) 0.54 specimen are greatly broadened. It is well known [19] that these solid solutions have a tendency to decomposition in the solid state. However, according to the X-ray data this specimen consists of a single phase. To explain the reason for the broadening of reflections (inhomogeneity, small grain size or microdeformations), the quantitative analysis of the profile of diffraction reflections was carried out using the pseudo-Voigt function (4.9). Analysis shows 144

Evaluation of the Size of Small Particles

that the width of all diffraction reflections greatly exceeds the width of the resolution function of the diffractometer. Broadening β h , caused by the inhomogeneity ∆x of the solid solution A 1–x B x , whose composition changes in the range x ± ∆x, is proportional to (sin 2 θ )/cos θ [10]:

sin 2θ ∆ x [degree] πλ h 2 + k 2 + l 2 cosθ 360

β h ( θ ) = 2.235 a' B1( x ) x = x

0

(4.19) or, measured in radians, is equal to

β h ( θ ) = 2.235 a' B1( x ) x = x

0

sin 2θ ∆ x [rad], λ h 2 + k 2 + l 2 cosθ 2

(4.20)

where a'B1 (x)≡da(x)/dx is the derivative of the concentration dependence of the lattice constant of the solid solution with respect to composition x. The broadening β s(2 θ ) ≡ 2 β s ( θ ), caused by the small particle size (crystallite, domain) is associated with the mean particle size ~ V 1/3 (V is the volume of the particle) by equation (4.6) which can be written in the form [9, 20]:

β s (θ ) =

K hkl λ 2 < D > cosθ

[rad]

or

β s (θ ) =

90 K hkl λ π < D > cosθ

[degree]. (4.21)

In a cubic crystal lattice, the crystallites have the size of the same order in three perpendicular directions. In this case, Scherrer ’s constant K hkl for reflections with different crystallographic Miller indices (hkl) of a cubic crystal lattice, may be calculated from the equation [21]:

K hkl =

6h

3

(h 2 + k 2 + l 2 )1 / 2 (6h 2 − 2 hk + kl − 2 hl )

.

(4.22)

Values of K hkl for reflections (hkl) of fcc lattice are presented in Table 4.1.

145

Nanocrystalline Materials Table 4.1 Scherrer’s constant K hkl for the diffraction reflections (hkl) from crystals with fcc lattice (hkl)

(111)

(200)

(220)

(311)

(222)

(400)

Khkl

1.1547

1.0000

1.0606

1.1359

1.1547

1.0000

(333)

(511)

(hkl)

(331)

(420)

(422)

Khkl

1.1262

1.0733

1.1527

1.1547 1.1088 for (333) and (511) reflections mean value is Khkl = 1.2113

(440) 1.0607

Deformation distortions and induced non-uniform displacements of the atoms from the lattice sites may form in the case of random distribution of the dislocations in the volume of the specimen. In this case, the atom displacements are determined by the superposition of displacements from every dislocation. This may be regarded as a local variation of interplanar distances. In other words, the distance between the planes continuously changes from d 0 – ∆d to d 0 + ∆d (d 0 and ∆d are the interplanar distance in the ideal crystal and the mean variation of the distance between the planes (hkl) in the volume V of the crystal, respectively). In this case, the quantity ε = ∆d/d 0 is the lattice microstrain which characterises the uniform strain averaged out with respect to the crystal. The diffraction maximum from the regions of the crystal with the changed interplanar distance appears at angle θ , which slightly differs from angle θ 0 for the ideal crystal. Therefore, the reflection broadens. The equation for the reflection broadening, caused by the microdeformations of the lattice, can be easily derived by differentiation of the Wullf–Bragg equation d = n λ /(2sin θ ); the result is ∆d/∆ θ =–(n λ /2)×(cos θ /sin 2 θ ) = –d/tan θ . The reflection broadening to one side of the maximum of the reflection, corresponding to the interplanar distance d, with the variation of the interplanar distance by +∆d is ∆ θ = –(∆d/d)×tan θ , and in the case of variation by –∆d, it is ∆ θ = (∆d/d)×tan θ . The total reflection broadening is equal to the sum of these broadenings and its value with respect to angle θ is:

β d ( θ ) = (2∆d/d)×tan θ = 2 ε tg θ [rad] β d ( θ ) = (360/ π ) ε tan θ [degree].

or (4.23)

Taking into account β d (2 θ ) ≡ 2 β d ( θ ), the strain broadening with respect to angle 2 θ is [9]:

146

Evaluation of the Size of Small Particles

β d (2 θ ) = 4 ε tan θ .

(4.24)

It should be mentioned that in (4.19)–(4.21) and (4.23), the broadening β is determined in degrees θ , and not 2 θ . The full width FWHM exp and the second momentum θ exp of every experimental diffraction reflection of the solid solution (ZrC) 0.46 (NbC) 0.54 are determined by approximation of the reflections by functions (4.9), where A = 1. Broadening β of the (hkl) reflections is determined from equation (4.18). The angular dependence of the second momentum θ R ( θ ) (Fig. 4.8a) of the resolution function of the X-ray diffractometer was determined in special experiments on annealed coarse-grained compounds with no homogeneity interval: a single crystal of hexagonal silicon carbide 6H–SiC, and on stoichiometric tungsten carbide WC. Large grain size, absence of strain distortions and the homogeneity of the composition of these specimens prevented broadening of the reflections. Comparison of the obtained values of θ exp ( θ ) with the resolution function θ R( θ ) made it possible to determine the angular dependence β exp ( θ ) of the experimental reflection broadening. Numerical analysis shows that the dependences, which are characteristic for the broadening associated with the inhomogeneity and with the size of the crystallites, are closest to the linearity, whereas there is no strain broadening (and randomly distributed dislocations) in the investigated specimens. Thus, it may be assumed that the observed broadening of the diffraction reflections is a superposition of only two factors β h and β s . Taking this into account, Fig.4.8 shows the dependences of the experimental broadening β exp ( θ ) of the diffraction reflections of (ZrC) 0.46 (NbC) 0.54 specimen on (sin 2θ ) / ( h 2 + k 2 + l 2 cosθ ) and K hkl sec θ . Since the observed broadening is the result of two independent mechanisms, it may be determined as a convolution of two functions. The first function corresponds to the broadening caused by inhomogeneity and the second function corresponds to the broadening caused by the small grain size. In this case, the total broadening can be determined from the equation

β = βh2 + βs2 .

(4.25)

Calculation shows that the separated contributions of the broadening β h , caused by inhomogeneity, and size broadening β s are

147

Nanocrystalline Materials

0.12

a

βexp

b

___

β (θ ) (degrees)

θ R (degrees)

0.14

θR = [(0.0052tg2θ + 0.0056)1/2] / (4√2ln2)

0.10 0.08 0.06 0.04

0.4

βh

0.3 0.2 0.1

0.02 20

40

60

80

0.1

θ (degrees) βexp

0.3 0.2

βs

0.1

1

0.3

d

β (θ ) (degrees)

β (θ ) (degrees)

c 0.4

0.2

(sin2θ )/[(h2+k2+l 2)1/2cosθ ]

2

0.4 0.3 0.2

βexp β = (βh2 + βs2)1/2 βh βs

0.1

10

20

30

40

50

60

θ (degrees)

Khklsecθ

Fig. 4.8. The second momentum θR( θ) of the resolution function (a) of the diffractometer and the separation of contributions to the broadening of diffraction reflections of the solid solutions (ZrC) 0.46(NbC) 0.54 : (b) dependence of the experimental broadening β exp on (sin 2 θ )/[h 2 +k 2 +l 2 ) 1/2cos θ ] and the separated contribution of inhomogeneous broadening βh; (c) dependence of experimental broadening βexp on Khklsecθ and separated contributions of size broadening β s ; (d) dependences of experimental β exp , size β s and inhomogeneous β h broadening on the diffraction angle θ and the approximation of the experimental broadening by superposition of the broadenings β h and β s in the form β = [( β h ) 2 +( β s ) 2 ] 1/2 . 2 2 2 2 linear functions of (sin θ ) /( h + k + l cosθ ) and K hkl sec θ , respectively (Fig. 4.8a, 4.8b). This confirms the presence of exactly these contributions. Figure 4.8c shows the angular dependences of the determined broadening β h ( θ ) and β s ( θ ) and approximation of the experimental broadening β ( θ ) by their superposition. The mean crystallite size and the degree of inhomogeneity ∆x of the solid solution (ZrC) 0.46 (NbC) 0.54 were calculated using equations (4.19) and (4.21) and the obtained values of broadening. According to calculation results, is equal to 70±10 nm and ∆x = 0.041. The calculation results show that the annealed solid solution

148

Evaluation of the Size of Small Particles

(ZrC) 0.46 (NbC) 0.54 is inhomogeneous and has a nanostructure. The grains differing in the content of zirconium and niobium and corresponding to the isostructural cubic phases of different composition, have a size of approximately 70 nm. This nanostructure is a decomposed solid solution with coherent precipitates of two isostructural cubic phases (ZrC) 0.42 (NbC) 0.58 and (ZrC) 0.50 (NbC) 0.50 with similar composition. Indeed, the diffraction reflections of the solid solution (ZrC) 0.46 (NbC) 0.54 are efficiently described by the superposition of two contributions, corresponding to the phase (ZrC) 0.42 (NbC) 0.58 and (ZrC) 0.50 (NbC) 0.50 . Thus, the diffraction study, in addition to obtaining standard information on the crystal structure, makes it possible to determine fine details of the microstructure. On the whole, the sequence of the diffraction experiments in determination of the grain (crystallite) size is the following: (1) measuring of the diffraction pattern of the reference substance, computational analysis of the profile of diffraction reflections, determination of the width, construction of the experimental dependence of the width of reflections on the diffraction angle and calculation of the approximating resolution function of diffractometer; (2) measuring of the diffraction pattern of the investigated substance, computational analysis of the profile of diffraction reflections and determination of the width of reflections; (3) determination of the reflection broadening for the investigated substance as a function of the diffraction angle; (4) separation of the contributions to the broadening, determined by the small particle size, inhomogeneity of the investigated substance, deformation distortions of the crystal lattice; (5) determination of the mean grain (particle, crystallite) size. Using the Wullf–Bragg condition, it may easily be shown that ∆d hkl /d hkl = –[cotan( θ hkl )]∆ θ ; on the other hand, ∆d hkl /d hkl = ∆ λ / λ , and therefore ∆ λ / λ = –[cotan( θ )]∆ θ . It then follows that the broadening β hkl (2 θ ) = 2(∆ λ / λ )tan θ hkl . We shall assume that the smallest reflection width is equal to the spectral width ∆ λ / λ ≈ 10 –3 of the reflection. Then, from the expression obtained it is easy to find the largest size of the crystallites which cause a measurable reflection broadening. In this case, the broadening β hkl = 2×10 –3 tan θ hkl and = λ /( β hkl cos θ hkl )=10 3 λ /[(2tan θ hkl )× (cos θ hkl )] = 10 3 l/(2sin θ hkl ) = 10 3 d hkl ≈ 200–300 nm. Thus, the diffraction method the particle size smaller than 300 nm to be determined. For more precise measurement of the width of the reflection it 149

Nanocrystalline Materials

is much more efficient to use the filtered-off radiation with a wavelength of λ 1 , because in this case the shape of the diffraction reflection is symmetric and simple for analysis because the reflection does not consist of a doublet. Figure 4.9 shows diffraction reflections (111) from coarsegrained titanium carbide TiC 0.62 with a grain size of approximately 5 µm and from the same titanium carbide subjected to severe plastic deformation by pressure torsion [22, 23]. The broadening of reflection indicates refining of the grain size. If the deformation contribution to broadening is neglected, the grain size in the deformed titanium carbide is equal to approximately 40 nm. Figure 4.10 shows X-ray diffraction patterns of metallic nickel with a grain size of approximately 2–10 µm and compacted nanocrystalline nickel with a grain size of less than 20 nm. The width of the reflections for coarse-grained nickel coincides almost completely to the smallest width that be measured in the diffractometer; the low intensity of reflection (111) is determined by the presence of a texture. It may be seen that the transition to the nanocrystalline state leads to a large broadening of the diffraction reflections.

 

7L& 

&RXQWV DUELWUDU\ XQLWV

QDQR7L& 

):+0 7L&



):+03,34 7L&













θ GHJUHHV

Fig. 4.9. Broadening of the diffraction reflection (111) B1 of the nanocrystalline titanium carbide TiC 0.62 produced by severe plastic deformation of the coarsegrained (with grain size ~5 µm) carbide TiC 0.62 [22, 23]: reflections of nanocrystalline and coarse-grained titanium carbide TiC 0.62 are indicated by the solid and broken lines, respectively; FWHM is the full width of reflection at half maximum. The grain size and the nanocrystalline carbide TiC 0.62 is 30–50 nm (radiation CuK α 1 ). 150

Evaluation of the Size of Small Particles 



FRDUVHJUDLQHG 1L

&RXQWV DUELWUDU\ XQLWV

QDQRFU\VWDOOLQH 1L





















θ GHJUHHV

Fig. 4.10. X-ray diffraction patterns of coarse-grained bulk Ni with a grain size of about 2–10 µm and compacted nanocrystalline Ni with a grain size of D < 20 nm. Low intensity of reflection (111) of coarse-grained Ni is caused by the presence of a texture. Diffraction reflections of nanocrystalline Ni are broadened strongly

The method of evaluation of the mean grain size from the broadening of diffraction reflections may be used for both bulk and powdered nanocrystalline materials. 4.3. SUPERPARAMAGNETISM, SEDIMENTATION, PHOTON CORRELATION SPECTROSCOPY AND GAS ADSORPTION The size of nanoparticles can be measured by magnetic measurements. In this case, the determination of the particle size is associated with the superparamagnetism effect. Ferromagnetic materials have a domain structure, which is energy-favourable because of the closure of magnetic fluxes inside the ferromagnetic. With a decrease of the size of the ferromagnetic particle, the closure of the magnetic fluxes inside the ferromagnetic is less advantageous from the energy viewpoint. When reaching some critical size D c , every particle contains only one domain and a further decrease in the size of the particles decreases the coercive force to zero as a result of transition to the superparamagnetic state. For typical ferromagnetics with a Curie temperature of 500– 1000 K, the disappearance of ferromagnetism and transition to the superparamagnetic state are possible when the particles become smaller than 2–20 nm [24]. 151

Nanocrystalline Materials

Superparamagnetism is interesting not only as a specific magnetic phenomenon but also as a non-destructive method of determination of the sizes, shape, concentration and composition of the particles of the precipitated magnetic phase, and also the size distribution of the particles. A relationship of superparamagnetism with the size of ferromagnetic particles is discussed in more detail in Section 5.4 where the magnetic properties of isolated nanoparticles are analysed. A suitable example of the application of magnetic measurements for determining the size of the nanoparticles are investigations [25– 28] in which copper with iron nanoparticles was studied taking into account the superparamagnetism effect. The starting copper was diamagnetic and contained a very small quantity of dissolved iron. The temperature and annealing dependences of the magnetic susceptibility of starting copper and of copper subjected to severe plastic deformation were measured. Analysis of these dependences shows that severe plastic deformation leads to the precipitation of iron particles previously dissolved in copper. The results show that the mean volume and size of the precipitated superparamagnetic particle of iron are ~1.8×10 –20 cm 3 and ~3 nm, respectively. The grain size of the produced submicrocrystalline copper smc-Cu is 130–150 nm. Determination of the particle size by sedimentation is based on measuring the time during which a particle travels the fixed distance S in a liquid medium with known viscosity η . Viscosity determines the force of resistance of the liquid (or gas) to the movement of a body. At a low rate of movement V = S/t and a small size of the body, the force of resistance to movement of the body with a mean size R is described by the well-known Stokes equation F = (4 π / α ) η RV = (4 π / α ) η RS/t

(4.26)

or (by a means of the volume v ~ R 3 ) F = (4 π / α ) η v 1/3 S/t .

(4.27)

In a general case, coefficient α depends on the shape of the body and has different values. For spherical bodies α = 2/3. For a steady rate of movement, the falling body is affected, in addition to the resistance force, by the force of gravity P = mg = v ρ g, where g is the acceleration of gravity, and the mass m of the body is equal to the product of the volume v by density ρ . To improve the 152

Evaluation of the Size of Small Particles

accuracy of calculations, a correction for the injecting Archimedes force is introduced. Then, the force causing the body to fall in the given medium (liquid or gas) is: F d = P – v ρ m = vg( ρ – ρ m ) ,

(4.28)

where ρ m is the density of the medium. From the equality of the force F d and the force of resistance to fall, F, it is easy to find the volume and averaged out size of the falling body: (4 π / α ) η v 1/3 S/t = vg( ρ – ρ m ) .

(4.29)

It then follows that

 (4π / α )ηS  v=   g (ρ − ρ m ) t 

3/ 2

(4.30)

and R~

(4π / α )ηS ~ k /t . g (ρ − ρ m ) t

(4.31)

For spherical particle F = 6 πη RS/t and F d = vg( ρ – ρ m ) = (4/3) π R 3 g( ρ – ρ m). Therefore, it follows from equality F = F d that

Rsph =

9ηS 2g (ρ − ρ m ) t .

(4.32)

Thus, the size of the falling particle in the sedimentation method is inversely proportional to the time during which the particle travels a fixed path. In currently available sedimentographs, the travel of the particles and their number is recorded by means of laser radiation. This technique is used to determine not only the size of separate particles but also the size distribution of the particles, i.e. the dispersion of the particle size. A disadvantage of the sedimentation method is that it cannot be used to measure the size of very small (less than 50 nm) particles. In addition to this, in order to obtain high precision of measurements, the liquid must efficiently 153

Nanocrystalline Materials

wet the surface of the particles. In the case of poor wetting, the surface tension force will maintain the small particles on the surface, and a gas shell will surround the large particles. During measurements it is also necessary to avoid coagulation of the separate particles which may greatly distort the results. Detailed description of sedimentation methods is given in monograph [29]. An efficient method of determination of the sizes of small particles is based on using of Brownian movement [30], on the one hand, and analysis of the spectral composition of the light scattered by the suspension or a colloid solution, on the other hand. A. Einstein invented this method of determination of the size of small particles. In fact, his first article [31] from a serious of studies into the theory of Brownian movement [31–34] is referred to as “Eine neue Bestimmung der Moleküldimensionen” (“New determination of the size of molecules”). It is interesting to note that this work was in fact an Einstein’s dissertation for the title of Doctor of philosophy. The study was presented in 1905 at the Natural Mathematics Section of the Higher Faculty of Philosophy of the Zurich University. Almost independently and at the same time as Einstein, the theory of Brownian motion was developed by Polish scientist M. Smoluchowski [35]. The size of small particles could be measured, using the wellknown Einstein equation, describing the Brownian motion of spherical particles in a liquid: R = k B T/(6 πη D diff ),

(4.33)

where h is the liquid shear viscosity, D diff is the diffusion coefficient of Brownian particles. At a known viscosity of the liquid, to determine radius R of the particles, it is necessary to know the diffusion coefficient D diff . The diffusion coefficient of Brownian particles is determined measuring the full width of a non-displaced (central) component in the spectrum of scattered light using an optical displacement spectrometer [36]: FWHM = 2D diff K 2 ,

(4.34)

where FWHM is the spectral line full width at half maximum, K is the variation of wave vector during light scattering. This method of determination of the size of small particles is known as photon correlation spectroscopy (PCS) [36–40]. Photon correlation spectroscopy is based on analysis of the 154

Evaluation of the Size of Small Particles

spectral composition of the light, scattered by the examined specimen. Photon correlation spectroscopy is used widely for measuring the sizes of submicrocrystalline and nanosized particles in transparent suspensions and colloid systems. Measurements are carried out using the radiation of a He–Ne laser with a wavelength of λ = 633 nm, laser power up to 10 mW. Light radiation is focused in the centre of a cuvette containing a liquid with a specimen (suspension). The cuvette is placed in a thermostat for temperature stabilisation. The light is scattered under the given angle θ and is received by a photoelectric multiplier. A multichannel correlator then measures the autocorrelation function G( τ ) of scattered light. Approximation of the autocorrelation function by the exponential dependence makes it possible to determine the width of the spectral line and then find the diffusion coefficient and the particle size. An increase in radiation power results in an improvement of the accuracy and accelerates the measurements. Industrial analysers of the particle size, using the PCS method, take measurements under several angles and make it possible to determine the size of the particles in the range from 3 nm to 3µm and also the size distribution of the particles. To determine the size of small particles in colloid solutions and suspensions, it is also efficient to use the mass distribution of particles using an ultracentrifuge. The method of separation of small particles by a means of a centrifugal force in an ultracentrifuge was developed the Swedish scientist T. Svedberg [41]. If the particle density is known, measuring the mass of the particle it is easy to find its volume and linear size. The gas adsorption method is used to determine the specific surface of the powder. The measured value of the specific surface may be used to estimate the mean particle size. To measure the specific surface, helium is passed through the prepared powder in a special chamber. Helium atoms are adsorbed by the particle surface. Helium-saturated powder is heated to remove the entire amount of adsorbed helium and the amount of adsorbed helium is determined from the variation of the mass. Assuming that helium atoms form a monolayer on the surface of the particles, from the volume of the adsorbed gas it is possible to determine the total surface area of the particles and the specific surface of the powder (in the units of area per units of mass, for example, m 2 g –1 ). If it is assumed that the size and shape of all the particles are the same, a simple relationship can be derived between the specific surface S sp and the linear particle size R. For example, if 155

Nanocrystalline Materials

the mass of a spherical particle is m = (4 π /3)R 3 ρ and the surface area s = 4 π R 2 , then its specific surface is S sp = s/m = 3/ ρ R .

(4.35)

Thus, as the specific surface increases, the particle size decreases. If a function describing the size distribution of the particles is available, then from the measured specific surface it is possible to determine the particle size, taking the size distribution function into account. The described methods of determination of the size of nanoparticles are used most widely.

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

7.

8.

9. 10.

11.

12.

G. Tomas, M. J. Goringe. Transmission Electron Microscopy of Materials (Wiley, New York 1990) 404 pp. J. C. H. Spence. High-Resolution Electron Microscopy 3 rd edition. (Clarendon Press, Oxford 2003) 400 pp. D. Shindo, K. Hiraga. High-Resolution Electron Microscopy for Materials Science (Springer, Tokyo 1998) 190 pp. G. Schmid. Chemical synthesis of large metal clusters and their properties. Nanostruct. Mater. 6, 15–24 (1995) Y. Sakka, S. Ohno. Hydrogen sorption–desorption characteristics of mixed and composite Ni–TiN nanopartices. Nanostruct. Mater. 7, 341–353 (1996) A. I. Gusev, A. A. Tulin, V. N. Lipatnikov, A. A. Rempel. Nanostructure of dispersed and bulk nonstoichiometric vanadium carbide. Zh. Obsh. Khimii 72, 1067–1076 (2002) (in Russian). (English transl.: Russ. J. General Chem. 72, 985–993 (2002)) A. A. Rempel’, A. I. Gusev. Nanostructure and atomic ordering in vanadium carbide. Pis’ma v ZhETP 69,436–442 (1999) (in Russian). (Engl. Transl.: JETP Letters 69, 472–478 (1999)) M. S. El-Eskandarany, M. Omori, T. Kamiyama, T. J. Konno, K. Sumiyama, T. Hirai, K. Suzuki. Mechanically induced carbonization for formation of nanocrystalline TiC alloy. Sci. Reports of Res. Inst. Tohoku Univ. (Sendai, Japan) 43, 181–193 (1997) M. A. Krivoglaz. Theory of X-ray and Thermal-Neutron Scattering by Real Crystals (Plenum Press, New York 1969) 405 pp. A. A. Rempel, A. I. Gusev. Preparation of disordered and ordered highly nonstoichiometric carbides and evaluation of their homogeneity. Fiz. Tverd. Tela 42, 1243–1249 (2000) (in Russian). (Engl. Transl.: Physics of the Solid State 42, 1280–1286 (2000)) A. I. Gusev, A. A. Rempel. Nonstoichiometry, Disorder and Order in Solids (Ural Division of the Russ. Acad. Sci., Yekaterinburg 2001) 580 pp. (in Russian) A. I. Gusev, A. A. Rempel, A. J. Magerl. Disorder and Order in Strongly Nonstoichiometric Compounds: Transition Metal Carbides, Nitrides and Oxides

156

Evaluation of the Size of Small Particles

13. 14. 15.

16. 17.

18. 19.

20.

21. 22.

23.

24. 25.

26.

27.

28. 29. 30.

(Springer, Berlin – Heidelberg – New York, 2001) 607 pp. B. D. Cullity. Elements of X-ray Diffraction (Addison-Wesley Publ., London 1978) 555 pp. B. E. Warren, B. L. Averbach, B. W. Roberts. Atomic size effect in the X-ray scattering by alloys. J. Appl. Phys. 22, 1493–1496 (1951) H. P. Klug, L. E. Alexander. X-ray Diffraction Procedures for Polycrystalline and Amorphous Materials (Wiley, New York 1954) 491 pp. (second edition: X-ray Diffraction Procedures (Wiley, New York 1974)) B. E. Warren. X-Ray Diffraction (Dover Publications, New York 1990) 381 pp. Defect and Microstructure Analysis by Diffraction (Intern. Union Crystallogr., Monographs on Crystallography. V.10) Eds. R. L. Snyder, J. Fiala, H. Bunge. (Oxford University Press, New York 1999) 808 pp. V. Ya. Shevchenko, A. E. Madison, Yu. I. Smolin. Peculiarities of diffraction on nanoparticles. Fizika i Khimiya Stekla 28, 465–476 (2002) (in Russian) S. V. Rempel’, A. I. Gusev. Phase Equilibria in the Zr–Nb–C Ternary System. Zh. Fiz. Khimii 75, 1553–1559 (2001) (in Russian). (Engl. transl.: Russ. J. Phys. Chem. 75, 1413–1419 (2001)) Ya. S. Umanskii, Yu. A. Skakov, A. N. Ivanov, L. N. Rastorguev. Crystallography, X-Ray Diffraction and Electron Microscopy (Metallurgiya, Moscow 1982) 632 pp. (in Russian) R. W. James. The Optical Principles of the Diffraction of X-rays (G. Bell & Sons Ltd., London 1954) 624 pp. A. A. Rempel, A. I. Gusev, R. R. Mulyukov. Preparation of nanocrystalline titanium carbide by severe plastic deformation. In: Ultrafine grained powders, materials and nanostructures. Proc. of Interdistrict Conf., Krasnoyarsk, December 17–19, 1996. Ed. V. E. Red’kin. (Krasnoyarsk State Technical University, Krasnoyarsk 1996) pp.131–132 (in Russian) A. I. Gusev. Phase equilibria, phases and compounds in the Ti – C system. Uspekhi Khimii 71, 507–532 (2002) (in Russian). (Engl. transl.: Russ. Chem. Reviews 71, 439–465 (2002)) S. V. Vonsovskii. Magnetism (Nauka, Moscow 1971) 1032 pp. (in Russian) A. A. Rempel, A. I. Gusev, S.Z. Nazarova, R. R. Mulyukov. Imputity superparamagnetism in plastically deformed copper. Doklady Akad. Nauk 347, 750–754 (1996) (in Russian). (Engl. transl.: Physics - Doklady 41, 152–156 (1996)) A. A. Rempel, S. Z. Nazarova. Magnetic properties of iron nanoparticles in submicrocrystalline copper. In: Advances in Nanocrystallization. Proceedings of the Euroconference on Nanocrystallization and Workshop on Bulk Metallic Glasses (Grenoble, France, April 21–24, 1998). Ed. A. R. Yavari. Materials Science Forum 307, 217–222 (1999). (Trans Tech Publications, Switzerland 1999) / J. Metastable Nanocryst. Mater. 1, 217–222 (1999) A. A. Rempel, S. Z. Nazarova, A. I. Gusev. Intrinsic and extrinsic defects in palladium and copper after severe plastic deformation. In: Structure and properties of nanocrystalline materials (Ural Division of the Russ. Acad. Sci., Yekaterinburg, 1999) pp.265–278 A. A. Rempel, S. Z. Nazarova, A. I. Gusev. Iron nanopatricles in severeplastic-deformed copper. J. Nanoparticle Researh 1, 485–490 (1999) C. Bernhardt. Particle Size Analysis. Classification and Sedimentation Methods (Kluwer Academic Publishers, Netherlands, Dordrecht, 1994) 448 pp. R. Brown. A brief account of microscopical observations made in the months on June, July, and August, 1827, on the particles contained in the pollen 157

Nanocrystalline Materials

31.

32.

33. 34. 35. 36. 37.

38. 39.

40.

41.

of plants; and on the general existence of active molecules in organic and inorganic bodies. Phil. Mag. 4, 161–173 (1828) A. Einstein. Eine neue Bestimmung der Moleküldimensionen. Inaugural Dissertation. Zürich Universität (Buchdruck. K. J. Wyss, Bern 1905) 28 pp.; Ann. Physik 19, 289–306 (1906) A. Einstein. Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Physik 17, 549–560 (1905) A. Einstein. Zur Theorie der Brownschen Bewegung. Ann. Physik 19, 371– 381 (1906) A. Einstein. Elementare Theorie der Brownschen Bewegung. Z. Elektrochem. 14, 235–239 (1908) M. von Smoluchowski. Zur kinetischen Theorie der Molekularbewegungen und der Suspensionen. Z. Physik 21, 756–780 (1906) Photon Correlation and Light Beating Spectroscopy. Eds. H. Z. Cummins, E. R. Pike. (Plenum Press, New York 1974) 584 pp. M. A. Anisimov, Yu. F. Kiyachenko, G. L. Nikolaenko, I. K. Yudin. Measurement of liquid viscosity and sizes of suspended particles by means of correlation spectroscopy of light beating. Inzhenerno-Fiz. Zh. 38, 651–655 (1980) (in Russian) Measurement of Suspended Particles by Quasi-Elastic Light Scattering. Ed. B. E. Dahneke. (John Wiley, New York 1983) 584 pp. Light Scattering and Photon Correlation Spectroscopy. Eds. E. R. Pike, J. B. Abbiss. (Kluwer Academic Publishers, Netherlands, Dordrecht, 1997) 470 pp. I. K. Yudin, G. L. Nikolaenko, V. I. Kosov, V. A. Agayan, M. A. Anisimov, J. V. Sengers. A compact photon-correlation spectrometer for research and education. Int. J. Thermophys. 18, 1237–1248 (1997) T. Svedberg. Colloid Chemistry: Wisconsin Lectures (Chemical Catalog Comp., New York 1924) 265 pp.

158

Properties of Isolated Nanoparticles and Nanocrystalline Powders

+D=FJAH# 5. Properties of Isolated Nanoparticles and Nanocrystalline Powders The physical properties of small atomic aggregates, referred to as clusters, small particles, isolated nanocrystals, have been determined in a large number of original experimental studies whose results have been generalised in a number of detailed reviews and monographs [1–12]. Consequently, this section can be restricted to discussing only the size effects associated with the structure of the nanoparticles and also some new experimental results obtained in recent years. 5.1 STRUCTURAL AND PHASE TRANSFORMATIONS The developed surface of isolated nanoparticles provides a significant contribution to their properties. The non-additive nature of thermodynamic functions, associated with the contribution of the interfaces and taken into account by introducing surface tension σ , leads to size effects of thermodynamic quantities. In the case of nanoparticles it is also necessary to take into account the dependence of surface tension on the particle size. The effect of surface energy is manifested in the thermodynamic conditions of phase transformations. Phases which do not exist in the given substance in the large particles may form in nanoparticles. With a decrease of the particle size, the contribution of the surface energy

FS =

Ñ∫ ó(n)ds

(where σ (n) is the surface tension which depends

on the direction of unit vector n, normal to the surface) to the free energy F = F V + F s (F V is the bulk contribution to the free energy) increases. If the phase 1 is stable in a coarse-grained (bulk) specimen at some temperature, i.e. F V(1) < F V(2) , then with a decrease in the particle size and with an increase in surface energy F s it may be shown that 159

Nanocrystalline Materials

F V ( 2 ) + F s ( 2 ) ≤F V ( 1 ) +F s ( 1 )

(5.1)

and the phase 2 will be stable in small particles. Since surface energy is a significant part of free energy and is high enough in comparison with bulk energy, equation (5.1) shows that deformation of the crystal which leads to decreasing of the surface energy may be efficient in decreasing the total energy of the system. A similar energy decrease may be realised by changing the crystal structure of the nanoparticles in comparison with a coarse-grained bulk specimen. The close-packed structures have the lowest surface energy. Therefore, the face-centered cubic (fcc) or hexagonal close-packed (hcp) structures are most suitable for nanocrystalline particles [1, 2]. This has also been confirmed by experiments. Electron diffraction examination [13] of niobium, tantalum, molybdenum and tungsten nanocrystals with a size of 5– 10 nm shows that they have an fcc or hcp structure, whereas in coarse-grained powders and in the bulk state these metals have a body-centered cubic (bcc) lattice. Nanoparticles of beryllium and bismuth contain cubic phases, although these elements have a hexagonal lattice in the bulk state [14]. The bulk gadolinium, terbium and holmium have an hcp structure. The authors of [15,16] investigated the structure of Gd, Tb and Ho particles with a size of 110 to 24 nm and observed traces of the fcc phase. They showed that the content of the fcc phase increases and the content of the hcp phase decreases with a decrease in the particle size of these metals. The hcp phase, which is typical of the bulk Gd specimens, was not found at all in Gd nanocrystals with a size of 24 nm. However, the author [4] has doubts on the accuracy of conclusions [15, 16] on the hcp – fcc transition, because the diffraction reflections, detected on the X-ray diffraction patterns of the Gd, Tb and Ho nanoparticles, could belong to low-temperature cubic modifications of oxides of these metals. A decrease in the particle size of certain elements (Fe, Cr, Cd, Se) resulted in a loss of crystal structure and appearance of an amorphous structure [14, 17]. It was noted in a review [12] that a decrease in the surface energy of particles may take place not only by means of a complete change of the crystal structure but also in the case of deformation of the structure. For example, small particles may have a multiply twinned structure which in bulk specimens exists only as a metastable structure. In particular, attention should be given to the structure of clusters, i.e. particles, which contain less than (1–2)×10 3 atoms. The results 160

Properties of Isolated Nanoparticles and Nanocrystalline Powders

of a large number of theoretical calculations show that in addition to the fcc structure, typical of the bulk crystal, clusters may have a crystallographic symmetry characterised by the fifth fold symmetry axis [18–21]. Two main assumptions are used in modelling the structure of clusters containing small numbers of atoms: a cluster should have close packing and should be stable from the energy viewpoint. It follows from the first assumption that clusters are constructed from the simplest stable atomic configurations and the fraction of tetrahedra among stable configurations should be large because the tetrahedron is a stable atom configuration with the smallest volume. Structural elements of the clusters are the tetrahedron, the octahedron, the cube, the cube–octahedron, the pentagonal pyramid, the icosahedron, etc. The smallest stable cluster with the symmetry axis of the 5 th order contains seven atoms and has the form of a pentagonal bipyramid, the next stable configuration with the symmetry axis of the fifth order is a cluster in the form of an icosahedron consisting of thirteen atoms. The stable configurations (isomers) of the cluster consisting n atoms are determined by those atomic coordinates which correspond to the minima of the surface of potential energy in (3n–6) dimensional space. Clusters with n > 10 have tens or even hundreds of isomers [21]. Examination of the relative stability of different structural modifications shows that icosahedral forms are more stable for clusters containing less than 150–300 atoms. The smallest icosahedron contains 13 atoms, 12 of which are located at equal distances around the central atom. The icosahedron of 13 atoms may be regarded as a figure consisting of 20 identical tetrahedra which have the common vertex and are connected together with common faces which are twinning planes. In icosahedral groupings, every k-th atomic shell (layer) contains (10k 2 + 2) atoms, and the total number of atoms of the icosahedral cluster is n = (2 N + 1) + 10

N

k2, Σ k =1

where k is the order number of

the atomic shell, N is the number of atomic shells. It should be mentioned that the icosahedral particles are characterised by a 6-angle profile on electron microscopic images. Each icosahedral particle has a particle-twin with a structure of a non-deformed fcc lattice. According to numerical calculations [19] the energy of a 13-atom icosahedral cluster is 17 % lower than the energy of an fcc cluster, and fcc clusters spontaneously transfer to 161

Nanocrystalline Materials

the icosahedral form. An increase in the number of atoms in a cluster leads to a rapid increase of the energy of elastic deformation, which is proportional to the volume. As a result, the increase in the elastic energy in a large cluster is larger than the decrease in the surface energy and it results in the destabilisation of the icosahedral structure. Thus, there is some critical size above which the icosahedral structure becomes less stable than cubic or hexagonal ones. Cubic and hexagonal structures are characteristic of nanoparticles larger than 10 nm. The application of a uniform hydrostatic pressure to a nanoparticle may increase the density of the structure. Under hydrostatic pressure, a nanocrystal of CdSe, having the wurtzite ZnS structure, acquires structure B1 [22]. The surface energy of a nanoparticle increases with a decrease in the nanoparticle size. Therefore, the pressure required for changing the crystal structure of the nanoparticles should also increase. This dependence of pressure on the size of the nanoparticles extracted from colloid solutions (or prepared by colloidal methods) was detected for CdSe [22] (Fig. 5.1), CdS, Si and InP [23,24]. The dependence of surface energy on the particle size predetermines the relationship between the melting point and the size of the nanoparticles. Let us consider a system which is a solid spherical isotropic particle, located in its own melt. If the Gibbs





5

97,38

*3D

&G6H



 







7 QP

Fig. 5.1. Size dependence of hydrostatic pressure p trans required for the transition of nanoparticles of CdSe from hexagonal (wurtzite type) structure to the B1 structure [22].

162

Properties of Isolated Nanoparticles and Nanocrystalline Powders

tension surface is introduced as the surface separating the two phases, then we shall have three subsystems: condensed phase 1, melt (phase 2) and interface 3. In equilibrium conditions, the total variation of the energy of these subsystems is equal to zero. This equality is fulfilled if the temperature and chemical potentials are the same in all subsystems, i.e. T 1 = T 2 = T 3 and µ 1 = µ 2 = µ 3 , and the pressure in phase 1 exceeds the pressure in phase 2 by the value 2 σ /r (Laplace pressure), determined by the curvature of the interface: (p 1 – p 2 ) = 2 σ /r.

(5.2)

Taking into account the equality of chemical potentials of the phases 1 and 2 and relationship (5.2), one obtains the well-known Thomson equation describing the dependence of the melting point of the particle, T melt (r), on its size (radius r) [T melt (r) – T melt ]/ T melt = –(v 1 /L)(2 σ /r),

(5.3)

where T melt and L are the temperature and heat of melting of a bulk solid, v 1 is the volume of 1 g of substance, i.e. the quantity reciprocal to density. Thomson’s equation (5.3) predicts the universal decrease of the melting point of the particles inversely proportional to their radius. For the ‘particle –melt’ system, equation (5.3) contradicts the initial assumption on the equilibrium of the solid particle with the surrounding medium. According to (5.3), when the system is heated the small particle should melt earlier than a melt of the bulk solid forms. In other words, any particle of a finite size should have a lower melting point than the bulk solid. It is clear that in this case the observed equilibrium of the crystal with a liquid is not possible. The invalidity of the Thomson’s equation is determined by the assumption made when deriving the equation. According to this assumption, the volume of the ‘solid–melt’ system is constant and the changes of the volume and mass of the phases are dependent in a simple way: the volume of the melt is proportional to the mass of the melt. Later, it was proposed to determine the melting point of small crystals as the temperature at which the solid and liquid spherical particles of the same mass are in equilibrium with their vapour [25]. Indeed, T melt (r) is the temperature at which the transfer of matter through the vapour from the solid to the liquid and vice versa does not take place in a mixture of solid and liquid particles with equal 163

Nanocrystalline Materials

masses. Using and developing the concept proposed in [25], the authors of [26–28] obtained the following equations for the equilibrium melting point T melt(r) of solid particles

Tmelt (r ) = Tmelt {1 −

2 [σ s − σ l ( ρ s / ρ l ) 2 / 3 ]} ρ s Lr

(5.4)

and particles coated with a layer of melt with thickness δ ,

Tmelt (r ) = Tmelt{1 −

ρ 2 σ sl σ l [ + (1 − s )]} , r ρl ρs L r − δ

(5.5)

where σ s , σ l , σ sl is the surface tension of the solid and liquid particles, and also at the interface of the solid and liquid phases; ρ s , ρ l are the densities of the solid and liquid particles. For the thermodynamic equilibrium conditions, the melting point is determined as a temperature at which the total free energies of the solid and liquid phases are equal to each other. Allowing for the surface energy in the equation for the total free energy, the author [29] proposed the following equation for the melting point of a spherical particle:

Tmelt (r ) = Tmelt {1 −

3 [σ s − σ l ( ρ s / ρ l ) 2 / 3 ]} ρ s Lr

(5.6)

This formula gives the smallest possible value of melting point T melt (r). There are other equations, which describe the decrease of the melting point of small particles with a decrease of their size. The equation (5.4)–(5.6), obtained by different authors for describing the size effect of the melting point of nanocrystalline particles, may be presented in the form Tmelt (r ) = Tmelt (1 − α / r )

(5.7)

where α is a constant which depends on the density, heat of melting and surface energy of the material. It may easily be seen that dependence (5.7) is similar to Thomson’s equation (5.3). Recently, it was proposed [30] to use expansion into a power series

T melt (r) = T melt (1 + α r –1 + β r –2 + ...)

164

(5.8)

Properties of Isolated Nanoparticles and Nanocrystalline Powders



6Q

%PHOW

.

   







7 QP



Fig. 5.2. Decrease in melting point T melt of Sn nanoparticles in relation to their reciprocal radius r –1 ; solid line corresponds to the melting point calculated from equation (5.5) [27].

for describing the experimental data for T melt (r), where α , β … are empirical constants. In experiments, a decrease in the melting point of small particles was detected in a large number of studies: Sn [27], Pb, In [28], Ag, Cu, Al [31], In [32], Au [33, 34], Pb, Sn [35], Pb, Sn, Bi, In, Ga [36–39], Ag, Au [40]. Electron diffraction examination of Sn nanoparticles with a diameter of 8–80 nm [27] showed a large deviation of the experimental dependence T melt (r) from the linear dependence T melt ~1/r. The linear dependence comes from Thomson’s equation which is wrong. Approximation of the results of [27] by equation (5.5) showed good agreement between experimental and calculated melting points (Fig. 5.2). Calculation was performed at the following values of the parameters of equation (5.5): ρ s = 7.18×10 3 kg m –3 , ρ l = 6.98×10 3 kg m –3 , σ l = 0.58 N m –1 , σ sl = 0.0622 N m –1 , L = 58.5 kJ kg –1 , δ = 3.2×10 –9 m, and T melt = 505 K. For tin, equation (5.5) at these values of the parameters has the form

 3.74 1  , Tmelt (r ) = 505 − 40 −   r − 3.2 r 

(5.9)

where the size of r is in nanometers. The size dependences of the relative melting point T melt (r)/T melt of Sn nanoparticles, determined 165

Nanocrystalline Materials



QDQR$X

%2

.

