Phase Transformations Examples from Titanium and Zirconium Alloys
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Phase Transformations Examples from Titanium and Zirconium Alloys
S. Banerjee and P. Mukhopadhyay Bhabha Atomic Research Centre, Mumbai, India
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Pergamon is an imprint of Elsevier Linacre House, Jordan Hill, Oxford OX2 8DP, UK Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands First edition 2007 Copyright © 2007 Elsevier Ltd. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email:
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Contents Foreword Preface Acknowledgements
xvii xix xxi
CHAPTER 1 Phases and Crystal Structures Symbols and Abbreviations 1.1 Introduction 1.2 Polymorphism 1.3 Phase Diagrams of Elemental Titanium and Zirconium 1.3.1 Introductory remarks 1.3.2 Titanium 1.3.3 Zirconium 1.3.4 Epilogue 1.3.5 Phase stability and electronic structure 1.3.6 Some features of transition metals 1.4 Effect of Alloying 1.4.1 Introductory remarks 1.4.2 Alloy classification 1.4.3 Titanium alloys 1.4.4 Zirconium alloys 1.4.5 Stability of titanium and zirconium alloys 1.5 Binary Phase Diagrams 1.5.1 Introductory remarks 1.5.2 Ti–X systems 1.5.3 Zr–X systems 1.5.4 Representative examples of Ti–X and Zr–X phase diagrams 1.6 Non-Equilibrium Phases 1.6.1 Introductory remarks 1.6.2 Martensite phase 1.6.2.1 Crystallography 1.6.2.2 Transformation temperatures 1.6.2.3 Morphology and substructure vii
3 3 4 4 7 7 9 10 11 13 18 21 21 21 21 23 24 26 26 27 29 29 43 43 44 44 47 48
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1.6.3 Omega Phase 1.6.3.1 Athermal and isothermal 1.6.3.2 Crystallography 1.6.3.3 Morphology 1.6.3.4 Diffraction effects 1.6.4 Phase separation in -phase 1.7 Intermetallic Phases 1.7.1 Introductory remarks 1.7.2 Intermetallic phase structures: atomic layer stacking 1.7.3 Derivation of intermetallic phase structures from simple structures 1.7.4 Intermetallic phases with TCP structures in Ti–X and Zr–X systems 1.7.5 Phase stability in zirconia-based systems 1.7.5.1 ZrO2 polymorphs 1.7.5.2 Stabilization of high temperature polymorphs 1.7.5.3 ZrO2 –MgO system 1.7.5.4 ZrO2 –CaO system 1.7.5.5 ZrO2 –Y2 O3 system References Appendix CHAPTER 2 Classification of Phase Transformations Symbols and Abbreviations 2.1 Introduction 2.2 Basic Definitions 2.3 Classification Schemes 2.3.1 Classification based on thermodynamics 2.3.2 Classifications based on mechanisms 2.3.3 Classification based on kinetics 2.4 Syncretist Classification 2.5 Mixed Mode Transformations 2.5.1 Clustering and ordering 2.5.2 First-order and second-order ordering 2.5.3 Displacive and diffusional transformations 2.5.4 Kinetic coupling of diffusional and displacive transformations References
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89 89 89 90 92 93 101 105 105 115 115 116 120 120 122
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CHAPTER 3 Solidification, Vitrification, Crystallization and Formation of Quasicrystalline and Nanocrystalline Structures List of Symbols 3.1 Introduction 3.2 Solidification 3.2.1 Thermodynamics of solidification 3.2.2 Morphological stability of the liquid/solid interface 3.2.3 Post-solidification transformations 3.2.4 Macrosegregation and microsegregation in castings 3.2.5 Microstructure of weldments of Ti- and Zr-based alloys 3.3 Rapidly Solidified Crystalline Products 3.3.1 Extension of solid solubility 3.3.2 Dispersoid formation in rapidly solidified Ti alloys 3.3.3 Transformations in the solid state 3.4 Amorphous Metallic Alloys 3.4.1 Glass formation 3.4.2 Thermodynamic considerations 3.4.3 Kinetic considerations 3.4.4 Microstructures of partially crystalline alloys 3.4.5 Diffusion 3.4.6 Structural relaxation 3.4.7 Glass transition 3.5 Crystallization 3.5.1 Modes of crystallization 3.5.2 Crystallization in metal–metal glasses 3.5.3 Kinetics of crystallization 3.5.4 Crystallization kinetics in Zr 76 Fe1−x Nix 24 glasses 3.6 Bulk Metallic Glasses 3.7 Solid State Amorphization 3.7.1 Thermodynamics and kinetics 3.7.2 Amorphous phase formation by composition-induced destabilization of crystalline phases 3.7.3 Glass formation in diffusion couples 3.7.4 Amorphization by hydrogen charging 3.7.5 Glass formation in mechanically driven systems 3.7.6 Radiation-induced amorphization 3.8 Phase Stability in Thin Film Multilayers
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3.9
Quasicrystalline Structures and Related Rational Approximants 3.9.1 Icosahedral phases in Ti- and Zr-based systems References CHAPTER 4 Martensitic Transformations Symbols and Abbreviations 4.1 Introduction 4.2 General Features of Martensitic Transformations 4.2.1 Thermodynamics 4.2.2 Crystallography 4.2.3 Kinetics 4.2.4 Summary 4.3 BCC to Orthohexagonal Martensitic Transformation in Alloys Based on Ti and Zr 4.3.1 Phase diagrams and Ms temperatures 4.3.2 Lattice correspondence 4.3.3 Crystallographic analysis 4.3.3.1 Morphology and substructure 4.3.3.2 Transition in morphology and substructure 4.3.4 Stress-assisted and strain-induced martensitic transformation 4.4 Strengthening due to Martensitic Transformation 4.4.1 Microscopic interactions 4.4.1.1 Lath boundaries 4.4.1.2 Twin boundaries and plate boundaries 4.4.2 Macroscopic flow behaviour 4.5 Martensitic Transformation in Ti–Ni Shape Memory Alloys 4.5.1 Transformation sequences 4.5.2 Crystallography of the B2 → R transformation 4.5.3 Crystallography of the B2 → B19 transformation 4.5.4 Crystallography of the B2 → B19 transformation 4.5.5 Self-accommodating morphology of Ni–Ti martensite plates 4.5.6 Shape memory effect 4.5.7 Reversion stress in a shape memory alloy 4.5.8 Thermal arrest memory effect 4.6 Tetragonal Monoclinic Transformation in Zirconia 4.6.1 Transformation characteristics 4.6.2 Orientation relation and lattice correspondence 4.6.3 Habit plane
241 248 252
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4.7
Transformation Toughening of Partially Stabilized Zirconia (PSZ) 4.7.1 Crystallography of tetragonal → monoclinic transformation in small particles References CHAPTER 5 Ordering in Intermetallics List of Symbols 5.1 Introduction 5.2 Theoretical Treatments 5.2.1 Alloy phase stability 5.2.2 Order–disorder transformations 5.2.2.1 Historical developments 5.2.2.2 Static concentration wave model 5.2.2.3 Cluster variation method 5.2.3 The ground states of the Lenz and Ising model 5.2.4 Special point ordering 5.2.4.1 BCC special points 5.2.4.2 HCP special points 5.2.4.3 FCC special points 5.2.5 Concomitant clustering and ordering 5.2.6 A case study: Ti–Al system 5.3 Transformations in Ti3 Al-based alloys 5.3.1 → D019 ordering 5.3.2 Phase transformations in 2 -Ti3 Al-based systems 5.3.3 Structural relationships 5.3.4 Group/subgroup relations between BCC (Im3m), HCP (P63 /mmc) and ordered orthorhombic (Cmcm) phases 5.3.5 Transformation sequences 5.3.5.1 Transformation sequence in the alloy Ti–25 at.% Al–12.5 at.% Nb 5.3.5.2 Transformation sequence in the alloys Ti–25 at.% Al–25 at.% Nb, Ti–28 at.% Al–22 at.% Nb and Ti–24 at.% Al–15 at.% Nb 5.3.6 Phase reactions in Ti–Al–Nb system 5.4 Formation of Zr3 Al 5.4.1 Metastable Zr3 Al (D019 ) phase 5.4.2 Formation of the equilibrium Zr3 Al (L12 ) phase 5.4.3 +Zr2 Al → Zr3 Al peritectoid reaction
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Phase Transformation in -TiAl-Based Systems 5.5.1 Structural relationship between 2 - and -phases 5.5.2 Phase reactions 5.5.2.1 Ti-34-37 at.% Al; → → 2 5.5.2.2 Ti-38-40 at.% Al; → 2 → 2 + 5.5.2.3 Ti-41-48 at.% Al; → + → 2 + 5.5.2.4 Ti-49-50 at.% Al; → 5.5.2.5 Ti49-50 at.% Al; → 5.5.3 Transformation mechanisms 5.5.3.1 Formation of the 2 + lamellar microstructure 5.5.3.2 Mechanism of the → massive transformation 5.5.3.3 Discontinuous coarsening of the lamellar 2 + microstructure 5.6 Site Occupancies in Ordered Ternary Alloys 5.6.1 Ordering tie lines 5.6.2 Kinetic modelling of B2 ordering in a ternary system 5.6.3 Influence of binary interaction parameters 5.6.4 B2 ordering in the Nb–Ti–Al system References CHAPTER 6 Transformations Related To Omega Structures List of Symbols 6.1 Introduction 6.2 Occurrence of the -Phase 6.2.1 Thermally induced formation of the -phase 6.2.2 Formation of equilibrium -phase under high pressures 6.2.3 Combined effect of alloying elements and pressure in inducing -transition 6.3 Crystallography 6.3.1 The structure of the -phase 6.3.2 The – lattice correspondence 6.3.3 The – lattice correspondence 6.4 Kinetics of the → Transformation 6.4.1 Athermal → transition 6.4.2 Thermally activated precipitation of the -phase 6.4.3 Pressure-induced → transformation 6.5 Diffuse Scattering
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6.6
Mechanisms of -Transformations 6.6.1 Lattice collapse mechanism for the → transformation 6.6.2 Formation of the plate-shaped induced by shock pressure in -alloys 6.6.3 Calculated total energy as a function of displacement 6.6.4 Incommensurate -structures 6.6.5 Stability of -phase and d-band occupancy 6.7 Ordered -Structures 6.7.1 Structural descriptions 6.7.2 Transformation sequences in Zr base alloys 6.7.3 Transformation sequences in Ti base alloys 6.7.4 Ordered -structures in other systems 6.7.5 Symmetry tree 6.8 Influence of -Phase on Mechanical Properties 6.8.1 Hardening and embrittlement due to -phase 6.8.2 Dynamic strain ageing due to -precipitation References CHAPTER 7 Diffusional Transformations List of Symbols 7.1 Introduction 7.2 Diffusion 7.2.1 Diffusion mechanisms 7.2.2 Flux equations: Fick’s laws 7.2.3 Self- and tracer-diffusion coefficients in -Zr and -Ti 7.2.4 Self- and tracer-diffusion coefficients in -Zr and -Ti 7.2.5 Interdiffusion 7.2.6 Phase formation in chemical diffusion 7.2.6.1 Phase nucleation 7.2.6.2 Phase growth 7.2.7 Diffusion bonding 7.3 Phase Separation 7.3.1 Phase separation mechanisms 7.3.2 Analysis of a phase diagram showing a miscibility gap 7.3.3 Microstructural evolution during phase separation in the -phase 7.3.4 Monotectoid reaction – a consequence of -phase immiscibility
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7.3.5 Precipitation of -phase in supersaturated -phase during tempering of martensite 7.3.6 Decomposition of orthorhombic -martensite during tempering 7.3.7 Phase separation in -phase as precursor to precipitation of - and -phases 7.4 Massive Transformations 7.4.1 Thermodynamics of massive transformations 7.4.2 Massive transformations in Ti alloys 7.5 Precipitation of -Phase in -Matrix 7.5.1 Morphology 7.5.2 Orientation relation 7.5.3 Invariant line strain condition 7.5.4 Interfacial structure and growth mechanisms 7.5.5 Morphological evolution in mesoscale 7.6 Precipitation of Intermetallic Phases 7.6.1 Precipitation of intermetallic compounds from dilute solid solutions 7.6.2 Precipitation in ordered intermetallics: transformation of 2 -phase to O-phase 7.7 Eutectoid Decomposition 7.7.1 Active eutectoid systems 7.7.2 Active eutectoid decomposition in Zr–Cu and Zr–Fe system 7.8 Microstructural Evolution During Thermo-Mechanical Processing of Ti- and Zr-based Alloys 7.8.1 Identification of hot deformation mechanisms through processing maps 7.8.2 Development of microstructure during hot working of Ti alloys 7.8.2.1 -alloys 7.8.2.2 + alloys 7.8.2.3 -alloys 7.8.2.4 Ti-aluminides 7.8.3 Hot working of Zr alloys 7.8.3.1 and near--Zr alloys 7.8.3.2 + alloys 7.8.3.3 -alloys 7.8.4 Development of texture during cold working of Zr alloys 7.8.5 Evolution of microstructure during fabrication of Zr–2.5 wt% Nb alloy tubes References
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CHAPTER 8 Interstitial Ordering List of Symbols 8.1 Introduction 8.2 Hydrogen In Metals 8.2.1 Ti–H and Zr–H phase diagrams 8.2.2 Terminal solid solubility 8.3 Crystallography and Mechanism of Hydride Formation 8.3.1 Formation of -hydride in the - and -phases 8.3.2 Lattice correspondence of -, - and -phases 8.3.3 Crystallography of → transformation 8.3.4 Crystallography of → transformation 8.3.5 Mechanism of the formation of -hydrides 8.3.6 Hydride precipitation in the / interface 8.3.7 Formation of -hydride 8.4 Hydrogen-Related Degradation Processes 8.4.1 Uniform hydride precipitation 8.4.2 Hydrogen Migration 8.4.3 Stress reorientation of hydride precipitates 8.4.4 Delayed hydride cracking 8.4.5 Formation of hydride blisters 8.5 Thermochemical Processing of Ti Alloys by Temporary Alloying With Hydrogen 8.6 Hydrogen Storage In Intermetallic Phases 8.6.1 Laves phase compounds 8.6.2 Thermodynamics 8.6.3 Ti- and Zr-based hydrogen storage materials 8.6.3.1 Ti-based hydrogen storage materials 8.6.3.2 Zr-based hydrogen storage materials 8.6.4 Applications 8.7 Oxygen Ordering In -Alloys 8.7.1 Interstitial ordering of oxygen in Ti–O and Zr–O 8.7.2 Oxidation kinetics and mechanism 8.8 Phase transformations in Ti-rich end of the Ti–N system References
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CHAPTER 9 Epilogue References
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Index
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Foreword The present volume looks at phase transformations essentially from a physical metallurgist’s view point, in consonance with the background and the research experience of the authors, and has some distinguishing features. Some, though not all, of these are enumerated in the following. Almost all types of phase transformations and reactions that are commonly encountered in inorganic materials, such as alloys, intermetallics and ceramics, have been covered and the underlying thermodynamic, kinetic and crystallographic aspects elucidated. It has generally been customary in metallurgical literature to draw examples from iron-based alloys for describing the characteristic features of different types of transformations, in view of the wide variety of transformations occurring in these alloys. The authors of this monograph have cited examples of all the phase transformations and reactions discussed from titanium- and zirconium-based systems and have successfully demonstrated that these alloys, intermetallics and ceramics exhibit an even wider range of phase changes as compared to ferrous systems and that the simpler crystallography involved renders them more suitable for developing a basic understanding of the transformations. Phase transformations are brought about due to changes in external constraints which include thermodynamic variables such as temperature and pressure. Till recently, the emphasis in metallurgical literature has been on the delineation of temperature-induced transformations. In this book, transformations driven by pressure changes, radiation and deformation and those occurring in nanoscale multilayers have also been brought to the fore, while accepting the pre-eminent position occupied by the temperature-induced ones. Order–disorder transformations, many of which constitute very good examples of continuous transformations, have been dealt with in a comprehensive manner. It has been demonstrated that first principles calculations of phase stability can yield meaningful results, consistent with experimental observations. Displacive transformations, both shear dominated (martensite, shock pressure induced omega) and shuffle dominated (omega), have been covered in a cogent manner. Some crystallographic bcc to hcp transformations, which occur by diffusional as well as by displacive modes, have been identified, compared and contrasted, in terms of the experimentally observable features which characterize them. The authors, who have a lifetime of experience in investigating phase transformations in zirconium and titanium alloys, have handled an ambitious project xvii
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by trying to bring diverse topics under the same cover. And they have certainly not failed in their endeavor. One could always point out that non-metallic systems have not been adequately represented in their treatment. However, in quite a few instances, they have compared phase transformations occurring in alloys, intermetallics and ceramics and have demonstrated that the underlying principles pertaining to all these systems are basically the same. The multidisciplinary and interdisciplinary interest in the area of phase changes have engendered a variety of approaches with regard to the study of phase transformations, each exhibiting some distinctive features. Physicists are interested primarily in the motivation or, in other words, the why of a transformation. They concern themselves mainly with higher order, continuous phase transitions occurring in simple, composition-invariant systems. Chemists, metallurgists and ceramists, by contrast, focus a major part of their attention on phase transformations (and phase reactions) involving alterations in crystal structure, chemical composition and state of order. Of great concern to metallurgists are the mechanisms, or the how, of such transformations. Phase changes of interest to geologists are similar to those encountered in metallic and ceramic systems but generally take place over much more extended temporal and spatial scales under extreme conditions of temperature and pressure. The present volume will be useful to students, research workers and professionals belonging to all these disciplines. In my judgment, the authors of this volume have done a commendable job while addressing phase transformations and phase reactions, drawing apposite examples from titanium-and zirconium-based systems, and have been able to produce a monograph which was not there but which should very much have been there. I congratulate them on this count. C.N.R. Rao, F.R.S. Linus Pauling Research Professor, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore, India
Preface Studies on phase transformations in metallic materials form a major part of physical metallurgy. The terms phase transitions and phase transformations are often used in an interchangeable manner in metallurgical literature although it is realized that the former generally refers to transitions between two phases having the same chemical composition while the latter spans a wider range of phenomena, including phase reactions leading to compositional changes. Having made this distinction, we would like to mention at the outset that the present volume deals with phase transformations. We started our respective research careers almost four decades back by looking into some phase transformations and phase reactions occurring in zirconium alloys. As we gathered more and more experience, we realized that these alloys, together with titanium alloys, exhibit nearly all types of phase transformations encountered in inorganic materials and that in this respect these are more versatile than even ferrous alloys. Moreover, the crystallographic features associated with the phase changes are often simpler in these systems, making them more suitable for providing a basic understanding of the relevant phenomena. In earlier days, some of the important issues in the area of phase transformations in alloys, intermetallics and ceramics pertained to the following: (1) crystallographic aspects of martensitic transformations, including the role of the lattice-invariant shear, in determining martensite morphology and substructure and the strengthening contribution of the latter; (2) distinguishing features of diffusional and displacive transformations and mechanisms of hybrid transformations; (3) analysis and synthesis of phase diagrams and the prediction of the sequence of phase transformations on the basis of phase diagram analyses; (4) spinodal decomposition leading to a homogenous phase separation process and the evolution of microstructure in systems exhibiting instability in respect of concentration and/or displacement waves of short and/or long wavelengths; (5) driving force, kinetics and mechanisms relevant to displacive phase transformations and the role of strain fluctuations and their localization in the nucleation of such transformations; (6) formation of amorphous structure in metallic materials, stability of the amorphous phase and the modes of crystallization on appropriate processing; (7) the effect of factors such as pressure, deformation and radiation on phase transformations. xix
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A number of research groups all over the world, including our group, responded to the challenges thrown by these issues. The background was well set as the information and knowledge accumulated on the basis of metallography observations (mainly at light microscopy levels), X-ray diffraction results and studies on kinetics had already provided a fair understanding of the mechanisms of different types of phase transformations. Theoretical developments such as the phenomenological theory of martensite crystallography, the thermodynamical theory of spinodal decomposition and the theory of the growth kinetics of precipitates had had noteworthy success in making quantitative predictions regarding many an aspect of solid state phase transformations. That was also the time when transmission electron microscopy emerged as a powerful technique for making observations, morphological as well as crystallographic, at a much higher resolution than hitherto available, enabling physical metallurgists to resolve a number of mechanism-related problems which had been raised on the basis of theoretical and experimental investigations carried out earlier. We are happy to state that each of the issues listed above has been addressed, in some manner or the other, by our colleagues and by us over the years to enhance our understanding and appreciation of these. If one scans today’s literature on phase transformations, one will find that most of these issues, though better comprehended than before, continue to be in the limelight. However, the experimental tools now available have enormously improved our ability to study phenomena at much higher levels of spatial as well as temporal resolution. This superior experimental capability, supplemented by tremendously enhanced computing power, is providing a much better understanding of phase transformation phenomena. We do hope that the readers of this volume will get a flavour of these advancements. The book is divided into nine chapters. The first of these provides some sort of an introduction to the various types of phase changes covered later on. The second chapter delineates different schemes of classification of phase transformations in a general manner. The following six chapters deal with specific types of transformations. An attempt has been made to elucidate the basic principles pertaining to the relevant transformations, in general terms, at the beginning of each of these chapters because we have felt that this would be pedagogically advantageous for developing a clear understanding of the subject. However, we have taken care to ensure that all the illustrative examples are drawn from titaniumand zirconium-based systems. The final chapter is in the nature of an epilogue. Srikumar Banerjee Pradip Mukhopadhyay
Acknowledgements This book reflects the totality of the experience gained by us during our research career which, in the formative years, was under the guidance of R. Krishnan in Metallurgy Division, Bhabha Atomic Research Centre (BARC). Our research has been almost entirely supported by this institute (BARC), where a sustained activity on the physical metallurgy of zirconium has remained in focus for nearly four decades. It is here that we have had the benefit of interacting with M.K. Asundi, V.S. Arunachalam, P. Rodriguez, B.D. Sharma, R. Chidambaram, C.V. Sundaram and C.K. Gupta over the years. Interactions with other major centres of physical metallurgy research in the country have also been of considerable help. In this connection, we would like to especially acknowledge the fruitful discussions on many aspects of phase transformations research with P.R. Dhar of Indian Institute of Technology (IIT), Kharagpur; S. Ranganathan and K. Chattopadhyay of Indian Institute of Science (IISc), Bangalore; T.R. Anantharaman, P. Rama Rao, P. Ramachandrarao and S. Lele of Banaras Hindu University (BHU), Varanasi; D. Banerjee and K. Muraleedharan of Defence Metallurgical Research Laboratory (DMRL), Hyderabad; and V.S. Raghunathan of Indira Gandhi Centre for Atomic Research (IGCAR), Kalpakkam. One of us (P. Mukhopadhyay) was introduced to ordering reactions in titanium aluminides by P.R. Swann at Imperial College, London, while the other (S. Banerjee) has had productive collaborations with R.W. Cahn and B. Cantor at University of Sussex, Brighton; M. Wilkens and K. Urban at Institut für Physik, Max-Planck Institut für Metallforschung, Stuttgart; and H.L. Fraser, R. Banerjee and J.C. Williams at the Ohio State University, Columbus. We also have had several occasions to imbibe pertinent ideas from H.I. Aaronson of Carngie-Mellon University, Pittsburgh; J.W. Cahn and L.A. Bendersky of National Institute of Standards and Technologies (NIST), Washington, D.C.; J.W. Christian of University of Oxford and V.K. Vasudevan of University of Cincinatti. We must acknowledge our indebtedness to the authors of many of the publications which have been instrumental in nurturing our understanding of the topics covered in this book. We have been extremely fortunate in having a continuous stream of bright colleagues in the course of our professional career. They have perhaps given us much more in terms of ideas and concepts than whatever advice and guidance we have been able to offer. We take this opportunity to list the names of some of those colleagues in the approximate sequence of our coming in contact with xxi
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them: S.J. Vijayakar, G.E. Prasad, L. Kumar, V. Seetharaman, E.S.K. Menon, M. Sundararaman, V. Raman, R. Kishore, U.D. Kulkarni, J.K. Chakravartty, G.K. Dey, K. Madangopal, D. Srivastava, R. Tewari, A.K. Arya, R.V. Ramanujam, J.B. Singh. Needless to say, this list is, by no means, complete. During the preparation of the manuscript of this book we received substantial help from many of our colleagues, notably G.K. Dey, D. Srivastava, A.K. Arya, R. Tewari, A. Laik, G.B. Kale, K. Bhanumurthy, R.N. Singh, S. Ramanathan and J.K. Chakravartty. We have received sustained secretarial assistance from M. Ayyappan and P. Khattar. P.B. Khedkar and A. Agashe have been mainly responsible for preparing the illustrations. We are grateful to Elsevier Publishers for their patience and readiness to help. Above all, we are greatly indebted to Robert Cahn, whose constant encouragement and occasional reprimands have contributed considerably to the completion of this work. He passed away at a time when this volume was in the proof-setting stage. His death has indeed created a void in the physical metallurgy community that will take a long time to be filled. To us it has been an irreparable loss, professional and personal. We dedicate this book to the memory of our parents and of Prof. Robert W. Cahn. Srikumar Banerjee Pradip Mukhopadhyay
Chapter 1
Phases and Crystal Structures 1.1 Introduction 1.2 Polymorphism 1.3 Phase Diagrams of Elemental Titanium and Zirconium 1.3.1 Introductory remarks 1.3.2 Titanium 1.3.3 Zirconium 1.3.4 Epilogue 1.3.5 Phase stability and electronic structure 1.3.6 Some features of transition metals 1.4 Effect of Alloying 1.4.1 Introductory remarks 1.4.2 Alloy classification 1.4.3 Titanium alloys 1.4.4 Zirconium alloys 1.4.5 Stability of titanium and zirconium alloys 1.5 Binary Phase Diagrams 1.5.1 Introductory remarks 1.5.2 Ti–X systems 1.5.3 Zr–X systems 1.5.4 Representative examples of Ti–X and Zr–X phase diagrams 1.6 Non-Equilibrium Phases 1.6.1 Introductory remarks 1.6.2 Martensite phase 1.6.3 Omega phase 1.6.4 Phase separation in -phase 1.7 Intermetallic Phases 1.7.1 Introductory remarks 1.7.2 Intermetallic phase structures: atomic layer stacking 1.7.3 Derivation of intermetallic phase structures from simple structures 1.7.4 Intermetallic phases with TCP structures in Ti–X and Zr–X systems 1.7.5 Phase stability in zirconia-based systems References Appendix
4 4 7 7 9 10 11 13 18 21 21 21 21 23 24 26 26 27 29 29 43 43 44 49 52 53 53 55 61 62 62 67 73
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Chapter 1
Phases and Crystal Structures Symbols A Cij C Cp e/a G: H P S T V Va p s ij bcc: fcc: hcp: -phase: -phase: m Ms Mf s s To AIP:
and Abbreviations Elastic anisotropy ratio (A = C44 /C ) Elastic stiffness modulus (elastic constant) Elastic shear stiffness modulus; shear constant; (C = C11 − C12 /2) Specific heat at constant pressure Electron to atom ratio Gibbs free energy (G = H − TS) Enthalpy Pressure Entropy Temperature Volume Atomic volume Piston velocity Shock velocity Thermodynamic interaction parameter between elements i and j Body centred cubic Face centred cubic Hexagonal close packed hcp phase in Ti- and Zr-based alloys bcc phase in Ti- and Zr-based alloys hcp martensite Orthorhombic martensite Generic martensite ( or ) Temperature at which martensite starts forming during quenching Temperature at which martensite formation is completed during quenching Temperature at which the m → reversion starts on up-quenching Temperature at which athermal phase starts forming during quenching Temperature at which the free energies of the parent () and product (m ) phases are equal. Ab initio pseudopotential 3
4
Phase Transformations: Titanium and Zirconium Alloys
ASA: ASW: FPLAPW: LAPW: LCGTO: LDA: LMTO: MC: MD: MT: NFE: QMC: QSD: TB:
1.1
Atomic sphere approximation Augmented spherical wave Full potential linear augmented plane wave Linear augmented plane wave Linear combination of gaussian type orbitals Local density approximation Linear muffin tin orbital Monte carlo Molecular dynamics Muffin tin Nearly free electron Quantum monte carlo Quantum structural diagram Tight binding
INTRODUCTION
Titanium (Ti), zirconium (Zr) and hafnium (Hf) are transition metals belonging to Group 4 (nomenclature as per the recommendations of IUPAC 1988) of the periodic table of elements. The interest in the metals Ti and Zr and in alloys based on them has gained momentum from the late 1940s in view of their suitability for being used as structural materials in certain rapidly developing industries; particularly, the aerospace and chemical industries in the case of Ti alloys and the nuclear power industry in the case of Zr alloys. Some important characteristics of these metals are listed in Table 1.1. It can be seen from this table that the electronic ground state configurations of these metals are Ar 3d2 4s2 and Kr 4d2 5s2 , respectively. The similarity in the dispositions of the outer electrons, i.e. the four electrons (two s electrons and two d electrons) outside the inert gas shells (M shell for Ti and N shell for Zr) is, to a large extent, responsible for the similarities in some of the chemical and physical properties of these two metals and as a corollary, in many aspects of their chemical and physical metallurgy, including alloying behaviour.
1.2
POLYMORPHISM
Apart from existing in solid, liquid and gaseous states, many elements exhibit a special feature: they adopt different crystal structures in the solid state under different conditions of temperature or pressure or external field. The transition from
Phases and Crystal Structures
5
Table 1.1. Some characteristics of elemental Ti and Zr. Property
Element Ti
Atomic number (Z) Number of naturally occurring isotopes Atomic weight Electronic ground state configuration Density at 298 K kg/m3 Melting temperature (K) Boiling temperature (K) Enthalpy of fusion ( Hf ) kJ/mol Electronegativity Metal radius (nm)
22 5 47.90
Ar 3d2 4s2 4510 1941 3533 16.7 1.5 0.147
Zr 40 5 91.22
Kr 4d2 5s2 6510 2128 4650 18.8 1.4 0.160
References: Froes et al. 1996, Kubaschewski et al. 1993, McAuliffe and Bricklebank 1994, Soloveichik 1994.
one modification (allotrope) to another is termed a polymorphous transformation or a phase transformation (transition). A phase transition is associated with changes in structural parameters and/or in the ordering of electron spins (Steurer 1996). It will be discussed in a later chapter that two basically different types of phase transitions may be encountered: first-order transitions and second-order (or higher order) transitions. A transition of the former type is associated with discontinuous changes in the first derivatives of the Gibbs free energy, G = H − TS, while a transition of the latter type is characterized by discontinuous changes in the second (or higher order) derivatives of the Gibbs free energy and there are no jumpwise changes in the first derivatives. In either type of transition, the crystal structure undergoes a discontinuous change at the transition point (e.g. transition temperature or transition pressure). It is not necessary to have a symmetry relationship between the parent and the product phases in a first-order transition. However, in a second-order transition a group/subgroup relationship can always be found in relation to the symmetry groups associated with the crystal structures of the two phases. Elemental Ti and Zr (and Hf) exhibit temperature induced as well as pressure induced polymorphism. The pertinent phases, transition temperatures and transition pressures are listed in Table 1.2 and Table 1.3. It can be seen from Table 1.2 that for both Ti and Zr, the high temperature phase, termed the -phase, has the relatively “open” bcc structure while the low temperature phase, termed the -phase, has the close packed hcp structure. The hcp structure of the -phase is, however, slightly compressed in the sense that the value of the axial ratio is smaller than the ideal value of 1.63. It has been pointed out (McQuillan 1963, Collings 1984) that the more open bcc structure has a higher vibrational entropy as compared to
6
Phase Transformations: Titanium and Zirconium Alloys
Table 1.2. Allotropic forms of elemental Ti and Zr at atmospheric pressure (Massalski et al. 1992) (Variable: temperature). Element
Phase
Temperature regime (K)
Enthalpy of transformation (kJ/mol)
Crystal structure
Ti
Alpha() Beta()
Up to 1155 1155–1943
4 174 2
hexagonal close packed body centred cubic
Zr
Alpha() Beta()
Up to 1139 (1136) 1139–2128
4 1033 9
hexagonal close packed body centred cubic
Note: The figures in parentheses are from Kubaschewski et al. 1993.
Table 1.3. Allotropic forms of elemental Ti and Zr at room temperature (Steurer 1996) (Variable: pressure) Element
Phase
Pressure regime (GPa)
Crystal structure
Ti
Alpha() Omega( )
Up to 2 >2
hexagonal close packed hexagonal
Zr
Alpha() Omega( ) Omega prime ( )
Up to 2 2–30 > 30
hexagonal close packed hexagonal body centred cubic
the close packed hcp structure and as a consequence of this, the free energy of a competing bcc lattice will decrease more rapidly than that of the hcp lattice with increasing temperature; a temperature will ultimately be reached at which the free energy of the former will be less than that of the latter so that the bcc form will be more stable. The -phase can be obtained from the -phase by the application of sufficient pressure in elemental Ti and Zr. Some crystallographic data pertaining to all these phases are presented in Table 1.4. The structure of the -phase has been determined to be either hexagonal, belonging to the space group P6/mmm ¯ (Silcock 1958), or trigonal, belonging to the space group P 3m1 (Bagariatskii et al. 1959), depending on the solute concentration. The equivalent positions in the unit cell of the structure are 000; 2/3 1/3 (1/2 − z); 1/3 2/3 (1/2 + z). For the ideal structure with hexagonal (P6/mmm) symmetry, z = 0 while 0 < z < 1/6 defines ¯ a non-ideal structure with trigonal (P 3m1) symmetry. There are three atoms in the unit cell. The axial ratio is close to 3/81/2 . The symmetry of the structure is high and as in the case of the simple hexagonal lattice, there are 24 point group operations (Ho et al. 1982). The packing density ( 0 57) associated with the hexagonal (hP3) structure of the -phase is lower than that for the bcc ( 0 68) and the hcp ( 0 74) structures. The occurrence of such an open structure in metals
Phases and Crystal Structures
7
Table 1.4. Crystal structures and lattice parameters of allotropic forms of elemental Ti and Zr (Massalski et al. 1992, Steurer 1996). Element
Ph
Crystal structure P
SN
PS
SG
Ti Va nm3 = 17 65 × 10−3
-Ti
Mg
A3
hP2
P63 /mmc
-Ti -Ti
W -Ti
A2 −
cI2 hP3
¯ Im3m P6/mmm
Zr Va nm3 = 23 28 × 10−3
-Zr
Mg
A3
hP2
P63 /mmc
-Zr -Zr
W -Ti
A2 −
cI2 hP3
¯ Im3m P6/mmm
W
A2
cI2
¯ Im3m
Lattice paramaters (nm)
Axial ratio
a = 0 29506 c = 0 46835 a = 0 33065 a = 0 4625 c = 0 2813
1 5873
a = 0 32316 c = 0 51475 a = 0 36090 a = 0 5036 c = 0 3109 −
1 0 0 6082 1 5929 1 0 0 617 −
Ph – Phase; P – Prototype structure; SN – Strukturbericht notation; PS – Pearson symbol; SG – Space group. Notes: 1. The lattice parameter values of - and - phase correspond to a temperature of 298 K. 2. The quantity Va refers to the atomic volume under ambient conditions.
with metallic d-bonding is somewhat unusual. Normally, the transition metals have close packed (fcc, hcp) or fairly close packed (bcc) structures. Open structures are common among the p-electron systems or the actinide elements (Duthie and Pettifor 1977, Skriver 1985). The stability of this phase has been attributed to the covalent bonding contribution from s-d electron transfer (Steurer 1996). In the case of Zr (and Hf), it has been found that on the application of substantially higher pressures (Table 1.3) the -phase transforms to the -phase, which has the bcc structure. Although a similar transformation has not been observed in the case of Ti, even at a pressure as high as 87 GPa, theoretical considerations indicate that this metal too would undergo such a transformation at still higher pressures (Ahuja et al. 1993, Steurer 1996). This issue is addressed in greater detail in a later chapter of this volume.
1.3
PHASE DIAGRAMS OF ELEMENTAL TITANIUM AND ZIRCONIUM
1.3.1 Introductory remarks From the point of view of the phase rule, a pure element represents a single component system which may exhibit different phases. The phase rule imposes
8
Phase Transformations: Titanium and Zirconium Alloys
the condition f + p = c + 2, where f is the number of degrees of freedom in the pressure–temperature–composition space, p is the number of phases and c the number of components. For an element under temperature and pressure conditions of interest, f = 3 − p. This implies that a single phase is represented by an area in the pressure–temperature plane (p = 1 f = 2), a two-phase mixture is represented by a curve (p = 2 f = 1), which may be termed a phase boundary or phase line, and a three-phase mixture by a point (p = 3 f = 0), generally known as a triple point. A single component phase diagram is essentially a plot of areas representing phases, which are demarcated by phase boundaries, in the pressure–temperature or the P–T plane. A typical phase diagram of an element will generally show a vapour phase, a liquid phase and one or more solid phases. The phase boundaries have to abide by a few thermodynamic rules. The entropy change ( S) and the volume change ( V) across a phase boundary are related to the slope of the boundary by the Clausius–Clapeyron equation: dP S = dT V
(1.1)
This slope can be positive or negative: S must be positive for increasing temperature by the second law of thermodynamics, but V can be either positive or negative. d2 P The second derivative, dT 2 , gives a measure of the curvature and can be expressed as (Partington 1957): 2 d2 P 1 d V dP d V dP Cp =− − (1.2) +2 dT 2 V dP dT dT dT T For relatively incompressible solids like the transition metals, the terms on the right-hand side are small with the result that the phase boundaries have very small curvature and look like straight lines over the experimentally available pressure ranges (Young 1991). Experimental work on pressure-induced phase transformations in transition metals has been somewhat limited because of their low compressibility; phase changes may occur only at very high pressures which are difficult to achieve. Shock wave experiments are at present the most effective means of studying the high-pressure phase diagrams of these metals (Young 1991). A shock wave is a disturbance propagating at a supersonic speed in the medium. One can imagine the shock to be arising from a piston which moves into the medium at a constant velocity p . The boundary between the compressed and the uncompressed material will move ahead of the piston with a certain velocity s , which is termed the shock velocity. The basic objective of shock wave experiments is to measure the velocities p and s
Phases and Crystal Structures
9
and to determine from them the thermodynamic state of the host material. For most materials, p and s bear a linear relationship. But at a phase boundary this relationship may break down and the s versus p plot may show a discontinuity (McQueen et al. 1970). The reason for this is that a steady shock wave needs a sound speed that increases with compression and that this requirement is violated by a firstorder phase transition, with the result that the shock wave breaks up into a lowpressure wave (representing the untransformed material) and a high-pressure wave (representing the transformed material). The detectors register the arrival of only the first (i.e. low-pressure wave) and the two-wave region appears as a flat segment of constant s on the s versus p plot; a third segment appears on the plot when the shock velocity in the transformed material exceeds that corresponding to the untransformed material (Young 1991). The appearance of discontinuities in the s –p plane is thus a good indication of the occurrence of a first-order phase transition. 1.3.2 Titanium As stated earlier in this chapter, elemental Ti exists as the hcp -phase at room temperature under atmospheric pressure. On raising the pressure, while keeping the temperature constant, Ti transforms to the hexagonal -phase at around 2 GPa pressure. The – phase boundary has been reported to have a negative slope (Zilbershteyn et al. 1975, Vohra et al. 1982). This transition is associated with a large hysteresis and the equilibrium phase boundary has not been determined accurately (Young 1991). Further compression at room temperature to pressures upto 87 GPa has not shown any phase other than the -phase until recently (Xia et al. 1990a,b). As indicated earlier, this point will be covered in a subsequent chapter. Under atmospheric pressure, the -phase transforms to the denser -phase (bcc) at 1155 K. The – phase boundary has been determined by high temperature, static pressure measurements (Bundy 1963, Jayaraman et al. 1963). The triple point at which the -, - and -phases meet occurs at about 9.0 GPa and 940 K (Young 1991). The – phase boundary has been experimentally determined upto a pressure of 15 GPa (Bundy 1963). No phase other than the -, - and –phases has been found in Ti. Shock wave experiments conducted on elemental Ti have shown a discontinuity in the s –p curve; it has been suggested that this may correspond to the – or – transition (McQueen et al. 1970, Kutsar et al. 1982, Kiselev and Falkov 1982). The experimentally determined pressure–temperature phase diagram of Ti is shown in Figure 1.1 (Young 1991). Linear muffin tin orbital (LMTO) calculations which take into consideration the hcp, bcc, and fcc structures have predicted the stability of the -phase for pressures up to 30 GPa (Gyanchandani et al. 1990). The disposition of the – boundary (Figure 1.1) is not inconsistent with the theoretical prediction that at 0 K the -phase is the equilibrium phase in the case of Ti.
10
Phase Transformations: Titanium and Zirconium Alloys Ti 2.5 Liquid
T ( × 103 K)
2.0
bcc ( β)
1.5
1.0 hcp (α)
hex ( ω)
0.5
0
0
6
12
18
P (GPa)
Figure 1.1. Experimentally determined temperature–pressure phase diagram for Ti.
1.3.3 Zirconium As in the case of Ti, elemental Zr exists as the hcp -phase at room temperature and pressure, while on pressurization at this temperature it gets converted to the hexagonal -phase at a pressure of about 2 GPa. In this case also, the – phase line exhibits a negative slope (Jayaraman et al. 1963, Zilbershteyn et al. 1975, Guillermet 1987). A precise determination of the equilibrium transition pressure has, however, not been possible due to the occurrence of hysteresis (Young 1991). Static pressure experiments at room temperature have established that the -phase transforms to a bcc phase ( ) at a pressure of 30 GPa (Xia et al. 1990a,b). This bcc phase has been found to be the same as the -phase. It has been mentioned earlier that under atmospheric pressure, –Zr transforms to –Zr at 1139 K. The – phase boundary for elemental Zr has been studied by high-temperature, static pressure experiments (Jayaraman et al. 1963, Zilbershteyn et al. 1973). The – boundary has been determined upto a pressure of 7.5 GPa (Jayaraman et al. 1963). The –– triple point has been found to occur at 975 K and 6.7 GPa. As mentioned earlier, the -phase appears to be identical to the -phase that occurs at room temperature under high pressures and this implies that the – phase boundary has to turn backward towards the T = 0 K axis at high pressures (Young 1991). Shock wave experiments conducted on Zr are reported to show a discontinuity in the s versus p curve as in the case of Ti and this has been interpreted as being suggestive of the occurrence of a
Phases and Crystal Structures
11
Zr 3
Liquid
T ( × 103 K)
2
bcc ( β)
1
hcp (α)
0
0
hex (ω)
2
4
6
8
10
P (GPa)
Figure 1.2. Experimentally determined temperature–pressure phase diagram for Zr.
phase transition (McQueen et al. 1970, Kutsar et al. 1984). The experimentally determined pressure–temperature phase diagram of Zr is shown in Figure 1.2. In the case of Zr, LMTO calculations predict that the – and – transitions should occur at pressures of 5 GPa and 11 GPa, respectively (Gyanchandani et al. 1990). 1.3.4 Epilogue The occurrence of the -phase at high pressures in elemental Ti and Zr and at room pressures in alloy systems such as Ti–V and Zr–Nb and the similarity of the and the structures have been interpreted as being indicative of the fact that the phase diagrams of Ti and Zr exhibit the phenomenon of s-d electron transfer (Sikka et al. 1982). Effecting an increase in the number of d-electrons, either by the application of pressure or by alloying with elements relatively richer in d-electrons, drives the structure towards the bcc structure characteristic of the next group of elements to the right (i.e. V or Nb). The specific form of the structure, which may be regarded as a hexagonal distortion of the bcc structure, may be related to the details of the Fermi surfaces (Myron et al. 1975, Simmons and Varma 1980). The crystal structures of Ti, Zr and Hf under pressure have recently been studied by Ahuja et al. (1993) by means of first principles, total energy calculations based the local density approximation. These calculations correspond to zero temperature
12
Phase Transformations: Titanium and Zirconium Alloys
but many of the results obtained by them, especially for Zr, are in good agreement with experimental observations made at room temperature. The observed crystal structure sequence: hcp hP2 → hP3 → bcc cI2 with increasing pressure has been validated for Zr and Hf and it has been predicted that the same sequence should apply to Ti. The equilibrium volumes obtained for Ti, Zr and Hf are 0.0160, 0.0222 and 0 0201 nm3 , respectively, which compare reasonably well with the experimental values of 0.0176, 0.0233 and 0 0223 nm3 for these metals. The calculated c/a values corresponding to the minimum total energy are also in good agreement with the experimental values. Some of the disagreement between the theoretical predictions and the room temperature experimental observations could be ascribed to thermal effects. For example, the calculations indicate that at the theoretical equilibrium volume, the hP3 structure is slightly more stable than the hP2 structure; but room temperature observations show that the reverse is true—a result that matches with the calculations at the experimental volume. An important point is that the calculations do show that the energy difference between the - and the -phases is small for both Ti and Zr, which is consistent with the fact that the pressure induced → transition can be brought about in these metals at moderately high pressures. The calculations of Ahuja et al. (1993) indicate that the charge density for the -phase has a substantial non-spherical component, reflecting covalent bonding. This is quite different from the chemical bonding prevailing in the fcc, hcp and bcc structures where the charge density is predominantly spherical around the atomic positions and flat in the intervening regions. The chemical bonding for these structures is metallic. However, despite the difference in the nature of the chemical bonds for the various structures, band filling arguments can be used, at least to some extent, to explain the crystallographic sequence encountered in these tetravalent metals. At zero temperature and sufficiently high pressures, all the three metals – Ti, Zr and Hf – are predicted to assume the bcc structure. Again, at zero pressure and high temperaturess, these elements are known to transform from the hcp to the bcc structure. There is thus the possibility that the two bcc regions in a pressure–temperature phase diagram will be in contact. A schematic phase diagram, pertinent to these metals, has been constructed by Ahuja et al. (1993) and is shown in Figure 1.3. These authors have also examined the issue of the stability of the bcc phase. They have shown that the tetragonal shear constant, C = C11 − C12 /2, has a negative value at zero pressure for the bcc structure. This corresponds to a mechanically unstable situation. However, the sign of C changes with increasing pressure. For the high pressure bcc phase, the calculated C values are all positive, in agreement with the observed high pressure bcc phase in Zr and Hf. This can be explained as an effect of s–d electron transfer; for example, the d-band occupation
Phases and Crystal Structures
13
Temperature
L
bcc
I
II
hcp
ω
bcc
Pressure
Figure 1.3. Schematic temperature–pressure phase diagram for the metals Ti, Zr and Hf. The bcc phase is mechanically unstable in the region I and mechanically stable in the region II at low temperatures.
of Zr increases under pressure, making it behave more like the element to its right, i.e. Nb, which has a bcc crystal structure. Even though the bcc structure, according to calculations, is mechanically unstable at zero pressure, the high temperature -phase of all the three metals is known to posses this structure. This can be explained in terms of the high entropy associated with the bcc structure. The -phase of these elements shows some anomalous properties including its well known anomalously fast diffusion behaviour. This behaviour might be related to the intrinsic mechanical instability associated with the value of the C parameter. Another possible explanation suggested for the anomalous diffusion behaviour invokes -embryos acting as activated complex configurations in the atom–vacancy exchange process (Sanchez and de Fotnaine 1975). The fact that the -phase is calculated to have a lower total energy than the -phase at the equilibrium volume for all the three metals lends support to such an interpretation. The mechanical instability of the bcc phase becomes less severe with increasing pressure in the sense that the value of C becomes less negative with decreasing volume. Therefore, as the pressure increases, a progressively lower temperature is needed to restore the stability of the bcc structure (Ahuja et al. 1993). 1.3.5 Phase stability and electronic structure The stability of phases, the dependence of this stability on parameters like temperature and pressure and the selection of phases that are actually observed and recorded in phase diagrams are determined by the result of the competition among several possible phases (and, therefore, structures) that could be stable in a given
14
Phase Transformations: Titanium and Zirconium Alloys
system. This competition is based on the respective values of the Gibbs free energy corresponding to the various pertinent phases and their variation with temperature, pressure, composition and parameters such as magnetic, electric or stress fields, dose rates of particle and photon irradiation, etc. A number of factors contribute to the enthalpy, H, and the entropy, S. A very important contribution to the entropy arises from the statistical mixing of atoms. There may be additional contributions from vibrational effects, clustering of atoms, distribution of magnetic moments, long range configurational effects, etc. The statistical mixing of atoms contributes to the enthalpy as well. These contributions are related to the interaction energies: those corresponding to nearest neighbour atoms, next nearest neighbour atoms and further distant atoms in a given structure. These interaction energies may arise from various origins – electronic, magnetic, elastic and vibrational. A formidable problem in the context of the assessment of phase stability is that the relative stability among the competing crystal structures is usually dictated by very small energy differences between large values of the cohesive energy. Apart from this, a correct prediction implies the prediction of the lowest free energy structure among the chosen structural alternatives. This, in turn, stipulates a prior algorithm to generate all probable structures. Even when all these difficulties are overcome, it is needed to incorporate the roles of variables like temperature and pressure in realistic terms. These are, indeed, difficult tasks. The success of a theory of phase stability is largely determined by its ability to make predictions that are consistent with experimental observations. There is a need to be able to calculate phase stability from “first principles” if the basic microscopic parameters that dictate the free energy of a phase are to be properly understood. It should also be possible to make use of such calculations for predicting phase diagrams in systems where the experimental determination of such diagrams is difficult. The understanding and prediction of phase stability in respect of disordered and ordered alloys in terms of electronic structure calculations constitute an area of considerable importance in materials science and significant progress has been made with regard to the “first principles” approach to the band theory of such materials (Massalski 1996). The computation of an alloy phase diagram from first principles implies its delineation from a knowledge of the electronic structure of the alloy. In a truly ab initio calculation, one begins with a periodic array of nuclei of charge Ze together with Z electrons per nucleus, and then solves the Schrodinger equation for the total energy of the system. When Z is small (e.g. for H, He and Li), it is possible to handle this problem by Quantum Monte Carlo (QMC) methods (Ceperley and Alder 1986) which are exact in principle. However, the QMC method is not yet practical for heavier atoms, and the development of the density functional theory and its computational version, the local density approximation (LDA), has been of
Phases and Crystal Structures
15
great value. Here the full many-body wave function is approximated as a product of one-electron functions, and the exchange–correlation energy is expressed as a function of the local electron density, nr, given by nr = k r 2 k r being the one-electron wave function for the occupied state k (Young 1991). In the density functional theory, the total energy of a system of nuclei and electrons is considered to be a unique functional of nr and is a minimum at the true ground state. The total energy, Et , is expressed as Et = E1 + E2 + E3 + E4 + E5 where the terms on the right-hand side represent the kinetic (E1 ), electron–nucleus (E2 ), electron–electron (E3 ), exchange–correlation (E4 ) and nucleus–nucleus (E5 ) energies. The different approaches used to solve the one-electron Schrodinger equation, with the imposition of the lattice periodicity (Bloch condition) as a boundary condition, have engendered a variety of band-structure methods; some of these are (Young 1991): ab initio pseudopotential (AIP); linear muffin tin orbital (LMTO); augmented spherical wave (ASW); linearized augmented plane wave (LAPW); full-potential LAPW (FPLAPW) and linear combination of Gaussian-type orbitals (LCGTO). The LMTO method, which has been extensively used, is based on some additional approximations. While the muffin tin (MT) potential implies that the atomic potential Vr is spherically symmetric within a sphere inscribed in the primitive unit cell and is constant in the interstitial region, the LMTO method brings in a further simplification by way of the atomic-sphere approximation (ASA), whereby the spherical potential is extended to the full atomic volumes, reducing the net interstitial volume to zero. The Bloch condition is implemented by effecting the cancellation of all neighbour wave functions within the atomic sphere (Skriver 1984). The ‘L’ in LMTO implies the approximation that the basis functions are made energy-independent; this permits the eigenfunctions to be obtained in a single diagonalization operation, speeding up the calculation enormously and thus contributing to the efficacy of the method, a major limitation of which is the restriction to high-symmetry crystal structures imposed by the ASA (Young 1991). The LMTO method has been used to predict the stability of different phases with regard to the pressure–temperature phase diagrams of many transition metals, including Ti and Zr. Total energy calculations based on the LDA, which use only atomic numbers as inputs, have been very successful in the estimation of 0 K ground state properties of the elements and of ordered compounds. In fact, the implementation of the LDA by many an investigator, combined with the development of efficient linear methods for studying the electronic structure of solids, has led to fully ab initio calculations of the total energy at 0 K of pure solids, relatively simple compounds and disordered alloys (Sanchez 1992). By making it possible to assess a wide range
16
Phase Transformations: Titanium and Zirconium Alloys
of physical properties quite close to the corresponding experimentally obtained values, these quantum mechanical total energy computations have provided very favourable evidence in support of the LDA method, which can be applied, together with appropriate statistical models, to address the difficult problem of alloy stability at non-zero temperatures. Even though the LDA method has been quite successful, it has some non-trivial limitations including the underestimation of band gap energies and the inability to predict narrow band Mott-transition phenomena (Young 1991). A more general method for calculating the equilibrium state of matter at finite temperatures is the quantum molecular dynamics method (Car and Parrinello 1989). In this approach, the LDA wave function is solved for a small number of nuclei in an arbitrary configuration and the Hellman–Feynman theorem is used for finding the net force on each nucleus; the nuclei are then moved in accordance with classical (Newtonian) dynamics and the LDA calculation is undertaken again for the new configuration of the nuclei. This approach has been found to be useful for arriving at band structures and bonding details in respect of solids and liquids at finite temperatures (Young 1991). In the context of statistical models, it is appropriate to make a mention here of the Monte Carlo (MC) (Binder 1986) and molecular dynamics (MD) (Hoover 1986) methods. Like QMC, these methods are exact in principle. Although it is possible to undertake direct calculation of free energy by MC, the technique is not yet very suitable for the determination of phase stability and accurate delineation of phase boundaries. As of now, the MD method also suffers from similar limitations. It is true that isobaric–isothermal ensemble versions of MC and MD have been successfully employed to predict the most stable crystal structures of certain solids (Parrinello and Rahman 1981), but these methods have found their most important use in providing a standard for comparing and refining approximate statistical mechanics models (Young 1991). Some of the aspects briefly outlined in the preceding paragraphs have been covered in greater detail in a subsequent chapter. It is to be noted that a major shortcoming of many of the ab initio phase diagram calculations concerns the inadequate treatment of local volume and elastic relaxations and the neglect of vibrational modes. Even in crystalline solids, atoms are in perpetual motion; they move from one lattice site to another by diffusion at non-zero temperatures and also vibrate about their equilibrium positions. In a multicomponent system like an alloy, a given lattice site is occupied by atoms of different species at different times. If a large atom replaces a small one, the environment of the lattice site responds by expanding. Likewise, when a small atom replaces a large atom, the neighbouring atoms relax towards the lattice site in question. It should be possible to address the accompanying strain fluctuations
Phases and Crystal Structures
17
within the same type of first principles framework that is pertinent to fluctuations in concentration. However, the treatment of local relaxations of this kind presents a very difficult problem and not many attempts appear to have been made to include this effect in first principles calculations of phase stability and phase diagrams (Sanchez 1992, Gyorffy et al. 1992). Apart from the direct quantum mechanical route, many semi-empirical schemes pertaining to phase stability have also been pursued, often with a good deal of success. Many of these schemes involve the construction of certain phenomenological scales on which various aspects of bonding and structural characteristics are measured (Raju et al. 1995). These scales include parameters like the electronegativity factor, the size factor, the coordination factor, the electron concentration (e/a) factor, the promotion energy factor, etc. that are used to systematize a variety of structural features. The resulting structure maps are essentially graphical representations of the relative structural stability of alloy phases. They are two-dimensional diagrams, constructed by using suitable alloy theory coordinates for sorting out different crystal structures that are compatible with a chosen alloy stoichiometry. The efficacy of these structure maps depends crucially on the appropriate choice of coordinates. What are needed are those “bond indicators” which are transparent in their physical content, are transferable in their applicability and have a bearing on the alloy formation situation in terms of a validated model (Raju et al. 1995). In the classical approach, the emphasis has been on the construction of physically simple and transferable coordinates that may systematize the observed trends in relation to the occurrence of alloy phases. The major limitations of the classical formalism lie in the linear dependencies among many of the different phenomenological scales and the absence of a microscopic model that connects one or more of these directly to a real space alloy physics (Raju et al. 1995). Quantum mechanical considerations have been invoked in order to tide over these deficiencies with the result that the classical coordinates have been replaced by what are known as quantum structural parameters and classical structure diagrams by quantum structural diagrams (QSD). There have been numerous applications of QSD to various classes of solids including intermetallics, quasicrystals, high Tc superconductors and permanent magnetic materials (Phillips 1991). Even though not all of these have served to elucidate the issue of structural stability of condensed phases, these have been very useful in ordering the vast available data base into certain systematics. There are, indeed, quite a few examples of QSD which have really enhanced the understanding of the physicochemical factors governing phase stability. Most of the existing models pertaining to phase stability, ranging from those offering detailed density maps and electronic parameters of alloys to the semiempirical ones, suffer from a major difficulty in the context of the construction
18
Phase Transformations: Titanium and Zirconium Alloys
of phase diagrams in that a theoretical treatment of the temperature dependence of energy is not straightforward and tractable (Massalski 1996). The calculations used for predicting enthalpy at 0 K (first principles calculations) or at some undefined temperature (semi-empirical models) are seldom able to furnish adequate information regarding the thermal behaviour of such enthalpies or the thermal entropy contributions to the free energy. The prediction of entropies, particularly for relevant metastable phases in phase diagrams, has to be realized for the utilization of the full potential of the theoretical methods of phase stability calculations. 1.3.6 Some features of transition metals Elements belonging to the family of transition metals, of which Ti and Zr are members, are generally characterized by certain interesting features. Some of these will be briefly covered in this section. Elements of Groups 3–10 in the periodic table constitute the transition metals which have in common that their d-orbitals (3d, 4d and 5d) are partially occupied. These orbitals are only slightly screened by the outer s-electrons, resulting in significantly different chemical properties of these elements going from left to right in the periodic table; the atomic volumes rapidly decrease with increasing number of electrons in the bonding d-orbitals, because of cohesion, and then increase as the anti-bonding d-orbitals get filled (Steurer 1996). Transition metals are characterized by a fairly tightly bound (and partially filled) d-band that overlaps and hybridizes with a broader nearly-free-electron (NFE) sp-band. The d-band (with a large density of states near the Fermi level) is well described within the tight-binding (TB) approximation by a linear combination of atomic d-orbitals and the difference in behaviour between the valence sp and d electrons arises from the d-shell lying inside the outer valence s-shell, thereby resulting in a small overlap between the d-orbitals in the bulk (Pettifor 1996). In general, the transition metals exhibit high densities, cohesive energies and bulk moduli, with some exceptions. These characteristics arise from strong d- electron bonding. Plots of molar volume, cohesive energy and bulk modulus against the number of d-electrons yield roughly symmetrical curves with extreme values approximately at the middle of the series (Young 1991). An exception to this trend occurs with the 3d magnetic elements. The values of these parameters for the transition elements are shown in Table 1.5. The general behaviour alluded to the above can be rationalized in terms of the Friedel model of transition metal d-bands (Harrison 1980). Cohesive energy versus group number plots for 3d, 4d and 5d transition metals are shown in Figure 1.4. The sequence of the observed room temperature (and pressure) crystal structures in the case of 3d, 4d and 5d transition metals is presented in Table 1.6. This
Phases and Crystal Structures
19
Table 1.5. Values of molar volume, cohesive energy and bulk modulus for transition metals (Young 1991). Z
Element
21 22 23 24 25 26 27 28 39 40 41 42 43 44 45 46 71 72 73 74 75 76 77 78
Sc Ti V Cr Mn Fe Co Ni Y Zr Nb Mo Tc Ru Rh Pd Lu Hf Ta W Re Os Ir Pt
Molar volume (m3 /M mol)
Cohesive energy (kJ/mol)
Bulk modulus (GPa)
15 00 10 64 8 32 7 23 7 35 7 09 6 67 6 59 19 88 14 02 10 83 9 38 8 63 8 17 8 28 8 56 17 78 13 44 10 85 9 47 8 86 8 42 8 52 9 09
376 0 467 0 511 0 395 0 282 0 413 0 427 0 428 0 424 0 607 0 718 0 656 0 688 0 650 0 552 0 376 0 428 0 619 0 781 0 848 0 774 0 788 0 668 0 564 0
54 6 106 0 155 0 160 0 90 4 163 0 186 0 179 0 41 0 94 9 169 0 261 0 − 303 0 282 0 189 0 47 4 108 0 191 0 308 0 360 0 − 358 0 277 0
observed sequence (hcp → bcc → hcp → fcc) indicates that close packed structures are preferred at either end of the series, while the more open bcc structure is preferred in the middle. Pettifor (1977) has carried out a TB orbital calculation and shown that the structure sequence across the series is the result of the filling of the d-band and that the s-p electron number is nearly constant. While this model correctly predicts the structure sequence hcp → bcc → hcp → fcc, it does not predict the structures of all the elements correctly. In the tight binding model, to a first-order approximation, the cohesive energy turns out to be independent of structure; the relative structural stability arises from small differences in band structure contribution to the total electronic energy, an adequate description of which calls for the inclusion of higher order moments for describing the density of states curve (Raju et al. 1996). A fully self consistent LMTO calculation leads to a still better agreement between theory and experiment (Skriver 1984).
20
Phase Transformations: Titanium and Zirconium Alloys 950
Cohesive energy (kJ/mol)
850 750
5d
650
4d 550 450
3d 350 250
3
4
5
6
7
8
9
10
Group number
Figure 1.4. Cohesive energy versus group number plots for 3d, 4d and 5d transition metals. Table 1.6. Crystal structures of d-transition metals at room temperature and pressure.
3d series 4d series 5d series
HCP
BCC
HCP
FCC
Sc Ti Y Zr Lu Hf
V Cr Fe Mn Nb Mo Ta W
Co Tc Ru Re Os
Ni Rh Pd Ir Pt
Note: The actual structure of Mn is complex though it is listed under bcc in this table.
A systematic theoretical study with regard to the phase transitions that can be expected to occur in unalloyed transition metals at ultra-high pressures has not yet been attempted. However, it is, in general, expected that the early transition metals will assume the structures of their right-hand side neighbours as the s–d electron transfer will lead to the filling of the d-band under pressure; for the later members of the series, pressure is expected to have the effect of emptying the d-band, thus reversing the earlier trend (Young 1991). Obviously, transition metal phase transitions can also be driven by alloying, whereby the number of electrons populating the d-band can be altered. Fairly general theoretical arguments suggest that alloys of transition metals with roughly half-filled d-bands exhibit ordering tendencies, while those with nearly empty or nearly full d-bands show clustering tendencies in the disordered state and thus tend to phase separate at low temperatures; this prediction appears to be borne out by a considerable body of experimental data, even though there are many exceptions to this rule (Gyorffy et al. 1992).
Phases and Crystal Structures
1.4
21
EFFECT OF ALLOYING
1.4.1 Introductory remarks In alloys based on Ti or Zr, a very important effect of an alloying element pertains to the manner in which its addition affects the allotropic -phase to -phase transformation temperature. Some elements stabilize the -phase by raising this temperature while some others lower it, thereby stabilizing the -phase. Elements which, on being dissolved in Ti or Zr, cause the transformation temperature to increase or bring about little change in it are known as -stabilizers. These elements are generally non-transition metals or interstitial elements (like C, N and O). Elements which, on alloying with Ti or Zr, bring down the transformation temperature are termed -stabilizers. These elements are generally the transition metals and the noble metals with unfilled or just filled d-electron bands. Among the interstitial elements, H is a -stabilizer. Unlike in pure Ti or Zr, in alloys the single phase and the single phase regions are separated by a two-phase + region in the temperature versus composition phase diagram. The width of this region increases with increasing solute content. The single equilibrium - to -phase transformation temperature associated with elemental Ti or Zr is replaced by two equilibrium temperatures in the case of an alloy: the -transus temperature, below which the alloy contains only the -phase, and the -transus temperature, above which the alloy contains only the -phase. At temperatures between these two temperatures, both the - and the -phases are present. 1.4.2 Alloy classification The allotropic transformation exhibited by Ti and Zr forms the basis of the classification of commercial alloys based on these metals. Such classification is effected on the basis of the phases present in these alloys at ambient temperature (and pressure). The relative proportions of the constituent phases are determined by the nature (-stabilizing or -stabilizing) and the amounts of the alloying elements. In the case of alloys, the - and -phases contain various amounts of the different alloying species in solid solution. 1.4.3 Titanium alloys Technical alloys of Ti, which are generally multicomponent alloys containing -stabilizing as well as -stabilizing elements, are broadly classified as alloys, + alloys and alloys. Within the second category, there are the subclasses “near ” and “near ” alloys, referring to alloys whose compositions place them near the / + or the + / phase boundaries, respectively.
22
Phase Transformations: Titanium and Zirconium Alloys
Unalloyed Ti and its alloys with one or more -stabilizing elements consist fully or predominantly of the -phase at room temperature and are known as alloys. The -phase continues to be the primary phase constituent of most of these alloys at temperatures well beyond about 1040 K (Froes et al. 1996). These alloys generally exhibit good strength, toughness, creep resistance and weldability, together with the absence of a ductile-to-brittle transition (Collings 1984). However, they are not amenable to strengthening by heat treatment. The compositions of + alloys are such that at room temperature they contain a mixture of the - and -phases. These alloys have one or more of - as well as -stabilizing elements as alloying additions. In general, + alloys possess good fabricability. They are very strong at room temperature and moderately so at high temperatures (Collings 1984). The relative volume fractions of the - and -phases in these alloys can be varied by heat treatment, which provides a handle for adjusting their properties. In -alloys, the -phase is stabilized by the addition of adequate amounts of -stabilizing elements and can be retained at room temperature. These alloys generally contain significant amounts of one or more of the transition metals V, Nb, Ta (Group 5) and Mo (Group 6). These “-isomorphous” alloying elements do not form intermetallic compounds through eutectoid decomposition of the -phase and are generally preferred to eutectoid forming -stabilizing elements such as Cr, Cu, Ni; however, elements of the latter category are sometimes added to (and + ) alloys for improving their hardenability and response to heat treatment (Froes et al. 1996). The strength of alloys is generally greater than that of + and -alloys. Moreover, they exhibit excellent formability (Wood 1972). But they have relatively high densities, are prone to ductile–brittle transition at low temperatures and generally possess inferior creep resistance as compared to and + alloys (Collings 1984, Froes et al. 1996). The archetypical -stabilizing and -stabilizing alloying additions to Ti are Al and Mo, respectively. It is useful to be able to describe a multicomponent Ti-based alloy in terms of its “equivalent” Al and Mo contents. The two pertinent expressions often quoted in this context (Collings 1994) are:
Al eq = Al + Zr /3 + Sn /3 + 10 O
Mo eq = Mo + Ta /5 + Nb /3 6 + W /2 5 + + V /1 25 + 1 25 Cr + 1 25 Ni + 1 7 Mn + 1 7 Co + 2 5 Fe where [X] indicates the concentration of the element X in weight per cent in the alloy. It can be seen that while Al and O are strong -stabilizers, Sn and Zr are relatively weak ones. It can also be seen that the efficacy of the transition elements
Phases and Crystal Structures
23
with regard to the stabilization of the -phase progressively increases in the order: Ta, Nb, W, V, Mo, Cr and Ni, Mn and Co, and Fe, the last being the strongest -stabilizer. It may be mentioned here that Ti can form extensive substitutional solid solutions with most of the elements with atomic size factor within about 20% and this fact has opened up many alloying possibilities for exploitation. Some examples of important commercial Ti base alloys are: Ti-5Al-2.5Sn ( alloys); Ti-8Al-1Mo-1V, Ti-6Al-2Sn-4Zr-2Mo (near alloys); Ti-6Al-4V, Ti-6Al-2Sn-6V, Ti-3Al-2.5V ( + alloys); Ti-6Al-2Sn-4Zr-6Mo, Ti-5Al-2Sn2Zr-4Cr-4Mo, Ti-3Al-10V-2Fe (near alloys); Ti-13V-11Cr-3Al, Ti-15V-3Cr3Al-3Sn, Ti-4Mo-8V-6Cr-4Zr-3Al, Ti-11.5Mo-6Zr-4.5Sn ( alloys). 1.4.4 Zirconium alloys Unlike Ti, Zr is not quite amenable to alloying. One of the reasons for this could be the relatively large size of the Zr atom. Most of the elements have very limited solubilities in -Zr, with a few exceptions such as Ti, Hf, Sc and O. By comparison -Zr is a much better solvent, but it is generally quite difficult to retain the -phase at room temperature in a metastable state by quenching (Froes et al. 1996). The occurrence of non-equilibrium phases in -quenched Ti- and Zr-based alloys has been dealt with in a later section. According to the exhaustive compilation made by Douglass (1971), the retention of the -phase during quenching has been found to be feasible in the binary Zr–Mo, Zr–Cr, Zr–Nb, Zr–U, Zr–V and Zr–Re systems. The minimum concentrations of alloying additions for complete retention of the -phase in the first four systems are 5 wt%, 7.2 wt%, 15 wt% and 20 wt% respectively. Retention of cent per cent -phase is not possible in the systems Zr–V and Zr–Re; alloys containing the maximum amounts of V or Re in solution at quenching temperatures as high as 1573 K have been found to contain the -phase in addition to the -phase (Petrova 1962). The retention of quite large volume fractions of a metastable, Zr-rich 1 -phase has been observed in relatively solute-lean alloys (Zr-2.5 wt% Nb and Zr-5 wt% Ta) belonging to the monotectoid Zr–Nb (Banerjee et al. 1976, Menon et al. 1978) and Zr–Ta (Mukhopadhyay et al. 1978, Menon et al. 1979) systems. The most common Zr alloys of commercial importance are the zircaloys, namely zircaloy 2: Zr-1.5Sn-0.1Cr-0.1Fe-0.1Ni, Cr + Fe + Ni not to exceed 0.38 wt%; zircaloy 4: Zr-1.5Sn-0.15Cr- 0.15Fe, Cr + Fe not to exceed 0.3 wt% and the Zr-2.5% Nb, Zr-1% Nb and Zr-2.5Nb-0.5Cu alloys. These alloys contain only small amounts of -stabilizing elements and are all basically -alloys, with the -phase as the predominant constituent phase.
24
Phase Transformations: Titanium and Zirconium Alloys
1.4.5 Stability of titanium and zirconium alloys The aspect of lattice stability or, in other words, of structural phase stability is an important issue with regard to pure metals like Ti and Zr and alloys based on these. It has been stated in an earlier section that the crystal structures of the three long periods of transition metals follow the sequence hcp → bcc → hcp → fcc as the group number increases from 3 to 10 (3d: Sc to Ni; 4d: Y to Pd; 5d: Lu to Pt). It appears that there is a correlation between the crystal structure and the group number in the case of the elemental transition metals and between the crystal structure and the average group number or the electron to atom (e/a) ratio in the case of alloys. The occurrence of correlations like this testifies to the fact that the electronic structure is a key factor in determining phase stability. The e/a ratio is a parameter which relates to many properties of binary transition metal alloys, particularly Ti–X alloys, where X represents a transition metal (Collings 1984). A qualitatively similar situation is obtained with Zr–X alloys also. However, a general and comprehensive theoretical explanation rationalizing the correlation between phase stability and electron concentration (which is the same as or is closely related to the e/a ratio) in the case of transition metal systems is still to evolve (Faulkner 1982). The issue of the stability of equilibrium phases in Ti (and Zr) alloys can also be addressed by adopting a thermodynamic approach (Kaufman and Bernstein 1970, Kaufman and Nesor 1973). In this approach, the energywise competition between the relevant phases is duly considered while assessing phase stability in unalloyed metals as well as in alloys. This quantitative thermodynamic approach has been used for the computation of phase diagrams pertaining to binary as well as multicomponent systems. It has been mentioned earlier in the context of Ti–X and Zr–X alloys that -phase stabilizers are generally non-transition or simple metals, while -phase stabilizers are generally transition metals and noble metals. Collings (1984) has put forward a qualitative explanation, based on electron screening considerations, with regard to the phase stabilizing action of -stabilizer and -stabilizer solutes. This is outlined in the following paragraphs. When a simple metal X is dissolved in Ti (or Zr), most of the electrons belonging to X atoms occupy states in the lower part of the band and only very few appear at the Fermi level. The d-electrons belonging to the host (solvent) tend to avoid the solute atoms and this leads to a dilution of the Ti (or Zr) sublattice. A consequence of this is to emphasise any pre-existing Ti–Ti (or Zr–Zr) bond directionality and thereby to preserve the hcp structure characteristic of Ti (or Zr). As more and more X atoms are added, the field of Ti- (or Zr)-like -stability is ultimately terminated, generally by the appearance of an intermetallic phase of the stoichiometry Ti3 X (or Zr 3 X), which is also based on or is closely related to the hcp structure.
Phases and Crystal Structures
25
Coming next to the case of -phase stabilization, one may first recall that the crystal structures of transition metals change from hcp to bcc as the e/a ratio increases from 4 to 6. Collings (1984) has pointed out that it is possible to rationalize this stabilization of the bcc structure within the framework of an electron screening model which stipulates that a high concentration of conduction electrons, by enhancing the screening of ion cores, may cause a symmetrical (i.e. cubic) crystal structure to be favoured. Thus an increase in the electron density (as in elements belonging to Groups 5 and 6) will tend to symmetrize the screening, thereby enhancing the stability of the bcc structure. The fact that the six d-transition metals belonging to Groups 3 and 4 undergo the hcp → bcc structural transformation at high temperatures indicates that symmetrization can also be accomplished through lattice vibrations (Collings 1984). Given this background, one can see that the addition of transition metals belonging to Groups 5–10 to Ti or Zr increases the electron density and as a consequence, stabilizes the bcc or -phase. Thus, such elements are -stabilizers. Ageev and Petrova (1970) have pointed out in the context of Ti alloys that the -stabilization brought about by transition metal solutes is more effective the farther they are from Ti in the periodic table and that for the retention of the -phase during quenching from the -phase field, the nature and the concentration of the -stabilizer has to be such that the value of the e/a ratio is at least 4.2. In the context of the stability of bcc transition metals, it has been shown (Fisher and Dever 1970, Fisher 1975) that the magnitude of the elastic shear modulus C , defined as C11 − C12 /2, can be used for comparing the stabilities of these metals and their alloys. A cubic monocrystal is characterized by three fundamental stiffness moduli, C11 C44 and C12 . The shear stiffness modulus, C , though made up of two fundamental moduli, is obtainable directly by experiment. The ultrasonic waves needed for the measurement of these moduli are (Collings 1984): a longitudinal wave in a <100> direction for C11 ; a transverse wave in a <100> direction, polarized along < 100 > or a transverse wave in a < 100 > direction, polarized along < 100 > for C44 ; and the other transverse wave in a < 100 > direction, ¯ > for C . Since C44 is governed by the transverse <100 > polarized along < 110 ¯ wave, <100> polarized, and C by the same wave, < 110> polarised, C = C44 in an isotropic cubic material. Collings and Gegel (1973) have studied the variation of the parameter C with the e/a ratio and have demonstrated that alloying Group 4 elements with elements occurring to the right of them in the periodic table enhances the stability of the bcc structure and that this effect is maximized at about e/a = 6 (for the elements Cr, Mo and W). They have also found that C almost vanishes at e/a = 4 1 and that this value corresponds to the compositional threshold for martensitic transformation. In an anisotropic cubic material, the extent of the departure from isotropy is indicated by the value of the so called Zener anisotropy
26
Phase Transformations: Titanium and Zirconium Alloys
ratio, A = C44 /C . While in simple bcc metals like Na, the values of A are quite large, these can be quite low for bcc transition metals; for example, for the Group 6 metals Cr, Mo and W, the values of A are 0.71, 0.72 and 1.01, respectively (Fisher 1975). Fisher (1975) has also pointed out that while the C44 shears are resisted primarily by nearest neighbour repulsion, the C shear depends mainly on the next nearest neighbour forces. The large values of C for bcc transition elements are thought to be a consequence of the cohesive contributions of the d-electrons. The parameter C appears to be interpretable as a bcc stability parameter. Thus, for the highly stable bcc transition metals of Group 6, C is about 1 5 × 1011 N/m2 but its values decrease rapidly with decreasing e/a ratio, approaching zero at room temperature for alloys which exhibit -phase instabilities or under a martensitic transformation at ordinary temperatures (Collings and Gegel 1973). When Ti (or Zr) is alloyed with transition metals of higher group numbers, the increasing stability of the -phase is reflected in a continuous lowering of the / + transus temperature. It is mentioned later in this chapter that in the case of -stabilized binary Ti alloys, two types of phase diagrams are encountered: -isomorphous and -eutectoid. Collings (1984) has pointed out that a general trend is that as the group number of the solute increases, there is a tendency for the phase diagram to change from the former to the latter type.
1.5
BINARY PHASE DIAGRAMS
1.5.1 Introductory remarks Binary Ti–X and Zr–X (X being any element other than Ti and Zr, respectively) phase diagrams exhibit multifarious forms and reflect various kinds of phase reactions. The equilibrium phases are the - and -phases and numerous intermetallic phases. These are the phases that are shown in the equilibrium phase diagrams. However, many non-equilibrium phases such as the martensite phase (hcp and orthorhombic), the -phase and a large number of metastable intermetallic phases also occur in binary Ti and Zr base alloys. Some of these will be covered in detail in the succeeding chapters. There have been many attempts to categorize Ti and Zr alloy phase diagrams, taking cognizance of the fact that basically there are two types of systems, namely -stabilized and -stabilized systems. As mentioned earlier, in the former case X is usually a non-transition or simple metal, while in the latter X is usually a transition or a noble metal. It has been suggested in the context of Ti–X systems that the regular solution thermodynamic interaction parameter, ij , is positive for -stabilized alloys, indicating a clustering tendency, and negative for -stabilized alloys, indicating an ordering tendency (Collings and Gegel 1975).
Phases and Crystal Structures
27
For a given element X, the differences in the nature of the binary Ti–X and Zr–X equilibrium phase diagrams generally arise from the relative inefficiency of -Zr and -Zr with regard to taking X in solid solution as compared to -Ti and -Ti, particularly when X is a substitutional element. 1.5.2 Ti–X systems Margolin and Nielsen (1960) have suggested that -stabilized Ti–X systems can be basically subdivided into three classes: (a) –-isomorphous systems where X is completely soluble in the - as well as -phases (e.g. Ti–Zr, Ti–Hf); (b) -isomorphous systems where X is completely soluble in the -phase and has limited solubility in the -phase (e.g. Ti–V, Ti–Mo) and (c) -eutectoid systems where X has a limited solubility in the -phase which decomposes eutectoidally into the -phase and an appropriate intermetallic phase, Tim Xn , on cooling. Depending on the kinetics of -phase decomposition, this class is further subdivisible into “active” (rapid, e.g. Ti–Cu, Ti–Ni) and “sluggish” (e.g. Ti–Cr, Ti–Mn) eutectoid systems. They have also suggested that -stabilized Ti–X systems can be subdivided into two categories, depending on the degree of -phase stabilization: (a) systems exhibiting a “limited” degree of -stability, where the -phase is related to the - and an appropriate intermetallic phase by a peritectoid reaction (e.g. Ti–B, Ti–Al); and (b) systems characterized by a “complete -phase stability” where the -phase can coexist with the liquid phase (e.g. Ti–N, Ti–O). An exhaustive classification scheme for binary Ti–X phase diagrams has subsequently been suggested by Molchanova (1965) who has classified the available equilibrium phase diagrams into three broad groups, each of which contains a few subgroups. This classification, as reported by Collings (1984), is shown below: Group I: Systems where X shows continuous solid solubility in the -phase Subgroup I (a): Complete solubility in the -phase (X: Zr, Hf) Subgroup I (b): Partial solubility in the -phase (X: V, Nb, Ta, Mo) Subgroup I (c): Partial solubility in the -phase and eutectoid decomposition of the -phase (X: Cr, U) Group II: Eutectic systems Subgroup II (a): Partial solid solubility in the - and -phases; eutectoid decomposition of the -phase (X: H, Cu, Ag, Au, Be, Si, Sn, Bi, Mn, Fe, Co, Ni, Pd, Pt) Subgroup II (b): Partial solid solubility in the - and -phases; peritectoid – transformation (X: B, Sc, Ga, La, Ce, Nd, Gd, Ge) Subgroup II (c): Extremely limited solid solubility in the - and -phases (X: Y, Th)
28
Phase Transformations: Titanium and Zirconium Alloys
Group III: Peritectic systems Subgroup III (a): Simple peritectic (X: N, O) Subgroup III (b): Partial solid solubility in the - and -phases (X: Re) Subgroup III (c): Partial solid solubility in the - and -phases; eutectoid decomposition of the -phase (X: Pb, W) Subgroup III (d): Partial solid solubility in the - and -phases; peritectoid – transformation (X: Al, C). In a simpler classification, Molchanova (1965) has suggested that binary Ti–X equilibrium phase diagrams can be divided into four categories: -isomorphous (including – isomorphous), comprising subgroups I (a), I (b) and III (b); -eutectoid, comprising subgroups I (c), II (a) and III (c); simple peritectic, comprising subgroup III (a); and -peritectoid, comprising subgroups II (b) and III (d). This classification scheme is shown in Figure 1.5 in which the legends , and stand for the -phase, the -phase and the pertinent intermetallic phase, respectively.
Binary Ti alloys
β-stabilized
α-stabilized
β-isomorphous
β-eutectoid
Simple peritectic
β-peritectoid
Solutes V,Zr,Nb,Mo, Hf,Ta,Re
Solutes Cr,Mn,Fe,Co,Ni,Cu Pd,Ag,W,Pt,Au H,Be,Si,Sn,Pb,Bi,U
Solutes N,O
Solutes B,Sc,Ga,La Ca,Gd,Nd,Ge Al,C
Ti
L+α
L
β
β
α
γ
L+
β
L+
L+β
L
β+γ
β+α
β
α
β+γ
α+
Ti
β
β
α
L L+γ β
α
β
α+
Temperature
L+
β+
L+β
L
α+γ
α+γ Ti Solute content
α
α+γ
Ti
Figure 1.5. A classification scheme for binary Ti–X equilibrium phase diagrams. The legends and stand, respectively, for the -phase, the -phase and the pertinent intermetallic phase.
Phases and Crystal Structures
29
It is to be noted that quite a few Ti–X systems, designated earlier as -isomorphous systems, are not so in reality (Massalski et al. 1992). Below a transus delineating the upper boundary of a region referred to as a “miscibility gap”, a homogeneous, single-phase -solid solution decomposes into a thermodynamically stable aggregate of two bcc phases, one Ti-rich (1 ) and the other solute-rich 2 → 1 + 2 . The former participates in a monotectoid reaction: 1 → + 2 , the monotectoid temperature and composition varying from system to system. Examples of Ti–X systems where such a monotectoid reaction occurs include Ti–V, Ti–Mo, Ti–Nb and Ti–W. 1.5.3 Zr–X systems It has been pointed out earlier that inspite of the similarity in the electronic and crystal structures of Ti and Zr (both of these transition metals belong to Group 4 of the periodic table of elements), the alloying behaviour of these elements exhibit noteworthy differences, largely due to the size factor. While one encounters the – isomorphous, -eutectoid and -stabilized types of equilibrium diagrams in Zr–X systems, -isomorphous type phase diagrams do not occur in these alloys. Alloying elements, X, which give rise to -isomorphous equilibrium phase diagrams with Ti, yield either -eutectoid (e.g. X: V, Mo, Re) or -monotectoid (e.g. X: Nb, Ta) types of equilibrium diagrams with Zr. For a Pauling valence of 4, the second Brillouin zone is the one most nearly filled for in the case of Zr. This zone for -Zr is bounded boththe -and -phases ¯ and 1120 ¯ planes and has a volume of 3.6 electrons per atom; by the 1012 ¯ side the excess electrons, 0.4 per atom, overlap into the third zone on the 1012 of the second zone (Luke et al. 1965). The second Brillouin zone for the -phase is bounded by 200 and 211 planes and has a volume of eight electrons per atom. The inscribed Fermi sphere accommodates 4.19 electrons per atom and does not touch the zone boundaries. The larger volume of the -phase second zone in comparison with the -phase zone implies that the -structure can accommodate more electrons and thus the solubility of some transition elements is greater in the - phase than in the -phase. 1.5.4 Representative examples of Ti–X and Zr–X phase diagrams In this section representative examples of a few types of Ti–X and Zr–X binary equilibrium phase diagrams will be introduced: Ti–Zr, Ti–Mo, Ti–V, Ti–Cr, Ti–Al, Zr–Nb, Zr–Fe, Zr–Sn, Zr–Al, Ti–N,Zr–H and Zr–O. The phase diagrams presented here are based on those appearing in Massalski et al. (1992). Subsequent updates have been published in respect of some of these binary systems. These updates have been referred to at appropriate places.
30
Phase Transformations: Titanium and Zirconium Alloys
The Ti–Zr system is an example of an – isomorphous system, while Ti–Mo and Ti–V constitute important examples of so called -isomorphous systems and form the basis of several commercial and + alloys. The Ti–Cr system is a typical -eutectoid system, while the -stabilizer related Ti–Al system is pertinent to several technical and + alloys. The Zr–Nb system, which relates to the important family of commercial Zr–Nb alloys, is a -monotectoid system. The Zr–Fe phase diagram exemplifies a -eutectoid system. The Zr–Sn and Zr–Al systems exhibit -phase stabilization. The former is very relevant with regard to important technical Zr alloys such as zircaloys, while the latter is germane to the Zr 3 Al intermetallic phase which has been considered as a potential nuclear reactor structural material. In all these cases, X is a substitutional solute. In the Ti–N, Zr–H and Zr–O systems, all of which are of technological importance, X is an interstitial solute. The Ti–Zr system (Figure 1.6) appears to be a truly isomorphous system, though perhaps not as close to an “ideal solution” situation as the Zr–Hf system. The equilibrium phases occurring in the Ti–Zr system are the liquid (L), Ti Zr, Ti Zr, -Ti, -Ti, -Zr and -Zr. Apart from these, the metastable (martensite) and -phases are also encountered. The special points of the Ti–Zr system are listed in Table 1.7 (Murray 1987, Massalski et al. 1992.)
Weight per cent Zr 0
10 20
30
40
50
60
70
80
90
100
2270
2128 K 2070
1943 K
L
1870
Temperature (K)
1813 K 1670 1470
(β-Ti, β-Zr)
1270
1155 K
1136 K
1070 870
~878 K (α-Ti, α-Zr)
670 0
Ti
10
20
30
40
50
60
Atom per cent Zr
Figure 1.6. Equilibrium phase diagram for the Ti–Zr system.
70
80
90
100
Zr
Phases and Crystal Structures
31
Table 1.7. Special points of the Ti–Zr system. Phase reaction
Type of reaction
L Ti Zr L Ti L Zr Ti Zr Ti Zr Ti Ti Zr Zr
Congruent Melting Melting Congruent Allotropic Allotropic
Temperature (K)
Composition (at.% Zr)
1813 ± 15 1943 2128 878 ± 10 1155 1136
38 ± 2 0 100 52 ± 2 0 100
Weight per cent Mo 0 10 20
30
40
50
60
70
80
90
100 2896 K
2870 2670
L
2470
Temperature (K)
2270 2070 1943 K
1870 1670
(β-Ti, Mo)
1470 1270
1155 K
~1123 K
1070
~968 K ~12 (α-Ti)
870 670 0
Ti
10
20
30
40
50
60
Atom per cent Mo
70
80
90
100
Mo
Figure 1.7. Equilibrium phase diagram for the Ti–Mo system.
In the Ti–Mo system (Figure 1.7), the equilibrium solid phases that are encountered are: the bcc (-Ti, Mo) solid solution, in which Ti and Mo are completely miscible above the allotropic transformation temperature of Ti (1155 K), the hcp -Ti (Mo) solid solution in which the solubility of Mo is restricted (maximum of about 0.4 at.%), -Ti, -Ti and Mo. This system exhibits a miscibility gap in (-Ti, Mo) and a monotectoid reaction: -Ti) (-Ti) + (Mo) (Terauchi et al. 1978), the monotectoid temperature being about 968 K. The metastable martensite (hcp and orthorhombic ) and -phases also occur in
32
Phase Transformations: Titanium and Zirconium Alloys
Table 1.8. Special points of the Ti–Mo system. Phase reaction
Type of reaction
L Ti L Mo Ti Mo Ti + Mo Ti Ti + Mo Ti Ti
Melting Melting Critical Monotectoid Allotropic
Temperature (K)
Composition (at.% Mo)
1943 2896 ∼ 1123 ∼ 968 1155
0 100 ∼ 33 (12) (0.4) ∼ 60 0
the Ti–Mo system. The special points of the Ti–Mo system are shown in Table 1.8 (Murray 1987, Massalski et al. 1992). The equilibrium phase diagram of the Ti–V system (Figure 1.8) also shows a miscibility gap in the bcc (-Ti, V) phase and a monotectoid reaction occurring at 948 K: (-Ti) (-Ti) + (V) (Nakano et al. 1980). Above 1155 K, Ti and V are completely miscible in the (-Ti, V) solid solution. The solubility of V in the hcp (-Ti) phase is restricted, with a maximum of 2.7 at.% V. The metastable phases, martensite ( or , depending on the V content) and , are
Weight per cent V 0
10
20
30
40
50
60
70
80
90
100
2183 K
2170 L
Temperature (K )
1970
1943 K 1878 K
1770
1570
(β-Ti, V) 1370
1170
1155 K
1123 K 948 K
970
(α-Ti) 770 0
Ti
10
20
30
40
50
60
Atom per cent V
Figure 1.8. Equilibrium phase diagram for the Ti–V system.
70
80
90
100
V
Phases and Crystal Structures
33
Table 1.9. Special points of the Ti–V system. Phase reaction
Type of reaction
L Ti V L Ti LV Ti V Ti + V Ti Ti + V Ti Ti
Congruent Melting Melting Critical Monotectoid Allotropic
Temperature (K)
Composition (at.% V)
1878 1943 2183 1123 948 1155
32 0 100 ∼ 50 (18) (2.7) (∼ 80) 0
also encountered in this system. The special points pertinent to the Ti–V system are listed in Table 1.9 (Murray 1987, Massalski et al. 1992). Subsequently, an update has been published by Okamoto (1993a) with regard to the Ti–V phase diagram. Figure 1.9 shows the Ti–Cr equilibrium phase diagram. The equilibrium condensed phases encountered are the liquid (L), the bcc (-Ti, Cr) solid solution, the hcp (-Ti) solid solution, the topologically close packed intermetallic phases -TiCr2 , -TiCr2 and -TiCr2 , and, of course, -Ti, -Ti and Cr. In a narrow temperature range below the congruent melting temperature, Ti and Cr are completely Weight per cent Cr 0
10
20
30
40
50
60
70
80
90
100
2270 2136 K 2070 L
1943 K
Temperature (K)
1870 1683 K
1643 K
1670 ~1543 K (β -Ti, Cr)
1470
γ-TiCr2
~1493 K
β -TiCr2 1270 1155 K (α -Ti)
1070
~1073 K 940 K
α -TiCr2
870 0
Ti
10
20
30
40
50
60
Atom per cent Cr
Figure 1.9. Equilibrium phase diagram for the Ti–Cr system.
70
80
90
100
Cr
34
Phase Transformations: Titanium and Zirconium Alloys
Table 1.10. Special points of the Ti–Cr system. Phase reaction
Type of reaction
L Ti Cr L Ti L Cr Ti TiCr 2 Ti Ti + TiCr 2 Ti + TiCr 2 TiCr 2 TiCr 2 TiCr 2 TiCr 2 TiCr 2 + Cr Ti Ti
Congruent Melting Melting Congruent Eutectoid Peritectoid Unknown Eutectoid Allotropic
Temperature (K)
Composition (at.% Cr)
1683 ± 5 1943 2136 ± 20 1643 ± 10 940 ± 10 ∼ 1493 ∼ 1543 ∼ 1073 1155
44 0 100 ∼ 66 (12 5 ± 0 5) (0.6) (∼ 63) (39) (∼ 63)(∼ 65) ∼ 65 to 66 (∼ 65) (∼ 66) (96) 0
miscible in the (-Ti, Cr) phase. The maximum solubility of Cr in the (-Ti) phase is 0.6 at.%. The martensitic and the -phase also form in this system. The special points germane to the Ti-Cr system are presented in Table 1.10 (Murray 1987, Massalski et al. 1992). In the Ti–Al equilibrium phase diagram, (Figure 1.10), the solid phases that appear are: the bcc (-Ti) and the hcp (-Ti) solid solutions, the ordered intermetallic phases, Ti3 Al (also referred to as 2 ), TiAl (also referred to as ), TiAl, Weight per cent Al 0
10
20
30
40
50
60
70
80 90 100
1970 1943 K
L
1770 ~1558 K
Temperature (K)
(β-Ti)
1570
δ
~1398 K
TiAl 1370
TiAl3 Ti3Al
1170
TiAl2
1155 K
α-TiAl3
(α-Ti)
970
938 K
933 K
(Al) 770 0
Ti
10
20
30
40
50
60
70
Atom per cent Al
Figure 1.10. Equilibrium phase diagram for the Ti–Al system.
80
90
100
Al
Phases and Crystal Structures
35
Table 1.11. Special points of the Ti–Al system. Phase reaction
Type of reaction
L Ti Al L + Ti TiAl L + TiAl L + TiAl3 L + TiAl3 Al L Ti L Al Ti + TiAl Ti Ti Ti3 Al Ti Ti3 Al + TiAl TiAl + TiAl2 TiAl2 + TiAl3 TiAl3 TiAl3 Ti Ti
Congruent Peritectic Peritectic Peritectic Peritectic Melting Melting Peritectoid Congruent Eutectoid Peritectoid Eutectoid Unknown Allotropic
Temperature (K)
Composition (at.% Al)
∼ 1983 ∼ 1753 ∼ 1653 ∼ 1623 938 1943 933 ∼ 1558 ∼ 1453 ∼ 1398 1513 ∼ 1423 ∼ 873 1155
11 (53) (47.5) (51) (73.5) (69.5) (71.5) (80) (72.5) (75) (99.9) (75) (99.3) 0 100 (43) (49) (45) ∼ 32 (40) (39) (48) (65) (70) (67) (71.5) (68) (75) 75 0
and TiAl3 , and the (Al) solid solution. The addition of Al to Ti stabilizes the (-Ti) phase relative to the (-Ti) phase. The maximum solubilities of Al in (-Ti) and (-Ti) are about 48 and 45 at.%, respectively while that of Ti in Al is around 0.7 at.%. The phase boundaries for the TiAl2 and phases are yet to be ascertained. The metastable martensitic phase also forms in the Ti–Al system. The special points of this system are indicated in Table 1.11 (Murray 1987, Massalski et al. 1992). Two updates (Okamoto 1993b, 1994) pertaining to the Ti–Al phase diagram have appeared later. The equilibrium phases encountered in the Zr–Nb system are: the liquid (L), bcc (-Zr, Nb), (-Zr) and (Nb) solid solutions and the hcp (-Ti) solid solution. The bcc (-Zr, Nb) solid solution exhibits a miscibility gap and a monotectoid reaction: (-Zr) ←−−→ (-Zr) + (Nb) occurs. The phase diagram (Abriata and Bolcich 1982, Massalski et al. 1992) is shown in Figure 1.11 and the special points pertinent to the system are listed in Table 1.12. The metastable martensite ( ) and -phases form in this system. The Zr–Nb phase diagram has subsequently been updated (Okamoto 1992). The equilibrium Zr–Fe phase diagram (Arias and Abriata 1988, Massalski et al. 1992) is shown in Figure 1.12. The equilibrium phases are: the liquid (L); the bcc terminal solid solution, (-Zr), in which the maximum solubility of Fe is about 6.5 at.%; the hcp terminal solid solution, (-Zr), in which Fe has a maximum solubility of 0.03 at.%; the four intermetallic phases, Zr3 Fe, Zr 2 Fe, ZrFe2 and ZrFe3 ; the high temperature bcc terminal solid solution, (-Fe), in which Zr has a maximum solubility of about 4.5 at.%; the fcc terminal solid solution, (-Fe)
36
Phase Transformations: Titanium and Zirconium Alloys Weight per cent Nb 0
10
20
30
40
50
60
70
80
90
100
2742 K
2770 L
2570
Temperature (K)
2370 2128 K
2170 1970
21.7
2013 K
1770
(β-Zr, β-Nb)
1570 1370
1261 K 60.6
1136 K
1170 970
0.6
893 ± 10 K
(α-Zr)
770 0
10
91.0
18.5
20
30
Zr
40
50
60
70
Atom per cent Nb
80
90
100
Nb
Figure 1.11. Equilibrium phase diagram for the Zr–Nb system.
Table 1.12. Special points of the Zr–Nb system. Phase reaction
Type of reaction
L Zr Nb L Zr L Nb Zr Nb Zr + Nb Zr Zr + Nb) Zr Zr
Congruent Melting Melting Critical Monotectoid Allotropic
Temperature (K)
Composition (at.% Nb)
2013 2128 2742 1261 893 ± 10 1136
21.7 0 100 60.6 (18.8) (0.6) (91.1) 0
which shows a maximum solubility of around 0.7 at.% Zr; and the low temperature bcc terminal solid solution, (-Fe), in which the maximum solubility of Zr is only about 0.05 at.%. Table 1.13 shows the special points relevant to the Zr–Fe system. Amorphous Zr–Fe alloys have been produced over a wide range of compositions by rapid solidification processing. The metastable -phase also forms in this system. An update of the Zr–Fe equilibrium diagram has appeared later (Okamoto 1993c). The assessed Zr–Sn phase diagram (Abriata et al. 1982, Massalski et al. 1992) is shown in Figure 1.13. In this diagram, there appears to be uncertainty regarding
Phases and Crystal Structures
37
Weight per cent Fe 0
10
20
30
50
40
60
70
80
90
100
2270
2128 K
2070
L
1946 K
1870
δ Fe
66.7
1755 K
Temperature (K)
1670 1470
1667 K
90.2 ~99.3 1630 K (γ-Fe)
(β-Zr)
1270 1070
1811 K
~95.5
1610 K
1247 K
1201 K
~6.5 1158 K 1003 K ~24.0
Zr2Fe
4.0 0.03
870
ZrFe2
670
n ag M ns a tr
~573 K
(α-Zr) 470
1048 K
Zr3Fe
1198 K ~99.9 1185 K 1043 K Magnetic trans (α-Fe) ZrFe3 ~548 K Magnetic trans
270 0
10
20
30
Zr
40
50
60
70
80
Atom per cent Fe
90
100
Fe
Figure 1.12. Equilibrium phase diagram for the Zr–Fe system.
Table 1.13. Special points of the Zr–Fe system. Phase reaction
Type of reaction
L Zr L Zr + Zr 2 Fe L ZrFe2 L ZrFe3 + Fe L Fe L + ZrFe2 Zr 2 Fe L + ZrFe2 ZrFe3 Fe L + Fe Zr Zr Zr Zr + Zr 3 Fe Zr + Zr 2 Fe Zr 3 Fe Zr 2 Fe Zr 3 Fe + ZrFe2 ZrFe3 + Fe Fe Fe Fe Fe Fe
Melting Eutectic Congruent Eutectic Melting Peritectic Peritectic Catatectic Allotropic Eutectoid Peritectoid Eutectoid Peritectoid Allotropic Allotropic
Temperature (K) 2128 1201 1946 1610 1811 1247 1755 1630 1136 1003 1158 1048 1198 1667 1185
Composition (at.% Fe) 0 ∼ 24 ∼ 6 5 31 66.7 (90.2) (75) (∼ 99 3) 100 (∼ 25) (66) (33.3) (86.7) (∼ 72 5) (75) (∼ 95 5) (90.8) (∼ 99 3) 0 (4) (0.03) (24) (∼ 6) (31) (∼ 25) (33.3) (26.8) (66) (75) (?) (∼ 99 95) 100 100
38
Phase Transformations: Titanium and Zirconium Alloys Weight per cent Sn 0
10
20
30
40
50
60
70
80
90
100
2470
2261 K
2270
2128 K 2070
L 1865 K
1670
17.0
19.1
(β-Zr)
1600 K
11.8
1415 K
1470
1255 K
(α-Zr)
Zr4Sn
870
ZrSn
1270 1136 1070
Zr5Sn3
Temperature (K)
1870
(β-Sn)
670
~505 K
505 K
470 0
10
Zr
20
30
40
50
60
70
80
Atom per cent Sn
90
100
Sn
Figure 1.13. Equilibrium phase diagram for the Zr–Sn system.
Table 1.14. Special points of the Zr–Sn system. Phase reaction
Type of reaction
L Zr L Zr + Zr 5 Sn3 L Zr 5 Sn3 L Sn L + Zr 5 Sn3 ZrSn2 Zr + Zr 5 Sn3 Zr 4 Sn Zr + Zr 4 Sn Zr Zr Zr Sn Sn
Melting Eutectic Congruent Melting Peritectic Peritectoid Peritectoid Allotropic Allotropic
Temperature (K)
Composition (at.% Sn)
2128 1865 2261 505 1415 1600 1255 1136 286
0 (19.1) (17) (40) 40 100 (79) (40) (66.6) (11.8) (40) (20) (4.9) (20) (7.3) 0 100
most of the liquidus and the entire region between about 30 and 50 at.% Sn. The special points pertaining to the Zr–Sn system are listed in Table 1.14. The metastable martensitic phase forms in this system. The equilibrium phases encountered in the phase diagram of the Zr–Al system (Massalski et al. 1992) shown in Figure 1.14 are: the liquid (L); the bcc (-Zr) and the hcp (-Zr) solid solutions, the ten intermetallic phases, Zr 3 Al, Zr 2 Al, Zr 5 Al3 ,
Phases and Crystal Structures
39
Weight per cent Al 0
20
10
30
40
50
60 70 80 100
2270 2128 2070
L 1753 K
1463 K ZrAl3
ZrAl
Zr4Al3
(α-Zr)
Zr3Al2
1213 K
Zr2Al
11.5
1136 1070
73.5
1261 K
ZrAl2
12.5
Zr2Al3
1270
Zr5Al4
1470
1548 K
1623 K
1523 K
Zr5Al3
26
1853 K
1863 K
49
37
Zr3Al
Temperature (K)
1803 K
1670 22.5
59
39
29.5
(β-Zr)
1918 K
1758 K
1668 K
1870
(Al) 934 K
870 0
Zr
10
20
30
40
50
60
Atom per cent Al
70
80
90
100
Al
Figure 1.14. Equilibrium phase diagram for the Zr–Al system.
Zr 3 Al2 , Zr 4 Al3 , Zr 5 Al4 , ZrAl, Zr 2 Al3 , ZrAl2 and ZrAl3 , and the fcc (Al) solid solution in which the maximum solubility of Zr is about 0.07 at.%. The addition of Al stabilizes (-Zr) relative to (-Zr) and the maximum solubilities of Al in these two phases are about 11.5 and 26 at.%, respectively. The special points of the Zr– Al system are shown in Table 1.15. Subsequently, three updates in respect of the Zr–Al phase diagram have appeared (Murray et al. 1992, Okamoto 1993d, 2002). The equilibrium condensed phases that occur in the binary Ti–N system are: the liquid (L), the terminal bcc solid solution (-Ti), the terminal hcp solid solution (-Ti), and the three stable nitride phases, Ti2 N, TiN and . Both the terminal solid solutions have wide ranges of composition. The dissolved N (-stabilizer) extends the stability regime of the -Ti phase to a temperature (2623 K) much above the melting point of elemental -Ti. Two of the nitride phases, Ti2 N and , are stable over narrow composition ranges while the third, TiN, exhibits stability over an extensive composition range. Figure 1.15 shows the Ti–N equilibrium phase diagram; the special points of this system are listed in Table 1.16 (Massalski et al. 1992). An update pertaining to the Ti–N phase diagram has appeared subsequently (Okamoto 1993e). Figure 1.16 (Zuzek et al. 1990, Massalski et al. 1992) shows the solid phases encountered in the Zr–H phase diagram. These are the bcc terminal solid solution
40
Phase Transformations: Titanium and Zirconium Alloys
Table 1.15. Special points of the Zr–Al system. Phase reaction
Type of reaction
L Zr L Zr + Zr 5 Al3 L + Zr 3 Al2 Zr 5 Al3 L + Zr 5 Al4 Zr 3 Al2 L Zr 5 Al4 L Zr 5 Al4 + Zr 2 Al3 L + ZrAl2 Zr 2 Al3 L ZrAl2 L ZrAl2 + ZrAl3 L ZrAl3 L + ZrAl3 Al L Al Zr Zr Zr + Zr 5 Al3 Zr 2 Al Zr + Zr 2 Al Zr 3 Al Zr + Zr 3 Al Zr Zr 5 Al3 Zr 2 Al + Zr 3 Al2 Zr 3 Al2 + Zr 5 Al4 Zr 4 Al3 Zr 5 Al4 Zr 4 Al3 + ZrAl Zr 5 Al4 + Zr 2 Al3 ZrAl
Melting Eutectic Peritectic Peritectic Congruent Eutectic Peritectic Congruent Eutectic Congruent Peritectic Melting Allotropic Peritectoid Peritectoid Peritectoid Eutectoid Peritectoid Eutectoid Peritectoid
Temperature (K)
Composition (at.% Al)
2128 1623 1668 1753 1803 1758 1868 1918 1763 1853 934 933 1136 1523 1261 1213 ∼ 1273 ∼ 1303 ∼ 1273 1548
0 (29.5) (26) (37.5) (∼ 37) (40) (37.5) (∼ 39) (44.4) (40) 44.4 (49) (44.4) (60) (∼ 59) (66.7) (60) 66.7 (73.5) (66.7) (75) 75 (99.97) (75) (99.93) 100 0 (22.5) (37.5) (33.3) (12.5) (33.3) (25) (9.2) (25) (11.5) (37.5) (33.3) (40) (40) (44.4) (42.9) (44.4) (42.9) (50) (44.4) (60) (50)
Weight per cent N 0
2
4
6
8
10
15
20
25
3770 3563 K 47.4
Temperature (K)
3270
L
2770
2623 K 20.5
15.2
28
2293 K 12.5 6.2
2270
4.0
1770
1943 K
TiN
(β-Ti) (α-Ti )
1270
23 1155 K
1323 K 33.3 1373 K 30 33 1073 K 39 34 Ti2N 37.5
δ′
770 0
Ti
5
10
15
20
25
30
35
Atom per cent N
Figure 1.15. Equilibrium phase diagram for the Ti–N system.
40
45
50
55
Phases and Crystal Structures
41
Table 1.16. Special points of the Ti–N system. Phase reaction
Type of reaction
Temperature (K)
Composition (at.% N)
L ←→ Ti L ←→ TiNa L + Ti ←→ Ti L + TiN ←→ Ti Ti + TiN + Ti2 N
Melting Congruent Peritectic Peritectic Eutectoid or Peritectoid Congruent Peritectoid (Probably) Allotropic
1943 ∼ 3563 2293 ± 25 2623 ± 25 1323 ± 60
0 47.4 (4.0) (12.5) (6.2) (15.2) (2.8) (20.5) (23) (30) (33)
∼ 1373 1073 ± 100
33.3 (34) (37.5) (39)
1155
0
TiN ←→ Ti2 Nb Ti2 N + + TiN Ti ←→ Ti a b
Observed under pressure >∼1 MPa. Occurrence if Ti + TiN + Ti2 N equilibrium is eutectoid.
Weight per cent H 1270
1136 K (β-Zr)
Temperature (K)
1070
δ
(α-Zr) 823 K
870
ε
5.93
~37.5
56.7
670
470
270 0
Zr
10
20
30
40
50
60
70
80
Atom per cent H
Figure 1.16. Equilibrium phase diagram for the Zr–H system.
(-Zr), which decomposes eutectoidally at 823 K at a H concentration of 37.5 at.%, the hcp terminal solid solution (-Zr) which exhibits a maximum H solubility of 5.9 at.% at 823 K and the hydride phases (fcc) and (fct). The Zr–O phase diagram (Abriata et al. 1986, Massalski et al. 1992) is shown in Figure 1.17. The equilibrium condensed phases are the liquid (L), the bcc terminal solid solution (-Zr), the hcp terminal solid solution (-Zr) and the oxide phases, -ZrO2−x (cubic, cF12), -ZrO2−x (tetragonal, tP6) and -ZrO2−x (monoclinic, mP12). The special points of the Zr–O system are shown in Table 1.17.
42
Phase Transformations: Titanium and Zirconium Alloys Weight per cent O 0
10
20
30
3270
2983 K L + G
P = 1 atm
2403 K
2070
2338 K
25
10 10.5
35 40
19.5
(α-Zr)
(β-Zr)
62
~1798 K
63.6 66.5
31.2
1670
~1478 K 29.8 1270
1136 K
(α′-Zr)
(α3″-Zr) 870 (α2″-Zr) (α1″-Zr)
~1243 K
29.1
66.7
~773 K 28.6 (α ″-Zr) 4
66.7
α-ZrO2-x
Temperature (K)
2470
~2650 K β-ZrO2-x
2243 K 2128 K
γ -ZrO2-x
L
2870
470 0
10
20
Zr
30
40
50
60
70
Atom per cent O
Figure 1.17. Equilibrium phase diagram for the Zr–O system. Table 1.17. Special points of the Zr–O system. Phase reaction
Type of reaction
L ←→ Zr
Melting
2128
0
L ←→ Zr
Congruent
2403 ± 10
25 ± 1
L ←→ ZrO2−x
Congruent
2983 ± 15
66.6
L ←→ Zr + ZrO2−x L + Zr ←→ Zr
Eutectic
2338 ± 5
40 ± 235 ± 162 ± 1
Peritectic
2243 ± 10
10 ± 0 519 5 ± 2 10 5 ± 0 5 ∼ 66 6 ∼ 66 6 ∼ 100
L + ZrO2−x + G ZrO2−x ←→ Zr + ZrO2−x ZrO2−x ←→ Zr + ZrO2−x ZrO2−x + ZrO2−x + G
Temperature (K)
2983 Eutectoid
ZrO2−x +ZrO2−x +G
∼ 1798
Composition (at.% O)
∼ 1478
63 6 ± 0 4 31 2 ± 0 5 66 5 ± 0 1 ∼ 66 5 29 8 ± 0 5 ∼ 66 5
∼ 2650
∼ 66 6 ∼ 66 6 ∼ 100
∼ 1478
∼ 66 6 ∼ 66 6 ∼ 100
ZrO2−x ←→ ZrO2−x
Congruent
∼ 2650
66.6
ZrO2−x ←→ ZrO2−x
Congruent
∼ 1478
66.6
Zr ←→ Zr
Allotropic
1136
0
Phases and Crystal Structures
1.6
43
NON-EQUILIBRIUM PHASES
1.6.1 Introductory remarks Phases such as the -, - and intermetallic phases mentioned earlier are equilibrium phases and the corresponding phase fields are delineated in equilibrium phase diagrams of the type described in the previous section. However, non-equilibrium or metastable phases, as distinct from equilibrium phases, are quite important in respect of many alloy systems, including those based on Ti and Zr. Equilibrium phase diagrams are usually developed by deducing the initial states of alloys which have been quenched from different temperatures to room temperature. But the quenching process may lead to the formation of non-equilibrium phases. Two important examples of such non-equilibrium phases in Ti–X and Zr–X systems are the martensite and the athermal -phases. Both these phases are formed through athermal displacive transformations. It will be seen in a later chapter that one way of classifying phase changes is to divide them into two broad classes: reconstructive and displacive (Roy 1973, Christian 1979, Banerjee 1994). Transformations of the former kind involve breaking of the bonds of atoms with their neighbours and re-establishment of bonds to form a new configuration in place of the pre-existing one. Such a process requires atomic diffusion comprising random atomic jumps and disturbs atomic coordination. Atomic movements in displacive transformations, on the other hand, can be brought about by a homogeneous distortion, by shuffling of lattice planes, by static displacement waves or by a combination of these. Cooperative movements of a large number of atoms in a diffusionless process accomplish the structural change in displacive transformations. Unlike the diffusional atomic jumps which are thermally activated, the displacive movements do not require thermal activation and cannot, therefore, be suppressed by quenching. A structural transition involving periodic displacements of atoms from their original positions can be described in terms of a displacement wave and the introduction of a displacement wave in the parent lattice requires coordinated atom movements in an athermal process; the athermal martensitic and -transformations can, respectively, be described in terms of long wavelength and short wavelength displacement waves (Banerjee et al. 1997). In the present chapter brief accounts of the martensite and the -phases and of phase separation in the -phase will be provided with reference to Ti–X and Zr–X alloys. A detailed coverage in respect of the same will be found in three of the subsequent chapters. The martensitic transformation, which is diffusionless and involves cooperative atom movements, proceeds by the propagation of a shear front at a speed that approaches the speed of sound in the material, leading to the formation of the metastable martensite phase. This transformation occurs in many alloy
44
Phase Transformations: Titanium and Zirconium Alloys
systems, including Ti–X and Zr–X systems, in which the major component exhibits allotropy. The -phase, which is an equilibrium phase in Group 4 metals (Ti, Zr, Hf) at high pressures, forms in several alloys based on these metals and also in many other bcc alloys at ambient pressure as a metastable phase. On rapidly quenching Ti–X and Zr–X alloys, X being an -stabilizing element, from the -phase field, the martensite phase, m , which has the hcp structure, is obtained. The situation is somewhat different when X is a -stabilizing element, such as a transition metal. During the process of rapid cooling from the -phase field, when a composition-dependent temperature (known as the martensitie start or Ms temperature) is crossed, the bcc -phase commences to transform spontaneously by the martensitic mode to the martensite phase m whose structure may be hcp ( ) or orthorhombic ( ), depending on the alloy composition. However, in the case of these alloys, another athermal process, namely, that associated with the formation of the athermal -phase, competes with the martensitic process. At any temperature compatible with the formation of both m and -phases, there is a narrow range of composition (or electron to atom ratio), just beyond the martensite formation regime, over which the athermal -phase forms from the parent -phase. If a s temperature, akin to the Ms temperature, is conceived as being associated with the start of athermal -phase formation, then one may visualize that the s locus lies above the Ms locus in the narrow composition range referred to above, if temperature is plotted against composition. In the composition regime of martensite formation, which lies to the left of this narrow range, the Ms locus lies above the s locus. Even though the -phase appears athermally on rapid quenching from the -phase field only over a narrow range of electron to atom ratio, this phase occurs over a broader composition range as a precipitation product of -phase decomposition. The typical structures exhibited by rapidly -quenched binary Ti–X or Zr–X alloys, X being a -stabilizing element, are indicated in the schematic shown in Figure 1.18. Beyond the + region (where these two phases coexist), the -phase is retained in a metastable (susceptible to decomposition on ageing) or stable manner on quenching. It may be noted that similar values of the electron to atom ratio (∼ 4 15) characterize the limit of the stability of the bcc -phase with respect to either of the two athermal transformations (Collings 1984). 1.6.2 Martensite phase 1.6.2.1 Crystallography The phenomenological crystallographic theories of the martensitic transformation are based on the concept that the interface between the martensite and the parent phases is macroscopically invariant. The central theme of these theories is that the total macroscopic shear consists of three components: (a) the lattice shear or the
Temperature
Phases and Crystal Structures
45
Ms
ωs
Concentration of X
I
II
III
IV
Figure 1.18. Schematic showing the Ms and s loci for a binary Ti–X or Zr–X system, X being a -stabilizing element, on rapid cooling from the -phase field. Region I corresponds to martensite (m ) formation; in regions II and III, the -phase co-exists with the athermal - and the aged -phases, respectively; in region IV only the -phase occurs in a metastable or stable state.
Bain strain which brings about the necessary change in the lattice (e.g. bcc to hcp); (b) a lattice invariant inhomogeneous shear which provides an undistorted plane; and (c) a rigid body rotation to ensure that the undistorted habit plane is unrotated as well. The inhomogeneous shear accompanying the martensitic transformation is instrumental in generating the martensite substructure which, in most cases, is too fine to be resolved under the light microscope. Transmission electron microscopy (TEM) techniques have been extensively used for resolving this substructure and for obtaining information regarding the orientation relationship, the habit plane and the nature of the inhomogeneous strain for individual martensite crystals. A unique feature of the → martensitic transformation in Ti and Zr is that the necessary lattice strains approximately satisfy the invariant plane strain condition. Because of this, the magnitude of the lattice invariant shear is comparatively small and it is relatively simple to characterize the substructure of the martensite in Ti–X and Zr–X alloys. There are a number of choices for relating the lattices of the parent () and product ( ) phases. The correct choice of lattice correspondence is generally made by selecting the one which involves the minimum distortion and rotation of the lattice vectors. In the case of the transformation in Zr, it has been suggested (Burgers 1934) that the 011 plane forms the basal plane 0001 , while the ¯ and 111 ¯ directions lying on that plane correspond to the close packed 111 ¯ ¯ close packed 1120 directions. This accounts for four of the six 1120 ¯ directions. directions; the remaining two are derived from the 100 and 100
46
Phase Transformations: Titanium and Zirconium Alloys
Having chosen this lattice correspondence, the next step would be to determine the magnitudes of the strains which would deform the distorted hexagonal structure into a regular one, having lattice parameters consistent with those of -Zr. If ao a and c refer to the lattice parameters of the - and -phases, respectively, the magnitudes of two of the principal lattice distortions, 1 (along 100 ) and 2 ¯ ) are given by 1 = a/ao and 2 = 3/2 21 a/ao . The distortion 3 along (along 011
011 is 1/21/2 a/ao where = c/a. On substituting the values for the lattice parameters at the transformation temperature, the magnitudes of the principal strains for pure Zr are seen to be as follows: 2% expansion along 011 , 10% expansion ¯ and 10% contraction along 100 . The situation is analogous in the case along 011 of Ti and the corresponding principal strains are 1% expansion, 11% expansion and 11% contraction, respectively, along the aforementioned directions. A pair of planes remains undistorted under the action of a homogeneous lattice strain if and only if one of the principal strains is zero and the other two are of opposite signs (Wayman 1964). A special feature of the martensitic bcc to hcp transformation in Ti and Zr is that the principal strain along the 011 direction is very small and the other two principal strains are of opposite signs. If the principal strain along the 011 direction, 3 , were zero, the lattice shear would have left a plane undistorted. Since 3 is very small in the case of Ti and Zr, it is not unreasonable to treat the transformation with the approximation that 3 is zero (Kelly and Groves 1970). It has been reported (Bagaryatskii et al. 1959, Flower et al. 1982) that the normally observed hcp ( ) structure of the martensite is distorted to an orthorhombic structure ( ) in many Ti–X systems, X being a transition metal, when the martensite is supersaturated beyond a certain limit. The orthorhombic distortion increases with increasing solute content. It has been noticed that the deformation induced martensite, mentioned later in this section, almost invariably has an orthorhombic structure (Williams 1973). This is not surprising when one considers the fact that this type of martensite can occur only in alloys which are so enriched in -stabilizing solutes that they are not transformed on -quenching. It has been demonstrated (Otte 1970) that the → transformation ¯ ¯¯ ¯ 111 ≡ 2112 involves the activation of the shear systems 112 2113 and ¯ 111 ≡ 1011 ¯ ¯¯ 101 [2113 . The habit plane associated with this transformation has been found to be very close to 334 (Williams 1973, Shibata and Ono 1977), although in some -stabilizing solute enriched Ti–X alloys 344 habit has also been reported (Liu 1956, Gaunt and Christian 1959, Hammond and Kelly 1970). The orientation relationship between the - and -phases has been ¯ > (approximately), which observed to be: 011 0002 ; < 111¯ > < 1120 is consistent with the approximate orientation relation deduced by Burgers with regard to the bcc → hcp transformation in elemental Zr (Burgers 1934).
Phases and Crystal Structures
47
In the case of the orthorhombic martensite, the orientation relationship has been reported to be: 100 and 010 inclined by about 2o from 001 and ¯ > (Hatt and Rivlin 1968). < 110 > , respectively; 001 < 110 1.6.2.2 Transformation temperatures The martensitic transformation is characterized phenomenologically by the assignment of several temperatures. The most common among these are: Ms , the temperature at which martensite starts forming during quenching; Mf , the temperature at which the transformation is completed; s , the temperature at which the m → reverse transformation starts during up-quenching (in Ti and Zr alloys); and To , the temperature at which the free energies of the parent and martensite ( and m ) phases are equal. If Ms and s are very close to each other, it is indicated that the driving force for the transformation is small and also that To can be taken to be the mean of these two temperatures (Collings 1984). A thermodynamic analysis (Kaufman 1959) of the → transformation in several Ti and Zr base alloys has shown that the To temperatures are about 50 K higher than the experimentally observed Ms temperatures. This implies that the supercooling (To –Ms ), necessary to initiate the martensitic transformation in these systems is relatively low. The change in free energy accompanying the transformation at the Ms temperature is significantly lower as compared to that in ferrous systems. At the Ms temperature, the chemical driving force necessary to start a martensitic reaction depends on the shear modulus of the alloy at the transformation temperature, the magnitude of the homogeneous shear associated with the transformation and the magnitude of the inhomogeneous shear. The strain energy associated with martensite formation is determined by the homogeneous lattice strains and the shear modulus while the surface energy corresponds to the energy of the interface between the parent and the product lattices. If the austenite–martensite reaction in ferrous systems is compared with the → transformation in Ti and Zr base alloys, it is found that although there is not much difference in the homogeneous strain values in the two cases, the shear modulus of ferrous alloys at the transformation temperature is much higher. Again, the energy associated with the parent–martensite interface can also be expected to be much smaller in the case of Ti- and Zr-based alloys because only a small amount of inhomogeneous shear is necessary to make the total strain an invariant plane strain in the case of the → transformation. These considerations indicate that the “back stress”, which arises from the strain and surface energies opposing martensite formation, is much smaller in Ti- and Zr-based alloys as compared to ferrous alloys. This explains why a small driving force is adequate for initiating martensite formation in the former. The chemical driving force which balances
48
Phase Transformations: Titanium and Zirconium Alloys
the “back stress” at the Ms temperature can be assisted by external stress, leading to stress-assisted or stress-induced or deformation-induced martensite formation. The Ms temperature is composition dependent. Again, the measured Ms temperature for a given alloy composition may exhibit a dependence on the rate of cooling (Jepson et al. 1970). In -stabilized Ti–X and Zr–X alloys, Ms increases with increasing solute content and may lie a little below the + / transus; in -stabilized alloys, Ms decreases with increasing concentration of X and always lies in the ( + ) field (Collings 1984). In dilute alloys of the latter type, the Ms temperature is relatively high and water quenching may not be sufficiently rapid for completely suppressing thermally activated atom movements, leading to some segregation of the solute atoms prior to the transformation and consequently to the retention of some -phase. If the solute content in the alloy increases, the Ms temperature decreases and the diffusional contribution is inhibited, with the result that a full transformation to the martensite phase comes about. It may be mentioned here that the quench rates necessary to achieve the structural transformation while preserving compositional homogeneity depend strongly on the nature of the alloying element, X, or more specifically, on its diffusion kinetics in the -phase. A quench rate that is adequate when X is an early transition metal such as V, Nb or Mo, may not be so when X is a late transition metal like Fe, Co or Ni. This is so because metals belonging to the latter category diffuse much faster in the -phase; for instance, in the context of diffusion in -Ti at 1273 K, it may be noted that the diffusion coefficients of Co and Mo are in the ratio 200:1 (Collings 1984). As the solute concentration increases further, a stage is reached where the Mf temperature drops below the temperature of the quenching bath; in this situation, the retention of some untransformed -phase again becomes feasible. 1.6.2.3 Morphology and substructure If dilute Ti–X and Zr–X alloys are quenched from the -phase field, maintaining an adequately fast cooling rate, one generally obtains a hcp martensite phase ( ) which is known as lath or packet or massive martensite and consists of relatively large, irregular packets or “colonies” which are populated by nearparallel arrays of much finer platelets or laths. No retention of the -phase occurs in a lath martensite. As the solute content increases, the average packet size and the average lath size decrease. Beyond a certain level of solute concentration, which depends on the nature of the solute, a transition occurs in the martensite morphology, resulting in the formation of plate or acicular martensite. In contrast to the arrangement of near-parallel units in the lath morphology, the martensite units form in various intersecting directions in the plate or acicular structure. Another important difference between the lath and the plate morphologies is that the size distribution in the latter case is much broader than in the former. This is essentially
Phases and Crystal Structures
49
due to the fact that in the plate morphology the martensite units continuously partition the parent -grain and as a result of this the space available for the growth of the plates belonging to the subsequent generations gets more and more limited. The transition from lath to plate morphology is not abrupt and the two may coexist over some range of composition. When the solute concentration is sufficiently high, the martensitic transformation may be incomplete and some -phase, which is usually trapped between the platelets of the acicular martensite, may be retained. Not far removed from the “ plus acicular martensite” quenched structure is the Widmanstatten arrangement consisting of groups of -phase needles lying with their long axes parallel to the 110 planes of the retained -phase. The term “substructure” of a martensite generally refers to the structure within the martensite unit as revealed under the transmission electron microscope (TEM). This substructure arises from (a) the lattice invariant component of the transformation strain which may be slip, twin or a combination of both; and (b) the post-transformation strain resulting from the accommodation effect. There has been considerable interest in characterizing the internal structure of martensite plates for determining the nature of the inhomogeneous shear participating in the transformation process, as envisaged in the phenomenological theory of the martensitic transformation. For this it is necessary to be able to identify and separate the inhomogeneities introduced by matrix constraints from those produced by the lattice invariant component of the transformation strain. Such a separation is not straightforward. In a twinned martensite plate, a set of transformation twins is expected to appear periodically at almost equal intervals within the plate; the ratio of the thicknesses of the twinned and the matrix portions should be consistent with the value predicted by the theory and the specific variant of the twin plane should be consistent with the observed habit plane. When a set of twins in a martensite plate satisfies all these conditions, the twins are taken to be transformation twins. In a dislocated martensite crystal, it is more difficult to separate the transformation induced dislocations from those introduced by post-transformation stresses. A rule of the thumb appears to be that only those dislocations which are arranged in regular arrays and are observed very frequently may be taken to have been produced by the inhomogeneous shear. Generally, a transition from the dislocated to the twinned substructure is found to occur with increasing concentration of alloying elements. 1.6.3 Omega phase 1.6.3.1 Athermal and isothermal It has been mentioned earlier that under ambient pressure, the -phase can occur in a metastable manner in alloys in which the -phase is stabilized with respect to the martensitic → m transformation. The composition range over which this phase may be encountered is a characteristic of the alloy system under consideration. It has also been indicated that this phase can be obtained either by rapidly quenching
50
Phase Transformations: Titanium and Zirconium Alloys
from the -phase field (athermal ) or as a product of thermally activated -phase decomposition (isothermal or aged ). The athermal → transformation is displacive, diffusionless and of the first-order and the -phase so obtained has a composition very close to that of the -phase. The thermally activated transformation, on the other hand, is accompanied by solute rejection by diffusional processes from the to the -phase and is thus partially replacive in nature. The athermal → transformation cannot be suppressed even by extremely rapid quenching and is completely and continuously reversible with negligible hysteresis. The special characteristics of this transformation also include the appearance of an extensive diffuse intensity distribution in diffraction patterns, with the maximum intensity located close to the positions of ideal -reflections, as a precursor to the transformation event and the stability of the dual phase + structure with extremely fine (∼1–4 nm) particles distributed in the -matrix along 111 directions (Banerjee et al. 1997). The number density of the particles is extremely large and this fact lends support to the contention that the transformation does not involve long range diffusion. The volume fraction of the isothermal -phase forming in the -matrix is a function of the reaction time. This dependence of the volume fraction on time arises essentially due to the diffusion controlled partitioning of the solute between solute lean and solute rich regions. The solute lean regions are eventually transformed to the -phase. The composition of the isothermal -phase corresponds to the maximum solubility of the solute in the -phase. Thus after prolonged ageing at temperatures lower than about 770 K, a metastable + state is attained, characterized at a given temperature by a fixed volume fraction and composition of each of the and terminal points (Hickman 1969). After sufficiently long ageing periods at 720–770 K, -phase precipitation can be expected. An early model of isothermal -phase development visualized an initial structural transformation of the lattice into and (as in the case of the athermal -phase in its pertinent composition regime), followed by an exchange of solute and solvent atoms across the / interface (Courtney and Wulff 1969). It has subsequently been suggested that initially a composition fluctuation occurs and this is followed by a structural → transformation within a solute lean zone, triggered by a longitudinal phonon with a 2/3 111 wave vector; it is the instability of the bcc lattice with respect to this disturbance that is responsible for the athermal transition (de Fontaine et al. 1971). 1.6.3.2 Crystallography The crystal structure of the -phase has been described in an earlier section. The orientation relationship between the and -phases has been determined by a large number of investigators and has been unanimously accepted as: 111 0001 ;
Phases and Crystal Structures
51
¯ 1210 ¯ 110 . It has been found that this orientation relationship is valid both for the athermal -phase and the isothermal -phase (Williams 1978). This relationship implies that there are four possible crystallographic variants of the -structure, depending on which one of the 111 planes is parallel to the (0001) plane. Again, for the same variant of the -structure, there are three 110 directions so that in all 12 variants of the -structure are possible. But since the basal plane of this structure has six-fold symmetry, the three variants for a given 111 plane will appear identical and, therefore, the contribution from only four variants will be seen in selected area diffraction (SAD) patterns. The lattice parameters, a and c of the structure and a of the (bcc) structure, are related as follows: a =
√
√ 2a c = 3/2a
1.6.3.3 Morphology Precipitates of the athermal -phase that evolve during rapid -quenching are very fine (<5 nm) and it is difficult to assign any well-developed geometrical shape to these particles which have a tendency to be aligned along 111 directions. The shapes of isothermal -phase precipitates are more readily discernible and generally two types of morphologies, ellipsoidal and cubic, are encountered, depending on the linear lattice misfit, V − V /3V , where V represents the unit cell volume divided by the number of atoms in the unit cell (Hickman 1969). If this misfit is small (< 0 5%), as is often the case when the solute is a 4d-transition metal such as Nb or Mo, the precipitate morphology is dominated by surface energy considerations, leading to an ellipsoidal shape (Hickman 1969). If the misfit is large (>1%), as is generally the case when the solute is a 3d-transition metal like V, Cr, Mn or Fe, the minimization of elastic strains in the cubic matrix dictates a cubic morphology (Hickman 1969, Blackburn 1970). 1.6.3.4 Diffraction effects Pronounced diffuse scattering has been observed in electron, X-ray and neutron diffraction patterns prior to the formation of the -phase in all -forming systems. These diffuse intensity patterns are closely associated with the non-diffuse (sharp) reflections corresponding to the crystalline -phase. In view of this close association, the diffuse intensity distribution has been attributed to non-ideal -structures. It has been mentioned earlier that the ideal hexagonal -structure is obtained when the parameter z has the value zero and the non-ideal trigonal -structure results if 0 < z < 1/6. Selected area electron diffraction patterns obtained from the truly athermal -phase are characterized by sharp spots and straight lines of intensity while broad reflections and either straight or curved diffuse lines of intensity
52
Phase Transformations: Titanium and Zirconium Alloys
(diffuse streaking) originate from the “diffuse” -phase (Collings 1984). A model proposed by Sass and co-workers (Dawson and Sass 1970, McCabe and Sass 1971, Balcerzak and Sass 1972) envisages an ensemble of -particles, 1–1.5 nm in diameter and 1.5–2.5 nm apart, arranged in rows along 111 directions. According to this model, clusters of such rows contribute to the sharp spots and straight lines of intensity, while the broad reflections and diffuse streaking arise from either individual rows of particles or isolated particles. It has been demonstrated that a transition from diffuse to sharp -reflections occurs in -quenched specimens in response to either decreasing solute content (Sass 1972) or decreasing temperature (de Fontaine et al. 1971). In both cases, curvilinear lines of diffuse intensity become straight and well defined. It has been pointed out (Williams 1973) that since the diffuse streaking tends to coincide with the positions of the -reflections when they are present, compositionwise there is no sharp line of demarcation separating the regions of athermal and diffuse . A soft phonon mechanistic model of the -phase reaction (de Fontaine et al. 1971) has been able to provide a rationalization, in terms of lattice dynamics, for the temperature and composition dependences of the athermal and diffuse -phases. After examining electron diffraction patterns belonging to several zones and considering the symmetry of the reciprocal lattices, de Fontaine et al. (1971) have constructed a three-dimensional model of the diffuse intensity which is distributed on quasi-spherical surfaces centred around the octahedral sites of the reciprocal of the bcc -lattice. These spheres of intensity touch all the 111 faces of the octahedra surrounding them. When this intensity distribution in the reciprocal space is sectioned to reveal the diffuse intensity pattern in a plane corresponding to any zone, the pertinent shifts of the diffuse intensity maxima from the positions of ideal -reflections and asymmetry in intensity distribution are manifested. The lattice dynamical model for phase stability, with special reference to the quenched -phase, has been further developed by Cook (1975). It does appear that the 23 111 soft mode, interacting with a lattice of composition and temperature dependent relative stability, is responsible not only for athermal but also for diffuse , which represents in varying degrees, dynamical fluctuations between the - and -phases (Collings 1984). 1.6.4 Phase separation in -phase Below a transus representing the upper boundary of a region in the equilibrium phase diagram known as a miscibility gap, a previously homogeneous single phase solid solution decomposes into a thermodynamically stable aggregate of two bcc phases, one solute lean and the other solute rich, designated respectively as the 1 - and 2 -phases. Two ideal examples of systems where such a → 1 + 2 decomposition occurs are the Zr–Nb and Zr–Ta systems, both of which exhibit
Phases and Crystal Structures
53
the monotectoid reaction 1 → + 2 . The equilibrium phase diagrams for these systems have many points of similarity with those of Zr–X and Ti–X (X being a transition metal) eutectoid systems; in the case of the latter, Ti is replaced by Zr and the intermetallic phase Tim Xn replaced by 2 . A bimodal free energy (G) versus solute concentration (X) curve is associated with either of the Zr–Nb and Zr–Ta systems, representing the occurrence, in equilibrium, of two solid solutions. This situation is somewhat different from that representing the coexistence of say, the -and -phases in equilibrium. While the latter situation is described in terms of two independent free energy parabolas, equilibrium phase separation, with the absence of structural change, has to be described by a continuous curve with two minima, separated by an intervening maximum, for which the second derivative of free energy with respect to solute concentration is negative. Although above the monotectoid temperature, both 1 and 2 are equilibrium phases, the former ceases to be an equilibrium phase below this temperature. However, the 1 -phase has been found to occur in a metastable manner at temperatures close to but lower than the monotectoid temperature in the Zr–Nb (Banerjee et al. 1976, Menon et al. 1978) as well as the Zr–Ta (Mukhopadhyay et al. 1978, Menon et al. 1979) systems. Phase separation in the -phase has also been reported in some Ti–X systems (X: Cr, V, Mo, Nb). In situations where the temperature (Williams et al. 1971) or the solute concentration (Williams 1973) is too high to be conducive for -phase precipitation, a solute lean bcc phase, designated as separates from the -phase. The → + phase separation reaction can be considered to be a clustering reaction characteristic of alloy systems which exhibit positive heats of mixing (Chandrasekaran et al. 1972) or similar manifestations of a tendency for the alloying constituents to unmix. It is interesting to note that -stabilizing elements such as Al, Sn and O, when added to Ti–V and Ti–Mo alloys in sufficient quantities, appear to increase the stability of the bcc lattice in that -phase formation is suppressed in favour of -phase separation (Williams 1971). Thus these solutes, which are certainly not -stabilizers in the conventional sense, can be regarded as stabilizers of the bcc lattice against the instability. Ageing in the + phase field, which lies just outside the + phase field, would eventually result in -nucleated -phase precipitation.