     

 7 QP

Fig. 5.3. Dependence of melting point T melt on the radius r of Au nanoparticles: the solid line is the melting point calculated from equation (5.5), the dashed line is the melting point of a macroscopic bulk specimen of Au [34].

in [27, 35], are in complete agreement in the range r = 10–40 nm. Previously, the same dependence T melt (r)/T melt on r was detected in [33] for nanoparticles of gold with a radius smaller than 40 nm. Equation (5.5) was used to describe the size dependences T melt (r)/ T melt for Pb, Sn, In, and Bi nanoparticles with a radius greater than 2 nm [37]. The results of electron diffraction examination of the dependence T melt (r) for gold particles with a radius greater than 1 nm [34] (Fig. 5.3) are described efficiently by both equation (5.4) and (5.5) because the accuracy of experimental measurements was insufficient for discovering a difference between the models (5.4) and (5.5). A large decrease of the melting point of Sn, Ga and Hg clusters with a size of ~1 nm, obtained in cavities of zeolites is described in [7, 41]. Specimens were produced by filling under pressure the cavities in zeolite with liquid metals. The largest decrease of the melting point of clusters of Sn, Ga and Hg was 152, 106 and 95 K, respectively, whereas no melting of clusters In, Pb and Cd was detected [41]. The very large (by several hundreds of degrees) decrease in the melting point was determined in [42] for colloid CdS nanoparticles with a radius of 1–4 nm (Fig. 5.4). In [43], equation (5.6) was used to calculate the dependence of the melting points of Al, Cu, Ni, and Ti nanoparticles on their 166

Properties of Isolated Nanoparticles and Nanocrystalline Powders

&G6



%PHOW

.





 









7 QP

Fig. 5.4. Dependence of melting point T melt on radius r of nanoparticles of cadmium sulfide CdS [42]. 

%PHOW

.





7L 1L &X $O

 

 7 QP



Fig. 5.5. Dependence of melting point T melt on reciprocal radius r –1 of nanoparticles of Al, Cu, Ni and Ti calculated from equation (5.6) using the parameters in Table 5.1 [43].

inverse radius 1/r (Fig. 5.5). The parameters of equation (5.6), used to calculate the dependences T melt (1/r), and coefficient a for equation (5.7) are presented in Table 5.1. The estimates obtained in [43] show that the melting point of a nanoparticle tends to zero when the nanoparticle radius drops below 0.5–0.6 nm. The majority of authors hold that melting of the nanoparticles starts from the surface because of the spatial inhomogeneity of the nanoparticle. In this case, the experimental size dependence T melt (r) should be described more efficiently by equation (5.5) which takes the presence of the liquid shell into account. However, it is shown in [35] that the same data are described quite accurately by 167

Nanocrystalline Materials Table 5.1 Parameters of melting point function (5.6) of nanoparticles [43]

(J m–2)

ρ l× 1 0 –5

α × 1 0 10 fo r e q ua tio n (5 . 7 ) (m)

0.926

0.865

0.894

4.43

1.592

1.320

1.310

1.250

4.07

17470

2.104

1.400

1.750

1.350

3.82

14150

1.797

0.910

1.500

0.868

5.80

M e ta l

T melt (K )

L (J mo l–1)

σs

ρ s× 1 0 – 5

(J m–2)

(mo l m–3)

Al

934

10700

1.032

Cu

1358

13050

Ni

1728

Ti

1943

σΙ

equation (5.4) which does not consider the liquid shell. A partial confirmation of the formation of the liquid shell is made by computer modelling of melting of gold particles [44]. According to [44], the liquid shell forms on the particles containing at least 350 atoms or larger. Surface melting was observed experimentally on films of Pb where melting of the surface started at a temperature equal to 0.75 of melting point T melt of bulk lead. The thickness of the molten layer increased with approach to T melt [45, 46]. Surface melting was also observed in the case of Ar [47], O 2 [48], Ge [49], and Ne [50]. The authors of [51] have proposed a different physical model of melting of the nanoparticles. According to [51], clusters with a given number of atoms are characterized by two temperatures T f and T melt . Temperature T f is the lower limit of the temperature of stability of the cluster in the liquid state. Temperature T melt is the upper temperature limit of stability of the cluster in the solid state. A set of identical clusters behaves as a statistical ensemble, which consists of solid and liquid clusters in a specific temperature and pressure range. The ratio of the number of solid and liquid clusters is equal to exp(–∆F/T), where ∆F is the difference of the free energies in the solid and liquid states. The equilibrium between the solid and liquid clusters is dynamic and each individual cluster transfers from the solid to liquid state and vice versa. Since the frequency of transition between the liquid and solid state of the cluster is low, equilibrium properties are certain for each phase. The results published in [51] were obtained using the theoretical analysis of the density of states of the cluster. The limiting temperatures correspond to reaching the maximum or minimum of free energy, i.e. appearing or loss of local stability by the phase. 168

Properties of Isolated Nanoparticles and Nanocrystalline Powders

Subsequent computer modelling [52] confirmed these conclusions. The behaviour of gold nanoparticles at T < T melt , observed by the authors of [53], was very similar to the model of melting proposed in [51]. Gold particles were deposited on a SiO 2 substrate. Au particles with a size of ~2 nm, excited by a beam of an electron microscope, transferred from the single crystalline structure to multiply twinned structure and vice versa. The lifetime of every structure was approximately 0.1 s. Interphase fluctuations were absent in particles larger than 10 nm. Earlier, analogous phenomena were observed by the authors of [54]. Analysis of the data obtained by various authors for the size dependence of the melting point of small particles shows that the melting points of bulk crystals and small particles with a size greater than 10 nm are almost the same. A strong decrease of the melting point, caused by the size effect, is detected when the nanoparticle size is below 10 nm. 5.2 CRYSTAL LATTICE CONSTANT The transition from bulk crystals or large particles to nanoparticles is accompanied by a change of the interatomic distances and crystal lattice constants [2–5, 12]. The main task is to determine whether the lattice constants decrease or increase with a decrease in the particle size, and also to determine that nanoparticle size behind which this change is measurable. The experimental data on the size dependence of the lattice constant reported in the literature are often ambiguous and contradictory. Analysing the variation of the lattice constant of the nanoparticles, it is important to take into account the possibility of transition from less dense bcc and hexagonal structures to more dense fcc structures accompanying by a decrease in the particle size, mentioned in Section 5.1. According to electron diffraction data [55], the hexagonal close-packed structure and the lattice constants, which are characteristic of bulk metals, remains unchanged with a decrease of diameter D of the particles Gd, Tb, Dy, Er, Eu and Yb from 8 to 5 nm. With a further decrease of the particle size, the lattice constants decrease rapidly. However, these changes are accompanied by a change of electron diffraction patterns, indicating a structural transformation, i.e. transition from the hcp to fcc structure, and not a decrease of the interplanar distances of the hcp lattice. In fact, investigation of nanoparticles of rare-earth metals by the X-ray diffraction technique showed a 169

Nanocrystalline Materials

, QP



QDQR$O





 





QP

Fig. 5.6. Dependence of lattice constant a on the diameter D of aluminium nanoparticles [57].

structural transition from the hcp to fcc lattice [15, 16]. Thus, to ensure reliable detection of the size effect on the lattice constant of the nanoparticles, it is also necessary to take into account the possibility of structural transformations. The effect of the size of nanoparticles on the lattice constant could be determined most reliably by investigating of substances with the fcc crystal lattice because the probability of a structural transition for the fcc solids is very low. Electron diffraction is one of the methods of determining the lattice constant of the nanoparticles. Analysis of the systematic errors of this method shows that only certain diffraction reflections could be used for accurate determination of the lattice constant of the nanoparticles. For example, it is recommended to use (220) reflection for cubic nanocrystals [56]. Consideration of the broadening of this diffraction reflection show that in Ag particles with a diameter of 3.1 nm and in Pt particles with a diameter of 3.8 nm the lattice constant decreases by 0.7% and 0.5% in comparison with bulk silver and platinum [56]. Electron diffraction using moire patterns shows that the variation of the diameter of Al particles from 20 to 6 nm decreases the lattice constant by 1.5% (Fig. 5.6) [8, 57, 58]. Previously [59] no decrease of the lattice constant of Al particles with a diameter of ≥ 3 nm was found. A decrease in the lattice constant from 0.405 nm for the bulk Al specimen to 0.402 nm for an Al nanoparticle with a diameter of 40 nm was detected by neutron diffraction [2]. 170

Properties of Isolated Nanoparticles and Nanocrystalline Powders

QDQR$J

QDQR$X



,,



∆  





 











QP

Fig. 5.7. Relative variation of the lattice constant ∆a/a in relation to the diameter D of gold and silver nanoparticles (2) [65].

The absence of the size dependence of the lattice constant was noted for particles of Pb and Bi with a diameter of ≥ 5 and ≥ 8 nm respectively [60], for Au particles with a diameter of 6– 23 nm [61, 62], and for Cu clusters with a diameter of ≥ 5 nm [63]. However, a decrease in the size of copper clusters to 0.7 nm decreased the lattice constant by 2 % in comparison with bulk metal [63]. Electron diffraction investigation [64] detected a small (~0.3 %) decrease of the lattice constant of an Au nanoparticles with a diameter of 2.5–14 nm. Compression of the lattice constant by ~0.1 % was established [65, 66] for the nanoparticles of Ag and Au with a diameter ranging from 40 to 10 nm (Fig. 5.7). The effect of the nanoparticle size on the lattice constant was noted not only for metals but also for compounds. A decrease in the lattice constant of submicrocrystalline titanium, zirconium and niobium nitrides with decrease in the particle size is described in [67–70]. Nitride powders were produced by the plasma chemical method. The dependence of the lattice constant a on the specific surface S sp of the titanium nitride submicrocrystalline powder is given in [70]: a(nm) = 0.42413–0.384×10 –8 S sp where S sp changes from 4×10 4 to 1×10 5 m 2 kg –1 . It should be mentioned that the dependence of the lattice constant on the size of the titanium nitride particles, established in [70], does not take into account the fact that the composition of the powders with different particle sizes differs. The nitrogen content in the nitride powder decreased with a decrease of the particle size in the powder. Unfortunately, the authors of [70] did not attempt to separate the effect of the 171

Nanocrystalline Materials

composition and the effect of particle size of titanium nitride on the lattice constant. The decrease of the lattice constant of the cubic zirconium nitride, explained in [68] by a decrease in the particle size of the powder, took place with a large change in the composition of the nitride. A large decrease of the lattice constant from 0.4395 nm for the bulk niobium nitride to 0.4382 nm for the NbN powder with a particle size of ~40 nm was found in [69]. The submicrocrystalline nitrides, produced by the plasma chemical method, usually contain a large (up to 7 at.%!) amount of impurity oxygen. The introduction of oxygen into the carbides and nitrides decreases their lattice constant [71]. The lattice constant of the cubic nitrides of group IV and V transition metals rapidly decreases with a decrease in the nitrogen content [72, 73]. Taking this into account, the conclusions made in [67–69] on the decrease of the lattice constant of the cubic nitrides with a decrease of the particle size cannot be regarded as reliable. In some cases, the lattice constant of the nanoparticles increases instead of decreasing [74, 75]. A decrease of the size of Si particles from 10 to 3 nm leads to an increase of the lattice constant by 1.1 % [76]. An increase of the lattice constant of CeO 2 oxide with a decrease of the particle size from 25 to 5 nm (Fig. 5.8) is found in [77]. It is possible that the increase of the lattice constant of the cerium oxide is caused by the adsorption of water, as observed in the case of MgO [78]. Thus, the experimental data on the size effect of the lattice constant of the nanoparticles are ambiguous. Above all, this ambiguity may be associated with the adsorption of the impurities by nanoparticles. In the case of compounds, which have a homogeneity interval, the different chemical composition of the coarse particles and nanoparticles could lead to ambiguity. Another possible reason for the ambiguity of the results are the structural transformations, caused by a decrease of the particle size. The next reason may be inaccurate measurement of the lattice constant. The most reliable results did not show any decrease of the lattice constant with a decrease of the particle size to 10 nm, whereas for smaller particles, the decrease of the interatomic distances in comparison with the bulk material is relatively realistic. This is confirmed by the experimental data on interatomic distances in metallic dimers (clusters consisting of two metal atoms). For dimers, these distances are smaller than those for the bulk metals. For example, the interatomic distances for clusters of Cu 2, Ni 2, Fe 2 172

Properties of Isolated Nanoparticles and Nanocrystalline Powders

&H2



,

QP















QP

Fig. 5.8. Dependence of the lattice constant a on the diameter D of nanoparticles of CeO 2 [77] .

are equal to 0.222, 0.2305 and 0.187 nm respectively, and the interatomic distances in these bulk metals are 0.256, 0.249 and 0.248 nm [79, 80]. Many authors assume that a decrease in the lattice constants of the nanoparticles is a consequence of the excess Laplace pressure ∆p = 2 σ /r generated by surface tension σ . According to the elasticity theory, the relative variation of the volume of the particle ∆V/V is proportional to ∆p, i.e. ∆V/V = –k T (2σ/r), where k T is isothermal compressibility. Since ∆V/V = 3(∆a/a), then ∆a/a = k/r, where k is the proportionality coefficient. However, the values of k for the same substance obtained by different authors greatly differ. In addition to this, in certain cases the expansion of the small particles is observed instead of compression. If the Laplace pressure compressed the nanoparticles, then only decreasing of lattice constant in nanoparticles would be their universal property. In [81–85], the decrease of the lattice constant of metallic particles was explained by the formation of thermal vacancies and by an increase of their concentration with a decrease in the particle size. The high concentration of vacancies was regarded as a consequence of uniform compression under the pressure ∆p = 2σ/r. The latter claim is doubtful. In fact, the increase of the concentration of vacancies in metals with increasing temperature is well known. The temperature and pressure in the free energy equation have opposite sign and, therefore, in a general case an increase of pressure should affect the vacancy concentration in the same manner as a decrease of temperature, i.e. it should lead to 173

Nanocrystalline Materials

a decrease (not increase) of the number of vacancies. In turn, a decrease of the concentration of vacancies, following the logic in [81–85] cannot result in a decrease of the lattice constant. The physical meaning of the Laplace pressure was analysed in [86]. According to [86] the Laplace pressure cannot cause uniform compression of solids. The Laplace pressure tends to change the form of a solid in such a manner as to ensure the minimum of its surface energy E S . For a liquid droplet it is assumed that E S is proportional to the surface area S, i.e. E S = σS, where surface tension σ is constant. The surface area of the droplet can be reduced by two methods: making the droplet spherical without decreasing the volume, or compress the droplet in order to decrease the surface area of even the spherical droplet. However, the phenomenological relationship E S =σS is valid only if the variation of the surface area takes place at a constant volume. This means that surface tension σ determines the equilibrium form of the surface of small particles but does not lead to their compression. A phenomenological approach was used by authors of [86] for the analysis of the Laplace pressure. According to [86], the Laplace pressure is a purely mathematical concept which makes it possible to treat the chemical potential of the atoms in a solid of finite size at true pressure p as a chemical potential of a solid of infinitely large size at pressure (p + ∆p), where ∆p is Laplace pressure. At the thermodynamic equilibrium, the shape of a small solid should ensure the minimum of surface energy. It is assumed that a particle with size D, surface area S and atomic density n 1 , i.e. phase 1, is in equilibrium with phase 2 at pressure p and temperature T. The number of atoms in the particle is N 1 and is proportional to n 1 DS or n 1 rS. In this case, the free Gibbs energy of the considered system is F = F 1 (p,T) + F 2 (p,T) + σ S = F 1 (p,T) + F 2 (p,T) + 2σN 1 /n 1 r. Differentiating F with respect to N 1 , it is possible to find the chemical potential of the particle µ 1 (S, p,T) ≡ µ 1 (D, p,T) = µ 1 (∞, p,T) + 2 σ /n 1 r, where µ 1 (∞, p,T) is the chemical potential of the specimen of an infinite size. Expanding µ 1 into a power series and limiting considerations to the terms of expansion of the first order, the authors of [86] obtained that µ 1 (S, p,T) = µ 1 (∞, p + ∆p, T). Thus, Laplace pressure ∆p makes it possible to express the chemical potential of a small particle by means of the chemical potential of a bulk specimen with S → ∞. When the dependence of the chemical potential of the small particle on the surface area S is taken into account explicitly, it is not required to introduce the Laplace pressure. 174

Properties of Isolated Nanoparticles and Nanocrystalline Powders

Analysis [86] shows that the Laplace pressure does not cause compression of the solids and, consequently, cannot be the reason for a decrease in the lattice constants of the nanoparticles. The most probable reason for the decrease of the lattice constant of the small particles in comparison with large particles or bulk material is the fact that interatomic bonds of the surface atoms are not compensated in contrast to the atoms located inside the particle. As a result of this, the distances between the atomic planes in the vicinity of the surface of the particle decrease because surface relaxation takes place. In fact, an atom in a surface layer has a smaller number of neighbours than in the volume because there are neighbours exclusively in a surface plane and under the surface plane. This disrupts equilibrium and symmetry in the distribution of the forces and masses and leads to a change of the equilibrium interatomic distances, shear deformation, smoothing of vertices and edges of a particle. Surface relaxation affects several surface layers and causes corrections in the volume of the particle of the order of D –1 (D is the particle size). According to [86–88], the largest surface relaxation takes place on the surface of nanoparticle, decreases from the surface to the centre of the particle and in certain conditions may be oscillating. Oscillating surface relaxation decreases in the direction to the centre of the particle and is associated with Friedel oscillations of the density of a degenerated electronic gas. The Friedel oscillations are caused by any defects which disrupt the translation symmetry of the crystal. In this case, the surface is such a two-dimensional defect. The Friedel oscillations are transferred to the lattice by means of electron-phonon interaction and lead to a change in the interplanar distances. According to [88], in the free electron model, the amplitude of Friedel oscillations decreases with increase of the distance from the surface. It should be mentioned that surface relaxation may not only lead to a decrease but also to an increase in the volume of the crystal depending on the lattice constant and the size of the crystal. Surface relaxation in nanoparticles is efficiently confirmed by measurements of the lattice constant of separate Al particles grown epitaxially on a substrate of single crystal MgO [8, 57]. Compression of the lattice was divided separated into two contributions. The first of them is compression of the lattice volume with a decrease of the size of Al nanoparticles. Second contribution is surface relaxation, i.e. a decrease of the lattice constant in the direction from the centre to the surface of the particle. 175

Nanocrystalline Materials

Unfortunately, the authors of [57] did not consider the interaction of epitaxial particles with a substrate, and this may have affected interpretation of the results obtained. According to [5], the main reason for the change of the interatomic distances and the lattice constants in the nanoparticles with a diameter smaller than 5 nm is a decrease in the number of atoms forming these particles. In fact, the restriction of the number of interacting atoms leads to a difference of the radial distribution of the atoms in the nanoparticles from that in bulk crystals [89]. The theoretical analysis of the dependence of the crystal lattice constant of a nanocrystalline substance on the nanoparticle size was carried out by Qin et al [90]. They considered a nanomaterial as a two-component system formed by discontinuous crystallites and a continuous matrix of grain boundaries containing highly disordered substance. According to the results of electron–positron annihilation [91, 92], the grain boundaries contain two types of free volume, namely monovacancies and vacancy clusters. Therefore, the density of substances at the grain boundaries is lower than the density of perfect crystallites not containing atomic defects. The excess volume of the grain boundaries is determined as ∆V = (V – V 0 )/V 0 , where V and V 0 are the molar volumes of defective and perfect crystallites, respectively. The grain boundary defects form a stress field. Under the effect of stresses, the atoms of the nanocrystallites deflect from their normal position in the lattice sites and this leads to a distortion of the lattice. If a is the mean lattice constant of the distorted crystals, and a 0 is the lattice constant of the perfect crystal, then the quantity a / a 0 characterises the deviation of the lattice constant of the nanocrystallite from the bulk value. According to [90], the value of a / a 0 is determined directly by parameters of the nanostructures such as the excess volume of the grain boundaries ∆V, the mean width of the grain boundaries ξ , and mean diameter D:

a 1 ξ (ξ + 2r0 ) ( 1 + ∆V − 1) , = a 0 2 D ξ + r0

(5.10)

where r 0 is the distance between the nearest adjacent atoms in a perfect lattice. In a general case, the excess volume of the grain boundaries ∆V is a function of grain size D. According to [93], the excess volume increases almost linearly with increasing grain size. Taking into 176

Properties of Isolated Nanoparticles and Nanocrystalline Powders

account the data [93], it is assumed in [90] that ∆V ~ D. In this case, equation (5.10) shows that a decrease of the grain size in a nanocrystalline material should result in an increase of the lattice constant. In [90] it is noted that the expansion of the crystal lattice takes place mainly in the vicinity of the grain boundaries, in a thin layer with a thickness of approximately (3 2 − 1)ξ / 2 . Thus, a decrease of a grain size in a nanomaterial should be accompanied by an increase of the lattice constant. This conclusion follows from the assumption that the excess volume of the grain boundaries increases with increasing grain size. However, the assumption is based on the data of a single study [93] and is not essential. It is well known that a decrease of the grain size accompanied by an increase of the relative number of atoms distributed at the grain boundaries and, consequently, it may be assumed that a decrease of the grain size will result in an increase of the excess volume of the grain boundaries, i.e. ∆V ~ D –1 . In this case, a decrease of the grain size should increase the lattice constant. However, if ∆V ~ D –2 or ∆V ~ D –3 , the lattice constant may be constant or decrease with a decrease of the nanoparticle size. Thus, theoretical analysis does not provide an unambiguous answer to the question of how should the lattice constant of the nanocrystalline substance vary in relation to the nanoparticle size. Evidently, the lattice constant may both increase and decrease with a decrease of the nanoparticle size. 5.3 PHONON SPECTRUM AND HEAT CAPACITY The main reason for the variation of the thermodynamic characteristics of nanocrystals in comparison with bulk materials is the variation of the phonon spectrum, i.e. the variation of the distribution function of frequencies of atomic vibrations (the term ‘frequency distribution function’ will be used). This is confirmed by the results of investigation [94] of single crystal Si and silicon powder by the inelastic scattering of slow thermal neutrons. The frequency distribution function, g( ω ), for a fine powder and bulk silicon greatly differ. Neutron scattering was also used for obtaining phonon spectra of MgO particles (D ~ 11, 16 and 23 nm), TiN particles (D ~ 30 nm) and bulk specimens of MgO and TiN [95– 97]. According to [19], the phonon spectrum of small particles does not contain low frequency modes which are present in the spectra of bulk crystals. The waves whose half-length exceeds the longest size D of the particle may form in the nanoparticles. Therefore, the phonon spectrum is restricted by some minimum frequency ω min ~ 177

Nanocrystalline Materials

c/2D (where c is the velocity of sound) on the lower limit of vibration frequency. There is no such restriction in bulk solids. The numerical value of ω min depends on the properties of the substance and the shape and size of the particles. It may be expected that a decrease of the particle size should displace the phonon spectrum to the range of high frequencies. Special features of the vibration spectra of the nanoparticles primarily reflect in heat capacity. The distribution of eigentones in the presence of restrictions on the side of lower limit of frequencies was discussed by the authors of [98, 99]. They proposed similar expressions for describing the number of eigentones n( ω ) of a rectangular particle taking into account its geometrical characteristics. The equation obtained in [99] for n( ω ) in a slightly modified form was used in [100] to describe the size effect on low-temperature heat capacity. According to [98], the frequency distribution function g( ω ) of the phonon spectrum of a small rectangular particle with edges L x , L y, L z , has the form

g (ω) = V ω2 / 2π 2 c3 + S ω / 8πc2 + L /16πc1

(5.11)

where V=L x L y L z , S=2(L x ,L y +L x L z +L y L z ), L=4(L x +L y +L z ) is the volume, surface area and total length of the edges; cl , ct are the velocities of propagation of the longitudinal and transverse elastic −j

−j

oscillations; c −j 1 = c l + 2c t is the effective velocity (here for transition from function g(ν), presented in [100], to function g( ω ) 1 g (ν = ω / 2π ) ). It should be 2π mentioned that the elasticity theory determines accurately only the

we use the expression

g (ω ) =

−1 −3 −3 quantity c3 = c l + 2ct corresponding to the effective velocity of sound in solid with a large solid V→∞, where the boundary conditions, determined by the presence of a surface, could be −1

neglected. The correct expression for c 2 , derived in [101–103] taking into account the boundary conditions in an accurate manner

∑ δ ( A – B ) divu + B (∂ u l

kl

l

k

+ ∂ k ul ) = 0 , (A + B) 1/2 = cl and B 1/2 =

c t , has the form

178

Properties of Isolated Nanoparticles and Nanocrystalline Powders

−1

c2 =

2c t4 − 3c t2 c l2 + 3c l4 c t2 c l2 (c l2 − c t2 )

.

(5.12)

Equation (5.12) takes into account the mode scrambling effect determined by the finite size of the particles [104]. Up to now, the correct expression for c1−1 is unknown. A spectral distribution similar to (5.11) but more accurate can be found in [105]. The total number of normal modes for a particle containing N atoms is

3N =

ω max

∫ g (ω )dω .

(5.13)

ω min

With allowance for (5.12) it follows from equation (5.13) that

ω max ≈ ω min

 18π 2 Nc 3   + V  

1/3

2 /3    18π 2 c 3  S − 2 /3  1 −   ( ) N ∆ +  144πc 2 N 1/3  V     (5.14)

where ∆(N –2/3 ) are the correction term of the order N –2/3 . Since ω = 2 πν , then at ω min = 0 equation (5.14) completely coincides with the analogous equation for ω max derived in [100]. Taking into account the boundaries of the phonon spectrum, the heat capacity of the small particle can be determined from

Cv =

ω max

∫

ω min

∂ε (ω , T ) g (ω )dω , ∂T

(5.15)

where ε (ω , T ) = (hω / 2)cotan (hω / 2k BT ) is the mean energy of the oscillator. According to [100], the heat capacity of a small particle at a temperature T → 0 and in the approximation ω min = 0 may be written in the form

( kB Lω max / 8π c1 )( kBT / hω max )I 2 × 3 2 C v (r ) = ( 4k BVω max / π 2 c 3 )(k BT / hω max ) 3 I 4 + (k B Sω max / 2πc 2 )

(k BT / hω max ) I 3 + 2

179

(5.16)

Nanocrystalline Materials ∞

where I m = (4m!/2

m+1

)

∑N N =1

–m

≡ (4m!/2 m+1 ) ζ (m) and ζ (m) is the

Riemann zeta-function (I 4 = π 4 /30, I 3 = 1.8031, I 2 = π 2 /6). One may assume that ω max = (18π 2 Nc 3 /V ) 1/3 , i.e. coincides with the maximum frequency for the bulk crystal. Then the first term in equation (5.16) represents Debye’s contribution (12 π 4 Nk B /5)(T/ θ D ) 3 to heat capacity and θ D = hω max / k B is the Debye temperature for the bulk crystal. Expression (5.16) can be written in the form C v (r) = a 3 VT 3 + a 2 ST 2 + a 1 LT ,

(5.17)

where a 1, 2, 3 are the positive constants. Equations (5.16) and (5.17) show that in the case of small particles, the heat capacity contains a contribution determined by the large surface area of these particles. Therefore, one may expect that the low-temperature heat capacity will increase and the Debye temperature will decrease with decreasing particle size. The analysis of the size effects of the phonon spectrum, carried out in [100, 101], is based on a quasi-continuum approach. The quantum approach [106–108] for calculating the frequency distribution function g( ω ) of a small particle with radius r, containing N atoms, is based on the expression

g (ω) = ∑ δ (ω − ωl , s ) ,

(5.18)

l,s

where δ is the energy interval between the adjacent permitted states; ω l, s = k l,s ct = ct a l′ ,s / r ; kl ,s is the wave vector; c t is the velocity of transverse elastic vibrations; a 'l , s is the s-th zero of the derivative of the spherical Bessel function of order l ; the degeneracy of kl ,s is (2 l + 1). According to [106], in the limit of long wave vectors k the total number of the modes for a spherical particle, which contains N atoms and has radius r, is

N = (2 / 9π)r 3 kn3 + (1/ 4 ) r 2 kn2 + ( 2 / 3π ) rkn ,

(5.19)

where k n is the boundary wave vector corresponding to the maximum frequency of vibrations, ω max , of the small particle. The terms in the right part of equation (5.19) take into account the 180

Properties of Isolated Nanoparticles and Nanocrystalline Powders

volume, surface and linear contributions. The molar heat capacity of a finite crystal with radius r has the form 3N

( hω j /k B T ) 2 exp( hω j /k B T )

j =1

[exp(hω j /k B T )] 2

C v ( r ) = 3 NA k B ∑

.

(5.20)

According to (5.19), the frequencies ω j depend on the particle size. The asymptotic expansion of heat capacity (5.20) using Poisson’s equation for the temperature range h ω min /k B > θ D , the upper integration limit in (5.22) is 1 >> θ D/T → 0, the integral D(x) → 13 ( θ D /T) 3 and heat capacity C v → 3R, i.e. tends to a limiting value determined by the Dulong– Petit law. According to the exact solution [106], the sum of the second and third terms in (5.21), i.e. the enhancement of the vibrational part of the heat capacity of the small particle with an allowance for the quantum size effect has the form

∆C = C v (r ) − C v = vm ∑ ∑ l s

3(2l + 1)k B 4πr

3

ξ2

exp ξ , (expξ − 1) 2

(5.23)

where ξ = hcal′ ,s / k B rT ; the summation in (5.23) is performed over all

s and l up to l max determined from the condition 181

l max

∑ (2l − 1) ≤ N l=0

.

Nanocrystalline Materials

From equations (5.21) and (5.23) and also from equations (5.16) or (5.17) it follows that at h w min /k B T 0 this difference is positive, i.e. ∆C > 0 (Fig. 5.9). The size effect of the vibrational (lattice) part of heat capacity was considered by the authors of [100, 105–108]. The electron subsystem of bulk metals in low- and high-temperature ranges provides a significant electronic contribution C el = γ e T to heat capacity. Estimation of the electronic heat capacity of the nanoparticles is complicated by discrete electronic energy levels which form as a result of a limited number of atoms. In the case of small particles and low temperatures, when the mean distance

% θ 

% θ 

Fig. 5.9. Temperature dependence of the heat capacity C: solid line is the heat capacity temperature dependence in according to the Debye theory, horizontal dashed line is the Dulong–Petit limit of heat capacity, and dashed curve is the deviation of heat capacity from the Debye theory, which is associated with the quantum size effect [107].

182

Properties of Isolated Nanoparticles and Nanocrystalline Powders

between the level is δ = h p F /2m * D > k B T (p F is the Fermi momentum, D is the particle size, and m* is the effective mass of the conduction electron), electronic heat capacity C el may greatly differ from that of bulk metal. The dependence C el (T) is determined by the distribution of energy levels. The linear dependence of the electronic heat capacity on temperature with the coefficient 2 γ e was obtained by the authors of [109, 110]. They used the 3 assumption on the random distribution of electronic levels. The theoretical analysis of the heat capacity in two-dimensional systems [111] showed that the electronic part of heat capacity is a linear function of temperature and the vibrational part is a quadratic function of temperature. This is in agreement with the conclusion made in [100, 105–108], according to which the temperature dependence of the heat capacity of small particles contains the quadratic term bT 2 determined by the surface contribution. For a bulk crystal with boundary wave vector k the authors of [112] transformed equation (5.19) and obtained the size dependence of the Debye temperature. If k = (6 π 2 /v) 1/3 is the boundary wave vector in a bulk solid and v = V/N is the atomic volume, then N = (2/9p)k 3 r 3 . Taking this into account, equation (5.19) may be written in the form γ *e =

k 3r 2 = r 2 kn3 + (9π / 8) rkn2 + 3kn

(5.25)

or, with the accuracy of the terms of the first order, k n = k(1 + ∆ k / k n)

k n θ D (r ) 3r 2 k 2 + (9π / 8)rk = ≈ 2 2 . θD k 3r k + (9π / 4)rk + 3

(5.26)

If the last term rk n in equation (5.19) is ignored, this gives a simple size dependence of the Debye temperature

k n θ D ( r ) 1 + (3π / 8rk ) = ≈ k 1 + (3π / 4rk ) . θD

(5.27)

Equation (5.27) may be written with a small decrease in accuracy in the form 183

Nanocrystalline Materials

θ D (r)/ θ D ≈ 1 – 3 π /(8rk) .

(5.28)

To estimate θ D (r) of a small particle of an arbitrary shape, having the volume V and the surface are S, equations (5.27) or (5.28) may be used taking the approximation r ≈ 3V/S into account:

θ D (r ) 1 + (πS / 8Vk ) ≈ 1 + (πS / 4Vk ) . θD

(5.29)

In [112], surface tension σ was additionally taken into account and equation (5.28) was transformed to the form

θ D (r)/ θ D ≈1 + [(2K σγ /r) – (3 π /8rk T )],

(5.30)

where γ is the Gruneisen constant and k T is isothermal compressibility. The dependence of the Debye temperature θ D (r) on the effective velocity of sound c may also be found using the expression proposed by the authors of [106] for the maximum vibration frequency ω max = a ′max c( r ) / r for the particle:

′ c( r ) / k B r . θ D (r ) = hω max / k B = ha max

(5.31)

As a rule, the Debye temperature θ D (r) for the nanoparticles is lower than θ D for the bulk materials. Allowing for this, equation (5.31) shows that the effective velocity of sound in the nanoparticles decreases with a decrease of the particle size and is proportional to r m , where m > 1. The changes in the phonon spectrum of the small particles should also affect the value of the root-mean-square dynamic atomic displacements ω

1 max ε (ω , T ) 〈u 〉 = g (ω )dω . Nm ω ∫ ω2 2

(5.32)

min

Heat capacity is one of the most extensively studied properties of nanoparticles. Let us discuss the results of investigation of the heat capacity of colloid nanoparticles of silver and gold. 184

Properties of Isolated Nanoparticles and Nanocrystalline Powders

Measurements were performed at very low temperatures from 0.05 to 10.0 K in the magnetic field B from 0 to 6 T [113]. At T < 1 K, the heat capacity of the nanoparticles of Ag (D = 10 nm) and Au (D = 4, 6 and 18 nm) is 3–10 times higher than the heat capacity of bulk specimens of silver and gold. The heat capacity of the largest Au particles (D = 18 nm) in the temperature range from 0.2 to 1.0 K almost coincides with that of the bulk specimen. With a decrease of the Au particle size from 18 to 6 nm an additional positive contribution to heat capacity initially increases and with a further decrease of the particle size to 4 nm it decreases but does not disappear and remains positive even for clusters Au 55 with a size of 1.5 nm. Measurements of the heat capacity of Ag nanoparticles with a size of 10 nm in the magnetic field with B = 6 T detected the quantum size effect: at T < 1 K, the heat capacity of Ag nanoparticles was lower and at T > 1 K it was higher than the heat capacity of bulk silver. In the absence of the magnetic field, the heat capacity of colloid nanoparticles of silver in the entire investigated temperature interval was higher than that of bulk Ag (Fig. 5.10). This experimental result is in good agreement with theoretical conclusions [107] on the quantum size effect of the heat

10-2 Ag nanoparticles

10-3

D = 10 nm B=0

C (J g-1 K-1)

B=6T

10-4

bulk Ag

10-5

10-6

10-7

0.1

1

10

T (K)

Fig. 5.10. Specific heat C versus temperature T of colloid Ag with D = 10 nm [113]: measurements were performed in magnetic fields B up to 6 T, dashed line corresponds to the specific heat of bulk silver.

185

Nanocrystalline Materials

capacity of the nanoparticles. Such an effect was not detected in the case of colloid particles of gold because their heat capacity becomes immeasurably small with increasing density of the magnetic flux. Measurements of the heat capacity of Pb nanoparticles with a diameter of 2.2, 3.7 and 6.6 nm and In nanoparticles with a diameter of 2.2 nm [114, 115] show that at T < 10 K the heat capacity C v (r) of nanoparticles is 25–75 % higher than the heat capacity C v of bulk metals. The maximum deviation ∆C = C v (r) – C v was observed in the temperature range 3–5 K. The sharp decrease of C v (r) at T ≤ 2 K was caused by the low-frequency truncation of the phonon spectrum as a result of the size effect. Results of [114, 115] were explained [106] by means of equation (5.23). Theoretical analysis [106] was performed for the case of a homogeneous spherical nanoparticle with a diameter of 2.2 nm, consisting of 184 atoms and accounting for the first 183 vibrational modes. The authors of [116] measure ∆C of vanadium nanoparticles with a diameter of 3.8 and 6.5 nm and palladium nanoparticles with a diameter of 3.0 and 6.6 nm, produced by vapour condensation. The heat capacity of vanadium particles at T < 10 K is determined in the main by the electronic contribution and the value of ∆C, determined by the size effect of the lattice heat capacity, is relatively low. An increase in the heat capacity of Pd nanoparticles in comparison with bulk palladium at 1.4 < T < 30.0 K (Fig. 5.11) is completely determined by the additional lattice contribution because the electronic heat capacity, regardless of the particle size, 

P- PRO 

%



 



.





3G









%









.

Fig. 5.11. Temperature dependences of the heat capacity of Pd nanoparticles with a diameter 3.0 nm (1) and 6.6 nm (2) and bulk palladium (3) [116].

186

Properties of Isolated Nanoparticles and Nanocrystalline Powders

is described by the normal linear law γ e T, and the coefficient γ e is the same as for bulk palladium. The similar size effect on the heat capacity of nanocrystalline powder of palladium with a mean particle size of 8 nm was observed in [117]. The temperature dependence of the heat capacity of nanocrystalline palladium n-Pd at 1 K ≤ T < 20 K was described by the function C(T) = aT + bT 2 + cT 3 , similar to equation (5.17) at a fixed value of r. The C(T) dependence of bulk palladium did not contain the quadratic term bT 2 . The coefficient of electronic heat capacity, γ e = a, of n-Pd was slightly lower, and the temperature coefficient of lattice heat capacity, c, was twice as high as the coefficients a and c for bulk palladium (Table 5.2). The results obtained in [117] are in good agreement with the data in [115] for the heat capacity of n-Pd. The heat capacity of bulk copper and nanocrystalline powders of Cu and CuO with a particle size of ~50 nm was investigated in temperature ranges 1–20 K and 300–800 K [118]. At a temperature below 20 K the heat capacity was described by the polynomial C(T) = aT + bT 2 + cT 3 (the values of the coefficient of the polynomial are presented in Table 5.2). The quadratic term bT 2 was presented only in the temperature dependence of the heat capacity of Cu nanoparticles. It should be mentioned that the coefficients of the linear and cubic terms of heat capacity of n-Cu were higher than the appropriate coefficients for the bulk copper (Table 5.2). At all temperatures (from 1 to 20 K and from 300 to 800 K) the highest heat capacity was recorded for the CuO nanopowder and Table 5.2 Coefficients of heat capacity polynomial C(T) = aT + bT 2 + cT 3

S p e c ime n

a (mJ mo l–1K –2)

b (mJ mo l–1K –3)

c (mJ mo l–1K –4)

Re fe re nc e s

Pd (b ulk )

9.7±0.2

0

0.10±0.03

11 7

na no - P d (D~ 8 nm)

8.5±0.2

0.10±0.03

0.20±0.03

11 7

Cu (b ulk )

0.68

0.01

0.051

11 8

na no – C u (D~ 5 0 nm)

1.03

0.32

0.066

11 8

C uO (D~ 5 0 nm)

0

0

0.410

11 8

187

Nanocrystalline Materials

the lowest heat capacity was found for bulk copper. The heat capacity of the Cu nanoparticles was from 1.2 to 2.0 times higher than that for bulk copper at temperature up to 450 K. With a further increase of temperature a growth of the Cu nanoparticles and a decrease of heat capacity to the values corresponding to bulk copper are observed. According to [119], the heat capacity of the nickel nanoparticles with a diameter of 22 nm is 2 times higher than the heat capacity of bulk nickel at 300–800 K. The C(T) dependence of n-Ni shows a weak diffuse exothermic effect at 380–480 K, associated with the recrystallisation of nickel particles, and a high endothermic peak with a maximum at 560 K determined by the magnetic phase transition. In bulk nickel, a weak endothermic peak, corresponding to the magnetic transformation, was recorded at 630 K. The authors of [120] used inelastic neutron scattering at 100– 300 K to study the phonon density of states of coarse-grained polycrystalline nickel and nanocrystalline nickel with a particle size of 10 nm in the form of a powder and a pressed compacted specimen with a relative density of 80 %. The largest size effect is an increase of the density of phonon states of n-Ni specimen in comparison with coarse-grained Ni at energies below 15 meV (Fig. 5.12). According to [120], the variation of the phonon spectrum of n-Ni is caused by the low density of substance in the grain boundaries. A decrease of the Debye temperature θ D , associated with a decrease of the particle size, was observed by many researches (Table 5.3). The relative values θ D (r)/ θ D were determined by calorimetric and diffraction methods. However, examination of small particles of Au and Fe (D = 5–7 nm) by the Mössbauer effect shows that the Debye temperature for these particles is the same as that for bulk crystals [126, 127]. Comparison of the lattice constant of small particles of Au and Fe with the relative intensity of scattered X-ray [128] also shows that the detected effects cannot be explained only by a decrease of the Debye temperature. According to [5], the contradiction of the experimental data for the Debye temperature of small particles indicates that it is necessary to take into account the oscillations of the clusters (metastable atomic groupings with increased local stability), which form a nanoparticle and have the symmetry different from that of the crystal. It is also necessary to take into account anharmonic effects, which are quite strong in the nanoparticles.

188

Properties of Isolated Nanoparticles and Nanocrystalline Powders Table 5.3 Size dependence of Debye temperature θ D (r) for small metal particles ( θ D is Debye temperature of a bulk metal) M e ta l

P a rtic le size (nm)

θ D( r ) / θ D

Re fe re nc e s

Ag

~ 20

0.75

121

Ag

10–20

0.75–0.83

122

Ag

15

0.735

123

Al

15–20

0.50–0.67

122

Au

2.0

0.69

124

Au

1.0

0.92

11 2

Au

10.0

0.995

11 2

In

2

0.80

11 5

Pb

2.2

0.87

11 5

Pb

3.7

0.90

11 5

Pb

6.0

0.92

11 5

Pb

20.0

0.84

125

Pd

3.0

0.64–0.83

11 6

Pd

6.6

0.67–0.89

11 6

V

3.8

0.83

11 6

V

6.5

0.86

11 6

 ,

  

E



Jω



PH9



   F

   













Kω π PH9







Fig. 5.12. Phonon density of states, g( ω ), of nanocrystalline compacted nickel Ni (a), non-compacted powdered nanocrystalline Ni (b) and coarse-grained bulk Ni (c) [120].