1.7
INTERMETALLIC PHASES
1.7.1 Introductory remarks A large number of intermetallic phases are encountered in binary Ti–X and Zr–X systems and these exhibit a variety of crystal structures. Many of these structures
54
Phase Transformations: Titanium and Zirconium Alloys
are derived from three simple crystal structures, namely, face centered cubic (fcc, Al), body centered cubic (bcc, A2) and hexagonal close packed (hcp, A3) structures, which are commonly associated with pure metals and disordered solid solutions. It may be noted that the formation of intermetallic phases does not appear to occur in these binary systems when X is an alkali or alkaline earth metal (with the exception of Be) or a transition metal belonging to Group 3 or Group 4. Likewise, no intermetallic phases form when X is a rare earth (RE) element. Ti–RE and Zr–RE phase diagrams are normally characterized by the absence of intermetallic phases, limited mutual solubilities in the solid state, and quite often, a miscibility gap in the liquid state. In general, in the intermetallic phase Zrm Xn , X is a transition metal from Group 5 (only V) to Group 10 or a simple (non-transition) metal from Group 11 to Group 16. In a similar manner in the intermetallic phase, Tim Xn , X is a transition metal from Group 6 (only Cr) to Group 10 or a non-transition metal from Group 11 to Group 16. In both Tim Xn and Zr m Xn families, X is more often a non-transition metal than a transition metal. However, this does not imply that the incidence of X being a transition metal is infrequent: close to a hundred binary intermetallic phases of this type have been reported. Almost all the important binary intermetallic phases that have been observed in Ti–X and Zr–X alloys, together with hydrides, borides, carbides, nitrides, oxides, phosphides and sulphides have been listed in Tables A1.1 and A1.2, respectively. The composition range, space group, Pearson symbol and strukturbericht designation associated with each of these phases have been incorporated in these tables. The nomenclatures of crystal structures, in terms of their strukturbericht designations and the corresponding Pearson symbols, are listed in Table A1.3 for ready reference. A survey of Table A1.1 and Table A1.2 shows that more than 80% of Tim Xn and Zrm Xn type intermetallic phases are almost equally distributed among three crystal classes: cubic, tetragonal and hexagonal. The orthorhombic system comes next (∼15%) while less than 5% belong to the rhombohedral and monoclinic systems. The unit cells of a majority (∼62%) of these phases are of the primitive type, followed by body centred (∼21%), face centred (∼10%) and base centred (∼7%) cells. The occurrence of body centred unit cells is most common in tetragonal phases, of face centred cells in cubic phases and of base centred cells in orthorhombic phases. The more frequently encountered structures in binary Ti–X intermetallics are the B2 (cP2, CsCl type), C11b (tI16, MoSi2 type), L12 (cP4, AuCu3 type), A15 (cP8, Cr 3 Si type), L1o (tP4, AuCu type) and D019 (hP8, Ni3 Sn type) structures. In the case of Zr–X intermetallics, these are the B2, C11b , C15 (cF24, Cu2 Mg type), D88 (hP16, Mn5 Si3 type), C16 (tI12, Al2 Cu type), C14 (hP12, MgZn2 type) and Bf (oC8, CrB type) structures.
Phases and Crystal Structures
55
Generally speaking, there are many intermetallics that can be put to a variety of uses. For example, there has been a considerable interest in developing strong alloys based on intermetallic phases for structural applications. However, such intermetallics are normally brittle and for this reason, their processing and application are difficult. But it has to be pointed out here that as a class of materials intermetallics, in which atomic bonding is at least partly metallic, tend to be less brittle than ceramics, where atomic bonding is mainly covalent or ionic in nature. Broadly speaking, alloys based on intermetallic phases are hard to deform plastically as compared to pure metals or disordered alloys because of their stronger atomic bonding and the resulting ordered distribution of atoms which gives rise to relatively complex crystal structures. The brittleness of intermetallics generally appears to decrease with increasing crystal symmetry and decreasing unit cell size. In view of this, intermetallics with relatively high crystal symmetry (e.g. cubic, such as B2, D03 , L12 , or nearly cubic, such as L1o , D022 , where a slight tetragonal distortion is present) are thought to have good potential for structural applications (Sauthoff 1996). In the context of Ti–X and Zr–X systems, the intermetallics that have been considered for structural applications include Ti3 AlD019 , TiAlL1o , TiAl3 D022 and Zr 3 AlL12 . Another potential application area pertains to hydrogen storage: certain Ti and Zr bearing Laves phase intermetallics show promise with regard to applications as hydrogen storage materials (Sauthoff 1996). 1.7.2 Intermetallic phase structures: atomic layer stacking The structures of many intermetallic phases can be considered to be formed by the sequential stacking of certain polygonal nets of atoms. These structural characteristics can be readily described by using specific codes and symbols, which can be very useful for a compact presentation and comparison of the structural features of different materials. Various notations have been devised for describing the stacking patterns (Pearson 1972, Ferro and Saccone 1996). Without going into details of these, only the Schlafli notation, P N , will be introduced here. In this notation, P N describes the characteristics of each node in the network in the following manner: the superscript N is the number of P-gon polygons surrounding the node. Thus P = 3 corresponds to a triangle, P = 4 to a rectangle or a square, P = 5 to a pentagon, P = 6 to a hexagon and so on. Some of the very commonly occurring nets are 36 (triangular, T net), 44 (square, S net), 63 (hexagonal, H net) and 3636 (kagome net or K net). These four types of nets are shown schematically in Figure 1.19(a). If a network has nodes which are not equivalent in terms of the polygons surrounding them, the net can be described by listing successively the different corners. For example, if a net is described as 32 434 + 33 42 (2:1), the implication is that in this net there are two types of nodes, 32 434 and 33 42 , and that they occur with a relative frequency of 2:1 (Figure 1.19(b)). A node of the first
56
Phase Transformations: Titanium and Zirconium Alloys T-net
S-net
N-net
K-net
(a)
32434 + 3342(2:1) (b)
Figure 1.19. (a) Schematic representation of a 36 (triangular) T net, a 44 (square) S net, a 63 (hexagonal) H net and a 3636 (kagome) K net of points. (b) The net shown is more complex and contains two types of nodes. This net can be described by the notation 32 434 + 33 42 (2:1). The implication is that these two types of nodes occur with a relative frequency of 2:1. A node of the first type (32 434) is surrounded, in the given order, by two triangles, one square, one triangle and one square, while a node of the second type (33 42 ) is surrounded by three triangles and two squares.
type is surrounded, in the given order, by two triangles, one square, one triangle and one square while a node of the second type is surrounded by three triangles and two squares. A close packed layer of atoms forms a 36 net composed of equilateral triangles. However, not all 36 nets of atoms correspond to close packed layers. To cite an example, the triangles of 36 nets of bcc 110 layers are not equilateral but have angles of 55o , 55o and 70o approximately. In the case of close packing (i.e. a 36 net comprising equilateral triangles), the nodes of one net lie over the centres of
Phases and Crystal Structures
57
the triangles of the nets immediately above and below. Such a situation is not obtained in respect of the stacking of the triangles of the 36 nets of bcc 110 layers (Pearson 1972). The morphologically triangular, 36 (close packed), hexagonal, 63 , and kagome, 3636, nets, together with those made up of squares, are of frequent structural occurrence. A 36 net can be subdivided into a 63 and a larger 36 net (the ratio of number of sites being 2:1) or into a kagome net and a larger 36 net (the ratio of number of sites being 3:1) (Pearson 1972). A primary classification of structures in terms of the stacking of (nearly) planar layers of atoms is quite instructive. For example, numerous structures can be formed by the stacking of T, H or K (triangle– hexagon) layer nets of atoms one over the other sequentially. It is a characteristic of each such layer that it can be positioned about one of three equivalent sites, A, B and C, and this leads to the possibility of varying the stacking sequence and/or the succession of the net types. Moreover, planes of atoms can be constituted of a combination of different layer networks (e.g. hexagonal plus triangular), each of which is occupied by a different chemical species. It is possible to derive a very large number of structure types by permuting the stacking and net sequences. Geometrically close packed (GCP) structures are obtained when the permutation involves only the stacking sequences of the equilateral triangular net. The number of possible structure types may be further increased by chemically ordering the component atoms on the triangular nets. Other structures may be generated by stacking together layer networks of atoms comprising only squares or squares along with triangles, pentagons and/or hexagons. The squares, pentagons or hexagons of one net may or may not be centred by atoms of nets above and below (Pearson 1972). It is interesting to see how several frequently encountered structures in respect of Ti–X and Zr–X intermetallics can be described in terms of the stacking of different types of layer networks of atoms. Before that a brief introduction to topologically close packed (TCP) structures will be provided in view of the fact that quite a few of these intermetallic phases have such structures. Octahedral and tetrahedral voids (Figure 1.20) are the two most common types of interstitial voids present in the simple spherically close packed (i.e. GCP) metallic structures (fcc and hcp). The former are larger and are surrounded by six atoms which form the corners of a triangular antiprism (octahedron). The latter, which are smaller, are enclosed by four atoms which are tetrahedrally disposed. The primitive unit cell of the hcp structure contains two atoms with coordinates (000) and ( 23 13 21 ). There are thus two atoms associated with each lattice point. If the axial ratio has the ideal value (c/a = 8/31/2 ), then the largest interstices (octahedral) have coordinates ( 13 23 41 ) and ( 13 23 43 ). There are two such interstices per unit cell. The next largest interstices (tetrahedral) occur at (00 38 ), (00 58 ), ( 23 13 81 ) and ( 23 13 87 ), there being
58
Phase Transformations: Titanium and Zirconium Alloys
a2
a2
120°
a1
a1 Atoms Octahedral voids Tetrahedral voids
Figure 1.20. This figure shows the locations of octahedral and tetrahedral voids in the hcp structure.
four such interstices per unit cell. The region around a tetrahedral void represents the densest packing of equal sized spheres; all topologically close packed structures are characterized by exclusively tetrahedral voids which may be geometrically imperfect because of the differences in the sizes of the component atoms (Sinha 1972). The coordination polyhedron of an atom is defined by the lines joining the centres of atoms in the shell of close neighbours around it. The coordination is twelve-fold in the cases of the fcc and hcp structures and the polyhedra formed by the twelve neighbours assume the shapes of a cubo-octahedron (fcc) and a twinned cubo-octahedron (hcp), respectively. In the case of the TCP structures yet another type of twelve-fold coordination polyhedron, in which all the faces are triangular, becomes important. This polyhedron is the icosahedron which has twenty faces in the shape of equilateral triangles and thirty edges which correspond to nearest neighbour distances. Each of the other two twelve-fold coordination polyhedra mentioned earlier has 24 edges. In the case of the icosahedron, the distance between the central atom and any atom on the polyhedron surface is around 10% smaller than that between the atoms on the surface. The atoms on the icosahedron surface are more close packed than in the fcc or hcp structures; however, because of the five-fold symmetry axis associated with the icosahedron, it is not possible to have a lattice like arrangement made up solely of icosahedra (Sinha 1972). The condition that only tetrahedral interstices may be present in a TCP structure brings in the requirement that besides a number of atoms having an icosahedral environment, certain others with higher coordination polyhedra around them must also be present; thus TCP structures are characterized by some or all of CN 12, CN 14, CN 15 and CN 16 polyhedra (Kasper 1956). All these Kasper polyhedra
Phases and Crystal Structures
59
have exclusively triangular faces and tetrahedral arrangement. The fractions of sites with different coordination numbers change from structure to structure. By way of illustration, let three TCP structures, be considered: A15 (cP8, Cr 3 Si type), C14 (hP12, MgZn2 type) and C15 (cF24, Cu2 Mg type); in the first, 25% of the sites correspond to CN 12 and 75% to CN 14 while in either of the other (Laves phase) structures, 67% of the sites correspond to CN 12 and 33% to CN 16 (Sinha 1972). It has been pointed out (Frank and Kasper 1958, 1959) that there is a significant consequence of there being, in a crystal structure, coordination polyhedra of CN 12, CN 14, CN 15 and CN 16 and exclusively tetrahedral voids and this is that the resulting structure is generally a layered structure. In fact, most of the TCP structures can be regarded as layered structures. The main atomic layers, referred to as primary layers, are tesselated and contain arrays made up of triangles, pentagons and hexagons. The triangular meshes in the primary layers correspond to the nearest neighbour atoms. Besides the primary layers, generally there are secondary layers in which the coordination does not correspond to nearest neighbours (Sinha 1972). The layer stacking has, of course, to be effected in such a manner that only tetrahedral interstices are present. It has been mentioned in the previous section that certain structures are quite frequently encountered in respect of Ti–X and Zr–X intermetallic phases. A brief account will now be presented to illustrate how these structures can be viewed in terms of the layer stacking sequence representation. The two atomic species considered will be designated as X and Z. First, the case of superstructures based on close packed layers stacked in close packing will be taken up. If X and Z are each on a 44 subnet in each close packed layer, then a family of polytypic structures with XZ stoichiometry and a rectangular arrangement of the components in close packed layers is obtained. An example of such a structure is the L1o (tP4, AuCu type) structure. This structure can also be described in terms of stacking alternate 44 layers of X and Z atoms in succession in the [001] direction. One can next consider the case of XZ3 stoichiometry in each close packed layer, with X atoms on a 36 subnet and Z atoms on a 3636 subnet. A family of polytypic structures with a triangular arrangement of X atoms is obtained. The L12 (cP4, AuCu3 type) and D019 (hP8, Ni3 Sn type) structures belong to this family. In the former, the close packed layers, which lie normal to the [111] direction, are stacked in the sequence ABCABC so that all layers are surrounded cubically. In the latter, the close packed layers are arranged in hexagonal ABAB , stacking. The D019 structure is a hexagonally stacked prototype of the L12 structure. It is thus a superstructure of the hcp (hP2) structure in the same way as the L12 structure is of the fcc (cF4) structure (Pearson 1972). Some superstructures based on bcc packing may now be considered. If one considers the XZ stoichiometry, with X atoms on 44 nets and Z atoms also on 44
60
Phase Transformations: Titanium and Zirconium Alloys
nets, one arrives at the B2 (cP2, CsCl type) structure. In this structure one species occupies the cube corners and the other the body centre, so that alternate layers of the two species occur along <100> directions. The X and Z atoms together form triangular nets parallel to 110 planes with X occupying one of the rectangular 44 subnets resulting from the geometry of the triangles of the 36 net, and Z occupying the other. Next, let the case of the XZ2 stoichiometry be taken up. If one considers 36 close packed layers in bcc [110] stacking with Z atoms on a 63 subnet and X atoms on a larger 36 subnet, one arrives at a family of polytypic structures XZ2 with close packed layers stacked in bcc sequence. The C11b (tI6, MoSi2 type) structure belongs to this family. The C16 (tI12, CuAl2 type) structure can be visualized as being made up of square-triangle, 32 434 nets of Z atoms at z = 0 and z = 21 which are oriented antisymmetrically with respect to each other; the squares in these Z layers which lie over the cell corners and basal face centre, are centred by a 21 44 net of X atoms at z = 41 and z = 43 (Pearson 1972). Even though many structures contain atoms in triangular prismatic coordination, there are some in which this is the main feature of the atomic arrangement. In one class of such structures, 36 and 63 nets of atoms, occupying the same stacking sequence, are stacked alternately in the “paired-layer” sequence (e.g. AaAa). The C32 (hP3, AlB2 type) structure can be regarded as the prototype of this family of structures and one of the simpler structures obtained from this structure is the Bf (oC8, CrB type) structure, which is made up of independent layers of triangular prisms of X atoms parallel to the (010) plane with the prism axes oriented in the [100] direction. The prisms are centred by atoms which form zigzag chains running in the [001] direction (Pearson 1972). An example of structures generated by the stacking of pentagon-triangle nets of atoms is the D88 (hP16, Mn5 Si3 type) structure. In the tetrahedrally close packed A15 (cP8, Cr3 Si type) structure of XZ3 stoichiometry, the X atoms form a bcc array and lines of Z atoms run throughout the structure parallel to the edges of the body centred cell formed by the X atoms. This structure is of the Frank–Kasper type and can be visualized as being formed by the alternate stacking of primary triangle-hexagon 32 62 + 3636 (2:1) layers and secondary 44 layers, with the result that each X atom is surrounded icosahedrally by 12 Z atoms and each Z atom is surrounded by four X atoms and 10 Z atoms in a CN 14 polyhedron with triangular faces. The Laves phase structures have XZ2 stoichiometry and belong to a family of polytypic structures in which three closely spaced 36 nets of atoms are followed by a 3636 kagome net parallel to the (001) plane when the structures are described in terms of a hexagonal cell. The former types of nets are stacked on the same sites as the latter. Alternatively, the Laves phases can be visualised
Phases and Crystal Structures
61
as having Frank–Kasper structures in which pentagon–triangle primary layers of atoms are stacked alternately, parallel to the (110) planes of the hexagonal cell, with secondary 36 triangular layers whose atoms centre the pentagons of the main layers (Pearson 1972). Thus, the C14 (hP12, MgZn2 type) structure is generated by stacking together pairs of primary pentagon–triangle 3535 + 353 (2:3) layers and secondary 36 layers parallel to the (110) plane. The cubic C15 (cF24, MgCu2 type) structure can also be regarded as being built up by stacking consecutively three triangular (36 ) layers and a kagome (3636) layer of atoms which lie in planes normal to the [111] direction in respect of the cubic cell. Each X atom is surrounded by a CN 16 polyhedron of 12 Z and four X atoms while each Z atom is icosahedrally enclosed by six X and six Z atoms. 1.7.3 Derivation of intermetallic phase structures from simple structures The structures of many intermetallics can be regarded as being derived from three simple structures, namely, fcc (A1), bcc (A2) and hcp (A3) structures, which are commonly associated with pure metals and disordered metallic solid solutions. The most common structures exhibited by binary intermetallic phases are listed in Table A1.4 (Ferro and Saccone 1996). Typical intermetallic phase structures derived from the fcc structure include L12 (cP4, AuCu3 type), C15b (cF24, AuBe5 type), L 12 (cP5, Fe3 AlC type), D022 (tI8, TiAl3 type), L1o (tP4, AuCu type), D1a (tI10, Ni4 Mo type), L11 , (hR32, CuPt type) and Pt2 Mo type (oI6) structures (Pitsch and Inden 1991, Sauthoff 1996). Generally these structures are cubic, tetragonal (often with an axial ratio close to unity), rhombohedral or orthorhombic. Examples of intermetallic phases with some of these structures in Ti–X and Zr–X systems are: TiCo3 , TiIr 3 , -TiNi3 , -TiPt3 , TiRh3 , TiZn3 , Zr 3 Al, ZrHg3 , Zr 3 In, ZrIr 3 , ZrPt 3 , ZrRh3 L12 ; ZrNi5 C15b ; TiAl3 , TiGa3 , -ZrIn3 D022 ; TiAl, -TiCu3 , TiGa, TiHg, Ti3 In2 , -TiRh, ZrHg (L1o ); and TiAu4 , -TiCu4 , TiPt8 D1a . Common intermetallic phase crystal structures derived from the bcc structure incldue B2 (cP2, CsCl type), B32 (cF16, NaTl type), D03 (cF16, BiF3 type), and L21 (cF16, Cu2 AlMn type) structures. (Pitsch and Inden 1991, Sauthoff 1996). Generally these structure are cubic. Examples of intermetallic phases with the B2 structure in Ti–X and Zr–X systems are: -TiAu, TiBe, TiCo, TiFe, -TiIr, TiNi, TiOs, -TiPd, -TiPt, -TiRh, TiRu, TiTc, TiZn, S-ZrCo, ZrCu, ZrIr, ZrOs, -ZrPt, -ZrRh, ZrRu and ZrZn. Intermetallics with other bcc-based structures are rare in these systems. Prominent among the intermetallic phase crystal structures derived from the hcp structure are: Bh (hP2, WC type), D019 (hP8, Ni3 Sn type), B19 (oP4, AuCd type), C49 (oC12, ZrSi2 type) and D0a (oP8, -Cu3 Ti type) structures. Generally, these structures are hexagonal or orthorhombic. Examples of intermetallic phases
62
Phase Transformations: Titanium and Zirconium Alloys
with these structures in Ti–X and Zr–X systems include Zr3 Se2 Bh ; Ti3 Al, Ti3 Ga, Ti3 In, Ti4 Pd, Ti4 Sb, Ti3 Sn, Zr 3 Co, ZrNi3 D019 ; -TiAu, -TiPd, -TiPt (B19); ZrGe2 , ZrSi2 (C49); and -TiCu3 , ZrAu3 D0a . 1.7.4 Intermetallic phases with TCP structures in Ti–X and Zr–X systems It has been mentioned earlier that quite a few of the intermetallic phases occurring in Ti–X and Zr–X systems have topologically closed packed (TCP) structures. These phases are mostly A15 (cP8, Cr 3 Si type) phases or Laves phases (C14, C15 or C36 structures). Examples of such phases are: Ti3 Au, Ti3 Hg, Ti3 Ir, Ti4 Pd (stoichiometric), Ti3 Pt, Ti3 Sb, Zr 3 Au, Zr 3 Hg, Zr 4 Sn, Zr 4 Tl (A15); -TiCr2 , TiFe2 , TiMn2 , TiZn2 ZrAl2 , -ZrCr2 , ZrMn2 , ZrRe2 , ZrRu2 , ZrTc2 (C14); TiBe2 , TiCo2 , -TiCr2 , -ZrCr2 , ZrFe2 , ZrIr 2 , ZrMo2 , ZrV2 , ZrW2 , ZrZn2 (C15); TiCo2 , -TiCr2 and -ZrCr2 (C36, hP24, MgNi2 type). The phase Zr 4 Al3 (hP7) is also a TCP phase the structure of which can be described either in terms of pentagon–triangle primary and 44 secondary nets parallel to the (110) plane, or with hexagon–triangle primary and 36 secondary nets parallel to the (001) plane (Pearson 1972). Some of the Laves phases mentioned above can absorb very significant quantities of hydrogen and, for this reason, are considered for applications as hydrogen storage materials. Reference must be made in this context to the phases ZrV2 , ZrCr 2 , ZrMn2 and TiCr 2 which exhibit high sorption capacities with hydrogen to metal ratio (H/M) values of 1.8, 1.3, 1.2 and 1.2, respectively (Sauthoff 1996). 1.7.5 Phase stability in zirconia-based systems Zirconia (ZrO2 )-based systems are among the most extensively investigated ceramics in so far as phase transformation studies are concerned. Not only do they exhibit interesting phase transformations, but also the properties of these ceramics can be engineered by suitably controlling the stability of different competing phases and by inducing phase transformations in a desired manner. In view of this, zirconia-based systems are pedagogically very appropriate systems for illustrating how phase transformations can be effectively utilized for controlling microstructure and, in turn, properties – mechanical, thermal, electrical and optical – of ceramics. Crystal structures and stability of different phases in zirconia ceramics are briefly described in this section. 1.7.5.1 ZrO2 polymorphs Pure ZrO2 exhibits three polymorphic forms under ambient pressure; these belong, respectively, to monoclinic, tetragonal and cubic crystal systems (Garvie 1970). The crystal structures and lattices parameters of these polymorphs are given in Table 1.18
Phases and Crystal Structures
63
Table 1.18. Crystal structures and lattice parameters of zirconia polymorphs. Phase
Crystal structure
Lattice parameters (nm)
Pearson symbol
Space group
m-ZrO2
mP12
P21 /c
t-ZrO2
tP6
P42 /nmc
c-ZrO2
cF12
¯ Fm3m
a = 0.5156 b = 0.5191 c = 0.5304 = 98 9o a = 0.5094 c = 0.5177 c/a = 1.016 a = 0.5124
(Stevens 1986, Massalski et al. 1992). The occurrence of an orthorhombic form of ZrO2 under high pressures has also been reported (Lenz and Heuer 1982). The monoclinic phase (generally designated as m-ZrO2 ) is stable upto about 1443 K where it transforms to the tetragonal phase (t-ZrO2 ) which is stable upto 2643 K; at still higher temperatures, the cubic phase (c-ZrO2 ) is encountered which is stable upto the melting temperature of 2953 K (Stevens 1986). Among these three phases, the monoclinic phase, has the lowest density. In m-ZrO2 , the Zr 4+ ion has seven-fold coordination with O ions, with a range of Zr–O bond lengths and bond angles. The OII coordination is close to tetrahedral with only one angle (134 3o ) differing significantly from the angle of the tetrahedron (109 5o ) while the OI coordination is triangular. The Zr ions are located in layers parallel to (100) planes, separated by OI and OII ions on either side. The average Zr–OI and Zr–OII distances are 0.207 and 0.221 nm, respectively (Stevens 1986). Figure 1.21 shows a schematic of the idealized ZrO7 polyhedron. Each Zr 4+ ion in t-ZrO2 is surrounded by eight O ions. There is some distortion in this eight-fold coordination due to the fact that while four of the O ions are at a distance of 0.2065 nm, in the form of a flattened tetrahedron, the other four are at a distance of 0.2455 nm in an elongated tetrahedron rotated through 90o (Stevens 1986). The high temperature cubic phase, c-ZrO2 , has the fcc fluorite (CaF2 ) type structure, in which each Zr 4+ ion is coordinated by eight equidistant O ions which are arranged in two equal tetrahedra. A layer of ZrO8 groups in c-ZrO2 is shown in Figure 1.22. 1.7.5.2 Stabilization of high temperature polymorphs An important concept which is often utilized in zirconia ceramics is to “alloy” pure ZrO2 with another suitable oxide to fully or partially stabilize high temperature polymorphs of ZrO2 to lower temperatures.
64
Phase Transformations: Titanium and Zirconium Alloys
a
a
Zr O
Figure 1.21. Schematic showing the idealised ZrO7 polyhedron pertinent to m-ZrO2 . I I I
II
II II
II
Zr O
Figure 1.22. This figure shows a layer of ZrO8 groups in c-ZrO2 .
The tetragonal to monoclinic transformation, which is martensitic in nature, is accompanied by a large (3–5%) volume expansion which is sufficient to exceed elastic and fracture limits even in relatively small grains of pure ZrO2 and can only be accommodated by cracking. A consequence of this is that the fabrication of large components of pure ZrO2 is not possible due to spontaneous failure on cooling.
Phases and Crystal Structures
65
The addition of cubic stabilizing oxides in appropriate amounts can permit the cubic polymorph to be stable over a wide range of temperatures: even from room temperature to its melting temperature. The oxides that are commonly used to form solid solutions with ZrO2 include MgO (magnesia), CaO (calcia) and RE oxides such as Y2 O3 (yttria) and CeO2 (ceria). These oxides exhibit extensive solid solubility in ZrO2 and are able to form fluorite type phases which are stable over wide ranges of composition and temperature. If the amount of stabilizing oxide added to ZrO2 is insufficient for complete stabilization of the cubic phase, then a partially stabilized zirconia (PSZ) is obtained rather than a fully stabilized form. The PSZ usually comprises a mixture of two or more phases. Both the cubic solid solution and the tetragonal solid solution are present and the latter may transform to the monoclinic solid solution on cooling. It may be mentioned here that the volume expansion associated with the tetragonal to monoclinic transformation may be used to advantage for improving toughness and strength. This aspect will be discussed in a later chapter. A relatively tough, partially stabilized zirconia ceramic, consisting of a dispersion of metastable tetragonal ZrO2 inclusions within large grains of stabilized cubic ZrO2 , can be derived by inducing a stress induced tetragonal to monoclinic transformation. An appraisal of the phase equilibria of zirconia with other oxide systems is very important with regard to the application of zirconia as an engineering ceramic. However, many difficulties are encountered while determining the equilibrium phase diagrams of even the simplest binary zirconia systems. First, the reactions in these systems at relatively low temperatures (<1680K) are somewhat sluggish owing to low diffusivity. Since diffusional reactions such as precipitation, eutectoid decomposition and ordering proceed very slowly, equilibrium is often difficult to achieve. Secondly, coherency strain between different phases may influence their relative stabilities in two phase regions and may be responsible for the retention of metastable phases. Finally, at high temperatures, the presence of impurities or non-stoichiometry may influence the equilibria. The salient features of binary phase diagrams of three important zirconia systems, namely, ZrO2 –MgO, ZrO2 –CaO and ZrO2 –Y2 O3 , are summarized in the following sections. 1.7.5.3 ZrO2 –MgO system The polymorphic transformations of pure ZrO2 , the eutectoid decomposition of the fluorite type solid solution into a tetragonal solid solution and MgO (the eutectoid temperature and composition being ∼1673 K and and ∼13 mole % MgO, respectively), and a very limited (<1 mole %) solubility of MgO in tetragonal ZrO2 at 1573K are some important data (Grain 1967) for the construction of the phase diagram which still remains somewhat tentative. The phase boundaries of
66
Phase Transformations: Titanium and Zirconium Alloys
the fluorite type solid solution field at high temperatures, the solubility of MgO in monoclinic ZrO2 and the occurrence of a eutectoid reaction in the tetragonal solid solution field still appear to be uncertain (Stubican 1988). In this system, the occurrence of a fluorite related ordered compound, Mg2 Zr 5 O12 (28.57 mole % MgO) has been established but not the regime of its stability. The formation of another metastable ordered compound, MgZr6 O13 (14.28 mole % MgO) has also been reported (Rossell and Hannink 1984). 1.7.5.4 ZrO2 –CaO system The ZrO2 -rich side of the ZrO2 –CaO system in in some ways similar to that of the ZrO2 –MgO system in that the terminal cubic as well as tetragonal solid solution phases decompose by eutectoid reactions. However, the solubility limits of CaO in both the tetragonal and monoclinic phases are considerably higher than those of MgO. Two ordered phases CaZrO9 ( 1 ) and Ca6 Zr 19 O44 ( 2 ) have been found to occur in this system and these are stable in the temperature ranges 1393 to 1502 ± 15 K and 1398 to 1528 ± 15 K, respectively. The CaZrO3 phase exhibits a polymorphic transformation at 2023 ± 30 K from a high temperature cubic to a lower temperature orthorhombic form. The presence of the two polymorphhs of the CaZrO3 phase and the ordered 1 and 2 phases introduces several phase reactions in the c-ZrO2 + CaZrO3 phase field. The ZrO2 rich side of the ZrO2 –CaO phase diagram (Stubican 1988) is shown in Figure 1.23. 3270
L L+φ
2770
Temperature (K)
C+L C
C + φ(c)
2270
2023 K 1770
T
C + φ(o) 1628 K φ φ2 + φ(o) φ1 2
T+C 1413 K
1270
T+M M 770
0
ZrO2
10
M + φ(o) 20
30
Mol % CaO
40
50
CaZrO3
Figure 1.23. The ZrO2 -rich side of the ZrO2 –CaO phase diagram. M, T and C stand, respectively, for the monoclinic, tetragonal and cubic (fluorite type) solid solutions and L for liquid; the symbols , (c), (o), 1 and 2 represent the phases CaZrO3 , CaZrO3 (cubic), CaZrO3 (orthorhombic), CaZrO9 and Ca6 Zr 19 O44 , respectively.
Phases and Crystal Structures
67
Temperature (K)
2670
2070
C C+T T
1470 M+T 870 M M+C M + ZYO 0
2
4
6
8
10
12
14
16
Mol % Y2O3
Figure 1.24. This figure shows the solid state phase relations in the ZrO2 rich end of the ZrO2 –Y2 O3 phase diagram. M, T and C stand, in that order, for the monoclinic, tetragonal and cubic solid solutions, while ZYO stands for the phase Zr 3 Y4 O12 .
1.7.5.5 ZrO2 –Y2 O3 system The low Y2 O3 region of the ZrO2 –Y2 O3 system is very important for technological applications. For this reason the subsolidus equilibria in this portion of the phase diagram have been examined extensively. Figure 1.24 depicts the phase relations in the ZrO2 -rich side of this system (Stubican 1988) subject to the possible limitations imposed by the fact that below ∼1500 K it is extremely difficult to attain equilibrium, even with reactive powders. The ordered compound, Zr 3 Y4 O12 , is stable up to 1655 ± 5 K. At still higher temperatures, it disorders to a fluorite-type solid solution.
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Phases and Crystal Structures
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APPENDIX Table A1.1. Crystal structures of important binary intermetallic phases in Ti–X systems (Massalski et al. 1992). Phase Ti2 Ag TiAg Ti3 Al TiAl Ti3 Al∗5 TiAl2 TiAl3 Ti3 Au TiAu (HT) TiAu (MT) TiAu (LT) TiAu2 TiAu4 TiB Ti3 B4 TiB2 TiBe2 TiBe3 Ti2 Be17 (HT) Ti2 Be17 (LT) TiBe12 TiBe TiC Ti2 C Ti2 Cd TiCd Ti2 Co TiCo TiCo2 TiCo2 TiCo3 TiCr 2 (LT) TiCr 2 (MT) TiCr 2 (HT)
Composition (at.% X) 33.3 48–50 22–39 48–69.5 58–63 65–68 75 25 38–58 49–50 50 66.7 79–82 49–50 56.1 65.6–66.7 66.7 75 89.5 89.5 92.3 ∼50 ∼32–48.8 ∼32–36 33.3 50 32.9–33.3 49–55 66.5–67 68.75–72 75.5–80.7 63–65 64–66 64–66
Space group
Pearson symbol
Strukturbericht designation
14/mmm P4/nmm P63 /mmc P4/mmm 14/mbm 141 /amd 14/mmm ¯ Pm3n ¯ Pm3m Pmma P4/nmm 14/mmm 14/m Pnma Immm P6/mmm ¯ Fd3m ¯ R3m P63 /mmc ¯ R3m 14/mmm ¯ Pm3m ¯ Fm3m ¯ Fd3m P4/mmm P4/nmm ¯ Fd3m ¯ Pm3m ¯ Fd3m P63 /mmc ¯ Pm3m ¯ Fd3m P63 /mmc P63 /mmc
tI6 tP4 hP8 tP2 tP32 tI24 tI8 cP8 cP2 oP4 tP4 tI6 tI10 oP8 oI14 hP3 cF24 hR12 hP38 hR19 tI26 cP2 cF8 cF48 tI6 tP4 cF96 cP2 cF24 hP24 cP4 cF24 hP12 hP24
C11b B11 D019 L1o D022 A15 B2 B19 B11 C11b D1a B27 D7b C32 C15 D2b B2 B1 C11b B11 E93 B2 C15 C36 L12 C15 C14 C36 (Continued)
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Phase Transformations: Titanium and Zirconium Alloys
Table A1.1. (Continued) Phase
Composition (at.% X)
Space group
Pearson symbol
Strukturbericht designation
Ti3 Cu Ti2 Cu TiCu Ti3 Cu4 Ti2 Cu3 TiCu2 TiCu4 TiCu4 TiCu∗3 Ti–Cu phase∗ TiFe TiFe2 Ti3 Ga Ti2 Ga Ti5 Ga∗3 Ti5 Ga∗4 TiGa∗ Ti3 Ga∗5 TiGa3 Ti5 Ge3 TiGe2 hydride hydride hydride∗ Ti3 Hg TiHg Ti3 In Ti3 In2 Ti3 Ir TiIr TiIr 3 TiMn2 TiMn4 Ti2 N TiN nitride Ti2 Ni TiNi TiNi3 TiNi∗3 Ti3 O Ti2 O TiO
25 33.3 48–52 57.1 60 66.7 78–80.9 ∼78–∼80.9 48–50.2 64.5–72.4 25 33.3 37.5 44.4 50 62.5 75 37.5 66.7 1.05–2.0 1.72–2.0 0.01–0.03 25 50 >21 25–27 35–57.5 73–77 60–70 81.5 ∼33 28 to > 50 ∼38 33.3 49.5–57 75 ∼83–88 ∼20–∼30 ∼25–33.4 34.9–55.5
P4/mmm I4/mmm P4/nmm I4/mmm P4/nmm Amm2 Pnma I4/m Pmmn P4/mmm ¯ Pm3m P63 /mmc P63 /mmc P63 /mmc 14/mcm P64 /mcm P4/mmm P4/mbm 14/mmm P63 /mcm Fddd ¯ Fm3m I4/mmm P42 /n ¯ Pm3n P4/mmm P63 /mmc P4/mmm ¯ Pm3n ¯ Pm3m ¯ Pm3m P63 /mmc ¯ R3m P42 /mnm ¯ Fm3m I41 /amd ¯ Fd3m ¯ Pm3m P63 /mmc ¯ Pm3m ¯ P 31c ¯ P 3m1 ¯ Fm3m
tP4 tI6 tP4 tI14 tP10 oC12 oP20 tI10 oP8 tP2 cP2 hP12 hP8 hP6 tI32 hP18 tP2 tP32 tI8 hP16 oF24 cF12 tI2 tP6 cP8 tP2 hP8 tP2 cP8 cP2 cP4 hP12 hR53 tP6 cF8 tI12 cF96 cP2 hP16 cP4 hP ∼ 16 hP3 cF8
L6o C11b B11 D1a D0a L1o B2 C14 D019 B82 D8m L1o D022 D88 C54 C1 L 2b A15 L1o D019 L1o A15 B2 L12 C14 C4 B1 Cc B2 D024 L12 B1
Phases and Crystal Structures
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Table A1.1. (Continued) Phase Ti3 O2 Ti2 O3 Rutile Anatase∗ Brookite∗ TiOs Ti3 P Ti5 P3 TiP TiP2 Ti4 Pb Ti4 Pd Ti2 Pd TiPd (HT) TiPd (LT) Ti2 Pd3 Ti3 Pd5 TiPd2 TiPd3 Ti–Pd phase TiPo Ti3 Pt TiPt (HT) TiPt (LT) Ti3 Pt 5 TiPt 3 Ti–Pt phase TiPt8 Ti5 Re24 Ti2 Rh TiRh (HT) TiRh (LT) Ti3 Rh5 TiRh3 TiRu TiS TiS2 TiS3 Ti4 Sb Ti3 Sb TiSb
Composition (at.% X) ∼40 59.8–60.2 ∼66.7 38–51 25 ∼36–∼ 39 48–50 66.7 ∼20 20 33.3 47–53 47–53 60 62.5 65–67 75 75–84 50 22–29 46–54 46–54 62.5 <75 75–81 89–98 82.8 33.3 ∼38–58 ∼38–58 62.5 73–78 45–52 ∼49.7 64.4–66.7 ∼75 >20.1–<25 25 50
Space group
Pearson symbol
Strukturbericht designation
P6/mmm ¯ R3c P42 /mnm I41 /amd Pbca ¯ Pm3m P42 /n P63 /mcm P63 /mmc I4/mcm P63 /mmc ¯ Pm3n I4/mmm ¯ Pm3n Pmma Cmcm P4/mmm I4/mmm P63 /mmc P4/mmm P63 /mmc ¯ Pm3n ¯ Pm3m Pmma Ibam P63 /mmc ¯ Pm3m I4/m ¯ I 43m I4/mmm ¯ Pm3m ¯ Pm3m Pbam ¯ Pm3m ¯ Pm3m P63 /mmc ¯ P 3m1 P21 /m P63 /mmc ¯ Pm3n P63 /mmc
hP ∼ 5 hR30 tP6 tI12 oP24 cP2 tP32 hP16 hP8 tI12 hP8 cP8 tI6 cP2 oP4 oC20 tP8 tI6 hP16 cP4 hP4 cP8 cP2 oP4 0I32 hP16 cP4 tI18 cI58 tI6 cP2 tP2 oP16 cP4 cP2 hP4 hP3 mP8 hP8 cP8 hP4
D51 C4 C5 C21 B2 D88 Bi C16 D019 A15 C11b B2 B19 ∼C11b C11b D024 L12 B81 A15 B2 B19 D024 L12 D1a A12 C11b B2 L1o L12 B2 B81 C6 D019 A15 B81 (Continued)
76
Phase Transformations: Titanium and Zirconium Alloys
Table A1.1. (Continued) Phase
Composition (at.% X)
Space group
Pearson symbol
Strukturbericht designation
TiSb2 Ti8 Se9 Ti3 Se4 Ti3 Si Ti5 Si3 TiSi
66.2–67.1 ∼52.9 55–57.6 25 35.5–39.5 50
TiSi2 Ti3 Sn Ti2 Sn Ti5 Sn3 Ti6 Sn5 (HT) Ti6 Sn5 (LT) TiTc Ti–Tc phase Ti5 Te4 TiTe Ti3 Te4 TiTe2 TiU2 TiZn15 TiZn3 TiZn2 TiZn Ti2 Zn
66.7 23–25 32.7–35.9 37.5 45.5 45.5 ∼50 ∼85 44.4 >40 55–59.2 ∼60–66.7 66.7 93.7 75 66.7 50 33.3
I4/mcm ¯ R3m C2/m P42 /n P63 /mcm Pmm2 Pnma Fddd P63 /mmc P63 /mmc P63 /mcm P63 /mmc Immm ¯ Pm3m ¯ I 43m I4/m ∼P63 /mmc C2/m ¯ P 3m1 P6/mmm Cmcm ¯ Pm3m P63 /mmc ¯ Pm3m I4/mmm
tI12 hP12 mC14 tP32 hP16 oP8 oP8 oF24 hP8 hP6 hP16 hP22 oI44 cP2 cI58 tI18 hP16 mC14 hP3 hP3 oC68 cP4 hP12 cP2 tI6
C16 D88 B27 C54 D019 B82 D88 B2 A12 ∼B81 C6 C32 L12 C14 B2 C11b
∗
Metastable phase; HT: High temperature phase; MT: Medium temperature phase; LT: Low temperature phase.
Table A1.2. Crystal structures of important binary intermetallic phases in Zr–X systems (Massalski et al. 1992). Phase
Composition (at.% X)
Zr 2 Ag ZrAg Zr3 Al Zr 2 Al Zr 5 Al3 Zr 3 Al2
33.3 50 25 33.3 37.5 40
Space group
Pearson symbol
Strukturbericht designation
14/mmm P4/nmm ¯ Pm3m P63 /mmc I4/mcm P42 /mmm
tI6 tP4 cP4 hP6 tI32 tP20
C11b B11 L12 B82 D8m
Phases and Crystal Structures
77
Table A1.2. (Continued) Phase
Composition (at.% X)
Zr 4 Al3 Zr 5 Al4 ZrAl Zr 2 Al3 ZrAl2 ZrAl3 Zr 3 Au Zr 2 Au ZrAu2 ZrAu3 ZrAu4 ZrB2 ZrB12 ZrBe∗ ZrBe2 ZrBe5 Zr 2 Be17 ZrBe13 ZrC Zr 2 Cd ZrCd3 Zr 3 Co
42.9 44.4 50 60 66.7 75 25 33.3 66.7 75 80 66.7–68 92.4 50 66.7 83.3 89.5 92.9 33–50 33.3 75 25
Zr–Co phase Zr–Co phase Zr–Co phase Zr–Co phase ZrCr 2 (HT) ZrCr 2 (MT) ZrCr 2 (LT) Zr 2 Cu ZrCu Zr 3 Fe Zr 2 Fe ZrFe2 ZrFe3 Zr 2 Ga Zr 5 Ga3 Zr 3 Ga2 Zr 5 Ga4 ZrGa
33.3 ∼50 >65 to ∼73 79.3 64–69 64–69 64–69 33.3 50 24–26.8 31–33.3 66–72.9 75 33.3 37.5 40 44.4 50
Space group
Pearson symbol
Strukturbericht designation
P 6¯ P63 /mcm Cmcm Fdd2 P63 /mmc I4/mmm ¯ Pm3n I4/mmm I4/mmm Pmmn Pnma P6/mmm ¯ Fm3m Cmcm P6/mmm P6/mmm ¯ R3m ¯ Fm3c ¯ Fm3m I4/mmm P4/mmm Cmcm P63 /mmc I4/mcm ¯ Pm3m ¯ Fd3m ¯ Fm3m P63 /mmc P63 /mmc ¯ Fd3m I4/mmm ¯ Pm3m Cmcm I4/mcm ¯ Fd3m ¯ Fm3m I4/mcm P63 /mcm P4/mbm P63 /mcm I41 /amd
hP7 hP18 oC8 oF40 hP12 tI16 cP8 tI6 tI6 oP8 oP20 hP3 cF52 oC8 hP3 hP6 hR19 cF112 cF8 tI6 tP4 oC16 hP8 tI12 cP2 cF24 cF116 hP12 hP24 cF24 tI6 cP2 oC16 tI12 cF24 cF116 tI12 hP16 tP10 hP18 tI16
Bf C14 D023 A15 C11b C11b D0a C32 D2f Bf C32 D2d D23 B1 C11b L6o E1a D019 C16 B2 C15 D8a C14 C36 C15 C11b B2 E1a C16 C15 D8a C16 D88 D5a Bg (Continued)
78
Phase Transformations: Titanium and Zirconium Alloys
Table A1.2. (Continued) Phase Zr 2 Ga3 Zr 3 Ga5 ZrGa2 ZrGa3 Zr 3 Ge Zr 5 Ge3 Zr 5 Ge4 ZrGe ZrGe2 hydride hydride hydride∗ Zr 3 Hg ZrHg ZrHg3 Zr 3 In Zr 2 In ZrIn ZrIn2 ZrIn3 (HT) ZrIn3 (LT) Zr 3 Ir Zr 2 Ir Zr 5 Ir 3 ZrIr ZrIr 2 ZrIr 3 ZrMn2 ZrMo2 ZrN Zr 2 Ni ZrNi ZrNi3 ZrNi5 ZrO2−x (HT) ZrO2−x (MT) ZrO2−x (LT) ZrOs ZrOs2 Zr 3 P
Composition (at.% X) 60 62.5 66.7 75 25 37.5 44.4 50 66.7 56.7–66.4 63.6 ∼1.0 25 50 75 25 33.3 50 66.7 75 75 25 33.3 37.5 48–53 66.7 70–81 60–79.2 60–67 >40 33.3 50 74–75.5 81.6–85.2 61–66.7 66.5–66.7 66.7 50 >61–∼70 25
Space group
Pearson symbol
Strukturbericht designation
Fdd2 Cmcm Cmmm I4/mmm P42 /n P63 /mcm P41 21 2 Pmma Cmcm ¯ Fm3m I4/mmm P42 /n ¯ Pm3n P4/mmm ¯ Pm3m ¯ Pm3m P4/mmm ¯ Fm3m I41 /amd I4/mmm I4/mmm ¯ I 42m I4/mcm P63 /mcm ¯ Pm3m ¯ Fd3m ¯ Pm3m P63 /mmc ¯ Fd3m ¯ Fm3m I4/mcm Cmcm P63 /mmc ¯ F 43m ¯ Fm3m P42 /nmc P21 /c ¯ Pm3m P63 /mmc P42 /n
oF40 oC32 oC12 tI16 tP32 hP16 tP36 oP8 oC12 cF12 tI6 tP6 cP8 tP2 cP4 cP4 tP2 cF4 t124 tI8 tI16 tI32 tI12 hP16 cP2 cF24 cP4 hP12 cF24 cF8 tI12 oC8 hP8 cF24 cF12 tP6 mP12 cP2 hP12 tP32
D023 D88 B27 C49 C1 L 2b A15 L1o L12 L12 L1o A1 D022 D023 C16 D88 B2 C15 L12 C14 C15 B1 C16 Bf D019 C15b C1 C43 B2 C14
Phases and Crystal Structures
79
Table A1.2. (Continued) Phase
Composition (at.% X)
ZrP (HT) ZrP (LT) ZrP2 Zr 5 Pb3 Zr 2 Pd ZrPd ZrPd2 ZrPd3 ZrPo Zr 5 Pt3 ZrPt (HT) ZrPt (LT) ZrPt3
50 50 66.7 37.5 33.3 50 66.7 75 50 37.5 50 50 75
Zr 3 Pu ZrPu4 ZrRe2 Zr 5 Re24 Zr 2 Rh ZrRh (HT) ZrRh (LT) Zr 3 Rh5 ZrRh3 ZrRu ZrRu2 Zr 2 S Zr 3 S2 ZrS
26 70–90 66.7 ∼82.8 33.3 50–62 > 50 62.5 72–82 48–52 66–68 33.3 40 50
ZrS2 ZrS3 Zr 3 Sb Zr 5 Sb3 ZrSb2 Zr 2 Se Zr 3 Se2 Zr 2 Se3 ZrSe2 ZrSe3 Zr 3 Si
66.7 75 25 36 66.7 33.3 40 60 64.9–66 75 ∼25
Space group
Pearson symbol
Strukturbericht designation
¯ Fm3m P63 /mmc Pnma P63 /mcm I4/mmm ¯ Fm3m I4/mmm P63 /mmc P63 /mmc P63 /mcm ¯ Pm3m Cmcm ¯ Pm3m P63 /mmc P6/mmm P4/ncc P63 /mmc ¯ I 43m I4/mcm ¯ Im3m ¯ Pm3m Cmcm ¯ Pm3m ¯ Pm3m P63 /mmc Pnnm ¯ P 6m2 ¯ Fm3m P4/nmm ¯ P 3m1 P21 /m I 4¯ P63 /mcm Pnnm Pnnm ¯ P 6m2 P63 mc ¯ P 3m1 P21 /m P42 /n I 4¯
cF8 hP8 oP12 hP16 tI6 cF4 tI6 hP16 hP4 hP16 cP2 oC8 cP4 hP16 hP3 tP80 hP12 cI58 tI12 cI2 cP2 oC32 cP4 cP2 hP12 oP36 hP2 cF8 tP4 hP3 mP8 tI32 hP16 oP24 oP36 hP2 hP8 hP3 mP8 tP32 tI38
B1 Bi C23 D88 C11b A1 C11b D024 B81 D88 B2 Bf L12 D024 C32 C14 A12 C16 A2 B2 L12 B2 C14 Bh B1 B11 C6 D0e D88 Bh C6 D0e (Continued)
80
Phase Transformations: Titanium and Zirconium Alloys
Table A1.2. (Continued) Phase Zr 2 Si Zr 5 Si3 Zr 3 Si2 ZrSi (HT) ZrSi (LT) ZrSi2 Zr 4 Sn Zr 5 Sn3 ZrSn2 ZrTc2 ZrTc6 Zr 3 Te Zr 5 Te4 ZrTe ZrTe2 ZrTe3 Zr 4 Tl U–Zr phase ZrV2 ZrW2 Zr 2 Zn Zr 3 Zn2 ZrZn ZrZn2 ∗
Composition (at.% X) 33.3 37.5 40 50 50 66.7 ∼20 33–∼40 66.7 66.7 85.7 25 44.4 50 55–66.7 75 20 22–37 ∼66.7 ∼66.7 33.3 39.5 50 66.7
Space group
Pearson symbol
Strukturbericht designation
I4/mcm P63 /mcm P4/mbm Cmcm Pnma Cmcm ¯ Pm3n P63 /mcm Fddd P63 /mmc ¯ I 43m ¯ R3m I4/m P63 /mmc ¯ P 3m1 P21 /m ¯ Pm3n P6/mmm ¯ Fd3m ¯ Fd3m I4/mmm P42 nm ¯ Pm3m ¯ Fd3m
tI12 hP16 tP10 oC8 oP8 oC12 cP8 hP16 oF24 hP12 cI58 hR12 tI18 hP4 hP3 mP8 cP8 hP3 cF24 cF24 tI16 tP20 cP2 cF24
C16 D88 D5a Bf B27 C49 A15 D88 C54 C14 A12 B81 C6 A15 C32 C15 C15 D023 B2 C15
Metastable phase; HT: High temperature phase; MT: Medium temperature phase; LT: Low temperature phase.
Table A1.3. Nomenclature of crystal structures: strukturbericht designations and corresponding Pearson symbols (Massalski et al. 1992). Strukturbericht designation
Prototype phase
Space group
Pearson symbol
Aa Ab Ac Ad Af Ag Ah Ai
Pa U Np Np HgSn6−10 B Po Po
I4/mmm P42 /mnm Pnma P421 2 P6/mmm P42 /nnm ¯ Pm3m ¯ R3m
tI2 tP30 oP8 tP4 hP1 tP50 cP1 hR1
Phases and Crystal Structures
81
Table A1.3. (Continued) Strukturbericht designation
Prototype phase
Space group
Pearson symbol
Ak Al A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16 A17 A20 Ba Bb Bc Bd Be Bf Bg Bh Bi Bk Bl Bm B1 B2 B3 B4 B81 B82 B9 B10 B11
Se Se Cu W Mg C (diamond) Sn In As Se C (graphite) Hg Ga Mn Mn I2 Cr 3 Si S P (black) U CoU AgZn CaSi NiSi CdSb CrB MoB WC TiAs BN AsS TiB NaCl CsCl ZnS (sphalerite) ZnS (wurtzite) NiAs Ni2 In HgS PbO CuTi
P21 /c P21 /c ¯ Fm3m ¯ Im3m P63 /mmc ¯ Fd3m I41 /amd I4/mmm ¯ R3m P31 21 P63 /mmc ¯ R3m Cmca ¯ I 43m P41 32 Cmca ¯ Pm3n Fddd Cmca Cmcm 121 3 P 3¯ Cmmc Pbnm Pbca Cmcm I41 /amd ¯ P 6m2 P63 /mmc P63 /mmc P21 /c Pnma ¯ Fm3m ¯ Pm3m ¯ F 43m P63 mc P63 /mmc P63 /mmc P31 21 P4/nmm P4/nmm
mP64 mP32 cF4 cI2 hP2 cF8 tI4 tI2 hR2 hP3 hP4 hR1 oC8 cI58 cP20 oC8 cP8 oF128 oC8 oC4 cI16 hP9 oC8 oP8 oP16 oC8 tI16 hP2 hP8 hP4 mP32 oP8 cF8 cP2 cF8 hP4 hP4 hP6 hP6 tP4 tP4 (Continued)
82
Phase Transformations: Titanium and Zirconium Alloys
Table A1.3. (Continued) Strukturbericht designation
Prototype phase
Space group
Pearson symbol
B13 B16 B17 B18 B19 B20 B26 B27 B29 B31 B32 B34 B35 B37 Ca Cb Cc Ce Cg Ch Ck C1 C1b C2 C3 C4 C6 C7 C8 C9 C10 C11a C11b C12 C14 C15 C15b C16 C18 C19 C21 C22
NiS GeS PtS CuS AuCd FeSi CuO FeB SnS MnP NaTl PdS CoSn SeTl Mg2 Ni CuMg2 ThSi2 PdSn2 ThC2 Cu2 Te LiZn2 CaF2 AgAsMg FeS2 (pyrite) Ag2 O TiO2 (rutile) CdI2 MoS2 SiO2 (high quartz) SiO2 ( crystobalite) SiO2 ( tridymite) CaC2 MoSi2 CaSi2 MgZn2 Cu2 Mg AuBe5 Al2 Cu FeS2 (marcasite) Sm TiO2 (brookite) Fe2 P
¯ R3m Pnma P42 /mmc P63 /mmc Pmma P21 3 C2/c Pnma Pmcn Pnma ¯ Fd3m P42 /m P6/mmm I4/mcm P62 22 Fddd I41 /amd Aba2 C2/c P6/mmm P63 /mmc ¯ Fm3m ¯ F 43m Pa3 ¯ Pn3m P42 /mnm ¯ P 3m1 P63 /mmc P62 22 ¯ Fd3m P63 /mmc I4/mmm I4/mmm ¯ R3m P63 /mmc ¯ Fd3m ¯ F 43m I4/mcm Pnnm ¯ R3m Pbca ¯ P 62m
hR6 oP8 tP4 hP12 oP4 cP8 mC8 oP8 oP8 oP8 cF16 tP16 hP6 tI16 hP18 oF48 tI12 oC24 mC12 hP6 hP3 cF12 cF12 cP12 cP6 tP6 hP3 hP6 hP9 cF24 hP12 tI6 tI6 hR6 hP12 cF24 cF24 tI12 oP6 hR3 oP24 hP9
Phases and Crystal Structures
83
Table A1.3. (Continued) Strukturbericht designation
Prototype phase
Space group
Pearson symbol
C23 C28 C32 C33 C34 C35 C36 C37 C38 C40 C42 C43 C44 C46 C49 C54 D0a D0c D0c D0d D0e D02 D03 D09 D011 D017 D018 D019 D020 D021 D022 D023 D024 D1a D1b D1c D1d D1e D1f D1g D13
Co2 Si HgCl2 AlB2 Bi2 Te3 AuTe2 (calaverite) CaCl2 MgNi2 Co2 Si Cu2 Sb CrSi2 SiS2 ZrO2 GeS2 AuTe2 (krennerite) ZrSi2 TiSi2 Cu3 Ti SiU3 Ir 3 Si AsMn3 Ni3 P CoAs3 BiF3 ReO3 Fe3 C BaS3 Na3 As Ni3 Sn Al3 Ni Cu3 P TiAl3 ZrAl3 TiNi3 MoNi4 Al4 U PdSn4 Pb4 Pt B4 Th Mn4 B B4 C Al4 Ba
Pnma Pmnb P6/mmm ¯ R3m C2/m Pnnm P63 /mmc Pbnm P4/nmm P62 22 Ibam P21 /c Fdd2 Pma2 Cmcm Fddd Pmmn I4/mcm I4/mcm Pmmn I 4¯ Im3¯ ¯ Fm3m ¯ Pm3m Pnma P421 m P63 /mmc P63 /mmc Pnma P63 cm I4/mmm I4/mmm P63 /mmc I4/m Imma Aba2 P4/nbm P4/mbm Fddd ¯ R3m I4/mmm
oP12 oP12 hP3 hR5 mC6 oP6 hP24 oP12 tP6 hP9 oI12 mP12 oF72 oP24 oC12 oF24 oP8 tI16 tI16 oP16 tI32 cI32 cF16 cP4 oP16 oP16 hP8 hP8 oP16 hP24 tI8 tI16 hP16 tI10 oI20 oC20 tP10 tP20 oF40 hR15 tI10 (Continued)
84
Phase Transformations: Titanium and Zirconium Alloys
Table A1.3. (Continued) Strukturbericht designation
Prototype phase
Space group
Pearson symbol
D2b D2c D2d D2e D2f D2g D2h D21 D23 D5a D5b D5c D5e D5f D51 D52 D53 D54 D58 D59 D510 D511 D513 D7a D7b D71 D72 D73 D8a D8b D8c D8d D8e D8f D8g D8h D8i D8k D8l D8m D81 D82
Mn12 Th MnU6 CaCu5 BaHg11 UB12 Fe8 N Al6 Mn CaB6 NaZn13 Si2 U3 Pt2 Sn3 Pu2 C3 Ni3 S2 As2 S3 Al2 O3 La2 O3 Mn2 O3 Sb2 O3 (senarmonite) Sb2 S3 Pt2 Zn3 Cr 3 C2 Sb2 O3 (valentinite) Al3 Ni2 Ni3 Sn4 Ta3 B4 Al4 C3 Co3 S4 Th3 P4 Mn23 Th6 ! CrFe Mg2 Zn11 Al9 Co2 Mg32 Al Zn49 Ge7 Ir 3 Ga2 Mg5 W2 B5 Mo2 B5 Th7 S12 Cr 5 B3 W5 Si3 Fe3 Zn10 Cu5 Zn8
I4/mmm I4/mcm P6/mmm ¯ Pm3m ¯ Fm3m I4/mmm Cmcm ¯ Pm3m ¯ Fm3c P4/mbm P63 /mmc ¯ I 43d R32 P21 /c ¯ R3c ¯ P 3m1 Ia3¯ ¯ Fd3m Pnma P42 /nmc Pnma Pccn ¯ P 3m1 C2/m Immm ¯ R3m ¯ Fd3m ¯ I 43d ¯ Fm3m P42 /mnm Pm3¯ P21 /c Im3¯ ¯ Im3m Ibam P63 /mmc ¯ R3m P63 /m I4/mcm I4/mcm ¯ Im3m ¯ I 43m
tI26 tI28 hP6 cP36 cF52 tI18 oC28 cP7 cF112 tP10 hP10 cI40 hR5 mP20 hR10 hP5 cI80 cF80 oP20 tP40 oP20 oP20 hP5 mC14 oI14 hR7 cF56 cI28 cF116 tP30 cP39 mP22 cI162 cI40 oI28 hP14 hR7 hP20 tI32 tI32 cI52 cI52
Phases and Crystal Structures
85
Table A1.3. (Continued) Strukturbericht designation
Prototype phase
Space group
Pearson symbol
D83 D84 D85 D86 D88 D89 D810 D811 D101 D102 E01 E07 E1a E1b E11 E21 E3 E9a E9b E9c E9d E9e E9e E94 F5a F01 F51 F56 H11 H24 H26 L 11 L 12 L 2b L 3 L1a L1o L11 L12 L2a L21 L22 L6o
Al4 Cu9 Cr 23 C6 Fe7 W6 Cu15 Si4 Mn5 Si3 Co9 S8 Al8 Cr 5 Al5 Co2 Cr 7 C3 Fe3 Th7 PbFCl FeAsS Al2 CuMg AgAuTe4 CuFeS2 CaTiO3 Al2 CdS4 Al7 Cu2 Fe Al8 FeMg3 Si6 Al9 Mn3 Si AlLi3 N2 CuFe2 S3 Fe3 W3 C Al4 SiC4 FeKS2 NiSbS CrNaS2 CuSbS2 Al2 MgO4 Cu3 VS4 Cu2 FeSnS4 Fe4 N AlFe3 C ThH2 Fe2 N CuPt3 AuCu CuPt AuCu3 CuTi AlCu2 Mn Sb2 Tl7 CuTi3
¯ P 43m ¯ Fm3m ¯ R3m ¯ I 43d P63 /mcm ¯ Fm3m ¯ R3m P63 /mmc Pnma P63 mc P4/nmm P21 /c Cmcm P2/c ¯ I 42d ¯ Pm3m ¯ I4 P4/mnc ¯ P 62m P63 /mmc Ia3¯ Pnma ¯ Fd3m P63 mc C2/c P21 3 ¯ R3m Pnma ¯ Fd3m ¯ P 43m ¯ I 42m ¯ P 43m ¯ Pm3m I4/mmm P63 /mmc ¯ Fm3c P4/mmm ¯ R3m ¯ Pm3m P4/mmm ¯ Fm3m ¯ Im3m P4/mmm
cP52 cF116 hR13 cI76 hP16 cF68 hR26 hP28 oP40 hP20 tP6 mP24 oC16 mP12 tI16 cP5 tI14 tP40 hP18 hP26 cI96 oP24 cF112 hP18 mC16 cP12 hR4 oP16 cF56 cP8 tI6 cP5 cP5 tI6 hP3 cF32 tP2 hR32 cP4 tP2 cF16 cI54 tP4
86
Phase Transformations: Titanium and Zirconium Alloys
Table A1.4. Binary intermetallic phases: most commonly exhibited structures. Structure type
Number of binary phases exhibiting the structure
Rank
Strukturbericht designation
Pearson symbol
Prototype phase
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
A1 A3 B1 A2 B2 L12 C15 D88 C14 C32 Bf D73 D2d D011 B81 C23 C1 L1o A15 C38 B27 C42 B82 C2 D8a
cF4 hP2 cF8 cI2 cP2 cP4 cF24 hP16 hP12 hP3 oC8 cI28 hP6 oP16 hP4 oP12 cF12 tP2 cP8 tP6 oP8 hP38 oI12 hP6 cP12 cF116 hR36
Cu Mg NaCl W CsCl AuCu3 MgCu2 Mn5 Si3 MgZn2 AlB2 CrB Th3 P4 CaCu5 Fe3 C NiAs Co2 Si CaF2 AuCu Cr 3 Si Cu2 Sb FeB Ni17 Th2 CeCu2 Ni2 In FeS2 Mn23 Th6 Be3 Nb
520 362 318 309 307 266 243 177 148 122 120 117 106 101 101 95 87 82 82 74 73 62 61 54 50 49 49
Chapter 2
Classification of Phase Transformations 2.1 Introduction 2.2 Basic Definitions 2.3 Classification Schemes 2.3.1 Classification based on thermodynamics 2.3.2 Classifications based on mechanisms 2.3.3 Classification based on kinetics 2.4 Syncretist Classification 2.5 Mixed Mode Transformations 2.5.1 Clustering and ordering 2.5.2 First-order and second-order ordering 2.5.3 Displacive and diffusional transformations 2.5.4 Kinetic coupling of diffusional and displacive transformations References
89 90 92 93 101 105 105 115 115 116 120 120 122
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Chapter 2
Classification of Phase Transformations Symbols G: T: Tc : S: P: H: Cp : : Tm : -phase: -phase: ∗ : e : A : Cij : Ms : V: Vi , Gid : Vd , Gdd :
2.1
and Abbreviations Gibbs free energy ( G = H-TS) Temperature Equilibrium transition temperature Entropy Pressure Enthalpy Specific heat at constant pressure Generalized order parameter Melting temperature hcp phase in Ti and Zr based alloys bcc phase in Ti and Zr based alloys Order parameter corresponding to maximum in free energy Equilibrium order parameter for a first order transition Chemical potential of component, A in the phase, Elastic stiffness modulus (elastic constant) Temperature at which martensite starts forming during quenching Specific volume Velocity and dissipated free energy associated with interface process Velocity and dissipated free energy associated with diffusion process
INTRODUCTION
The study of phase transformations is of interest to metallurgists, geologists, chemists, physicists and indeed to all scientists concerned with the states of aggregation of atoms. Due to the multidisciplinary interest in this subject, a wide variety of nomenclature, sometimes even misleading, has been introduced in the literature for the characterization of different types of phase transformations. It is not uncommon that different sets of terminologies are used in different disciplines for describing essentially similar phase transformations which, in a generalized manner, can be defined as a change in the macrostate of an assembly of interacting atoms or molecules as a result of some variation in the external constraints. The diversity of scientific interest and the complexity of the possible interactions between individual atoms of the assembly naturally lead to many different 89
90
Phase Transformations: Titanium and Zirconium Alloys
approaches for the study of phase transformations. Physicists primarily focus their attention on higher-order continuous phase transitions in single-component systems such as magnetic, superconducting and superfluid transitions. In contrast, metallurgists and chemists are mainly concerned with phase transformations (which include phase reactions) involving changes in crystal structure, chemical composition and order parameter (both long and shortrange). Phase transformations encountered by geologists, though quite similar to those observed in metallic and ceramic systems, usually occur over much more extended time and length scales under extreme conditions of pressure and temperature. Phase transformations also occur in organic materials such as polymers, biological systems and liquid crystals. Many of the relevant concepts developed for inorganic systems have parallels in organic systems. However, no attempt will be made in this chapter to compare and contrast phase transformations in organic and inorganic systems as the nature of atomic interactions responsible for the transformations is quite different in these two classes of materials. Alloys, intermetallics and ceramics form a group of materials in which phase transformations can be discussed on a common conceptual basis and, therefore, a single classification scheme can be used for appropriately grouping different types of transformations in these systems. As mentioned earlier, Ti- and Zr-based systems, which include alloys, intermetallics and ceramics, exhibit nearly all possible types of phase transformations and, therefore, serve as excellent examples for studies on phase transformations in inorganic materials in general. Phase transformations can be classified on the basis of different criteria, namely, thermodynamic, kinetic and mechanistic (Christian 1965, Roy 1973, Rao and Rao 1978). A comparison of the characteristic features of different types of transformations is presented in this chapter with a view to providing a coarse-brush picture of these in a generalized manner. The chapters which follow will describe these transformations more elaborately, taking illustrative examples from Ti- and Zr-based systems.
2.2
BASIC DEFINITIONS
In order to resolve some of the confusion and controversy which are of a semantic nature a summary of some basic definitions is presented here. A phase is a portion of a system bounded by surfaces and with a distinctive and reproducible structure and composition. Within a single phase, minor fluctuations in structure and/or in composition can occur. One phase can be distinguished from a second phase if at the contacting surface there is a sharp (within one or two atom layers) first-order change in composition and/or structure and hence properties.
Classification of Phase Transformations
91
The two terms phase transformation and phase transition are often used interchangeably. Sometimes a distinction between them is implied but rarely specified. In a paper entitled “A synchretist classification of phase transitions", Rustum Roy (1973) has addressed the controversy which exists in the literature in this regard. Generally the word phase transition is restricted to transitions between two phases which have identical chemical compositions, while the term phase transformation covers a wider spectrum of phenomena which include phase reactions leading to compositional changes. In the metallurgical literature, phase transformations include precipitation of a second phase, , of a different crystal structure and chemical composition from the parent phase ( → + ), and having the same crystal structure but different chemical compositions, eutectoid decomposition ( → + ) and many such processes which, in the chemistry literature, are grouped as phase reactions (Rao and Rao 1978). A more subtle point concerns the meaning of identical chemical composition. The equilibrium point defect concentration may be different in two polymorphs. Though in a strict sense they cannot be considered as identical in chemical composition, transformations between such polymorphs are usually classified as composition-invariant transformations. In considering equilibria between two phases, the requirement of reversibility must be taken into account. Several relationships pertaining to equilibria between two phases can be explained using the free energy versus temperature plot of a single component system (Figure 2.1). The liquid (L) to crystal (A) transition, Equ
ilibr Gla ium ss
A′
tas
tab
oo
rc
le A
TA/B
led liq
Crystal B
pe
Me
Su Cry
stal
uid
Free energy
Tg
A
T L/A
Enantiotropic
Liq
Monotropic
T2
uid
T1
Temperature
Figure 2.1. Free energy versus temperature plots showing phase transformations in a singlecomponent system. The differences between monotropic and enantiotropic transitions and between stable equilibrium and metastable equilibrium transitions are highlighted.
92
Phase Transformations: Titanium and Zirconium Alloys
L A, at the melting point, TL/A , and the crystal A to crystal B transition, A B, at the transition temperature, TA/B , are stable equilibrium transitions. The transition between a metastable phase A and another metastable phase A , A (metastable) A (metastable), can also be an equilibrium transition and can be grouped along with the former two cases as being enantiotropic, i.e. reversible and governed by classical thermodynamics. In contrast, when we consider transitions which can be represented by vertical lines in this diagram, such as A (metastable) → B (stable) at T1 and Glass → A (stable) at T2 , the reversibility criterion is not met. These irreversible transformations, defined as monotropic transitions, proceed only in one direction and it is not possible to establish an equilibrium between the parent and the product phases. Polymorphic transformations are generally defined as those which involve a structural transition without a change in the chemical composition. Sometimes these transformations are also referred to as congruent processes. There are, however, several examples, such as the transformation of crystalline oxygen to crystalline ozone and transformations of position isomers, which satisfy the aforementioned definition of a polymorphic transition, but cannot even be considered as phase transitions. This is because ozone and oxygen, in the phase rule sense, are two different substances (or components) which survived even the solid → liquid → vapour transitions while preserving their individuality. Similarly each position isomer is an individual component and, therefore, isomeric transitions cannot be considered as phase transitions. In view of this, the definition of polymorphic transformations needs to be restricted to transformations involving phases with different crystal structures which are part of a single component system. In multicomponent metallic alloys and intermetallics, chemical composition-invariant crystallization is a good example in which the parent phase transforms to the product without allowing any partitioning of the constituent elements (or components) between the two phases. In this sense, the system behaves as if it is a single-component system.
2.3
CLASSIFICATION SCHEMES
There are several ways in which phase transformations can be classified, based on thermodynamic, kinetic and mechanistic criteria. A single classification scheme may not be adequate to include all types of transformations encountered in all varieties of materials. In this chapter, an attempt is made to evolve a classification scheme which is applicable to phase transformations in metals, alloys, intermetallics and ceramics.
Classification of Phase Transformations
93
2.3.1 Classification based on thermodynamics Ehrenfest (1933) proposed a classification based on the successive differentiation of a thermodynamic potential, usually the Gibbs free energy function, with respect to an external variable such as temperature or pressure. The order of a transformation is then given by the lowest derivative which shows a discontinuity at the transition point. In the generalized sense, for an nth order transition
n G
T n
n−1 G
T n−1
= 0 P
=0
at
T = Tc
(2.1)
where G represents Gibbs free energy and Tc is the equilibrium transition temperature. It is to be noted that the Ehrenfest classification can be used only for equilibrium transitions of a single component system. Substituting n = 1 and 2 in Eq. (2.1), at T = Tc we get for first-order transitions
G = 0
G
T
and for second-order transitions H
G = 0 = − S = −
T P Tc
= − S = P
2 G
T 2
P
H Tc
(2.2)
Cp 1
H = = = 0 Tc
T P Tc (2.3)
A comparison between a first- and a second-order transition can be made in schematic plots of different thermodynamic quantities as functions of temperature (Figure 2.2). First-order transitions are characterized by discontinuous changes in entropy, enthalpy and specific volume. The change in enthalpy corresponds to the evolution of a latent heat of transformation, and the specific heat at the transition temperature, as a consequence, is effectively infinite. In contrast, second-order transitions are characterized by the absence of a latent heat of transformation (as H, S and V do not undergo a discontinuous change at Tc ) and a high specific heat at the transition temperature. There are experimental results which show that in some instances of second-order transitions the specific heat at Tc exhibits infinity rather than a finite discontinuity. A true second-order transition is, therefore, defined as one showing a finite discontinuity in the second derivative of the Gibbs function while a so-called lambda point transition exhibits an infinity. Though the Ehrenfest classification examines the presence of a discontinuity in the nth derivative of the Gibbs function for deciding the order of a transition, in the modern literature transitions with n ≥ 2 are grouped
94
Phase Transformations: Titanium and Zirconium Alloys First order
G
Second order
S
G L TM
T
Tc
T
Tc
T
Tc
T
L H
q
S TM
H
T CP
CP
TM
T Temperature
Figure 2.2. Changes in the thermodynamic quantities, free energy, enthalpy and specific heat, at the transition temperatures corresponding to first-order and second-order phase changes.
together as “higher-order transitions” which are characterized by a continuous first derivative, G/ T P = 0, followed by either a discontinuity or infinity for “higher” derivatives. In a multidimensional plot of free energy against temperature, pressure, etc. each phase can be represented by a well-defined surface, as illustrated in Figure 2.3. The equilibrium transformation conditions between two phases are then defined by the intersection of two such surfaces. Moreover the free energy surface for a given phase may be extrapolated into conditions where that phase is not in thermodynamic equilibrium, and the difference in free energy, which is represented by the separation of the free energy surfaces corresponding to the two phases, can be regarded as the driving force for a first-order transformation from one phase to the other. This concept of a metastable phase is not readily applicable to a second-order transformation where it is more appropriate to consider that there is a single continuous free energy surface. Most of the phase transformations encountered in metallic systems are of the first-order type. Ferromagnetic ordering and some chemical ordering processes are examples of higher-order transitions in metallic systems. These transitions can be represented in “mean field” descriptions of cooperative phenomena where the respective order parameters continuously decrease to zero as the temperature
Classification of Phase Transformations
95
β
Free energy
α
e
ur
ss
e Pr
α β Temperature
Figure 2.3. Free energy surfaces for two phases, and , as functions of pressure (P) and temperature (T ). The projection of the line of intersection of the two surfaces on the P–T plane represents the – phase boundary in the P–T phase diagram.
is raised to the transition temperature (Curie temperature or the critical ordering temperature), as shown in Figure 2.4. Any transition which can be described in terms of a continuous change in one or more order parameters can be treated in terms of a generalized Landau equation (Landau and Lifshitz 1969) which states
Tc
η
η Tc
T Second order (a)
T First order (b)
Figure 2.4. Order parameter () versus temperature (T ) plots for (a) second-order and (b) first-order transitions.
96
Phase Transformations: Titanium and Zirconium Alloys
that close to the critical temperature the free energy difference, G, between finite and zero values of the order parameter, , may be expanded as a power series: G = A2 + B3 + C4 + · · ·
(2.4)
the coefficients A, B, C, etc. being functions of pressure and temperature. The fundamental differences between the first- and higher-order transitions can be explained on the basis of the corresponding Landau plots. For higher-order transitions, the free energy must be an even function of which means that B = 0 (and similarly the coefficients of the odd-powered terms of are zero). Figure 2.5(a) shows the G versus plots for higher-order transitions at different temperatures, both above and below Tc . When T > Tc , the system exhibits a single stable equilibrium at = 0 which corresponds to a positive value of A. As the temperature approaches Tc , the curvature ( 2 G/ 2 at = 0 gradually decreases and as the temperature is lowered below Tc , the curvature as well as the value of A become negative. This essentially means that the system becomes unstable at T = Tc and any infinitesimal fluctuation in the order parameter leads to a lowering First order
Second order
T ≈ Tc Tc > T
η
0
T > Tc Free energy
Free energy
T > Tc
T ≈ Tc Tc > T > Ti
η
0
η = ηc Ti > T
–ve +ve Tc = Ti (a)
Tc >> Ti (b)
Figure 2.5. Free energy as a function of order parameter () for (a) second-order and (b) firstorder transitions. In the case of second-order transitions, the parent phase becomes unstable, ( 2 G/ 2 ) < 0 at = 0 at the transition temperature, Tc , which is the same as the instability temperature, Ti . For some first-order transitions an instability temperature, Ti (which is much lower than the equilibrium transition temperature, Tc ), can be identified where the parent phase becomes unstable at = 0.
Classification of Phase Transformations
97
of the free energy. The negative curvature of the G versus plot also implies that with an increase in the order parameter, the free energy progressively decreases, finally reaching the stable equilibrium positions defined by the minima present in the plots corresponding to T < Tc . The two minima corresponding to the positive and negative values represent two equivalent states associated with antiphase domains of the same ordered structure. Landau plots for a first-order transition are shown in Figure 2.5(b). In this case, the value of B in the Landau expansion (Eq. (2.4)) is not zero. At the transformation temperature, the G versus curve shows two minima, one at = 0 and the other at = c , the two minima being separated by a free energy barrier. At = 0 the system is not unstable as the curvature remains positive at this point at T = Tc . A continuous increase in the order parameter, therefore, will initially raise the free energy which will drop only after the peak of the free energy hill is crossed. Since the system as a whole is not unstable either at Tc or at temperatures close to but below Tc , a gradual transition of the system in a homogeneous manner to the free energy minimum at or near = c is not possible. A phase transition under such a situation can initiate only if localized portions of the system are activated to cross the free energy barrier to reach a point beyond ∗ where can grow further spontaneously. The formation of such localized product phase regions (where has nearly reached the c value) is known as nucleation. The product nuclei remain separated from the parent phase by sharp interfaces and the phase transition proceeds through the growth of these nuclei. The presence of two free energy minima separated by a free energy hill near the equilibrium in the case of a first-order transformation brings out its characteristic features, namely, the coexistence of the parent and the product phases and the discrete nature (involving nucleation and growth) of the transformation. In contrast, all higher-order transitions, by definition, are homogeneous in the sense that the parent and the product phases cannot be distinguished at any stage of the transition and there is no question of having an interface between the two phases. It is interesting to note that a discussion on Landau’s theory, which is strictly concerned only with the equilibrium state, has led us to consider continuous vis-à-vis discrete transformations. Continuous or homogeneous transitions are those in which the parent phase as a whole gradually evolves into the product phase without creating a localized sharp change in the thermodynamic properties and the structure in any part of the system. Such a process can occur only when the system becomes unstable with respect to an infinitesimal fluctuation which leads to the transition and the free energy of the system continuously decreases with the amplification of such a fluctuation. All higher-order transitions, by definition, satisfy the condition of homogeneous/continuous transformation at equilibrium. In contrast, all first-order
98
Phase Transformations: Titanium and Zirconium Alloys
transitions at equilibrium are discrete transitions which necessarily involve nucleation and growth. When we consider Landau plots for first-order transitions at temperatures far below Tc (Figure 2.5(b)) we notice that at temperatures below the instability temperature, Ti , the curvature of the G versus plot becomes negative at = 0 and the free energy continuously drops with increase in , finally reaching the value corresponding to c . Therefore below Ti a first-order transition can also proceed in a continuous mode. It must be emphasized here that the occurrence of an instability temperature is not universal for all first-order transitions. Only in rather a limited number of cases can first-order transitions be described in terms of a Landau representation. Spinodal clustering, spinodal ordering and displacement ordering processes are some examples in which continuous first-order transitions are encountered in conditions far from equilibrium. Equilibrium phase diagrams showing a miscibility gap correspond to solid solutions which exhibit a clustering tendency. The boundary of the equilibrium two-phase field, 1 + 2 , in the phase diagram (Figure 2.6) is determined by equating the chemical potentials of the two components, A and B, in the two phases, 1 and 2 , in equilibrium at a given temperature, T1 : 1 2 = AB AB
(2.5)
α A
Temperature
α1
X1
B
T1
X 2 α2
Equilibrium solvus Coherent solvus Coherent spinodal
C D
Atom fraction (X)
Figure 2.6. Schematic phase diagram for a clustering system which remains a homogeneous solid solution in region A. A phase separation process occurs by a discrete nucleation and growth mechanism in regions B and C. Spinodal decomposition occurs in region D.
Classification of Phase Transformations
99
The concentrations, X1 and X2 , at which the tie line at T1 intersects the miscibility gap correspond to those of 1 and 2 . A homogeneous solid solution decomposes into an incoherent mixture of 1 and 2 as it is brought from the region A to the region B. Such a decomposition reaction, involving the creation of sharp interfaces between the two phases, is undoubtedly of the first order. A coherent mixture of the two phases can exist in the region C defined by the coherent miscibility gap. The coherent spinodal region D, fully residing within the region C, defines the concentration–temperature field where the phase separation process can initiate by introducing long wavelength (relative to interatomic distances) concentration fluctuations and a continuous amplification of such fluctuations. Though the process is continuous, the transformation is of the first order except at the point where the coherent spinodal touches the coherent solvus, where it may be considered to be of second order. The driving force for the phase separation arising from the negative curvature of the free energy (G)–concentration (X) plot (( 2 G/ X 2 )< 0 in the spinodal region) is opposed by the gradient energy and the coherency strain energy. All these factors and a correction factor for thermal fluctuations determine the wave number for which the amplification rate is the highest. Conceptually a chemical ordering process in which the ordered superlattice can be created only by replacement of atoms in the lattice of the disordered phase can be described in a manner similar to the spinodal clustering process. In the case of continuous ordering, concentration modulations with wavelengths of the order of the interatomic spacing need to be introduced. For an ordering system, the effective gradient energy is negative, and many of the predictions are opposite to those of spinodal decomposition. The amplification factor is negative beyond a critical wavelength and the maximum amplification corresponds to a wavelength equal to a small lattice vector. As shown in Figure 2.7(a) and (b), continuous ordering can be envisaged for both first-order and second-order reactions. Any change in the lattice dimensions due to ordering introduces a third-order term in the Landau equation (Eq. (2.4)) which makes the transformation first order. Continuous ordering in the first-order case requires finite supercooling below the coherent phase boundary. De Fontaine (1975) has also distinguished spinodal ordering from continuous ordering. In the former, the early stages of ordering are characterized by the ordering wave vectors which maximize the amplification factor but the amplification of these does not evolve the equilibrium ordered structure. In true continuous ordering, the equilibrium ordered structure continuously evolves from a low-amplitude quasihomogeneous concentration wave. De Fontaine (1975) has also examined the Landau–Lifshitz symmetry rules for determining whether a specific ordering wave vector qualifies to be a candidate for a second-order transformation.
100
Phase Transformations: Titanium and Zirconium Alloys First order
Second order
Temperature
αd
αd Equilibrium phase boundary
A
A
Coherent phase boundary B
B
αo
C
Coherent instability boundary, T i
D
X
X (a)
(b)
Figure 2.7. Schematic phase diagram for an ordering transition of (a) second order and (b) first order in which continuous ordering occurs under supercooling below the instability temperature, Ti . Region A represents the phase field where the disordered solid solution, d , is stable while Region B corresponds to the stability domain of two-phase mixture, o + d . Region C corresponds to stability domain of ordered o . In the case of a second-order process the d → o transition occurs in a continuous manner below the equilibrium phase boundary. In contrast, continuous ordering is possible in Region D only below the coherent instability boundary, Ti , in some first-order transitions (where the symmetry elements of the ordered structure form a subset of those of the parent disordered structure).
A product phase can evolve from the parent phase through a continuous displacement of the parent lattice. Two types of displacement, namely, a homogeneous lattice deformation and a relative displacement of atoms within unit cells (often called shuffles), can take place, either singly or in combination. The lattice deformation produces a change in volume and external shape and it seems very improbable that this could be accomplished continuously or homogeneously unless the principal lattice strains are very small (within the limits of linear elastic strain). Most martensitic transformations in metals and alloys involve much larger values of lattice strains and are, therefore, not candidates for continuous transformations. However, the possibility of a continuous displacive transformation has to be considered if the lattice deformation is very small. Transformations in some ferroelectric crystals of low symmetry are believed to be of the second order and occur by the progressive development of an instability in a dynamic plane displacement which becomes, at Tc , a static wave extending through the crystal. The approaching soft-mode instability is indicated as a pretransition effect above the transition temperature by a reduction in an appropriate lattice stiffness. Low and reducing values of the shear constant, 21 (C11 − C12 ), are
Classification of Phase Transformations
101
found in the parent phase as Ms is approached from higher temperatures in many noble metal bcc alloys and in fcc In–Tl; but in other martensitic transformations, especially those in steels, there are no anomalies of this kind and, therefore, these transformations are clearly of the first order. The athermal bcc → transformation, which is envisaged as a pure shuffle transition, is another example of a continuous displacive transformation which is a first-order transformation from the consideration of symmetry rules. Though it is possible to accomplish this transformation by a continuous amplification of a displacement wave, the observed fine particle and dual phase ( + ) morphology suggests that -particles form in a quasiperiodic manner through heterophase fluctuations which have the form of ellipsoidal wave packets of displacement wave (Cook 1974). 2.3.2 Classifications based on mechanisms Buerger (1951) has introduced a classification based on mechanisms, namely, reconstructive and displacive transformations. In the metallurgical literature, however, a mechanistic classification groups transformations into (a) nucleation and growth and (b) martensitic types. The introduction of the term nucleation and growth in this context has created considerable confusion, as all first-order transformations including martensitic transformations require the nucleation and the growth steps. In the current literature, usage of such confusing nomenclature is avoided and a mechanistic classification designates the two classes as (a) diffusional and (b) displacive transformations. The former corresponds to the reconstructive transformation in which atom movements from the parent to the product lattice sites occur by random diffusional jumps. This implies that near neighbour bonds are broken at the transformation front and the product structure is reconstructed by placing the incoming atoms at appropriate positions which results in the growth of the product lattice. In contrast, atom movements in a displacive transformation can be accomplished by a homogeneous distortion, shuffling of lattice planes, static displacement waves or a combination of these. All these displacive modes involve cooperative movements of large numbers of atoms in a diffusionless process. Displacive (which includes martensitic) transformations initiate by the formation of nuclei of the product phase, and the growth of these nuclei occurs by the movement of a shear front at a speed that approaches the speed of sound in the material under consideration. In order to differentiate the mechanisms of atom movements across the transformation front, Christian (1965, 1979) has compared the movements involved in diffusional and displacive transformations with civilian and military movements, respectively. In the latter case, if the atoms are labelled in the parent lattice, the coordination between the neighbours can be shown to be essentially retained in
102
Phase Transformations: Titanium and Zirconium Alloys Shear F
F
E
E
D
D
C
C
B
B A
G
H
I
A
J
G
H
I
J
(a) Shear
N′
N M
P′
O′
P O
M′ Q
R
Q′
S
(b)
Figure 2.8. Schematic illustration of lattice correspondence in a two-dimensional lattice. Though the parent and product lattices in both (a) and (b) are identical, the lattice correspondences in the two cases can be distinguished if the dots representing atoms occupying the lattice sites can be labelled. It is through the establishment of the lattice correspondence that the nature of the homogeneous shear and shuffle, if required, can be identified.
the product lattice, though the bond angles undergo changes. This point is illustrated in Figure 2.8(a) which shows how a set of atoms (labelled A, B, C, etc.) decorating the parent lattice changes to the product lattice. The existence of a lattice correspondence implies that a vector in the parent lattice, defined by the sequence of atoms ABCD , becomes a vector in the product lattice with the atoms arranged in the same sequence, although the spacing between them gets altered to match the product lattice dimensions. Such a transformation can be viewed as a homogeneous deformation of the parent lattice (a simple shear in the
Classification of Phase Transformations
103
case of the two-dimensional illustration in Figure 2.8(a), shear directions being shown by arrows). The importance of lattice correspondence in determining the shear can be illustrated by Figure 2.8(b) which shows identical arrays of spots representing the parent and the product structures but with a different lattice correspondence. The atomic rows MNO and MQR are shown in both the parent and the product structures. It is evident from this drawing that the product structure is derived by a combination of a homogeneous deformation and a shuffle. The shear, as indicated by arrows, transforms the rectangle MOPQ in the parent to a parallelogram M O P Q in the product and the atom N is shifted to its new position by a shuffle. The best experimental evidence for the inheritance of the atomic coordination through a displacive transformation is provided by the observation that the chemical order present in the parent structure is fully retained in the product structure. A similar correspondence also exists for crystallographic planes. A relationship of this kind in which straight lines transform to straight lines and planes to planes is described mathematically as an affine transformation. Physically it may be considered as a homogeneous deformation of one lattice into the other. The correspondence associates each vector, plane and unit cell of the parent with a corresponding vector, plane and unit cell of the product. In general, the corresponding lattice vectors and the spacings of corresponding lattice planes are not equal in the two structures, and the angular relation between any pair of lattice vectors in the parent structure is not preserved in the product. It is to be noted that the lattice correspondence does not by itself imply any orientation relation between the phases, since the transformation may involve a rigid body rotation of the product structure with respect to the parent structure. In diffusional transformations, such lattice correspondences are not present. Even in those cases of diffusional transformations in which the chemical compositions of the parent and the daughter phases are identical and a strict orientation relationship exists between them, random jumps of atoms from the parent to the product lattice positions do not permit the lattice correspondence to be preserved. Such composition-invariant diffusional transformations proceed by atomic jumps across the advancing transformation fronts which separate the parent and the product phases. In order to understand the basic difference between displacive and diffusional transformations let us again consider labelling the atoms as A, B, C, D, etc. in the parent lattice and the same set of atoms as A , B , C , D etc. in the product lattice in Figure 2.9(a) and (b), respectively. The transformation front has been shown to advance by a single atomic layer. This schematic drawing shows that the sequence in which these atoms were placed before the transformation is not the same as that in the product lattice. This can happen if each atom breaks the bonds with its neighbours in the parent lattice and shifts to a new position
104
Phase Transformations: Titanium and Zirconium Alloys
F′
F
E′
E
D′
D C B
C′ B′
A (a) Transformation front F′
F
D′
E
E′
D
C′
C
A′
B
B′ A (b)
Figure 2.9. Schematic diagram of atom movements across the transformation front in a (a) displacive transformation; (b) diffusional transformation.
which corresponds to a lattice point in the product structure. In this manner, the transformation boundary proceeds towards the parent phase, converting the parent to the product phase. The jumps of atoms A, B, C, etc. are random and are not correlated with those of their neighbours, unlike the case of displacive transformations. Since diffusional transformations involve the breaking of bonds between neighbouring atoms and the reconstruction of bonds to form the product phase structure, they are also known as reconstructive transformations.
Classification of Phase Transformations
105
2.3.3 Classification based on kinetics Phase transformations are also grouped in terms of the kinetics of the process. The most important distinguishing kinetic criterion is the requirement of thermal activation. First-order transformations necessarily occur by the nucleation of the product followed by its growth by the propagation of the interface between the parent and the product phases. The movement of the interface can be either thermally activated or athermal. The atom transfer process across the interface is thermally activated in the case of the former while it does not require the assistance of thermal fluctuations in the latter. The kinetic classification, originally introduced by Le Chatelier (Roy 1973), divides phase transformations into two main groups: (a) rapid or nonquenchable and (b) sluggish or quenchable. Transformations belonging to the former class are so fast that the parent phase, which is stable at high temperatures (or high pressures), cannot be retained by a rapid quench to ambient conditions. In contrast, sluggish transitions are slow to the extent that the high-temperature (or highpressure) phase can be retained metastably on quenching. The basic idea behind this classification scheme also centres around the requirement of thermal activation. This classification, however, suffers from the limitation that the experimental ability to rapidly change temperature and pressure is continuously improving and transformations which are grouped as “non-quenchable” today may become “quenchable” tomorrow. A truly displacive transformation occurs through the passage of a glissile interface which is essentially a displacement (or shear) front, the movement of which is not assisted by thermal activation. Such transformations are non-quenchable irrespective of the quenching rate employed.
2.4
SYNCRETIST CLASSIFICATION
The fundamental parameters on the basis of which a phase transformation is classified are thermodynamic, mechanistic and kinetic. A syncretist classification scheme has been introduced by Roy (1973) by taking all these aspects into account. Figure 2.10 shows a three-dimensional matrix with the x-, y- and z-axes representing the mechanistic (structural), thermodynamic and kinetic parameters, respectively. Along the x-axis the two major classes, namely, diffusional (reconstructive) and displacive transformations, are separated by a “mixed” class of transformations which have attributes of both displacive and diffusional transformations. Examples of each of these are available in Ti- and Zr-based systems. Martensitic transformations of the bcc () phase of pure Ti, Zr and of alloys based on these metals have been discussed in detail in Chapter 4 which also deals
106
Phase Transformations: Titanium and Zirconium Alloys
z
Thermally activated Athermal
Intermediate Rapid
Kinetic
Slow
y
ic
m
na
y od
rm
e Th
≠ t ΔV er rs Fi ord nd a ΔH
0
d ion ith V ixe it w , Δ M rans ect ΔH et ff all pr e sm
=0 d on der ΔV c d r Se o an
ΔH
Displacive
Mechanistic (Structural)
Mixed
Diffusional
x
Figure 2.10. Syncretist classification scheme of phase transformations based on mechanistic (structural), thermodynamic and kinetic criteria.
with martensites in NiTi-based intermetallics and ZrO2 -based ceramics. A host of diffusional transformations such as precipitation, amorphous-to-crystalline phase transformations, massive transformation, eutectoid phase reactions have also been encountered in these systems and they are discussed in Chapters 4 and 7. Displacive transformations can be further divided into different subgroups, depending on whether the transformation is dominated by lattice strains (martensitic transformation) or by shuffles (e.g. omega transition and ferroelectric transitions). The → transition which is frequently observed in several Ti- and Zr-based systems is unique with respect to lattice registry in three dimensions, pretransition effects and transformation product morphology. A detailed account of the -transformation is presented in Chapter 7. The classifications based on thermodynamic criteria, represented along the y-axis of Figure 2.10, divide phase transformations into first-order, higher-order and “mixed” transformations. The distinctions between these classes of transformations are illustrated in Figure 2.11 which depicts the variation of thermodynamic quantities such as specific volume (V ), enthalpy (H) and entropy (S) at and near the transition temperature. In a first-order transition, there is a step change in these quantities at this temperature and there is no need for V , H and S of one phase to show a tendency to gradually approach the value corresponding to the other phase as the transition temperature is approached. A second-order transition is characterized by a gradual change in V , H and S and the absence of a step change in
Classification of Phase Transformations
107
Specific volume Enthalpy Entropy T
T
T
Temperature (a) First order
(b) Mixed
(c) Higher order
Figure 2.11. Changes in specific volume (V ), enthalpy (H) and entropy (S) at and near the transition temperature in (a) second-order and (b) and (c) first-order transitions; (b) shows a “mixed” character, exhibiting pretransition effects as well as step changes at the transition temperature.
these parameters at the transition temperature. The mixed situation is encountered in a large number of transitions such as ferroelectric and ferromagnetic transitions. In such mixed transformations, though a pretransition second-order-like effect is observed, there is a finite discontinuous jump in the value of these thermodynamic quantities at the transition temperature. Considering kinetics as the third variable, phase transformations can be grouped into thermal and athermal classes. All true displacive transformations are athermal which cannot be suppressed by quenching. In contrast, reconstructive or diffusional transformations are invariably thermally activated and, therefore, such transformations are, in principle, suppressible on quenching. The required quenching rate for suppressing a diffusional transformation, however, varies depending on the incubation period involved. Having discussed different schemes of classification of phase transformations in alloys, intermetallics and ceramics, we will now examine how a given phase transformation can be classified on the basis of thermodynamic, kinetic and mechanistic criteria. A classification tree (Figure 2.12) can be constructed by addressing appropriate questions at different levels. The first question to raise is whether the transformation is homogeneous (or continuous) or discrete. Higher-order transitions are continuous while first-order transitions are discrete at the equilibrium transition temperature, Tc . Some firstorder transitions exhibit an instability temperature, Ti (Ti << Tc ), below which the transformation initiates in a continuous manner. Homogeneous transformations can further be subdivided depending on the nature of fluctuations (waves), the amplification of which represents the progress of the transformation, and the
108
Phase Transformations: Titanium and Zirconium Alloys
Phase transformations (Thermodynamic) Higher order
First order
Necessarily
Homogeneous continuous
Below instability temperature in selected cases
Discrete
Nucleation (Kinetic) Thermally activated
Athermal
Stable particles of new phase Growth (Mechanistic) Diffusional reconstructive
Long range
Displacive
Short range
Thermally activated atom transport
Bulk diffusion controlled
Interface diffusion controlled
Solute partitioning
Cellular Homogeneous precipitation and heterogeneous Eutectoid precipitation decomposition
Lattice strain dominated
Shuffle dominated
Cooperative movements of atoms, athermal
Interface diffusion controlled
Martensitic transformations
Displacive omega transformations
Composition invariant Composition invariant
Polymorphic crystallization Massive transformation
fcc → bcc, bct in Fe alloys
Chemical ordering
bcc → hcp in Ti, Zr alloys
Recrystallization
Figure 2.12. Classification tree for phase transformations.
bcc → ω in Ti, Zr alloys
Classification of Phase Transformations
109
Table 2.1. Lattice wave descriptions of homogeneous transformations. Nature of wave
Long wavelength
Short wavelength
Replacive Displacive Electric Magnetic
Spinodal clustering Martensitic Ferroelectric Ferromagnetic
Spinodal ordering Omega Antiferroelectric Antiferromagnetic
wavelength associated with these waves. Table 2.1 lists different types of homogeneous transformations which can be distinguished on the basis of the nature of the wave and the wavelength. Discrete transformations, by definition, occur by the nucleation and growth process in which a small particle of the product phase (or a phase structurally and chemically close to the product phase) appears in the parent matrix. The two phases are separated by a sharp interface which gradually moves towards the parent phase, converting it to the product phase. Discrete transformations are subdivided into diffusional (or reconstructive) and displacive, based on the mechanism of atom transfer across the advancing transformation front. When the atom jump across the interface occurs by random diffusional jumps, the transformation is designated as a diffusional transformation. A subset of diffusional transformations is the replacive transformation which is accomplished by replacement of atoms in the lattice of the parent phase without destroying the parent lattice. Good examples of such transformations are those chemical ordering reactions which lead to the formation of a superlattice, without involving lattice reconstruction but rearrangement of atoms in the parent lattice. Phase transformations in which the chemical composition of the parent phase is not inherited by the product naturally involve partitioning of different components through long-range diffusion. Identification of such transformations as belonging to the category of diffusional transformations is rather trivial. Those cases where both the parent and the product phases have an identical composition, for example, the polymorphic transformation of a single-component system, the massive transformation of a multicomponent alloy or the martensitic transformation, can be grouped into the two categories, namely, displacive and diffusional, depending on the mechanism of atom transport across the transformation front. The mechanisms of atomic movements are difficult to observe through direct experiments. Therefore, the growth mechanism is inferred from a variety of crystallographic, morphological and kinetic observations such as presence of lattice correspondence, orientation relation, shape of the product phase, habit plane, macroscopic shape deformation, inheritance of atomic order and thermally activated or athermal nature of the growth process. These experimental observables
110
Phase Transformations: Titanium and Zirconium Alloys
are good indicators for deciphering the mechanisms of atomic movements and in a great majority of cases the growth mechanisms established from these observations have received universal acceptability. There are, however, some cases for which unanimity regarding the mechanism of atomic movements has not been arrived at. There is also a continued debate as to whether the operation of a shear mechanism in a transformation can be inferred from the existence of lattice correspondence and from the fact that the habit plane obeys the invariant plane strain condition. Some of these issues are discussed in later chapters of this book. The nature of the nucleation step in discrete transformations needs to be examined for recognizing the class of a given transformation. For example, a martensitic transformation can have either an athermal or a thermally activated nucleation step. In the former case, the overall transformation assumes an athermal character, while in the latter the volume fraction of the martensite grows with time sigmoidally at a constant temperature between the start (Ms ) and finish (Mf ) temperatures associated with martensite formation (Figure 2.13). Athermal nucleation is also encountered in diffusional transformations such as crystallization of metallic glasses. In this case, the nuclei of crystalline phases present in the asquenched amorphous matrix are activated for growth during the crystallization process. Diffusional transformations can be further subdivided into different categories based on the diffusion lengths of the different atomic species required to accomplish the transformation. In composition-invariant transformations, such as polymorphic transformations of single-component systems, polymorphic crystallization, massive transformation and partitionless solidification, the random
Isothermal
Athermal (Burst)
Athermal
T1
Martensite (%)
Ms
Martensite (%)
Martensite (%)
~100%
Ms > T1 > Mf
Mf Temperature (a)
Temperature
Time
(b)
(c)
Figure 2.13. (a) and (b) Volume fraction transformed as a function of temperature in the case of athermal nucleation of martensite. (c) Increase in volume fraction transformed with time at a constant temperature, T (Ms > T > Mf ), resulting from thermally activated nucleation of martensite.
Classification of Phase Transformations
111
diffusive atom movements occur only across the advancing transformation front. The distances involved in the diffusion process are in the range of the nearest neighbour atomic distances. These transformations are, therefore, designated as short-range diffusional transformations. The partitioning of atomic species occurs between the product phases mainly by interfacial diffusion in cases such as eutectoid decomposition, eutectic crystallization and cellular precipitation, resulting in a two-phase lamellar product. The interlamellar spacing can vary from a few nanometres to a few micrometres. These transformations can be grouped into the category of intermediate-range diffusional transformations. Sometimes classification is made on the basis of the nature of diffusion – whether bulk or interfacial – which dominates in the transformation process. If one draws a comparison between the eutectic and the eutectoid decompositions, one can see that in the former case diffusion is mainly in the liquid phase ahead of the transformation front whereas in the latter case, partitioning of different components occurs primarily at the transformation front. As indicated earlier, displacive transformations are those which can be accomplished by introducing a lattice deformation in the parent lattice. In this class of transformations, a perfect lattice correspondence is maintained and chemical order, if present in the parent phase, is retained in the product phase. A very wide range of transformations are grouped in this class, which covers softmode ferroelectric and ferroelastic transitions in materials such as SrTiO3 and BaTiO3 , the omega transition in Ti and Zr alloys, shear transformations in (bcc) phases in noble metal alloys and martensitic transformations in intermetallics such as nickel aluminides and nickel titanides, cubic-to-tetragonal or cubic-toorthorhombic transitions involving small lattice strains and classical systems of iron-based alloys. Since the characteristics of all of them are not the same, they have been further subdivided into different groups using different criteria for classification. In recent literature, Cohen et al. (1979), Delaye et al. (1982) and Christian (1990) have proposed these classification schemes which are summarized in Table 2.2. It has been mentioned earlier that a displacive transformation involves a homogeneous strain of the parent lattice which is accompanied by atomic shuffles (relative displacements of atoms within unit cells) in some cases. The relative contributions of the lattice strain and the shuffle can be used as important criteria for grouping martensites into two classes, namely, the shuffle dominated and the lattice strain dominated transformations. There are a number of examples of displacive transformations which exhibit pretransition softening of elastic moduli. Amongst these, Ni–Al alloys, containing 30–50% Al, constitute the most widely studied systems; they show weak to moderate first-order character. The fact that the high temperature 2 -phase
112
Phase Transformations: Titanium and Zirconium Alloys
Table 2.2. Classification of displacive transformations. Criteria Magnitude of shuffle and of homogeneous lattice strain (Cohen et al. 1979)
Presence of precursor mechanical instability (Delaye et al. 1982)
Structural basis (Christian 1990)
Classification Shuffle dominated
Ferroelectric Ferromagnetic Omega
Lattice strain dominated
Martensitic: high strain energy controls morphology and kinetics Quasimartensitic: low strain energy; can occur continuously
No mechanical instability as Ms is approached Strongly first order
Allotropic changes in pure elements Transformations in primary solid solutions
Moderate indications of mechanical instabilities Moderate first order
-phases of noble metal alloys and Ni-based shape memory alloys
Marked mechanical instability Weakly first order, second order
Cubic → tetragonal Cubic → orthorhombic with small lattice strains
Between close packed layer structure
fcc, hcp, 9R, 18R, 4H, etc.; including monolithic and orthorhombic distortions
Between fcc, bcc and derived structure Between bcc, hcp and derived structure Between cubic and tetragonal and slight distortions
(CsCl structure) prepares itself for the transformation as the temperature is lowered towards the martensitic start (Ms ) temperature is well reflected in X-ray and electron scattering as well as in acoustic measurements. The entire 4 0 transverse acoustic phonon branch (corresponding to the shear modulus C in the limit → 0) is unusually low and the energy decreases considerably (though not to zero) at certain wave vectors as the temperature approaches Ms . This partial softening and the evolution of diffuse scattering due to elastic strain along the 0 directions indicate the existence of localized fine-scale displacement patterns. In the Ni625 Al375 alloy, the presence of localized displacements, which are remarkably similar to that required for the formation of the 7R martensite (the
Classification of Phase Transformations
113
stacking sequence of close packed planes being ABAC), has been observed in highresolution electron microscopy images just prior to the transformation (Tanner et al. 1990). As the parent phase approaches the Ms temperature, the density of such microregions deformed by lattice strains increases. The nucleation event in such cases can be viewed as a collapse or growth of microdomains associated with non-uniform lattice strains into a macrodomain of a size larger than the critical size of a nucleus and of homogeneous and nearly appropriate lattice strain. On the question of precursors in martensitic transformations, there are conflicting observations reported in different systems. Martensitic transformations can, therefore, be divided into different subclasses based on the nature and extent of precursor mechanical instability. Strongly first-order martensitic transformations do not show any softening of elastic constants as the system approaches the Ms temperature. In contrast, there are systems where moderate or marked indications of softening of elastic constants are present in the vicinity of the Ms temperature. Generally these “mixed” transformations, exhibiting partial mode softening, are associated with small lattice strains. Displacive-type structural transitions accompanying ferroelectric transitions in BaTiO3 can be cited as examples. These are associated with the displacement of a whole sublattice of ions of one type relative to another sublattice. The perovskite structure with a generalized composition ABO3 consists of a three-dimensionally linked network of BO6 octahedra, with A ions forming AO12 cuboctahedra to fill the spaces between BO6 octahedra. In view of these topological and geometrical constraints, there are only three structural degrees of freedom: (a) displacement of cations A and B from the centres of their cation coordination polyhedra, AO12 and BO6 , respectively; (b) distortions of the anionic polyhedra, coordinating A and B atoms; and (c) tilting of the BO6 octahedra about one, two or three axes. The first of these is the most important for the occurrence of ferroelectricity, since a separation of the centres of positive and negative charges corresponds to an electric dipole moment. Structural phase changes in BaTiO3 with temperature are shown in Figure 2.14. In cubic paraelectric BaTiO3 , both Ba and Ti have zero displacements, with perfectly regular polyhedra of coordinating oxygen ions. The tetragonal, monoclinic (orthorhombic) and rhombohedral forms, which are all ferroelectric, are associated with displacements of the ionic species and distortions of the polyhedra. In tetragonal BaTiO3 , both AO12 and BO6 are elongated along the c-axis, as c/a = 10098. Displacements of 9.68 pm for the Ba2+ ion and 11.50 pm for the Ti4+ ion along the tetragonal axis are responsible for creating a dipole moment with the polarization vector along the same direction. Smaller displacements of the oxygen ions contribute to distortions, with the four oxygen ions in the BO6 octahedron being perpendicular to the tetragonal axis displaced by 3.63 pm, in
114
Phase Transformations: Titanium and Zirconium Alloys Cubic At 130°C; a 1 = a 2 = a 3 = 4.009 Å
Tetragonal
a2
a3
a1
130°C a 1 = a 2 = 3.992 Å
At 0°C
c
a2 a1
c = 4.035 Å Monoclinic a 1 = a 2 = 4.013 Å At c = 3.976 Å –90°C β = 98° 51′
0°C
a3
Rhombohedral
β a1
Orthorhombic c′
α
–90°C a3 a2
α b′
c
a′
a ′ = 5.667 Å At b ′ = 5.681 Å –90°C c ′ = 3.989 Å
α
a1 a 1 = a 2 = a 3 = 3.998 Å At –90°C α = 89° 52.5′
Figure 2.14. Sequence of phase transformations which occur in BaTiO3 at 130, 0 and −90 C. Unit cell dimensions and the orientation of the polarization vector are also indicated.
the opposite direction to the Ti4+ displacement. In the rhombohedral structure, displacements and distortions are correlated. In this case, displacements are parallel to the threefold axis which passes through two opposite triangular faces of the octahedron. The orthorhombic structure, by virtue of its lower symmetry, presents a wider range of polyhedral distortions. As is illustrated in Figure 2.14 the structure of BaTiO3 does not remain stable over the whole temperature range below the first ferroelectric Curie point and it transforms successively into lower symmetry variants, namely, cubic → tetragonal → monoclinic (orthorhombic) → rhombohedral. The vector directions of polarization are also indicated within the unit cells of these structures (Eric Cross 1993). Christian (1990) has grouped martensitic transformations in subcategories based on the structures of the parent and the product phases, as listed in Table 2.2.
Classification of Phase Transformations
2.5
115
MIXED MODE TRANSFORMATIONS
The classification scheme discussed so far makes an attempt to assign a given transformation to a specific category based on thermodynamic, kinetic and mechanistic criteria. It must be emphasized that there exist several phase transformations in real systems which do not fall exclusively in a single category. These transformations, which exhibit characteristics of different classes of transformations, are often called “mixed mode” or “hybrid” transformations (Banerjee 1994). Some of these cases are briefly discussed here for the purpose of illustration. From thermodynamic considerations one can cite cases which show pretransition effects similar to those of second-order transitions and at the same time a sharp discontinuity of thermodynamic functions at the transition temperature. Though these have been designated as “mixed” in Figure 2.11 the sharp discontinuity at the transition temperature makes them first-order transitions as per the thermodynamic definition. Pretransition effects in these often arise due to quasistatic structural fluctuations (modulations in chemical composition or displacement) or modulations in the polarization of electric or magnetic vectors associated with lattice points. The interplay between more than one homogeneous phase transformations can be illustrated by (a) concomitant ordering and clustering processes and (b) sequential operation of spinodal clustering and magnetic ordering. 2.5.1 Clustering and ordering The formation of an ordered intermetallic phase from a supersaturated dilute solid solution often requires concomitant ordering and clustering. The conditions for either simultaneous or sequential operation of clustering and ordering processes have been identified (Kulkarni et al. 1985, Khachaturyan et al. 1988, Soffa and Laughlin 1989) in terms of instabilities associated with the clustering wave vector (k close to 000) and the ordering wave vector (k terminating at one of the special points of the reciprocal space). Let us consider an fcc solid solution which experiences the influences of 100 ordering instability and <000> clustering instability. The following situations can arise and the transformation sequence is accordingly selected: (a) The disordered solid solution is initially unstable with respect to 100 ordering but metastable against clustering. Ordering of the solid solution to an optimum level can introduce a tendency towards phase separation in the optimally ordered structure. (b) The disordered solid solution simultaneously exhibits 000 clustering and 100 ordering tendencies. Both the processes can proceed simultaneously, their relative kinetics determining the rates of progress of the two processes. (c) The peak instability temperature for 000 clustering is higher than that for 100 ordering. In this situation, clustering occurs first, creating solute-rich regions within which the ordering process sets in once the condition of ordering
116
Phase Transformations: Titanium and Zirconium Alloys
T1 αp
Tricritical point
αp
αf
Paramagnetic
Ferromagnetic P Q
A
αf
R
Composition (%B) (a)
Free energy
Temperature
Tc (X )
A
Composition (%B) (b)
Figure 2.15. (a) A phase diagram showing a two-phase region introduced by a ferromagnetic ordering. (b) Free energy–concentration diagram at T1 showing the introduction of a spinodal clustering region in the ferromagnetic phase.
instability is fulfilled. Such coupled clustering–ordering processes are discussed in Chapter 7 in connection with phase transition sequences in Zr–Al and Cu–Ti alloys. Higher-order transitions like magnetic ordering can also induce a clustering tendency in a solid solution, resulting in the appearance of multicritical points in the phase diagram. Allen and Cahn (1982) have discussed these issues in great detail. Let us consider a binary fcc system of components A and B, where A is ferromagnetic and the Curie temperature, Tc (X), of the A–B solid solution changes with composition in the manner shown in Figure 2.15. In the absence of the magnetic contribution, the system behaves like an ideal solution while with the introduction of the magnetic contribution, the free energy of the -phase is reduced from that corresponding to paramagnetic p to that of ferromagnetic f . At temperatures below the tricritical point, a two-phase region emerges between f and p and two spinodes can also be identified in the free energy–composition plot (Figure 2.15(b)). At a temperature T1 an alloy at the point Q experiences a clustering instability and initially decomposes spinodally to two ferromagnetic phases, the one enriched in the non-magnetic component eventually undergoing a ferromagnetic → paramagnetic disordering. At the points P and R, the alloys are metastable with respect to spinodal clustering and, therefore, the paramagnetic phase, p , in the former and the ferromagnetic phase, f , in the latter can form only by nucleation and growth. 2.5.2 First-order and second-order ordering There are some instances where a system exhibits simultaneously a second-order and a first-order chemical ordering tendency. The ordering process in Ni4 Mo can
Classification of Phase Transformations
117
be cited in this context (Banerjee 1994, Arya et al. 2001). The Ni4 Mo alloy, quenched from the high-temperature disordered (fcc) phase field, exhibits (Spruiell and Stansbury 1965, Ruedl et al. 1968) a short-range ordered (SRO) state characterized by diffraction intensity at 1 21 0 fcc positions and a complete extinction of intensity at 210fcc positions in the reciprocal space. The 1 21 0 reflections do not coincide with the superlattice reflections of the equilibrium Ni4 Mo (D1a ) structure (Figure 2.16). While the SRO state consists of heterospace fluctuations in the form of concentration wave packets of size 2–5 nm, with wave vectors 1 21 0 , the equilibrium D1a structure is associated with wave vectors, 15 420. The two competing superlattice structures, as shown in Figure 2.16(c) and (d), respectively, can be described in terms of 420 planes of all Ni (N) and all Mo (M) atoms in the stacking sequences of MMNNMMNN and MNNNNMNNNN, respectively. While the 1 21 0 ordering fulfils all the symmetry criteria for a second-order transition, the 15 420 ordering is necessarily a first-order transition. In order to examine the relative strengths of the two ordering tendencies, namely, the first order 15 420 and the second order 1 21 0 , the free energy (F ) surfaces, as functions of the respective order parameters, 1 21 0 and 15 420, have been calculated using first principles thermodynamic calculations (Arya et al. 2001). The instability of the system with respect to fluctuations corresponding to the two order parameters can be determined by examining the curvature ( 2 F/ 2 ) of the F versus plots 1 1 at = 0 along the two directions, 1 2 0 and 5 420. With decreasing temperatures, T1 , T2 , T3 and T4 , the following four situations arise, the corresponding free energy surfaces being depicted in Figure 2.17(a)–(d). (1) Tc (D1a ) < Tc (1 21 0) < T1 : positive curvatures for both 1 21 0 and 15 420 ordering, implying stability of the disordered state, = 0. curvature (2) Tc (D1a ) < T2 < Tc (1 21 0): negative curvature for 1 21 0 and positive for 15 420 ordering, implying instability of the system for 1 21 0 ordering and no tendency towards D1a ordering. (3) Ti (D1a ) < T3 < Tc (D1a ), < Tc (1 21 0): negative curvature for 1 21 0 and positive curvature for 15 420 ordering at = 0 but a dip in the free energy with respect to the latter (D1a ordering) near 15 420 = 08. This implies that the system 1 experiences simultaneously tendencies towards 1 2 0 ordering (second order) and D1a ordering (first order). (4) T4 < Ti (D1a ) < Tc (D1a ) < Tc (1 21 0): negative curvatures along both 1 21 0 and 1 420 ordering implying 5 1 that 1the system is unstable with respect to the development of both 1 2 0 and 5 420 In this situation, homoge ordering. neous ordering is feasible for both 1 21 0 and D1a ordering. A mixed state consisting of concentration waves with wave vectors ranging from 1 21 0 to 1 420 is encountered on the path of the ordering process. This mixed state 5
118
Phase Transformations: Titanium and Zirconium Alloys
(b)
(a)
Period
M
N N M M
N p: 0
1
2
3
4
N2 M 2 (420) -Unit Cell
N2 M2 -Tile
(c) M
M 4 N3 2 1 M
N p: 0 1
2 3
4
5
N4 M
(420)
N4 M
(d)
Figure 2.16. Electron diffraction patterns corresponding to (a) the “short-range ordered” structure characterized by 1 21 0 reflections and complete extinction of 210 reflections and (b) the D1a ordered structure in the Ni–25 at.% Mo alloy. Real lattice descriptions of the fcc-based superstructures in terms of stacking of 420 planes in the [001] projections and static concentration waves are shown in (c) for 1 21 0 ordering and in (d) for D1a -Ni4 Mo ordering with wave vector 15 420. The sequences of Ni (N) and Mo (M) layers of (420) planes and subunit cell clusters are also shown.
Classification of Phase Transformations 6
8
6
T1 > T2 > T3 > T4
4
F ord (K)
4
T1
2
2
T2
T1
0
T2 0
ηc
–2
T3
ηc
T3
–2
–4
T4
T4
–4 0.0
119
0.2
0.4
0.6
0.8
1.0
–6 0.0
0.2
0.4
0.6
0.8
1.0
η /ηmax Figure 2.17. The ordering free energy of the Ni–25 at.% Mo alloy, exhibiting the 1 21 0 and the 1 420 ordering tendencies, plotted as a function of order parameters for the corresponding ordering 5 wave vectors at four different temperatures, T1 , T2 , T3 and T4 , respectively, pertinent to the situations described in the text.
is characterized by diffraction (Figure 2.18) which show a spread of patterns diffracted intensity linking 1 21 0 and 15 420 positions and by the presence of subunit cell clusters (or motifs) representing 1 21 0 and 15 420 ordered structures (as illustrated in Figure 2.16) in lattice resolution images (Figure 2.18).
(a)
(b)
Figure 2.18. Microstructure and diffraction pattern corresponding to mixed 1 21 0 and 15 420 ordering: (a) diffraction pattern showing intensity distribution linking 1 21 0 and 15 420 spots; (b) high-resolution electron micrograph showing motifs of D1a and N2 M2 structures (as schematically illustrated in Fig. 2.16 (c) and (d)).
120
Phase Transformations: Titanium and Zirconium Alloys
2.5.3 Displacive and diffusional transformations Phase transformations are classified as displacive and diffusional on the basis of the nature of atom movements across the advancing transformation front. One can envisage transformations which occur by a coupling between a displacive and a diffusional mode of atomic movements. The formation of ordered -structures from the disordered parent bcc -phase can be cited as an appropriate example of a mixed diffusive/displacive transformation in which the bcc lattice is transformed into the hexagonal -structure by a periodic displacement of lattice planes while the decoration of the -lattice by different atomic species occurs through diffusional atom movements. These two processes can as well be designated as displacive and replacive ordering, respectively, and the overall process can be viewed as a superimposition of a displacement wave and a concentration wave on the bcc lattice (Banerjee et al. 1997). This mode of transformation (discussed in Chapter 6) is encountered in several bcc Ti and Zr alloys, leading to the formation of a wide variety of ordered -structures. Displacive and diffusional atom movements can also be coupled through kinetic considerations in several cases, one of the best examples being the formation of -hydride precipitates in either the - or the -phase matrix alloys – a topic discussed in detail in Chapter 8. The formation of the -hydride phase from either the - or the -phase involves a shear transformation of the parent lattice accompanied by partitioning of hydrogen atoms (discussed in Chapter 8). The latter process being exceptionally rapid, the displacive lattice shear and hydrogen partitioning can occur nearly concurrently. Hydride formation in Zr- and Ti-based alloys can be compared with bainite formation in steels. 2.5.4 Kinetic coupling of diffusional and displacive transformations Olson et al. (1989) have analysed the kinetics of a transformation process in which the product phase forms with a partial redistribution of the interstitial element during non-equilibrium nucleation and growth. The rate at which the advancing transformation front moves depends both on its intrinsic mobility and on the ease with which the interstitial element diffuses ahead of the moving interface. The intrinsic mobility is related to the process of structural change across the moving interface. The growth, involving partial supersaturation in which local equilibrium is not established at the interface, seems to be an unstable process. This is because a perturbation in composition towards equilibrium would lead to a reduction in free energy which would drive the system to attain a local equilibrium. Stability of the non-equilibrium growth mechanism can, however, be brought about by another process, such as structural transformation across the interface, occurring in series. A displacive structural transition involving movement of a glissile interface and
Classification of Phase Transformations
121
Xl
γ
Gdd
G id
Carbon →
Free energy
α
γ
X
Xα
Xα
Xl Xm
X
Carbon concentration
Distance → (b)
(a)
Figure 2.19. (a) Free energy–concentration plots for ferrite () and austenite () showing the free energy component dissipated in structural change at the interface, Gid , and the free energy component dissipated in diffusive movements of interstitial atoms, Gdd , ahead of the transformation front; (b) shows concentration distribution in the - and -phases.
diffusive movements of interstitial atoms ahead of the interface are, therefore, modelled as coupled processes resulting in the bainitic transformation in steels. Both these processes dissipate the net free energy, G (as indicated in Figure 2.19) which is made up of Gid and Gdd amounts dissipated in the interface process and the diffusion process, respectively. The interfacial velocities can be calculated for the two processes and expressed as Vi = Gid and Vd = Gdd
(2.6)
where and are response functions relating the velocity to the appropriate dissipation. For the process to be kinetically coupled, the interface velocity, V = Vi = Vd . The interface velocity is calculated on the basis of thermally activated motion of dislocations which constitute the ferrite/ austenite glissile interface and is found to be comparable with the diffusion field velocities computed for different levels of interstitial supersaturation. It may be noted that the two types of thermally activated events which are coupled in this treatment operate on widely differing size scales. The manner in which the unit processes of the displacive and the diffusional aspects of the transformation couple at the microscopic level can be understood in terms of the discontinuous nature of the thermally activated interfacial motion. During the “waiting time” prior to a thermally activated event, the solute partitioning in the vicinity of the interface can cause a steady increase in the local interfacial driving force till it reaches a threshold value where the interface is driven to its next position of temporary halt. The movement between these positions occurs through a diffusionless “free glide” motion.
122
Phase Transformations: Titanium and Zirconium Alloys
This coupled transformation model has been applied to bainitic transformations in steels. The model predicts an increasing interfacial velocity and supersaturation during growth with decreasing transformation temperature, while the nucleation velocity passes through a maximum, giving C-curve kinetics. In this context, it should be mentioned that there is a different view point on the mechanism of the bainitic transformation in which atom transport across the interface is considered to occur through diffusional random jumps. The displacive process, involving coordinated atomic jumps from a parent lattice site to a predestined site in the product lattice, has not been accepted as a requirement for a bainitic transformation in the diffusionist view which has been summarized in an excellent manner by Reynolds et al. (1991).
REFERENCES Allen, S.M. and Cahn, J.W. (1982) Bull. Alloy Phase Diagrams, 3, 287. Arya, A., Banerjee, S., Das, G.P., Dasgupta, I., Saha-Dasgupta, T. and Mookerjee, A. (2001) Acta Mater., 49, 3575. Banerjee, S. (1994) Solid → Solid Phase Transformations (eds W.C. Johnson, J.M. Howe, D.E. Laughlin and W.A. Soffa) TMS, Warrendale, PA, p. 861. Banerjee, S., Tewari, R. and Mukhopadhyay, P. (1997) Prog. Mater., Sci., 42 (1–4), 109. Buerger, M.J. (1951) Phase Transformations in Solids, Wiley, New York, p. 183. Christian, J.W. (1965) Physical properties of martensite and bainite, Special Report 93, Iron and Steel Institute, London, p. 1. Christian, J.W. (1979) Phase Transformations, Vol. 1, Institute of Metallurgists, London, p. 1. Christian, J.W. (1990) Mater. Sci. Eng., A127, 215. Cohen, M., Olson, G.B. and Clapp, P.C. (1979) Proceedings of International Conference on Martensite, MIT Press, Cambridge, MA, p. 1. Cook, H.E. (1974) Acta Metall., 22, 239. de Fontaine, D. (1975) Acta Metall., 23, 553. Delaye, L., Chandrasekaran, M., Andrade, M. and Van Humbeck, J. (1982) Solid– Solid Phase Transformations (eds H.I. Aaronson, D.E. Laughlin, R.F. Sekerka and C.A. Wayman) TMS, Warrendale, PA, p. 1429. Ehrenfest, P. (1933) Proc. Acad. Sci. Amst., 36, 153. Eric Cross, L. (1993) Ferroelectric Ceramics., (eds Nava Setter, Enrico L. Colla and Birkhaeuser Basel), Switzerland. Khachaturyan, A.G., Lindsey, T.F. and Morris, J.W. (1988) Metall. Trans., 19A, 249. Kulkarni, U.D., Banerjee, S. and Krishnan, R. (1985) Mater. Sci. Forum, 3, 111. Landau, L.D. and Lifshitz, E.M. (1969) Statistical Physics, Pergamon Press, Oxford. Olson, G.B., Bhadeshia, H.K.D.H. and Cohen, M. (1989) Acta Metall., 37, 381. Rao, C.N.R. and Rao, K.G. (1978) Phase Transitions in Solids, McGraw Hill, New York.
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Reynolds, W.T., Jr, Aaronson, H.I. and Spanos, G. (1991) Mater. Trans. JIM, 32, 737. Roy, R. (1973) Phase Transitions (ed. L.E. Cross) Pergamon Press, Oxford, p. 13. Ruedl, E., Delavignette, P. and Amelinckx, S. (1968) Phys. Status Solidi, 28, 305. Soffa, W.A. and Laughlin, D.E. (1989) Acta Metall., 37, 3019. Spruiell, J.E. and Stansbury, E.E. (1965) J. Phys. Chem. Solids, 26, 811. Tanner, L.E., Schryvers, D. and Shapiro, S.M. (1990) Mater. Sci. Engi., A127, 205.
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Chapter 3
Solidification, Vitrification, Crystallization and Formation of Quasicrystalline and Nanocrystalline Structures
3.1 Introduction 3.2 Solidification 3.2.1 Thermodynamics of solidification 3.2.2 Morphological stability of the liquid/solid interface 3.2.3 Post-solidification transformations 3.2.4 Macrosegregation and microsegregation in castings 3.2.5 Microstructure of weldments of Ti- and Zr-based alloys 3.3 Rapidly Solidified Crystalline Products 3.3.1 Extension of solid solubility 3.3.2 Dispersoid formation in rapidly solidified Ti alloys 3.3.3 Transformations in the solid state 3.4 Amorphous Metallic Alloys 3.4.1 Glass formation 3.4.2 Thermodynamic considerations 3.4.3 Kinetic considerations 3.4.4 Microstructures of partially crystalline alloys 3.4.5 Diffusion 3.4.6 Structural relaxation 3.4.7 Glass transition 3.5 Crystallization 3.5.1 Modes of crystallization 3.5.2 Crystallization in metal–metal glasses 3.5.3 Kinetics of crystallization 3.5.4 Crystallization kinetics in Zr 76 Fe1−x Nix 24 glasses 3.6 Bulk Metallic Glasses 3.7 Solid State Amorphization 3.7.1 Thermodynamics and kinetics 3.7.2 Amorphous phase formation by composition-induced destabilization of crystalline phases 3.7.3 Glass formation in diffusion couples 3.7.4 Amorphization by hydrogen charging 125
128 128 128 135 140 141 145 150 152 153 153 157 157 159 165 171 176 180 182 184 185 187 192 200 205 212 215 220 220 225
3.7.5 Glass formation in mechanically driven systems 3.7.6 Radiation-induced amorphization 3.8 Phase Stability in Thin Film Multilayers 3.9 Quasicrystalline Structures and Related Rational Approximants 3.9.1 Icosahedral phases in Ti-and Zr-based systems References
226 229 237 241 248 252
Chapter 3
Solidification, Vitrification, Crystallization and Formation of Quasicrystalline and Nanocrystalline Structures
List of Symbols Gl/s : Free energy of the liquid/solid phase l/s : Chemical potential of the liquid/solid phase cl/s : Composition of the liquid/solid phase l/s : Activity coefficient for the liquid/solid phase Tl/s : Liquidus/solidus temperature Kl/s : Thermal conductivity in the liquid/solid phase ko : Partition coefficient representing the ratio of the slopes of the liquidus and solidus lines : Wavelength of a perturbation : Amplitude of a perturbation : Rate of growth/decay of a perturbation Gc : Composition gradient of the liquid phase Dl : Diffusion coefficient of the solute tf : Local solidification time Ms : Martensitic start temperature for the phase Tmi : Melting point of the ith component
Hfi : Heat of fusion of the ith component I: Rate of nucleation of a crystalline phase Z: Frequency of the atomic jumps across the interface A∗ : Surface area of the critical nucleus N ∗ : Number density of critical nuclei W ∗ : Work done to form a critical nucleus
Gv : Volume free energy change for the formation of critical nucleus : Induction time f : Fraction of the transformed volume Q: Activation energy of a given process Tp : Temperature at which transformation rate reaches the peak value u: Growth rate ˜ D: Interdiffusion constant
Hv : Enthalpy of vacancy formation 127
128
Phase Transformations: Titanium and Zirconium Alloys
kB : Boltzmann constant ± : Frequency of ordering + or disordering − jumps Z : Number of nearest neighbor -sites around an -site
3.1
INTRODUCTION
In this chapter, transformations involving liquid, amorphous, nanocrystalline and quasicrystalline phases are discussed. Many of these transformations occur under conditions far removed from equilibrium. With the continuous development of non-equilibrium processing techniques, an increasing number of novel transformation products, some of them possessing exotic properties, have been discovered in the recent past. Ti and Zr alloys have had their due share in this exciting scientific development of production of metastable microstructures by techniques such as liquid and vapour state processing, mechanical attrition, interdiffusion and radiation processing. Phase transformations associated with the liquid, amorphous, quasicrystalline and nanocrystalline states are discussed in this chapter by citing examples taken from Ti- and Zr-based alloys.