189

Nanocrystalline Materials

5.4 MAGNETIC PROPERTIES The special features of the magnetic properties of the nanoparticles are associated with the discrete electronic and phonon states. One of these special features is the oscillation dependence of the susceptibility of the nanoparticles of paramagnetic metals on the strength of the magnetic field H. In addition to this, Curie paramagnetism may greatly overlap Pauli paramagnetism of metals due to a small size of the nanoparticles. The results of theoretical and experimental investigations of the magnetic properties of nanoparticles of paramagnetics are considered in reviews [10, 11]. The effect of the degeneration of the electronic states on the magnetic susceptibility of the small particles of paramagnetic metals taking into account the even or odd number of electrons in them was discussed in [109, 110, 129]. In weak magnetic fields µ p H 20 kOe with a decrease of temperature below 70–80 K, the susceptibility of clusters Hg 13 increased in accordance with the Curie law to high paramagnetic values ( χ = 1 cm 3 g –1 at H = 40 kOe), although bulk mercury is a diamagnetic. According to [137, 138], the magnetic susceptibility of Na clusters in zeolite is also governed by the Curie law, even in strong magnetic fields. Variation of the magnetic susceptibility of Ag clusters in zeolite in accordance with Curie–Weiss law in the temperature interval from 4 to 300 K was detected in [139]. The increase of paramagnetic susceptibility of the Mg nanoparticles (D ~ 3 nm) in comparison with bulk magnesium and a large decrease of susceptibility of the nanoparticles at T → 0 are noted in [140]. According to the authors of [12], these experimental results are explained by the fact that the very small clusters and nanoparticles of these metals do not have metallic properties, because their outer s-electrons are localised on atoms. Therefore, a normal exchange interaction forms between the atoms in the clusters. The clusters and nanoparticles of metals lose the metallic properties with decreasing size. For example, investigation of photoemission from clusters Pt 6 [141] and tunneling phenomena in Fe 13 clusters with a volume of 0.15 nm 3 (D ~ 0.5 nm) [142] shows that these clusters are not metals (although the Fe 35 cluster already have metallic properties). According to [143], mercury clusters, containing from 20 to 70 atoms, are characterised by a transition from the Van-der-Waals crystal to the metal. In [117] the magnetic susceptibility of nanocrystalline palladium particles (D = 8 nm) and bulk palladium was measured in the temperature range of 1.8–300.0 K. In the entire temperature range, nano-Pd and bulk Pd are paramagnetics, a decrease in temperature increases susceptibility. The χ (T) dependence of bulk palladium at T ≈ 80 K shows a weak diffuse maximum which was not found on the similar dependence in the case of nano-Pd. At T > 20 K and up to 300 K, the susceptibility of nano-Pd is 20–25 % lower than that of bulk palladium. According to [117], the absence of a maximum on the χ (T) dependence of Pd nanoparticles indicates a large difference of the electronic energy spectra of nano-Pd and bulk Pd in the vicinity of the Fermi energy. The results of magnetic measurements [117] cause certain doubts because the temperature dependence of the susceptibility of bulk palladium greatly differs from that recorded in reliable and accurate experiments [144, 145]. The anomalies of magnetic susceptibility of the nanoparticles were manifested in investigations by the EPR method. According 191

Nanocrystalline Materials

to [146], a decrease in the size of the nanoparticles should reduce the width of the EPR lines and this effect should be detected for particles smaller than 10 nm. However, examination by the EPR of small particles of Na with a size from 600 to 2 nm [147, 148] showed a reverse dependence: with a decrease in the size of sodium particles, the width of the EPR lines increased. A large broadening of the EPR lines of the Gd nanoparticles (D ~ 10 nm) in comparison with bulk Gd was detected by the authors of [149]. The nanostructured state affects the properties of ferromagnetics. Ferromagnetic materials have a domain structure which forms as a result of minimization of the total energy of the ferromagnetic in the magnetic field. According to [150], this energy includes the exchange energy, which is minimum for the case of parallel electron spins; the energy of the crystallographic magnetic anisotropy, determined by the presence in the crystals of axes of easy and hard magnetization; magnetostriction energy, associated with the change of the equilibrium distances between the lattice sites and the length of the domain; magnetostatic energy, linked with the existence of magnetic poles both inside the crystal and on its surface. Closure of the magnetic fluxes inside the domains, distributed along the axes of easy magnetization, decreases the magnetostatic energy, whereas any disruptions of the homogeneity of the ferromagnetic (especially, interfaces) increase its internal energy. With a decrease in the size of the ferromagnetic, the closure of the magnetic fluxes inside the ferromagnetic is less and less advantageous from the viewpoint of energy. Whilst the ferromagnetic particles have a multi-domain structure, its interaction with the external magnetic field is reduced to the displacement of the boundary layer (wall) between the domains. With approach of the ferromagnetic particles to the single-domain state, a coherent rotation of the majority of magnetic moments of the separate atoms becomes the main mechanism of remagnetisation. This is inhibited by the anisotropy of the shape of the particles, crystallographic and magnetic anisotropy. When reaching a certain critical size D c, the particles consist of single domain and this accompanied by an increase in the coercive force H c to the maximum value (for remagnetising the single-domain spherical particle by coherent rotation it is necessary to apply a reverse magnetic field, i.e. maximum coercive force H c = 2K/I s , where K is the anisotropy constant and I s is saturation magnetisation). According to [151], the largest size of the single-domain particles of Fe and Ni does not 192

Properties of Isolated Nanoparticles and Nanocrystalline Powders

exceed 20 and 60 nm respectively. A further decrease of the particle size leads to a large decrease of the coercive force to zero as a result of transition to the superparamagnetic state. The critical linear size of the particles, D c , corresponds to a disappearance of ferromagnetism at a temperature below the Curie point owing to thermal fluctuations of the orientation of the magnetic moment. Taking into account the Heisenberg’s uncertainty principle it was shown in [150] that the critical linear size of the particles is about 1 nm. Indeed, if the size of a ferromagnetic particle is equal to some value δ 0 , then the momentum p of the electron, which freely propagates in the volume of the particle, has the uncertainty ∆p. Heisenberg’s uncertainty principle shows that ∆p ≈ h / δ 0 . Part of the electron energy, determined by the limited size of the particles, is equal to ∆ ε 0 = (∆p) 2 /2m e ≈ h 2 /2m e δ 0 2

(5.33)

or, taking into account the values of h and m e ∆ ε 0 ≈ 6.1×10 –39 /δ 0 2 ,

(5.34)

where energy ∆ ε 0 is measured in J, and size δ 0 is measured in meters. Energy ∆ ε 0 has a disordering effect on the magnetic moments, similar to the effect of thermal vibrations. With disruption of the homogeneity of magnetisation, a correction must be introduced to the energy of exchange interaction. This correction is maximum when the magnetisation vector changes its direction to reversed at distances of the order of the distance between the adjacent metallic atoms, i.e. at distances at the order of lattice constant a. The physical meaning of the correction is that the exchange energy tends to maintain the isotropy of magnetisation at any disruption. In other words, the exchange energy is the energy of magnetic ordering. The maximum correction of the exchange max energy is ∆ε exch ≈ AV / a 3 , where A is the exchange energy, V is the volume of the solid. Complete disruption of the isotropy of magnetisation and disorientation of the magnetic moments take place at Curie temperature T C when the spontaneous magnetisation of max should be ferromagnetic decreases. Therefore, the correction ∆ε exch equal to or slightly lower than the thermal energy k BT CV/a 3 . From this it follows that the exchange energy is

193

Nanocrystalline Materials

A ≈ k BT C .

(5.35)

Equating the disordering energy ∆ ε 0 (5.34) and the ordering energy of exchange A (5.35), it is possible to evaluate the critical linear size δ 0 of the ferromagnetic particle at which the ferromagnetism disappears at all temperatures because of disordering of the magnetic moment under the effect of energy ∆ ε 0 :

δ 0 ≈ 2×10 –8 T C–1/2 [m].

(5.36)

According to (5.36), for ferromagnetics with a Curie temperature of 500–1000 K the critical size of the particle at which ferromagnetism disappears and transition to the superparamagnetic state takes place, is ~1 nm. The exchange energy is slightly lower than k B T C and, therefore, quantity d 0 might be slightly higher, as indicated by the estimation using equation (5.36). For typical ferromagnetics, the transition to the superparamagnetic state is possible when the particle size becomes smaller than 1–10 nm. Analysis of the literature data on the dependence of coercive force H c on the mean size of the ferromagnetic particles [4] confirms the increase of H c with a decrease in the particle to some critical size. The maximum values of H c are obtained for particles of Fe, Ni and Co with a mean diameter of 20–25, 50–70 and 20 nm, respectively. These values are close to theoretical estimates of D c of single-domain particles [151]. As regards a decrease of H c at D < D c, it may be associated not only with the superparamagnetism effect but also with other magnetic properties of the surface layer. For example, if the anisotropy of the surface layer is reduced, the layer will be remagnetised in weaker fields and facilitate the magnetisation of the entire nanoparticle [151]. The dependence of the relative residual magnetisation I r /I s (here I s is saturation magnetisation of bulk metal) on the size of particles of Fe, Co and Ni also pass through a maximum in the vicinity of values of D c for these metals [4]. A decrease of saturation magnetisation with a decrease of the size of Fe, Ni and Co nanoparticles in ferromagnetic alloys was detected in many studies [152–159]. The authors of [4, 152–156] treat the decrease of I s as the result of oxidation of the surface layer of metallic nanoparticles, whereas in [157–159] the decrease of I s was explained directly by the size effect. The authors of [159] have investigated the magnetic properties of spherical particles of iron with a diameter of 40–100 nm at 194

Properties of Isolated Nanoparticles and Nanocrystalline Powders

temperatures at 4.2–300 K in fields with a strength of up to 25 kOe. Particles were suspended in paraffin and their volume concentration was 0.01. Examination by nuclear gamma resonance shows that the iron particles are not oxidised. Measurements of the coercive force of particles of different sizes at 4.2, 77 and 300 K showed a distinctive maximum of H c at D ~ 24 nm. According to [159], this maximum is determined by the superposition of two process: increase of H c in transition of the particles to the singledomain state and the appearance of superparamagnetism in singledomain particles when they reach the critical size. The saturation magnetisation I s, even for the largest particles of iron (D ~ 98 nm), was smaller than saturation magnetisation of bulk iron; with a decrease of the particle size from ~ 40 to 35 nm, I s decreases and with a further decrease of size the saturation magnetisation remains constant. The maximum of the I r /I s ratio is observed for particles with a size of 24 nm (I r is a residual magnetisation). According to [159], transition of Fe particles from ferromagnetic to superparamagnetic state is observed when particle size is ≤ 24 nm. Investigations of the saturation magnetisation of bulk Ni and nanocrystalline powder of Ni (D = 12, 22 and 100 nm) at 10–300 K [119] show that with a decrease of the particle size to 12 nm, I s is almost halved in comparison with bulk nickel. At temperatures below 50 K for nanoparticles of Ni with D < 50 nm, the magnetic hysteresis loop was asymmetric. According to [119] the displacement of the hysteresis loop and a decrease in I s are associated with a presence of the surface oxide shell and are determined by the anisotropy of the exchange interaction of ferromagnetic nickel with the antiferromagnetic oxide NiO, which forms the shell of nanoparticles. On the whole, at present there is no common view regarding the reasons for the change in the saturation magnetisation of ferromagnetic nanoparticles. It is important to mention the results of investigation of compacted nanocrystalline nano-Ni with a mean grain size of 70–100 nm [160]. The saturation magnetisation of compacted nano-Ni was approximately 10 % lower than that of coarse-grained nickel; the same was also reported for submicrocrystalline nickel produced by deformation and heat treatment [161]. To explain this effect, the authors of [162] assumed that the atoms distributed in the vicinity of grain boundaries, being in the non-equilibrium state, are dynamically more active than the atoms in the grains and form a grain-boundary phase. The difference in the magnetic properties of the grain195

Nanocrystalline Materials

boundary phase in relation to the properties of the phase inside the grain is the reason for a decrease of the saturation magnetisation of submicrocrystalline nickel. It may be possible that the decrease of the saturation magnetisation of nanoparticles may be associated not only with oxidation of the surface or with the size effect, but also with the special state of the surface layer of isolated nanoparticles or powder nanoparticles. At the same time, the presence of the developed surface of nanoparticles itself is a consequence of their small size. Recently, the dependence of the coercive force on the size of nanoparticles of Fe, Ni and nanoparticles of Fe 0.91 Si 0.09 alloy was studied by the authors of [163, 164]. The nanocrystalline powders of Fe, Ni and Fe 0.91 Si 0.09 alloy with a minimum particle size of 8, 12 and 6 nm, respectively, were produced by ball milling for 380, 350 and 180 hours. The magnetic measurements showed that a decrease in the size of iron nanoparticles from 80 to 8–10 nm is accompanied by an increase in coercive force H c by almost a factor of 3. The dependence of coercive force on the size of particles nano-Ni showed a maximum corresponding to nanoparticles with a diameter of 15–35 nm; with a decrease in the particle size from 15–12 nm, the value of H c rapidly decreases almost 5 times. Saturation magnetisation I s of nano-Ni particles (D = 10 nm) was 37 % higher than that of bulk nickel. However, this may be associated with the appearance of an impurity of 15 at.% Fe as a result of milling. A decrease in the size of nanoparticles of Fe 0.91 Si 0.09 from 40 to 6 nm is accompanied by an increase of the coercive force 5 times. The size and temperature dependences of the coercive force of cobalt nanoparticles with nitrided and oxidised surfaces were investigated in [165]. The size of Co nanoparticles was equal to 15–60 nm. The coercive force of the nitrided and oxidised particles of Co increased with a decrease of temperature from 240 K and 200 K for particles with a size of ~ 10 nm and 30–50 nm respectively. The highest value H c ≈ 2 kOe was recorded at 5 K for nanoparticles with a diameter of 34 nm. According to [165], oxidation leads to a larger increase of H c of the cobalt nanoparticles than nitriding. It should be mentioned that the increase of H c as a result of oxidation of the Fe and Co nanoparticles was detected previously in [166–168]. A large (up to ~ 800 K) increase of the Neel temperature was found in bcc Cr nanoparticles with a diameter of 38–75 nm [169], although bulk chromium is antiferromagnetic with a Neel 196

Properties of Isolated Nanoparticles and Nanocrystalline Powders

temperature of 311 K. Interesting results were obtained in examining the magnetic properties of nanoparticles of hematite α -Fe 2 O 3 [170]. In the normal state, hematite is an antiferromagnetic. Measurements show that with a decrease in the particle size from 300 to 100 nm, the magnetic susceptibility of nanocrystalline hematite does not change and equal to the magnetic susceptibility of the bulk crystal. A decrease in the particle diameter from 100 to 20 nm resulted in a rapid increase of magnetic susceptibility. Analysis of this spontaneous magnetisation of the nanoparticles [171] carried out in the approximation of the molecular field, shows the presence of the size dependence of Curie temperature. According to [171], a decrease in the Curie temperature becomes noticeable for particles with a size of D < 10 nm; for particles with D = 2 nm, a decrease in T C in comparison with bulk metal does not exceed 10 %. However, examination of the thermodynamics of superparamagnetic particles by the Monte-Carlo method does not detect any dependence of the Curie temperature on the particle size [172]. In fact, the transition of the nanoparticles from the superparamagnetic state to paramagnetic state is smooth, without any sharp point of magnetic transformation. Measurements of the Curie temperature of nanoparticles of nickel (D = 2.1–6.8 nm) [173], saturation magnetisation and the Curie temperature of Fe films with a thickness of ≥1.5 nm [174], saturation magnetisation of Fe nanoparticles (D = 1.5 nm) [175] and Co nanoparticles (D = 0.8 nm) [176] show that these values coincide (within the measurement error range) with those for bulk metals. According to [4, 5], the Curie temperature of the ferromagnetic particles with a decrease of their size to 2 nm does not differ from that of bulk metals. However, the authors of [177] detected a decrease of T C by 7 and 12 % for nickel nanoparticles with a diameter of 6.0 and 4.8 nm, respectively. It should be noted that the superparamagnetism phenomenon greatly complicates the investigation of the size dependences of the coercive force, saturation magnetisation and Curie temperature of the ferromagnetic nanoparticles. A decrease in the size of the single-domain particle leads to the transition from ferromagnetic state to superparamagnetic state. Thermal fluctuations may cause rotation of magnetic moments, if the mean thermal energy k B T is equal to or higher than the anisotropy energy of E = KV, where K is the constant of anisotropy and V is the volume of the particle. The total magnetisation of the particle, which is achieved in an external magnetic field with 197

Nanocrystalline Materials

sufficient strength for saturation, becomes equal to zero after switching the field off during relaxation time τ r . In the model of discrete orientations [178], the relaxation time is

τ r = τ 0 exp( KV / k BT ) .

(5.37)

If measurement time τ m is greatly shorter than the relaxation time τ r , the particle retains the initial ferromagnetic state. Otherwise, when the measurement time τ m is longer than τ r , the thermal fluctuations completely disorientate magnetic moments and the particle will behave a superparamagnetic one. The transition from the ferromagnetic to the superparamagnetic state takes place at a certain blocking temperature T = T B , for which τ r = τ m . Taking equation (5.37) into account, the blocking temperature is

TB = KV /[ k B ln(τ m / τ 0 )] .

(5.38)

A nanoparticle of a ferromagnetic material which has the volume V, at T < T B behaves as a ferromagnetic, and at T > T B is in the superparamagnetic state. For the given temperature, the condition τ r = τ m also determines critical volume V B (blocking volumes): a nanoparticle with V < V B is in the superparamagnetic state, a nanoparticle whose volume is larger than critical (V > V B ) is a ferromagnetic. Estimates [150] show that for typical ferromagnetics or ferrimagnetics at 100 K the critical volume is 10 –27 –10 –23 m 3 and corresponds to nanoparticles with linear size smaller than 1–15 nm. Superparamagnetism is found in nanoparticles (D ≤ 10 nm) of Ni in matrixes of silicagel [179] and Pb [80]; Co in the Cu matrix [181] and in Hg [182]; Fe in Hg [175, 182] and in β -brass [183]. The experimental data on superparamagnetism have been reviewed in detail in [4, 5]. Therefore, only recent experimental investigations will be discussed briefly in this section. Detailed investigation of the magnetic properties of cobalt nanoparticles with a diameter of 1.8 to 4.4 nm, produced by deposition from a colloid solution, was carried out in [184]. The magnetic properties were measured using a SQUID-magnetometer in the temperature 2–300 K in a field with a strength of up to 50 kOe. At 300 K the Co nanoparticles were superparamagnetic. The variation of the blocking temperature T B from 22 to 50 K with an increase of the particle size from 1.8 to 4.4 nm was described 198

Properties of Isolated Nanoparticles and Nanocrystalline Powders

by the dependence T B = KV/30k B , i. e. by the function (5.38). Using the dependence T B (V) and experimental results for T B and the particle size, the authors of [134] found the size dependence of anisotropy constant K: the anisotropy constant increases with a decrease in the particle size and in the entire examined range 1.8 ≤ D ≤ 4.4 nm it is larger than K of bulk fcc cobalt. The size dependence of coercive force H c was measured at 10 K at which the nanoparticles of all sizes were ferromagnetic. Increase in H c with an increase of the size of nano-Co particles completely corresponds to the behaviour of single-domain particles. The size dependences of T B , K, H c of the cobalt nanoparticles are in good agreement with similar data for nanoparticles of other ferromagnetic metals. A different situation exists in the case of magnetisation. Measurements show that at T = 2 K the nanoparticles of Co do not reach magnetic saturation even in a field of 55 kOe. Therefore, the values of saturation magnetisation I s were obtained by the extrapolation of the dependence I(1/H) to an infinitely large field, i.e. 1/H → 0. The value of I s increases with a decrease in the size D, and I s of particles with D < 3.3 nm is higher than I s of bulk cobalt. The saturation magnetisation of the smallest Co particles (D = 1.8 nm) is 20 % higher than I s of bulk cobalt. An increase of the magnetic moment of the cobalt atom in the nanoparticles is predicted theoretically by the authors of [185, 186] and is observed in experiments [187] on cobalt clusters. The nanocrystalline powder of γ -Fe 2 O 3 (D ~ 4–7 nm) was synthesised by the plasma chemical method using a microwave generator [188]. The magnetic measurements show that nanoparticles of γ -Fe 2 O 3 are superparamagnetic with a blocking temperature of T B ≈ 80 K. With a decrease T < T B the particles of γ -Fe 2 O 3 behave as a ferrimagnetic, their residual magnetisation increases, reaching the maximum at 20 K, and then decreases. The size dependence of the blocking temperature of the γ -Fe 2 O 3 nanoparticles with a size from 3 to 10 nm, distributed in a polymer matrix, is determined in [189]. The dependence T B (V) is close to linear and is described by a function of type (5.38). The blocking temperature of the particles with a volume of ~100 nm 3 (D ~ 4–5 nm) is equal to ~75 K and is in good agreement with the results [188]. The superparamagnetism of Fe nanoparticles, distributed in a copper matrix, was investigated in [180]. Starting copper, containing approximately 0.01 at.% of dissolved Fe, was diamagnetic in the entire examined temperature range 300–1225 K. The temperature 199

Nanocrystalline Materials

dependence χ (T) of starting copper is in good agreement with the data [191]. As a result of severe plastic deformation of starting copper by equal channel angular pressing (the true logarithmic degree of deformation e was 3.5), the authors obtained submicrocrystalline copper smc-Cu with a grain size of 130–150 nm and observed precipitation of iron particles, previously dissolved in copper. Magnetic measurements were performed in a vacuum of 1.3×10 –3 Pa (10 –5 mm Hg) on a high-sensitive magnetic balance in a field with an induction of 8.8 kGs. To understand the results, it is important to know the measurement procedure: heating from 300 K to annealing temperature T; holding the specimen for 1 h at this temperature and measurement of the susceptibility at the end of holding cycle; cooling from annealing temperature to 300 K and measuring the susceptibility at 300 K; heating to the next annealing temperature, and so on (Fig. 5.13). The annealing temperature was varied from 300 to 1225 K with a step of 25 K. The measurements, made directly at the annealing temperature, relate to the temperature dependence of susceptibility and will be henceforth denoted by χ (T). The susceptibility measurements, carried out after annealing and subsequent cooling to 300 K, refer to the annealing curve and will be henceforth designated as χ (300, T). The results of measurements are shown in Fig. 5.14. The measurements show that the susceptibility of submicrocrystalline copper is higher than that of starting copper. In addition to this, it was observed that the susceptibility of smc-Cu is inversely proportional to the magnetic field strength H. This indicates the presence of a ferromagnetic impurity of iron in the 

%

.

    







 9 KRXUV

Fig. 5.13. Sequence of magnetic susceptibility measurements in annealing experiments: (  ) measurements in situ (at annealing temperature T), ( l ) measurements after cooling from annealing temperature to 300 K [192]. 200

Properties of Isolated Nanoparticles and Nanocrystalline Powders   

χ  FP J 



 











 





%





.

Fig. 5.14. Magnetic susceptibility χ of nanocrystalline copper (n-Cu) matrix with iron impurity in a magnetic field of H = 8800 Gs [190]: (  ) temperature dependence c(T); (l) annealing dependence χ(300, T); (¢) reverse temperature trend of susceptibility, corresponding to susceptibility of copper with 0.01 % of dissolved iron impurity

specimen. It is known that dissolved iron is precipitated from copper during rolling [193]. In [190], the precipitation of iron particles, previously dissolved in copper, was initiated by severe plastic deformation. The annealing dependence χ (300, T) in the vicinity of the nanotransition temperature T n ≈ 425 K (transition of copper from submicrocrystalline to coarse-grained state) exhibits a pronounced abrupt increase of susceptibility (Fig. 5.14). Over the temperature interval from 450 to 650 K the χ (300, T) curve does not virtually change. As the annealing temperature is raised further, the χ (300, T) curve increases gradually, passes through a maximum at a temperature of 975 K and then decreases drastically to diamagnetic values, corresponding to the susceptibility of copper free from ferromagnetic impurities (Fig. 5.14). The susceptibility χ (300, T) of smc-Cu ceases to depend on H after annealing at T > 1200 K. Measurements of the temperature dependence of susceptibility, χ (T), revealed a large decrease of χ even at a temperature under 425 K. After a slight increase of the susceptibility in the range 425–475 K, the temperature curve descends to diamagnetic values and at a temperature of 850 K transforms to the temperature dependence of the susceptibility of copper free from ferromagnetic impurities. The dependence of χ on the field strength H disappears at T > 850 K. 201

Nanocrystalline Materials

The reverse temperature trend of the susceptibility from 1225 to 300 K (Fig. 5.14) corresponds to the susceptibility of copper with 0.01 at.% of iron impurity dissolved in it [194]. There is no magnetic field strength dependence of susceptibility in this case. One of the most interesting experimental results discovered in [190, 195–197] is a jump on the annealing and temperature dependences of magnetic susceptibility at a temperature of T n ≈ 425 K, corresponding to the transition of submicrocrystalline copper to coarse-grain copper. Is the detected jump associated with the change in the susceptibility of copper proper? The main contributions to the magnetic susceptibility of bulk crystalline copper are the diamagnetism of atomic cores, the Pauli spin paramagnetism and the Landau diamagnetism of conduction electrons. The sum of these contributions for copper is negative and, therefore, copper is a diamagnetic. The weak quadratic temperature dependence of susceptibility [191] is determined by the Pauli contribution. In the considered case, the sign of the susceptibility of smc-Cu is positive as a result of the precipitation of iron particles. The lowest susceptibility of submicrocrystalline copper (Fig. 5.14, dependence χ (300, T)) could be a consequence of the lower density of states on the Fermi level and the lower effective mass of conduction electrons. However, this should not result in any large change of the temperature dependence of the susceptibility. Therefore, the difference in susceptibility values on the annealing and temperature curves of smc-Cu after the nanotransition (∆ χ = χ (300, T) – χ (T) at T ~ 500 K), cannot be explained only by change in the state of copper. The analysis carried out in [190, 195–197] showed that the jump of χ on the annealing and temperature dependences of susceptibility at 425–450 K is probably not associated with the change in the susceptibility of copper but with a variation of the magnetic contribution from the iron impurity, precipitated in the form of nanoparticles at joints of copper grains. If one assumes that the jump on the dependences χ (T) and χ (300, T) at the nanotransition temperature in smc-Cu is associated with a change in the magnetic contribution by an impurity, then one can subtract from χ (T) the susceptibility of copper, χ Cu (T), and thereby can determine the contribution of the ferromagnetic phase, χ Fe (T), to susceptibility (Fig. 5.15). This phase may be a surface phase or bulk phase. If this is a two-dimensional surface phase, it can lie on the boundary of two grains. If the phase is a threedimensional one, it is most likely to be located at the joints of three 202

Properties of Isolated Nanoparticles and Nanocrystalline Powders 





%.0



χ0  FP J 

%3





CHECK SYMBOLS IN CAPTIOM 

 







%





.

Fig. 5.15. Approximation of the susceptibility versus temperature curve for the iron impurity superparamagnetic phase [190]: (1) and (2) are the variations of susceptibility after the nanotransition with and without allowance for the dissolution of Fe impurity; (3) variation of susceptibility before the nanotransition; ( o ) value of susceptibility of the superparamagnetic impurity at 300 K after the nanotransition in copper. The vertical dashed lines show the region of the nanotransition near temperature T n , and also Curie temperature TCFe of crystalline iron.

or more grains. Let us consider in detail the interpretation of χ Fe (T) dependence, proposed by the authors of [190, 197]. Interpretation was carried out in the approximation of precipitation of iron particles of the same size and the independence of Curie temperature T C of iron on the nanoparticle size. For conventional ferromagnetics, the temperature dependence of the susceptibility in saturating magnetic fields at low temperature is not so strong as that observed experimentally (Fig. 5.15) [190]. A strong dependence at low temperature is possible in the case of the superparamagnetism of precipitated iron particles. Expressed in dimensionless units, the superparamagnetic contribution χ sp , at temperature T in magnetic field H may be represented in the form [150]:

χ sp = nspVsp

M s (T )  Vsp M s (T ) H  , L  k BT H  

(5.39)

with L = [coth(x) – 1/x] being the Langevin function, n sp the number of superparamagnetic particles in the unit volume, V sp the volume of the superparamagnetic particle, M s (T) the saturation 203

Nanocrystalline Materials

magnetisation of the ferromagnetic phase at temperature T. The temperature dependence of the saturation magnetisation for the crystalline ferromagnetic phase, M s (T), is determined by solving the equation

 M (T )TC M s (T ) = tanh s M s (0)  M s (0)T

  , 

(5.40)

where M s (0) is saturation magnetisation at T = 0 K. For crystalline iron M s (0) = 1740 Gs and T C = 1043 K [198]. To the quoted saturation magnetisation M s (0) = 1740 Gs corresponds the iron atomic magnetic moment 2.22µ B (µ B is the Bohr magneton). The ferromagnetic contribution disappears at a temperature of 850 K, which is much lower than the Curie temperature of conventional polycrystalline iron (Fig. 5.14, 5.15). This may be due to complete equilibrium dissolution of the ferromagnetic impurity in copper even at a temperature of 850 K. According to the phase diagram [193], the undissolved iron concentration in copper decreases and the dissolved iron concentration, c Fe (T), in copper decreases with an increase of temperature in accordance with the equation c Fe (T) = c Fe (0) – Cexp(–E m/k B T) ,

(5.41)

where c Fe (T) is the relative atomic concentration of dissolved Fe at 0 K or the concentration at 300 K, which is virtually equal to this concentration; C is a constant; E m is the energy of mixing. According to [193] C = 43, E m/k B = 9217 K or E m = 0.79 eV. To go from the dimensionless bulk susceptibility χ sp and atomic concentration c Fe over to the mass susceptibility of the ferromagnetic phase, whose concentration varies owing to dissolution, one can use the relation

χ Fe (T ) = χ sp (T , nsp ,Vsp ) cFe (T )

AFe , ACu ρ Fe

(5.42)

where A Cu = 63.55 and A Fe = 55.85 are the atomic weights of copper and iron, ρ Fe = 7.86 g cm –3 is the density of iron. Approximating the temperature dependence of susceptibility χ Fe (T) in the temperature range from 425 to 1043 K by equation (5.42), 204

Properties of Isolated Nanoparticles and Nanocrystalline Powders

with allowance for equations (5.39)–(5.41), and assuming the Curie temperature equal to 1043 K and the mixing energy equal to E m = 0.79 eV, the authors of [190, 197] obtained a good agreement with the experimental data (Fig. 5.15, curve 1). Curve 1 also passes at 300 K through the point labelled £ which correspond to χ (300, T) values in the annealing temperature range 450–600 K and would lie on the temperature curve if there was no susceptibility jump at the nanotransition in copper. Curve 2 (Fig. 5.15) was plotted without allowing for the dissolution of iron, i.e. with the ferromagnetic impurity concentration remaining unchanged up to the Curie temperature. The fitting (Fig. 5.15, curve 1) gave the values of the volume of the superparamagnetic particle, V sp = 1.8×10 –20 cm 3 , and the number of particles, n sp = 5.7×10 14 cm –3 , after nanotransition, and also constant C = 0.4. This value of constant C is approximately 100 times lower than that in [193] and shows that the rate of dissolution of iron, obtained by authors [190, 197], is higher than that indicated in [193]. The high rate of dissolution of iron is a consequence of the fact that the precipitation of nanoparticles of Fe and their existence in the copper matrix are thermodynamically non-equilibrium. If the dissolution of iron in copper up to a temperature of 650 K is ignored, then the relative volume concentration of the superparamagnetic impurity, n sp V sp , in copper before and after nanotransition is the same. Consequently, in accordance with (5.39), the superparamagnetic contributions at 0 K are the same too. Assuming that the Curie temperature does not depend on the size of superparamagnetic particles and that the susceptibility of copper remains unchanged during the transition, the experimental data for the susceptibility prior to the nanotransition may be approximated (Fig. 5.15, curve 3). Approximation shows that prior to the nanotransition, the volume of the superparamagnetic particles was a factor of 1.62 smaller and the mean particle size a factor of (1.62) 1/3 = 1.17 smaller than after the transition. The difference in the temperature dependences of susceptibility before and after the nanotransition comes from the increase in the mean size of the superparamagnetic particles from 2.8 to 3.3 nm. The volume of copper per 1 superparamagnetic impurity particle is V = 1/n sp . It is possible to determine the linear size of the copper particle. It turns out that for each superparamagnetic particle there is one copper particle 128 nm in diameter prior to the nanotransition and a particle 150 nm in diameter after the transition. These sizes are similar to the grain size of copper in this specimen prior to and 205

Nanocrystalline Materials

after the transition. Therefore, one may assume that the impurity particles are distributed in copper uniformly and there is one iron particle per every copper grain. An iron particle may be located, for example, in the space where several grains are contacted. When the copper grains coarsen, the number of grain joining points decreases and the impurity iron atoms have to diffuse over the surface of copper grains to the points that remain. The impurity nanoparticles already present coarsen and their number decreases. Similar processes also take place at higher annealing temperature, when the copper grain growth continues. In the temperature 450–600 K, the annealing dependence χ (300, T) (Fig. 5.14) is almost constant. This means that the state of the superparamagnetic impurity, i.e. the number and size of the particles, does not change when the specimen is heated in this temperature range and subsequently cooled. For the temperature dependence χ (T) this is confirmed by the results of calculation (Fig. 5.15, curves 1 and 2) which shows that the dissolution of iron at 450–600 K is negligible. The increase of susceptibility χ (300, T) (Fig. 5.14) by the amount ~1×10 –7 cm 3 g –1 , observed after annealing at temperatures between 650 and 975 K, is partially associated with an increase in the size of superparamagnetic particles in copper cooled to 300 K and with an increase in the contribution from the impurity at 300 K. However, this factor may be responsible for an increase of χ (300, T) by an amount of ~2× 10 –8 cm 3 g –1 only. The remaining increase in the susceptibility should be associated with other factors, for example, the smaller saturation magnetisation M s of nanoparticles in comparison with that of a bulk crystal, or with the precipitation of a larger quantity of ferromagnetic phase when the specimen is cooled. According to [190, 195–197], the decrease of susceptibility in the temperature range 1000–1225 K is detected only at a high cooling rate of the specimen and is a result of quenching of the high temperature state, in which the whole of the ferromagnetic impurity is dissolved in copper. If cooling after annealing is slow, the iron impurity has a chance to transfer to the ferromagnetic phase and the decrease in susceptibility χ (300, T), observed in Figure 5.14, is absent after the maximum. The investigations [190, 195–197] show that the measurement of magnetic susceptibility is an information-carrying method of examining the behaviour of ferromagnetic nanoparticles in a diamagnetic matrix. The presence of the matrix prevents the rapid growth of the nanoparticles at a temperature of structural relaxation 206

Properties of Isolated Nanoparticles and Nanocrystalline Powders

of the ferromagnetic polycrystal and, at the same time, greatly increases the temperature range of existence of the nanocrystalline state of the ferromagnetic. 5.5 OPTICAL PROPERTIES The scattering and absorption of light by the nanoparticles has a number of special features in comparison with macroscopic solids [199]. Experimentally, these special features are most evident when examining a large number of particles. For example, colloid solutions and granular films may be extensively coloured as a result of the specific optical properties of nanoparticles. A classical object for investigating the optical properties of the dispersed media is gold. Even Faraday paid attention to the similarity of the colour of a colloid solution of gold and a gold film and assumed the dispersion state of the gold film. In absorption of light by fine-grained metal films the visible part of the spectrum contains absorption peaks which are absent in the spectra of bulk metals (in metals the optical absorption by conduction electrons takes place in a wide wavelength range). For example, the granular films of gold particles with a diameter of 4 nm have a distinctive maximum absorption in a range of λ =560– 500 nm [200, 201]. The spectra of absorption of nanoparticles Ag, Cu, Mg, In, Li, Na, K also had maxima in the optical range [4, 202]. Another special feature of the granular films is a decrease of their absorption in transition from the visible to infrared part of the spectrum in contrast to continuous metal films in which the absorption of radiation increases with increasing wavelength λ [4, 201, 203–207]. The size effect of the optical properties are important for the nanoparticles whose size is considerably smaller than the wavelength λ and does not exceed 10–15 nm [9, 199]. The differences in the absorption spectra of nanoparticles and bulk metals are caused by the differences of their dielectric permittivity ε = ε 1 + i ε 2 . The dielectric permittivity of the nanoparticles with a discrete energy spectrum depends on both the particle size and on radiation frequency. In addition to this, the value of dielectric permittivity does not depend monotonically on frequency but oscillates as a result of transitions between electronic states [208]. The minimum number of particles, which requires for the experimental investigation of the optical properties, is at least 207

Nanocrystalline Materials

10 10 . It is not possible to produce 10 10 –10 13 particles of the same size and shape and, consequently, in experiments in practice these oscillations are smoothed out. Nevertheless, even the value ε averaged out for the ensemble of particles differs from the value of dielectric permittivity of the bulk substance. According to [208, 209], the imaginary part of dielectric permittivity is inversely proportional to particle radius r

ε 2 ( ω ) = ε ∞ 2 ( ω ) + A( ω )/r ,

(5.43)

where e ∞2 ( ω ) is an imaginary part of the dielectric permittivity of the macroscopic crystal and A( ω ) is some function of frequency. Experimental results [210, 211], obtained on particles of gold with r = 0.9–3.0 nm at a constant wavelength λ = 510 nm, confirm the dependence ε 2 ~ 1/r. The particle size also determines the width of the absorption band and the shape of the low-frequency edge of the absorption band. The broadening of the absorption band of light by the gold and silver nanoparticles with a decrease of the size is observed by the authors [210, 212, 213]. The shift of the resonance peak of absorption of light is another size effect. The free path length of the electron in metallic particles, whose diameter is smaller than the free path length of electrons l∞ in the bulk metal, is equal to particle radius r [4, 5]. In this case, the effective relaxation time τ ef may be represented in the form

τ−ef1 = τ−1 + vF / r ,

(5.44)

where τ = l∞ /v F is the relaxation time in bulk metal; v F is the velocity of the electron with the Fermi energy. If interband transitions are neglected and the movement of only free electrons takes into account [214], then

ε1 = 1 −

ω2P , ω12 + 1/ τef

(5.45)

where ω p = 4 π Ne 2 /m * is the plasma frequency; N, e, m* is the concentration, charge and effective mass of free electron. In Mie’s theory [215] the maximum of light absorption is reached on the condition ε m = – ε 1 ( ω 1 ); taking this into account, for very small 208

Properties of Isolated Nanoparticles and Nanocrystalline Powders

particles with τef−1 ~ vF / r equation (5.45) gives the expression for the resonance frequency

 ω p2 vF2   ω1 =  − 2  1 + 2 ε r  m 

1/2

.

(5.46)

According to (5.46), the resonance frequency decreases with a decrease in the particle size, i.e. the absorption band should be displaced to the low-frequency range. The red shift of the resonance peak of absorption of light with a decrease in the particle size is predicted by theory [216]. On the other hand, quantummechanical calculations [208, 217] predict an increase in the frequency of the resonance peak, i.e. the blue shift of the absorption band with a decrease in the nanoparticle size. Experimental results obtained for the displacement of the frequency of resonance absorption in relation to the nanoparticle size are also contradicting. In [218–220] with a decrease in the size of Ag particles from 10 to 1 nm there was strong red shift of the absorption peak. According to the data [221–223] the position of the absorption peak of silver and gold particles with a diameter of 2.5–10.0 nm does not depend on the particle size. The blue shift of the absorption peak of Ag nanoparticles with a decrease of their size to 1–2 nm was established in [208, 212, 213, 224]. In [225, 226] it is shown that both blue and red shifts can be observed. It depends on the degree of ‘smearing’ of the electron cloud on the surface of the particle. For transition from one effect to another it is sufficient to change slightly the size of the region of diffusion ‘smearing’ of the electrons. According to [224, 226], the width of the absorption band of light is a complicated function of the particle size and reaches a maximum in the vicinity of D ≈ 1.1 nm. Recently, special attention has been paid to the investigations of the size effect on the optical and luminescent properties of semiconductor substances, because optical absorption is one of the main methods of studying the band structure of semiconductors. In semiconductors, the energy of interatomic interactions is high. Therefore, when describing the electronic properties, the macroscopic semiconductor crystal might be regarded as a single large molecule. The electronic excitation of semiconductor crystal may leads to the formation of a weakly bonded electron–hole pair, 209

Nanocrystalline Materials

i.e. Mott–Vanie exciton. The region of delocalisation for such the exciton is considerably greater than the lattice constant of the semiconductor. A decrease of the size of the semiconductor crystal to the value comparable with the size of the exciton affects the semiconductor properties. Thus, the specific properties of semiconductor nanoparticles are determined by the fact that the size of the nanoparticle is comparable with the Bohr radius of the excitons, r ex ≈ n 2 h 2ε /µ ex ε 2 , in the macroscopic crystal (here µ ex = m e m h /(m e + m h ) is the reduced mass of the exciton; m e and m h are the effective masses of the electron and the hole; n = 1, 2, 3…). For semiconductors, the Bohr radius of the exciton changes in a wide range: from 0.7 nm for CuCl to 10 nm for GaAs. The energy of the electronic excitation of a molecule is usually higher than the energy gap (width of the forbidden band) in the macroscopic semiconductor. It follows from this that when going from a crystal to a molecule, i.e. with a decrease in the particle size, there should be a range of particle size in which the energy of the electronic excitation changes smoothly from a lower to a higher value. In other words, a decrease in the size of the semiconductor nanoparticles should be accompanied by displacement of the absorption band to the highfrequency range. A manifestation of this effect is the blue shift of the exciton band of absorption of semiconductor nanoparticles with a decrease in their size [227–231]. In the most widely studied semiconductor CdS, the blue shift of the absorption band was found for nanoparticles with D ≤ 10–12 nm. The effect of the size of the nanoparticles on the optical spectra was found for many types of semiconductor [228–243]. For a macroscopic crystal, the exciton energy E includes two contributions. First of them is the width of the forbidden band E g (the energy difference between the lower energy of conduction band and the upper energy of valence band), reduced by the binding energy of the electron and the hole, i.e. the effective Rydberg energy E Ry = µ ex e 4 /2n 2 h 2 . The second contribution is kinetic energy of the centre of masses of the exciton. For a semiconductor nanoparticle with radius r, the second contribution is equal to n 2 p 2 h2 /2 µ ex r 2 [227] and is inversely proportional to the square of the radius. A more rigorous analysis [229, 244] of the effect of the nanoparticle size on the exciton energy and taking into account the Coulomb interaction between the electron and the hole give the following expression:

210

Properties of Isolated Nanoparticles and Nanocrystalline Powders

E = E g − 0.248E Ry + (n 2 π 2 h 2 /2µ ex r 2 ) − (1.78e 2 /ε r ) .

(5.47)

The sum of the first and third terms in (5.47) is the effective width of the forbidden band. Equation (5.47) shows that a decrease in the particle size should be accompanied by an increase of the effective width of the forbidden band. In particular, this broadening effect is observed for CdTe nanoparticles: when going from a bulk crystal to nanoparticles with a diameter of 4 and 2 nm, the effective width of the forbidden band increased from 1.5 eV to 2.0 and 2.8 eV respectively [245]. An increase in the width of the forbidden band of the very fine dispersed powder Si 3 N 4 in comparison with a bulk crystal was detected in examination of infrared and fluorescent emission spectra [246]. The excitation energy of the exciton is E = h ω where ω is the frequency of incident light. Therefore it follows from equation (5.46) that with a decrease in the size of the nanoparticles, the lines of the optical spectrum should be shifted to the high-frequency range. This shift of the absorption bands by 0.1 eV in the spectra of nanoparticles of CuCl (D = 31, 10 and 2 nm), dispersed in glass, was observed in [228]. As an example, Fig. 5.16 shows the optical spectra of absorption of CdSe [243]: with a decrease of the diameter of the CdSe nanoparticles, the absorption bands are shifted to the range of higher energies, i.e. blue shift is observed. To a first approximation, %  .