3.2
SOLIDIFICATION
Solidification and melting are strong first-order transformations which are of tremendous importance in various technological applications such as ingot casting, foundry casting, single crystal growth and welding. An understanding of the mechanism of solidification is very important for predicting how different parameters such as temperature distribution and cooling rate influence microstructure, alloy partitioning and mechanical properties of cast and fusion welded materials. The objective of this section is to introduce some of the concepts which will be needed for the discussions in later sections dealing with rapid solidification, amorphization and devitrification. 3.2.1 Thermodynamics of solidification As in the case of any other transformation, solidification cannot proceed at equilibrium. Depending on the extent of departure from the equilibrium condition, a hierarchy of different conditions has been identified by Boettinger and Biloni (1996), the solidification rate increasing as the system is driven away from equilibrium. These conditions are listed as follows in the order of increasing departure from equilibrium.
Solidification, Vitrification and Crystallization
129
(1) Full diffusional equilibrium is established globally. Under this condition, there are no gradients of chemical potential and temperature in the system. The compositions of the liquid and the solid phases attain the equilibrium values. The solidification process cannot continue after this condition is attained. (2) Chemical equilibrium is established locally at the liquid/solid interface. The compositions of the liquid and solid phases at the interface are given by the equilibrium phase diagram, any correction due to the interface curvature being taken into account. (3) Local interfacial equilibrium is established between the liquid and a metastable solid phase. Such a situation arises when the equilibrium phase cannot nucleate or grow fast enough to compete with the metastable phase. The compositions of the solid and liquid phases at the interface are given by the pertinent metastable phase diagram. (4) Local equilibrium condition is not established at the liquid/solid interface. Here the temperature and compositions at the interface are not given by either equilibrium or metastable phase diagrams. The condition of local equilibrium, whether stable or metastable, is that the chemical potentials of the components in the liquid and the solid phases are equal across the liquid/solid interface. Condition (4) relates to a situation where the rapidly moving interface does not permit the chemical potentials of the components to equalize across the interface. The rapid growth rates which result under large supercooling can trap the solute into the freezing solid at levels exceeding the equilibrium value for the corresponding liquid composition prevailing at the interface. If one considers only the chemical potential of the solute, it increases upon being incorporated in the freezing solid by a process called solute trapping. However, to make this process thermodynamically possible, a decrease in the chemical potential of the solvent, leading to a net decrease in the free energy, becomes essential (Baker and Cahn 1971). Let us consider the thermodynamics of solidification in a binary system comprising the components A and B, which is represented by the free energy–composition plots, Gl and Gs (Figure 3.1), corresponding to the liquid and solid phases at a temperature which is between the solidus and the liquidus temperatures. In order to determine the composition range of solids which can form from a liquid of composition co , a tangent is drawn to the Gl curve at co . This tangent intersects the Gs curve at two points, cs1 and cs2 . The Gs curve between cs1 and cs2 remains below the tangent, indicating that it is thermodynamically possible to form a solid in this composition range from the liquid of composition, co . At temperatures above the liquidus, tangents to any point on the Gl curve do not intersect the Gs curve and at the liquidus temperature, the tangent to the Gl curve touches the Gs
130
Phase Transformations: Titanium and Zirconium Alloys μsB(c∗n) Tl
Gs
c (To)
A
μ l (co)
P
μsA(c∗s)
Q
μsA(c∗n)
Gl
M
N
cs*
cn* A
μ l (co) cs1
A
cs(eq)
cs2
co
μsB(c∗s)
c l(eq)
Atom fraction of B
B
Figure 3.1. Free energy–concentration plots, Gl and Gs , of the liquid and the solid phases, respectively, at a temperature between liquidus (Tl ) and solidus (Ts ). From thermodynamic considerations, a liquid of composition, co , can form a solid of any composition between cs1 and cs2 .
curve only at one point which gives the equilibrium solid composition, cs eq. If we define the chemical potentials of the components A and B in the liquid and the solid phases at the interface as Al , Bl , As and Bs , respectively, the free energy change during solidification, G, is given by
G = As − Al 1 − cs∗ − Bs − Bl cs∗
(3.1)
where cs∗ is the composition of the solid phase being separated. Considering that the liquid phase is homogeneous in composition, Al and Bl correspond to the liquid composition, co , and are given by the intercepts made by the tangent to Gl at co on the free energy axes corresponding to pure A and pure B, respectively. The free energy change associated with the formation of the solid of composition cs∗ is represented by the drops MN and PQ shown in Figure 3.1 for two different values of cs (cn∗ and cs∗ ). It is to be noted that for cs = cs∗ , solidification involves a lowering of the chemical potentials, A and B , of both the components. In contrast, for cs = cn∗ , A = Al − As < 0 but B = Bl − Bs > 0. Based on the free energy–composition plots for the liquid and the solid phases at a given temperature, the domains of all possible solid compositions, which are allowed to form from thermodynamic considerations, can be presented in an isothermal plot of the solute content of the liquid versus that of the solid at the interface (Figure 3.2). At a given temperature, the liquidus composition gives the
Solid composition at interface, atom fraction of B (cs)
Solidification, Vitrification and Crystallization
O
Y c (To)
X cs (eq)
Slope 1 ΔG > 0
cs2
B
Δμ > 0
ΔG = 0
Slope 1 Δμ A = 0 Slope k Δμ B = 0
E
Δμ A > 0 Δμ A < 0 ΔμB < 0
131
Equilibrium P
cs1 cl(eq)
Liquid composition at interface, atom fraction of B (c l)
Figure 3.2. The domain OXYEP of all possible solid compositions that can form from various liquid compositions. This can be divided into three distinct regions, the shaded region OEP where chemical potentials of both A and B decrease on solidification, the region above the line OE where solute trapping occurs and the region below the line PE where solvent trapping occurs (after Baker and Cahn 1971).
maximum solute content of the liquid at the interface from which solidification can occur, and in case solidification occurs without any solute partitioning, the solid inherits the composition of the liquid. The condition of partitionless solidification is met at a point where Gl and Gs curves intersect (Figure 3.1(b)) (i.e. where the integral molar free energies of the two phases are equal). The locus of these points defines the To line. The domain of all possible solid compositions that can form from various liquid compositions at a given temperature can be defined by the curve OXYEP. The maximum limit of solute concentration in the solid is given by the point Y , which is fixed by the point of intersection of the cTo line and the line OY (slope = 1) representing identical compositions of the liquid and the solid phases. The point E corresponds to the maximum limit of solute concentration in the liquid, given by the liquidus composition, cl eq, and the equilibrium solidus composition, cs eq. At the boundary of the OXYEP domain, the condition of the change in the integral molar free energy for solidification being zero ( G = 0) is satisfied. As the liquid composition is changed to lower the solute content, a tangent intersects the Gs curve at two points. This situation is illustrated in Figure 3.1 by the tangent at co which intersects Gs at cs1 and cs2 , and solids of any composition between these two limits can solidify from the liquid of co composition. At the composition,
132
Phase Transformations: Titanium and Zirconium Alloys
cTo , where Gl and Gs intersect, the range of solid compositions spans from zero (point P) to cTo (point Y ). The domain OXYEP can be divided into three regions. In the shaded region OEP, the chemical potentials of both A and B decrease on solidification, while the chemical potential of only one of the components decreases outside this region. The overall free energy change favours solidification in the entire OXYEP domain, but outside the region OEP, one of the components enters the solid phase with an increase in the chemical potential. Such a process is termed solute trapping (in the region OXYE) or solvent trapping (below the line PE, where A > 0 but
B < 0. For a quantitative description of the solute trapping concept, let us consider a simple case of a dilute binary solution for both liquid and solid. Then the chemical potentials are given by Henry’s law for the minor component: Bs = Bs + RT ln s cs
(3.2)
Bl = Bl + RT ln l cl
(3.3)
where Bsl and sl are related constants which depend on the temperature and reference state. Under equilibrium conditions, where cs = cs eq and cl = cl eq, Bs − Bl = Bs − Bl + RT ln
s cs eq =0 l cl eq
(3.4)
The change in chemical potential of the minor component across the solidification front can, therefore, be expressed as
B = RT ln
cs cl eq cs eqcl
(3.5)
In terms of the distribution coefficient, k, at the interface, k = cs /cl ,
B = RT lnk/keq
(3.6)
The straight line OE with a slope keq (in Figure 3.2) represents the equilibrium condition in which the minor component experiences no change in chemical potential. When k > keq, in the region above the line OE, B > 0, which corresponds to solute trapping. For the major component, Raoult’s law holds, and the change in the chemical potential of the component A on solidification can be written as
Solidification, Vitrification and Crystallization
A = RT ln
133
1 − cs 1 − cl eq 1 − cs eq1 − cl
(3.7)
The straight line PE, of slope 1 − cs eq/1 − cl eq = 1, through the equilibrium composition point cs eq, cl eq in Figure 3.2 represents the equilibrium condition. Solvent trapping occurs below the line PE where A > 0. The G = 0 curve is given by the condition 1 − cs A + cs B = 0
(3.8)
and the curve passes through the points O, X, Y , E and P. Thermodynamics places general restrictions on the composition limits of the solidifying phases, but it does not specify the solid composition under a given supercooling ( T ) and solidification rate (V , the velocity of the solid/liquid interface). Boettinger (1982) has shown the allowable composition ranges of the solid superimposed on phase diagrams of binary systems (Figure 3.3(a) and (b)). The shaded regions in these diagrams indicate thermodynamically allowed solid compositions that may be formed from a liquid of composition co at various temperatures. The To curve gives the highest temperature at which partitionless solidification co = cs can occur. Figure 3.3(b) shows the case where the To curve plunges and partitionless solidification is not permitted for a liquid of composition co . These curves can be used to determine the limit of the extension of solubility obtainable by rapid quenching of the liquid phase. If the To curve plunges to a
co L To
Temperature
Temperature
α
co L To a
Atom fraction of B
Atom fraction of B
(a)
(b)
Figure 3.3. Thermodynamically allowed solid compositions which can form from a liquid of composition co are shown by the shaded regions in two schematic phase diagrams. The To curve gives the highest temperature at which partitionless solidification (co − cs ) can occur. While in (a), partitionless solidification of a liquid of composition co is possible, in (b) where the To curve plunges, partitionless solidification is not permitted for the same liquid composition (after Boettinger 1982).
134
Phase Transformations: Titanium and Zirconium Alloys
very low temperature, as in Figure 3.3(b), the -phase with solute contents beyond the To curve cannot be formed from the melt. In fact, for phase diagrams with a retrograde solidus, the To curve plunges sharply resulting in a limited extension of solubility. Eutectic systems with plunging To curves are good candidates for easy formation of metallic glasses. This point is elaborated in Sections 3.4.1 and 3.4.2. In contrast, alloys with To curves which are only slightly depressed below the liquidus curves (as shown in Figure 3.3(a)) make good candidates for extension of solubility and are unlikely ones for glass formation. In the analysis of a solidification problem, the condition of local equilibrium is often invoked. This assumption is valid whenever the deviation from equilibrium, expressed as T , cs − cs eq, or cl − cl eq, is small compared to the total temperature and composition ranges pertinent to the solidification process. This is indeed so for most solidification problems that involve rather low velocities of the liquid/solid interface. Let us now consider the steady state plane front condition. We impose a velocity V on the liquid/solid interface and assume that, after an initial transient, the temperature and composition profiles and the position of the interface move with the velocity V . Under this condition, the composition of the solid, cs , must be equal to the overall composition, co , of the alloy. Because cs = co , the process has some aspects of “partitionless” solidification, but it includes the situation in which a liquid layer of a different composition (cl = co ) remains ahead of the advancing solidification front (Figure 3.4). Under steady state conditions, this layer remains unchanged with time and may be hard to detect. It will have a thickness of about Tm
ko < 1
Solid
Liquid
Composition
→
Tl (co)
Initial transient (I) →
Temperature
co
Ts (co)
Initial transient (II) Final transient (III)
Δco co ko co
co/ko
Composition
co ko Δco co c o ko Distance (b)
(a)
Figure 3.4. (a) A schematic phase diagram showing the liquidus and solidus temperatures corresponding to an alloy of composition co . (b) The composition profile of the liquid and the solid in the vicinity of the solidification front under steady state solidification. ko is the equilibrium partitioning ratio.
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135
Dl /V where Dl is the diffusivity in the liquid. The thickness of the layer turns out to be less than 1 m when V exceeds 1 cm/s. The possibility of establishing steady state conditions can be examined by redrawing in Figure 3.1 the domains of thermodynamically allowable liquid interface compositions at different temperature – composition regions, defined, respectively, as Regions I, II and III. In Region I, the point B1 , the maximum solute concentration in the solid, remains below co , and therefore, it is not possible to establish the steady state from thermodynamic considerations. This is essentially because G > 0 for all possible values of cl . In Region II, co lies between BII and EII , implying that steady state solidification is possible, provided the composition of the liquid remains within the band marked in the figure. The solid formed remains metastable with respect to partial remelting. There is, however, a diffusional instability for steady state solidification in Region II. A downward fluctuation in the solid composition (cs < co ) will lead to a shift of the liquid composition to the right (cl increasing). Since the solid that is forming is below the average composition co , excess solute is rejected to the liquid, making the liquid further enriched in the solute. This may result either in the break-up of the plane front interface or in a reduction in the interface temperature. Steady state solidification, though thermodynamically possible, is diffusionally unstable. In Region III, steady state growth is not only thermodynamically possible but also stable. If solidification starts at the point EIII , cs > co and the liquid will be depleted of excess solute and the system will spontaneously leave the point EIII and settle on the horizontal line cs = co for partitionless solidification. 3.2.2 Morphological stability of the liquid/solid interface In the previous section, it has been assumed that the solid–liquid interface is microscopically planar. Under this condition, the composition profile induced in the solid varies only in the direction of growth. A planar interface may become unstable to small changes in shape even if the heat flow remains unidirectional. The stability of the liquid–solid interface during solidification is considered here, first for pure metals and then for alloys. In pure metals, solidification can be described in terms of the latent heat being conducted away from the liquid–solid interface, i.e. dT dT = Kl + vLv (3.9) Ks dx s dx l where Ksl are the thermal conductivities of and dT/dxsl are the temperature gradients in the solid and the liquid, respectively, Lv is the latent heat of fusion per unit volume and V is the velocity of the liquid–solid interface. The morphological
136
Phase Transformations: Titanium and Zirconium Alloys
stability of this interface is governed by the sign of the temperature gradient in the liquid ahead of the transformation front. When the solid grows into a superheated liquid, i.e. for a positive gradient, dT/dxl > 0, the interface remains stable. This can be understood from the following argument. If a small protrusion of solid develops at the plane solidification front due to a local increase in V , the temperature gradient in the liquid ahead of the protrusion will increase, while that in the solid will decrease. Consequently, more heat will be conducted into the protruding solid and less away, resulting in a decrease in the growth rate in this localized region compared to that in the planar region. As a consequence, the protrusion will disappear. The process in which a perturbation of the planar morphology dies down when the latent heat is extracted through the solid is schematically illustrated in Figure 3.5(a).
Solid
Liquid Solid
Liquid
Tm Heat flow
Tm Heat flow T
T
v
v x
x
Solid
Solid
Liquid
Liquid
Heat flow
Heat flow
Isotherms (a)
(b)
Figure 3.5. Temperature distribution during solidification (a) for extraction of heat through the solid and (b) for heat flow into the liquid.
Solidification, Vitrification and Crystallization
137
During the solidification process, in which the solid grows into a supercooled liquid, the gradient Gl = dT/dx ahead of the interface is negative. If a protrusion forms on such an interface, the negative temperature gradient becomes more negative and, therefore, heat is removed more effectively from the tip of the protrusion than from the surrounding flat regions. A perturbation created on the interface, therefore, tends to grow with time, indicating an inherent instability of the solidification front, as shown in Figure 3.5(b). The instability of the solid–liquid interface is responsible for developing arms on crystals nucleated in a supercooled liquid. These arms grow along crystallographic directions of easy heat transfer and during their growth create secondary and tertiary arms, resulting in a dendritic structure. Dendrites in pure metals are known as thermal dendrites to distinguish them from those forming in alloys primarily due to the constitutional supercooling phenomenon, which is described in the following by considering the case of one-dimensional movement of the solidification front in a binary alloy of composition, co , as shown in the corresponding phase diagram Figure 3.4(a). As in this case the solidification process leads to a partitioning of the solute preferentially towards the liquid phase (partition coefficient, the ratio of the slopes of the liquidus and solidus lines, ko < 1), the liquid ahead of the solidification front becomes solute enriched. After the initial transient, a steady state is established when the liquid in contact with the solidification front attains a composition, co /ko , and the solid/liquid interface reaches the solidus temperature, Ts co . Under this condition, a local equilibrium is established at the solidification front. Tiller et al. (1953) have expressed the composition of the liquid ahead of the solidification front in terms of the solute diffusion coefficient, Dl , in the liquid, the distance, z, from the interface and the velocity, v, of the interface: 1 − ko −vz cl = co 1 + exp (3.10) ko Dl The profile of the solute concentration ahead of the solidification front is shown in Figure 3.6(a) while the liquidus temperature of the solute-enriched region in front of the solidification front is depicted in Figure 3.6(b). The corresponding liquidus temperature for the composition in front of the interface is given by 1 − ko −vz Tl z = Tm + ml co 1 + exp (3.11) ko Dl where ml is the slope of the liquidus line. Figure 3.6(b) also shows three possible profiles of the actual temperature. These are labelled as cases (a), (b) and (c). For the case (a), the actual temperature remains above Tl z, which means constitutional supercooling does not occur ahead of the solid–liquid interface.
138
Phase Transformations: Titanium and Zirconium Alloys a
Slope Gc Solute Temperature gradient, Gs
Solid
c
Temperature
Temperature
Interface
Tl (z) b
Temperature gradient in liquid, Gl
Liquid Distance, z
Distance, z
(b)
(a)
(c)
Figure 3.6. (a) Solute concentration, (b) temperature profile ahead of the solidification front for a system with k0 < 1; z is measured from the solid/liquid interface and (c) early stages of the development of morphological instability of liquid–solid interface as revealed in dendrites of the rapidly solidified Zr 545 Cu20 Al10 Ni8 Ti75 alloy frozen in the amorphous matrix (magnification = 10000X).
Case (b) represents the situation where the actual temperature profile, Tz, is tangent to the Tl z line at z = 0, while for case (c), the Tz line remains below Tl z for some distance ahead of the solid–liquid interface where the condition of constitutional supercooling prevails. The case (b) essentially depicts the limiting condition between the presence and the absence of constitutional supercooling ahead of the solidification front, and this condition can be obtained by equating the slopes of the Tl z and Tz lines at z = 0 which yields Gl ml co ko − 1 ≥ V Dl ko
(3.12)
for constitutional supercooling and the resulting instability in the liquid–solid interface to occur. This criterion for constitutional supercooling, which was obtained by Tiller et al. (1953), serves as a model to understand the major cause of the morphological instability of the solid–liquid interface, but it does not yield any information about the size scale of the modulation developing on the solid–liquid interface. The analysis of morphological stability of the moving solid–liquid interface has been originally reported by Mullins and Sekerka (1963) and many assumptions
Solidification, Vitrification and Crystallization
139
of the original theory have subsequently been relaxed, as summarized by Sekerka (1986). An outline of the analysis of the morphological stability of the solid–liquid interface can be presented as follows. A perturbation of amplitude, , and wavelength, , is introduced on a flat solid– liquid interface growing in the z-direction. For a two-dimensional z x analysis, the perturbed surface can be represented as z = expt + 2ix/
(3.13)
where is the rate of growth (or decay) of the perturbation. The value of is determined by solving the steady state heat flow and diffusion equations with appropriate boundary conditions for small values of (linear theory). The planar interface is stable if the real part of is negative for all values of and = 0 will define the condition of the stability/instability transition which is given by the following equation: G − ml Gc c +
4 2 Tm =0 2
(3.14)
where is the surface energy and G, the conductivity weighted temperature gradient, is given by G=
Ks Gs + Kl Gl Kl + Ks
(3.15)
Kl and Ks are the conductivities of liquid and solid, respectively. Gc , the composition gradient in the liquid, can be obtained from Eq. (3.10) as Gc =
vco ko − 1 ko Dl
(3.16)
The parameter, c , can usually be set to unity. However, c may deviate significantly from unity under rapid solidification conditions. In general, c is given by c = 1 +
2ko
1 − 2ko − 1 +
4Dl 2 1/2
(3.17)
V
The stability of the solid–liquid interface is determined by the sign of the left hand side of equation (3.14). If it is positive, the interface is stable with respect to the perturbation introduced. The first term G has a stabilizing influence for
140
Phase Transformations: Titanium and Zirconium Alloys
a positive temperature gradient. For a single component material, this is the only term which is present. Therefore, in such a case, morphological instability can set in only during the growth of the solidification front into a supercooled liquid (negative temperature gradient). The second term in Eq. (3.14) represents the effect of solute diffusion in the liquid and being negative has always a destabilizing influence. The third term, which involves capillarity, has a stabilizing influence for all wavelengths, the minimization of the total surface energy being the motivating factor. This factor becomes more prominent at short wavelengths and, therefore, acts as a balancing force against any reduction in the wavelength of the modulation of the solid–liquid interface. The undulation at the solidification interface in case of a Zr–Al alloy is shown in Figure 3.6(c). 3.2.3 Post-solidification transformations Microstructures of as-solidified alloys based on Ti and Zr are influenced by the solute migration resulting from the solidification process. The first phase to solidify in these alloys in the (bcc) phase which undergoes subsequent solid state phase transformations depending on the local chemical composition (Figure 3.7). This point is explained in Section 3.2.5 by using a hypothetical binary phase diagram for an alloy with a -stabilizing element. Enrichment of the liquid with stabilizing solutes causes a local depression of the Ms temperature. During
Figure 3.7. Bright-field microstructure showing martensitic structure cutting across the cellular boundaries. The martensite has formed from the phase which was the first phase to solidify from the liquid.
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141
Table 3.1. Sequences of transformations during solidification processing in Ti- and Zr-based alloys. Alloy system
Sequence of phase transformations
Zr (with Si and O impurities) Zr – 1 at.% Nb Zr – 8.5 at.% Nb Zr – 27 at.% Al Ti – 50 at.% Al Zr3 Al–Nb
L → directly L L L L L
→ → → → → → + → Zr 2 AlB82 + → → 2 → 2 + → → + → + Zr 5 Al3 → Zr 2 Al
post-solidification cooling, some regions encounter a martensitic ( → ) or a Widmanstatten ( → ) transformation, depending on the prevailing cooling rate. Regions which are enriched in the -stabilizing solute beyond a certain level either retain the phase fully or transform into a + structure. The martensitic and the retained (with or without dispersion) structure is superimposed on the dendritic or cellular structure in the final microstructure, with the local chemical composition determining the nature of the post-solidification transformation. The solute enrichment process in the interdendritic regions can also induce other phase reactions such as the formation of an ordered intermetallic phase, → Zr 2 Al B82 , or a peritectoid reaction, + Zr 5 Al3 → Zr 2 Al, or → 2 → 2 + (as in TiAlbased alloys). Transformation sequences during solidification processing have been determined in some limited studies on Ti- and Zr-based alloys. Table 3.1 summarizes the results in a few representative cases. 3.2.4 Macrosegregation and microsegregation in castings The solidification theory discussed in the preceding section is utilized in assessing the extent of macro- and microsegregation in ingots and other castings. Macrosegregation causes non-uniformity in alloy composition that occurs over large distances, while microsegregation is over distances comparable to the dendrite arm spacing. The extent of microsegregation is expressed in terms of the segregation ratio (= local maximum solute concentration/local minimum solute concentration) or the volume fraction of the non-equilibrium secondary phase which forms as a result of segregation. As the solidification front grows into the liquid metal pool, the dendrite arms isolate the liquid into microscopic pools in the mushy zone (Figure 3.8). The region between two adjacent dendrite arms can be taken as a characteristic volume since the dendrite spacing, d, in an alloy under a given cooling condition remains quite uniform. By applying the equilibrium partition ratio, ko , the interface composition
142
Phase Transformations: Titanium and Zirconium Alloys Solid + liquid (mushy zone)
Solid
Liquid
λ = 0 (Dendrite spine) Solid
λ = λ i (Solid–liquid interface)
λ Liquid
λ = d /2 (Midpoint between two dendrites)
xR
xr
Figure 3.8. Schematic diagram showing dendritic areas of the solid phase growing into the mushy zone.
of the solid and the liquid phases can be estimated to be ko co and ko cl where co is the average composition and cl is the liquidus composition at the non-equilibrium solidus, as shown in Figure 3.9. The redistribution of the solute within the solid dendrite by diffusion can be computed from the material balance equation cl∗ − cs∗ dfs =
fs 0
Ds
2 cs df dt + 1 − fs dcl∗ x2 s
(3.18)
where cl∗ and cs∗ are the compositions at the interface of the liquid and the solid, respectively, fs is the weight fraction solid within the volume element and Ds is the diffusion coefficient of the solute in the base metal. The solute redistribution due to diffusion can be computed by substituting values for diffusion coefficient, dendrite arm spacing, solidification time and by employing numerical techniques described in literature (Brody and Flemings 1966). An estimate of the extent of microsegregation can be made by evaluating the parameter a=
4Ds tf d2
(3.19)
where tf is the local solidification time which is defined as the difference in time between the passing of the liquidus and solidus isotherms for a given point in the ingot. For a 1, diffusion in the solid is negligible and microsegregation is maximum, whereas for a 1 microsegregation is negligible. Since d = Ctfn 03 < n < 05 microsegregation does not change with the cooling rate when n = 05.
(3.20)
Temperature
Solidification, Vitrification and Crystallization
143
Tl
T s′ c ok o
c ′I
co Composition
Solid
Solid + liquid
Liquid
Temperature
Tl
T s′
cR
xT
Liquid composition
Distance
c ′I co
cR
xT Distance
Figure 3.9. Temperature–composition diagram showing interface composition of the solid (ko co ) and the liquid phases (ko cl ) where co is the average composition and cl is the liquidus composition at the non-equilibrium solidus. The liquid composition profile ahead of the solidification front is also shown.
Brody and David (1970) have compared the microsegregation parameters of several binary Ti alloys. It is noted that the extent of microsegregation increases with an increase in the freezing range and with a steep slope of the liquidus. Ternary and more complex alloys are expected to show more extensive microsegregation whenever the addition of alloying elements results in a lowering of the freezing range. Macrosegregation arises out of the bulk movement of solute rich liquid (for ko <1) in the mushy zone. Mechanical stress, either applied or thermally induced, may cause opening up of tears in the nearly solid regions. The tears may be filled then by the liquid from the neighbourhood which will have a much higher
144
Phase Transformations: Titanium and Zirconium Alloys
concentration of the solute. This mechanism leads to the formation of films of high solute content that run roughly parallel to isotherms. The flow of soluteenriched liquid to feed the solidification shrinkage is another mechanism of macrosegregation. Consumable electrode arc melting is the most extensively used technique for making Ti and Zr alloy ingots. The nature of microsegregation in such ingots can be visualized by considering the movement of the liquid pool during the growth of an ingot. Figure 3.10 shows a schematic of the consumable electrode arc melting process in which a liquid pool and a mushy zone traverse upwards as the electrode is progressively consumed. At the start of the cast, the operation will not be at a steady state which is attained after the liquid zone moves to a certain extent. The steady state growth, during which the size of the liquid pool and mushy zone remains constant, may be disrupted if the heat input to the ingot suddenly changes. This can happen due to a physical discontinuity in the electrode or to a shorting of the arc. As a result of a decrease in the heat flux, the local cooling rate increases and a semicontinuous band of positive segregation develops parallel to the isotherms. Such defects in the ingot are called “freckles”. During the growth of the ingot, the shape and size of the liquid pool and mushy zone change. Initially they remain thin and nearly horizontal. At steady state, the liquid pool and mushy zone are larger and deeper at the centre than at the sides. Liquid in the mushy zone at the centre will flow not only axially but also radially to feed metal to the sides for compensating solidification shrinkage. The centre would generally experience a negative segregation while the outer regions a positive segregation (Brody and David 1970).
Electrode
↓
↓
↓
Liquid pool Liquid pool + solid zone (mushy) Solid region Starter pad Cooled metal mould
Figure 3.10. Schematic drawing showing consumable electrode arc melting; liquid, mushy and solid zones at steady state and the directions of liquid movement for feeding solidification shrinkage (both radial and axial) are indicated.
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145
The following mechanisms of macrosegregation in ingots produced by consumable arc electrode melting have been identified: (1) Flow of solute-enriched liquid into hot tears. (2) Interruption of steady state feeding of liquid metal for compensating solidification shrinkage. (3) Radial segregation resulting from transient conditions at start-up or near completion of a cast or at other times when the operating conditions are significantly changed. 3.2.5 Microstructure of weldments of Ti- and Zr-based alloys Welding is observed to severely alter the microstructure and consequently the mechanical properties of Ti- and Zr-based alloys. Usually properties such as fatigue life, fracture toughness and tensile ductility are adversely affected in the fusion zone and heat-affected zones of weldments. As the name suggests, the fusion zone refers to the region which undergoes melting and subsequent solidification. For an or an + alloy, a solid state transformation, leading to a conversion into the full phase, occurs during heating prior to melting. During the cooling down stage, the first solidifying phase is and as the temperature falls below the /( + ) transus, the -phase starts forming. It is because of these four (two in the solid state and the other two involving melting and solidification) transformation steps that the microstructure of the fusion zone undergoes substantial alteration. The zone known as the heat-affected zone is the one which goes through only solid state transformations, such as from or + to followed by to or + . Since this region does not undergo the melting–solidification cycle, microstructural changes are introduced only through solid state transformations including recrystallization and grain growth. Microstructural modifications which occur in the fusion and heat-affected zones of weldments due to phase transformations are primarily governed by the local alloy composition and the heating/cooling rate imposed. Since the diffusivity in the liquid phase is quite high, it is reasonable to assume that the liquid pool created during welding corresponds to the average alloy composition unless a filler material of a different composition is used. There is, however, a possibility of picking up interstitial elements such as O, N and H in the molten pool as these elements have high solubilities in the base metals. In order to avoid the picking up of interstitial contaminants, welding of Ti and Zr alloys is performed invariably in a protective environment (either inert gas cover or vacuum). The solidified microstructure, as revealed in an optical microscopy examination of the fusion zone, shows either a dendritic or a cellular structure, depending on the growth rate and the local thermal gradient in the liquid. For revealing the solidified
146
Phase Transformations: Titanium and Zirconium Alloys
microstructure, a special etching technique known as “dendrite decoration” is used in which the oxide formed on the sample surface during etching is retained. Solidification in the fusion zone begins by the epitaxial growth of partially melted grains at the interface of the molten pool. The grains grow into the liquid metal pool. Concurrently grains are nucleated at the upper molten pool surface and interfaces of these grains in contact with the liquid pool tend to develop a morphological instability leading, generally, to a cellular structure. This variation of the structure from the cellular at the upper surface to a fully dendritic in the lower surface of the fusion zone arises due to a substantially lower value of G/R (where R is the cooling rate) at the lower surface compared to that at the upper. As the transition from cellular to dendritic growth occurs as a function of position in the fusion zone, the observed growth direction shows an increased dependence on the shape of the receding weld pool. Near the lower surface, a single variant of the dendritic structure extends nearly from the heat-affected zone to the weld centre line, suggesting that the growth direction deviates to a significant extent from the maximum thermal gradient. Closer to the upper weld surface, the solidification process adheres more strictly to the maximum thermal gradient; changes in growth direction occur both by secondary nucleation of more favourably oriented grains and by deviation in the growth direction from the crystallographically favoured one. The 100 directions are those which correspond to the maximum growth rate. In the vicinity of the lower weld surface, the growth of dendrites occurs predominantly along 100 directions, often at the expense of deviating from the maximum thermal gradient. If the deviation is too large, dendritic growth continues along more favourably oriented secondary arms or by nucleation of more favourably oriented grains. Near the upper weld surface, where the value of G/R is much higher, the growth is more stable and a cellular solidification structure is generally produced. The cellular structure often becomes curved so that the growth direction adheres to the maximum thermal gradient. The next point which needs to be addressed is the segregation of alloying elements. In order to evaluate the segregation tendency of a given alloying element, the parameter 1/k = −ml /Dl 1 − ko can be used, where ml is the liquidus slope, Dl is the diffusivity in the liquid and ko is the partitioning ratio. This parameter, a scaling factor in instability equations for the movement of the solidification front, provides a measure of the extent to which a given alloying element promotes the breakdown of a planar liquid/solid interface. Large values of ko correspond to alloying elements which promote interface instability, while small values favour planar interface movement. Table 3.2 lists the ko parameter for different alloying elements in binary alloys of Ti and the relative concentrations of these elements which can cause an interface breakdown at a given value of G/R normalized to the element with largest ko (Gould and Williams 1980). Experiments have shown that out of
Solidification, Vitrification and Crystallization
147
Table 3.2. Values of the parameters K and critical required concentrations at a given G/R for the various alloying additions. l K = −m 1 − k Dl C − s
Alloying addition
C∗ ∗ CFe
wt% − cm2
123 × 106 980 × 105 879 × 104 702 × 104 235 × 104
Fe Cu V Al Sn
1 1.26 14.0 17.5 52.3
the alloying elements compared in Table 3.2, Cu has a much stronger tendency to segregate as compared to V, Al and Sn, consistent with the predictions based on the ko parameter. However, Fe shows an anomalously low segregation tendency. We have, so far, considered the solidification morphology, though the final microstructure of the fusion zone is developed by a superimposition of solid state transformations on the compositionally graded solidification product. The heat-affected zone also undergoes compositional changes due to redistribution of alloying elements between the and the phases. During the final cooling stage, a variety of phase transformations occur within the -phase, depending on the local alloy composition and the cooling rate. The possible phase transformations in the -phase can be indicated in a schematic phase diagram (Figure 3.11). Let us
Ms(α′) 1
2
Temperature
β 3
Ms(ω)
4
α
Ti
c1
c2
c3 c4 Atom Fraction of c
Figure 3.11. Schematic phase diagram indicating possible phase transformations in the -phase.
148
Phase Transformations: Titanium and Zirconium Alloys
consider an alloy (either Ti–X or Zr–X) with the average concentration of X being c1 . During welding, the fusion zone in the liquid state will attain a homogeneous concentration x1 , provided there is no pick-up of alloying elements from the filler material or environment. As discussed earlier, the solidified -phase will exhibit a range of compositions within the fusion zone, depending on the extent of segregation occurring during solidification. The heat affected zone can be subdivided into several regions, A, B, C, etc. on the basis of the temperature each region has seen during welding. Region A (represented by point 1 in Figure 3.11) corresponds to going into the full -phase field. Depending on the cooling rate imposed, this region will either transform into the martensitic phase or will exhibit a Widmanstatten
+ structure. Region B (represented by point 2) enters into the + phase field and the -phase, during cooling, transforms either martensitically to or by a diffusional process to the two-phase + structure. The final microstructure of region B, therefore, consists of a primary and a transformed (either or ( + )) phase mixture. The volume fractions of the primary and the transformed phases will depend on the temperature at which the – equilibration results in partitioning of alloying elements, -stabilizing elements preferentially migrating to the -phase (and similarly -stabilizing elements to the -phase). This causes further stabilization of the -phase. With increased -phase stability corresponding to a lower equilibration temperature (as in the case of point 3 in Figure 3.11), the -phase transforms into the + structure on cooling (region C). Finally, in region D, very small volume fraction of the -phase present in the intervening space of primary is retained on cooling (point 4 in Figure 3.11). In this manner, the local composition of the alloy, as dictated by the solidification process or by the – equilibration process in the fusion and the heat-affected zones, determines the microstructure. As the peak temperature reached during the welding process decreases from the melting temperature at the centre line of the weld to the ambient, the regions A, B, C and D will appear successively with increasing distance from the centre line. The thickness of each zone, however, will depend on the energy input during welding, the thermal gradient and the duration of the thermal exposure at elevated temperatures. The foregoing discussion on microstructure development in the heat-affected zone assumes attainment of equilibrium which rarely takes place during the short duration of thermal excursion in welding. As a consequence, an incomplete partitioning of the alloying elements in the and the phases occurs and correspondingly subsequent phase transformations take place. The development of microstructure in the fusion and the heat-affected zones of a weldment is strongly influenced by the thermal history of the various regions, the sizes of which are primarily determined by welding parameters such as power density, traverse speed and the thermal properties of the material. Figure 3.12 shows
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149
Figure 3.12. Welding microstructures produced in Ti–6Al–4 V by laser welding.
the microstructure produced in Ti–6Al–4 V by laser beam welding (LW). The notable differences between gas-tungsten arc welding (GTAW) and LW processes have been found to be as following: (a) a deeper penetration can be achieved by LW compared to GTAW at comparable values of power input and traverse rate; (b) Both the heat-affected and the fusion zones associated with GTAW welds show Widmanstäatten -plates, indicating a diffusional → transformation while these zones produced in LW consist mainly of the martensitic -phase. The deeper penetration in laser welding (and also in electron beam welding) is the consequence of the mechanism of energy transfer, commonly known as the key hole effect. As the high power density beam falls on the workpiece, vaporization occurs at the surface of the workpiece creating a hole. The hole acts as a radiation trap, or black body, enhancing the coupling of the beam energy into the work piece. Vapour is ejected from the key hole at a very high speed, dragging with it molten material and stabilizing the molten walls of the key hole. Such a process leads to the formation of a fusion zone with a high depth-to-width ratio. As the energy input is much smaller than that required in the case of GTAW to achieve the same level of penetration, the thickness of the heat-affected zone also remains restricted in laser welding. A larger pool of molten metal and a larger heat-affected zone are responsible for a relatively slower cooling rate in these zones in GTAW welds as compared to laser welds. This is reflected in the microstructure produced in these zones: →Widmanstatten transformation in GTAW welds as against martensitic → transformation in laser welds.
150
3.3
Phase Transformations: Titanium and Zirconium Alloys
RAPIDLY SOLIDIFIED CRYSTALLINE PRODUCTS
Rapid solidification processing (RSP) techniques, which induce extension of solid solubility (thereby enlarging the alloying range), refinement of microstructure, suppression of segregation during solidification and dispersion of precipitating phases in a finer spatial scale have been applied to Ti and Zr alloys only to a limited extent. Because of the high reactivity of these alloys, RSP techniques involving containment by crucibles, nozzles, etc. cannot be used. Special techniques, viz., plasma rotating electrode process, inert gas atomization, melt extraction from molten pool or pendant drop, laser or electron beam surface melting, cold hearth arc melt spinning and hammer and anvil splat quenching, are adopted for rapid solidification of these alloys. These techniques are illustrated in Figure 3.13 and their important features are described. The plasma rotating electrode technique (Figure 3.13(a)) consists of local melting of a rotating consumable electrode of the alloy to be processed, by striking an arc with a tungsten electrode. The molten metal is thrown out of the pool, by the action of the centrifugal force, in the form of atomized particles which undergo solidification in the flight path from the centre to the periphery of the chamber maintained in an inert gas environment. Powder particles produced in such a process are typically 50–200 m in diameter and show a dendritic structure. Cooling rates achieved in this process remain in the range of 103 –105 K/s. This method is used for the production of rapidly solidified Ti alloy powders in commercial scales. The process can be modified by using a high-intensity laser beam heat source in place of the arc. The laser heat source allows the introduction of a high-velocity inert gas jet to supplement centrifugal atomization and increases the rate of heat extraction. The inert gas atomization technique involves cold hearth melting and bottom ejection through a refractory metal-skull nozzle, Figure 3.13(b). Typically, powders in the size range of 200–400 m of Ti alloys can be produced using this technique in kilogram scale in a single run. Melt extraction is performed by extracting a solidified filament from a molten pool, either contained in a crucible or created by local heating of the melt stock, Figure 3.13(c). A cold rotating wheel is used for extracting the heat and ejecting the solidified filament. This method is usually adopted in the laboratory, but some derivatives of this have the potential for scaling up. Rapid solidification can be effected by scanning an intense laser or electron beam on the surface of finished alloy components, Figure 3.13(d). This requires a suitable control of the energy density of the incident beam and of the interaction time between the beam and the material so that a thin layer is melted and subsequently rapidly solidifies, heat from the molten layer being extracted by the bulk of the
Solidification, Vitrification and Crystallization B
1
A
151 B
2
A
3 4 B
C A
D (a)
(b)
Melt stock Heat source Wiper Solid filament
(c)
(d) 1 2 3 5 4
(e)
Figure 3.13. RSP techniques (a) plasma rotating electrode, (b) inert gas atomization technique, (c) melt extraction/pendant drop, (d) laser/electron beam surface melting and (e) hammer and anvil splat quenching.
component. Very intimate melt–substrate contact is achieved which results in high cooling rates during and after solidification. Cold hearth arc melt spinning involves the ejection of a stream of liquid alloy from a cold hearth arc melted pool on to a rapidly spinning metallic wheel. The ribbon produced is thin (about 15–50 m thick) and a cooling rate of about 105 K/s can be achieved. Hammer and anvil splat quenching is carried out by squeezing a liquid droplet between impacting piston and anvil heat extracting surfaces, Figure 3.13(e). The solidification rate can be as high as 107 K/s but the process can produce only milligram scale quantity in a single run.
152
Phase Transformations: Titanium and Zirconium Alloys
The rapid solidification techniques outlined here are all suitable for the processing of Ti and Zr alloys. RSP techniques have been extensively used for modifying the microstructure of crystalline Ti alloys, mainly with the aim of controlling the microstructure of Ti alloy powders which have several industrial applications. Powder metallurgy applications for Zr alloy products are, however, quite limited, and therefore, not many investigations pertaining to the effect of RSP on the microstructure of crystalline Zr alloys have been reported. 3.3.1 Extension of solid solubility One of the objectives of rapid solidification processing is to extend the limit of solid solubility. The extent to which such a metastable extension of solubility is effected depends on the rate of quenching during solidification, the increment in the chemical potential of the solute element above the equilibrium value and the diffusivities of the solute in the liquid and the solid phases. Ti and its alloys usually solidify as a bcc -Ti phase which may either be retained at room temperature or may undergo solid state phase transformations, depending on the relative stability of the -phase. Table 3.3 shows the solubility limits (in atom percent) of some alloying elements under equilibrium and rapid solidification conditions (Rowe et al. 1987). The metastable solubility extension shown in Table 3.3 refers to the solubility limits at the solidification temperature. With lowering of temperature, most of these supersaturated solid solutions reject solutes which get precipitated in the form of second phase dispersions which can be very effective in improving high temperature strength and creep resistance of rapidly solidified Ti alloys.
Table 3.3. Solubility limits (in at.% of some alloying elements under equilibrium and rapid solidification conditions. Element
Matrix
B C Si Nd Y Gd Er
Binary Binary ∼10 at.% Zr Binary ∼8 Al Binary ∼5 Al
EBSQ, electron beam splat quenching.
Maximum equilibrium solubility in the -phase
Extended solubility in -phase in RSP alloys
0.5 3.1 5.0 ∼0.3 ∼0.3 ∼0.3 ∼0.3
6.0 10.0 6.0 >1.0 >2.0 >0.5 >1.5
RSP technique
Splat Splat Splat EBSQ Laser EBSQ Splat
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153
3.3.2 Dispersoid formation in rapidly solidified Ti alloys Attempts to disperse hard refractory compounds in Ti alloys have not been quite successful in the ingot metallurgy of these alloys. This is due to the fact that stable compounds segregate to a very significant extent during solidification at the conventional ingot cooling rates. Rapid solidification processing has, therefore, been employed in several Ti alloys with the aim of producing a finely dispersed structure. Rapidly solidified Ti alloys with additions of (a) Er or (b) Ce and S have shown very fine (50–100 nm) dispersions of Er2 O3 , CeS and Ce4 O4 S3 . These dispersed phases have shown excellent resistance against coarsening at temperatures as high as 1223 K. Compounds of Ti with the metalloid elements C, B, Si and Ge have high melting points and good chemical stability. These compounds are, however, less stable in the Ti alloy matrix than the rare earth oxides and sulphides. Rapidly solidified Ti alloys containing C show a distribution of fine spherical TiC precipitates while those containing B produce a fine dispersion of needle-shaped TiB precipitates. Ti alloys with Si and Ge, in the rapidly solidified condition, contain fine dispersions of Ti5 (Si,Ge)3 . Both carbides and silicides coarsen rapidly at 973 K while TiB remains stable upto about 1073 K. Rapid solidification processing is also applied for alloys in which eutectoid decomposition takes place. The cooling rate required for solidification without any significant segregation is lower in the case of eutectoid-forming alloys than in rare earth or metalloid-containing alloys. A higher volume fraction of the second phase makes the eutectoid-forming alloys (containing one or more of the alloying elements Cr, Mn, Fe, Ni, W and Cu) suitable candidates for high strength alloys which can retain their strength up to intermediate temperatures. The upper temperature limits of the stability of intermetallic compounds present in eutectoid alloys restrict their high temperature applications. 3.3.3 Transformations in the solid state The influence of the rapid solidification treatment on the subsequent phase transformations in the solid state has not been studied extensively. Inokuti and Cantor (1979) have reported refinement of the martensitic structure in rapidly solidified Fe based alloys. The refinement is attributed to the small size of austenite grains forming from the liquid phase which, in turn, limits the size of the martensite plates. Banerjee and Cantor (1979) have reported the microstructure produced in unalloyed Zr and Zr–Nb alloys by rapid quenching from the liquid state. The possibility of the formation of the -phase directly from the liquid phase, skipping the intermediate equilibrium -phase, has been examined. The thermodynamic feasibility
154
Phase Transformations: Titanium and Zirconium Alloys 0
–100
L (1700 K)
↓
Tα
α (Zr)
β (Zr)
L (Zr)
–300
–500
α (Zr) Tα
β (1135 K)
1000
β (Zr)
L (Zr)
Tβ
L (2125 K)
↓
–400
↓
G (kJ/mol)
–200
2000 3000 Temperature (K)
4000
Figure 3.14. Schematic free energy versus temperature plots for the liquid, the - and the -phases for pure Zr. The temperatures T/l and T /l correspond to those at which the liquid/ and the liquid/ equilibrium are established.
of such a process is illustrated schematically in Figure 3.14. The evidence of a direct L → transformation has been recorded in unalloyed Zr samples contaminated with Si and O. The cellular structure of the -phase with Si-enriched intercellular region (Figure 3.15(a)) observed in rapidly solidified samples points to the fact that the -phase cells originated directly from the liquid. The extent of supercooling required for making such a direct L → transformation possible can be reduced by alloying elements which enhance the relative stability of the -phase in comparison with the -phase. Since both Si and O are strong
-stabilizers, contamination of these elements in Zr is expected to raise the Tl/ temperature and thereby reduce the extent of supercooling required for the direct solidification. In alloys containing a -stabilizing element, two types of displacive transformations, namely the → martensitic and the → transformations, are encountered. Which of these processes is selected by a given alloy is determined by the alloy composition. As has been explained in Chapter 1, the → transformation operates in compositions where Ms is higher than Ms and vice versa. The transition from the structure to the + structure is noticed in Zr–Nb
Solidification, Vitrification and Crystallization
(a)
155
(b)
(c)
Figure 3.15. Microstructures developed in rapidly solidified Zr alloys undergoing post-solidification phase transformation: (a) cellular morphology of the -phase which formed directly from the liquid. Intercell boundary regions are richer in Si and O. (b) Propagation of -martensite laths across the boundaries of -cells which formed as a solidification product. (c) + microstructure in Zr-5.5 wt% Nb alloy.
samples quenched from the -phase field (solid state quenched) when the Nb level exceeds ∼7 at.%, the composition at which Ms line intersects the Ms line (Figure 1.18). Banerjee and Cantor (1979) have shown that in rapidly solidified Zr–Nb samples, this transition is shifted to a lower (∼5.5%) level of Nb. This preference for → transformation over the → martensitic transformation under rapid solidification is attributed to the retention of a higher concentration of vacancies which are known to stabilize the -like defects prior to the transformation (Kuan and Sass 1976). Figure 3.15(b) and (c) shows the martensitic structure and the + structure, respectively, in rapidly solidified Zr–Nb alloys. It may be noted that the laths in (b) are cutting across the intercell boundaries, implying that the concentration profiles across the cell boundaries are such that Ms remains above the quenching temperature all along the growth path of martensite laths.
156
Phase Transformations: Titanium and Zirconium Alloys
Figure 3.16. Dark-field micrograph of Zr-27 at.% Al alloy showing cellular structure leading to Al enrichment at cell boundaries where higher density of Zr2 Al particles are seen.
In a study on a rapidly solidified Zr-27 at.% Al alloy, Banerjee and Cahn (1983) investigated the sequence of transformation events which led to the formation of the ordered Zr2 Al phase (B82 structure) in this alloy. The analysis of the morphological features (Figure 3.16) of the transformation products led to the conclusion that the following sequence of transformation events occurred: (1) Formation of a supersaturated (bcc)-phase mainly by a partitionless solidification process. The limited alloy partitioning in some localized areas led to the formation of a cellular structure where the cell boundaries were decorated by a higher number density of Zr2 Al particles as shown in Figure 3.16. (2) Spinodal decomposition of the supersaturated -phase (Zr–Al) during continuous cooling subsequent to solidification. This process resulted in the formation of a compositionally modulated structure with modulations along the elastically soft 100 directions which, in turn, produced Al-rich cuboids of about 20 nm size. (3) A combined chemical and displacement ordering within the Al-rich cuboids resulting in the formation of the Zr2 Al structure – details of which are discussed in Chapter 6. (4) → transformation in the Al-depleted regions in the intervening space between the cuboids.
Solidification, Vitrification and Crystallization
3.4
157
AMORPHOUS METALLIC ALLOYS
Amorphous metallic alloys or metallic glasses have emerged as a new class of engineering materials after vitrification of metallic alloys by using the technique of ultrarapid quenching of molten alloys has become possible. These materials, which do not have any long-range crystalline order but retain metallic bonding, exhibit several interesting properties emanating from their unique structure which is isotropic and homogeneous in the microscopic scale. Extremely high hardness and tensile strength, exceptionally good corrosion resistance and very low magnetic losses in some soft magnetic materials are some of the attractive properties associated with amorphous metallic alloys. There are three technologically important classes of amorphous alloys, namely (a) the metal–metalloid alloys, such as Fe–B, Fe-Ni-P-B and Pd–Si, (b) the rare earth–transition metal alloys, such as La–Ni and Gd–Fe, and (c) the alloys made up of a combination of early and late transition metals such as Ti–Cu, Zr–Cu, Zr–Ni and Nb–Ni. With the availability of metallic glasses, several important issues concerning the stability of these metastable phases and the kinetics of their thermal decomposition have attracted the attention of the phase transformation research community. It will be shown in this chapter that the formation and the decomposition of Ti- and Zr-based metallic glasses offer some unique opportunities for studying several aspects of phase stabilities and transformations in metallic glasses. These include glass-forming abilities (GFAs), diffusion mechanisms, modes and kinetics of crystallization and formation of bulk metallic glasses. A major advantage of studying these systems is that a number of binary metal–metal alloys based on Ti and Zr are amenable to easy glass formation. 3.4.1 Glass formation The three-dimensional lattice arrangement of atoms in a crystalline solid is destroyed as it melts. In the liquid state, the long-range order both translational and orientational of the crystalline solid is not retained as the atoms vibrate about positions which are rapidly and constantly interdiffusing. Melting being a strongly first-order transition, thermodynamic quantities such as specific volume, enthalpy and entropy undergo a discontinuous change at the melting temperature as a crystalline solid transforms into a liquid. At temperatures above the melting point, a liquid is in a state of internal equilibrium and its structure and properties are independent of its thermal history. A low viscosity, which is essentially the inability to resist a shear stress, characterizes the liquid state. On cooling, a liquid transforms into a crystalline solid under the equilibrium cooling condition. Vitrification of a liquid is possible only when the liquid is cooled at a rate sufficiently rapid to escape a significant degree of crystallization so that the “disordered” atomic
158
Phase Transformations: Titanium and Zirconium Alloys
configuration of the liquid state is frozen in. Glass formation is easy in a number of non-metallic systems such as silicates and organic polymers. The nature of bonding in these systems places severe limits on the rate at which crystalline order can be established during cooling. Thus the melt solidifies into a glass even at low cooling rates (often less than 10−2 K/s). Metallic melts, in contrast, have non-directional bonding which allows a very rapid rearrangement of atoms into the crystalline state. Hence very high cooling rates need to be imposed for forming metallic glasses by avoiding crystal formation. Cooling rates exceeding 105 K/s are necessary for the formation of metallic glasses in several binary and ternary alloy systems based on Ti and Zr. In recent years, several Zr-based alloys with a number of components have been found to be amenable to vitrification at substantially slower cooling rates (Inoue 1998). This has opened up the possibility of obtaining metallic glasses in the bulk form. The formation and properties of bulk metallic glasses are discussed in Section 3.6. The process of vitrification of a liquid under non-equilibrium cooling can be compared with the equilibrium crystallization process in plots of viscosity, , and thermodynamic quantities such as specific volume, V , and specific heat, Cp , against temperature (Figure 3.17). While the liquid to crystal transformation is accompanied by a step change in these properties, a progressive change in viscosity and enthalpy precedes the vitrification event as the liquid is undercooled below the equilibrium melting temperature, Tm . It is evident from these plots that vitrification is possible only if the equilibrium crystallization process is avoided. The cooling rate, therefore, needs to be sufficiently high so that insufficient time for nucleation and/or growth does not permit the formation of the crystalline phase to a detectable level. Although the driving force for nucleation continuously increases with the extent of undercooling, the rapid increase in viscosity is responsible for decreasing atomic mobility and thereby effecting the kinetic suppression of crystallization. Eventually, the atomic configuration of the liquid becomes homogeneously frozen at the glass transition temperature, Tg . This structural freezing to the amorphous state is, by convention, considered to occur when the viscosity reaches a value of 1013 poise. Since the atomic configuration of the amorphous state does not correspond to a unique equilibrium structure, Tg and the glass structure are both cooling rate dependent, variations in the latter resulting in glasses with different states of structural relaxation. Figure 3.17(a) shows the glass transition temperatures, Tg1 and Tg2 , for two glasses, G1 and G2 , forming under different rates of quenching. The GFAs of different metallic systems have been assessed in terms of both relative thermodynamic stabilities of the amorphous and the equilibrium and metastable crystalline phases which compete to form during cooling and kinetic factors which determine the critical cooling rate necessary for avoiding the crystallization process. The thermodynamic and kinetic criteria for glass formation
Solidification, Vitrification and Crystallization 5
G2 q2 = 1 Ks–1 G1
15
Crystal (X)
0
Tg2
CP
10
–5
100
C px
40
L
→
C gp
60
→
–10 0
C lp
80
→
Cp /gtw K
glass, G Tg1
log τ
log η →
q1 = 105 K s–1
5
159
20 Tm
1/Tm
0
4
2 1/T →
100 200 300 400 500 600 700 800
Temperature K (b)
(a)
V cm3 mol–1
L 9.0 G1
Tg1 Tg2 ↓
↓
Tm
X
G2 8.5 500
1000
Temperature (c)
Figure 3.17. (a) Viscosity () as a function of reciprocal temperature showing the liquid to glass transitions at Tg1 and Tg2 for two different cooling rates, q = 1 and 105 K/s. The liquid to crystal, (X), transformation is also shown in the same figure. (b) Specific volume, V , and (c) specific heat, Cp , as a function of temperature showing step changes at the liquid to glass transition. CXp , Cgp and CLp refer to crystalline, glassy and liquid phases, respectively.
have been discussed in the following section with special reference to Ti- and Zr-based systems. 3.4.2 Thermodynamic considerations Pure metals are extremely difficult to vitrify under the conditions of rapid solidification which typically attain a cooling rate of about 106 K/s. Thin sections of
160
Phase Transformations: Titanium and Zirconium Alloys
splat quenched foils of Ni with dissolved gaseous impurities to a level of about 2% have been reported to vitrify. This, however, requires a cooling rate as high as 109 –1010 K/s, which is estimated to be attainable in thin sections (<100 nm) of splats. The difficulty in the vitrification of pure metals can be ascribed to the poor stability of the amorphous phase with respect to the equilibrium crystalline structure and the ease of spontaneous crystallization. The latter is expected to require very little structural adjustment and consequently a very low thermal activation. From the assessment of GFAs of a number of alloys, it has been observed that GFA is enhanced substantially in several systems with increasing additions of solute elements in certain composition ranges. The origin of enhanced GFA can be traced to factors which can be grouped into thermodynamic and kinetic factors. This section deals with the relative thermodynamic stability of the liquid/ amorphous phase with respect to the competing crystalline phases. The alloy systems in which glass formation occurs most readily are those which exhibit either one or more deep eutectics or a steep decrease in the liquidus temperature, Tl , with increasing levels of solute content. Phase diagrams of a number of Ti- and Zr-based alloys (e.g. Ti–Cu, Ti–Be, Zr–Cu, Zr–Ni, Zr–Ni–Cu) show such characteristics. Let us take the example of the Zr–Ni binary system for examining the relative stabilities of different competing phases. Figure 3.18 shows the free energy–concentration (G–c) plots for the ordered intermetallic phases and the supercooled liquid (amorphous) phase. Such free energy–concentration plots have been constructed on the basis of an analysis of the phase diagram (Katgerman 1983) and by using the computed values of formation energies of intermetallic phases (Miedema 1980). Sharply bent U-shaped G–c plots (shown as vertical lines in Figure 3.18) for the intermetallic phases reflect the stability of these phases within narrow composition limits. The G–c plots for the liquid and the phases intersect at a point where the integral molar free energies of these phases are equal (Figure 3.19(a)). This marks the composition limit up to which partitionless solidification is thermodynamically possible. The locus of these intersection points at different temperatures defines the To c line, the shaded area below, which defines the region of stability of the -phase with respect to the liquid phase under the condition of partitionless solidification (Figure 3.19(b)). Since rapid solidification at a sufficiently high cooling rate does not permit partitioning of the alloying elements between the parent liquid and the crystalline product, the composition of the liquid is inherited by the supersaturated crystalline phase. The To c curve can, therefore, be considered as the polymorphic melting curve of a multicomponent system. The addition of solute atoms to a pure metal, specially in the case where the solute and the solvent atoms differ in size and in chemical character, makes the diffusive rearrangement of atoms a necessary step during solidification into
Solidification, Vitrification and Crystallization
161
20 16
650 K 12 8 4 0.1 0.2
0.3
0.4
0.5 0.6
0.7 0.8 0.9 1.0
ΔG kJ/gm – atom
0 4
A
8 12 16 20
40
Zr
Zr7Ni10
36
ZrNi Zr9Ni11
32
c (at.% Ni) →
Zr8Ni21 Zr2Ni7 ZrNi5
28
Zr2Ni
24
Ni
Figure 3.18. Free energy ( G) versus composition (c) plot at 650 K corresponding to the amorphous phase and ordered intermetallic phases in the Zr–Ni system.
a crystalline phase. As a consequence, glass formation becomes easier. The glass transition temperature, Tg , generally increases with solute concentration, as shown in Figure 3.19(b). Hypothetical G–c plots consistent with the phase diagrams of Ti–Be and Ti–Fe systems (Figure 3.20) show that a metastable equilibrium between the liquid (amorphous) phase and the -phase can be established at a sufficiently large supercooling provided, the formation of equilibrium phases is suppressed. The point of intersection of the To c and Tg c lines indicates the minimum solute concentration limit for an alloy to vitrify. An alloy having a solute content exceeding this limit cannot solidify into the crystalline solid through a partitionless solidification as it cools down to Tg , where the viscosity of the liquid rapidly rises to about 1013 poise. A thermodynamic basis of glass formation can thus be explained by considering the competition between the partitionless polymorphic solidification of the liquid to supersaturated on one hand and the liquid to glass transition on the other. It must be emphasized here that the avoidance of the formation of the equilibrium crystalline phases, namely the equilibrium or the
162
Phase Transformations: Titanium and Zirconium Alloys
γ γ
L
G β
L β
L1
L2
S2
S1
c2
c1
c→
c→ (a)
L
→
T1
T
Te β T2 S2
L2
C2
Tg
T3 To A
L+γ
S1 C1 L1
ce
β+γ
c (at fraction B) → (b)
Figure 3.19. (a) Schematic G–c plots and (b) corresponding phase diagram representing the equilibrium between the liquid, and intermetallic phases. The integral molar free energies of the liquid and the phases are equal at c1 and c2 , corresponding to T1 and T2 . The locus of such points defines the To line.
equilibrium intermetallic phases, cannot be explained on a thermodynamic basis. The suppression of nucleation and growth of equilibrium crystalline phases are essentially controlled by kinetic factors which are discussed in Section 3.4.3. Metallic glasses are generally grouped into the following major categories, depending on the constituent elements: (1a) Late transition metals (TL) alloyed with metalloids (M): TL includes Group VIIB, Group VIII and Group IB noble metals while M includes Si, B and Sb. Typical examples of this category are Fe–B, Ni–B, Fe-Ni-Co-B, Pt–Sb glasses. In several binary systems, deep eutectics are encountered at compositions in the range of 13–25% M. Many of the amorphous alloys in this category are of technological importance because of their soft ferromagnetic properties which arise due to the presence of a combination of Fe, Co and Ni.
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163
m-TiBe TiBe2 β
L (T3) L (T3) L (T2) L (T1)
→
→
L (T2)
G
β
G
TiFe
L (T1)
α α
ce Ti
20
↓
40
c (at.%Be) (a)
60
80
Ti
20
↓
ce 40 c (at.%Fe)
60
80
(b)
Figure 3.20. Hypothetical G–c plots consistent with the phase diagrams of Ti–Be and Ti–Fe systems.
(1b) Early transition metals (TE) alloyed with metalloids (M): Early transition metals in this class of amorphous alloys include Ti, Zr and Nb which form eutectics at compositions in the range of 15–30% of metalloids such as Si and B. (2) Lanthanide metals (RE) alloyed with late transition (TL) and Group IB metals: Many of these systems, exhibiting deep eutectics, are easy glass-forming alloys. Examples of systems of this type include La78 Ni22 Gd–Fe32−50 and Gd–Co40−50 . (3) Early transition metals (TE) alloyed with late transition metals (TL): The addition of a TL (or a Group IB) metal generally leads to a sharp decrease in the liquidus temperature, Tl , which finally drops down to a low eutectic temperature. The Zr–Ni system, which has been discussed earlier, represents this category of potential glass formers which includes Ti–Cu, Zr–Cu, Zr–Fe, Zr–Co, Zr–Ni–Fe and several other Ti- and Zr-based systems. In all the categories of easily glass-forming alloys, a high GFA is almost invariably associated with either the presence of one or more deep eutectics or a sharp drop in the liquidus temperature. Both these aspects of phase diagrams strongly indicate an enhancement of the stability of the liquid phase with increasing additions of alloying elements in the relevant composition range. The free energy difference between the supercooled liquid and the competing equilibrium crystalline phase is thereby substantially reduced.
164
Phase Transformations: Titanium and Zirconium Alloys
The extent of lowering of Tl is usually expressed in terms of a reduced glass temperature, Tgr , defined as the ratio, Tg /Tl . The lower is the Tl and the higher is the Tg , the higher is the GFA. The highest known values of Tgr for metallic glasses are in the range of 0.66–0.69. Free energy, Gc, plots at different temperatures for the competing crystalline and amorphous phases are indicative of the relative stabilities of these phases. Though the free energy of the supercooled liquid (glass) is brought down substantially by alloying additions in potentially glass-forming alloys, the Gc curve for the liquid/amorphous state still remains at a level higher than that corresponding to the equilibrium crystalline phases. It is, therefore, essential that kinetic factors play a major role in the vitrification process by suppressing the nucleation and limiting the growth of crystals. It is also clear from Gc plots (e.g. Figure 3.19(a)) that the stability of the liquid phase is not necessarily maximum at the eutectic compositions. In fact it will be shown later using kinetic arguments that GFA is the highest in some systems not at the eutectic composition but at the composition corresponding to one of the intermetallic phases. Various semiempirical methods have been proposed for relating GFA with the depression of Tl below some mean value. Marcus and Turnbull (1976) have proposed a normalized parameter T/Tlo , where T = Tlo − Tl is expressed as the deviation of Tl from the ideal solution liquidus temperature, Tlo , which is given by Tlo =
HfA TmA
HfA − R ln1 − cTmA
(3.21)
where HfA and TmA are the heat of fusion and melting point, respectively, of the solvent metal and c is the mole fraction of the solute. Large positive values of this parameter indicate a large deviation from the ideal solution behaviour which corresponds to an ordering tendency and a high GFA. It has been pointed out by Zielinski et al. (1978) that a combination of two factors, namely the unlike atom bond strength, expressed by the magnitude of the excess negative enthalpy of mixing, Hm , and the depression of the melting point, (Tlo −Tl ), can be very effective in categorizing different alloy systems into readily glass-forming (RGF) and non-glass-forming alloys. This is to be expected since a deep depression in the liquidus temperature is invariably associated with a strong interaction on mixing of the alloy components. Free energy plots, Gc, for the Zr–Ni system (Figure 3.18) clearly illustrate this point. The Gc plot for the liquid phase shows sharp drops with increasing addition of Ni (up to about 50%) in Zr in the vicinity of compositions where a series of intermetallic phases, which are all chemically ordered structures, are present. The atomic radius ratio, rA /rB , where A represents
Solidification, Vitrification and Crystallization
165
the smaller size atom in an A–B alloy, is another parameter which is seen to have a strong correlation with GFA. A survey of a number of alloys shows that the alloys which readily form an amorphous phase correspond to an rA /rB ratio of less than 0.85. 3.4.3 Kinetic considerations The essential requirement for the formation of a glassy state in an alloy during cooling from the liquid state is that the equilibrium crystalline phases are not allowed to form to a detectable level prior to the rapid rise in the viscosity finally leading to vitrification. A quantitative estimation of the cooling rate required for achieving the condition of avoidance of crystal formation can, therefore, be made by considering the kinetics of the nucleation and growth of crystalline phases which compete with glass formation. For a glass to form, one of the following conditions has to be satisfied: (a) complete suppression of nucleation of crystalline phases; (b) limited nucleation of crystals but no significant growth of these, resulting in a distribution of quenched-in nuclei of crystals in an amorphous matrix; (c) nucleation of a very few isolated crystalline nuclei which grow to relatively large sizes but the number density of these crystals embedded in the amorphous matrix is too low to be detected by X-ray diffraction (XRD). Condition (a) is rarely satisfied in metallic glasses including Ti- and Zr-based ones. Even in a process such as laser glazing, where the cooling rate of the thin layer (<10 m) of the surface is very high, it is difficult to suppress the nucleation of crystals completely. A certain number density of quenched-in nuclei is invariably present, and therefore, glass formation is essentially controlled by the fact that these nuclei do not get an opportunity to grow. Recent high-resolution electron microscopy work (Savalia et al. 1996) has clearly demonstrated the presence of such quenched-in nuclei in several Zr-based amorphous alloys. In some instances, a limited number of large-size crystals have been detected in metallic glass samples, suggesting that condition (c) is operative in such cases. An estimation of the critical cooling rate required for the avoidance of crystal nucleation can be made for both homogeneous nucleation (either steady state or transient) and heterogeneous nucleation. Theories of nucleation originally developed with regard to the formation of liquid drops in a supersaturated vapour have been extended later to condensed phases. The rate of nucleation, I, of a crystalline phase from the liquid/amorphous matrix phase is given by I = ZA∗ N ∗
(3.22)
where the factor Z is related to the frequency of atoms jumping across the interface between the liquid/amorphous matrix and the critical nuclei and is expressed
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Phase Transformations: Titanium and Zirconium Alloys
in terms of the diffusion constant, D, the jump frequency , and the jump distance, , as Z = N 2 /3 = 6N 2/3 D/2
(3.23)
and A∗ and N ∗ are, respectively, the surface area and the number density of a critical nucleus which contains N atoms. Volmer and Weber (1926) proposed that the concentration of the critical nucleus, N ∗ , is given by N ∗ /N = expW ∗ /kT
(3.24)
where W ∗ is the work done to form the critical nucleus and is related to the volume free energy change, Gv , associated with the formation of the critical nucleus, the surface energy required for creating the surface of the nucleus and the elastic strain energy resulting from the volume changes that occur during the transformation (Figure 3.21). W ∗ can be expressed in terms of the specific surface energy, , the volume free energy change and the strain energy density, E, as −W ∗ = A − Gv V + EV
(3.25)
where A and V represent the surface area and volume of the critical nucleus.
Work done for nucleation (w)
Aγ = K 1r 2
Interfacial energy
w*
O
r
r* w ΔGv = K 2r 3 Volume free energy
Figure 3.21. Volume free energy change ( Gv ), interfacial energy and the net work done (W ) for creating a nucleus of radius r.
Solidification, Vitrification and Crystallization
167
The diffusion coefficient, D, can be expressed as D = Do exp−Q/kT for solids, where Q is the activation energy for diffusion and D = kT/3 for liquids (Stokes–Einstein equation, where is the diameter of the diffusing molecule and can be taken as equal to the jump distance). Combining Eqs. (3.22)–(3.25), the nucleation rate, I, can be written as I = 6A∗ N 5/3 Do /2 exp−Q + W ∗ /kT
(3.26)
I = 6A∗ N 5/3 D/2 expW ∗ /kT
(3.27)
for solids and
for liquids. The preexponential factors in Eqs. (3.26) and (3.27) are usually in the range of 1035 –1040 /cm3 s. For a spherical nucleus of radius r, the work done, W , for creating a nucleus can be expressed as 4 W = 4r 2 − r 3 Gv − E 3
(3.28)
The variation of W with r for a homogeneous nucleus is schematically shown in Figure 3.21 and W ∗ and r ∗ for the critical nucleus can be obtained by finding out the maximum of the W versus r plot. Differentiation of Eq. (3.28) yields 2
Gv − E
(3.29)
163 3 Gv − E2
(3.30)
r∗ = and W∗ =
In the case of the nucleation of a solid phase in a liquid matrix, the strain energy contribution can be neglected and Eq. (3.30) can be reduced to W∗ =
163 3 Gv 2
(3.31)
Combining Eqs. (3.27) and (3.31), the nucleation rate of crystals in a liquid matrix can be expressed in terms of the driving force, Gv , as 163 1035 exp (3.32) I= 3 Gv 2 kT
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Phase Transformations: Titanium and Zirconium Alloys
where is the viscosity of the liquid. Some time is required for establishing the steady state rate of nucleation. The nucleation frequency rises sigmoidally with time from zero to the steady state value, following the relation I = Io exp− /t
(3.33)
where Io is the steady state rate of nucleation and is the induction time. Kaschiev (1969) derived an equation for the induction time by assuming that transport was controlled by jumps across the interface, which on modification leads to the following form for comparison with experiments, = 482 /V Gv 2
(3.34)
Nucleation occurring before the attainment of the steady state, that is, within the induction time , is known as transient nucleation. Substitution of appropriate values in Eq. (3.34) yields an induction time of several microseconds. Since glass formation in metallic systems requires a quenching rate of about 106 K/s, nucleation events occurring within microseconds also play an important role in vitrifying a molten alloy. The steady state nucleation rate and the rate of transient nucleation are both dependent on the degree of undercooling, which strongly influences the values of Gv . The volume free energy change, Gv , can be expressed in one of the following three forms:
Gv =
Hf T Tm
1 − Tm + 1 + T T + Tm
T
Hf T Cp T 2 1− −
Gv = Tm 2T 6T
Hf T
Gv = Tm
(3.35) (3.36)
(3.37)
where = T lnT/Tm . Dubey and Ramachandrarao (1984) have pointed out that Eq. (3.37) is the most appropriate for the estimation of Gv as a function of undercooling as compared to Eqs. (3.35) (Jones and Chadwick 1971) and (3.36) (Thompson and Spaepan 1979). Savalia et al. (1996) have plotted the estimated Gv against temperature in plots A, B and C in Figure 3.22 as obtained from Eqs. (3.35), (3.36) and (3.37), respectively. It is to be noted that as per Eq. (3.37), the estimated Gv increases with the extent of undercooling, reaches a maximum and then decreases with further increase in undercooling. The reduced value of Gv at high undercooling,
Solidification, Vitrification and Crystallization
169
9 × 103
ΔGC (J/mol)
A
5 × 103
B C
2 × 103 3 × 102
7 × 102
103
Temperature (K)
Figure 3.22. Estimated GC (J/mol) as a function of temperature (K). Curve A corresponds to Eq. (3.35) (Jones and Chadwick 1971), curve B corresponds to Eq. (3.36) (Thompson and Spaepan 1979) and curve C to Eq. (3.37) (Dubey and Ramachandrarao 1984).
corresponding to Eq. (3.37), as compared to those obtained from Eqs. (3.35) and (3.36), leads to a substantial reduction (2–3 orders of magnitude) in the nucleation frequency. During rapid solidification, an undercooled liquid can vitrify only if both homogeneous and heterogeneous nucleation – either transient or steady state – are essentially bypassed. How the number density of nuclei increases with time under these four conditions is schematically shown in Figure 3.23. Transient nucleation times can be calculated by using the formulations proposed by Kelton et al. (1983, 1986) and Ghosh et al. (1991). Savalia et al. (1996) have calculated the transient
Number of nuclei
a
b c d Transient nucleation
Time
Figure 3.23. Schematic diagram showing the variation of number density of nuclei with time during rapid solidification. Curves a and b are for transient homogeneous and heterogeneous nucleation, respectively. Curves c and d are for steady state homegeneous and heterogeneous nucleation, respectively.
170
Phase Transformations: Titanium and Zirconium Alloys 15.0
Log (t n)
10.0
5.0
0.0
–5.0
–10.0
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Reduced temperature (T /T m)
Figure 3.24. Transient nucleation time as a function of reduced temperature (T/Tm ) for the ternary Zr 76 Fe16 Ni8 alloy.
nucleation time, tn , as a function of the reduced temperature (T/Tm ) using the approach of Kelton et al. (1983, 1986) and the experimental data on enthalpy changes occurring during solidification and crystallization, Hf =13 kJ/mol and
Hc = 598 kJ/mol, respectively, for the ternary Zr 76 Fe16 Ni8 alloy (Figure 3.24). The estimated value of the transient nucleation time being comparable to the incubation period, the role of transient nucleation cannot be neglected. In fact, in this specific case, the estimated value of the transient nucleation time has been found to be larger than the estimates made earlier (Ghosh et al. 1991) due to lower values of Gv . It may be emphasized here that a longer transient nucleation time implies a reduced contribution of steady state nucleation to the overall crystal nucleation process in a situation of rapid solidification which corresponds to a very short solidification time. Transient nucleation time, therefore, is of considerable significance under rapid solidification and a longer transient nucleation time will imply a better GFA. The rate of steady state homogeneous nucleation is given by Eq. (3.32), indicating the influence of the viscosity, , of the liquid which can be expressed in terms of the Vogel–Fulcher–Tamman equation (Busch et al. 1998): T = exp (3.38) T −T where and are constants and T is the temperature where the excess configurational entropy of the free volume is zero. By substitution of appropriate values of
Solidification, Vitrification and Crystallization
171
20 15 10 5
Log I ss
0.0 –5 –10 –15 –20 –25 –30 –35 –40 0.2
0.3
0.4
0.5
0.6
0.7
0.8
Reduced temperature (T /T m)
Figure 3.25. Steady state nucleation frequency as a function of the reduced temperature (T/Tm ) for Zr 76 Fe16 Ni8 alloy.
these quantities ( = 546 × 1011 Ns/m2 ; = 182216 K; T = 2484 K) obtained on the basis of the analysis of Battezzatti and Greer (1989), the viscosity of the molten alloy Zr 76 Fe16 Ni8 has been calculated as a function of temperature (Savalia et al. 1996). Using Eqs. (3.32) and (3.38), the steady state nucleation frequency has been obtained for Zr 76 Fe16 Ni8 as a function of the reduced temperature, T/Tm (Figure 3.25). A comparison of calculated values of the steady state nucleation frequency for different alloy compositions reveals their relative GFAs. In systems where homogeneous nucleation frequencies are very low, heterogeneous nucleation dominates. Glass formation in such systems is possible at somewhat slower cooling rates, provided heterogeneous nucleation sites are eliminated. The Pd40 Ni40 P20 alloy is one such system which was known about three decades back as amenable to vitrification at a cooling rate of 103 K/s in the absence of heterogeneous nucleation. With the discovery of bulk metallic glasses in multicomponent Zr alloys in the recent past, it is now demonstrated that avoidance of homogeneous and heterogeneous nucleation of crystals and thereby formation of glass are possible in several alloy systems at much slower cooling rates (typically 100 K/s). 3.4.4 Microstructures of partially crystalline alloys Rapid solidification of an alloy at a cooling rate close to the critical rate necessary for avoiding nucleation of crystals develops a condition under which the crystal
172
Phase Transformations: Titanium and Zirconium Alloys
formation and the vitrification processes are closely competitive. As has been mentioned earlier, the total avoidance of crystal nucleation is difficult to achieve in most of the potential glass forming alloys, and therefore, a large majority of predominantly glassy alloys do contain a distribution of crystalline particles. In this section, the morphologies of crystalline particles embedded in the amorphous matrix have been examined with a view to gaining an understanding of the nucleation and growth processes of such crystals which are preserved in the frozen-in condition in the early state of their growth. Studies on partially crystalline Zr alloys have been reported in Zr–Ni, Zr–Fe and Zr–Fe–Ni systems by Dey and Banerjee (1985, 1986, 1998) and in the Zr–Be system by Tanner and Ray (1979). Partially crystalline structures can be grouped into the following two categories, essentially based on the size of the crystalline particles: (a) the structure containing a negligible volume fraction of extremely fine crystalline particles or quenched-in nuclei which did not get an opportunity to grow to any significant extent and (b) partially crystalline structures consisting of a distribution of crystalline particles of a micrometre size range in an amorphous matrix, maintaining a low volume fraction of the crystalline phase; such a structure can form when nucleation of a few crystals and their limited growth are possible within the time available prior to vitrification but the nucleation frequency is so low that the crystalline particles remain few and far between. The presence of quenched-in nuclei in the amorphous matrix is not detectable in XRD and electron diffraction patterns which show the broad maxima corresponding to the first, second and third peaks in the radial distribution function. High-resolution electron microscopy of alloys, which had been characterized to be fully amorphous by diffraction experiments, has revealed the presence of small domains of crystalline particles. Figure 3.26 shows one such example in which lattice fringes associated with quenched-in nuclei are seen in the amorphous Zr–Ni matrix. The measured lattice spacing of about 0.25 nm matches closely with d110 of the (bcc)-Zr phase. This is not unexpected as the crystalline phase most likely to form in this alloy is the -phase which can even inherit the composition of the liquid phase. The frequency of composition invariant homogeneous nucleation in Zr76 Fe16 Ni8 has been estimated in Section 3.4.3. An undercooling to the extent of T/TM = 06, as per this estimation, produces a homogeneous nucleation frequency of about 1010 m−3 /s. Therefore, a limited number of nuclei are created even under a rapid cooling rate of 105 K/s. These quenched-in nuclei remain very small in size as they do not get an opportunity to grow prior to the vitrification of the matrix. The presence of such quenched-in nuclei has a strong influence on the kinetics of crystallization as will be discussed in Section 3.5.3.
Solidification, Vitrification and Crystallization
173
Figure 3.26. Small crystalline particles distributed within the amorphous matrix as revealed by high-resolution electron microscopy (HREM).
Partially crystalline structures in Zr 3 Fex Ni1−x alloys produced under different cooling rates show a distribution of several crystalline phases in the amorphous matrix, as revealed by XRD. Since the cooling rate during melt spinning drops as one traverses from the side which comes in contact with the copper wheel (heat extracting surface) to the air side, the phases present in the melt-spun ribbon in the vicinity of one side are sometimes different as compared to those on the other side. The results of phase analyses of alloys of different compositions are summarized in Table 3.4. TEM investigations on these partially crystalline structures show that while single isolated crystals are present in some cases, crystal aggregates of special morphologies are often encountered. An aggregate morphology which is frequently observed consists of a central core crystal surrounded by peripheral crystals (Figure 3.27(a)). The central core exhibits both spherical and dendritic Table 3.4. Phase analysis of rapidly solidified Zr–Fe–Ni alloys. Alloy composition
Nature of phases present
Zr 76 Fe24
Zr 3 Fe
Zr 76 Fe20 Ni4
Zr 3 Fe
Zr 76 Fe16 Ni8
Zr 3 Fe
Zr 76 Fe12 Ni12
Zr 3 Fe
Zr 76 Fe8 Ni16
Zr 3 Fe, Off-stoichiometric Zr 2 Ni
Zr 76 Fe4 Ni20
Zr 3 Fe, Off-stoichiometric Zr 2 Ni
Zr 76 Fe8 Ni24
Off-stoichiometric Zr 2 Ni
174
Phase Transformations: Titanium and Zirconium Alloys
1.0 μm
0.2 μm (a)
(b)
0.2 μm (c)
Figure 3.27. Morphology of crystals in amorphous matrix in partially crystalline alloys. (a) An aggregate morphology consisting of a central core crystal surrounded by peripheral crystals, (b) heterogeneous nucleation of peripheral crystals around the core crystals is occasionally suppressed, under the rapid cooling condition of piston–anvil splat quenching, as is revealed from the presence of -crystals embedded in the amorphous matrix and (c) “sunflower” morphology.
morphologies. Electron diffraction and energy dispersive spectroscopy (EDS) have identified the core crystals to be of the -phase supersaturated with Fe and Ni, both of which are strong -stabilizers. The peripheral crystals are of the Zr 3 Fe Ni phase in alloys with higher levels in Fe (c > 05) and of the Zr 2 Ni phase in alloys with higher Ni contents (c < 05).
Solidification, Vitrification and Crystallization
175
The defect-free dendritic morphology of the core crystals of the supersaturated -phase suggests that they form from the liquid phase and not from the amorphous phase. The composition of the core crystals indicates that alloy partitioning occurs only to a limited extent during solidification. The enrichment of -stabilizing elements in the core crystals is responsible for the retention of this phase even at room temperature. The peripheral crystals are nucleated heterogeneously on the surface of the core crystals in either of the following two situations. (a) Nucleation of peripheral crystals from the liquid phase can occur heterogeneously on the surface provided by core crystals. As the core crystals grow with a limited solute partitioning, the composition of the liquid ahead of the solidification front gradually changes. Finally, a stage is attained in which the formation of -phase becomes less preferred than the formation of intermetallic phases such as Zr 3 Fe Ni or Zr 2 Ni due to either thermodynamic or kinetic considerations. (b) Nucleation of peripheral crystals can occur after the vitrification of the matrix. As the core crystals grow in continuously cooling liquid phase, the viscosity of the surrounding liquid rises sharply, finally leading to vitrification. The peripheral crystals subsequently nucleate and grow in the amorphous matrix. Depending on the prevailing cooling rate at the interface of the core crystals, fresh nucleation of intermetallic phases can take place. Under the rapid cooling condition of piston–anvil splat quenching, heterogeneous nucleation of peripheral crystals around the core crystals is occasionally suppressed as is revealed from the presence of -crystals embedded in the amorphous matrix (Figure 3.27(b)). In contrast, peripheral crystals of Zr 3 Fe Ni and Zr 2 Ni are often seen around core crystals in melt-spun alloys. The interfaces between the core crystals and the surrounding amorphous matrix act as heterogeneous nucleation sites on which Zr 3 Fe Ni or Zr 2 Ni crystals nucleate (Figure 3.27(a)). As mentioned earlier, the crystallization products in Zr 3 Fex Ni1−x alloys are Zr 3 Fe Ni and Zr 2 Ni for alloys with x > 05 and x < 05, respectively. In the former case, a partitionless crystallization occurs as is reflected in faceted crystal/amorphous interfaces. Crystals of Zr 3 Fe Ni phase, which has a Re3 B type structure, tend to form a special orientation relationship with the core -crystals. Different crystallographic variants, which are twin related, are often found to form adjacent to each other, sharing the twin interface between them. The resultant morphology of the crystal aggregate, as illustrated in Figure 3.27(c), is described as the “sunflower” morphology. Electron diffraction patterns from individual “petals” and dark field imaging have revealed that the opposite petals have the same orientation and that each “petal” is twin related with the two adjacent petals. The growth of crystal aggregates with “sunflower” morphology is encountered quite frequently in Zr 3 Fex Ni1−x alloys with x > 05 in which polymorphic crystallization can occur. The growth of such three dimensionally symmetric aggregates can occur only if the matrix is fully
176
Phase Transformations: Titanium and Zirconium Alloys
isotropic as in the case of crystallization of an amorphous phase. From the consideration of the symmetry relation between the parent and the product phases, the amorphous to crystalline phase transformation is analogous to crystal formation from a liquid or vapour phase. This explains why the morphology of the crystal aggregates in the amorphous matrix often bears a similarity with the crystalline products from liquid or vapour phase. 3.4.5 Diffusion The thermal stability of metallic glasses is a subject of vital concern as one desires to produce glasses which will retain their amorphous nature at as high a temperature as possible. The current limit is about 1300 K, for some W-based glasses. Sometimes the amorphous phase is used as an intermediate which can be transformed into crystalline phases of desired grain structures. Such an approach is often adopted for the production of nanocrystalline structures using the amorphous phase as a precursor state. Phenomena such as diffusion, structural relaxation and crystallization need to be discussed for assessing the thermal stability of metallic glasses. A clear understanding of the changes occurring in metallic glasses during heat treatments is dependent on the elucidation of diffusion parameters and mechanisms. There have been very few direct measurements of diffusion in amorphous alloys. The experimental difficulties are considerable since the diffusion distance is very small for the accessible temperatures where crystallization can be avoided. When diffusion distances are in the range of 100 nm, techniques which permit composition analysis at a very high depth resolution need to be employed for measuring the concentration-depth profiles. These include Auger Electron Spectroscopy (AES) with sputter etching, Rutherford Backscattering Spectroscopy (RBS) and Secondary Ion Mass Spectroscopy (SIMS). Some indirect methods for the measurement of the diffusivities are also employed. In one of the early diffusion experiments, Gupta et al. (1975) determined the diffusivity of Ag in Pd81 Si19 . The surface of the alloy was sputter-deposited with 110Ag radioactive isotope. After diffusion annealing, the surface was sputter-etched. The Ag concentration was determined, from the radioactivity of the material removed, as a function of etching depth. One of the problems with such experiments is that different points on the surface are sputtered at different rates. Birac and Lesueur (1976) used a neutron beam, and the (n, ) reaction of 6 Li nuclei, for measuring its diffusion in Pd80 Si20 . Cahn et al. (1980) succeeded for the first time in measuring the self-diffusion of B in Fe40 Ni40 B20 . A layer of the same composition, containing 10 B and 11 B in the ratio, 96:4, was sputtered onto the metallic glass, which contained the natural isotopes in the ratio 20:80. After diffusion annealing, the surface was sputtered and 10 B/11 B ratio was measured by using SIMS.
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177
Several indirect methods for the measurement of diffusivity are available. For example, it is possible to use the rate of primary crystallization. The diffusivities of B and C in Fe-B-C alloys have been deduced from measurements of the crystal growth rate (Koster and Herold 1980, Koster 1983). They obtained a diffusion coefficient, D, of 2 × 1019 m2 /s and an activation energy of 180 kJ/mol. These values led them to surmise that B diffuses as a substitutional atom rather than as an interstitial atom. This method has been applied to Fe40 Ni40 P14 B6 by Tiwari et al. (1981). There have been some attempts, by Taub and Spaepan (1979), to evaluate the diffusion coefficient from viscosity data. In the Pd–Si system, the diffusivity of gold in the as-quenched glass has been reported to be some orders of magnitude larger than the value of D as the glass is made to relax by annealing at a temperature below but close to Tg . These results strongly suggest that frozen-in ‘defects’ in metallic glasses, which are present in the as-quenched state, play an important role in the mechanism of diffusion. Let us focus our attention on Ti- and Zr-based metallic glasses which are primarily grouped under the class of metal–metal amorphous alloys with early and late transition metals as constituents. As has been indicated in Section 3.4.2, a fairly large number of binary alloys of this type are easy glass formers. Interpretations of diffusion data obtained in such systems are expected to be straightforward as complications due to multicomponent interactions will be absent. Measured values of diffusion coefficients, D, of different diffusing species in binary metal–metal amorphous alloys show an Arrhenius type dependence on temperature (Figure 3.28) when the data are replotted as log D against reciprocal of normalized temperature, Tg /T . This observation is indicative of the fact that for a given diffusing species and a given amorphous alloy, a single thermally activated diffusion mechanism remains operative over the entire temperature range studied. The influence of the atomic size of the diffusing species on the diffusion constant can be seen from the diffusivity data for Cu, Al, Au and Sb in a given metallic glass, Zr61 Ni39 . At any given temperature, it has been observed that DCu > DAl > DAu > DSb which is consistent with the fact that rCu < rAl < rAu < rSb where r is the respective atomic radius. The D values for Cu were found to be higher than the corresponding values for Al by about an order of magnitude in the temperature range of 556–621 K (Sharma and Mukhopadhyay 1990). The values of the activation energy, Q, for diffusion evaluated on the basis of the observed Arrhenius type temperature dependence of D, were found to be 1.33 ± 0.17 eV for Cu and 1.68 ± 0.13 eV for Al. The corresponding values of the preexponential factor, Do , were 10−757±146 and 10535±175 m2 /s, respectively. This comparison also reveals that the activation energy of diffusion in a given amorphous alloy increases with increasing atomic size of the diffusing species. Such a rule is expected to be
Phase Transformations: Titanium and Zirconium Alloys
Diffusion Coefficient, D (m2 s–1)
178
10–18
10–22
10–26 1.0
1.2
1.4
Tg/T
Figure 3.28. Diffusion coefficients, D, of different diffusing species in binary metal-metal amorphous alloys as a function of normalized temperature Tg /T showing an Arrhenius type dependence on temperature.
valid only in cases where the diffusing species have more or less similar chemical interactions with the amorphous matrix. There have been a number of investigations to find out whether the diffusivities of a given species in an alloy in crystalline and amorphous states are significantly different. Contradictory results have been reported from these investigations. Valenta et al. (1981) have shown significantly slower diffusion of P and Fe in Fe40 Ni40 P14 B6 when the amorphous alloy is crystallized. Contrary to this, Akhtar et al. (1982a,b) have shown a much faster diffusion of Pt in Ni33 Zr67 after crystallization. Such contradictory results stem from the fact that there is a wide variety of crystallization mechanisms which result in a variety of crystal structures, phase distributions and grain sizes in the crystallized products. Diffusion data obtained from the homogeneous amorphous phase cannot be compared with that obtained in the crystallized product of the same material. Such a comparison is somewhat meaningful only in cases where the amorphous phase crystallizes into a single phase crystalline state with a large grain size (as in the case of polymorphic crystallization described in Section 3.5.1). Structural changes within the amorphous phase induced by heat treatments causing relaxation, plastic deformation and irradiation are expected to bring about changes in diffusivity in metallic glasses. Cantor and Cahn (1983) have reviewed the experimental data to arrive at the conclusion that diffusivity is very sensitive to relaxation in amorphous alloys which are very rapidly cooled through Tg .
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Diffusion data reported in Zr61 Ni39 in the temperature range of 551–621 K show that a relaxation heat treatment does not affect diffusivity significantly (Sharma and Mukhopadhyay 1990). Autorelaxation of the glass during the cooling down from Tg appears to have reduced the influence of the subsequent relaxation heat treatment. Akhtar et al. (1982a,b) have reported the diffusion coefficient of Au in amorphous Zr67 Ni33 at four different temperatures using as-quenched, relaxed and plastically deformed specimens. At each temperature, the diffusivity in the deformed condition is found to be higher than that in the as-quenched condition, the latter in turn being higher than that in the relaxed condition. Cahn et al. (1982) have observed that Au diffusion coefficients in amorphous Zr64 Ni36 decrease after the amorphous alloy is irradiated with fast neutrons. They have argued that the irradiation-enhanced chemical short-range order causes a reduction in atomic volume with a corresponding reduction in the diffusion coefficient. Diffusion of small size H atom in Ti- and Zr-based metallic glasses assumes a great significance because of the possibility of H storage in some of these materials. The activation energy of H diffusion in Zr67 Pd33 has been reported to be 0.25 eV between 270 and 365 K, increasing to about 0.7 eV in the temperature range of 430–490 K. Similar characteristics have been observed in amorphous Ti–Cu alloys. Diffusion of H in hydrogenated Ti–Cu and Zr–Pd amorphous alloys is between one and two orders of magnitude faster than in the corresponding crystalline hydrides. Cantor and Cahn (1983) considered various experimental and theoretical information when available regarding diffusivities in amorphous alloys for arriving at possible atomistic mechanisms of diffusion in these systems. Since an Arrhenius type relation has been found to be valid for the temperature dependence of diffusivity in amorphous alloys, it is attractive to consider whether atomic models of diffusion in crystalline materials can also be applied to amorphous alloys. This approach can be further justified in view of the fact that the local arrangements of atoms in and the densities of amorphous and crystalline alloys are somewhat similar. The atom-vacancy exchange process is known to be the most important mechanism for both self- and impurity diffusion in crystalline alloys. In the absence of a reference lattice, a vacancy in an amorphous alloy can be defined as an empty space in the amorphous structure of atomic or near atomic dimensions. Several investigations have been made with a view to examining the stability of vacant sites of atomic dimensions in an amorphous alloy. From modelling work, it has been shown that if an atom is removed from the dense random packed structure of an amorphous alloy, atomic vibrations quickly redistribute the excess space
180
Phase Transformations: Titanium and Zirconium Alloys
over a large volume. This tendency of smearing the excess volume created by the removal of an atom does not allow the presence of a near atomic size vacant space in the amorphous structure. The structure and size of soft sphere dense random packed models of amorphous alloy structures show that most interstitial sites are surrounded by distributed tetrahedral and octahedral groups of atoms with more tetrahedral and fewer octahedral sites than in an equivalent close-packed crystal. The size distribution in Figure 3.29 shows that a small fraction of octahedral interstices have sizes in the range of 0.6–0.7 of the atomic diameter, and these large interstices can be considered as vacancies in the amorphous structure. 3.4.6 Structural relaxation The structure and properties of a glass depend on the quenching rate. When a glass is annealed, its structure will first relax to that of a glass formed at lower cooling rates and ultimately tend towards that of an “ideal” glass. Egami (1983) used energy dispersive XRD methods and showed that two types of change occurred during relaxation. One is related to the topological short-range order. During the process of atomic movement, the tetrahedra per se are not affected by their relative configurational change. A higher degree of topological short-range order is established by a highly collective phenomenon involving the cooperation of a number of atoms. The second change is related to the chemical short-range ordering during which site interchange of atoms takes place.
Fraction of sites
0.3
Tetrahedral
0.2
octahedral – fcc interstitial radii
Radius of vacant sites in amorphous structure
0.1
0.2
0.4
0.6
Vacant site radius/atomic radius
Figure 3.29. Size distribution of vacant sites in amorphous structure.
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Further studies have indicated that relaxation mechanisms can be divided into reversible and irreversible types. Reversible changes appear to be associated with changes in the chemical short-range order. Kurusmovic and Scott (1980) have shown that the Young’s modulus of the Fe40 Ni40 B20 glass may be cycled reversibly between the values which are characteristic of different annealing temperatures. The analogy with short-range order in crystalline alloys is particularly striking. The irreversible change appears to be associated with a change in the topological short-range order. During structural relaxation, the alloy becomes denser. Significant changes occur in many properties such as the electrical resistivity, magnetic anisotropy, Curie temperature, elastic modulus and mechanical properties. Some changes are beneficial, while others are detrimental. The changes in magnetic and mechanical properties are discussed below. Luborsky (1983) have reported remarkable improvements in the magnetic properties of Fe–Ni-based glasses after stress-relief annealing for 2 h at 100 K below their glass transition temperature. These changes include an increase in remanence, a decrease in the saturation field and a change in the Curie temperature. However, if the treatment leads to crystallization, then the magnetic properties deteriorate. Hence, it is important to ensure that the compositions of metallic glass ferromagnets are chosen so as to have good thermal stability. The Curie temperature can be determined either by magnetic measurements or by locating the appropriate thermal anomaly in a differential scanning calorimeter (DSC) run. Egami (1983) found that cycling Fe27 Ni53 P14 B6 repeatedly between 523 and 573 K caused the Curie temperature to cycle between 368 and 375 K. These reversible changes are due to a change in the position of Fe and Ni atoms and the consequent alteration in the chemical short-range order. Further exploration of the use of low-temperature annealing in bringing about such beneficial changes in magnetic properties appears to be desirable. One of the attractive properties of metallic glasses is their large ductility in bending and compression. In many cases, low-temperature annealing treatments lead to the loss of this ductility. There have been several investigations of this temper embrittlement phenomenon, and it appears to be dependent on the composition. In the case of Fe40 Ni40 P14 B6 , a heat treatment at 373 K for 2 h leads to embrittlement. AES has been used for demonstrating that segregation of P occurs during structural relaxation which leads to embrittlement (Walter and Bertram 1978, Walter 1981). In general, metal–metal glasses and Cu60 Zr40 appear to be immune to this type of embrittlement. An interesting suggestion with regard to the embrittlement tendency has come from the group of Davies (1978). Alloys which eventually crystallize as an fcc phase do not exhibit the embrittling tendency, whereas alloys which crystallize
182
Phase Transformations: Titanium and Zirconium Alloys
into a bcc or hcp phase become brittle on relaxation. This implies that the structural groupings of the crystalline phases responsible for embrittlement can be partially observed in the amorphous phase. This might also explain why embrittlement occurs within a certain composition range in a given system. 3.4.7 Glass transition The nature of the glass transition is still a matter of controversy. Experimental evidence and theoretical models suggest the glass transition to be a first-order phase transition, based on the free volume approach, a higher order phase transition or no phase transition at all, e.g. kinetic freezing. However, there is a general agreement that the maximum undercooling of a liquid is limited to the isentropic temperature in order to avoid the paradoxical situation described by Kauzman (1948) where the configurational entropy of the supercooled liquid becomes smaller than the configurational entropy of the ordered equilibrium phase. Consequently, as long as crystallization can be prevented, the undercooled liquid will freeze to a glass close to the ideal glass transition temperature, Tgo , where the entropy difference between the liquid and the equilibrium crystalline phase would vanish. In reality, the glass transition sets in at a temperature somewhat above Tgo . Depending on the deviation of the glass transition temperature, Tg , from the ideal glass transition temperature, Tg , the glass attains different relaxation states. Only at an infinitely slow cooling rate, if the liquid is vitrified (by avoiding crystallization), the liquid to glass transition occurs at Tgo and the resulting glass attains a fully relaxed state. Such a transformation can be considered as a second-order transition at Tgo . Under realistic cooling rates, a glass having excess entropy and consequently not in the fully relaxed state forms at Tg . Depending on the extent of relaxation of the product glass, the glass transition temperature measured from experiments varies with the imposed cooling rate, a higher cooling rate yielding a higher value of Tg , as shown in Figure 3.17 in which Tg1 and Tg2 are the glass transition temperatures for the cooling rates, G1 and G2 (G1 > G2 ), respectively. The glass transition event is also encountered during heating experiments. Continuous heating experiments in a DSC often show an endothermic event prior to the large exothermic event of crystallization. A thermogram obtained during heating of the Zr-35 at.% Ni glass at a heating rate of 5 K/min clearly reveals the endothermic event attributable to the glass to liquid transition preceding the crystallization event (Figure 3.30). Taking into account experimental values of the specific heat of a stable and highly undercooled liquid and measured values of the enthalpy of crystallization of the amorphous alloy, the undercooled liquid at Tg is found to exhibit only a very small excess entropy in comparison with the stable crystalline phase. Specific heat, thermal expansion and Mossbauer spectroscopy data on several fully
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Heat flow exothermal 50 000 mW
↓
150
250
350
400
Temperature, °C
Figure 3.30. DSC thermogram of Zr-35 at.% Ni glass obtained at a heating rate of 5 K/min showing the endothermic nature of the glass to liquid transition.
relaxed amorphous alloys reveal that the glass transition can be approached under internal equilibrium conditions (Tg approaching Tgo and becoming independent of the heating rate). Usually amorphous alloys form in composition ranges where the heat of mixing between the components is negative (strong ordering tendency). However, there are instances where a positive deviation from the ideal solution behaviour is noticed in narrow composition ranges where the amorphous phase tends to separate into two phases, both having amorphous structure. Such a system is expected to exhibit two glass transition temperatures. Experimental observations of two glass transition events and of a phase separated microstructure of the amorphous alloy have led Tanner and Ray (1980) to infer the presence of a two-phase amorphous structure in the Zr36 Ti24 Be40 alloy. Using an atom probe microscope, Grunse et al. (1985) have shown the presence of spatially extended concentration waves in as-quenched Ti50 Be40 Zr10 . Observations on structural relaxation and localized fluctuations in structure and composition of metallic glasses have prompted a model of glass transition based on the heterogeneous glass in terms of density and concentration. In this model, it is visualized that a glass consists of liquid-like regions of large free volume or high local free energy and solid-like regions with small free volume or low
184
Phase Transformations: Titanium and Zirconium Alloys τ*
(a)
Relaxation time (τ)
↓
Relaxation time (τ)
↓
T1 > Tg
Frequency
Frequency
T2 < Tg
(b)
Figure 3.31. Schematic diagram showing relaxation spectra at (a) temperatures above Tg and (b) temperatures below Tg .
free energy. Each region undergoes a transition at a frequency much smaller than the Debye frequency (∼1013 s−1 ) between local energy minima corresponding to different configurational states, the relaxation time, i, being proportional to exp− i /kB T ) where i is the energy barrier between these states. The relaxation spectra at temperatures below and above Tg are schematically shown in Figure 3.31. At T1 > Tg , the whole spectrum lies to the left of the time of measurement ∗ (for example, 30 s), so that the whole system undergoes frequent configurational transformations and is liquid-like. With lowering of temperature below Tg , (T2 < Tg ), the whole spectrum shifts to longer times such that relaxation times for most of the regions are greater than ∗ , and the isolated liquid-like regions are small in volume fraction and are embedded in the rigid solid matrix. At such temperatures, localized, short-range structural relaxation can occur, leading to glass transition in these small domains. In the close vicinity of Tg , the peak of the distribution of the relaxation time is located near ∗ , leading to a long-range, cooperative structural relaxation which causes a rapid decrease in the relaxation time and an accompanying rise in the viscosity.
3.5
CRYSTALLIZATION
Several experimental techniques have been used to monitor the crystallization of metallic glasses. Among these, DSC and TEM have proved to be particularly useful. In the case of DSC, crystallization gives rise to distinct exothermic peaks. The heat of crystallization can be measured and is found to be of the order of 40% of the heat of fusion of the alloy, the remaining enthalpy having been extracted from the liquid during quenching. Using electron microscopy, the morphology
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and structure of crystals can be established and the mechanism and kinetics of crystallization can be followed. Heating of an amorphous alloy leads to several other changes apart from relaxation. These include glass–liquid transition, phase separation and crystallization. Not many studies have been made on the glass–liquid transition. Although phase separation into two amorphous phases is a well-documented event in the case of oxide glasses, metallic glasses appear to undergo phase separation in only rare instances. Evidence for phase separation has been reported in Pd74 Au8 Si18 by Chou and Turnbull (1975) and in B40 Ti24 Zr36 by Tanner and Ray (1979). DSC plots indicate the occurrence of two glass transition temperatures. Banerjee (1979) reported a spinodal decomposed amorphous microstructure in the Zr-24% Fe alloy. Piller and Haasen (1982) have used the sensitive atom probe field ion microscope in order to demonstrate that Fe40 Ni40 B20 decomposes into two amorphous regions, one having composition corresponding to (Fe,Ni)3 B and the other being a B-deficient region. 3.5.1 Modes of crystallization From symmetry rules, it can be shown that the amorphous to crystalline phase transition is necessarily a first-order transition. This is consistent with the observed nucleation and growth processes encountered by different investigators studying crystallization. As there is no periodic arrangements of atoms in the parent amorphous structure, there is no possibility of achieving a lattice correspondence between the parent and the product structures. Therefore, the occurrence of crystallization via “military” atom movements is ruled out. Hence, the transition is expected to occur essentially by diffusional atom movements. Depending on the diffusion distances involved in the crystallization process, one can broadly classify the mechanisms of crystallization into three broad categories, namely (a) polymorphic crystallization, (b) eutectic crystallization and (c) crystallization involving long-range diffusion – primary followed by eutectic or primary followed by polymorphic crystallization. These three modes of crystallization are schematically illustrated in Figure 3.32. (a) Polymorphic crystallization (A → ): When the compositions of the parent amorphous phase and of the product crystalline phase are the same, the crystallization process involves diffusional atomic jumps across the advancing transformation front. This situation is analogous to that in massive transformation or in “partitionless solidification” processes and is illustrated in Figure 3.32 for the alloy composition c1 . Koster and Herold (1980) have termed this type of crystallization as “polymorphic crystallisation”.
Phase Transformations: Titanium and Zirconium Alloys
Free energy (G)
186
Gβ
GA
Gα
c1 c4 c6
A
c2 c3
c5
c7
B
c, atom fraction of B A
A α
α
A′
c→
c5 c1
A
α
c2 A
Distance → A (c1)→ α (c1)
Polymorphic (c 1)
E
α E
A
α
E
α
c4 Distance → A → α (c4) + A′ (c5)
Distance → A′ (c5) → α (c6) + β (c 7)
Primary + Eutectic (c 2)
c3
E A
A
Distance → A (c3) → α (c6) + β (c7) Eutectic (c 3)
Figure 3.32. Schematic representation of different modes of crystallization: polymorphic, primary followed by eutectic and eutectic occurring in amorphous alloys of compositions given by c1 , c2 and c3 , respectively. Free energy changes which motivate the crystallization process are indicated by arrows in the free energy – concentration (G–c) plots corresponding to the amorphous (A) and the two crystalline phases, and . In polymorphic crystallization, the amorphous alloy of composition c1 transforms into the -phase of the same composition. In primary crystallization, phase of composition c4 forms first from the amorphous alloy of composition c2 ; the composition of the latter gradually changes to c5 which finally decomposes into an eutectic mixture (E) of and . In eutectic crystallization, the amorphous phase of composition c3 directly transforms into the eutectic mixture (E) of and . Concentration profiles across a crystalline particle is shown below each of the schematic micrographs.
(b) Eutectic crystallization (A → + ): The partitioning of alloying elements into two crystalline phases (illustrated in Figure 3.32 for the composition c3 ), which are forming simultaneously from the parent amorphous phase in a cellular transformation, requires diffusion at or near to the transformation front. Eutectic/eutectoid decomposition and cellular precipitation are the analogous phase transformations in crystalline systems.
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Eutectic crystallization, which has been encountered in several systems such as Fe–B, Fe–Ni–B, Mo–Ni, and Zr–Fe, occurs at a relatively low rate as compared to polymorphic crystallization. Decomposition of an amorphous phase into a mixture of a crystalline and a second amorphous phase, in a cellular transformation mode, appears to be possible in a system in which the amorphous phase has a tendency towards phase separation (or unmixing). However, in the case of metallic glasses, such a transformation, which is analogous to the monotectoid reaction, has not been encountered. (c) Primary crystallization followed by eutectic or polymorphic crystallisation: During the formation of a crystal having a composition which is different from that of the parent amorphous phase, a long-range diffusion field is established ahead of the transformation front. Primary crystallization, as designated by Koster and Herold (1980), involves such a process, the kinetics of which are generally controlled by the mechanism of long-range atom transport in the amorphous matrix. The composition of the matrix amorphous phase may finally transform via one of the many possible phase reactions. In the case of Fe–B, the amorphous matrix gradually attains the Fe75 B25 composition and then transforms into the Fe3 B phase via a polymorphic crystallization process. By analogy with various liquid-to-crystal phase reactions, one can visualize many possible reactions, such as the following: (a) Primary crystallization followed by a peritectic/peritectoid reaction (A→ (primary) + A → ): There exists a possibility of this transformation sequence which has not been reported in any metallic glass system. (b) Primary crystallization followed by an eutectic reaction (A → (primary)) + A → (primary) + + (eutectic): A and A are amorphous phases having different compositions, and and are different crystalline phases as shown in Figure 3.32 for composition c2 . The appearance of more than one exothermic peak in DSC thermograms appears to originate from such successive phase reactions (Figure 3.33).
3.5.2 Crystallization in metal–metal glasses Metal–metal glasses have not received the same degree of attention as metal– metalloid glasses, although this situation appears to be changing with increasing interest in these glasses. Metal–metal glasses present several features of interest. Unlike metal–metalloid glasses, which are generally restricted to compositions of around 20 at.%, metal–metal glasses can be formed over a wider range of compositions. Both metal–metalloid and metal–metal glasses can be prepared around compositions corresponding to deep eutectics. In addition, metal–metal
188
Phase Transformations: Titanium and Zirconium Alloys Eutectic or polymorplic
·
Rate of heat flow (H )
Primary Crystallization Glass transtion
2 Tg
Tx Temperature
Polymorplic or eutectic crystallization
Tg
1 T x1
T x2
Temperature Primary followed by eutectic or polymorplic crystallization
Figure 3.33. Differential scanning calorimetry thermograms showing, dH/dt, the rate of heat flow versus temperature, T , at a constant rate of heating (dT /dt = constant) for (a) polymorphic and eutectic crystallization and (b) primary followed by eutectic or polymorphic crystallization. The endothermic event of glass transition at Tg and the exothermic events of crystallization at Tx are indicated. While for polymorphic and eutectic crystallization a single exothermic event is observed, as shown in (a), two distinct thermal events are noticed in primary crystallization followed by either eutectic or polymorphic crystallization as shown in (b).
glasses can form at compositions which correspond to stoichiometric compounds having high melting points. There are also indications of structural differences. Polyhedral packing, characterized as Kasper polyhedra, appears to be a dominant structural feature in metal–metal glasses. Systems which have been investigated in detail include Cu–Zr, Ni–Zr, Fe–Zr, Ni–Nb, Ti–Be–Zr and Mg–Zn. In order to illustrate the type of investigations made on these alloys, the Ni–Zr and Fe–Zr systems are used as examples in the discussion which follows. The equilibrium diagram for the Ni–Zr system contains four well-defined eutectics at 8.8, 36.3, 63.5 and 75.9 at.% Zr. Glasses are formed at compositions which correspond to these eutectics as well as at compositions which correspond to the equilibrium intermetallic phases. The results of Dong et al. (1981) and Dey et al. (1986) with regard to four alloys are described below. The Ni365 Zr635 alloy corresponds to an eutectic between the Zr2 Ni and ZrNi phases. Crystallization of this amorphous alloy involves two steps: primary crystallization of Zr2 Ni crystals, followed by the formation of ZrNi crystals. The activation energy for the nucleation of primary crystals has been reported to be 500 kJ/mol, and the diffusion controlled growth of these crystals of average diameter, d, has been shown to be characterized by the relationship, d t1/2 where t is the annealing time.
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The Ni333 Zr667 alloy exhibits polymorphic crystallization, which results in the formation of Zr2 Ni crystals. The activation energy associated with the crystallization process has been determined to be about 320 kJ/mol from both DSC experiments performed in the continuous heating mode and measurement of crystal size as functions of time and temperature. Dey et al. (1986) have observed the presence of very closely spaced planar faults within the Zr2 Ni crystals (Figure 3.34). The Zr72 Ni28 alloy crystallizes into an off-stoichiometric Zr2 Ni phase which has a C16 (tetragonal) structure. The crystals exhibit a dendritic morphology but a closer examination reveals a spherulitic morphology. The activation energy for growth has been measured to be 180 kJ/mol. DSC experiments carried out on Zr76 Ni24 have revealed two exothermic peaks. The first has been attributed to the formation of the hcp -phase, identified by TEM studies. Using peak shift observations obtained from DSC runs made at different heating rates, the activation energy associated with the primary crystallization event has been determined to be 310 kJ/mol. This value agrees closely with that observed in Ti50 Be40 Zr10 , where primary crystallization into has been reported. As the -phase is expected to be solute lean, the rate-controlling process is identified as being the diffusion of the solute element in the amorphous matrix. The second step in the crystallization process could be either an eutectic reaction, leading to simultaneous formation of and Zr2 Ni, or a polymorphic reaction, leading to the formation of Zr2 Ni. The activation energy associated with the second step has been found to be 180 kJ/mol (Table 3.5). Such a low value of the activation energy is consistent with the occurrence of a eutectic or a
Figure 3.34. Bright field TEM micrograph showing the presence of very closely spaced planar faults within Zr2 Ni crystal.
190
Composition
Sequence (mode)
Bulk or surface
Zr76 Fe24
(1)
Bulk
Zr76 Fe24 Ni4
(2)
Bulk
Zr76 Fe16 Ni7
Tg 80 K/min
T1 (K) 20 K/min
Tp (K) 20 K/min
E/En /Eg (kJ/min)
T (K)
n/n1 /n2
650.0
653.5
656.0
272.0 (E)/545.0 (En )/168.0 (Eg )
626.0
3.10 (n)
641.0
653.0
655.0
286.0 (E)
631.0
2.71 (n)
639.0
650.7
652.0
278.0 (E)
634.0
2.65 (n)
Zr76 Fe12 Ni12
(3)
Bulk
627.0
646.0
648.0
275.0 (E)
633.0
2.55 (n)
Zr76 Fe8 Ni16
(4)
Surface
624.0
634.0
647.0
274.0 (E)
631.0
Zr76 Fe4 Ni20
(5)
Bulk
624.0
634.0
647.0
236.0 (E)
634.0
Zr76 Ni24
(7)
Bulk
650.0
652.0
654.0
271.0 (E) 410.0 (En )
631.0
2.21 (n1 )/ 2.73 (n2 ) 1.98 (n1 )/ 4.00 (n2 ) 3.20 (n)
(1) A → Zr3 Fe (polymorphic); (2) A → Zr3 (Fe,Ni) + A (primary) → + Zr2 Ni (eutectic); (3) A → Zr3 (Fe,Ni) (polymorphic); (4) A → Fe rich (Fe,Ni)3 , Zr(Ll2 ) + A (primary); (5) A → Zr3 (Fe,Ni); (6) A (primary), A → Zr2 Ni + (eutectic); (7) A → Zr2 Ni + (eutectic).
Phase Transformations: Titanium and Zirconium Alloys
Table 3.5. Crystallization sequence (mode), glass transition temperature, Tg , crystallization temperature, Tx , peak crystallization temperature, Tp , activation energies for (i) overall crystallization, E; (ii) nucleation, En , and (iii) growth, Eg , isothermal annealing temperature, T and Avarami exponent for (i) single-step process (n) and (ii) two-step processes (n1 and n2 ).
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polymorphic crystallization process, both of which involve only short-range atom transport at the transformation front for the distribution of the solute into the two product phases. However, Dong et al. (1981) have reported single step polymorphic crystallization. Such differences can arise due to variations in the initial conditions of the amorphous phase (e.g. the extent of relaxation which the glass has undergone during the melt-spinning operation). Koster and Herold (1980) have reported that polymorphic crystallisation occurs in Fe40 Zr60 , Fe30 Zr70 and Fe24 Zr76 glasses. The activation energy associated with the growth of crystals has been found to be about 170 kJ/mol. The resulting crystals in the aforementioned alloys are FeZr2 , FeZr2 and FeZr3 respectively. Dey and Banerjee (1986) have carried out DSC experiments employing both isothermal holding and continuous heating runs and have found that the activation energy associated with the overall crystallization process in the Fe24 Zr76 glass is 270 kJ/mol. TEM studies have revealed the presence of Zr3 Fe crystals (orthorhombic Re3 B structure), with planar boundaries separating the crystalline and amorphous phases. This is possible because of the absence of any long-range atom transport during the polymorphic crystallization process. Crystal aggregates of a fascinating shape have been observed in both partially crystalline as-melt-spun tapes and crystallized samples of fully amorphous Zr76 Fe24 glass. These crystal aggregates, which presumably had formed at a very early stage in the crystallization process, consisted of six petals originating from a central spherical crystal, thus giving rise to a “sunflower”-like appearance (Figure 3.35).
Figure 3.35. Bright field TEM micrograph showing crystal aggregate having a “sunflower” morphology.
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Phase Transformations: Titanium and Zirconium Alloys
Crystallographic analyses of each of these petals and of the core have permitted the orientation relationship to be determined. The petals consisted of an ordered phase crystals of Zr3 Fe which can be viewed as being an ordered structure based on
-Zr, while the core had a bcc structure. The formation of such an agglomerate is energetically favourable because the unique orientation relationship of the adjacent crystals permits the formation of low energy interfaces between them. 3.5.3 Kinetics of crystallization It has been mentioned earlier that the amorphous to crystalline transition (devitrification) occurs in different modes such as polymorphic, eutectic and primary followed by eutectic. The kinetics of the process is, therefore, governed by the mode which operates in a given system. Devitrification, being a strongly first-order transition, occurs by the nucleation of crystals in the amorphous matrix followed by the growth of these nuclei by the movement of the crystal/amorphous interface, resulting in a progressive consumption of the matrix amorphous phase. The overall kinetics of devitrification is determined by the number density of quenched-in nuclei, the rate of thermally activated fresh nucleation and the rate of growth of the crystalline phase. If the nucleation rate is so high at the early stages of the transformation that all the quenched-in nuclei are consumed before appreciable growth occurs and thermally activated fresh nucleation is limited, the number density of crystalline particles will remain more or less constant and their sizes will remain uniform during the growth process. On the other hand, when fresh nucleation continues along with growth, a distribution of particle sizes will result. As mentioned earlier, the composition of a growing crystal or the average composition of a two-phase nodule, which are the products of polymorphic and eutectic crystallization respectively, remains the same as that of the amorphous matrix during the growth process. Therefore, in these cases, there is no longrange concentration field ahead of the growing crystals. In contrast, if the growing crystalline phase has a composition different from that of the amorphous matrix (as in the case of primary crystallization), a long-range diffusion field is created ahead of the crystal/amorphous interface. It is possible to identify the mode of crystallization from the analysis of the kinetics of the overall process (consisting of nucleation and growth) using the Johnson, Mehl and Avrami (J–M–A) formulation (Burke 1965) which relates the actual fraction, f , of the transformed volume with t, the duration of the transformation: f = 1 − exp−Ktn
(3.39)
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193
where n, known as the Avrami exponent, assumes different values for different mechanisms and geometries of the growing crystals, as discussed later in this section. The temperature dependence of K is given by the Arrhenius equation: Q (3.40) K = Ko exp − RT where Q is the activation energy of the process. Data on the fraction transformed at a given temperature for different durations of the transformation can be analysed to obtain the value of n for an isothermal crystallization process in a glass. It may be noted that it is not possible to determine unequivocally the nucleation and growth behaviour of a crystallization process from data on the time dependence of the transformed volume only, as is often attempted. It is essential to have additional knowledge about the process, best gained from direct microscopic observations on the growing crystals as a function of time. Table 3.6 shows the values of the Avrami exponents, n, for different types of nucleation and growth transformations. DSC is often used for studying the crystallization kinetics mainly because of the precise control of temperature and heating rates and the high sensitivity of recording events of heat release associated with this technique. Both continuous heating and isothermal holding experiments are carried out for gaining information regarding the crystallization kinetics. The methodology of this analysis has been discussed later in this section for illustrating typical cases representing polymorphic, eutectic and primary crystallization. In the context of devitrification of metallic glasses, it is to be noted that during continuous heating experiments, crystallization occurs at temperatures only slightly higher than the glass transition temperature, Tg . Since structural relaxation of the Table 3.6. Values of Avrami exponent, n, for different types of nucleation and growth transformation. Geometry
Nucleation rate
n
Interface controlled Plate Cylinder Sphere Sphere
Rapid, depleting Rapid, depleting Rapid, depleting Constant
1 2 3 4
Long-range diffusion controlled Sphere Sphere Cylinder Plate
Rapid, depleting Constant Rapid, depleting Rapid, depleting
3/2 5/2 1 1/2
194
Phase Transformations: Titanium and Zirconium Alloys
glass occurs during heating it up to Tg , causing substantial reductions in atomic transport rates, the nucleation and growth rates of crystals depend not only on the temperature but also on the thermal history. In continuous heating calorimetric studies, when the temperature is increased linearly with time, the thermal effects of glass transition and devitrification may overlap, making the analysis of the results very difficult. It is important, therefore, to select such systems for studies on devitrification kinetics for which the kinetic crystallization temperature is several degrees celsius above Tg . In isothermal kinetics studies, the occurrence of a suitably long incubation period prior to detectable transformation makes it convenient to record the thermal evolution data on a stable baseline. Isothermal DSC results can be analysed to obtain the fraction transformed, ft, as a function of time. The zero time is defined by the instant when the isothermal holding temperature is reached. The total heat evolution due to crystallization is reflected in the exothermic peak observed during the isothermal holding at a given temperature. The measurements of the area under the heat evolution curve up to different time periods give the ft values which, when presented in a (J–M–A) plot of ln− ln1 − f ) versus ln t, gives a straight line fit, the slope of the straight line giving the J–M–A exponent, n. This method of analysis of results of isothermal kinetics experiments in DSC has been illustrated later with examples of polymorphic crystallization in binary Zr76 Fe24 and primary crystallization in Zr67 Ni33 glasses. Even though the interpretation of isothermal kinetics data is straightforward in principle, a number of problems are encountered in practice. It is often difficult to maintain the baseline flat over the entire length of the transformation time. The variation in the specific heat (Cp ) of the amorphous and crystalline phases contributes to the baseline shift, which can be computed by taking into account the Cp value of the mixture of the amorphous and the crystalline phases prevailing at different stages of the transformation. The temperature range over which isothermal experiments can be conducted is limited: if the temperature is too high, the transformation may start even before the isothermal holding is reached and will certainly overlap instrumental transients; if it is too low, the rate of reaction is so sluggish that the fraction transformed cannot be determined accurately. For these reasons, and also because of its greater speed and convenience, non-isothermal DSC (such as continuous heating experiments) is often practiced while conducting studies on kinetics of crystallization. This technique is also useful in cases where the crystallization process exhibits more than one exothermic peak, suggesting the occurrence of more than one crystallization event, such as primary followed by eutectic crystallization. In a continuous heating experiment, the differential power required for maintaining the temperature of a sample and a reference material is measured as a function of temperature, the heating rate being kept constant.
Solidification, Vitrification and Crystallization
195
Essentially such an experiment yields the rate of enthalpy change of the sample undergoing a first-order phase transition as a function of temperature. An outline of the reaction kinetics of a phase transformation under a constant heating rate condition, as worked out by Kissinger (1957), is presented here. The kinetics of a solid state reaction can be described by the equation df = A1 − f n exp−Q/RT dt
(3.41)
where df /dt is the rate of change in the fraction transformed, n is the order of the reaction and Q is the activation energy. As the temperature is raised at a constant rate, = dT/dt, the transformation rate, df /dt, rises to a maximum value at T = Tp and then falls due to a continuous reduction in the untransformed volume, (1 − f ), which becomes zero at the completion of the process. By differentiation of Eq. 3.41 d df Q df n−1 = − An1 − f exp−Q/RT 2 dt dt RT dt (3.42) d df AtT = Tp =0 dt dt Therefore, the maximum transformation rate and Tp , the temperature at which this rate reaches the peak value, are related by Q = An1 − f n−1 exp−Q/RTp RTp2 This can be reduced to
Tp2
(3.43)
= A exp−Q/RTp
(3.44)
A is a temperature-independent factor, provided the fraction transformed corresponding to the peak transformation rate remains the same at all temperatures. Henderson (1979) has shown that the peak in df /dt occurs at f = 063 for J–M–A kinetics and linear heating. Equation (3.44) suggests that a plot of ln /Tp2 versus 1/Tp (Kissinger plot) can be expected to yield a linear fit, the activation energy of the overall transformation process being given by the negative slope of the plot. Experimental determination of the activation energy of the overall crystallization process can, therefore, be made by recording DSC thermograms at different heating rates ( ), which give Tp values for different . The usefulness of the Kissinger method for studying kinetics of crystallization is illustrated in Section 3.6.1 by considering examples of polymorphic, eutectic and primary + eutectic crystallization modes in Zr–Fe–Ni glasses.
196
Phase Transformations: Titanium and Zirconium Alloys
The activation energy, Q, determined from the Kissinger peak shift method under the continuous heating condition refers to the activation energy of the overall process which includes both nucleation and growth. Ranganathan and Heimendahl (1981) have proposed a methodology for separately determining the activation energies associated with the nucleation and growth processes, using experimental data from isothermal kinetics studies. For considering the kinetics of the nucleation and growth processes separately, let us take the case of a constant nucleation rate, I, which can be expressed as
Q I = Io exp − n RT
(3.45)
where Qn is the activation energy for nucleation, which is a sum of W ∗ , the energy required to form a critical nucleus and Qd , the activation energy of diffusion. Qn can be determined from the slope of a plot of ln I against 1/T . TEM examinations of samples aged for different durations at a few selected temperatures can provide data on the nucleation density as a function of time. The activation energy for nucleation of crystals can be determined using such data analysed on the basis of Eq. 3.45. The measurements of the size of crystals (for polymorphic and primary crystallization) or of nodules (for eutectic crystallization) can provide data which can be used for obtaining the activation energy for growth. The radius, r (or a representative linear dimension), of crystals or nodules growing with time, t, following a linear growth law r = A1 t
(3.46)
is pertinent to polymorphic and eutectic crystallization, while the growth is parabolic r = A2 Dt1/2
(3.47)
for primary crystallization which is usually bulk diffusion controlled. The growth rate, u= dr/dt, can be written as
Qg u = uo exp − RT
(3.48)
where Qg is the activation energy for growth, which can be determined from the slope of the plot of ln u against 1/T .
Solidification, Vitrification and Crystallization
197
An illustrative example of such a kinetic analysis of nucleation and growth processes separately from the experimental data obtained from isothermally treated samples is shown in Figure 3.36. Isothermal holding of metallic glass samples in a DSC close to the crystallization temperature, and monitoring the rate of heat evolution as a function of time, gives data on the fraction transformed as a function of time which can be analysed using the J–M–Avrami formulation (Eq. 3.39) for obtaining the Avrami exponent. The method of this analysis for polymorphic crystallization in Zr-33 at.% Ni glass is shown in Figure 3.37. Let us now consider the relationship between the activation energies for the overall process (Q) and for the nucleation (Qn ) and the growth (Qg ) steps for the following situations: Case (i): Nucleation rate, I = 0 and linear growth, u = constant. This situation arises when a fixed number (N ) of quenched-in nuclei operate and no fresh thermally activated nucleation occurs. A linear growth rate is a characteristic feature of polymorphic and eutectic crystallization in which atom movements are essentially confined to the vicinity of the transformation front. If the growing particles (for polymorphic) and nodules (for eutectic) are assumed to be spherical, the fraction transformed, f , can be expressed as 4 3 3 (3.49) f = 1 − exp − Nu t 3 From Eqs. 3.39, 3.40, 3.48 and 3.49, we get the Avrami exponent, n = 3 and Q = 3Qg . In the case of two dimensionally growing particles, i.e. discs with fixed thickness, the same analysis could be applied with n = 2 and Q = 2Qg . For the case of one-dimensional growth (needles) n = 1 and Q = Qg . Case (ii): Constant nucleation rate, I > 0 and linear growth, u = constant. In this case, the fraction transformed can be written for spherical particles as (3.50) f = 1 − exp − u3 It4 3 Substituting the values of I and u from Eqs. 3.45 and 3.48, we obtain Qn + 3Qg 4 3 t f = 1 − exp − Io uo exp − 3 RT Therefore Q = Qn + 3Qg and the Avrami exponent, n = 4.
(3.51)
198
Phase Transformations: Titanium and Zirconium Alloys 50
5.0
623 K 40
4.0
Max diameter (μm)
623 K
623 K Number of nuclei (μm)–3
Zr76Fe24 Zr76Ni24
Zr76Fe24 Zr76Ni24
30
623 K 20
623 K
10
0
100
200
3.0
593 K 2.0
1.0
623 K
0
580 K
603 K 623 K 613 K 590 K 600 K 610 K
0.0
300
0
100
200
Time (min)
Time (min)
(a)
(b) 1.0
300
Zr76Fe24 Zr76Ni24
0.0
–1.0
Max crystal size (μm)
0.8
605 K 595 K
0.6
In ISS In G
615 K
–2.0
–3.0
ISS
0.4
G ISS
–4.0
0.2
G 0
4
8
12
16
Time1/2 (min1/2)
(c)
20
24
–5.0 135
150
160
170
1/T (10–3 K–1)
(d)
Figure 3.36. Kinetic analysis of nucleation and growth processes from the experimental data obtained from isothermally treated samples at different temperatures: (a) number of nuclei as a function of time, (b) maximum diameter as a function of time, (c) crystal size as a function of time in case of primary crystallization and (d) plots of nucleation rates and growth rates against 1/T for Zr76 Fe24 and Zr76 Ni24 . Activation energies of the nucleation and growth processes are determined from these plots.
Solidification, Vitrification and Crystallization
199
0.0 679 K
1.0
677 K
675 K 673 K
–0.1 672 K
677 K 675 K
679 K
673 K 672 K
–0.2 –0.3
0.9
–0.4
0.8
Log (– log(1 – x ))
–0.5 0.7
x
0.6 0.5 0.4
–0.6 –0.7 –0.8 –0.9 –1.0
0.3
–1.1 –1.2
0.2
–1.3 0.1 0
–1.4 1
2
3
4
5
6
7
8
9
–1.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Time (min) (a)
Log t (b)
Figure 3.37. (a) Fraction transferred as a function of time for polymorphic crystallization in Zr-33 at.% Ni. (b) Plots drawn to estimate the Avrami exponent using the fraction transformed data.