2SWLFDO GHQVLW\ DUELWUDU\ XQLV

 QP  QP

 QP  QP  QP





















 H9 Fig. 5.16. Low-temperature (T = 10 K) optical linear-absorption spectra of CdSe nanocrystals [243]. Mean particle diameters are shown in figure. The spectra have been scaled for clarifying. 211

Nanocrystalline Materials

the energy of the maximum of the absorption band is inversely proportional to the square of the radius, r 2 , of the CdSe particles. The large width of the absorption bands (~0.15 eV or 1200 cm –1 ) is determined by the dispersion of the nanoparticle size, i.e. deviation of the particle diameter from the mean value by ± 5 %. In fact, the broadened absorption spectra are observed even when investigating the most monodispersed specimens. In other words, the so-called inhomogeneous broadening is observed. Therefore, in [243] the femtosecond photon–echo technique is used to probe the dynamics of quantum-confined excitons in nanocrystalline CdSe. This made it possible to exclude the inhomogeneous broadening and determine the ‘homogeneous’ line width accurately corresponding to the given particle size. Consequently, it was shown that a decrease in the nanoparticle diameter increases the width of absorption lines (Fig. 5.17, curve 4). The authors of [243] separated three contributions in the homogeneous absorption line width. The most significant contribution (Fig. 5.17, curve 1) to the line width is caused by the elastic scattering of radiation from imputity and crystalline defect sites. This contribution depends of the nanoparticle size (or more accurately, on the effective surface area of scattering, proportional 7

 .



 







FP



∆



 











'!





QP

Fig. 5.17. Measured homogeneous line width, ∆, of optical linear-absorption spectra of CdSe nanocrystals as a function of diameter at a temperature of 15 K [243]. The three most important contributing mechanisms to this line width ∆ are also displayed: ( ¡ ) elastic scattering contribution, ( • ) phonon broadening due to coupling of lowfrequency vibrational modes, (∇) lifetime broadening. The solid line shows the size dependence of the total line width ( W ) and displays the sum of three contributions. 212

Properties of Isolated Nanoparticles and Nanocrystalline Powders

to the ratio S/V, where S is the surface area and V is the volume of the particle) and does not depend on temperature. The second contribution (Fig. 5.17, curve 2) is determined by the coupling to a heat bath of low-frequency vibrational modes of the nanocrystal. This contribution depends strongly on temperature and causes line broadening, which increases linearly with increasing temperature. The phonon broadening, due to low-frequency modes, provides a significant contribution (up to 20–35 %) to the homogeneous line width not only at high but even at low temperatures. The third contribution to the line width (Fig.5.17, curve 3) is the smallest. It is associated with the lifetime to that corresponds the fast decay of the initial state into some other electronic configuration which is less strongly coupled to the grand state. The change in the state strongly depends on the particle size as a result of exciton trapping to a localized surface state. If the trapping is driven by a simple overlap between an interior wave function and a localised surface state, then the trapping rate should vary roughly as the ratio of the surface area of the particle to its volume, i.e. S/V. The recombination of light-generated charges leads to the luminescence of nanoparticles. Examination of the luminescent spectra of nanoparticles ZnO [247], ZnS [248, 249], CdS [250–253], and CdSe [254, 255] also shows a blue shift, i.e. the shift of the spectra to the short wave range with a decrease in the particle size. For the given nanoparticle size, the decay time of luminescence depends on the wavelength and decreases with an increase in the energy and a decrease of the wavelength of the emitted light quantum. The dependence of the lifetime of the excited state on the wavelength λ of luminescence is determined by contribution of the Coulomb interaction between the electron and the hole to the energy of the emitted light quantum h ω = 2 π h c/ λ [256]: 2 π h c/ λ = E min – (D h – D e ) + (e 2 / ε r eh ) ,

(5.48)

where E min is the minimum energy of excitation of luminescence of the nanoparticle with a radius r, D h and D e are the depth of the traps of the hole and the electron, r eh is the distance between the electron and the hole. The electron–hole pairs with small distance r eh in tunneling recombination of the holes and electrons emit the light with a faster rate and with a smaller wavelength λ than the pairs with high r eh .

213

Nanocrystalline Materials

References 1. 2.

3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

14. 15.

16.

17.

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R. Rademann, B. Kaiser, U. Even, F. Hensel. Size dependence of the gradual transition to metallic properties in isolated mercury clusters. Phys. Rev. Lett. 59, 2319-2321 (1987) C. J. Kriesmann, H. B. Callen. The magnetic susceptibility of the transition elements. Phys. Rev. 94, 837-844 (1954) J. E. van Dam, O. K. Andersen. Temperature dependence of the palladium susceptibility. Solid State Commun. 14, 645-647 (1974) A. Kawabata, R. Kubo. Electronic properties of fine metallic particles. II. Plasma resonance absorption. J. Phys. Soc. Japan. 21, 1765-1772 (1966) M. A. Smithard. Size effect on the conduction electron spin resonance of small sodium particles in sodium azide NaN 3 . Solid State Commun. 14, 411415 (1974) D. Gordon. Conduction-electron spin resonance in small particles of sodium. Phys. Rev. B 13, 3738-3747 (1976) M. Rappaz, A. Chatelain, L.A. Boatner. EPR investigation of size effects in the crystal field of small dielectric particles. J. de Physique - Colloque C2 38, Suppl. No 7, C2-105 - C2-108 (1977) S. V. Vonsovskii. Magnetism (Nauka, Moscow 1971) 1032 pp. (in Russian) E. I. Kondorskii. Micromagnetism and remagnetization of quasi-single-domain particles. Izv. AN SSSR. Seriya Fizicheskaya 42, 1638-1645 (1978) (in Russian) A. Tasaki, S. Tomiyama, S. Iida, N. Wada, R. Uyeda. Magnetic properties of ferromagnetic metal fine particles prepared by evaporation in argon gas. Japan. J. Appl. Phys. 4, 707-711 (1965) Yu. I. Petrov, Yu. I. Fedorov. Electromagnetic properties of colloidal suspension of nickel in paraffin. Zh. Tekhnich. Fiz. 37, 726-728 (1967) (in Russian) A. E. Ermakov, O. A. Ivanov, Ya. S. Shur, R. M. Grechishkin, G. M. Ivanova. Magnetic properties of monocrystalline nickel powders. Fiz. Metall. Metalloved. 33, 558-563 (1972) (in Russian) I. N. Shabanova, A. E. Ermakov, V. A. Trapeznikov, Ya. S. Shur. Dependence of saturation magnetization of nickel aerosols on particle surface state studied by electron spectroscopy method. Fiz. Metall. Metalloved. 38, 314-322 (1974) (in Russian) W. D. Corner, P. A. Mundell. Properties of ferromagnetic micropowders. J. Magn. Magn. Mater. 20, 148-157 (1980) A. E. Petrov, V. I. Petinov, I. V. Plate, E. A. Fedorova, M. Ya. Gen. Magnetic properties of small aerosole particles of cobalt. Fiz. Tverd. Tela 13, 15731577 (1971) (in Russian) A. E. Petrov, V. I. Petinov, V. V. Shevchenko. Magnetic properties of small aerosole Ni particles at 4.2-300 K. Fiz. Tverd. Tela 14, 3031-3036 (1972) (in Russian) A. E. Petrov, A. N. Kostygov, V. I. Petinov. Magnetic properties of small spherical particles of iron at a temperature 4.2-300 K. Fiz. Tverd. Tela 15, 2927-2931 (1973) (in Russian) R. Z. Valiev, R. R. Mulyukov, Kh. Ya. Mulyukov, L. I. Trusov, V. I. Novikov. Curie temperature and saruration magnetization of nickel with submicrongrained structure. Pis’ma v ZhTF 15, 78-81 (1989) (in Russian) A. V. Korolev, A. I. Deryagin, V. A. Zavalishin, R. I. Kuznetsov. Peculiarities of magnetic state of highly deformed polycrystalline ultrafine-grained nickel. Fiz. Metall. Metalloved. 68, 672-678 (1989) (in Russian) R. Z. Valiev, Ya. D. Vishnyakov, R. R. Mulyukov, G. S. Fainstein. On the

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cadmium sulfide particles. Colloids Surf. 55, 121-136 (1991) M. V. Rama Krishna, R. A. Friesner. Exciton spectra of semiconductor clusters. Phys. Rev. Lett. 67, 629-632 (1991) H. Weller. Quantized semiconductor particles: A novel state of matter for materials science. Advanced Mater. 5, 88-95 (1993) D. M. Mittleman, R. W. Schoenlein, J. J. Shiang, V. L. Colvin, A. P. Alivisatos, C. V. Shank. Quantum size dependence of femtosecond electronic dephasing and vibrational dynamics in CdSe nanocrystals. Phys. Rev. B 49, 1443514447 (1994) Y. Kayanuma. Quantum-size effects of interacting electrons and holes in semiconductor microcrystals with spherical shape. Phys. Rev. B 38, 97979805 (1988) T. Raih, O. I. Misic, A. J. Nozik. Synthesis and characterization of surface-modified colloidal cadmium telluride quantum dots. J. Phys. Chem. 97, 11999-12003 (1993) Chi-mei Mo, L. Zhang, C. Xie, T. Wang. Luminescence of nanometer-sized amorphous silicon nitride solids. J. Appl. Phys. 73, 5185-5188 (1993) S. Hotchandani, P. V. Kamat. Charge-transfer processes in coupled semiconductor systems. Photochemistry and photoelectrochemistry of the colloidal CdS - ZnO system. J. Phys. Chem. 92, 6834-6839 (1992) W. G. Becker, A. J. Bard. Photoluminescence and photoinduced oxygen adsorption of colloidal zinc sulfide dispersions. J. Phys. Chem. 87, 48884893 (1983) Hyeong Chan Youn, S. Baral, J. H. Fendler. Dihexadecyl phosphate, vesiclestabilized and in situ generated mixed cadmium sulfide and zinc sulfide semiconductor particles: Preparation and utilization for photosensitized charge separation and hydrogen generation. J. Phys. Chem. 92, 6320-6327 (1988) P. V. Kamat, N. M. Dimitrijevic, R. W. Fessenden. Photoelectrochemistry in particulate systems. 6. Electron-transfer reactions of small cadmium sulfide colloids in acetonitrile. J. Phys. Chem. 91, 396-401 (1987) K. R. Gopidas, P. V. Kamat. Photoinduced charge transfer processes in ultrasmall semiconductor clusters. Photophysical properties of CdS clusters in Nafion membrane. Proc. - Indian. Acad. Sci. (Chem. Sci.) 105, 505512 (1993) T. Raih, O. I. Misic, D. Lawless, N. Serpone. Semiconductor photophysics. 7. Photoluminescence and picosecond charge carrier dynamics in cadmium sulfide quantum dots confined in a silicate glass. J. Phys. Chem. 96, 46334641 (1992) Y. Nosaka, K. Tanaka, N. Fujii. Laser-irradiation effect on poly(vinyl alcohol) films doped with nanometer-sized CdS particles: Ablation and thied harmonic generation. J. Appl. Polym. Sci. 47, 1773-1779 (1993) N. M. Dimitrijevic. Electron-transfer reactions on cadmium selenide colloids as studied by pulse radiolysis. J. Chem. Soc. Faraday Trans.1 83, 1193-1201 (1987) K. R. Gopidas, P. V. Kamat. Photophysical behavior of ultrasmall cadmium selenide semiconductor particles in a perfluorosulfonate membrane. Mater. Lett. 9, 372-378 (1990) N. Chestnoy, T. D. Harris, R. Hull, L. E. Brus. Luminescence and photophysics of cadmium sulfide semiconductor clusters: The nature of the emitting electronic state. J. Phys. Chem. 90, 3393-3399 (1986)

226

Microstructure of Compacted and Bulk Nanocrystalline Materials

+D=FJAH$

6. Microstructure of Compacted and Bulk Nanocrystalline Materials The difference in the properties of nanocrystalline and coarsegrained polycrystalline substances is associated with the small size of crystallites and extremely developed interfaces of the nanocrystalline substances. The interfaces can include up to 50 % of atoms of the nanocrystal. At present, many researchers of nanocrystalline compacted materials assume that the physical properties of these materials (in particular, mechanical properties) are determined primarily by the large area and special structure of the interfaces [1,2]. For this reason, the examination of nanocrystalline compacted substances is concentrated mainly on the analysis and describing of the special features of the structure of grain boundaries. Before we proceed to discuss the microstructure of nanocrystalline materials, i.e. polycrystalline substance, let us define more exactly some notions, which will be used below. Three types of the grain contact are possible in a polycrystalline substance. They include contact surfaces, contact lines, and contact points. Surfaces of two grains, which contact one another, are called interfaces. A contact line may represent a common line for three or more adjacent grains. The contact line of three grains is called triple junction. The triple junctions are most often appearing contact lines among different contact lines, which exist in polycrystals. Usually about six grains meet at a contact point. The boundary of a grain is its surface. Grain boundaries, which are seen in metallographic slides, represent the section of interfaces by the slide plane. A triple point is the section of the triple junction (contact line of three grains) by a plane. 227

Nanocrystalline Materials

6.1 INTERFACES IN COMPACTED MATERIALS The density of nanocrystalline materials, produced by different methods of compacting nanopowders [3–13] can reach from 70–80 % to 95–97 % of theoretical density. In the simplest case, a nanocrystalline material, consisting of the atoms of the same type, contains two phases (components) differing in the structure [14]: ordered grains (crystallites) with a size of 5–20 nm, and intergranular boundaries up to 1.0 nm wide (Fig. 6.1). All crystallites have the same structure and differ only in the crystallographic orientation and sizes. The structure of the intergranular boundaries is determined by the type of atomic interaction (metallic, covalent, ionic) and by mutual orientation of the adjacent crystallites. The different orientation of adjacent crystallites results in a small decrease of the density of the substances at the intergranular boundaries. In addition to this, the atoms at the intergranular boundaries have a different short-range order in comparison with the atoms in the crystallites. In fact, Xray and neutron diffraction investigations of nanocrystalline compacted nc-Pd [15, 16] show that the density of the substance at the intergranular boundaries is 20–40 % lower than that of coarse-grained Pd, and the coordination number of the atoms, belonging to the intergranular boundary, is smaller than the

Fig. 6.1. Computed two-dimensional model of atomic structure of a nanostructured material [14]: ( ™ ) atoms of crystallites; ( l ) atoms of interfaces are displaced by more than 10 % of interatomic distance from the corresponding lattice sites; all atoms are chemically identical, atomic structure of all crystallites is identical too. The computations are performed by modeling the interatomic forces by the Morse potential.

228

Microstructure of Compacted and Bulk Nanocrystalline Materials

coordination number of the atoms in the normal crystal. The width of the intergranular boundaries, determined by different methods for different nanocrystalline compacted materials, varies from 0.4 to 1.0 nm [17–20]. According to initial modeling considerations [7, 21, 22], the structure of intergranular substance is characterised by a random distribution of the atoms and by the absence of not only the longrange but also short-range order. According to the authors of [7, 21, 22], this structure is a gas-like structure, taking into account not the mobility of the atoms but only their distribution (Fig. 6.1). Experimental confirmation of some disordering of the intergranular substance in nanomaterials, produced by compacting of nanopowders, has been provided by the results of diffraction investigations [21, 22]. At the same time, the latest investigations show [23–27] that the structure of the interfaces in nanomaterials is similar to that in conventional coarse-grained polycrystals and the degree of order in the distribution of the atoms at the interfaces is considerably higher in comparison with previous assumptions [7, 14–16, 21, 22]. The application of high-resolution electron microscopy shows [28] that in nanomaterials, like in coarse-grained polycrystals, the atoms of the interfaces are subjected to the effect of only two adjacent crystallites. Pores were detected only in triple points and not along the entire length of the interfaces; the density of the atoms in the intergranular boundaries was almost the same as in the crystallites. The data on the relatively high degree of order in the distribution of atoms at the grain boundaries in specimens of nanocrystalline compacted nc-Pd are reported by the authors of [17, 18, 22, 24]. In [25, 26], analysis of experimental data on X-ray diffraction and X-ray absorption spectroscopy (EXAFS) of nanocrystalline substances was carried out using the function of the radial distribution of atomic density

ρ (r) = x l ρ l (r) + (1 – x l ) ρ

gb

,

(6.1)

where r is the interatomic distance; x l is the fraction of atoms occupying the sites of the crystal lattice; ρ l (r) is the atomic density distribution function of the nanocrystal, in which the atoms of the external layers of the crystallite are distributed in the sites of the lattice; gb is the atomic density of the grain boundaries in which all atoms are distributed in random positions, not coinciding with the 229

Nanocrystalline Materials

sites of the crystal lattice. The value of coefficient x l can be determined if one knows the experimental atomic density distribution function ρ (r). For this purpose, using the function ρ (r), it is necessary to construct the dependence of the relative coordination number Z i / Z iideal (Z i is the experimental number of the atoms in the i-th coordination sphere; Z iideal is the coordination number of the ith coordination sphere for the perfect crystal) on the interatomic distance r. The limiting value of Z i / Z iideal at r = 0 corresponds to the value x l , i.e. the fraction of the atoms occupying the sites of the crystal lattice. Figure 6.2 shows the function of the radial distribution of the atomic density in a specimen of nanocrystalline compacted nc-Pd, aged at room temperature for 4 months. The distribution function ρ (r) was calculated [25] on the basis of the experimental data obtained by X-ray diffraction. The analysis carried out in [25] using the function ρ (r) shows that in the nc-Pd specimens, aged at room temperature for several months or on the same specimens, subjected to additional annealing at 973 K after aging, the ratio Z i / Z iideal at r = 0, i.e. the value of coefficient x l is equal to unity (Fig. 6.3a, 6.3b). In the nc-Pd specimens, investigated not later than ten days after compacting, 8–14 % of the atoms were not located in the lattice sites (Fig. 6c) and the degree of short-range







   

ρ 7

QP









































7 QP

Fig. 6.2. Atomic density distribution function ρ (r) for an aged nanocrystalline compacted nc-Pd specimen [25]: highest peaks of eight coordination spheres are numbered. 230

Microstructure of Compacted and Bulk Nanocrystalline Materials 1.10

Zi /Ziideal

1.05

1.10

a

b

c

1.05

1.00

1.00

0.95

0.95

0.90

0.90

0.85

0.85

0.80

0.80

0.75

Zi /Ziideal = -0.17ri + 1.009

Zi /Ziideal = -0.06ri + 0.996 Zi /Ziideal = -0.07ri + 0.861

0.70

0.75 0.70

0.2 0.4 0.6 0.8 1.0 1.2 1.4 ri (nm)

0.2 0.4 0.6 0.8 1.0 1.2 1.4 ri (nm)

0.2 0.4 0.6 0.8 1.0 1.2 1.4 ri (nm)

Fig. 6.3. Relative coordination number Z i / Z iideal versus interatomic distance r i for nanocrystalline compacted nc-Pd [25]: (a) specimen aged without annealing, (b) same aged specimen after several annealing runs, the last at 973 K, (c) as-prepared specimen nc-Pd.

order was very low. The results show that immediately after production of the compacted specimens, the grain boundaries in ncPd are in the non-equilibrium state with a low short-range order. This state is unstable even at room temperature and during 120– 150 days changes to a more ordered state with an increase of the crystallite size from 12 to 25–80 nm (Fig. 6.4) [26]. The calculations carried out in [25–27] show that the coordination number of the atoms, distributed at the interfaces in aged nc-Pd, is close to that in coarse-grained crystalline Pd. Examination of the short-range order in nanocrystalline compacted nc-Pd and crystalline coarse-grained Pd by the EXAFS method [29] showed that the functions of the radial distribution of atomic density ρ (r) are identical. The coordination number for the first coordination sphere in the as-prepared and annealed (at 370 K) specimens of nc-Pd was 5–6 % lower than that for the coarse-grained Pd. This is in agreement with the data [25] (Fig. 6.3 a,c). According to [29], the reduced coordination number of the first coordination sphere in nc-Pd is a consequence of the thermodynamically non-equilibrium state of the substance and of the presence of lattice vacancies in the specimen. Examination of compacted specimens of nanocrystalline iron nc-Fe (produced in high vacuum) with a mean size of crystallites of 10 nm [30] shows that 95±5 % of all atoms are located in the sites of the bcc lattice. In an earlier study [22], the authors did not find any significant short-range order in the distribution of the atoms at the grain 231

Nanocrystalline Materials



 QP

      









7LPH GD\V Fig. 6.4. Variation of grain size 〈D〉 with specimen age at room temperature for several nc-Pd specimens [26].

boundaries of nc-Fe. In [30] it is shown that unusual results [22] are associated with the oxidation of the surface of crystallites: in the specimens of nc-Fe, produced in high vacuum with subsequent exposure in air, only 72±5 % of the atoms occupied the sites of the bcc lattice of iron, and the majority of the rest iron atoms (~23 %) forms an amorphous oxide phase, and only a small part of the iron atoms (~5 %) are located in the lattice sites of crystalline oxide phase. Investigation of the short-range order in nanocrystalline compacted cobalt nc-Co [31] with a mean crystallite size of 7 nm shows that the specimens contained ~70 % of the disordered amorphous phase and ~30 % of the ordered crystalline phase. The authors of [31] noted that the disordered phase, located at the grain boundaries, has no special features typical of the disordered gaslike phase. The relative content of the disordered phase in nc-Co overestimates evidently because the specimens of nc-Co were partially oxidised (this has been reported by the authors themselves); in addition, in processing of the experimental EXAFS spectra, the presence of lattice defects and free volumes were not taken into account. In fact, the interfaces of nanocrystalline compacted material may 232

Microstructure of Compacted and Bulk Nanocrystalline Materials

Fig. 6.5. Two-dimensional schematic model of a nanocrystalline material with submicroscopic free volumes as detected by positron lifetime spectroscopy [32]: vacancy-like free volumes (with the positron lifetime τ 1 ) in the interfaces, nanovoids (agglomerates of about 10 vacancies) as triple junctions ( τ 2 ), and large voids ( τ 3 ) of the size of missing crystallites.

contain three types of defects [32]: single vacancies; vacancy agglomerates or nanovoids, located at triple points of the crystallites; large voids in the place of absent crystallites (Fig. 6.5). These defects represent structural elements of the interfaces with reduced density. Ignoring the presence of free volumes leads to large errors in the determination of the volume fraction of the interfaces in the nanocrystalline materials. For example, examination of nc-Pd by the method of small-angle neutron scattering [33] and subsequent processing of the experimental data, disregarding porosity, resulted in an incorrect conclusion according to which the volume fraction of the crystallites and the interfaces in nc-Pd are equal to 0.3 and 0.7, respectively. According to the estimations [33], the relative density of intergranular substance was only 50 %. In a later examination of ncPd subjected to high degree of compacting [30] it is found that the density of the second phase reaches to zero. This means that the voids but not the interfaces with reduced density are scattering objects. Comparison of the data [33, 34] and own results enabled the authors of [26] to conclude that small-angle scattering may provide information 233

Nanocrystalline Materials

on the atomic structure of interfaces, but may not provide information of the free volumes in the interfaces. 6.2. STUDY OF NANOCRYSTALLINE MATERIALS BY MEANS OF POSITRON ANNIHILATION TECHNIQUE Positron annihilation [32, 35] is the most efficient, sensitive and reliable method of studying free volumes in nanocrystalline compacted and nanocrystalline bulk materials. This method is most promising for examining the electronic structure of solids with point (zero-dimensional), linear (one-dimensional) and volume (threedimensional) defects. The method is sensitive to a very small content of defects in the solid, from 10 –6 to 10 –3 defects per atom. Because of the trapping by defects, the positrons are efficiently used for analysis of the interfaces in nanostructured substances. The free path (diffusion) of a positron in a defect-free ideal crystalline solid is approximately 100 nm, i.e. it is longer than the size or particles or grains of the nanocrystalline material. Therefore, after thermalisation, i.e. slowing down, the positron is usually trapped at the interfaces. In addition, some of the positrons can be trapped in triple junctions of adjacent crystallites, voids and in the volumes of missing nanocrystallites (Fig. 6.5) [32]. This circumstance providing a unique possibility of solving one of the most complicated and interesting problems of nanomaterials, i.e. understanding the structure of the interface. In fact, the structure of the interface together with the small grain size determine most the properties of the nanomaterial. Other direct methods, including high-resolution transmission electron microscopy and diffusion of atoms, are far less suitable for investigation of the interfaces. Although electron–positron annihilation is a modern method of examining the defects of solids, including nanocrystalline materials, this method is not known widely to many material scientists. Therefore, its specific features, possibilities and applications will be discussed briefly here. Positron annihilation makes it possible to determine the characteristics of the electronic system of perfect crystals and, at the same time, is sensitive to imperfections of especially small sizes in the solid, such as vacancies, vacancy clusters and free volumes up to 1 nm 3 . There are three methods of electron–positron annihilation: positron lifetime, angular correlation of annihilation radiation, and coincident Doppler broadening of positron–electron annihilation radiation. If the measurement of positron lifetime gives 234

Microstructure of Compacted and Bulk Nanocrystalline Materials

information on the electron density in the site of positron annihilation, the two other methods provide the data on the distribution of momenta of electrons, with which the positrons are annihilated. Further, all three methods may be subdivided into two groups. The first group of methods uses slow positrons, which make it possible to investigate of the substance at a small depth below the surface. Methods of second group use fast positrons, which penetrate to a larger depth (up to 50 µm) and give the data on the type, concentration and distribution of defects in the entire volume of the solid. In most cases, nanomaterials are studied using fast positrons because they are efficiently in providing information on the structure of the interfaces. It is common practice to use radioactive isotope 22 Na, 44 Ti or 58 Co having the maximum energy of emitted positrons equal to 0.54, 1.50 and 0.47 MeV [36], respectively. According to [37], the linear coefficient of absorption of the positron in the solid is:

α=

(16 ± 1) ρ 1.43 E max

[cm –1]

(6.2)

Substituting the maximum energy of the positron E max (in MeV) and the density ρ (in g cm –3 ) into equation (6.2), it is possible to determine x = α –1 , i.e. the penetration depth of a positron in a nanosubstance at a given intensity of the beam. The isotopes used in most cases emit positrons with a maximum energy of approximately E max = 0.5 MeV, and the density of nanomaterials ρ is from 5 to 10 g cm –3 . Taking this into account, equation (6.2) shows that the positron penetrates in a nanosubstance to a maximum depth from 200 to 300 µm (in this case, the intensity of the positron beams weakens e 6 times), and the main information is received from a depth of 25–50 µm (intensity weakens of e times). The positron emitted from a radioactive source (emitter) is thermalised after penetrating in a solid, i.e. rapidly loses its velocity and energy, which decreases to the value k B T corresponding to the temperature T of the crystal. The thermalisation time is about 1– 5 ps and is negligibly small in comparison with the positron lifetime in a solid. Because the grain size of nanomaterials is not larger than 40 nm, after thermalisation the positrons are uniformly distributed over the volume of many grains located at a large depth from the crystal surface. Thanks to the small size of the grains, the 235

Nanocrystalline Materials

fraction of the positrons thermalised near the surface of the grains, coincides in the order of magnitude with the fraction of the positrons thermalised inside the grains. A thermalised positron starts to diffuse through the nanomaterial in the so-called ‘free’ (delocalised) state and annihilates from this state in the characteristic time τ f of approximately 100 ps. This value of the positron lifetime in the free state (or ‘free’ lifetime) is characteristic of metals [38] and many compounds, for example, carbides [39]. A positron can move in defect-free solids to a distance of about 100 nm during the time τ f . This estimate follows from the experimental data concerning the diffusion coefficient of positron D 0+ = 1–2 cm 2 s –1 in such metals as aluminium, copper, molybdenum, silver and in silicon carbide SiC [40, 41] where positrons move to a distance of 100–180 nm and 70±10 nm respectively during their free lifetime. If a substance is not defectfree then positron could be trapped by defect. If the distance between defects is close to the diffusion length of positron, some of the positrons will annihilate in free state and some of positron will annihilate in trapped (localised) state. In this case the positron lifetime spectrum consists of two components. If the distance between defects is very short in comparison with the diffusion length of positron, all the positrons will be trapped by defects. This case is called ‘saturation’ and leads to a one component positron lifetime spectrum. Since the grain size in nanomaterials is smaller than the diffusion length of the positron in a defect-free grain, almost all positrons may reach the grain surface and, consequently, the interface. In this case the data mostly on defects in the interfaces and in triple junctions could be obtained. If the interior of the grains contains defects which trap positrons, only some of the positrons reach the grain boundaries and, consequently, the data on intragranular defects in nanomaterials may be obtained. According to the results of a large number of investigations, the positron lifetime τ in a substance is determined by the number of valence electrons and not by the total number of electrons. This takes place owing to the fact that a positively charged positron escapes the positively charged atom nuclei. Therefore, almost all positrons (approximately 95–98 %) annihilate with valence electrons in the internuclear space. The authors of [38, 42, 43] established a correlation between the normalised rate of positron annihilation

λ ∗ = ( λ − λ ∞) / λ ∞

(6.3) 236

Microstructure of Compacted and Bulk Nanocrystalline Materials

and the normalised volume density of valence electrons

η = zVB / V .

(6.4)

In equations (6.3) and (6.4), λ ∞ = 2.00×10 9 s –1 ; λ = 1/ τ ; z is the number of valence electrons, participating in annihilation and corresponding to one formulae unit of the substance occupying a volume V; V B = 6.208×10 –31 m 3 is the volume of a sphere with the Bohr radius. For metals, the correlation between the normalised rate of positron annihilation, λ *, and the normalised volume density of valence electrons, η , is nonlinear [42]:

λ ∗ = (24.8 η ) 0.81 .

(6.5)

For the transition metal oxides [43], a deviation from this dependence, associated with the participation in annihilation of d electrons, is found. In carbides, in which the carbon vacancies are surrounded by the atoms of transition metals, the relationship between the normalised rate of annihilation (6.3) and the volume density of d electrons, calculated from equation (6.4), is close to the dependence (6.5) but is linear [38] val λ * = 17.9 ηM .

(6.6)

The essence of the correlations (6.5) and (6.6) is that as the volume density of the valence electrons increases, the positron lifetime becomes shorter. In addition to this, these correlations make it possible to estimate quantitatively the free positron lifetime and the lifetime of the positrons in vacancies, if the atomic and crystal structure and chemical composition of the substance are known. From the fundamental studies of positron annihilation in metals [38], semiconductors [44], nanocrystals [45, 46], amorphous materials [47, 48] and ceramics [38, 43] it is known that the positrons are trapped by vacancies on the metallic and nonmetallic sublattices, vacancy clusters, grain boundaries, dislocations, etc. The interstitial atoms, which also represent defects in a solid, usually do not trap positrons because of the large positive charge of the atomic nucleus. After trapping by a defect, a positron annihilates from the localised state in a time exceeding τ f . The 237

Nanocrystalline Materials

electron density in the defect is smaller than the electron density in the interatomic space of the defect-free crystal, and, consequently, the positron lifetime in the defect is longer than the free lifetime. The positron lifetime depends strongly on the size of the free volume. As the free volume increases, the positron lifetime also increases. Interesting investigations were carried out in [49] where a specific relationship was found between the positron lifetime τ and the number of vacancies, N V , in the free volume of silicon Si

τ = C + AN V/(B + N V) ,

(6.7)

where A = 266.57 ps, C = τ f = 218 ps, and B = 4.60. By replacing the number of vacancies N V by the free volume V, equation (6.7) may be simplified and presented in the following approximate form: Si

τ [ns] = 0.22 + 1.4V [nm 3] .

(6.8)

Although this relationship is obtained for the pure element substance silicon, it may be used to evaluate the size of the free volume V at the interfaces of nanocrystalline materials. For this purpose, the equation should be presented in the following form:

τ [ns] = τ f [ns] + 1.4V [nm3] ,

(6.9)

where τ f is the positron lifetime in a coarse-grained defect-free substance or a defect-free single crystal. In addition to the possibility of investigating the size of the free volume, the positron annihilation makes it possible to carry out the chemical analysis of the atomic environment of the free volume [50–52]. This possibility has appeared recently as a result of the rapid development and reduction in the cost of high-resolution equipment for detecting γ -quanta. Relatively cheap and easy to operate γ -quanta detectors with a high energy resolution of about 1–2 keV for the γ -quanta energies close to 511 keV have made it possible to develop a two-detector method of Doppler broadening of the electron–positron annihilation radiation. To obtain the distribution of electron momenta up to very high values of electron momentum, it is necessary to take special coincident measurements using two detectors. This coincident method makes it possible to increase the signal–noise ratio and decrease the effect of

238

Microstructure of Compacted and Bulk Nanocrystalline Materials

background on the spectrum. The Doppler experiments provide data on the distribution of the momentum of electrons annihilating with the positron. The thermalised positron has momentum approaching zero and, consequently, the shift of the energy of annihilation γ -quanta is determined exclusively by electron momenta. High electron momenta are determined by the core electrons of the atoms and, consequently, the profile of the distribution at high momenta is a unique ‘fingerprint’ of the orbitals of the core electrons of the element and makes it possible to identify chemical elements, which are located in the nearest surrounding of the site of positron annihilation, i.e. in the surrounding of the free volume. Calculations or numerical processing of the profile of the experimental Doppler spectrum are possible if we know the threedimensional electron momentum distribution. The momentum distribution may be determined using the Schrödinger equation and first principal quantum calculations, but this is a very time-consuming and difficult operation which does not always lead to success in the case of complicated defective substances and nanocrystalline materials. Analysis of the chemical environment of the defects may be carried out using a semi-empirical model proposed in [53]. This model is based on the possibility of measuring the mean kinetic energy of core electrons, E k, and on comparison of this energy with the tabulated values of binding energy E B, which are available at the present time for the electronic orbitals of almost all elements of the Periodic Table of Elements. These measurements are possible in experiments with the angular correlation of annihilation radiation or in Doppler experiments. The mean kinetic energy of the electrons of a specific orbital and the binding energy of this electron in the atom are compared using the virial theorem whose consequence is the equality E B = E k. For calculating the Doppler spectrum, all electrons may be divided into valence and core electrons. In the valence electrons, i.e. those which take part in chemical bonding, it is necessary to distinguish delocalised and localised electrons. The delocalised electrons are quasifree electrons of metallic bond, and localised ones are the electrons of the covalent and ion bonds, and also the electrons of the core shells. The positively charged positrons are pushed away from the nuclei of the atoms because they also have a positive charge. Consequently, the probability of the presence of a positron in the internuclear space is maximum. However, the probability of a positron penetrating into the region of core electrons is not equal to zero and amounts to several percent. The basic information on the wave function of the positron is obtained from the experimental spectra in which the contribution of 239

Nanocrystalline Materials

every component is proportional to the overlapping integral of the wave function of the electron of the appropriate orbital and the positron wave function. In the description of the valence electrons in the model of free (delocalised) electrons, the wave function of the gas of free electrons is a Bloch wave with a continuous spectrum of momenta within the limits of the Fermi sphere. The square of the positron wave function in the region of the valence electrons is close to a constant and, therefore, the Doppler broadening spectrum [53] is a truncated inverted parabola

  p 2  I p ( p z ) = hp 1 −  z   f ( p z − p F ) ,   p F  

(6.10)

where p z is the z-th component of the electron momentum, h p is the height of the parabola, p F is the truncation momentum of the

1, if | p z | ≤ p F 0, if | p z | > p F

parabola or the Fermi momentum, f (| p z | − p F ) = 

is the Heaviside function. This approach makes it possible to determine by experiments the Fermi energy E F = (p F ) 2 /2m 0 . Already in studies of non-stoichiometric carbides of transition metals [53] it was established empirically that the high-energy part of the spectrum of the angular correlation of annihilation radiation is efficiently described by a Gaussian. The spectra of the angular correlation and of Doppler broadening give the same physical data on the momentum distribution and, consequently, the distribution of electron momenta in the Doppler broadening spectra may be represented in the form of a Gaussian:

 p2 I g ( p z ) = hg exp − z 2  2σ 

 ,  

(6.11)

where h g is the height of the Gaussian, σ is the second momentum of the Gaussian. In accordance with the results of [54], from the parameter σ it is possible to transfer directly to the binding energy using equation E B = (2 σ ) 2 /2m 0 (m 0 is the electron rest mass). To determine the type of chemical environment of the defect, the 240

Microstructure of Compacted and Bulk Nanocrystalline Materials

Doppler spectrum is expanded into the truncated inverted parabola and several Gaussians

I ( p z ) = I p ( p z ) + ∑ I gi ( p z ) ,

(6.12)

i

where i is the number of the Gaussians, which is equal to the number of different orbitals of the electronic shells, participating in annihilation with the positrons. The parabola corresponds to the contribution from positron annihilation with free electrons, and each Gaussian corresponds to the contribution from annihilation of positrons with a specific orbital of the core electrons. For transition metals and compounds of these metals, the application of this model is complicated by hybridization of s, p and d bands. Consequently, only part of the valence electrons can be described by the truncated inverted parabola; the remaining valence electrons are usually described by a Gaussian. Because of Coulomb repulsion of the positron from the atomic nuclei, the contribution of the core electrons to the spectra of the electron momentum distribution does not exceed several percent. Therefore, the core components can be separated only in the range of high electron momenta in which the contribution from the valence electrons is negligibly small. The experimental spectrum N(p z ) widens as a result of the finite resolution of the detectors. Taking into account the instrumental resolution function of the spectrometer, R(p z ), the experimental spectrum N(p z ) can be presented as a convolution of the true spectrum I(p z ) and resolution function R(p z ):

N ( pz ) =

∞

∫ R( p ′ − p z ) ⋅ I ( p z )dp ′ .

−∞

(6.13)

The resolution function is a Gaussian. Taking this into account and integrating equation (6.13) with the use of functions (6.10)–(6.12), the following dependence is obtained in [55]:

N( pz ) =

hp ( p F2 − σ R2 )  ( p z − p 0 ) 2    p F − ( p z − p 0 )   p F + ( p z − p 0 )   erf  1 −  + erf   + 2 2 2  σR 2 σR 2 2 pF ( p F − σ R )        

241

Nanocrystalline Materials

+

hpσ R   ( pF − p z + p0 ) 2  p p p + − [ ( )] exp − +  z F 0 p F2 2π  2σ R2  

 ( p − p )2  hgi σ gi  ( p + p z − p 0 ) 2   0 z + [ p F − ( p z − p 0 )] exp − F + exp −  + BG,  ∑  2 2 2σ R2  2(σ R + σ gi )     i σ R2 + σ g2i

(6.14) where erf ( x) =

2 x 2 ∫ exp(−t )dt is the error function integral, p 0 is π 0

the momentum corresponding to maximum of the Doppler spectrum, σ R is the second moment of the instrumental resolution function, BG is the random coincidence background. The application of this model requires fitting of the parameters by the weighted least squares method, i.e. assumes the minimisation of the function:

χ =∑ 2

j

[ N ( p z j ) − N exper ( p z j )]2 N exper ( p z j )

.

(6.15)

It is now possible to transfer directly to the analysis of investigations of nanocrystalline materials, carried out using the previously described methods of positron annihilation. The positron annihilation was used for the first time for examining the special features of the structure of nanocrystalline materials by the authors of [56] who investigated the vacancies in the nickel nanoparticles (D~15 nm), measuring the position lifetime. Examination by the positron annihilation revealed the existence of vacancies and nanovoids in nanocrystalline metals Al, Cu, Mo, Pd, Fe and Ni, in nanocrystalline silicon Si and zirconium oxide ZrO 2 [32, 35, 38, 57– 60]. The results of these investigations show that the position lifetime spectra usually contain three components with the intensities I 1 , I 2 , and I 3 , = 1 – I 1 – I 2 , to which the lifetimes τ 1 , τ 2 and τ 3 correspond (Table 6.1, see also Fig. 6.5). The first two lifetimes τ 1 and τ 2 with the relative intensities I 1 and I 2 dominate the position lifetime spectrum and the long-lived component τ 3 appears with only a small intensity I 3 . In the nanocrystalline metals, lifetime τ 1 is close in value to the positron lifetime τ 1V in lattice monovacancies of coarse-grained metals and, consequently, τ 1 is regarded as the positron lifetime in vacancies 242

243

4 4 4

191±1 158±1 145±1

–τ(ps) (na no c rysta lline )

–

187±1 158±1 151±1

τ (p s ) (a mo rp ho us)

148±1 145±1 11 4 (F e 3S i)

c rysta lline )

–τ(ps) (coarse- grained

* In a d d itio n, the ' fre e ' life time s τ f in d e fe c t- fre e c rysta ls, a s we ll a s p o sitro n life time in mo no va c a nc ie s τ iV a nd in a gglo me ra te s o f i va c a nc ie s ( τ iV) a re give n; D is the c rysta llite (gra in) size

C o 33Zr67 F e 90Zr10 F e 73.5C u1N b 3S i13.5B9

Allo y

– –

– – 0.21 –

0.79 0.31

– 14500±600

415±6 378±1

189±2 199±2

20 10

2 2

N iZr ZrO 2

– – 3 2 1 ( i= 9 ) – – 2 7 2 ( i= 8 ) [ 6 6 ] –

– – 1 8 0 [3 8 ] – – 2 7 2 [4 4 ] – 1 0 8 [6 5 ] 1 0 8 [6 5 ] 1 0 3 [3 8 ] – – 2 1 9 [4 4 ] 1 0 8 (P d ) [6 5 ] 1 4 2 [3 8 ] 1 7 5 [4 3 ]

0.10 0.04 0.84 0.17 0.12 0.79 0.84 0.90 0.96 0.16 0.83 0.88 0.20 0.16

– – – – – 3330±300 –

359 398 345±2 297±9 330±9 4 2 2 ± 11 351±3

202 172 204±9 171±2 161±1 314±35 170±8

86 76 10 ≥100 200 10 9

1 3 2 3 3 1 2

– –

1 0 8 [6 5 ]

0.60

161

29

0.40

4 2 2 ( i= 1 3 ) 3 3 4 ( i= 1 0 ) 3 7 6 ( i= 1 3 ) – – – – –

11

* τ iV (p s) [6 3 ]

–

10

∗ τ 1 V( p s )

316

412±7 337±6 363±2 299±40 3 11 ± 1 3 347±5 465 365

1

9

* τ f (p s )

2 5 1 [3 8 ] 1 7 5 [3 8 ] 1 8 0 [6 4 ] 1 7 9 [3 8 ] – 11 5 [6 0 ] – –

8

I2

1 6 8 [3 8 ] 1 0 6 [3 8 ] 9 4 [3 8 ] 11 2 [3 8 ] – 1 0 8 [3 8 ] 1 0 8 [3 8 ] 1 0 8 [3 8 ]

7

I1

0.41 0.75 0.72 0.44 0.59 0.66 0.41 0.53

6

τ 3 (p s )

0.58 0.20 0.28 0.43 0.40 0.33 0.59 0.47

5

τ 2 (p s )

1970±160 900±40 530±70 470±50 600±200 1080±170 – –

4

τ 1 (p s )

253±4 161±8 74±1 175±10 182±8 182±5 2 11 186

40 10 12 20 14 12 20 20

1 1 1,2 1 2 1 1 1

Al Fe Ni Cu Cu Pd P d (is me a sure d in va c uum) P d (is me a sure d a fte r e xp o sure to a ir) P d (c o mp a c tio n a t a p re ssure o f 2 GP a a nd a t a te mp ra ture of 373 K ) P d (b a ll milling) Pd Mo Cu Ni Si P d 3F e

3

D (nm)

2

P re p a ra tio n me tho d

1

Ma te ria l

Table 6.1 Positron lifetimes τ 1 , τ 2 , τ 3 and relative intensities I 1 , I 2 in nanocrystalline solids after compaction of crystallites prepared by evaporation (1) or sputtering (2), in nanostructured metals prepared by severe plastic deformation (3) and moderate annealing, as well as the mean positron lifetime τ in nanostructured alloys after crystallization of the amorphous alloys (4) [32, 35, 60–62]

Microstructure of Compacted and Bulk Nanocrystalline Materials

Nanocrystalline Materials

in the interfaces. The size of these vacancies (‘vacancy-like’ free volumes) corresponds to 1–2 missing atoms. The affiliation of these free volumes to the interfaces, and not to the crystallites, is confirmed by the fact that the lifetime τ 1 is observed even after annealing nanocrystalline metals at a temperature higher than the annealing temperature of lattice monovacancies. The positron lifetime τ 2 characterises the positron annihilation in threedimensional lattice vacancy agglomerates (nanovoids) whose size approximately corresponds to 10 missing atoms. The positron lifetimes τ 1 and τ 2 in submicrocrystalline metals Cu, Pd, Ni (Table 6.1) are similar to the values τ 1 and τ 2 for the appropriate nanocrystalline metals. However, in the case of submicrocrystalline metals, the contribution of the second component I 2 to the positron lifetime spectrum is considerably lower because the size of the crystallites in the submicrocrystalline metals is larger than that in the nanocrystalline metals. The long positron lifetime τ 3 corresponds to the positron annihilation in nanopores, i.e. large free volumes whose size is close to the size of missing crystallite (see Fig. 6.5). The annealing of nano-Pd at 700 > T > 400 K increases the mean positron lifetime τ as a result of joining of the separate free volumes and of increasing their size. This process of structural relaxation of the interfaces is accompanied by an increase of the density of substance of the interfaces. At higher annealing temperatures the crystallites grow and at T > 1200 K, the mean crystallite size is already larger than the free path length of the free positron. Consequently, the contribution of the free volumes of the interfaces to the positron annihilation process decreases and the mean lifetime τ is shortened to the value corresponding to the free positron lifetime τ f in coarse-grained metals. Let us consider in detail the investigations of Fe–Si–Nb nanocrystalline alloys [67] which threw light on the possibilities of the positron annihilation method in the examination of nanomaterials. This work indicates how the positron annihilation can be used for detailed precision investigations of the complicated and interesting processes, which take place on the microscopic atomic level in the nanomaterials at elevated temperatures. In [67], using the combination of the positron annihilation and Xray diffraction analysis, the authors investigated the formation of thermal vacancies and of the long-range order of the D0 3 type in the disordered (Fe 3Si) 95 Nb 5, produced by mechanical alloying [68]. Using radioactive 58 Co isotope as a source of positrons, it was 244

Microstructure of Compacted and Bulk Nanocrystalline Materials

possible to measure the positron lifetime in situ at high temperatures and observe the formation of vacancies in the volumes of the grains and at the interfaces. The two-detector Doppler spectroscopy has detected Nb at the interfaces and triple points and made it possible to clarify the mechanism of stabilisation of the nanocrystalline state. The Fe–Si–Nb system was chosen because of the unique combination of the high thermal stability of the nanocrystalline state and a low vacancy formation enthalpy in D0 3 -ordered Fe 3 Si crystallites. Previous studies of FINEMET industrial alloy, produced by crystallisation from the amorphous state, showed that a high diffusivity is the result of a high thermal vacancy concentration in Fe 80 Si 20 nanocrystallites [48]. In [67], it is shown that a special feature on the temperature dependence of the mean positron lifetime, detected at 800 K, appears as a result of competition of two processes: (1) positron trapping at nanovoids, and (2) positron trapping in thermal vacancies. Analysis of the behaviour of the positron lifetime using a simple model, considered previously, made it possible to calculate the vacancy formation enthalpy in nanocrystalline Fe 3Si silicide. In [67], attention was also given to the correlation between the ordering process and the variation of diffusivity as a result of the modification of the local atomic distribution, i.e. the short-range order. The (Fe 3 Si) 95 Nb 5 nanocrystalline powder was prepared by mechanical attrition [68] with a Spex 8000 laboratory mixer/mill. A mixture of elemental powders with composition 71.25 at.% iron, 23.75 at.% silicon and 5 at.% niobium was milled in an inert argon atmosphere at room temperature for 48 hours. The mill and the milling balls were made of a hard alloy based on tungsten carbide. The ball-to-powder weight ratio was 4:1. Specimens with a diameter 5 mm and 1 mm thick for the investigations by electron–positron annihilation were compacted from the produced powder at room temperature under uniaxial pressure of 1.5 GPa. For investigations, radioactive isotope 58 Co was used as the positron source. This isotope in the form of liquid chloride 58 CoCl 2 was deposited on the surface of one specimen. Because the specimen was porous, it was partially impregnated by cobalt chloride. After drying the specimen, cobalt was reduced from the dichloride in a H 2 atmosphere at a temperature of 483 K for 2 hours. The specimen with the 58 Co positron source was stacked between two other identical specimens. The prepared sandwich was placed in a thin-walled Fe container and then container with 245

Nanocrystalline Materials

specimens sealed off in a quartz ampoule under high vacuum. The iron container, characterised by a short positron lifetime (approximately 100 ps), was used to ensure that in the course of experiments it would be possible to control the absence of contamination of the outer surface of sandwich by the positron source. In the case of such contamination part of positrons should annihilate with an iron. The component, responsible for positron annihilation with iron (if such a component appears), would be easily separated because it is considerably shorter than the other components of the spectrum of the investigated substance. In the absence of the iron container and in case of contamination of the outer surface of sandwich, annihilation will take place with quartz, which has a long positron lifetime. The component of the lifetime spectrum, responsible for annihilation of positrons with quartz, cannot be separated from the components of the spectrum of the investigated material, because they are very similar in value. The positron lifetime was measured by means of a fast-slow γγ -spectrometer with a time resolution (full width at half maximum FWHM) of 215 ps. The minimum number of coincidence counts in each spectrum was 0.5×10 6 . A series of 2 hour annealing cycles with simultaneous measurement of the positron lifetime was carried out in situ in the spectrometer at temperatures T an = 623, 713, 833 and 1023 K. After each annealing, the positron lifetime spectra were recorded at room temperature. The evolution of the structure of the specimens upon annealing was studied by X-ray diffraction method. Attention was given to the determination of the parameters of the structure such as the volumeweighted mean grain size D V , the root mean square (rms) microstrains 〈ε {2hkl }〉 1 / 2 , the long-range order parameters S D 03 , S B2 and lattice constant a. X-ray investigations were carried out at room temperature in a Siemens D500 X-ray diffractometer, using CuKα radiation and a secondary graphite monochromator. A Rietveld-like analysis of the X-ray spectra was performed, fitting each Bragg reflection profile with two Voigt functions for the K α 1 and K α 2 lines with the same width. The intensities of these doublets were calculated from structure factors using the kinematic scattering theory and the appropriate intensity factors [69]. The long-range order parameters were determined for a compound with the composition Fe 3 Si. The vacancies were not taken into account because the concentration of vacancies was considerably smaller than the concentration of antistructural defects (antisite) for both 246

Microstructure of Compacted and Bulk Nanocrystalline Materials

partially and completely ordered states. It was assumed that the long-range order parameters depend linearly on the site occupancies and change from the maximum values S D 03 = 1 and S B2 = 1/2 to the minimum values S D 0 = S B2 = 0 for the completely disordered 3 state. The long-range order parameters were determined from the ratio of the intensities of superstructure reflections {111} and {200} to the intensity of the structural reflection {220}. The intensity of 2 the {111} reflection is proportional to S D 03 and is determined only by the phase D0 3 , whereas the intensity of the {200} reflection depends on the long-range order parameters of both phases and is 2 proportional to ( S D 03 + 2S B 2 ) . Consequently, it was possible to determine the long-range order parameters for both phases D0 3 and B2. From the line widths, the grain size and rms microstrains 〈ε {2hkl }〉 1 / 2 were determined with taking into account the anisotropy of elastic distortions and adding the broadening of superstructure reflections {111}, {200}, etc. by boundaries of ordered domains [70]. To obtain information on the chemical environment of the atomic free volumes where positrons are annihilated, the electron momentum distribution was measured. These measurements were performed by the coincident Doppler broadening method at room temperature. The Doppler broadening spectra were analysed using the previously considered model of description of the electron momentum distribution (see the equations (6.13), (6.14) and (6.15)). The separation of the spectra into components showed that in the range of the electron momentum from 18×10 –3 m 0 c to 24·10 –3 m 0 c (or in the energies of γ-quanta from 4.5 to 6 keV) the spectra are determined mainly by the core electrons, and in the range from 0 to 10·10 –3 m 0 c the spectra are determined by the valence electrons. In the intermediate range, the contributions from both valence and core electrons are important. In order to observe the special features of spectra with the naked eye, i.e. without preliminary numerical analysis, the Doppler broadening spectra could be normalised to the same area and after that divided by the reference spectrum. The reference spectrum is usually measured with high statistics on pure silicon Si specimen (see Fig. 2 in [67]). The temperature dependences of the structural parameters and the mean positron lifetime τ , measured at ambient temperature, are shown in Fig. 6.6. The as-milled material is characterised by a small grain size D V = 15 nm. This is in good agreement with the results of examination by transmission electron microscopy [68], a

247

Nanocrystalline Materials

high microstrain 〈ε {2220}〉 1 / 2 = 1.1 % and a very low parameter of D0 3 long-range order. The lattice constant a is higher than that for ordered Fe 75 Si 25 (a 0 = 0.565 nm [71]). The reduced lattice constant of the ordered phase is often found in a disorder–order phase transformations in metallic alloys [72]. The decrease is usually related to different sizes of the atoms, which may result in denser packing in the case of ordered state. However, lattice constant a = 0.5710 nm, measured for ball-milled and most disordered (Fe 3 Si) 95 Nb 5, is even greater than that for the milled and disordered 













$



$ 





QP



,

$   $  

,













!;

 





 



ε











ε



!;

QP









τ

B

SV



 







%,3 . Fig. 6.6. Long-range order parameters S D 03 and S B2 , lattice constant a, volume2 1/ 2 weighted mean grain size D V , root mean square microstrain 〈ε {220}〉 , and mean positron lifetime τ for n-(Fe 3 Si) 95 Nb 5 at ambient temperature after 2 h annealing at T an [67]. 248

Microstructure of Compacted and Bulk Nanocrystalline Materials

Fe 75 Si 25 (0.5663 nm [73] or 0.5695 nm [74]). Extrapolation of the lattice constant of the solid solution with the Si content smaller than 10 % [75] by the lattice constant of the solution with a Si content of 25 % also gives a low value of 0.5699 nm. Thus, the enhanced lattice constant indicates that in addition to disordering, partial dissolution of niobium atoms with a larger atomic radius also takes place in the compound. After annealing the specimen at 483 K for 2 hours, a small decrease is observed in the lattice constant and in the microstrain 〈ε {2220}〉 1 / 2 , while the grain size D V is unchanged. The mean positron lifetime τ = 221±2 ps results from the combination of two components τ 1 = 171 and τ 2 = 375 ps with relative intensities of I 1 = 75 % and I 2 = 25 % (Table 6.2). The value τ 1 coincides with the positron lifetime for monovacancies in ferromagnetic α -Fe ( τ 1V = 175 ps [76]) and in iron silicide Fe 3 Si with a structure D0 3 ( τ 1V = 175 ps) [77]) and therefore characterises the positron lifetime in free volume of the size of one missing atom. Component τ 2 corresponds to the positron lifetime in nanovoids of the size of 10–15 missing atoms [63] (void diameter 0.5–0.8 nm). Both positron lifetime components reflect general features of spectra of nanocrystalline metals [57]. The value τ 1 relates to the positron trapping in structural vacancies in interfaces or in non-equilibrium thermal vacancies in the crystallites, and τ 2 is associated with nanovoids located at the intersections of interfaces of close-packed nanocrystallites. The absence of a lifetime component smaller than the free lifetime τ f indicates saturation trapping of positrons by defects. Saturation takes place because the mean positron diffusion length in metal crystals (L + ≈ 100 nm) is considerably larger than the crystallite size; therefore, the positrons during diffusion reach the interfacial traps with a high probability. Table 6.2 Mean positron lifetime τ and component analysis in starting specimen (Fe 3 Si) 95Nb 5 and the same specimen after annealing at the temperature T an = 1023 K for 2 h. I 1 and I 2 represent the relative intensities of the two components with lifetimes τ 1 and τ 2 , respectively [67]

T an(K )

τ– (p s)

τ 1(p s )

τ 2(p s )

I1(%)

I2(%)

483 1023

222±2 225±2

171±1 174±1

375±2 370±3

75.0±0.5 74±1

25.0±0.5 26±1

249

Nanocrystalline Materials

The electron momentum distribution shows that positrons annihilate in the vicinity of niobium atoms [67]. This is indicated by the profile of the electron momentum distribution, which resembles the profile of pure Nb to a greater extent than the electron momentum distribution profile of Fe or iron silicide Fe 3Si (see Fig. 2 in [67]). This shows that either Nb atoms are segregated to the interfaces or the vacancies in the crystallites are surrounded by niobium atoms. A temperature of T an = 623 K is characterised by a large decrease of the mean positron lifetime from 221 to 215 ps. This decrease takes place as a result of structural relaxation at interfaces associated with annealing of strains in crystallites and with the start of segregation of niobium atoms at interfaces. It leads to a large decrease of the lattice constant a. Upon annealing at T an = 623 K, a small degree of the long-range order, similar to that in the as-milled state, is retained. Nearly complete ordering of the type D0 3 ( S D 03 ≈ 0.9) is obtained after annealing at T an = 713 K. Indeed, the shift of the superfine magnetic field measured by Mössbauer spectroscopy [68, 78], shows that dissolved niobium leave the crystallite body after annealing for 1 hour at 723 K. Consequently, both processes (ordering and Nb segregation) take place at this temperature. It leads to a further decrease of the lattice constant a (by about 0.9 %) and of the microstrain 〈ε {2220}〉 1 / 2 (see Fig. 6.6). After annealing at T an = 833 and 1023 K, the complete long-range order is established and, according to the decrease in the width of the superstructure reflection {111}, all antiphase domains, in particular those, which determined by the displacement vector 100 /2, are annealed out. In this stage, the rms displacement decreases and the lattice constant a reaches a value of 0.5658 nm. Assuming that all Nb atoms are segregated at the interfaces, this value corresponds to a composition Fe 76 Si 24 [73], which is in agreement with the energy-dispersive X-ray analysis (EDX) [68]. At a high temperature of 800–1000 K, the most evident mechanism of structural relaxation is grain growth. On the basis of the lower thermal stability of Fe 3 Si without Nb additions it may be concluded that the segregation of niobium at interfaces inhibits grain growth. After annealing at 1023 K, when the mean grain size D V reaches ~110 nm, there are no the component corresponding to the free positron lifetime. Since the width of the instrumental resolution function, determined on a polycrystalline silicon nitride Si 3 N 4 , corresponds to a crystalline size of 100–200 nm, the value 250

Microstructure of Compacted and Bulk Nanocrystalline Materials

D V ≈ 110 nm is highly indeterminate. The saturation trapping of the positrons means that L + ≈ 100 nm is still lower than the grain radius D V /2, and confirms the results of X-ray diffraction analysis. It should be mentioned that the mean positron lifetime τ and the ratio of the components in the annealed and starting state are practically identical (see Table 6.2), although the crystallite size differs greatly. In accordance with the conventional interpretation according to which the nanovoids are distributed mainly at lines of intersection of interfaces, the ratio I 2 /I 1 should greatly decrease with increasing crystallite size. Evidently, a decrease in the value of I 2 /I 1 is compensated by annealing of quenched thermal or interfacial vacancies and, consequently, the intensity of I 1 decreases. Interfacial vacancies may be associated with a presence of niobium atoms at interfaces, because there are no interfacial vacancies in pure Fe 3Si. In addition to this, additional nanovoids may appear during crystallite growth as a result of agglomeration of the vacancies. This increases the intensity of I 2 . This is fulfilled for pure iron, processed by ball milling. Two-component analysis of the Doppler spectra confirms the hypotheses according to which the observed increase in the mean positron lifetime τ after annealing at T an = 1023 K takes place due to a decrease of I 1 or increase of I 2 , and not because of the change in the size of the vacancy clusters. It is important to note that after annealing at 823 K only small changes were found in the electron momentum distribution. This is in agreement with the previously described positron annihilation with electrons of Nb atoms segregated in the interfaces or in structural vacancies, surrounded by niobium atoms. The most important result of [67] is the unusual behaviour of the mean positron lifetime at high temperatures (T > 800 K). As shown in Fig. 6.7, the mean lifetime τ has a maximum in the vicinity of 800 K and then rapidly decreases with increasing temperature. The measured dependence τ (T ) is equilibrium and is fully reversible with a subsequent decrease or increase of temperature. Analysis of the positron lifetime spectra shows that no new component with a short lifetime, which could be used to explain the decrease in τ , forms. A decrease in τ correlates with an increase in intensity I 1 as a result of a decrease in intensity of I 2 . With increasing temperature, the probability of positron trapping by the vacancies, formed in thermodynamic equilibrium inside the crystallites, increases. Since the positron lifetime in equilibrium thermal vacancies, τ V, T , is approximately equal to that in interfacial vacancies, τ V, S , then the intensity of the shorter lifetime component τ 1 increases and that of component τ 2 decreases. This ensures the 251

Nanocrystalline Materials  

%2 

B τ (SV



























%,3 .

Fig. 6.7. Mean positron lifetime τ in n-(Fe 3 Si) 95 Nb 5 [67] measured in isothermal conditions with the following sequence: (∆) 2 h annealing at T an = 1023 K was performed; (  ) the specimen was cooled to 293 K and a series of data was measured at temperature increasing up to 1003 K; ( £ ) further data were taken at decreasing temperature. This sequence proves the full reversibility of the observed temperature dependence. The solid line is a fit to the whole set of data according to the threestate model of positron trapping (Eq. 6.16). The inset shows the same data together with an extension of the fitting curve to higher temperatures.

reversibility of equilibrium dependence τ (T ) . The chemical environment of interfacial vacancies is enriched by niobium, while the chemical environment of thermal vacancies formed inside the crystallites is niobium poor. Therefore, the electron momentum distribution at high temperatures should change and the profile of the spectrum should approach that of pure coarsegrained Fe 3 Si. According to the above interpretation, the temperature dependence of τ can be described by the combination of the temperature behaviour of three different traps: (1) Interfacial vacancies (vacancy-size free volumes at the interfaces) with a positron trapping rate σ V,S C V,S , which is independent of temperature and is described [79] in the form: σ V, SC V, S = 6α / 〈 D〉 , where C V, S is the concentration of interfacial vacancies, 〈D〉 is the grain size, and α is the trapping coefficient of grain boundaries. If α = 4.2×10 2 m s –1 , as determined in [80], and 〈D〉 =110 nm from X-ray diffraction analysis, then the trapping rate σ V,S C V,S = 2.3×10 10 s –1 is obtained. (2) Nanovoids with a trapping rate σ void C void = (1 + β (T– 293))( σ void C void ) T=293 K , which increases linearly with temperature. 252

Microstructure of Compacted and Bulk Nanocrystalline Materials

The value of (σ void C void )T = 293 K can be determined from the intensity I 2 σ void C void ratio I = σ C by substituting the values I 1 and I 2, measured 1 V, S V, S

T = 293 K, and the value of σ V, SC V, S calculated above. As a result, a trapping rate (σ void C void )T = 293 K = 0.8×10 10 s –1 is obtained. (3) Thermodynamically equilibrium thermal vacancies with a σ V, T specific trapping rate and a concentration F F F F − C V, T = exp( S V, / k ) exp( H / k T ) , where H and S V, T B V, T B V, T T are the effective vacancy formation enthalpy and entropy, respectively, and k B is the Boltzmann constant. In a three-state transition-limited model without detrapping, the mean positron lifetime may be represented as

τ =

τ V, Tσ V, T C V, T + τ V, Sσ V, SC V, S + τ voidσ void C void σ V, T C V, T + σ V, SC V, S + σ void C void

.

(6.16)

Dependence (6.16) with the above values of σ V, SC V, S and (σ void C void )T =293 K was used to fit the temperature dependence of the mean positron lifetime after annealing a specimen at 1023 K. In approximation, the positron lifetimes τ V, T = τ V, S = τ 1 = 174 ps and τ void = τ 2 = 370 ps, determined by the two-component analysis of the spectrum obtained at room temperature (Table 6.2), were kept constant. The free parameters of the fit are the temperature F coefficient β tr , combined preexponential factor σ V, T exp( S V, T / kB ) , F and vacancy formation enthalpy H V, T . Good agreement between the modeling curve and the experimental data (Fig. 6.7) is ensured by the temperature coefficient β tr of the positron trapping rate by nanovoids, which describes the low temperature part of the curve. This coefficient β tr is equal to (1.0 ± 0.1)×10 –3 . This value must be compared with β tr ≈ 8×10 –3 , determined for neutron irradiated molybdenum with nanovoids of the size of 2.6 nm [81]. Taking into account the fact that in the present case the nanovoid diameter is 0.5–0.8 nm. Since the coefficient β tr depends linearly with the square of the void diameter [82], the two results are in good agreement and confirm that the temperature dependence of a trapping rate, σ void , is the principal reason for the observed linear increase of τ in the temperature range from 300 to 600 K. Positron trapping in the nanovoids restricts the positron trapping in 253

Nanocrystalline Materials

thermal vacancies, and in (Fe 3Si) 95 Nb 5 this takes place up to higher temperatures than in neutron-irradiated molybdenum, as a result of a higher concentration of the nanovoids in (Fe 3 Si) 95 Nb 5 alloy. In other words, the positron trapping in nanovoids remains transitionlimited. Table 6.3 compares these vacancy formation parameters, obtained by approximation, with those of coarse-grained Fe 79 Si 21 with D0 3 superstructure, Fe 75 Si 25 [77] and Fe 76 Si 24 [38, 68]. The F vacancy formation enthalpy H V, T = 1.1 eV for nanocrystalline (Fe 3Si) 95 Nb 5 coincides with that of other silicides, enriched in iron. This coincidence confirms the validity of the used model of describing temperature dependence τ (T ) . This model predicts that the mean positron lifetime initially increases and then decreases with increasing temperature towards the limiting value τ V, T when positron trapping at thermal vacancies becomes the completely dominant process. The inset in Fig. 6.7 shows the modeling curve τ (T ) extrapolated to higher temperatures: the dependence τ (T ) should reach the limiting value τ = τ V, T for T ≥ 1400 K. This temperature is close to the melting point T melt = 1500 K of iron silicide. However, in this temperature range the grain growth should be so fast that it is not possible to verify this assumption for the nanocrystalline state. In order to determine the main reason for ordering of (Fe 3 Si) 95 Nb 5 , let us consider the process of atom diffusion. Complete ordering of the absolutely disordered state requires mutual rearrangement of at least half of the atoms of silicon and iron in the crystallite. The jump of a single vacancy takes place more often than the jump of a single atom. The maximum distance to sink and source of atoms or vacancies in nanocrystals is very short and, F Vacancy formation enthalpy H V, T and preexponential factor / k B ) in nanocrystalline n-(Fe 3 Si) 95 Nb 5 [67] and in coarse-grained or monocrystalline Fe 75 Si 25 , Fe 76 Si 24 and Fe 79 Si 21

Table 6.3

F σ V, T exp(S V, T

Ma te ria l n- (F e 3S i)95N b 5 F e 76S i24[8 3 , 8 4 ] F e 75S i25[7 7 ] F e 79S i21[7 7 ]

HFV.T ( e V)

σ V.T e xp (S FV.T/k B)

1.1±0.2 1.06±0.04 0.77±0.08 1.1±0.06

0.5 2.2* 0.5 4.4

(1 0 16 s –1)

* σ V.T= 6 . 6 × 1 0 14 s–1 a nd S FV.T= 3 . 5 k B ta k e n fro m Re fs. [8 3 , 8 4 ] re sp e c tive ly 254

Microstructure of Compacted and Bulk Nanocrystalline Materials

consequently, only a small part of the atoms should move in order to reach the equilibrium concentration of thermal vacancies (for example, interfaces are a sink of vacancies in nanocrystals). Therefore, the equilibrium diffusion properties must be extrapolated to a very short diffusion length. It may be expected that ordering in the nanocrystallites is completed when all atoms make a few jumps over a diffusion length of about 1 nm. The evaluated diffusion lengths of either Fe or Si at 650 K (the mean temperature of the ordering process) in the absolutely disordered alloy with the structure A2 is slightly longer than the ‘ordering length’ equal to 1 nm (Table 6.4). Activation enthalpy Q for D0 3 short-range ordering in nanocrystalline (Fe 3 Si) 95 Nb 5 is equal to 2.7 eV [78]. This value is higher than the activation enthalpy estimated for the disordered state with a structure A2 (see Table 6.4). In strongly deformed Fe 3 Si, the activation enthalpy of short range ordering is 2.0 eV [68]. Partial ordering of B2 type (see Fig. 6.6) or the presence of niobium in grains may be the reason for the fact that the ordering rate in nano(Fe 3 Si) 95 Nb 5 is lower than expected. The diffusion length of Fe atoms in the D0 3 -ordered state is considerably longer than that of Si atoms, which is less than 1 nm (see Table 6.4). This indicates that the ordering of partially ordered superstructure D0 3 is controlled by the diffusion of Si rather than by the diffusion of Fe. In other words, the movement of Si atoms is a limiting factor of ordering. Assuming that the frequency factor D 0 of Si diffusion in D0 3 -Fe 3 Si is equal to 0.19 m 2 s –1 (see Table 6.4) and the activation enthalpy, measured for D0 3 short-range ordering, is equal 2.7 eV, Table 6.4 Diffusion activation enthalpies Q, frequency factors D 0 and diffusion length 4Dt an calculated for an annealing time t an = 2 h in different Fe–Si materials [67] F e – S i ma te ria ls 59

F e in D0 3- F e 76S i24 [8 4 ] Ge in D0 3- F e 76S i24 * [8 5 ] 59 F e in A 2 - F e 76S i24 * * S i in A 2 - F e 76S i24* * * 71

Q (eV)

D0 (m2 s–1)

4Dt an at 650 K

1.64 ± 0.04 3.23 ± 0.04 2.2 2.2

11..33++−000.5..54–0×.4×10 10−4–4 ×1−01–1 1.9++−000...996–0×.610 8×10–4 3×10–3

850 0.02 14 28

(nm)

* Diffusio n le ngth o f S i is c lo se to tha t o f Ge * * Va lue s Q a nd D0 a re o b ta ine d b y e xtra p o la tio n o f d iffusio n o f iro n a to ms in F e 1- xS ix with A 2 struc ture a t lo w c o nte nt o f S i up to x = 0 . 1 Ra tio Q/T melt = 0 . 0 0 1 5 e V K –1 is d e te rmine d fro m F ig. 1 4 in Re f. [8 5 ] using the me lting te mp e ra ture T melt = 1 5 0 0 K o f F e 76S i24. At a te mp e ra ture lo we r tha n C urie te mp e ra ture the a c tiva tio n e ntha lp y Q a nd fre q ue nc y fa c to r D0 sho uld b e to so me e xte nt la rge r. * * * Ap p a re ntly, d iffusio n c o e ffic ie nt o f S i a to ms in a d iso rd e re d F e 76S i24 is 2 – 5 time s la rge r tha n tha t fo r F e a to ms

255

Nanocrystalline Materials

it can be calculated that the diffusion length of Si atoms is 2.5 nm at 650 K. This value is very close to the ‘ordering length’. In conclusion it should be noted that the model, in which ordering is controlled by the movement of Si atoms and by the formation of thermal vacancies, is in good agreement with the diffusion data for coarse-grained iron silicide Fe 3 Si. Thus, in [67] the method of electron–positron annihilation was used to study the ordering and formation of vacancies in nanostructured (Fe 3 Si) 95 Nb 5 . The states from the as-milled disordered to completely D0 3-ordered with a crystallite size of about 100 nm were investigated. The superstructure D0 3 forms after annealing for 2 hours at 713 K. The kinetics of the ordering is controlled by the diffusion mobility of both elements Si and Fe in the disordered phase, but the diffusion mobility of silicon is smaller than that of iron. The presence of niobium slows down the ordering process in comparison with the kinetics of ordering in pure Fe 3 Si. The activation enthalpy, which is equal to 2.7 eV [78] and is higher than the activation energy for disordered Fe 3 Si, makes it possible to explain the experimental results of [67]. Structural ordering leads to thermal stability of the nanomaterial in which rapid self-diffusion of iron takes place as a result of a small vacancy formation enthalpy. In particular, by this relationship Fe 3 Si differs from compound FeAl with B2 structure in which a small vacancy formation enthalpy is not accompanied by the jumps of Fe atoms to the closest sites of the crystal lattice. The formation of thermodynamically equilibrium vacancies in nanocrystallites of nanostructured (Fe 3Si) 95 Nb 5 compound has been confirmed by the reproducible equilibrium temperature dependence of the mean positron lifetime τ (T ) . The competing positron trapping at nanovoids and thermal vacancies results in a maximum of the dependence τ (T ) in the vicinity of 800 K. This behaviour is caused by the unique combination of the high thermal stability of nanocrystalline state in respect to the grain growth, which ensures by the segregation of Nb atoms at interfaces, and by a low vacancy formation enthalpy. The data for the positron lifetime are described efficiently by the three-state trapping model with a low vacancy F F formation enthalpy H V, T = (1.1±0.2) eV; this value of H V, T completely corresponds to the results obtained for coarse-grained Fe 3 Si. Thus, the increased thermal stability of the nanocrystalline state in relation to grain growth is associated with the segregation of niobium atoms at interfaces of the nanograins and with atomic 256

Microstructure of Compacted and Bulk Nanocrystalline Materials

ordering. Because of low diffusion mobility, stabilisers for other nanostructured materials should include either transition metals of group IV (Ti, Zr) or group V (V, Ti), or their carbides, nitrides and oxides. It should be mentioned that such a phenomenon of structural modification as atomic ordering, controlled by diffusion, was found in nanocrystalline vanadium carbide. In future, the understanding of atomic mechanisms, determining the processes of grain growth and atomic ordering will lead to goal-oriented synthesis of nanomaterials with the required microstructure. Recently, a new experimental method, the time-differential length change, has been proposed for studying point thermal defects [86]. The method is based on contact-free measurement of the change in the length of the investigated specimen as a function of time t at a sudden change of temperature T. The cylindrical specimen with a ring-shaped ledge in the centre is spark cut from the ingot. One end of the specimen is hardly fixed in a cooled specimen holder. The ring-shaped ledge in the form of a step is located at a distance l = 20 mm from the outer end of the specimen (Fig. 6.8). The surface of the ledge is parallel to the surface of the outer end of the specimen, and both surfaces are polished. The change in specimen length l as a function of time t is measured in the experiment. The change of length is recorded by a two-beam Michelson laser interferometer, which radiation is reflected from parallel polished surfaces of the outer end and the ledge.

20 mm

40 mm

specimen laserinterferometer

cooled specimen holder

Fig. 6.8. Sketch of the experimental setup. The length of the specimen, which is suspended together with the furnace in a vacuum recipient, is measured by the two laser beams on the polished parallel planes on the specimen front and ledge [86].

257

Nanocrystalline Materials

Determination by this method of the vacancy formation enthalpy, the activation enthalpy and the migration enthalpy of vacancies in Fe 55 Al 45 showed that the results are in good agreement with the values obtained previously by the positron lifetime spectroscopy [87]. In [88] it was reported that the method described in [86] is a methodological innovation in studying thermal vacancies. On the whole, examination of positron annihilation in nanocrystalline compacted metals and alloys gave the following results: 1. The positron lifetime in nanocrystalline metals is longer than the lifetime τ f of free positrons. 2. The fraction of positrons, trapped by the vacancies, increases with increasing compaction pressure applied during preparation of the nanocrystalline solids; this means that an increase of the compaction pressure increases the interfacial area. 3. The positrons are trapped by monovacancies, nanovoids (vacancy complexes) and also nanopores whose size is close to that of the missing crystallite. 4. The free vacancy-like volumes, trapping the positrons at low temperatures, belong to the interfaces and not the crystallites. 5. The positron trapping by dislocations of the crystallites is unlikely to take place because the plastic deformation of metals leads to a smaller change in the positron lifetime than the preparation of nanocrystalline metals by compaction of nanopowders. 6.3 STRUCTURAL FEATURES OF SUBMICROCRYSTALLINE METALS PREPARED BY A SEVERE PLASTIC DEFORMATION At present it is clear that the model of the gas-like structure does not describe to the actual structure of the interfaces in nanocrystalline materials. An alternative of gas-like structure model is the assumption on the non-equilibrium interfaces having high energy because of the presence of dislocations directly at the interfaces and non-compensated disclinations at triple points. The long-range stress field of non-equilibrium interfaces is characterised by a strain tensor whose components inside the grain are proportional to r –1/2 (r is the distance to the grain boundary). Consequently, the stress field leads to the formation of elastic distortions of the crystal lattice, with the maximum distortions near the interfaces. This model is proposed in [89–94] when studying the

258

Microstructure of Compacted and Bulk Nanocrystalline Materials

submicrocrystalline materials produced by different methods of severe plastic deformation. The results obtained by electron microscopy show that the main special feature of the structure of submicrocrystalline materials is the presence of randomly misoriented grain boundaries. These grain boundaries are in non-equilibrium state. The as-prepared submicrocrystalline metals and alloys before annealing are characterised by the presence of extinction contours along grain boundaries. Extinction contours indicates high elastic stresses [89, 90, 95–99]. Since the dislocation density inside the grains is considerably lower than at the interfaces, the non-equilibrium interfaces become the main sources of elastic stresses. After annealing, many grains become completely free from dislocations, the extinction contours disappear and a banded contrast, typical of the equilibrium state, appears at the grain boundaries. The latter indicates that the relaxation of these boundaries have taken place. As an example, Fig. 6.9 shows the microstructure of submicrocrystalline palladium, produced by severe plastic deformation with using torsion under quasi-hydrostatic pressure, and its variation after annealing for 1 hour at different temperatures [98]. Plastic deformation of the starting coarse-grained palladium was affected at room temperature in air. As a result of severe rotation of Pd, the true logarithmic degree of deformation e was 7.0. After severe

Fig. 6.9. Microstructure of submicrocrystalline palladium Pd after annealing for one hour at different temperatures [98]: (a) 475 K, (b) 505 K, (c) 535 K, (d) 575 K, (e) 795 K, (f) 855 K.

259

Nanocrystalline Materials

plastic deformation, the specimens acquired a finely divided dislocation-tangle-laden structure with misoriented fragments of mean size of about 150 nm. The volume density of dislocations at the boundaries of individual grains was as high as ρ V b = 4×10 11 cm 2 . After annealing at 475 K, a submicrocrystalline structure arises in the specimen, which has grains of a mean size of about 200 nm (Fig. 6.9a). From approximately 40–60 % of the grains lattice dislocations were removed, while the dislocation density within the rest of grains reached 2.5×10 10 cm –2 . The intragranular bending extinction contours and the boundary diffusion contrast indicate that the grain boundaries are out of equilibrium; the volume density of grain boundary dislocations ρ V b is equal to 1.10×10 11 cm –2 . After annealing at 505 K (Fig. 6.9b) the mean grain size increased to 300 nm, and the fraction of grains free from lattice dislocations increases by a large amount and ranges up to about 60 or 70 %. Annealing at 535 K (Fig. 6.9c) leads to non-uniform grain growth. The grain size distribution becomes bimodal with the mean grain size of 850 nm. About 30 to 40 % of grains have non-equilibrium boundaries and a mean size of 350 nm. The rest of the grains are approximately 1100 nm in size and nearly half of them are characterised by a banded contrast common for the equilibrium state. The presence of the contrast indicates relaxation of these boundaries. The dislocation density ρ V b in the non-equilibrium boundaries is 6×10 8 cm –2 . After annealing at 575 K (Fig. 6.9d), the grain size increases to 2.2 µm. As previously, dislocations are observed at the grain boundaries. In individual grains (their fraction is about 10–20 % of all grains) the lattice dislocation density amounts to (0.3–1.5)×10 9 cm –2 , while in the remaining grains ρ V b is low and does not exceed 2×10 7 cm –2 . This suggests that a large number of grains have already been cleaned to remove dislocations. Annealing at 795 K (Fig. 6.9e) results in an increase of the grain size to 4.5 µm. The interior of the grains is virtually free from dislocations. Approximately half of the grain boundaries remain in non-equilibrium state with a dislocation density of ρ V b ≈ 4×10 9 cm –2 . After annealing at 855 K (Fig. 6.9f ) the grain size amounts to as much as 9 µm and half of the grains do not contain any dislocations. However, the situation reverses dramatically for the remaining grains: they contain tangles of lattice dislocations with a density of up to 3×10 9 cm –2 . These dislocations form a cellular structure with a cell size from 0.5 to 2.0 µm. As a result of further annealing, grain growth continues and defects disappear completely 260

Microstructure of Compacted and Bulk Nanocrystalline Materials

from the interior of the grains. After annealing at 1075 K the mean grain size is about 20 µm. The results of investigation of the microstructure of submicrocrystalline Pd [98] show that annealing is accompanied by a two-stage increase in the grain size. The first substantial increase in the grain size occurs at a temperature slightly above 475 K. The second stage of grain growth starts at temperature above 795 K, i.e. at a temperature higher than the secondary (collective) recrystallisation temperature. The experimentally observed local distortions of the crystallite lattice near their boundaries confirm the presence of elastic stresses at the interfaces [17]. Examination of submicrocrystalline Fe by Mössbauer spectroscopy showed [10] that experimental spectrum is a superposition of two spectra corresponding to two different states of the iron atoms. One of them (the state of the Fe atoms in crystallites) coincides with the state of the iron atoms in coarse-grained α -Fe. The second component of the experimental spectrum reflects the special state of the iron atoms at the interfaces, although the crystalline structure of the grains and their boundaries was identical. According to [89, 90], the difference in the parameters of the superfine structure of the Mössbauer spectra of submicrocrystalline iron is caused by higher dynamic mobility of the interfacial atoms. The results obtained by electron–positron annihilation [62] (see Table 6.1) point to some similarity of the microstructure of nanocrystalline and submicrocrystalline materials; in particular, these materials contain free volumes of the same type. In addition to relaxation of the interfaces, the annealing of submicrocrystalline materials is accompanied by grain growth, and an abrupt jump-like change in the properties of submicrocrystalline metals is observed after annealing at the same temperature when grain growth starts. This is confirmed by the results of investigation of the evolution of the microstructure and properties of submicrocrystalline Ni and Ni 3 Al alloy [101]. Submicrocrystalline nickel with a mean grain size of approximately 100 nm was produced by means of severe plastic deformation carried out by torsion under high pressure. An increase of annealing temperature of submicrocrystalline Ni to 450 K was accompanied by a slow decrease of residual electrical resistance ρ 4.2 and microhardness H V at an almost unchanged grain size (Fig. 6.10). Annealing at 500–525 K led to a rapid decrease of electrical resistance and microhardness due to the beginning of rapid grain 261











  









+'

ρ

QP



'!



1L

*UDLQ VL]H 



ρ µΩ FP

+DUGQHVV





+'

*3D

Nanocrystalline Materials



'!

  





%





.

Fig. 6.10.Evolution of residual electrical resistance ρ 4.2, microhardness H V and grain size 〈D〉 in submicrocrystalline nickel Ni as a function of annealing temperature T [101].

growth. With a further increase of annealing temperature the slow growth of the grains of submicrocrystalline Ni continues and there are small changes of electrical resistance and microhardness. According to [101], the changes of electrical resistance and microhardness in relation to annealing temperature are associated directly with an increase in the grain size and are correlated only slightly with internal stresses. A model of the grain boundaries [91–94, 102–104], which takes into account the presence of dislocations and disclinations, makes it possible to evaluate quantitatively the magnitude of developed stresses and the excess energy of the interfaces, and also crystal volume change caused by the excess elastic energy. According to [92], an interface represents a disordered dislocation networks, and the rms strains ε disl , formed at an interface and related to the unit area of the interface, is equal:

ε disl ≈ 0.23b[( ρ /D)log(R/2b)] 1/2 ≈ 0.13b[ ρ V log(R/2b)] 1/2 .

(6.17)

Excess energy γ ex, disl , caused by external dislocations and related to the unit area of the interface, is

γ ex, disl = Gb 2 ρ log(R/2b)/[4 π (1 – ν )] . 262

(6.18)

Microstructure of Compacted and Bulk Nanocrystalline Materials

In equations (6.17) and (6.18) b is the Burgers vector of dislocations, D is the grain size in the nanocrystal, ρ and ρ V ≈ 3 ρ /D are the linear and volume density of dislocations, and R is the grain size in a coarse-grained polycrystal, G is the shear modulus, and ν is Poisson coefficient. High internal stresses result in changes in the volume of the submicrocrystalline material. The change of volume due to disordered dislocation networks is ∆V/V ≈ 0.13b 2 ρ V log(R/2b) .