This case also applies to the primary recrystallization process of cold worked metals and alloys. Case (iii): Nucleation rate, I = 0 and parabolic growth, r = A2 Dt1/2 When N number of spherical quenched-in nuclei are operating, the fraction transformed at time, t, is given by 4 3 3/2 15Qd f = 1 − exp − NA2 Do exp − (3.52) 3 RT Following a similar procedure as the one for cases (i) and (ii), we get Q = 3/2Qd and n = 3/2. For a two dimensionally growing particle, n = 2/2 = 1 and Q = Qd Case (iv): Constant nucleation rate I > 0 and parabolic growth, r = A2 Dt1/2 For three-dimensional growth of spherical particles in duration t, 4 8 t f = ∫t=0 I A32 D3/2 t3/2 dt = IA32 D3/2 t5/2 3 15
(3.53)
200
Phase Transformations: Titanium and Zirconium Alloys
3.5.4 Crystallization kinetics in Zr76 Fe1−x Nix 24 glasses Detailed kinetics studies on the crystallization process in binary Zr–Fe, Zr– Ni and ternary Zr–Fe–Ni glasses have revealed the characteristics of different modes of crystallization. A number of investigations (Buschow 1981, Dey and Banerjee 1985a,b, Dey et al. 1986, Ghosh et al. 1991) have established that while the Zr 76 Fe24 glass crystallizes polymorphically to the Zr 3 Fe phase (body centred orthorhombic, Re3 B type structure), the Zr 76 Ni24 glass decomposes into a mixture of the hcp and the Zr 2 Ni (body centred tetragonal) phases on crystallization. As Ni substitutes for Fe in ternary alloys, Zr 76 Fe1−x Nix 24 , (x = 0 4 8 12 16 20 24), the crystallization products are expected to change as the equilibrium Zr 3 Fe and Zr 2 Ni phases have limited solubilities in respect of Ni and Fe, respectively. The partitioning of Ni and Fe atoms among the growing crystalline phases during crystallization is also expected to have a significant influence on the kinetics and mechanisms of the crystallization process. The work of Dey et al. (1998) has demonstrated how ternary additions influence the crystallization behaviour of Zr 76 Fe1−x Nix 24 alloys. In view of the fact that the common modes, namely polymorphic, eutectic, primary followed by eutectic and surface crystallization, are all encountered in this system, the results of this work are summarized here for highlighting the different kinetic features which characterize these modes of crystallization. The number and the nature of the thermal events accompanying crystallisation in these glasses are shown in DSC thermograms (Figure 3.38(a)), all of which correspond to a constant heating rate of 10 K/min. The glass transition temperature, Tg , the crystallization start temperature, Tx , and the peak transformation rate temperature, Tp , are listed along with the sequence and the mode of crystallization in Table 3.6. The progress of crystallization, as detected during isothermal holding in a DSC, in all these alloys is shown in Figure 3.38(b). Some special features of these exotherms need special mention: (a) the presence of a well-defined and sufficiently long incubation period prior to crystallization in glasses of the following compositions: Zr 76 Fe24 , Zr 76 Fe20 Ni4 , Zr 76 Fe16 Ni8 , Zr 76 Fe12 Ni12 and Zr 76 Ni24 ; (b) very short incubation periods for the glasses Zr 76 Fe8 Ni16 and Zr 76 Fe4 Ni20 ; and (c) alloys grouped in (a) show symmetric exotherms while those grouped in (b) exhibit very broad asymmetric exotherms. Phase analysis by XRD and electron microscopy and diffraction on specimens crystallized at temperatures in the range of 573–673 K for different durations have provided information regarding the identity of the crystalline phases formed and the mode of crystallization as well as the nucleation density and growth rate of crystalline particles as functions of temperature and time. These are complementary to kinetics data obtained from the bulk samples. The occurrence of surface crystallization could be detected from metallographic examinations of cross-sections
Rate of heat flow
Solidification, Vitrification and Crystallization
201
1 3 2 4
5
6 7
600
620
640
660
680
700
720
Temperature (K)
Rate of heat flow
(a)
631 K 7
4
631 K
5 6
00
05
10
3
2
620 K 15
632 K
1
633 K 630 K
632 K 20
25
30
35
40
45
50
Time (min)
(b)
Figure 3.38. (a) DSC thermogram showing the number and the nature of the thermal events accompanying crystallization in Zr 76 Fe1−x Nix 24 glasses at compositions (i) Zr 76 Fe24 , (ii) Zr 76 Fe20 Ni4 , (iii) Zr 76 Fe16 Ni8 , (iv) Zr 76 Fe12 Ni12 (v) Zr 76 Ni24 , (vi) Zr 76 Fe8 Ni16 and (vii) Zr 76 Fe4 Ni20 ; (b) DSC thermogram showing the progress of crystallization during isothermal holding Zr 76 Fe1−x Nix 24 glasses.
of partially crystallized specimens and by carrying out XRD of specimens before and after the removal of surface layers of about 10 m thickness. Taking into account the results obtained from microscopy and DSC experiments, the following inferences could be drawn. DSC thermograms for the alloys Zr 76 Fe8 Ni16 and Zr 76 Fe4 Ni20 , which do not show long incubation periods, are associated with the surface crystallization process which precedes bulk crystallization. The product phase resulting from surface crystallization has been identified to be an ordered cubic phase with the L12 structure (lattice parameter: 0.420 nm) and an approximate composition of Zr 3 (Fe,Ni). The exotherms for these two glasses show a limited overlap of two peaks – the first
202
Phase Transformations: Titanium and Zirconium Alloys
one being asymmetric, characteristic of the primary crystallization of Zr3 (Fe,Ni) occurring on the surface and the second symmetric peak corresponding to the predominant eutectic crystallization occurring in the bulk. The J–M–A analysis of the kinetic data which requires a deconvolution of the two peaks yields the Avrami exponents, n1 and n2 , for the two processes. The Zr 76 Fe24 glass crystallizes polymorphically to yield the equilibrium Zr 3 Fe phase. The observed Avrami exponent of 3 corresponds to an interface-controlled growth of a spherical transformed product (Zr 3 Fe crystals in the present case). As mentioned earlier, polymorphic crystallization is a composition invariant process in which the kinetics are controlled by the mechanism of short-range atom transport across the crystal/amorphous matrix interfaces. The activation energy obtained from the J–M–A analysis of the isothermal kinetics corresponds to the overall kinetics including both nucleation and growth; activation energies for these steps can be determined separately by measurement of the number density and the size of largest crystals formed in specimens annealed at different temperatures for different durations. TEM examinations of samples which have undergone different extents of crystallization provide such informations (plotted in Figure 3.36 for both the Zr 76 Fe24 and the Zr 76 Ni24 glasses). The Zr 76 Ni24 glass crystallizes in an eutectic mode to produce a mixture of the
-Zr and the Zr 2 Ni phases. Nodules of this two-phase mixture nucleate and grow to consume more and more of the amorphous matrix. No long-range diffusion field is created around these growing nodules. As a consequence, these nodules grow until they come in contact with adjacent nodules. No amorphous matrix is retained at this stage. The observed Avrami exponent, n = 32, is again consistent with an interface-controlled growth of crystalline aggregates in three dimensions. Though there is a partitioning of solutes at the interface between the amorphous matrix and the two product phases, the absence of a long-range diffusion field ahead of the growing particle makes the process interface controlled. Figure 3.36(a) and (b) shows that the rates of nucleation and growth remain constant (linear growth) with time for both polymorphic (in the case of Zr 76 Fe24 ) and eutectic (in the case of Zr 76 Ni24 ) crystallization. For such cases, the temperature dependence of the growth rate, dr/dt, can be expressed as dr/dt = exp−Qg /RT 1 − exp− G/RT
(3.54)
where r is the radius of the growing crystal at time t, G is the change in the chemical free energy per mole accompanying crystallization, Qg is the activation energy for growth, is the characteristic frequency and is the interface width. Since crystallization experiments are carried out at temperatures where glasses are supercooled to a great extent, G << RT and the growth rate approximates to
Solidification, Vitrification and Crystallization
203
dr = exp−Qg /RT dt
(3.55)
The slopes of plots of lndr/dt versus 1/T (Figure 3.36(d)) yield the activation energy for the growth process. The activation energy values obtained from measurements of isothermal growth rates of crystalline particles are found to be different from those corresponding to the overall kinetics, as determined from the fraction transformed as a function of time (isothermal, J–M–A analysis) and from the Kissinger peak shift analysis of data obtained from constant heating rate experiments. As explained earlier, this difference arises due to the fact that the overall crystallization process includes both the thermally activated formation of fresh nuclei and the growth of the quenched-in nuclei as well as of the continuously forming nuclei. In some cases, crystallization occurs in two distinct steps as reflected in two well-separated peaks in the DSC thermograms recorded in continuous heating experiments (Figure 3.39). Quenching of samples from different temperatures, selected between the first and the second peak temperatures, arrest the crystallization reaction at different stages, and microstructural investigations on these samples reveal that the first peak is associated with a primary crystallization event which is often followed by either a polymorphic or a eutectic crystallization step. The two-stage crystallization occurs only when the growing crystals during the primary event do not inherit the composition of the parent amorphous phase; in such a situation, solute rejection and the creation of a long-range diffusion field ahead of the growing crystal/amorphous interface take place. The dendritic
Heating rate 10 K/min Zr60 Ni40
Heat flow
↓
720
↓
T C1
T C2
730
740
750
Temperature
760
770
↓
710
↓
Figure 3.39. DSC thermogram obtained in a continuous heating experiment showing two-step crystallisation.
204
Phase Transformations: Titanium and Zirconium Alloys
morphology of the primary crystals is one of the manifestations of the solute rejection process. In the case of crystallization of Zr 76 Fe4 Ni20 glass, the primary crystallization product is the Zr 3 (Fe,Ni) phase. The growth of these crystals in the Zr76 Fe4 Ni20 glass matrix will necessitate a rejection of Ni atoms from the growing crystals to the matrix. How the radius, r, of such crystals changes with time, t, can be expressed in the following form: √ (3.56) r = Dt where is a dimensionless constant depending on the crystal shape and D is the interdiffusion constant. The value of D, determined from measurements on the growth rate of Zr3 (Fe,Ni) crystals in a Zr76 Fe4 Ni20 matrix, is 35 × 10−18 m2 s at a temperature of 595 K. This value is close to that reported for diffusion of Ni in Zr-rich Zr–Ni glasses (Sharma et al. 1989), indicating that long-range diffusion of solute elements is the rate-limiting step of diffusional growth of crystals forming during the primary crystallization event. The Avrami exponent determined for the first step (n1 = 198) and for the second step (n2 = 400) is also consistent with the two modes – primary for the first and eutectic for the second crystallization events, the products of eutectic crystallization being Zr2 Ni and -Zr. The crystallization behaviour of the ternary Zr76 (Fe1−x Nix )24 system also brings out the fact that the Zr3 (Fe,Ni) phase can accept Ni atoms to a considerable extent. As a consequence, polymorphic crystallization is possible in the range of amorphous alloys corresponding to x < 12. On the other hand, the Zr2 Ni phase, having a very limited solubility for Fe, rejects Fe atoms into the amorphous matrix during its growth. The mode of crystallization in amorphous Zr76 Fe20 Ni4 depends on the crystallization temperature. At T < 623 K, the Zr3 Fe phase forms by primary crystallization while at higher temperatures, 623 K < T < 673 K, the Zr3 (Fe,Ni) phase forms by polymorphic crystallization. The temperature dependence of the solubility of Ni in the Zr3 (Fe,Ni) phase appears to be responsible for this behaviour. It is to be noted that the activation energy for polymorphic growth in a large number of Zr-based metal–metal glasses is of the same order as that associated with the diffusion of metal atoms in these amorphous alloys (Dey and Banerjee 1999). This is unlike the case of polymorphic crystallization in metal–metalloid glasses where the activation energy for growth is much larger than that for diffusion. This implies that the atom transport mechanism at the crystallization front is quite different for metal diffusion and metalloid diffusion in such amorphous alloys. In the former case, a group of atoms get involved in the elementary activation process, while metalloid diffusion occurs primarily by a single, atom-free volume exchange process.
Solidification, Vitrification and Crystallization
205
An analogy can be drawn with regard to polymorphic crystallization, partitionless solidification and massive transformation. The common features of these transformations are (a) a rapid movement of the transformation front, under the influence of a large driving force created by a high degree of undercooling and (b) the absence of composition difference across the front. Perepezko and Massalski (1972) have suggested that in a massive transformation the breakdown of the parent structure occurs within a diffuse region, a part of which is expected to be amorphous. The thickness of the transformation front, as estimated from the measured growth rate for polymorphic crystallization in a typical Zr-based glass, has been found to be about 2 nm, which compares favourably with that associated with massive transformation in alloys.
3.6
BULK METALLIC GLASSES
The formation of metallic glasses in several Ti- and Zr-based binary and ternary alloys, as discussed in previous sections, requires a minimum critical cooling rate above 105 K/s. In recent years, a number of multicomponent alloys have been identified, which can be amorphized at a cooling rate range as low as 10−1 – 102 K/s. The sample thickness which can be amorphized in these alloys can be increased to several centimetres. These alloys are generally known as bulk amorphous alloys which are characterized by a higher value (in the range of 0.60–0.75) of the reduced glass transition temperature, Tg /Tm . Furthermore, these multicomponent amorphous alloys have a much wider supercooled region before crystallization, i.e. the difference Tx − Tg is as large as 50–100 K. Table 3.7 lists a few Zr-based bulk metallic glasses and the corresponding values of Tg /Tm and Tx − Tg . It has been pointed out by Inoue (1998) that bulk metallic glasses are generally associated with (a) multicomponent (more than three components) alloy compositions, (b) large differences in the atomic sizes of constituent atoms (above 12%) and (c) negative heats of mixing. These empirical criteria for easy glass formation can be rationalized in terms of thermodynamic (relative stabilities of amorphous and crystalline phases) and kinetic considerations. As in the case of binary and ternary amorphous alloy compositions, the increased tendency of glass formation results from a low value of the free energy, GL−X , of the transformation in respect of the liquid to crystalline phase. This corresponds to a low heat of fusion, Hf , and a high value of the entropy of fusion Sf . The configurational entropy of the system being dependent on the number of possible microscopic states, a multiplicity of components leads to an increase in Sf . A high value of the reduced glass transition temperature, on the other hand, relates to a low Hf .
206
Phase Transformations: Titanium and Zirconium Alloys
Table 3.7. Table showing Tg Tx and Tx − Tg for some Zr-based glasses. Tg (K) Tx (K) Tx = Tx − Tg (K)
S. No.
Alloy
Method of preparation
1. (I)
Zr65 Cu75 Ni10 Al75
Mechanical alloying of GFC mixture
650
746
96
MA of elemental powder mixture
638
694
56
MA of GFC mixture
672
794
122
MA of elemental powder mixture
647
711
64
–
760
–
2. (I)
3. (II)
Zr57 Ti5 Cu20 Ni8 Al10
Zr64 Ni36 (60 h) milling time
MA
4. (II)
Zr60 Ni25 Al15 (60 h)
MA
725
797
72
5. (II)
Zr63 Ni10 Cu175 Al175 C2 (30 h) MA
665
747
82
6. (III)
(Zr065 Al007 Ni010 Cu0175 )100−x Wx for 2% W
–
658
720
62
7. (III)
(Zr065 Al007 Ni010 Cu0175 )100−x Wx for 4% W
–
637
706
59
8. (IV)
Zr34 Ni55 Al11
835
853
18
9. (IV)
Zr10 Ni65 Al25
797
826
29
(melt spinning) can
10. (IV) Zr30 Ni55 Al11 Y4
be cast into bulk
11. (IV) Zr24 Ni55 Al11 Y10
rod (1 mm diameter) fully
756
798
42
12. (IV) Zr20 Ni55 Al11 Y14
comprising amorphous phase
736
781
45
13. (IV) Zr17 Ni55 Al11 Y17
717
771
54
14. (IV) Zr12 Ni55 Al11 Y22
696
733
37
15. (V) Zr65 Al10 Ni10 Cu15
656
756
105 K
16. (V) Zr65 Al10 Ni10 Cu15 Be125
689
818
142 K
17. (VI) Zr54 Cu46 (heating rate 5 K/min)
642.6
689.5
The ease of glass formation essentially depends on the avoidance of crystal nucleation and growth during the solidification process. The homogeneous nucleation and growth rates of crystalline phases are discussed in Section 3.5. An examination of Eqs. 3.32 and 3.35 reveals that a significant reduction in the nucleation and growth rates can be achieved by increasing the viscosity, , of the liquid and the heat of fusion, Hf , and by lowering the liquid/crystal interfacial energy, , and the entropy of fusion, Sf .
Solidification, Vitrification and Crystallization
207
Furthermore, the reason for the high GFA of a particular multicomponent alloy can be investigated from the point of view of interatomic distances. The importance of atomic size differences on the GFA is revealed from the observation that the multicomponent bulk metallic glass forming alloys consist of elements with significant differences in atomic sizes (above 12%). A combination of negative heat of mixing and large difference in atomic sizes is expected to promote a high random packing density of the constituent atoms in the supercooled liquid. Corresponding to such a structure, an increased difficulty in atomic rearrangements can cause a significant reduction in the atomic diffusivity. The formation of a higher degree of dense random packed structure has also been confirmed from the XRD profiles obtained on Zr–Al–Ni amorphous alloys. The reported changes in the atomic distances and coordination number corresponding to different atomic pairs occurring during the amorphous to crystalline transformation show that these changes are insignificant for Ni–Zr and Zr–Zr pairs, while the coordination number for Zr–Al pairs undergoes a drastic change on crystallization. This is indicative of the requirement of a large-scale reorganization of Zr–Al pairs during the crystallization event. It is because of this reason that Al addition in Zr–Cu-based glasses has a pronounced influence on Tx (= Tx −Tg ) as shown in Figure 3.40. The reasons for the attainment of the high GFA for some ternary and multicomponent alloys have been summarized schematically in Figure 3.41. The relation between the GFA and Tx can be demonstrated clearly in the case of the ternary Zr–Al–Ni system. Figure 3.42 shows the contours of equal values of Tx in the ternary diagram. The Tx value is maximum (77 K) for the alloy Zr60 Al15 Ni25 ; Tx values of above 50 K are obtained in a wider composition range, as indicated in Figure 3.42. Having examined a large number of alloy compositions, Inoue (1998) has identified the generic alloy composition, Zr65 Al75 TM275 , which is associated with the largest Tx values (about 1.5 times of that for other bulk glass forming systems). Based on this, a number of multicomponent alloys have been formulated which are good candidates for making metallic glasses in the bulk form. The alloy Zr65 Al75 Cu25 Co10 Ni15 is an example of this set of bulk glass forming alloys. DSC experiments on the crystallization of three glass compositions in the family of Zr65 Al75 TM275 have revealed that crystallization occurs in a single exothermic event (Figure 3.43), with H ranging from 3.13 to 3.74 kJ/mol. Prior to the crystallization event, all these alloys exhibit an endothermic reaction occurring over a wide temperature range below Tx . It has also been shown that the amorphous phase in these alloys transforms gradually to the supercooled liquid state over a wide composition range, followed by crystallization at about 750 K. Inoue (1998) has examined a number of five-component alloys Zr65−x Al10 Cu15 Ni10 Mx where M is Ti, Hf, V, Nb, Cr and Mo, for finding out suitable
208
Phase Transformations: Titanium and Zirconium Alloys 800
Zr65AIx Cu35–x
T x,T g/K
750
700
650
Tx
600
ΔT x (= T x – T g)/K
Tg 100
50
0
5
10
15
20
at. % Al
Figure 3.40. Plots showing effect of Al addition in Zr–Cu-based glasses on Tx , Tg and Tx (= Tx − Tg ) (after Inoue, 1998).
compositions having a strong tendency for the formation of the glassy phase in bulk. For the convenience of experimentation, they have scanned arc-melted ingots of a cross-section of approximately 100 mm2 for determining the area fraction of the ingot which remains in the amorphous phase in the as-cast condition, the amorphous area fraction being a good measure of the GFA. The results of these experiments have shown that the glass forming tendency in the arc-melted ingots is not always related to Tx , which reflects the thermal stability of the supercooled liquid. This is evident from the easy GFA in alloys Zr-Al-Cu-Ni-M (M = Ti, Nb or Pd) in which Tx is rather small, ranging between 47 and 65 K. This apparent anomaly can be explained by considering the fact that arc-melted ingots always contain a number of crystalline nuclei in the region which remains in contact with the watercooled copper mould. Nucleation of crystals is, therefore, not a limiting factor for crystal formation under this condition. It is the difficulty of the crystal growth process which is responsible for the vitrification of the molten mass. It is, therefore, inferred that the addition of Ti, Nb or Pd to Zr-Al-Cu-Ni alloys is useful in suppressing the growth of crystallites.
Solidification, Vitrification and Crystallization
209
The constituent elements have large negative heats of mixing and large atomic size ratios (above 10%)
Formation of amorphous phase with a higher degree of dense random packed structure
Increase of solid/liquid interfacial energy
Difficulty of atomic rearrangement of the constituent elements on a longrange scale
Supression of nucleation of crystalline phases
Supression of crystal growth
Figure 3.41. Schematic diagram summarizing the reasons for the attainment of the high glass forming ability for some ternary and multicomponent alloys. Ni
10
90
20
80 10
0 Al 60 (at.% ) 50 40 3 70
70 60 50 30 20
20
80
ΔT x
) 30 .% 0 (at Ni 50 4 60
70
90
ΔT x ( = T x – T g) (K)
Al 10 20 30 40 50 60 70 80 90 Zr Zr (at.%)
Figure 3.42. Zr–Al–Ni ternary diagram in which the contours of equal values of Tx are shown (after Inoue, 1998).
Attempts of substituting Zr by Y to a large extent have been quite successful. Alloys with compositions in the range Zr60−x Yx Al15 Ni25 (x = 15–30) have been found to be easy glass formers. The crystallization process of these amorphous alloys is quite special as they exhibit two glass transitions and two supercooled
210
Phase Transformations: Titanium and Zirconium Alloys 0.67 K/s
·
Rate of heat flow (H)
Zr65Al7.5Ni5Cu22.5
↓
Zr65Al7.5Ni10Cu17.5
Zr65Al7.5Ni15Cu12.5 ↓ Tg
500
↓ Tx
600 700 Temperature (K)
800
Figure 3.43. DSC thermograms showing the crystallization of three glass compositions in the family of Zr65 Al75 TM275 (after Inoue, 1998).
liquid regions before the completion of crystallization. Figure 3.44 shows the DSC plots in which the first stage glass transition Tg1 , the first stage exothermic crystallization Tx1 and the second stage glass transition Tg2 , are indicated. The distinct splitting of the glass transition and crystallization reactions into two stages appears to originate from the tendency of phase separation of Zr-rich and Y-rich regions. The separation of the two amorphous phases is also revealed from the modulated contrast exhibited by the as-quenched Zr-Y-Al-Ni glasses. On annealing at temperatures between Tx1 and Tx2 , a microstructure containing a fine dispersion of Y-rich crystalline particles in the Zr-rich glassy matrix is produced. Homogeneous precipitation of the nanoscale Y-rich crystalline particles has a significant influence on the viscosity and atomic diffusivity of the remaining amorphous phase which becomes enriched in Zr and depleted in Y. The increased viscosity (to an extent of about 30%) enhances the thermal stability of the remaining amorphous phase as is reflected in the wide supercooled region ( Tx ∼ 100 K). The crystallization products resulting from metallic glasses based on Zr are quite similar to those appearing in the melt-spun binary alloys described earlier. The crystalline phases resulting from the amorphous Zr–Al–Cu alloys are Zr2 Al (orthorhombic), ZrAl (bct), Zr2 Al (B82 ) and Zr2 (Al,Cu). However, the crystallization kinetics for alloys with large Tx have been found to be different from those corresponding to binary amorphous alloys with low Tx . This point can be illustrated using the example of the crystallization behaviour of the Zr65 Al75 Cu275
Solidification, Vitrification and Crystallization
211
0.67 K/s Tg1 ↓562
Zr45Y15Al15Ni25
Rate of heat flow (H ) →
↑562 Tx1
Tg1 ↓562
Zr33Y21Al15Ni25
↑813 Tx2 Tg2 ↓763
⋅
↑562 Tx1 Tg1 ↓562
Zr33Y27Al15Ni25
↑821 Tx2 Tg2 ↓772
↑562 Tx1
Zr35Y30Al15Ni25
Tg1 ↓562
500
600
↑812 Tx2 Tg2 ↓775
↑562 Tx1 400
Tg2 ↓775
↑805 Tx2 700
800
900
Temperature (K)
Figure 3.44. DSC thermograms of Zr60−x Yx Al15 Ni25 (x = 15–30) amorphous alloys showing the first stage glass transition, Tg1 , the first stage exothermic crystallization, Tx1 , and the second stage glass transition, Tg2 (after Inoue, 1998).
alloy which has a Tx of 90 K. J–M–A analysis of the crystallization kinetics of Zr65 Al75 Cu275 reveals that the Avrami exponent (n value) is not constant over the entire temperature range covering the amorphous solid and the supercooled liquid. The value of n increases from 3 to 3.7 with increasing annealing temperature, Ta (Figure 3.45). Inspite of the fact that the crystalline phases forming due to annealing at temperatures ranging from 663 K (
212
Phase Transformations: Titanium and Zirconium Alloys 4.0
Avarami exponent (n )
Zr65Al7.5Cu27.5
3.5
↓
3.0
Tg
2.5
660
680
700
720
740
Ta(K)
Figure 3.45. Avrami exponent (n) as a function of annealing temperature (Ta ) obtained during the crystallization of Zr65 Al75 Cu275 alloy (after Inoue, 1998).
the activation energy for nucleation has been estimated to be 400 kJ/mol in the former and 230 kJ/mol in the latter matrix. The influence of the matrix is also noticed on the morphology of the crystalline products. While Zr2 Cu crystals formed in the amorphous solid have been found to be nearly spherical with a smooth interface, a dendritic morphology is associated with the crystals forming from the supercooled liquid (Inoue 1988). The existence of the supercooled liquid over a wide temperature range ( Tx ∼ 90 K) is attributed to the retardation of the growth of the Zr2 (Cu,Al) phase, which requires redistribution of Al around Zr in a highly dense random packed structure. Such a dense random packing arises due to significantly different atomic sizes and strong attractive bonding between the three constituent elements. The high degree of dense random packing is also responsible for a high value of the liquid– solid interface energy, which causes a retardation in the precipitation of crystalline phases.
3.7
SOLID STATE AMORPHIZATION
The transition from crystal to glass can be compared with the melting process since the glass can be considered thermodynamically as a frozen, highly undercooled liquid. In spite of a longstanding interest in the phenomenon of melting, a generally
Solidification, Vitrification and Crystallization
213
accepted theory of the transition between a crystalline and a liquid phase has not been developed yet even for a single component system. All melting theories are based on some form of instability or catastrophe which are either vibrational, elastic, isochoric, defective or entropic in nature. These theories of melting envisage a homogeneous melting process though it is well established that melting is a firstorder transition which occurs by the creation of a well-defined interface between the liquid and the crystalline phases. Heterogeneous nucleation on external and internal surfaces plays an important role in the melting process. A considerable degree of superheating of crystals is possible if effective heterogeneous nucleation sites are eliminated. While considering melting and freezing of a pure metal, Fecht et al. (1989) have indicated the limits of supercooling of liquids and of superheating of crystals. In analogy to the Kauzman temperature, Tgo , where the excess entropy of the undercooled liquid vanishes, one can identify a temperature where the entropy of the superheated crystal would become equal to that of the liquid. It has been shown that this isentropic limit of stability of the solid phase for pure Al is located at 1.38 Tm . The isoentropic limits of stability of superheated crystals and of supercooled liquids for pure Al are shown in Figure 3.46 (a) and (b). It may be noted here that before reaching the isentropic limits during supercooling or superheating, a succession of catastrophe barriers are encountered. Those include a thermodynamic barrier (equilibrium melting and crystallization) and a kinetic barrier (glass formation and shear instability for melting). Experimental data on specific heat and thermal expansion of a system under supercooled and superheated conditions are required for the estimation of these barriers. In the absence of such data, a realistic estimation of these limits is difficult to make. However, the general
Tg TI Tm
Free energy change
Atomic specific heat
20 15
10
Undercooled liquid
5
Crystal
TI Tm
Liquid
0
Temperature (K)
Temperature (K)
(a)
(b)
Figure 3.46. Isoentropic limits of stability of (a) supercooled liquids and (b) superheated crystals for pure Al.
214
Phase Transformations: Titanium and Zirconium Alloys
observation that the superheating of crystals above their melting points is much more restricted than supercooling of liquid is due to the pronounced asymmetry between the two processes as indicated in Figure 3.46(b). Let us now consider the case of the melting of a binary alloy in which there exists a possibility of redistribution of solutes between the crystalline and the liquid phases. In this case, the kinetics of melting for crystalline solid solutions depend on the imposed heating rate. While equilibrium melting occurs under low heating rates, superheating above the equilibrium solidus temperature can be achieved at high heating rates. In case the extent of superheating is high enough to exceed the To temperature (where the integral molar free energies of the crystalline and the liquid phases are equal), the crystalline phase can undergo a partitionless melting process. The crystal to amorphous transition, which is necessarily a non-equilibrium process, can be induced under certain constraints which prevent the formation of equilibrium phases. This point can be explained with the help of a polymorphous phase diagram which becomes relevant under the circumstance where the partitioning of the solute is not possible. Figure 3.47 shows the equilibrium phase diagram (drawn by broken lines) superimposed on a polymorphous phase diagram which indicates the stability domains of equilibrium and of metastable above the To line. One must use this diagram in conjunction with a time scale. This time scale should be long enough to permit ergodic sampling of a set of microstates
Metastable α above T o s
Equilibrium α
Metastable α below T o
Liquid To
o
Tg
T* C* Concentration
Glass ↓
Temperature
Ti
Figure 3.47. Equilibrium phase diagram (drawn by dashed lines) superimposed on a polymorphous phase diagram which indicates the stability domains of equilibrium and of metastable above the To line.
Solidification, Vitrification and Crystallization
215
belonging to the metastable phases but short enough so that partitioning of the components in different phases is prevented. The situations under which solid state amorphization is possible can be grouped into the following two broad categories: (1) composition induced destabilization and vitrification of crystalline alloys under isothermal conditions, and (2) amorphization under externally driven conditions. While glass formation in diffusion couples and hydrogen-induced vitrification belong to the former category, radiation-induced and mechanically driven amorphization represent the latter. 3.7.1 Thermodynamics and kinetics The stability/instability criteria of a solid with respect to melting include the effects of both shear strain and hydrostatic pressure. The conditions of stability of a solid require the elastic moduli (both shear and bulk moduli) to be positive, =>0 ! TCi
(3.57)
1 P =B>0 V V TCi
(3.58)
where , ! and are shear stress, strain and modulus, respectively, and B is the bulk modulus. The elastic constants of a superheated crystal, (= C11 − C12 /2, isothermal shear modulus), and B (= C11 + 2C12 /3, isothermal bulk modulus), vanish at the critical temperatures, T and TB , respectively, which are above the melting temperature, Tm . Experimental data on elastic moduli as function of temperature (Tallon and Wolfenden 1979), when extrapolated to temperatures above Tm , show that the instability sets in at T ∼ 16Tm (Figure 3.48). The increase in thermal disorder as the temperature is raised leads to an ultimate shear instability of a metastable superheated crystalline solid. Other types of disorder, namely defects and chemical disorder, can also contribute in bringing about the shear instability. Thorpe (1983) has theoretically studied the mechanical stability of a model crystal, consisting of balls of equal mass M connected with nearest neighbour balls by springs with force constant K. Beginning with the perfect crystal, the springs are gradually removed from random locations. Using computer simulation, the shear and bulk moduli are calculated as functions of the fraction of springs removed. The results of this theoretical work have shown that both and B vanish where a critical fraction of springs are removed (Figure 3.49).
216
Phase Transformations: Titanium and Zirconium Alloys
8
Tm
Elastic moduli 1010 Pa
B 6
4
μ
2
200
Instability
600
1000
1400
1800
Temperature
Elastic moduli (arb. unit)
Figure 3.48. Variation of elastic moduli as a function of temperature.
B
μ
Fraction of removed springs
Figure 3.49. Shear and bulk modulus as function of fraction of springs removed.
Egami and Waseda (1984) have considered a binary solid solution containing atoms of two different sizes. By carrying out a simple analysis of local strain effects using an elastic continuum approach, they have shown that such a solution becomes topologically unstable when the concentration of the smaller atoms (A atoms)
Solidification, Vitrification and Crystallization
217
reaches a critical concentration, cA∗ , which depends on the ratio of atomic sizes r = RA /RB , where RA and RB are the atomic radii of the two types of atoms. Considering the instability arising out of a critical level of strain disorder, the concentration level at which the solid solution becomes topologically unstable has been found to be cA∗ = 2R3B /R3B − R3A + higher order terms
(3.59)
Let us now examine the free energy – concentration diagram of a system at a temperature below Tg (for the alloy compositions under consideration). Figure 3.50 shows such a diagram which depicts the condition of metastable equilibrium between the crystalline -phase and the amorphous phase (a). The presence of the equilibrium intermetallic phase is denoted here with a dotted line. If the formation of the intermetallic phase is kinetically prevented, nucleation of the amorphous phase becomes thermodynamically possible only when the concentration of B atoms in the -phase, cB exceeds the limit cB1 . The maximum free energy change associated with the nucleation of the amorphous phase from an -phase having a composition given by cB is shown by the vertical line Gm . Such a nucleation process, which is facilitated at heterogeneities like high-energy grain boundaries, involves the partitioning of B atoms preferentially towards the nucleating amorphous phase. If, however, the -phase is enriched to a composition where cB > co , a massive (i.e. partitionless) → a transformation becomes possible.
α
Free energy
a
ΔG m
α
c1B
cB co c B2
A
CB
B
Figure 3.50. Free energy–composition diagram showing metastable equilibrium between the crystalline -phase and the amorphous phase.
218
Phase Transformations: Titanium and Zirconium Alloys
As discussed earlier in Section 3.4.1, the phase diagrams of typical glass forming alloys are characterized by steeply plunging To -lines, as seen for Zr–Ni and Zr–Cu alloys. For such alloys, a generic non-equilibrium phase diagram can be developed, neglecting the kinetically excluded intermetallic compounds, for illustrating the possible thermodynamic states of a metastable system constrained to be a single phase. For alloys with large negative slopes for the To lines, the To line must cross the ideal glass transition line Tg∗ at a certain composition, c∗ . Under this condition, a triple point (c∗ , T ∗ ) is defined in respect of the supersaturated crystal, undercooled liquid and ideal glass (Figure 3.51). The entropic instability line, Tis , against melting (shown by broken line) should pass through the triple point since the entropy difference between the crystal and the liquid vanishes at this point. This polymorphous phase diagram (Figure 3.51) essentially depicts that the following two conditions are satisfied: G = H − T S = 0 (condition for polymorphic melting) and S = 0 (Kauzman condition) at the triple point. This also shows that the composition-induced disorder reduces the polymorphic melting temperature of the crystalline solid solution to the ideal glass transition temperature. The slope of the To line, dTo /dc = − G/c/ S at the triple point approaches infinity. Fecht et al. (1989) have predicted the triple point for -Zr supersaturated with Ni to be 638 K and 11.5 at.% Ni. The To line extends below the triple point as a
Temperature
T
s i
To
Liquid
T g* Crystal
T* C*
Glass
Composition
Figure 3.51. Schematic diagram indicating a triple point (c∗ , T ∗ ) as defined in respect of the supersaturated crystal, undercooled liquid and ideal glass.
Solidification, Vitrification and Crystallization
219
straight line with infinite slope as long as non-ergodicity prevails and the Kauzman argument holds. Below the triple point, the transition between the crystal and the glass is isentropic and, therefore, truly continuous in volume as long as the metastable constraints are maintained. The presence of non-equilibrium lattice defects such as vacancies and anti-site defects play a major role in providing the constraint under which the melting temperature of the crystal can be considerably reduced. These defects, which have very low mobility at relatively low processing temperatures, remain frozen in the lattice. If one considers the vacancy as a second component in the system, one can draw a phase diagram like the one shown in Figure 3.52. The free energy difference, G = H − T S, between a liquid and a single crystal for pure metals can be realistically estimated to be (Fecht et al. 1989)
G = 7 Sf TTTm + 6T
(3.60)
and for glass forming alloys by
G = 2 Sf TTTm + T
(3.61)
where T is the undercooling below the melting point, Tm , and Sf the entropy of fusion. The increase in free energy of the crystalline phase can be expressed as
Gv = cv H v − T S v + kB T cv ln cv + 1 − cv ln1 − cv
T/ Tm
1.0
(3.62)
ΔG = 0 Liquid
0.8
Crystal 0.6
T∗
0.4
ΔS = 0
Glass
0.2
c °v
ΔH = 0 0
0.02 0.02
0.04
0.06
0.08
0.10
Vacancy concentration (c v)
Figure 3.52. Phase diagram illustrating the role of non-equilibrium lattice defects such as vacancies and anti-site defects in providing the constraint under which the melting temperature of the crystal can be considerably reduced.
220
Phase Transformations: Titanium and Zirconium Alloys
Combining these equations, the decrease in melting temperature of the defective crystal can be expressed as a function of defect concentration. 3.7.2 Amorphous phase formation by composition-induced destabilization of crystalline phases There are a number of experimental results to demonstrate that a crystalline material can be transformed into an amorphous one by progressively introducing alloying elements. It has often been noticed that a crystalline phase is destabilized when loaded with some specific alloying elements to a level exceeding a certain threshold. Introduction of alloying elements can be effected by several means, for example, (a) by isothermal annealing of diffusion couples, (b) by mechanical alloying, (c) by introducing hydrogen by diffusion and (d) by ion implantation. All these treatments, which are essentially isothermal but are implemented under chemically non-equilibrium conditions, can lead to the formation of amorphous phases. It is the excess chemical energy associated with the initial configuration which permits the glassy state to be adopted and retained as a metastable product. Some experimental results pertaining to the aforementioned treatments will now be described, and glass formation will be rationalized in terms of thermodynamics and kinetics of the pertinent process. 3.7.3 Glass formation in diffusion couples The early observations on glass formation in diffusion couples were reported in initially crystalline multilayers of Au–La (Schwartz and Johnson 1983) and in samples of Si coated with a thin film of Rh (Herd et al. 1983). The formation of an amorphous layer in the reaction zone of binary diffusion couples has been observed in a number of systems in which one of the components is Zr or Ti. These are the well-known glass forming systems such as Zr–Cu, Zr–Ni, Zr–Co, Zr–Fe and Ti–Ni. A variety of techniques have been employed for detecting the amorphous phase. Serial sectioning of diffusion couples near the reaction zone provides samples for a plan view examination by XRD and TEM, while cross-sectional TEM reveals the presence and distribution of different layers forming at the reaction zone. The composition profile is determined by using electron proble microanalysis and Rutherford backscattering. The presence of the amorphous phase can also be detected by the observation of a crystallization event during heating in a DSC. Let us examine some of the reported experimental results in order to understand the thermodynamics and kinetics of the formation of amorphous layers in the reaction zones of diffusion couples. Cross-sectional TEM studies on Zr–Ni diffusion couples by Newcomb and Tu (1986) have shown the presence of a well-defined planar interlayer of an amorphous phase in diffusion couples reacted at 573 K for durations of 1.5 and 4 h. The formation of the ordered intermetallic ZrNi phase has
Solidification, Vitrification and Crystallization
221
Kirkendal voids
Amorphous interlayer
NiZr
20000
Zr
–10000 –20000
= Equilibrium compounds
550 K
10000
ΔG (kJ/mol)
Ni
HEX FCC
BCC
–30000 –40000
Amorphous –50000 –60000
Ni
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
Zr
cZr
(a)
(b)
Figure 3.53. (a) Schematic diagram showing the distribution of different phases and of Kirkendal pores in a Zr–Ni diffusion couple reacted at 573 K for 12 h, (b) free energy concentration diagram in Zr–Ni system.
been observed when the reaction time is extended to 12 h. A schematic illustration (Figure 3.53(a)) shows the distribution of different phases and of Kirkendal pores in a Zr–Ni diffusion couple reacted at 573 K for 12 h (Newcomb and Tu 1986). Free energy–concentration plots for the different competing phases in this system (Figure 3.53(b)) can be used for illustrating two possible mechanisms for the formation of the amorphous layer. The normal downhill interdiffusion process is expected to produce a hcp solid solution of Ni in Zr and an fcc solid solution of Zr in Ni. The amorphous phase can either nucleate heterogeneously at the interface, the local metastable equilibrium being dictated by tangent construction on the free energy curve for the terminal solid solution or by solute enrichment of the solid solution to an extent where it reaches beyond the co limit (when the system is forced below the To temperature corresponding to the composition, co ) leading to polymorphic vitrification. In order to find out whether a solid solution phase in the Ni–Zr system can become unstable with respect to the Gibbs criterion, it is necessary to determine the minimum level of solute content which allows a massive or polymorphic vitrification. For the solution of Ni in hcp Zr, the To cN i ) line reaches 0 K at cNi = 022. The criterion based on the local strain, as proposed by Egami (1983), gives the composition limits for instability as cNi = 026 and cZr = 013 for the hcp and fcc solid solutions, respectively. From these estimations of instability limits, it appears possible that the destabilization of a solid solution phase can indeed occur, leading to a polymorphic vitrification process. It should, however,
222
Phase Transformations: Titanium and Zirconium Alloys
be noted that the nucleation of an amorphous phase by heterogeneous nucleation becomes thermodynamically feasible at much lower levels of the solute content in terminal solid solutions. The importance of the heterogeneous nucleation of an amorphous phase in diffusion couples can also be realized from the observation that an amorphous phase does not form in a couple made by depositing Ni on a single crystal of Zr. The question of polymorphic vitrification vis-a-vis nucleation of an amorphous phase of a composition given by the local metastable equilibrium condition has been addressed by Bhanumurthy et al. (1988, 1989) while analysing the results pertaining to Zr–Cu diffusion couples. These experiments have shown that an interface reaction between Zr and Cu at a temperature of 873 K results in the formation of an ordered intermetallic phase, Zr2 Cu, whereas at a temperature close to 600 K, an amorphous layer forms in the reaction zone. At such low temperatures, the formation of the ordered intermetallic phase is fully suppressed. An examination of hypothetical free energy – concentration plots for the , the and the liquid phases (Figure 3.54) reveals that the metastable solubility of Cu in the -Zr lattice is considerably extended when intermetallic phase formation is suppressed. The equilibrium conditions between the different competing phases are shown by the common tangents AB (between hcp and Zr2 Cu), CD (between
and the amorphous a-phase), EF (between and the bcc -phase) and finally JK (between and L). The points of intersection in the free energy – concentration
T~600 K
H Ga
Free energy (G)
Gα
G Zr2Cu
IJ F
A C E
Gβ M
N T
AB: G α, G Zr2Cu
K
CD: G α, G a
D P
α
ET: G at E EF: G α, G β
B
JK: G β, G a 0.1
0.2
0.3
0.4
0.5
Atomic fraction (Cu)
Figure 3.54. Free energy – concentration plots for , , Zr2 Cu and the liquid phases.
Solidification, Vitrification and Crystallization
223
curves for different phases mark the limits at which polymorphic transformations between them become thermodynamically possible. For the -phase to become amenable to polymorphic vitrification, the Cu enrichment of this -phase should go beyond the point H. Before attaining such a high Cu concentration, the supersaturated -phase becomes amenable to a composition invariant massive to transition, which has not been observed in the reaction zone of Zr–Cu diffusion couples. Microanalysis of the different phases present in the reaction zone has shown that the maximum concentration of Cu in the -phase, which lies in contact with the amorphous layer, is about 4 at.%. Based on these observations, it has been inferred that the formation of the amorphous phase in the reaction zone of Zr–Cu diffusion couples occurs by its nucleation from the -phase, enriched to a Cu content of about 4 at.% and its subsequent growth by the coalescence of independently nucleated amorphous regions. Once the -phase is enriched in Cu to a level shown by the point E, the maximum free energy change for the nucleation of the amorphous phase is given by the drop NP where P is given by a tangent drawn on the Ga plot which is parallel to ET. A continuous amorphous layer eventually develops which grows in thickness by consuming the adjacent -phase layer till the supply of Cu atoms gets restricted by the formation of Kirkendal pores on the Cu-rich side. The Cu level required for the destabilization of the -Zr lattice, on the basis of the Egami criterion, has been estimated to be about 28 at.%. This lends further support to the contention that the amorphous layer forms in the Zr–Cu system not by the destabilization of the -lattice but by the nucleation of a Cu-rich amorphous phase which can establish a metastable equilibrium with an
-Zr–Cu alloy containing about 4 at.% Cu. The kinetics of the one-dimensional growth of the amorphous interlayer in the reaction zone of a binary diffusion couple can be described by a set of coupled differential equations (Johnson 1986): c 2 c ˜ =D t x2 ˜ D ˜ −D
(3.63)
dx c x1 = 1 − c1 1 x dt
(3.64)
dx c x2 = c2 2 x dt
(3.65)
dx2 = K2 c2 − c2o dt
(3.66)
dx1 = K1 c1 − c1o dt
(3.67)
224
Phase Transformations: Titanium and Zirconium Alloys dX2 dt
dX1 dt 1
GLASS
2
X
1.0
c10 c 20
c1 0 K1 ~ D
μ1
K2
μ2
X1
X2
Figure 3.55. Schematic diagram showing the concentration profile of metal 1 and chemical potential profile of metal 1 and metal 2.
˜ is the interdiffusion constant in the amorphous phase, c(x) is the concenwhere D tration profile of metal 1 in the amorphous phase, c1o and c2o are the concentrations of metals 1 and 2, respectively, in the amorphous phase which are in equilibrium with pure metal 1 and pure metal 2 (as shown in Figure 3.55); x1 and x2 give the positions of the interfaces separating the amorphous layer and the metals 1 and 2, K1 and K2 are the kinetic response parameters at these interfaces and c1 and c2 are abbreviated forms of cx1 and cx2 . ¯ Introducing some simplifying assumptions such as the interdiffusion constant D being independent of composition and the response parameters K1 and K2 being linear and being related as K1 /K2 = c1o /1 − c2o
(3.68)
The equations have been solved numerically and for long times (t → ), ˜ 2 + 2aDt ˜ x2 = −D/K
(3.69)
Solidification, Vitrification and Crystallization
225
where a is a constant of order unity. For short times (t → 0) the solution is in the following form: x2 = constantK2 t + negligible higher order terms
(3.70)
These results predict a linear growth law at short times (for a thin amorphous interlayer) and a shifted t1/2 law in the limit of long times. This means that when ¯ the amorphous layer is thinner than the characteristic length, l= D/K, the growth is interface controlled while a diffusion controlled growth mechanism operates when the amorphous interlayer is much thicker than l. Analysis of experimental data on the growth of amorphous interlayers in diffu˜ the interdiffusion constant, sion couples of Ni–Zr, Co–Zr and Ni–Hf has yielded D, ˜ values as a function of temperature at T < Tg . Johnson (1986) has shown that D match very closely with the diffusion constant for impurity diffusion of Ni in the amorphous Ni67 Zr33 alloy. This observation points to the fact that the interdiffusion process is strongly dominated by the migration of the smaller atoms of the late transition metals (Ni in the case of Zr–Ni) and practically no migration of Zr atoms. The formation of Kirkendal voids along the interface separating Ni from the amorphous layer is a direct evidence that Ni is the moving species. The void formation is responsible for reducing interfacial contact and ultimately for cutting off the supply of Ni atoms. At this stage, the growth of the amorphous layer terminates. Experiments have shown that the growth of amorphous interlayers can lead to a layer thickness of 100–200 nm without any accompanying formation of crystalline intermetallic compounds. Since this thickness is much larger than the size of critical crystalline nuclei, the avoidance of crystalline nucleus formation during interdiffusion annealing for a time scale of about 104 s appears improbable. The fact that only the late transition metal atoms are mobile with virtually no migration of Zr atoms can perhaps explain why nucleation of ordered intermetallic compounds does not occur during the growth of the amorphous interlayer.
3.7.4 Amorphization by hydrogen charging Yeh et al. (1983) have reported that a bulk polycrystalline partially ordered fcc solid solution of Zr075 Rh025 composition transforms to an amorphous phase when hydrided by exposure to hydrogen gas at temperatures between 425 and 500 K. This reaction can be expressed as Zr 075 Rh025 cryst + H2 gasZr 075 Rh025 H114 amorphousT < 500 K
(3.71)
226
Phase Transformations: Titanium and Zirconium Alloys
At higher temperatures (T > 500 K) the equilibrium product forms as per the following equation: Zr 075 Rh025 cryst + H2 gasZrH2 cryst + RhcrystT > 500 K
(3.72)
TEM observations have revealed that at T < 500 K, amorphization proceeds by the nucleation of glassy zones along the grain boundaries of the crystalline starting material, followed by the expansion of these amorphous regions into the grain interiors. The boundary between the crystalline and the amorphous regions is sharp, suggesting a strong first-order transition. The entire process proceeds much as melting would proceed in a polycrystalline sample. X-ray studies have indicated that the fcc Zr075 Rh025 solid solution dissolves some hydrogen prior to transforming to the glassy phase. This suggests that with hydrogen entry, the free energy of the alloy is raised above that of the glassy phase. As a consequence, the superheated crystalline phase transforms into an amorphous phase. The temperature threshold, 500 K, below which amorphization occurs, is dictated by the relative values of the diffusion constants of H and the metal atoms. Above 500 K the equilibrium product consists of two crystalline phases, namely fcc Rh and fcc ZrH2 . For the formation of such a product, metal atom redistribution must occur by thermally activated diffusion over a length scale at least of the order of the respective critical nuclei sizes of the two crystalline phases. This is apparently not possible below 500 K. The temperature dependence of the chemical rate constants of Eqs. 3.71 and 3.72 is expected to be considerably different as these processes are controlled by hydrogen and metal atom diffusion, respectively. It is this difference in temperature dependence that enforces a kinetic constraint on the separation of the two crystalline phases and instead allows the hydrogen-charged Zr75 Rh25 alloy to undergo amorphization in a manner similar to melting. 3.7.5 Glass formation in mechanically driven systems High energy ball milling can lead to glass formation from elemental powder mixtures as well as by amorphization of intermetallic compound powders. Solid state amorphization by high energy milling has been demonstrated in a number of Ti- and Zr-based and other alloy systems such as Ni–Ti, Cu–Ti, Al–Ge–Nb, Sn– Nb, Ni–Zr, Cu–Zr, Co–Zr and Fe–Zr. The process of ball milling is illustrated in Figure 3.56. Powder particles are severely deformed, fractured and mutually cold welded during collisions of the balls. The repeated fracturing and cold welding of powder particles result in the formation of a layered structure in which the layer thickness keeps decreasing with milling time. A part of the mechanical energy accumulates within these powder particles in the form of excess lattice defects which facilitate interdiffusion between the layers. The continuous reduction in the diffusion distance and the enhancement in the diffusivity with increasing milling
Solidification, Vitrification and Crystallization
227
A Powder particle B
Hard ball
Figure 3.56. Schematic diagram illustrating the process of ball milling.
time tend to bring about chemical homogeneity of the powder particles by enriching each layer with the other species being milled together. The sequence of the events that occur during milling can be followed by taking out samples from the ball mill at several intervals and by analysing these powder samples in respect of their chemical composition and structure. Let us describe one such experiment in which elemental powders of Zr and Al were milled in an attritor under an Ar atmosphere. Elemental powders of Zr and Al of 99.5 purity, when milled in an attritor using 5 mm diameter balls of zirconia as the milling media and keeping the ball to powder weight ratio at 10:1, showed a progressive structural change as revealed in XRD patterns (Figure 3.57(a) and (b)). Diffraction peaks associated with the individual elemental species remained distinct upto 5 h of milling at a constant milling speed of 550 rpm. All particles and the balls appeared very shiny in the initial stages. With increasing milling time, the particles lost their lustre, the 111 and 200 peaks of fcc Al gradually shrunk and the three adjacent low-angle ¯ 0002 and 1011, ¯ became broader. After peaks of hcp -Zr, corresponding to 1010, about 15 h of milling, XRD showed only -Zr peaks which shifted towards the high angle side, implying a decrease in the lattice parameters resulting from the enrichment of the -Zr phase with Al. After 20 h of milling, all Bragg peaks except one broad peak close to the {1010} peak disappeared. Powders milled for 25 h showed an extra reflection corresponding to a lattice spacing of 5.4 nm, which matches closely to a superlattice reflection of a metastable D019 (Zr3 Al) phase. On further milling, the powders transformed into an amorphous phase. The sequence of structural evolution could be described as -Zr + Al −→ -Zr (Al) solid solution + Al −→ nanocrystalline solid solution + localized amorphous phase −→ Zr3 Al (D019 ) + -Zr (Al) solid solution + amorphous phase −→ bulk amorphous phase.
Phase Transformations: Titanium and Zirconium Alloys
Zr (002)
Zr (101) a–5 h b – 10 h c – 15 h d – 20 h
a – 25 h b – 30 h c – 45 h
Intensity (A.U.)
Zr (100) Al (111) Zr (102) Al (200)
a
Intensity (A.U.)
228
b c d 22
26
30
34
38
42
46
50
16
20
24
28
32
2–θ
2–θ
(a)
(b)
36
40
44
48
Figure 3.57. XRD patterns showing a progressive structural change for different times when elemental powders of Zr and Al of 99.5 purity were milled in an attritor using 5 mm diameter balls of zirconia with a ball to powder weight ratio of 10:1.
The mechanism of solid state amorphization during mechanical alloying has been studied on the basis of experimental observations made on several alloy systems. One of the probable mechanisms, based on local melting followed by rapid solidification, has not found acceptance as evidence of melting could not be seen in experiments. The example of ball milling of elemental Zr and Al powders has demonstrated that the amorphisation process is preceded by the enrichment of the -Zr phase to a level of approximately 15 at.% Al. The solute concentration progressively changes during milling. The various stages encountered in the course of amorphization can be explained in terms of schematic free energy versus concentration plots for the , the metastable D019 , and the amorphous phases (Figure 3.58). With increasing degrees of Al enrichment, the free energy of the interface region gradually moves along the path 1-2 (Figure 3.58). Once the concentration crosses the point 2, it becomes thermodynamically feasible to nucleate the Zr3 Al phase which has the metastable D019 structure. Although the equilibrium Zr3 Al phase has the L12 structure, it has been shown (Mukhopadhyay et al. 1979) that the metastable D019 structure is kinetically favoured during the early stages of precipitation from the -phase. This is not unexpected as the hcp
Solidification, Vitrification and Crystallization
229
Zr3 Al (DO19)
α
1
Amorphous
Free energy
2 3
4
3′
2′
Zr
25 Aluminium content (at.%) →
Figure 3.58. Schematic free energy – concentration plots in Zr–Al system for the , the metastable D019 and the amorphous phases illustrating the various stages encountered in the course of amorphization.
structure and the D019 structure (which is an ordered derivative of the former) follow a one-to-one lattice correspondence and exhibit perfect lattice registry. With further Al enrichment, as the concentration crosses the point 3, nucleation of the amorphous phase becomes possible. It is to be emphasized that the change in composition occurs gradually from the interface to the core of the particles, with the result that the amorphous phase starts appearing at interfaces while the core remains crystalline. As the Al concentration in the powder particles crosses point 4, each particle can turn amorphous by a polymorphic process. The observed sequence of solid state amorphization in the case of ball milling of elemental Zr and Al powders suggests the occurrence of amorphization by a lattice instability mechanism which is brought about by solute enrichment of the -phase beyond a certain limit (point 4 in Figure 3.58). 3.7.6 Radiation-induced amorphization It was discovered in the early 1980s that some intermetallic compounds undergo a crystalline to amorphous (C −→ A) transition under irradiation by energetic particles. It was also recognized that the C −→ A transition results from displacement damage and not from ionization damage. Displacement damage occurs due to the momentum transfer from the projectile particles (incident electron, ion or neutron) to atoms occupying lattice sites in the target material. For example, if one considers an elastic collision between an incident particle of mass m and an atom of atomic weight M, the struck atom can receive the maximum energy,
230
Phase Transformations: Titanium and Zirconium Alloys
Emax = 4Mm/En M + m2 , where En is the energy of the incident particle. While an 1 MeV neutron can transfer several hundred keV energy to a target atom, a lighter particle such as an electron of the same energy (1 MeV) is capable of transferring only some tens of eV energy to a target atom. The threshold energy, Ed , for displacing an atom from its lattice site is in the range of 20–80 eV ( 25 eV for Cu) and so an electron of 1 MeV energy can displace only one or two atoms from lattice sites. In contrast, a 1 MeV neutron can impart an energy of as much as 200 000 eV to a Cu atom. Such an atom, called a primary “knock-on”, causes further damage by displacing secondary, tertiary, etc. “knock on”s. The damage structure produced by a single energetic electron will, therefore, consist of a single (or two) vacancy – interstitial pair, the separation between the vacancy and the interstitial being dictated by the length of the replacement collision sequence. Figure 3.59(a) and (b) shows schematically the defect production process due to an incident energetic electron. In contrast, a high-energy neutron (or an accelerated ion) produces a row of primary “knock-on”s, each of which triggers a series of displacements in its path, as shown in Figure 3.59(c). The cascade of displacements finally terminates where the energy transferred to the target atoms falls below the threshold energy for atomic displacement. At these termination points, the displaced atoms deposit their energies by thermal excitation of neighbouring atoms. Such a thermal excitation has a very short life span (10−12 s) and is known e
e V
V I
V
I
I (a)
(b)
Ion
(c)
Figure 3.59. Schematic diagrams showing the defect production process due to an incident energetic electron ((a) and (b)). In contrast, a high-energy neutron (or an accelerated ion) produces a row of primary “knock-on”s, each of which triggers a series of displacements in its path (c).
Solidification, Vitrification and Crystallization
231
as a thermal spike. The region over which structural change occurs due to a cascade of displacements is known as a “displacement cascade”, the size of which is determined by the mass and the energy of the projectile particle and the mass and the threshold displacement energy of the target atoms. Since a large number of atoms are expelled from the core of the cascade, this depleted region contains a high density of vacancies while the periphery of the cascade gets enriched in interstitials. Often interstitials are produced at the end of a replacement collision sequence chain which propagate along close packed directions from the core to the periphery of the cascade. The brief description of the radiation damage processes, pertinent to irradiation by electrons and relatively heavy particles (such as neutrons and ions), given here provides a background for gaining an understanding of why and how radiation damage induces the crystal to amorphous transformation primarily in intermetallics. It has been observed that some compounds undergo amorphization while others remain crystalline under similar irradiation conditions. The observed difference in the susceptibility to amorphization under irradiation has led to the identification of several empirical criteria which promote amorphization. (1) Directional bonding such as ionic and covalent bonding is a requirement as evidenced from the fact that pure metals (with the exception of Ga) and disordered solid solutions cannot be amorphized under irradiation (Cahn and Johnson 1986). The melting point of Ga at ambient pressure is anomalously low (Tm = 302 K, the heat of fusion being 0.6 kcal/mol). This corresponds to a very small difference in the free energies of the crystalline and amorphous states of Ga at low temperatures. The small requirement of enthalpy for the crystal to amorphous transition can be met by the energy stored in the form of point defects which are produced under irradiation. Semimetals such as Si, Ge and Bi are also amenable to amorphization under irradiation – an observation which is consistent with this criterion. (2) Intermetallic compounds which exhibit narrow solubility ranges in the phase diagram tend to amorphize under irradiation, while those with wide solubility ranges remain crystalline. Though this criterion is not universal, one can argue that a narrow solubility range of composition corresponds to a high energy being associated with the anti-site defects created during irradiation. A large energy storage through these defects may eventually lead to amorphization. (3) The presence of deep eutectics and of a number of line compounds in the same region of a phase diagram is indicative of a relatively high stability of the liquid phase and of an inclination towards chemical ordering in the system. These thermodynamic features are usually associated with a tendency for amorphization not only under irradiation but also during rapid solidification (see Section 3.4). Binary alloys with constituents from the early and the late transition metals such
232
Phase Transformations: Titanium and Zirconium Alloys
as Zr–Ni, Zr–Fe, Zr–Cu, Ti–Ni and, Ti–Cu exhibit phase diagrams which satisfy this criterion and are known to be amenable to amorphisation under irradiation. Since the amorphous state is metastable with respect to the unirradiated crystalline state, the occurrence of a crystal to amorphous (C → A) transformation under irradiation is possible only if the increase in free energy, Girr , due to irradiation is greater than the difference between the free energies, GC→A , of the crystalline and amorphous phases, i.e.
Girr > GC→A
(3.73)
Some of the energy input from the incident radiation must, therefore, be stored permanently (or for a time comparable to the amorphization time) in the material. Since most of the irradiation energy is dissipated as heat, a question arises as to the mechanism by which enough energy could be accumulated in the lattice for fulfilling the above criterion. There are two important mechanisms of energy storage within an irradiated material: first, by accumulation of point defects, both vacancies and interstitials, much in excess of their equilibrium concentrations, and second, by the creation of anti-site defects produced by random displacements and by replacive collision sequences. Let us now examine the relative merits of these two conceivable energy storage mechanisms. One can estimate the critical vacancy concentration required to raise the enthalpy of a metallic crystal at T = 0 K by an amount equal to the heat of fusion, Hf , at the melting temperature, Tm . Doyama and Koehler (1976) have shown that the critical vacancy concentration, cv , required for raising the free energy of a crystal at T = 0 K to the free energy of an amorphous phase is given by cv = Hv / Hf 0008
(3.74)
where Hv is the enthalpy of formation of a vacancy. The question that now arises is whether the steady state vacancy concentration in a crystal can be raised to a level as high as 8 ×10−3 under irradiation. The steady state vacancy concentration under irradiation depends on the rate of production of vacancy – interstitial pairs, the rate at which they recombine and the rate of disappearance of vacancies at sinks such as surfaces, interfaces and dislocation loops. It has been found that even at low temperatures and at high displacement rates (∼103 displacements per atom), the steady state vacancy concentration does not exceed about 10−3 . This is primarily because of the high probability of recombination of vacancies and interstitials as their concentrations increase with the displacement rate. Even when the enthalpy of formation of interstitials (whose steady state concentration remains at one or two orders of magnitude lower than
Solidification, Vitrification and Crystallization
233
that of vacancies) is added to that of vacancies, the free energy of the crystalline phase of a pure metal cannot be raised above the free energy of its amorphous phase by introducing point defects to a realistic maximum concentration. The second mechanism of storing energy in the crystal lattice exposed to irradiation is by the creation of anti-site defects. Obviously such a mechanism can operate only in chemically ordered structures. Let us consider the example of the B2 (CsCl type) ordered structure. Here the lattice can be viewed as comprising two interpenetrating simple cubic sublattices, denoted as and . In the fully ordered condition, the -sites are occupied by only A atoms and the -sites are occupied by only B atoms. Each -site is surrounded by eight nearest neighbour -sites and vice versa. In such a structure, vacancies in - and - (denoted by and ) sites can be distinguished and their exchanges with atoms lead to a change in the order parameter, S. Under irradiation, a high steady state vacancy concentration promotes atom– vacancy exchange process, as shown below: A + v A + v
(3.75)
B + v B + v
(3.76)
A + B A + B
(3.77)
Equation 3.77 shows the exchange of A and B atoms from the to -sites and vice versa, leading to disordering in the forward reaction and to ordering in the backward reaction. The overall reaction, however, involves two steps corresponding to two successive atom–vacancy site exchanges as shown in Eqs. 3.75 and 3.76. The energy barriers for the reactions of A and B atoms with vacancies on the and sublattices in a fully ordered (order parameter, S = 1) and a fully disordered (S = 0) alloy are shown in Figure 3.60. Four different types of atom– vacancy interchange processes are indicated. In a completely disordered alloy, such exchanges do not lead to any change in the energy of the system and therefore in the activation barrier, Em , since such jumps are symmetric. In the partially or fully ordered alloy sublattice, changes of vacancies contribute to an alteration in the degree of order and lead to a decrease or increase in the energy of the system by an amount U . By linear interpolation, one obtains for the contribution at the saddle point, one half of this energy. The jump frequency can, therefore, be written as U ± = o exp − Em ∓ kB (3.78) 2
234
Phase Transformations: Titanium and Zirconium Alloys
Em
u
S=0 S=1
Aβ + Bα = Aα + Bβ
Figure 3.60. Schematic diagram showing the energy barriers for the reactions of A and B atoms with vacancies on the and sublattices in a fully ordered (S = 1) and disordered (S = 0) alloy.
where kB denotes Boltzmann’s constant. The lower value of the activation energy holds for a vacancy jump in which an atom changes from a wrong to the right sublattice, i.e. for an ordering jump ("+). A higher value of activation energy refers to a disordering jump ("−). The overall rate of change of order in an irradiation environment can be written as (Banerjee and Urban 1984) a sum of three terms: dS/dt = dS/dtc + dS/dtr + dS/dtt
(3.79)
where (dS/dt)c and (dS/dt)r refer to the disordering rate due to the replacement collision sequence and the random defect annihilation process, respectively, while (dS/dt)t is the rate of change of order parameter arising from the thermally activated exchange of A and B atoms from the and sites and vice versa: dS/dtt = K+ cA cB − K− cA cB
(3.80)
where cA , c , etc. represent concentration of A in -site and B in -site, respectively, and the rate coefficients K+ (ordering) and K− (disordering) correspond to the reverse and forward reactions shown in Eq. 3.77 and are expressed as K± =
±Z cv Z cv Z cv + Z cv
where Z is the number of nearest neighbour -site around a -site.
(3.81)
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235
0.010
0.3
T = 580 K
T = 1120 K 0.2
570
0.005
1130
dS/dt →
dS/dt →
0.1 0
1140
0
560
–0.1
553 540
–0.005 –0.2 –0.3
1150 0
0.1
0.2
Order parameter (S ) → (a)
0.3
–0.010
0
0.5
1.0
Order parameter (S ) → (b)
Figure 3.61. dS/dt versus S plots showing the influence of irradiation in creating a high concentration of anti-site defects for (a) thermal disordering and (b) irradiation disordering of a B2 alloy.
The influence of irradiation in creating a high concentration of anti-site defects (in other words, chemical disordering) can be illustrated by comparing the dS/dt versus S plots (Figure 3.61) for (a) thermal disordering and (b) irradiation disordering of a B2 alloy. While the disordering temperature, Tc , is 1140 K (above which dS/dt is negative for all values of S) under thermal disordering condition, the disordering temperature under irradiation Tc∗ is 553 K as shown in Figure 3.61(b). For details on kinetics of order–disorder transformation in alloys under irradiation, readers may refer to Banerjee and Urban (1984). The creation of anti-site defects (in other words, chemical disordering) plays a very important role in irradiation-induced amorphization, and the contribution of point defects is relatively less important. However, a quantitative estimation of the contributions of these components requires experiments using different types of radiations that have different replacement to displacement ratios and modelling of the irradiation-induced microstructural evolution using chemical rate equations as well as a molecular dynamics approach. Extensive experimental results on irradiation-induced amorphization of the Laves phase ZrCr Fe2 precipitates in a zircaloy-2 matrix under electron, ion and neutron irradiation are available. Some of the important results are summarized here with a view to making comparisons of the efficacy of different types of radiation with regard to bringing about amorphization: (1) Amorphisation occurs with all the three types of radiation when the irradiation (electrons, ions, neutrons) temperature is below a critical temperature, Tc . The radiation doses required for amorphization under 1.5 MeV electron, 127 MeV Ar ion and neutron irradiation are shown in Figure 3.62.
236
Phase Transformations: Titanium and Zirconium Alloys
Electron
Dose (dpa)
60
Ion
Neutron
40
20
0
200
300
400
500
600
Temperature (K)
Figure 3.62. Irradiation dose (dpa) as a function of temperature (K) quantifying the radiation doses required for amorphization under 1.5 MeV electron, 127 MeV Ar ion and neutron irradiation.
Crystalline fraction Electron
1.00
0.50
Neutron 0.00
0.00
0.50
1.00
Fraction of irradiation time
Figure 3.63. Crystalline fraction as a function of irradiation time showing the sharp drop in the degree of crystallinity with electron irradiation.
(2) Under electron irradiation, amorphization occurs homogeneously within the entire volume of the precipitates. The degree of crystallinity drops sharply, as shown in Figure 3.63. Since electron irradiation is carried out on thin foils (10–20 nm thickness), interstitials migrate to the surface swiftly, allowing vacancy supersaturation to build up in the centre of the foil so that there is a substantial accumulation of defect energy within the foil. Chemical disordering also contributes towards amorphization.
Solidification, Vitrification and Crystallization
237
(3) Under neutron irradiation, the amorphization of ZrCr Fe2 precipitates initiates from the precipitate–matrix interface, suggesting that cascade (or ballistic) mixing in the thin layer close to this interface is responsible for amorphization. The ballistic mixing of the two phases at the interface can bring about a significant departure from stoichiometry which causes a large increase in the free energy of this thin layer. With the increase in radiation dose, the amorphous layer gradually propagates towards the core of the precipitates, as shown in Figure 3.63.
3.8
PHASE STABILITY IN THIN FILM MULTILAYERS
When thin films are deposited on suitable substrate surfaces, they can exhibit crystal structures which are metastable with respect to those associated with the same materials in bulk form. Such metastable structures have been documented in literature for many systems including metal/metal and metal/semiconductor systems. Experimental observations on multilayers include instances where either one or both layers can exist in the metastable state. With the increase in introduction of multilayered nanostructures in a variety of applications, it is pertinent to examine the reasons for the shift in the relative stabilities of the relevant phases with variation in the thicknesses and in the thickness ratio of the constituents. Once again it is seen that the knowledge gained from recent researches on Ti- and Zr-based multilayer systems has provided an insight into this important issue. To illustrate this point, experimental observations on Ti/Al multilayers reported by Fraser and coworkers (Ahuja and Fraser 1994a,b, Banerjee et al. 1996) are summarized here. An examination of multilayered structures comprising alternate Al and Ti layers, each of several nm thickness, has revealed that in systems in which individual layers are of equal thickness and the unit bilayer thickness ≥ 20 nm, both metals assume their stable structures, namely fcc for Al and hcp for Ti. However, for < x nm, both metals have the hcp structure and for intermediate thickness ranges (x < y < ) both show the fcc structure. Thus, with increase in , the Al layer transforms once from hcp to fcc, while the Ti layer transforms twice: from hcp to fcc and then back to hcp. This behaviour of Ti is unexpected in two ways: first because it transforms from its stable structure (hcp) to a metastable structure (fcc) upon increase in thickness, and second, the transition from fcc/fcc to hcp/fcc multilayers occurs upon increase in thickness with a decrease in the misfit at the interfaces. The observations showing structural changes in multilayers have been rationalized by Banerjee et al. (1999) in the manner outlined here. A bilayer “unit system” comprising two layers and two interfaces (Figure 3.64) is defined and the specific free energy of this “unit system” is expressed as
238
Phase Transformations: Titanium and Zirconium Alloys
A B A B A
Figure 3.64. A schematic diagram showing a bilayer “unit system” comprising two layers and two interfaces.
g = 2 + GA fA + GB fB
(3.82)
where g is normalized by the area of the interface, is the change in interfacial energy, Gi = G (metastable) − G (stable)) and fi are, respectively, the allotropic free energy change per unit volume of the reference phase for and the volume fraction of the metal i in the reference bilayer. Equation (3.81) describes a thermodynamic potential surface that varies as a function of two independent variables, f and . The specific free energy of this biphase system, as described in Figure 3.65(a), can be represented by a surface in the f − −1 space, and the equilibrium structure will be the one with the lowest specific energy of formation, g. The transformation from one biphase configuration to another (as e.g. Tifcc + Alfcc → Tihcp + Alfcc ) occurs when the g surface for a given combination of biphase (e.g. Tifcc + Alfcc ) intersects the g surface for a different biphase (e.g. Tihcp + Alfcc ). In this manner, the stability regimes of different biphase configurations can be depicted in biphase diagrams. It may be noted here that the interfacial energy term, , includes a chemical term arising out of dissimilar metals bonding at interfaces as well as a structural term due to the disregistry of the two contiguous lattices at the interfaces. The strain energy and the bulk chemical free energy terms are included in Gi which is volume dependent. Let us consider the case of the Al/Ti multilayers studied by Ahuja and Fraser (1994a,b) who reported a transition of Ti from the bulk, stable hcp form to a metastable fcc form below a critical value of = ∗fcc/fcc and both Al and Ti becoming hcp below another value of = ∗hcp/hcp . These two transformations of the biphase (Al/Ti) system can be depicted conveniently by a constant volume fraction cut of the g f surface, i.e. a plot of g versus , as shown in Figure 3.65(b). For an arbitrarily large value of , the lowest free energy of
Solidification, Vitrification and Crystallization
239 fcc/fcc ΔGAl/Al
fcc/fcc
0.25
hcp/hcp fcc/hcp
(2)
0.2
(ii) (iii)
fcc/fcc
0.1
2Δγ hcp/hcp
hcp/hcp
0.05 0.0 0.0
hcp/fcc
2Δγ fcc/fcc
0.15
(1)
0.2
fcc/hcp
0.4
hcp Ti + fcc Al fcc Ti + fcc Al
hcp Ti + fcc Al
(3)
0.6
ΔGTi / Ti
(i)
0.0 Δg
1/λ,λ = bilayer thickness (nm)
hcp/hcp
0.3
0.8
λ∗hcp/hcp
1
λ∗fcc/fcc
Volume fraction of Ti2 fTi
Bilayer thickness, λ
(a)
(b)
a
B A
Ti
B A
Al
1 nm
Ti (c)
Figure 3.65. (a) The specific free energy of a biphase system represented by a surface in the f − −1 space; (b) g versus plot showing a transition of Ti from the bulk, stable hcp form to a metastable fcc form below a critical value of = ∗fcc/fcc and both Al and Ti becoming hcp below another value of = ∗hcp/hcp ; and (c) HRTEM image showing the Al/Ti multilayered sample, with the thickness of the Ti and Al layers being 5.0 and 20 nm, respectively. The beam direction is parallel ¯ to <1120> (after R. Banerjee et al.).
the system is achieved if both metals assume their stable structures, i.e. GTi =
GAl = 0 (line (i) in Figure 3.65(b)). Below = ∗fcc/fcc , the lowest free energy of the system corresponds to a bilayer configuration in which Ti has adopted a metastable (fcc) structure (line (ii) in Figure 3.65(b)). Below a still lower value of = ∗hcp/hcp , the energy of the system is minimized by the transition of both
240
Phase Transformations: Titanium and Zirconium Alloys
metals to the hcp structure (line (iii) in Figure 3.65(b)). The stability diagram shown in Figure 3.65(b) provides a physical basis for the formation of metastable phases in nanolayered materials. In the hcp Ti + hcp Al regime, Ti remains in its stable state, making GTi = 0. For this regime, Eq. 3.82 is then reduced to
g = 2 hcp/hcp + GAl fAl
(3.83)
This means the intercept and the slope of the line (iii) are 2hcp/hcp and GAl fAl , respectively. Based on a similar argument, the intercept and the slope of the line (ii) are 2fcc/fcc and GTi fTi . The line (i) corresponds to the combination of hcp Ti and fcc Al, both in their respective stable states, implying that , GAl and
GTi are all equal to zero. It is, therefore, evident that the metastable states such as fcc Ti and hcp Al are stabilized in nanolayered structures due to the negative values of which more than compensate for the increase in G resulting from the formation of the metastable phases. Figure 3.65(b), which is consistent with experimental observations on the hierarchy of biphase stability, originates from the following inequalities:
GAl < GTi
and
hcpTi/hcpAl < fccTi/fccAl < 0
(3.84)
Figure 3.65(b) is constructed for a fixed value of the volume fraction, fTi . Biphase diagrams for cases where both the volume fraction and the bilayer thickness are variable can be constructed in the fi − −1 space. This is illustrated for Ti/Al multilayers in Figure 3.65(a). In the absence of coherency, the boundaries in this type of biphase diagrams are straight lines. Non-linear dependence of composite moduli on volume fraction will introduce curvature in the variation of 1/, with f in the biphase diagram. The biphase diagram for the Al/Ti multilayers has been constructed on the basis of the slopes of the lines (1), (2) and (3) respectively, which are given by GTi /2 fcc/fcc , GAl /2 hcp/hcp and ( GAl + GTi /2 hcp/hcp − fcc/fcc ). Experimental data points are superimposed on the biphase diagram of the Al/Ti multilayered system shown here. The symbols shown in the inset represent fcc Al/fcc Ti, hcp Al/hcp Ti and fcc Al/hcp Ti multilayers observed for different values of fTi and . These experimental data points are consistent with the biphase diagram. Application of the biphase diagram concept in predicting the structure of several multilayer systems such as Co/Cr, Zr/Nb and Ti/Nb has been successful (Thompson et al. 2003). A HRTEM image (Figure 3.65(c)) shows the Al/Ti multilayered sample with the thickness of the Ti and Al layers being 5.0 and 2.0 nm, respectively (Banerjee et al. 1996).
Solidification, Vitrification and Crystallization
3.9
241
QUASICRYSTALLINE STRUCTURES AND RELATED RATIONAL APPROXIMANTS
Quasicrystalline structures can be defined as those with long-range aperiodic order and crystallographically forbidden rotational symmetries (e.g. 5-, 8-, 10- and 12fold rotation axes). The observation that certain intermetallic compounds exhibit sharp diffraction peaks displaying the “non-crystallographic” icosahedral rotational point group has generated a great deal of excitement. It is well known that the translational periodicity of atoms allows only certain rotational operations about an axis which bring the arrangement back into registry with the unrotated assembly. For three-dimensional periodic crystals, the allowed rotation operations are two-, three-, four- and sixfold, about appropriately chosen axes. Taken together with other operations such as translations, reflections and inversions, these point group operations define all of the 230 space groups. It was generally believed that only periodic arrangements of atoms can produce sharp diffraction peaks. The discovery of a quasicrystalline structure in a rapidly quenched Al–Mn alloy, schematic diffraction patterns from which are shown in Figure 3.66(a), has laid this myth (Shechtman et al. 1984) to rest. It is now realized that the occurrence of sharp Bragg diffraction peaks does not require the presence of long-range periodic translational order, but rather of long-range positional order, which may or many not be specified by a periodic function in three dimension. This point can be explained in one dimension by considering the Fibonacci sequence: 1 1 2 3 5 8 13 21 34# # # where every term of the series is generated by the addition of the two immediately preceding terms. Let us consider two translation vectors S (short) and L (long) along a given direction and generate a series using an algorithm in which S is replaced by L and L is replaced by LS in every successive series. It may be noted that the repeat period grows as per the Fibonacci sequence of numbers (as shown in Figure 3.67). The ratio of the number of L to the number of S segments√changes in recursive steps and finally converges to the “golden mean” ( = 1 + 5/2 1618) as the repeat period grows to infinity. This illustrates how an array of two segments, S and L, can be created in such a way that translational symmetry is absent even when the one-dimensional sequence is extended upto infinity. Penrose (1974) has shown that by using two specially shaped tiles, as designated by a kite and a dart (Figure 3.68), it is possible to cover a plane with fivefold symmetry. There exist “matching rules” for the construction of the Penrose lattice. The ratio of the sides of these tiles (kite and dart) is given by , as shown in Fig. 3.68. Some important properties of the Penrose pattern are (a) orientational order, (b) quasiperiodic translational order and (c) self-similarity.
242
0τ21 10.81°
°
79.2
13.28°
29°
τ2τ41
58.
°
(6)
.37
37
20.91°
31.72
(5)
63.43
13.28° (4)
20.91° 1τ20 (1)
(2)
010
10.81° 1 τ 0 (3)
(a)
(b)
Figure 3.66 Schematic diffractive pattern from a quasicrystalline phase showing (a) fivefold, threefold and twofold symmetries and the observed angles between the corresponding zone axes and (b) stereogram showing matching of observed symmetry elements with those of an icosahedron.
Phase Transformations: Titanium and Zirconium Alloys
13τ + 1τ
Solidification, Vitrification and Crystallization A • • PERIOD: 1 • • A PERIOD: 2
A
B •
A
A • A/B = 1/0 = 00
•
A
•
•B •
A
• B • A/B = 1/1 = 1
(b)
• • A A/B = 2/1 = 2
(c)
(d)
A
•
243 (a)
A • B • • PERIOD: 3
A
•
A
•
B
A • B • • PERIOD: 5
A
•
A
•
B
A • B • • A/B = 3/2 = 1.5
B A • • • PERIOD: 8
A
•
A
•
B
B • A • • A/B = 5/3 = 1.66
H
T
A
• (e)
Figure 3.67. Fibonacci sequence of numbers.
T 1 H 72°
144°
1 72°
1 36°
H
1 36°
T
216°
τ
τ 72°
τ
τ 72°
T
H
(a)
(b)
Figure 3.68. Schematic diagrams showing two specially shaped tiles (a) kite and (b) dart. The ratio, , of the sides of these tiles (kite and dart) is also shown in the figure.
Mackay (1982) has extended Penrose tiling to three dimensions (3D) and has demonstrated that by making use of a pair of acute and obtuse rhombohedral tiles (as shown in Figure 3.69), filling of space with fivefold symmetry is possible. The interesting feature of 2D and 3D Penrose lattices is that the Fourier transform of their structures gives rise to sharp diffraction peaks displaying icosahedral symmetry for the latter. Quasilattices can be constructed with any arbitrary orientational symmetry and arbitrary quasiperiodicity. The existence of octagonal (Wang et al. 1987), decagonal (Bendersky 1985, Chattopadhyay et al. 1985) and duodecagonal (Ishimasa et al. 1985) quasicrystalline structures in various alloy systems has been established by experiments.
244
Phase Transformations: Titanium and Zirconium Alloys
(b) (a) (c)
(d)
(e)
Figure 3.69. Penrose tiling in three dimension (3D) demonstrating filling of space with fivefold symmetry by using a pair of acute and obtuse rhombohedral tiles.
The diffraction patterns shown in Figure 3.66(a) show fivefold, threefold and twofold symmetries and the observed angles between the corresponding zone axes. The observed symmetry elements matched with those of an icosahedron as illustrated in the stereogram shown in (b). An inspection of the sequence of diffraction spots along a radial direction of the fivefold pattern in Figure 3.66 shows that the ratio of the distances from the origin to any two bright spots is an irrational number within reasonable experimental error. For icosahedral √ quasicrystals, this irrational number is some power of the golden mean, = 1+ 5/2, which arises
Solidification, Vitrification and Crystallization
245
from the geometries of icosahedra, pentagons and decagons. Though translational symmetry is not present in these patterns, there exists an inflation symmetry. For example, the diffraction patterns in Figure 3.66 can be expanded or contracted by a factor of 3 to yield patterns indistinguishable from the originals. In view of the observed icosahedral symmetry of the diffraction patterns from some quasicrystalline structures, the indexing of these patterns has been carried out on the basis of six real space vectors defined by vectors, ei , that point from the centre to the vertices of an icosahedron (as illustrated in Figure 3.70). The reciprocal lattice vectors are then defined by G1 = 6i=1 ni ei ⎡ ⎤ 1 0 ⎢ −1 ⎡ ⎤ 0 ⎥ ⎢ ⎥ i ⎢ ⎥ 1 0 1 ⎥ ⎣ ⎦ ⎢ j ei = √ (3.85) 0 1 ⎥ 1 + 2 ⎢ ⎢ ⎥ k ⎣ 0 −1 ⎦ 0 1 − where ei are the real space basis vectors, ni are integers and go is a constant which determines the scale of the diffraction pattern. Unlike in the case of periodic crystals where the diffraction pattern can be related to the lattice parameter, a, by a relation of the type 2/a, no single fundamental length, go , can be chosen for diffraction patterns from quasicrystals ab initio. This is also evident from the presence of the inflation symmetry. The reciprocal space of the icosahedral quasicrystal, instead of having a regular lattice of intensity maxima as observed for crystals, can have peaks arbitrarily close to any given peak by taking integer linear combinations of fundamental basis vectors. Thus the reciprocal space of the icosahedral quasicrystal is uniformly 1
5
6
4
6
4
2
3
5
2 3 ei⊥
i eN
(a) 1 (b)
Figure 3.70. Schematic diagrams showing six real space vectors defined by ei that point from the centre to the vertices of an icosahedron.
246
Phase Transformations: Titanium and Zirconium Alloys
dense. Within the Landau theory, one expects a decreasing hierarchy of peak intensities as the number of reciprocal lattice star vectors required to arrive at the peaks increases. Since the peaks close to any given peak are obtained only by the higher generation number, the intensities associated with these peaks are very feeble and are not distinguishable from the background. Peaks associated with lower indices are strong enough to produce a diffraction pattern with a discrete set of spots/reflections. For an icosahedral quasicrystalline structure, each reciprocal lattice vector requires six indices for indexing as has been expressed in Eq. 3.85. The advantage of describing a three-dimensional quasiperiodic structure using a six index system (which corresponds to a six-dimensional space) arises from the fact that a projection from a higher dimensional periodic lattice points on to a lower dimensional space can generate either a periodic or a quasiperiodic lattice, depending on the orientation of the projected space. This point can be explained in a simple manner by taking the example of a projection from a 2D space to a 1D space. Let us consider a 2D square lattice which is projected on a set of two perpendicular directions, designated as g1 and g2 . If the direction g1 is drawn from any lattice point of the 2D structure along a rational direction (defined by tan = m/n, where is the angle between g1 and the X-axis of the 2D lattice), the g1 line will intersect the lattice point with coordinates (n m) and will periodically intersect lattice points (2n 2m), (3n 3m), etc. resulting in a periodic one-dimensional structure. In contrast, if tan is an irrational number, the g1 , line starting from the origin will not intersect any other lattice point. For projecting lattice points of the 2D structure on to the line g1 , we can arbitrarily select a strip indicated by a pair of broken lines parallel to g1 as shown in Figure 3.71. Projections of lattice points lying within this strip on g1 produce an array of points which is quasiperiodic. Two segments, short and long, appear along g1 , but the sequence L S L L S L S L L S # # # is such that one cannot identify a unit which has a repeated periodic appearance. If the strip width is increased more number of spots will appear on the projected line g1 , making the line uniformly dense as the strip width is enlarged to infinity. However, it can also be seen that two points which are very close on the projected line will appear only if the strip width is increased to a very large extent, which means that these points arise from two points (in 2D lattice) which are widely separated along the g2 direction. A Fibonacci sequence in 1D can be created by projecting a 2D square lattice on a g1 line which has an inclination, tan = . In case tan is chosen to be a rational quantity, the projection will result in a periodic sequence. A particular set of crystalline approximants, the Fibonacci rational approximants, is obtained when the ratio of two consecutive numbers p and q of the Fibonacci sequence is
Solidification, Vitrification and Crystallization
247
g⊥ g ⏐⏐
Figure 3.71. Schematic diagram indicating that projection of lattice points of 2D structure on to the line g1 . This can be done by arbitrarily selecting a strip marked by a pair of dashed lines parallel to g1 .
chosen (tan = q/p), where q/p = 1/1 2/1 3/2 5/3 8/5 138 21/13# # #
(3.86)
It is to be emphasized that the rational ratio is not restricted to the Fibonacci series since non-Fibonacci rational approximants have also been observed experimentally. The compositional similarities between quasicrystals and their respective approximants suggest similarities in their local atomic structures, substantiated by similarities in physical properties. Approximants are important for studies on the formation and stability of quasicrystals since they are amenable to established theoretical tools. Reversible transformations between quasicrystals and related crystalline approximant structures have been encountered in some cases in which the structural relationship between them could also be established. The projection method for the 1D case can be extended to 2D and 3D and can be used for the construction of both the real lattice and the reciprocal lattice. The 3D projected lattice represents the icosahedral quasilattice. Just as the reciprocal lattice of a crystalline structure is generated by a basis of three vectors, the icosahedral diffraction pattern is generated by a set of six reciprocal lattice vectors because of its incommensurate nature. This means that all the g vectors in the reciprocal space of an icosahedral structure can be expressed in terms of linear combinations of the basis vectors of the reciprocal space. In this method, a 6D periodic reciprocal lattice is projected orthogonally on to a suitably oriented 3D subspace.
248
Phase Transformations: Titanium and Zirconium Alloys
For the icosahedral alloys, characterized by six fivefold axes, the 3D structure results from the projection of points of a 6D hypercubic lattice that are contained in a 3D acceptance domain (analogous to the strip drawn with a pair of broken lines for the 2D to 1D projection as shown in Figure 3.71) appropriately oriented with respect to the 6D lattice. This acceptance domain is a triacontahedron in the 3D space which is orthogonal to the physical space. For the icosahedral phase, the orientations of the acceptance domain and the physical space are specified by a 6 × 6 orientation matrix given by : 1 −1 0 0 1 0 0 1 1 0 1 −1 − 0 (3.87) M=√ 0 1 1 0 2 3 + 2 − 1 − 0 0 − 1 0 0 1 − −1 The upper three row vectors give the 6D coordinates of the three vectors which define the physical space while the lower three rows refer to vectors spanning the orthogonal space. The analogues of the L and S segments generated on the projection line from the points lying within the strip are oblate and prolate rhombohedra (Figure 3.69) which are the constituent tiles of the 3D Penrose lattice. The edge length of these rhombohedra, ar , plays a role analogous to the lattice constant for periodic crystals and is, therefore, called the quasilattice constant. As discussed in the case of the 2D to 1D projection, rational approximants can be constructed from the 6D hypercubic lattice by a suitable selection of the acceptance domain. Fibonacci rational approximants are obtained when is replaced by a rational ratio q/p. Elser and Henley (1985) first demonstrated that the cubic -AlMnSi structure can be obtained by a 1/1 rational projection from the same atomic decoration of the 6D hypercubic lattice used to define the icosahedral phases. They also made the first quantitative assessment of the similarities between the bcc (or close to the bcc structure) in the case of the -AlMnSi phase and the structure of the icosahedral phase. This analogy brings out the fact that the atomic arrangements of the related crystalline phases can be constructed from the same building blocks which generate the icosahedral phases. The building blocks may be taken either as icosahedral clusters of atoms or as a set of two types of rhombohedral bricks which may also be derived from the decomposition of the icosohedral clusters. 3.9.1 Icosahedral phases in Ti- and Zr-based systems Quasicrystalline structures have been reported most extensively in Al-based alloys. Icosahedral phase formation is now known to be quite common in Ti alloys
Solidification, Vitrification and Crystallization
249
also, with Ti-based icosahedral phases constituting the second largest class of quasicrystals (Kim and Kelton 1995). Alloys containing Ti and 3d transition metals from V to Ni have received the maximum attention. Quasicrystalline phases have been reported in many of these alloys in the rapidly solidified condition. In most cases, the microstructure produced consists of a finely distributed mixture of quasicrystalline and crystalline phases. The presence of Si and O in these alloys often plays a crucial role in stabilizing the icosahedral phase. A representative list of alloys based on Ti and Zr in which the formation of icosahedral phases has been reported is given in Table 3.8. A number of alloys in which both Ti and Zr are present have attracted considerable interest (Kelton et al. 1994, Kim and Kelton 1995, 1996) due to their strong tendency for icosahedral phase (i-phase) formation. Some of these (such as Ti–Zr–Fe alloys) show localized diffuse scattering and significant diffraction spot shape anisotropy while some others (such as Ti–Zr–Ni alloys) are more ordered, as reflected in the sharp diffraction spots obtained from the icosahedral phases in these alloys.
Table 3.8. Icosahedral phases in Ti- and Zr-based alloys Alloy composition (approximate)
Processing/ stability
Extent of phason disorder
Quasilattice parameter (nm)
Ti-TM-Si-O
RSP/MS
Very high
0.47–0.48
Ti–Zr–Fe
RSP/MS
Very high
0.485–0.488
Ti53 Zr27 Ni20
RSP/MS
Very low
0.512
Ti53 Zr27 Co20
RSP/MS
Moderate
0.510
Ti415 Zr415 Ni17
AI/S
Very low
0.517
Ti63 Cu25 Al12
C/MS
Low
–
Zr65−70 Cu12−17 Ni10−11 Al75
C/MS
Low
–
Zr65−70 Cu10−15 Ni10−13 Pd7−10
C/MS
Low
–
Zr65 Cu125 Ni10 Al75 M5
C/MS
Low
–
M = Ag, Pd, Au, Pt RSP, rapid solidification processed; AI, annealed ingot; C, crystallized glass; S, Stable; MS, Metastable.
Reference Libbert and Kelton (1995) Kim and Kelton (1995) Zhang et al. (1994) Kim and Kelton (1996) Kim et al. (1997) Koster et al. (1996a) Koster et al. (1996b) Murty et al. (2000a) Murty et al. (2000b,c)
250
Phase Transformations: Titanium and Zirconium Alloys
As can be seen from Table 3.8 the icosahedral phases in Ti-based systems can be grouped into two classes. Those belonging to the first are associated with a smaller quasilattice constant (∼0.48 nm) and show arcing of diffraction spots and diffuse intensity distribution, suggesting the presence of phason disorder to a considerable extent. In contrast, members of the second group have larger quasilattice constants (> 0.51 nm) and exhibit sharp diffraction spots indicating a higher degree of phason order. Out of the Ti-based icosahedral phases listed in Table 3.8, the i-phase in only Ti415 Zr415 Ni17 , which forms during annealing of arc-melted ingots, is stable. In all the other cases, the i-phase is metastable, forming during rapid solidification and disappearing during subsequent annealing. The microstructures of most of Ti-based icosahedral phases are similar. Fine particles of the i-phase are usually found dispersed in the amorphous matrix of rapidly solidified alloys of compositions listed in Table 3.8. These particles also appear during the early stages of crystallization of amorphous alloys. In a number of observations, i-phase particles have been found to be surrounded by the -Ti (bcc) phase. Several crystalline approximants of the i-phase are found to coexist in partially crystallized samples. The occurrence and the structure of these phases are briefly discussed in the following paragraphs. The -1/1 rational approximant, a large unit cell bcc phase (lattice parameter = 1.31 nm), consisting of Mackay icosahedra packed face to face along the <111> cubic direction, is frequently observed in Ti-Mn-Si and Ti-Cr-Si alloys. This phase is believed to be the appropriate approximant to the Ti-3d TM-Si icosahedral phases. The -phase, a face-centred orthorhombic phase with a large unit cell (a = 320 nm, b = 266 nm, c = 104 nm) appears frequently with the i-phase in Ti-MnSi, Ti-Mn-Fe-Si and Ti-Cr-Si alloys. Based on TEM studies, it has been inferred that the -phase can be constructed structurewise from Mackay icosahedra packed with their vertices aligned along the c-direction of the unit cell. A large unit cell fcc phase (a = 1.12–1.17 nm), presumably with the Ti2 Ni structure, has been found in many Ti-Zr-Fe samples. Usually the fcc phase and the i-phase are found in different regions of the sample, suggesting that these two phases evolve directly from the liquid phase under different conditions of cooling. A hcp phase with a = 0.515 nm and c = 0.837 nm has been detected in TiZr-Fe alloys containing relatively low concentrations of Ti. Energy dispersive spectroscopy has revealed the composition of this phase to be Ti41−49 Zr21−29 Fe26−31 with small amounts of Si (Kim and Kelton 1995). This phase is isostructural with MgZn2 type Laves phases. The observed intensity modulation in the diffraction patterns from this Laves phase is similar to that associated with the i-phase, suggesting a similarity in the local atomic arrangements in the two.
Solidification, Vitrification and Crystallization
251
As mentioned earlier, the i-phase is often seen to be surrounded by the (bcc)-phase. In addition to the prominent bcc reflections, spots at 1/3 and 2/3 positions corresponding to the fundamental bcc reflections are recorded. These additional spots suggest the presence of the -phase in the -matrix. The presence of the -phase is not unusual in -Ti alloys which contain sufficient amounts of -stabilizing elements. Unlike in the cases of Ti-TM-Si-O and Ti-Zr-Fe alloys, the i-phase forming in the Ti-Zr-Ni system is stable. This has been demonstrated conclusively by forming the i-phase by annealing as cast ingots of a Ti45 Zr38 Ni17 alloy which initially contained only a C14 Laves phase and a hexagonal solid solution phase. The crystalline approximant which forms along with the i-phase in Ti-Zr-Ni is the bcc W-phase (which is a 1/1 rational approximant). The composition range of the W-phase is Ti40−50 Zr31−42 Ni16−19 . XRD peak intensities from this phase are quite distinct from those pertaining to the -1/1 approximant described earlier. Based on the powder diffraction data and the number of atoms per unit cell (168.5, estimated from measured density), Kim et al. (1997) have inferred that the W-phase has a Bergman type structure similar to that encountered in Al-Li-Cu and Al-Mg-Zn alloys. In this context, it is worth mentioning that two basic cluster types, both having icosahedral symmetry, are used in describing atomic positions in icosahedral phases. The Mackay cluster, as shown in Figure 3.72 is a double-shell icosahedral cluster, with atoms decorating the vertices of the inner and outer icosahedra and the midpoints of the edges of the outer icosahedron (Mackay 1962). While the
-1/1 rational approximant in Ti-TM-Si-O alloys is based on the Mackay cluster, the W-phase in Ti-Zr-Ni alloys is based on the Bergman cluster which is also a double-shell icosahedral cluster; however, the midpoints of the faces of the outer Al Mn
Figure 3.72. The Mackay cluster – a double-shell icosahedral cluster with atoms decorating the vertices of the inner and outer icosahedra and the midpoints of the edges of the outer icosahedron.
252
Phase Transformations: Titanium and Zirconium Alloys
icosahedron are occupied, instead of the edge centres as in the case of the Mackay cluster. The i-phases in these two types of systems can thus be grouped into two distinct classes, the former containing Mackay clusters and the latter constituted of Bergman clusters. Kim et al. (1997) have classified i-phases based on a correlation between the measured quasilattice constant, aq , and the atomic separation, as , calculated from the measured i-phase densities. By this method, all three Bergman type i-phases, including i-(Al-Li-Cu), i-(Al-Mg-Zn) and i-(Ti-Zr-Ni) have the ratio aq /as 2, while for i-phases which have Mackay type clusters, aq /as = 185.
REFERENCES Ahuja, R. and Fraser, H.L. (1994a) J. Electron. Mater., 23, 1027. Ahuja, R. and Fraser, H.L. (1994b) J. Metals, 46, 35. Akhtar, D., Cantor, B. and Cahn, R.W. (1982a) Acta Metall., 30, 1571. Akhtar, D., Cantor, B. and Cahn, R.W. (1982b) Scr. Metall., 16, 417. Baker, J.C. and Cahn, J.W. (1971) Solidification, American Society for Metals, Metals Park, OH, p. 23. Banerjee, S. (1979) Report submitted to University of Birmingham. Banerjee, S. and Cahn, R.W. (1983) Acta Metall. Mater., 31, 1721. Banerjee, S. and Cantor, B. (1979) Proc. Int. Conf. Martensite, Boston, p. 195. Banerjee, S. and Urban, K (1984) Phys. Status Solidi (a), 81, 145. Banerjee, R., Ahuja, R. and Fraser, H.L. (1996) Phys. Rev. Lett., 76, 3778. Banerjee, R., Zhang, X.D., Dregia, S.A. and Fraser, H.L. (1999) Acta Mater., 47 (4), 1153. Bhanumurthy, K., Dey, G.K. and Banerjee, S. (1988) Scr. Metall., 1395. Bhanumurthy, K., Dey, G.K., Banerjee, S., Khera, S.K., Asundi, M.K. (1989) Praman – J. Phys, 32, 289. Bendersky, L.A. (1985) Phys. Rev. Lett., 55, 1461. Birac, C. and Lesueur, D. (1976) Phys. Status Solidi, A36, 247. Blatter, A. and Von Allmes, M. (1985) Phys. Rev. Lett., 54, 2103. Boettinger (1982) Rapidly Solidified Crystalline and Amorphous Alloys (eds B.H. Kear and B.C. Griessen) Elsevier, North Holland, NY, p. 15. Boettinger, W.J. and Biloni, H. (1996) Physical Metallurgy, 4th edition (eds P. Haasen and R.W. Cahn) North Holland, Amsterdam, p. 669–842. Brody, H.D. and David, S.A. (1970) Science, Technology and Application of Titanium (eds R.I. Jaffee and N.E. Promisel) Pergamon, Oxford, p. 21. Brody, H.D. and Flemings, M.C. (1966) Trans. AIME, 239, 615. Burke, J. (1965) The Kinetics of Phase Transformations in Metals, Pergamon, Oxford, p. 36. Busch, R., Bakke, E. and Johnson, W.L. (1998) Acta Mater., 46, 4725 Buschow, K.H.J. (1981) J. Less Common Metals, 79, 243.
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Chapter 4
Martensitic Transformations 4.1 Introduction 4.2 General Features of Martensitic Transformations 4.2.1 Thermodynamics 4.2.2 Crystallography 4.2.3 Kinetics 4.2.4 Summary 4.3 BCC to Orthohexagonal Martensitic Transformation In Alloys Based on Ti and Zr 4.3.1 Phase diagrams and Ms temperatures 4.3.2 Lattice correspondence 4.3.3 Crystallographic analysis 4.3.4 Stress-assisted and strain-induced martensitic transformation 4.4 Strengthening Due to Martensitic Transformation 4.4.1 Microscopic interactions 4.4.2 Macroscopic flow behaviour 4.5 Martensitic Transformation in Ti–Ni Shape Memory Alloys 4.5.1 Transformation sequences 4.5.2 Crystallography of the B2 → R transformation 4.5.3 Crystallography of the B2 → B19 transformation 4.5.4 Crystallography of the B2 → B19 transformation 4.5.5 Self-accommodating morphology of Ni–Ti martensite plates 4.5.6 Shape memory effect 4.5.7 Reversion stress in a shape memory alloy 4.5.8 Thermal arrest memory effect 4.6 Tetragonal Monoclinic Transformation in Zirconia 4.6.1 Transformation characteristics 4.6.2 Orientation relation and lattice correspondence 4.6.3 Habit plane 4.7 Transformation Toughening of Partially Stabilized Zirconia (PSZ) 4.7.1 Crystallography of tetragonal → monoclinic transformation in small particles References
260 261 261 266 277 280 281 282 289 294 324 326 329 335 339 340 342 342 345 347 352 356 360 362 362 363 366 369 372 373
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Chapter 4
Martensitic Transformations
Symbols F: T: P: To : Ms : : as : f : a : : : Ep : E : Ms : i : i : Bi : R: S: P: E: x: x: L, B, T :
: kB : MB : Mf : R: Tan : : eijT : li :
and Abbreviations Helmoltz free energy Temperature Pressure Equilibrium transformation temperature Martensite start temperature Molar volume Surface area Chemical free energy change Applied stress tensor Microscopic strain Interfacial energy per unit area Total energy dissipated in plastic flow Elastic modulus of parent phase Stress required for martensitic transformation Principal strain with direction Principal distortions in the principal direction Bain strain matrix Rigid body rotation matrix Total shape strain matrix Lattice invariant shear matrix Total distortion matrix Composition Distance Plate dimensions (length, breadth and thickness) Twin fraction Boltzman constant Martensite burst temperature Martensite finish temperature Gas constant Magnitude of shear Poison ratio Stress free transformation strain Direction cosine of the position vector 259
260
Phase Transformations: Titanium and Zirconium Alloys
V: Uj (s): DSA: IPS: LIS: b: Ms : Md : : : : p : Txy : h: : :
1 2 : stat : m : ∗: A∗ : T: As : Af : : T :
4.1
Volume of the inclusion Displacement at point x Degree of self-accommodation Invariant plane strain Lattice invariant shear Burgers vector Temperature above plastic yield starts after martensitic transformation Stress required for stress-assisted martensite nucleation Difference between the flow stresses of and Flow stress of Flow stress of True plastic strain Stress required to move a dislocation out of a small angle boundary Spacing between the dislocations Shear modulus Number of dislocations Geometric slip distances for and Independent component of flow stress Athermal component of flow stress Thermal component of flow stress Activation area Homogeneous deformation matrix Austenite start temperature Austenite finish temperature Reversion stress Reversion temperature
INTRODUCTION
Martensitic transformations take place in numerous materials. Evidences of their occurrence have been found in several pure metals such as Fe, Co, Hg, Li, Ti, Zr, U and Pu, in many ferrous and non-ferrous alloys and in several oxides and intermetallic compounds such as ZrO2 , BaTiO3 , V3 Si, Nb3 Sn, NiTi and NiAl. Some years ago, the word “martensite” was used solely to describe a microconstituent in quench hardened steels. Bain (1924) put forward a mechanism for the transformation of the face centred cubic austenite to the body centred tetragonal martensite
Martensitic Transformations
261
in steels in which the structural change was considered to be brought about by a homogenous deformation of the parent lattice. It was implicit in this description that the transformation did not involve random or diffusive atom movements and that it resulted from only small relative displacements of neighbouring atoms. The fact that a similar mechanism is operative in a large number of solid state phase transformations has led to a proliferation of the use of the terms “martensite” or “martensitic” in a much wider sphere. As indicated in Chapter 3, martensitic transformations are grouped in the general class of displacive transformations and belong to that subset which involves the operation of a lattice deformation. The characteristic features of martensitic transformations are described in the following section to provide a background for discussions on martensitic transformations in alloys, intermetallics and ceramics based on Ti and Zr.
4.2
GENERAL FEATURES OF MARTENSITIC TRANSFORMATIONS
Martensitic transformations are characterized by a number of thermodynamic, kinetic, crystallographic and mechanistic features. Experimental observables pertaining to each of these are needed to unequivocally qualify a transformation to be martensitic in nature. This section is devoted to a brief discussion on these aspects. 4.2.1 Thermodynamics The first and foremost condition of a martensitic transformation is that the product phase inherits the composition of the parent phase. For a single component system like a pure metal, the driving force for the transformation to occur can be represented with the help of Helmholtz free energy (F ) versus temperature (T ) or pressure (P) plots. Taking the example of pure Fe, the chemical free energy change, F , accompanying a transformation from the austenite to the ferrite phase, can be expressed as follows: F = −1202 + 263 × 10−3 T 2 − 154 × 10−6 T 3 cal/mol 200 K < T < 900 K F = −1474 + 340 × 10−3 T 2 − 200 × 10−6 T 3 cal/mol 800 K < T < 1000 K (4.1) While the former expression is due to Kaufman and Cohen (1956, 1958), the latter was proposed by Owen and Gilbert (1960). This change in free energy
262
Phase Transformations: Titanium and Zirconium Alloys 1800
ΔF α→γ = 1202–2.63 × 10–3T 2 + 1.54 × 10–6T 3 Fe
ΔF Fe α→γ (cal/mol)
1600 1400 1200 1000 800 600 400
0
100
200
300
400
500
600
700
800
Temperature (K)
Figure 4.1. The free energy change accompanying the → in pure Fe as a function of temperature.
keeps on increasing with a lowering of the temperature from the austenite/ferrite equilibrium transition temperature, 1183 K (Figure 4.1). When a multicomponent system is considered, the chemical driving force is given by the drop in the free energy of the system as the parent phase transforms into the product, retaining the initial chemical composition. In this sense, the system behaves as if the transformation is occurring in a single component system. This point can be illustrated by taking the example of the Fe–Ni alloy system, the phase diagram and the free energy – composition diagrams which are shown in Figure 4.2(a) and (b), respectively. The equilibrium condition between the austenite and the ferrite phases can be identified by constructing a common tangent which locates the compositions of the two phases in equilibrium. When such partitioning of the alloying element is suppressed by a rapid quench, the composition, xo , can be defined at which the integral molar free energies of the two phases are equal at To . A composition-invariant transition from austenite to ferrite is thermodynamically possible only below this To temperature. The composition dependence of To is superimposed on the phase diagram in Figure 4.2(a). Martensitic transformations, like any other first-order transformation, do not start at To , where F = 0 but are initiated when some supercooling is provided. The temperature at which a martensitic transformation “starts” is known as the Ms temperature. The difference, To − Ms , indicates the extent of supercooling required
+1200
x=0 x = 0.05
+1000
–1000
x = 0.10 x = 0.15
+800
–800
Ms
x = 0.35
+200
–400 –200
0
0
–200
+200
–400
+400
–600
+600 400
600 800 1000 1200
Temperature (K) (a)
Ms ½ (Ms + As) γ
800 700 600
α+γ
Ad
500 400
α
α+γ
300
As
200
As
1100
900
–600
ΔF γ→α′ = F α′– Fγ Temperature (K)
ΔF α→γ = F γ – Fα′
+400
x = 0.25 x = 0.30
1200
1000
x = 0.20
+600
263
–1200
cal/mol
cal/mol
Martensitic Transformations
Md
To (calc.)
200 100 00
Fe
10
20
30
40
50
x (Ni at. %) (b)
Figure 4.2. (a) Experimental and theoretical determination of To in the Fe–Ni system. (b) Chemical free energy change accompanying the martensite transformation in the Fe–Ni system.
to initiate the transformation. For a number of ferrous alloys, this difference is about 200 K, while for the alloys based on Ti and Zr, it is much lower (≈50 K). The requirement of supercooling arises from the necessity of overcoming the following energy components which oppose the transformation: (a) the interfacial energy between the martensite and the parent matrix, (b) the elastic energy stored in the martensite – parent assembly to accommodate the shape change and the volume change accompanying the transition, (c) the energy dissipated in plastic deformation of both the martensite and the parent phases and (d) the driving force required for the rapid propagation of the martensite interface. Apart from the chemical free energy change, F , an applied stress can contribute to the driving force for a martensitic transformation. This is amply demonstrated in several alloy systems where the transformation can be induced at a temperature higher than the Ms temperature by applying stress. This effect can be attributed to the interaction of the applied stress field with the shape strain involved in the formation of a martensite plate. Provided the interaction has the correct sign, the formation of a plate will relieve the potential of the applied stress field. The formation of a small region of martensite in the presence of a stress field will release a small amount of mechanical energy, which may be positive or negative depending on the nature of the stress field and the orientation of the plate.
264
Phase Transformations: Titanium and Zirconium Alloys
Considering the driving forces arising due to the chemical free energy change and the applied stress and the restraining forces associated with the four factors listed earlier, the following energy balance expression can be written for the initiation of a martensitic transformation: vf + A1 ≥ As + vA2 E 2 + Ep
(4.2)
where the volume and the surface area of the martensite plate are denoted by v and as , respectively, f is the free energy change per unit volume of the martensite, a and are tensors representing, respectively, the applied stress and the macroscopic strain associated with the transformation, is the interfacial energy per unit surface area of the plate, E is the elastic modulus of the parent phase, Ep is the total energy dissipated in plastic flow and in imparting a high velocity to the martensite interface and A1 and A2 are dimensionless geometrical factors. The specific interfacial energy, , between the martensite and the matrix depends on the extent of coherency at the interface. Since atom transfer across the transformation front (interface) occurs through coordinated and highly disciplined atom movements (like regimented movements during a change in a military formation), the maintenance of coherency at the interface becomes a necessary condition. The presence of an array of dislocations at the interfaces arises out of a geometrical necessity as will be discussed in a later section. The changes in the shape and in the specific volume associated with the formation of a martensite plate of a given geometry in the matrix result in the development of a strain, both within the plate and in the matrix. The partitioning of the strain between the two phases, however, depends on the respective values of their elastic moduli. The strain so developed is accommodated either by an elastic deformation of the assembly or by a combination of plastic flow and elastic deformation. The latter situation prevails when the accommodation stress developed exceeds the flow stress in either of the phases. The driving force for the martensitic transformation must exceed the corresponding restraining force for the growth of the transformation product. The difference between the driving and the restraining forces is utilized in moving the interfacial dislocations. For a conservative movement of dislocations, the Peierls stress and the other internal stresses opposing their motion need to be overcome. The growth velocity of martensite interfaces, measured by the rate of change of electrical resistance of samples undergoing a martensitic transformation, has been found to be about one-third the velocity of elastic waves in the parent phase. This growth velocity was also found by Bunshah and Mehl (1953) to be essentially constant at all temperatures between 73 and 293 K for both types of martensites which show athermal and isothermal kinetics for overall growth. The facts that the growth
Martensitic Transformations
265
velocity is very high even at cryogenic temperatures and that it is independent of temperature suggest that the growth process is athermal. Rapid growth is commonly encountered when the transformation is driven by large driving forces and is thus adiabatic. The interface can, therefore, accelerate rapidly up to its limiting velocity, which is of the same order as the velocity of crack propagation or of twin formation. Such a growth of an isolated plate can cause plastic deformation in the matrix which, in turn, results in the loss of coherency at the interface and in the nucleation of fresh plates in the adjoining untransformed regions. The growth of the primary plate ceases at this point. Once the coherency at the interface is lost, it is not possible to reactivate its motion by changing the driving force (either by heating/cooling or by deformation). In contrast to the scenario described above, a martensite plate can reach a thermoelastic equilibrium when it assumes its full size under a given condition of temperature and applied stress. This can happen if the driving force (having chemical as well as mechanical components) exactly balances the restraining force arising from the surface energy and the elastic strain energy. The basic requirements for attaining a thermoelastic equilibrium are, therefore, that the elastic stress limits in the parent and the product phases should be high and that the shape strain associated with the transformation should be small. As the strain energy builds up with the growth of a plate, a thermoelastic equilibrium is established when the plate assumes a certain critical size. In such a situation, the interface retains complete coherency and is amenable to movement in either direction, leading to the growth or the shrinkage of the plate, depending on the magnitude of the driving force. As pointed out earlier, a supercooling to the extent of To −Ms is needed to induce spontaneous nucleation of martensite plates. Martensite plates can, however, be nucleated at temperatures higher than Ms if additional driving force is provided by an applied stress. The influence of such an applied stress on the martensitic transformation can be explained by using a schematic diagram (Figure 4.3) which was originally presented by Olson and Cohen (1972). It can be seen that the stress required for martensite formation increases linearly as the temperature rises from Ms to Ms ; beyond this point, plastic deformation of the parent phase sets in. In the temperature range, Ms < T < Ms , the applied stress complements the chemical driving force which decreases linearly with increasing temperature. Martensite nucleation in this temperature range is stress-assisted. At temperatures higher than Ms , the elastic driving force derived from the applied stress is inadequate to satisfy the requirement for martensitic nucleation. As the applied stress exceeds the flow stress of the parent phase, plastic deformation causes the creation of fresh martensite nuclei and the formation of strain-induced plates. The features shown in Figure 4.3 have been explained in detail in Section 4.3.4.
266
Phase Transformations: Titanium and Zirconium Alloys
b
Str tra ain-i n s nd for uc ma ed tio n
σ2
a
Str tra ess-a ns for ssis ma ted tio n
Applied stress
c
σ1
Ms
T1
0.2% pro of au of stress stenit e
T2 Msσ Temperature
Md
Figure 4.3. Schematic diagram showing the critical stress for martensite formation in a typical ferrous alloy as a function of temperature.
4.2.2 Crystallography The crystal geometry associated with martensitic transformations in various systems has been found to be governed by invariant plane strain (IPS) considerations which will be discussed in this section. The validity of the IPS criterion in predicting the transformation geometry is so overwhelming that sometimes a transformation is identified as martensitic purely on the basis of this geometrical criterion. This approach, however, is currently being questioned since some diffusional transformations have also been shown to exhibit geometrical features predictable from IPS considerations. In this section, the essential points concerning the phenomenological theory of martensite crystallography, developed independently by Wechsler et al. (W-L-R)(1953) and by Bowles and Mackenzie (B-M) (1954), will be discussed. The important geometrical features of martensitic transformations are listed below: (1) The formation of a martensite plate in a grain of the parent phase creates upheavals (surface relief) on a polished reference surface of the parent grain. This is illustrated in a schematic drawing in Figure 4.4 which shows the macroscopic shear produced in a parent crystal in which a martensite plate is formed. Observations on the displacement of reference lines drawn on the surface of the crystal indicate that all reference straight lines are transformed
Martensitic Transformations
267
D M4
C P
M3
P4
M2 P3
A
e nit
Ma r
P1
B
ste Au M ten sit e
M1
P2
P
ite
ten
s Au
Figure 4.4. The shape deformation due to formation of a martensite plate. Surface M1 M2 M3 M4 remains plane and tilted about M1 M2 and M3 M4 . The straight line AD marked on austenite is transformed into ABCD, where the segment BC within the martensite plate remains a straight line after the transformation. There is no discontinuity at points B and C, which are at the martensite– austenite interface, indicating that the interface is undistorted and unrotated.
into straight lines and all reference planes into planes in the product martensite. This implies that the transformation strain is linear and, therefore, can be expressed in the form of a matrix. Such a transformation is described mathematically as an affine transformation. The fact that no discontinuity is produced at the interface plane separating the martensite plate and the matrix indicates that the interface plane (habit plane) is an undistorted and unrotated plane (invariant plane). (2) The habit plane which is seen to be characteristic of a specific transformation is generally irrational. (3) A precise reproducible orientation relation is invariably present between the parent and the martensite crystals, as revealed from diffraction experiments. (4) Martensite plates very often contain a periodic arrangement of internal twins. The concept of lattice strain which came from the suggestion of Bain (1924) is illustrated schematically in Figure 4.5 wherein the fcc austenite is converted to the body centred cubic (bcc) ferrite by a single “upsetting” process in which the dimensions of the fcc unit cell are altered to those of the bcc unit cell by a homogeneous deformation of the parent lattice requiring only small shifts in the atom positions. The three vectors chosen to define unit cells in this description are mutually perpendicular before and after the lattice transformation. In general,
268
Phase Transformations: Titanium and Zirconium Alloys [101]A → [111]M
X3, X′3
(101)A → (112)M
X′2
X1
X2
X′1 (a)
ao
c
a o / √2 (b)
a (c)
Figure 4.5. Lattice correspondence and lattice deformation for the fcc to bct austenite–martensite transformation in Fe alloys.
in a homogeneous deformation, it is always possible to select three mutually perpendicular vectors (say, X1 , X2 and X3 ) which remain perpendicular after the deformation, and these are called the principal axes of deformation. When a volume of the parent phase, represented by a unit sphere, is subjected to a homogeneous strain, it is transformed into an ellipsoid. The construction of strain ellipsoids (Figure 4.6) illustrates the conditions for the homogeneous strain to have at least one plane undistorted. When the principal strains associated with the homogeneous strain are all positive or all negative, the strain ellipsoid does not intersect the unit sphere at all, implying that not a single vector remains undistorted by the homogeneous strain. Such a situation is shown in Figure 4.6(a). For a plane to remain undistorted, the necessary and sufficient conditions are that one of the principal strains should be zero while the other two should be, respectively, positive and negative. If the principal strains, 1 , 2 and 3 , are such that 1 is zero,
Martensitic Transformations X3
269 X3
B′
η3 = l+ε3
A′ A
B
l
η2 = l+ε2
(a)
X2
X2
(b)
Figure 4.6. Deformation of a unit sphere into an ellipsoid by homogeneous lattice strain (Bain strain) (a) 1 2 3 > 1 and (b) 1 = 1, 2 < 1, 3 > 1. The details are explained in the text.
2 is negative and 3 is positive (as illustrated in Figure 4.6(b)), the strain ellipsoid will touch the sphere at the point of intersection of the X1 axis with the unit sphere and will intersect the sphere at two points A and B on the plane containing the X2 and X3 axes. The planes defined by OA × X1 and OB × X1 vectors remain undistorted though they are rotated from their original positions, defined by the planes OA × X1 and OB × X1 vectors. An examination of the Bain strain necessary for the deformation of the parent lattice into the product lattice reveals that, in general, the Bain strain or lattice strain alone does not satisfy the aforementioned conditions which ensure at least one undistorted plane. Moreover, the macroscopic shape strains measured from the surface relief observations in several martensites do not match with the respective Bain strains. It is because of these two factors that the concept of a second shear was invoked in the martensite crystallography. While the Bain strain is responsible for bringing about the change in the lattice, the second shear, observed as the lattice invariant shear (LIS), superimposed on the Bain strain makes the total shear satisfy the undistorted plane condition. The necessity of a second shear can be explained by citing a specific example. The lattice or Bain distortion, B1 , necessary for transforming the (fcc) austenite
270
Phase Transformations: Titanium and Zirconium Alloys
(with lattice parameter ao ) into the (bct) martensite (with lattice parameters a and c) can be expressed in terms of a matrix: ⎡
1 B1 = ⎣ 0 0
0 2 0
⎤ 0 0⎦ 3
(4.3)
√ where 1 = 2 = a 2/ao and 3 = c/ao Substituting the lattice parameter values for a carbon steel, one finds that a tensile strain of 12% in all directions perpendicular to the c-axis (X3 -axis, marked in Figure 4.5) and a compression of 17% along the c-axis are required for upsetting the lattice from fcc to bct. It is obvious that this lattice deformation cannot satisfy the condition for having an undistorted plane. Therefore, the total macroscopic shear, which is experimentally shown to be an IPS, must consist of additional components which, in conjunction with the lattice shear, satisfy the IPS condition. The same conclusion was arrived at, before the phenomenological crystallographic theory (W-L-R and B-M) was introduced, through an elegant experiment by Greninger and Troiano (1949). They experimentally determined the magnitude of the macroscopic shear from observations on the surface relief produced due to the martensitic transformation in an Fe–22% Ni–0.8% C alloy. They noticed that the experimentally measured macroscopic shear, when applied to the parent austenite lattice, did not generate the martensite lattice. In order to account for the observed difference between the macroscopic strain and the lattice strain, an LIS (either slip or twinning) has been introduced as a component of the total strain. The phenomenological theory of martensite crystallography is based on the postulate that the habit plane (the interface separating the parent and the product phases) is not an atomistically flat plane which remains invariant on a microscopic scale during the transformation. Misfits between the two structures develop and the accumulated misfits periodically get corrected to establish an average or macroscopic fit. The essence of the theory can be described by a set of schematic drawings (Figure 4.7). Let us consider the transformation of the two-dimensional lattice shown in Figure 4.7(a) to that shown in Figure 4.7(b), the corresponding unit cells being indicated by thick lines. The required lattice strain which brings about the change in the lattice also produces a shape strain; this is reflected in the rotation of the vector AB in the parent lattice to the vector A B in the product lattice. The magnitude of the vector A B can be brought back to the magnitude of the vector AB without changing the product lattice by the introduction of an LIS either by slip or by twinning. The geometries associated with these options are illustrated in Figure 4.7(c) and (d), respectively. In the case of the LIS being provided by slip, the product martensite plate consists of a single
Martensitic Transformations
271
B′
B
A Initial crystal (a)
A′
After lattice deformation (b)
B′
B′
A′ Lattice deformation followed by slip shear (c)
A′ Two lattice deformations leading to twin related regions (d)
Figure 4.7. Schematics showing (a) untransformed crystal, (b) after undergoing a lattice deformation, (c) the additional effect of a slip shear and (d) crystal having alternately twined regions. (c) and (d) show that a combination of lattice deformation and lattice invariant deformations (slip or twin) can make the habit plane an invariant plane.
variant of the martensite crystal. However, if the LIS is provided by twinning, two twin-related martensite variants form within a single martensite plate. In order to bring the vector A B into coincidence with the original vector AB, an additional rigid body rotation is necessary. The total macroscopic shape strain (S), which has to satisfy the IPS criterion, is, therefore, conceptually divided into components, namely the lattice strain (B) which is responsible for changing the parent lattice into the product lattice, the LIS (P), which, on being superimposed on the lattice shear, establishes an undistorted plane, and a rigid body rotation (R), which ensures that the undistorted plane is unrotated as well, S = RPB. In the case of twinning as the LIS, it is necessary to satisfy another symmetry criterion. The two twin-related orientations in the product phase evolve from a single parent phase crystal. It is, therefore, necessary that crystallographically equivalent lattice strains are operative in the adjacent regions which transform into a pair of twin-related product orientations of the product crystals. The formation of such a configuration is also expected from the consideration of symmetry breaking in a phase transformation. If the number of symmetry elements of the parent crystal gets reduced due to a transformation process, there is a general tendency for the restoration
272
Phase Transformations: Titanium and Zirconium Alloys B2
Mirror plane
φ2
B2 B D1
C2 C1
φ1 2
C2 C
B1
C1 A2
A1
1
A2 A1
D2 D B1
Figure 4.8. Schematics showing restoration of the symmetry in a macroscopic sense through the creation of a number of crystallographic variants.
of the symmetry in a macroscopic sense through the creation of a number of crystallographic variants. This can be illustrated in a two-dimensional construction (Figure 4.8) in which a parent square lattice ABCD is transformed into two equivalent rectangular lattices, A1 B1 C1 D1 and A2 B2 C2 D2 . When these two rectangular regions are rotated to bring them into coincidence along their diagonals, A1 C1 and A2 C2 , a twin is created where the twin plane is derived from a mirror plane in the parent crystal. In fact, the mirror symmetry of the parent crystal on this plane is lost due to the transformation, and the formation of the twinned product crystals tends to restore, at least partially, the lost mirror symmetry. The relative volumes of the two orientations, usually expressed in terms of the ratio of the thicknesses of the adjacent twins, are determined by the requirement of the lattice invariant deformation necessary to satisfy the IPS condition. Referring back to the transformation described in Figure 4.5, the lattice (Bain) distortions, B1 , associated with the two adjacent twin-related variants can be represented in the (i1 j1 k1 ) and (i2 j2 k2 ) principal axes systems respectively by the matrices ⎡ ⎡ ⎤ ⎤ 0 0 1 0 1 0 B1 = ⎣ 0 2 0 ⎦ and B2 = ⎣ 0 3 0 ⎦ (4.4) 0 0 3 0 0 2 These two matrices can then be expressed in the axis system of the parent crystal, (i j k), by the standard similarity transformation procedure, which involves
Martensitic Transformations
273
rotations of the axes systems from the basis of the martensite crystal to that of the parent crystal. The Bain strains, B1 and B2 , can be represented in the axis system (i j k) as B1 and B2 , respectively: ⎡ ⎤ ⎡ + ⎤ 2 − 1 1 + 2 2 − 1 1 2 0 0 ⎢ 2 2 ⎥ ⎢ 2 ⎥ 2 ⎢ ⎥ ⎢ ⎥ ⎥ and B2 = ⎢ 1 + 2 2 − 1 0 3 0 ⎥ B1 = ⎢ ⎢ ⎥ ⎢ 0⎥ ⎢ ⎥ ⎣ 2 ⎦ 2 ⎣ − 1 + 2 ⎦ 2 1 0 0 0 3 2 2 (4.5) In order to bring the two adjacent regions into twin-related orientations, it is necessary to introduce rigid body rotations, 1 and 2 , to the regions marked 1 and 2, respectively, in Figure 4.8. Therefore, 1 and 2 describe the rotations of the principal axes of the pure distortions in regions 1 and 2 relative to an axis system fixed in the untransformed parent phase. Figure 4.9 shows an arbitrary vector r (represented by the straight line OV) in the parent phase which becomes a zigzag line OA B C D U V in the twinned V
2 1
V′
U′
2 OV = r 1 2 C′
OV′ = r′ D′
1 (1–x )
2 A′ 1
B′ (x )
O
Figure 4.9. Schematic appearance of internally twinned martensite minor and major regions, which undergo the lattice deformation along different but crystallographically equivalent principal axes.
274
Phase Transformations: Titanium and Zirconium Alloys
martensite crystal where the fractional thicknesses of the two constituent variants 1 and 2 are (1 − x) and x, respectively. The vector r is transformed into the vector r , the latter being an average of the segments OA , A B , B C U V . Thus r is the vector sum OV = OA + A B + B C + · · · · · + U V and can be expressed in terms of the pure lattice distortions and rigid body rotation as
or where
r = 1 − x1 B1 + x2 B2 r
(4.6)
r = Er
(4.7)
E = 1 − x1 B1 + x2 B2
(4.8)
The total distortion matrix, E, when it operates on any vector in the parent lattice, produces the corresponding vector in the transformed twinned martensite. Since the habit plane is an undistorted and unrotated plane, any vector lying on this plane will satisfy the following condition: Er = r
(4.9)
The rigid body rotations in the regions 1 and 2, as given by 1 and 2 , bring the planes (represented by AC and A2 C2 in Figure 4.8) derived from the mirror plane into coincidence. A rotation (2 = 1 ), which gives the relative rotation between 1 and 2 , can be defined and the total macroscopic shear can be expressed as E = 1 1 − xB1 + xB2 = 1 G
(4.10)
where G = 1 − xB1 + x B2 The macroscopic distortion thus has the following three components: 1 , a rigid body rotation: G, a fraction of the twin shear; and the Bain strain, B1 . The matrix algebra problem then reduces to an eigenvalue problem for vectors in the habit plane with solutions (if they exist) only for certain values of x, fraction of the twin shear (or the magnitude of the LIS). From a knowledge of the lattice parameters of and the lattice correspondence between the parent and the product lattices and of the twinning system, it is possible to predict the indices of the macroscopic habit plane, the orientation relationship and the twin thickness ratio, which are all experimental observables. The success of the phenomenological crystallographic theory is well documented in the extensive literature wherein the predicted habit planes in different systems have been shown to match closely with those experimentally determined (Table 4.1).
Martensitic Transformations
275
Table 4.1. Comparison of experimentally determined and theoretically computed crystallographic parameters of martensites in different systems. System
Habit plane Exp.
fcc–bcc Fe–30.9Ni
0.1656
Orientation relationship
Comp.
Exp.
0.1848
0.7998
0.7823
0.5771
0.5948
Comp.
(111)f ∧(011)b ¯ f ∧(011) ¯ b 1¯ 12
0.3
0.54
2.2
1.67
¯ f ∧ 1¯ 11 ¯ b 101
−24
−362
(111)f ∧(011)b ¯ f ∧(011) ¯ b 1¯ 12
0.86 ± 0.10
0.83
¯ f ∧ 1¯ 11 ¯ b 101
4.42 ± 0.10
−444
<1
15
2
1.9
−25
−27
Breedis and Wayman (1962)
Diff 1.8
fcc–bcc
0.1910
0.1562
Fe–24.5Pt
0.7599
0.7404
Ejsic and Wayman (1967)
0.6214
0.6537
Diff 2.56
Fe–22Ni–8C
0.8208
0.8027
(111)f ∧(011)b ¯ f ∧(011) ¯ b 1¯ 12
Greninger and Troiano (1981)
0.5472
0.5691
¯ f ∧ 1¯ 11 ¯ b [101]
fcc–bcc
fcc–hcp Au–47Cd Liberman et al. (1955)
0.1642
0.696 −0686 0.213
0.1783
0.6968
(001)c ∧(001)o
0
2.4
−06810
[111]c ∧[011]o
0
18
0.2250 <15
The restriction regarding the selection of the system of lattice invariant deformation in a twinned martensite plate (the twinning plane being necessarily derived from a mirror plane of the parent crystal) does not apply when the LIS occurs by slip. Despite the physical difference between the two types of inhomogeneous shear (LIS, slip and twin), the macroscopic strain due to the transformation can be described by the same formal theory. There is, however, no simple criterion which can be used for selecting the slip system operating as the LIS in a given martensitic transformation. A comparison of the magnitudes of the LIS required for different possible slip systems to satisfy the IPS conditions reveals that the operating slip system is often associated with the smallest shear magnitude. Returning to the general theory, it must be emphasized that the various components of strain described as the constituent parts of the overall macroscopic deformation are not to be considered to be associated with separate physical steps. It seems probable that at the interface each atom moves directly to the nearest site of the product lattice, and the break-up of different shear components as described in the theory is not real in either a spatial or a temporal sense.
276
Phase Transformations: Titanium and Zirconium Alloys
The formulation developed by Bowles and Mackenzie (1954) is slightly different from that due to Wechsler et al. (1953) outlined earlier. Bowles and Mackenzie considered that the shape deformation, E, combined with a “complementary” or “invisible” lattice deformation, H, could accomplish the change in structure. The macroscopic effects of H are not detectable because these are counterbalanced by an opposite lattice invariant deformation, G , which is assumed to be a simple shear. The total lattice deformation is thus factorized into an IPS E and a simple shear H so that 1 B = HE
(4.11)
The left hand side of Eq. (4.9) represents the combination of the rigid body rotation and the Bain strain. The total lattice strain, 1 B, is a product of two invariant plane strains and, therefore, satisfies the condition of an invariant line strain. As mentioned earlier, the lattice invariant deformation G produces a shape change equal and opposite to that due to H (G H = I, the unit matrix). The operation G on Eq. (4.9) leads to G B = G HE = E
(4.12)
which is essentially equivalent to Eq (4.7) provided G 1 = 1 G. The matrices G and G represent the same lattice invariant deformation applied, respectively, before and after the rotation 1 . Bowles and Mackenzie introduced an additional variable, namely the dilation parameter, . The condition of the habit plane being an IPS was relaxed and the habit plane was allowed to dilate uniformly to a small extent (up to about 2%) without permitting any rotation of the vectors lying in the habit plane. The success of the phenomenological crystallographic theory in predicting the orientation relationship between the parent and the product structures, the habit plane resulting from the transformation and the magnitude of the lattice invariant deformation (twin fraction in a twinned martensite plate), has been remarkable, as illustrated in Table 4.1. The fact that there is a geometrical necessity of an LIS in most martensitic transformations is reflected in the presence of a fine structure of martensite plates, consisting of internal twins, stacking faults or arrays of dislocations at the martensite/parent interface. The magnitude of the lattice invariant strain can be determined experimentally in twinned martensite plates by measuring the ratio of the thicknesses of adjacent twinned components. Martensite plates in which the LIS occurs by slip often contain dislocation debris which decorate the slip plane. Though the operating slip plane and the slip direction can be tentatively identified from the geometry and the Burgers vector of these dislocations, the magnitude of the LIS is not measurable from experimental observations.