(6.19)

Similar equations for a system of disclinations, formed at the junction of several grains are derived in a two-dimensional model of the polycrystal with square grains [102]:

ε discl ≈ 0.1〈Ω 2〉 1/2 ,

(6.20)

γ ex,discl =[G〈Ω 2 〉Dlog2]/[16 π (1 – ν)] ,

(6.21)

where 〈Ω 2 〉 is the rms power of the disclinations. According to [105], the power of junction disclinations is 1–2°, and 〈Ω 2 〉 1/2 ≈ 0.03. The volume dislocation density ρ V for submicrocrystalline materials is 3×10 15 m –2 [43]. Taking these values into account, for submicrocrystalline aluminium Al with a grain size of 10 nm the excess energies are γ ex,disl ≈ 0.3 J m –2 and γ ex,discl ≈ 0.06 J m –2 . An increase in the volume, caused by the presence of the dislocations, is ∆V/V ≈ 4×10 –4 . Since the change in the volume is proportional to elastic energy and γ ex,discl is up to five times smaller than γ ex,disl , then the increase of the volume caused by the presence of disclinations, will be ~0.8×10 –4. The total increase in the volume of submicrocrystalline Al is ∆V/V ≈ 4.8× 10 –4 . This is half of the experimental value ∆V/V ≈ 9×10 –4 [106]. Evidently, the additional increase in the volume is associated with the formation of vacancies during deformation. For submicrocrystalline copper with a grain size D = 200 nm, ρ V ≈ 3×10 15 m –2 and 〈Ω 2 〉 1/2 ≈ 0.03, estimates obtained using equations (6.17)–(6.21) give: ε disl ≈ 7.5×10 –3 , ε discl ≈ 3×10 – 3 , total internal elastic strain ε = ( ε 2disl + ε 2discl ) 1/2 ≈ 8×10 – 3 , γ ex,disl ≈ 0.41 J m –2 , γ ex,discl ≈ 0.09 J m – 2 , and total excess energy γ ex = γ ex,disl + γ ex,discl ≈ 0.5 J m –2 . The resultant value of internal elastic strain ε is in satisfactory agreement with the magnitude of internal strains in submicrocrystalline copper, which was determined by X-ray 263

Nanocrystalline Materials

diffraction [93]. The model [94, 102, 103] assumes an important role of disclinations formed at triple junctions of the grains. Special interests are splitting of grain boundary disclinations [103, 104]. The splitting of disclinations at triple junctions decreases the elastic energy of the system. This splitting of a disclination into smallpower disclinations may be of the volume type (Fig. 6.11b); in this case, the local amorphisation of the triple junction region takes place. Another variant is the splitting of a disclination into three rows of small-power disclinations located in the boundaries of adjacent grains (Fig. 6.11c). Line splitting of grain boundary disclination to a number of small-power disclinations, located along the grain boundary (Fig. 6.11e), is also possible. According to [103, 104], a decrease of the elastic energy of the initial grain boundary disclination becomes greater with an increase of the number of new disclinations formed. In the case of line splitting, the largest decrease of the elastic energy is reached as a result of the formation of two disclinations, located at the maximum permissible distance from each other, i.e. at the distance equal to the grain boundary length. The splitting of the disclinations in the interfaces of the nanocrystalline materials is an effective channel of relaxation of the elastic energy. Disclination splitting is accompanied by a change in the structure of interfaces (appearance of stacking faults),

a

b

d

c

e

Fig. 6.11. Splitting of a triple-junction disclination (a) and of a grain boundary disclination (d) [103]: (b) splitting of initial triple-junction disclination into the “circle” ensemble of small-power disclinations and formation of local amorphous region; (c) the initial triple-junction disclination splits into three rows of smallpower disclinations located in grain boundaries; (e) line splitting of initial grain boundary disclination into the row of small-power disclinations

264

Microstructure of Compacted and Bulk Nanocrystalline Materials

decreases the probability of nucleation of microcracks in the vicinity of the interface and stimulates grain boundary diffusion. Thus, in addition to the small grain size and large area of the interfaces, the presence of a long-range field of elastic stresses is one of the main special features of nanocrystalline materials. Main concepts of the microstructure of nanocrystalline materials are based to a large extent on the results of X-ray investigation of the lattice parameters, internal stresses and atomic displacements. In comparison with the coarse-grained materials, X-ray diffraction patterns of nanocrystalline materials are characterised by a larger width of diffraction reflections, some changes of diffraction reflection shape and also shift of diffraction reflections. Broadening of the diffraction reflections is caused by the small grain size, microdeformations, stacking faults of the crystal lattice, and inhomogeneity (non-uniform composition) of substance (see Section 4.2) The shape and intensity of reflections depend on the atomic displacements. Shift of the reflections indicates changes in the lattice constant. In the general case, scattering intensity of a crystal can be presented [107–110] as the sum I(q) = I 0 (q)exp(–2M) + I DAD (q)[1 – exp(–2M)] + I D (q),

(6.22)

where I 0 (q) is the intensity of structural reflections in the absence of atomic displacements; q denotes the diffraction vector (|q| ≡ q = (2sinθ)/λ); exp(–2M) ≡ exp(2π iqu j )] is the Debye–Waller factor allowing for attenuation of structural reflections as a result of static and dynamic (thermal) atomic displacements. Second term I DAD (q)[1 – exp(–2M)] in (6.22) denotes the intensity of diffuse scattering due to displacement u j of atoms from crystal lattice sites. I D(q) is the diffuse scattering intensity resulting from the difference between atomic scattering factors and correlations in the mutual arrangement of atoms, i.e. short-range order. In other words, I D (q) is the intensity of diffusion scattering by the disordered crystalline solid solution. According to [109, 110], the intensity I D(q) can be presented as the sum of the intensity of Laue white noise and the diffusion scattering intensity, which is due to short-range order. Important information on special features of the structure is provided by the diffuse scattering which is the sum of three contributions: the diffusion scattering caused by atomic displacements (second term in equation (6.22)); diffusion scattering by the disordered crystalline solid solution, I D(q); and the diffuse scattering 265

Nanocrystalline Materials

due to the disordered distribution of atoms in amorphous substance. As a result of amorphisation, some of the atoms leave the crystal lattice sites; it leads to decrease of the intensity of structural reflections from I 0 (q) to [I 0 (q) – ∆], and appearance of additional contribution, which equals ∆, into diffuse scattering. Atomic displacements cause a monotonic increase in the intensity of the diffuse scattering with increase in reflection angle θ , and the absence of the order in the atom distribution leads to monotonic decrease in the intensity of diffuse scattering. The diffuse scattering, which is associated with a short-range of the type of ordering or phase separation, modulates the Laue white noise, i.e. results in periodic changes in diffuse scattering intensity [109, 110]. According to the experimental data, a decrease in the grain size of nanocrystalline substances may result both in a decrease [111– 114], and in an increase [114–116] of lattice constants a, b, c and the volume V of the unit cell of the crystal lattice. For example, for cubic metal such as Cr and Pd, the unit cell volume increases with decreasing of the mean grain size [115]. In nanocrystalline nc-Se, a decrease in the grain size is accompanied by an increase of the lattice constant a of the unit cell whereas the lattice constant c remains unchanged; in accordance with this, the volume of the unit cell of nc-Se increases from 0.0819 to 0.0823 nm 3 with a decrease of the grain size from 70 to 12.5 nm [116]. The largest increase of the lattice constant a and the unit cell volume V of nanocrystalline selenium is observed when the grain size D becomes smaller than 15 nm. However, a decrease of the volume and lattice constant (or constants) is more likely. This can be observed as a result of compression (reduction) of crystallite when the crystallite size is smaller than 10 nm. The observed increase of the unit cell volume and the lattice constant is in all likelihood a consequence of the adsorption and dissolution of impurities by the surface of crystallites, as in the case of isolated nanoparticles (see Section 5.2). The broadening of diffraction reflections is found for all nanocrystalline materials. In Section 4.2 it is shown that broadening β s , associated with a small crystallite size 〈D〉, is described by the equation

β s (2 θ ) ≡ 2 β s ( θ ) = K hkl λ /(〈D〉cos θ ) ,

(6.23)

where K hkl is the Scherrer’s constant whose value depends on the shape of the particle (crystalline, domain) and on the Miller indices 266

Microstructure of Compacted and Bulk Nanocrystalline Materials

(hkl) of diffraction reflection. Deformation broadening β d , caused by stacking faults, is calculated from the equation:

β d(2 θ ) = 4A〈 ε 2〉 1/2 tg θ ,

(6.24)

where A is a constant approximately equal to unity in the case of uniform distribution of dislocations in the crystallite. In equations (6.23) and (6.24), broadening β is expressed in radians. Thanks to a different dependence of the size and deformation broadening on the order of diffraction reflection it is possible to separate these contributions using the pairs of reflections (hkl) differing only in the order of reflection. In this case the total diffraction reflection broadening β is

β ≈ 0.5[ β s + ( βs2 + βd2 ) 1/2 ] .

(6.25)

It is assumed that size broadening β s does not depend on the index l. The separation of the size and deformation broadening of the diffraction reflections shows that the rms deformation in nanocrystalline Al, Ru, Pd, Cu, AlRu [23, 112, 117, 118) is equal to 1–3 % and is considerably greater than that in coarse-grained metals. Analysis of diffraction measurements in order to determine the size of crystallites and the level of deformation is described in detail in Section 4.2. The results of X-ray diffraction investigations of nanocrystalline materials are examined to some extent in [119]. Modeling of the X-ray diffraction pattern of nanocrystalline materials [120–123], which takes into account the grain size, distortions of the crystal lattice, the thickness and structure of interfaces, is of interest. Modeling was carried out using the kinematic theory of X-ray scattering. A polycrystal containing 361 cubic crystallites was investigated; the length of the crystallite edge was equal to Na (a is the constant of the unit cell). The size of the crystallites varied by varying N. When calculating atomic displacements it is assumed that all atoms of the external layer of the crystallites are displaced from the positions of the ideal lattice in a random manner and the displacements varies from 0 to 0.5b where b = a / 2 is the Burgers vector. Calculations carried out in [120, 121] show that atomic displacements on the surface of crystallites lead only to a decrease of the diffraction reflection intensity but have no effect on the 267

Nanocrystalline Materials

shape, width and position of diffraction reflections. A decrease in the crystallite size results in a large broadening of diffraction reflections. The effect of the long-range field of elastic stresses on the parameters of diffraction reflection was modeled by changing the linear dislocation density ρ at the interfaces. An increase of dislocation density from 0 to 0.1 and 1.0 nm –1 resulted in broadening of the reflections and their displacements to the range of high angles θ . At ρ = 0.1 nm –1 the size broadening is dominant when the crystallite size is smaller than 30 lattice constants of the unit cell (D < 30a). For larger crystallites, the main contribution to the diffraction reflection broadening comes from the elastic distortions of the crystallite lattice, caused by dislocations at the interfaces. From this, it follows that the effect of interfacial elastic stresses on the microstructure of nanocrystalline materials decreases with an decrease of the crystallite size. 6.4 NANOSTRUCTURE OF DISORDERED SYSTEMS Investigations of glasses and amorphous metallic alloys, carried out since 1985, show that the disordered materials possess a peculiar nanostructure. The results of diffraction and electron microscopic investigations [124–134], discussed in previous sections, confirm the nano-heterogeneous structure of amorphous alloys. Similar conclusions on the nano-heterogeneous structure of glasses and amorphous substances were made independently on the basis of investigations of low-energy vibrational spectra and the properties determined by the spectral distribution of elastic vibrations. The results of these investigations will be examined briefly. The vibrational spectra of various disordered systems such as glasses and amorphous solids greatly differ from those of normal crystals. The density of vibrational states of crystals in the lowenergy range is described by the Debye law

ω2 g D (ω ) = 9 N 3 , ωD

(6.26)

where ω D is the maximum frequency for the Debye function of the frequency distribution. In contrast to the crystals, at energies lower than 1 K the spectra of glasses and amorphous substances are characterised by 268

Microstructure of Compacted and Bulk Nanocrystalline Materials

a constant density of vibrational states, and in the energy range from 2 to 10 meV (from 15–25 to 120–125 K) there is an excess density of vibrational states in comparison with Debye density g D( ω ). This excess density of state is found in all glasses and is manifested in low-energy spectra of inelastic neutron scattering, low-frequency spectra of Raman scattering, in infra-red adsorption spectra, and in low-temperature heat capacity and heat conductivity. According to modeling assumptions [135–139], vibrational excitations, responsible for the excess density of states in disordered substances, are localised in a region containing from several tens to hundreds of atoms and having the size from 1 to several nanometers. Thus, the low-energy special features of phonon spectra of disordered materials indicate that the structure of amorphous substances and glasses are characterised by a presence of a regions with characteristic spatial scale of order of several nanometers. In other words, amorphous substances and glasses have a nanostructure. Indeed, comparison and analysis of the vibrational spectra of nanoparticles and the experimental results on the phonon spectra and low-temperature heat capacity of nanoparticles and nanomaterials (see Section 5.3) with the observed special features of low-energy spectra of disordered glassy substances allow us to note the following. In the energy range lower than 15 meV the density of vibrational states and the heat capacity of nanocrystalline and disordered glassy substances are higher than these for the coarsegrained crystals. In the low-frequency region the vibrational spectra of nanocrystalline and glassy materials differ mainly by the form of the excess density of states. In the glasses the excess density of vibrational states has the form of a peak (Fig. 6.12), whereas in nanocrystalline materials with a particle size of about 10 nm it has the form of a kink (see, for example, Fig. 5.12 for the density of phonon states g( ω ) of n-Ni). Low-energy quasilocal excitations in glassy materials and their crystalline analogues As 2S 3 , SiO 2 and Mg 70 Zn 30 were studied by the methods of inelastic incoherent neutrons scattering and Raman scattering [139]. The inelastic neutron scattering provide direct information on the density of vibrational states (see Fig. 6.12). The excess density of vibrational states ∆g( ω ) = g( ω ) – g D ( ω ), where g D ( ω ) is the Debye density of vibrational states, was determined from the experimental values of the velocity of sound. For all three disordered materials the excess density of states, ∆g( ω ), has the form of a peak whose maximum corresponds to some frequency 269

Nanocrystalline Materials  



,

 





θ   DUELWUDU\ XQLWV

 











.









  









 PH9 Fig. 6.12. The density of vibrational states in (1) crystalline and (2) glassy materials [139]: (a) As 2 S 3 , (b) SiO 2 , (c) Mg 70 Zn 30 .

ω max and energy E max and exceeds g D ( ω max ) by a factor of 2–6. For the glasses of different chemical composition, with different type of the short-range order and with different type of chemical bond (planar structure in As 2 S 3 , covalent bonds in SiO 2 , dense packing in Mg 70 Zn 30 metallic glasses) the dependences of the reduced excess density of states ∆g′ = ∆g( ω )/∆g( ω max ) on the reduced energy E/E max have the same shape. The same spectral shape of the reduced excess density of states, ∆g′, indicates the universal nature of the structural features responsible for the appearance of lowfrequency anomalies in vibrational spectra of different amorphous materials. 270

Microstructure of Compacted and Bulk Nanocrystalline Materials

In the low-frequency range, the Raman scattering spectra of glasses are characterised by the presence of a wide peak, which is not found in the spectra of appropriate crystals. Analysis of the spectra shows that the low-frequency peak is associated with the first order light scattering at vibrational excitations, which governed by Bose statistics [139]. The characteristic frequency of the maximum of the boson peak is in the range of acoustic vibrations and for different materials covers the range from 1/3 to 1/7 of the Debye frequency ω D. This means that the characteristic length of localisation of vibrational excitations, which scatter light with the appearance of a boson peak, is equal to several interatomic distances. The boson peak is a reflection of the excess density of vibrational states in the Raman spectrum. In the reduced coordinates, the spectral form of the boson peak for different glasses, produced by melt cooling, is the same [140]. The best function, which describes the experimental spectra of the excess density of vibrational states and the boson peak in glasses and contains the least number of fitting parameters, is the logarithmico–normal function,  ln 2 (ω / ω max )  ∆g ' (ω ) = exp −  2σ 2  

(6.27)

In the reduced coordinates, function (6.27) is determined by only one dimensionless parameter: the dispersion σ , which is equal to the same number, 0.48±0.05 [141], for all low-molecular glasses. Thus, a new universal parameter appears in the physics of glasses, i.e. the distribution dispersion, σ , associated with the basic characteristics of the nanostructure (note that the same logarithmico–normal function (2.1) is also used to describe the size distribution of nanocrystalline particles, produced by evaporation and condensation). This enabled the authors of [139] to link the excess density of vibrational states with the presence in glasses of the regions having the characteristic length (radius) with the nanometer scale. It is assumed that low-energy vibrational excitations, responsible for the excess density of states, are localised on nano-heterogeneities of the structure. This is confirmed, in particular, by the results of lowfrequency Raman scattering study of glasses whose matrix contains grown clusters of a different chemical composition with a size of several nanometers [142, 143]. The authors of [142] have been 271

Nanocrystalline Materials

investigated photochromic glasses with SiO 2–B 2O 3 matrix containing clusters of silver halide. The cluster size depends on the annealing time of specimens and varies in the range from 4 to 8 nm. The Raman scattering spectra of the starting glass contain boson peak typical of the spectra of all glassy materials (Fig. 6.13). Annealing of specimen led to an appearance of a new peak only in the lowfrequency part of the spectrum: namely, if the glass matrix contains clusters in the amount not smaller than 2 %, then after annealing the spectrum of inelastic scattering contain an additional band (see Fig. 6.13). The difference of the spectra of the annealed and starting specimens corresponds to the spectrum of surface vibrational modes of clusters; it should be noted that the experimental Raman scattering spectrum contained only main modes with the frequency

ω T = 0.8v t /D ,

(6.28)

,__ *,* >ω3ω  @ DUELWUDU\ XQLWV

where v t is the transverse velocity of sound, D is the diameter of a spherical cluster, ω T is the frequency of the main torsion mode. According to the results of [143], the identical spectrum of glasses













ω

FP







Fig. 6.13. Low-frequency Raman scattering spectra for photochromic glasses with SiO 2 –B 2 O 3 matrix containing clusters of silver halide [139]: (1) starting specimen, (2) annealed specimen with clusters of silver halide, (3) calculated spectrum of contribution of acoustic vibrational excitations.

272

Microstructure of Compacted and Bulk Nanocrystalline Materials

Al 2 O 3 ·MgO with MgCr 2 O 3 –MgAl 2 O 3 clusters contained only spherical vibrational modes with the frequency ω S = 0.8v l /D ,

(6.29)

where v l is the longitudinal velocity of sound. The difference between the first and second case is associated with the different ratio of the elastic constants of the cluster and matrix: in photochromic glasses the matrix, which is ‘harder than the cluster’, suppresses spherical vibrations. Analysis carried out in [144] shows that the low temperature heat conductivity of glasses with clusters has a plateau whose position correlates with the cluster size. According to [144], the Ioffe–Regel criterion for the phonon localisation is fulfilled in the region of the plateau, i.e. λ ~ l, where l is the free path length (in the case of strong scattering it is determined by the size of the heterogeneity of the structure), and λ is the phonon wavelength. Comparing these data with the results of measurements of heat conductivity in glasses, where localisation scale corresponds to the correlation length of the structure, the authors of [144] found that for glasses the correlation length is 1–3 nm. Direct calculations of the excess low-energy density of vibrational states in elastic medium with fluctuating elastic constants was carried out by the authors of [145] in the framework of the theory of perturbation with respect to small fluctuations. Calculation showed that the elastic constant fluctuations, having a correlation radius R c ≈ 1–2 nm, leads to the appearance of an excess density of states in the low frequency ( ω ~ v/R c ) range. It can be shown that any rational elastic constant correlation function, decreasing with distance, results in the displacement of part of the highfrequency vibrational modes to the low-frequency part of the spectrum, thus forming an excess density of the vibrational state. As already mentioned, the spectrum of excess density of vibrational states is efficiently approximated by the logarithmico–normal function (6.27) with the dispersion σ = 0.48. If the excess density of states is caused by vibrational excitations, localised on nanometer heterogeneities of the structure, the frequency of quasilocal vibrations, ω , is linked with the heterogeneity size, D, by the relationship ω = Kv/D, where K is the constant of the order of unity. This means that the size distribution of nano-heterogeneities can also be described by the logarithmico–normal function similar to (6.27), with the dispersion σ : 273

Nanocrystalline Materials

 ln 2 ( D / D0 )  F ( D) = exp − . 4σ 2  

(6.30)

In (6.30) the value D 0 is the most probable size of the nanoheterogeneity. The low-energy features of vibrational spectra of glasses, associated with the presence of nano-heterogeneities, may have a strong effect on the properties of glasses not only at low but also high temperatures, up to the solidification temperature of glass. These properties include those for which the effect of the lowenergy density of vibrational states is enhanced in comparison with the region of the spectrum near the Debye frequency. For example, the contribution of low-energy phonons to the magnitude of rms thermal vibrations of the atoms is increased in proportion to the inverse square of the frequency of vibrations. Consequently, as shown in [146], the presence in glasses of an excess density of vibrational states, which is equal to ~10 %, increases the amplitude of thermal vibrations of the atoms by 30–40 % in comparison with the crystalline material having the same temperature. The proposed [139] universal form of the low-energy vibrational spectrum of glasses, glassy and amorphous substances indicates that structure of these substances contains heterogeneities of the nanometer size. In this chaos and disorder with which the structure of amorphous materials and glasses is usually associated, there is a universal spatial scale typical of glasses of different nature (dielectric, semiconductor, metallic). According to [139], the presence of nanoregions in the disordered materials may have the same important role for the theory of glassy and liquid states as the presence of the unit cell for the theory of the crystal structure.

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Microstructure of Compacted and Bulk Nanocrystalline Materials

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M. A. Krivoglaz. Theory of X-ray and Thermal-Neutron Scattering by Real Crystals (Plenum Press, New York 1969) 405 pp. H. P. Klug, L. E. Alexander. X-ray Diffraction Procedures for Polycrystalline and Amorphous Materials (Wiley, New York 1954) 491 pp. (second edition: X-ray Diffraction Procedures (Wiley, New York 1974)) A. I. Gusev, A. A. Rempel. Structural Phase Transitions in Nonstoichiometric Compounds (Nauka, Moscow 1988) 308 pp. (in Russian) A. I. Gusev, A. A. Rempel, A. J. Magerl. Disorder and Order in Strongly Nonstoichiometric Compounds: Transition Metal Carbides, Nitrides and Oxides (Springer, Berlin - Heidelberg – New York, 2001) 607 pp. J. Balogh, L. Bujdosó, G. Faigel, L. Gránásy, T. Kemény, I. Vincze, S. Szabó, H. Bakker. Nucleation controlled transformation in ball milled FeB. Nanostruct. Mater. 2, 11-18 (1993) J. A. Eastman, M. Fitzsimmons, L. J. Thompson. The thermal properties of nanocrystalline palladium from 16 to 300 K. Philosoph. Mag. B 66, 667696 (1992) X. Y. Qin, X. J. Wu, L. F. Cheng. Exothermal and endothermal phenomena in nanocrystalline aluminium. Nanostruct. Mater. 2, 99-108 (1993) M. L. Sui, K. Lu. Variation in lattice parameters with grain size of nanophase Ni 3 P. Mater. Sci. Engineer. A 179-180, 541-544 (1994) J. A. Eastman, M. Fitzsimmons, L. J. Thompson, A. C. Lawson. Diffraction studies of the thermal properties of nanocrystalline Pd and Cr. Nanostructr. Mater. 1, 465-471 (1992) H. Y. Zhang, K. Lu, Z. Q. Hu. Formation and lattice distortion of nanocrystalline selenium. Nanostruct. Mater. 6, 489-492 (1995); Y. H. Zhao, K. Zhang, K. Lu. Structure characteristics of nanocrystalline element selenium with different grain sizes. Phys. Rev. B 56, 14322-14329 (1997) E. Hellstern, H. J. Fecht, F. Zhu, W. L. Johnson. Structural and thermodynamic properties of heavily mechanically deformed Ru and AlRu. J. Appl. Phys. 65, 305-310 (1989) G. W. Neiman, J. R. Weertman, R. W. Siegel. Mechanical behavior of nanocrystalline Cu and Pd. J. Mater. Res. 6, 1012-1027 (1991) I. V. Alexandrov, R. Z. Valiev. Study of nanocrystalline materials by X-ray analysis. Fiz. Metal. Metalloved. 77, No 6, 77-87 (1994) (in Russian) . I. V. Alexandrov, R. Z. Valiev. Computer simulation of X-ray diffraction patterns of nanocrystalline materials. Philosoph. Mag. B 73, 861-872 (1996) I. V. Alexandrov, R. Z. Valiev. X-ray pattern simulation in textured nanostructured copper. Nanostruct. Mater. 6, 763-766 (1995) K. Zhang, I. V. Alexandrov, K. Lu, R. Z. Valiev. Structural characterization of nanocrystalline copper by means of X-ray diffraction. J. Appl. Phys. 80, 5617-5624 (1996) R. Z. Valiev, I. V. Alexandrov, W. A. Chiou, R. S. Mishra, A. K. Mukherjee. Comparative structural studies of nanocrystalline materials processed by different techniques. Materials Science Forum 235-238, 497-506 (1997) A. Inoue, H. M. Kimura, K. Sasamori, T. Masumoto. Ultrahigh strength of rapidly solidified Al 68-x Cr 3 Ce 1 Co x (x = 1, 1.5 and 2 %) alloys containing an icosahedral phase as a main component. Mater. Trans. Japan. Inst. Met. 35, 85-94 (1994) A. Inoue. Preparation and novel properties of nanocrystalline and nanoquasicrystalline alloys. Nanostruct. Mater. 6, 53-64 (1995) J. Kwarciak, L. Pajak, J. Lelatko. Crystallization kinetics of iron-cobalt-

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silicon-boron (Fe,Co) 78 Si 9 B 13 glasses. Z. Metallkunde 79, 712-715 (1988) N. I. Noskova, E. G. Ponomareva, V. A. Lukshina, A. P. Potapov. Effect of rapid crystallization on the properties of Fe 5 Co 70Si 15B 10. Nanostruct. Mater. 6, 969-972 (1995) N. I. Noskova, E. G. Ponomareva. Structure, strength, and plasticity of nanophase Fe 73.5 Cu 1 Nb 3 Si 13.5B 9 alloy: I. Structure. Fiz. Metal. Metalloved. 82, No 5, 163-171 (1996) (in Russian). (Engl. transl.: Phys. Metal. Metallogr. 82, 542-548 (1996)) N. I. Noskova, E. G. Ponomareva, M. M. Myshlyaev. Structure of nanophases and interfaces in multiphase nanocrystalline Fe 73 Ni 0.5 Cu 1 Nb 3 Si 13.5 B 9 alloy and nanocrystalline copper. Fiz. Metal. Metalloved. 83, No 5, 73-79 (1997) (in Russian). (Engl. transl.: Phys. Metal. Metallogr. 83, 511-515 (1997)) N. I. Noskova, E. G. Ponomareva, V. N. Kuznetsov, A. A. Glazer, V. A. Lukshina, A. P. Potapov. Crystallization of amorphous Pd-Cu-Si alloy in creep condition. Fiz. Metal. Metalloved. 77, No 5, 89-94 (1994) (in Russian) N. I. Noskova, E. G. Ponomareva, I. A. Pereturina, V. N. Kuznetsov. Strength and plasticity of a Pd–Cu–Si alloy in the amorphous and nanocrystalline states. Fiz. Metal. Metalloved. 81, No 1, 163-170 (1996) (in Russian). (Engl. transl.: Phys. Metal. Metallogr. 81, 110-115 (1996)) A. Inoue, H. M. Kimura, K. Sasamori, T. Masumoto. Structure and mechanical strength of Al-V-Fe melt-spun ribbons containing high volume fraction of nanoscale amorphous particles. Nanostruct. Mater. 7, 363-382 (1996) K. Yamauchi, Y. Yoshizawa. Recent development of nanocrystalline soft magnetic alloys. Nanostruct. Mater. 6, 247-254 (1995) B. V. Zhalnin, I. B. Kekalo, Yu. A. Skakov, E. V. Shelekhov. Phase transformations and magnetic properties variation during nanocrystalline state formation in amorphous iron base alloy. Fiz. Metall. Metalloved. 79, No 5, 94-106 (1995) (in Russian) V. K. Malinovskii, V. N. Novikov, P. P. Parshin, A. P. Sokolov, M. G. Zemlyanov. Universal form of the low-energy (2 to 10 meV) vibrational spectrum of glasses. Europhys. Lett. 11, 43-47 (1990) M. I. Klinger. Low-temperature properties and localized electron states of glasses. Uspekhi Fiz. Nauk 152, 623-652 (1987) (in Russian) U. Buchenau, Yu. M. Galperin, V. L. Gurevich, H. R. Shober. Anharmonic potentials and vibrational localization in glasses. Phys. Rev. B 43, 50395045 (1991) J. E. Graebner, D. Golding. Phonon localization in aggregates. Phys. Rev. B 34, 5788-5890 (1986) V. K. Malinovskii, V. N. Novikov, A. P. Sokolov. On nanostructure of disordered substances. Uspekhi Fiz. Nauk 163 (5), 119-124 (1993) (in Russian) V. K. Malinovsky, A. P. Sokolov. The nature of boson peak in Raman scattering in glasses. Solid State Commun. 57, 757-761 (1986) V. K. Malinovskii, V. N. Novikov, A. P. Sokolov. Log-normal spectrum of low-energy vibrational excitations in glasses. Phys. Lett. A 153, 63-66 (1991) V. K. Malinovsky, V. N. Novikov, A. P. Sokolov, V. G. Dodonov. Low-frequency Raman scattering on surface vibrational modes of microcrystals. Solid State Commun. 67, 725-729 (1988) E. Duval, A. Boukenter, B. Champagnon. Vibration eigenmodes and size of microcrystallites in glass: observation by very-low-frequency Raman scattering. Phys. Rev. Lett. 56, 2052-2055 (1986) C.Yu. Clare, J. J. Freeman. Thermal conductivity and specific heat of glasses.

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145.

146.

Phys. Rev. B 36, 7620-7624 (1987) V. K. Malinovsky, V. N. Novikov, A. P. Sokolov. Structural difference of two glasses of amorphous semiconductors. J. Non-Cryst. Solids 114, 6163 (1989) V. K. Malinovsky, V. N. Novikov. The nature of the glass transition and the excess low-energy density of vibrational states in glasses. J. Phys.: Condens. Matter 4, L139-L144 (1992)

283

Nanocrystalline Materials

+D=FJAH% 7. Effect of the Grain Size and Interfaces on the Properties of Bulk Nanomaterials When interpreting the experimental results, obtained for bulk nanomaterials, it is important to be able to separate grain boundary (associated with interfaces) from volume (associated with small grain size) effects. This problem is far from being solved because at present investigation of bulk nanomaterials is still in the stage of collecting experimental results. For this reason, the level of understanding of the structure and properties of the bulk nanocrystalline materials is considerably lower in comparison with isolated nanoparticles. The properties of bulk nanomaterials in relation to the particle size and the state of the grain boundaries were discussed in reviews [1, 2]. 7.1 MECHANICAL PROPERTIES Because of the application of bulk nanocrystalline materials in practice, it has been necessary to investigate in detail their hardness, strength, elasticity, plasticity and other mechanical properties. In the group of the mechanical properties of nanocrystalline materials, special attention must be given to the very high hardness. Hardness characterises the resistance of materials to plastic deformation during indentation of a harder body into it. Diamond is used generally as an indentor material. When measuring hardness by the Vickers method, the effects associated with the difference in the elastic properties of the materials are almost completely 284

Effect of Grain Size and Interfaces on Properties of Bulk Nanomaterials

excluded because the size of the indentation is measured after removing the stress, i.e. in the absence of elastic loading. The experimentally measured values of hardness are subjected to the effect of various secondary factors, such as the non ideal surface of the material, deviation from the perpendicularity of the surface of the material and the axis of the indentor, incorrect selection of the loading time and the load, and also the presence in the material of pores, voids or free volumes. In the main the hardness of the material is determined by yield limit σ y . The grain size has a strong effect on microhardness; this effect has been studied extensively on metals, alloys and ceramics with a grain size D greater than 1 µm. According to the Hall–Petch law [3, 4]:

σ y = σ 0 + k y D – 1/2 ,

(7.1)

where σ 0 is the internal stress preventing the movement of dislocations and k y is a constant. At a temperature T/T melt < 0.4– 0.5 (T melt is the melting point), hardness H V (Vickers microhardness) is linked with the yield limit σ y by empirical relationship H V/ σ y ≈ 3 [5]. This leads to the following size dependence of hardness H V ≈ H 0 + kD –1/2 ,

(7.2)

where H 0 and k are constants. Deformation is carried out by diffusion sliding, then according to [6] at a low temperature T/T melt the deformation rate ε ≡ dε /dt is

dε = BσΩδd dif /k B TD 3 , dt

(7.3)

where B is a proportionality coefficient; σ is applied stress, Ω is the atomic volume, δ is the thickness of the grain boundary, d dif is the coefficient of grain boundary diffusion. Equations (7.1)–(7.3) show that a decrease in the grain size should result in a strong change in the mechanical properties. In particular, equations (7.1) and (7.2) predict hardening of the material with a decrease in the grain size D. At the same time, equation (7.3) shows that at the nanometer size of grain, diffusion sliding becomes important even at room temperature, greatly increasing the deformation rate. Thus, the effect of the grain size on the strength properties of a nanocrystalline material is ambiguous 285

Nanocrystalline Materials

and depends on the ratio between the change of the yield limit and the deformation rate. In addition, it is important to take into account the possible increase of the grain boundary diffusion coefficient d dif with a decrease in the grain size. At 300 K, the microhardness of nanocrystalline materials is 2– 7 times higher than H V of coarse-grained materials. The literature data on the size dependence of the microhardness of nanocrystalline materials are relatively contradicting. The effect of the grain size of the microhardness of nanocrystalline metals (copper and palladium) was investigated for the first time in [7]. The grain size of Cu and Pd was varied by annealing. The results show that a decrease in the grain size of coarse-grained copper from 25×10 3 to 5×10 3 nm is accompanied by an increase in microhardness. Microhardness of nanocrystalline copper n-Cu (D ~16 nm) was ~2.5 times higher than that of copper with a grain size of 5×10 3 nm, but with a decrease in the grain size of n-Cu from 16 to 8 nm H V decreased by ~25%. The decrease of H V was also found in the case of a decrease in the grain size of n-Pd from 13 to 7 nm. The decrease of the microhardness of n-Cu and n-Pd was explained [7] by the considerably higher (in comparison with coarse-grained metals) deformation rate ε  ≡ dε /dt . A decrease in the H V value of nanocrystalline alloys Ni–P, TiAlNb, TiAl, NbAl 3 with a decrease in the grain size from 60–100 to 6–10 nm was reported in [8–11]. According to the authors, this effect was determined by the increase in the contribution of diffusion mobility in the deformation with decreasing grain size D. An increase in H V with a decrease of the grain size of n-Cu, nFe and n-Ni was reported in [12–15]. An interesting fact was reported by Fougere et al [16]: an increase or decrease in H V depends on the method of changing the grain size. An increase in H V with a decrease in grain size D is observed generally when hardness is measured on a series of asprepared specimens differing in the grain size. If microhardness H V is measured on a single specimen, that is successively heated to produce ever-increasing grain sizes, then dependence H V (D) is different. In this case, an increase in grain size D up to some critical value leads to a rise in H V ; the following increase in D is accompanied by decreasing H V and corresponds to the Hall–Petch law. For example, initial annealing at 569 K of nanocrystalline compacted nc-Pd for 60–90 min leads to the grain growth and an increase of hardness by 7–11%; longer annealing, which leads to further grain growth, is accompanied by a decrease of hardness 286

Effect of Grain Size and Interfaces on Properties of Bulk Nanomaterials







QP  











+'

*3D

QF&X 



 







 

 QP  

Fig. 7.1. Dependence of the microhardness H V of bulk nanocrystalline copper nc-Cu on D –1/2 (D is the grain size) [16]: the growth of grains of nc-Cu was achieved by increasing annealing time at 423 K (dashed arrow indicates the direction of increase of annealing time).

(here and further abbreviation “nc” means nanomaterials prepared by compaction of nanocrystalline powders). The microhardness of nanocrystalline compacted nc-Cu increased by 4–5% in the first 20– 30 minutes of annealing at 423 K, and then decreased (Fig. 7.1) [16]. The results of measurements of the hardness of nanocrystalline metals Ag, Cu, Pd, Se, Fe and Ni on different specimens with different grain sizes were generalised in [17]. According to [17], if several independently produced specimens of a nanocrystalline metal have the grains of different sizes, the hardness of these specimens increases with a decrease of D to 4–6 nm; the dependence of H V on D –1/2 is governed by the Hall–Petch law (7.2). The hardness of the intermetallic compounds and alloys in which the grain size D was varied by annealing depends in a more complicated manner on D: initially an increase in D leads to an increase in H V and then decreases H V [17]. In other words, in the case of grain growth as a result of annealing, the Hall–Petch law (decrease of H V with increasing D) was fulfilled only at D > 12– 20 nm. According to [18], the decrease in the hardness of nanocrystalline materials, detected in the range of low values of D, may be a consequence of the fact that the volume fraction of triple points 287

Nanocrystalline Materials

becomes larger than the volume fraction of the grain boundaries. According to [17, 19] the contradiction of experimental data for the size dependence of the hardness of nanomaterials may be a consequence of different structures of the interfaces. In view of this the results of studies [20, 21] of the microhardness of submicrocrystalline Al 98.5 Mg 1.5 alloy are of interest. They show that even at unchanged grain size D ≈ 150 nm, a transition from completely non-equilibrium interfaces to less non-equilibrium interfaces, resulted from annealing, strongly affects the microhardness. According to electron microscopic data, consecutive annealing of submicrocrystalline alloy Al 98.5 Mg 1.5 (D ~150 nm) at 400 K results in a relaxation of grain boundaries and their gradual transition to an equilibrium state, although the grain size remains unchanged. The relaxation of grain boundaries as a result of annealing was accompanied by a decrease in microhardness from 1.7 to 1.4 GPa. The ambiguous fulfillment of the Hall–Petch law was also detected for nanocrystalline alloys produced by crystallisation of Fe–Si–B, Fe–Co–Si–B, Fe–Cu–Nb–Si–B, Pd–Cu–Si amorphous alloys. Crystallisation of these amorphous alloys results in the precipitation of highly dispersed phases with a grain size of several nanometers. For example, in Fe–Si–B alloys, the amorphous phase showed precipitation of the grains of the bcc phase α -Fe(Si) and Fe 2 B boride. In cobalt-rich Fe–Co–Si–B alloys, α -Co and β -Co, Co 2 Si, Co 2 B (or Fe,Co) 2 B) and α -Fe may precipitate in the amorphous phase [22, 23]. In nanocrystalline Fe–Cu–Nb–Si–B alloys, the amorphous matrix contains distributed particles of the main precipitated bcc phase α -Fe(Si) with a size of 5–15 nm and copper clusters [24–26]. Particles of Pd, Pd 5 Si and Pd 9 Si 2 with a size of 5 to 30 nm are detected in nanocrystalline alloys Pd–Cu– Si [27, 28]. The phase composition of the alloys and the grain size of the precipitated phases depend on the heat treatment and/or deformation treatment conditions. The authors of [28] studied the dependence of microhardness of nanocrystalline alloys, produced by crystallization, on the grain size of precipitated phases. The grain size was changed mainly by variation of the annealing temperature. The dependences of the microhardness H V of nanocrystalline alloys on D –1/2 (D is the size of grains (particles) of the nanocrystalline precipitated phase) measured in [28], are represented in Fig. 7.2. It is seen that the Hall–Petch law holds for all alloys in the range of D from ~10 to ~100 nm or more. At a grain size of D < 10 nm, the Hall–Petch 288

Effect of Grain Size and Interfaces on Properties of Bulk Nanomaterials 







 

+'

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QP





Fig. 7.2. Dependence of microhardness H V on the grain size D of phases, precipitated in nanocrystalline alloys, produced by crystallisation of amorphous alloys [28]: (1) Fe 7.35 Cu 1 Nb 3 Si 13.5 B 9 , (2) Fe 81 Si 7 B 12 , (3) Fe 5 Co 70 Si 15 B 10 , and (4) Pd 81 Cu 7 Si 12 .

law is fulfilled only for the Fe 73.5 Cu 1 Nb 3 Si 13.5 B 9 alloy. For other alloys, a decrease in the grain size from ~10 nm to ~4 nm is accompanied by a decrease in microhardness. An increase in microhardness in accordance with the Hall–Petch law was observed for nanocrystalline alloy Ni 75 W 25 [29] with a decrease in the mean grain size from 1000 to ~10 nm; a further decrease of the grain size resulted in a decrease in microhardness. Starting amorphous alloy Ni 75 W 25 was produced by electrodeposition, and the grain size was varied during crystallisation of the amorphous alloy by a means of annealing at different temperatures [29]. It should be mentioned that for the nanocrystalline alloys, discussed in [28, 29] (with the exception of Fe 73.5 Cu 1 Nb 3 Si 13.5 B 9 ) the dependence of microhardness on the grain size, varied by a means of annealing, was the same as in [16, 17]. Control of the structure of nanocrystalline and nanoquasicrystalline materials, produced by crystallisation of amorphous alloys is an efficient means of obtaining high tensile strength combined with good malleability. In Section 3.3 we considered briefly the special features of the structure of rapidly solidified Al– Cr–Ce–Co amorphous aluminium alloys because of their high tensile strength σ f . The high strength of these alloys is determined by the precipitation of aluminium-coated icosahedral nanoparticles in the amorphous phase. The authors of [30] investigated the structure and strength properties of amorphous alloys of the Al–V–Fe system, 289

Nanocrystalline Materials

differing in a high volume content of the nanosised amorphous particles. The Al 98–x V x Fe 2 and Al 96–x V 4 Fe x alloys in the form of ribbon were prepared by a single roller melt-spinning technique at a temperature from 1373 to 1423 K with a circumferential rate of 40 m s –1 . Rapidly cooled alloys Al 96 V 4 , Al 95 V 3 Fe 2 and Al 93 V 5 Fe 2 contain only the fcc phase, enriched in aluminium and the icosahedral phase. In contrast to them, the structure of quenched Al 94 V 4 Fe 2 alloys is a two-phase mixture of non-homogeneously distributed nanoregions with a size of ~7 nm for the aluminium reach fcc phase and ~10 nm for the amorphous phase. The volume fractions of the amorphous and fcc phases in these alloys are ~60% and ~40%. The particles of the amorphous phase are surrounded by the fcc phase, but there is no distinctive boundary between the phases. The composition of the amorphous phase is Al 94 V 4 Fe 2 and that of the Al-rich fcc phase is Al 93 V 5 Fe 2 . The similar compositions of the phases indicate that the redistribution of elements between them is repressed during rapid solidification. Investigation of the dependence of tensile strength σ f and microhardness H V on the composition of rapidly quenched alloys Al 98–x V x Fe 2 and Al 96–x V 4 Fe x [30] shows that the highest values σ f = 1350–1400 MPa and H V = 0.47–0.48 MPa correspond to Al 94 V 4 Fe 2 alloy. On the basis of comparison of the structure of alloys of different composition and their mechanical properties, the authors of [30] assumed that the transition from the icosahedral to the amorphous phase increases σ f and H V . This means that the nanoparticles of the amorphous phase act as a hardening phase. In addition, the value of σ f tends to increase with a decrease of the distance between the particles and an increase in the volume fraction of the hardening amorphous nanoparticles. Differential thermal analysis and high temperature X-ray analysis of quenched Al 94 V 4 Fe 2 alloy show that the crystallisation of the amorphous phase starts at 580 K. During heating the crystallisation takes place as the ‘amorphous phase → icosahedral phase → (Al + Al 11 V)’ transition. Thus, the experimental studies [22–30] show that the structure of amorphous alloys (metallic glasses) immediately after producing and, even more so, after crystallisation is characterised by the presence of nanoregions, which correspond to different phases. In other words, these alloys have a nano-heterogeneous structure. Of special interest are alloys of Fe and Ni containing Al, Ti, Cr, V, Mo, Co and W in various combinations, and also from 5 to 12 at.% B and from 0 to 7.5 at.% C, P and Si. The additivities of B, 290

Effect of Grain Size and Interfaces on Properties of Bulk Nanomaterials

C, P, and Si support the formation of a glassy (amorphous) state [31–33]. The amorphous ribbons of these alloys are produced by melt spinning and then the ribbons are consolidated by extrusion or hot isostatic pressing. During consolidation, the alloys completely crystallise with the formation of a nanocrystalline structure. For example, the Ni 59 Mo 29 B 12 and Ni 56 Mo 23 Fe 10 B 11 alloys have a matrix based on nickel with the uniform distribution of highly dispersed particles of Ni 2 Mo and Ni 4Mo intermetallics and large (0.5–2.0 µm) boride particles NiMoB 2 . These alloys have higher fracture toughness and higher hardness than cutting steels, especially at a temperature > 600 K. The microstructure and properties of these alloys depend greatly on the conditions of producing of the bulk compacted material. Aging of alloys also increase their hardness. The nanostructure of Ni 81 Si 10 B 9 alloy is similar to the nanostructure of aluminium-based alloys: the nanoparticles of nickel with a size of 10–20 nm are uniformly distributed in the amorphous metallic matrix [34]. Ni–Si–B nanocrystalline alloys are stronger than completely amorphous alloys with the same composition. Measurements of the microhardness H V of nanomaterials at 300 K after annealing at a series of temperatures T are used to determine relaxation temperatures. The effect of annealing temperature on H V and the structure of submicrocrystalline copper with a mean grain size of 200–300 nm was studied in [35] and the copper with a grain size of 130–160 nm was investigated by the authors of [36–38]. The microhardness of starting and submicrocrystalline copper, studied in [36–38], was 1.2 and 1.4 GPa, respectively. Annealing at T < 400 K was found to have no appreciable changes in the structure and H V . An abrupt decrease of microhardness of submicrocrystalline copper by a factor of 2 to 3 is observed after annealing at 425–450 K (Fig. 7.3) and is associated with grain growth, the relaxation of internal stresses and the lowering of the dislocation density, i.e. with an irreversible transition from the submicrocrystalline to the coarse-grained state. The microhardness of submicrocrystalline Pd with a mean grain size of 150 nm, produced by twisting under a quasi-hydrostatic pressure, was 2.1 GPa [38–41]. Annealing resulted in a decrease in the microhardness of submicrocrystalline Pd, and a rapid decrease of H V started after annealing at 475 K (Fig. 7.4). The sudden decrease of H V by almost a factor of 3 after annealing at 475– 600 K is associated, according to electron microscopic examination, with grain growth and partial annealing of dislocations. The microhardness of submicrocrystalline titanium with a mean 291

Nanocrystalline Materials

 &X

*3D



+'













%



.