Martensitic Transformations
277
Martensite morphology is yet another important observable which can be characterized through detailed metallographic investigations. The word “morphology” in the context of martensites refers to the shape of individual martensite units and the nature of the assemblage of a group of neighbouring martensite units. Factors such as the minimization of the strain energy, the maintenance of a glissile coherent interface and the non-isotropic growth of martensitic products are responsible for a large majority of martensite crystals assuming the shape of lenticular plates as in mechanical twinning. However, there are many alloy systems in which the martensite/parent interface is essentially planar and thus the martensite unit is slab-like (rather than than lens-like). There are also examples of what is known as the single interface transformation, as in Au–Cd (Lieberman et al. 1957) and In–Tl (Basinski and Christian 1954) alloys in which an internally twinned martensite bicrystal can be induced to grow under a temperature gradient imposed on a single crystal of the parent phase. In such a situation, it is easier to propagate a single interface of the martensite unit which nucleates first than to repeatedly nucleate fresh martensite units. In a number of ferrous alloys, generally, with low contents of substitutional and interstitial solutes and exhibiting correspondingly high Ms temperatures, the martensite units are lath-like. If the length, the breadth and thickness of a martensite unit are designated as L, B and T , respectively, then L = B >> T for plates and L > B > T for laths. Apart from the differences in the relative external dimensions, the lath and the plate morphologies differ with respect to the nature of the assembly of the martensite units. While a plate morphology is characterized by groups of martensite units of differing orientation and habit plane variants, the distinctive feature of the lath morphology is the occurrence of groups of near parallel martensite units which are separated from each other by small angle boundaries. In some alloys, adjacent martensite laths exhibit a twin relation. The grouping of martensite units in some typical patterns is motivated essentially by their tendency towards self-accommodation. The overall strain energy of the system can be substantially reduced by an appropriate grouping of martensite units as is reflected in the formation of energetically favourable polydomain morphologies in several martensites. The issue of self-accommodation is of primary importance in shape memory alloys and will be addressed in detail in Section 4.5.5. 4.2.3 Kinetics Being a first-order-type transition, martensitic transformations occur by the nucleation and growth process. The overall kinetics of the transformation in most cases are athermal. This can be represented in a plot of fraction transformed versus temperature (Figure 4.10(a) and (b)). A transformation on cooling begins at the Ms
278
Phase Transformations: Titanium and Zirconium Alloys
~100%
T1
Martensite (%)
Martensite (%)
Martensite (%)
100%
MS Mf
Mf
Temperature (a)
Ms > T1 > Mf
Mb
Temperature (b)
Time (c)
Figure 4.10. Kinetics of the martensite transformation as a plot of fraction transformed versus temperature for athermal martensite (a), athermal burst martensite (b) and overall transformation kinetics for isothermal martensite (c).
temperature; the extent of the transformation progressively increases with lowering of temperature and it is completed, finally attaining the complete transformation at the temperature Mf , which is known as the martensite finish temperature (Figure 4.10(a)). In some cases of athermal martensitic transformation, the volume fraction transformed at Ms shows a sharp rise in a burst, as shown in Figure 4.10(b); therefore, the start temperature is designated as burst temperature (Mb ). The time taken to reach the indicated fraction transformed at any given temperature between Ms and Mf is very short, and longer holding at the same temperature does not result in further increase in the fraction transformed. In another variety of martensitic transformations, overall isothermal characteristics are exhibited in the overall transformation kinetics (Figure 4.10(c)). In these cases, the volume fraction of the martensitic phase keeps on increasing with time at any given temperature between Ms and Mf . In such cases, although the growth of martensite units occurs by the characteristic athermal movement of glissile interfaces, the process of nucleation is thermally activated. Experimental studies on martensite nucleation have been carried out quite extensively in systems which show the isothermal behaviour. From such studies, it has been established that a homogeneous nucleation process through thermal activation cannot account for the observed nucleation at very low temperatures (even at temperatures approaching 0 K). Kaufman and Cohen (1956) invoked the presence of pre-existing martensite embryos to explain the observed nucleation phenomenon. The same anomaly is encountered if one considers the nucleation of martensite in an ideally perfect parent crystal. Assuming the shape of a martensite nucleus to be a thin oblate spheroid and choosing realistic values of the chemical free energy change due to the transformation, the surface energy
Martensitic Transformations
279
and the strain energy of the assembly of the nucleus and the matrix, the nucleation barrier (F ) can be estimated to be about 5 × 103 eV per nucleation event. This corresponds to about 105 kT at temperatures where nucleation is experimentally observed. This indicates that the thermal energy is much too small for homogeneous nucleation to occur. The postulation of pre-existing embryos which could act as heterogeneous sites for martensite nucleation was recognized to resolve this anomaly. In the early stages of the development of the theory of martensite nucleation, these embryos were conceived as being structurally similar to the martensite phase (such as bcc embryos in fcc austenite in ferrous alloys). However, no experimental evidence in support of the presence of such embryos could be obtained. Though there is a general concurrence on the requirement of heterogeneous nucleation in martensitic transformations, a precise structural description of the heterogeneities at which nucleation occurs is still not available. Olson and Cohen (1976) proposed a general mechanism of martensite nucleation by faulting of groups of existing dislocations. For the fcc → bcc transformation, martensite nucleation can be considered in terms of the splitting of a group of dislocations which form a parallel array, one above the other, in the parent phase. Movement of partial dislocations subsequent to dissociation produces a martensite nucleus bounded by a coherent interface. There is an alternative approach to seeking an answer to the problem of martensite nucleation. In a number of systems, martensitic transformations are preceded by precursor phenomena, usually known as “premartensitic” effects. Elastic moduli of the parent phase in several systems “soften” prior to the transformation. Clapp (1973) proposed a “strain spinodal” approach according to which a localized soft phonon mode may operate near a lattice defect, resulting in the nucleation of the martensite phase. The possibility of heterophase fluctuations aided by elastic interactions with pre-existing dislocations to produce martensite has also been considered. There have been only a few investigations in which the growth velocities of martensite plates have been experimentally measured. Two types of martensites have been encountered, one grows very rapidly and the other at a much slower pace; the former is termed “umklapp” and the later “schiebung”. From “in situ” monitoring of resistivity, the growth velocity of the martensite interface has been determined to be 1100 m/s for “umklapp” and 10−4 m/s for “schiebung”, respectively. The rapid movement of the martensitic interface is driven by the free energy difference between the parent and the martensitic phases. The interface can be visualized as being semicoherent in nature. The movement of the coherent segments is not opposed by any reactive force, while the movement of the interfacial dislocations involves Peierls stress due to
280
Phase Transformations: Titanium and Zirconium Alloys
lattice friction and other internal stresses opposing their motion. This movement may be thermally activated if the interaction of the internal strain fields with the stress fields of dislocations is short range, but usually the internal friction stress acts as a long-range stress and, therefore, an athermal movement of the interface is necessary. The dislocations at the interface which acts as the transformation front have two functions: to accommodate the misfit between the lattices of the parent and the martensite phases and to generate the LIS by their propagation. In fact, there are three possible criteria for the selection of the interfacial array of mismatch dislocations: (a) the criterion of the minimum interfacial energy, (b) the criterion of the minimum force required to move the array and (c) the criterion of the fulfilment of the requirement of LIS. Since all the criteria cannot always be satisfied by the same set of dislocations, the third criterion is often chosen for modelling the martensite interface. The fact that in a majority of systems martensite interfaces propagate very rapidly in an athermal manner suggests that the relationship between the velocity of the interface and the force causing its movement (which is derived from the free energy difference between the parent and the product phases at the transformation temperature) contains an instability. It is likely that the instability is due to the fact that the driving force required to nucleate a martensite plate is much greater than that required for its growth. 4.2.4 Summary Various features of martensitic transformations have been briefly presented in the foregoing sections. The experimental observables on the basis of which one can identify a transformation to be martensitic have also been mentioned in the course of the presentation. Based on these considerations, the atomistic mechanism of martensitic transformations has been conceived to involve jumps of atoms from the parent lattice sites to the product lattice sites in a coordinated or disciplined manner by maintaining a lattice correspondence. If one could label the atoms along a vector in the parent lattice, one would observe that the same atoms occupy sites along a vector in the product lattice in the same sequence. In the same way, a labelled plane in the parent becomes a similarly labelled plane in the product. This is what, as illustrated in Figure 4.5, is known as a lattice correspondence. The maintenance of a lattice correspondence is possible only if atomic jumps from a specific lattice site of the parent to a corresponding lattice site of the product are predestined. This is evidenced from the fact that the chemical order of the parent phase is inherited by the martensite phase. Table 4.2 summarizes the distinguishing features of martensitic transformations.
Martensitic Transformations
281
Table 4.2. Characteristic features of martensitic transformation. 1. 2. 3. 4. 5. 6.
7.
4.3
Coordinated/disciplined jumps of atoms from parent lattice sites to product lattice sites Strict lattice correspondence between the parent and the product lattices Strict orientation relation between the parent and the product lattices Occurrence of surface tilts representing the macroscopic shears associated with martensite plates Inheritance of the chemical composition and the state of atomic ordering from the parent to the martensitic phase Transformation through a nucleation and growth process, the nucleation step being either athermal or thermally activated, while the growth process is invariably athermal. The growth of martensite plate by a rapid movement of a glissile coherent interface in a manner similar to propagation of a shear front
BCC TO ORTHOHEXAGONAL MARTENSITIC TRANSFORMATION IN ALLOYS BASED ON Ti AND Zr
Pure Ti and Zr transform martensitically from the high-temperature (bcc) phase to the low-temperature (hcp) phase on quenching from the -phase field, provided the cooling rate exceeds a certain critical value. There have been only a few experimental investigations of the critical cooling rate for martensitic transformation in these pure metals. In order to suppress the competing diffusional transformation (massive → transformation), a quenching rate of several hundred degrees celsius per second is necessary in the case of these metals (if the total interstitial content is less than 200 ppm). Once the massive → transformation is bypassed, the same structural change occurs through the martensitic process with the attainment of the required supercooling at the Ms temperature which is about 50 K lower than the equilibrium → transition temperature. The Ms temperature for alloys of Ti and Zr is a function of the alloy composition. The Ms temperature of any given alloy is determined by the - or the -stabilizing tendencies and the amounts of the alloying elements present in it. There are three possible athermal transformation products in -quenched dilute alloys of Ti and Zr. These are the hcp and the orthorhombic martensites and the athermal phase which has a hexagonal crystal structure. The mechanism of the → transformation is quite different from those of the → or the → martensitic transformations and has been discussed in a separate chapter. The orthorhombic martensitic phase, which forms in alloys containing large concentrations of -stabilizing elements, can as well be considered as a distorted hexagonal phase,
282
Phase Transformations: Titanium and Zirconium Alloys
the orthorhombic distortion being introduced by a high level of supersaturation of alloying elements. The crystallography of the transformation can also be described in general terms for the bcc to orthorhombic structure, the latter encompassing the hcp structure as a special case. 4.3.1 Phase diagrams and Ms temperatures The range of alloy compositions over which the different athermal products, ,
and , form in a binary system based on Ti or Zr can be illustrated in a schematic isomorphous phase diagram (Figure 4.11). The Ms temperature for the martensitic transformation and the start temperature, s , as functions of xB , the atom fraction of a -stabilizing alloying element, are superimposed on this schematic phase diagram.
Ms
Tβ/α
Mf
T1
} for β → α′, α′′
Temperature
T2
ωs for β → ω
T4
α
T5
x1
Ti Zr
x2
x ′2
x3
xB Atom fraction of alloying element β-Quenched
α′
α″
β+ω
β
α″ + β + ω
α + β Alloy
Alloy classes
β Alloy
α Alloy
Figure 4.11. Basis for the classifications of commercial Ti and Zr alloys into alloys, + alloys and alloys.
Martensitic Transformations
283
On quenching from the -phase, alloys of different compositions exhibit different athermal transformation products. With reference to Figure 4.11, which is a modified version of Figure 1.18, alloys in the composition range 0 > xB > x1 produce (hcp) martensite, while those in the range x1 > xB > x2 transform into orthorhombic martensite, x1 defining the level of supersaturation at which orthorhombic distortion sets in. The plots corresponding to Mf and s as functions of xB intersect at xB = x2 . This implies that for alloy compositions xB < x2 , the martensitic transformation reaches completion during a quenching operation before the s temperature is encountered and therefore the product is fully martensitic. The temperature gap between Ms and Mf in most of the Ti- and Zr-based alloys is very small (in the range of 25 K, as reported in a few systems). Because of this reason, an incomplete martensitic transformation is not frequently observed. In the composition range of x2 < xB < x2 , the quenched structure consists of martensitic plates along with some untransformed -phase in which the -phase is finely distributed. In the composition range of x2 < xB < x3 , the quenched product contains a distribution of athermal -particles in the -matrix. The reason for the formation of a dual phase + structure in preference to a fully -structure will be discussed in the chapter on transformation. Quenching from the ( + ) phase field results in duplex microstructures. Depending on the temperature of equilibration in the ( + ) phase field, a wide variety of microstructures can be produced. This point can be illustrated by taking the example of the alloy composition, x4 . Table 4.3 lists the product phases in this alloy when quenched from the different temperatures, T1 , T2 , T3 and T4 as marked in Figure 4.11 which also shows the tie lines at these temperatures. Figure 4.11 could serve as a basis for the classification of commercial Ti alloys into , + and alloys. Since Zr alloys are almost exclusively used for structural applications in nuclear reactors, concentrated alloys do not find much use due to their high thermal neutron absorption cross-sections. Therefore, such a classification scheme is not in vogue for Zr alloys. However, from physical metallurgy considerations, Table 4.3. Microstructures produced in the alloy of composition x4 (as indicated in Figure 4.11) on quenching from different solutionizing temperatures. Solutionizing temperature
Microstructure
T1 T2 T3 T4 T5
Fully martensitic Primary + Primary + + Primary + + Primary +
284
Phase Transformations: Titanium and Zirconium Alloys
the same classification scheme is also applicable for Zr alloys. The classification scheme is briefly described in the following. Alpha alloys are those which on equilibration at temperatures close to 873 K consist of the single-phase hcp ( ) structure. Alloys which on quenching from the phase field retain the -phase with or without a distribution of the -phase are grouped as alloys. The composition range between those of the alloys and the alloys is covered by the ( +) alloys. These three composition ranges are marked in the schematic phase diagram in Figure 4.11. The composition ranges corresponding to these three classes of alloys can be better defined in ternary alloys where the stabilities of the and the -phases are balanced by suitable alloying additions. The Ti-Al-V system can be chosen as a good example. The pseudobinary phase diagram (Figure 4.12) shows an enlarged -phase stability region due to the presence of the strong -stabilizing element Al. A ternary composition Ti–6% Al–3.5% V marks the limit of the -alloys, while a complete suppression of the → martensitic transformation on quenching requires an addition of about 15% V in a Ti alloy containing 6% Al. This composition (6% Al, 15% V) marks the low V limit for the -alloys. As indicated earlier, the composition range between 3.5 and 15% V corresponds to the + class of alloys. It may be noted that a -quenching treatment given to the ( + ) alloys produces a fully martensitic ( or ) structure. On subsequent tempering, the -phase (and/or the pertinent intermetallic phases) can be precipitated in the tempered martensitic matrix of these alloys.
4 wt% Al vertical section
Temperature (K)
1000
900
800
700
0
6.9Al 93.1Tl 0V
1
2
3
4
5
6
7
Atomic percent V
Figure 4.12. A pseudobinary diagram of Ti-Al-V system.
8
9 10
6.9Al 83.0Tl 10.1V
Martensitic Transformations
285
As mentioned in Section 4.2.1, thermodynamic studies on martensitic transformations essentially involve the determination of the relative stabilities of the parent and the product phases and of the To temperature at which the two phases of identical composition possess the same value of the integral molar free energy. Using the values of the free energy change, F → , associated with the to phase transformation for pure Zr and Ti, Kaufman (1959) determined the To line for the isomorphous binary Ti–Zr system. An outline of this thermodynamic treatment is given here to illustrate the special case where both the pure metals involved exhibit similar allotropic transformations and have complete solid solubility in both the high- and the low-temperature phases. The free energies for the and the phases, F and F , in Ti–Zr solid solutions can be expressed as F = 1 − xFTi − xFZr − Fex + RTx ln x − 1 − x ln1 − x
(4.13)
F = 1 − xFTi − xFZr − Fex + RTx ln x − 1 − x ln1 − x
(4.14)
and
where FTi Zr are free energies of the pure components and Fex are the excess free energies of mixing for the and the phases; x is the fraction of Zr in the Ti–Zr alloy. The condition for equilibrium between the two phases is given by the following identities at any given temperature T :
F Ti x = F Ti x
and
F Zr x = F Zr x
(4.15)
where F Ti x and F Ti x are partial molar free energies or chemical potentials of Ti in the solid solution of compositions x and x representing the compositions of the and the phases in equilibrium at T (represented by tie lines drawn on the phase diagram in Figure 4.13. Equations (4.11) – (4.13) yield 1 − x Fex Fex
→
FTi + RT ln x − Fex − x x = Fex − x (4.16) 1 − x x x and FZr → + RT
x Fex Fex
x − Fex + 1 − x x (4.17) ln = Fex + 1 − x x x
The martensitic transformation being composition invariant, the chemical driving force for the → transition for a Ti–Zr alloy of composition x is given by F → = 1 − xFTi → + xFZr → + Fex →
(4.18)
286
Phase Transformations: Titanium and Zirconium Alloys xβ
xβ
1100
β (bcc)
Temperature (K)
To (calculated) 1000
α+β
xα
xα
900
Ms (Duwez (1951))
800
α (hcp)
700 0
10
20
30
Ti
40
50
60
70
80
Atomic percent Zr
90
100
Zr
Figure 4.13. Ti–Zr phase diagram showing the position of Ms temperature for Ti–Zr alloys.
At the To temperature, F → = 0, while at the Ms temperature, the value of F → is adequate to provide the surface and the strain energies necessary for initiating the transformation. The free energy differences between the allotropes
and of pure Ti and Zr, expressed as FTi → and FZr → , respectively, have been evaluated in Section 4.1. In order to determine the excess free energies of mixing for the and the phases, Fex can be expressed in a power series expansion: Fex = x1 − xAo + A1 x + A2 x2 + A3 x3 + · · ·
(4.19)
where the coefficients Ai are temperature dependent. By choosing equilibrium conditions at different temperatures, which means substituting in Eqs. (4.14) and (4.15) the free energy values for different temperatures and the corresponding x and x from the phase diagram, several values of the coefficients Ai upto the nth order term can be determined in a generalized manner; this procedure is known as analysis of phase diagrams (Rudman 1970). In the case of the Ti–Zr system, Kaufman (1959) has shown that the approxima = x1 − xAo and Fex = x1 − xBo , tion of using only the zeroth order term, (Fex
Martensitic Transformations
287
can also yield reasonable values of To . With these approximations, Eqs. (4.14) and (4.15) are reduced to 1 − x = x 2 A − x 2 B 1 − x
(4.20)
x = 1 − x 2 A − 1 − x 2 B x
(4.21)
FTi → + RT ln and FZr → + RT ln
Since the Ti–Zr isomorphous phase diagram shows a minimum, at any given temperature of the / equilibrium, two tie lines and correspondingly two sets of x and x values, one on the Ti-rich side and the other on the Zr-rich side, can be obtained. Values of B − A, calculated from the Ti-rich and the Zr-rich sides, have been found to be in good agreement; these values can be expressed at temperatures between 1100 and 810 K: B − A = −2340 + 126T cal/mol
(4.22)
Consequently, Eq. (4.16) can be written explicitly for the Ti–Zr system as F → = 1 − xFTi → + xFZr → − x1 − x2340 − 126T cal/mol (4.23) Equation (4.21) can be used for calculating the To temperature as a function of x (the results are presented in Figure 4.13) and for computing F → as a function of T for different compositions of Ti–Zr alloys. Kaufman’s plots for F → versus T for different alloy compositions, superimposed with experimentally determined Ms temperature values (Figure 4.14), indicate that a chemical driving force of about 50 cal/mol is required for initiating the → transformation martensitically. This corresponds to a supercooling (To − Ms ) of about 50 K. A comparison of the chemical free energy requirements for martensitic transformations in alloys based on Ti and Zr with those in ferrous alloys indicates that this requirement is much smaller in the case of the former (50 cal/mol as against 300 cal/mol for ferrous alloys). This suggests that the restraining forces comprising strain energy and surface energy associated with the martensites in Ti and Zr alloys are considerably smaller compared to those associated with ferrous alloy martensites. This point will be dealt with while discussing the relative values of lattice strains. In binary phase diagrams involving Ti or Zr on one side and a -stabilizing element B (such as Mo, Nb, Ta or V) on the other, the Ti/Zr-rich side can be approximated as a isomorphous system. The chemical driving force for the
288
Phase Transformations: Titanium and Zirconium Alloys x = 0.2
x = 0.3
Titanium-rich
x = 0.1
ΔF α′ → β/ ~40 Cal/mol Ms
Difference in free energy ΔF α′→β (cal/mol)
+100
Ms
0
To –100
(ΔF
α′→β
x = 0.0
= 0) x = 0.4
x = 0.5 x = 0.8
Zirconium-rich
x = 1.0
x = 0.9
ΔF
α′→β
/ ~60 Cal/mol Ms
+100
Ms
0
To –100
(ΔF α′ → β = 0) x = 0.6
700
800
900
x = 0.7 1000
1100
1200
Temperature (K)
Figure 4.14. The chemical driving force for martensitic transformation in Ti–Zr alloys as a function of composition and temperature.
→ transformation in moderately dilute solutions (x < 007 x < 015 x < 020) can be approximately written (Kaufman and Cohen 1956, Kaufman 1959) as x x − 2x + x 2 1 − x
→
→
→ FTiZr + RT ln = 1 − xFTiZr − xRT ln − F x x 2 1 − x (4.24) The calculated To versus x plots and experimental Ms versus x plots (Duwez 1951, 1953) are superimposed on the phase diagrams of Ti–Mo and Ti–V systems in Figure 4.15. The fact that the Ms line lies 25–50 K below the To line is consistent with the general trend of the thermodynamics of martensitic transformations in these systems. Experimental values of Ms temperatures for various alloys based on Ti are plotted in Figure 4.15.
Martensitic Transformations Calculated To
1200
Ti – Ta 1100
β
Ti – W β
α+β
α+β
1000
α
To To
900
Ms Temperature (K)
289
Ms
800
Sotubility ≈ 0.2 at 973 K Ti – V
1200
1100
Ti – Nb
β
β
1000
To
α
900
α+β
α
α+β
To 800
Ms
Ms 0
5
10
15
0
5
10
15
20
Atomic percent alloying element
Figure 4.15. BCC and HCP phase relations in Ti-based alloys. Calculated To –x curve is compared with the observed Ms values.
4.3.2 Lattice correspondence The lattices of the parent and the product phases can be related in a number of ways. The correct choice of the lattice correspondence is generally made by selecting one which involves the minimum distortion and rotation of the lattice vectors. The choice made by Burgers (1934), as illustrated in Figure 4.16, shows that the basal plane of is derived from an {011} -type plane and that [011] and [100] directions transform into [0110] and [2110] directions, respectively. The close-packed directions [111] and [111] lying on the 110 plane transform to ¯ directions are derived two close-packed <1120> directions. The other <1120> from <100> directions.
290
Phase Transformations: Titanium and Zirconium Alloys [011]β// [0001]α
(0001)α
[011]β// [0110]α [111]β// [1210]α
[111]β// [1120]α
[100]β// [2110]α
Figure 4.16. The distorted closed-packed hexagonal cell (hcp), derived from the parent bcc lattice.
As mentioned earlier, martensitic transformations in these systems result in the formation of either an hcp or an orthorhombic structure, the former √ being a special case of the latter structure with the ratio of lattice parameters b/a = 3. In general, the orthohexagonal axes system can be used for describing the martensite crystallography covering both → and → transformations. The lattice correspondences between the bcc and the hcp and between the bcc and the orthorhombic structures are depicted in Figure 4.17. Six crystallographically equivalent lattice correspondences between the orthohexagonal and the bcc lattices are described and labelled as variants 1–6 in Table 4.4. The lattice (Bain) distortion B associated with this transformation is given by ⎤ 1 0 0 B = ⎣ 0 2 0 ⎦ 0 0 3 ⎡
(4.25)
where 1 = 23 a /a 2 = a /a and 3 = 1/2 c /a The substitution of the lattice parameter values for pure Zr shows that the lattice strains are approximately 10% tensile, 10% compressive and 2% tensile,
Martensitic Transformations
291
[001]
C
[001]β
[010]β
c
[100]β
[111]β // [1210]α
[100]o
a a3 a1
[011]β// [0110]α
a2
[111]β // [1120]α [010]
Cubic (β lattice)
Figure 4.17. The lattice correspondence between bcc and orthohexagonal cells for the bcc → hcp transformation. The primitive hcp cell is defined by the vectors a1 , a2 and c and the orthohexagonal cell by a= a1 , b= a1 + 2a2 and c. The broken lines show the position of the bcc unit cell. Table 4.4. Correspondence between orthohexagonal and cubic cell (oRc ). Variant
[1 0 0]o
[0 1 0]o
[0 0 1]o
1 1 3 4 5 6
[1 [0 [0 [0 [0 [1
¯ c [0 1 1] [1¯ 1 0]c [1 1 0]c [1¯ 0 1]c [1 0 1 ]c [0 1 1]c
[0 [1 [1 [1 [0 [0
0 0 0 1 1 0
0]c ¯ c 1] ¯ c 1] 0]c 0]c 0]c
1 1 1¯ 0 1 1¯
1]c 0]c 0]c ¯ c 1] 1]c 1]c
respectively, along 1 , 2 and 3 directions. It may be noted that the lattice strains in this case very nearly satisfy the IPS condition. The deviation from the IPS condition arises only from the 2% tensile strain in the direction perpendicular to the basal plane.
292
Phase Transformations: Titanium and Zirconium Alloys
An approximate analysis of the crystallography of the martensitic transformation
can be performed by neglecting the 2% strain in the direction along [011]
[0001] (Kelly and Groves 1970). The Bain distortion, Ba , then reduces to ⎡ ⎤ 09 0 0 Ba = ⎣ 0 11 0 ⎦ (4.26) 0 0 0 The construction of the strain ellipsoid for the corresponding distortion is illustrated in Figure 4.18, which shows that the two vectors OP and OQ have not been distorted by the Bain distortion but have been rotated from their initial positions OP and OQ, respectively. Since there is no distortion in the direction perpendicular to the plane of the paper, the pair of vertical planes containing the vectors OP and OQ also remain undistorted. This follows from the theorem that a plane remains undistorted if three non-collinear vectors lying in that plane remain unchanged in length or, in other words, if the lengths of two vectors lying in the plane together with their included angle remain unchanged. The total strain requires a rigid body rotation which rotates either OP or OQ to its earlier position. If one chooses the vertical plane passing through OQ as the habit plane, a clockwise rigid body rotation is required for bringing the undistorted
[110]β 10% Tensile
X
P′
X′
P
[011]β
10% Compression [011]β
O
Q Q′
Y Y′
Figure 4.18. Strain ellipsoid construction for bcc to hcp lattice deformation in Ti and Zr alloys.
Martensitic Transformations
293
plane to its original position. Let the coordinates of the point Q be (x y). Operation of the Bain distortion, Ba , brings the point Q to Q, the latter having the coordinates (0.9 x, 1.1 y). The condition
OQ = OQ
(4.27)
or x2 + y2 = 09 x2 + 11 y2 yields x/y = 105. The habit plan which contains OQ and [011] will be at an angle tan−1 105= 465 with the [100] direction. It is interesting to note that even in this case, where no inhomogeneous lattice invariant deformation has been introduced, the habit plane is not a rational plane. The choice of the other vertical plane passing through OP as the habit plane gives the second solution which is crystallographically equivalent to the former. This is a manifestation of the fact that the plane (100) , i.e. the vertical plane passing through [011] , is a mirror plane of the parent bcc structure. The next step is to determine the orientation relationship between the and the phases. The approximate Bain strain, Ba , and the rigid body rotation maintain the (0001) plane parallel to the (011) plane, while the rotation brings the [111] direction close to the [211] direction (within 1.5 ). The orientation relationship, therefore, can be described as (0001)
(011) ; [110
[111 This is widely known as the Burgers orientation relation which should be differentiated from the Burgers lattice correspondence given by Figure 4.16. It may also be noted that the Burgers orientation relation renders the (112) plane ¯ plane. This planar correspondence will be shown nearly parallel to the (1100) to be of great significance in deciding the / interface plane in diffusional transformations in Ti- and Zr-based alloys. Though the lattice strain along the 3 direction is non-zero (about 2% extension), for most of the → martensitic transformations studied in Ti- and Zr-based alloys, the predictions of the habit plane and of the orientation relation from this approximate analysis are not far from those experimentally observed. This is why the approximate analysis is quite instructive in arriving at a general understanding of the transformation geometry. In this context, the work of Bywater and Christian (1972) may be cited in which a suitable alloy, Ti–22% Ta, was chosen in which the lattice strain along the 3 direction is indeed zero. The absence of internal twins and of any dislocation substructure in the martensite plates in this alloy experimentally validated the absence of lattice invariant shear in this case. If the extension along the [110]
[0001] direction, which is present in the Bain distortion for most of these alloys, is not neglected, no plane remains undistorted on the application of the Bain distortion. An LIS then becomes a necessity to
294
Phase Transformations: Titanium and Zirconium Alloys
make the total shear satisfy the IPS condition. The crystallographic analysis of a specific case is discussed in the following section as an illustrative example. 4.3.3 Crystallographic analysis The → transformation in a Zr alloy (Zr–2.5% Nb) is chosen as the illustrative example for demonstrating the steps of the crystallographic analysis. Lattice deformation: Using the Burgers correspondence and substituting the lattice parameter values of the and the phases in the Zr–2.5% Nb alloy (a = 03211 nm, c = 05115 nm and a = 03577 nm) in Eq. (4.23), the lattice deformation matrix can be expressed as
089768 00 00
109942 00
B01 =
00 (4.28)
00 00 101213 where B01 is the strain matrix of correspondence variant 1 of Table 4.5. This Bain distortion is on the basis of an axes system defined by the principal strain directions (x along [011]
[100]o , y along [011]
[010]o and z along [011]
[001]o directions). The same Bain distortion can be expressed in the axes system of the bcc crystal by using the similarity transformation for variant 1. ⎡ ⎤ 089768 00 00 105578 −004365 ⎦ Bc1 = O Rc B01 O Rc −1 = ⎣ 00 (4.29) 00 −004365 105578 Table 4.5. Bain strain matrices (Bc ) of bcc to hcp transformation in Zr–2.5 Nb alloy in the cubic basis for the six possible correspondence variants of the martensitic phase. Variant 1
2
3
Bain strain ⎡ 0 0 ⎣ 0 0 ⎡ 0 ⎣ 0 0 0 ⎡ − ⎣ − 0 0
= 089768; = 105578;
Variant ⎤ ⎦
4
⎤ ⎦ ⎤ 0 0 ⎦
= −004365.
5
6
Bain strain ⎤ ⎡ 0 ⎣ 0 0 ⎦ 0 ⎡ 0 − ⎣ 0 0 − 0 ⎡ 0 0 ⎣ 0 − 0 −
⎤ ⎦ ⎤ ⎦
Martensitic Transformations
295
In a similar manner, the Bain distortion for any other martensite variant can be determined, all on the basis of the axes system of the parent bcc lattice: the corresponding matrices are given in Table 4.5. The Bain distortion, when applied to a unit sphere of the parent phase, produces an ellipsoid. The intersection of the ellipsoid and the unit sphere defines the locus of the position vectors, r, which remain undistorted on the application of the Bain distortion. The undistorted vectors define the Bain cone which can be obtained from the condition
Br 2 = r 2
(4.30)
The initial and the final positions of the Bain cone for the variant 1 are shown in the stereogram in Figure 4.18. Lattice invariant shear: The next step in the crystallographic analysis is to identify the mode and the system of invariant deformation which can occur either by slip or by twinning. The lattice invariant deformation is determined in such a way that in combination with the Bain distortion it will maintain a plane of zero distortion. The system of a lattice invariant deformation (simple shear) can be defined by the shear plane normal, m, and the shear direction, l. For a simple shear the vectors which remain undistorted in length lie on two planes, K1 and K2 , as shown in Figure 4.19; K1 is the shear plane and K2 is the second undistorted plane which makes an angle of 90 ± with the shear plane, before and after the shear operation, being related to the magnitude of shear by the equation g = 2 tan
(4.31)
g
g /2 g /2
After shear
Before shear K2
α
Ko α
K 2′
K1
n1
Figure 4.19. Schematic showing that vectors lying on planes K1 and K2 remain unchanged in length after application of a simple shear.
296
Phase Transformations: Titanium and Zirconium Alloys
For a habit plane solution to exist, it is necessary that the traces of the undistorted planes K1 and K2 (the plane K2 after the operation of the shear) must intersect the initial Bain cone, Bi . This is due to the fact that the vectors defined by the points of intersection remain undistorted on the operation of either the Bain distortion or the LIS. Therefore, they are not distorted by the total shear as well. The above criterion has been expressed as l /m restriction by Bilby and Crocker (1961). They have given the following two inequalities for examining whether a given system of shear qualifies for being an LIS in a given transformation: m21 1 − 22 1 − 32 + m22 1 − 32 1 − 12 + m23 1 − 12 1 − 22 ≤ 0
(4.32)
l12 12 1 − 22 1 − 32 + l22 22 1 − 32 1 − 12 + l32 32 1 − 12 1 − 22 ≤ 0 (4.33) where [m1 m2 m3 ] is the direction normal to the shear plane and [l1 l2 l3 ] is the shear direction defined with reference to the axes system defined by the directions of the principal stress components 1 , 2 and 3 . In case the LIS occurs by twinning, the two twin components necessarily maintain crystallographically equivalent lattice correspondences. This imposes additional criteria for the selection of the system of LIS. For type I twinning, the twinning plane K1 should be derived from a mirror plane of the parent crystal, while for type II twinning, the direction of the twinning shear 2T should be derived from a diad of the parent crystal. Based on the lattice correspondence for variant 1, it can be seen that some of the variants of {100} and {110} mirror planes are transformed into {1012} and {1011} planes which qualify to be twinning planes of the LIS as they satisfy the ‘l’ and the ‘m’ criteria (Table 4.6). Mackenzie and Bowles (1957) classified the transformation on the basis of the operating twinning system for the LIS, designating {1012} twinning for class A and {1011} twinning for class B transformations. It is also seen from Table 4.7 that some of the mirror plane variants of the parent crystal are transformed into mirror planes of the martensite crystal (e.g. (100) and (011) transform into (2110) and (0001) , respectively), and therefore, these planes cannot be twin planes of the martensite crystal. In case the LIS occurs by slip, there is no restriction that the shear plane has to be derived from a mirror plane of the parent crystal. Otte (1970) has examined the suitability of a large number of shear systems for the LIS and has identified, on the basis of low values of the magnitude of the required shear, the following as the most probable shear systems: {1101} < 2113> , {0110} < 2110> . Predictions of crystallographic theory: Let us first consider the bcc to orthohexagonal transformation in which the LIS occurs by twinning on the
Martensitic Transformations
297
Table 4.6. The result of Bilby and Crocker criterion (1 and m) for all the possible shear systems in the product hcp martensitic phase for correspondence variant 1. Sr. No.
1(a) 1(b) 1(c) 1(d) 1(e) 1(f) 2(a) 2(b) 2(c) 2(d) 2(e) 2(f) 3(a) 3(b) 3(c) 3(d) 4(a) 4(b) 4(c) 4(d) 5(a) 5(b) 5(c) 5(d) 6(a) 6(b) 6(c) 7(a) 7(b) 7(c) 8(a) 8(b) 8(c)
bcc
hcp
Direction
Plane
Direction
Plane
[111] [111] [111] [111] [111] [111] ¯ [111] ¯ [111] ¯ [111] ¯ [111] ¯ [111] ¯ [111] ¯ [011] ¯ [011] ¯ [011] ¯ [011] [010] [010] [010] [010] [100] [100] [100] [100] [113] [113] [113] ¯ [113] ¯ [113] ¯ [113] ¯ [311] ¯ [311] ¯ [311]
¯ (110) ¯ (011) ¯ (101) (112) (121) (211) (011) ¯ (101) (110) (121) ¯ (112) ¯ (112) (011) (111) (211) (311) (101) (001) (100) (102) (011) (010) (001) (012) ¯ (110) ¯ (121) ¯ (121) (110) ¯ (211) (121) (011) (112) ¯ (121)
¯ [21¯ 13] ¯ [21¯ 13] ¯ [21¯ 13] ¯ [21¯ 13] ¯ [21¯ 13] ¯ ¯ [2113] ¯ [1210] ¯ [1210] ¯ [1210] ¯ [1210] ¯ [1210] ¯ [1210] ¯ (0110) ¯ (0110) ¯ (0110) ¯ (0110) ¯ [0111] ¯ [0111] ¯ [0111] ¯ [0111] ¯ [21¯ 10] ¯ ¯ [2110] ¯ [21¯ 10] ¯ [21¯ 10] ¯ [1213] ¯ [1213] ¯ [1213] ¯ [1¯ 123] ¯ [1¯ 123] [1-23] ¯ [1100] ¯ [1100] ¯ [1100]
¯ 1) ¯ (110 ¯ (0110) ¯ (101¯ 1) ¯ (1121) ¯ (1211) ¯ (2112) (0001) ¯ (101¯ 1) ¯ (1011) ¯ (1013) ¯ (1013) ¯ (1013) (0001) ¯ (21¯ 14) ¯ (21¯ 12) ¯ (3034) ¯ (1101) ¯ (0112) ¯ (21¯ 10) ¯ (2314) (0001) ¯ (0112) ¯ (0112) ¯ (0114) ¯ 1) ¯ (110 ¯ (112¯ 1) ¯ (1010) ¯ (1011) ¯ (1100) ¯ [1¯ 123] (0001) ¯ (1121) ¯ (1121)
l
m
<0.0 <0.0 <0.0 <0.0 <0.0 <0.0 <0.0 <0.0 <0.0 <0.0 <0.0 >0.0 <0.0 <0.0 <0.0 <0.0 <0.0 <0.0 <0.0 <0.0 >0.0 >0.0 >0.0 >0.0 <0.0 <0.0 <0.0 <0.0 <0.0 <0.0 >0.0 >0.0 >0.0
<0.0 <0.0 <0.0 <0.0 <0.0 <0.0 <0.0 <0.0 <0.0 <0.0 <0.0 >0.0 < 0.0 < 0.0 < 0.0 < 0.0 <0.0 <0.0 <0.0 <0.0 >0.0 >0.0 >0.0 >0.0 <0.0 <0.0 <0.0 <0.0 <0.0 <0.0 >0.0 >0.0 >0.0
(1101)
(101) plane. The lattice correspondence of the major twin component is chosen to be that of variant 1. The minor twin component thus turns out to be variant 3 (as indicated in Table 4.8). The Bain deformation matrices for the major and the minor twin components of the martensitic plate can be expressed as
298
Phase Transformations: Titanium and Zirconium Alloys
Table 4.7. Possible twin planes in the hexagonal product phase derived from mirror planes of the parent bcc phase. Mirror plane
Variant 1
Variant 2
Variant 3
Variant 4
Variant 5
Variant 6
(100) (010)
(21 10) Mirror (0112)
(0112)
(0112)
(0112)
(0112)
(0112)
(0112) (2110) Mirror (0110) Mirror (0001) Mirror (1101)
(21 10) Mirror (011 2)
(0112)
(1101)
(1011)
(101 1)
(1011)
(1101)
(1101)
(1101)
(2110) Mirror (0001) Mirror (0110) Mirror (1011)
(21 10) Mirror (01 12)
(21 10) Mirror (011 2)
(001)
(01 12)
(110)
(1011)
(110)
(1101)
(101)
(101 1)
(1101)
(1011)
(011)
(0001) Mirror (0110) Mirror
(1101)
(1101)
(0111) Mirror (0001) Mirror (101 1)
(1011)
(101)
(0001) Mirror (0110) Mirror (1011)
(1011)
(1011)
(1101)
(1101)
00 10558 −00437
⎤ 00 −00437 ⎦ 10558
(011)
⎡
08977 ⎣ 00 B1 = 00 ⎡
10558 ⎣ B3 = 00437 00
00437 10558 00
⎤ 00 00 ⎦ 08977
(1101) (0110) Mirror (0001) Mirror
(4.34)
(4.35)
The crystallographic problem can be described in terms of a simple schematic construction (Figure 4.20). A vector r in the parent crystal is transformed into a vector r as a result of a total macroscopic shear (E) comprising Bain distortions in the two twin-related regions and the corresponding rotations, as given by Eqs. (4.6) and (4.7). For r to lie on the habit plane, the following condition needs to be satisfied: r = 1 1 − xB1 + xB3 r = 1 F r where F = 1 − xB1 + xB3
(4.36) (4.37)
Table 4.8. Crystallographic parameters of martensites in different Zr- and Ti-based alloy systems. x
Structure
Ms
OR
Habit
– 0.75 13 10 Vc
hcp hcp hcp
800 600–270 –
B – B
8, 9, 12 – –
Ti–Nbd
0.25 at.%Nb 35 Nbe 0–22 Ta 22–36 Tag 2.4–5.9 Cr
hcp Ortho ( ) hcp ortho ( ) hcp
871–212 175 – – 320–67
– Ba – – –
– – – – {334}
5.5–18.7 at.%Cri 4 Moj 6–10 Mo 6 Mo 11 Mol 11, 12.5 Mom 15 Mon 4.3–5.2 mm 5 mmp 3 Fe 2–5.45 Ni 0.56–8 Cu 8 al 8Al-1Mo-2V 4Al-3Mo-1Vv
hcp + fcc hcp
hcp hcp hcp hcp hcp hcp hcp + fcc hcp + w hcp hcp (fcc) hcp hcp
– 973 883 600 340 340
IC-S (fcc)
–
Ti–Taf Ti–Crh
Ti–Mok
Ti–Moo Ti–Feq Ti–Nir Ti–Cus Ti–Alt Ti-Al-Mo-Vu
OR: Orientation relation. a Williams et al. (1954), Erickson et al. (1969). f Yamane and vedg (1966). k Ka (1967).
u
Weing and Machlin (1954). Erickson (1969).
370 680–540 740–570 – – –
McMillan et al. (1967).
c
Menon et al. (1980).
d
g
Bywater and Christian (1972). Liu (1956).
h
Erickson (1969). Gaunt and Christian (1959).
i
l
m
n
r
Barton et al. (1960).
s
q v
Liu (1956), Nishiyama et al. (1966). Williams and Blackburn (1967).
(1011)T (111)T (1011) (0001)
– B –
– (8, 9, 12)4 (3, 4, 4)4
B B – – A – – –
{334}{344}
(1011)T
{334} – {1011} – – –
(1011)T – (1011)T – (1011)T (1011)T
C Hammond (1972).
Liu (1956), Oka (1967). Hammond and Kelly (1969). Williams et al. (1970a).
e Baker (1971), Bagaryatski et al. (1958). j Davis et al. (1979). o Hammond and Kelly (1969), Liu (1956). t Armitage (1965).
299
p
b
3˜ 00
(1011)T (1011)T (1011)T ) (21 1 3) Twin ratio 1:3 – – (1011)T (111)T {1011}TW (1102)TW
Martensitic Transformations
Tia Ti–Vb
LIS
300
Phase Transformations: Titanium and Zirconium Alloys r
r′
x
1
Before transfarmation
1–x
2
1
Major twin fraction
Minor twin fraction
After transfarmation g
B
2 B′
C
C′
1
E
F 2 A B
Figure 4.20. Schematic diagram of an internally twinned martensite. The complementary twining shear transforming original vector, r , in the cubic crystal to vector r averaged over the two twin regions (major and minor twin fractions).
The rotation matrix, , for twinning on (1¯I01) for the variant 1 can be obtained from Euler’s formula (Wayman 1964) as reproduced in the following: ! p1 − p2 q1 − q2 = U tan p1 + p2 q1 + q2 2
(4.38)
where p1 , q1 and p2 , q2 are the initial and the final positions of two vectors lying in the unrotated plane, with the axis of rotation being U and the angle of rotation being !; and ! have been evaluated for the present case as ⎤ 098655 004079 015831 ⎥ ⎢ = ⎣ 003744 099856 004079 ⎦ 015975 003474 098655 ⎡
(4.39)
and ! = 9658 Substituting the values of B1 , B3 and in Eq. (4.34), the macroscopic shape distortion, E, can be expressed in terms of the twin fraction, x. In general, F , which corresponds to a fraction of the twin shear is a non-symmetric matrix which can be expressed as the product of a symmetric matrix, Fs , and a rotation matrix, ".
Martensitic Transformations
301
Furthermore, a rotation matrix can be found which diagonalizes the symmetric matrix Fs to Fd . Thus the macroscopic shape distortion, E, can be written as E = " Fd ∗
(4.40)
where Fd is a function of x. In this expression for E, all the matrices excepting Fd are rotation matrices and Fd is a diagonal matrix with diagonal elements 1 ,
2 and 3 (eigenvalues). An axes system with basis d(id , jd , kd ) can be defined in which the distortion is diagonal (axes system defined by the eigenvectors). In this axes system (basis d), the distortion of the rd vector can be written as rd = Fd rd
(4.41)
Those vectors which remain undistorted satisfy the following condition:
rd = rd = Fd rd
(4.42)
21 − 1Xd + 22 − 1Yd + 23 − 1Zd = 0
(4.43)
which is equivalent to
where Xd , Yd and Zd are the indices of a vector rd which is one of the undistorted vectors. As mentioned earlier, the distorted ellipsoid can intersect the unit sphere on an undistorted plane only if the distortion along one of the principal strain directions is zero. This means that one of the elements is unity and Eq. (4.40) reduces to a quadratic equation in x, the twin thickness ratio. Solving this equation, two crystallographically equivalent values of x are obtained, of which the second value corresponds to (1 − x). From Eq. (4.40), the two x values obtained are 0.19535 and 0.80465. Substituting the value of x in Eq. (4.34), the matrix F is determined to be ⎤ 092544 00 02716 ⎥ ⎢ 105578 −002797 ⎦ F = ⎣ 001568 −003324 −004365 102254 ⎡
(4.44)
The eigenvalues (principal strains) of the matrix F are given by
1 1 = 092546, 2 2 = 1000 and 3 3 = 107936. It is, therefore, seen that the condition of zero distortion plane is satisfied for a thickness ratio of 4.11903.
302
Phase Transformations: Titanium and Zirconium Alloys
The eigenvectors, Vi , corresponding to these eigenvalues and the rotation matrix, , are given by ⎡ ⎢ ⎢ =⎢ ⎣
V1 099779 −006302 −002122
V2 −001582 −053496 −084473
V3 −006459 −084252 053477
⎤ ⎥ ⎥ ⎥ ⎦
(4.45)
Experimental determination of crystallographic parameters and comparison with theoretical predictions: Gaunt and Christian (1959) examined six-sided crystal bar specimens (refined by Van Arkel iodide process) for determining the orientation relationship and the habit plane of martensites forming in pure Zr (with a very low total interstitial content). Since the material deposition in the iodide refining occurs at ∼1573 K, the crystal gets deposited in the high temperature (bcc) phase which transforms completely into the martensitic -phase during cooling. In the absence of the parent -phase in the specimens examined, the orientations of large-sized -grains were determined from the shear ridge markings present on the sample surfaces. These ridges were found to be traces of the 110 planes of the pre-existing -phase. The -orientation thus determined was consistent with the external shape of the crystal bar. The orientations of large -grains, recognized by polarized light microscopy, were determined from Laue diffraction patterns using microbeam X-ray. The ¯
[111] ¯ was established Burgers orientation relationship (0001)
(110) ; [21¯ 10] within the accuracy attainable (±2 ). It was not possible to measure the shape deformation for Zr. The experimentally determined habit plane, as reported by Gaunt and Christian (1959), for pure Zr is (072, 051, 0.46) ±2 . This experimental result is compared with those obtained from computations based on the Bowles– Mackenzie formulation (1957), the Wechsler–Lieberman–Read formulation (1957) and the approximate analysis (described in Section 4.3.2). The Burgers (1934) mechanism of transformation implicitly assumed the same correspondence as described in Section 4.3.2 and accomplished most of the length ¯ [111 ¯ changes normal to the [110] direction by means of a shear on the (112 system; the remaining adjustments were not considered in detail. Since the principal strain in the [110] direction is very small, it is attractive to retain Burgers assumption that this direction is exactly parallel to the hexagonal c-axis and that the requisite strain is taken elastically at the interface and subsequently relaxed plastically. As has been shown in Section 4.3.2, the approximate analysis of the bcc to hcp transformation can be worked out, neglecting the ∼2% distortion along the [110] direction. The invariant line in the (110) plane (the shear direction)
Martensitic Transformations
303
¯ is readily found to be ∼[223], and the habit plane, defined by this line and by ¯ [110] , is (334 . Although the approximate theory does not describe the transformation crystallography adequately, it is useful in showing the physical significance of the experimental results, which tends to be obscured by the mathematics of the Bowles–Mackenzie theory. Thus the observations that the orientation relation is close to the Burgers relation, that the habit plane is nearly normal to the hexagonal basal plane and that the shape deformation is nearly simple shear are all the consequences of the small principal strain parallel to the c-axis. Experimental determination of habit plane and shape strain can be made more accurately if the parent -phase can be retained at room temperature in a metastable state on quenching, the orientations of several -grains are precisely determined by Laue diffraction, and martensite plates are introduced either by stress or by quenching to cryogenic temperatures below Ms . It is desirable to have martensite plates of sufficiently large size present in a matrix of well-characterized grains for orientation determination by diffraction. Stress-induced martensite plates offer an additional advantage, as they can be induced at room temperature and, therefore, the martensite/matrix habit can be examined without subjecting the sample to any change in temperature. Gaunt and Christian (1959) determined the habit plane, the direction and magnitude of the macroscopic shear and the orientation relationship in a Ti–12.5% Mo alloy in which stress-induced martensite was produced. Experimentally determined habit plane poles and shear directions are plotted in Figure 4.21(a), which shows that the habit plane corresponding to the stress-induced martensite is close to 344 and the shear direction is close to a great circle through the 111 and 011 poles. These results, when compared with those obtained in thermally induced martensite of Ti–11% Mo (Figure 4.21(b)) bring out the important difference between them. In the case of Ti–11% Mo, the habit plane pole is close to 334 and the shear direction lies close to a great circle passing through the 111 and 001 poles. Since the microstructure of martensite was investigated by Gaunt and Christian (1959) only at the light microscope level, information regarding the martensite substructure was not reported. Subsequently a large number of alloys have been investigated using TEM. Table 4.8 summarizes the results of martensite crystallography as reported by several investigators. It may be noted that with increase in the concentration of -stabilizing elements the formation of 344 habit martensite is promoted. The observation showing 344 habit for martensite plates in Ti–12.5% Mo in contrast to 334 habit in Ti–11% Mo has been interpreted by Bowles and Mackenzie (1957) in terms of maximum resolved shear stress.
304
Phase Transformations: Titanium and Zirconium Alloys 110
100 011 2 3
101
6
H
S
1 4
36 4 1 5 2
111
5
100
111 001
101
010
(a) 110
S 3
100
111 101 111
2 1 4' B 4
011 H 2 53 001 B 4 111
S
1
(b)
Figure 4.21. Stereographic projections showing the habit plane and shear directions for (a) stressinduced martensite in Ti–12.5% Mo alloy and (b) thermally induced martensite in Ti–11% Mo alloy.
Later investigations by Hammond and Kelly (1969) and Banerjee and Krishnan (1971) have shown that the 344 habit, which is the macroscopic habit plane between the twinned martensitic -plate and the -matrix, is made up of two segments, each corresponding to 334 planes. It will be shown in a later section that ¯ large primary plates containing periodic 1011 internal twins generally follow the Bowles–Mackenzie ( − +) solution under which both the twin components match the Burgers orientation relation very closely. Each of these twin segments tend to maintain the IPS condition by forming a habit segment of 334 type. 4.3.3.1 Morphology and substructure Morphologies and substructures of martensites have been studied in a number of Ti- and Zr-based-alloys. On the basis of the observations made in these studies, martensite morphologies can be classified into several groups which are described in this section. The underlying principles which control the evolution of a given
Martensitic Transformations
305
type of morphology are also discussed. As has been explained later, the morphology of martensite is often linked with the internal structure (substructure) of martensite crystals. It is for this reason that the discussion on morphology, which deals with the nature of assembly of martensite crystals, and on substructure, which reflects the operation of the LIS, is placed in the same section. The factors which play significant roles in controlling the morphology of martensite products are the following: (1) The symmetry elements of the parent and the martensite crystals which dictate the polydomain morphology. (2) The strain coupling between martensite crystals in a given region, resulting in autocatalytic nucleation and self-accommodation. (3) The sequence of formation of different generations of martensite crystals which controls the progressive partitioning of the grains of the parent phase. (4) The nature of the operating LIS. Polydomain morphology. The formation of a polydomain structure of martensite is clearly a consequence of the fact that the different variants of the martensite phase have equal probability of being generated from a single parent grain. The structures of all the variants are related by the symmetry operation of the parent crystal which is not a symmetry operation of the martensite crystal. The method of determining the number of variants on the basis of group theoretical considerations is described in Table 4.4. For the present discussion, we can use Table 4.4 where all the possible variants and their lattice correspondences with the parent crystals are listed. An examination of the relative orientations of these variants reveals that many of them are mutually twin related (Table 4.9). Strain coupling – self accommodation: The equivalence of strain energy minimization and IPS deformation in explaining the formation of internally twinned martensite plates has been discussed in Section 4.3. The process of the minimization of the strain energy does not remain confined to only within an individual Table 4.9. The combination of major and minor twin correspondence variants related by the possible twinning planes in the hcp Zr–2.5Nb alloy. Twinning (0112)H (1101)H (1101)H (1011)H (1011)H
Major minor twin fraction 1-6 1-4 1-3 1-2 1-5
2-3 2-5 2-1 2-6 2-4
3-2 3-1 3-4 3-5 3-6
4-5 4-2 4-6 4-1 4-3
5-4 5-6 5-3 5-2 5-1
6-1 6-3 6-5 6-4 6-2
306
Phase Transformations: Titanium and Zirconium Alloys
plate. In a group of martensite plates, mutual strain coupling can significantly reduce the overall strain energy. This is known as self-accommodation. A method for an approximate estimation of the effectiveness of self-accommodation for different groups of martensite plates, developed by Madangopal et al. (1993, 1997), Srivastava et al. (1993, 2000) and Srivastava (1996) is summarized here. The shape strain associated with ellipsoidal martensite plates causes an elastic strain field in the surrounding parent phase. The elastic strain fields of the different variants then interact with each other in such a manner that a maximum reduction in the strain energy density of the assembly of the plates and the matrix is achieved. Plates belonging to a “cluster” are, therefore, “bonded” by the magnitude of the strain energy reduction. The Eshelby expression (1957) for determining the displacement Ux at a point x at a distance r caused by an ellipsoidal inclusion is given in the (i, j, k) axes system by Figure 4.22. Uj x =
VeijT gijk 8#1 − $r 2
(4.46)
where V is the volume of the inclusion, $ is the poison ratio, eijT is the stress-free transformation strain which is the shape strain in a martensitic transformation, and Point x (r, l1, l2, l3)
Centre of interacting strain fields
B (eT1 )
C (e1T) A (e1T)
Figure 4.22. Schematic illustration of the interacting strain fields of three point forces. Strain energy density is determined from the displacement measured at a point far from X.
Martensitic Transformations
307
gijk = 1 − 2$%lj lk + %ik lj − %ij lk + li lj lk li lj and lk being the direction cosines of the position vector of the point x. The above expression is valid when the martensite plate is assumed to be point force and the distance r is large. The strain components due to the displacement Uj x can be derived from the following relation: eij =
Uij + Uji 2
(4.47)
where Uij = Ui / xj The stress at a point x can be computed from the strain eij by using the following relation: ij = eij + 2eij
(4.48)
where the elastic constants = c12 and = c44 . The strain energy at the point x is then given by 1 Eij = ij eij 2
(4.49)
Integrating the above expression over the r, l1 , l2 and l3 space, the total strain energy density E due to a single martensite plate can be determined to be 2 1 D (4.50) E = 2 8#1 − $ where D = c1 eij2 + c2 eji2 + c3 eii ejj + c4 eij eji and the coefficients cj are c1 = 160 + 248& c2 = 125 + 198 c3 = 150 + 60& c4 = 250 + 396 The strain energy density for a group of martensite plates can be computed by considering the superimposition of the strain fields due to each of them. Since the distance between the plates within a group is small compared to r, the strain fields of the interacting plates can be considered at the same point (i.e. at the origin). The net displacement due to a group of n martensite plates (illustrated in Figure 4.22) is given by Ujn x =
Vgijk 'eijT 8#1 − $r 2
(4.51)
308
Phase Transformations: Titanium and Zirconium Alloys
It can be seen that the summation term in the above expression is the average shape strain matrix of all the interacting martensite plates. The degree of self-accommodation (DSA) can now be defined as DSA =
E 1 − E n 100 E 1
(4.52)
where E 1 and E n correspond to the strain energy density due to an isolated plate and a group of n plates, respectively. The values of DSA for different possible groups of martensite plates can be computed for assessing the DSA achieved in different groups. Some of the simplifying assumptions used in the aforementioned formulation for calculating the DSA are the following: (1) The volume fractions of the different variants interacting in a group have been considered to be identical. Further reduction in the strain energy is possible by optimizing the volume fractions of different variants in a manner similar to fulfilling the IPS criterion by a suitable adjustment of the twin thickness ratio in an internally twinned martensite plate. (2) The nature of the interfaces between the variants has not been taken into account. In spite of these approximations, an evaluation of DSA for different groups of martensite variants provides a basis for the selection of those variants which group together in a self-accommodating manner. This point will be discussed further while comparing the predicted grouping with those frequently encountered in experiments. The overall martensite morphology develops through a sequence of transformation events. The first-generation plates fragment the parent grain by spanning across the entire grain. Martensite plates forming in later generations are progressively smaller as the space available in the fragmented parent phase for the growth of plates continuously decreases with the progress of the transformation. The crystallographic constraints such as the adherence to habit planes and the self-accommodation tendency remain operative throughout. This causes nearly self-similar pattern generation as illustrated in Figure 4.23. Hornbogen (1989, 1990) has pointed out the fractal nature of the formation of martensite as shown in Figure 4.23, which illustrates pattern generation through successive stages of fragmentation. The development of such a pattern, however, does not occur in the lath martensite structure in which a group of laths stacked in a parallel array are arranged within each packet.
Martensitic Transformations
309
Figure 4.23. A Schematic showing self-similarity pattern in martensitic transformation.
Hcp martensites in Ti- and Zr-based alloys exhibit several morphologies which can be broadly classified into the lath and the plate types. In the former, martensite units are grouped together in parallel arrays within a packet and several such packets make up the volume of the parent -grain. In contrast, the plate morphology is characterized by acicular plates forming along different variants of the habit plane, continuously partitioning -grains, as schematically illustrated in Figure 4.24. A more detailed description of these morphologies is given in the following sections.
(a)
(b)
Figure 4.24. Schematic diagram showing acicular plates forming along different variants of the habit plane, continuously partitioning -grains.
310
Phase Transformations: Titanium and Zirconium Alloys
Lath morphology. As mentioned earlier, the lath morphology consists of several martensite packets within each parent -grain. These packets are resolvable under the light microscope. Under polarized light each packet exhibits a separate contrast, suggesting that the orientation within a given packet is nearly the same. The fact that each packet is made up of a number of laths which are stacked in a parallel array is revealed from surface relief observations and from the microstructure revealed by TEM. Figure 4.25 shows the typical lath morphology under different imaging conditions. Williams et al. (1970a,b, 1973) have reported a lath martensite morphology in dilute Ti–Cu alloys, while a similar morphology of martensites in dilute Zr–Nb and Zr–Ti alloys has been reported by Banerjee and Krishnan (1971, 1973). TEM investigations on these martensites have clearly revealed that a group of laths present within a single packet belong to the same orientation variant, notwithstanding the small angle tilt boundaries between the adjacent laths. Contrast analysis of dislocations constituting lath boundaries has shown that these dislocations have
type Burgers vectors. The misorientation between the adjacent laths has been found to be 1.0–1.5 from the observed spacings of these dislocations. The alloys which exhibit the lath morphology are all associated with high Ms temperatures (above 923 K), and therefore, the transformation is complete and no retained -phase is left in the quenched product. The determination of the habit plane has, therefore, been carried out in such alloys by assuming the Burgers orientation relationship. The reported habit plane indices for pure iodide Ti, 8912 (Williams et al. 1954) and for pure iodide Zr, 334 (Gaunt and Christian 1959) have been obtained from back reflection Laue patterns taken from individual packets, each containing a group of parallel laths. The orientation of the parent -grain has been worked out on the basis of the Burgers orientation relation, with different variants being operative in different packets within a prior -grain. Single surface trace analysis of parallel laths seen within the packets has indicated the habit plane indices to be close to 334 . Later investigations based on TEM observations have confirmed the same indices for the lath martensite habit plane in Ti (Williams 1973) and in Zr (Banerjee and Krishnan 1971, 1973) alloys. With increasing additions of -stabilizing elements, the Ms temperature of the alloys decreases and this is reflected in the change in the martensite morphology. In general, dilute alloys with Ms temperatures above about 923 K exhibit a packet morphology, each packet containing a stack of laths. Two types of orientation distributions of laths in a given packet have been reported. In one case, laths belonging to the same orientation variant are stacked together with a small angle boundary separating the adjacent laths, while in the other alternate laths are twin related. The packets of the latter type with twin-related laths are encountered more frequently in alloys
Martensitic Transformations
311
(a)
(b)
(c)
(d)
(e)
(f)
(g) (i)
(ii)
Figure 4.25. (a and b) Optical micrographs showing massive martensite in pure zirconia. TEM micrograh showing (c) large lath with irregular interface in a Zr-0.5% Nb alloy (d) laths with straight interface. (e and f) Bright and dark field micrographs showing alternately twin related laths in a Zr-2.5% Nb alloy. (g) Self-accommodating 3-plate clusters of martensite plates of “indentation morphology”. (i) General view of microstructure showing the presence of large number of 3-plate clusters when viewed along the <111> direction. (ii) Schematic drawing showing the arrangement of the 3-plate clusters. (iii) SAD patterns from each of the interface regions (A-B, B-C, C-A) separating the individual martensite in a 3-plate cluster. (iv) Keys to SAD patterns obtained from individual martensite crystal in a 3-plate cluster (in (iii)) and their orientation with respect to their neighbouring crystals. (v) Relative orientation of plates A, B, and C are shown in the [011] stereogram. Basal planes and the habit plane traces ( HA, HB and HC) of A, B and C are marked. The 1120H directions of all the crystals are aligned along the 111C direction. The habit plane poles A, B and C, as marked, are located on the outer circle and on the dotted great circles GB and GC respectively.
312
Phase Transformations: Titanium and Zirconium Alloys
(iii) (iv) 1103 0002 1101
A (011)b //(0001)M (111)b //(1120)M
1101
1101 B
1101
1101 A
B (101)b //(0001)M (111)b //(1120)M
1103 0002 1101
C 1103 1101 1101
A B
C
C
0002 1101 1103
A 1101 0002
0002
C
1101
1101 1101 1101
1101 1103
(v)
111
433
FOIL PLANE TRACE
334 011
334 334
433
101
0111 0110
1103 1101 1101
1210 100
101 1101 A /C 6(+)/2(–) (0001)B HB 433 1011 0111 1103 111 001
111 1120
0002
1100
2110
1010
C 1101 (110)b //(0001)M (111)b //(1120)M
110 111 (1120)
B,C
343
1103 343 1011 433 1102 HC (0001)C A/B 6(+)/4(–) 110 1101 2201
GC
343
(0001) 011 B/C 4(+)/2(–) (1101)B HA 334 1105 111
001 1210
0110
GB
1010
343 2110
1100 433
111
Figure 4.25. Continued
such as Ti–5.3% Cu (Zangvil et al. 1973) and Zr–2.5% Nb (Srivastava et al. 2000) in which the Ms temperature is brought down close to 923 K. In most cases laths contain dislocations and are free of periodic internal twins. This suggests that the lattice invariant shear operative within laths is slip and that the dislocations present are debris of the slip process. The < c + a > Burgers vector of these dislocations and the alignment of their line vectors along the traces of 10¯l1 planes support this contention. The dislocation structure and the lath interface structure undergo significant alterations during cooling from the relatively high transformation temperature. It is for this reason that lath boundaries and the internal dislocation structure characteristic of the martensitic transformation are often not preserved well in the final microstructure.
Martensitic Transformations
313
Plate morphology. The plate morphology is characterized by martensite plates, parallel sided or lenticular, forming along all the habit plane variants and dividing the parent phase grains by a sequential partitioning process. The fractal nature of this morphology is evident from the self-similarity of assemblies of martensite plates forming in successive generations. The first-generation plates span/extend across the entire grain of the parent phase. The partitioned grain is then amenable to transformation in the next stage. As a consequence, the second-generation plates are shorter in length and further partition the parent grain. The next generation of plates form within these fragmented untransformed spaces and the process continues till the transformation is complete (in some cases pockets of the parent phase remain untransformed). The pattern generated by a group of martensite plates belonging to one generation remains essentially the same though the dimensions of the plates decrease in every successive generation, maintaining the self-similarity of the structure. Detailed crystallographic analysis has been primarily carried out on martensite plates belonging to the earlier generations mainly because they are large and well formed and, therefore, better suited for the analysis of orientation relations and habit plane indices. A survey of the substructure of primary martensite plates in ¯ twinning is frequently observed within these Ti and Zr alloys reveals that 1011 plates (Figure 4.26). However, a number of primary martensite plates remain only dislocated. Quantitative data on the frequency of twinning in primary plates are not available, but the incidence of twinning is seen to increase with a lowering of the Ms temperature. Martensite plates forming in later generations are found to be less frequently twinned. Such a tendency is presumably related to the fact that the later-generation plates form in a matrix which is substantially work hardened by the stresses resulting from the plastic accommodation of the earlier-generation plates.
(a)
(b)
Figure 4.26. Bright field TEM micrographs showing plate martensites: (a) internally twined (thick twins) martensites and (b) internally twined martensite where the minor twin fraction is very thin.
314
Phase Transformations: Titanium and Zirconium Alloys
Hammond and Kelly (1969) were the first to point out that the internally twinned martensite plates in Ti–Mn alloys could be grouped into two classes. The first group was characterized by a large spacing of thick 10¯l1 twins. The thicknesses of the major and the minor orientations were sufficiently large to produce a clearly resolvable zigzag habit which was made up of habit segments of the two twin-related variants present within a single martensite plate. The ratio of the thicknesses of the minor and the major twins was seen to be 1:4, a value which closely matched with that predicted from the magnitude of the LIS calculated from the phenomenological theory. Similar zigzag habit has also been observed in several Ti and Zr alloy martensites where the thicknesses of the major and the minor twin orientations are large (minor twins thicker than 100 nm). The second group of plates was characterized by the presence of very thin twins stacked within the plates (Figure 4.26). The average value of the ratio of the thicknesses of the minor and the major twin variants was much smaller compared to that predicted from the phenomenological theory. This was suggestive of the fact that an additional component of the lattice invariant shear (possibly slip) was operative within these plates. An analysis of habit plane variants of the two types of internally twinned plates (containing thick and thin twins) indicates that there exists an important difference between them in terms of the orientations of their minor twin components. In Bowles–Mackenzie theory, four non-equivalent habit plane solutions, denoted by ( ± ±), are obtained in a class A transformation, where class A corresponds to the LIS on 1011 planes which are derived from the 110 -type mirror planes of the parent bcc phase. For the lattice correspondence given in Figure 4.16 and for the lattice invariant deformation on the (1101) plane, the orientation relations corresponding to the ( − +) and the ( + +) solutions, as represented in the stereogram in Figure 4.27, are obtained by a rigid body rotation around the [0001] direction in either the clockwise or the anticlockwise direction. The most important difference between these two solutions arises from the fact that the (1101) plane on which the LIS acts remains nearly parallel (an angular deviation of 2 ) to the (011) mirror plane for the class A ( − +) solution, whereas the (¯l101 twin plane gets separated from the (110) plane from which it is derived by an angle of about 11 for the class A ( + +) solution. This means that the twin plane remains nearly parallel to the mirror plane in the former case even after the application of the rigid body rotation. In such a situation both the twin components follow the Burgers orientation relation very closely. The growth of a twinned martensite plate involves the creation of successive twin components as the plate lengthens. As a particular variant grows, the combined effect of the rigid body rotation and the Bain strain (R.B) tends to establish an invariant plane between the parent and the growing martensite variant. The deviation of R.B from
Martensitic Transformations 100
111
1011
1101
111
111 1210
101
1120
011
011
0002
101
011 0110 0110 101 110
1011
101
110
1120
0210 111
0110 011
011 0002
011
101
0210
111 1210
1101
1011
110
0110
111
2110 100
2110
1120
315
110 1011
1101
1120 111
111 2110
100
100
2110
Figure 4.27. Stereographic projection showing the orientation relations corresponding to class A ( −+) and class A( ++) solution of Bowles and Mackenzie formulation. These two orientations are derived from the clockwise and anticlockwise rotations along [110]
11001 .
the IPS condition causes a strain build-up which gets periodically corrected by the creation of a second variant at regular intervals. In the case of a plate which belongs to the ( − +) solution, both the twin components follow the Burgers orientation relation very closely, and therefore, the R.B combination operating within each twin component nearly satisfies the IPS condition. As a consequence, these two twin components can grow to a significant extent (the thickness of the minor twin variant exceeding 100 nm), maintaining independent habit segments with the parent phase. In the ( + +) case, the twin plane (¯l101 does not remain parallel to the (110) mirror plane subsequent to the rigid body rotation. The major twin orientation in this case satisfies the Burgers orientation relation and the IPS condition quite closely. However, the minor twin variant ( 2 ) is significantly away from the Burgers orientation relation, the (0001) plane being 9 away from (011) . The IPS condition is, therefore, not obeyed even approximately within the minor twin variant. The growth of such a variant would cause a substantial build-up of strain, resulting from the deviation from the IPS condition. This would restrict the growth of the 2 component. The observed stack of very thin twins in the martensite plates belonging to the ( + +) solution lends support to this contention. Since the requirement of LIS is not met in these plates in which the twin thickness ratio is much smaller than the predicted value of 1:4, the operation of an additional component of LIS in the form of slip appears essential. A periodic array of < c + a > dislocations along a different variant of 10¯l1 planes within
2 components is suggestive of the additional lattice invariant deformation.
316
Phase Transformations: Titanium and Zirconium Alloys
Substructure resulting from atomic shuffles: The phenomenological crystallographic theory discussed so far takes into account the fact that the total macroscopic shape strain ratifies the IPS condition. The necessity of introducing a lattice invariant deformation in order to achieve this condition in a great majority of alloys has also been explained. The lattice invariant deformation can be manifested either by creating a stacking of internal twins or by repeated slipping at periodic intervals. The former results in the production of the internally twinned martensite, while in the latter debris of dislocations left behind are responsible for a dislocated substructure. The lattice (Bain) deformation shown in Figure 4.28(a)–(c) transforms the orthorhombic unit cell whose dimensions (a, b and c) match those of the product martensite. However, a Bain strain does not bring all the atoms into the right positions consistent with the hexagonal symmetry of the product phase. An additional atomic shuffle is required for achieving the right atomic positions. This point was taken into account in the Burgers model for the bcc → hcp transformation. As shown in Figure 4.28, the bcc → hcp transformation can be accomplished by a shear which converts the (011) plane into the (0001) plane and a shuffle of the atoms at O locations in the figure into the positions B or C for obtaining the hcp stacking. It is this choice in the direction of shuffle (either O → B or O → C) which allows the formation of domains with different stacking sequences. Figure 4.29(a) indicates the stacking sequence for a perfect hcp structure while Figure 4.29(b) shows the stacking sequence in a crystal containing two domains, corresponding to the two alternative shuffle directions. Atoms in the A layers remain unaltered while in Figure 4.29(b) a domain boundary is created by introducing different shuffles in the two halves of the crystal. Such a domain boundary is created ¯ partial dislocation. The fault in Figure 4.29(c) can be by the passage of a 13 1010 converted to that of Figure 4.29(b) by the removal of a C layer of atoms at the arrow, by ¯ + 1 [0001] → 1 [2023] ¯ insertion of a 21 [0001] partial. The resulting reaction 13 [1010] 2 6 indicates that the faults in Figure 4.29(c) are associated with a displacement vector ¯ . However, since the fault boundary is created by the random shuffling of of 16 2023 atoms, there is no necessity for partial dislocations to be present. A typical substructure of martensite laths and plates, primary as well as secondary, both dislocated and twinned, consists of fine striations within the laths and plates. Such a substructure was recorded by Ericksons et al. (1969), Hammond and Kelly (1970) and Knowles and Smith (1981a,b) in Ti–Mn and Ti–Cr binary alloys. Selected area diffraction from any martensite plate and both shows streaking which has been characterized by Banerjee and Muralidharan (1998) by detailed tilting experiments. It has been observed that the plane of diffracted intensity (which causes streaking in diffraction patterns along different zone axes) extends along a plane nearly perpendicular to the martensite habit plane (as shown in a stereographic projection in Figure 4.30. This observation is also consistent with
Martensitic Transformations
317
[111] (011)
(211)
70.53°
(a) A A
A
(0001)
A B
C
O
57°
A
A A
(b)
(c)
Figure 4.28. (a) A basic cell for the hcp structure outlined within five bcc unit cells, before the transformation. Shear along [111] transforms this cell to a hexagonal unit cell. The choice in the direction of shuffle to obtain the hexagonal structure. (b) Homogeneous shear transforms this cell into a hexagonal unit cell and (c) shows a choice (OB or OC) in the direction of a shuffle to obtain the hcp structure (after D. Banerjee and Muraleedharan, 1998).
the fact that the fine-scale substructure observed in Ti martensites shows striations approximately perpendicular to the habit plane Figure 4.31. Tilting experiments have also established from the invisibility criteria that the displacement vector asso¯ or 1 (2023]. ¯ This conclusion ciated with the observed striations is either 13 [1010] 6 has been arrived at on the basis of the observation that the striations are visible ¯ only with g = n 1100 + m0001 when n = 0. The micrographs in Figure 4.31
318
Phase Transformations: Titanium and Zirconium Alloys (a)
(b) Fault with 1 6 [2023] displacement vector A C A C Faulting created A by wrong shuffle B A B Sh A C ea [1 r b A 01 y C 0] A C B A B A
A B A B A B A B A
1 2
1 3
Ins er t [00 ion 01 of ]
hcp stacking
(c)
Figure 4.29. (a) The stacking along the [0001] direction in a hcp structure. The fault in the stacking ¯ shear. created by alternate choices of shuffle and (b) the fault created by a 1/3 [1010]
(100) (1120)
(1010)
(2110) Na
Nb (0110) (434) HABIT (111)
(011)
(1100) (011)
← plane of intensity
[1213]
(1210) [2113]
(0001) (011) [1123]
(1210) (111)
[4223] (011) (1100)
(0110)
(2110) (111)
(1010)
(1120) (100)
Figure 4.30. A stereographic projection of the Ti and the bcc-Ti phases in the Burgers relationship (open circles). The direction of the streaking measured from bright field micrographs with different zones and the line drawn through them indicates the plane of intensity distribution resulting in the streaking (after D. Banerjee and Muraleedharan, 1998).
Martensitic Transformations
319
Figure 4.31. Micrographs showing three-dimensional domains and their boundaries (after D. Banerjee and Muraleedharan, 1998).
also suggest that the substructure does not merely consist of planar faults which extend across the martensite plates and laths but is more closely associated with three-dimensional domains and their boundaries. The domain structure has a close similarity to the antiphase domain structure in ordered alloys though the substructure is seen in the martensitic -phase which is not ordered. The domain structure is not isotropic in the sense that the bounding surfaces of the domains are predominantly along basal planes with some segments being present along prism and pyramidal planes. Based on the observations made on the details of the substructure of Ti alloy martensites, Banerjee and Muralidharan (1998) came up with a schematic diagram (Figure 4.32) showing the domain structure in relation to a section of a martensite plate. The domains have been named as “stacking domains”, the displacement vector associated with the boundaries separating [00
01
(00
01
)
M
id
rib
]
(2
11
0) Habit pla
ne
[12
10
]
Figure 4.32. A schematic diagram of the domain structure in relation to a section of a martensite plate (after D. Banerjee and Muraleedharan, 1998).
320
Phase Transformations: Titanium and Zirconium Alloys
adjacent domains being that of a stacking fault in the fundamental hcp structure (unlike a fault vector in the superlattice of an ordered structure). The presence of strain contrast in the images of the domain boundaries has also been established. The origin of domain formation in a martensitic structure can be considered in the following manner. When a thin sheet of martensite (of the order of a few unit cells in dimension) forms by the expansion of a shear loop which transforms a specific set of 110 planes into the 0001 planes, the required atomic shuffles for accomplishing the structural change occur in a random fashion. This process creates a given domain. It is not possible to ascertain whether the shuffle occurs at the advancing transformation front or within the product after the manifestation of the shape strain within that small volume. With further propagation of the shear front, contiguous domains form by randomly selecting one of the shuffle directions within a domain. Since there is no change in the nearest neighbours across a (0001) interface between two adjacent domains, the (0001) surfaces become the predominant bounding surfaces of these domains. Short segments of domain boundaries form along prismatic and near prismatic planes to close the domain volume as columns parallel to the basal plane, extending across the plate thickness (as shown in schematic drawing in Figure 4.32). 4.3.3.2 Transition in morphology and substructure In the literature on the martensite in Fe-based alloys, a number of terminologies have been used for describing the morphology and the substructure of the martensite product. Various characteristics, such as the nature of the assembly of martensite units which includes variant distribution and stacking pattern, the extent of self-accommodation, the nature of LIS and the transformation (Ms ) temperature are taken into account in grouping the martensite products and assigning suitable terminologies for describing them. Krauss and Marder (1971) have made a survey and have reported that there is a definite preference for the use of the terms: lath martensite and plate martensite for the two broad classes of martensites, the characteristic features of which are given in Table 4.10. These attributes which have been identified essentially from a large body of experimental data on Fe-based martensites can as well be adopted for classifying martensites in Ti- and Zr-based alloys. As pointed out earlier, martensites in these alloys can be broadly grouped into the lath and the plate types. The facts that based on the same characteristic features, martensites in Ti- and Zr-based alloys can be classified into these two groups and that a transition is morphology and substructure can be introduced by suitably choosing the alloy composition suggest that there is something fundamental in this morphological transition. The crystallography of the transformations, fcc → bcc or bct in ferrous alloys and bcc → hcp or orthorhombic in Ti and Zr alloys, being quite distinct, there must be some common issues (not related to
Martensitic Transformations
321
Table 4.10. Characteristic features of two classes of martensitic products in Fe-based alloys. Attributes
Lath martensite
Plate martensite
Shape of the martensite unit, length, width and thickness given by a, b and c
a > b > c, parallel sided
a ≈ b c, acicular or lensshaped
Nature of stacking of of martensite units
A group of laths of nearly identical habit plane stacked in a parallel array within a packet: (a) martensite variants, (b) close orientations in a group and (c) alternate laths of twin-related variants
Multiple variants present along intersecting habit planes in the same microscopic domain
Mesoscopic morphology
Packets of martensite plates, several packets filling up the entire austenite grain
Fractal morphology arising from repeated partitioning of the austenite grains by martensite plates of different habit variants – smaller martensite plates filling up the partitioned austenite volume created by martensite plates of previous generation
Internal structure arising from the lattice invariant shear
Generally dislocated martensite (only in a limited cases twinning is observed within lath martensites)
Frequently encountered internally twinned plates
Transformation temperature and sequence of transformation events
Relatively high Ms temperature dilute alloys
Relative low Ms temperature Move concentrated alloys
Kinetics
Slowly growing martensite volume fraction with increasing cooling
Rapidly growing martensite volume fraction with increasing cooling
Mechanism
Martensite laths of successive generations weakly strain-coupled
Strong strain coupling of martensites of successive generations
Type
Schiebung martensite
Umklapp martensite
the crystallographic symmetries and the magnitude of the (Bain strain) which dictate the transition in morphology and substructure of martensites. The factors which appear to have strong influences on the morphology and substructure are (a) transformation temperature, (b) self-accommodation of a group of martensite
322
Phase Transformations: Titanium and Zirconium Alloys
plates (c) plastic flow of the soft parent phase in partially accommodating the transformation strain and (d) autocatalytic nucleation of martensite plates. These factors are again interdependent. With decreasing Ms temperature, the flow stress of both the high-temperature parent phase and the low-temperature martensitic phase increases substantially. This enhances the strain energy accumulation due to the formation of a single martensite plate. The requirement of self-accommodation, therefore, becomes more important as the Ms temperature is lowered. The morphological transition from the lath to the plate type in several ferrous alloys is accompanied by a change in substructure from dislocated to twinned. The coincidence of morphological transition (lath to plate) and substructural transition (dislocated to twinned) has also been noticed in Ti- and Zr-based martensites (Banerjee and Krishnan 1973). It is still difficult to conclude whether the transition in one is instrumental in bringing about the other transition. Figure 4.33(a) shows the Ms temperature as a function of alloy content in several binary alloys of Ti and Zr. It is seen that as the Ms temperature is brought down below about 923 K, both morphological and substructural transitions occur. In fact, the correlation between the transition in morphology and substructure and the Ms temperature is universally valid for Ti and Zr systems whereas the correlation is not so perfect in case of ferrous alloys. Let us now examine the plausible reasons for a transition in morphology and in substructure with lowering of the Ms temperature. When a martensite lath forms at sufficiently high temperatures where the matrix -phase remains relatively soft, the shape strain associated with the formation of a lath is primarily accommodated by the plastic flow of the matrix. No significant stress field is generated around the first-formed laths and therefore, stress-assisted nucleation of other martensite variants around the first-generation laths remains quite restricted. The surfaces provided by the first-formed laths, however, act as favourable nucleation sites for new generation laths. Those orientation variants are favoured which have a habit plane close to that of the first-formed laths and whose orientations are either close to or twin related with the former. These tendencies are responsible for creating a packet of parallely stacked laths. As far as the selection of the LIS is concerned, a lower critical resolved shear stress for slip compared to that for twinning promotes the operation of the slip mode at high temperatures. This point is illustrated in Figure 4.33(b). With the lowering of the Ms temperature, the stress field created by the firstgeneration plates assists the nucleation of several martensite variants in the vicinity of the former. However, the principle of self-accommodation allows the nucleation and growth of those plates which permit minimization of the stress field associated with the assembly of plates. The grouping of variants is invariably controlled by
Martensitic Transformations
323
900
Lath-dislocated martensite
Temperature (°C)
800 700 600
Zr
Ti
Ta
500
Cu
Nb
Cr
400
Acicular (plate) twinned martensite
300 200 Zr
5
10
15
20
20
15
10
5
Ti
Atom % (a) Slipped
Stress
Twinned
Twinning
Slip Temperature (b)
Figure 4.33. (a) Plots of the Ms temperature versus the alloy content in several binary alloys of Zr and Ti showing the necessary level of alloy additions to bring about a change in the morphology and the substructure. (b) A schematic diagram illustrating the variation of the CRSS for slip and for twinning as a function of temperature. The diagram suggests that a lowering of the Ms temperature would result in a slip to twin cross-over in the operating mode of inhomogeneous shear.
the principle of self-accommodation, leading to a polydomain morphology with least stored elastic energy. The elastic bonding of one plate with its neighbouring plates is the primary driving force for the generation of the plate morphology. Low transformation temperatures also promote the occurrence of twinning which predominates at least in the primary martensite plates in plate morphology.
324
Phase Transformations: Titanium and Zirconium Alloys
4.3.4 Stress-assisted and strain-induced martensitic transformation The chemical free energy change which acts as the driving force for initiating a martensitic transformation can be supplemented by applied stress and/or plastic strain. As has been mentioned earlier, the Ms temperature defines the extent of supercooling (To −Ms ) which is required for the provision of the adequate chemical free energy change which can meet the surface and the strain energy requirements for martensite nucleation. It is because of this reason that the martensite start temperature is raised above Ms when external stress or plastic deformation is introduced in the parent phase. The interrelationships among applied stress, plastic strain and martensitic transformation have been extensively studied in Fe-Ni-C and Fe-Ni-Cr-C alloys by Bolling and Richman (1970a,b). They have defined a temperature, Ms (lying above Ms ) below which plastic yielding under applied stress is initiated by the onset of martensitic transformation and above which plastic flow of the parent phase precedes the nucleation of martensite. Olson and Cohen (1972) have provided a schematic representation of the interrelationships between applied stress, plastic strain of austensite and the stress required for martensitic nucleation (Figure 4.3) at temperatures above Ms . In the temperature range Ms < T < Ms , the stress required for martensite nucleation increases with temperature and follows a temperature dependence consistent with the elastic stress requirement for martensite nucleation; the phenomenon in this temperature range is, therefore, described as stress-assisted martensite nucleation. With increase in temperature within this range Ms < T < Ms , the stress requirement increases in a linear fashion as the driving chemical free energy change decreases approximately linearly with temperature. In the temperature range Ms < T < Md , the stress required for stress-assisted nucleation exceeds the yield strength of austenite, and therefore, the austenite initially yields by slip, causing extensive dislocation activity. With the accumulation of plastic strain (and dislocation multiplication), localized strain centres are created where martensite nucleation becomes energetically feasible. The yield strength of austenite marks the lower boundary of the stress required for strain-induced nucleation, as shown in Figure 4.3. To summarize the overall situation, the chemical driving force is sufficient at Ms for the pre-existing nucleation sites or embryos in the austenite to become operative without the application of stress. At temperatures between Ms and Ms , in the stress-assisted nucleation regime, such nucleation can occur only with the aid of applied stress. Here the required initiating stress is in the elastic range but increases with increasing temperature because of the concomitant decrease in the chemical driving force. At temperatures above Ms , plastic straining of the austenite results in the creation of martensite nuclei in the regime of straininduced nucleation. With increasing temperature above Ms , further reduction in
Martensitic Transformations
325
chemical driving force necessitates increased plastic strain in order to produce detectable amounts of transformation. This requires the initiating stress to rise above the yield stress of the austenite. As a result, plastic flow of the austenite, leading to extensive dislocation multiplication, occurs during the onset of plastic deformation. The nucleation of martensite becomes favourable at locations where strain accumulations exceed a certain threshold value. The formation of stress-assisted martensite is reflected in the appearance of a deviation from the linear elastic behaviour at a stress value lower than the yield stress value of the parent phase. Figure 4.34 illustrates the stress–strain plot for a Ti alloy in which stress-assisted martensite formation commenced at the point marked by an arrow. As mentioned earlier, martensite plates were introduced by applying external stress in alloys of suitable compositions, e.g. Ti–12.5% Mo (Gaunt and Christian 1959), Ti-V (Menon et al. 1980), in which the -phase could be retained in a metastable state by quenching. In such alloys, the Ms temperature in invariably below the room temperature and the martensite formed is primarily of the internally twinned plate type. TEM examination of these martensites has revealed the presence of periodic twins within these martensite plates (Figure 4.35). Since these martensites contain a fairly high level of -stabilizers (in order to lower the Ms temperature below room temperature), they often exhibit orthorhombic distortion, though it is not essential that all stress-assisted or strain-induced martensites
Engineering stress (MPa)
1000
800
FC
WQ
600
400
↓
200
+α –β heat-treated β heat-treated 0 0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Engineering strain
Figure 4.34. Stress–strain plot for a Ti alloy in which stress–assisted martensite formation commenced at the point marked by an arrow. While furnace cooled (FC) samples do not exhibit stressinduced martensite formation, water quenched (WQ) samples show deviation from linear elastic deformation at a low stress value (pointed by an arrow) corresponding to the formation of stressinduced martensite.
326
Phase Transformations: Titanium and Zirconium Alloys
200 nm
200 nm
(a)
(b)
OM
β
OM
OM
750 nm
(c)
Figure 4.35. TEM micrographs showing martensitic plates with periodic twins.
in Ti- and Zr-based systems will be orthorhombic in nature. It may also be noted that stress-assisted martensite has been encountered only in a limited number of Zr-based alloys, the Zr-Mo-Al alloy system being an example in which clear evidence of stress-induced martensite has been reported.
4.4
STRENGTHENING DUE TO MARTENSITIC TRANSFORMATION
It is now well established that the strengthening due to the martensitic transformation arises from various factors such as solid solution hardening, precipitation hardening, order hardening and hardening due to lattice defects and fine crystallite size. Several attempts have been made to determine quantitatively the individual contributions of these factors towards the overall strengthening of martensites. Excellent reviews by Christian (1971) on ferrous martensites and by Warlimont and Delaye (1974) on non-ferrous martensites treat the subject in a comprehensive manner. Extensive research on the strength of ferrous martensites has demonstrated
Martensitic Transformations
327
that the major contribution to the strength of Fe–C martensites arises from the solid solution strengthening brought about by C atoms in the supersaturated body centred tetragonal phase and that the defect structure introduced by the martensitic transformation is responsible only for a relatively small increase in the strength level. That part of the strength which arises from solution hardening could be imparted to the base metal by introducing C atoms in excess of the solubility limit even by some alternative methods such as sputter deposition (Dahlgren and Merz, 1971) and thus cannot be attributed to the martensitic transformation per se. In fact, the strengthening due to a fine-scale precipitation or due to Snoek ordering of C atoms is also essentially connected with the C supersaturation in the martensite. Apart from solid solution hardening, a substantial part of the strength of martensites in several Cu-based alloys is derived from the ordering of substitutional atoms. In other words, the inheritance of a high level of solute concentration or an ordered arrangement of atoms from the parent phase is primarily responsible for the strengthening of martensites in Fe- and Cu-based alloys. Heat treatments involving the martensitic transformation have been found to be quite effective in strengthening Ti- and Zr-base alloys. The + alloys in these systems are amenable to the formation of the martensitic phase on quenching from the phase field. Standard heat treatments of these alloys involve a quenching treatment followed by a tempering treatment which results in the best combination of strength, ductility and fracture toughness. Unlike in the case of the Fe–C system, the hcp martensites in Ti- and Zr-based alloys are neither supersaturated with interstitial elements nor possess any distorted crystal structure. It is, therefore, interesting to examine the extents to which different factors are responsible in imparting strength to Ti and Zr alloy martensites. Banerjee et al. (1978) have investigated the strengthening mechanisms operative in martensites in binary Ti–Zr alloys. The isomorphous Ti–Zr system provides a unique opportunity in which the strengthening contributions of the solid solution and the martensite substructure can be separated. While quenching produces the martensitic structure, a slow cooling from the -phase field results in the formation of a Widmanstatten structure. The chemical composition (including the interstitial content) and the crystal structure corresponding to the and phases so formed being identical, the difference between the flow stress values of
and can be considered to arise from the differences in the defect structures of these products. The martensitic structure contains more closely spaced interfaces separating adjacent laths, plates and twins and a higher density of dislocations as compared to the slow-cooled Widmanstatten structure of the same composition. The as well as the phases in this isomorphous system possess the equilibrium composition. Since there is no tendency of phase separation in either of these phases, the possibility of fine-scale precipitation, usually pertinent to the Fe–C
328
Phase Transformations: Titanium and Zirconium Alloys
martensite, can be ruled out. The → transformation attains completion both during the quenching and the slow cooling operations, and thus no interference from the retained parent phase is present. A comparison of the true stress versus true strain curves associated with the and structures of a given alloy directly yields the strengthening contribution, , which can be attributed solely to the martensitic substructure, being defined as the difference between the flow stresses of the and the structures, = − x , for a given value of the true plastic strain, p . The assumption which is implicit in this treatment is that the contributions of the various pertinent factors to the strength are additive. In order to quantify the microstructural difference between the Widmanstatten
and the martensitic (either dislocated lath or twinned plate morphology) phases, the average spacing between the partitioning interfaces in these has been measured for binary Zr–Ti alloys of different compositions (Table 4.11). While in the case of the Widmanstatten structure, the average thickness of the plates gives the mean free spacing between the partitioning interfaces, for the lath martensite structure, it is the average thickness of the laths and for the internally twinned plate martensite, it is the average twin thickness. The average packet size for the lath martensite structure has also been recorded as the packet boundaries are the effective barriers against the passage of slip or twin deformation. This point is discussed in a later section. The value of , which is a measure of the strengthening contribution solely of the martensitic structure, has been found to be strongly dependent on the alloy composition as seen in Figure 4.36. This is not unexpected in view of the fact that the morphology and substructure undergo a drastic change from the dislocated lath to the twinned plate martensite as the Ti content exceeds a limiting value of about 15% in Zr–Ti alloys. The sharp change in the value at around this composition strongly indicates that the twinned plate martensite structure provides a more effective barrier against dislocation motion. The value has also been plotted against the true plastic strain (Figure 4.37) for different alloy compositions. The sharp rise in with increasing plastic strain in the case of twinned plate Table 4.11. Average size of the and the units in Zr–Ti alloys. Average width of Widmanstatten plates ((m) Zr–5Ti Zr–10Ti Zr–15Ti Zr–20Ti
81 11.5 3.1 18
Average spacing between partitioning interfaces ((m) 0.8 0.4 0.2 0.15
Average packet size in the lath structure ((m) 60 25 – –
Martensitic Transformations
329
0.010 40
0.005 0.015 and 0.020
400
0.002
36
Transition in morphology and substructure lath → plate 32 dislocated → twin
300
24 200
20
MN/m2
Δσ (kg/mn2)
28
16 12 100 8 4 0
Zr
10
20
30
Ti atom percent
Figure 4.36. The rapid rise in corresponds to the composition range over which the martensite structure changed from the dislocated lath to the twinned plate type. The values of the plastic strain at which flow stress values were measured are indicated.
martensites suggests that this substructure exerts a powerful influence on the work hardening process. In contrast, the work hardening rate remains somewhat less sensitive to the plastic strain for dislocated lath martensites in pure Zr and in Zr–5% Ti and Zr–10% Ti alloys. 4.4.1 Microscopic interactions We will now examine the effectiveness of different types of interfaces in offering resistance to an approaching deformation front (either slip or twin). 4.4.1.1 Lath boundaries It has been observed (Banerjee et al. 1978) that a slip band can propagate through the small angle boundaries separating adjacent laths contained within a packet. Slip deformation can be induced in a thin foil sample by beam heating in the TEM. A dynamic experiment has shown that a set of dislocations can easily penetrate through the lath boundaries, the path followed by each dislocation being marked by a pair of slip traces which suffers a small angular deviation at the lath boundary. The stress required to push a dislocation out of a small angle boundary, made up of an array of parallel edge dislocations lying on a plane normal to the slip
330
Phase Transformations: Titanium and Zirconium Alloys 700
700
σ (α′) σμ (α′)
400
σ (α)
300
σ∗ (α′)
100
0.02
0.02
σμ (α′)
500
Stress (MN/m2)
Stress (MN/m2)
500
0
0.02
σμ (α)
300
σμ (α)
200
σ∗ (α)
100
0.02
σ (α)
400
σ∗ (α′)
0
0.02
0
0.02
0.04
(a) 900
Zr –15% Ti
σ (α′)
σ (α)
500
σμ (α) 400
Stress (MN/m2)
600
Stress (MN/m2)
0.08
Zr –20% Ti
800 700
σ (α′)
600
σμ (α′)
σμ (α′)
700
0.06
(b) 900
800
σ∗ (α)
Plastic strain
Plastic strain
500
σ (α)
400
σμ (α) 300
300
200
200
σ∗ (α′)
σ∗ (α)
100 0
Zr –10% Ti
600
600
200
σ (α′)
Zr – 5% Ti
0.02
0.04
0.06
0.08
σ∗ (α′)
σ∗ (α)
100 0
0.0
0.02
0.04
Plastic strain
Plastic strain
(c)
(d)
0.06
0.08
Figure 4.37. The value plotted against the true plastic strain for different alloy compositions.
Martensitic Transformations
331
direction, x, is given by xy =
035b 2h1 − $
(4.53)
where h is the spacing between the dislocations, is the shear modulus and $ is the Poisson ratio. The stress xy can be generated at the head of a pile-up of n dislocations where n = xy , being the applied stress. Since n, and L, the average lath thickness, are related by #L b
(4.54)
0352 b2 2#Lh1 − $
(4.55)
n= Eq. (4.51) reduces to )2 =
Substituting typical values for L= 1m, h (= 40 nm) and b (= 032 nm), the magnitude of the applied stress, , required to push a dislocation out of the lath boundary can be estimated to be about /5000. Such a small value of justifies the easy penetration of glide dislocations through the lath boundaries. TEM of deformed lath martensites has also shown that deformation twins too can propagate through the lath boundaries which separate laths of nearly identical orientations. The packet boundaries separating laths of two distinct orientation variants have been found to be quite effective in stopping slip and deformation twin bands. It is, therefore, the packet size which is more important in deciding the strengthening contribution of the martensitic structure in dislocated lath martensites in Ti- and Zr-based alloys. Since the magnitude of the LIS is rather small in these alloys, the density of dislocations in these martensites is not very large. A high Ms temperature (above 923 K) is also responsible for a partial annihilation and rearrangement of dislocations resulting the LIS and the plastic accommodation. As a result, the strengthening contribution of dislocations in the lath martensite structure is not very significant. 4.4.1.2 Twin boundaries and plate boundaries A large proportion of the partitioning interfaces in the internally twinned plate martensite structure is constituted of 10¯I1 twin interfaces. Some of these are boundaries separating adjacent twins within single plates while some others are located between contiguous martensite plates obeying twin-related variants of the orientation relation.
332
Phase Transformations: Titanium and Zirconium Alloys
Contrary to expectations, the increase in strength associated with the transition from a dislocated lath to a finely twinned substructure in steel martensites is not very large. Had the twin interfaces in steel martensites acted as effective barriers against dislocation motion, the internally twinned plate morphology would have been about three times as strong as lath martensites, consistent with an order of magnitude refinement of the mean free slip length. Experiments have shown twinned martensites to be only 8–30% stronger than lath martensites in ferrous systems. This discrepancy has been explained by Kelly and Pollard (1969) on the basis of the Sleeswyk–Verbraak mechanism (1961) of the passage of a/2 <111> slip dislocations through the coherent twin interfaces. The effectiveness of 1011 -type twin interfaces in resisting the motion of dislocations in a twinned martensite plate in a Ti or a Zr alloy can be evaluated ¯ by considering the interaction of the different variants of slip in the 1010 ¯ < 1210 > system with a specific 1I01 twin interface. Yoo (1969) has expressed the dislocation reaction at a twin interface in a general form as Xhklm X∗ hklt ± nbt
(4.56)
where X is the Burgers vector of a dislocation gliding on the (hkl)m slip plane of the matrix which, after encountering a twin interface, gets converted to a new dislocation in the twin crystal with a Burgers vector X∗ (hkl)t which can glide on the (hkl)t plane of the twin orientation. While crossing the twin boundary, the dislocation creates a twin dislocation with Burgers vector nbt where bt is the Burgers vector of a unit twin dislocation. Using the relevant transformation matrices for transforming the Burgers vectors and Miller indices of the corresponding slip systems, the interaction of three variants of the first-order prismatic slip, {101¯ 0} <1210> , with a specific (11¯ 01) twin interface can be described by the following dislocation reactions: 1/32110m 0110m → 1/25146t 1343t − bt
(4.57)
1/31120m 1100m → 1/31120t 1103t
(4.58)
1/31210m 1010m → 1/21546t 3143t + bt
(4.59)
Out of these possible interactions, only in the second case can a dislocation with ¯ propagate through the (11¯ 01) twin interface without changing its b = 1/3 [1120] Burgers vector. The line of intersection of the (1¯ 100) slip plane and the (11¯ 01) ¯ m direction and, therefore, the dislocation line assumes twin plane defines the [1120] a perfect screw character as it encounters the twin plane; and as a consequence of this, it is amenable for cross slip on the (1¯ 01)t plane of the twin crystal. This
Martensitic Transformations
333
mechanism has been illustrated schematically in Figure 4.38(a). Passage of slip deformation in the other two cases is quite unlikely as the indices of the slip plane and direction are irrational and a twinning dislocation is left behind at the interface. Thus for these two slip systems, the (11¯ 01) twin plane acts as totally opaque. If one considers that the orientations of martensite crystals are nearly randomly distributed, the probability of the twin interface being completely opaque towards a first-order prismatic slip is 2/3. For the remaining 1/3 probability, the entry of 1/3 [1120] dislocations through a cross slip mechanism (illustrated in
(1101) Twin plane
ξ
b (1100)M
(1100)T
(a) (1122)
21°
3]
[112
(0112)
(1011)
1]
[011
(1011)
(2111)T
(0112)T
(b)
Figure 4.38. (a) A schematic showing the mechanism for cross slip on the (1¯ 01)t plane of the twin crystal (b) Schematics showing the entry of 1/3[1120] dislocations through a cross slip mechanism.
334
Phase Transformations: Titanium and Zirconium Alloys
Figure 4.38(b)) is possible. However, the stress required to make the dislocation glide on the new slip plane in the twin segment would be different, depending on the Schmidt factors for the slip systems in the matrix and the twin segments. The analysis of the interaction of the most favourable prismatic slip mode with {101¯ 1} twin interfaces shows that for a majority (2/3 probability) of the orientations of the stress axis a single twin segment would behave as a plateshaped crystal strongly bounded by a pair of non-deformable interfaces such that neither can the slip dislocations penetrate these nor can there be any sliding along the coherent twin interface. The deformation of a thin crystal of thickness in the range of 50–200 nm, bounded by non-deformable twin interfaces, results in the generation of a strain gradient and as a consequence an array of geometrically necessary dislocations is created. Ashby and Johnson (1969) has shown that during the deformation of plastically inhomogeneous materials, geometrically necessary dislocations are produced and the dislocation density is proportional to /bL where and b are the magnitudes of the shear and the Burgers vector, respectively, and L is the spacing between the non-deformable interfaces. Banerjee et al. (1978) have shown that geometrically necessary dislocations indeed accumulate in Zr–Ti martensites subjected to plastic deformation. These dislocations are seen to be lined up along the twin boundaries and their density increases with lowering of the mean free slip length, L. The possibility of the passage of deformation twins cutting across transformation twins needs be considered in these hcp alloys in which different deformation twin systems are operative, particularly at low temperatures and at high strain rates. The most common deformation twin systems in these alloys are {101¯ 2} ¯ <1123 ¯ > . The possibility of the <1¯ 011> , {1121} < 2113> and {1122} intersection of different variants of these deformation twins across a stack of (101¯ 1) transformation twins can be examined by the invoking the following plastic compatibility conditions proposed by Cahn (1953): (a) the traces of the crossing twin and the secondary twin in the composition (K1 ) plane of the crossed twin should be parallel and (b) the direction, the magnitude and the sense of the shear should be identical in the crossing and the crossed twins. If these plastic compatibility conditions are fulfilled, a deformation twin cannot be impeded by twin interfaces. From such an analysis, it can be seen that the following twin shears can pass through a set of (101¯ 1) transformation twins: (011¯ 2) [01¯ 11] , (11¯ 02) [1¯ 101] , (1122) [1123] and (21¯ 1¯ 2) [21¯ 1¯ 3] These types of intersections have been reported in the microstructure of the deformed martensite in Zr–15% Ti and Zr–20% Ti alloys (Banerjee et al. 1978). Figure 4.39 illustrates a few cases of intersection of deformation twins and transformation twins.
Martensitic Transformations
(a)
335
(b)
(c)
Figure 4.39. Examples of intersection of deformation twins and transformation twins.
4.4.2 Macroscopic flow behaviour The influence of partitioning interfaces, which are mostly impenetrable to dislocations, on the macroscopic flow behaviour of Zr–Ti alloys with respect to initial yielding as well as work hardening has been examined in this section. The accumulation of geometrically necessary dislocations in the vicinity of partitioning interfaces, observed in deformed Zr–Ti martensites, points to the fact that the flow behaviour of martensites can be rationalized in terms of Ashby’s one parameter work hardening theory for plastically inhomogeneous materials. According to this theory, the work hardening associated with a given microstructure characterized by a mean free slip length, , is expected to be governed by geometrically necessary dislocations if the density of these is much larger than that of statistically stored dislocations. Under such circumstances, the stress–strain plot would assume a parabolic shape given by the following relation: 1/2 b 1/2 (4.60) = 0 + C1 EM
where 0 and are the tensile stresses at strain levels zero and , respectively, C1 is a dimensionless constant which has been estimated to be 0.35 ± 0.15 for
336
Phase Transformations: Titanium and Zirconium Alloys
plate-like barriers (Ashby, 1971), b is the Burgers vector, M is Taylor’s orientation factor and E is the Young’s modulus of a random polycrystalline sample. The applicability of Ashby’s one parameter work hardening theory to the plastic flow behaviour of Zr–Ti martensites has been demonstrated by the linear nature of the plots of versus 1/2 (Figure 4.40(a)). It can be seen that each flow curve can be fitted with two straight line segments, the first segment corresponding to strain values less than about 1% and showing a much larger work hardening rate than Zr – 15% Ti(α′) 800
σ0
σ (MN/m2)
400 Zr – 10% Ti(α′) Zr – 15% Ti(α)
600 500
Zr – 10% Ti(α)
400
Zr – 5% Ti(α′)
Δσ0 (MN/m2)
700
500
σ1 – σ0
Zr – 5% Ti(α)
300
200
300 Zr(α′) 200
Zr(α)
100 0
100
1
ε1 2
Slope 0.12 MN × m 0
0
5
10
20
15
ε
1
2
25
30
0
[(λ 2)
–1
× 102
(a)
1
2
– (λ1)
–3
2 –1
2
] (μm)
2
3 –1
2
(c) 1.8 1.6
E
σ1 – σ0
× 103
1.4 1.2 1.0 0.8 0.6 0.4 0.2
Slope 0.19 0
1
2
3
4
( bε1) λ
½
5
6
7
8
× 103
(b)
Figure 4.40. (a) A plot of versus 1/2 , (b) a plot of 1 − 0 /E vs b/ 1/2 / and (c) a plot 1/2 1/2 between 0 and 2 − 1 .
Martensitic Transformations
337
the second segment. In martensitic structures, the initial flow showing a high work hardening rate can be attributed to the localized flow of some regions of high stress concentration, while the second segment corresponds to the parabolic stress–strain curve characteristic of the deformation regime where work hardening is essentially controlled by the short-range interaction between gliding dislocations and the steadily increasing geometrically necessary dislocations. This has been confirmed by demonstrating a linear relation (Figure 4.40(b)) between the two dimensionless quantities (1 − 0 /E and (b/ 1/2 , where 1 is the stress corresponding to a plastic strain 1 , 0 is the extrapolated value of the true stress at = 0 (as shown Figure 4.40(b) and is the mean free slip length between the partitioning interfaces which are impenetrable to dislocations. The parameter corresponds to the average width of Widmanstatten plates for the slow-cooled structure and to the average packet size and twin spacing in the lath and the twinned martensite, respectively. Substituting M = 4, a value proposed by Armstrong et al. (1962) for hcp metals, the theoretical slope of a plot of (1 − 0 /E versus (b1 / 1/2 can be estimated to be 0.17 which is quite close to that experimentally obtained. This analysis points to the fact that the enhanced work hardening in internally twinned Zr–Ti martensites is due to the presence of closely spaced partitioning interfaces (which are predominantly {101¯ 1} twin boundaries). The 0 values, obtained form the extrapolation shown in Figure 4.40, closely match with the 0.2% yield stress values for both the martensitic and the Widmanstatten structures. In order to examine whether the influence of the geometric slip distance, , on the yield stress of these structures follows the Hall– Petch relation, 0 values for a given alloy for the and structures, represented by 0 ( ) and 0 , respectively, can be expressed as follows: & 0 = i + KX −1/2 1
0 = i + KX −1/2 2 0 = 0 − 0 = KX −1/2 − −1/2 2 2
(4.61)
In these equations, i is the friction stress corresponding to zero strain value, KX is the Hall–Petch constant and the geometric slip distances for the and the structures are 1 and 2 , respectively. It is implicit in Eq. 4.59 that the parameter i has nearly the same values for the and the structures for a given alloy composition in spite of the difference in the dislocation densities in these. The Hall–Petch constant KX depends on alloy composition mainly through its dependence on the shear modulus, . Since the shear moduli for Zr and Ti are not much different, KX is not expected to alter significantly with alloy composition in the binary Zr–Ti system. This justifies a linear relation between − −1/2 ), as shown in Figure 4.40(c). The slope of this plot has been 0 and ( −1/2 2 1
338
Phase Transformations: Titanium and Zirconium Alloys
found to be close to the reported value of the Hall–Petch constant (Ramani and Rodriguez 1970) in Zr alloys at 300 K. The applicability of the Hall–Petch relation for estimating the increase in 0 due to the introduction of the martensitic structure demonstrates that the refinement of the grain size factor essentially controls the martensite strengthening in this system. The flow stress, , can be expressed in terms of several components as per different criteria chosen, viz., thermally activated and athermal, dependent on or independent of grain size, plastic strain and solute content. Banerjee et al. (1978) have examined how the martensitic structure influences the different components of the flow stress of Zr–Ti alloys. On the basis of the preceding analysis, the flow stress, , can be expressed as = 1 + stat + KX 1/2 + Cb/ 1/2
(4.62)
where i is the friction stress, C represents the constant term C1 EM −1/2 (see Eq. 4.48) and stat is the -independent component of the flow stress arising from the statistically stored dislocations accumulated during plastic flow. The contribution of stat in the case of Zr–Ti martensites has been shown to be very small and can be neglected. Stress relaxation experiments have been used for measuring the athermal component, M , of the flow stress, and the thermal component, ∗ , of the flow stress has been obtained by subtracting from the total flow stress, (from the relation, = + ∗ ). The values of the activation area, A∗ , associated with the thermally activated flow in both Widmanstatten and martensitic structures for these alloys have also been determined from the stress relaxation data. The salient features of the results arrived at are as follows: (1) The contribution of the martensitic structure (for both the dislocated lath and the twinned plate martensites) to strengthening is mainly through an increase in the athermal component, , of the flow stress. (2) The thermal component, ∗ , of the flow stress is essentially independent of the Ti concentration in these Zr–Ti alloys, suggesting that the strengthening due to the substitutional solid solution arises mainly from an increase in the long-range athermal stress. This is consistent with the results obtained in a number of Zr- and Ti-based substitutional alloys (Conrad et al. 1972). (3) The value of ∗ is independent of plastic strain, implying that dislocation– dislocation interactions do not constitute the rate-controlling mechanism of thermally activated deformation. (4) The values of the activation area for both the and structures in Zr– Ti alloys remain within a narrow band between 25b2 and 35b2 where b2 is
Martensitic Transformations
339
the Burgers vector. This implies that the rate-controlling mechanism of the thermally activated flow remains unaltered with changes in the Ti content of these alloys and with the introduction of the martensite structure. The values of the activation area in the case of these martensites closely match with that reported for annealed Zr containing about the same levels of interstitials (about 600 ppm O2 and 60 ppm N2 ). Based on the results of stress relaxation experiments, it could be inferred that the rate-controlling thermally activated deformation mechanism is not altered by the introduction of the martensitic structure and even by the addition of substitutional alloying elements. This is consistent with the findings of Conrad et al. (1972) who have demonstrated that substitutional alloying elements and / phase distribution do not affect the rate-controlling deformation mechanism in Ti-based alloys. They have suggested that a mechanism involving dislocations overcoming the interstitial solute obstacles is rate controlling during low temperature deformation of Ti alloys. However, there is some controversy over this issue, and an alternative view is that the Peierls–Nabarro barrier, which is increased by interstitial additions, is the rate-controlling obstacle. The fact that ∗ is not affected by the introduction of the martensitic structure and the extent of plastic strain indicates that ∗ is contained within the friction stress i (Eq. 4.59), which can be resolved into two components, ∗ and j (X). The former is essentially controlled by the interstitial level while the latter, a part of the athermal stress, , is independent of and and depends on the nature and the extent, X, of substitutional alloying additions. The total flow stress can, therefore, be expressed as = ∗ + j X + KX −1/2 + Cb/ 1/2
(4.63)
While the increments in the initial yield stress and the work hardening rate due to the introduction of the martensite structure are given by the terms KX −1/2 and Cb/ 1/2 , respectively, the increments in the initial yield strength due to interstitial and substitutional solid solution hardening are reflected in the terms, ∗ and j (X), respectively.
4.5
MARTENSITIC TRANSFORMATION IN Ti–Ni SHAPE MEMORY ALLOYS
Near-equiatomic Ti–Ni alloys and many of their multicomponent derivatives are well known for exhibiting the shape memory effect and pseudoelasticity. In these
340
Phase Transformations: Titanium and Zirconium Alloys
Ti–Ni-based alloys, three martensitic phases occur. The fully annealed binary Ti– Ni alloys transform from the B2 parent phase directly to the monoclinic B19 phase. In contrast, thermomechanically treated Ti–Ni alloys transform in two steps, namely the B2 phase to the trigonal R-phase (Goo and Sinclair 1985) and the R-phase to the monoclinic B19 phase (Wang et al. 1968). Ternary Ti–Ni alloys containing a few percentage of Fe or Al undergo the same sequence, B2 → R → B19 , of transformations (Mitsumoto and Honma 1976). Ti-Ni-Cu alloys, in which Cu substitutes Ni by 5–15 at.%, also transform in two steps; but in this case, the B2 phase first transforms into an orthorhombic B19 phase which finally transforms into the monoclinic B19 phase (Nam et al. 1990). The basic crystallography associated with these transformations has been discussed in the first part of this section. Some of the important microstructural characteristics of these martensites, which are essentially responsible for imparting the shape memory property, are also highlighted. The microstructural feature which is common to all shape memory alloys is such that the martensite plates are spatially organized in a very effective (Saburi and Nenno 1982) accommodating manner. The issue of self-accommodation in the martensite microstructure has been introduced earlier in the context of martensite morphology. The importance of both macroscopic self-accommodation and microscopic self-accommodation in the martensitic microstructure of Ni–Ti alloys, as reported recently by Madangopal (1997), has also been discussed in this section. The shape memory effect is intimately linked with some associated phenomena such as thermoelastic equilibrium, pseudoelasticity and thermal arrest memory effect. An account of these related phenomena is included in this section mainly to explain the interrelationship of these in terms of the microstructural response of shape memory alloys to alterations in the applied stress and temperature. 4.5.1 Transformation sequences The transformation temperatures and the sequence of transformation of Ni–Ti shape memory alloys are known to be sensitive to the chemical composition of the alloy, the memory imparting heat treatment, the prior cold work and the externally applied stress. Differential scanning calorimetry (DSC) and resistivity measurements have been mainly used for detecting the transformation sequences in Ni–Ti alloys with varying compositions and prior thermomechanical histories. A fully annealed, near-equiatomic NiTi alloy shows only a single peak in the DSC thermogram (Figure 4.41) during either a continuous heating or a continuous cooling experiment, indicating that both the forward B2 → B19 and the reverse B19 → B2 transformations occur in a single step. The same material, however, after being cold worked to a level of about 15% reduction in thickness, exhibits a two-step transformation during cooling, the first and the exothermic peaks representing the B2 → R and the R → B19 transformations. The transformation during heating
Martensitic Transformations
R/R293 K
(a)
341
Ms
1.4
Af
Mf 1.0
TR As
(b)
Heat flow Endothermic exothermic
1.6 × 10–3 J/s
Af Heating
Cooling TR
Ms 170
220
270
320
370
Temperature, T (K)
Figure 4.41. DSC thermogram showing transformation temperatures and the sequence of transformations in a Ni–Ti shape memory alloy.
still occurs in a single step, giving rise to a single endothermic peak. The temperature gap between the B2 → R and the R → B19 transformations increases with an increase in the extent of the prior cold work. The appearance of the R phase during the cooling of cold-worked NiTi alloys can be detected by monitoring electrical resistivity as a function of temperature or during in situ cooling of a sample in a TEM. Electron diffraction patterns, obtained from samples comprising the B2 phase at a temperature close to the B2 → R transition, show diffuse intensity with maxima located at reciprocal lattice positions which divide the B2 reciprocal lattice vectors in three equal segments. The intensity maxima at the 1/3 positions remain very weak and dark field micrographs using such reflections fail to reveal any observable contrast. However, as the sample is cooled further, plates of the R
342
Phase Transformations: Titanium and Zirconium Alloys
phase appear progressively and these plates get organized in a self-accommodating configuration which is described later. In situ experiments have also indicated that R-phase plates can nucleate at relatively small strain centres, such as single dislocations. As these plates grow and gradually fill up the B2 grain, sharp 1/3 reflections appear in the diffraction patterns. These reflections can be indexed on the basis of the trigonal crystal structure of the R phase. On further cooling, the R phase transforms into the B19 phase. Crystallographic features of these two martensitic transformations are discussed in the next section. 4.5.2 Crystallography of the B2 → R transformation The lattice correspondence between the B2 and the R phases is given in Table 4.12 which lists the relative dispositions of the four crystallographic variants of the R phase with respect to the parent B2 phase. Self-accommodation is achieved in this transformation by an assembly of the variants which is illustrated in Figure 4.42. While between variants 1 and 2 and between variants 3 and 4 there ¯ R type compound twinning, that between variants 2 and exists a relation of {1122} 3 and between variants 1 and 4 is of {1121}R type compound twinning. As shown ¯ R -type twin planes are derived from {100}B2 type in Figure 4.42(b), the {1121} mirror planes and the {1122}R type twin planes are derived from {110}B2 -type mirror planes of the parent phase. It is also evident from Figure 4.42 that six kinds of equivalent arrangements are possible (two for each of the <001> axes of the parent phase) among the four variants generated from a single parent crystal. Two distinct morphologies, as illustrated in the light micrograph in Figure 4.42, are possible, one characterized by a stack of twin-related bands and the other by a herring bone morphology in which the four variants meet along an invariant line parallel <001>B2 axis. In both cases, the planar interfaces separating adjacent variants are all twin planes. 4.5.3 Crystallography of the B2 → B19 transformation The lattice correspondence between the B2 structure and the orthorhombic B19 structure is shown in Figure 4.43, in which i , j and k represent the orthorhombic Table 4.12. Lattice correspondence between B2 (parent) and R phases. Variant
[100]R
[010]R
[001]R
(001)R
1 2 3 4
¯ p [121] [211]p [121]p [211]p
¯ p [112] ¯ ¯ p 112 [112]p ¯ p [112
[111]p ¯ p 111 ¯ p [1¯ 11] ¯ p [111]
[111]p ¯ p [111] ¯ p [1¯ 11] ¯ p [111]
Martensitic Transformations
343
1 4 1 4 1 1 2 1 2 1 2 1 2 1
4 4 1 2 23 4 1 42 23 4 4 1 2 2 3 (a) [001] [010]
[100] (110) (010)
(110)
4 3
2 4 (b)
Figure 4.42. (a) Schematic of an optical micrograph of self-accommodation of the R phase. The variant numbers are assigned on the basis of the lattice correspondence described. (b) Threedimensional arrangement of self-accommodating R-phase variants.
axes while i, j and k correspond to the cubic axes. The k -axis can be made parallel to one of the cube axes, and in such a case, the i and j axes can be oriented in two distinct manners. Therefore, six possible correspondence variants are present. The relative orientations of these six variants, numbered 1–6 with respect to the axes system of the parent B2 crystal, are indicated in Table 4.13. The homogeneous deformation matrix, T , which is associated with the B2 → B19 transformation is given by ⎤ ⎡ a 0 0 √ ⎥ ⎢ T = 1/ao ⎣ 0 b/ 2 0 ⎦ (4.64) √ 0 0 c/ 2
344
Phase Transformations: Titanium and Zirconium Alloys Ni atoms k,k ′
Ti atoms
j
i′
j′ i′
aj ′ k,k ′
β
bj′ cj′
Figure 4.43. The lattice correspondence between the B2 structure and the orthorhombic B19 structure.
Table 4.13. B2-martensite lattice correspondence. Variant
[100]m
[010]m
[001]m
1 1 2 2 3 3 4 4 5 5 6 6
[100]c ¯ c 100 [100]c ¯ c 100 [010]c ¯ c 010 [010]c ¯ c 010 [001]c ¯ c 001 [001]c ¯ c 001
[011]c ¯ c 01¯ 1 ¯ c 011 ¯ c 011 ¯ 101c ¯ c 101 [101]c ¯ 1 ¯ c 10 ¯ c 110 ¯ c 110 [110]c ¯ c 1¯ 10
¯ c 011 ¯ c 011 ¯ c 01¯ 1 ¯ ¯ c 011 [101]c [101]c ¯ c 101 ¯ c 101 ¯ c 101 [110]c ¯ c 110 ¯ c 110
Martensitic Transformations
345 (110)P
(100)P
1
6
(010)P
3 4
2 5
[001]
[100]
(101)P
2
4 6
[010]
(a)
(001)P
(011)P
(b)
Figure 4.44. Self-accommodation model of B19 phases.
where ao is the lattice parameter of the B2 structure while a, b and c are the lattice parameters of the B19 structure. Two types of twin relations, namely (111) [211]O type I and (011) [01¯ 1]O compound twins, exist between variants which come in contact along a planar surface. The subscript O refers to the orthorhombic B19 structure. Four 111 O -type twin interfaces (derived from parent 110 B2 planes) and six 011 O -type twin interfaces (derived from 100 B2 planes) separate adjacent variants in the self-accommodating assembly of martensite variants produced in this transformation. Self-accommodation model of B19 phases is shown in Figure 4.44. 4.5.4 Crystallography of the B2 → B19 transformation The B2 → B19 transformation can be conceptually divided into two components, namely the B2→ orthorhombic B19 and the B19→ monoclinic B19 transformations. As discussed in the previous section, six possible variants of lattice correspondence are possible between the B2 and B19 structures. The lattice shear which transforms the orthorhombic structure into the monoclinic structure is a simple shear acting on the (100)O plane along the [001]O direction. Depending on ¯ O direction, the sign of this shear, i.e. whether it is along the [001]O or the 001 two monoclinic variants can be generated from each of the orthorhombic variants; these monoclinic variants are designated as 1,1 , 2,2 , 3,3 , 4,4 , 5,5 and 6,6 . Therefore, a total of 12 variants can form from a single B2 crystal and their relative orientations are also indicated in Table 4.13. The homogeneous deformation matrix, T , which converts the parent B2 crystal into the monoclinic martensite is given by √ ⎤ a 0 c cos / 2 √ ⎥ ⎢ 0 T = ao ⎣ 0 b/ 2 ⎦ √ 0 0 c sin / 2 ⎡
(4.65)
346
Phase Transformations: Titanium and Zirconium Alloys
where defines the angle between the a- and the c-axes. Considering the case of a specific variant, 6 , the Bain deformation B6 can be calculated by incorporating the necessary axis transformation matrix R6 : ⎡ ⎤ 0 1 1 R6 = ⎣ 0 −1 −1 ⎦ (4.66) −1 0 0 B6 can be expressed as B6 = R6 T R6 −1 and can be shown to be ⎤ ⎡ i j 0 ⎥ ⎢ (4.67) B6 = ⎣ j i 0 ⎦ l l k √ √ where j = 2/4 amo −b + c sin , k = c cos /2 amo , l = a/ao and i = 2/4 ao b + c sin Substituting the following values of lattice parameters (Miyazaki et al. 1983), ao = 03015 nm, a = 02889 nm, b = 04120 nm, c = 04662 nm, and = B6 can be evaluated as ⎤ ⎡ 10213 00550 00 ⎥ ⎢ (4.68) B6 = ⎣ 00550 10213 00 ⎦ ‘00907 ‘00907 09582 Bain deformations for the other variants can be computed using the appropriate axis transformation matrices. The LIS in these martensites invariably occurs by twinning. As will be discussed later, the propagation of twin interfaces has a pivotal role to play in the process of shape recovery. The basis of the selection of the twinning systems and their experimental identification are briefly covered here. In order to satisfy the condition of equivalent lattice correspondences for the two twin components of a martensite plate, it is necessary that either the twin plane (K1 plane) is derived from a mirror plane of the parent crystal or the 2 direction is derived from a diad of the parent. Examining all the possible twin systems, Madangopal and Banerjee (1992) have arrived at the conclusion that only ¯ M type I, the conjugate [112]M type II, (011)M type four twin systems, viz., 111 I and the conjugate [011]M type II can qualify to be the lattice invariant shear system in this case. Examples of type I and type II twins in the Ni–Ti martensite are illustrated in Figure 4.45. Since type I and type II twins are conjugate to each other, the twinning shear is the same in the two twinning modes. In type I twinning
Martensitic Transformations
347
Figure 4.45. TEM micrographs showing the (a) type I and (b) type II twinning as the lattice invariant shear.
the K1 plane, being of rational indices, will be fully coherent. In contrast, the irrational K1 planes of type II twins generally consist of rational ledges and steps (Figure 4.45). In this respect, type I twin interfaces are energetically favourable, and as a consequence, the mean spacing between the twin interfaces is usually much smaller for type I twins than that for type II twins. Early TEM observations on the as-quenched Ni–Ti monoclinic martensites have ¯ M type I and/or {001}M compound twins (Otsuka shown the presence of 111 et al. 1971, Gupta and Johnson 1973). Knowles and Smith (1981a,b) have detected <011M type II twins and have pointed out that the {001}M compound twins do not yield any habit plane solution on the basis of the phenomenological crystallographic theory. The presence of {001}M compound twins in Ni–Ti martensites has been attributed to the plastic accommodation effect. The operation of different LIS systems in successive generations of martensites appears to be responsible for the occurrence of such a variety of internal twins in this system. 4.5.5 Self-accommodating morphology of Ni–Ti martensite plates It has been shown earlier that self-accommodating arrangements of different orientation variants of martensite crystals can minimize the strain energy of the assembly. Mutual interactions of the elastic strain fields of individual martensite plates need to be examined in determining the extent of self-accommodation for a given
348
Phase Transformations: Titanium and Zirconium Alloys Table 4.14. The 24 habit plane variants (HPV) and the matching correspondence variant-combinations (CVC). HPV
CVC
HPV
CVC
HPV
CVC
1+ 1− 1 + 1 − 2+ 2− 2 + 2 −
1-2 1-2 1 -2 1 -2 2-1 2-1 2 -1 2 -1
3+ 3− 3 + 3 − 4+ 4− 4 + 4 −
3-4 3-4 3 -4 3 -4 4-3 4-3 4 -3 4 -3
5+ 5− 5 + 5 − 6+ 6− 6 + 6 −
5-6 5-6 5 -6 5 -6 6-5 6-5 6 -5 6 -5
The first and second numerals designate the major and minor twin-related regions in the habit plane variant, respectively.
group of martensite plates. In the case of the martensitic transformation from the B2 to the monoclinic (B19 ) phase, 12 variants of lattice correspondence designated as 1,1 , 2,2 , 6,6 , as described in Table 4.14, are possible. Each of these orientation variants has two possible directions of lattice invariant shear, making a total of 24 habit plane variants of martensite plates designated as 1+, 1−, 1 +, 1 −, 2+ 6 +, 6 −. The signs (+) and (−) represent the two equivalent lattice invariant shears which can operate in opposite directions for a given correspondence variant. The correspondence variant combinations present in each type of martensite plate (i.e. a given habit plane variant) are listed in Table 4.14. The DSA accomplished in different groups of martensite plate variants has been examined in detail by Madangopal et al. (1991, 1993, 1994) Madangopal and Banerjee (1992) and Banerjee and Madangopal (1996). Possible groups of 2-, 3- and 4-variants have been considered using a simplifying assumption that the volume fraction of each of the variants in a group is the same. Table 4.15 shows the computed values of DSA corresponding to several 2-, 3- and 4-variant groups of martensite plates. TEM observations on the morphology of the Ni–Ti martensitic structure have revealed that the primary plates which form first are frequently configured in a “V”-shaped pattern (Figure 4.46(a)) which can be described as a 2-variant group. Primary plates which are large, some of them spanning across the prior B2 grains, are also widely spaced. The identification of component variants of primary plates making a “V”-shaped pattern has brought out the fact that they invariably belong to a 2-variant group which corresponds to a high DSA (typically 3+ and 6 + having a DSA value of 83.4%). Frequent occurrence of such two plate groups suggests that the formation of a primary plate promotes the nucleation of a specific second plate variant in its vicinity in such a manner that the stress field
Martensitic Transformations
349
Table 4.15. The degree of self-accommodation (DSA) for the possible strain coupled plate groups of the (−) habit plane variant of the Ni–Ti martensite. Coupling axis
Plate group
Tension ¯ c 1¯ 11
a. 1− + 4 − + 5−
34.1
b. 1− + 1+ + 4 +
26.7
¯ c 011 ¯ c 001 Compression ¯ c 111
DSA
c. 1− + 1+ + 5−
25.4
1− + 1+ + 1 + + 1 −
49.0
a. 1− + 1 + + 2− + 2 +
64.4
b. 1− + 2− + 3 + + 4 +
85.7
a. 1− + 3 + + 6 +
89.9
b. 1− + 1+ + 3 +
71.1
c. 1− + 1+ + 6 +
55.4
101c
1− + 2 + + 5− + 6 +
50.1
100c
a. 1− + 1 + + 2− + 1 +
69.4
b. 1− + 1+ + 1 + + 1 −
49.0
Figure 4.46. TEM micrograph showing (a) V-shape configured two variant combination of Ni–Ti martensite and (b) three variant Ni–Ti martensite grouping.
created by the first aids the nucleation of the second. The formation of a third variant 6 −, as is shown in Figure 4.46(b), results in a further improvement in self-accommodation. Primary plates partition the parent B2 grain progressively and a self-similar scaled down structure forms within the partitioned volume in successive stages in pursuit of achieving a higher DSA (thereby reducing the strain energy density of the assembly of martensite plates and the matrix of the
350
Phase Transformations: Titanium and Zirconium Alloys
untransformed B2 grain). It is because of the tendency of self-accommodation and of the thermoelastic nature of the martensite plates in Ni–Ti- based systems the martensite plates of successive generations remain elastically coupled. In a system like this, the parent to martensite transformation attains completion below Mf without leaving any untransformed B2 phase. Madangopal (1997) has shown that the regions left untransformed at a late stage of the transformation process are filled up by tetrahedral groups of three variant martensite plates as shown in Figure 4.47. Detailed crystallographic analysis has revealed that this group consists of three internally twinned plates, A, B and C, the correspondence variants present in each of the plates being, A: 1, 2 ; B: 5 , 6 and C: 4, 3. The habit plane variants for the plates A, B and C have been identified as 1−, 5 − and 4+, respectively. It is interesting to note that the interfaces between the plates A, B and C are 110 -type mirror planes of the parent B2 crystal. The orientation analysis of three plate clusters has also shown that the A–B, A–C and B–C interfaces are ¯ A , 11¯ 1 ¯ A and 111 ¯ C type I twin planes (corresponding to the major twin 111 component of each of the plates. It has also been reported (Madangopal 1997) that in some cases both major and minor twin variants of a plate find twin orientations across a given intervariant plane (as in the A–B case). In other cases (as in the A–C and the B–C cases) though the major twin orientations across the intervariant interface are mutually twin related, the minor twin orientations are not twin related. In such situations the minor twin orientations taper off as they approach the intervariant interfaces which remain twin related all along the plane of contact. The shape memory phenomenon which has been described in the following section requires microstructural reversibility of the polydomain configuration of the martensitic structure. This can be attained only if the interfaces separating adjacent variants remain coherent and glissile which ensures that both forward and backward movements of these interfaces can be induced by suitable changes in temperature and/or in applied stress. The coherency between adjacent variants can
Figure 4.47. TEM micrograph showing self-accommodating tetrahedral group of three variant martensite plates in Ni–Ti martensite.
Martensitic Transformations
351
be easily established when they are related by relations derived from a mirror plane or a diad of the parent crystal. The intervariant interfaces, A–B, A–C and B–C, in the group of three variants described earlier satisfy this condition. Microstructures of Ni–Ti martensites have shown the following types of interfaces preponderantly: ¯ type I and [011] type II twins within internally twinned martensite plates (a) (111) (Figure 4.45) and (b) between two adjacent internally twinned plates in which the major twin fractions have the same orientation while the minor twin fractions are not mutually twin related. In such cases, there is no “boundary” between the adjacent plates (Figure 4.48(a)), while the minor twin fractions taper as they approach the habit plane making a line contact between non-twin-related variants (Figure 4.48(b)). In case neither of the twin components of a plate have a twin relation with those of a neighbouring plate, a new “bridge” variant is created between the plates (Figure 4.48(c)). The bridge variant establishes twin relation on either side. An examination of relative orientations of a given variant with all other variants brings out the fact that in a majority of combinations a twin relationship between adjacent plates can be established. However, in several cases, tapering of the minor twin fraction or formation of a bridge variant is necessary for making the intervariant interface a twin plane. Table 4.16 shows the orientation relations
Figure 4.48. Bright field micrograph showing (a) no boundary between adjacent plate, (b) tapering of minor twin fractions as they approach habit plane making a line contact between non-twin-related variants and (c) neither of twin components of a plate have twin components with neighbour plates, a new bridge variant is created, (d) schematic of (c).
352
Phase Transformations: Titanium and Zirconium Alloys
Table 4.16. Results of the 13 unique variant interactions of the 1+ variant of NiTi martensite. HPV
CR
Interface plane/bridge variant
group (g)/plate (p) clustering∗
Remarks
1− 1 + 1 −
– T T
2 + 2 −
T T
Boundaryless combination (100)c //(100)m1 (100)c //(100)m1 //(100)m2 ¯ c // (001)m1 or 011 ¯ c //(011) ¯ m1 010 ¯ c //(0,11) ¯ m1 //(011)m2 (010)
(g-g) (g-g) (p-p) (p-p) (g-g) (p-p)
T T T T B B B B
¯ m2 ¯ c //(0.721¯ 1) (0.72 01) ¯ ¯ m2 (110)c //(111)m1 //(11¯ 1) ¯ c //(11¯ 1) ¯ m1 //(111)m2 (1 10) ¯ c //(11¯ 1) ¯ m1 (101) (101)c //(111)m , 1 Possible bridge variant 1 + Possible bridge variant 1 − Possible bridge variant 2 − Possible bridge variant 1 −
(p-p) (g-g) (g-g) (g-g) (g-g) (g-g) (g-g) (g-g) (g-g)
Tapering of minor TF Tapering of minor TF Single interface Tapering of minor TF Tapering of minor TF Single interface derived from conjugate type 1 mode Tapering of minor TF Single interface Single interface Taper twin Taper twin ** **
3 − 4 − 6 + 5− 3− 5 + 3+ 4+
HPV, habit plane variant; CR, crystallographic relationship; T, twin variant; B, bridge variant; TF, twin fraction; ∗ the (p-p) and (g-g) combinations are assuming {100}c clustering, ∗∗ can also form twin interface by tapering of the major twin fraction.
between the 1+ variant with 13 other habit plane variants with which intervariant twin interfaces can be formed. 4.5.6 Shape memory effect The shape memory effect and the pseudoelastic deformation in alloys exhibiting the martensitic transformation can be described using a set of hypothetical stress– strain plots corresponding to different temperatures (Figure 4.49). Let us consider a tensile test sample of an alloy exhibiting the shape memory effect which is being subjected to straining in a hard testing machine, with the stress developed in the sample being continuously recorded. When the testing is carried out at a temperature T1 > Md , the stress–strain plot shows elastic deformation up to a high stress value, followed by a limited plastic flow and finally a sudden brittle fracture. The parent B2 phase is stable with respect to any phase transformation at T1 , and thus the observed deformation behaviour is consistent with that expected from a brittle ordered intermetallic compound. At the testing temperature, T2 Md > T2 > Af , the B2 phase is unstable with respect to the stress-induced martensitic transformation which is induced when the stress attains a threshold value indicated by the point 1 on the stress–strain plot. The plastic flow from 1 to 2 corresponds to the formation of increasing volume fractions of the martensite. While unloading the
Martensitic Transformations
353
M f > T3 T1 > Md
Md > T2 > Af Pseudoplastic deformation
8 8′
2 1
Stress
3 4
6
5
Recoverable pseudoplastic strain
d a
c
7 10
9
b Str
ain
Mf M s ure t a r pe em
s Af A
T
Figure 4.49. A schematic drawing showing shape memory effect and pseudoelastic deformation in alloys exhibiting martensitic transformations.
sample, the stress falls from 2 to 3 in a manner similar to elastic unloading. At the point 3, the volume fraction of the stress-induced martensitic starts decreasing and the stress–strain plot follows the path indicated by 3 → 4. The closed stress–strain hysteresis loop indicates that the stress-induced martensitic structure completely reverts back to the parent phase during unloading. The non-linear deformation which is recoverable on unloading is known as pseudoelastic deformation. The stress–strain plot at T3 T3 < Mf shows a deviation from elastic deformation at a fairly low stress value, resulting in a deformation plateau (indicated by 5 → 6). Unloading from the point 6 shows an elastic unloading to the point 7. The plastic strain at the point 7, however, is recoverable by heating the sample to a temperature higher than Af and hence is known as pseudoplastic strain. The recovery of the pseudoplastic strain starts on heating to a temperature above Af . This strain recovery (shape recovery) process of a pseudoplastically deformed material, when subjected to a heating cycle to go through the parent phase, is known as the shape memory effect. In case deformation at T3 is continued beyond the point 6, a second stage of linear elastic deformation is encountered. The stress rises substantially to reach the point 8 where a deviation from linearity is noticed. Specimens fracture at 9
354
Phase Transformations: Titanium and Zirconium Alloys
after plastic flow occurs to a limited extent. Unloading from a point such as 8 results in elastic recovery of strain in a linear manner (8 to 9). On subsequent heating above Af , the strain recovery occurs only to a limited extent (9 to 10). Components of some shape memory alloys, after being subjected to a number of straining and thermal cycles (for strain recovery), acquire a two-way memory. In such cases, the component (or the specimen) remains in two states of strain (or two shapes) at two temperatures, one above Af and the other below Mf . The two-way shape memory is also denoted by a strain – temperature cycle, abcd, as shown in Figure 4.49. The schematic drawing (Figure 4.49) shows in an idealized manner different phenomena, namely pseudoelastic deformation (or rubber-like behaviour), pseudoplastic deformation, shape memory and two-way memory effects. Intensive research on these phenomena during the last four decades has established the sequence of structural changes which occur during these processes and are described in detail in several reviews (Delaey et al. 1974, Schetky 1979, Wayman 1980, Tadaki et al. 1988). Physical processes accompanying these individual steps are not exactly identical for all the shape memory alloys. In spite of the differences between the systems, one can attempt an oversimplified description of the processes responsible for the aforementioned phenomena. Such an attempt is made in the following paragraphs. At a temperature between Af and Md , the latter being the upper temperature limit for the stress-assisted martensitic transformation, the austenite to martensite transformation can be induced when the chemical driving force for the transformation is augmented by the applied mechanical stress. The deviation from the linear elastic behaviour is noticed at a value of stress which is adequate for initiating the stress-assisted transformation. With a further increase in stress, the volume fraction of the martensite phase increases, the mechanical work done on the system by the applied stress being entirely used up for the creation of the metastable martensitic phase. The martensitic phase forming under this condition remains in thermoelastic equilibrium, and it is, therefore, possible to reverse the process (i.e. to induce the martensite to austenite transformation) once the stress level is lowered. The unloading path, as shown by 2-3-4, consists of pure elastic unloading from 2 to 3 followed by the martensite to austenite reversion from 3 to 4. The loop, therefore, represents the hysteresis in the stress–strain relation with respect to the stress-assisted martensitic transformation. Deformation of the fully martensitic structure, at temperatures below Mf , shows pseudoplastic flow (from point 5 to 6) which can be completely recovered by the thermal cycle as described earlier. This can be contrasted with the usual plastic flow of metals and alloys by slip in which the slipped regions of crystalline grains are shifted to new positions having identical atomic surroundings. Since the slip process
Martensitic Transformations
355
is not reversible, there is no tendency of the slipped regions to retrace the deformation path. The slip mechanism cannot, therefore, explain the recoverable pseudoplastic deformation. The starting structure, being fully martensitic, consists of an aggregate of different variants of plates which are self-assembled in a manner such that the strain energy of the assembly is minimized. It has also been shown that a majority of the intervariant interfaces satisfy a twinning relationship across them. On application of an external stress, some of the orientation variants grow at the expense of their neighbouring orientations. Under the given system of the external stress, some variants are favourably oriented for growth while others are not. Though pseudoplastic flow can be accomplished by such a reorientation process which does not revert during unloading, the self-accommodation achieved in the virgin martensitic structure is disturbed. A thermal cycle through the austenite phase brings back the self-assembly and in that process nullifies the pseudoplastic strain. A component of a shape memory alloy, when cycled a number of times through the process of pseudoplastic deformation and shape recovery, develops a twoway memory. Such a component assumes two shapes given by two states of strain corresponding to two temperatures, one below Mf and the other above Af . The two-way memory arises due to accumulation of residual plastic deformation in the material during the course of repeated thermal cycling. The locked-in residual stress eventually builds up to such a level that it can drive pseudoplastic deformation without the introduction of any externally applied stress. The shape recovery occurs in the usual manner during the heating-up step. In the foregoing paragraphs, it has been brought out that the martensitic transformation plays the pivotal role in bringing about shape memory and related phenomena. It is also true that all systems, metallic or ceramic, in which martensitic transformations occur do not exhibit the shape memory effect with a significant extent of shape recovery. It is, therefore, important to identify the criteria which must be fulfilled for a system to display shape memory up to a few percentage of pseudoplastic strain. These criteria are as follows. (a) The first and foremost condition is that the system must display a thermoelastic martensitic transformation. This condition essentially implies that the magnitude of transformation strain is not large enough to cause irreversible plastic flow in either the matrix or the martensitic phase. (b) Martensite plates in the product microstructure must form a self-accommodating assembly. The minimization of the overall strain energy results in a strain coupling between different martensite plates. (c) Interfaces between adjacent martensite plates should be such that they can propagate either way without losing atomic registry. Such a condition is met if a great majority of these interfaces satisfy a twin relationship across these interfaces. (d) Atomic ordering of both the austenite and the martensite phases favours the occurrence of the shape memory effect, though this is not an essential requirement. The presence
356
Phase Transformations: Titanium and Zirconium Alloys
of atomic ordering restricts the number of variants for the martensitic transformation and also increases the threshold stress at which thermoelastic reversibility is lost due to plastic deformation. The criteria which are listed here are based on Ni–Ti, In–Tl, Au–Cd and Cubased shape memory alloys, which were the systems in which this phenomenon was reported initially. In recent times, this phenomenon has been observed in several other alloys such as intermetallics and ceramic systems. A number of Fe-based alloys including Fe-Ni-C, Fe-Mn-Si, Fe-Mn-Si-Cr-Ni and Fe-Ni-Co-Ti have shown shape memory effect to somewhat restricted recoverable plastic strain values. Currently, the search for shape memory alloys with higher recovery temperatures and larger hysteresis is being pursued in several laboratories. Intermetallics based on Ni–Al and Ti–Pd show good promise. 4.5.7 Reversion stress in a shape memory alloy The performance of a shape memory device depends not only on the maximum limit of the recoverable strain but also on the stress against which the strain recovery can occur. The reversion stress r is defined as the stress developed in a sample constrained against recovery during heating to a temperature higher than Af . Data available on reversion stress and on its dependence on prior plastic strain and temperature have been very limited. The method of the measurement of the reversion stress in shape memory alloys and the criteria which determine the upper limit or r have been reported by Madangopal et al. (1988). The experiments involved (a) deforming fully martensitic tensile test samples of Ni–Ti shape memory alloys to a plastic strain p using a hard tensile testing machine at a temperature below Mf , (b) unloading the sample to just above the zero load level, (c) arresting the crosshead of the testing machine at this position, (d) rapid heating of the sample to a temperature Tr Tr > Af by introduction of a hot water bath and recording the rise in stress as the sample underwent the reversion process and (e) unloading from the maximum stress level, r , maintaining the sample temperature at Tr . These steps are schematically illustrated in Figure 4.50(a). Thermal stresses introduced due to differential thermal expansion of the sample and the loading assembly have been determined to be 15 MPa, which is much smaller compared to the measured r values (100–500 MPa). The influence of p and Tr on r and its saturation value is shown in Figure 4.50(b). Figure 4.51(a) shows r values attained for different levels of initial plastic strain and the unloading paths (pseudoelastic) after attainment of r . The shape of this unloading curve closely matches with the pseudoelastic unloading curve (outer envelope of the stress–strain hysteresis loop shown in Figure 4.51(a). This suggests that the thermoelastic equilibrium attained after the development of r in a constrained recovery process is similar to (close to but not identical
Martensitic Transformations
M d > T2 > A f
Stress (σ) →
σr
357
T1 < Mf
Stress (ε) → (a)
Reversion stress (MPa), σr→
600
Equiatomic Ni Ti SMA As ~ 323 K Ms ~ 318 K
353 K 369 K 378 K 398 K 0 423 K
00 0
500
0 0 400
0 300
0 200
100
1
2
3
4
5
6
Constraint (%), εp →
7
8
(b)
Figure 4.50. (a) Schematic stress–strain plot showing the different steps involved in the measurement of the reversion stress. (b) The variation of the reversion stress with temperature and prestrain.
358
Phase Transformations: Titanium and Zirconium Alloys Tr ~ 313 K
Stress (MPa) →
350 300 250 200 150 100 50 0
0
0.5
1
1.5
2
2.5
3
Strain (%) → (a) 322 K
300
σr →
Stress (MPa) →
318 K
200
313 K 308 K 303 K
100
– Crosshead Stationary only chart movement – Thermal perturbations 10
Strain (%) → (b)
Figure 4.51. (a) The unloading path from superimposed on the correspondence pseudoelastic loading–unloading loop. (b) The change in with thermal perturbation.
with) that prevailing in the unloading path of the pseudoelastic stress–strain loop. Therefore, it is surmised that r is nothing but the stress threshold which marks the start of the reversion of stress-induced martensite (M) into austenite (B2). The fact that a thermoelastic equilibrium is reached after the attainment of r is elegantly demonstrated in an experiment in which growth of martensite phase and a consequent drop in stress have been recorded as the sample temperature is slightly lowered by spraying cold water on the sample (Figure 4.51(b)). The important observations listed above point to the fact that on the attainment of r , the M → B2 transformation gets arrested and a thermoelastic equilibrium is established. The stress value at which the stress-induced martensite starts reverting back to the parent austenite phase is generally designated as As which is marked
Martensitic Transformations 500
Tr > As
σMs
Stress (MPa), σ Ms,σAs,σr →
400
359
σ
σr σMs
σAs
σAs
ε 300
200
100
Ms 0
As
283 287
Af 300
310
320
330
Temperature (K) Tr →
Figure 4.52. The variation of As , Ms and r∗ with temperature.
on the pseudoelastic unloading path in Figure 4.52. This stress value essentially corresponds to the stress at which the reverse M → B2 transformation commences during the unloading process. On the other hand, as the temperature is raised above Af in the constrained condition, pseudoplastically deformed martensite progressively transforms to the austenite (B2) phase and the elastic stress developing in the system (consisting of a mixture of M and B2 phases) gradually rises. This reversion stress, however, cannot rise above that corresponding to the M → B2 thermoelastic equilibrium. This explains why r attains a maximum value of r∗ which equals As . It is possible to predict r∗ by using a modified Clausius–Clapeyron equation relating As and the reversion temperature, Tr : dr∗ Hp =− dTr T
(4.69)
where H is the change in enthalpy for the B2→ M transformation, is the maximum pseudoplastic strain, T is the Af temperature and p is the density of the B2 phase.
360
Phase Transformations: Titanium and Zirconium Alloys
4.5.8 Thermal arrest memory effect A new phenomenon named thermal arrest memory effect (TAME) reported in Ni–Ti shape memory alloys is briefly discussed here. When the martensite to parent (M → P) transformation is arrested for a short duration at a temperature, TA , before completion of the transformation (i.e. As < TA < Af and cooled down to a temperature below Mf , the martensite to parent (M → P) transformation is halted during subsequent heating at the same temperature, TA . This means that the system remembers the temperature at which the transformation was initially arrested in the preceding heat treatment cycle. The phenomenon can be seen clearly in continuous heating and cooling experiments in a differential scanning calorimeter (DSC). DSC thermograms illustrating the exothermic (P→M) and the endothermic (M→P) transformations are shown in Figure 4.53(a). The steps involved in the experiment are described below. (Arrest in the martensite to parent M→P transformation) Sequence A (Figure 4.53(b)) Step 1: The M→P transformation is arrested at a temperature
NiTi Thermal cycle:
1
ΔH, heat flow rate (arbitrary units)
+
2
260 K
3
A Thermal arrest during P-M
4 360 K
M→ P
Thermal arrest
–
B during M-P
M← P B A
–
(a)
280 K 1
2
+
+
360 K
Thermal arrest 337.1 K
1
2
Thermal arrest
4
275 K
360 K
(b) 3 360 K
280 K
– 3
4 (c)
Temperature
Figure 4.53. DSC thermogram exhibiting the effect of interrupting the M-P and P-M transformations in Ni50 Ti50 alloy.
Martensitic Transformations
361
TA = 3371 K, between As and Af . Step 2: On cooling from 337.1 K to a temperature below Mf , the parent phase generated in step 1 is again transformed to martensite. Step 3: During heating in the next cycle, the M→P endotherm splits into two, centered at TA . Step 4: During cooling from a temperature above Af , the original exothermic peak for the P→M transformation is obtained. (Arrest of the parent to martensitic (P→M) transformation) Sequence B [Figure 4.53(c))] Step 1: The P → M transformation is interrupted at TA (= 314.8 K) between Ms and Mf . Step 2: On heating to a temperature above Af , the partially transformed martensitic microstructure reverts back to the parent phase. Step 3: On subsequent cooling to a temperature below Mf , the original exothermic peak is obtained without showing any effect of the thermal arrest in step 1. Step 4: During heating the original exothermic peak for the complete M→P transformation is observed. These observations clearly demonstrate that a thermal arrest during the M→P transformation is remembered by the system during the next heating cycle. Such is not the case for a thermal arrest during the P→M transformation. However, in cold-worked samples, TAME is observed for arrest of both P→M and M→P transformations. A detailed study of the influence of various factors like supercooling, premartensitic R-phase transformation, repeated thermal arrest of TAME has been reported by Madangopal et al. (1994). Taking into account all these observations, the thermal arrest memory effect has been rationalized as follows: The split in the M→P transformation, as seen in Figure 4.53(b), step 3, implies that the M→P transformation stops for a short duration at the arrest temperature TA in the previous cycle. As discussed earlier, martensite plates tend to minimize the elastic strain energy of the system by self-accommodation. Plates belonging to different generations (forming at different stages of the transformation) are elastically coupled. An interruption in the transformation, therefore, results in dividing the population of martensite plates into two distinct groups. At TA , the temperature of arrest, the system consists of a mixture of the martensite and parent phases. On subsequent cooling, martensite plates transforming from this austenitic region form a distinct (second) group. The M→P transformation for these two groups of martensite does not remain linked during the second cycle. The reverse transformation of the second population occurs first and only after the completion of the transformation of this group, the M→P transformation of the other group commences. This sequence results in the introduction of a break in the endotherm corresponding to the M→P transformation. The thermal arrest memory effect is directly linked with the self-accommodation process which binds one generation of martensite plates with the next by elastic coupling. In case the self-accommodation in a small group of three or four plates is very effective, the magnitude of the unaccommodated strain will be small,
362
Phase Transformations: Titanium and Zirconium Alloys
and consequently the extent of elastic coupling between generations of martensite plates will also be small. Such a situation prevails in Cu-Zn-Al martensites where self-accommodation of neighbouring martensite plates in a group is as high as 98%. TAME is, therefore, observed rather weakly in this system.
4.6
TETRAGONAL MONOCLINIC TRANSFORMATION IN ZIRCONIA
4.6.1 Transformation characteristics Polymorphic transformations in pure zirconia (ZrO2 ) have been briefly described in Chapter 1. This transformation, first detected by Ruff and Ebert (1929) using high-temperature X-ray diffraction, has been extensively studied using thermal analysis, X-ray and electron diffraction, light and electron microscopy and electrical resistivity measurements. Wolten (1963) was the first to suggest that the monoclinic to tetragonal transformation is martensitic in nature. The important characteristics of this transformation are summarized in the following. (1) The high-temperature tetragonal phase cannot be retained on quenching to room temperature. (2) The transformation is athermal. (3) The observed growth rate of monoclinic platelets has been found to be consistent with a velocity of the transformation front approaching that of sound in solids (Fehrenbacher and Jacobson 1965). The transformation kinetics, as measured from thermal analysis experiments, show a burst-like behaviour (Maiti et al. 1972) during the reverse transformation (i.e. monoclinic to tetragonal). (4) The transformation exhibits a large thermal hysteresis (Figure 4.54); the forward and the backward transitions occurring during cooling and heating experiments show transition temperatures in the vicinity of 1123 and 1323 K, respectively (Figure 4.54). (5) The monoclinic to tetragonal transformation is accompanied by the sudden appearance of surface relief which corresponds to the formation of tetragonal plates during heating experiments. (6) A strict orientation relationship exists between the parent and the product phases, as described in detail in the next section. The product phase, either plate-shaped or lenticular-shaped, always forms along specific habit planes. The lattice correspondence derived from the observed orientation relationship suggests that the transformation can be accomplished by atomic movements, primarily of O atoms, to an extent smaller than the interatomic distance and minor shifts of Zr atoms.
Martensitic Transformations
363
100 Decrease temperature
Percent tetragonal phase
80 Increase temperature 60
40
20
0
600
700
800
900
1000
1100
1200
Temperature (°C)
Figure 4.54. Monoclinic to tetragonal transformation exhibiting a large thermal hysteresis during cooling and heating.
All these experimental observations strongly point to the fact that both the forward and the backward transformations can occur by a martensitic mode when the cooling or the heating rate exceeds a certain critical value. 4.6.2 Orientation relation and lattice correspondence There has been some initial disagreement in literature on the exact orientation relation between the monoclinic and the tetragonal phases (Bailey 1964, Wolten 1964, Smith and Newkirk 1965, Patil and Subbarao, 1967). The experimental difficulties due to the high temperature of the transformation have been mainly responsible for this discrepancy among results initially reported by these different investigators (Bansal and Heuer 1972). They devised an ingenious experiment in which single crystals of monoclinic ZrO2 were grown from a fluxed melt at temperatures below the tetragonal to monoclinic transformation temperature. These single crystal monoclinic samples were heated to temperatures above the As temperature for the monoclinic to tetragonal transformation and were subsequently cooled down to a temperature between the Ms and Mf temperatures. X-ray diffraction patterns were recorded in the initial monoclinic state, after the first reverse transformation and after cooling down below Ms . A comparison of the X-ray diffraction patterns obtained from the sample in these three stages yielded the orientation relation between the two phases. These orientation relations are consistent with the possible
364
Phase Transformations: Titanium and Zirconium Alloys
(a)
LC A
at
at
bm at
LC B
ct
cm am
at
bm
am
at
LC C
at
cm
at
bm
am cm
at (b)
Figure 4.55. (a) Tetragonal and monoclinic structures and (b) three possible lattice correspondences consistent with the orientation relation of the tetragonal and the monoclinic structures of ZrO2 .
lattice correspondences which can be derived from an inspection of the tetragonal and the monoclinic structures Figure 4.55(a) of ZrO2 . Three distinct lattice correspondences can be envisaged as illustrated in Figure 4.55(b). If a right-handed screw convention is adopted for the unit cell axes, three lattice correspondences, designated as types A, B and C, can be defined, depending on which monoclinic axis, am , bm or cm , is parallel to ct , the c-axis of the tetragonal crystal (suffixes “m” and “t” referring to monoclinic and tetragonal structures). The type A lattice correspondence gives rise to the orientation relationship: (001)m
100 t and [100]m
< 100 >t ; which has been experimentally observed only for the first reverse (monoclinic → tetragonal) transformation (Bansal and Heuer 1974). For the type A lattice correspondence, no surface relief has been
Martensitic Transformations
365
noticed, suggesting that this correspondence is not operative in the case of martensites forming on the surface of monocrystalline samples. The type B lattice correspondence leads to the following two orientation relationships, designated as B-1 and B-2: B-1 * 100m
010t & 010m
001t & 001m 9 to 100t ie 001m
100t B-2 * 100m 9 to 010t & 010m
001t & 001m
100t ie 100m
010t These orientation relations are observed when the crystal is heated to the fully tetragonal phase field and subsequently cooled down to 1273 K (1000 C) where it is partially transformed into the monoclinic phase. The type C lattice correspondence results in the following two orientation relations, C-1 and C-2: C-1 * 100m
100t & 010m
010t & 001m 9 to 001t ie 001m
001t C-2 * 100m 9 to 100t & 010m
010t & 001m
001t ie 100m
100t These orientation relations have been encountered in monoclinic martensite plates forming in the later stages of the transformation (during cooling down from 1273 K 1000 C to room temperature). The directions and magnitudes of the principal distortions associated with the three possible lattice correspondences between tetragonal and monoclinic zirconia can be determined by analytical geometry, as illustrated in Figure 4.56. The principal distortions, 1 , 2 and 3 , are defined along the orthonormal axes parallel to the [100]t , [010]t and [001]t directions, respectively. The problem is two dimensional as one principal axis is obtained directly by inspection (Figure 4.55(b), 3
bm ). The other two principal distortions lie in the plane normal to the 3 direction. The method of determination of principal distortions shows the changes in the relevant tetragonal plane. The
Ym sin β
Yt
Ym
Y Xt X Tetragonal plane
Xm
Xm
(a) Expansion
(b) Shear monoclinc plane
Figure 4.56. Schematic showing various steps of martensitic transformation.
366
Phase Transformations: Titanium and Zirconium Alloys
Table 4.17. Computed principal strains for three lattice correspondences (LC). LC A
LC B
LC C
1
Direction Magnitude
0 08267 05627 1,0956
0 07383 −06745 0.9337
07860 −06183 0 0.9428
2
Direction Magnitude
0 −05627 08267 0.9287
0 06745 07383 1.0897
06183 07860 0 1.1045
3
Direction Magnitude
100 1.0128
100 1.0128
001 0.9896
distortion of this plane can be factorized into (a) a change in dimensions (expansion/contraction) and (b) a change in shape (simple shear). The calculated values for the principal strains for the three lattice correspondences are given in Table 4.17 (lattice parameters and principal strains). 4.6.3 Habit plane Bansal and Heuer (1974) and Kriven et al. (1981) have analysed the crystallography of the tetragonal to monoclinic transformation in zirconia for the lattice correspondences designated as type B and type C. The systems of lattice invariant shear considered for this analysis are the following: For the type B lattice correspondence: 1¯I0t 001t 110t 1¯I0t 100t 001t 100t 011t For the type C lattice correspondence: 1¯I0t 001t 1¯I0t 110t 10¯It 010t 111t 1¯I0t These shear systems, expressed in terms of tetragonal indices, correspond to slip and twin shears of the monoclinic system. Type A lattice correspondence, being the least likely to be operative in view of the highest lattice strains involved, has not been considered here. Experimental observations support the existence of the correspondences B and C. There is some evidence to indicate that the correspondence B is favoured in the transformation in large grains of the tetragonal phase (Hugo et al. 1988). On the other hand, when the transformation occurs in small particles or fibres, correspondence C is almost always observed (Muddle and Hanmik 1986). The application of the Bowles–Mackenzie (1954) theory with regard to the correspondences B and C yields solutions for habit plane indices, directions and magnitudes of shape strains, a detailed account of which has been presented by Kriven et al. (1981) (Table 4.18).
Martensitic Transformations
367
Table 4.18. Predicted habit planes and shape strains. Lattice invariant shear system (010 [101]
¯ (011) [011]
¯ (101)[101]
¯ ¯ (111) [011]
¯ (010 [100] Twin system
Habit plane indices (PI)
Direction of shape strain D1
Magnitude of shape strain M1
−0615 −0334 0.730 ¯ ∼(2¯ 12) −0103 −0988 0.122 ¯ ∼(1¯ 92)
−0067 −0984 0.160 ¯ ∼[06¯ 1] −0620 −0281 0.716 ¯ ∼[7¯ 10]
0.120
−0049 −0999 0.011 ¯ ∼(010) −0976 −0219 0.001 ¯ ∼(5¯ 10)
−0990 −0144 −0000 ¯ ∼[7¯ 10] 0.028 −1000 0.010 ¯ ∼[010]
0.159
0.018 −1000 −0002 ¯ ∼(010) 0.960 0.280 −0011 ∼(720) 0.056 0.996 −0075 ∼(010) −0978 −0206 −0013 ¯ 102) ¯ ∼(15
−0979 −0203 0.011 ¯ ∼[5¯ 10] −0097 0.995 0.003 ∼[010] 0.992 0.126 0.018 ∼ [810] 0.026 −0997 0.074 ¯ ∼[010]
0.164
−0383 −0810 0.455 ¯ ∼(1¯ 21) −0383 0.810 −0455 ¯ ∼(1¯ 2¯ 1)
−0379 −0801 −0452 ¯ ∼ [1¯ 2¯ 1] −0379 −0801 −0452 ¯ ∼[1¯ 2¯ 1]
0.052
0.120
0.159
0.164
0.170
0.170
0.52
(Continued)
368
Phase Transformations: Titanium and Zirconium Alloys
Table 4.18. (Continued) Lattice invariant shear system (110) [001]
Habit plane indices (PI) −0178 −0047 1.006 ¯ ∼(106) −0855 0.483 0.195 ¯ ∼(952)
Direction of shape strain D1 −0859 0.494 0.134 ¯ ∼[741] −0132 −0076 0.966 ¯ ∼[107]
Magnitude of shape strain M1 0.115
0.115
Kelly and Ball (1986) and Kelly (1990) has pointed out that the grouping of martensitic variants which results in the formation of the important morphological features of zirconia martensites can be explained on the basis of habit plane predictions for different lattice correspondences. For both the correspondences B and C, a number of different LIS systems lead to two groups of habit plane. The first is in the region of (140)t for correspondence B (or (410)t for correspondence C). Depending on the correspondence adopted, the predicted orientation relationship is B-2 or C-2 within about 1 and in both the cases twin-related variants are predicted with a junction plane corresponding to (100)t . The twin relationship, though not precise, is normally less than 05 from an exact twin -relationship. The configuration of the twin-related variants illustrated in Figure 4.57 is produced for the B type lattice correspondence with the LIS on (011)t [211]t . It is to be noted that a similar configuration with two variants nearly twin related across the (100)m junction can also be generated when a different LIS system is chosen. The habit plane solutions for these cases are of the types {130}t and {140}t . The other group of solutions gives a habit plane which, for the correspondence B, is 2 or 3 from (100)t or (001)t and for the correspondence C is in the vicinity of (106)t (i.e. 3–12 from (001)t or (001)m ). The predicted orientation relationship is B-1 or C-1, respectively, and in both cases the theory predicts variants that are twin related about (001)m within 0.2–07 . An example of the twin-related configuration corresponding to the lattice correspondence B is schematically shown in Figure 4.57. A habit plane solution in the vicinity of (671)m or (761)m is predicted for a restricted choice of LIS systems, the magnitude of the shear being higher than those for the two sets of solutions discussed earlier. However, there is one significant exception, for correspondence C and for a LIS system of 110t ¯I12t a habit plane.
Martensitic Transformations (130)
T
369
or (14
0)T
16°
0) or (14 T (130) T
(a) Near (001)M i.e. (100)T (100)T//(001)M Near (001)M i.e. (100)T
(b)
Figure 4.57. The configuration of twin-related variants for B type lattice correspondence and ¯ t (after P.M. Kelly, 1990). LIS on (011)t [211]
4.7
TRANSFORMATION TOUGHENING OF PARTIALLY STABILIZED ZIRCONIA (PSZ)
Phase diagrams of binary oxide systems, discussed in Chapter 1, show that the two transition temperatures (corresponding to the cubic → tetragonal and tetragonal → monoclinic transformations) are lowered by alloying ZrO2 with CaO, MgO, Y2 O3 and rare earth oxides. Partial stabilization of ZrO2 , which results in a structure containing a mixture of the cubic and the monoclinic (or the tetragonal) phases, leads to an improvement in the thermal shock resistance and the toughness of zirconia ceramics. Reductions in the thermal expansion coefficient and in the volume change associated with the tetragonal → monoclinic phase transformation are factors which contribute towards the improved thermal shock resistance of PSZ (Subbarao 1981) (Table 4.19). The toughening, however, has been attributed Table 4.19. Lattice parameters (in nm) and thermal expansion coefficients of zirconia as a function of temperature. Temperature ( C)
am
bm
cm
956 (meas.) 0.51882 0.52142 0.53836 81 217 950 (calc.) 0.51881 0.52142 0.53835 81 22 Thermal 1.031 0.135 1.468 expansion coeff. (nm × 10− 6 C−1 )
Temperature ( C) 1152 (meas.) 950 (calc.)
at
ct
0.51518 0.52724 0.51485 0.52692 1.160 1.608
370
Phase Transformations: Titanium and Zirconium Alloys
to a stress-induced martensitic transformation (Garvie et al. 1975, Porter and Heuer 1977, Evans and Cannon 1986, Kelly 1990). The martensitic transformation has two beneficial effects in this context. First, the transformation plasticity can partially relieve the applied stress; in other words, the energy of the transformation can directly contribute towards an increase in the fracture energy. Second, the strain associated with the transformed region may help counter the high tensile stresses at the tip of the advancing crack and resist its further propagation. Thermal processing of PSZ to achieve the maximum strength and toughness has been extensively studied and the results are summarized in several reviews (Pascoe et al. 1977, Nettleship and Stevens 1987, Evans and Heuer 1980, Subbarao 1981, Kelly and Rose 2002). The principle of microstructural engineering of PSZ for enhancing fracture toughness is based on distributing the tetragonal (and partly, the monoclinic) phase particles in the matrix of the cubic phase. The second phase may appear at grain boundaries during the sintering of zirconia, suitably alloyed for controlling the relative stabilities of the competing phases. Intragrain dispersion of the second phase occurs either during cooling from a temperature above the solvus line or during post-sintering heat treatments. Intragranular precipitates assume an ellipsoidal shape and are aligned along {100}c habits in order to minimize the strain energy of the precipitate-matrix assembly. The optimum size of the precipitate particles has been reported to be about 0.2 (m, as particles of still smaller sizes retain the tetragonal symmetry while larger particles transform into the monoclinic phase spontaneously. The role of metastable tetragonal precipitates in a cubic matrix in toughening PSZ has first been noted by Garvie et al. (1975) and later elucidated by Porter and Heuer (1979). It has been shown that all the precipitate particles within several micrometres of a crack possess a monoclinic symmetry, whereas all other particles are tetragonal. This suggests that the stress field near the crack tip causes the particles to transform into the monoclinic phase. By this mechanism, the elastic stress field at the crack tip can be dissipated through the formation of stress-induced martensite in the metastable tetragonal precipitates which also lose coherency in the process. Further extension of the crack, therefore, requires the application of an additional stress. Thus the stress-induced tetragonal to monoclinic transformation within the fine precipitates essentially operates as a crack blunting mechanism which is schematically illustrated in Figure 4.58. The retention in the fine (smaller than 0.2 (m) particles of the metastable tetragonal phase, when they are distributed in the cubic ZrO2 matrix, can be explained in terms of a much larger value of the strain energy associated with the cubic–monoclinic assembly in comparison with that associated with the cubic– tetragonal assembly (Porter and Heuer 1979). This strain energy difference more
Martensitic Transformations
371
Critical crack
Figure 4.58. Tetragonal to monoclinic transformation in zirconia with volume expansion resulting in microcracks around the particle; cracks propagating into the particle get deviated leading to increased fracture resistance. Table 4.20. Transverse rupture strength and fracture toughness of zirconia.
Tetragonal + cubic ZrO2 Monoclinic + cubic ZrO2 (overaged at 1400 C) Cubic ZrO2 (solution – annealed at 1850 C, 4 h)
Transverse rupture strength (MPa)
KI C (MN/m3/2
650 250
≈7.1 3.7
245
2.8
than compensates for the difference in the chemical free energy between the monoclinic and the tetragonal phases in the case of fine particles. A comparison of the strength and the fracture toughness of zirconia ceramics in three different microstructural states (Table 4.20) clearly indicates that the presence of the metastable tetragonal phase is directly responsible for enhancing the fracture toughness and transverse rupture strength quite substantially. Such a conclusion can also be arrived at from the observation that a nearly fully tetragonal material, prepared by fine particle technology and comprising a ZrO2 –Y2 O3 solid solution, shows a high strength at room temperature whereas a ZrO2 –CeO2 solid solution, in which the tetragonal phase is stable at room temperature, is relatively weak. The latter material exhibits enhanced strength and toughness at liquid nitrogen temperature, where the tetragonal phase is metastable. The phenomenon of toughening of PSZ by a stress-induced martensitic transformation can be compared with transformation-induced plasticity in TRIP steels, in spite of the fact that in the latter case the transformation occurs in the entire austenite matrix while in the former only the fine dispersed precipitates take part in the martensitic transformation (Figure 4.58).
372
Phase Transformations: Titanium and Zirconium Alloys
4.7.1 Crystallography of tetragonal → monoclinic transformation in small particles The mechanism of transformation toughening described in the preceding section essentially requires the fulfilment of the following conditions: (1) The strain energy associated with the transformation should be of such a magnitude that particles below a certain critical size are retained in a metastable tetragonal state. (2) It should be possible to trigger a stress-induced martensitic transformation in these fine particles. (3) The volume change associated with the transformation should be positive so that it could counter the tensile stresses at the tips of advancing cracks. (4) The magnitude of the macroscopic shear associated with the transformation should be large for facilitating the occurrence of the stress-induced transformation. At the same time, ceramics being incapable of tolerating large shear strains, the overall shear strain associated with a single tetragonal particle should not be very high. These conflicting requirements can be met if a number of variants of the monoclinic martensite plates forming within a particle belong to a self-accommodating group. In that case, a major part of the shear strain associated with a single martensite variant can be neutralized by those associated with the other variants present in the group. The crystallography of the martensitic transformation in fine particles has been studied by Kelly (1990) mainly to identify the possible self-accommodating groups of martensite variants which can form within these fine particles (Figure 4.59). t mt m
[001]m [001]t
Strain = 4.12%
[100]m [100]t
Strain // [010]
Strain = 0.51%
= 0%
(a)
[001]m [001]t
Strain =0.97%
Strain // [010]
[100]m [100]t Strain = 2.65%
= 0%
(b)
Figure 4.59. The four possible arrangements of twin-related variants together with the range of strain values predicted for the distortion.
Martensitic Transformations
373
It is important to note that TEM studies on PSZ have established (Kelley 1990) that each of the fine tetragonal particles undergoes a stress-induced transformation, leading to the formation of an array of alternate bands, twin related on either (100)m or (001)m planes. These twins do not correspond to the LIS. Instead, these bands are twin-related variants which have shape strains that lead to the cancellation of the shear component when the two variants are of equal thickness. The large shear component associated with the formation of the first variant creates the “back stresses” which induce the formation of the twin-related variant with the opposite shear strain. The accommodation of shear strain by the combination of these twinrelated variants is responsible for the creation of the zig-zag arrangements of the variants.
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Smith, D.K. and Newkirk, H.W. (1965) Acta Cryst., 18, 983. Srivastava, D. (1996) Ph.D Thesis, Indian Institute of Science, Bangalore. Srivastava, D., Madangopal, K., Banerjee, S., and Ranganathan, S. (1993) Acta Metall. Mater., 41, 3445. Srivastava, D., Mukhopadhyay, P., Banerjee, S. and Ranganathan, S. (2000) Mater. Sci. Eng., A288, 101. Subbarao, E.C. (1981) Advances in Ceramics Vol. 3, Science and Technology of Zirconia (eds. A.H. Heuer and L.W. Hobbs) Am. Ceramic Soc., Columbus, OH, p. 1. Tadaki, T., Otsuka, K. and Shimizu, K. (1988) Annual Rev. Mater. Sci. (Ed. R.S. Huggins), 18, 125. Wang, F.E., DeSavage, B.F. and Buchler, W.J. (1968) J. Appl. Phys., 39, 2166. Warlimont, H. and Delaey, L. (1974) Prog. Mater. Sci., 18, 117. Wayman, C.M. (1964) Crystallography of Martensitic Transformations, The Macmillan Company, New York. Wayman, C.M. (1980) J. Metals, 32, 129. Wechsler, M.S., Liberman, D.S. and Read, F.A. (1953) Trans AIME, 197, 1503. Weing, S. and Machlin, E.S. (1954) Trans. AIME, 200, 1280. Williams, J.C. (1973) Proc. Titanium Science and Technology, Vol. 3, Plenum Publishing, New York, p. 1433. Williams, J.C. and Blackburn, M.J. (1967) Trans. ASM, 60, 373. Williams, J.C. and Hickman B.S. (1970) Metall. Trans., 1, 2648. Williams, A.J., Cahn, R.W. and Bawelt, C.S. (1954) Acta Metall., 2, 117. Williams, J.C., Taggart, R. and Polonis, D.H. (1970a) Metall. Trans., 1, 265. Williams, J.C., Pollonis, D.H. and Taggart, R. (1970b) The Science, Technology and Applications of Titanium (eds R.I. Jaffee and N. Pfomisel, Pergamon Press, London, p. 733. Wolten, G.M. (1963) J. Am. Ceram. Soc., 46, 418. Wolten, G.M. (1964) Acta Crystal., 17, 763. Yamane, T. and Vedg, J. (1966) Acta Metall., 14, 438. Yoo, M.H. (1969) Trans. Metall. Soc. AIMME, 245, 2051. Zangvil, A., Yamamoto, S. and Muratami, Y. (1973) Metall. Trans., 4, 467.
Chapter 5
Ordering in Intermetallics 5.1 Introduction 5.2 Theoretical Treatments 5.2.1 Alloy phase stability 5.2.2 Order–disorder transformations 5.2.3 The ground states of the Lenz and Ising model 5.2.4 Special point ordering 5.2.5 Concomitant clustering and ordering 5.2.6 A case study: Ti–Al system 5.3 Transformations in Ti3 Al-based alloys 5.3.1 → D019 ordering 5.3.2 Phase transformations in 2 -Ti3 Al-Based systems 5.3.3 Structural relationships 5.3.4 Group/subgroup relations between BCC (Im3m), HCP (P63 /mmc) and ordered orthorhombic (Cmcm) phases 5.3.5 Transformation sequences 5.3.6 Phase reactions in Ti–Al–Nb system 5.4 Formation of Zr3 Al 5.4.1 Metastable Zr3 Al (D019 ) phase 5.4.2 Formation of the equilibrium Zr3 Al (L12 ) phase 5.4.3 +Zr2 Al → Zr3 Al peritectoid reaction 5.5 Phase Transformation in -TiAl-Based Systems 5.5.1 Structural relationship between 2 - and -phases 5.5.2 Phase reactions 5.5.3 Transformation mechanisms 5.6 Site Occupancies in Ordered Ternary Alloys 5.6.1 Ordering tie lines 5.6.2 Kinetic modelling of B2 ordering in a ternary system 5.6.3 Influence of binary interaction parameters 5.6.4 B2 ordering in the Nb–Ti–Al system References
380 383 384 386 397 401 407 412 416 416 417 421 424 428 432 436 437 439 441 443 443 446 451 458 458 460 462 464 465
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Chapter 5
Ordering in Intermetallics
List of Symbols k ij : Short-range order parameter for the ij-pair in the kth coordination shell B : Bulk modulus of a superstructure EF : Fermi energy k : The kth volume expansion coefficient for Etot Etot : Total electronic energy of a superstructure Ecoh : Cohesive energy of a superstructure : Formation energy of a superstructure Eform : A superstructure : A cluster : Long-range order parameter Jt : Effective cluster interaction for the cluster of type t jk : The kth volume expansion coefficient for J
k: A wave vector in the reciprocal space kB : Boltzmann constant NEF : Density of states at the fermi level r: Atomic occupation probability at position r pni : Site occupation operator for a species i at a site n k: Amplitude of the concentration wave with wave vector k Probability of a -point cluster to occur in configuration
: i : A site operator which can take value +1−1, for a component A (B) in a binary alloy Ti− : Instability temperature below which the disordered phase is unstable w.r.t. spontaneous ordering Ti+ : Instability temperature above which the ordered phase is unstable w.r.t. spontaneous disordering Teq : Equilibrium temperature T0 : Temperature corresponding to equal free energies of two phases Tcs : Conditional spinodal temperature for the ordered phase V0 : Equilibrium volume of a superstructure 379
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Phase Transformations: Titanium and Zirconium Alloys
t : wij : G: F: H: G: H: kF kR :
F R :
5.1
Correlation function for a -point cluster of type t Interaction potential between ‘ij pair of atoms Gibbs free energy Helmholtz free energy Enthalpy change for a given process Space group of a given crystalline phase Subgroup of a given space group G Rate constants for the forward and backward reactions, respectively Pre-exponential factors for the forward and backward reactions, respectively
INTRODUCTION
In an alloy which can be approximated as an ideal solid solution, the distribution of different atomic species in the lattice is nearly statistical. The interaction energies, Vij , between atomic pairs, ij, are such that there is no preference for the formation of bonds between either the like or the unlike atoms. When a solid solution exhibits a tendency to deviate from the ideal behaviour, one can visualize two alternatives. The like atoms tend to cluster together when there is a preference for the formation of bonds between like atoms. The second alternative is that the atoms arrange themselves in an ordered array where specific atomic species occupy specific lattice sites. The clustering and ordering tendencies are, respectively, associated with a positive and a negative deviation from ideality, as the free energy of the respective systems deviates in the positive or in the negative direction from the free energy of the ideal solid solution. The phenomenon of ordering in an alloy can be described in terms of the simple example of the ordering of a bcc (A2) structure to a CsCl type (B2) superlattice (Figure 5.1(a)). Let us consider an equiatomic alloy of the elements A and B in which the atoms are distributed in the bcc lattice. Considering only the nearest neighbour interactions, one can intuitively see that if VAB , a measure of the attractive interaction between the unlike atoms, is much stronger than the interaction between like atoms, VAA and VBB , there will be a tendency for A and B atoms to, respectively, select the body centre and the corner positions of the unit cell of the bcc lattice or the other way round. In case a perfect order is established, all A atoms will have only B atoms as their first near neighbours and vice versa. When the temperature of such an alloy is raised, the entropy factor plays an increasingly important role and above the order–disorder transition temperature, Tc , the distribution of atoms in the lattice sites becomes random. The competition between
Ordering in Intermetallics
B32
B2 (a)
381
D03 (b)
(c)
Figure 5.1. bcc-based ordered intermetallic phases which are the ground state superstructures under the first and the second nearest neighbour (NN) pair approximation.
the entropy factor, which tries to randomize the system at elevated temperatures, and the enthalpy factor (resulting from interatomic interactions), which tends to organize the atoms in an ordered array, dictates the state of the order. When the formation enthalpy of an ordered structure is very large, the state of order may persist even up to the melting temperature, and in such cases an order to disorder transition is not encountered in the solid state. Ti- and Zr-based alloys are essentially based on the hcp and bcc lattices, corresponding to the two allotropic modifications of these elements. Studies on the order–disorder transition in these alloys, therefore, include an examination of the possible superlattice structures based on the bcc and hcp lattices, the determination of ground states (energetically favourable superlattice structures at 0 K) and investigations regarding the mechanism of order evolution in these systems. The interest in order evolution is centred around the question as to whether the process occurs in a homogeneous manner by the development of a concentration modulation with appropriate wave vectors, followed by a gradual amplification of such modulations or by a heterogeneous process involving the nucleation of nearly perfect ordered particles in the disordered matrix, followed by the growth of these particles. In the context of Ti and Zr alloys, the disorder–order transition has been studied in detail in only a few systems. The most important among the structural transitions involved in these are: (a) (hcp)→ D019 in the Ti–Al, Ti–Sn and Ti–Ga systems in the vicinity of the Ti3 X (X = Al, Sn, Ga), and (b) (bcc) → B2 in the Ti–Al and Zr–Al systems. Theoretical treatments for the determination of thermodynamic parameters of ordering processes and experimental results which illustrate the essential features of these processes have been reviewed in this chapter.
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Phase Transformations: Titanium and Zirconium Alloys
Phase transformations (including phase reactions) involving ordered phases in the aluminides of Ti and Zr have attracted attention ever since the emergence of these intermetallics as possible structural materials. Chronologically Zr3 Al was one of the first intermetallics to be considered as a structural material. The prospect of using stoichiometric Zr3 Al as a pressure tube material in pressurized heavy water nuclear reactors (PHWRs), originally conceived by Schulson (1977), was quite bright because of the exceptionally good resistance of this material against irradiation creep, which happens to be one of the life-limiting factors under the prevailing service conditions. The presence of atomic order could be regarded as being essentially responsible for the superior irradiation creep resistance of this material. The intermetallic was designed on the basis of the maxim that ordering generally reduces the rates of migration of vacancies and interstitials, thereby enhancing the probability of annihilation of these defects. This, in turn, reduces the irradiation creep rate under a given stress. Some other virtues of this material, namely, a low thermal neutron absorption cross-section, promising mechanical properties and good corrosion resistance in water in the temperature, pressure and radiation environment present in the PHWRs make it eminently suitable for the proposed application. The reason for which Zr3 Al was not finally accepted for pressure tube application was the tendency of disordering, leading even to amorphization, of this alloy under irradiation and its poor room temperature ductility. The equilibrium structure of Zr3 Al is L12 (Cu3 Au type) and this phase evolves through a peritectoid reaction. A classical order–disorder transition cannot be studied in this system as Zr3 Al cannot be congruently disordered, unless radiation disordering is resorted to. Several interesting aspects of phase transformations, such as the peritectoid reaction between the and Zr2 Al phases to form Zr3 Al, the formation of a metastable Zr3 Al phase (D019 structure) during precipitation from the supersaturated (hcp) matrix of non-stoichiometric Zr–Al alloys and the cellular reaction leading to the formation of an + Zr 3 AlL12 lamellar aggregate, have been discussed in this chapter. The story of the development of Ti aluminides is even more exciting. The need for the development of suitable aerospace materials for use in the temperature range of 770–920 K has been felt for bridging the gap between the temperature ranges pertinent to conventional high -Ti alloys on one hand and Ni-based superalloys on the other. The binary intermetallic Ti3 Al (2 -phase) has a specific modulus and a stress rupture resistance comparable to those of superalloys but the complete absence of ductility at room temperature comes in the way of its being accepted as a structural material. Blackburn and Smith (1978) were the first to demonstrate room temperature ductility in a Ti3 Al-based composition (Ti–24% Al–11% Nb) and this was followed by wide-ranging investigations on phase transformations
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383
in and deformation behaviour of a large number of Ti aluminides. These studies have revealed a variety of phase transformations in multicomponent Ti aluminides, one of the most exhaustively studied systems being the ternary Ti–Al–Nb. The constituent phases, the sequences and mechanisms of transformations and the mechanical properties are quite distinct in different composition regimes for the Ti–Al–Nb system. Based on these considerations, the transformation processes can be discussed in two distinct groups of alloys, one based on Ti3 Al and the other based on TiAl ( phase, L10 structure). The Ti3 Al-based alloys can again be categorized on the basis of their -stabilizer content. Important alloys belonging to the first category are Ti–24Al–11Nb (Blackburn and Smith 1978) and Ti-25Al-8Nb-2Mo-2Ta (Marquardt et al. 1989). They contain -stabilizers in the range of 10–12 at.%. Alloys with about 14–17 at.% -stabilizers, including Ti-24Al-(14-15)Nb (Blackburn and Smith 1978) and Ti24Al-10Nb-3V-1Mo (Blackburn and Smith 1989), can be grouped into the second category. The third category of alloys, containing 25–30 at.% -stabilizers, has been developed more recently (Rowe 1991, Gogia et al. 1993). Phase reactions and mechanical properties associated with these three categories of Ti3 Al-based alloys are quite different and, therefore, the development of microstructure in these groups of alloys has been discussed separately. The phase equilibria and solid-state phase transformations in -TiAl alloys, with the Al content varying between 40 and 50 at.%, have also attracted considerable attention in the recent years. This is primarily because of the fact that -TiAl-based alloys offer an attractive combination of properties, namely, low density, substantial high temperature strength and good creep as well as oxidation resistance. These alloys, at room temperature, usually consist of the 2 (D019 ) and (L10 ) phases. The 2 -phase disorders into the hcp -phase above about 1420 K. The occurrence of the eutectoid reaction → 2 + , the existence of unique orientation relationships connecting the three phases and the sensitivity of phase transformation modes to the cooling rate are some of the factors responsible for the development of many interesting microstructures in -TiAl alloys. An attempt has been made in this chapter to rationalize the variety of microstructures reported in -TiAl alloys in terms of some plausible phase transformation schemes.
5.2
THEORETICAL TREATMENTS
In this section, we will present some state-of-the-art theoretical tools to study phase stability and ordering in alloys. Since most commercial alloys are complex multiphase mixtures, it is important to have knowledge of all the possible alloy phases that can occur in a given system: terminal solid solutions, possible ordered
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Phase Transformations: Titanium and Zirconium Alloys
intermetallic phases and other disordered solid solutions. Hence, the study of alloy phase stability and order–disorder transformations is a major area of research today. 5.2.1 Alloy phase stability More than 70 years ago, Hume-Rothery (1967) laid the foundations for the study of alloy phase stability in relation to the electronic structure. He pointed out the connection between the observed crystal structures and the electron concentration, which is related to the valence electron difference between the constituent elements and to the composition. Earlier theories of stability of alloy structures have identified the important role of the density of states (DOS) curve for correlating the extent of phase stability with the electron concentration (e/a). The DOS is defined as the number of electronic states per atom (or volume) with energies between certain values: NEo =
dS 3 4 E=Eo Ek
(5.1)
where the integration is carried out over a constant energy surface in k-space between Eo and Eo + E and is the cell volume. The first attempt in this direction was made by Jones (1962) in terms of relative electronic DOS curves between any two simple alloy phases (1 and 2 ). He tried to explain the phase competition between the - and the -phases in the Cu–Zn system in terms of a rigid band model for the conduction electrons. The gist of the argument was that, within the low e/a range corresponding to the -phase, the total DOS for the -structure is higher than that for the -structure which follows it, because of the prominent peak in the DOS curve associated with the 111 zone contact with the Fermi surface. Thus the conduction electrons can be accommodated within a lower total energy than that for the -phase. As e/a increases, the 110 peak in the DOS curve of the -phase structure becomes prominent, making it a more favourable structure. Thus, at alloy compositions at which the Brillouin zone boundary is just touched, a given crystal structure should be particularly favoured. These compositions correspond to those of Hume-Rothery phases. Experiments have proved that in many metals, the Fermi surface is very different from that of a free electron sphere, giving birth to the nearly free-electron theory. The deep potential energy wells at the centres of atoms play a very small role. In 1959, the pseudopotential theory (Harrison 1966, 1973) took roots; in this, it was considered that as a conduction electron passes through the central region of an ion in a metal, it not only experiences a strong electrostatic attraction to the nucleus but also a strong repulsion from the core electrons. In some metals
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385
like Na, Mg and Al, these two opposite contributions almost cancel, so that the ions in these metals appear to be nearly transparent to the conduction electrons. It thus becomes possible to replace the true potential in the Schrödinger equation by a pseudopotential in which the Pauli repulsion is introduced to cancel out most of the coulomb potential of the nucleus (which can be treated using perturbation methods on the free electron theory). The present status of the electronic structure methodologies is attributed to three important breakthroughs during the last 30 years or so. The first was the introduction of Hohenberg–Kohn–Sham density functional theory (DFT) (Hohenberg and Kohn 1964, Kohn and Sham 1965) in the mid-1960s, when it was rigorously proved that the complicated and unmanageable many-electron problem can be transformed into an effective one-electron problem which, in turn, can be solved within the so-called local density approximation (LDA) (Jones and Gumarsson 1989). The DFT is based on the quantum statistical approach. It does not attack the many-body problems frontally, but it possesses a certain conceptual simplicity for which it has emerged as the most successful tool in describing the ground state properties of inhomogeneous electronic systems. The second breakthrough came in 1975, when linear methods for solving the one-electron band structure of solids was introduced by Andersen (1975). The LAPW (linear augmented plane wave) (Andersen 1975, Jansen and Freeman 1984) and the LMTO (linear muffin-tin orbital) (Andersen et al. 1993) methods and associated methods like the ASW (augmented spherical wave) method are the most widely used today for ab initio (i.e. without any a priori assumption regarding interatomic interactions) investigation of materials. The third breakthrough came in 1985, when Car and Parrinello (1985) proposed a recipe for ab initio molecular dynamics (MD) simulation of atomic aggregates, where “forces” acting on atoms are evaluated within the framework of DFT. This DF-MD method has since been extensively used for studying various kinds of clusters, nanoparticles, liquids and amorphous solids, although the underlying plane wave pseudopotential makes its applications restricted mainly to s–p bonded systems (Payne et al. 1992). In contrast to ordered solids, the calculation of any physical property of a disordered alloys requires configuration averaging over all possible configurations. Most experiments measure configurationally averaged properties. The configuration averaging is usually achieved in the framework of the coherent potential approximation (CPA) (Soven 1967, Taylor 1967). The essence of the CPA is to replace the array of random potentials by an effective energy-dependent coherent potential (z). The scattering properties of z are then determined self-consistently (in a single site mean-field sense), from the requirement that an electron travelling in an infinite array of z undergoes, on average, no further scattering upon replacement of any z with the actual potential. So CPA is a
386
Phase Transformations: Titanium and Zirconium Alloys
self-consistent prescription which allows one to obtain an effective non-random Hamiltonian. CPA, coupled with first principles band structure methods in the framework of LDA, provides a starting point for the ab initio calculation of the electronic structure of disordered alloys. Though CPA provides reliable results, there are many situations where the single site approximation inherent in CPA begins to fail: as in cases where clustering effects become important (e.g. in the impurity bands of split band alloys, like the Zn band in a Cu-rich CuZn alloy), where short-range order dominates leading to ordering or segregation and where local lattice distortion because of size mismatch of the constituents leads to essential off-diagonal disorder (as in CuPd and CuBe alloys). Recently the augmented space recursion (ASR) method which involves the recursion (Haydock 1980) method in the augmented space (Hilbert space ( ) + Configurational space ()) (Mookerjee 1973a,b, 1987) is gaining importance to handle disordered alloys. The recursion method (Haydock 1980) offers an alternative to the band structure method. This method is suitable for computing the local properties which are related to the diagonal elements of the resolvent, where it offers a large computational advantage over the band structure methods. Moreover, the recursion method is based on real space and so does not require lattice periodicity for its operation, in contrast to the band structure methods. As a result, both ordered and disordered systems as well as systems with broken translational symmetry (such as surfaces) can be treated within the recursion method. The augmented space formalism, introduced by Mookerjee (Andersen et al. 1993), is a novel and conceptually attractive method for the calculation of the configurationally averaged Green function of a disordered material. In this method, one transforms the Hamiltonian describing a given disordered system to an ordered Hamiltonian whose Green function matrix elements correspond to appropriate configurational averages of the Green function of the original disordered system! The new ordered Hamiltonian is said to be in an augmented space which can be described as the direct product of the Hilbert space, , spanned by the original Hamiltonian with a configuration or disorder space, , which spans all possible configurations of the system. 5.2.2 Order–disorder transformations The thermodynamics of solid solutions consists of contributions from electronic, displacive, vibrational, magnetic and configurational degrees of freedom. The theoretical model presented here assumes that the configurational degree of freedom can be separated from the other degrees of freedom and that change in the latter effects can be neglected in certain types of phase transformations or phase changes, called coherent phase transformations. These coherent phase transformations are
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387
defined as those where the basic lattice framework of the crystalline solid solution is conserved, i.e. the two phases differ from each other only in the chemical distribution of the constituent elements. Hence chemical rearrangement of atomic species is allowed along with displacements from the average lattice positions, but any discontinuous change of the lattice framework, leading to dislocations, incoherent interfaces, free surfaces and grain boundaries, is not allowed. 5.2.2.1 Historical developments As far back as 1925, Lenz and Ising (Temperley 1972) introduced a model to study the thermodynamics of magnetic systems. They proposed to associate a magnetic moment at each lattice site of a given lattice. These magnetic moments can have only two components, up or down, along a given direction, and a moment interacts with only its nearest neighbours. The pairs of parallel moments have an energy −J , whereas pairs of antiparallel moments have an energy +J . It is then quite easy to calculate the energy of a given configuration of a finite system. It is not always possible to obtain explicit expressions for the thermodynamic quantities of interest when the size of the system increases to infinity. Various approximations have been proposed for the solution of the Lenz and Ising model in three dimensions (3D). The resemblance of this model to a binary substitutional alloy in which each site is occupied by either the A or the B component was immediately evident, with a site-occupation operator replacing the magnetic moment. The same analogy can be extended to interstitial alloys where each interstitial site is either occupied or vacant, disregarding the host matrix. This model, within certain approximations, has been shown to contain the essential ingredients to account for various types of phase transitions and various topologies of phase diagrams. The thermodynamic state of a system can be easily obtained by finding the proper expression for the partition function = exp−E/kB T, where signifies a configuration and the summation is over all possible configurations. The basic problem is then to calculate the partition function and its derivatives with respect to external fields, from which we can obtain the free energy and all averages of interest. Except in a few cases, it is not possible to perform these calculations. One has, therefore, to make some plausible approximations. The most useful ones are the so-called mean field (or molecular field) approximation (MFA) (de Fontaine 1979, Ducastelle 1991). Mean field theories (MFTs) are, in general, derived from variational principles and have been shown to suffer from serious drawbacks, particularly close to second order phase transitions, which are characterized by the fact that long-range order (LRO) parameter is a continuous function of temperature, vanishing at the critical temperature, Tc . At temperatures near Tc , the singularities of the thermodynamic functions are not correctly reproduced. In such cases, it is necessary
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Phase Transformations: Titanium and Zirconium Alloys
to treat the fluctuations in the medium. The celebrated renormalization group theory (Ma 1976, Toulouse and Pfeuty 1977) has been developed in the 1970s to account for this critical regime. This theory is specially designed to identify the universal characteristics of phase transitions. However, at temperatures away from Tc , its practical implementation is not very easy and also not very reliable. Since most of the order–disorder transitions concerning the 3D Lenz and Ising model seem to be of the first order, i.e. the LRO parameter is discontinuous at Tc , transition occurs before the fluctuations have increased too much in the disordered phase. In such a situation, MFTs are expected to give reasonably accurate results. The simplest MFT (called the Bragg–Williams approximation) (Bragg and Williams 1934, 1935, Wilson and Kogut 1974), applied to the Lenz and Ising model, easily predicts the occurrence of order–disorder transitions and provides a description of the disordered and ordered states. This is a single-site approximation in which each atom is assumed to be embedded in a mean field. Due to its simplicity, this theory is successful in many respects, but quantitatively it is rather inaccurate. The critical temperature are found to be much too high, the short-range order is rather badly treated, and the phase diagrams can even be qualitatively wrong in the case of frustrated lattices. In this theory, correlation between sites or the local statistical fluctuation was totally neglected. In view of this defect, Bethe (1935) introduced the concept of short-range order. This is to improve the approximation from a single site in the mean field to a pair of sites in a molecular field. As expected, a great difference from the result of the B–W approximation appears near the critical point. In particular, the results showed the existence of short-range order above Tc , which decays as the temperature is raised. In addition, the critical point was lowered in comparison to the B–W approximation for the same value of pair interaction, J2 . Generally the better is the degree of approximation, the lower is the Tc , and thus, the Tc is taken as a measure of the degree of approximation. Since the exact solution of the 3D Lenz and Ising problem cannot be obtained, the most important source of exact information concerning this problem lies in a series expansion of the partition function and other derived properties at low and high temperatures. An important example of a high-temperature series expansion is due to Kirkwood (1938), who devised an ingenious method to evaluate the partition function in a systematic way by a series expansion in terms of the moments of probability distributions. Kirkwood evaluated these moments down to the third order. The series expansion approaches can be classified into two types. In the first, a particular selection of terms which can be readily evaluated is summed to provide closed form approximations similar to other approximate methods. In this
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connection, the method of Rushbrooke and Scions (1955) and that of Yvon (1948) and Fournet (1967) may be cited. In the second kind of approach, the known terms of the expansion are used to assess its asymptotic behaviour. This approach enables one not only to estimate the true critical point but also to closely investigate the critical behaviour. These methods involve cumbersome mathematical calculations, without adding much physical insight. 5.2.2.2 Static concentration wave model Landau and Lifshitz (1969, 1980) developed some very useful concepts related to order–disorder transitions based on the MFT. Since their theory was based on abstract group theoretical arguments, it could not be exploited by the metallurgists initially. However, these concepts have been widely used by some Russian scientists like Krivoglaz (Krivoglaz and Smirnov 1964) and Khachaturyan (1978, 1983; static concentration wave (SCW) model). Elsewhere the theory has been applied mostly to displacive soft mode transitions. The SCW model is in fact a B–W model in an arbitrary number of LRO parameters and has been derived from Landau’s general analysis of order–disorder reactions. It is useful for analysing order–disorder systems in which the ordered phases are crystallographic derivatives of their respective parent phases (coherent phases). In the SCW model, the crystal is modelled by a rigid lattice onto which “average atoms”, or atoms whose properties are assumed to be averaged according to the occupancy of particular sites, are superposed. The internal energy is assumed to be derived only from pairwise interactions between atoms, and the entropy is taken to be equal to the sum of the entropies of the average atoms. In this model, the occupation probability, (r), at a given site, r(p), is expanded in a Fourier series, i.e. it can be represented as the sum of SCWs whose amplitudes are the Fourier coefficients and whose wave vectors determine the superstructure period: r =
N
ke−ikhrp
(5.2)
h=1
where (k)s are the Fourier coefficients given by k =
N 1 reikhrp N p=1
(5.3)
The first summation is over the N lattice points of the periodic crystal and the second is over the N points of the first Brillouin zone. The wave vector k(h) and the lattice vector r(p) are defined by
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Phase Transformations: Titanium and Zirconium Alloys
rp = p a ( = 1 2 3; p are integers, summation implied) kh = 2h b (h = m /N , m = 0, ±1 ±2 ) where a and b are lattice translation vectors and primitive translation vectors of the reciprocal lattice such that a · b = . The concentration wave amplitudes k corresponding to the wave vector k that generates the ordering instability are expressed in terms of normalized (with respect to c) LRO parameter, n , via the following relation, k = n c
(5.4)
The normalized order parameter ( n ) is related to the standard order parameter ( ), as n = / max , where max is the maximum order parameter attainable at a given composition. For a disordered phase, (r) is independent of r and is equal to the atomic fraction, c, of the solute. As an example, we will briefly illustrate the binary L12 ordering in an fcc solid solution using the SCW model. The L12 structure is one in which all the facecentred positions are occupied by one type of atoms and all the corner positions are occupied by the other type of atoms. For the L12 phase, all the (k) values vanish, except (000), (100), (010) and (001) (Figure 5.2). The value of (000) is just the bulk (or solute) concentration, c, and the other three amplitudes are equal. The occupation probability for each sublattice can then be calculated to give ∗ ∗ ∗ r = c + e−i100 r + e−i010 r + e−i001 r
(5.5)
where = 100 = 010 = 001 and 000 = c. The exponents in the above equation have only two values: −i and −i2 depending on whether the position vector r lies in a 200 or a 100 plane with respect to the origin, respectively. The atoms are situated on the fcc lattice such that they lie either (a) in all three 100 planes or (b) in one 100 and two 200 planes, so that (r) assumes only two values: (1) c + 3, if the atom is in a corner position and (2) c − , if it is in a face-centred position. There are 41 N corner atoms and 43 N face-centred atoms. The internal energy, in the pair approximation up to an arbitrary coordination shell, is given by E=
N N Jkhkh∗ kh 2 h=1
(5.6)
Ordering in Intermetallics
391
(001)
(100)
(010)
Figure 5.2. The L12 structure (AB3 stoichiometry) viewed as superposition of three {100} concentration waves. Filled (open) circles represent A (B) atoms, respectively.
where the star (∗) indicates the complex conjugate of the amplitude of the corresponding concentration wave, and Jk, the Fourier transforms of the pair interactions, are given by1 Jk = 1/N
N
Jreikhrp
(5.7)
p=1
Hence, for the L12 structure, the internal energy is E=
N J000c2 + 3J1001002 2
(5.8)
The expression for the configurational entropy is given as S = kB
N
rp ln rp + 1 − rp ln1 − rp
p=1
1
The dependence of r and k on p and h are assumed if not shown explicitly.
(5.9)
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Phase Transformations: Titanium and Zirconium Alloys
which, in terms of sublattice probabilities, can be expressed as S = kB
=1
NS ln + 1 − ln1 −
(5.10)
where is the total number of sublattices, while and NS are the occupation probability and the number of atoms on the th sublattice, respectively. In the SCW model, the internal energy is already given in terms of the order parameter (Eqs. (5.6) and (5.8)) or the amplitude of the concentration wave. The LRO parameter can, most comfortably, be substituted in the expression for the entropy, as will be seen now. The expression for the configurational entropy, after substitution of Eq. (5.5), is 1 S = −NkB c ln c + 1 − c ln1 − c − c − k lnc − k 2 1 (5.11) + c + k lnc + k 2 where kB is the Boltzmann constant. Hence the Helmholtz free energy, F , given by F = E − TS
(5.12)
is obtained as a function of T c and on substituting the normalizing relation (5.4). The free energy of the L12 ordered phases is given as F L12 T c n =
1 k T J000 + 3Jk 2n c2 + B 3 c1 − n ln c1 − n 2 4 + 3 1 − c1 − n ln1 − c1 − n + c1 + 3 n lnc1 + 3 n + 1 − c1 + 3 n ln1 − c1 + 3 n
(5.13)
5.2.2.3 Cluster variation method The most powerful approximation, now, is the cluster variation method (CVM) (Kikuchi 1951, Sanchez and de Fontaine 1978, Morita 1984, Fontaine 1994) which includes the correlation between a few sites in some clusters. The CVM of statistical mechanics was originally proposed as an improved approximation for the solution of the Lenz and Ising model. It is, in fact, a hierarchy of approximations
Ordering in Intermetallics
393
ranging from the simplest Bragg–Williams–Gorsky single-site (one-point) approximation to a very elaborate one considering clusters of lattice points of varying size, shape and complexity. Therefore the CVM is regarded as a natural way of generalizing the MFT for the solution of the 3D Lenz and Ising model used for the study of configurational thermodynamics of alloys. In the CVM, the statistical thermodynamics of alloys is described in terms of atomic configurations of clusters of lattice sites. The state of partial order in alloys is described in terms of multisite correlation functions, as will be described below. Detailed accounts of the subject are provided in many books and review articles such as those due to Ducastelle (1991), de Fontaine (1979, 1994), Morita et al. (1994), Inden and Pitsch (1991), Mohri et al. (1985). The linkage between quantum mechanics and statistical mechanics leading to the first principles configurational thermodynamics of alloys comes via the configurational energy expression which is expanded in terms of effective pair/multisite (cluster) interactions (ECI). These configurationally averaged effective interactions are calculated using ab initio methods. Traditionally there have been two approaches for obtaining these ECIs. The first approach is to start with the electronic structure calculations and the total energy determination of some judiciously selected ordered superstructures (configurations) of a given alloy system and expand the energy in terms of ECIs. The ECIs can then be calculated by the inversion of the expansion – the Connolly–Williams inversion method (IM) (Connolly and Williams 1983, 1984). The other approach is to start with the disordered phase, set up a perturbation in the form of concentration fluctuation and study whether the alloy can sustain such a perturbation. This approach includes the generalized perturbation method (Ducastelle and Gautier 1976), the embedded cluster method (Gonis et al. 1984) and the concentration wave approach (Gyroffy and Stocks 1983), all of which are based on the CPA. The direct configuration averaging (DCA) method (Dreyssé et al. 1989) consists of obtaining the effective pair interactions (EPI) for each random configuration of the disordered alloy using the recursion technique and then performing the configurational averaging over a (selected) number of configurations. In the ASR (Mookerjee 1973a,b) along with the orbital peeling (OP) (Mookerjee 1987) technique, the recursion is applied in the augmented space and the averaging is performed over all the possible configurations of the disordered alloy to obtain the EPIs. The present treatment is based on the article by Inden and Pitsch (1991). Consider a crystalline system with N lattice sites and having M constituents. These constituents can be atomic species like A, B, C (or can be “vacancies” also) such that i Ni = N , where Ni is the number of atoms of type i. A particular distribution of these constituents on the lattice sites defines a configuration. A configuration is specified by site operators, n (n = 1 2 N ), which may take
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Phase Transformations: Titanium and Zirconium Alloys
values (numerical labels) i (i = 1 2 M), corresponding to the occupation of the site n by the constituent i. For a M-component system, the site operator can take the values M − 1 0 −M − 1, e.g., for a binary system, n corresponding to the lattice site n can take the two values ±1, depending on whether the site n is occupied by component A (+1) or by component B (−1). Any configuration is then specified by a N -dimensional vector = 1 2 N . In total, there are M N different configurations, as each of the N sites can be occupied in M different ways. Any function of n (including n itself) is called a configurational variable. A second operator pni , called the site occupation operator, which allows to count the number of sites n, occupied by the same type of atom i, for taking averages, is defined as follows: pni
=
1 0
if an atom of type i occupies site n otherwise
The cluster approximation: It is much more convenient to consider the configurations of much smaller units called clusters in place of the N -point lattice which is quite large. A cluster is defined by a set of lattice points 1, 2, …, and a configuration on this cluster is given by = 1 2 (Figure 5.3). On a -site cluster there are, in total, M M M N ) configurations. The configurations of the N -point system can then be classified into groups with the same number of clusters N with the configuration . The probability, , of a -point cluster to occur in a configuration is given by = N /N
(5.14)
b a
c
d
Figure 5.3. The irregular tetrahedron (IT) cluster (abcd) approximation for the bcc lattice.
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395
where N is the number of equivalent -site clusters contained in the system. These fractions are called cluster probabilities or reduced density matrices. They specify the configuration in the -point cluster approximation. The thermodynamic functions derived by the CVM depend on the probabilities of this cluster and also on the probabilities of all the subclusters which can be derived from that of the largest cluster by partial summations. The correlation functions: We characterize the atomic configuration in terms of correlation functions (s) by considering averages over equivalent clusters. An -point ( ≤ ) correlation function, i , for the cluster type i, can be defined as i =
1 i
N
1 2 · · ·
(5.15)
where the summation is taken over all the possible configurations of the cluster of type i. These correlation functions, for a perfectly ordered state, can be determined by inspection. However, these correlations serve as variables with respect to which the free energy of the system is minimized to arrive at the equilibrium state of order (or partial order). In order to determine the stability of a configuration at T = 0 K and in the equilibrium with respect to exchange of atoms, one has to minimize the internal energy, E G T V !A !B , in the grand canonical scheme, given as a function of temperature (T ), volume (V ) and chemical potential (!i ). The grand canonical internal energy (E G ) is obtained from the canonical energy, E, by its Legendre transformation: E G T V !A !B = ET V NA NB +
M
!i Ni
(5.16)
i=1
Generally, one considers a constant volume system and the configurational part of the internal energy is separated. The configurational energy, EcG , in the simplest approach, is expressed in terms of pairwise interactions (Vij ) up to an arbitrary coordination shell as 1 k k Z Vij EcG = N 2 k ij
ij 0k + N
!i
i 0
(5.17)
i
where Zk is the coordination number for the kth coordination shell and position “0” stands for any position in the crystal taken as the origin. The CVM entropy (S) is evaluated according to S = kB ln , where is the number of arrangements which can be formed for given values of the correlation
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Phase Transformations: Titanium and Zirconium Alloys
functions. In the CVM, we take correlations only up to the basic cluster size () and therefore, the thermodynamic probability, , is the number of possible arrangements of this basic cluster, corrected to take account of the overlapping clusters. We will not derive the expression for the CVM entropy here. Interested reader is referred to some of the excellent articles mentioned above. In brief, the CVM entropy takes the form S = −kB N
=1
m a
ln
(5.18)
where the sum runs over all the subclusters, , of the basic cluster, . The number of -clusters per lattice point, m , and the Kikuchi–Barker coefficients, a , are to be derived by geometrical considerations of the lattice. The cluster probabilities,
, can be expressed as
1 (5.19) v
= 1+ 2
≤ where the summation is over all the subclusters, , of the cluster, ; v are a sum of -order products of spin variables , the structure of which depends upon the symmetry of the cluster in question. As an example of what has been discussed above, we briefly present here the CVM treatment for the equiatomic B2 (CsCl type) structure which is a bcc-based superstructure (see Figure 5.1). The B2 structure has two sublattices corresponding to the corner and the body-centred positions, respectively. The cluster approximation selected is irregular tetrahedron cluster as shown in Figure 5.3. Let us represent the basic cluster tetrahedron by (abcd), denoting the four vertices. In a perfectly ordered B2 structure, the cluster sites (a) and (b) will be occupied by type “A” atoms ( 1 = 1) and sites (c) and (d) will be occupied by type “B” atoms ( 2 = −1). Now the enumeration of different subclusters of this basic cluster gives: (1) (2) (3) (4) (5) (6) (7) (8)
point cluster (type 1): a/b point cluster (type 2): c/d NN pair cluster (type 1): (ac), (bc), (ad) and (bd) NNN pair cluster (type 2): (ab) NNN pair cluster (type 3): (cd) isosceles triangular clusters (type 1): (abc), (abd) isosceles triangular clusters (type 2): (acd), (bcd) irregular tetrahedron cluster: (abcd).
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397
The subcluster probabilities corresponding to the above clusters are then given as (here we have used short notations: i ≡ a , j ≡ b , k ≡ c and l ≡ d ), 1 1
= 21 1 + i1
2 1
= 21 1 + k1
1 2
= 41 1 + i1 + k1 + ik2
2 2
= 41 1 + i + j1 + ij2
3 2
= 41 1 + k + l1 + kl2
1 3
= 18 1 + i + j1 + k1 2 + ik + jk2 + ij2 + ijk3
2 3
= 18 1 + j1 1 + k + l1 + jk + jl2 + kl2 + jkl3
1 4
=
1
2
1
2
1
2
1
2
3
1
1
2
2
1
3
1
2
1 2 1 2 3 1 1 + i + j1 + k + l1 + ik + il + jk + jl2 + ij2 + kl2 16
+ ijk + ijl3 + ikl + jkl3 + ijkl4 1
2
1
On substituting the above expressions for cluster probabilities in the CVM free energy expression (Eqs. 5.17 and 5.18), one obtains free energy (F = E − TS) as a function of T , !i s and set of eight independent correlation functions with respect to which the F is to be minimized. 5.2.3 The ground states of the Lenz and Ising model The determination of ground states of the Lenz and Ising model is much easier than evaluating the free energy of a system at finite temperatures. Many exact results have been obtained (Allen and Cahn 1972, de Fontaine 1979, Sanchez et al. 1982, Kanamori and Kaburgi 1983, Ducastelle 1991) when the interactions are short range. This is based on a method of linear inequalities that allows the determination of the most stable ordered structures as functions of the concentration and the strength of the pair interactions. In practice, it is very difficult to include more than fourth or fifth nearest neighbour interactions. If the range of interactions is finite, only a finite number of ordered structures can be found. The correlation functions characterizing the atomic configurations can only vary within a restricted range of numerical values due to certain consistency conditions. This restricted range of values is called the existence domain or the
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Phase Transformations: Titanium and Zirconium Alloys
configuration polyhedron outside which the values do not define actually existing atomic configurations. The dimensions of the configurational space depend either on the number of correlations or equivalently on the size of the largest cluster. These existence domains are most useful for an analysis of ground states. We give examples of the calculation of existence domain for the bcc and hcp lattices under the first and second nearest neighbour pair interactions. The first step is to select a basic cluster which determines the dimension of the configuration space. We select the irregular tetrahedron abcd (Figure 5.3) as our basic cluster which accounts for up to the second nearest neighbour pair interactions. The configurational variables are then the point correlation function x1 = 1 ; pair correlation functions, x2 = 1 2 x3 = 1 3 ; 3-point correlation function, x4 = 1 2 3 ; and the tetrahedron correlation function, x5 = 1 2 3 4 . ijkl Because of the constraint that 4 ≥ 0, we are led to the following consistency relations expressed in the matrix form as (Ducastelle 1991, Inden and Pitsch 1991) ⎡
⎤ 1 4 4 2 4 1 ⎢ 1 2 0 0 −2 −1 ⎥ ⎢ ⎥ ⎢ 1 0 0 −2 0 1 ⎥ ⎢ ⎥ ⎢ 1 0 −4 2 0 1 ⎥ ⎢ ⎥ ⎣ 1 −2 0 0 2 −1 ⎦ 1 −2 4 2 −4 1
⎤ 1 ⎢ x1 ⎥ ⎢ ⎥ ⎢ x2 ⎥ ⎢ ⎥≥0 ⎢ x3 ⎥ ⎢ ⎥ ⎣ x4 ⎦ x5 ⎡
If these inequalities are taken as equalities, these equations define a simplex in the five-dimensional configurational space. All the vertices correspond to existing atomic configurations. It is well known that the minimum of energy will occur for atomic configurations corresponding to the vertices of the configuration polyhedron. Its projection onto the subspace (x1 x2 x3 ) defines the configuration polyhedron (Figure 5.4). The coordinates of the vertices are given in Table 5.1. The corresponding crystallographic structures are shown in Figure 5.1 and the pertinent crystallographic data are tabulated in Table 5.2. Similarly ground state analysis can be carried out for the hcp lattice under the octahedron–tetrahedron cluster approximation (Figure 5.5), which includes up to third nearest neighbour pair interactions and gives rise to 14 correlation functions defined as follows: x1 = 1 , x2 = 1 2 , x3 = 1 6 , x4 = 3 6 , x5 = 1 6 7 , x6 = 1 3 6 , x7 = 1 2 6 , x8 = 2 3 4 , x9 = 1 2 3 4 , x10 = 1 2 3 6 , x11 = 1 2 6 7 , x12 = 2 3 6 7 , x13 = 1 2 3 6 7 , x14 = 1 2 3 6 7 8 . The ground state superstructures for the hcp lattice corresponding to the vertices of the 14-dimensional configuration polyhedron are listed in Table 5.3. A few of them are also depicted in Figure 5.5.
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399
x3
x2
1 3
x1
2
4
Figure 5.4. Projection of the configuration polyhedron corresponding to the irregular tetrahedron of the bcc lattice. The vertices designated correspond, respectively, to (1) B2, (2) D03 , (3) D03 and (4) B32 ground state ordered structures.
Table 5.1. Coordinates of the vertices of the configurational polyhedron corresponding to the minimum in energy under the first and second nearest neighbour (NN) pair approximation. Configuration Pure A (A2) A3 B, AB3 D03 ) AB (B2) AB (B32) Pure B (A2)
x1
x2
x3
x4
x5
1 ± 21 0 0 −1
1 0 −1 0 1
1 0 1 −1 1
1 ± 21 0 0 −1
1 −1 1 1 1
For the fcc lattice, the octahedron–tetrahedron cluster approximation (Figure 5.6) takes into account up to the second nearest neighbour pair interactions. The configurational polyhedron corresponding to the octahedron is a simplex in the nine-dimensional space corresponding to all possible multiatom interactions. All subclusters equivalent in the fcc lattice are also equivalent in the octahedron. If we project this simplex onto the space (x1 , x2 , x3 ) corresponding to point, first and second nearest neighbour correlation functions, only the 10 vertices survive. For the tetrahedron inequalities the corresponding polyhedron is a prism. This prism
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Phase Transformations: Titanium and Zirconium Alloys
Table 5.2. Crystallographic data for various bcc-based ground state superstructures under the first and the second NN pair approximations. Structures bcc-based
Compositional formulae
Space group symbol (no.)
Wyckoff positions
Multiplicity
Transformed basis
A (a)
2
a = a100
A (a) B (b)
1 1
a1 = a100 a2 = a010 a3 = a001
A2
A
B2
AB
¯ (225) Im3m ¯ (221) Pm3m
B32
AB
¯ (227) Fd3m
A (a) B (b)
8 8
a1 = a200 a2 = a020 a3 = a002
D03
A3 B
¯ (225) Fm3m
A (c) A (b) B (a)
8 4 4
a1 = a200 a2 = a020 a3 = a002
The transformed basis vectors are given in terms of the vectors of the bcc lattice.
(a) Unit cell
(f) B19
(b) Octahedron (c) Tetrahedron
(g) D019
(d) CuPt
(e) A2B
(h) D0a
Figure 5.5. A few ordered structures which are the ground state superstructures under the first and the second nearest neighbour interaction approximation for the hcp lattice. Large circles are in {00n} planes and small circles are in {00n + 21 } planes. (a) Unit cell of hcp structure, (b) octahedron, (c) tetrahedron, (d) CuPt type, (e) A2 B, (f) B19, (g) D019 and (h) D0a .
cuts the octahedron polyhedron. The final polyhedron is shown in Figure 5.7. It has 15 vertices out of which only 9 vertices are really distinct. These vertices are described in Table 5.4 and the corresponding superstructures are also shown in Figure 5.8; relevant crystallographic data are listed in Table 5.5.
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Table 5.3. Ground state structures of the hcp lattice corresponding to the vertices of the 14-dimensional configurational polyhedron. Concentration
Prototype
A AB AB AB A3 B A3 B A2 B A2 B A2 B A5 B A4 B3
Mg AuCd CuTe WC Ni3 Sn -Cu3 Ti
Designation A3 B19 Bh D019 D0a
B2 NdRh3 Si2 Zr B2 NdRh3
C49
b a c f
e
d g
Figure 5.6. The tetrahedron (abcd)–octahedron (bcdefg) (TO) cluster approximation for the fcc lattice.
5.2.4 Special point ordering A wide range of phenomena related to order–disorder and magnetic transitions can be explained using the symmetry properties of the pair potentials (Vij ). One such example is the Landau theory of continuous phase transitions. The free energy of an alloy system for different wave vectors corresponding to appropriate concentration fluctuations has been shown by Khachaturyan (1978) and de Fontaine (1979) to be a useful parameter in the study of the instabilities associated with ordering phase transitions. The roles that symmetry and wave vectors, terminating at special points of the reciprocal lattice, play can be illustrated as follows. If a symmetry element of the space group in k-space is located at a point h, the vector representing the gradient, h Vh, of an arbitrary potential energy function Vh at that point must lie along or within the symmetry element. If two or more symmetry elements intersect at the point h, one must necessarily have h Vh = 0
(5.20)
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Phase Transformations: Titanium and Zirconium Alloys x3
1 2
x2 8
3 4 7 5 x1 6
9
Figure 5.7. Dual configurational polyhedron for the fcc lattice under the tetrahedron–octahedron cluster approximation. The vertices designated correspond, respectively, to (1) pure A, (2) L12 , (3) D022 , (4) A5 B, (5) Pt2 Mo (Immm), (6) A2 B, (7) A2 B2 , (8) L10 and (9) L11 . Table 5.4. Ground state structures of the fcc lattice corresponding to the vertices of the nine-dimensional configurational polyhedron. Configurations
x1
x2
x3
A
1 2 3 1 2 1 2 1 3
1 1 3
1 1 3
0
1
A5 B L12 D022 Pt2 Mo
0 1 9
2 3 1 9
A2 B
1 3
0
1 3
L10
0
1 3
1
A2 B2
0
1 3
1 3
L11
0
0
1
Ordering in Intermetallics
L12
D022
L10
A2B2
403
L11
A2B (Immm )
Figure 5.8. fcc-based ordered intermetallic phases which are the ground state superstructures under the first and the second NN pair approximation.
At these so-called special points, the potential energy function, Vh, represents an extremum regardless of the choice of the pair interaction energies. Thus special points play an important role in the search for lowest energy ordered structures. The special points are, however, quite insufficient for a complete description of the ground states which can be treated conveniently in the real space as demonstrated in the last section. Among the most important applications of special points is the study of the onset of short wavelength instabilities in alloys. Such instabilities may take place at a high supercooling below a first order transition temperature, frequently bearing no symmetry relation to either the low or the high temperature phases. This particular metastable ordering mechanism, known as spinodal ordering, has been observed in several alloy systems (e.g. Ni–Mo alloys). The points which differ by a lattice vector of a reciprocal lattice are considered equivalent. In the case of simple structures with a single atom per unit cell, it is sufficient that two symmetry elements intersect at special points. These special points are listed in crystallographic tables. They are always located at the surface of the Brillouin zone. The “star” of a special point vector k is obtained by applying all the rotations and rotation-inversion of the space group on the vector k. All these vectors of a star are also considered equivalent.
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Phase Transformations: Titanium and Zirconium Alloys
Table 5.5. Crystallographic data for various fcc-based ground state superstructures under the first and the second NN pair approximation. Structures bcc-based
Compositional formulae
Space group symbol (no.)
Wyckoff positions
Multiplicity
Transformed basis
A1
A
¯ (225) Fm3m
A (a)
4
a = a100
L10
AB
P4/mmm (123)
A (a) B (d)
1 1
a1 = 21 a110 a2 = 21 a110 c = a001
L11
AB
¯ (166) R3m
A (a) B (b)
1 1
a1 = 21 a110 a2 = 21 a101 c = a222
A2 B2
A2 B2
I41 /amd (141)
A (a) B (b)
4 4
a1 = a010 a2 = a001 c = a200
Pt2 Mo
A2 B
Immm (71)
A (i) B (a)
4 2
a = 21 a110 b = a001 c = 21 a330
A2 B
A2 B
B2/m (12)
A (i) A (c) B (a)
4 2 2
L12
A3 B
¯ (221) Pm3m
a = 21 a552 b = a220 c = 21 a110
A (c) B (a)
3 1
a = a100
D022
A3 B
I4/mmm (139)
A (d) A (b) B (a)
4 2 2
a1 = a010 a2 = a001 c = a200
Ni4 Mo type
A4 B
I4/m (87)
A(h) B(a)
8 2
a1 = 21 a310 a2 = a130 c = 002
The transformed basis vectors are given in terms of the vectors of the fcc lattice.
5.2.4.1 BCC special points The special points of the bcc structure are located at the points ", H, P and N of the Brillouin zone (Figure 5.9). The corresponding stars are given in Table 5.6 (Ducastelle 1991). Here we have a single structure per star. The domain of stability in the case of nearest neighbour and next nearest neighbour interactions is shown in Figure 5.10. The star 21 21 0 does not appear which is understood by noting that the corresponding structure has not been obtained as a possible ground state which requires up to third NN interactions. The only ground state that does not appear is the D03 structure which can be constructed by a superposition of 100 and 21 21 21 waves.
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kZ
H Γ
kY
N
H
kX
Figure 5.9. The Brillouin zone for the bcc lattice where various special points have been marked. Table 5.6. The special points and stars of the bcc structure. k-Vector star
Members
000
000
"
100
1 1 1
100 1 1 11 1 1
H
B2
P
B32
N
AB
2 2 2
1 1 0 2 2
2 2 2
Brillouin zone points
2 2 2
1 1 1 1 1 1 0 202 02 2 2 2 1 1 1 1 1 1 0 202 02 2 2 2
Ordering structure
V2 〈
111 〉 222 (a)
(b)
V1 〈000〉
〈100〉
Figure 5.10. The domains of stability of the stars for the bcc lattice in the first (V1 ) and the second (V2 ) nearest neighbour interaction plane. The lines designated (a) and (b) correspond, respectively, to (a) 2V1 − 3V2 = 0 and (b) 2V1 + 3V2 = 0.
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Phase Transformations: Titanium and Zirconium Alloys
5.2.4.2 HCP special points The analysis of the hcp structure is much more complex as it has two atoms per unit cell and the space group of the structure (P63 /mmc) is not symmorphic. There are six special points for the hcp lattice, viz., " (000), M ( 21 00), K ( 13 31 0), L ( 21 0 21 ), H ( 13 31 21 ) and A (00 21 ). Using symmetry arguments, it can be shown (Ducastelle 1991) that some of these special points are irrelevant for the hcp structure. The three relevant special points are ", M and H (Table 5.7). At the points " and M, there are two extremes and we must combine the concentration waves in the two sublattices in two different ways. These combinations are obtained from the eigenvectors of the potential energy matrix V k. The result is that at " and M, we have simple modes which may be called acoustic and optical modes as in the theory of lattice vibrations. If we let the origin of sublattice 1 be at (000) then the origin of sublattice 2 is at 23 13 21 . The stars are tabulated in Table 5.7 and the structures corresponding to the pure modes are described below (Figure 5.11). Table 5.7. Important special points and stars of the hcp structure. k-Vector star
Members
000
1 00 2 1 1 1
000 1 1 1 1 00 0 2 0 2 2 0 2 1 1 11 1 1
332
33 2
Brillouin zone points " M H
332
V2
(b)
〈000〉 ; 〈 1 00〉 2 II I
(a)
〈 〈000〉
11 0〉 44
V1
I 1 〈 00〉 2 II
Figure 5.11. The domains of stability of the stars for the hcp lattice in the first (V1 ) and the second (V2 ) nearest neighbour interaction plane. The lines designated correspond, respectively, to (a) V1 − 2V2 = 0 and (b) V1 + V2 = 0. In the hatched region, Vh is minimum at 41 41 0 , which is not a special point.
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Star 000 : The acoustic mode 000 I corresponds to the segregation process and the optical mode 000 II corresponds to an alternate stacking of A and B triangular planes (CuPt-type structure) Star 21 00 : Here we regard 21 00 I and 21 00 II as modes corresponding to the CuPt type and MgCd (B19) type structures. Also we obtain the D019 structure by considering the acoustic combination of k1 = 21 00, k2 = 0 21 0 and k3 = 21 21 0. There is no structure corresponding to the optical mode. Star 13 31 21 : There is no structure associated with this star. In the case of nearest neighbour interactions, V (k) is never a minimum at H. When the second nearest neighbour interaction is introduced, the absolute minimum of V (k) in the range 0 < V2 < 21 V1 occurs at the point [ 41 41 0 ] that does not correspond to a special point but rather is a “non-special” point. 5.2.4.3 FCC special points The special points of the fcc structure are located at the points ", X, W and L of the Brillouin zone (Figure 5.12). The corresponding stars are given in Table 5.8 (Ducastelle 1991). The domain of stability in the case of nearest neighbour and next nearest neighbour interactions is shown in Figure 5.13. 5.2.5 Concomitant clustering and ordering The thermodynamic analysis of concomitant clustering and ordering was presented by Kulkarni et al. (1985) and Soffa and Laughlin (1988, 1989). The latter theoretically studied various barrierless reaction mechanism for the transition
kZ
L Γ
X kX
kY W
Figure 5.12. The Brillouin zone for the fcc lattice where various special points have been marked.
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Phase Transformations: Titanium and Zirconium Alloys Table 5.8. The special points and stars of the fcc structure. k-Vector star
Members
000
000
"
100
1 120
100 010 001 1 1 1 1 2 0 2 01 01 2 1 21 0 21 01 01 21 0
X
L12 , L10
W
A2 B2
L
L11
1 1 1 2 2 2
1 1 1 2 2 2
1 1 1 2 2 2
1 1 1 2 2 2
1 1 1 2 2 2
Brillouin zone points
Ordering structure
V2 〈
111 〉 222 (a)
(b) 1 〈1 0〉 2 〈000〉
V1 〈100〉
Figure 5.13. The domains of stability of the stars for the fcc lattice in the first (V1 ) and the second (V2 ) nearest neighbour interaction plane. The lines designated (a) and (b) correspond, respectively, to (a) V1 − 2V2 = 0 and (b) V1 + V2 = 0.
“fcc disordered solid solution → ordered state having L12 structure” by employing the graphical thermodynamics approach and the SCW model. Soffa and Laughlin have expounded the idea of ordering and clustering reactions in constructing the schematic free energy versus composition plots which illustrate different situations. The stability/instability of a solution with respect to change of some parameter, e.g. an appropriate order parameter or the site occupancy, is determined by the second derivative of the free energy (F ) with respect to that variable. A solution is considered stable, critical or unstable according to whether F > 0, F = 0 or F < 0, respectively. The equation F = 0 describes the point at
Ordering in Intermetallics
409
which an instability develops in the solution. While the presence of an instability can be determined by examining the sign of the second derivative of free energy with respect to appropriate order parameter, the equilibrium phase boundaries are given by the principle of common-tangents on the free energy–concentration curves for the ordered and the disordered phases. The instability lines on the temperature (T )–concentration (c) plane are defined as follows: Ti− : the line below which the solid solution is unstable with respect to congruent ordering Ti+ : the line above which the ordered solid solution becomes unstable with respect to spontaneous congruent disordering Tcs : the line below which spinodal clustering instability develops only after the system undergoes ordering to a certain extent (conditional spinodal) Ts : the line below which the disordered phase becomes unstable with respect to spinodal clustering. The two equilibrium phase boundaries (Teq s) define the ordered, disordered and the two-phase regions. The four instability lines Ti− , Ts , Ti+ and Tcs are categorized in terms of which phase forms the instability: the disordered () or the ordered () phase. Accordingly, Ti− and Ts are the instabilities of the disordered phase, being the points at which the disordered phase becomes unstable with respect to ordering and clustering, respectively. The other instabilities, T+i and Tcs , are the points at which the ordered phase becomes unstable with respect to spontaneous disordering and clustering, respectively. The computation of the disordered phase instabilities is straightforward and can be done exactly, while the ordered phase instabilities are somewhat more complex and can be determined from the graphical thermodynamic approach. The relative positions of these instability lines, as schematically illustrated in the phase diagram (Figure 5.14), identify the conditions under which different transformation sequences are possible from thermodynamic considerations. In the cases where barrierless mechanisms are operative, the phase transformation mechanism will be determined by which of the instability lines have been crossed on quenching. Much research has been dedicated to the study of complex instability reactions that result in simultaneous ordering and phase separation in alloys. Laughlin and Soffa compared these results (Soffa and Laughlin 1989) to phenomena observed in several systems involving → L12 phase transition. It is possible for a miscibility gap to develop in the L12 phase at a temperature where the curvature of the free energy is negative. It is equally possible to have a miscibility gap develop in the disordered phase by the development of a negative
410
Phase Transformations: Titanium and Zirconium Alloys
Ti + B′
B
Ti –
C
Ts
Temperature
Temperature
A D Ts E Ti –
Tcs
Composition
Composition
(a)
(b)
Figure 5.14. The instability diagrams which schematically illustrate the relative positions of all the four instability lines (Ti− , Ti+ , Tcs and Ts ) superimposed on the phase diagram. The two equilibrium phase boundaries are drawn with thick lines. (a) shows ordering instability line (Ti− ) at higher temperatures compared to clustering instability line (Ts ) while in (b) Ts is above Ti− . These lines segment the phase diagram into several domains in which different reaction mechanisms are operative. Domain A: → + (nucleation and growth). Domain B: → spinodal ordering → + . Domain C: → simultaneous ordering and clustering. Domain B : → spinodal ordering → spinodal clustering. Domain D: → spinodal clustering → spinodal ordering within solute-enriched regions → + . Domain E: simultaneous ordering and clustering.
curvature along that branch of the free energy. Depending on the temperature at which these develop, a variety of phase diagrams can be produced. They gave results for both first order and second order ordering reaction cases. In all, five different phase diagrams (shown in Figure 5.15; Table 5.9), along with the instability lines that would be obtained in each case are: (a) the classical phase separation case, (b) the first-order ordering case, (c) a monotectoid case, in which a miscibility gap develops in the ordered phase, (d) one in which a miscibility gap develops in the disordered phase that is metastable with respect to the ordering reaction and (e) a syntectoid case, in which a miscibility gap develops in the disordered phase at high temperatures and an ordering reaction occurs at low temperatures. An alloy of composition co is in a single phase disordered region (Figure 5.15(a)) at higher temperatures (above the miscibility gap maximum) and within a
Ordering in Intermetallics
C1 A
X
α
α2
Coherent spinodal α1 + α2 C0
II + III
Ti
+
α
β
Ti
–
V IV
C2
Composition
Ti
+
Composition
Composition
(a)
II + III – Ti Vβ
IV
Ax By B
Ax By
I
I
Temperature
α1
α Miscibility gap I Chemical spinodal
Temperature
Temperature
α0
411
(c)
(b) T cs
X
I
Temperature
II + III
Ti
+
β
IV + V VI + VII
–
VII + IX
Ti
+
β
Ti
–
VIII VI
Ax By
Composition
(d)
Ti
Temperature
α
I
VI
Ax By Composition
(e)
Figure 5.15. The phase diagrams produced by systematically varying the curvatures of the free energy curves for the ordered and disordered phases. The phase diagram (c) is not observed in the SCW model. (a) A simple phase separation case, (b) a simple ordering diagram, (c) monotectoid diagram, (d) ordering with metastable miscibility gap diagram and (e) a syntectoid case. The stability regions have also been shown (see Table 5.9, after Soffa and Laughlin 1989).
two-phase ordered region at lower temperatures. The two phases differ from one another only in composition but have identical crystal structure, i.e. isostructural decomposition, whereas in the cross-hatched region in Figure 15.5(b), the homogeneously ordered phase is unstable with respect to phase separation. The Ti+ is the instability line with respect to disordering upon heating. In Figure 15.5(c), the cross-hatched region is the region of thermodynamic instability with respect to phase separation. The Laughlin and Soffa approach has the advantage of simplicity and generality; it does not rely on the assumption of any particular model. It has the disadvantage that the thermodynamic consistency is not built-in, i.e. there are consistency rules for construction of free energy curves that have not been examined.
412
Phase Transformations: Titanium and Zirconium Alloys Table 5.9. The characteristic barrierless reactions operative in different stability regions as predicted by the SCW model (Simmons 1992). Region
Reaction
Region
Reaction
I III
No reaction No reaction
II IV
V
→
VI
VII
→
VIII
IX
→ + →
X (T > Tconsolute ) (T < Tconsolute )
No reaction → → + → → → + → → + → → + → + →
≡ disordered phase and ≡ ordered phase.
5.2.6 A case study: Ti–Al system As an illustration of what has been discussed so far, we will present here a summary of the results obtained on the Ti–Al alloy system using the first principles configurational thermodynamic approach by Asta et al. (1992, 1993). The Ti–Al system is an incoherent system as the different stable/metastable ordered phases that appear across the full composition range are the superstructures of either the hcp (corresponding to -Ti) or the fcc (corresponding to Al) lattice. Prior to the work of Asta et al., several ab initio electronic structure calculations had been carried out on the Ti–Al system (Fu 1990, Hong et al. 1991) to study the energetics as well as the structural and mechanical properties of stable and metastable stoichiometric compounds at 0 K. But the effect of variation of temperature and composition on these properties was not studied. Asta et al. (1992) have used the local density based full potential linear muffin tin orbital (FP-LMTO) method to calculate the ground state cohesive properties of all the stable and metastable phases under the tetrahedron–octahedron cluster approximation for both fcc and hcp lattices (Table 5.10). The ECI were calculated using the Connolly–Williams inversion method and the CVM free energies were used to determine the phase stabilities as functions of temperature, composition and order parameter to arrive at the Ti–Al phase diagram depicting the stability regimes of various stable and metastable phases and the pertinent transition temperatures. Their results showed that of all the structures considered, only hcp-Ti, fcc-Al, D019 -Ti3 Al, L10 -TiAl and D022 -TiAl3 were energetically stable in agreement with the experimental results. The remaining phases were found to be metastable. Of
Ordering in Intermetallics
413
Table 5.10. Cohesive properties of some of the intermetallics of Ti–Al system. System
Lattice parameter (a, (c/a)) (Å)
Bulk modulus (GPa)
Al
3.985 4.046 2.864 2.951 5.649 5.761 3.935 4.005 3.790 3.848
84 82 106 105 126 120–145 128 160–176 118 —
Ti Ti3 Al TiAl TiAl3
(1.000) (1.000) (1.617) (1.588) (0.809) (0.807) (1.012) (1.016) (2.240) (2.234)
Formation energy (kJ/mol) 0.000 0.000 −287 −253 −420 −3650 −419 −307 to −374
In the first row of each column are the calculated results as obtained by Asta et al. using the FP-LMTO method; in the second row are the experimental results.
these metastable phases, L12 -TiAl3 was found to be in close competition with D022 -TiAl3 phase. Further, the hcp disordered alloys were noted to be stable as compared to fcc disordered solid solutions up to 50 at.% Al. The Al-rich and equiatomic compounds were shown to undergo large structural relaxations even though the size mismatch between the atomic constituents was small. The calculated CVM phase diagrams for the fcc- and hcp-based Ti–Al alloys showed that ordered fcc-based alloy phases are more stable than hcp-based alloy phases with respect to corresponding disordered solid solution. Consequently there is a stronger tendency for atomic ordering in fcc-based alloys which was attributed to the differences in the electronic structures between fcc- and hcp-based alloy phases. At Al-rich compositions, the L10 and the D022 phases were found to be strongly ordered to well above their experimentally observed melting temperatures. Moreover the phase field of D022 -structured TiAl3 was predicted to be much narrower than that for L10 -TiAl which is in agreement with the experimental phase diagram. The theoretical transition temperature for the D019 -Ti3 Al phase was about 600 C too high; this discrepancy was attributed to the neglect of contributions from vibrational and electronic excitations to the free energy. The phase relationships in the Ti–Al system have been subjected to many modifications in the recent past, partially because of differences in the impurity levels in the alloys studied and in the interpretation of the observed microstructures. There have been major discrepancies concerning the extension of the (-Ti) field with respect to temperature. These modifications are also reflected in the various calculations published in the literature. Kauffmann and Nesor (1978) used very simple descriptions in their calculations: the disordered solution phases were
414
Phase Transformations: Titanium and Zirconium Alloys
described as quasiregular solutions and the intermetallic compounds Ti3 Al, TiAl and TiAl3 were assumed to be stoichiometric. Within the limitations of the model descriptions used, the general features of the then accepted phase diagram could be reproduced. Murray (1987) used the subregular solution model to describe the disordered phases and assumed TiAl to be stoichiometric. Gros et al. (1988) calculated the Ti-rich part of the system. The liquid and (-Ti) phases were described as quasiregular solutions and (-Ti) and Ti3 Al were described in terms of the sublattice model as one single phase undergoing an order–disorder transition. The calculated partial Gibbs free energies of Al and Ti in (-Ti) agreed within 5–12% of the experimental values. Within the constraints used, reasonable agreement was obtained for the (-Ti) and Ti3 Al phase boundaries. More complete calculations of the Ti–Al system, taking into account the entire phase diagram and the ranges of homogeneity of the ordered intermetallic phases, were carried out later by Murray (1988). He described the disordered solution phases as quasiregular solutions, used the Bragg–Williams approximation for the ordered Ti3 Al and TiAl compounds and assumed TiAl3 to be stoichiometric. In order to improve the agreement between the calculated and experimental phase diagrams, Murray found it necessary to relax the conditions given by the Bragg– Williams approximation for the Gibbs free energy for these compounds. Later, Hsieh et al. (1987) and Lin et al. (1988) included the TiAl2 and Ti3 Al7 phases in the calculated phase diagram by using a quasisubregular solution model. Recently, a thorough thermodynamic assessment of the Ti–Al system has been undertaken by Zhang et al. (1997) in which three different analytical descriptions have been used to describe the three different types of phases occurring in the system: the stoichiometric compounds, the disordered solution phases and the ordered intermetallic compounds which have homogeneity ranges. A least square technique has been used to optimize the thermodynamic quantities pertinent to the analytical description, using the experimental data available in the literature. We will give here a brief account of the approach used by Zhang et al. (1997) for the assessment of the Ti–Al phase diagram in order to illustrate the semi-empirical methodology adopted. For an accurate description of these phases, the Gibbs free energy (G) must be expressed as an analytical function of the thermodynamical variables of interest, viz., composition, temperature and pressure. The Bragg–Williams model is inadequate to account for the phase reactions/transitions in this system because of its inability to account for local correlations or short-range ordering effects. Zhang et al., therefore, have used different models for the ordered and disordered phases. They have considered nine phases in their calculations: the disordered solution phases, liquid, -Ti, -Ti and Al; and the ordered intermetallic compounds Ti3 Al, TiAl, TiAl2 , Ti2 Al5 and TiAl3 . The TiAl2 and Ti2 Al5 phases have been assumed
Ordering in Intermetallics
415
to be stoichiometric, while the wide ranges of homogeneity exhibited by Ti3 Al and TiAl have been taken into account in the analytical model. The free energies corresponding to the stoichiometric compounds TiAl2 and Ti2 Al5 are described by o o GoTi + xAl GoAl + Gf G = xTi
(5.21)
where xio are the mole fractions of component i and Goi are the respective reference states. The term Gf represents the Gibbs free energy of formation. These G parameters are either constant or are linear functions of temperature. The disordered solution phases, liquid, -Ti, -Ti and Al are described as random mixtures of Ti and Al: xi ln xi + xTi xAl Go + G1 xTi − xAl (5.22) G = xTi GoTi + xAl GoAl + RT i
where Go and G1 are the coefficients of the terms of the excess Gibbs energy. This model is called the quasisubregular solution model. The ordered intermetallic compounds Ti3 Al, TiAl and TiAl3 exhibit an appreciable range of homogeneity and are described by the sublattice or Wagner–Schottky model. These compounds are considered to consist of two sublattices which are occupied by Ti and Al atoms only in the perfectly ordered state. The partially ordered regions are described by assuming the formation of substitutional atoms on each sublattice. The analytical description is 1 2 1 o o ni ln n1i + RT ni ln n2i − RT Ni ln Ni1 G = xTi GTi + xAl GAl + RT i
−
i
Ni2 ln Ni2 + Gf +
i
i
n2i G2i + n1Ti n1Al G1o + G11 n1Ti − n1Al
i
+ n2Ti n2Al G2o + G21 n2Ti − n2Al + n2Ti n1Al G12 Here we have xi = n1i + n2i and N k = nkTi + nkAl where nki are the mole fractions of component i on sublattice k and N k are the site fractions of sublattice k. Gf is the free energy of formation of the perfectly ordered phase at the stoichiometric composition. Gko are the coefficients of polynomial interaction terms between atoms on the same sublattice and G12 is that between substitutional atoms on the different sublattices. The quantities nki are calculated by minimizing the Gibbs free energy for given xi . The optimization and calculation programmes developed by Lukas et al. (1982) have been used for these calculations of the Ti–Al system.
416
5.3
Phase Transformations: Titanium and Zirconium Alloys
TRANSFORMATIONS IN TI3 AL-BASED ALLOYS
5.3.1 → D019 ordering The 2 -phase based on the stoichiometry Ti3 Al has a D019 (hP8) structure and P63 /mmc symmetry. The structure is characterized by an atomic arrangement on the close packed (0001) planes (Figure 1.20) which ensures that Al atoms have only Ti atoms as their first near neighbour. A hexagonal stacking ABABAB of such planes produces the D019 structure. An examination of the symmetry rules for a second order ordering process in the context of → D019 transformation reveals that the first Landau–Lifshitz condition (symmetry elements of the product phase having a subset of that of the parent) is fulfilled. This means that replacement of atoms in an -lattice can produce a D019 structure. The D019 structure can be formed by introduction and amplification of a concentra tion wave with wave vector star of k 21 00 . This wave vector terminates at the special point of the reciprocal space fulfilling the second Landau–Lifshitz condition. The third condition, however, is not satisfied as the sum of three variants of the wave vector star (k1 + k2 + k3 ) results in a reciprocal lattice vector of the parent hcp structure. Therefore → D019 transformation is not a candidate for a second order ordering process. With a high degree of supercooling, at T< Ti , Ti being the instability, a short wave length concentration wave with k= 21 00 , can amplify at least to a limited extent to generate a partially ordered D019 structure. The formation of the Ti3 X precipitates having the D019 structure is seen in a number of binary and ternary alloy systems such as Ti–Al, Ti–Ga, Ti–Al–Ga, Ti–Al–Sn, Ti–Al–In. It is important to note that the mismatch between the lattice dimensions between the parent and the ordered Ti3 X precipitates are different in different alloy systems and correspondingly the morphology of precipitates varies from one system to the other. The extent of mismatch between the matrix and Ti3 X D019 precipitates in different systems are listed in Table 5.11. Table 5.11. Matrix–precipitate mismatch along a and c directions for Ti–Al alloys. Alloy system
Ti–Al Ti–Al–Ga Ti–Al–Sn Ti–Al–In
Matrix–precipitate mismatch Along a (%)
Along c (%)
0.83 0.99 0.20 0.58
0.35 0.51 0.15 0.21
Ordering in Intermetallics
417
5.3.2 Phase transformations in 2 -Ti3 Al-based systems The development of engineering alloys based on Ti3 Al (2 ) was initiated with the primary objective of ductilizing the brittle binary Ti3 Al intermetallic by the introduction of -stabilizing elements, particularly Nb. As mentioned earlier, Ti3 Albased alloys can be classified into different groups on the basis of the extent of additions of -stabilizers. The relative stabilities of three phases, namely, the 2 (D019 ), o (B2) and O (orthorhombic) phases, play a key role in determining the phase equilibria in these alloys. The thermal history, however, remains as the other major factor in controlling the mode and the sequence of phase transformations and the microstructure developed. The literature on Ti3 Al-based alloys is overwhelmingly dominated by that on ternary Ti–Al–Nb alloys. The trends in the phase transformations in Ti3 Al-based alloys are, therefore, discussed in this chapter with reference to the ternary Ti–Al–Nb system. Figure 5.16 shows (a) pseudobinary phase diagram (Ti–27.5% Al with increasing Nb) and (b) the isothermal section at 1173 K of the ternary phase diagram of the Ti–Al–Nb system (Banerjee and Rowe 1993). Phase fields in which a single phase, 2 , o or O, exists are shaded and two- and three-phase fields are marked. The composition ranges associated with the three classes of alloys, containing different levels of -stabilizers, which have been studied in detail, are indicated by circles, inscribed with the legends I, II and III. This 1173 K isotherm of the ternary phase diagram suggests that alloys belonging to class I consist of a mixture of the 2 - and o -phases while those belonging to class II made up of a mixture of the o - and O-phases. The class III alloys of intermediate compositions exhibit a three-phase (2 + o + O) microstructure.
30
Nb
10
βo + O II
50Ti Al 20
20
Al
O + βo O
40
1000
Al
O
10
βo
25
α2 + βo α2
Ti–27.5Al
60Ti
O + βo + α2
30
1100
900
Nb
III
βo (B2)
Al
α + βo α + α2
Nb
α
20
1200
Temperature (°C)
I βo + α2
α2
(b) 30
Atomic % Nb (a)
Figure 5.16. (a) Pseudobinary phase diagram (Ti–27.5% Al with increasing Nb) and (b) the isothermal section at 1173 K of the ternary phase diagram of the Ti–Al–Nb system.
418
Phase Transformations: Titanium and Zirconium Alloys
Since the compositions of all the three classes of alloys fall nearly along the line of 25 at.% Al, a pseudobinary section along the Ti3 Al–Nb line (Figure 5.16(a)) can conveniently be used to show the possible phase reactions in these alloys. A comparison can be drawn between this pseudobinary phase diagram and a binary phase diagram of Ti alloyed with a -stabilizing element (Figure 6.1). The similarity between them essentially arises due to their -isomorphous nature which restricts the composition range of the /2 phase field. Another point of similarity is reflected in the tendency for the formation of an orthorhombic structure at relatively high levels of -stabilizer addition. In the case of the binary Ti–X (X being a -stabilizer like V, Mo, Nb) system the orthorhombic phase ( , orthorhombic martensite) is an extension of the hcp -martensite. In comparison, the orthorhombic O-phase in the Ti3 Al–Nb phase diagram can be considered as an extension of the Ti3 Al (D019 ) phase with an orthorhombic distortion. This argument will be further elaborated when the crystallography of these phases is considered. The essential difference between the binary Ti–X and the pseudobinary Ti3 Al–Nb phase diagrams is due to the fact that there is a strong tendency towards chemical ordering in the latter case. In Ti3 Al-based alloys with low Nb contents, the terminal solid solution experiences a strong tendency to order into the D019 structure while alloys more enriched with Nb are under the influence of a strong tendency towards B2 ordering at elevated temperatures and O-phase ordering at lower temperatures. The presence of these ternary ordering tendencies is responsible for the introduction of additional phase reactions in this system. Having discussed the general tendencies related to phase stabilities in Ti3 Albased alloys, let us examine the experimental observations made on this system. Banerjee (1994a,b), in his excellent review, has constructed a diagram in which experimental observations on phase transformation processes in this system have been summarized. Figure 5.17 is essentially a reproduction of the same diagram in which the data reported in a number of research papers have been assimilated (Banerjee et al. 1988; Kestner-Weykamp et al. 1989, 1990; Bendersky et al. 1991; Kattner and Boettinger 1992; Muraleedharan et al. 1992a,b; Rowe et al. 1992). At low Nb levels (<7.5 at.% Nb), quenching from the -phase field results in a martensitic transformation, leading to the formation of hcp -martensite which undergoes an ordering process to yield the 2 -phase. The → 2 ordering reaction cannot be suppressed on quenching and this is clearly revealed in the observation of fine D019 domains within the martensite laths in quenched alloys containing up to 7.5 at.% Nb (Figure 5.18(a)). The Ms temperature for the → transformation is rapidly depressed beyond 7.5 at.% Nb (Strychor et al. 1988). This is because neither can the required chemical ordering occur during the martensitic transformation nor can the ordered o -phase martensitically transform into the 2 - or the O-phase. The occupancy of lattice sites in the parent (o ) and product (2 or O) structures
Ordering in Intermetallics 1200
α
Liquid
α+β
α2 + β
α2
1100
α2 + β o
βo to α2/0 Widanstatten
1000
419
α2 + βo + 0
900
O 800
Tempered martensite 700
βo to 0 Massive
βo to α2 Massive
Temperature (°C )
βo + 0
600
500
βo to α-type 400
0
Ti3Al
5
10
15
20
25
Nb (Atomic %)
Figure 5.17. Ti–Al–Nb pseudobinary phase diagram summarizing experimental observations on phase transformation reactions in this system.
are not consistent with a martensitic transformation. The -phase in higher Nb alloys (7.5–25 at.% Nb), therefore, is retained on quenching (Strychor et al. 1988), although it exhibits instabilities with respect to B2 ordering and also to a variety of displacement ordering processes. The quenched-in -phase shows a domain structure resulting from the bcc to B2 ordering (Figure 5.18(b)). In addition to the presence of B2 superlattice reflections, extensive streaking along < 110 > and < 112 > directions is noticed (Figure 5.18(c)). These instabilities have been ¯ < 110 ¯ > and attributed to transverse displacement waves which are of the 110 ¯ 112 < 111 > types. Strychor et al. (1988) have pointed out that the former is associated with a precursor for the → martensitic transformation while the latter originates from the tendency for the occurrence of the → # transition. A tweed contrast observed in the quenched-in o (ordered ) phase (Figure 5.18(d)) is consistent with the presence of transverse displacive waves and this tendency is noticed over the entire range of Nb concentration from 7.5 to 25 at.%. A relatively slow cooling from the -phase induces diffusional phase transformations in alloys containing 7.5–25 at.% Nb, leading to a mixture of 2 + o ,
420
Phase Transformations: Titanium and Zirconium Alloys
(a)
(c)
(b)
(d)
Figure 5.18. (a) Fine D019 domains within the martensite laths in quenched alloys containing up to 7.5 at.% Nb, (b) quenched-in -phase shows a domain structure resulting from the bcc to B2 ordering, (c) in addition to the presence of B2 superlattice reflections, extensive streaking along < 110 > and < 112 > directions is shown and (d) a tweed contrast as observed in the quenched-in o (ordered ) phase (after Strychor et al. 1988).
2 + o + O or O + o phases. The parent o -phase undergoes a decomposition into either 2 + o or O + o phase mixtures which is followed by the subsidiary phase reaction, 2 → 2 + O or 2 + o → O. The precipitation of the 2 -or the O-phase results in a lath, lamellar or mosaic morphology, depending on the alloy composition and the cooling rate. The Burgers orientation relationship between the hexagonal or the orthorhombic product and the parent cubic phase is invariably maintained. The volume fraction of the primary 2 -phase (which subsequently transforms to the O-phase) decreases with increasing Nb content and remains confined to grain boundaries as the alloy composition approaches 25 at.% Nb. The morphology of the product, at the level of light microscopy, bears a significant
Ordering in Intermetallics
421
resemblance to that observed in conventional + Ti alloys slowly cooled from the -phase field. An increase in the cooling rate results in a refinement in the size of the packet (which is constituted of a group of laths stacked in a parallel array) and in the formation of a fine basket weave morphology. As mentioned earlier, quenching from the -phase field yields an ordered martensitic structure (2 ) in alloys containing up to 7.5 at.% Nb and leads to the retention of the o -phase in alloys containing 7.5–25 at.% Nb. Tempering of the martensite results in a rapid growth of the B2 domains (Sastry and Lipsitt 1977) and causes -phase precipitation and recrystallization to a limited extent (Martin et al. 1980). The retained o -phase, on ageing, decomposes to #-related structures at temperatures below about 770 K. The transformation from the B2 structure to a variety of ordered #-structures has been discussed in detail in Chapter 6. Ageing of the quenched-in o -phase in the composition range of 7.5–11 at.% Nb induces a massive transformation in the 2 -phase. Alloys with higher Nb contents (11– 25 at.%) undergo a o → O transformation on ageing; this can occur either by a massive transformation producing equiaxed O-phase grains or by a Widmanstatten precipitation of O-phase plates which exhibit a martensite-like substructure. These two decomposition modes can operate in parallel or sequentially with the equiaxed O-phase grains finally consuming all the O-plates (Bendersky et al. 1991; Muraleedharan et al. 1992a,b). The O-phase subsequently decomposes into the equilibrium 2 - and/or the o -phases. 5.3.3 Structural relationships The phases which are present in Ti3 Al–Nb alloys (Nb content up to 30 at.%) are all based on two basic structures: bcc at high temperatures and hcp at low temperatures. While the bcc and hcp structures are related by the well-known Burgers lattice correspondence (discussed in detail in Chapter 4), the hcp → 2 (D019 ) and the A2 → o (B2) transformations involve replacive ordering in which the crystallographic axes of the product remain parallel to those of the parent. A structural description of these ordering processes has been given in earlier sections. With increasing Nb content the hexagonal symmetry of the 2 -phase gets distorted to orthorhombic symmetry (O-phase) in a manner quite similar to the structural change associated with the transition from hexagonal ( ) to orthorhombic ( ) martensite. The lattice correspondence between the and the (Bagariatskii) is essentially the same as the Burgers correspondence as can be seen by changing from the hexagonal to the orthorhombic axes system. The martensite phase is disordered and has a Cmcm space group with four atoms per unit cell. The O-phase in the ternary Ti–Al–Nb system corresponds to the stoichiometry of Ti2 AlNb where Nb atoms occupy distinctive sublattices (Banerjee et al. 1988, Mozer et al. 1990). The A2 BC structure shown in Figure 5.19 is
422
Phase Transformations: Titanium and Zirconium Alloys
Figure 5.19. The A2 BC structure, a prototype of the structure of the Ti2 AlNb phase (O-phase), is closely related to the D019 structure.
the prototype of the Ti2 AlNb phase (O-phase) and is closely related to the D019 structure. Crystallographic data pertaining to the ordered A2 BC structure are given in Table 5.12. In this structure, the atoms can occupy a range of positions without destroying the symmetry and the stacking sequence of the O-phase. The ranges Table 5.12. Atomic positions of Ti, Al and Nb atoms in the orthorhombic O-phase. Atom
Positions (x y z)
Ti
0.2310, 0.7690, 0.2310, 0.7690, 0.7310, 0.7310, 0.2690, 0.2690,
0.9041, 0.0959, 0.0959, 0.9041, 0.4041, 0.5959, 0.4041, 0.5959,
0.2500 0.7500 0.7500 0.2500 0.2500 0.7500 0.2500 0.7500
Al
0.0000, 0.0000, 0.5000, 0.5000,
0.1630, 0.8367, 0.6633, 0.3367,
0.2500 0.7500 0.2500 0.7500
Nb
0.0000, 0.0000, 0.5000, 0.5000,
0.6357, 0.3643, 0.1357, 0.8643,
0.2500 0.7500 0.2500 0.7500
Ordering in Intermetallics
423
of atomic coordinates for each equivalent position are also given in the table. The atomic sites of the Ti2 AlNb unit cell proposed by Mozer et al. (1990) lie within these ranges and this structure √ can be considered to be a special case of the A2 BC structure with b/a less than 3 and c/a less than the ideal value. The similarity between the D019 structure and the O-phase structure can be seen by comparing the basal plane projection of the former with the (001) plane projection of the latter (Figure 5.20). Comparing the first, second, third and fourth near neighbour bond lengths in the D019 structure with a non-ideal c/a ratio (close to that of Ti3 Al) and the corresponding bond lengths in the Ti2 AlNb phase, Singh et al. (1994) have shown that these structures have a close similarity. It may be noted that the O-phase structure involves further ternary ordering of the 2 -phase with Ti, Al and Nb atoms predominantly occupying three different types of sites (Banerjee et al. 1988, Mozer et al. 1990). In view of the structural relations associated with these phases, the relevant structural changes can be described in terms of the following:
C
01 (001)
02 (001)
A
C
A 0.970 nm
α2 [0001]
A
B
B
C
B
[2110]
1.001 nm
[010]
[0110]
(1) distortion of {110} planes and changes in their interplanar spacings (2) shuffles, or relative displacements of neighbouring {110} planes, and
[100] Ti2AlNb 0.578 nm Atom Layer A
0.595 nm Al
Ti
Nb
Angle ABC
α2
01
02
60° 63° 65°
Layer B. c /2 above
Figure 5.20. A comparison between the D019 and the O-phase structures as elucidated by the basal plane projection of the former with the (001) plane projection of the latter.
424
Phase Transformations: Titanium and Zirconium Alloys
(3) chemical ordering that changes the occupancies of Ti, Al and Nb atoms among the lattice sites. The main advantage of describing the phase transitions in this system using a common framework is that a single thermodynamic potential can be identified as a continuous function of a set of order parameters which describe the three types of structural change. Bendersky et al. (1994) have considered the group/subgroup symmetry relationship with respect to the transformations in this system and have identified the possible sequence of crystallographic transitions. 5.3.4 Group/subgroup relations between BCC (Im3m), HCP (P63 /mmc) and ordered orthorhombic (Cmcm) phases A continuous phase transition of first or higher order requires that the symmetry of the product phase is a subgroup of that of the parent phase. This is, in fact, the Landau–Lifshitz rule I for a transition to qualify as being of second order (refer to Chapter 2). Usually the low-temperature phase has a lower symmetry than the high-temperature phase, and a transition involving a lowering of symmetry is known as ordering while one associated with an increase in symmetry corresponds to disordering. The approach pertinent to determining the possible sequence of transformations is based on successive stages of symmetry reduction. The necessary information regarding symmetry relations is contained in the space group tables of the International Tables for Crystallography (1991). On the basis of this information, the individual steps of a transition can be predicted by considering the maximal subgroup relation between the parent and the product phases in any given step. A subgroup H of the space group G is called a maximal subgroup of G if there is no subgroup L of G such that H is a subgroup of L. This approach helps in the identification of all symmetry reduction steps though it is not necessary that each of these steps must really occur in a transformation sequence. There are no apparent subgroup relations between structures with cubic and hexagonal symmetries, G1 and G2 , respectively, due to the presence of noncoinciding threefold <111> cubic and sixfold [0001] hexagonal symmetry axes. A connection between these symmetries can be arrived at by introducing an intermediate structure with space group Gt which is either a supergroup of both the structures, G1 and G2 , or a subgroup of both the structures. A supergroup, constituting a group union of G1 and G2 , does not exist as both the bcc and hcp structures exhibit very high symmetry. However, a subgroup Gt can be found as the intersection group of G1 and G2 In particular, considering the disordered bcc and hcp structures with Im3m and P63 /mmc space groups, respectively, and taking ¯ [1120] ¯ , into account the Burgers orientation relationship, (110) 0001 ; [111]
Ordering in Intermetallics
425
between them, the intersection group Gt is found to be the orthorhombic space group Cmcm, with its c-axis parallel to the [110] direction. The Cmcm space group, with an appropriate choice of Wyckoff sites, can represent a structure which is close to hcp but differs from it in symmetry and in the relative positions of atoms on the basal plane (Figure 5.21). The Cmcm structure can also be considered as the bcc structure distorted by relative shifts of the (110) planes. Bendersky et al. (1991) have proposed sequences of transitions that connect the higher symmetry cubic and hexagonal space groups to the lower symmetry orthorhombic space group (Figure 5.22). These sequences include all known equilibrium phases observed in Ti3 Al–Nb alloys with Nb contents of less than 30 at.%. The sequences shown in Figure 5.22 show two branches, one starting from the disordered bcc structure and the other from the ordered B2 structure. The directions of symmetry reduction are indicated by the arrows, and the indices of symmetry reduction between two neighbouring subgroups are indicated by numbers in square brackets within these arrows. The index of a subgroup is the ratio of the number of symmetry elements in a group to that in the subgroup. These indices give the number of lower symmetry variants (domains) that can be generated when a transition from a high-symmetry to a low-symmetry structure occurs. Inclined arrows indicate symmetry changes due to atomic site (Wyckoff position) changes, leaving the occupancy fixed, i.e. displacive ordering. Vertical arrows, on the other hand, A
E
B A fA2
fC2
B
fS1 fA1
C
b
fC1
[010] fS2
D
F
C
[100]
a
Figure 5.21. The structure corresponding to Cmcm space group, with an appropriate choice of Wyckoff sites, can represent a structure which is close to hcp but differs from it in symmetry and in the relative positions of atoms on the basal plane.