Fig. 7.3. Dependence of microhardness of submicrocrystalline Cu, measured at 300 K, on annealing temperature T [37].

3G



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%





.

Fig. 7.4. Dependence of the microhardness of submicrocrystalline Pd, measured by 300 K, on annealing temperature T [40].

grain size of 150 nm, produced by equal-channel angular pressing with subsequent cold rolling, was 2.9 GPa [42]. This is twice the microhardness of starting coarse-grained titanium. Annealing of submicrocrystalline titanium for 1 hour at temperatures of up to 573 K has no effect on the microhardness and the grain size (Fig. 7.5). A larger decrease of H V , which is caused by the recrystallisation and grain growth, starts at an annealing temperature 292

Effect of Grain Size and Interfaces on Properties of Bulk Nanomaterials

HV (GPa)

2.8

2.4

2.0

1.6

400

600

800

1000

(K)

Fig. 7.5. Dependence of microhardness H V of submicrocrystalline titanium on annealing temperature T: (•) values of HV after annealing for 1 h, (o) values of H V after annealing for 3 hours [42]. Dashed lines indicate the temperatures of the start and the end of recrystallisation. The accuracy of measurement of microhardness is +5 %.

higher than 600 K and continues up to 873 K. After annealing at 773 K, the mean grain size increased to 1–2 µm and after annealing at 973 K the grain size reached 15 µm. A further increase of the annealing temperature to 1145 K resulted in a smooth decrease of microhardness to 1.5 GPa, which is close to the microhardness of coarse-grained titanium, equal to 1.3–1.4 GPa [43]. Longer, 3 hour annealing at T < 573 K had no effect on microhardness, and annealing at a temperature from 573 to 873 results in a decrease in H V (Fig. 7.5). This means that in the temperature range from 573 to 873 K, which corresponds to the recrystallisation region, submicrocrystalline titanium is in the non-equilibrium state and does not transfer to the equilibrium state during annealing for 1 hour. Annealing of specimens of nc-Ag (D ≈ 10 nm), prepared by compaction of nanoclusters, is accompanied by a gradual decrease in microhardness from 1.5 to 0.3–0.5 GPa; no sudden drop in H V is observed [44]. The decrease in H V and increase of the grain size start after annealing at 370 K. In contrast to nc-Ag, annealing of the specimens of nanocrystalline oxide nc-MgO (D ≈ 10 nm) up to a temperature of 870 K is not accompanied by any change in microhardness which is equal to 2.5 ± 0.2 GPa at 870 K [45]. Different behaviour of the microhardness of nanocrystalline metal (nc-Ag) and nanocrystalline ceramics (nc-MgO) upon annealing 293

Nanocrystalline Materials

QF0J2



+'

*3D





QF$J  





%



.

Fig. 7.6. Variation of the microhardness H V of nanocrystalline compacted specimens of silver nc-Ag and magnesium oxide nc-MgO after annealing [44, 45].

(Fig.7.6) indicates the much higher structural stability of nanocrystalline ceramics and the conservation of grain size in such a material up to 800–900 K. It is interesting to note that annealing of nc–MgO at 870 K < T ≤ 1070 K increased microhardness by ~30%. Microhardness measurements are used as a method of certification of nanocomposites. In [46], industrial powders of iron and copper were milled in a ball mill. Milling in argon for 12–20 hours resulted in the formation of a Cu 1–x Fe x powder mixture with a mean grain size of 10–20 nm. The nanopowder was subjected to preliminary cold pressing and then to hot pressing at a pressure of 0.64 GPa; hot pressing temperature was 723–773 K. The produced nanocomposite Cu 1 –x Fe x (0.15 ≤ x ≤ 1.0) has homogeneous distribution of the grains of iron and copper with a size of 25–30 and 45–30 nm, respectively. The microhardness of nc-Cu and ncFe was 4 times higher than that of coarse-grained copper and iron. The microhardness of the Cu 1 -x Fe x nanocomposite in the entire composition range was 40–50% higher than the expected additive microhardness H V of the mechanical mixture. The largest increase of H V was found for the nanocomposite of 70 % Fe. The similar increase of H V was observed in Cu–W and Cu–Nb nanocomposites [47, 48]. According to [46], the increase of H V in the Cu–Fe nanocomposite is determined by the formation of interfaces with an elevated dislocation density, since Cu and Fe have different structures (fcc and bcc, respectively). 294

Effect of Grain Size and Interfaces on Properties of Bulk Nanomaterials

Consideration of the set of the currently available experimental data for the hardness of metallic and ceramic nanocrystalline materials shows that the hardness increases with a decrease in the grain size to some critical value D crit , and at D < D crit hardness decreases. The factors determining the critical grain size are not clear. It cannot as yet be reliably confirmed that the values of hardness, measured on the compacted nanomaterials, correspond to the hardness of completely dense (pore-free) nanomaterials. Evidently, the measured hardness is greatly affected by the pores, microcracks and other special features of the macrostructure, associated with the method of producing the nanocrystalline material. The deviation of the size dependence of the hardness of nanocrystalline materials from the Hall–Petch law (7.2) is explained by various reasons. In particular, it has been assumed that in crystals at the grain size D < D crit the formation of flat dislocation arrays is restricted and this should lead to a decrease of hardness and strength. The authors of [49] assumed that annealing of the nanomaterial results in relaxation of the grain-boundary structure and a decrease in the excess grain-boundary energy. It leads to an anomalous decrease in hardness with a decrease in the grain size. According to [50], the change of hardness (or yield limit) with the grain size is described by the dependence H V(D) ~ D n , where, in contrast to the Hall–Petch relationship, n ≠ –1/2, and changes from –1/4 to –1. Each value of the exponent n corresponds to a specific mechanism of interaction of dislocations with the grain boundaries. It is also assumed that the value n depends on the type of crystal lattice. A simple phenomenological model, describing the dependence of hardness on the grain size, is proposed by the authors of [51]. The model is based on the assumption [52] according to which the nanomaterial contains an intragranular crystalline phase and an amorphous grain–boundary phase. If the crystalline phase is governed by the Hall–Petch law, and the grain-boundary phase has constant hardness H gb , the hardness H of the nanocrystalline material may be treated as the superposition of two contributions:

H = v(H 0 + kD −1/2) + (1 − v)Hgb ,

(7.4)

where v = (D – s) 3 /D 3 is the volume fraction of the crystalline phase, s is the thickness of the grain boundary phase, H gb = G/12, and G is the shear modulus of the coarse-grained bulk material. 295

Nanocrystalline Materials

However, dependence (7.4) does not have an extremum and does not describe the transition from one type of the size dependence of hardness to another one in the vicinity of D crit . The combined mechanism of plastic deformation of nanocrystalline materials is proposed in [53]; plastic shear in large grains takes place by the dislocation model, and in fine grains by means of the vacancy mechanism of grain-boundary sliding. Thus, a significant role in the considered models of plastic deformation of nanomaterials [49–51, 53] is played by grainboundary sliding. The authors of [2], examining the size effect of the mechanical properties of nanomaterials, made the following conclusion: plastic deformation of nanocrystalline materials always starts with grain-boundary microsliding irrespective of the method of preparation. Microsliding takes place by the same mechanism as the process of shear deformation in the amorphous state because the structure of the grain boundaries in nanomaterials can be described completely or partially by the model of the amorphous state. Measurements of the velocities of longitudinal and transverse ultrasonic waves in submicrocrystalline copper as a function of the annealing temperature made it possible to estimate the elasticity modulus E and shear modulus G [54]. The grain size of submicrocrystalline copper prior to annealing was 200–400 nm. Annealing was carried out in a temperature range of 373–623 K with a step of 25–50 K with holding for 1 hour at each temperature. The values E and G of the starting submicrocrystalline copper were 10–15% lower than those of coarse-grained copper. Earlier, the reduced (by ~30%) value of the elasticity modulus was observed in nanocrystalline nc–Pd [12, 55]. At an annealing temperature 423–

&X





















%





.

296

*



*3D

*3D



Fig. 7.7. Effect of annealing temperature T on elasticity modulus E and shear modulus G of submicrocrystalline copper (elasticity moduli were measured at 300 K) [54].

Effect of Grain Size and Interfaces on Properties of Bulk Nanomaterials

456 K the values of E and G increased in jumps (Fig. 7.7). The authors of [54] are explained these variations in the elasticity moduli by a change in the structural state of the grain boundaries: in the specimens of submicrocrystalline copper with a grain size of ~200 nm the grain boundaries were in non-equilibrium state and had excess energy. Annealing at T ≥ 423 K resulted in relaxation of the grain boundaries. Using the data [54], the authors of [56, 57] evaluated the elasticity modulus of the grain boundaries. For an equilibrium grain boundary with a thickness of 1 nm the elasticity eq eq = 0.16 E and Ggb = 0.12G . Similar estimates showed modulus are Egb

that the thickness of non-equilibrium grain boundaries in submicrocrystalline metals may reach 4 nm. In [58] it is shown that a decrease in the elasticity modulus of copper, subjected to severe plastic deformation, is evidently associated with both the formation of a crystallographic structure and with the decrease in the grain size. However, for the nanocrystalline materials with a grain size smaller than 5–10 nm, the fraction of atoms, located in the grain boundaries, is very large. According to [2], these materials have atomic structure which is closed to amorphous one. The elasticity modulus of amorphous materials is approximately half the modulus of similar coarsegrained crystalline materials [59]. Therefore, one may expect that the elastic properties of nanomaterials with a grain size of 5–10 nm or smaller should be considerably lower. This is in agreement with the estimates [60] obtained by the molecular dynamics methods: a decrease in Young’s modulus E, shear modulus G and uniform compression modulus K of copper with a decrease in the grain size to several nanometers is 25, 50 and 80 %, respectively. The effect of temperature on elasticity modulus E and the yield limit of submicrocrystalline copper produced by compacting nanoparticles (D ~ 80 nm) and by the equal-channel angular pressing (D ~ 100 nm), was studied in [61, 62]. The copper, compacted from a nanopowder under a pressure of 1 GPa, had a porosity of ~11 %. The elasticity modulus E of compacted copper increases after annealing at T ≤ 470–520 K and rapidly decreases after annealing at 670 K as a result of grain growth and a simultaneous increase in the porosity of the compacted copper. As the annealing temperature is raised, the elasticity modulus of plastically deformed copper rapidly increases in the 370–500 K and 720–870 K temperature intervals. The yield limit decreases with increasing annealing temperature. The jump of elasticity modulus E after 297

Nanocrystalline Materials

annealing at 370 K was explained by the authors of [61] by the pinning of grain-boundary dislocations by point defects and by the lowering of dislocation mobility as a result of a change in the structure of the interfaces. The second jump was explained by the disappearance of the grain-boundary phase as a result of grain growth. On the whole, the results in [54, 55, 58–62] show that the unusual elastic properties of the submicrocrystalline materials are determined not as much by the small grain size as by the state of the interfaces. Grain refinement is the well-known method of increasing the strength properties of materials. Studying of the stress–strain diagrams for compacted specimens of nanocrystalline Pd (D = 5– 15 nm) and Cu (D = 25–50 nm) [12] shows that the yield limit σ y of nanocrystalline metals is 2 to 3 times higher than that of coarsegrained metals. A large increase of σ y of the submicrocrystalline alloys of magnesium in comparison with coarse-grained alloys was detected by the authors of [63]. In nanocrystalline alloys Fe–Si– B, Fe–Co–Si–B, Fe–Cu–Nb–Si–B and Pd–Cu–Si, produced by crystallisation from the amorphous state, a decrease in the size D of precipitated particles of dispersed phases from 30–35 to ~10 nm is accompanied by an increase in the yield limit, whereas with a further decrease of D the yield limit decreased [28]. The tensile strength of nanocrystalline metals is 1.5 to 8 times higher than that for coarse-grained metals [12, 17]. According to [12], the main reasons for the increase in the strength of nanocrystalline metals may be high energy of formation and movement of dislocations, caused by the small grain size or/and high residual stresses. The changes in the strength properties of the materials, caused by the transition to the nanocrystalline state, require detailed study. In particular, this relates to multiphase nanocrystalline alloys, produced by crystallisation from the amorphous state [64]. An important problem is the damping of vibrations of metallic materials. The improvement of the damping properties reduces the harmful effect of cyclic loads, which are the cause of many accidents and failures; decreases the noise caused by the vibration of mechanisms; raises the accuracy of measuring devices owing to the damping of vibrations. Investigations [65–69] of the amplitude dependence of internal friction in submicrocrystalline copper show that as the size of the crystallites decreases and as the deviation of grain boundaries from the equilibrium state increases, the internal friction background and damping properties of the material increase. 298

Effect of Grain Size and Interfaces on Properties of Bulk Nanomaterials

For example, in submicrocrystalline copper with a mean grain size of ~200 nm, that has been annealed at 423 K, the background level is 3 to 5 times higher than that in the coarse-grained specimens and 2 to 3 times higher than that in grey cast iron (the background level for grey cast iron is a conventional boundary for heavy damping). The temperature, at which internal friction begins sharp rise, in submicrocrystalline copper decreases by approximately 100 K in comparison with that of coarse-grained copper. In addition, at 475 K a pronounced maximum appears in the temperature dependence of internal friction. The internal friction in the temperature range from 240 to 475 K is approximately 3 times higher for submicrocrystalline copper than that for coarse-grained specimens. All listed features are related to the difference between the elasticity modulus of the grains and of grain boundaries [70, 71]. The difference in the modulus makes it possible to examine the submicrocrystalline material as a heterogeneous material for the propagation of elastic vibrations. Consequently, the submicrocrystalline material shows considerable scattering of elastic vibrations resulting in an increase in the damping properties. It should be mentioned that the nanocrystalline and submicrocrystalline materials combine enhanced strength [16, 61, 72] and damping [65– 71] properties. In conventional materials, the strength properties decrease with increasing damping properties. The effect of the simultaneous increase of the damping and strength properties, discovered in submicrocrystalline copper, was confirmed by the studies of stainless steel [73, 74]. It is shown that the formation of the submicrocrystalline structure in the steel increases the internal friction background and the yield limit approximately 4 times. The superplasticity of ceramic nanomaterials should also be mentioned. For a long time, superplasticity was only a dream of materials scientists studying the processes of shaping and deformation of ceramics. Superplasticity is characterised by an exceptionally large relative elongation or shrinking of the material under tensile loading [75]. This phenomenon was demonstrated for the first time in 1934 on an example of the elongation of Sn–B alloy by more than a factor of 20 [76]. The superplasticity of ceramics was discovered for the first in 1985 on polycrystalline tetragonal oxides ZrO 2 stabilised with yttrium oxide Y 2 O 3 [77]. Later, superplasticity was observed in the two-phase composite ceramics Si 3 N 4 /SiC [78], and in other ceramic materials. Superplasticity is very important for producing ceramic ware by 299

Nanocrystalline Materials

shaping, solid-phase sintering or hot pressing at relatively low temperatures. Superplasticity leads to a high accuracy of the dimensions of ceramic ware of vary complicated shapes with inner cavities and surfaces with a varying curvature. According to [79], the superplasticity of ceramics manifests itself most vividly when the grains are smaller than 1 µm. With increasing temperature the grain size should remains unchanged as long as possible. For example, in nanocrystalline compacted magnesium oxide nc-MgO, the grain size remains almost constant when annealed up to 800–900 K [45]. In zirconium oxide ZrO 2 , grain growth with increasing temperature is suppressed by small additions of Y 2 O 3. In two-phase ceramics based on silicon nitride and silicon carbide, the grain growth of the matrix phase is suppressed due to precipitation of grains of the second phase. Factors increasing the plasticity of ceramics, also include the high-angle misorientation of the grain boundaries and the presence of a small amount of an amorphous grain-boundary phase [80]. In the nanocrystalline state, some ceramic materials (for example, TiO 2, [81]) become plastically deformable already at room temperature. In [82], special features of the behaviour of various mechanical properties such as microhardness and elasticity modulus with decreasing grain size were investigated using the statistical model of an ensemble of grain-boundary defects. The ensemble of defects of the type of microshear and microcracks in developed deformation stages has clear features of collective behaviour. The concentration of these defects is very high and reaches 10 13 –10 14 cm –3 . Therefore, the reason for the appearance of cooperative effects may be regarded as a thermodynamic reason. At the same time, each of the elementary defects (grain boundary or microcrack) is in a general case a thermodynamically non-equilibrium system. The evolution of ensembles of grain-boundary defects depends on the distribution of the nuclei of defects and by the interaction between defects. The solution of the Fokker–Planck equation for the distribution function of the nuclei of defects leads to a conclusion on the different behaviour of the dependence of the tensor component p zz on applied stress σ zz at different values of some dimensionless parameter δ . Analysis [82] shows that there are three different regions of solutions separated by asymptotics δ c and δ *. These solutions determine qualitatively different reactions of the polycrystal to the increase of the concentration of defects, which is caused by loading, in relation to the grain size. According to [82], the region of the values δ > δ * with a stable distribution 300

Effect of Grain Size and Interfaces on Properties of Bulk Nanomaterials 

QDQR3G



5 - PRO





.

FRDUVHJUDLQHG 3G

 QDQR&X 

FRDUVHJUDLQHG &X

 



%





.

Fig. 7.8. Effect of the nanostructured state on the temperature dependence of heat capacity C p (T) of copper and palladium [83]: ( F) nanocrystalline compacted ncPd and (∆) coarse-grained palladium Pd; ( l )nanocrystalline compacted nc-Cu and ( ™ ) polycrystalline coarse-grained copper Cu.

of the grain-boundary defects corresponds to a reaction of materials with a very small grains, i.e. nanocrystalline materials. In particular, the variation of the grain size in the vicinity of δ * may be manifested in the different slope of the Hall–Petch dependences, and in a rapid change of the elasticity modulus (see, for example, Fig. 7.1 and 7.7). 7.2 THERMAL AND ELECTRIC PROPERTIES The surface and size effects which observe in the phonon spectrum and in behaviour of heat capacity of nanoparticles, are studied in detail (see Section 5.3). The theoretical analysis and calorimetric investigations show that in the temperature range 10 K ≤ T ≤ θ D the heat capacity of nanopowders is from 1.2 to 2 times higher than that of the coarse-grained bulk materials. The increased heat capacity of the nanopowders is determined both by the size effect and by the very large surface area, which introduces an additional contribution to heat capacity. In contrast to the nanoparticles, investigations of the heat capacity of nanocrystalline bulk materials are limited to several studies. The heat capacity C p of nanocrystalline compacted specimens of nc-Pd (D = 60 nm) and nc-Cu (D = 8 nm), prepared by compacting nanoclusters, was measured in the temperature range from 150 to 300 K [83]. The relative density of the nc–Pd specimens was equal to 80 % and that of the nc-Cu specimens was equal to 90 % of the density of pore-free polycrystalline coarse-grained palladium Pd and 301

Nanocrystalline Materials

copper Cu. Measurements revealed that the C p of nc-Pd and ncCu specimens are 29–53 % and 9–11 % higher than the heat capacity of conventional polycrystalline Pd and Cu, respectively (Fig. 7.8). When nc-Pd was heated at T = 350 K, an exothermic effect was observed but the grain size remained unchanged or increased to 10 nm. The heat capacity of nc-Pd heated to 350 K, was found to exceed the heat capacity of coarse-grained palladium by 5 %. The authors of [83] assumed that the observed elevated heat capacity is caused by the ‘looser’ structure of the interfaces. This explanation is not plausible because it has been established that the structure of the grain boundaries in the compacted nanomaterials contains free volumes with the size of monovacancy or divacancy, but the effect of this free volumes is not large enough in order to explain excess of heat capacity. One of the explanation of excess could be impurities in palladium. Indeed, when studying the heat capacity of nc-Pt, the authors of [84] concluded that at a temperature ~300 K a large part of the excess heat capacity of the compacted nanomaterials is due to the excitation of the impurity hydrogen atoms. Impurity hydrogen is often present in nanomaterials, prepared by condensation of nanoclusters in an inert gas and by subsequent compacting. For example, the high solubility of hydrogen in grain boundaries of ncPd is noted in [85, 86]. According to [87], hydrogen dissolves primarily in the nc-Pd grains rather than in the interfaces. The low temperature heat capacity of the bulk nanocrystalline compacted copper nc-Cu with a grain size of 6.0 or 8.5 nm in the temperature range from 0.06 to 10.0 K is 5 to 10 times higher than that of coarse-grained copper [88]. The largest increase in heat capacity was detected in the specimens of nc-Cu with the smaller grains. The increase in the heat capacity of nc-Cu at T > 1 K may be caused by the fact that the weakly bonded atoms on the surface of grains behave as Einstein’s linear oscillators and surface vibrational modes appear in the phonon spectrum. According to estimate [88], each 6 th to10 th surface atom (depending on the grain size), is such an oscillator. In [89], inelastic neutron scattering at 100–300 K was used to study the density of phonon states g( ω ) in an n-Ni nanopowder, in a nanocrystalline compacted specimen of nc-Ni with a relative density of 80 % and in coarse-grained nickel. The grain size in both n-Ni and nc-Ni was about 10 nm. The most pronounced size effect is the increase in the density of phonon states g( ω ) of both n-Ni and nc-Ni specimens in comparison with coarse-grained nickel in 302

Effect of Grain Size and Interfaces on Properties of Bulk Nanomaterials

the energy range below 14 meV (Fig. 5.12). Calculations carried out using the data for the density of phonon states show that the heat capacity of nc-Ni at T ≤ 22 K is 1.5 to 2 times higher than the heat capacity of coarse-grained nickel. According to [89], the change in the phonon spectrum and high heat capacity of nc-Ni are caused by the contribution of the grain boundaries with reduced density of substance. In [89] it is also noted that the excess heat capacity of the compacted nanomaterials at room temperature is most likely due to the impurity of hydrogen atoms, whose vibrations are excited at T ≥ 300 K. Measurements of the temperature dependence of the heat capacity of compacted specimens of nanocrystalline nickel nc-Ni with a mean grain size of ~70 nm [90] show that at T ≤ 600 K ncNi has higher heat capacity in comparison with coarse-grained nickel. According to [70, 90], the high heat capacity of nc-Ni is caused by the contribution of the grain-boundary phase with a reduced Debye temperature and higher (by 10–25 %) heat capacity in comparison with the coarse-grained metal. The measurements of the temperature dependence of the heat capacity of amorphous, nanocrystalline and coarse-grained selenium Se in the temperature interval from 220 to 500 K [91] showed a small increase in the heat capacity of nanocrystalline n-Se in comparison with a coarse-grained Se at T < 375 K. The heat capacities of amorphous and n-Se coincide within the measurement error. Bulk nanocrystalline n-Se was prepared by crystallisation from the amorphous state in order to exclude the effect of distortions of the structure and also gaseous and other impurities on heat capacity. Table 7.1 shows the heat capacity of several substances in the Table 7.1 Comparison of heat capacity C p (J mol –1 K –1 ) of the nanocrystalline, amorphous and coarse-grained polycrystalline states of different materials [83] State nanocrystalline

Material

Pd Cu Ru Ni0.8P0.2 Se

synthesis method*

crystalline size D(nm)

Cp

1 1 2 3 3

6 8 15 6 10

37 26 28 23.4 24.5

Amorphous Cp

Coarse- grained Cp

27 23.4 24.7

25 24 23 23.2 24.1

T

Reference

(K)

s

250 250 250 250 245

83 83 72 92 91

*(1) compaction of ultrafine powders prepared by evaporation, (2) ball milling, (3) crystallization from the amorphous state

303

Nanocrystalline Materials

nanocrystalline, amorphous and coarse-grained states. A large difference in heat capacity in comparison with a coarse-grained state is observed for the specimens prepared by compacting of the nanopowders. On the other hand, this difference is very small and does not exceed 2 % for the specimens produced by crystallisation from the amorphous state. It may be assumed that the main part of the excess heat capacity of the bulk nanomaterials is determined by the large area of the interfaces, structural distortions and impurities. Temperature measurements of the crystal lattice constant of selenium [93] also made it possible to determine the dependence of the coefficient of volume thermal expansion, α V, on the crystallite size D. Coefficient α V of nanocrystalline selenium increases by approximately 30 % with a decrease in the crystallite size D from 45 to 10 nm. To explain the anomalies of low temperature heat capacity, the authors of [94] proposed a model of a bulk nanocrystalline material in which all the grains are rhombohedrons and are of the same size. The model cell consists of 8 such grains (Fig. 7.9). In modeling, the grain size D, which is determined as the diameter of a spherical particle with the same number of atoms, was assumed to be equal to 1.1, 2.0 and 2.8 nm. Interatomic interactions were described by the Lennard–Jones potential. Calculations of the density of vibrational states, g( ν ), showed that in comparison with a perfect fcc crystal

2 2

1

2

1

2 1

1

1

2

1

2

2

2 1

2

2

2 1

2

1

z y x

Fig. 7.9. Idealised 3-dimensional space-filling polycrystal model [94]. The 3Dperiodic simulation cell contains 8 identically shaped rhombohedral grains. There are two types of grains, which are indicated by 1 and 2. For each type, all the surfaces of the grains are crystallographically equivalent. The eight grains are connected by 24 identical asymmetric tilt grain boundaries.

304

Effect of Grain Size and Interfaces on Properties of Bulk Nanomaterials 

Jν







 





ν





7+]

Fig. 7.10. Density of the vibrational states, g(v) for the idealised model nanocrystal with grains of size D = 1.1 nm (solid line) and for a fcc perfect monocrystal, which contains 500 atoms (dashed line) [94]. 

  QP

∆& ;



N





  QP 

  QP  







%





.

Fig. 7.11. Excess heat capacity ∆C V for the idealised model nanocrystal as a function of temperature T and grain size D [94].

consisting of 500 atoms, g( ν ) of the model nanocrystal (D = 1.1 nm) is extended into both low- and high-frequency regions (Fig. 7.10). The majority of additional low- and high-frequency vibrational modes are localised at the grain boundaries. According to the calculations, the heat capacity of the nanocrystal is higher than that of the ideal fcc crystal; the difference of their heat capacities ∆C increases with a decrease in the grain size (Fig. 7.11). The excess heat capacity of the nanomaterial is determined by low-frequency vibrational modes related to the grain boundaries. The contribution of high-frequency vibrations 305

Nanocrystalline Materials

to the anomalous increase of the heat capacity of the nanocrystal is very small. Theoretical analysis of the internal energy and excess heat capacity of nanocrystalline materials was also carried out in [95] using a formalism equivalent to the mean-field approximation. According to [95], the excess heat capacity ∆C in the low temperature range is a linear function of temperature; when T ≤ J, the difference ∆C has a wide maximum (J is the energy parameter describing the interaction of atoms, each of which has two equilibrium positions). In the simplest case, according to the Gruneisen equation, the coefficient of thermal expansion, α , is proportional to the heat capacity C V . Taking this into account, it may be expected that nanocrystalline material should have a higher coefficient α in comparison with coarse-grained material. In fact, nc-Cu with a mean crystallite size of 8 nm has a coefficient of thermal expansion α = 31×10 –6 K –1 , which is twice as large as the value α = 16× 10 –6 K –1 for coarse-grained copper [96, 97]. To detect the effect of the grain boundaries on the coefficient of thermal expansion, the authors of [98] measured the thermal expansion of rolled copper foils with a grain size of 17 µm and of polycrystalline copper with a grain size of 19 mm. The thermal expansion coefficient of the copper foil was higher than that of coarse-grained copper. According to [98], the high value of α is caused by the fact that the grain boundaries have a considerably higher thermal expansion coefficient than that of the crystallites: for grain boundaries α gb = (40–80)×10 –6 K –1 , i.e. 2.5 to 5 times higher than the value of α of coarse-grained copper. It should be mentioned that the copper foil, studied in [98], have been produced by the method which was similar to the method of production of such submicrocrystalline materials in which the atoms in the interfaces have an enhanced mobility. The size dependence α (D) of nanocrystalline alloy Ni 0.8 P 0.2 was studied in [99]. Nanocrystalline specimens were produced by crystallisation of a ribbon of Ni 0.8 P 0.2 amorphous alloy at seven different annealing temperatures from 583 to 693 K. As-crystallised Ni–P specimens contain two phases: a solid solution of phosphorous in Ni with a fcc structure and compound Ni 3 P with a bodycentered tetragonal structure. The mean grain size of the precipitated phase Ni 3 P in relation to annealing temperature was 7.5–127 nm. Measurements show that as the grain size decreases from 127 to 7.5 nm, the coefficient of linear thermal expansion 306

Effect of Grain Size and Interfaces on Properties of Bulk Nanomaterials



1L3 





α  . 



  











QP

Fig. 7.12. Linear thermal expansion coefficient α of nanocrystalline Ni 0.8 P 0.2 alloy versus the grain size D of the Ni 3 P phase [99]

increases from (15.5±1.0)×10 –6 K –1 to (20.7±1.5)×10 –6 K –1 (Fig. 7.12). The linear thermal expansion coefficients of coarse-grained Ni–P alloy (D ≥ 10 µm) and Ni–P amorphous alloy of the same composition are 13.7×10 –6 and 14.2×10 –6 K –1 , respectively. It is clear that the value of α of the nanocrystalline alloy is higher than that of the coarse-grained and amorphous alloys. The authors of [99] represented coefficient α nc for the nanocrystal in the form

αnc = α in f in + α c(1 − f in) ,

(7.5)

where α in and α c are the linear thermal expansion coefficients of interfaces and crystallites, f in = c/D is the volume fraction of the interfaces, c = 1.9 is a constant, and D is the crystallite size. Calculations using the experimental data showed that with a decrease of D the difference ( α in – α c ) = ( α nc – α c )/f in rapidly decreases. For example, at D = 100 nm the ratio α in / α c is equal to 12.7, i.e. the thermal expansion coefficient for the interfaces is ten times higher than that for the crystallites. For a nanocrystal with a grain size of several nanometers in diameter α in / α c = 1.2. According to [99], the large decrease of the value of ( α in – α c ) with a decrease in the grain size may be a consequence of the increase in the density of the interfaces and/or of the decrease of lattice constant of nanometer crystallites. The latter seems a more likely reason. 307

Nanocrystalline Materials





N- PRO



*

















QP

Fig. 7.13. Variation of the molar Gibbs free energy G of a binary nanocrystalline solid solution (alloy) α – β with crystallite size D at fixed p, T, and fixed concentration of second component, x β = 0.05. The dotted line denotes the Gibbs free energy of the monocrystalline binary solid solution with the same x β ; this Gibbs free energy is independent on the grain size and is equal to 4 kJ mol –1 [102] .

The high heat capacity and thermal expansion coefficient for the nanocrystalline compacted materials indicate that such nanomaterials are thermodynamically unstable. It is shown in [100] using nc-Pd as an example that the structural state of the nanocrystalline compacted material immediately after preparation is thermodynamically non-equilibrium (see Section 6.1). General problems of thermodynamics and segregation for nanocrystalline binary solid solutions (alloys) are discussed in [101– 104]. The effect of segregation at the grain boundaries on the thermodynamic stability of solid solutions, which have a large segregation heat, is considered in [101, 102]. According to [101, 102], if specific grain-boundary energy σ in a binary solid solution is negative, this solid solution is metastable. This occurs only at rather low temperatures. If σ < 0, the dependence between the free Gibbs energy G and the crystallite size D becomes non-monotonic and exhibits a pronounced minimum. Fig. 7.13 displays a model G(D) dependence for the polycrystalline solid solution α – β with the concentration of the second component x β = 0.05. For this solid solution σ = –1.7 J m –2 . Calculation is performed for a temperature of 600 K. As is seen, the G(D) dependence (solid line) has a free energy minimum, which corresponds to the range of D from 10 to 20 nm. The free energy G ≈ 4 kJ mol –1 of the monocrystalline solid solution with the same composition is shown in Fig. 7.13 as a 308

Effect of Grain Size and Interfaces on Properties of Bulk Nanomaterials

dashed line. When the grain size is rather large, the free energy of a polycrystalline alloy asymptotically tends to the free energy G ≈ 4 kJ mol –1 of the monocrystalline alloy. The results obtained in [101, 102] reveal that the dependence of the free Gibbs energy on the solid solution composition changes both quantitatively and qualitatively when the grain size of a polycrystalline alloy decreases to a nanometer range. The problem of thermal stability of bulk nanocrystalline compacted materials was studied in [105]. In this study, the time dependence of electromotive force (emf) E of high-purity nc-Pd with a mean grain size of 11 and 18 nm was measured at 613 K. For the reference electrode, coarse-grained palladium with a grain size of ~20 µm was used. Measurements show that the emf of the nanocrystalline specimens is negative, rapidly increases in the first 4–5 minutes and then increases slowly asymptotically approaching zero. Because of the large area of the grain boundaries, the exchange electrochemical reaction in nc-Pd takes place at the grain boundaries at a high rate; in this case, the emf E of the nanomaterial is directly related to the thermodynamic characteristics of grain boundaries by a simple relationship ∆G gb = –|z|FE, where z is the valence of the palladium ion, F is the Faraday constant, and ∆G gb is the Gibbs energy of the interfaces. Taking this into account, the negative emf corresponds to the positive Gibbs energy of nanocrystalline palladium in comparison with coarse-grained palladium; this means that nc-Pd is thermally unstable at elevated temperatures. According to [105], the rapid increase of the emf in the first stage of measurement is caused by the relaxation of interfaces; the subsequent slow approaching of the emf to zero is due grain growth. Similar behaviour of emf, associated with the relaxation of interfaces and grain growth, was observed in calorimetric measurements of nanocrystalline Pt [106]. The emf after relaxation of the interfaces was equal to –36, –7 and –4 mV for nc-Pt with a grain size of 11, 18 and 20 nm, respectively. Thus, as the grain size decreases, the thermodynamic stability of the material also decreases. The thermal instability of the nanomaterial is caused primarily by the non-equilibrium state of the grain boundaries. The greatly developed interfaces and the high defect concentration cause the extensive scattering of current carriers in nanomaterials. The large increase of the specific electrical resistivity of nanocrystalline Cu, Pd, Fe, Ni and the different alloys with a decrease in the grain size was reported by many 309

Nanocrystalline Materials

researchers. Investigation of the temperature dependence of the electrical resistivivity of compacted nanomaterials is used to characterise the state of grain boundaries and to determine the relaxation temperature. Specific electrical resistivity ρ of nc-Cu (D = 7 nm) in the temperature range 0 < T ≤ 275 K is 7 to 20 times lower than that of conventional coarse-grained copper [107]. At T ≥ 100 K the specific electrical resistivity ρ of coarse-grained Cu and nc-Cu increases linearly with increasing temperature, but the temperature coefficient of resistivity, ∂ ρ /∂T, for nc-Cu is equal to 17×10 –9 Ω cm K – 1 and is higher than ∂ ρ /∂T = 6.6×10 –9 Ω cm K –1 of conventional copper. Analysis of the experimental dependences ρ (T) of nanocrystalline and coarse-grained copper showed that the coefficient of electron scattering, r, at the grain boundaries in nc-Cu is equal to 0.468 at 100 K and is equal to 0.506 at 275 K, while for coarse-grained copper r = 0.24, which is lower by a factor of two. This difference is a consequence of the different width and structure of the grain boundaries in nanocrystalline and coarsegrained copper. The temperature dependence of the coefficient r of nc-Cu is caused by a high thermal expansion coefficient of the grain boundaries: according to [96, 97], α gb = 66×10 –6 K –1 . Kai [107] believes that the high electrical resistivity ρ and temperature coefficient ∂ ρ /∂T of nc-Cu are caused primarily by electron scattering at the grain boundaries. Another reason for the high electrical resistivity of nc-Cu may be the short mean free path λ of electrons: for nc-Cu λ ≈ 4.7 nm, and for coarse-grained copper λ ≈ 44 nm. Investigation of the resistivity of polycrystalline cobalt films with a thickness in the range from 2 to 50 nm demonstrated that the value of ρ is almost independent on temperature, decreases with increasing film thickness and is higher than that of bulk cobalt [108]. According to [108], the temperature coefficient of resistivity, which is close to zero, and the high specific electrical resistivity of nanocrystalline Co films are a consequence of partial localisation of the electrons when the grain diameter becomes shorter than the electron mean free path. Localisation affects the electrical conductivity more strongly than the increase in the scattering of charge carriers at the interfaces because it leads to a decrease in the concentration of charge carriers. As a result, a decrease in the crystallite size increases the degree of localization, decreases the concentration of charge carriers and, hence, increases specific electrical resistivity. The specific electrical resistivity of submicrocrystalline Cu, Ni and 310

Effect of Grain Size and Interfaces on Properties of Bulk Nanomaterials

Fe, produced by equal-channel angular pressing, was investigated in [109–111]. The mean grain size of submicrocrystalline metals was 100–200 nm. At 80 K the specific resistivity ρ of submicrocrystalline copper was almost twice the value of ρ of coarse-grained copper. The high electrical resistivity of submicrocrystalline copper is caused by a high coefficient of electron scattering, r, at non-equilibrium grain boundaries: in submicrocrystalline copper r = 0.29–0.32 instead of r = 0.24 for equilibrium grain boundaries in coarse-grained copper. According to [112], the increase in coefficient r is due to distortions of translation symmetry caused by long-range stress fields and by the dynamically excited state of atoms in the grain-boundary phase. Annealing at 420–470 K results in a rapid decrease in ρ ; with a further increase in annealing temperature, specific electrical resistivity ρ slowly decreases [109, 110]. According to the results of microstructural study, the rapid decrease in ρ as a result of annealing at 420–470 K is caused by the relaxation of grain boundaries and by their transition from the stressed non-equilibrium state to equilibrium state. The subsequent slow decrease in ρ is a consequence of grain growth. According to the results [111], the specific electrical resistivity ρ of submicrocrystalline Cu, Ni and Fe at 250 K is 15, 35 and 55 % higher than that of the same coarse-grained metals; the temperature coefficients of the electrical resistivity of submicrocrystalline and coarse-grained Cu, Ni and Fe differ only slightly. In [111], measurements of the temperature dependences of the thermal emf of submicrocrystalline copper, nickel and iron were also taken. The absolute value of the thermal emf of submicrocrystalline metals is lower than that of coarse-grain metals; for Cu and Fe thermal emf is positive, and for Ni it is negative. The observed change of the transport properties of the submicrocrystalline metals with an increase in temperature from 20 to 270 K was explained by extensive electron scattering at the grain boundaries [111]. 7.3 MAGNETIC PROPERTIES The effect of the nanocrystalline state on the magnetic properties of paramagnetics is studied in [38–41, 113] on an example of palladium. Polycrystalline Pd with crystallites several micrometers in size is characterised by a unique electronic structure, which is extremely sensitive to ferromagnetic impurities or to the effect of external pressure. This prompted the authors of [113] to assume that the formation of the submicrocrystalline structure in palladium 311

Nanocrystalline Materials

may affect the electronic structure and magnetic susceptibility. Submicrocrystalline Pd was produced from coarse-grained Pd by severe plastic deformation with using torsion under quasi-hydrostatic pressure; this resulted in a true logarithmic degree of deformation e = 7. The density of submicrocrystalline Pd coincided with the density of starting palladium and did not change after annealing at temperatures of 300 to 1200 K. This indicates the absence of porosity of submicrocrystalline Pd. The grain size of submicrocrystalline Pd, determined by X-ray diffraction and electron microscopy, was 120–150 nm. Magnetic susceptibility χ was measured by the Faraday method with an accuracy of ±0.05×10 –6 cm 3 g –1 in a vacuum of 1.3×10 –3 Pa (10 – 5 mm Hg). The procedure and sequence of magnetic measurement is described in Section 5.4 and is shown in Fig. 5.13. The magnetic susceptibility of starting and submicrocrystalline Pd did not depend on the magnetic field strength H; this indicates the absence of ferromagnetic impurities in the specimens. Measured temperature dependence χ (T) and annealing dependence χ (300,T) of the magnetic susceptibility of starting and submicrocrystalline Pd in the temperature range from 300 to 1225 K are shown in Fig. 7.14  

χ .  FP  J 

3G

 

   







%





.

Fig. 7.14. Magnetic susceptibility of submicrocrystalline and coarse-grained palladium [40]: (1) and (2) are annealing χ (300, T) and temperature χ (T) dependences of the susceptibility of submicrocrystalline Pd, respectively; (3) and (4) are annealing χ (300, T) and temperature χ (T) dependences of the susceptibility of starting coarsegrained palladium, respectively. Annealing dependences χ (300,T) of susceptibility (curves 1 and 3) were measured at 300 K after annealing at temperature T and subsequent cooling to 300 K.

312

Effect of Grain Size and Interfaces on Properties of Bulk Nanomaterials

(measurements of χ made directly at annealing temperature T after holding at this temperature for 1 h relate to the temperature dependence χ (T); measurements of susceptibility, made at 300 K after a specimen had been annealed at temperature T and then cooled to room temperature, relate the annealing dependence χ (300,T)). At T < 825 K, the magnetic susceptibility χ (300,T) of submicrocrystalline Pd considerably exceeds the susceptibility of starting palladium, which is independent of the annealing temperature. As a result of annealing at 825–1025 K, the susceptibility of submicrocrystalline Pd first drops and then decreases slowly to the value of χ corresponding to the starting palladium. The temperature dependence χ (T) for submicrocrystalline Pd (curve 2 in Fig. 7.14) does not contain similar sharp transition. With an increase of temperature, the susceptibility of submicrocrystalline Pd decreases and smoothly changes to the dependence χ (T) of starting palladium; starting from T = 725 K, the temperature dependences χ (T) of submicrocrystalline and starting Pd almost completely coincide. The main contribution to the susceptibility of polycrystalline palladium is due to the Pauli spin paramagnetism of conductivity electrons χ p [114]. There is strong Stoner enhancement of χ p by a factor of 10 to15 [115], which reflects many-particle effects of electron interaction. Another important feature of palladium is the presence of a high and narrow (~0.3 eV) peak near the Fermi energy [115]. This peak is responsible for the high density of electron states (~2.3 eV –1 atom –1 ) at the Fermi energy. The narrow density of states peak determines the sensitivity of the properties of palladium to the Fermi energy. In the low-temperature region, approximately down to 50 K [116], the magnetic susceptibility of palladium increases as T 2 and is described by the Pauli temperature dependence common to fermions

χ p (T) = χ p (0)[1 + (1/6) π 2 ν 0 k B T 2 ] ,

(7.6)

where χ p (0) is the magnetic susceptibility at T = 0 K with allowance for the Stoner enhancement, ν 0 is a constant which depends on the density of states on the Fermi level N(E F) and also its first and second derivatives with respect to energy. As the temperature is raised, the susceptibility of Pd passes through a pronounced maximum in the range of 50 to 100 K and then rapidly decreases [116, 117]. The authors of [39–41, 113] investigated the high-temperature region T ≥ 300 K, in which the 313

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contribution (7.6) to susceptibility decreases continuously and there is no maximum on the experimental dependence χ (T). Statistical processing results for temperature dependences of the magnetic susceptibility χ (T) for starting and submicrocrystalline palladium, measured in [39–41, 113], shows that at T ≥ 775 K these curves are adequately described by the Curie relation χ=

C 1 N A µ B2 p 2 = , T 3 ρ k BT

(7.7)

where χ is the magnetic susceptibility of the mass unit, C is the Curie constant, N A is the Avogardo number, µ B is the Bohr magneton, ρ is the density of the substance, and p is the effective number of Bohr magnetons per atom. At T ≥ 775 K the Curie constants for starting and submicrocrystalline Pd coincide within the limits of the measurement error and equal 1945±10 cm 3 K g –1 . The effective magnetic moment, calculated from the Curie constant C using equation (7.7), was µ eff = p µ B = 0.44µ B . The gradual and continuous conversation of susceptibility from the Pauli dependence to the Curie dependence, detected in a wide temperature range from 50 to 775 K, is associated with the thermal excitation of electrons and ‘smearing’ of the Fermi level. This conversation may be viewed as an analogue of the transition in the distribution of electrons from the Fermi–Dirac statistics to the classical Maxwell–Boltzmann statistics. The susceptibility of Boltzmann electrons is not associated with the density of states on the Fermi level, but is determined only to the localised electron magnetic moments being non-compensated. For this region, the magnetic susceptibility of Pd at high temperatures is not sensitive to the effects influencing the density of states. This is confirmed by the coincidence of the temperature dependences of susceptibility χ (T) for the starting and submicrocrystalline palladium at T > 775 K, detected in [39–41, 113]. The most interesting result [39–41, 113] is the large difference, by 8 %, in the magnetic susceptibility of submicrocrystalline and starting coarse-grained palladium, observed at 300 K. This difference also remains after annealing submicrocrystalline Pd at T < 825 K. According to [39, 40], the observed difference of χ cannot be related with the variation of the volume content of the grain boundaries and their transition from the stressed nonequilibrium state to the equilibrium state. Indeed, according to the 314

Effect of Grain Size and Interfaces on Properties of Bulk Nanomaterials

electron microscopic data and the results of measurement of microhardness (see Fig. 7.4), the most considerable grain growth, the decrease in the density of dislocations and grain boundary relaxation take place after annealing at T < 800 K. The volume fraction of the grain boundaries in submicrocrystalline palladium varies by more than 10 times as a result of annealing in the temperature region of 300 to 800 K, and the lattice dislocations density changes by three orders of magnitude. However, this is not reflected in the behaviour of susceptibility which starts to decrease only at an annealing temperature of T > 810 K (Fig. 7.14, curve 1). According to [39–41], intragranular vacancy complexes are the most probable defect type affecting the behaviour of the susceptibility. In [118] it is found that in nanocrystalline n-Pd (D = 5–10 nm) the vacancies agglomerate into complexes, which are less mobile than monovacancies and may exist up to temperatures higher than 400 K. The effect of intragranular vacancy complexes on the magnetic susceptibility of submicrocrystalline palladium may be a consequence of the change in the density of electron states at the Fermi energy. As already mentioned, in palladium the Fermi energy lies on the slope of a very narrow and high peak of the density of states N(E) [115]. The appearance of c v vacancies in the palladium lattice leads to emptying of n ec v states in the conduction band (n e = 10 is the number of electrons in the conduction band per one palladium atom). If c v vacancies are introduced the Fermi energy decreases by ∆E v, then the number of empty states may be represented as EF

∫

EF −∆ Ev

N( E )dE = ne cv

(7.8)

In order to determine the value of ∆E v , the authors of [40] used the numerical data [115] for N(E F ), N'(E F ) and N''(E F ) and expansion N(E) in a Taylor series to the terms of the second order. According to calculation results, the Fermi energy must decrease by ∆E v = 0.014 eV for the enhancement of susceptibility and, hence, the density of states by a factor of 1.08 at T = 0 K. (in [39, 40] the given value ∆E v = 0.14 eV is a misprint). Taking ∆E v value into account and using equation (7.7) it was found that the vacancy concentration, that guarantees the required increase in N(E F ) and 315

Nanocrystalline Materials

a decrease of the Fermi energy by ∆E v ≈ 0.014 eV, is equal to 0.003 vacancies per atom. This vacancy concentration, 0.3 at.%, can be achieved quite easily by severe plastic deformation because according to the equation [119] c v ≈ [exp(e)–1]×10 –4

(7.9)

the concentration of vacancies at e = 7 is considerably higher. At T > 500 K, the ‘smearing’ of the density of states near the Fermi energy by the value ~k B T becomes comparable with the width of the narrow peak of the density of states near the Fermi energy. As a consequence of this, the effect of vacancies on the magnetic susceptibility of Pd at high temperatures is very low. This explains the absence of a drop in the temperature dependence χ (T) of submicrocrystalline palladium. Thus, the high susceptibility of submicrocrystalline palladium is associated with the excess concentration of vacancy clusters. The recovery of the susceptibility of submicrocrystalline palladium to the values, characteristic of coarse-grained palladium, is caused by the condensation of vacancies and the annealing of dislocation tangles at T > 825 K. However, the results [120] on the positron lifetime in submicrocrystalline palladium contradict to a certain extent the conclusions [39, 40] on the effect of vacancy clusters on the magnetic susceptibility of palladium. According to [120], the positron lifetime spectrum of submicrocrystalline palladium contains two components with lifetimes of τ 1 ≈ 167 ps and τ 2 ≈ 280–330 ps. The first component is characterised by a high intensity (approximately 95 %) and the intensity of the second one is around 5 %. The value of lifetime τ 1 indicates the trapping and annihilation positrons in lattice vacancies, the second component with a longer lifetime τ 2 corresponds to the annihilation of the positrons in vacancy clusters with a volume of 6–12 atoms. According to [120], the vacancy clusters are observed in submicrocrystalline palladium only to a temperature of T ≤ 455 K, and are annealed at higher temperatures. Thus, although the results in [120] confirm the assumption [40] on the presence of vacancy clusters, which affect the electron–energy spectrum of submicrocrystalline palladium near the Fermi energy, the problem of the temperature stability of these complexes has not as yet been completely solved. In recent years, there has been special interest in the producing of submicrocrystalline titanium [121–125]. It is expected that the physical properties of submicrocrystalline and coarse-grained 316

Effect of Grain Size and Interfaces on Properties of Bulk Nanomaterials

titanium should be greatly differ, as reported for other substances [1, 126, 127]. In fact, the application of severe plastic deformation (equal-channel angular pressing and uniform forging) allows to reduce the grain size in titanium to several hundreds nanometers [121–125] and allows to detect changes in the mechanical properties. A large difference of magnetic susceptibility of coarsegrained and submicrocrystalline titanium is observed by the authors of [42]. The starting material was hot-rolled polycrystalline titanium with a hcp structure (a = 0.29494 nm, c = 0.46844 nm). The mean grain size in starting Ti was 15 µm, the density 4.505 g cm –3. The content of impurities in starting titanium was (wt.%): 0.32 Al, 0.18 Fe, 0.12 O, 0.07 C, 0.04 N, and 0.01 H. To analyse the possible effect of ferromagnetic impurities, reference specimen from nondeformed superpure titanium containing 1.75×10 –4 wt.% Fe, 1×10 –6 wt.% Co and 10×10 –4 wt.% Ni was used. Submicrocrystalline titanium was produced from titanium rod with a diameter of 40 mm by two-stage processing. The first stage of severe plastic deformation consisted of 8 consecutive cycles of warm equal-channel angular pressing at a temperature 670–720 K, the angle between the channel was 90° [124, 128]. After each cycle, the specimens were rotated through 90° around the longitudinal axis, and after 4 cycles additionally through 180° around the axis normal to the longitudinal axis. After first deformation stage, the specimens had a homogeneous structure in both the longitudinal and cross sections. The mean grain size was 300 nm. In the second stage of processing, the deformed titanium was subjected to cold rolling in circle–oval–rhomb–circle gauges; the achieved degree of deformation was 93 % [128]. Rolling was carried out in the direction parallel to the pressing axis. Rolling resulted in a decrease in the grain size from 300 nm to 150 nm. Rollers and setup for equal-channel angular pressing were produced from a ferromagnetic tool steel. Equal-channel angular pressing and rolling were carried out in air over a short period of time (the duration of each cycle did not exceed 30 s) and, consequently, there was no oxidation or inflow of impurities into the volume of the specimen as a result of diffusion. In addition, the specimens for measurements were cut out from the centre of the volume of deformed titanium so that particles of the roller material could not penetrate into titanium. It should be mentioned that such cutting of the specimen does not prevent the transfer of particles of die or plunger into the specimen during equal-channel angular pressing because there is constant transfer of the surface of the specimen, 317

Nanocrystalline Materials

which is in contact with die, inside the specimen. Thus, regardless of the measures taken, a penetration of ferromagnetic impurities into the paramagnetic specimen cannot be excluded. Structural certification of the specimens was carried by two methods: transmission electron microscopy (JEM-100B) of thin foils and X-ray diffraction in CuKa 1,2 radiation with a determination of diffraction reflection broadening. Microhardness measurements (with a load of 100 g) were taken to analyse the thermal stability and structural transformations during annealing. Magnetic susceptibility was measured using the same procedure, which was employed for studying the magnetic susceptibility of submicrocrystalline palladium and copper [36–40] and was described previously (see also Section 5.4 and Fig. 5.13). Heating, annealing and cooling of specimens were carried out in situ directly in magnetic susceptibility balance. Annealing temperature was varied from 300 to 1043 K in 50 K steps, holding time of the specimen at annealing temperature was 1 h. The microstructure of the produced submicrocrystalline titanium in the cross and longitudinal sections is shown in Fig. 7.15. In the cross section, the grains are equiaxial and their mean size is 150 nm; in the longitudinal section, grains have a subgrain structure and are elongated in the rolling direction, which coincides with the pressing direction. Both the sections are characterised by the presence of high- and low-angle grain boundaries and by a large lattice dislocation density of 10 14 –10 15 m –2 . The azimuthal smearing of spot reflections on electron diffraction patterns indicates the presence of high internal stresses. Analysis of the broadening of the diffraction reflections

Fig. 7.15. Microstructure of submicrocrystalline titanium produced by equal-channel angular pressing and cold rolling [42]: transverse (a) and longitudinal (b) sections in relation to the rolling axis. The rolling axis coincided with the pressing axis and the titanium rod axis.

318

Effect of Grain Size and Interfaces on Properties of Bulk Nanomaterials (10 10) and (2020) shows that the size of the regions of coherence

scattering is 70–90 nm and the magnitude of elastic distortions, <e 2 > 1/2 , in the cross section is 0.23 % [129]. X-ray investigation also shows that as a result of deformation treatment, submicrocrystalline titanium is characterised by a distinctive crystallographic texture [130]: the basal plane (0001) is located preferentially in the plane parallel to the pressing axis (Fig. 7.16). The density of submicrocrystalline titanium was 0.2 % lower than that of starting coarse-grained titanium. This indicates the absence of any extensive porosity, characteristic of bulk specimens of titanium, produced by compaction of nanopowders [131]. Regardless of the high dislocation density in submicrocrystalline titanium, it may be assumed that they introduce a negligibly small free volume into the specimen. Scanning and transmission electron microscopy confirm the absence of porosity in titanium. Therefore, reduced density may be associated either with an increase of the interatomic distances near the grain boundaries [131] or with vacancies and vacancy clusters formed as a result of severe plastic deformation. The results of measurement of the magnetic susceptibility of the specimen after the first stage of deformation, i.e. after warm multiple equal-channel angular pressing, are shown in Fig. 7.17. In accordance with the results obtained by transmission electron microscopy, the microstructure of the specimen is isotropic, the grains are almost spherical, and the mean grain size is approximately 300 nm. The temperature dependence χ (T) for

Fig. 7.16. Pole figures for three different sections of submicrocrystalline titanium specimen [42]: (1) direction, which is perpendicular to the rolling direction, (2) rolling direction. Directions [1 0 1 0] , [0 0 0 1] , and [1 0 11] are normal to the plane of the figure. 319

Nanocrystalline Materials

VXEPLFURFU\VWDOOLQHVPF7L 7L VXEPLFURFU\VWDOOLQH FRDUVHJUDLQHG7L7L FRDUVHJUDLQHG

χ  FP J 







 









L , Fig. 7.17. Magnetic susceptibility of starting coarse-grained and submicrocrystalline titanium [42]: ( l ) and ( ¢ ) denote annealing dependences χ (300, T); (  ) and ( £ ) denote temperature dependences χ (T). The temperature range, in which irreversible processes of structural relaxation and recrystallisation take place, is indicated by the vertical dashed lines.

starting titanium in the studied temperature range is in good agreement with literature data [132]. The susceptibility of submicrocrystalline titanium at room temperature is 5 % higher than that of starting titanium and is equal to 3.3×10 –6 cm 3 g –1 . The annealing dependence of susceptibility χ (300, T) of submicrocrystalline titanium remains unchanged up to a temperature of 673 K. At T > 673 K, a smooth irreversible decrease of susceptibility to the values corresponding to starting titanium is observed. Similar transition to lower values of susceptibility in the temperature range 673–823 K is detected on the temperature dependence of susceptibility χ (T). In the temperature range from 300 to 673 K the susceptibility of submicrocrystalline titanium exceedes that of starting titanium by the same value 0.15×10 –6 cm 3 g –1 . At T > 823 K the temperature dependences χ (T) of submicrocrystalline and starting titanium coincide almost, i. e. complete recovery of the values of magnetic susceptibility takes place. A different situation is found in the case of the magnetic susceptibility of submicrocrystalline specimens after the second stage of plastic deformation, i.e. after rolling. Since rolling results in anisotropy of the structure, two specimens are investigated. One specimen is cut along the rolling axis and denotes as smc||-Ti. The 320

Effect of Grain Size and Interfaces on Properties of Bulk Nanomaterials

second specimen, denoted by smc⊥-Ti, is cut across the rolling axis. It is known that the magnetic susceptibility of titanium single crystal has an anisotropy. For example, the susceptibility of pure titanium at room temperature along the [0001] direction is 3.35× 10 –6 cm 3 g –1 , and across this direction it is 3.07×10 –6 cm 3 g –1 [133]. As a result of anisotropy, the difference in susceptibility for different orientations of the specimen reaches 9%, i.e. it is comparable with the effects found in submicrocrystalline titanium. Comparison of Figs. 7.17 and 7.18 shows that the values of susceptibility in the longitudinal and transverse sections of submicrocrystalline titanium are higher than the susceptibility of starting coarse-grained titanium. The difference between the susceptibilities of the longitudinal and transverse directions (Fig. 7.18, smc||-Ti and smc⊥-Ti specimens) corresponds to an anisotropy of 9 %. This shows that the additional contribution to susceptibility, appeared as a result of pressing and rolling, does not depend on the crystallographic direction and anisotropy of the microstructure. The recovery of susceptibility to the values of χ for coarsegrained titanium in the specimen without rolling takes place in the temperature range 670–820 K (Fig. 7.17), and for the specimen with rolling it takes place in the temperature range 727–935 K (Fig. 7.18). It should be mentioned that there is no complete recovery of susceptibility for the rolled specimen. The region of the recovery 



VXEPLFURFU\VWDOOLQH VPFB7L _ VXEPLFURFU\VWDOOLQH VPF__7L

χ  FP  J 













 









L ,

Fig. 7.18. Magnetic susceptibility of submicrocrystalline titanium, cut out along (smc||-Ti) and across (smc⊥−Ti) rolling axis: (l) and (¢) denote annealing dependences χ (300, T); (  ) and ( £ ) denote temperature dependences χ (T). The temperature range of structural relaxation and recrystallisation of submicrocrystalline titanium is indicated by dashed lines.

321

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temperature for the specimens with and without rolling is shifted to high values in comparison with the recrystallisation temperature range 573–873 K, determined from microhardness measurements (Fig. 7.5). This means that the variation of the susceptibility of submicrocrystalline titanium as a result of deformation and annealing is determined to a greater extent by the changes in the electronic structure of titanium and is associated only indirectly with recrystallisation. In principle, there are two main mechanisms of increasing the magnetic susceptibility of submicrocrystalline titanium: impurity mechanism and internal mechanism. The impurity mechanism may be realised by introducing ferromagnetic and paramagnetic impurities during plastic deformation. In addition to this, during plastic deformation the impurities, presented in starting titanium, may precipitate or be activated, as in the case of submicrocrystalline copper [37]. The observed increase of the susceptibility of submicrocrystalline titanium by 5 % cannot be explained by intensive mass transfer of the paramagnetic substance from the rollers or die to the specimen. If it is assumed that the paramagnetic susceptibility of the transferred substance is twice that of titanium, the observed increase of χ may take place when 5 wt.% of this substance is transferred into the specimen. Since annealing of submicrocrystalline titanium without rolling leads to the disappearance of the addition to the susceptibility without any change of the mass of the specimen, this impurity mechanism can be unambiguously excluded. The submicrocrystalline titanium, subjected to rolling, does not show any disappearance of the additional susceptibility after annealing. In this specimen, in addition to the increase in susceptibility as the result of the change in the structural state, there is also an impurity contribution to the susceptibility. Let us consider the internal mechanism of increasing susceptibility. This mechanism cannot be realised by a means of changes in the paramagnetic Pauli contribution, as takes place in submicrocrystalline palladium [113], because electron spectrum of titanium has no singularities near the Fermi energy. The absence of this contribution is also confirmed by the slope of the temperature dependences of susceptibility of submicrocrystalline specimens (Fig. 7.17 and 7.18); for all cases the slope corresponds to the Pauli paramagnetic contribution of pure non-deformed titanium. According to [42], the most probable reason for the increase of the magnetic susceptibility after plastic deformation is 322

Effect of Grain Size and Interfaces on Properties of Bulk Nanomaterials

the variation of Van-Fleck paramagnetism. Van-Fleck paramagnetism is independent of temperature and appears as a result of disruption of the symmetry of electron shells of atoms due to structural heterogeneity, stresses and distortions of the crystal lattice. All these factors are found in titanium after plastic deformation. The recovery of susceptibility at recrystallisation temperatures when the structural heterogeneity, stresses and distortions are no longer found in submicrocrystalline titanium, is indirect confirmation of the relationship of the discussed change of χ with Van-Fleck paramagnetism. The majority of studies into the magnetic properties of bulk nanocrystalline materials have been carried out on ferromagnetic metals and alloys. Saturation magnetisation I s , Curie temperature T C and coercive force H c of nanocrystalline compacted nickel nc-Ni (D = 10 nm) and single crystal of nickel are studied in [134]. Magnetic measurements were taken in the temperature from 5 to 680 K in magnetic field up to 5.5 T. At T < 300 K the magnetisation of nc-Ni is lower than I s of the Ni single crystal. Measurements of the temperature dependence of magnetisation of nc-Ni in a field of 0.17 T show that at temperature of 510–545 K magnetisation rapidly decreased by ~20 %. The authors of [134] assume that this decrease is associated with the ferromagnetic–paramagnetic transition in the interfacial substance and, consequently, the Curie temperature T C for the grain-boundary phase of nc-Ni is equal to 545 K. Further heating of nc-Ni is accompanied by grain growth from 15 to 48 nm and leads to a decrease or complete disappearance of magnetisation at a Curie temperature of 630 K, corresponding to T C of coarse-grained nickel. In cooling from 650 to 450 K, the transition to the ferromagnetic state takes place at 630 K and then magnetisation smoothly increases without any special features. The values of magnetisation in cooling are higher than that of in heating of nc-Ni. According to [134], at T = 0 K magnetisation of nc-Ni was 0.52µ B per atom in contrast to coarsegrained nickel, for which I s (0) = 0.6µ B atom –1 . Coercive force H c of nc-Ni in the temperature range from 100 to 300 K remains unchanged and equal to ~10 Oe. The most interesting result [134] is the decrease of the Curie temperature of nc-Ni. Therefore, the same group of authors investigated later [135] again the magnetic properties of nc-Ni, paying attention to oxygen impurity. They found that the previously detected non expected decrease of the magnetisation of the nc-Ni at 510–545 K is due to the presence of 323

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oxygen impurity and, consequently, there is no justification for changes of the Curie temperature of nanocrystalline nickel. In fact, rough estimation shows that every impurity atom of oxygen decreases magnetic moment of the Ni particle by the value corresponding to the magnetic moment of a single atom of crystal nickel. The decrease of T C of nc-Ni (D = 70–100 nm) by 30–40 K in comparison with conventional coarse-grained nickel was reported by the authors of [90]. This result was obtained by scanning calorimetry and by measurements of the temperature dependence of saturation magnetisation. Measurements of the dependence I s (T) show that at 300 K the magnetisation of nc-Ni (D = 100 nm) is ~10 % lower than that of coarse-grained nickel (D ≈ 1 µm). Compacted specimens of nc-Ni were produced in [90] by pressing the nanopowder in air and, finally, contained a large amount of oxygen. Taking into account the results [135] and the high sensitivity of the magnetic properties of nickel to the oxygen impurity [136], it may be assumed that the effects found in [90], are a consequence of the contamination of nanocrystalline nickel with oxygen and are not associated directly with the nanocrystalline state of nickel. A small (by ~3 %) decrease of the magnetisation of submicrocrystalline nickel, produced by torsion under quasi-hydrostatic pressure, was noted in [137]. The grain size of submicrocrystalline Ni was assumed to be equal to 100–200 nm. The temperature dependence χ (T) of submicrocrystalline nickel is characteristic of ferromagnetics. During the first heating of plastically deformed nickel specimens, the decrease of susceptibility on approaching the Curie temperature was smooth. In efficiently annealed specimens of submicrocrystalline nickel the approach to T C was accompanied by a rapid decrease of χ , the same as in the case of non-deformed nickel in the ferromagnetic→ paramagnetic transition range. After consecutive annealing, the susceptibility of deformed nickel increases to the values of χ corresponding to starting non-deformed nickel. The observed anomalies of the magnetic properties of submicrocrystalline nickel are explained by the authors of [137] by the fact that part of the smallest crystallites of submicrocrystalline nickel are in the superparamagnetic state. In this case, submicrocrystalline nickel should be treated as a heterogeneous material whose susceptibility is the superposition of the susceptibilities of the ferromagnetic and superparamagnetic components. The explanation proposed in [137] is relatively doubtful. The authors of [137] assume that the grain boundaries are in the amorphous state and form a paramagnetic shell around the grains 324

Effect of Grain Size and Interfaces on Properties of Bulk Nanomaterials

of nickel. This shell isolates the grains from each other. However, the results of a large number of experiments show that the interfaces even in materials with considerably smaller grains retain the crystalline structure (see Section 6). In addition to this, amorphous nickel is in the paramagnetic state only at T > 530 K and can be hardly an efficient magnetic insulator at a mean grain boundary width of 3 nm. The superparamagnetic behaviour of Ni may be observed on particles smaller than 10–15 nm [138] (see also Section 5.4). In submicrocrystalline nickel with a mean grain size greater than 100 nm, the fraction of grains with D < 10 nm is negligible. It is shown in [135] that magnetisation of nanocrystalline nickel, containing impurity oxygen, increases after annealing. The comparison of the results [135] with the data [137] on a variation of the magnetic susceptibility of submicrocrystalline nickel after several consecutive annealing cycles indicate that the effects, observed in [137], are associated with the presence of oxygen impurity in submicrocrystalline nickel. Contamination of nickel with oxygen may have taken place during plastic deformation, which was carried out in air. Investigation of the temperature dependence of coercive force H c of Ni–Cu alloys [139] showed that severe plastic deformation has no effect on the Curie temperature of nickel, whereas the coercive force H c of submicrocrystalline Ni–Cu alloy and Ni is several times higher than H c of the starting alloy and coarsegrained nickel. Investigations of the microstructure and magnetic hysteresis of submicrocrystalline Ni and Co [140, 141] corroborate that the coercive force of the plastically deformed ferromagnetics is several times higher than that of starting metals. It is shown in [140] that the annealing of submicrocrystalline nickel at T ≤ 470 K decreases the coercive force with the grain size remaining almost unchanged. Annealing at higher temperature is accompanied by a simultaneous decrease of H c and an increase in the grain size. This shows that the high coercive force of the submicrocrystalline metals and alloys is due equally to the stressed non-equilibrium state of the interfaces, on the one hand, and to the small grain size, on the other hand. Relaxation of the interfaces as a result of annealing and grain growth decreases the value of H c . In [142] the structure of the interfaces in nc-Fe (D = 10-15 nm) was studied by the methods of the magnetic after-effect and magnetic saturation. The magnetic after-effect is the time dependence of the magnetic susceptibility after demagnetising. 325

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Annealing of nc-Fe at T = 350–500 K resulted in irreversible changes in the magnetic after-effect spectrum; at the same time, the time dependence of the magnetic after-effect was observed. According to [142], similar changes are caused by the redistribution of the atoms and redistribution is associated with a decrease in free volumes at the interfaces. After annealing nc-Fe at 600 K, the magnetic moment per single iron atom at a temperature of 5 K, increased from 2.0µ B to 2.2µ B , i.e. to the value corresponding to coarse-grained α -Fe. This means that in nc-Fe the local distribution of the atoms at the interfaces slightly differs from that in coarsegrained iron. In the last decade, special interest was attracted by ferromagnetic amorphous alloys (metallic glasses) based on Fe with additions of Nb, Cu, Si, B and based on Co or Fe–Co with additions of Si and B, and also by alloys of the systems Fe–M–C, Co–M– C, Ni–M–C (M = Zr, Hf, Nb, Ta). Crystallisation of these amorphous materials yields nanocrystalline alloys with a grain size of 8 to 20 nm. Alloys obtained characterise by unique magnetic properties. Changes in the structure and properties of these and other metallic alloys, associated with transition from the amorphous state to the nanocrystalline state, are discussed in particular in [143]. Crystallisation of the amorphous alloys takes place at a low mobility of the atoms, which favours to a greater extent the formation of crystallites than their growth, i.e. it supports the formation of the nanocrystalline structure. The nanocrystalline alloys of the Fe–Cu–Nb–Si–B system (for example, Fe 73.5 Cu 1 Nb 3 Si 13.5 B 9 ), which are called FINEMET, are best known. These alloys are soft magnetic materials with a very low coercive force, comparable with H c of amorphous alloys based on cobalt, and by high magnetic saturation close to that in Fe-based amorphous alloys [24, 144]. The development of the nanostructure in the amorphous alloy assumes a combination of a high rate of formation of crystallisation centres and a low rate of growth of such centres. In FeCuNbSiB alloys, the presence of copper in Fe–Cu–Nb–Si–B alloys increases the number of crystallisation centres and favours their uniform distribution over the volume, Nb inhibits the growth of grains, and Si supports the formation of the bcc phase α –Fe(Si). Annealing of the amorphous alloy at 740–820 K leads to the precipitation of crystallites of an ordered solid solution α -Fe(Si). The crystallites of α -Fe(Si) have a size of 10–15 nm, contain up to 13–19 at.% of Si and are separated by a thin layer of the amorphous phase (Fig. 326

Effect of Grain Size and Interfaces on Properties of Bulk Nanomaterials

7.19) [144, 145]. The presence of copper decreases the activation energy of crystallisation and facilitates the nucleation of the bcc phase α -Fe(Si) [146]. Crystallisation at a higher annealing temperature leads to the formation of boride phases. Precipitation of one or another phase depends on the relation between the annealing temperature and annealing time: the amount of the bcc phase increases and that of the amorphous phase decreases as the annealing temperature and annealing time increase. The highest magnetic permeability µ p and largest magnetic saturation was found for alloys with a high content of the bcc phase. These alloys are prepared by annealing at 780–820 K for 1 hour. In conventional ferromagnetic alloys, the grain growth leads to a decrease in coercive force. According to [144], the coercive force for the nanocrystalline alloys of the systems Fe–Cu–M–Si–B (M = Nb, Ta, W, Mo, Zr, V) is proportional to the square of the grain size, i.e. H c ~ D 2 . The Fe 73.5 Cu 1 Nb 3 Si 13.5 B 9 alloy with a mean grain size of ~10 nm has a very low coercive force H c ≈ 0.5 A m –1 . The high sensitivity of magnetic permeability, coercive force, saturation magnetisation, magnetostriction and other magnetic properties to the alloy microstructure was the reason for intensive investigations of the conditions of crystallisation of amorphous alloys [144–156]. In [25, 157] it is shown that preliminary deformation (~6%) of a ribbon of Fe 73.5 Cu 1 Nb 3 Si 13.5 B 9 amorphous alloy by rolling with subsequent annealing for 1 hour at 813–820 K leads to an additional decrease of the grain size from 8–10 to 4–6 nm. Low-temperature annealing of the alloy at 723 K for 1 hour and subsequent shortterm annealing for 10 seconds at 923 K resulted in the formation of a nanocrystalline structure with a grain size of 4–5 nm. The phase composition of the alloys, produced by these methods, was the same as after conventional crystallisation at 810–820 K. Other soft magnetic nanocrystalline alloys, produced by crystallisation from the amorphous state, are also available. Alloys Fe–M–F, Fe–M–B, Fe–M–N and Fe–M–O (M = Zr, Hf, Nb, Ta, Ti) (Fig. 7.19) with a mean grain size of 10 nm have a saturation magnetisation of 1.5–1.7 T, permeability µ p = 4000–5000 and low (< 10 –6) magnetostrictions [144, 158–160]. Investigation of Fe–M–B (M = Zr, Hf, Nb) nanocrystalline alloys showed [161] that their magnetic properties can be improved by an increase of the heating rate to temperature at which crystallisation annealing is carried out. For example, the magnetic permeability µ p of Fe 90Nb 7 B 3 alloy, annealed for crystallisation at 923 K for 1 hour, was 2400 and 29000 at a heating rate of 0.008 and 3.3 K s –1 , 327

Nanocrystalline Materials

Fig. 7.19. Schematic image of microstructure of typical nanocrystalline soft magnetic alloys produced by crystallisation from the amorphous phase [144].

respectively; the coercive force of this alloy at the same heating rates was 20 and 5 A m –1 , respectively. The high heating rate resulted in a narrow size distribution of the grains of the precipitated highly dispersed bcc phase and the decrease of the mean grain size: at a heating rate of 0.042 K s –1 the mean grain size of the bcc phase was 19.8 nm, and at a heating rate of 3.3 K s –1 it was already 13.3 nm. The coercive force H c of nanocrystalline alloys Fe 78–93 M 5–11 B 2–11 (M = Zr, Hf, Nb) with the grain size D < 35 nm is proportional to the D 5/2 and rapidly increases with increasing grain size; at 35 ≤ D ≤ 100 nm coercive force H c ≈ 1200–1300 A m –1 and is independent of the grain size. Small amounts of alloying additions to Fe–M–B alloys leads to an additional improvement of the magnetic properties [161]. For example, the magnetic permeabiliy of Fe 84 Nb 7 B 9 alloy, annealed at 923 K for 1 hour at a heating rate of 3.3 K s –1 , was ~34000; the addition of a small amount of gallium to this alloy and crystallisation in the same conditions resulted in the formation of Fe 83 Nb 7 B 9 Ga 1 nanocrystalline alloy with a magnetic permeability of 38000 at a frequency of 1 kHz. The soft magnetic properties of alloys of Fe–Cu–Nb–Si–B and Fe– M–B systems (where M = Zr, Hf, Nb or Ta) produced by rapid quenching of ribbons, are unstable at high temperatures. At the same time, in recording magnetic heads it is necessary to use thin-film soft magnetic materials with thermal stability sufficient to retain their properties during high temperature joining with the substrate. These requirements are satisfied by nanostructured composite materials of the Fe–M–C, Co–M–C and Ni–M–C systems (M is the transition metal of group IV or V) [143, 159, 162]. Films of amorphous alloys 328

Effect of Grain Size and Interfaces on Properties of Bulk Nanomaterials

are deposited by spraying and then crystallised at ~700 K. The obtained nanocrystalline metallic matrix consists of Fe grains with a size of ~10 nm and nanoparticles of MC carbide with a size of ~1–4 nm are distributed in matrix (mainly in triple points) [163]. Most attention has been given to the alloys of the Fe–Ta–C system, which have high thermal stability and retain the nanocrystalline structure up to 1000 K. For comparison, it should be mentioned that in nanocrystalline nickel, which does not contain carbide, grain growth already starts at ~350 K [164]. The high thermal stability of the nanostructure is caused by the pinning of the grain boundaries of α Fe by nanoparticles of tantalum carbide TaC. In crystallisation of Fe– Ta–C alloys crystallites of α -Fe are form in the first stage; during growth of these crystallites, tantalum and carbon are precipitated into the amorphous phase and form (at the equiatomic ratio) dispersed nanosized precipitates of stoichiometric carbide TaC. If there is an excess of tantalum or carbon, other compounds may form. According to [165], the best soft-magnetic properties were obtained for Fe 81.4 Ta 8.3 C 10.3 alloy. Crystallisation of the amorphous alloys makes it possible to produce not only soft-magnetic nanomaterials but also hardmagnetic nanocrystalline materials with high coercive force. In [157] it is shown that annealing of amorphous soft-magnetic (H c ≤ 40 A m –1 ) alloys Fe 81 Si 7 B 12 and Fe 60 Cr 18 Ni 7 Si x B 15–x (x = 3 or 5) for 1 hour at 823 K increases H c by a factor of 125 to 700. Crystallisation for 1 hour at 873 K or rapid crystallisation for 10 seconds at 923 K of the amorphous soft-magnetic alloy Fe 5 Co 70 Si 15 B 10 with H c < 1 A m –1 makes it possible to produce nanocrystalline alloys with a mean grain size of 50–200 nm and H c = 3200 A m –1 , or with a mean grain size of 15–50 nm and H c = 8800 A m –1 , respectively [23, 166]. These nanocrystalline alloys have a high residual magnetization. Increase in the coercive force is associated with the precipitation of highly dispersed crystalline phases; cubic α -Co phase has the highest coercive force. The high-coercivity state of the alloy, produced by rapid crystallisation, is thermally stable and remains unchanged after annealing at 673 K. According to [23, 166] the increase of the coercive force of a rapid crystallised alloy in comparison with that of a slowly crystallised alloy is a consequence of the precipitation of anisotropic single-domain bcc particles of α -Fe with high saturation magnetisation, on the one hand, and of a decrease in the grain size of α -Co, on the other hand. If the grain size is smaller than the width of the magnetic domain wall, the increase in the 329

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8. Conclusions Three groups of technologies have ensured the scientific and technical advances in the first half of the XXI century: electronics and computer technology, biotechnology and nanotechnology. It is believed that the development of electronics and computer technology will reach the maximum activity in the years 2010–2015, and the contribution of biotechnology, which started in 1968–1973, will be largest in the period from 2025 to 2035, whereas nanotechnology will become the main driving force of scientific and technical advances in 2045–2055. The essence of nanotechnology is the ability to work at the atomic, molecular and supramolecular levels, in the length scale of about 1–100 nm range, in order to create, manipulate and use materials, devices and systems that have novel properties and functions because of the small scale of their structures [1]. In some situations, the length scale under which the novel phenomena and properties develop may be less than 1 nm or larger than 100 nm. Control of matter on the nanoscale means tailoring the fundamental structure, properties, processes and functions exactly on the scale where the basic properties of solids are defined. Nanotechnology includes integration of nanoscale structures into larger material components, systems and architectures that could be used in most industries, healthcare systems and environment. At present, the development of nanotechnology and the production and application of nanomaterials have not yet reached a maximum. Prior to 1990, there were no specialised scientific journals on nanomaterials and nanotechnology. At present, approximately 20 international scientific journals are concerned exclusively with nanomaterials and nanotechnology. Among them are such journals as Fullerene Science and Technology, International Journal of Nanoscience, Journal of Micromechanics and Microengineering, 340

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Number of publications in SCI

Journal of Nanoparticles Research, Journal of Nanoscience and Technology, Journal of Metastable and Nanocrystalline Materials, Journal of Physical Chemistry B, NanoLetters, Nanostructured Materials, Nanotechnology, Physica E: Low-dimensional Systems and Nanostructures. In addition to these, articles on nanomaterials and nanotechnology are being published by all materials science journals, journals for condensed matter physics and for colloid chemistry. In addition to the number of specialised scientific and technical journals, two obvious indicators of the interest in nanomaterials and nanotechnology are the number of published scientific studies and the number of patents. The first is an efficient indicator of scientific activity, and the latter indicates the possibility of application of scientific results. Figure 8.1 shows the change in the number of publications and patents on nanomaterials and nanotechnology between 1980 and 1998 [2]. The data on the worldwide number of publications have been extracted from the database of the Science Citation Index (SCI), and the nanopatents are those recorded by the European Patent Office (EPO) in Munich. As indicated by Fig. 8.1, in the period from 1981 to 1985 the number of articles was small but gradually increased from year to year. A large increase in the number of publications was recorded in 1986, and in the period from 1989 to 1998 the number of studies published

Fig. 8.1. Annual number of publications and patents in nanomaterials and nanotechnology from 1981 to 1998 world-wide [2]. The number of publications includes all nanotechnology-related articles published world-wide and covered by the Science Citation Index (SCI) database. The nanopatents are those filed at the European Patent Office (EPO).

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every year increased from 1000 to 12000. A large volume of the cumulated experimental data on nanomaterials and nanotechnology is used as a reliable base for generalisation of the fundamental and applied results in review articles and monographs (see, for example [3–11]). A clear confirmation of the interest in nanomaterials and nanotechnology is also the dynamics of growth of financial investment into this complex branch of scientific and applied developments. In the last 15–20 years of the 20 th Century, 340 million dollars was spent throughout the world in the development of new processes and materials associated with the ‘nano’ concept. In addition to this, as a result of the development of nanomaterials and nanotechnology, additional investment was made into electronics, chemical pharmaceutics, catalysis, aerospace industry and toolmaking worth 300, 180, 100, 70 and 2 million dollars respectively. Thus, the total expenditure on scientific and applied investigations into nanomaterials already at the end of the twentieth century exceeded 1 billion dollars. In the USA in 2000, 2001, 2002 and 2003, government investment in scientific research programmes into nanomaterials and nanotechnologies was 270, 424, 697 and 774 million dollars, and the investment by the venture innovative capital reached 500, 800, 1000 and 1200 million dollars, respectively. In the USA in 2004, government investment in scientific investigations into nanomaterials should reach 847 million dollars. National programmes on nanomaterials in other countries are highly extensive: in 2002 in Japan, the appropriate expenditure was 800 million dollars, in the European Union countries 300 million dollars, and in the rest of the world 400 million dollars. It should be mentioned that there has been a gradual displacement of scientific and applied interest from the development of nanomaterials to the construction of devices and systems utilising the nanocrystalline state effects. The range of investigations into biological nanomaterials and bionanotechnology is being expanded continuously. The broader perspective of qualitative improvements, which nanotechnology will bring in the society, cannot be underestimated. Nanoscale science and engineering promise restructuring almost all industries towards the next industrial revolution, reshaping our intellectual comprehension of nature, and assuring the quality of life. Nanotechnology is seen as an emerging technology of the present century because of the importance of the control of matter at nanoscale on almost all technologies from information and medicine to manufacturing and environment. 342

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The book which you have just read is concerned mainly with nanocrystalline solids. New investigations in the last decade have greatly expanded the knowledge of the effects associated with the size of grains (crystallites) in solids. For a long time, main attention has been concentrated on examination of small particles and nanoclusters, which properties are intermediate between the properties of isolated atoms and the polycrystalline solids. A development of the methods of producing bulk materials with an unusually fine-grained structure, in which the grains are of the nanometer sizes, has enabled transfer to examining the structure and properties of the solid in the nanocrystalline state. At present, the main methods of producing bulk nanocrystalline materials are [6, 7, 11]: compacting of isolated nanoclusters, produced by evaporation and condensation; deposition from colloid solutions or decomposition of precursors; crystallisation of amorphous alloys; severe plastic deformation; ordering of strongly non-stoichiometric compounds and solid solutions [7, 12–28]. Each of these methods has advantages and disadvantages and none of them is universal because they can be used most efficiently for a certain range of substances. Analysis of the currently available experimental results shows that an important role in the nanocrystalline solid is played not only by the grain size (as in case of the isolated nanoparticles) but also by the structure of the interfaces (grain boundaries). In fact, grain boundaries in bulk nanomaterials, produced by different methods, is characterised by large differences. For example, in nanomaterials produced by severe deformation, the grain boundaries are characterised by a high dislocation density, and in nanomaterials produced by crystallisation from the amorphous state, the grain boundaries may be quasi-amorphous or may have a greatly distorted crystalline structure. All these special features must be taken into account in the interpretation of the properties of bulk nanomaterials. The effect of the interface on the structure and properties is especially strong in nanomaterials produced by compacting or severe plastic deformation. In these materials, immediately after production, the interfaces are in the non-equilibrium stressed state and have excess energy. The relaxation of non-equilibrium interfaces in nanocrystalline metals and alloys takes place spontaneously even at room temperature and in most cases is accompanied by grain growth [22, 29]. Ceramic oxide nanomaterials are more stable than metallic ones; their structure and grain size may remain almost constant even after annealing at 600–800 K [30]. 343

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The properties of nanocrystalline metals and alloys, especially those produced by compacting nanoclusters, are very sensitive to the oxygen impurity. The unusually large area of the interfaces results in high chemical activity of nanocrystalline metals. A large part of the unexpected results, obtained in the period up to 1992, after subsequent examination proved to be the consequence of contamination of nanocrystalline metals by impurity oxygen; in the case of nanocrystalline Pd, also contamination by hydrogen is possible. Of special importance for theoretical comprehension of the experimental results, obtained on isolated nanoparticles and bulk nanocrystalline materials, is the separation of the surface, grainboundary (associated with the interfaces) and volume (associated with the size of particles, crystallites, or grains) effects. At present, this problem is far from being solved. On the whole, the understanding and explanation of the structure and properties of isolated nanoparticles is considerably higher in comparison with bulk nanocrystalline materials. This is associated with the fact that dispersed systems and nanoclusters have already been studied since the beginning of the XX th century, whereas bulk nanomaterials became a subject of investigation only after 1982. Undoubtedly, the active investigations of bulk nanocrystalline materials will make it possible to eliminate the existing delay in the next 10–15 years. Analysing the state of material science investigations, it is possible to define four stages of the “life” of materials: proposal of a concept, intensive investigations, increase in the production volume and decrease in the production volume. Evidently, bulk nanocrystalline materials are in the stage of intensive investigations. It may be expected that in the near future the intensity of studying the nanocrystalline materials will increase. The most important directions will evidently be the deep examination of the microstructure, separation of effects determined by the particle size and the interfaces, determination of the conditions of stabilisation of the nanostructure ensuring the retention of the properties at elevated temperatures, development of models for adequate theoretical description of the nanocrystalline state. The new stable nanocrystalline materials will be developed on the basis of multi-component systems and not only on the basis of metals. Compounds of metals with oxygen, nitrogen and carbon, having a high melting point and high thermal stability, will become the main components of nanocrystalline materials in the near future. Using of these compounds will make it possible to develop nanomaterials 344

Conclusions

characterised by stable operation and no changes in their properties throughout the entire servicelife. In particular, the oxides, nitrides and carbides of metals are expected to manifest their best properties in the world of nanomaterials. Detailed investigations of nanocrystalline materials will result in the development of such new sciences as physics and chemistry of nanocrystalline solids so that it will be possible to construct a strong bridge between nanomaterials and nanotechnology. References 1. 2.

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