426
Phase Transformations: Titanium and Zirconium Alloys
Im3m (A2) [2]
[3] I4/mmm
Pm3m (B2) [3]
[2]
[2]
P4/mmm
Pmmm [2]
[2]
P 63 /mmc (A3) [2]
[2]
[4]
Cmcm (A20)
Cmmm [2]
[2]
P 63 /mmc (D019)
Pmmc (B19) [2]
[3]
Cmcm (A2BC)
Figure 5.22. The sequences of transitions connecting the higher symmetry cubic and hexagonal space groups to the lower symmetry orthorhombic space group. The directions of symmetry reduction are indicated by the arrows, and the indices of symmetry reduction between two neighbouring subgroups are indicated by numbers within these arrows.
indicate symmetry changes due to changes in atomic site occupancy, i.e. chemical ordering. Slight adjustments in site positions and occupancies due to new atomic environments will accompany chemical and displacive ordering, respectively. The sequences of possible structural transitions (Figure 5.22) are arrived at from the consideration of maximal subgroup relations. The atomic correspondence between these structures are depicted in Figure 5.23. The I4/mmm and Fmmm structures are obtained by homogeneous straining of the parent cubic lattice, A2 ¯ (Im3m). Similarly the P4/mmm and Cmmm structures correspond to a homoge¯ neously distorted B2 structure (Pm3m). While a tetragonal distortion along a cubic <100> direction brings about the first step of the transition from the A2 and the B2 structures, the second step is associated with different distortions along two orthogonal cubic <011> directions. The overall orthorhombic distortion of the cubic structure, if not accompanied by ordering, is probably unstable for materials with isotropic metallic bonding. Therefore these structures are not expected to exist as metastable states but rather represent a homogeneous strain accompanying the subsequent symmetry reduction by shuffle displacements (from Fmmm and Cmmm to Cmcm and Pmma, respectively). The Cmcm and Pmma space groups which correspond, respectively, to the Strukturbericht A20 (-U prototype) and B19 (AuCd prototype) structures can be obtained by heterogeneous shuffles of pairs of (110) planes of either a disordered
Ordering in Intermetallics
427
Im3m (A2)
011
a1
Cmcm (A20)
b
100
P 63 /mmc (A3)
a2
a
Ti, Al, Nb
Ti, Al, Nb
011
a1
Al, Nb
a2
Al Ti, Nb
Ti
100
a
Pm3m (B2)
b
Al, Nb
P 63/mmc (D019)
Ti
b
– Lower layer
Al Nb
– Upper layer Pmma (B19)
a
Ti
Cmcm (A2BC)
Figure 5.23. The sequences of transitions showing two branches, one starting from the disordered bcc structure and the other from the ordered B2 structure.
or an ordered cubic structure. The amplitude of the shuffle displacement wave is reflected in the y-coordinates of Wyckoff positions: 4c (0, y, 41 ) for the Cmcm structure and 2e ( 41 , y1 , 0); 2f ( 41 , y2 , 21 ) for the Pmma structure. For some special values of Wyckoff position’s y-coordinates and/or lattice parameters, a structure of higher symmetry can be generated. Such a high symmetry structure, hexagonal P63 /mmc, is generated from the disordered Cmcm structure 1 when the shuffles are such that √ the Wyckoff position parameter y is 3 and the ratio of lattice parameters, b/a, is 3. Such a symmetry enhancement is not possible for the orthorhombic Pmma structure due to the chemical order inherited from the B2 structure. On the other hand, no thermodynamic barrier exists for the Cmcm (A20) to P63 /mmc structural transition as the latter structure is associated with a higher entropy (due to its higher symmetry) while the internal energies, estimated on the basis of first near neighbour interaction energies, of the two structures are nearly the same. The structure with the lowest symmetry, the O-phase, has the Cmcm space group and ternary ordering on the Wyckoff positions 4c1 4c2 and 8g. This ternary ordering is responsible for making the translations in the O-phase structure twice as much as those in the binary ordered B19 structure. The O-phase structure can be obtained by ordering either the Pmma (B19) or the P63 /mmc (D019 )
428
Phase Transformations: Titanium and Zirconium Alloys
structure. The D019 structure can be obtained by binary ordering of the disordered hcp (A3) structure and could correspond to a metastable intermediate phase. 5.3.5 Transformation sequences The group–subgroup relations suggest the following three transformation sequences, designated as A, B and C: ¯ A$ Im3mA2
12 CmcmA20 1 P63 /mmcA3 −−→ − → contd P63 /mmcA3 4 P63 /mmcD019 3 CmcmO − → − → ¯ ¯ B$ Im3mA2 2 Pm3mB2 12 PmmaB19 2 CmcmO − → −−→ − → ¯ C$ Im3mA2 12 CmcmA20 2 PmmaB19 2 CmcmO −−→ − → − → The numbers in the square brackets give the number of variants possible at each of the transition steps. It must be emphasized here that the A2 → O transformation can also occur in a single step provided the transformation mode is reconstructive. However, coherent transformations proceed in steps, following one of the sequences mentioned above. Each transient phase may exist over a temperature range between the upper and lower critical temperatures (or temperatures of phase instability for first order transformations). The number of variants in each transition is equal to the index of the subgroup (as indicated within the square brackets). For a sequence of transitions the number of variants of the lowest symmetry phase (with respect to the highest symmetry phase) will be equal to the product of the indices for each step. The creation of the expected number of variants at each transition state generates a hierarchy of domain structures. It is from the domain structure that the exact sequence of steps followed during the transformation can be identified. The microstructures resulting from the sequences, A, B and C, will all consist of the same O-phase but with different hierarchies and types of interfaces, and the nature of the interfaces, whether rotational, translational or mixed, will be determined by the group–subgroup relation. Each symmetry reduction step necessarily leads to the formation of more than one variant of the lower symmetry phase, thereby restoring the lost symmetry partially in the macroscopic sense. As mentioned earlier, the number of variants in each transition is equal to the index of the subgroup, indicated within the square brackets in the sequences. For the complete sequence of transitions the number of variants of the lowest symmetry O-phase is given by the product of the indices corresponding to each of the steps involved in the sequence. This means that the
Ordering in Intermetallics
429
sequence A results in 12 × 1 × 4 × 3= 144 variants while the sequences B and C result in 482 × 12 × 2 or 12 × 2 × 2 variants. A variety of domain interfaces can be created between adjacent domains. A domain interface will remain stress free if it is planar and if the self-strains on both sides are compatible, so that no discontinuity of displacements occurs at the interface. Bendersky et al. (1991) have examined the expected distribution of domains arising out of the different sequences, A, B and C, of the transformation and have listed the types of interfaces which can form between domains produced in different transitions (Table 5.13). The microstructural development corresponding to the three paths, A, B and C, has also been predicted on the basis of the domain configuration expected to be generated in the successive steps of the transformation. The three paths differ primarily as to whether the hexagonal symmetry phases or the B19 phase occur at an intermediate stage of transition. All the three paths finally lead to somewhat similar microstructures consisting of domains of the orthorhombic phase, although the microstructure at the intermediate steps are significantly different. For the transformation sequences A and C the first step, A2 → A20, is a displacive transition and involves the creation of a polytwin structure. In contrast, the first step of the sequence B, namely, the A2 → B2 transition, involves pure chemical ordering and produces only antiphase boundaries (APB). Polytwins are separated from their neighbours by planar boundaries, while the isotropic APBs, which are usually curved surfaces, separate either interconnected or closed volume domains. Table 5.13. List of interfaces between domains in different group/subgroup transitions. Group/subgroup
Type of interface
¯ ¯ Im3mA2 → Pm3mB2 ¯ → CmcmA20 Im3m
Translational APB Rotational twins Translational with stacking fault mixed twin/translational No new interface Translational APB Rotational (compound twins) Rotational twins Translational with stacking fault mixed twin/translational Translational APB Translational APB
CmcmA20 → P62 /mmcA3 P63 /mmcA3 → P63 /mmcD019 P63 /mmcD019 → CmcmO ¯ → PmmaB19 Pm3m CmcmA20 → PmmaB19 PmmaB19 → CmcmO
The interfaces are described by the domain generating symmetry operation of lowest symmetry.
Phase Transformations: Titanium and Zirconium Alloys
A3
D019
O
B2
B19
O
A20
B19
O
→
430
→ →
A2
Figure 5.24. Schematics of development of microstructure by the creation of anisotropic planar interfaces (due to the associated stacking fault nature), isotropic APBs (resulting from pure chemical ordering) and secondary polytwin boundaries (as in the case of the D019 → O phase transition).
The B2 → B19 transition in the sequence B is responsible for the introduction of a polytwin structure. Further development of microstructure by the creation of anisotropic planar interfaces (due to the associated stacking fault nature), isotropic APBs (resulting from pure chemical ordering) and secondary polytwin boundaries (as in the case of the D019 → O phase transition) is schematically illustrated in Figure 5.24 The transformation paths predicted from group–subgroup relations pertain only to congruent phase transformations in which a single phase transforms into another single phase without involving a partitioning of the alloying elements. Such transformations in multicomponent systems can occur under equilibrium conditions only at special compositions, known as consolute points, or in second order transitions. However, under sufficient undercooling, a partitionless transformation can be induced. Bendersky and Boettinger (1994, Bendersky et al. 1994) have shown that the predictions made closely match the observed transformation sequences in some specific ternary Ti–Al–Nb alloys. The salient features of their experimental observations are summarized in the following, mainly in order to illustrate how observations on the domain distribution can lead to the identification of the transformation path. 5.3.5.1 Transformation sequence in the alloy Ti–25 at.% Al–12.5 at.% Nb The observations made on the needle-shaped transformation product in this alloy correspond to a situation where a displacive transition takes place in the disordered
Ordering in Intermetallics
431
A2 structure. Therefore, either of the sequences A or C can be expected to be operative. Bendersky and Boettinger (1994, Bendersky et al. 1994) have discussed the morphological features of the needle-shaped product formed in this alloy when cooled from 1373 K at the rate of 400 K/s and the displacement vectors associated with the domain boundaries contained within these needles. Even though diffraction patterns obtained from individual needles can be indexed in terms of one of the three variants of the D019 structure, the splitting of diffraction spots is indicative of the occurrence of orthorhombic distortions in the basal plane of the D019 phase, suggesting the presence of the O-phase. Two types of interfaces, namely, ¯ > and planar wavy, isotropic APBs with the displacement vectors, R = 16 < 1120 1 ¯ stacking fault-like defects with R = 4 0110, have been detected within these nee¯ > APBs points towards the occurrence of the dles. The presence of the 16 < 1120 disordered A3 phase as an intermediate step prior to the D019 ordering. Out of the ¯ > faults, only one variant has been actually three possible variants of the 41 < 1010 observed. This deviation from hexagonal symmetry suggests that the preferred direction (arising from orthorhombic symmetry) existed prior to the appearance of hexagonal symmetry, which is consistent with the initial step of sequence A. Finally the presence of O-phase domains, related to each other by hexagonal symmetry, is attributable to the occurrence of the last step of sequence A. 5.3.5.2 Transformation sequence in the alloys Ti–25 at.% Al–25 at.% Nb, Ti–28 at.% Al–22 at.% Nb and Ti–24 at.% Al–15 at.% Nb In both these alloys, Bendersky and Boettinger (1994) and Muraleedharan et al. (1992a,b) have provided evidences for the occurrence of A2 → B2 ordering prior to the transition to the close packed structure. This means sequence B is expected to be operative in these alloys which show self-accommodating plate-like domains. The observed lattice correspondence, [001]o [011]c and [100]o [100]c (o and c referring to the orthorhombic and the cubic phases, respectively), is consistent with the group–subgroup relations discussed earlier. This gives six rotational variants of the orthorhombic phase (either B19 or O), each with its basal (001)o plane parallel to one of the six {110}c planes of the cubic structure. Accommodation of transformation strains requires a small mutual rotation between the contacting variants by the creation of stress-free interfaces. The measured misorientation between adjacent plates has been shown to be consistent with that calculated from the lattice strains (Bain strain) required for the B2 → O transformation. An analysis of the displacement vectors associated with the domain boundaries has revealed the presence of two types of domain boundaries. The first type, with a wavy APB appearance, is associated with the displacement vector R = 21 [010]o (or the 21 [100]o vector which is equivalent due to the C-centring of the Cmcm space
432
Phase Transformations: Titanium and Zirconium Alloys
group of the O-phase). APBs of this type are expected to form as a result of the B19 → O transformation in which the unit cell parameters, a and b, are doubled. The second type of domain boundaries show a distinct faceted appearance and correspond to the displacement vectors, R = 41 [012] or 41 [010]. These interfaces are expected to be produced during the B2 → B19 transition. The homogeneous (Bain) strain corresponding to the B2 → B19 transition is also responsible for the observed orientations of the six twin variants of the orthorhombic or the hexagonal structure. In view of these observations, it could be concluded that these alloys go through sequence B during relatively fast cooling from 1373 K. As mentioned earlier, the sequences predicted from the group–subgroup relations and those observed in the alloys discussed in this section correspond to the situation of congruent (partitionless) transformations. Transformation sequences pertaining to a situation where the cooling conditions allow alloy partitioning and phase reactions are discussed in the following section. 5.3.6 Phase reactions in Ti–Al–Nb system Phase reactions involving partitioning of the alloying elements can be understood using the proposed section (Banerjee et al. 1990) of the Ti–Al–Nb system (Figure 5.16) in the vicinity of the composition Ti2 AlNb. This pseudobinary section is along the composition line joining Ti3 Al and Nb3 Al. This section has later been modified and enlarged to some extent (Figure 5.25) by Bendersky and Boettinger (1994, Bendersky et al. 1994) by taking into account the following factors: (1) The disordered -phase definitely exists between the and the 2 -phases in pure Ti3 Al and the -phase field narrows down with increasing additions of Nb. β
Temperature (K)
1300
βo α
1200 1100
α2 + β
1000 900
α 2 + βo
α2
+β
+O
O
o
α2 α2 + O
800 700
10
Ti3Al
Ti–25Al–15Nb 20
Atomic % Nb
30
Nb3Al
Figure 5.25. The pseudobinary section, enlarged and modified, along the composition line joining Ti3 Al and Nb3 Al for the phase reactions involving partitioning of the alloying elements of the Ti–Al–Nb system in the vicinity of the composition Ti2 AlNb.
Ordering in Intermetallics
433
(2) The → o ordering transition qualifies to be a second order transition and therefore, the and o -phase fields need not be separated by a two-phase field. (3) The maximum stability of the o -phase is assumed to be centred around the Ti2 AlNb composition. This is because the two sublattices of the B2 structure of Ti–Al–Nb alloys are preferentially occupied by Ti atoms and by a mixture of Al and Nb atoms, respectively. The maximum order is, therefore, likely to be exhibited at a composition where the atomic fraction of Ti atoms is equal to the sum of the atomic fractions of Al and Nb atoms. The maximum in the ordering curve is taken to occur at about 1673 K (Bendersky and Boettinger 1989). (4) The → , → 2 and 2 → O transitions are all first order transitions; therefore, the , , 2 and O single phase fields are separated by two-phase fields. (5) The ternary phase diagram essentially depicts the phase stability regimes of the three phases, (2 ), (o ) and O as shown in Figure 5.25. The two-phase fields, 2 + o o + O and 2 + O, remain separated by a three-phase field, 2 + o + O. A variety of phase reactions is possible on cooling an alloy through the threephase field. These include the eutectoid reaction, o → O + 2 , the peritectoid reactions o + O → 2 and o + 2 → O, and the precipitation reactions, o → o + O, o → o + 2 and 2 → 2 + O. Out of these, the reactions that occur are determined by the alloy composition and the relative positions of the three phase isotherms at different temperatures. A systematic study of phase reactions in the alloy, Ti–25 at.% Al–15 at.% Nb, which passes through the three-phase field, has been reported by Muraleedharan et al. (1992a,b). The results of this study are summarized here (Figure 5.26) to illustrate the variety of phase reactions possible in such an alloy: (1) Quenching (cooling rate exceeding 100 K/s) from a temperature above 1383 K results in the retention of the o -phase, the /o transition temperature being around 1403 K. The → o ordering reaction cannot be suppressed by quenching. (2) Quenching from temperatures between 1253 and 1383 K produces a microstructure consisting of equiaxed 2 -grains in a o -matrix. Equilibration at a temperature below but close to 1253 K, followed by quenching, results in a three-phase microstructure comprising the 2 - and o -phases and a blocky O-phase. The transus temperatures bounding different phase fields for this alloy have been identified from these observations and are indicated by the
434
Phase Transformations: Titanium and Zirconium Alloys 1200
Ti–25Al–16Nb
T-T-T curves, continuous cooling β B2 α2 + B2
B2 → α2 + B2
Temperature (°C) →
1000 α2 + α′2 + O
α2 + B2→O α2 + B2 + O
800
O + B2 B2 → O + B2
10°C/s 4°C/s AC AQ
0.7°C/s FC I
0.1°C/s FC II
0.02°C/s FC III
600
10
102
103
104
Time (s) →
Figure 5.26. Phase reactions in the three-phase field in Ti–24 at.% Al–15 at.% Nb ternary alloy.
intersections of the dotted vertical line corresponding to the alloy composition with the o /2 + o and 2 + o /2 + o + O transus lines. (3) Continuous cooling at slower rates (10–0.02 K/s) results in the formation of a primary phase – either O or 2 – depending on the cooling rate. Phase reactions induced under different cooling rates are indicated in the schematic T-T-T diagram shown in Figure 5.27. The kinetics of the o → 2 + o and the o → O + o reactions are such that faster cooling rates result in the -phase transforming directly to the O-phase, skipping the formation of the 2 -phase, even though the 2 + o and 2 + o + O phase fields are crossed during the continuous cooling process. (4) In case the 2 -phase is present as a primary decomposition product, a variety of secondary reactions such as 2 → 2 +O and 2 +o → O are possible. The fine multivariant O-phase plates observed at the periphery of the 2 -plates, which satisfy the 2 /O orientation relation can be identified as the product of the former phase reaction. On the other hand, the monolithic O-phase rim produced around the 2 -plates corresponds to the product of the peritectoid reaction between the 2 - and o -phases. The various microstructures that can be produced by continuous cooling in the Ti–24 at.% Al–15 at.% Nb alloy under different rates of cooling are shown in Figure 5.27 to illustrate the variety of phase reactions observed.
Ordering in Intermetallics
435
Figure 5.27. Various microstructures produced by continuous cooling in the Ti–24 at.% Al–15 at.% Nb alloy under different rates of cooling (after Banerjee, 1994a).
Ageing of the -quenched alloys in the O + o and the 2 + o phase fields produces 2 - or O-phase precipitates in the form of Widmanstatten laths in a o matrix, thin films of the retained o -phase being left behind at the lath interfaces. Samples aged after subtransus solution treatments show changes with regard to the primary 2 -as well as the metastable o -phases. Ageing in the O + o phase field results in the decomposition of the o -phase through two different kinetically competitive modes which operate simultaneously; the first is a conventional continuous precipitation of the O-phase in the o -phase while the second results in o -precipitation within O-phase grains. The mechanism of the o → O transformation has been discussed in detail in a later section. Based on the phase diagram (Figure 5.25) discussed in this section, a To versus composition diagram (Figure 5.28) can be constructed, To being the temperature where the parent and the product phases have the same molar free energy. Such diagrams are useful in rationalizing composition-invariant transformations, which have been described in Section 5.3.5. A schematic drawing (Figure 5.29) showing the free energies of the /o , /2 /O phases as functions of composition has been constructed (Bendersky and Boettinger 1994, Bendersky et al. 1994). This diagram can be viewed as the superposition of the free energy–composition plots for ordering transformations in the bcc-based and hcp-based phases.
436
Phase Transformations: Titanium and Zirconium Alloys β (bcc)
1300
α (hcp)
1100
α2 (D019)
B2
900
α
1000
β→
β→
To (°C)
1200
βo (B2)
O α2 → O →
θ → B2 9 B1
B2
800 700
Ti3Al
10
20
30
Atomic % Nb
Nb3Al
Figure 5.28. To versus composition diagram (based on the phase diagram in Figure 5.25) to rationalize composition-invariant transformations.
Free energy
α
2
Alloy 2
α
Alloy 1
β βo B19
O
Ti3Al
10
20
Atomic % Nb
30
Nb3Al
Figure 5.29. A schematic drawing showing the free energies of the bcc-based /o and hcp-based /2 /O phases as functions of composition.
5.4
FORMATION OF ZR3 AL
The equilibrium Zr3 Al phase has the L12 (cP4) structure. The availability of several slip systems in the L12 structure, which is essential for meeting the Von Mises criterion for ductility of a polycrystalline material, made it attractive to consider Zr3 Al as an intermetallic for structural applications. The equilibrium Zr3 Al phase differs from the equilibrium Ti3 Al phase in the following subjects: (a) Unlike Ti3 Al (D019 structure), the Zr3 Al does not form by chemical ordering of the terminal
Ordering in Intermetallics
437
solid solution (hcp -phase). (b) The Zr3 Al phase forms through a peritectoid reaction, + Zr 2 Al → Zr3 Al (refer phase diagram given in Chapter 1). (c) The Zr3 Al phase is a line compound which does not tolerate any significant deviation from the stoichiometric composition. 5.4.1 Metastable Zr3 Al (D019 ) phase Mukhopadhyay et al. (1979) explored the possibility of forming a metastable Zr3 Al phase with the D019 structure by quenching an alloy of composition Zr–14 at.% (4.6 wt%) Al from the -phase field. Water-quenched thin (0.5 mm thickness) samples of thin alloy showed a lath martensite structure. Selected area diffraction experiments have shown the presence of diffuse intensity maxima at D019 superlattice positions apart from the hcp reflections. On subsequent ageing, these superlattice reflections have been found to intensify and sharpen. Dark-field images with these superlattice reflections have shown the presence of 5–10 nm microdomains of the D019 phase distributed in the martensite matrix (Figure 5.30). This observation has clearly established that in a supersaturated Zr–Al solid solution (with hcp structure), a metastable D019 phase can form. This is quite expected as the development of the D019 structure in an hcp lattice can be achieved by replacive ordering. Figure 5.31 shows how by an introduction of a concentration wave with the wave vector, k= a2 1120 , the D019 structure can be formed in an hcp lattice. As has been mentioned earlier, the wave vector satisfies two of the three
Figure 5.30. (a) Dark-field image showing the presence of 5–10 nm microdomains of the D019 phase distributed in the martensite matrix and (b) corresponding D019 diffraction pattern along the (0001) direction.
438
Phase Transformations: Titanium and Zirconium Alloys r
ye
La
0
L 1
0.5
2
2d1120 t1 < t2
4
A layer B layer
Al Zr Al Zr
Al fraction
3
t1
0.25
t2
p=0
1
2
3
4
¯ planes. Figure 5.31. (a) Atomic arrangement of the D019 structure showing compositions of {1120} (b) Development of concentration wave as a function of time (t) leading to the formation of D019 structure from the parent hcp phase (average composition Ti–25 at.% Al).
Landau–Lifshitz conditions for the A3 → D019 transformation to be a second order transition. This is indeed a fit case for a continuous ordering under a high degree of supercooling. Figure 5.31 illustrates the evolution of the A3 → D019 ordering by formation and amplification of a concentration wave with k= a2 1120 in an alloy of stoichiometric composition, i.e. Zr–25 at.% Al. Supersaturation of the -phase to such an extent is difficult, if not impossible. In the Zr–14 at.% Al alloy, investigated by Mukhopadhyay and Banerjee (1979), the attainment of near-stoichiometric composition has been achieved by a spinodal clustering process which has resulted in a concentration modulation of wavelength of about 20 nm. It is reported that the first step of the transformation has been the spinodal clustering which resulted in a concentration-modulated structure along 1010 direction, the elastically soft direction in the -phase. The regions getting enriched with Al subsequently undergo an ordering process by developing and amplifying a short wave length concentration wave. The overall process can, therefore, be considered as a superposition of spinodal clustering followed by continuous ordering. Such a transformation sequence can occur if the system at the ageing temperature is initially unstable with respect to a long wave length (20 nm) concentration fluctuation and, after attainment of a concentration within the ordering spinodes, becomes unstable with respect to the A3 → D019 ordering (within the Al-enriched regions). The condition under which such a concomitant ordering and clustering processes can occur is shown in the schematic free energy–concentration plots (Figure 5.32) corresponding to the disordered () and the ordered (, D019 ) phases. An alloy of composition C1 (which lies between the spinodes of free energy–concentration plot for the disordered -phase) is unstable with respect to spinodal clustering (concentration modulation with wave vector, k = 000 ). This process leads to the development of alternate Al-rich regions which in turn becomes unstable with respect to → D019 ordering. With the gradual amplification of 21 1120 concentration wave, the order
Ordering in Intermetallics
439
T3
Free energy →
D
A
A
A x
x
Composition (% B) →
Figure 5.32. A schematic free energy–concentration plot depicting the condition under which concomitant ordering and clustering processes can occur corresponding to the disordered, , and the ordered, D019 , phases.
parameter in these regions increases finally leading to nucleation of Zr3 Al particles of D019 structure. These ordered particles decorate the alternate Al-enriched layers, as shown in Figure 5.30. 5.4.2 Formation of the equilibrium Zr3 Al (L12 ) phase The equilibrium Zr3 Al (L12 ) evolves from the -quenched structure containing a distribution of fine 2 (Zr3 Al with D019 structure) particles in the -matrix. The emergence of the equilibrium Zr3 Al phase has been found to occur through a cellular precipitation process. The nucleation of the two-phase cells ( + Zr3 Al) invariably occurs at the lath boundaries of the -quenched structure. These cells grow by propagation of the cell boundaries towards the interior of laths (Figure 5.33). Some of the general characteristics of cellular precipitation can be illustrated by this example. These are listed as follows: (1) Nucleation of a cell (two-phase region containing a group of and Zr3 Al (L12 ) lamellae, each with a specific orientation) invariably occurs at the grain boundaries. In the present example, the starting structure is that of a lath martensitic -matrix with a distribution of fine Zr3 Al (D019 ) precipitates. Majority of the lath boundaries are small-angle boundaries separating laths of close orientations.
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Phase Transformations: Titanium and Zirconium Alloys
(a)
(b)
(c)
Figure 5.33. Micrographs showing a cellular precipitation process. (a) The nucleation of the twophase cells ( + Zr 3 Al) invariably occurs at the lath boundaries of the -quenched structure. (b) Growth of the cells by propagation of the cell boundaries towards the interior of laths. (c) Twophase ( + Zr 3 Al) transformation product.
(2) The -orientation (1 ) of the cell matches with one of the neighbouring -lath (1 ), while the Zr3 Al (L12 ) orientation maintains exact orientation relationship with 1 . (3) A cell so related grows into the adjacent lath (orientation 2 ). Similarly a cell with 2 -orientation grows into the 1 -lath. Such a process essentially makes the original flat lath boundary into an undulated boundary which acts as the transformation front, the propagation of which accomplishes the transformation of the + Zr 3 Al (D019 ) region into + Zr 3 Al (L12 ) cells (Figure 5.34). (4) In general, where a supersaturated phase decomposes into a cellular producer, the transformation can be described as a phase reaction, → + , where is the precipitating phase. (5) The process of partitioning of the alloying elements between the two product phases occurs primarily of the transformation front. The diffusion field is, therefore, related to the lamellar spacing of the product structure.
Figure 5.34. Micrograph showing periodic spacing of interfacial dislocations at /Zr3 Al boundaries and faults within Zr3 Al lamellae.
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Interfacial Dislocations
α
(0002)α // (111) Zr3Al
α
Zr3Al
m c
Figure 5.35. A schematic drawing showing the misfit dislocations along the interface between and Zr3 Al (L12 ) phases.
(6) The diffusion distance being somewhat small and the diffusion being enhanced due to the presence of the interface (transformation front), the kinetics of cellular precipitations are relatively fast. (7) In the present case, where the + Zr 3 Al (D019 ) structure decomposes into the equilibrium +Zr 3 Al (L12 ) structure, it is imperative that the metastable Zr3 Al (D019 ) precipitates get dissolved ahead of the transformation front leading to an increase in the level of supersaturation in the -phase which breaks up into the equilibrium + Zr 3 Al (L12 ) phase mixture at the transformation front. As mentioned earlier, the orientation relationship between the and the Zr3 Al (L12 ) phases is maintained within the cells. The lamellar product, therefore, contains primarily two types of semicoherent interfaces characterized by arrays of misfit dislocations. The degree of misfit along the c-direction of the phase being small, the spacing between misfit dislocations is quite large. A schematic drawing (Figure 5.35) shows the misfit dislocations along the interface between and Zr3 Al (L12 ) phases. 5.4.3 +Zr2 Al → Zr3 Al peritectoid reaction The phase diagram (Figure 6.30) shows that the equilibrium Zr3 Al (L12 phase) results from a peritectoid reaction, + Zr 2 Al → Zr3 Al. The Zr2 Al (B82 structure) can be viewed as an ordered #-phase, as described in detail in Chapter 6. Due to the close lattice matching between and Zr2 Al phases, the latter grows in the -matrix as equiaxed particles as an alloy is cooled in the + Zr 2 Al phase field.
442
Phase Transformations: Titanium and Zirconium Alloys 1300
← Tp
Temperature (K)
1200
← Tα 1100
1000
80%
900
50% 5% 800
0.1
1
10
100
1000
Time (h)
Figure 5.36. The proposed time-temperature-transformation (T-T-T) diagram showing that the rate of the peritectoid reaction is fastest at around 888 C, where a Zr2 Al particle of about 3 !m diameter gets fully transformed within about 8 h.
The average size of Zr2 Al particles depends on the rate of cooling through the two-phase field and varies from 8–10 !m in diameter in larger ingots to about 1 !m in the smaller ones. When the + Zr 2 Al phase mixture is allowed to react at a temperature below about 975 C, the two phases undergoes a peritectoid reaction resulting in the formation of the Zr3 Al (L12 ) phase. Schulson (1975) and Schulson and Graham (1976) have studied the peritectoid reaction in detail. The time-temperature-transformation proposed by them (Figure 5.36) shows that the rate of the peritectoid reaction is fastest at around 888 C, where a Zr2 Al particle of about 3 !m diameter gets fully transformed within about 8 h. Since peritectoid reaction requires the presence of two reacting phases, the reaction invariably initiates at the interface. The reaction product, in this case, Zr3 Al “envelopes” the Zr2 Al particles and provides a barrier for the reaction to proceed rapidly. The subsequent growth of the product layer requires diffusion of the fast moving species (in this case Al) across the product layer. Schulson and his coworkers have clearly shown the presence of the envelopes of Zr3 Al phase around Zr2 Al particles and have shown that growth of the product phase occurs through long-range diffusion-controlled migration of Zr2 Al/Zr3 Al and Zr3 Al/-Zr interfaces in opposite directions. Correspondingly, the transformation time is strongly dependent on the size of the Zr2 Al particles in the starting microstructure.
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5.5
443
PHASE TRANSFORMATION IN -TiAl-BASED SYSTEMS
Two-phase -TiAl-based alloys offer an attractive combination of properties, namely, low density, as well as good creep resistance, high-temperature strength and oxidation resistance. It is because of the excellent combination of several desirable properties that these alloys are being considered as candidate materials for high-temperature applications in aerospace industries, particularly in gas turbine engines. Monolithic -TiAl is very brittle at room temperature. Several attempts have been made to improve the room temperature ductility of TiAl-based alloys by controlling the alloy composition and by tailoring the microstructure through phase transformations. The evolution of microstructure in these alloys is somewhat unique and is strongly influenced by the orientation relation between the participating phases. The two-phase lamellar structure, which is exhibited by these alloys when appropriately heat treated, is responsible for imparting to them an attractive combination of mechanical properties. The optimization of the mechanical properties requires a proper control of the sizes and volume fractions of the lamellar colonies and the interlamellar spacing. Another important aspect of these alloys is their microstructural stability, which determines the service life of components made from these during exposure to elevated temperatures. Phase transformation studies in these alloys, therefore, address two main issues: (a) the evolution of microstructure through different sequences and mechanisms of transformation and (b) the stability at elevated temperatures of the microstructure so obtained. 5.5.1 Structural relationship between 2 - and -phases In binary Ti–Al alloys in the composition range of 38–48 at.% Al, two ordered intermetallic phases, 2 -(Ti3 Al) and -(TiAl), coexist in equilibrium at temperatures below about 1270 K. The D019 structure of the 2 -phase has been described earlier. The -TiAl phase possesses a tetragonal L10 structure. The lattice parameters of the 2 -phase (with 38 at.% Al) and the -phase (with 48 at.% Al) are: 2 $ a = 057 nm% c = 046 nm% c/a = 0803 $ a = 040 nm% c = 040 nm% c/a = 1020 The axial ratio of the tetragonal unit cell of the -phase being very close to unity, the structure of the -phase can be approximated to be cubic and the transformation between the 2 - and the -structures can be considered as that between an ordered hcp and an ordered cubic structure. Figure 5.37 also shows the atomic arrangement and the interatomic distances on the basal (0001) plane of the D019 structure and on the close packed (111) plane of the L10 structure. It may
444
Phase Transformations: Titanium and Zirconium Alloys (0001) of α2-phase
(111) of γ-phase
Ti atom
Al atom
Figure 5.37. Atomic arrangement and the interatomic distances on the basal (0001) plane of the D019 structure and on the close packed (111) plane of the L10 structure.
be noted that the interplanar spacings of the (0002)2 and the (111) planes are very close, the difference being smaller than 0.5%. Again, the nearest neighbour distances in the close packed planes in these two phases differ only by about 2%. The near-perfect lattice matching between the close packed planes of the 2 (D019 ) and (L10 ) phases is the determining factor which controls the morphology of the lamellar morphology of the 2 + structure. A strict orientation relation between 2 and , which is invariably maintained in the lamellar 2 + structure, is revealed from the diffraction pattern shown in Figure 5.38. This orientation ¯ > < 110 > , first reported by Blackburn relationship, (0001)2 {111} ; < 1120 2 (1967), conforms to the commonly observed orientation relationship in fcc to hcp (002)γ
T
(2200)α
(002)γ
2
(220)γ
(2202)α
T
2
(111)γ (0002)α
(111)γ
000 2
(111)γ
T
γ fundamental γT fundamental α2 fundamental
(111)γ
T
(220)γ
γ superlattice γT superlattice α2 superlattice
Figure 5.38. Diffraction pattern showing orientation relation between 2 and , in the lamellar ¯ 110
¯ . 2 + structure. Zone axis: 1120
2
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445
structural transitions. The diffraction pattern (Figure 5.38) shows reciprocal lattice sections of two twin-related variants, the twinning plane being the specific {111} plane which is parallel to the basal plane of 2 . The lamellar aggregate of the - and 2 -phases can also be viewed as a stacking of close packed layers in the cubic ABCABC sequence over a certain distance corresponding to the thickness of the -plate, followed by a hexagonal stacking sequence of ABAB which builds up the 2 -structure. The diffraction pattern shown in Figure 5.38 suggests that a group of neighbouring lamellae contains regions not only corresponding to ABCABC stacking but also to ACBACB stacking, the two -regions being twin related across the {111} plane. Twin-related -plates can either remain in contact along the twin plane or may be separated from each other by 2 -plates. Single -plates are also seen to be divided into ¯ and [011] ¯ ordered domains which are generated due to the fact that the [110] directions of the L10 structure are not crystallographically equivalent. The possible orientation variants of 2 + structure are shown in Figure 5.39.
110
101
011
101
101
OR1
[011]
110
011
[101]
1120
110
011
011 OR2
OR6 1210
2110
011
α2
[110]
110
101
110
OR3
OR5
(a)
110
101
011
101 Ti Al
OR4 = γI (ABC-type stacking) = γII (ACB-type stacking)
(b)
Figure 5.39. The possible orientation variants of 2 + structure.
446
Phase Transformations: Titanium and Zirconium Alloys
5.5.2 Phase reactions The relevant portion of the binary Ti–Al phase diagram, which is based on the work of Perepezko and Mishruda (1993), is shown in Figure 5.40. It can be seen from the phase diagram and the transformation pathways indicated by vertical dashed lines that the 2 - and the -phases can evolve through one of the pathways listed below: (1) (2) (3) (4) (5)
→ → 2 → 2 → 2 + → + → 2 + → → 2 +
The phase evolution sequences reported in Ti–Al alloys containing about 35– 50 at.% Al (Jones and Kauffman 1993, Yamabe et al. 1995) are discussed in the following. 5.5.2.1 Ti-34-37 at.% Al; → → 2 As these alloys are cooled from the -phase field to the 2 -phase field, one of the three transformation processes takes place depending on the cooling rate.
1600 L α+L
β+L
1500
Temperature (°C)
β 4
α+β
1400 1
2 α
1 3
1300
1100
α+γ
α + α2
1200 α2
γ
5
α2 + γ
1000
35
40
45
50
Atomic % Al
Figure 5.40. A portion of the binary Ti–Al phase diagram depicting the transformation pathways, as indicated by vertical lines, for evolution of 2 - and the -phases.
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447
These are: precipitation of either - or 2 -plates in the -matrix involving longrange diffusion, composition-invariant massive transformation and diffusionless martensitic transformation. Precipitation of the -phase in the -phase matrix produces a basket weave microstructure akin to that commonly encountered in ( + ) Ti alloys. In contrast, a faster cooling induces a composition-invariant massive transformation in which massive -grains with jagged boundaries are produced. Al being a strong stabilizer, the To for the → transformation is about 1523 K. At such a high temperature the kinetics of massive transformation are expected to be very rapid. It is, therefore, difficult to suppress this massive transformation by any solid state quenching technique. The → 2 ordering temperature is also very high (between 1373 and 1473 K) and, therefore, the ordering process immediately follows the formation of the -phase, which forms either as Widmanstatten -plates or as massive -grains. 5.5.2.2 Ti-38-40 at.% Al; → 2 → 2 + The alloys which follow the transformation sequences (2) and (21 ) generally belong to hypo- and hyper-eutectoid compositions, respectively. Before discussing these transformation sequences it is worthwhile to consider a few special features of the eutectoid reaction → 2 + . As has been discussed earlier, the 2 -phase (D019 structure) is an ordered version of the -phase (hcp, A3 structure). The compositional proximity of the -phase of eutectoid composition (39.5 at.% Al) and the equilibrium 2 -phase (38 at.% Al) suggests that the extent of supersaturation required for a congruent ordering to occur is only marginal. This ordering transformation, occurring at a temperature in the range of 1373–1473 K, proceeds very fast without the requirement of any significant undercooling. The formation of the -phase from the -phase requires a change in the stacking sequence though the lattice matching of the two structures on (0001)2 (111) planes is excellent. In the case of a composition-invariant → transformation the structural change can be brought about by the passage of Shockley partials on every alternate close packed plane. This mechanism will be discussed in the next section. Such a lattice correspondence makes the kinetics of the lengthening of -plates quite fast but the kinetics of their thickening is rather slow. The overall kinetics of -phase formation is, therefore, much slower than that of the → 2 transition. In view of this, it is quite unlikely for the high-temperature -phase to decompose by a pearlitic mode in which the two product phases simultaneously emerge from the parent phase across the incoherent transformation front. The non-pearlitic modes of eutectoid decomposition which can operate in hypo- and hyper-eutectoid Ti–Al alloys are outlined in the transformation pathways (2) and (21 ), respectively.
448
Phase Transformations: Titanium and Zirconium Alloys
An alloy in the composition range Ti-38-40 at.% Al, when cooled from the -phase field to a temperature below the eutectoid, first crosses the To temperature for the → 2 ordering. As explained earlier, the equilibrium → 2 + eutectoid decomposition is kinetically quite unfavourable in comparison with the pathway → 2 → 2 + . The decomposition of the -phase in a pearlitic mode has, therefore, not been encountered in experiments. The fact that → 2 ordering precedes the formation of -plates is revealed from the observation that -plates cut across APBs in the 2 -phase. A typical time-temperature-transformation diagram for alloys containing 38–40 at.% Al has been schematically illustrated in Figure 5.41(a) which depicts the kinetics of the competing processes on a relative scale. Since the precipitation of the -phase from the 2 -phase involves an incubation period of nearly 100 s, it is possible to suppress the transformation by a rapid quench which produces a metastable 2 -structure. On subsequent ageing -precipitation occurs within the 2 -grains, producing a lamellar 2 + structure.
5.5.2.3 Ti-41-48 at.% Al; → + → 2 + In these hypereutectoid alloys, -phase precipitation in the -phase is the primary step that occurs during slow cooling from the -phase field. Again the same lamellar + morphology, dictated by the Blackburn orientation relation, is exhibited. Subsequently the -regions undergo the ordering process during cooling through the To temperature for → 2 ordering. At a somewhat faster cooling rate, the step comprising -phase precipitation in the -phase is skipped; the → 2 ordering occurs first which is followed by the decomposition of 2 into the 2 + lamellar mixture. During a still faster cooling, the formation of the -phase can be completely suppressed, resulting in the retention of a metastable 2 -product in the as-quenched condition (as shown in Figure 5.41(b). However, suppression of -phase formation by quenching is not possible in alloy compositions close to the -phase field. A competing → massive transformation comes into play in such alloys. For example, an alloy containing 48 at.% Al, in the as-quenched condition, shows only a small volume fraction of lamellar 2 + regions while the major constituent of the microstructure comprises large -grains which contain stacking faults, twins and sometimes APBs. The nature of these defects has been analysed in detail mainly for ascertaining whether the → transformation goes through an intermediate disordered fcc phase. This point will be taken up later while discussing the mechanism of the massive → transformation, which was first reported by Wang and Vasudevan (1992) and later confirmed by Denquin and Naka (1995) and Zhang and Loretto (1995). Figure 5.41(c) shows a time-temperature-transformation
Temperature (°C)
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449
Eutectoid α → α2
α → α2 + γ
α2 → α2 + γ
α2 + γ (fine γ plates)
α2 (thermal APBs)
Log time (a) α /α + γ
Temperature (°C)
1200
α2 → α + γ
1090
Eutectoid α /α + α2
α2 → α2
Metastable extension
α2 → α2 + γ α2 (thermal APBs) α2 + γ (lamellar)
Log time (b) 1500
Temperature (°C)
b
Tα
1400
γLamellar
TO
1300
γFeathery
1200
γMassive
Tγ
1100 1000
100°C/s 200°C/s 300°C/s
500°C/s
900
1000°C/s
400°C/s
800
0.1
1
10
Time (s)
(c)
Figure 5.41. (a) A schematic time-temperature-transformation (T-T-T) diagram for alloys containing 38–40 at.% Al depicting the kinetics of the competing processes on a relative scale. (b) T-T-T diagram showing kinetics of (i) → + and (ii) → 2 → 2 + processes and (c) T-T-T diagram indicating a kinetic competition between the processes, → + (lamellar or feathery) and → (massive).
450
Phase Transformations: Titanium and Zirconium Alloys
diagram indicating a kinetic competition between the processes → + and → (massive). 5.5.2.4 Ti-49-50 at.% Al; → Alloys in this narrow composition range can transform into a single phase structure under the equilibrium condition. Two distinct types of product microstructure have been encountered; one is described as Widmanstatten colony and the other as massive (Yamabe et al. 1995). In the former case, the original -grains are divided into or -like regions with different orientations. Subsequently these regions transform into a lamellar microstructure containing primarily -plates. The mechanism of and the motivation for the creation of -regions of different orientations have not been identified so far. The second mechanism of the → transformation, namely, massive transformation, operates at faster cooling rates under which Widmanstatten colony formation is suppressed. 5.5.2.5 Ti49-50 at.% Al; → When alloys in this composition range, after solutionizing in the single phase field, are brought to the + phase field, -plates precipitate. The cubic symmetry of the parent -phase allows the formation of different variants of -plates along the {111} habit. Some illustrative examples of different transformation products in -TiAl-based alloys are shown in Figure 5.42. The formation of the lamellar microstructure and of the massive -phase are the two most important phase transformation phenomena encountered in the -TiAl alloys. In addition, discontinuous coarsening of the lamellar structure plays a major role in the evolution of microstructure. The suggested mechanisms underlying these phenomena are summarized in the following section.
100 μm
(a)
10 μm
(b)
Figure 5.42. Illustrative examples: (a) massive m products in Ti–49 at.% Al alloy and (b) two-phase lamellar 2 + structure observed in light microscopy.
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5.5.3 Transformation mechanisms 5.5.3.1 Formation of the 2 + lamellar microstructure The overall transformation process for the → 2 + decomposition can be viewed as consisting of the ordering reaction → 2 and the precipitation reaction → . Ordering precedes or follows precipitation depending on whether the decomposition temperature is below or above the To temperature for ordering. In both the cases, the ordered 2 -superlattice is created on the hcp lattice of the -phase by the required atomic replacements. The lattice correspondence between the A3 and D019 structures, in which their respective basal planes remain parallel, allows the formation of only a single rotational domain. However, APBs corresponding to translational domains associated with displacement vectors of the ¯ > are formed. Such APBs within the 2 -phase are usually known type 13 < 1120 as thermal APBs. The precipitation of the -phase from the -phase involves both structural and chemical changes. The former is accomplished mainly by a transition in the stacking sequence from the hexagonal to the cubic type and the latter requires preferential Al enrichment of the regions where -precipitates appear. These precipitates invariably maintain lattice registry with the parent crystal and the precipitate plates have a strong tendency to form into a group. The kinetics of the lengthening of the -plates along the (0001)2 habit is much faster than that associated with their thickening. Groups of -plates often appear in the vicinity of grain boundaries and occasionally within the /2 grains. The habit plane being the basal plane, only one habit plane of -variant can form within a given 2 -grain. With the growth of these -precipitates, every single grain of the parent /2 structure gets transformed into a lamellar 2 + structure with a fairly uniform interlamellar spacing. The nucleation of -precipitates in a group and the uniform spacing between the -lamellae suggest that the formation of -plates and their growth do not take place individually but in a cooperative manner, the concentration field ahead of the edge of the plates being responsible for deciding the lamellar spacing. The most important difference between this mechanism and that of a cellular growth (as in the case of discontinuous precipitation or pearlitic transformation) lies in the fact that the lamellar product is not separated from the parent phase by an interface (Figure 5.43). This is because of the close lattice matching between the advancing -lamellae and the parent /2 matrix across the transformation front. This point is illustrated schematically in Figure 5.43. Transmission electron microscopy (TEM) studies have revealed some important features of the nucleation and growth steps of -phase precipitation. Blackburn (1970) and Sundararaman and Mukhopadhyay (1979) have shown that the nucleation of the -phase occurs heterogeneously, solely by the dissociation of a ¯ > dislocations on the basal plane. The stacking faults enclosed by the < 1120 6 2 a ¯ > partials can be considered as single layer nuclei of the -phase. < 1010 6 2
452
Phase Transformations: Titanium and Zirconium Alloys
→
→
→
(a)
(b) Transformation front Alloy partitioning α /α2 Replacement ordering α /γ transformation chemical ordering + change in stacking sequence
γ
α2
(c)
Incoherent boundary No lattice site correspondence Cellular transformation
(d)
→
(e)
(f)
Figure 5.43. Schematics of difference between mechanism of lamellar and of a cellular growth as in the case of discontinuous precipitation or pearlitic transformation.
Preferred sites for such nucleation are grain boundaries, subgrain boundaries and individual dislocations. The density of these faults has been found to increase as the cooling rate from temperature above the 2 / + 2 transus is decreased. The growth of these nuclei to -plates can be compared with the hcp → fcc transformation in so far as the stacking sequence is concerned. The passage of a ¯ > partial dislocations over every alternate basal plane produces the < 1010 6 2 cubic stacking sequence. One of the possible modes by which a single layer stacking fault can grow into a -plate of several tens of nanometre thickness is by the operation of a pole mechanism in which a Shockley partial dislocation is anchored to a sessile [0001]2 pole dislocation. The rotation of the Shockley partial will change the stacking sequence, and during rotation the partial climbs along the pole through a distance equal to twice the interplanar spacing; thus each
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453
rotation causes the product phase to thicken by two atom layers. Mukhopadhyay (1976) has provided some evidences for the presence of spiralling Shockley partial dislocations on the basal plane of the 2 -phase. Partitioning of Al atoms between the 2 - and -phases must occur through diffusive jumps in order to allow the -plates to attain the equilibrium composition. The movement of Shockley partials and the Al enrichment of the faults possibly take place concurrently, as the latter is expected to reduce the stacking fault energy. No doubt it is difficult for individual atoms to join the -lamellae on the broad face which is a highly stable low-energy planar interface between the 2 and -lamellae. In contrast, lateral growth of -lamellae through the movement of Shockley partials on the habit plane by the so called “terrace-ledge-kink” mechanism is a plausible mode in which the appropriate composition change of a growing lamella is ensured by the transfer of atoms on to kinks which provide favourable sites for atomic attachment. The fact that a combination of shear and diffusional processes is operative for the growth of -lamellae has been clearly revealed by high-resolution electron microscopy studies (Huh et al. 1990) on the 2 / interfaces which show a lateral ¯ > dislocations and by atom probe investigagrowth by the passage of a6 < 1010 2 tions of Denquin and Naka (1995) showing the sluggish nature of the diffusional growth. The lamellar -precipitates formed during fairly rapid cooling exhibit non-equilibrium partitioning of Al in the 2 - and -phases, the former remaining supersaturated in Al and the volume fraction of the latter being much larger than that corresponding to the equilibrium condition. The compositions of these two phases and their volume fractions approach the equilibrium values on ageing. It is to be noted that the results of the atom probe experiments indicate the absence of any concentration gradient around the 2 / interfaces; this suggests that the lamellar growth is controlled by the ledge mechanism rather than by a classical long-range diffusional process. The growth rate slows down with decreasing Al supersaturation of the matrix which reduces the driving force for the ledge-kink motion on the broad face. 5.5.3.2 Mechanism of the → massive transformation Ti–Al alloys (48–50 at.% Al) which are quite close to the -phase field, when fast cooled from the -phase field, transform into the single -phase. This transformation takes place without any change in composition and the morphology of the product phase has several features which suggest that the transformation is of the massive type. Wang and Vasudevan (1992) were the first to report the occurrence of such a massive transformation in this system and this observation has been confirmed in a number of more recent studies (Jones and Kauffman 1993, Zhang and Loretto 1995, Denquin and Naka 1995, Wang et al. 1998). The
454
Phase Transformations: Titanium and Zirconium Alloys
-phase produced by this massive transformation is generally designated as m in order to distinguish this phase from the lamellar -phase. The quenched structure of alloys containing 48–50 at.% Al often shows a mixed morphology with massive m -regions coexisting with lamellar 2 + regions. The orientation of the 2 -lamellae in a given region gives the orientation of the prior -grain, as the -and 2 -structures are related by a unique orientation relation. Attempts aimed at finding out a possible orientation relation between the prior -grain and the m -product have not been successful (Denquin and Naka 1995), implying that no specific orientation relation exists between them. The interfaces separating adjacent grains of the m -phase and those between m - and matrix 2 phases have been found to be quite jagged and it is clear that such a morphology cannot result from a shear-like process in which a planar contact between the parent and product phases is expected. The massive m -grains consist of a large number of faulted domains. The defects present in these domains are of various types: stacking faults lying on all the four variants of {111} planes, microtwins and APBs which do not adhere to any crystallographic plane. The displacement vectors associated with these APBs correspond to a2 <011> vectors. A detailed contrast analysis of the defects in the m -phase (Wang et al. 1998) has revealed that both 21 <110 and 21 <101 unit dislocations are present in this phase, the latter being linked by highly curved non-conservative APBs. It has also been shown that wide stacking faults, which are created by the dissociation of 21 <101> unit dislocations, lie on {111} planes and are bound by Shockley partials of all possible types. Hug (1988) has examined the possibilities of the formation of different types of stacking faults in the L10 structure of -TiAl and have identified the following three types: complex stacking faults (CSF), superlattice intrinsic stacking faults (SISF) and superlattice extrinsic stacking faults (SESF). A CSF can be considered to be a superimposition of an SISF and an APB. The dislocation dissociation reactions which can result in the formation of these stacking faults are 1 ¯ 1 ¯ 1 ¯ ¯ 110 → 211 + SF + 12 1 2 6 6 1 1 101 → 101 + APB + 101 → 2 2 1 ¯ 1 1 ¯ 1 112 + SISF + 211 + APB + 112 + CSF + 211 6 6 6 6 Because of the extremely high APB energy of -TiAl, the [101] dislocations are expected to be present as pairs of narrowly spaced 21 [101] dislocations linked by
Ordering in Intermetallics
455
APBs, each 21 [101] dislocation further dissociating into Shockley partials connected by a SISF or a CSF. Observations on deformed -TiAl have shown that the separations between the partial dislocations bounding SISFs and APBs are less than 6 and 4.5 nm, respectively. In contrast to these narrow separations observed in deformed -TiAl, Shockley partials of all types have been found to be dissociated, creating stacking faults of about 400 nm width, in the m -phase. Such a wide separation of Shockley partials in m can arise if these stacking faults are inherited from the parent -phase. However, this possibility is ruled out as faulting in that case would have been confined only to that {111} plane variant which is derived from the basal plane of the parent -structure. A second possibility is that these wide stacking faults are created during the → m transformation. However, it appears quite unlikely that these faults form in the ordered L10 structure either during the transformation or during subsequent cooling in view of the high APB and CSF energies. The presence of APBs and three variants with the m -phase has prompted Zhang et al. (1996) to propose the following intermediate step in the massive transformation: (massive transformation) → disordered fcc phase (ordering transformation) → m . The existence of widely separated dissociated 21 <110 unit dislocations and 21 <101 unit dislocations and the presence of all possible {111} stacking faults bounded by widely separated Shockley partials are perhaps better evidence for the occurrence of the intermediate step in which the disordered fcc phase is formed. The APBs which are intimately attached to these dislocations are essentially by-products of the ordering reaction from the fcc to the L10 structure. The question as to whether an orientation relation between the parent hcp and the product fcc phases exists at the time of nucleation has not been adequately answered. Heterogeneous nucleation of the massive product at grain boundaries has been observed (Denquin and Naka 1995). It is possible that the nuclei of the massive product form at grain boundaries, by creating a low-energy interface between the massive product and one of the grains. The highly coherent character of this interface will not allow it to propagate within the parent grain. The interface can, however, grow into the neighbouring grain with which no special orientation relation exists. The absence of a specific orientation relation between the m -and the adjoining 2 -regions (which represent the orientation of the parent -grain) can, therefore, be explained even if a strict orientation relation does exist in the nucleation stage. The growth of a massive product occurs by thermally activated short-range jumps of atoms from the lattice sites of the parent to those of the product. The atomic jumps take place across the transformation front which is either an incoherent interface or a terraced surface of good atomic fit, interspersed with ledges. While a curved and jagged interface of the product grains characterizes the former
456
Phase Transformations: Titanium and Zirconium Alloys
type, a planar interface is expected in the case of the latter type of transformation front. The observed curved and generally irregular interfaces between the m domains within a colony, between colonies and between adjacent m and matrix 2 -regions suggest that the massive transformation in the Ti–Al system takes place through a mechanism involving atomic jumps across incoherent interfaces. 5.5.3.3 Discontinuous coarsening of the lamellar 2 + microstructure The stability of the lamellar structure during exposure to elevated temperatures is a matter of serious concern in view of the intended high-temperature applications of -TiAl-based alloys. The driving force for microstructural changes during such exposure can arise either from chemical free energy changes resulting from changes in the volume fractions and compositions of the constituent phases or from the reduction in the total interfacial energy of the system. In the case of the lamellar 2 + microstructure (Figure 5.44), a combination of both the factors is expected to contribute towards microstructural changes during exposure to elevated temperatures. This is because the lamellar structure contains a very large area of interphase interfaces and the constituent phases may not have the equilibrium compositions and volume fractions at the exposure temperature. There have been a number of investigations on the microstructural stability of the lamellar 2 + structure, two of the important references in this regard being those of Denquin and Naka (1995) and Ramanujan et al. (1996). It has been shown that coarsening of the lamellar structure takes place in either a continuous or a discontinuous mode. Continuous lamellar dissolution has been shown to be
Figure 5.44. Lamellar 2 + structure produced on cooling a hypereutectoid Ti–Al alloy in a transformation sequence → + → 2 +. Orientation distributions are marked on the micrograph (cf. Figure 5.39).
Ordering in Intermetallics
457
initiated by the formation of steps on the broad faces of lamellae and to progress by subsequent step migration. While continuous coarsening of the lamellar structure by progressive dissolution of some -lamellae occurs at temperatures in the vicinity of 1073 K, discontinuous coarsening becomes operative at still higher temperatures. The discontinuous coarsening process involves the propagation of - or 2 grain boundaries. When such boundaries move towards the primary fine lamellar structure, the fine structure is consumed and a much coarser lamellar structure develops behind the advancing front. This observation clearly suggests that the growing - or 2 -grain is different in orientation from the primary grain (containing the fine lamellar structure) which gets consumed in the process. The primary lamellar structure consisting of planar, extremely flat and highly coherent interfaces results from the -lamellae through the ledge mechanism. The solute supersaturation in the 2 / phase is primarily responsible for providing the driving force for the thickening of the -plates. The thickening process slows down and becomes more and more difficult with progressive reduction in the extent of supersaturation. The continuous coarsening of the structure requires a competitive diffusion-assisted growth between two lamellae while maintaining the volume fractions of the and 2 / phases unchanged. The driving force for such a coarsening process essentially arises from the reduction in the total interface energy. However, the highly coherent nature of the interface renders solute flow between two lamellae extremely difficult. In contrast, the discontinuous coarsening process takes place in the vicinity of grain boundaries where the solubility is higher than that in the matrix. Grain boundaries are, therefore, endowed with sufficient mobility. Discontinuous coarsening can thus be visualized as a process involving dissolution of primary -lamellae at the advancing grain boundary front and reprecipitation of coarser -lamellae behind this front. The orientation relationship between the /2 and the -lamellae is maintained in the discontinuously coarsened structure, though the lamellar morphology is considerably irregular compared to the primary structure. This essentially means that the interfaces between the /2 and the -lamellae are not maintained flat on an atomic scale over a long distance. The formation of exceptionally flat interfaces in the primary lamellar structure has its origin in the movement of Shockley partials. The irregularity observed in the discontinuously coarsened structure suggests that propagation of Shockley partials is not a determining factor in its formation. Ramanujan et al. (1996) have examined the possibility of refining and stabilizing the primary lamellar structure by W and B additions to -TiAl alloys. They have demonstrated that not only is the lamellar spacing reduced but the structure also becomes more resistant against coarsening when these alloying elements are introduced. Similarly, Cr and Nb additions have also been found to be beneficial in this context. The increased stability of the lamellar structure in these alloys
458
Phase Transformations: Titanium and Zirconium Alloys
could be related to a possible reduction in the interfacial energy or to a higher thermodynamic stability of the 2 -phase.
5.6
SITE OCCUPANCIES IN ORDERED TERNARY ALLOYS
The degree of ordering has a very important role to play in determining the mechanical properties, particularly ductility of ordered intermetallic compounds. It has been demonstrated in several cases that ternary alloy additions cause improvement in ductility in several ordered alloys either by dispersion of a ductile second phase at grain boundaries of the ordered phase or by altering the distribution of constituent atoms in the sublattice sites of the ordered structure. The latter process can be modelled on the basis of the chemical rate theory by computing the rates of atomic exchanges between different sublattices in a given superlattice. Experimentally site preferences in ordered compounds can be determined using the technique of atom location by channelling enhanced microanalysis (ALCHEMI). In this section a model for computing site occupancies in ternary B2 compounds is presented and the predictions of the model are compared with available experimental results in the ternary Ti–Nb–Al system. 5.6.1 Ordering tie lines Some of the most commonly studied intermetallic alloys such as FeAl and NiAl possess the B2 (CsCl) structure which can be described in terms of two interpenetrating simple cubic lattices with lattice displaced with respect to the other by the vector 21 21 21 . These two interpenetrating lattices, referred to as the - and the sublattice, are occupied exclusively by the A atoms and the B atoms, respectively, in a fully ordered AB alloy. In the B2 structure, every -site is surrounded by eight first near neighbour -sites and vice versa. The fact that all nearest neighbour bounds are between unlike atoms and that the atomic coordination around both the sublattice sites are identical made it possible to study the evolution of order from order parameter = 0 to 1, without encountering any cluster frustration. The occupancies of the two sites and in a ternary alloy with constituent atoms A, B and C can be conveniently represented graphically in a ternary composition triangle. The composition of the - and the -sites can be plotted as two points in this triangle as shown in Figure 5.45. The line joining these two points is called the ordering tie line (OTL). When the alloy is completely disordered, i.e. the concentration A, B and C in the - and the -sublattices is the same, which can be represented by a single point (the average composition of the alloy) in the composition triangle. As the alloy is progressively ordered, the concentrations
Ordering in Intermetallics
459
A
P
θ
ι xA, xB, xC
B
Q C
Figure 5.45. The composition triangle showing the composition of - and -sites and the ordering tie line.
at the - and the -sites which depart from the average composition can be represented by two points P and Q in the composition triangle. The line joining P and Q is defined as the OTL. This representation, first introduced by Hou et al. (1996), must be differentiated from a thermodynamic tie line in a ternary phase diagram. While the latter joins the points corresponding to the compositions of two different phases in equilibrium at a given temperature, the OTL joins the points corresponding to the compositions of two sublattices of the same ordered phase. The B2 ordering in a ternary system can be described using a single concentration wave vector along the <111> direction of the cubic lattice and two independent order parameters. The two order parameters, 1 and 2 , in the SCW approach (Khachaturyan 1983) are defined as differences between the occupancies of A and B atoms on the two sublattices in the B2 structure. If xij is the occupancy of i atoms on the j sublattice, the order parameters, 1 and 2 , can be defined as 1 = xAB − xA , 2 = xBB − xB . These two order parameters are related to the two independent degrees of freedom of the OTL, its length, l, and orientation, & (Figure 5.45). It can be shown from the √ geometry of OTL that √ 1 and 2 are related to l and & as 1 = 2l sin &/ 3 2 = −lcos & + sin &/ 3. From the relationship between l, 1 and 2 , it is clear that l corresponds to the amplitude of the < 111 > concentration wave in the SCW formalism. With increasing temperature the value of l decreases and reaches zero at the order– disorder transition temperature, Tc . The length of the OTL is a measure of the extent of ordering in alloy and is geometrically constrained to lie within the ternary triangle. Therefore, the maximum ordering is achieved when the OTL is extended to a stage when one of the sublattice composition touches either a point on the binary edge or a unary corner of the ternary triangle. The second degree
460
Phase Transformations: Titanium and Zirconium Alloys
of freedom, &, is a measure of the relative preferences of the three elements, A, B and C on the two sublattices and can be considered as the polarization of the < 111 > concentration wave responsible for the B2 ordering. It should be noted that & is not equal to zero at Tc and therefore does not characterize the transition. 5.6.2 Kinetic modelling of B2 ordering in a ternary system The model proposed in this paper is based on the Bragg–Williams (1934) theory of order–disorder transition using the MFA. Dienes (1955) has shown that the Bragg–Williams theory can be derived from kinetic arguments using chemical rate theory for estimating the rates of exchanges of atomic species A, B and C in the two sublattice sites, and . There are three possible exchange reactions, as shown below, which contribute to the ordering/disordering processes: A + B = A + B
(a)
B + C = B + C
(b)
A + C = A + B
(c)
Consider a ternary B2 alloy with the average composition, xi , where xi is the atomic fraction of element i. Let the concentrations at the - and the -sublattices be xi and xi , respectively. The reaction rate for each of the three exchange reactions shown above can be written in terms of a difference between the forward and the reverse reaction rates. As an example the reaction rate, R1 for (a) can be expressed as R1 = kF · xA · xB − kR · xA · xB where kF and kR are the rate constants for the forward and reverse reactions, which can be expressed in terms of Arrhenius type relationship as follows: Em − H/2 kF − F · exp − RT and E + H/2 kR − R · exp − m RT H is the enthalpy change associated with the atomic exchange process between the right site and the wrong site, as depicted schematically in Figure 3.18 and
Ordering in Intermetallics
461
Em is the migration energy of the atoms in the disordered state. As shown in Figure 3.18, the activation barrier in the disordered alloy is symmetric while that for an ordered alloy is asymmetric with a favourable bias for the exchange process which leads to ordering. F and R are the pre-exponential factors including activation entropies associated with the forward and the reverse reactions, respectively. The enthalpy changes H1 involved in the exchange reaction (a) can be written in terms of the nearest neighbour binary interaction parameters wAB , wAC and wBC between the unlike atom pairs A–B, A–C and B–C, respectively: (5.23) H1 = 2wAB xA xB − xA xB + 2wAC xA xC − xA xC = 2wBC xB xC − xB xC
(5.24)
The binary interaction parameter wij is defined as follows: wij = Vij − 21 Vii + Vjj , where Vii , Vjj and Vij are the bond enthalpies of i–i, j–j and i–j bonds, respectively. The enthalpy changes for reactions (b) and (c) can be written in an analogous manner as follows: (5.25) H2 = 2wBC xB xC − xB xC + 2wAC xA xC − xA xC = 2wAB xB xC − xB xC H3 = 2wAC xA xB − xA xB + 2wAB xA xC − xA xC = 2wBC xB xC − xB xC
(5.26) (5.27) (5.28)
The sublattice concentrations in a B2 alloy with an average composition xi can be expressed in terms of the length l and the orientation & of the OTL as follows: √ xA = xA − l sin '/ 3 (5.29) √ xA = xA + l sin '/ 3
(5.30)
√ xB = xB + lcos ' + sin '/ 3/2
(5.31)
√ xB = xB − lcos ' + sin '/ 3/2
(5.32)
√ xC = xC − lcos ' − sin '/ 3/2
(5.33)
462
Phase Transformations: Titanium and Zirconium Alloys 1 0.9 0.8 0.7
l
0.6 0.5
R2 = 0
R3 = 0
R1 = 0
0.4 0.3 0.2
R1 = R2 = R3 = 0
0.1 0
20
40
60
80
100
θ (degrees)
120
140
160
180
Figure 5.46. Three binary interaction parameters, wij , and the rates of reactions (R1 , R2 and R3 ) plotted as function of l and &.
√ xC = xC + lcos ' − sin '/ 3/2
(5.34)
Using the equations and substituting the values of three binary interaction parameters, wij , the rates, R1 , R2 and R3 , of all the three reactions (a), (b) and (c) can be expressed as functions of l, & and T . Under the equilibrium condition at a given temperature R1 = R2 = R3 = 0, as shown in Figure 5.46. The values of l and & will give the equilibrium position of the tie line and the two terminating points of the OTL will define the fractional occupancies of A, B and C in - and -sublattice positions. 5.6.3 Influence of binary interaction parameters The nature of the interactions, wAB , wBC and wAC , can be either ordering (unlike bonds favoured relative to like bonds, w < 0 or clustering (like bonds favoured relative to unlike bonds, w > 0). How the orientation OTL is controlled by the nature and relative strength of the interaction parameters is illustrated in the following case studies: Case I: wAB < 0
wBC = wAC = 0
The OTL, calculated for the composition xA = xB = xC = 033, shows the OTL lying parallel to the AB line. Since A–C and B–C interactions are similar and are much weaker than the strong A–B interaction, the OTL assumes the orientation as indicated in Figure 5.47(a). This implies that, if the element C is added to
Ordering in Intermetallics A
A
Strongly ordering
0.0
C
B
0.0
Strongly ordering
(a)
(b)
A
A
Strongly ordering
0.0
B
0.0
0.0
B
463
Strongly ordering
C
(c)
Strongly ordering
0.0
B
C
Strongly ordering
C
(d)
Figure 5.47. (a) The OTL, calculated for the composition, xA = xB = xC = 033, showing the OTL lying parallel to the AB line for A–C and B–C interactions is similar and much weaker than the strong A–B interaction; (b) Similar to (a) with strongly ordering tendency between B and C atoms; (c) in a case where the binary B–C and A–C interactions are made strongly ordering and of equal magnitude (−0.65 eV), the calculated OTL for the alloy composition, xA = xB = xC = 033, lies along the perpendicular bisector of the AB side of the triangle; (d) by keeping the same interaction parameters, if the average composition of the alloy is changed, the OTL slope changes; and for the equiatomic composition alloy, the stronger A–C interaction causes the OTL to tilt towards a direction nearly parallel to the AC line.
the binary B2 ordered AB alloy, it will occupy the and -sublattices in equal fractions: Case II: wAB = 0
wBC = wAC ≤ 0
In a case where the binary B–C and A–C interactions are made strongly ordering and of equal magnitude (−065 eV), the calculated OTL for the alloy composition, xA = xB = xC = 033, lies along the perpendicular bisector of the AB side of the triangle (Figure 5.47(b)). Since both B–C and A–C ordering tendencies are equal, the OTL assumes a symmetrical orientation with respect to the vertice C. This implies that one sublattice is occupied primarily by C atoms while the other is equally shared by A and B atoms.
464
Phase Transformations: Titanium and Zirconium Alloys
Keeping the same interaction parameters, if the average composition of the alloy is changed, the OTL slope changes as illustrated in Figure 5.47(c). As the composition is made higher in A, the OTL rotates towards A. This is consistent with the tendency for maximizing the strongly ordering bonds. A similar effect is seen when the average composition is made richer in B: Case III: wAB = 0
wBC < 0 wAC < wBC
This case is examined with an example in which interaction parameters for the A–B, B–C and A–C pairs are 0, −013 and −065, respectively. For the equiatomic composition alloy, the stronger A–C interaction causes the OTL to tilt towards a direction nearly parallel to the AC line (Figure 5.47(d)). Again with a change in the average composition the tendency for the OTL to point towards a vertice is noticed. 5.6.4 B2 ordering in the Nb–Ti–Al system The model described has been applied to several alloy compositions in the Nb– Ti–Al system in which experimental data on site occupancy are available. Since the length of the OTL is a measure of the order parameter, one can obtain an order parameter (l/lmax ) versus temperature plot (Figure 5.48) from which the transition temperature, Tc , can be determined. The model described in the preceding section has been applied for predicting OTLs for the five ternary alloys in the Nb–Ti–Al system. The binary interaction parameters, wij (in kJ/mol), used for the calculations are as follows (Banerjee et al. 2001): wTi−Nb = +5 wTi−Al = −70 wNb−Al = −43. For this ternary system, the Ti–Al and Nb–Al interactions are strongly ordering, whereas the Ti–Nb interaction is weakly clustering. This
Order parameter (l )
0.3 0.25
Nb–40Ti–15Al
0.2
To = 1370 K
0.15 0.1 0.05 0
800
1000
1200
1400
Temperature (K)
Figure 5.48. Plot of an order parameter (l/lmax ) versus temperature from which the transition temperature, Tc , can be determined.
Ordering in Intermetallics Ti
Ti
900 K
1200 K
40 Ti
40 Ti
25 Ti
465 Ti
40 Ti
25 Ti
25 Ti
10 Ti
10 Ti
Nb
Al (a)
Nb
Al (b)
Nb
Al (c)
Figure 5.49. A comparison of the OTLs for this ternary Ti–Al–Nb system in which the Ti–Al and Nb–Al interactions are strongly ordering, whereas the Ti–Nb interaction is weakly clustering.
system would, therefore, be close to Case II as described earlier. Experimental determination of TLs for different alloy compositions has been made using the ALCHEMI technique and the experimental results reported by Hou et al. (1996) on 10Ti, 25Ti and 40Ti, by Banerjee et al. (1987) on Ti–24 at.% Al–11 at.% Nb and on Ti–25 at.% Al–25 at.% Nb are compared with calculated OTLs in Figure 5.49. As can be seen the agreement between the calculated and experimental OTLs is quite good. Possible factors which are responsible for the differences between calculated and experimental results are the errors in the interaction energies and the uncertainties in the equilibration temperatures.
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Phase Transformations: Titanium and Zirconium Alloys
Martin, P.L., Lipsitt, H.A., Nuhfer, N.T. and Williams, J.C. (1980) Titanium 80, Science and Technology Vol. II (eds H. Kimura and O. Izumi) TMS-AIME, Warrendale, PA, p. 1245. Mohri, T., Sanchez, J.M. and de Fontaine, D. (1985) Acta Metall., 33 (7), 1171. Mookerjee, A. (1973a) J. Phys., C6, 1340. Mookerjee, A. (1973b) PhD Dissertation, University of Cambridge, UK. Mookerjee, A. (1987) J. Phys., C17, 1511. Morita, T. (1984) J. Stat. Phys., 34, 319. Morita, T., Suzuki, M., Wada, K. and Kaburagi, M. (eds) (1994) Prog. Theor. Phys., Suppl. 115. Mozer, B., Bendersky, L.A., Boettinger, W.J. and Rowe, R.G. (1990) Scr. Metall., 23, 2363. Mukhopadhyay, P. and Banerjee, S. (1979) Phase Transformations, Inst. of Metallurgists, London, 2, 11–41. Mukhopadhyay, P., Raman, V., Banerjee, S. and Krishnan, R. (1978) J. Mater. Sci., 13, 2066. Mukhopadhyay, P., Raman, V., Banerjee, S. and Krishnan, R. (1979) J. Nucl. Mater., 82, 227. Muraleedharan, K., Gogia, A.K., Nandy, T.K., Banerjee, D. and Lele, S. (1992a) Metall. Trans. A, 23, 401. Muraleedharan, K., Nandy, T.K., Banerjee, D. and Lele, S. (1992b) Metall. Trans. A, 23, 417. Murray, J.L. (1987) Phase Diagrams of Binary Titanium Alloys, American Society for Metals, Metals Park, OH, p. 12. Murray, J.L. (1988) Metall. Trans. A, 19A, 243. Payne, M.C., Teter, M.P., Allan, D.C., Arias, T.A. and Joannopoulos, J.D. (1992) Rev. Mod. Phys., 64, 1045. Perepezko, J.H. and Mishurda, J.C. (1993) Titanium ’92: Science and Technology (eds F.H. Fores and I. Caplan) TMS, Warrendale, PA, p. 563. Ramanujan, R.V., Maziasz, P.J. and Liu, C.T. (1996) Acta Mater., 44, 2611. Rowe, R.G., Banerjee, D., Muraleedharan, K., Larsen, M., Hall, E.L., Konitzer, D.G. and Woodfield, A.P. (1992) Proc. 7th World Conference on Titanium, San Diego, CA. Rushbrooke, G.S. and Scions, I. (1955) Proc. Roy. Soc., A230, 74. Sanchez, J.M. and de Fontaine, D. (1978) Phys. Rev., B17, 2926. Sanchez, J.M., Gratias, D. and de Fontaine, D. (1982) Acta Crystallogr., A38, 214. Sastry, S.M.L. and Lipsitt, H.A. (1977) Metall. Trans. A, 8, 1543. Schulson, E.M. (1975) J. Nucl. Mater., 56, 38. Schulson, E.M. (1977) J. Nucl. Mater., 66, 322. Schulson, E.M. and Graham, D.B. (1976) Acta Metall., 24, 615. Simmons, J.P. (1992) PhD Thesis, Carnegie Mellon University, Pennsylvania. Singh, A.K., Nandy, T.K., Mukherjee, D. and Banerjee, D. (1994) Philos. Mag. A, 69, 701. Soffa W.A. and Laughlin D.E. (1988) Physical Properties and Thermodynamic Behaviour of Minerals, NATO ASI Series C, Vol. 225, p. 213. Soffa, W.A. and Laughlin, D.E. (1989) Acta Metall., 37, 3019.
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Soven, P. (1967) Phys. Rev., 156, 809. Strychor, R., Williams, J.C. and Soffa, W.A. (1988) Metall. Trans. A, 19, 225. Sundararaman, M. and Mukhopadhyay, P. (1979) Metallography, 12, 87. Taylor, D.W. (1967) Phys. Rev., 156, 1017. Temperley, H.N.V. (1972) Phase Transitions and Critical Phenomena, Vol. 1, Chapter 6 (eds C. Domb and M.S. Green) Academic Press, New York. Also, Stanley, H.E. (1971) Introduction to Phase Transitions and Critical Phenomena, Clarendon Press, Oxford. Toulouse, G. and Pfeuty, P. (1977) Introduction to the Renormalization Group Theory and to Critical Phenomena, Wiley, New York. Wang, P. and Vasudevan, V.K. (1992) Scr. Metall. Mater., 27, 89. Wang, P., Kumar, M., Veeraraghavan, D. and Vasudevan, V.K. (1998) Acta Mater., 46, 13. Wilson, K.G. and Kogut, J. (1974) Phys. Rep., 12, 75. Yamabe, Y., Takeyama, M. and Kikuchi, M. (1995) Gamma Titanium Aluminides (eds Y.W. Kim, R. Wagner and M. Yamaguchi) The Mineral, Metals and Materials Society, Warrendale, PA. Yvon, J. (1948) Cahiers Phys., 31, 32. Zhang, X.D. and Loretto, M.H. (1995) Philos. Mag. A, 71, 421. Zhang, X.D., Li, Y.G., Kaufman, M.J. and Loretto, M.H. (1996) Acta Mater., 44, 3735. Zhang, F., Chen, S.L., Chang, Y.A. and Kattner, U.R. (1997) Intermetallics, 5, 471.
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Chapter 6
Transformations Related to Omega Structures 6.1 Introduction 6.2 Occurrence of the -Phase 6.2.1 Thermally induced formation of the -phase 6.2.2 Formation of equilibrium -phase under high pressures 6.2.3 Combined effect of alloying elements and pressure in inducing -transition 6.3 Crystallography 6.3.1 The structure of the -phase 6.3.2 The – lattice correspondence 6.3.3 The – lattice correspondence 6.4 Kinetics of the → Transformation 6.4.1 Athermal → transition 6.4.2 Thermally activated precipitation of the -phase 6.4.3 Pressure-induced → transformation 6.5 Diffuse Scattering 6.6 Mechanisms of -Transformations 6.6.1 Lattice collapse mechanism for the → transformation 6.6.2 Formation of the plate-shaped induced by shock pressure in -alloys 6.6.3 Calculated total energy as a function of displacement 6.6.4 Incommensurate -structures 6.6.5 Stability of -phase and d-band occupancy 6.7 Ordered -Structures 6.7.1 Structural descriptions 6.7.2 Transformation sequences in Zr base alloys 6.7.3 Transformation sequences in Ti base alloys 6.7.4 Ordered -structures in other systems 6.7.5 Symmetry tree 6.8 Influence of -Phase on Mechanical Properties 6.8.1 Hardening and embrittlement due to -phase 6.8.2 Dynamic strain ageing due to -precipitation References
474 475 475 479 481 484 484 485 488 490 491 492 494 495 499 499 504 506 509 516 518 518 522 530 533 534 536 536 539 550
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Chapter 6
Transformations Related to Omega Structures
List of Symbols : Phase with hexagonal close packed crystal structure. : Phase with body centred cubic crystal structure : Phase with hexagonal crystal structure : Martensite phase with hexagonal close packed crystal structure : Martensite phase with orthorhombic crystal structure Ms ( ): Martensitic start temperature Ms (): Start temperature for athermal omega x: Composition x: Distance G: Free energy per mole G→ : Free energy difference between the , from the -phase T : Temperature To : Temperature at which the free energy of the two phases are equal S: Difference in the entropy of the two phases R: Gas constant P: Pressure a , c : Lattice parameters of the -phase a : Lattice parameters of the -phase K : Wave vector in the reciprocal space associated with the -phase : Reduced wave vector : Order parameter : Interplanar coupling parameter U: Displacement vector
o : Critical resolved shear stress
: Time constant for -precipitation m : Dislocation density ˙ : Macroscopic strain rate l: Mean free spacing
: Anti-phase boundary energy 473
474
f: b: r: : E:
6.1
Phase Transformations: Titanium and Zirconium Alloys
Volume fraction of the -particles Burger’s vector Average radius of -particles Yield stress Young’s modulus
INTRODUCTION
The -phase, which is an equilibrium phase in Group 4 metals at high pressures, forms in several alloys based on Ti, Zr and Hf and also in many other bcc alloys as a metastable phase. The occurrence of this phase was first reported by Frost et al. (1954) who found that aged Ti–Cr alloys containing ∼8% Cr were unexpectedly brittle. Subsequent X-ray diffraction patterns obtained from these alloys showed the presence of a second phase which was designated as the -phase. The formation of the -phase was extensively studied, initially because of its deleterious effects on mechanical properties, later because of its influence on such physical properties as superconductivity and most importantly because the → transformation exemplifies an interesting class of phase transformations. In recent years the structures of several ordered intermetallics based on Ti and Zr have been shown to be chemically ordered derivatives of the -structure. The lattice collapse mechanism of the → transformation, which will be described in detail in this chapter, therefore finds a much wider application in the context of the formation of these chemically ordered -structures. The -phase was encountered as an equilibrium phase in elemental Ti and Zr under static high pressures by Jamison (1963, 1964). The equilibrium phase boundaries in the pressure–temperature plane have been established from a number of high pressure studies, an excellent review of which is provided by Sikka et al. (1982). The crystallography of the → transition has provided some clues pertaining to the mechanism of this transition. The role of dynamic pressures (shock loading) in the → transition has also been investigated. The diffuse scattering features associated with the → transition have prompted intense theoretical and experimental activities including inelastic neutron scattering experiments for studying the soft phonon mode behaviour. Pretransition effects observed prior to the → transformation have attracted the attention of many research groups and a structural description of the pretransition state has now emerged which is consistent with phonon dispersion relations, observed diffuse scattering and high resolution electron microscopy observations. The electronic basis of the stability of the -phase has been elucidated by Sikka et al. (1982). They have shown that the athermal → transformation in alloys
Transformations Related to Omega Structures
475
of Ti and Zr requires a certain level of occupancy of the d-shells. They have also demonstrated the equivalence of effects of pressure and alloying for achieving a certain d-shell occupancy which causes the stabilization of the -phase.
6.2
OCCURRENCE OF THE -PHASE
As mentioned earlier, the -phase appears as a metastable phase in several binary and multicomponent alloys of Ti and Zr which contain the required levels of -stabilizing elements. In the case of the pressure-induced transformation, the equilibrium -phase can be generated from the -phase by the application of adequate static pressure. The → transformation is associated with a large hysteresis and because of this the -phase generated by the application of pressure can be retained in a metastable state at ambient pressure and temperature. The formation of thermally induced as well as pressure-induced -phase is discussed in the following sections, making use of the relevant phase diagrams. 6.2.1 Thermally induced formation of the -phase The -phase appears in Ti and Zr alloys when the stability of the -phase is enhanced by adding -stabilizing elements like V, Nb, Mo, Ta, Fe, Cr, Ni, etc. Binary phase diagrams with these elements belong to -isomorphous, -eutectoid and -monotectoid systems, as classified in Chapter 1. Figure 6.1(a) shows the Ti (or Zr) rich side of a typical binary phase diagram on which the metastable stability regime of the -phase is superimposed. The Ms temperatures corresponding to the → martensitic transformation (Ms ( )) and the athermal → transformation, (Ms ()) are also plotted against the concentration of the -stabilizing element. The composition, x1 , at which the Ms ( ) and the Ms () lines intersect, defines the composition limit up to which the → martensitic transformation occurs on quenching an alloy from the -phase field. An alloy having a composition, x, where x2 > x > x1 , on quenching from the -phase field, exhibits the formation of the athermal -phase which inherits the composition of the parent -phase. This metastable product undergoes compositional changes during subsequent ageing when -particles depleted in the -stabilizing element form, replacing the athermal -particles. The composition, x2 , corresponds to the limiting composition for the formation of the athermal -phase. Beyond this composition, the -phase is retained on quenching. The composition of the aged -phase is given by the metastable boundary between the - and the + phase fields. Schematic free energy versus composition plots corresponding to the -, the - and the -phases (Figure 6.1(b)) illustrate the relative stabilities of these phases. Construction of
476
Phase Transformations: Titanium and Zirconium Alloys
x
x
Equilibrium α /β Metastable ω /β Ms (α) Ms (ω) Athermal ω Thermal ω
x
x
x
↑
T
x
a
c
e f
d b
x
T1
x
298 K
↑ Gα
x2
x3
Gω
x
↑
x1
(a) T1
Gβ
G
c a
d
b
(b)
x→
Figure 6.1. (a) Schematic equilibrium phase diagram of binary alloys (Ti–V, Zr–Nb) from Groups 4 (solvent) and 5 (solute) (marked by solid lines). Superimposed metastable phase boundaries have also been shown by broken lines. Cross-over between Ms () and Ms () curves mark the composition range beyond which the -phase forms in preference to the martensites ( ) phase upon quenching (shaded region in the figure). (b) Schematic free energy curves for the -, - and the -phases. The letters ‘a’ and ‘b’ shown in the figures correspond to the compositions of and in equilibrium; ‘c’ and ‘d’ correspond to the compositions of and in metastable equilibrium and ‘e’ and ‘f’ correspond to Ms ( ), Ms (), respectively.
common tangents which represent the equilibria (stable or metastable) between (i) the - and the -phases and (ii) the - and the -phases defines the compositions corresponding to the end points of the tie lines within the + and the + phase fields. The composition limits for the formation of the -phase can be indicated with reference to the schematic phase diagram shown in Figure 6.1(a). The athermal → transformation occurs in alloys with compositions in the range x1 <x<x2 , while aged forms on isothermal treatments in the composition range x1 <x<x3 .
Transformations Related to Omega Structures
477
In general, the martensitic or phase does not coexist with the -phase, although there are some recent reports which show the presence of -particles in the retained -regions coexisting with martensitic -plates. The competition between the martensitic → and the → transformations has been studied in Ti–Nb alloys as a function of the alloy composition and of the severity of quench (Moffat and Larbalesiter 1988). They have found that the martensitic -phase forms in alloys containing upto 25 at.% Nb when the quenching rate is in excess of 300 K/s. Under slower quenching rates (∼0.3–3 K/s) the -phase precipitates in alloys containing up to 50 at.% Nb. Only alloys containing 60 at.% or more Nb have been found to retain the -phase fully upon quenching. While the lower concentration limit (25 at.% Nb) defines the point where the Ms ( ) curve interests the Ms () curve, the higher concentration limit (60 at.% Nb) corresponds to the point where the boundary of the metastable ( + ) phase field approaches room temperature. The Zr–Nb alloy system has been studied in detail with reference to the -transformation. The Ms ( ) and the Ms () curves in this system intersect at a Nb concentration of about 7 at.%, the Ms () curve comes down to room temperature at nearly 18 at.% Nb and correspondingly the metastable boundary of the ( + ) phase field approaches room temperature at a concentration level of about 30 at.% Nb (Figure 6.2). Table 6.1 shows the composition limits within which athermal and thermally activated -formation occurs in several alloy systems. The composition x( /) at which the Ms ( ) curve intersects the Ms () curve in a binary alloy system can be estimated from the concentrations corresponding to the boundaries
1136 K
β
Temperature (K)
α+β
β1 + β2
893 K
Ms(α ′)
18.5 β +ω 673
Ms(ω)
α′ 0
Zr
β +ω 10
1073
298
Metastable β 20
30
Atomic % Nb
Figure 6.2. Binary phase diagram of the Zr–Nb system showing the superimposed Ms () and Ms () lines. Composition ranges in which martensitic , athermal in the -matrix and metastable -structure are formed on -quenching are indicated.
478
Phase Transformations: Titanium and Zirconium Alloys
Table 6.1. Comparison of calculated and observed compositions where Ms ( ) and Ms () lines intersect in different Ti- and Zr-based alloys. Alloy system
Ti–V Ti–Cr Ti–Mn Ti–Fe Zr–Nb
Onset of athermal -formation (at.%) compositions at which Ms ( ) and Ms () intersect Calculated
Observed
16 8 6 5 9
14 6 4 4 7
Maximum solute content (at.%) of aged -formation observed
25–30 18–20
30–33
of two phase fields, either the equilibrium ( + ) field or the metastable ( + ) field. The slope of the To versus x plot, where To is the temperature at which the chemical free energies per unit volume of the parent and the product phases are equal and x denotes the composition (in at.% solute), can be related to the changes in the free energy of the solution G→ at xo and the changes in the entropy of the solvent, S → , due to the change in phase from to either or as:
→ dTo /dxx→ = Gx→ /STiZr o o
(6.1)
The above expression is based on the regular solution model. At low solute concentrations
→ Gx→ = RT lnx /x + STiZr T o
(6.2)
which reduces to
= RT lnx /x Gx→ o
(6.3)
The second term on the right hand side of Eq. (6.2) is much smaller compared to the first term. The simplifying assumptions and some data which are used for the estimation of the composition x ( /) at which the quenched product changes from the to the -structure in binary alloys of Ti and Zr are listed below: (a) The heats of transformation for the → transformation in Ti and Zr are taken from Kubaschewski et al. (1993) as 4.2 kJ/g mole for Ti and 3.9 kJ/mole for Zr which yield the values of S → as 0.74 for Ti and 0.85 for Zr.
Transformations Related to Omega Structures
479
(b) The steep slopes of the / transus in the pressure–temperature phase diagrams of pure Ti and Zr indicate that the differences in entropy between the - and the -phases in these metals are very small. In view of this S → is taken to be approximately equal to S → . (c) The To versus x plot is assumed to be linear. (d) The composition at which the Ms curves for the → and the → transformations intersect is taken to be the same as the point of intersection of the respective To lines. This implies that the supercooling (To − Ms ) required for the two transformations is assumed to be equal. This approximation is valid as the extents of supercooling required for these transformations are quite small due to a relatively small contribution of the strain energy associated with the nucleation of either of the product phases in these systems. The composition, x( /) can then be expressed as
x / =
→ → − TTiZr TTiZr
dTo→ /dx − dTo→ /dx
(6.4)
The difference between the / and the / transition temperatures for pure Ti and Zr being about 400 K, one obtains by substituting Eq. (6.1) in Eq. (6.4)
− G→ x / = S → 400/G→ xo xo
(6.5)
Using the Eqs. (6.3) and (6.5) the composition, x( /), at which the quenched product changes from to has been calculated and the result has been compared with the experimental value, wherever available (Table 6.1). 6.2.2 Formation of equilibrium -phase under high pressures High-pressure resistivity studies on pure Zr first showed a phase transition at a hydrostatic pressure of about 6.0 GPa (Bridgman 1948a,b). Subsequently, Jayaraman et al. (1963) detected transformations in Ti at 8.0–9.0 GPa and in Zr at 5.0–6.0 GPa using the same technique. The transformation was initially thought to be the → transformation, as the -phase was supposed to be the denser phase at that time. Jamison (1963) was the first to identify the high-pressure phase to be one with a simple hexagonal structure (-phase). Jamison also pointed out the difficulties in determining the equilibrium / phase boundary in Ti and Zr, as a large hysteresis is associated with the transformation. It has been observed (Jamison 1964) that the -phase, once formed by the application of a high pressure, does not revert back fully to the -phase on removal of the pressure. The metastable
480
Phase Transformations: Titanium and Zirconium Alloys
retained -phase remains in association with the equilibrium -phase at ambient pressure in samples which have been subjected to a pressure treatment. A complete transformation of the -phase into the -phase requires heating Ti to 380 K and Zr to 470 K for several hours after the removal of pressure. Using the equilibrium transformation data compiled by Kutsar (1975) for the → , → and → transformations, Sikka et al. (1982) have constructed a schematic pressure–temperature phase diagram (Figure 6.3) for the Group 4 transition metals. The triple point coordinates (Pt , Tt ), the equilibrium → transformation temperature at 0.1 MPa, T ( → ), the equilibrium / transformation pressure at 293 K, Po (/), and the → reversion temperature at 0.1 MPa, T ( → ), for Ti, Zr and Hf are indicated in Table 6.2. The hysteresis observed in the → transition is represented in Figure 6.3 by a pair of broken lines corresponding to the completion of the → and the → transitions. The fact that
Equilibrium lines Hysteresis lines
Temperature (K)
α
β
β
T
(Pt , Tt) α ω
ω
α α
T
ω
293
α
ω
Poα – ω
Pressure (GPa)
Figure 6.3. Schematic pressure–temperature phase diagram of Group 4 transition metals. Equilibrium lines are shown by solid lines and the values of the triple points are given in Table 6.2 for all the Group 4 metals. The broken line represents the hysteresis in the transformation from one phase to another. This hysteresis could be due to the presence of interstitial solutes in the pure metals.
Table 6.2. Calculated values of triple points for the Group 4 transition metals.
Pt (GPa) Tt (K) T ↔ (K) P ↔ (GPa) T ↔ (K)
Ti
Zr
Hf
9 1100 1155 2 380
6 975 1135 2.2 470
30 1800 2030 21 –
Transformations Related to Omega Structures
481
Table 6.3. Calculated values of the thermodynamic parameters required for various transformations in Group 4 transition metals. Ti
Zr
Hf
→
→
→
→
→
→
→
→
→
V (cm3 /mol)
−0032 −0036 −0052
0.18
0.14
−009
−029 −019 −02
0.1
−016 +01 −002
−045
0.43
S (J/mol/K)
3.68 3.64
−159
5.18
3.55
−125
4.81
3.34 2.51
2.93
5.85
dT /dp (K/GPa)
8 −26 −9 −10
90
−24
160
38 −7
150
70
27 112
−26
21
the -phase is partially retained at ambient conditions after a pressure treatment has also been depicted in this figure as the → hysteresis line intersects the temperature axis at 0.1 MPa at a temperature higher than room temperature. Additional data regarding the volume and entropy changes associated with different phase transitions and the slopes of various phase boundaries in the P–T plane are listed in Table 6.3. The → transition has also been studied in dynamic (shock) experiments involving pressure pulses of microseconds duration. Phase transitions under such conditions are usually revealed in the observed discontinuities in the plots of shock velocity (Us ) versus particle velocity (Up ). McQueen et al. (1970) have noticed the discontinuities in the Us versus Up plots for Ti (17.5 GPa, 370 K), Zr (26.0 GPa, 540 K) and Hf (40.0 GPa, 725 K). The transition pressures under shock loading conditions have been found to be substantially higher than those reported for static pressure investigations. An extensive hysteresis in the / transition has also been noticed in shock pressure experiments. 6.2.3 Combined effect of alloying elements and pressure in inducing -transition As pointed out in the preceding two sections, the -phase can be generated in Ti, Zr and Hf systems either by the addition of adequate amounts of a stabilizing alloying element or by the application of pressures sufficient for inducing the -transition. The combined effect of alloying elements and pressure has been studied by only a few investigators (Afronikova et al. 1973, Ming et al. 1980, Vohra et al. 1981, Dey et al. 2004). High pressure studies on Zr–Nb,
482
Phase Transformations: Titanium and Zirconium Alloys
the Ti–Nb and the Ti–V alloy systems have brought out the following common features: (a) The application of pressure promotes the formation of the -phase in the composition range where the -quenched structure is martensitic ( ). The composition range over which the -phase is present is, therefore, wider under pressure. (b) The -start pressure is lowered with increasing concentration of a -stabilizing element up to a certain point beyond which the -start pressure rises with further increase in solute concentration. The results of high-pressure studies on Ti–V alloys up to a pressure of 25.0 GPa are shown in the pressure–composition phase diagram in Figure 6.4. It may be noted that the boundaries of the different phase fields do not correspond to the equilibrium condition as these results are compiled from X-ray diffraction and resistivity data which were obtained from samples maintained under pressure, whereas transmission electron microscopy results pertain to pressure-treated samples which were brought back to ambient conditions. The features of the pressure–composition phase diagram in Figure 6.4 could be rationalized in terms of a set of hypothetical free energy versus concentration plots, as shown in Figure 6.5. Thus the essential features of experimental observations, such as the extension of the composition range of -formation and the variation in the -start pressure with concentration, could be explained on the basis of these
Pressure, GPa
20
ω
ω+β β
10
α+ω α Ti
10
20
30
40
50
V, Atomic %
Figure 6.4. Pressure–composition phase diagram of the Ti–V system at room temperature. As could be noticed from the figure, addition of V in Ti reduces the pressure for → phase transformation. Upon comparing this figure with Figure 6.2, a similarity between the trends in the transition temperature and transition pressure could be noticed.
Transformations Related to Omega Structures α
ω
α
α A
P = Po
P = P1 T = 300 K
β
↑ F
β
α+β
α α+ω
Composition →
α
ω
T = 300 K
β
↑ F
483
B
β
β+ω
Composition →
A
P = P2
α
T = 300 K
B
P = P3 T = 300 K
ω β ↑ F
α
A
α+ω ω
ω+β
Composition →
ω
↑ F
β
β
ω B
A
β ω+β Composition →
B
Figure 6.5. Hypothetical free energy curves as a function of composition and pressure. These curves are drawn based on the phases observed at various pressures and alloy compositions.
hypothetical free energy–concentration plots. Under the equilibrium condition a system possessing such concentration dependence of free energies for three competing phases would be expected to exhibit an eutectoid phase reaction, as shown in Figure 6.6. Though experimental confirmation of an eutectoid phase reaction in a -forming system in the pressure–concentration plane is still not available, the observed metastable Ti–V phase diagram (Figure 6.4) and a generalized version of the same (Figure 6.6) bring out the basic trends in relative phase stabilities as functions of pressure and concentration. Of late the formation of the -phase was observed in alloys upon shock loading (Hsiung and Lassila, 2000). These alloys under normal conditions do not show the formation of the -phase. Recently, Dey et al. (2004) have shown presence of in the Zr–20 Nb alloy upon shock loading. The morphology of the -phase has exhibited plate-like morphology. In both these studies the plane of contact between the - and the -phases was the
484
Phase Transformations: Titanium and Zirconium Alloys
P3
ω
Pressure
↑ P2
β+ω
α+ω
P1
α P0
Composition
↑
A
β α+β
B
Figure 6.6. A generalized version of the phase diagram constructed on the basis of the free energy plots shown in Figure 6.5. This phase diagram suggests the presence of an eutectoid phase reaction in the pressure–composition phase diagram.
{112} plane of the -lattice. Such observations showed that the -structure can also be visualized as a defect in the bcc lattice akin to twinning (Hsiung and Lassila, 2000).
6.3
CRYSTALLOGRAPHY
The crystallography of the → transformation has provided the essential clue to the transformation mechanism and is, therefore, a subject which has been investigated in great detail. Soon after the -phase was first reported, a controversy arose regarding the crystal structure of this phase. The structure of the -phase is described in the following section in which a summary of the early controversy is also given. The lattice correspondences between (i) - and -phases and (ii) - and -phases are discussed subsequently. 6.3.1 The structure of the -phase There has been a serious controversy regarding the crystal structure of the -phase. Early studies suggested that the -phase had a complex bcc unit cell with a lattice parameter three times that of the -phase. In fact, selected area diffraction (SAD) patterns taken from the + structure corresponding to all zones excepting the
Transformations Related to Omega Structures
485
<100> and the <111> zones, invariably show extra diffraction spots which divide all bcc reciprocal lattice vectors into three equal segments. It has subsequently been recognized that the occurrence of all the variants of -particles which are distributed in the -matrix gives rise to composite diffraction patterns which possess the symmetry of the matrix. The -structure has later been determined to be either hexagonal (belonging to the space group D16H (P6/mmm) (Silcock et al. 1955) or trigonal (belonging to the ¯ space group D33D (P3m1) (Bagaryatskiy et al. 1955). Both these structures can be generated from the parent bcc structure by simple atom movements, as discussed in the following section. The discrepancy between the reported hexagonal and trigonal structures of the -phase has also been resolved in a study carried out by Sass and Borie (1972) who have shown that increasing solute (-stabilizer) concentration causes the sixfold symmetry characteristic of D16H to degenerate into the threefold symmetry characteristic of D33D . 6.3.2 The – lattice correspondence The lattice correspondence between the - and the -structures not only establishes the crystal structure of the -phase but also provides a clue to the transformation mechanism. The orientation relationship between the two phases has been determined in a large number of investigations and has been unanimously described as ¯ > < 1120 ¯ > 111 0001 < 110 The relation between the lattice geometries of the two phases can be best illustrated in a diagram showing the stacking sequence of the (222) atomic planes of the bcc structure (Figure 6.7(a)). The ABCABC stacking sequence of these atomic planes generate the open bcc structure with the threefold axes along the <111> direction. The -lattice can be created by collapsing a pair of (222) planes to the intermediate position, leaving the next plane undistorted, and collapsing the next pair of planes and so on (Figure 6.7(b)). When the collapse is complete a sixfold rotation symmetry is created around the specific <111> direction along which lattice collapse occurs (Figure 6.7(c)). In case the collapse is incomplete, the trigonal symmetry is not lost, and the resulting -structure is associated with a trigonal structure with the space group D33D . The - and the trigonal structures can be formed by introducing a displacement wave in the parent bcc () structure. This point is illustrated in Figure 6.7(d) where a displacement wave with the wave vector, K= 2/3 < 111 >, can create the -structure when the amplitude of the wave is sufficient for moving {222} planes by a distance 1/2{222}. When the amplification of the wave is smaller, the resulting structure is trigonal. On the basis of the lattice correspondence, shown in Figure 6.7, the lattice parameters of
486
Phase Transformations: Titanium and Zirconium Alloys [111]
[0001] Layer No. 3 2 1.5 1 0
[011]β [110]β
(222)
[101]β
[1210]
(0001)
[2110]
ω-hexagonal
β bcc (a)
(b)
2 Layer 1 Layer 0 Layer [2110] (c)
Displacement Up
[1210]
+ – 0
1
2
3
4
5
6
(d)
Figure 6.7. Planar arrangement of the {222} atomic planes in the bcc lattice; (a) the ABCABC sequences of atomic plane arrangements could be noticed (planes are marked as 0,1,2,3 ). (b) The planar arrangement of the (0001) atomic planes in the -lattice. As could be observed the -lattice could be obtained from the bcc lattice by collapsing a pair of planes (1 and 2, as shown in figure) in the middle (marked as 1.5) while keeping the third one (0 and 3) undisturbed. (c) Completely collapsed {222} planes form a sixfold symmetry whereas an incomplete collapse forms a threefold symmetry along the <111> axis. (d) The displacement of {222} planes could also be visualized in terms of a sinusoidal displacement wave where an upward movement is seen as a positive displacement and a downward movement as a negative displacement.
the -structure (a and c√ ) can be expressed in terms of the bcc lattice parameter, √ a :a = 2 a and c = 3/2.a . A comparison of the lattice parameters of the -phase computed from the a value of the corresponding -phase and those experimentally observed shows that the divergence between the two is negligibly small for the athermal → transformation (Table 6.4). A similar comparison between the corresponding lattice dimensions of the -phase and the aged -phase, which is in metastable equilibrium, reveals that the isothermal → transition is associated with a linear contraction of about 5% (volume contraction ∼15%). Since there are four sets of <111> directions, there are four possible crystallographic variants of the -structure in a given bcc parent crystal. The matrix representation of the lattice correspondence for all the four variants is shown in
Transformations Related to Omega Structures
487
Table 6.4. The calculated and experimentally determined values of the lattice parameters of the -phase encountered in different alloy systems. Alloy system
Calculated values of lattice parameters (nm)
Experimentally determined values of lattice parameters (nm)
Difference (per cent)
Zr–Nb
a = 0.5003 c = 0.3064
a = 0.5019 c = 0.3089
a = 031 c = 082
Zr–Nb–Cu
a = 0.5003 c = 0.3064
a = 0.5029 c = 0.309
a = 052 c = 085
Ti–Nb
a = 0.4596 c = 0.3031
a = 0.4627 c = 0.2836
a = 067 c = 643
Ti–V
a = 0.4596 c = 0.28146
a = 0.460 c = 0.282
a = 008 c = 019
Table 6.5. Correspondence matrices relating the -(bcc) and the -(hexagonal) structures. Variant
Orientation relationship
1
¯ [12 ¯ 10] ¯ (111) (0001) ; [110]
2
¯ [12 ¯ 10] ¯ ¯ (0001) ; [1¯ 10] (111)
3
¯ [12 ¯ 10] ¯ ¯ (0001) ; [110] (1¯ 11)
4
¯ 10] ¯ ¯ (0001) ; [110] [12 (111)
Correspondence ⎡ 2 2¯ ⎣0 2 2¯ 0 ⎡ 0 2¯ ⎣ 2 2¯ 2¯ 0 ⎡ 2¯ 2 ⎣ 0 2¯ 2¯ 0 ⎡ 0 2 ⎣ 2¯ 2 2¯ 0
matrix [R] ⎤ 1 1⎦ 1 ⎤ 1¯ 1⎦ 1 ⎤ 1¯ 1¯ ⎦ 1 ⎤ 1 1¯ ⎦ 1
Table 6.5. Within the same orientation variant of the -structure, associated with a given <111> variant of lattice collapse, three distinct subvariants can be created, depending upon whether the A, B or C plane remains as the uncollapsed plane. These subvariants are schematically illustrated in Figure 6.8. An -region within the bcc phase remains coherent with the parent, as seen from the small difference in their lattice dimensions. However, given the total number of variants and subvariants (4 × 3 = 12), the elastic strains (arising from the atomic displacements and the slight change in volume) associated with different particles interact with one another. This elastic interaction is a controlling factor which governs the arrangement of -particles in the -matrix.
488
Phase Transformations: Titanium and Zirconium Alloys ω -Subvariants A B C A B C A B C A
A
B
C
Uncollapsed plane
Figure 6.8. Schematic presentation of the translational subvariants of the -phase produced due to the random collapse of different {222} planes of the bcc lattice. Regions of the different subvariants are marked in the figure.
6.3.3 The – lattice correspondence Diffraction experiments have established that the following two orientation relations exist between the - and the -phases: Orientation Relation I (OR I) ¯ <1120> ¯ <1101> ¯ 0001 0111 Orientation Relation II (OR II) ¯ <1120> ¯ 0001 0001 1120 On the basis of these observed orientation relations two distinct mechanisms, involving either the direct → route or the two stage → → route, have been proposed for the transformation from the to the -phase. Silcock et al. (1955) first proposed a model for the formation of the -lattice directly from the lattice which leads to the orientation relation described in OR II. The shift in atomic positions required for the direct → transformation is shown in Figure 6.9. The (112¯ 0) plane is generated from the (0001) plane by displacement of alternate hexagons on the (0001) plane by 0.148 nm (for Ti). The atoms marked a, b, c and d shift to positions a , b , c and d . In addition, a contraction of ∼4.7% along ¯ directions are required ¯ and an expansion of ∼4.5% along the [1100] the [1120] for the generation of the -unit cell. Usikov and Zilbershtein (1973) have suggested that the → transformation occurs via the intermediate -structure which is unstable at the pressures and temperatures under consideration. This suggestion is based on the fact that the
Transformations Related to Omega Structures
489
0.7667 nm
d′
0.295 nm
d
c′
a′
c
a
b′
b
[1120]α ⎥⎪ [0001]ω [1100]α ⎥⎪[1100]ω (0001)α⎥⎪(1120)ω
¯ plane of the Figure 6.9. Lattice correspondence between the - and the -phases. The (1120) -phase is generated from the (0001) plane of the -phase by shifting the atoms at a, b, c, d to a , ¯ and expansion along [1100] ¯ direction is also needed b , c , d positions. Contraction along [1120] to generate the -unit cell.
orientation relation observed by them (OR I) can be described in terms of the product of the correspondence matrices R and R which pertain to the → and the → transformations, respectively. They have shown that all the observed orientation relations between the - and the -phases, as revealed from diffraction patterns, are consistent with the correspondence matrix associated with OR I. Later TEM studies (Vohra et al. 1980, 1981) on high-pressure treated Ti–V alloys, subjected to high pressures where alloying addition promotes -phase formation, have confirmed the presence of the -phase along with the -phase in the matrix of -phase grains. This observation provides a direct evidence of the occurrence of the -phase as an intermediate state during the → transition. The fact that the stability of the -phase is enhanced by V additions appears to have been responsible for the retention of this intermediate phase. Orientation relations have also been determined in samples in which the -phase has been made to form by the application of shock pressure. Kutsar et al. (1990) have reported that the / orientation relation in shock-treated Zr matches with OR II. In some recent investigations Song and Grey (1994, 1995) have observed a new orientation relation, as described below, between the - and the shock-induced -phases: ¯ 1100 ¯ <1123> ¯ 0001 1011 On the basis of this observation they have proposed a different transformation mechanism according to which the -phase forms directly from the -phase. Jyoti
490
Phase Transformations: Titanium and Zirconium Alloys
]
d 1120 [
j
e
k 10
]
12 0°
(2.5725)
[10
f
c
(2.5725)
h
j
(2.5725)
g k
(2.5725)
e
(2.5418)
f
i b
d
a
(0001)α (a)
b
(2.5418)
j
e (+ 4219)
k
h
(2.1098)
f h
(– 4219)
a
i
d [11
g
(2.5418)
(2.5725)
]
01
3] 12 c
[1
(2.5418)
109 °
0] 10 c [1
(2.9533)
b
(011)β (b)
a
i
(1011)ω (c)
Figure 6.10. Schematic presentation of (a) the (0001) plane of the -structure (b) the (011) plane ¯ plane of the structure. Filled circles represent atoms on the plane of the and (c) the (1011) whereas open circles represents atoms away from the planes. The distance of separation is given in parenthesis (in Å). A reference set of atoms is marked on each projection. As could be noticed very small movement of atoms j, k, h and g on (0001) plane is needed to obtain either the -planar or the -phase. A close resemblance suggests a possible path of the → phase transformation may have the -phase as the metastable phase as an intermediate path. Results are more inclined to direct → transformation (for detail see text).
et al. (1997) have examined SAD patterns belonging to a larger number of zone axes for determining the orientation relation between the - and the -phases and have attempted a harmonization of all the conflicting results reported earlier and have demonstrated that nearly all the orientation relations reported earlier can be expressed as subsets of OR I. They have considered the possibility of the formation of the -phase through an intermediate -structure and have compared the crystallographic results with those expected from the direct formation of the -phase from the -phase. A schematic representation of the atomic arrangements pertinent to either of the routes is shown in Figure 6.10. Small variation in the positions of atoms marked in the figure generates all the three structures indicating a close resemblance among these structures.
6.4
KINETICS OF THE → TRANSFORMATION
The → transformation has been found to proceed under two different conditions, namely, during quenching from the -phase field and during isothermal holding at a temperature below about 773 K. The kinetic characteristics of these two cases correspond to an athermal and a thermally activated process, respectively. A comparison between these two modes of -phase formation is made in the following two sections.
Transformations Related to Omega Structures
491
6.4.1 Athermal → transition On quenching an -forming alloy from the -phase field to a temperature below Ms (), the athermal → transformation can be initiated. The initiation of the athermal -phase is characterized by diffuse intensity distribution in diffraction patterns, as is illustrated in Figure 6.11. The athermal nature of the transformation has been established through ultrahigh quench rate (∼11 000 K/s) experiments (Bagaryatskiy and Nosova 1962). The facts that the transformation is not
00021
110
00011 000
(a)
002
(b)
(c)
Figure 6.11. (a) Diffuse intensity patterns obtained on the selected area diffraction patterns from the <110> of the bcc lattice. (b) Schematic drawing exhibiting deviation from the 1/3, 2/3 position from the rel vector [112]. (c) Diffuse intensity distributions in various zones.
492
Phase Transformations: Titanium and Zirconium Alloys
suppressible even under such quenching conditions and that it occurs at temperatures where the diffusivity of both self and solute atoms is negligibly small suggest that the transformation can occur without any thermal activation. The diffuse intensity distribution gradually changes to sharp -reflections as the temperature is progressively lowered below Ms (), suggesting an increase in the volume fraction of the -phase and a growth of individual -particles. The complete reversibility of the transformation is yet another evidence for its athermal nature. All these observations suggest that this transformation can be classified as a displacive transformation. Particles of the -phase that form on quenching do not exhibit a plate-like morphology or a surface relief effect which are both characteristic features of a martensitic transformation. It is for these reasons that the athermal → transformation is not categorized as a martensitic transformation in spite of its athermal and composition-invariant character. 6.4.2 Thermally activated precipitation of the -phase Precipitates of the -phase form in a -phase matrix on ageing at temperatures below about 773 K, which is the upper cut-off temperature for -phase formation by ageing in most of the -solid solutions. The progressive increase in the volume fraction and the growth of -particles with an increase in the isothermal holding time are indicative of a thermally activated transformation mechanism. Time-temperature-transformation (T-T-T) plots pertaining to isothermal -precipitation have been experimentally generated in several alloys. Figure 6.12 shows two such plots corresponding to Zr–Nb alloys. The characteristic C-shape of these T-T-T plots is consistent with the well-known dependence of the incubation period on the driving force and the diffusivity which change in opposite directions with a lowering of temperature. An in-situ study of -precipitation under 1 MeV electron irradiation in a highvoltage electron microscope has revealed the real time changes (Figure 6.13) in diffraction patterns and has demonstrated the progressive sharpening of -reflections from the initial diffuse intensity distribution. This observation is remarkably similar to the evolution of sharp -reflections from the diffuse intensity distribution which has been reported to occur in the athermal → transformation as the temperature is progressively lowered. The lattice correspondence between the - and the “aged ” structures has also been found to be the same as that recorded in the case of the athermal → transformation. These features suggest that isothermal -precipitation occurs by an atomistic mechanism which is essentially the same as that operative in the athermal → transformation. The thermally activated component of the overall transformation mainly involves a solute-partitioning process which accompanies the lattice collapse mechanism
Transformations Related to Omega Structures 900
α + β Zr + βNb α BY DILATOMETRY
Temperature ~°C
β 800
ω + β Zr α + β Zr
700
β/α + β Zr
600
493
α + β/α + β Nb
α + β Zr
α + β Zr + βNb
500
400
β
ω + β Zr
300
200 100
ωr 0.2
1h 0.5
1
2
5
10
102
5
2
1 day
6h 2
5
103
4 days
2
5
104
Time ~ Min
(a) 900 800
β ω + βZr α + βNb+ β βI/βI + βII
700
Temperature ~°C
α + β Nb + β Zr α + βNb
β + βI + βII
βI + βII/α + βNb 600
β + βNb + β
500
β
α + βNb + β Zr
α + βNb
α + β Zr
400
ω + β Zr 300 200
1h 100
0.2
0.5
1.0
2
5
10
2
5
6h 10
2
4 days
1 day 5
10
2
5
10
Time ~ Min
(b)
Figure 6.12. T-T-T diagrams for (a) Zr–12% Nb alloys and (b) Zr–17.5% Nb alloys. Increasing Nb concentration retards the formation of the -phase from 3 h at 300 C for Zr–12% Nb to 2 days at 300 C for Zr–17.5% Nb alloy.
494
Phase Transformations: Titanium and Zirconium Alloys 300 K
t =0 s 450 K
t=0 s
60 s
300 s
120 s
240 s
Figure 6.13. Selected area diffraction patterns from the -phase regions after irradiating with 1 MeV electrons for different time periods. Gradual transformation of the diffuse intensity to the -reflections could be noticed.
(as discussed in a later section). It may be noted from Figure 6.13 that the -transformation can occur even at ambient temperature due to the condition of enhanced diffusion obtained under 1 MeV electron irradiation. 6.4.3 Pressure-induced → transformation It has been noticed that the transformation pressure, Ps→ , determined by different research groups, shows a fairly high scatter for pure Ti (from 2.8 to 9.0 GPa) and for pure Zr (2.2 to 6.0 GPa) (Sikka et al. 1982). This large scatter has been attributed to variations in the pressure exposure time, the starting microstructure, the impurity content and the hydrostatic component of the applied pressure. The influence of the exposure time at a constant pressure at ambient temperature on the → transition has been systematically studied and the time dependence of the growth of the -volume fraction has been established. The rate of the -transformation as a function of pressure shows a peak (Figure 6.14) (Zilbershtein et al. 1975, Vohra 1978) analogous to that observed in nucleation rate versus temperature plots pertaining to thermally activated martensitic transformations (isobaric). The time-dependent nature of the pressure-induced → transformation suggests that there exists a barrier, possibly that of the nucleation step, which can be overcome by thermal activation. The nucleation step involves the growth of quasistatic -embryos to the critical size by thermally assisted diffusion processes. After attaining the critical size, the nuclei grow spontaneously in an athermal manner. The observed peak in the rate-versus-pressure plot is consistent with the fact that the driving force increases while the diffusivity decreases with increasing pressure.
Transformations Related to Omega Structures x x 3
x
x
x
x
2
x
xx x x
x
1
0
2.0
xx
ω
6.0
10.0 2.0
x
x
xx xx α ω ↑
x x x x x x P Mα S
↑
Rate of transformation arbitrary unit
4
495
P MS 6.0
xx
x x 10.0
P (GPa) (a)
(b)
Figure 6.14. The rate of -phase transformation with pressure shows peaking in between (a) 6–7 GPa in the case of Ti and (b) in between 5–6 GPa in the case of Zr. Theoretically calculated start pressures for the → transformation are also shown.
6.5
DIFFUSE SCATTERING
The -forming systems, in certain composition ranges, invariably exhibit diffuse intensity distribution in electron, X-ray and neutron diffraction patterns. The complexity of these patterns, though far from being fully understood, has induced many research groups to study this unusual displacive phase transformation. The essential features of the experimental observations regarding the diffuse intensity distribution and the soft phonon behaviour associated with the -transformation are summarized here. Diffracted intensity from the athermal -phase is often diffuse, centred away from the “ideal ” positions in the reciprocal space and can show pronounced curvature and asymmetry. These effects are predominantly manifested as the alloy system is moved away from the region of relative -stability in the phase diagram along either the concentration or the temperature axis. The nature of the diffuse intensity in electron diffraction patterns is shown in Figure 6.11. The diffraction pattern corresponding to the [110] zone axis (Williams et al. 1973) clearly shows that the intensity maxima in the diffuse intensity distribution are located at positions √ slightly away from those expected for the ideal -structure (c = 3/2.a and a = √ 2.a , with the lattice correspondence described earlier). For the ideal -structure the (0001) and the (0002) reflections should occur at 1/3 and 2/3 of the distance between the (000) and the (222) reflections, respectively. The line drawn along the <112> direction through bcc spots in Figure 6.11 shows where the reflections corresponding to the ideal -structure should be located. The observed (0001) reflection is displaced from the ideal position away from the (000) spot, while the
496
Phase Transformations: Titanium and Zirconium Alloys
(0002) reflection is displaced towards the (000) spot. Dawson and Sass (1970) have shown that the extent of this deviation depends on the stability of the -phase with respect to the -phase which, in turn, is decided by the alloy composition and the departure from the Ms () temperature. The observed diffuse intensity in electron diffraction patterns corresponding to [110] and [102] zone axes appears in the form of quasicircular arcs (de Fontaine et al. 1971). In-situ experiments carried out using a cooling stage in an electron microscope have shown that the diffuse intensity gradually transforms into sharp -reflections as the sample is cooled to temperatures sufficiently below the Ms () temperature. Reheating of samples has shown a gradual disappearance of the -reflections and the reappearance of diffuse patterns. The complete reversibility of the process has been demonstrated by the experiments of de Fontaine et al. (1971). The possible occurrence of multiple scattering has been considered for explaining the appearance of diffuse streaks, especially the quasicircular arcs. Multiple scattering will no doubt alter the relative streak intensities, but it cannot account for the presence of the streaks themselves. Examining several zones of electron diffraction patterns and considering the symmetry of the reciprocal lattice, de Fontaine et al. (1971) have constructed a three dimensional model of the diffuse intensity (Figure 6.15) which is distributed on quasispherical surfaces centred around the 002
020
000
110
200
Figure 6.15. Three-dimensional model of the intensity distribution of the diffuse intensity in the reciprocal space of the bcc unit cell. The sphere of diffuse intensity touches the octahedra of {111} faces which surrounds it.
Transformations Related to Omega Structures
497
octahedral sites 100, 111, , of the fcc lattice (reciprocal of the bcc -lattice). These spheres of intensity touch all the {111} faces of the octahedra surrounding them. Below the Ms () temperature, -reflections appear in the form of discs lying on {111}∗ reciprocal lattice planes at the points of tangency of the intensity spheres with the faces of the octahedra. The coordinates of these points are 1/3 <111>, 2/3 <111> etc., which can be identified as -reflections. With lowering of temperature the spherical surfaces on which the intensity is distributed become the octahedra themselves which are seen as streaks along traces of {111} planes. Neutron diffraction experiments (Sikka et al. 1982) have also shown that with increasing solute content, the -reflections become diffuse and are shifted away from ideal -positions in the reciprocal space. Figure 6.16, which shows the (011) reciprocal lattice section for a Zr–20% Nb alloy, clearly demonstrates this point. The shifts correspond to an increase in the -wave vector from q = 2/a )
h′ 3 0.0
4.0
4 8 12
1.0
2.0
3.0 4.0
3.0
3.0
h3
2.0
h′ 1
2.0
1.0
1.0
[111]β
0.0
↑
↑
0.0
–3.0
–2.0
–1.0
0.0
[211]β 1.0
2.0
3.0
h1
Figure 6.16. Neutron diffraction intensity plots showing distribution of the diffuse intensity on the (011) reciprocal lattice section of the Zr–20% Nb alloy (Keating and Laplace 1974).
498
Phase Transformations: Titanium and Zirconium Alloys
(2/3,2/3,2/3) to q + q. The magnitude of q, which is a measure of the extent of incommensuration with respect to the ideal -lattice, is seen to be dependent on the solute concentration and the extent of undercooling below the Ms () temperature. However, in some systems, cooling to a temperature as low as 5 K does not change the diffuse intensity, suggesting the presence of the incommensurate structure even at such low temperatures. It has been noticed that with the application of pressure the -reflections get sharper and the structure tends to become commensurate. It has been noticed that longitudinal phonons having a wave vector K which is close to that for an “” phonon (K = 1/3 [222]) exhibit a sharp central peak flanked by a pair of weak diffuse peaks. The diffuse peaks are representative of the dynamic phonon structure whereas the central peak represents a static phonon. This observation can be interpreted in terms of a mechanism involving the propagation of a large amplitude dynamic phonon. As it propagates, it structurally degenerates into two alternate configurations, and anti- (described in Section 6.6) which have a large difference in free energy (Cook 1973). Large amplitude phonons are thus “pinned” in a metastable state at the low free energy -positions. The lattice collapse mechanism which is implicit in this interpretation will be discussed in detail in Section 6.6. The experimental longitudinal (L) and transverse (T) phonon dispersion curves for -Zr are shown in Figure 6.17 (Stassis et al. 1978). The pronounced dip in the longitudinal [111] branch at the reduced wave vector, m (= Km d/) 0.7, which is slightly away from the ideal , is indicative of a tendency of this mode Γ
N
H
Γ
P
25
25
[110]
[001]
[111] 20
Energy (meV)
20 L
L
L
15
15 T
10
10
bcc Zr T = 1423 K
5
0.5
0.3
0.1 0
0.2
0.6
T 5
1.0
0.8
0.6
0.4
0.2
0
Reduced wave vector
Figure 6.17. Experimental phonon dispersion curves for bcc Zr. A pronounced dip in the longitudinal [111] branch (marked as L) at a wave vector ∼0.7 could be noticed. This wave vector is slightly away from the wave vector needed for ideal which is 0.66 (Stassis et al. 1978).
Transformations Related to Omega Structures
499
to soften up. It is to be noted that the softening is not due to the extent that the system becomes unstable with respect to the development of this longitudinal displacement wave. Stassis et al. (1978) have also observed that the intensity of diffuse peaks as well as the extent of their shifts from the ideal -positions are temperature dependent.
6.6
MECHANISMS OF -TRANSFORMATIONS
As discussed in the preceding sections, the -phase can form in Ti and Zr base alloys in the following situations: (a) on application of hydrostatic pressure to the -phase – either by a static pressure or by a shock pressure (b) on quenching from the -phase field (c) on ageing the metastable -phase (d) on application of shock to the -phase (in -stabilized alloys such as Zr–20% Nb) The mechanisms involved in -phase formation in these cases are not the same. However, the lattice collapse mechanism, which converts the bcc structure to the hexagonal -structure, appears to be a common feature in all these cases and is, therefore, discussed first. 6.6.1 Lattice collapse mechanism for the → transformation The → transformation has been studied in detail and well documented in literature. The key experimental observations of these studies are summarized in the following: (1) The → transformation exhibits strong pretransition effects in terms of profuse diffuse scattering (e.g. Figure 6.11) and lattice softening (Sass 1972). (2) The → transformation cannot be suppressed by quenching. In the -quenched alloys the resultant -phase always remains finely divided typically less than 5 nm in diameter. Generally the shape of these -particles is ellipsoidal. All the possible four variants (corresponding to four <111> directions) of -precipitates remain equally distributed in the -matrix. High-resolution electron microscopy (HREM) has confirmed the presence of all variants. (3) The -particles grow on ageing and during their growth, they gradually assume a cuboidal morphology with the cube faces parallel to {100} planes. Isothermal
500
Phase Transformations: Titanium and Zirconium Alloys
ageing can also produce ellipsoidal -phase particles in concentrated alloys. The latter process is usually preceded by phase separation in the bcc -phase, resulting in solute rich and solute depleted -regions. (4) Under application of shock deformation, -plates have been produced (Hsiung and Lassila 2000, Dey et al. 2004) in the -phase. The product -phase assumes a plate shape and the transformation is reported to have similarities with the martensitic transformation. (5) The → transformation, resulting from thermal ageing, is accompanied by a reduction in the specific volume (typically about 5% (Hickman 1969)). Based on the above experimental features, the resultant -phase can be classified in the following categories: (1) Fine ellipsoidal particles athermally produced on quenching. Morphology of the particles suggests that there is no volume change associated with the → transformation. (2) Cuboidal or ellipsoidal -particles (20–50 nm), produced during isothermal ageing by a transformation resulting from diffusional alloy partitioning followed by lattice collapse. The transformation is accompanied by a volume change. (3) Plate shaped resulting from the shock pressure induced → transformation. The lattice correspondence between the - and the -structures, as described in Section 6.3.2, clearly shows that the -structure can be created from the -structure by collapsing a pair of adjacent (222) planes (e.g. 1 and 2 in Figure 6.7(b)), leaving the “0” and “3” planes undisplaced. This results in a transition from the ABCABC stacking of the bcc structure (in Figure 6.7(a), these three layers are depicted as “0 1 2”.) into the AB AB stacking of the -structure (shown as “0 1.5 3” in Figure 6.7(b)), where B planes correspond to the collapsed position located midway between B and C planes. A view along the [111] direction shows how the threefold rotation symmetry changes to a sixfold symmetry where the collapse is complete (Figure 6.7(c)). The ordered sequence of the displacement of the {111} type planes (Figure 6.7(d)) which causes the → transition can be represented by a displacement wave with wavelength = 3 d222 , the corresponding wave vector, K , being equal to 2/3 <111>∗ . The development of the longitudinal displacement wave (Figure 6.7(d)) along the [111] direction can, therefore, formally describe the transformation. The merit of this description lies in the fact that it can be applied to partial collapse situations by varying the amplitude of the displacement wave, the amplitude being directly related to the order parameter, . The →
Transformations Related to Omega Structures
501
transition in Ti and Zr alloys which occurs on quenching from the -phase field has been found to be associated with the following attributes: (a) athermal character; (b) presence of pretransition diffuse intensity associated with structural fluctuations, the diffuse intensity peaks being slightly away from the perfect -peaks; (c) stability of a two phase microstructure comprising fine particles of the -phase distributed in a -phase matrix immediately below the Ms () temperature. The following mechanism of the athermal → transition, based on the work of de Fontaine (1970), de Fontaine et al. (1971) and Cook (1973), takes these essential features into account while developing a Landau theory of displacement ordering for this transition. As shown in Figure 6.7(d), the → transition can be described in terms of the introduction of a displacement wave with the wave vector K = 2/3 <111>∗ in the bcc structure. A planar lattice model, as depicted in Figure 6.18, is very
(222) planes i=0
1
2 3
4
5
6
(a)
bcc
(d)
(b)
ω
(e)
[111]
Up
+ –
(f) Anti-ω
1
2
3
4
5
6
0
1
2
3
4
5
6
+ Up
(c)
0
–
Figure 6.18. Two possibilities of the collapse of the {222} planes are shown: (a) represents arrangement of the {222} planes, viewed on edge, in the bcc lattice. One possibility of collapse is shown in (b), where a two-plane collapse structure is shown (planes 1 and 2, 4 and 5), while in (c) a three-plane collapse structure is shown (marked as 2, 3, 4). The two-plane collapse produces the -structure whereas the three-plane collapse produces an anti- structure. (d) A collapse region in the bcc lattice showing an -region marked by an ellipse. This schematic provides a clue for the ellipsoidal morphology of the -phase. (e) and (f) are the wave representations of the - and anti- structures. The final positions of the planes are shown by broken lines.
502
Phase Transformations: Titanium and Zirconium Alloys
useful in visualizing the process. It can be seen from the illustrations in Figure 6.18 that the -structure is produced by the -wave while a wave with an equal but negative amplitude (hence, negative ), represented in Figure 6.18(f), produces the anti-omega (−) structure in which both B and C layers are displaced towards the A layer. Needless to say, the anti- configuration in which three planes come close together due to the collapse mechanism will be energetically unstable. The fact that the anti- configuration is not equivalent to the configuration both in structure and in free energy suggests that the → transition cannot be of a second (or higher) order. Cook (1973) has put forward this symmetry argument to justify the attribution of a first-order character to this transition which has been experimentally vindicated by the observation of the occurrence of the two phase + structure. Considering the planar lattice, the free energy change associated with the lattice collapse can be expressed in the form 1 1 1 G = ij ui uj + ijk ui uj uk + ijkl ui uj uk ul + 2 3 4
(6.6)
where ij , the interplanar coupling parameters, are defined by the appropriate derivatives of the free energy with respect to the planar displacements which are denoted by ui , uj and are respectively normal to the planes i, j , numbered in sequence from the original at i = 1. The coupling parameters in Eq. (6.6) are not independent but are linked by the energy translational and rotational invariance relations (Leibfried and Ludwig 1961). Substituting into Eq. (6.6) the deformation u1 = 0 u2 = −u3 = a 3/12 ui+3 = ui (6.7) where a is the lattice parameter and the Landau order parameter for the displacement ordering in the → transition, the free energy change can be expressed after rearrangements in terms of the standard Landau expression G = A2 + B3 + C4
(6.8)
Schematic free energy versus order parameter plots (Figure 6.19) can show the relative stabilities of the - and the -structures at different temperatures. As is expected in a first-order transition, the stability of the -structure is brought about by a negative value of the third-order constant (B) which is responsible for lowering the free energy below zero at T < To (). With reduction in temperature, the magnitude of the third-order constant (B) increases and as temperature approaches Tm , a metastable state appears at a positive . With further reduction in temperature, the potential well of the metastable state deepens and the minimum appears
Transformations Related to Omega Structures
503
T > Tm ΔF
T > Tm
To(ω) < T < Tm T = To(ω)
η T < To(ω)
Figure 6.19. Schematic presentation of the free energy of two ( and ) phases as a function of order parameter. The asymmetric nature of the free energy curves represents the first-order nature of the -phase transformation. At temperature Tm , a local minimum free energy appears which deepens as temperature of the system reduces in the temperature range Tm >T >To . In this temperature range the system temporarily attains the -structure with life time larger than the inverse of vibration frequency. For temperatures lower than To (), the -phase is more stable than the bcc phase.
towards the value corresponding to ideal -structure as shown by dotted line in Figure 6.19. As temperature reaches below Tm , a true minimum in free energy is attained and the -structure becomes stable with respect to the bcc structure. The minima correspond to the metastable states in the temperature range T T > To (). The strongly anharmonic behaviour of Zr with respect to the displacement ordering with the wave vector 2/3 <111>∗ has been demonstrated by first principles calculations of Ho and Harmon (1990). This point is discussed in the following section. The same conclusion can be arrived at from the fact that the K
504
Phase Transformations: Titanium and Zirconium Alloys
wave, acting alone, can produce a non-vanishing third-order term, as shown by the summation rule: K + K + K = 222, a reciprocal lattice vector. 6.6.2 Formation of the plate-shaped induced by shock pressure in -alloys The lattice collapse mechanism of the → transformation is pure shuffle transformation and does not envisage any macroscopic shape strain. When such a mechanism is operative, the product and the parent phases can maintain complete coherency along the entire interface. For such a transformation the strain energy minimization by a proper choice of the product morphology is not important. However, when a macroscopic shape strain is superimposed on the → transformation, choice of the interface is restricted by the criteria of minimization of the interfacial energy and the strain energy associated with the transformation. For example, in the martensitic transformation, the strain energy contribution dominates and the invariant plane strain (IPS) condition is satisfied, the surface energy term dictates the selection of habit planes. In several diffusional transformations also the invariant line (IL) vector, lying along the habit plane, which defines the primary growth direction, is essentially due to the minimization of the strain energy associated with the transformation. In the similar fashion, if there is a volume change involved during the → transformation and if the external stress involved has a shear component, the choice of the interface is not completely random. In fact, it has been shown under these circumstances the product -phase has plate-shape morphology and habit plane between the - and the -phases can be predicted using phenomenological theory of martensite (Dey et al. 2004). Using the phenomenological theory of martensitic transformation, Dey et al. (2004) could derive the Bain strain and also the macroscopic shape strain matrix. The predicted habit plane was very close to the experimentally determined (121) habit plane (Figure 6.20(a)–(d)). The observation of {112} planes as the habit planes of the -phase is suggestive of the fact that the → transformation involves a shearing of the -lattice along a {112} plane. A bcc structure can be described by a sixlayer packing sequence of {112} planes, as shown in Figure 6.21. Hatt and Roberts (1960) have examined the possibility of generating the -structure by gliding an {112} plane and have shown that a suitable sequence of glides can indeed produce the → transformation. Figure 6.21 shows that a macroscopic shear on an {112} plane along an <111> direction, superimposed with atomic shuffles, can produce the -structure. Since the macroscopic deformation is a simple shear, the invariant plane, which is the contact plane between the two phases, is the shear plane itself. In this case, the lattice invariant shear operates for the macroscopic strain to satisfy the invariant plane strain condition. It is possible to define a homogeneous shear on the (112) plane with a magnitude of cot /2 = 0767, where
Transformations Related to Omega Structures
505
Figure 6.20. (a) SAD pattern from a Zr–20% Nb alloy showing the presence of the -phase. The occurrence of diffuse intensity indicates the tendency for the -phase transformation, (b) plateshaped morphology of the -phase, (c) interface between a plate of the -phase and the -matrix is lying along a <112> direction and (d) composite SAD pattern showing the presence of a single -variant.
¯ and the [110] directions. The [1¯ 10] ¯ is defined as the angle between the [1¯ 12] direction (2 ) remains undistorted as the result of application of this shear, which ¯ direction is perpendicular to the plane of is applied in such a way that the [1¯ 12] shear. Using the well-known geometrical relation of homogeneous deformation, which is very commonly used in the case of deformation twinning, the magnitude of shear can be determined. The homogeneous deformation comprising the shear part alone produces a lattice, which is indistinguishable from the -parent lattice. When atomic shuffles similar to those shown in Figure 6.21 are superimposed on the homogeneous deformation, the -structure is generated. The lattice
(111)β⎥⎪(0001)ω
0 1 2 3 4 5 6
Atomic plane no.
ω
[110]
Displacement
[111] [111]
β
[001]
Phase Transformations: Titanium and Zirconium Alloys [112]
506
[112]
β
[111]
Figure 6.21. Planar stacking of the {112} planes of the bcc lattice (shown by open circles). If the shuffling of the atoms is superimposed during the shearing of the {112} planes, the -structure (shown by the filled circles) is produced. The orientation relationship between the two structures remain same as shown in Figure 6.7. The two lattices have also been shown. In order to bring out the equivalence between the shear model and the wave model, the displacement wave has also been shown in the inset.
correspondence between the - and -lattice remains the same as that encountered when a longitudinal displacement wave is introduced in the -lattice. This equivalence, generated by (i) a combination of shear and shuffle and (ii) a longitudinal displacement wave, can be appreciated from Figure 6.21. Since any specific {112} plane contains only one <111> direction, a single orientational variant of the -structure can be produced within a single {112} plate. In short, the deformation produced by a shock wave in the bcc lattice develops a tendency for the shear of the {112} planes where the formation of the -structure occurs when the shuffle process is superimposed on the shear deformation. The axis of the shear direction decides the choice of the variant for the <111> direction contained in the specific {112} plane and, therefore, plate-shaped single variants of the -phase emerge in the shock deformed -phase. 6.6.3 Calculated total energy as a function of displacement It was believed earlier that the -phase occurred exclusively in Ti, Zr and Hf alloys. However, more recently -structures have been reported in other systems as well (de Fontaine, 1988). Nevertheless, these Group 4 metals are the only pure
Transformations Related to Omega Structures
507
metals in which the -phase is observed. This suggests that the stability of the -phase is directly related to the electronic structure of these Group 4 transition metals, all three of which exhibit an allotropic bcc → hcp transition. The fact that the collapsed plane in the -structure has the same two-dimensional structure as the basal plane of graphite (Group 14) is also indicative of the inherent stability of the collapsed plane configuration for these systems with some similarity in the electronic structure. Ho et al. (1984) have evaluated the total energy as a function of the displacement corresponding to the longitudinal 2/3 <111> phonon for Mo, Nb and bcc Zr on the basis of first principles calculations. The basic procedure for using total energy calculations to study a particular phonon mode includes first principles band structure calculations taking as inputs the atomic number, the crystal structure, the lattice parameter and the atomic displacement corresponding to a particular “frozen phonon”. This involves the assumption of the adiabatic condition which implies that the motion of ions is so much slower than the motion of electrons that at each instant of a given ionic displacement the electrons are in the ground state defined by the instantaneous ionic positions. The large difference between the electronic and the ionic masses makes this assumption quite valid. To study a particular phonon the ionic positions are fixed at various displacements from their equilibrium positions in the direction corresponding to the phonon eigenvector (polarization). A fully self-consistent band structure calculation has been performed by Ho et al. (1984) for each frozen-in condition and the total potential energy has been determined, combining the electronic energy and the ion–ion coulomb energy. The total energy, expressed in terms of the displacement, has been obtained not only for small displacements where harmonic contributions predominate but also for large displacements which correspond to structural transitions. Experimentally determined phonon dispersion curves (Figure 6.22(a)) for the <111> longitudinal branch for three bcc metals, Mo, Nb and Zr (high temperature bcc phase) have been compared (Stassis et al. 1978). Though these elements are neighbours in the periodic table, their phonon frequencies are remarkably different. The pronounced dip seen near 2/3 <111> for Zr is close to that required for inducing the → transition. Frozen phonon calculations for this mode have shown that the total energy versus displacement curves (Figure 6.22(b)) for Mo and Nb are close to quadratic, implying an essentially harmonic behaviour. The curve for Zr is strongly anharmonic and the minimum in the energy occurs when two of the (111) planes collapse together to form the -phase. It is seen that the ground state energy of bcc Zr rises with displacement on the positive- side and after reaching a maximum drops down to a lower minimum at a displacement of 0.5 corresponding to the ideal -structure, while on the negative displacement (anti-omega structure) side the energy rises very sharply. The results of the first
508
Phase Transformations: Titanium and Zirconium Alloys 8 L [111] 7
Nearly harmonic 0.010
Energy (Ry/atom )
ν (THz )
6 5 4 3 2
Mo
Strongly anharmonic
Zr 0
1
kω
Nb
Ideal ω
km bcc
0 0.2
0.4
0.6
ξ (a)
0.8
–0.2
0
0.5
Displacement of (222) planes (b)
Figure 6.22. (a) Phonon dispersion curves for Mo, Nb and bcc Zr showing pronounced dip for Zr at k = k . (b) Calculated total energy as a function of displacement corresponding to the longitudinal (2/3, 2/3, 2/3) phonon.
principles total energy calculations, therefore, are consistent with the Landau plots for the → transition which have been discussed in the preceding section. These calculations also reveal the influence of large atomic displacements on the local charge density. For Zr it has been shown that a transfer of d-like charge from uncollapsed (A type) planes to s-p like charge in collapsed (B and C type) planes occurs which is unlike the calculated behaviour of charges in Nb and Mo. Thus s-p like bonding tends to develop in the collapsed planes of Zr (and other Group 4 metals), mimicking the bonding of atoms in the basal plane of graphite. The calculated total energy, as a function of displacement, shows that the bcc phase is unstable with respect to the → transformation. However, in pure Ti and Zr this transformation cannot be induced without the application of hydrostatic pressure. This is due to the relative stability of the hcp -phase in these systems. The phonon description of the bcc → hcp transition has been given earlier in Chapter 3. The addition of bcc alloying elements (V, Mo, Nb, Cr, Fe, etc.) to Ti and Zr promote the formation of the athermal -phase. This is incidental because these alloying elements actually destabilize the -phase with respect to the -phase but apparently they do so not as strongly as they destabilize the -phase with respect to the -phase. This point is discussed in Section 6.6.4 where -phase stability is rationalized in terms of d-band occupancy by electrons.
Transformations Related to Omega Structures
509
6.6.4 Incommensurate -structures The lattice collapse mechanism described earlier can account for the first-order nature and the athermal displacement ordering character of the → transition. Landau plots (G versus ) at different temperatures (Figure 6.19) can also provide a description of the increase in the incidence of interplanar collapse with decreasing temperature and the presence of pretransition effects. The perfect -structure, however, cannot explain why the diffuse peak positions are not located exactly at the -reflections and why a fine dispersion of -particles in the -matrix invariably forms in the as-quenched condition. There have been some approaches involving incommensuration of the -structure which attempt to explain the offset of the diffuse peaks and the duplex fine particle ( + ) morphology. Cook (1975) has argued that the offset of the diffuse intensity peaks has its origin in the modulation of the “ideal-” structure by a wave with wave vector, q = Km − K , where Km , as defined earlier, is the wave vector associated with the maximum of diffuse intensity. He has used a formalism, as outlined here, in which the Fourier transform (denoted by K-representation) of the free energy is considered. The displacements, ui , are substituted by a Fourier series: (6.9) ui = UK expiKxi K
in which xi is the distance from the origin to the ith plane of the undistorted planar lattice and UK is the Fourier amplitude given by UK = 1/N
N
ui exp−iKxi
(6.10)
i=1
where N is the total number of planes and K is an allowed wave vector in the first Brillouin zone. The expression for the free energy change, on the substitution of Eq. (6.10) in Eq. (6.7), becomes G = NF2 + F3 + F4 +
(6.11)
where F2 , the purely harmonic portion of the free energy, and the third order anharmonic term F3 are respectively given by F2 =
1 KUK2 2 K 2
(6.12)
F3 =
1 K K UKUK K + K + K g 3! KK K 3
(6.13)
510
Phase Transformations: Titanium and Zirconium Alloys
In these expressions, 2 (K) and 3 (K) are appropriate K-dependent secondand third-order coefficients, which are essentially the Fourier transforms of the second-order and the third-order interplanar coupling parameters: the term K + K + K g is a delta function which is zero unless the sum (K + K + K ) is equal to a reciprocal lattice vector, g, along an <111> direction. The complex amplitudes U are proportional (in modulus) to the displacements of the atomic planes which are essentially Fourier transforms of the second-order and third-order interplanar coupling parameters: (6.14) 2 K = ii+j expiKxj j
3 K =
j
ii+jj+l expiKxj iKxl
(6.15)
l
For the trivial solution of zero amplitudes for all K, as is the case for the bcc structure, the free energy change G, assumes the minimum of value zero. If the system behaves in an essentially harmonic fashion (i.e. the contributions of F3 , F4 , etc. are negligible), there is no other minimum in G as an increase in the amplitude raises the value of the free energy change parabolically. Body-centred cubic metals such as those belonging to Groups 5 and 6 (e.g. Nb and Mo) exhibit such harmonic behaviour (Figure 6.22). The first principles electronic structure calculations for bcc Zr, discussed in Section 6.6.3, clearly demonstrate a strongly anharmonic behaviour, as revealed by the presence of a deep second minimum away from the zero amplitude first minimum (Figure 6.22). A similar behaviour is also expected in the cases of Ti and Hf. As mentioned earlier, the harmonic contribution (arising out of the second-order force constant, 2 , to the free energy change) is minimized when the wave vector is Km while the anharmonic contribution (arising from the thirdorder term) can further reduce the free energy change through interactions between displacement waves of different wave vectors lying in between Km and K . The fluctuation thus created results from a competition between the second- and the third-order force constants. Cook (1975) has demonstrated this fluctuation mechanism by considering the growth of athermal displacement modulation represented by a sinusoidal standing wave ui = A sinK + qxi = A cos qxi sin K xi + A sin qxi cos K xi
(6.16)
with the amplitude A being related to the Landau order parameter, , by the relation A = a /6
(6.17)
Transformations Related to Omega Structures
511
Longitudinal displacements
When qxi is small, the first term gives the ideal -structure. As xi increases to a value of /q, the anti- structure develops. The presence of a spectrum of wavelengths from K to Km (= K + q) will generate wave packets of - and anti- structures as shown in Figure 6.23(a). The anti- structure, however, is
+ 0 –
5 1
2
3
6
7
8 9
4
10
Distance in (111) interplanar units (a) ω
–ω
ω
2π/Δk (b) G
G
G
T < To(ω ) Km
Kω
K = Km
K = Kω K = Km
K = Kω A
(β + ω ) Dual structure
A∗ (c)
Aω A
T << To(ω) ΔGω < ΔGω + β
A
Kω < K < Km
ΔGω + βΔGω
K = Km
δΔGω,β (d)
(e)
Figure 6.23. (a) The accumulated phase difference caused by the difference between the Km wave vector and K wave vector. The positive displacement forms the -structure, whereas the negative displacement forms the anti- structure. (b) Flipping of phonon causes the region of anti- to get converted into one of the variants of . (c) Represents a situation where temperature is above To (). This plot can be taken as the reciprocal space representation of Figure 6.19. A local minimum in free energy allows the system to attain a dual phase structure as it is always energetically more favourable in comparison to a completely single phase (shown by a line joining the two minima). (d) For temperatures lower than To (), the -phase becomes more stable than the bcc phase but for intermediate positions of the amplitude, a dual phase structure is again more stable than a single phase. (e) For temperatures far lower than To (), -phase is more stable with respect to either of the phases. However, at these temperatures very low diffusion rates do not allow transformation to reach completion.
512
Phase Transformations: Titanium and Zirconium Alloys
not expected to survive due to the high energy associated with it. This causes a “phonon flipping” which converts the anti- regions into -regions, resulting in the formation of a row of -wave packets (Figure 6.23(b)). The spacing d , between the adjacent -particles (or wave packets) will, therefore, be d = /q
(6.18)
Since the longitudinal <111> phonon is responsible for the formation of this structure, -particles are lined-up along <111> directions. Substituting the q value, the spacing of -particles can be determined to be about 2.5 nm which is in close agreement with the interparticle spacing experimentally determined by Dawson and Sass (1970). As mentioned earlier, the athermal → transformation has never been found to reach completion. Invariably athermal -particles remain distributed in the matrix in the quenched product. Schematic Landau plots can be used for explaining the stability of this dual phase + structure. A possible explanation for such incomplete transformation of the -phase to the -phase can be given using a reciprocal space (k-space) representation of free energy as a function of amplitudes and wave vectors associated with the dual phase ( + ). As discussed earlier, the transformation associated with Km leads to secondorder phase transformation and will, therefore, correspond to a minimum harmonic energy. The plot corresponding to Km is drawn with a curvature smaller than that for the K vector in the vicinity of the origin. At larger amplitudes due to the influence of third-order term, the free energy of the K vector becomes smaller. The wave vector associated with dual phase having a metastable state lying between origin (representing bcc structure) and A (amplitude associated with the -structure) will have lower energy than the single phase as shown in Figure 6.23(c). At extremely low temperatures (Figure 6.23(e)), thermodynamically single phase become more stable in comparison to the dual phase structure. However, at such low temperatures due to kinetic reasons, completion of phase transformation appears difficult. At intermediate temperatures, that is, at lower super cooling, T < To () (Figure 6.23(d)), it can be shown that two phase + structure corresponds to a lower value of free energy in comparison with that of a -structure. Sanchez and de Fontaine (1977) have proposed a different model for explaining the appearance of curved diffuse intensity streaks characteristic of the → transformation. By considering the transformation of the observed diffuse intensity distribution they have calculated a plausible atomic displacement field which is found to have a transverse width of a few lattice parameters. The displacement patterns in the longitudinal direction exhibit a variation in sign and magnitude along
Transformations Related to Omega Structures
513
a given <111> atomic row. Displacements in adjacent rows also vary in a similar manner but are out of phase. Incommensuration of these types is schematically shown in Figure 6.8 which shows that along a given <111> direction three subvariants, designated as 1 , 2 and 3 , corresponding to the location of the uncollapsed plane on A, B or C atomic layers, respectively, can form. The basis of the various incommensuration models proposed lies in the nature of the stacking of different subvariants (Borie et al. 1973, Kuan and Sass 1976, Pynn 1978, Horovitz et al. 1978). Borie et al. (1973) have considered a specific sequence of subvariants, namely, 1 , 3 , 2 , 1 , to be present along a given <111> direction. This sequence, which corresponds to a phase shift of 4/3 in the displacement wave from one subvariant to another, can correctly predict the shift in the diffuse intensity. However, this model does not explain the diffuse intensity observed near the bcc positions, the presence of the untransformed -phase and the reason for the occurrence of the particular ordered sequence of subvariants. The model of Kuan and Sass (1976) assumes the presence of a vacancy which introduces a phase shift from one variant to the other. This model suffers from the drawback that the predicted shift for the (2/3 2/3 2/3) reflection is opposite to that observed experimentally. Horovitz et al. (1978) have proposed the presence of a small -phase region between successive -phase variants. These -layers can be considered as solution walls which bring about the phase shift from one -variant to the next. Such discommensurations are quite common in models of long period structures. In this model the variation in the lattice parameters of the - and the -phases has also been considered and the peak shift has been attributed to the phase shift as well as the density shift, the latter arising from the small variation in the lattice dimensions of - and -regions. If the difference in the lattice constants is very small, the -cluster is completely locked into the -matrix and a commensurate structure is produced as is the case for alloys having Ms () temperatures above ∼573 K. When the mismatch is large, solution appears in the ground state, partially unlocking the cluster from the -matrix. Such a situation is encountered in alloys with Ms () temperatures close to room temperature and as a consequence the reflections move away from the commensurate positions. HREM has recently been employed for studying the commensurate and incommensurate structural configurations in -phase forming alloys. The -regions in the -matrix can be easily recognised in HREM images as the members of an atom pair which is a pair of atoms that are pushed into the collapsed plane come too close to be resolved in the image along an <110> direction and they jointly produce a relatively bright dot contrast. How the configuration of the -regions develop in the -matrix is schematically shown in Figure 6.24(a) using the structural relationship between the - and the -phases. Schryvers and Tanner (1990) have shown
514
Phase Transformations: Titanium and Zirconium Alloys β
ω
0.425 nm
[112]
[001]
0.285 nm
[110]
Z=0 [111]
Z = 1/2
[110]
25
0.4 nm
Figure 6.24. (a) Schematic presentation of formation of -lattice from the -lattice through the collapse mechanism and (b) HREM image showing the above two phases.
that commensurate -domains, formed in an aged Ti–8% Mo alloy indeed exhibit bright dots arranged in a rectangular lattice, the short dot spacing (0.285 nm) being ¯ direction and the large dot spacing (0.425 nm) along either the [111] or the [111] ¯ being along [112] or [112], respectively. Tilting experiments have shown that the different variants of commensurate fill in the entire volume without leaving any
Transformations Related to Omega Structures
515
-region untransformed. Figure 6.24(b) shows a high-resolution image obtained from the Zr–Nb alloy. The presence of the -region could easily be identified from the matrix -regions. For imaging the structure of incommensurate -regions in the -matrix, as quenched samples of Ti–8 Mo and Ti–15% Mo alloys have been examined by Schryvers and Tanner (1990). HREM images of the -phase when viewed along [110] direction have shown presence of strings of dots along a line vector slightly deviated from the K111 . Figure 6.25(a) shows the arrangement of atoms in regions where -like defects are observed in the bcc lattice. As could be noticed from the figure, because of such arrangements the line joining the elliptical dots, representing incommensurate , form a vector deviating from the <111> direction of bcc. Figure 6.25(b) is a high-resolution image from a region showing diffuse intensity pattern (shown as inset in the figure) where series of dots can observed. Joining these dots forms a line which is nearly 10 degree away from the <111> direction of the bcc.
(a) 0.23
(110)
nm
→ ← β→ γ← γ
α
α β γ α
[111] (b)
k
111
[112] ki 10°
k [111] (c)
(d)
Figure 6.25. (a) Incommensurate -structure produced by partial collapse of row of atoms causing an angular deviation of 10 from k = 111 of the bcc lattice and (b) high-resolution image taken from the [110] zone axis showing short strings of white dots of incommensurate .
516
Phase Transformations: Titanium and Zirconium Alloys
6.6.5 Stability of -phase and d-band occupancy The occurrence and stability of the -phase have initially been explained in terms of the concept of the Fermi surface touching the Brillouin zone (BZ) boundary in the reciprocal space by treating both sp and d electrons within the framework of the nearly free electron approximation. These arguments are essentially similar to the Hume-Rothery rules and are based on the fact that it is costly in terms of energy to add further electrons to an alloy, once the filled states reach the zone boundary. Then the crystal makes a phase change to another crystal structure in which more electrons can be filled before the Fermi surface touches the BZ boundary. Luke et al. (1968) have assigned an electron/atom (e/a) ratio of 4.06–4.14 for the occurrence of the -phase from experimental data. However, as pointed out by Hickman (1969), the range of stability of the -phase in alloys cannot be predicted from the e/a ratio alone. The position of the alloying element in the periodic table is also an important factor. It is to be emphasized that Fermi surfaces in transition metals are far from spherical and that the electronic density of states departs considerably from that given by the free electron theory because of the d-band and the s-d hybridization effects. Therefore, the stability of the -phase in Ti and Zr base alloys cannot be predicted accurately from the free electron theory. The importance of the d-band occupancy in determining the equilibrium crystal structure of transition metals has been pointed out by Friedel (1969). Pettifor (1977) has shown that the number of sp electrons remains essentially constant (∼1 e/a) across the transition metal series and it is only the d-band occupancy that changes from one element to the other. It may be noted that the equilibrium crystal structures of transition metals show the definitive series hcp → bcc → hcp → fcc as the d-shell is progressively filled with electrons with the exception of the magnetic elements Mn, Fe and Co and of La, the element at the beginning of the rare-earth series. This suggests that the cohesive energy of transition metals is primarily governed by the bonding contribution of the d-band (Udbond ). The equilibrium and the metastable phase diagrams which delineate the composition ranges of stability of -, - and -phases in Zr or Ti base alloys are shown in Figure 6.1. Hypothetical free energy-composition diagrams (Figure 6.1) can be converted into free energy versus d-band occupancy plots. With increasing additions of -stabilizing elements (V, Nb, Mo and Ta) the d-band gets progressively filled, as shown in Figure 6.26. The metastable -phase appears in a certain composition range which is defined by the cross-over points, Nd1 and Nd2 where the free energy curves for the - and the -phases, respectively, intersect that for the -phase. The minimum in the free energy for the -phase is located at Ndo . This construction reiterates the point that for a composition range corresponding to Nd1 to Nd2 , the -phase is more stable than either the - or the -phases
Transformations Related to Omega Structures
Free energy
α
517
ω
β
α–ω ΔFo
A
0 2 ΔN1d ΔN d ΔN d
B
ΔNd (Change in d-band occupancy)
Figure 6.26. Energy curve as function of d-band occupancy. The -phase is more stable in comparison with - and -phases for electron concentration in d-band falling between N1d and N2d .
individually but is metastable with respect to the equilibrium mixture of the - and the -phases. Sikka et al. (1982) have computed the Ndo values for different -forming alloys from the experimentally obtained compositions of the aged -phase in these alloys (Table 6.6). It may be noted that the athermal → transformation is thermodynamically favourable in the range Nd1 –Nd2 where the drop in free energy is the maximum for the composition invariant → transformation. The fact that the → transformation occurs under high pressures implies that the free energy gap, G− , as indicated in Figure 6.1 is closed by the application of o pressure. This can be explained from the fact that the elements belonging to the first half of the transition metal series show large transfer of electrons from the s-band to the d-band on application of pressure. Physically this can be attributed to Table 6.6. The composition of the -phase in the pseudo-equilibrium state during ageing and the corresponding equivalent change in d-band occupancy for various -forming alloy systems. Alloy
Alloying content in aged (at.%)
Ndo in aged (electron/atom)
Ti–V Ti–Cr Ti–Mn Ti–Fe Ti–Nb Ti–Mo Zr–Nb
13.8 ± 0.3 6.6 ± 0.2 5.1 ± 0.2 4.3 ± 0.2 9±2 4.3 ± 0.4 10–11
0.138 ± 0.003 0.130 ± 0.004 0.153 ± 0.006 0.172 ± 0.008 0.090 ± 0.020 0.086 ± 0.008 0.100–0.110
518
Phase Transformations: Titanium and Zirconium Alloys
the large sp-core in the case of transition metals. Under compression the bottom of the s-band (Bs ) shows a rapid rise because of its increased kinetic energy as these electrons are pushed into the core region where they are repelled by orthogonality effects. This increase in Bs reduces the spacing between the s-and the d-bands and thereby gives rise to an increase in the d-band occupancy (Nd ). Vohra (1979) has calculated the extent of the s → d transfer (Ndc ) at the transition pressures for Ti, Zr and Hf. The calculated values, listed in Table 6.6, show that the application of 2.0 GPa pressure on Ti will lead to the transfer of 0.0246 e/a from the s- to the d-band. This value is not adequate for meeting the requirement of the → transformation as indicated by the Ndo values for different alloys (Table 6.6).
6.7
ORDERED -STRUCTURES
The displacive transformation mechanism of the → transformation has been discussed in the preceding sections of this chapter. It has also been argued that the transformation can be viewed as a displacement ordering process which can be accomplished by the introduction of a periodic displacive wave in the bcc structure. This section is devoted to transformations which involve a combination of both displacive and replacive ordering. The possible superlattice structures which can evolve from the bcc structure are described in Chapter 5. We shall now examine the mechanism of the formation of intermetallic phases which can be best described as chemically ordered -structures. 6.7.1 Structural descriptions The structures of some equilibrium and metastable intermetallic phases are such that they can be described as chemically ordered -phases. The fact that the lattice correspondences between these structures on the one hand and the parent bcc structure on the other, are essentially the same as that between the - and the -structures also validates this description. The possibility of the involvement of the lattice collapse mechanism in the formation of such chemically ordered intermetallic structures has been examined purely from the consideration of the lattice relationships between the parent bcc and the intermetallic structures. The B82 structure (hexagonal, P63 /mmc) of the equilibrium Zr2 Al, Ti2 Al and Zr2 Sn phases and of some ternary phases is illustrated in Figure 6.27 in which the stacking sequence of this structure is compared with that of the bcc structure. It is clear from this figure that the → B82 structural transition would require a combination of the collapse of the layers (1, 2; 4, 5; etc.) and a chemical ordering. The fact that such a lattice correspondence indeed exists in several real systems
Transformations Related to Omega Structures (a)
(b)
(c)
519
(d)
c 6 5 4 3 2 1 0
Z2
a1 a2 B2 Pm3m 1a 1b
Z1
ω′
ω″
P3m1 1a, 2d2 1b, 2d1
P3m1 1a 1b 2d1
B82 P63/mmc 2a 2c 2d
2d2
Figure 6.27. Formation of the ordered -structures. (a) The (222) planar stacking of the ordered bcc structure. (b) Onset of the collapse structure producing a lower symmetric phase with the same number of Wyckoff points. The structure is named as . (c) A structure ( similar to the structure with a higher number of Wyckoff points. (d) The B82 structure is higher symmetry structure attained only after the collapse is complete in the bcc lattice.
has been demonstrated (Banerjee and Cahn 1983, Strychor et al. 1988). Based on this correspondence the lattice parameters (a and c) of the B82 phase can be expressed in terms of the lattice parameter, a , of the -phase as: aB82 =
√
2a
cB82 = 6d222 =
√
3a
Substituting the extrapolated values of a for Zr–33 at.% Al and for Ti–33 at.% Al, the lattice parameters of Zr2 Al and Ti2 Al work out to be the following: For Zr2 Al : a = 0.4852 nm; c = 0.5942 nm For Ti2 Al : a = 0.4580(3) nm; c = 0.5520(4) nm While examining the B2→ B82 transformation in ternary Ti–Al–Nb alloys, Bendersky et al. (1990a,b) have encountered a metastable intermediate phase, , which is of a lower symmetry than either of the initial B2 and the final B82 structure. The -phase is characterized by partially collapsed (111) planes and is reordered relative to the B2 phase. The occurrence of a partially collapsed structure essentially indicates a strong coupling between the chemical (replacive) and the displacive ordering processes. If the starting structure is B2, as described in
520
Phase Transformations: Titanium and Zirconium Alloys
Figure 6.27(a) with the atomic layering sequence A-B-A-B-A-B-A the -collapse operation produces a sequence A-B/A-B-A/B-A. Such a structure is designated (Bendersky et al. 1990a,b). The hexagonal c-axis of the ordered -structure is twice as long as that of the disordered -phase. ¯ A diffusional B2→ transition yields a P3m1 crystal structure with four distinct Wyckoff sites (1a, 1b, 2d1 and 2d2 ) but only two distinct site occupancies (as in the B2 structure) (Figure 6.27(b)). Thus the -phase is inherently unstable with respect to changes in chemical order that lead to four rather than two (Figure 6.27(c)) distinct site occupancies because the inherited B2 occupancies are characteristic of ¯ a higher symmetry than P3m1. To summarize, the -structure is the one which can form by the operation of a -collapse mechanism on the B2 structure inheriting the chemical order on the collapsed planes while in the -structure the collapsed planes do not have an ordered arrangement of atoms (Figure 6.27(d)). The structure to be examined next is the D88 structure (prototype Mn5 Si3 ). The equilibrium Zr5 Al3 and the metastable Ti5 Al3 (Bendersky et al. 1990b) phases exhibit this structure with a hexagonal unit cell which possesses 18 Wyckoff positions (Table 6.7). The stacking sequence and the lattice correspondence in the D88 structure are compared with those in the B2 structure in Figure 6.28. This figure also depicts the lattice correspondence between these two structures. Based on this correspondence the lattice parameters of the D88 -phase are related to that of the -phase (B2 structure) as aD88 =
√
6a
cD88 = 6d222 =
√
3a
Table 6.7. Wyckoff positions, site occupancy, interplanar spacings and the relative peak intensities for Zr5 Al4 (Ga4 T5 types) and Zr5 Al3 (Mn5 Si3 ) phases. Prototype structure
Wyckoff position/occupancy
hkl
d-spacing (nm) ( = 01544 nm)
Relative intensity
Ga4 Ti5 (Zr5 Al4 )
2(b) 0 0 0 Al 4(d) 1/3 2/3 0 Zr 6(g1 ) x1 0 1/4 Zr 6(g2 ) x2 0 1/4 Al x1 = 0.23, x2 = 0.60
100 110 200 111 002
0.727 0.420 0.363 0.343 0.297
25 0 62.5 62.5 –
Mn5 SI3 (Zr5 Al3 )
2(b) 0 0 0 vacancy 4(d) 1/3 2/3 0 Zr 6(g1 ) x1 0 1/4 Zr 6(g2 ) x2 0 1/4 Al x1 = 0.23, x2 = 0.60
100 110 200 111 002
0.727 0.420 0.363 0.343 0.297
0 66 100 66
Transformations Related to Omega Structures
521
Zr atom Al atom Vacancy [111]
[0001]
B2
Layer No.
D88
6 5 5,4 4 3 2 2,1 1 [211] B2 [121]
B2
[2110] [1120]
(111) B2 B2
0 D88
D88 (0001) D8 8 D88 – Zr5Al3
Figure 6.28. Lattice correspondence between the ordered bcc (B2) and the D88 structures. The collapse of 222 planes of the bcc structure (designated as 1 and 2 or 4 and 5) at the intermediate positions (designated as 2,1 or 5,4) would produce the D88 structure.
In the case of Zr5 Al3 , the substitution of the corresponding a (= 0.3455 nm) values yields the a and c parameters of the D88 phase as 0.8464 nm and 0.5984 nm, respectively, which are fairly close to the experimentally obtained lattice parameters (a = 0.845 nm and c = 0.5902 nm) of the equilibrium Zr5 Al3 phase. Figure 6.28 shows that the D88 structure can also be generated from the B2 structure by a combination of lattice collapse and chemical ordering. The pairs of {222} planes which collapse in the → B82 and in the → D88 structure are essentially same. In case of the D88 -structure an ordered array of vacancies is necessary in those B2 planes which remain undisplaced during the collapse. This is essential for meeting the stoichiometric requirement. The difference between the B82 and the D88 structures lies in whether the 2(b) positions are occupied by Zr atoms or are vacant (Figure 6.29). The fact that two vacancies for 18 atom positions per unit cell are required for the formation of the D88 structure necessitates the retention of a high concentration of vacancies in the parent B2 structure and an ordered arrangement of these vacancies. If the 2(b) positions are filled with Al atoms, a stoichiometry of Zr5 Al4 , having a Ga4 Ti5 structure, is attained. Bendersky et al. (1990a,b) have reported such a structure in ternary Ti–Al–Nb alloys. Rather
522
Phase Transformations: Titanium and Zirconium Alloys [112] // [1120]D88
[121] // [1210]D88 aD = 0.845 nm ⊥
⊥
0
Cv →
1
D88
B82 ⊥
⊥
[110] // [2110]B82
√3aB = 0.847 nm
[101] // [1210]B82
(111) Atom Vacancy
0.497 nm
Zr atom Al atom
1,2 Layer 0 layer
Figure 6.29. A schematic representation of the basal planes of the D88 and the B82 structures in accordance with their relationship with the bcc matrix. Difference between the two structures lies in the arrangement of the vacancies on the basal plane. The plan view of the collapsed plane is drawn on scale at the bottom of the figure. As could be noticed from this figure, the atoms could be accommodated without introducing any substantial strain.
recently the Zr5 Al4 phase with a hexagonal structure has been included in the phase diagram (Massalski et al. 1986). 6.7.2 Transformation sequences in Zr base alloys Phase transformation sequences have been studied in binary Zr–Al (near Zr–25at.% Al) alloys (Banerjee and Cahn 1982, 1983, Banerjee et al. 1997) and in ternary Zr3 Al–Nb alloys (Tewari et al. 1999). Rapid solidification was employed for processing the alloys used in these studies for suppressing the equilibrium phase reactions. A portion of the binary phase diagram of the Zr–Al system is shown
Transformations Related to Omega Structures
523
wt% Al 10%
50% 2000
Zr4Al3
L
Zr3Al2
Zr5Al3
1900
1700
X
1350 →
1300
β
1100
975
Temperature °C
1500
ZrAl
900
700
Zr2Al
Zr3Al
α
500 0
Zr
20
40
At.% Al
Figure 6.30. Zr-rich side of the binary phase diagram of the Zr–Al system showing phase fields of the pertinent intermetallic phases.
in Figure 6.30 in which the vertical broken lines indicate the compositions of the alloys for which the transformation sequence is discussed in this section. In all these alloys (Zr–25% Al, Zr–27% Al, Zr–Al–Nb) the rapidly quenched structure shows a retained supersaturated -phase. This observation suggests that Al, when present in Zr at a level of about 25%, tends to stabilize the bcc -phase. The fact that the -phase field extends considerably at the eutectic temperature (1623 K) supports this contention. It is interesting to note that in more dilute alloys (Al content less than about 10%) Al acts as an -stabilizer and the / transition temperature increases with Al additions. The retained -phase in the aforementioned alloys in the quenched condition exhibits a tendency towards phase separation, leading to the evolution of a spinodally decomposed structure (Figure 6.31). The formation of the compositionally modulated structure and the amplification of such modulation with ageing have
524
Phase Transformations: Titanium and Zirconium Alloys
Figure 6.31. (a) Rapidly solidified Zr–Al and Zr–Al–Nb alloys showing modulations along <100> directions on ageing at 723 K for 20 h. SAD pattern shows asymmetric intensity spread around 110 reflections along [100] and [010] directions. These are consistent with the modulations observed in the dark-field images (b–e) taken from reflections as marked in the key to SAD.
been demonstrated (Tewari et al. 1999). Typically, a modulation wavelength of about 20 nm along the elastically soft <100> directions has been observed in the quenched alloys. An order of magnitude estimate of the time constant, tc , of the spinodal decomposition can be made using the following relationship (Cahn 1970): 1/tc = 2KM4m where M is the atomic mobility, K is the gradient energy coefficient and m is the wave number of the fastest growing concentration wave. An approximate value of K is given by K = Nv kB Tcoh 2/m 2
Transformations Related to Omega Structures
525
where Nv is the number of atoms per unit volume, kB the Boltzmann constant and Tcoh the coherent spinodal temperature, an upper bound estimate for which is about 1500 K for the alloys being considered here. It can be shown that for the spinodal decomposition to occur within 10−5 s (during the rapid quenching treatment) the diffusion constant, DMkB T , needs to be of the order of 10−11 m2 /s. This value is about two orders of magnitude larger than those reported for the diffusion of several solutes in -Zr at about 1500 K(Douglass 1971). Such anomalous fast diffusion is not unexpected in these systems for the following reasons: (1) the retention of excess vacancies in rapidly quenched alloys may be responsible for an enhanced diffusivity and (2) anomalous diffusion can be promoted due to the -forming tendency in these alloy systems. This point has been discussed earlier in Section 6.5. Since the formation temperatures of ordered -phases in these alloys are very high, the concentration of -embryos is expected to be large at temperatures in the range of 1300–1500 K where the spinodal decomposition appeared to have occurred. An estimation of enhanced diffusivity due to -like embryos (Banerjee and Cahn 1983) yields a diffusivity of 10−11 m2 /s in this temperature range. The spinodal decomposition of the -phase leads to the formation of a compositionally modulated structure, the directions of the modulations being along the <100 > directions which are elastically soft directions in bcc Ti and Zr alloys. Composition modulations along these three mutually perpendicular directions result in the creation of Al-rich cuboids at the nodes of the concentration waves. These Al-rich nodes remain separated from neighbouring nodes by Aldepleted -phase regions. A schematic description of such a structure is given in Figure 6.32. The next step in the transformation sequence is the formation of either the Zr2 Al (B82 ) or the Zr5 Al3 (D88 ) phase in the Al-rich nodes. These two phases form under equilibrium conditions through a peritectoid and an eutectoid reaction, respectively. The morphological features exhibited by these intermetallic phases, as observed in rapidly solidified alloys such as Zr–27% Al, Zr–25% Al and Zr3 Al–3% Nb (Banerjee and Cahn 1983, Tewari et al. 1999) and as illustrated in Figure 6.33 clearly indicate that they are not products of the equilibrium phase reactions. The features based on which the transformation mechanism has been proposed are as follows: (a) Each cuboidal solute-rich region develops into a single domain of one of the ordered structures (either B82 or D88 ), depending on the alloy composition.
526
Phase Transformations: Titanium and Zirconium Alloys
Zr2Al–3Nb
Ageing time = 600 s (a)
Ageing time = 4 h (b)
D019 in Zr2Al laths
Zr2Al
Zr2Al Ageing temperature 1123 K
Zr3Al (L12)
β
100 010 (c)
(d)
Zr2Al
Zr2Al Ageing temperature 1223 K
β
100
β
010
Figure 6.32. Schematic representation of the microstructures developed upon ageing rapidly solidified Zr–Al–Nb alloys. (a) and (b) represent microstructures developed at temperatures lower than the temperature of second peritectoid reaction, as shown in Figure 6.30, whereas (c) and (d) represent microstructures developed at temperatures higher than the temperature of second peritectoid reaction.
Figure 6.33. Rapidly solidified alloy microstructures showing solute-rich regions with cuboidal morphology.
Transformations Related to Omega Structures
527
(b) All the possible crystallographic variants are present in the quenched structure. This is also revealed by the fact that the superlattice reflections corresponding to the different crystallographic variants are of nearly equal intensity. This indicates that though the probabilities of the appearance of different variants are the same, each individual cuboidal region comprises only a single variant. (c) The B82 as well as the D88 particles exhibit lattice correspondences which can be arrived at by the lattice collapse mechanism discussed earlier. (d) Diffraction patterns show diffuse intensity distributions characteristic of partially collapsed -structures in the Al-depleted regions separating the B82 or D88 particles. (e) The size of the B82 and the D88 particles correspond to the wavelength of the compositional modulation. In the event of the formation of B82 or D88 particles by a classical nucleation and growth process, one would expect homogeneous nucleation of all the variants to occur under the large supercooling provided by rapid quenching. This would result in the formation of several variants of B82 or D88 particles within a single cuboidal Al-rich region. Contrary to this, each cuboidal region has been found to be converted into a single variant particle of the B82 or D88 phase. Taking into account all the observed features of the transformation as listed earlier, it has been suggested that the transformation within each of these cuboids is driven by an instability, leading to the development of a concentration wave and a displacement wave. The lattice correspondences between the - and the Zr2 Al (B82 ) phases or between the - and the Zr5 Al3 (D88 ) phases, as described in Section 6.7.1, suggest that the → Zr2 Al and → Zr5 Al3 transformations in stoichiometric alloys can be viewed as a superimposition of concentration and displacement waves which can be described in the following manner. The periodic displacement wave which collapses pairs of {222} planes (p = 1 2 4 5 ) to intermediate positions, keeping the preceding and the succeeding planes (p = 0, 3, 6 ) undisplaced, can be represented by a stationary longitudinal displacement wave (Banerjee and Cahn 1983): Up = Ad sin Kd · xp
(6.19)
where Up is the displacement of the pth plane and Ad is the amplitude of the displacement wave. This is essentially the same as the displacement wave associated with the → transformation where in the complete collapse corresponds to Ad =0.577 d222 . Kd is the wave vector (=2/ ; being the wavelength) and xp is the distance from the origin to the pth plane. The concentration wave which needs
528
Phase Transformations: Titanium and Zirconium Alloys
to be superimposed on the displacement wave for accomplishing the → Zr2 Al (B82 ) transformation can be expressed as 6 2n 2n 1 1 cos p − 1 + cos 5 − p Cp = + Ac 3 6 n=1 6 6
(6.20)
where Cp is the concentration of Al on the pth (222) plane, Ac is the amplitude of the concentration wave and the wave vector, Kc = 2/6 d222 The → Zr5 Al3 (D88 ) transition requires the superimposition of a vacancy concentration wave (Eq. 6.21) on the two waves described by Eqs. (6.19) and (6.20): Cv =
1 2 2 + Av cos 3 3 3
(6.21)
where Cv is the concentration of vacancies and Av is the amplitude of the vacancy concentration wave. The Landau order parameter, , can be so defined that = 0 represents the completely disordered bcc structure while = 1 corresponds to the fully ordered structure (B82 or D88 ) in which both the displacement and the replacement ordering are fully accomplished. The parameter can then be related to the amplitudes of the displacement and the concentration waves. Though the → B82 and the → D88 transitions can be viewed as processes which can proceed by progressively increasing the order parameter, it can be easily shown from symmetry considerations that they do not qualify to be second-order transitions. The free energy versus order parameter plots (Figure 6.19) proposed by Cook (1975) for explaining the fine particle morphology and the dual phase structure, as discussed in Section 6.6.4, illustrate that at a temperature below To , the instability temperature, the free energy of the system continuously drops with increase in order parameter. Under such a situation the transformation can occur homogeneously without involving the nucleation of the product phase. The system as a whole can proceed towards the product structure progressively reducing its free energy all along. The bulk transformation of the Al-rich cuboidal regions into the Zr2 Al (B82 ) and Zr5 Al3 (D88 ) phases suggests that a homogenous transformation occurs in these cases. Each of the cuboidal regions picks up a fluctuation corresponding to one of the crystallographic variants and that specific fluctuation grows in amplitude leading to the formation of the intermetallic structure through a hybrid mechanism consisting of displacive and chemical ordering. A random selection of the variant of the initial fluctuation results in the appearance of different variants of the ordered structures with equal probabilities. In order to reach the instability temperature, a high degree of supercooling is essential. The rapid quenching employed in the
Transformations Related to Omega Structures
529
experiments discussed here appears to be adequate in bringing about the requisite supercooling. The gradual enrichment of the atomic layers corresponding to p = 0, 4, 8 in the smaller Al atoms (and also in vacancies in the case of the D88 structure) is expected to facilitate the collapse of the adjacent layers p = 0 and p = 1 on one hand and p = 4 and p = 5 on the other. It is, therefore, attractive to envisage that the two fluctuations, one in concentration and the other in displacement, would amplify simultaneously. It is difficult to determine experimentally whether the two processes are fully coupled or they occur in succession. For a homogeneous transformation to occur, it is more likely that the two fluctuations grow simultaneously, one facilitating the other. The as-quenched structure consists of cuboidal particles separated from each other by Al-depleted regions within which the -phase is either fully retained or partially transformed into the mixed and athermal -structure. The size of the -particles within these Al-depleted regions is much smaller (2–5 nm) than that of the cuboidal particles. These fine particles also exhibit the ellipsoidal morphology characteristic of the athermal -phase. The evolution of the equilibrium structure from the rapidly solidified metastable structure has been studied in both Zr–27 at.% Al and Zr3 Al–3 at.% Nb alloys (Tewari et al. 1999). In the former case the retained -regions first get transformed into a mixed + structure. Subsequently the equilibrium Zr3 Al (L12 structure) phase forms through a peritectoid reaction between the Zr2 Al (B82 ) and the -phases. The phase transformation sequence in the latter alloy has revealed a very interesting transformation sequence. The cuboidal D88 particles, on ageing at a temperature of 1123 K, get converted into the B82 structure within a period of 600 s. Each D88 particle has been found to be transformed into a single B82 particle, maintaining the lattice correspondence shown in Figure 6.27. This observation can be explained by a mechanism which invokes the diffusion of Zr atoms into the Zr5 Al3 (D88 ) particles, causing the required changes in the stoichiometry and structure through the elimination of structural vacancies. A rough estimate indicates that a diffusion coefficient of 10−17 m2 /s is required for the complete transformation of a 20-nm sized D88 particle into a B82 particle within the period indicated. These B82 particles can be retained in the aged structure if the ageing treatment is carried out at temperatures between the equilibrium peritectoid temperatures corresponding to the following reactions: (1) + Zr5 Al3 → Zr2 Al (1523 K) (2) + Zr2 Al → Zr3 Al (1248 K)
530
Phase Transformations: Titanium and Zirconium Alloys
Ageing at temperatures below the second peritectoid temperature initially produces the Zr2 Al (B82 structure) phase; however, subsequently the metastable Zr3 Al (D019 structure) and the equilibrium Zr3 Al (L12 structure) phases emerge in succession. The orientation relationship between the B82 and the D019 structures has been found to be (0001)H {1120}B ; < 1¯I00>H <11¯I00>B where the subscripts B and H, respectively, stand for the B82 and the D019 structures (Tewari et al. 1999). This observed orientation relation shows that the lattice correspondence between the B82 and the D019 structures compares reasonably well with that between the and the -phases observed in the → transition induced by the application of high pressures, as discussed in Section 6.3.3. The B82 /D019 structural relationship suggests that the D019 structure can be created by the introduction of Zr atoms ¯ plane of the latter from which the (0001) plane in the B82 structure. An (1120) of the former emerges is shown in Figure 6.34. The positions marked by grey circles indicate the vacancies in the B82 structure which are filled up by Zr atoms in the D019 structure. The transformation sequence, Zr5 Al3 (D88 )→ Zr2 Al(B82 )→ Zr3 Al(D019 ), can, therefore, be considered as a process in which an open structure is gradually being filled by an influx of Zr atoms, progressively changing the composition and finally attaining a close packed configuration. Such a process can occur as long as the chemical potential of Zr in the adjoining -matrix is higher than that in the intermetallic phase, which finally attains the stoichiometry of Zr3 Al. The metastable D019 structure of the Zr3 Al phase subsequently transforms into the equilibrium L12 structure through a congruent transformation. This transformation can be accomplished by a change in the stacking sequence from the hexagonal ABABAB to the cubic ABCABCABC stacking of the close packed atomic layers which individually satisfy the Zr3 Al stoichiometry. The low-stacking fault energy (Holdway and Staton-Bevan 1986) (2 mJ/m2 ) of Zr3 Al, which can be taken as a measure of the free energy difference between its L12 and D019 forms, is consistent with the fact that the metastable D019 phase appears first when Zr3 Al precipitates from a supersaturated hcp Zr–Al solid solution (Mukhopadhyay et al. 1979). The whole sequence of formation of various phases has been schematically presented in Figure 6.35. 6.7.3 Transformation sequences in Ti base alloys The sequences of phase transformations involving ordered -phases have been studied primarily in the ternary Ti–Al–Nb system by Strychor et al. (1988), Bendersky et al. (1990a,b) and Bendersky (1994). The pseudo-binary phase diagram of the Ti3 Al–Nb system shows the composition range over which the -phase undergoes a martensitic transformation following which an ordering process leads to the formation of the ordered 2 -phase (D019 structure), as discussed earlier. In Ti3 Al–Nb alloys rich in
Transformations Related to Omega Structures
531
aH = 0.604 nm cB = 0.592 nm Distortion = 1.82%
[0001]B//[1120]H
[1100]B//[1100]H
√3 aB = 0.8476 nm √3 aH = 1.043 nm Distortion = 18.73%
CH = 0.493 nm aB = 0.489 nm Distortion = 0.82%
B type atoms A type atoms A type atoms missing in B82
¯ > plane of the B82 structure. A systematic Figure 6.34. Schematic drawing of a prismatic <1120 placement of Zr atoms on this plane can produce the basal plane of the D019 structure (shown by broken lines). The subscripts H and B, respectively, stand for the D019 and the B82 structures.
Nb, the → transformation is suppressed on quenching from the high temperature -phase field, and the -phase undergoes ordering to assume the B2 structure. The quenched B2 structure exhibits a tweed morphology with modulations along the <110> directions. The corresponding diffraction patterns show strong streaks along <110> rel vector with intensity maxima at the 1/2 <110> positions. It may be noted here that the modulations observed in Zr3 Al-based alloys are along the soft <100> directions and have been ascribed to modulations is chemical composition. The <110> modulations appear in several bcc alloys such as Ni–Al, Ni–Ti and so on and have been identified to be the result of an incommensurate shear of the {110} <1¯I0> type which can as well be described in terms of a transverse displacement wave with <110> wave vector and <1¯I0> polarization vector
532
Phase Transformations: Titanium and Zirconium Alloys Zr3Al – Nb Alloys Normal solidification liquid
Rapid solidification liquid
~102K/s Solute partitioning (Nb,Al atoms) Constitutional supercooling Dendritic morphology Interdendritic region (Nb-rich phase) Dendritic region (Nb-lean phase) Orthorombic phase β – Zr Stabilized/partially stabilized
~105 K/s Partitionless solidification (β ) Spinodal decomposition (composition fluctuations)
Solute rich regions
Solute lean regions
β – Zr Stabilized/partially stabilized
ω collapse + ordering
D88 phase Ageing treatment
ω -collapse + Ordering
Below peritectoid Above peritectoid B82 phase B82 Phase
B82 Phase
D019 Phase
L12 Phase
Figure 6.35. Schematic presentation of the formation of various phases in the Zr3 Al–Nb alloys.
(Strychor et al. 1988). It has been shown that the extinction of the streaks and striations observed under different diffracting conditions is consistent with {110} <1¯I0> shear strains and that the 1/2<110> satellite reflections, which commonly occur at the intersections of the streaks are associated with the transverse displacement wave. The observed lattice shears correspond to atomic movements required for the transformation of the bcc lattice into a 2H orthohexagonal martensite lattice. Since resolvable martensite crystals could not be imaged, Strychor et al. (1988) have concluded that such pseudo-hexagonal regions are too fine to be resolved.
Transformations Related to Omega Structures
533
In addition to the previously mentioned B2 reflections, diffracted intensity maxima at -positions have been observed in quenched Ti3 Al–Nb alloys suggesting the occurrence of a longitudinal displacement wave as well. Reflections that appear at half of the normal distance of -reflections from the origin have also been detected. This suggests that the chemical ordering present in the B2 structure is inherited in the -structure. Ageing of a Ti–27.8 at.% Al–11.7 at.% Nb alloy at 673 K has been found to result in a gradual degradation in the tweed structure and an enhancement in the intensities of -reflections. Dark-field imaging using these reflections has shown the presence of fine -like particles. Convergent beam electron diffraction has confirmed the B82 structure of these -like particles observed in the aged samples. Bendersky et al. (1990a,b) have shown that the B2 phase in a Ti–37.5 at.% Al–12.5 at.% Nb alloy transforms into the B82 phase via a metastable intermediate -phase. This alloy, when quenched or slow cooled (∼400 K/s) from 1673 K and 1373 K, exhibits a microstructure containing coarse (10–1000 m) D019 and L12 precipitates in a matrix of the transformed B2 phase. The latter consists of a mixture of fine domains (∼0.1 m) of trigonal - and hexagonal B82 -phases. Prolonged (26 days) ageing at 973 K results in a complete replacement of ! domains by B82 -domains, which show a substantial coarsening. These observations have established the B2 → → B82 transformation sequence in alloy with the nominal stoichiometry of Ti4 Al3 Nb. 6.7.4 Ordered -structures in other systems Ordered -structures have also been reported in several other alloy systems such as Cu–Zn (Prasetyo et al. 1976), Ni–Al (Georgopolous and Cohen, 1976, 1981; Reynaud, 1977) and maraging steels (Servant et al. 1987). Diffuse scattering, characteristic of -related phases, has been observed in all these systems. Table 6.8 lists the -related phases in several alloy systems and some important features associated with them. Georgopoulos and Cohen (1981) have tested several theories that have been proposed for the -transformation and have shown that Cook’s phenomenological model (1974) based on the Landau theory predicts a much weaker diffuse intensity at the 4/3 <111> rel points than is experimentally observed in an Al–46.2 wt% Ni alloy. In contrast, they have found a satisfactory agreement between the theoretically calculated and the experimentally measured diffuse scattering intensities when they consider the model of Kuan and Sass (1976) which invokes heterogeneous nucleation of -particles around point defects. The requirement, for this process, of about 1% quenched-in vacancies in Zr and Ti alloys has also been relaxed due to the suggestion that substitutional atoms too can act as nucleation sites for -particles.
534
Phase Transformations: Titanium and Zirconium Alloys
Table 6.8. Summary of - and -related phases observed in various alloys as reported in literature. Alloy
Phases observed in the system (Symmetry of the phase is written in the bracket)
Comments
Zr3 Al–Nb
¯ ¯ (Im3m), B2 (Pm3m), B82 (P63 /mmc), D88 (P63 /mcm)
Sequence of formation of phases has been established and shown in the symmetry tree in Figure 6.36.
Ti–Al–Nb
¯ , B2, (P3m1), B82 , D88
Sequence of formation of phases has already been established.
Cu-based alloys
¯ , D03 (Fm3m), Ordered ,
Formation of ordered -phase from D03 phase has been shown.
Co–Ga
,
High vacancy concentration leads to the formation of -phase.
Ni–Al
, B2, B82 ,
B82 structure in the Ni-rich Ni2 Al phase and -structure in the Ni3 Al2 phase have been observed.
Maraging steel
¯ , B2, Fe7 Mo6 (R3m), , B82
The formation of these various phases has been reported.
6.7.5 Symmetry tree It is useful to examine the subgroup/supergroup symmetry relations, as given in the International Tables of Crystallography (Hahn 1987), that describe transformation sequences involving a combination of displacive and replacive ordering. Bendersky et al. (1990a,b) and Tewari et al. (1999) have constructed a symmetry tree for displaying the symmetry relationships between the different phases encountered in the transformation path. On the basis of these symmetry relations, inferences can be drawn on the following points: (i) whether a transformation is of the first or of a higher order; (ii) how many rotational and translational variants can be created or destroyed in a transformation and (iii) which intermediate phases are likely to be metastable. The International Tables of Crystallography list “maximal nonisomorphic subgroups” which are the space groups of possible product phases that may form if the symmetry reduction does not include an intermediate step. This point can be ¯ to the explained by taking the example of the transition of the B2 structure (Pm3m) ¯ -structure (P3m1); the latter is not listed as a maximal non-isomorphic subgroup of the former. The symmetry reduction accompanying the B2→ transition
Transformations Related to Omega Structures
535
¯ ¯ → P3m1, ¯ involves the sequence Pm3m→ R3m and both the steps correspond to listed supergroup–subgroup relations. The index of symmetry reduction (or increment) is an integer (shown in square brackets in the symmetry tree) which is equal to the number of symmetry elements in the supergroup divided by the number in the subgroup. If the index is an odd integer then the transition is necessarily of the first order (except at an isolated point) and if it is an even integer, then the transition can be of either the first or a higher order. The index also corresponds to the number of domains/variants that may be created in the product phase from a single crystal of the parent phase. ¯ ¯ destroys the fourfold symmeThe first step of symmetry change, Pm3m→ R3m try of the B2 structure by effecting a homogeneous rhombohedral distortion along an <111>B2 direction which is the direction of lattice collapse and is selected as ¯ → P3m1 ¯ the threefold axis in the trigonal -type phase. The second step, R3m selects which of the (111) planes collapse to form double layers and which remain undisplaced. The symmetry index [4] corresponding to the first step gives the number of rotational variants associated with the rhombohedral distortion while the index [3] gives the number of translational subvariants associated with each rotational variant. The → B82 transition involves an increase in symmetry: P63 /mmc is a ¯ supergroup P3m1. Usually an increase in symmetry is associated with an increase in entropy and so the low entropy phase is expected to be the low temperature phase as well. The isothermal supergroup formation in the → B82 transition, therefore, suggests that the -phase is metastable. This is also indicated from the fact that the → B82 transition proceeds from a more ordered -phase to a less ordered B82 phase. ¯ indicates that the B2 (Pm3m ¯ The fact that P63 /mmc is not a subgroup of Pm3m → B82 (P63 /mmc) transition must be strongly of the first order. The observed transformation path, B2→ → B82 , traverses a state of minimum symmetry ¯ ¯ and P63 /mmc. This space group that is a subgroup of both Pm3m ( , P3m1), of minimum symmetry is, in fact, the intersection of the parent and the product groups. It is worth noting that the formation of the -phase as an intermediate metastable phase provides a continuous structural path for the B2→ B82 transition.The existence of such a continuous path provides the basis for a common thermodynamic function for all the phases encountered. The occurrence of vacancy ordering in the D88 structure modifies the symmetry path, leading to the formation of the D88 phase which has a lower symmetry than that of the B82 phase. Elimination of structural vacancies results in the conversion of the D88 structure into the B82 structure. The sequence of such phase transformation which is dictated by symmetry consideration has been shown in Figure 6.36. It can be noticed from the figure that there are multiple choices
536
Phase Transformations: Titanium and Zirconium Alloys Chemically ordered
Chemically disordered Im3m (A2:bcc) Homogeneous disordering
Chemical ordering [6]
Chemical ordering [2]
Fm3m D03:BiF3)
Rm3m (B2:CsCl)
R3m
Homogeneous distortion [4]
Homogeneous distortion ω -Collapse Chemical ordering [4] [2], c′=2c R3m P3m1 (Trigonal ω ) Completion of collapse [2]
Chemical ordering
ω -Collapse
ω -Collapse [4]
[3]
[2], c′ = 2c
P6/mmm (ω Ti)
P3m1 (ω ′,ω ″ ) Vacancy ordering
P63 /mcm (D88:Mn5Si3)
R3m
Disordering in single layer
Ordered ω -Structure
Disordering in single layer [2] P63/mcm (D88:Ni2In)
Disordering Disordered ω
Figure 6.36. Symmetry tree showing the sequence of the phase transformations of the -related phases.
available on each mode. The transformation from one phase to another is, therefore, dictated by the external parameters.
6.8
INFLUENCE OF -PHASE ON MECHANICAL PROPERTIES
The mechanical properties of -Ti alloys undergo drastic changes with the precipitation of the -phase: a large increase in the yield stress, which is invariably accompanied by a significant reduction in ductility, is observed. Gysler et al. (1974) have studied the deformation behaviour of -Ti–Mo alloys and have shown that extensive slip localization occurs with -precipitation. 6.8.1 Hardening and embrittlement due to -phase As mentioned earlier in this chapter, the presence of the -phase in -quenched alloys based on Ti and Zr is often inferred from the high hardness values
Transformations Related to Omega Structures
c
16
Load [103 N]
537
b 12
8
4
a
0
2
6
4
8
10
12
Elongation [%]
Figure 6.37. Engineering load-elongation curves of the Ti–11% Mo alloy for (a) as-quenched specimen, (b) specimen aged at 623 K for 3 h and (c) specimen aged at 623 K for 10 h. Increase in yield stress and sharp drop in ductility upon ageing is directly related with the increase in the volume fraction of the -phase.
exhibited by these. A comparison of the stress–strain diagrams (Figure 6.37) of the -quenched Ti–15% Mo alloy after ageing under different conditions clearly indicates that an ageing treatment which causes -precipitation can significantly increase the yield strength of the alloy though the ductility is severely impaired. Ageing treatments to produce -precipitation restores the ductility to a limited extent. One of the important characteristics of -embrittlement is that the fracture surface shows a ductile dimple-type morphology even when the material shows no macroscopic ductility. The microvoid coalescence mechanism, which gives rise to the dimple morphology, appears to be operative on a microscopic scale along the fracture surface inspite of the occurrence of brittle fracture. Gysler et al. (1974) have studied the microstructures of deformed -Ti–Mo alloys and have reported that dislocations in the deformed material are primarily arranged along well-defined slip bands. Dark-field imaging using -reflections has revealed that these bands are devoid of -particles. The destruction of -particles in the vicinity of slip bands essentially implies that these particles get sheared due to the passage of a fairly large number of dislocations through these particles. The stress required for the cutting of -particles has been estimated by using the following relation (Gleiter and Horbogen 1965) 3
1
1
1
"o = 102 2 G− 2 b−2 r 2 f 2
(6.22)
538
Phase Transformations: Titanium and Zirconium Alloys
where the increase in the critical resolved shear stress, "o , is expressed in terms of the shear modulus, G, of the matrix, the Burgers vector of dislocations, b, the anti-phase boundary energy and the average radius, r, and the volume fraction, f , of -particles. Gysler et al. (1974) have plotted the yield stress, = 2"o against r 1/2 f 1/2 , taking the experimental values obtained on a Ti–11% Mo alloy and have arrived at a value of to be 300 kJ/m2 (Figure 6.38). Since the anti-phase boundary energy, , for the -particles, determines the specific interaction force between these particles and mobile dislocations; this high value of shows that the shearing of -particles is, indeed, a very difficult process. However, the stress requirement for the alternative mechanism involving dislocations bypassing closely spaced -particles has been shown to be even higher than that for particle shearing (Figure 6.38). The observed hardening due to -precipitates can, therefore, be rationalized in terms of the stress required for the shearing of -particles along the slip bands. The formation of sharp slip bands in -alloys containing a distribution of densely populated, fine, aged -particles can also be viewed as a consequence of the particle shearing process. Once a set of leading dislocations glide along a slip plane, cutting across the -particles on their path, a soft -free channel is created. Subsequent deformation in these soft channels can occur without dislocations experiencing such high resistance on the slip path. Such -free channels have been observed in deformed samples and from trace analyses of these slip bands it has been shown
σ0.1 [MNm–2]
1800
1500
1200
900
600
0
1
2
3
r½ f½ [Ž]
Figure 6.38. Yield stress (#01 ) of the Ti–11% Mo alloy aged at 623 K as a function of size (r 1/2 ) times volume fraction (f 1/2 ) of the -particles. Circles represent the calculated stress for the bypass mechanism.
Transformations Related to Omega Structures
539
that in most cases they occur along the {110} planes and less frequently along the {112} and the {123} planes. The Burgers vectors associated with the dislocations along these planes have been found to be of the a/2 <111> type. The observed decrease in macroscopic ductility with increasing ageing time (Figure 6.37) can be explained qualitatively in terms of an increased tendency for inhomogeneous slip. Plastic flow remains confined within a limited number of slip bands, the points where they mutually interest or they meet the grain boundaries becoming the sites of crack nucleation. The limitation of plastic flow to narrow slip bands causes high stress concentrations and favours crack nucleation and growth at low macroscopic plastic strains. The local deformation within the slip bands is quite pronounced, as revealed from the large offsets observed at the intersections of slip bands with other planar features. A survey of the fracture surface topography of -alloys as a function of ageing time, when aged at temperatures below about 773 K, shows the following trends: (a) The fracture surfaces of -solutionized samples, which contain very fine athermal -particles and which exhibit quite high macroscopic ductility (about 10% elongation) and a reasonably homogeneous plastic flow, are characterized by large dimples with an average spacing of about 50 m. (b) As the percentage elongation comes down to about 4–5% as a consequence of -precipitation due to ageing, the average dimple size drops down to about 1–2 m. (c) As the macroscopic ductility falls to a negligible level, fracture propagates predominantly along grain boundaries, as revealed in low-magnification micrographs. Each grain boundary facet, at higher magnifications, shows the presence of steps corresponding to offsets created by slip bands and a distribution of fine (1–2 m) dimples. The extensive microscopic flow along the fracture path is well demonstrated inspite of nearly zero macroscopic ductility. 6.8.2 Dynamic strain ageing due to -precipitation Recent studies (Banerjee and Naik 1996) on the plastic flow behaviour of a -Ti alloy (Ti–15% Mo) and of an intermetallic (Nb–Ti–Al (Grylls et al. (1998))) in the temperature range 575–775 K have shown that very pronounced serrated yielding occurs in these systems over a certain range of strain rates. Such non-linear plastic behaviour is often connected with a plastic instability arising out of dynamic interactions of mobile dislocations with individual solute atoms, their clusters or fine precipitates. The phenomenon of dynamic strain ageing, leading to serrated yielding, is known in the literature as the Portevin–Le Chatelier (PLC) effect. The occurrence of the PLC effect in substitutional alloys has generally been attributed to a
540
Phase Transformations: Titanium and Zirconium Alloys
periodic locking and unlocking of dislocations by individual solute atoms. Since this process involves long range diffusion of solute atoms, enhanced diffusivity due to plastic flow has been invoked for explaining the kinetics of the process. Numerous experimental observations, showing the presence of a critical strain, c , preceding the appearance of serrated yielding, have provided support to the diffusivity enhancement model. The extensive literature on this subject has been reviewed elegantly by Estrin and Kubin (1995). The occurrence of PLC effects has been treated on the basis of the collective movement of a group of dislocations by Schoeck (1984) and of the propagation of localized deformation bands by Schlipf (1992). There have also been reports in which the locking is considered to be effected not by individual solute atoms but by clusters of solute atoms (Onodera et al. 1983, 1984). The dynamic strain-ageing phenomenon due to -precipitation assumes a special significance, as this is an example which demonstrates that the PLC effect can also result from the dynamic interaction of a group of dislocations directly with precipitates. Experimental evidences (Banerjee and Naik 1996) on the basis of which this new mechanism of the PLC effect has been established are summarized in this section. The stress–strain plots for -quenched Ti–15% Mo alloy samples tested at different temperatures in the range 455–775 K at a strain rate of = 131×10−4 s−1 are shown in Figure 6.39 which also includes a magnified portion of the loadelongation plot illustrating the characteristic C-type serrated flow, as per the classification scheme proposed by Brindley and Worthington (1970). The ranges of strain over which the serrated flow behaviour is observed are indicated on the flow curves schematically by short vertical segments representing the extents of the load drop. The temperature–strain rate regime corresponding to serrated yielding is represented in the strain rate versus reciprocal temperature plot (Figure 6.40). The shaded region in this figure, which represents the regime of serrated flow, is confined within a low temperature and a high temperature limiting lines, the former being temperature dependent and the latter being athermal. The characteristic features of dynamic strain-ageing, such as (i) the presence of a hump in the descending part of the yield stress versus temperature plot and (ii) a negative strain rate sensitivity, are observed under the conditions corresponding to the occurrence of serrated yielding. The critical strain, c , is seen to decrease with temperature; this is a behaviour which is known to be normal in the PLC effect literature. Tensile specimens unloaded after the occurrence of a few load drops show the presence of macroscopic deformation bands (known as PLC bands) across the width of the specimens. An one-to-one correspondence between the number of load drops and the number of PLC bands is seen in the early stages of serrated flow. A magnified view of PLC bands shows fine slips bands which are not
Transformations Related to Omega Structures
541
Strain rate = 1.31 × 10–4 S–1
Load
5 kg
0.01 nm 50 MPa
Stress
455 K Elongation 525 K
300 K 575 K
675 K
625 K
725 K
775 K
5% Strain
Figure 6.39. Stress–strain plots for the -quenched Ti–15% Mo samples tested at different temperatures (as indicated alongside of the flow curves) at a strain rate of 131 × 10−4 s−1 . Serrated flow indicated by short vertical lines on the flow curves is observed in the temperature range 575–725 K. Arrows are marked at the onset of serrated flow. The inset shows a magnified portion of a load-elongation plot.
perfectly straight and are confined within -grains. These slips bands which are not observed outside the PLC bands, create offsets when they intersect each other or when they meet a grain boundary. TEM examination of PLC bands shows “deformation bands” on a still finer scale. These bands are essentially channels devoid of -precipitates. Figure 6.41 shows PLC bands, deformation bands and -free channels. The occurrence of serrated flow and the magnitudes of load drops can be controlled by changing the stability of the -phase with respect to -precipitation at the testing temperature. This has been demonstrated by altering the composition of the matrix either by giving a prior ageing treatment or by choosing an alloy of a different composition. The influence of prior ageing on the amplitude of serrations in the Ti–15% Mo alloy is shown in Figure 6.42. Prior ageing at 675 K for 1 h and 168 h which produces a Mo-lean aged -phase makes the -matrix richer in Mo. The magnitude of stress drops is seen to decrease with increasing ageing time which corresponds to increasing Mo concentration in
Phase Transformations: Titanium and Zirconium Alloys Temperature (K) 773 723 673 623 573 523 498 473 453
65.5
Strain rate × 10 (s–1)
No serration Serration flow
0.5
26.2
0.2
13.1
0.1
Serration flow 6.55
0.05
2.62
0.02
1.31
0.01
0.653
Cross-head speed (cm/min)
542
0.005 1.2
1.4
1.6
1.8
2.0
2.2
1/T × 103 (K–1)
Figure 6.40. Strain rate-temperature region in which serrated flow is observed in the Ti–15% Mo alloy.
the matrix. The serrated flow can be fully suppressed by giving a prior ageing treatment at 875 K for 1 h, which produces -precipitates and causes further Mo enrichment of the matrix. An alloy containing 25% Mo does not show serrated yielding in the entire range of temperature and strain rate scanned in Figure 6.43. The fact that -precipitation is responsible for the dynamic strain-ageing phenomena in -forming systems is also revealed from static ageing experiments in which the cross-head motion of the testing machine is arrested after the start of serrated flow and the load is continuously recorded as a function of time. This test is similar to a stress relaxation test (Gupta and Li 1970) normally carried out for the determination of the activation parameters of plastic flow. During such tests it has been noticed that the stress initially relaxes and subsequently rises. As the stress value reaches the upper envelope of the flow curve, a sudden drop in stress is observed. This is followed by an increase in stress upto the upper threshold value when a second drop in stress occurs. Such repeated load drops continue to occur, giving rise to a serrated appearance of the stress-versus-time plot. This static
Transformations Related to Omega Structures
543
(a)
(b)
(c)
(d)
(e)
(f)
Figure 6.41. Microstructures of tensile specimens unloaded after the occurrence of a few load drops. (a) Microstructure showing the presence of deformation bands within PLC bands, (b) deformation band within a -grain. Termination of deformation of these bands at the grain boundary can be noticed. (c) and (d) are Bright-field TEM micrographs showing parallel arrangement of dislocations within a deformation band. (e) Reduced density of particles within the imaged band and (f) restoration of -particles within a deformation band after ageing.
T = 575 K
(a) (b)
Stress
(c) (d) 20 MPa
0.2% Strain
Figure 6.42. The influence of prior ageing treatment on the amplitude of serration is seen from parts of the flow curves at 575 K of Ti–15 Mo alloy, heat-treated at different ageing conditions, (a) 575 K for 1 h, (b) 675 K for 1 h, (c) 675 K for 168 h and (d) 875 K for 1 h.
Phase Transformations: Titanium and Zirconium Alloys
50 MPa
544
Stress
Strain rate = 1.31 × 10–4 S–1
5% Strain
Figure 6.43. Stress–strain curve of the Ti–25% Mo alloy. Absence of serrated yielding could be noticed from the figure.
ageing phenomenon can be understood by considering the different components of strain which contribute towards the total strain, tot , of the sample ( tot = 0, when the cross-head motion is stopped). tot =
d
/E + p + Cf + M = 0 dt
(6.23)
The modulus, E, of the specimen is given by (Eo +E), where Eo is the modulus at the start of relaxation, E the difference between the moduli of the - and the -phases and f the volume fraction of the -phase formed during time t; p is the plastic strain of the specimen due to relaxation. Since the → transformation is associated with a 14% volume contraction, there is a tendency for contraction of the specimen due to the dynamic -formation as given by Cf , where C is a constant related to the volume change due to the transformation. The last term in Eq. (6.23), M, is the strain contribution due to the elastic deformation of the testing machine, M, which is related to the stiffness of the machine. The volume fraction, f , of the -phase after time t can be written as f = fo 1 − et/"
(6.24)
where " is the time constant of -precipitation and fo the equilibrium volume fraction of the -phase. The relation between the plastic strain rate p and can be written as = 1 + K log p / o
Transformations Related to Omega Structures
545
Table 6.9. Input parameters used for simulating the stress-versus-time plot during on-load ageing (Figure 6.44). Elastic modulus, Eo Modulus difference, E Machine stiffness Equilibrium -volume fraction, fo Time constant for -formation,
50 000 MPa 20 000 MPa 50 000 N/mm 10% 50 s [Figure 6.44] 100 s [Figure 6.44] 490 MPa
Threshold stress for formation of ∗ deformation bands, 1
where K is inversely proportional to the activation volume of the process and o is the maximum strain rate at a given temperature at which serrated flow is observed. Substitution of reasonable values (as indicated in Table 6.9) yields a -versus-t plot (Figure 6.44) showing an initial stress relaxation, followed by a rise in stress and then repeated stress drops, as encountered experimentally. The oversimplified treatment presented here qualitatively explains the relaxation behaviour due to the usual plastic flow and the competing rise in stress resulting from specimen contraction due to dynamic -precipitation. When the latter process
530 520 510 500
Stress (MPa)
490 480
(a)
T0 = 50 s
(b)
T0 =100 s
470
500 490 480 470 460 450
100
200
300
400
500
600
700
Time (s)
Figure 6.44. Computed stress-versus-time plot for static ageing under load experiment. The parameters used for computation are given in Table 6.9.
546
Phase Transformations: Titanium and Zirconium Alloys
dominates, there is a net rise in stress due to on-load static ageing. When the stress finally reaches the threshold value for the load drop, deformation bands are nucleated. The repeated occurrence of load drops in the stress–time plot can, therefore, be explained in terms of a combination of a stress relaxation process and an -ageing process. Repeated yielding is encountered in alloys, in general, during deformation by repeated mechanical twinning, stress-assisted martensitic transformation and dynamic strain ageing (PLC effect). In the -forming systems discussed here, no deformation twins or martensite plates are detected within the PLC bands. Further, the serrated yielding process in -forming systems is thermally activated whereas deformation twinning and martensite formation are athermal processes. Contrary to the observations made in -forming systems, the latter processes are favoured at lower temperatures. Dynamic strain-ageing in an alloy can occur through either of the following mechanisms: (a) by solute atom–dislocation interactions to form a solute atmosphere around the dislocations and (b) by the establishment of chemical and/or displacement order or fine scale precipitation. Both these mechanisms can lead to pinning of dislocations and under those test conditions where locking of mobile dislocations and unlocking of pinned dislocations become competitive the alloy exhibits repeated yielding. In solute atom–dislocation interactions, solute atoms migrate rapidly to form an atmosphere around freely moving dislocations resulting in a drag on them. The stress required to move dislocations thus increases until they are able to free themselves from the solute atmosphere. The mechanism involving pinning of dislocations by either interstitial or substitutional solute atoms is not appropriate in the context of serrated yielding in -forming systems in view of the following observations: a) Serrated flow behaviour can be completely or partially suppressed in these systems by stabilizing the -phase with respect to the → transformation. b) The upper temperature limit of the strain rate–temperature regime for serrated flow is independent of the strain rate and is dictated by the dissolution temperature of the -phase. Based on the experimental observations described in the preceding paragraphs on serrated yielding in -forming systems (Ti–Mo alloys, Nb–Ti aluminides), Banerjee and Naik (1996) have proposed the following mechanism.
Transformations Related to Omega Structures
547
A sharp load drop can be exhibited in a stress–strain plot when a sudden burst of plastic flow is triggered. It is envisaged that such sudden bursts of plastic flow can occur when deformation bands are created by the shearing of -particles within these bands. Microstructural observations reveal that the localization of sudden plastic flow occurs in three different length scales. In the finest scale, a single -free channel is created by the passage of leading dislocations which facilitates the passage of an avalanche of trailing dislocations through the same soft channel. A typical -free channel is seen to be a few nanometres wide. A group of such channels constitute a deformation band, typically a few m thick. Such deformation bands remain confined within the grains, but plastic flow is triggered in adjoining grains by the nucleation and growth of fresh deformation bands. In this manner plastic compatibility between neighbouring grains is maintained. The density of deformation bands decreases as one moves from the centre to the edge of a macroscopic deformation band (PLC band, typically a few millimetres wide) of inhomogeneous plastic flow. The sudden plastic flow is, therefore, caused by the creation of -free channels, the grouping of these channels in deformation bands and the propagation of such a deformation process from one grain to its neighbouring grain. The triggering step involves the movement of an ensemble of dislocations in a material containing a distribution of fine -particles which are sheared and even destroyed with the passage of the leading dislocations. It is this shear localization process which is responsible for the exceptionally poor ductility of a material with the + aged- microstructure. As shown by Gysler et al. (1974) extensive plastic flow along soft -free channels results in large offsets at grain boundaries leading to the nucleation of microvoids and subsequently fracture at very low macroscopic strains. The same microstructural state can exhibit a reasonably high ductility when the testing temperature is raised to the range 550–750 K where serrated yielding is observed. This is possible because at these temperatures the dynamic formation of -precipitates harden the soft channels, arresting plastic flow within them. In the serrated flow regime, plastic flow within -free channels cannot continue upto the point of microvoid nucleation as the dynamic restoration of -particles gradually pin the mobile dislocations, the density of which decreases with time between two successive load drops. The PLC effect can be understood in terms of the density of mobile dislocations by considering the following expression relating the macroscopic strain rate, ˙ , the mobile dislocation density, m , the Burgers vector b and the velocity, $, of mobile dislocations: ˙ = m b$
(6.25)
548
Phase Transformations: Titanium and Zirconium Alloys
The average velocity of moving dislocations in a material containing a distribution of dislocation pinning obstacles can be expressed as $ = l/tw + tf ≈ l/tw
(6.26)
where l is the mean free spacing of the obstacles, tw is the average waiting time for dislocations at obstacles and tf (<< tw ) is the time of flight of dislocations from one obstacle to the next. Schoek (1984) has argued that an instability of plastic flow (accompanied by a negative strain rate sensitivity) occurs in the neighbourhood of a critical value, p , of the mobile dislocation density (Figure 6.45) where tw /tp = m / p
(6.27)
tw being the average value of the waiting time and tp , the average ageing time for pinning a dislocation. The same argument can be invoked in describing the plastic flow within the localized deformation bands in -forming systems. When the kinetics of -formation are favourable, the time (tp ) required to pin dislocations by restoration of -particles along the -free channels becomes smaller than the waiting time tw . Under this condition, as plastic flow continues within a -free channel immediately after the propagation of an avalanche of dislocations through it, more and more dislocations become pinned by fresh -particles. Consequently, the stress required to sustain the strain rate gradually rises. The flow within a channel at a constant strain rate can be schematically represented by a “local” stress–strain plot (Figure 6.46) in which the flow stress, 2 , corresponds to the high mobile dislocation density ( 2 ) within a channel immediately after its creation while the threshold stress for shearing of -particles is given by 1 , with the corresponding mobile dislocation density 1 . Following the stress drop, (1 –2 ), as m decreases
σe ε2 ε1
ρp
1
ρp
2
ρm
Figure 6.45. The change of effective stress (e ) plotted against the density of moving dislocations m = ˙ /lbtw for two constant strain rates ˙ 1 and ˙ 2 > ˙ 1 .
Transformations Related to Omega Structures
σ
σ1 σ2
549
ρ1 ρ2 ε
Figure 6.46. Schematic stress () versus strain ( ) diagram within a localized deformation band; 1 represents the threshold stress required for triggering a deformation band.
Figure 6.47. Multiscale deformation process shown in form of pyramidal structure. Lowest portion of the pyramid shows macroscopic PLC bands in a sample (in the scale of mm) unloaded after the appearance of two load drops in the stress–strain plot. Scanning electron micrograph showing the -grain structure revealed within the PLC band. Presence of finer deformation bands (in the scale of 10 s of m) within the grains inside a PLC band could be noticed in the next block. Parallel arrangement of the dislocations within a -grain and nanometre wide -free channels were observed in the same region at much finer scale (∼nm) as shown in two upper blocks. In the top block of the pyramid is the lattice collapse mechanism operative at the unit cell level.
550
Phase Transformations: Titanium and Zirconium Alloys
from 2 to 1 with increasing degree of pinning, the stress rises and finally attains the threshold stress, 1 , which triggers either the creation of fresh channels or the reactivation of channels which have earlier experienced a similar burst of dislocation mobilization. Successive operation of these processes leads to a serrated flow in which the maximum and the minimum stress envelopes are given by 1 and 2 , respectively. The activation energy of the ageing process can be evaluated from the plot of ln against 1/T , where o is the maximum strain rate at the temperature T where serrated flow is observed. The experimental value of 125 kJ/mol for Ti–15% Mo is consistent with that for the diffusion of solute atoms (Mo) in -Ti (125 kJ/mol). Since the segregation of solute atoms occurs as a precursor to displacement ordering for the restoration of -particles, the thermal activation required for the restoration process is essentially the same as that for diffusion. In summary, it is to be emphasized that the aged -particles, which render the -alloys exceptionally brittle at room temperature, are responsible for a serrated plastic flow at temperatures where dynamic -formation can kinetically compete with the particle shearing process. As has been pointed out earlier, the general phenomenon of strain ageing is related to a competitive process of pinning and mobilization of plastic flow. While in a majority of alloy systems, solute atom–dislocation interactions are responsible for causing this phenomenon, in forming alloys, pinning of dislocations by fine -particles and creation of soft -free channels followed by a burst of plastic flow along these channels, result in periodic serration in the stress–strain plot. The multiscale deformation during serrated yielding could be summarized in the form of a pyramid shown in Figure 6.47.
REFERENCES Afronikova, N.S., Degtyareva, Y.F., Litvin, Y.A., Rabinkin, A.G. and Skakov, Y.A. (1973) Soviet Physics Solid State, 15, 746. Bagaryatskiy, Y.A., Nosova, G.I. and Tagunova, T.V. (1955) Dokl. Akad. Nauk. USSR, 105, 1225. Bagaryatskiy, Y.A. and Nosova G.I. (1962) Fiz. Metal. Metalloved., 13, p. 415. Banerjee, S. and Cahn, R.W. (1982) Solid-Solid Phase Transformations, (eds H.I. Aaronson et al.) TMS, Warrendale, PA, USA, 1005. Banerjee, S. and Cahn R.W. (1983) Acta Metall., 31, 721. Banerjee, S. and Naik, U.M. (1996) Acta Mater., 44, 3667. Banerjee, S., Tewari, R. and Mukhopadhyay, P. (1997) Prog. Mater. Sci., 42, 109.
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Bendersky, L.A., Boettinger, W.J., Burton, B.P., Biancaniello, F.S. and Shoemaker, C.B. (1990a) Acta Metall., 38, 931. Bendersky, L.A., Burton, B.P., Boettinger, W.J. and Biancaniello, F.S. (1990b) Scripta Metall., 24, 1541. Bendersky, L.A. (1994) Proceedings of International Conference on Advances in Physical Metallurgy, ICPM-94 (Ed. S. Banerjee and R.V. Ramanujan) 216. Brindley, B.J. and Worthington (1970) Metall. Rev., 4, 101. Bridgman, P.W. (1948a) Proc. Am. Acad. Art Sci., 76, 55. Bridgman, P.W. (1948b) Proc. Am. Acad. Art Sci., 76, 71. Bridgman, P.W. (1952) Proc. Am. Acad. Art Sci., 81, 165. Borie, B., Sass, S.L. and Andreassen, A. (1973) Acta Crystallogr., A29, 585. Cahn, J.W. (1970) Critical Phenomena in Alloys, Magnets and Superconductors (ed. R.E. Mills, E. Ascher and R.I. Jaffee) McGraw Hill, New York. Cook, H.E. (1973) Acta Metall., 21, 1445. Cook, H.E. (1974) Acta Metall., 22, p. 239. Cook, H.E. (1975) Acta Metall., 23, p. 1041. Dawson, C.W. and Sass, S.L. (1970) Metall. Trans., 1, p. 2225. de Fontaine, D. (1970) Acta Metall., 18, p. 275. de Fontaine, D., Paton, N.E. and Williams, J.C. (1971) Acta Metall., 19, 1153. de Fontaine, D. (1988) Metall. Trans., 19A, 169–175. Dey, G.K., Tewari, R., Banerjee, S., Jyoti, G., Gupta, S.C., Joshi, K.D. and Sikka, S.K. (2004) Acta Mater., 52, 5243–5254. Douglass, D.L. (1971) The Metallurgy of Zirconium, International Atomic Energy Agency, Vienna. Estrin, Y. and Kubin, L.P. (1995) Spatial Coupling and Prerogative Plastic Instability in: Continuum Models for Materials and Microstructure, Editor H.B. Muhlhaus, John Wiley and Sons, pp. 395–450. Friedel, J. (1969) The Physics of Metals (ed. J.M. Ziman), Chapter 9, Cambridge University Press, Cambridge. Frost, P.D., Parris, W.M., Hirsh, L.L., Doig, J.R. and Schwartz, C.M. (1954) Trans. Am. Soc. Metals, 46, 231. Georgopoulos, P. and Cohen, J.B. (1976) Acta Metall., 24, 1053. Georgopoulos, P. and Cohen, J.B. (1981) Acta Metall., 29, 1535. Gleiter, H. and Hornbogen (1965) Phys. Status Solidi, 12, 235. Grylls, R.J., Banerjee, S., Perugulam, S., Wheeler, R. and Fraser, H.L. (1998) Intermetallics, 6, 749. Gupta, I. and Li, J.C.M. (1970) Mater. Sci. Eng., 6, 20. Gysler, A., Lutjering, G. and Gerold, V. (1974) Acta Metall., 22, 901. Hahn, T. (1987) International Tables For Crystallography, Volume A, Space Group Symmetry, Published by The International Union of Crystallography, The Netherlands. Hatt, B.A. and Roberts, J.A. (1960) Acta Metall., 8, 575. Hickman, B.S. (1969) J. Mater. Sci., 4, 554. Ho, K.M., Fu, C.L. and Harmon, B.N. (1984) Mater. Phys. Rev. B, 29, 1575. Ho, K.M. and Harmon, B.N. (1990) Mater. Sci. Eng. A, 127, 155.
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Silcock, J.M., Davies, M.H. and Hardy, H.K. (1955) Symposium on the Mechanism of Phase Transformations in Metals, Institute of Metals, London, 93. Song, S.G. and Grey, G.T. III, (1994) High Pressure Science and Technology-1993 (eds. S.C. Schmidt , J.W. Flows, G.S. Samara and M. Ross) AIP Press, New York, 251. Song, S.G. and Grey, G.T. III, (1995) Philos. Mag., 71, 275. Stassis, C., Zarestky, J., Wakabayashi, N. (1978) Phys. Rev. Lett., 41, 1726. Strychor, R., Williams, J.C. and Soffa, W.A. (1988) Metall. Trans., 19A, p. 225. Tewari, R., Mukhopadhyay, P., Banerjee, S. and Bendersky, L.A. (1999) Acta Mater., 24, 1307. Usikov, M.P. and Zilbershtein, V.A. (1973) Phys. Status Solidi, 19A, 53. Vohra, Y.K. (1978) J. Nucl. Mater., 75, 288. Vohra, Y.K. (1979) Acta Metall., 27, 1671. Vohra, Y.K., Sikka, S.K., Menon, E.S.K. and Krishnan, R. (1980) Acta Metall., 28, 683. Vohra, Y.K., Sikka, S.K., Menon, E.S.K. and Krishnan, R. (1981) Acta Metall., 29, 457. Willaims, J.C., de Fontaine, D. and Paton, N.E. (1973) Metall. Trans., 4, 2701. Zilbershtein, V.A., Chistotina, N.P., Zharov, A.A., Grishina, N.S., and Estrin, E.I. (1975) Fiz., Metal Metalloved, 39, 445.
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Chapter 7
Diffusional Transformations 7.1 Introduction 7.2 Diffusion 7.2.1 Diffusion mechanisms 7.2.2 Flux equations: Fick’s laws 7.2.3 Self- and tracer-diffusion coefficients in -Zr and -Ti 7.2.4 Self- and tracer-diffusion coefficients in -Zr and -Ti 7.2.5 Interdiffusion 7.2.6 Phase formation in chemical diffusion 7.2.7 Diffusion bonding 7.3 Phase Separation 7.3.1 Phase separation mechanisms 7.3.2 Analysis of a phase diagram showing a miscibility gap 7.3.3 Microstructural evolution during phase separation in the -phase 7.3.4 Monotectoid reaction – a consequence of -phase immiscibility 7.3.5 Precipitation of -phase in supersaturated -phase during tempering of martensite 7.3.6 Decomposition of orthorhombic -martensite during tempering 7.3.7 Phase separation in -phase as precursor to precipitation of - and -phases 7.4 Massive Transformations 7.4.1 Thermodynamics of massive transformations 7.4.2 Massive transformations in Ti alloys 7.5 Precipitation Of -Phase in -Matrix 7.5.1 Morphology 7.5.2 Orientation relation 7.5.3 Invariant line strain condition 7.5.4 Interfacial structure and growth mechanisms 7.5.5 Morphological evolution in mesoscale 7.6 Precipitation of Intermetallic Phases 7.6.1 Precipitation of intermetallic compounds from dilute solid solutions 7.6.2 Precipitation in ordered intermetallics: transformation of 2 -phase to O-phase
558 560 560 562 564 566 570 578 584 587 589 597 603 606 609 616 618 623 623 626 632 633 642 643 648 655 657 657 662
7.7 Eutectoid Decomposition 7.7.1 Active eutectoid systems 7.7.2 Active eutectoid decomposition in Zr–Cu and Zr–Fe system 7.8 Microstructural Evolution During Thermo-Mechanical Processing of Ti- and Zr-based Alloys 7.8.1 Identification of hot deformation mechanisms through processing maps 7.8.2 Development of microstructure during hot working of Ti alloys 7.8.3 Hot working of Zr alloys 7.8.4 Development of texture during cold working of Zr alloys 7.8.5 Evolution of microstructure during fabrication of Zr–2.5 wt% Nb alloy tubes References
670 675 676 683 684 687 691 701 706 710
Chapter 7
Diffusional Transformations
List of Symbols JABv : Atomic flux of species A, B and vacancy c or C: Concentration of solutes x: Distance D: Diffusion coefficient or diffusivity D0 : Pre-exponential constant for diffusivity Q: Activation energy for diffusion T : Temperature Tm : Melting temperature − → v : Velocity of lattice plane and transformation front ˜JA J˜B : Interdiffusion flux of A and B : Width of intermediate phase region in a diffusion couple G: Total free energy change Gc : Chemical free energy change : Surface energy change W : Mechanical work due to volume change accompanying formation of a new phase at the interdiffusion couple w: Width of diffusion zone Kp : Penetration constant Qp : Activation energy for diffusion-controlled growth HXS : Excess enthalpy of a solid solution (deviation from enthalpy of ideal solid solution) G: Gibbs free energy A1 (C1 ): Chemical potential of the component A in the 1 -phase having solute (B) concentration, C1 : Wave length of concentration modulation in spinodally decomposed solid solution G : Gradient energy of the solid solution with concentration modulation
: Fractional change in lattice parameter, a, per unit composition change E: Young’s modulus : Poisson’s ratio Vm : Molar volume 557
558
Phase Transformations: Titanium and Zirconium Alloys
g c : R∗ : Tc : Cc : K: Cij : S: Ts∗ : TS : Tmono : g: : M:
7.1
Free energy density of solid solution of composition c Critical radius of a cluster Critical temperature of the miscibility gap Critical composition of the miscibility gap Gradient energy coefficient Elastic constant Entropy Coherent spinodal temperature Chemical spinodal temperature Monotectoid temperature Diffraction vector excited Distance of a side band from the main reflection Atomic mobility
INTRODUCTION
Diffusion of atoms plays a dominant role in a variety of phase transformations in alloys, intermetallics and ceramics. As pointed out in Chapter 2, diffusional transformations can be further classified into several types, primarily on the basis of the diffusion distances involved in these. One may ask whether there is any special reason for studying diffusional transformations in Ti- and Zr-based systems in the context of gaining an understanding of this class of transformations, in general. The answer to such a question is strongly affirmative in view of the following points: (1) Nearly all types of diffusional transformations are encountered in Ti- and Zr-based metallic and ceramic systems. (2) As will be shown in this chapter, the → transformation in Ti- and Zr-based alloys essentially follows the Burgers orientation relation, irrespective of whether the transformation is diffusional or martensitic. Therefore, these systems offer a unique opportunity for making a comparison between the characteristic features of transformations occurring by martensitic and diffusional mechanisms. Crystallographic and morphological details, thermodynamic and kinetic information and a knowledge of the nature of interface structures in relation to the → transformation are all available for both martensitic and diffusional transformations. A comparative study of these, therefore, leads to a better understanding of the mechanism of such transformations in general.
Diffusional Transformations
559
(3) The / interfaces in Ti- and Zr-based alloys have proved to be considerably more amenable to detailed TEM investigations than such interfaces in many other alloy systems. (4) Diffusion of substitutional atoms in the -phase exhibits an anomalous behaviour, due to which some of these transformations are seen to occur at unusually fast rates. Such transformations are somewhat unique and are not generally encountered in other alloy systems. (5) Precipitation reactions in ceramics have been studied most exhaustively in those based on ZrO2 . It is, therefore, very suitable as a prototype system for studying diffusional transformations in ceramics. (6) The issue of the participation of shear in several hybrid diffusional–displacive transformations has attracted renewed attention in current literature. Transformations in some Ti- and Zr-based systems exhibit novel features and provide new insight in connection with such hybrid processes. The wide variety of diffusional transformations in these systems and the depth of understanding achieved through numerous studies undertaken on them have made this field quite fascinating as will be illustrated in this chapter. The first section of this chapter focuses on the phenomenon of diffusion in the - and the -phases, including anomalous diffusion in the latter, and interdiffusion in different alloy systems. The general principles pertaining to the formation of intermetallic compounds and intermediate phases, either equilibrium or metastable, in the interdiffusion zones of diffusion couples of dissimilar metals have been discussed. The applications of these general principles in rationalizing the formation of intermediate phases in diffusion couples between Zr or Ti on one side and Al, Ni or Fe on the other have been addressed. The tendency towards phase separation in the -phase field is a controlling factor in a number of phase reactions such as spinodal decomposition in the -phase, monotectoid reactions, tempering of martensite and even in - or -precipitation in a -matrix. All these aspects will be covered under the section on phase separation. The precipitation of -plates in the -matrix has been a subject of a number of studies, primarily due to the adherence to the orientation relation between the two phases and to the simple nature of the / interface. Different morphological variations of the -plates are described and the mechanisms involved in their formation are discussed. The importance of the invariant line strain in dictating the growth direction has been duly emphasized. Massive transformations occur in a number of alloys based on Ti and Zr. The fact that short-range atomic jumps across the advancing transformation front is the operative mechanism for a massive transformation has been well illustrated in Ti-based systems. The composition range over which massive transformations are
560
Phase Transformations: Titanium and Zirconium Alloys
thermodynamically feasible has also been discussed in this chapter. Precipitation of intermetallic phases from the supersaturated -phase has been dealt with in terms of the precipitate morphology and orientation relation. The precipitation of an O-phase in a B2 matrix has been covered in some detail. O-phase plates of different crystallographic variants intersect at several points where offsets are created. The presence of offsets gives an impression that a shear mechanism takes part in the formation of the O-phase plates. The origin of the offsets in a diffusional transformation has been explained by invoking the concept of the “lattice-site correspondence” which exists between the matrix and the precipitate phases. The eutectoid decomposition in these systems can be classified into two distinct groups, namely, the “slow” and the “active” eutectoid decomposition. While the former has the characteristic features of pearlitic reactions (as observed in steels and in Cu–Al alloys), the mechanism associated with the latter involves atom transport across the transformation front by maintaining lattice correspondence simultaneously between the parent and the two product lattices.
7.2
DIFFUSION
Diffusion is essentially the mass flow process by which atoms (or molecules) change their positions relative to their neighbours under the influence of thermal energy and gradients which can be a concentration gradient, a magnetic or an electrical field gradient, a stress gradient or a combination of these. Diffusion plays a pervasive role in bringing about non-athermal phase transformations in the solid state and constitutes an irreversible process. There are two types of approaches for studying diffusion phenomena: macroscopic and microscopic. In the macroscopic approach, fluxes are the observables. A formal expression for these fluxes is obtained as functions of thermodynamic forces and of parameters which are referred to as phenomenological coefficients. In the microscopic approach, the fluxes are estimated by considering atomistic mechanisms wherein the parameters used are the jump frequencies and atomic distances. Unlike the phenomenological coefficients, which are nothing but proportionality constants, these frequencies have a tangible physical connotation. In fact, a knowledge of the underlying atomic mechanisms is required for a proper description of diffusion phenomena in both the formalisms. 7.2.1 Diffusion mechanisms In crystalline solids, the atoms occupy well-defined equilibrium positions. However, they do oscillate about their equilibrium positions with a mean amplitude that increases with increasing temperature. There are infrequent occasions
Diffusional Transformations
561
when the relative oscillation of a local group of atoms is such that one of the atoms is ejected out of its original site and gets transplanted in an adjoining equilibrium site. The fraction of an atom’s oscillations which may lead to such a jump is miniscular, even at temperatures close to the melting temperature of the host material; however, such jumps do occur and make diffusion possible. Bocquet et al. (1996) have summarized the possible mechanisms of diffusion in “dense structures” in the following manner: (1) Exchange mechanisms: One can think of two exchange mechanisms: direct and cyclic. In the former, two neighbouring atoms exchange their positions. However, this mechanism is unlikely to operate in dense structures because it involves large distortions, leading to unrealistically large activation energies. In the latter, a group of atoms exchange their positions simultaneously. In this process, the energy involved is substantially lower than that associated with direct exchange. However, due to the constraints imposed by the requirement of a collective motion, this mechanism is unlikely to operate. There are, as of now, no experimental evidences for the operation of either of these exchange mechanisms in crystalline metallic materials. However, in liquid and amorphous metallic materials such cooperative motions are more likely to operate. (2) Mechanisms involving point defects: The presence of an optimum number of point defects, the number being dictated by the temperature, lowers the free energy of a crystal. For this reason, under thermal equilibrium a crystal always contains point defects such as a monovacancies, vacancy aggregates (divacancies, trivacancies, etc.) and interstitials. The occurrence of these defects facilitates the movement of atoms without producing very large distortions. It is to be noted that generally the enthalpy of formation of an interstitial is much larger than that of a vacancy while the converse applies with respect to the enthalpy of migration of these point defects. In the interstitial mechanism of diffusion, atoms move from one interstitial site to another. Generally small interstitial atoms, such as those of H, B, C, N, O, diffuse through the lattice by this mechanism. In the somewhat more complex interstitialcy mechanism, atoms move from an interstitial site to a substitutional site and vice versa. This mechanism becomes important under non-equilibrium conditions brought about, for example, by irradiation or plastic deformation, when an equal number of vacancies and interstitials are created and contribute to diffusion. The equilibrium vacancy concentration increases with temperature. In metals and alloys, at temperatures approaching the melting temperature, vacancies occupy about 1–10 out of 10 000 atomic sites. This small number of vacancies facilitates the movement of atoms and, therefore, the process of diffusion. The vacancy mechanisms of diffusion constitute the more prevalent operating mechanisms and
562
Phase Transformations: Titanium and Zirconium Alloys
involve jumps to nearest neighbour (NN) sites. Jumps to next nearest neighbour (NNN) sites are also to be taken into account, particularly for bcc crystals, where the NN and NNN distances do not differ greatly. Apart from monovacancies, vacancy aggregates, particularly divacancies, can also contribute to the diffusion process. The role of divacancies becomes more important at high temperatures since the ratio of the number of divacancies to that of monovacancies generally increases with increasing temperature. The vacancy mechanisms are particularly relevant while considering the diffusion of substitutional solutes. (3) Mechanisms involving extended defects: Apart from point (zero-dimensional) defects, in crystalline materials linear (one-dimensional) defects like dislocations and planar (two-dimensional) defects like grain boundaries, interfaces, surfaces are present. These defects can be considered to be disordered regions in which atomic migration is relatively easy. This implies that diffusion can occur more readily via such defects as compared to diffusion occurring in the bulk of the crystal. Diffusion along these preferred paths is faster and is known as short circuit diffusion. In addition to these mechanisms, some less common mechanisms of diffusion have been invoked for rationalizing diffusion behaviour under special circumstances. For example, a mixed mechanism has been proposed to account for unusually fast diffusion. In this mechanism, it is envisaged that the fast diffusing solute dissolves in the host lattice substitutionally as well as interstitially and that mass transport is dictated by a mixed vacancy and interstitial mechanism involving vacancy–interstitial pairs. This mechanism is operative in the diffusion of noble metals in alkali metals (Santos and Dyment 1975). 7.2.2 Flux equations: Fick’s laws Diffusion takes place by atomic jumps occurring throughout the material and this has to be related to measurable, macroscopic parameters such as flux (J), or concentration (c), when one considers mass flow under a concentration gradient. In this context, flux equations, expressed in the form of Fick’s laws of diffusion, are very important. Fick’s first law can be used to describe mass flow under steady-state conditions and is identical in form to Fourier’s law for heat flow under a constant temperature gradient or to Ohm’s law for current flow under a constant electric potential gradient. Assuming for simplicity that the concentration varies only along the x-direction, this law can be expressed as J = −D
c x
(7.1)
Diffusional Transformations
563
where D is the diffusion coefficient or diffusivity and (c/x) is the concentration gradient along the diffusion direction, x. A partial derivative is used to recognize the fact that the gradient can vary with time. The negative sign on the right-hand side of Eq. (7.1) indicates that mass flow occurs down the concentration gradient. The diffusion coefficient, D, depends on the nature of the diffusing species, the matrix in which it is diffusing and the temperature. Under the condition of steadystate flow, the flux, J, is independent of time and remains the same at any crosssectional plane along the diffusion direction. A schematic representation of the concentration–penetration profile under steady-state flow is shown in Figure 7.1. When D is independent of concentration, the profile is ideally a straight line. However, when D is a function of concentration, the profile is such that the product D(c/x) is a constant. As long as steady-state flow occurs, the profile remains unchanged over time, in either case. Fick’s second law is an extension of the first law to conditions of non-steadystate flow. In these cases, at any given instant, the flux is not the same at different cross-sectional planes along the diffusion direction, x. Apart from this, the flux changes with time at the same cross-section; in other words, the concentration– penetration profile changes with time. Fick’s second law for unidirectional mass flow under non-steady-state conditions can be expressed as c c = D t x x
C2 B
C1 A
→ J
(7.2)
dc dx (b) (a)
J →
X X2
X1 (a) (b)
Steady state Non-steady state
Figure 7.1. Schematic representation of diffusion flux (concentration–penetration profile) under (a) steady-state diffusion and (b) non-steady-state diffusion.
564
Phase Transformations: Titanium and Zirconium Alloys
In case D is independent of concentration and thus of x, 2 c c =D 2 t x
(7.3)
In geometrical terms, the interpretation of Eq. (7.3) is straightforward; 2 c/x2 is essentially the curvature of the concentration–penetration profile. Thus, if c is concave upward in a given region, the concentration increases with time in that region. Empirically it is found that the diffusion coefficient, D, varies exponentially with temperature and an Arrhenius type of relationship is valid: D = D0 exp −Q/RT
(7.4)
where D0 is a pre-exponential constant and Q is the activation energy for diffusion. A plot of ln D versus 1/T should yield a straight line. 7.2.3 Self- and tracer-diffusion coefficients in -Zr and -Ti Self-diffusion coefficients of pure Zr and self- and impurity-diffusion coefficients of a few elements in Ti are plotted in Figures 7.2 and 7.3, respectively. It could be seen that diffusion in -Zr shows an unusual temperature dependence and the Arrhenius plot indicates a negative curvature which could be due to different modifications of the vacancy mechanism. The pertinent factors could be (a) a reduction in the self-diffusion enthalpy as a result of lattice softening prior to the transformation to the -phase; (b) a temporary trapping of vacancies at impurity atom sites; and (c) rapid diffusion due to the presence of impurity atom–vacancy pairs. It has been concluded that the rapid pair formation mechanism controls self-diffusion in the -Zr phase (Frank, 1989). Unlike in the case of Zr, Ti exhibits a normal self-diffusion behaviour in the -phase. The large volume of data available on the solute and self-diffusion in -Ti and -Zr has been reviewed by Hood (1993). Recently reviewed data pertaining to some of these diffusivities, published in Defects and Diffusion Forum (1999), are reproduced in Figures 7.4 and 7.5. The main issues addressed in these studies are: (a) the atomic size effect, (b) the anisotropy of diffusion in the hcp lattice and (c) the effects of impurities. There is a clear correlation between the atomic radius of the diffusing element and the diffusion coefficient in both -Zr and -Ti. It is observed that large D values are associated with small radii of impurity atoms. It also appears that diffusion along the a- and c-axes in -Zr as well as -Ti is generally different. It has been observed that the diffusion coefficient for Fe, Cr and Ni along the c-axis in -Zr is about three times larger than that
Diffusional Transformations
565
↑
T (K)
1773
1273
973
773
–11
10
β-Zr(bcc)
10–13
10–15
D T (m2/ s)
↑
10–17
α-Zr(hcp) 10–19
10–21
T αβ
10–23
4
6
8
10
12
↑
104/T (K–1)
Figure 7.2. Arrhenius plots of the tracer self-diffusion coefficient DT in -Zr and -Zr (after Frank 1989).
along a direction perpendicular to this axis while for self-diffusion, the ratio of the diffusion coefficients parallel and perpendicular to the c-axis is around 1.3 (Hood 1974). Accurate measurements with regard to this anisotropic behaviour are very important for Zr in view of its use as a fuel cladding material in nuclear reactors. It should be emphasized in this context that a factor of two in the diffusion anisotropy produces a significant effect on the sink biases and microstructural changes through the so-called “diffusion anisotropy difference” (DAD) in materials during irradiation (Sizmann 1978). There is a large scatter in the impurity and self-diffusion coefficients reported in literature, presumably due to the varying purity of the materials used. It has been realized that single crystals and pure materials are essential for conducting experiments if reliable and reproducible data are to be obtained. In general, the vacancy mechanism appears to operate over the whole temperature range in both -Zr and -Ti. It has been suggested that monovacancy as well
566
Phase Transformations: Titanium and Zirconium Alloys ↑
T (K)
1473
1273
β(bcc)
10–9
10–10
P
D (m2/ s)
Melting point
Co
10–13
873
α(hcp)
Ni
10–11
10–12
1073
Phase transition
10–8
1873 1673
Fe Mn Co
Ti
Fe Co Ni
10–14
P
P
Ni Fe
10–15
Mn 10–16
Mn
10–17
10–18 5.0
Ti
6.0
7.0
8.0
9.0
10.0
11.0
12.0
104/T (K–1)
Figure 7.3. Temperature dependence of self- and impurity-diffusion of a few elements in both parallel () and perpendicular (⊥) directions to the c-axis in Ti (after Hood 1993).
as divacancy mechanisms contribute to the overall diffusion, with an increasing participation from the latter at higher temperatures (Seeger 1997). 7.2.4 Self- and tracer-diffusion coefficients in -Zr and -Ti The self- and tracer-diffusion coefficients for Zr, Ti and Hf, both in the - and the -phases, on a normalized temperature scale, Tm /T (Tm being the melting temperature), for some selected bcc metals and impurity-diffusion coefficients for Fe, Co and Ni are presented in Figure 7.6. It could be seen that the diffusion coefficients for Zr, Ti and Hf are larger in the -phase than in the -phase and that this behaviour is more pronounced at low temperatures. It could further be
Diffusional Transformations
567
Ti alloys 10–12
D (m2/s)
10–14
Au Co Ga In Si Zr Hf Ni Pb Sn
10–16
10–18
10–20
10–22 0.5
1
1.5
2.5
2
3
3.5
103/ T (K)
Figure 7.4. Diffusivity of various elements in -Ti (after Fisher 1999).
Zr alloys 10–10 10–12
Cr in Zr Cr in Zr–2Sn Cr in Zr–2.5Nb Hf in Zr O in Zr Zr in Zr–3.2Ag Hf in Zr–1Nb Hf in Zr–2.5Nb
D (m2/s)
10–14 10–16 10–18 10–20 10–22 10–24 0.5
1
1.5
2
2.5
3
103/ T (K)
Figure 7.5. Diffusivity of various elements in the -phase of Zr alloys (after Fisher 1999).
seen that the impurity-diffusion coefficients of Fe, Co and Ni are a few orders of magnitude larger than self-diffusion coefficients in -alloys. The major deviation from normal diffusion behaviour, generally termed anomalous diffusion, observed in several bcc metals and alloys, is characterized by the following features: (a) non-linearity in plots of ln D versus 1/T; (b) very low
568
Phase Transformations: Titanium and Zirconium Alloys
10–10
Co in β–Zr
β–Hf
Fe in β–Zr –12
10
D (m2/s–1)
Zr Ti 10–14
Na 10–16
K Ta
10–18
V W Mo Cr
Li
Nb
10–20 1.0
1.4
1.8
Tm / T
Figure 7.6. Comparison of self-diffusion values of selected bcc metals and of very fast impurity diffusion in bcc metals on a normalized temperature scale (after Petry et al. 1989).
values of the pre-exponential factor, D0 ; and (c) very low values of the activation energy, Q. Extensive work has been carried out with a view to understand the anomalous diffusion behaviour of -Zr, -Ti and of some other bcc metals (Herzig and Kohler 1989, Mundy 1992). Several bcc metals, such as V, Nb, Ta, Mo, W, which do not undergo any allotropic transformation, also show slight curvatures in the Arrhenius plots if the temperature range of the study is extended. The anomalous diffusion behaviour of bcc metals is attributed to the various defect structures present. Some of the models associate these defects with the lattice stability of the bcc phase. The occurrence of the / transformation in Ti- and Zr-based systems can be related to the mechanical instability of the bcc lattice at the transition temperature, arising from a drop in the elastic constant. As experimental evidences for lattice softening prior to the / transition have not been obtained, this model has not found general acceptance. A model for anomalous diffusion in the -phase of Ti- and Zr-based alloys due to the presence of -embryos has been proposed by Sanchez and de Fontaine (1978).
Diffusional Transformations
569 B3
C2 B3
C1 V
1/3
1/2
2/3
B2
I
C2
C1
B1
C3 I V
B2
B1 C3
V
1/3
1/2
2/3
I
Figure 7.7. Saddle point configurations and activation energy plot for diffusion by a vacancy mechanism in the bcc structure.
The path of the atom–vacancy exchange process, as schematically illustrated in Figure 7.7, shows that an atom goes through two saddle point configurations which resemble a subunit cell cluster of the -phase. As an atom moves along a <111> direction to occupy a vacant position, it goes through centroids of triangular arrangements of atoms marked B1, B2, B3 and C1, C2, C3. These two positions mark the atomic arrangements for the two saddle points. As described earlier, the -phase can be created from the -phase by collapsing adjacent 222 layers, keeping the third 222 layer undisplaced. Therefore, from structural considerations, the saddle point atomic configuration can be viewed as an embryonic atomic cluster representing the -phase. In systems which have a tendency for the formation of the -phase, there is a finite probability of encountering such -embryos in the -phase; the higher the stability of , the higher the number density of such -embryos. The presence of saddle point configurations of activated complexes for diffusion, therefore, enhances the diffusivity. An estimate of the population of -embryos, as a function of temperature, can be correlated with enhanced diffusivity in these systems. At relatively high temperatures, where the -phase becomes unstable, the population of -embryos is reduced and as a result the diffusivity tends to attain normal values. Kohler and Herzig (1988) have proposed a model based on soft phonon without involving the presence of static -embryos. It is well known that the phonon spectra in these systems exhibit soft modes with wave vectors q = 2/3 <111> and q = 1/2 <110>. While the former is associated with the → transformation, the latter relates to the → transformation. Softening of the modes essentially leads to a lowering of the energy barrier associated with atomic displacements.
570
Phase Transformations: Titanium and Zirconium Alloys
These modes, being of low frequency, contribute to large fluctuations in the displacement coordinate, leading to small migration enthalpies and correspondingly high diffusion coefficients. Recently Seeger (1997) has drawn attention to the weakness of this model by pointing out that the soft mode hypothesis is concerned only with the migration enthalpy and not with the enthalpy of vacancy formation. 7.2.5 Interdiffusion As discussed in the previous sections, self- or impurity-diffusion relates to mass transfer of a single species under the influence of its own concentration gradient and usually the concerned species only diffuses without effectively altering the composition of the host lattice in which it is diffusing. The situation is different when two blocks of materials with different compositions interact with each other at elevated temperatures. The species from both the blocks diffuse into each other. The diffusion process occurring in such a case is referred to as an interdiffusion or a chemical diffusion process. For simplicity let us consider the two blocks 1 and 2 of solid solutions of the A–B binary system of different compositions, as shown in Figure 7.8. Let block 1 comprise a solid solution relatively rich in B and block 2 be made up of a solid solution rich in A. If these blocks, placed in contact with each other, are heated to an elevated temperature and held at the temperature for sufficiently long times for diffusion to occur, atoms of the species A from the A-rich solid solution will diffuse into the B-rich solid solution and B atoms will diffuse from the B-rich solid solution to the A-rich solid solution by a chemical diffusion process. Such a chemical diffusion process takes place in order to homogenize the composition. Basically the process occurs to decrease the free energy of the system and the driving force arises from A→ ←B
A atom B atom
2
1
A-rich solid solution
B-rich solid solution
Figure 7.8. Diffusion couple between A-rich and B-rich solid solutions of an A–B binary system.
G2 →
G3
571
G1
→
Molar free energy (G ) →
Diffusional Transformations
G4 A
1 2 Concentration of B-atoms (C B) →
B
Figure 7.9. Schematic molar free energy diagram for the A–B binary solid solution system.
the chemical potential gradient. A schematic molar free energy diagram for the A–B binary solid solution system is shown in Figure 7.9. The compositions of the blocks 1 and 2 are marked on the concentration axis. G1 and G2 are the molar free energies at these concentrations. Before the diffusion process starts, the total free energy of the system is given by G3 . The molar free energy of the homogeneous alloy is given by G4 . The A and B atoms will diffuse to lower the free energy to G4 . The diffusion process stops with the equilibration of the chemical potential and the attainment of system equilibrium. During the interdiffusion process, the species diffuse via the vacancy mechanism. The resultant concentration profile at the end of diffusion is shown in Figure 7.10. The rate at which each species exchanges with vacancies is its own intrinsic diffusion coefficient. In general, the intrinsic diffusion coefficient of each species will be different. As atoms jump via vacancies, the flux of atoms and vacancies are equal but in opposite directions. Of the two atomic species A and B in the A–B binary system, if the intrinsic diffusivity of B is higher than that of A, there will be a net vacancy flux in the direction opposite to that of B atoms. The intrinsic fluxes of A and B atoms can be calculated by considering their intrinsic diffusivities and concentration profiles. The fluxes of these species A, B and of vacancies are represented by vectors and are shown in Figure 7.10. The relation between these fluxes is given as − → − → − → JA = JB − Jv
(7.5)
The net flux of vacancies cause the movement of lattice planes with respect to the laboratory fixed frame of reference. The vacancy flux, Jv , gives the velocity of the lattice planes and is given as Co v where Co is the number of atoms per unit
572
Phase Transformations: Titanium and Zirconium Alloys
(a)
CA Marker plane O
X
JB Jv
JA
JB J
JA
(b)
O (c) dJ v O dx Vacancies created
Vacancies destroyed
Figure 7.10. Concentration profile of A atoms in the A–B binary solid solution system.
volume and is independent of composition. Hence the velocity of lattice planes is given as → − → − → v = JA − JB Co −
(7.6)
This shifting of lattice planes with respect to the laboratory fixed frame of reference is referred to as Kirkendall effect: such shifting was first observed by Kirkendall in his famous experiment on the Cu–Zn system using Mo marker wires. The flux of vacancies requires the creation of lattice sites on one side of the marker plane and destruction of the same on the other side, as shown in Figure 7.10. As seen from the region, where no diffusion takes place or in other words, as seen from the inertial laboratory fixed frame of reference, the interdiffusion flux of the component A, JA is equal and opposite to that of the component B, JB , i.e. JA = −JB
(7.7)
The interdiffusion flux is composed of two components, viz., the intrinsic diffusion flux and the velocity of the lattice planes through which diffusion is taking place.
Diffusional Transformations
573
The nature of the diffusion zone during interdiffusion depends upon the nature of the phase diagram of the system and on the composition of the end members of the diffusion couple. If the ranges of composition covered by the starting alloys are within the range of solid solubility, the diffusion zone is of a single phase. If, however, the ranges of composition of the end members of the diffusion couple encompass intermediate phases, as depicted in the phase diagram, the diffusion zone consists of a number of bands corresponding to the intermediate phases. The concentration profile in single phase diffusion couple is a smooth sigmoidal curve between the two compositional limits of the end members of the couple. A typical concentration profile is shown in Figure 7.11; the composition of one end of the couple is c− and that of the other is c+ . As discussed earlier in this section, the lattice planes move during interdiffusion. Due to this the original interface cannot be established for the evaluation of the flux or the diffusion coefficients.
Concentration (C) →
C+
Slope = (dc /dx )
C′
C–
Matano interface
C′
Shaded area = C–
∫ x dc
Distance (x ) →
Figure 7.11. Boltzmann–Matano method for evaluation of chemical diffusion coefficient in a binary system.
574
Phase Transformations: Titanium and Zirconium Alloys
Before the start of diffusion, the flux across the original weld interface is zero. After diffusion commences, fluxes move on both sides of this plane. The Matano interface is defined as the plane in the diffusion zone across which the net flux is zero. This corresponds to the original interface, in general. In a case where there is a molar volume change, a shift is there between the Matano interface and the original interface. The flux at a given plane with concentration c is given as the area under the concentration profile and is expressed as c+ x dc. Mathematically the Matano interface is described as c− x dc = 0 where c− and c+ are concentrations in the right and left endments of the diffusion couple (Figure 7.11). The interdiffusion coefficient is generally a function of the concentration and hence of the position x. Fick’s second law in such cases is written as c dc = D (7.8) dt x x The solution of this differential equation cannot be found analytically. The interdiffusion coefficients are evaluated using the Boltzmann–Matano method. This method is exclusively employed to evaluate interdiffusion coefficients in binary as well as ternary systems. According to this, the interdiffusion coefficient, ), at a concentration c can be given as D c c = 1 dx D c x dc 2t dc c
(7.9)
where t isthe duration of the diffusion anneal, dc/dx is the concentration gradient c at c and c x dc is the area under the concentration penetration profile up to the concentration c . The terms used in Eq. (7.9) are indicated in Figure 7.11. As an illustrative example, we consider single phase diffusion in the Zr–Al system, studied by Laik et al. (2002). The diffusion couples were made with pure Zr on one side and a Zr–2.8 wt% Al alloy on the other side. W markers were placed at the original weld interface to monitor the movement of lattice planes. The couple was annealed at 1203 K so that both the end members of the diffusion couple were within the solubility range at that temperature. Figure 7.12 shows the microstructure of the resulting diffusion zone. The figure also shows the position of the W marker at the end of the diffusion process. The corresponding concentration–penetration profile of Al across the diffusion zone is shown in Figure 7.13 which also marks the position of the Matano interface and the marker plane. The interdiffusion coefficients were evaluated at various concentrations. Intrinsic diffusion coefficients of Zr and Al were also evaluated from the marker movement velocity.
Diffusional Transformations
575
Figure 7.12. Microstructure of diffusion zone between pure Zr and Zr–7.8 at.% Al alloy.
100
Al Zr Matano interface
Marker position
6 96 4 94
Concentration of Al (at.%)
Concentration of Zr (at.%)
98
8
2 92
Pure Zr 0
50
Zr–7.8% Al 100
150
200
250
0
Distance (μm)
Figure 7.13. Concentration profile across diffusion zone between pure Zr and Zr–7.8 at.% Al alloy.
576
Phase Transformations: Titanium and Zirconium Alloys 2000
1943
1900
L
1800
Temperature (K ) →
1700 1600
β
1500 1400 1300 1200
1162
1100
(Ag)
α
A2B (Ti2Ag)
1000
γ
AB (TiAg) T1
900 800 0
A (Ti)
10
20
30
40
50
60
70
Concentration of B (Ag) in at.% →
80
90
100
B
Figure 7.14. Phase diagram of A–B (Ti–Ag) binary system with intermediate phases.
The diffusion zone in a multiphase diffusion couple is characterized by bands corresponding to intermediate phases shown in the phase diagram. Figure 7.14 shows the phase diagram of a binary A–B (Ti–Ag) system. It shows terminal solid solutions along with intermediate phases. Intermediate phases may have some solubility ranges or may be line compounds. If a diffusion couple made between pure A (Ti) and pure B (Ag) is annealed at an elevated temperature, T1 , the intermediate phases stable at the annealing temperature appear in the diffusion zone as bands. Figure 7.15(a) shows the layers of intermediate phases in a schematic microstructure of the multiphase system. The concentration–penetration profile across the diffusion zone (Figure 7.15(b)) is not continuous and has steps corresponding to the intermediate phases forming in the system. The step is flat and of a constant concentration for a line compound (A2 B). The step has a concentration range corresponding to the solubility range of the intermediate phase (AB). The concentrations at the phase boundaries generally match well with those given in the phase diagram. The interdiffusion coefficients at various concentrations are evaluated from concentration profiles employing the Boltzmann–Matano method. Generally the interest in such cases lies in the interdiffusion coefficients in the intermediate phase
Diffusional Transformations A
α
A2B
AB
577
γ
B
(a)
Concentration of A (CA) →
γ
AB A2B
A
α Distance (x) → (b)
Figure 7.15. (a) Layers of intermediate phases in A–B (Ti–Ag) binary system at temperature T1 . (b) Schematic concentration profile in A–B(Ti–Ag) binary system at temperature T1 .
alone. The interdiffusion coefficient values do not change much with concentration in the intermediate phase. Hence a single value of the interdiffusion coefficient is calculated for the intermediate phase. A modified Boltzmann–Matano–Heuman (Heuman 1952) method is used to evaluate the interdiffusion coefficient in respect of an intermediate phase. It is expressed as c1/2 = 1 d D x dc 2t c c−
(7.10)
where d is the width and c is the solubility range to a pertinent intermediate phase and c1/2 is its average composition as shown in Figure 7.16. An inspection of Eq. (7.10) shows that the method cannot be used for line compounds since c tends to infinity. As discussed in Section 7.2.6, diffusion takes tends to zero and D place to lower the free energy of the system, and the chemical potential or activity gradient is the driving force. In most situations, activity and concentration have a monotonic relationship and it is easy to measure the concentration and hence it is usually used to evaluate diffusion coefficients. Garg et al. (1999) have described thermodynamic interdiffusion coefficients employing the activity gradient. The method is quite useful for the evaluation of diffusion coefficients of line compounds. In a line
578
Phase Transformations: Titanium and Zirconium Alloys
C+
→
d
C
C–
C½
ΔC
x→
Figure 7.16. Boltzmann–Matano–Heuman method for evaluation of diffusion coefficient in an intermediate phase in a binary system.
compound, even though there is no concentration gradient, there is a finite activity gradient and, therefore, thermodynamic interdiffusion coefficients can be evaluated. As an illustrative example, let us consider the multiphase diffusion in the Zr–Ni system (Bhanumurthy et al. 1990). The Zr–Ni phase diagram exhibits eight intermetallic compounds. Diffusion couples made between pure Ni and Zr were annealed at 1133 K. The microstructure of the diffusion zone, as shown in Figure 7.17, indicated the formation of only five layers of (NiZr2 , NiZr, Ni10 Zr7 , Ni7 Zr2 and Ni5 Zr) intermediate phases which are stable at this annealing temperature. 7.2.6 Phase formation in chemical diffusion As mentioned earlier in Section 7.2.5, the nature of the diffusion zone in chemical diffusion depends on the nature of the phase diagram. The product may be a solid solution or an intermediate phase. From thermodynamic considerations alone, all the intermediate phases shown in the phase diagram which are stable at the temperature of diffusion annealing are expected to appear in the diffusion zone. Figure 7.14 shows the phase diagram of a binary system A–B (Ti–Ag). Diffusion annealing at the temperature T1 would form the intermediate phases A2 B and AB which are stable at T1 . However, it is observed that sometimes none of the pertinent phases or only a few of these appear in the diffusion zone. For example, Kale et al. (1980) have reported that chemical diffusion in the Fe–Ti system in the temperature range of -Fe does not lead to the formation of any of the pertinent
Diffusional Transformations
579
Figure 7.17. Microstructure of diffusion zone in Zr–Ni diffusion couple annealed at 1133 K. The intermetallic compounds are marked as (1) NiZr2 , (2) NiZr, (3) Ni10 Zr7 , (4) Ni5 Zr2 and (5) Ni5 Zr.
intermediate phases (TiFe or TiFe2 ) in the diffusion zone. Bhanumurthy et al. (1990) have reported the formation of a few phases only in the Zr–Ni system. The lack of correspondence in the phases formed in the diffusion zone and those shown in the equilibrium phase diagram is due to the fact that quasi-equilibrium conditions prevail during diffusion. Hence the nature of the phases formed in the diffusion zone also depends on the type of sources of flux, i.e. the type of diffusion couple. If the diffusion couple is made between two components whose thicknesses are much larger than the width of the diffusion zone, the couple is called an infinite diffusion couple and the sources of both the components are inexhaustible. In such cases, all the intermediate phases depicted in the phase diagram may appear simultaneously as a series of parallel layers in the same order as shown in Figure 7.15(a). If, however, the thickness of at least one of the components is smaller than the width of the diffusion zone, the couple is called semi-infinite and the source of the “thin” component is exhaustible. In such cases, all the intermediate phases do not form simultaneously. The first compound formed grows until the exhaustion of the thin component and then the subsequent compound appears. Let us consider that the component A is thin and exhaustible. At time t = t1 , the intermediate phase A2 B may form and grow till A is exhausted, as shown in Figure 7.18. Subsequently diffusion between A2 B and B will take place and intermediate phase AB will form at the interface.
580
Phase Transformations: Titanium and Zirconium Alloys A2B
A
A2B
A
B
AB
B
B
t = t1
t =0
t = t2
Figure 7.18. Sequential formation of intermediate phases in A–B binary system in a semi-infinite diffusion couple.
7.2.6.1 Phase nucleation The formation of a new phase as a result of diffusion is associated with the free energy change of the system. The free energy change has two components: (a) chemical free energy change and (b) non-chemical free energy change. Let us consider the formation of the intermediate phase AB in a diffusion couple made between A and B, as shown in Figure 7.19. Let x be the width of the region comprising the intermediate phase AB. Due to the formation of AB the original interface between A and B is eliminated and two new interfaces between A and AB and B and AB are created. The free energy of the system decreases when the intermediate phase forms as its free energy of formation is negative. If Gv is the free energy of formation per unit volume, then the decrease in free energy is given by Gv x, considering unit cross-sectional area of the couple. Non-chemical free energy is brought into play due to the creation of new interfaces and to the mechanical work done consequent to the formation of the new phase. If 1 , 2 and 3 are the interfacial energies for the interfaces between A and B, A and AB and B and AB, respectively, then the total interfacial energy γ1
A
B
γ2
γ3
A
AB
B
Figure 7.19. Formation of an intermediate phase AB in a diffusion couple between A and B.
Diffusional Transformations
581
change per unit area can be given as = 2 + 3 − 1
(7.11)
The total free energy change of the system per unit area can be expressed as G = Gv x + + W
(7.12)
where W is the mechanical work done due to volume change. The phase AB would nucleate if G is negative. The free energy change, G, will be always negative if the non-chemical component is negative. In case the non-chemical component of the free energy is positive, the total free energy change of the system G will depend upon the relative magnitudes of the chemical and non-chemical components of free energy. In the case of a phase for which the non-chemical component has a large positive value, the total free energy change G will be positive up to a certain thickness x∗ of the phase. The total free energy change will be negative when the thickness of the phase is larger than the critical thickness x∗ . The phase will nucleate if this critical thickness x∗ is of the order of the lattice parameter and will not nucleate if x∗ is much larger than the lattice parameter. 7.2.6.2 Phase growth The nucleated phase grows if the kinetics of growth is favourable. Let us consider the growth of the nucleated phase AB in the diffusion couple made between A and B, as shown in Figure 7.20. We assume DA DB ; the phase AB forms and grows as the species A diffuses through AB into B. The component A, with a flux of J1 , reacts with B to form AB as follows: A J1 + B → AB
A
AB → J1
(7.13)
B → J2
Figure 7.20. Growth of an intermediate phase AB formed in a diffusion couple between A and B.
582
Phase Transformations: Titanium and Zirconium Alloys
The phase AB decomposes as the species A from AB diffuses into B with a flux J2 : AB → A J2 + B
(7.14)
The net increase in the width of the phase AB can be expressed as Co
dxAB = J1 − J2 dt
(7.15)
Free energy (G) →
where xAB is the width of the phase AB and Co is as described in Eq. (7.6). The equation suggests that the growth of this phase depends upon the net flux across the phase. The conditions under which a subsequent phase nucleates and grows are different from those pertaining to the first phase. Let us consider that the phase AB has grown in a diffusion couple between A and B. The driving force for the formation of the next phase A2 B will be changed due to the presence of the AB phase. Figure 7.21 shows the free energy values for the AB and A2 B phases in the A–B system. After the phase AB is formed, the free energy of the A–AB system is given by P . Hence the driving free energy for the nucleation of A2 B is given by P Q which is lower than the free energy change PQ, associated with the compositioninvariant transition from the corresponding solid solution (P) to the intermetallic compound (Q). As illustrative examples we consider the formation of intermediate phases during diffusion in the two systems Ti–Fe and Zr–Al.
A
P P′ Q
A2B
AB
B
Concentration of B (C B) →
Figure 7.21. Schematic free energy diagram for the intermediate phases AB and A2 B in the binary system A–B.
Diffusional Transformations
583
Atomic per cent iron 0
10
2000
20
30
50
60
70
80
90
100
L
1800
1700 1590 54.1
1400
1358
1200
24.7
75.4
86
91.3
51.3
←(γ Fe)
TiFe2
(βTi)
1562
68.2
(αFe)
1600
TiFe
Temperature (K) →
40
1943
1155
Magnetic transition
1000 868
800
← (αTi)
17
600 0
10
Ti
20
30
40
50
60
70
80
90
100
Concentration of Fe (wt%) →
Fe
Figure 7.22. Equilibrium phase diagram of Ti–Fe system.
The equilibrium phase diagram of the Ti–Fe system is shown in Figure 7.22. It depicts two intermediate phases, namely, TiFe2 and TiFe. The interdiffusion in this system has been studied by Kale et al. (1980). They have reported that none of these phases appears during diffusion annealing of couples comprising pure Fe and Ti or even of incremental couples made between pure Fe and the TiFe phase, when annealed in the -Fe region. But in the case of Fe/Ti–Fe diffusion couples annealed in the -Fe range, the formation of TiFe2 is observed. The phases do not grow in the -Fe range as the flux entering these phases are lower than those leaving them, as shown in Table 7.1. The formation of TiFe2 takes place when the Fe flux entering the phase is higher than that leaving it (Table 7.1).
Table 7.1. Flux of Fe atoms in Fe–Ti system Couple
-Fe/-Ti -Fe/TiFe
Phase
TiFe2 TiFe TiFe2
Flux atoms (m2 /s) Entering
Leaving
84 × 1010 14 × 1012 145 × 1010
14 × 1012 20 × 1014 286 × 107
584
Phase Transformations: Titanium and Zirconium Alloys 0
–20
α-Zr
–30
–50 0.0
ΔGZrAl
–40
2
ΔGZr Al 2 3
Molar free energy, G (kJ/mol)
–10
ZrAl3 ZrAl2
Zr2Al3
0.2
0.4
0.6
0.8
1.0
Concentration of Al (mole fraction)
Figure 7.23. Molar free energy diagram for formation of various phases in Zr-Al system.
The Zr–Al phase diagram indicates the presence of 10 intermediate phases. Interdiffusion in this system has been studied by Laik et al. (2004). They have reported that ZrAl3 forms first and subsequently Zr2 Al3 forms in the diffusion zone. The phase which has a higher diffusivity grows first. In the absence of diffusivity data, a simple relationship of self-diffusivity with melting temperature can be extended to intermediate phases also. Accordingly, ZrAl3 , which has a lower melting point and hence a higher diffusivity, grows first. The molar free energies of formation of various phases in the Zr–Al system have been plotted in Figure 7.23. When the ZrAl3 phase first forms, a local equilibrium between the -Zr and ZrAl3 is attained and the free energy of the system is given by the common tangent between the two corresponding curves. It is seen from the figure that the free energy of formation of Zr2 Al3 is more negative compared to that of ZrAl2 , indicating the subsequent formation of Zr2 Al3 . 7.2.7 Diffusion bonding Diffusion bonding is a technique which employs the solid state diffusion of atoms as a main process for the development of a joint. Diffusion bonding involves keeping the work pieces to be joined in close contact under moderate pressure and heating the assembly at an elevated temperature for a certain duration.
Diffusional Transformations
585
The bonding occurs in three stages. In the first stage, the materials yield and creep. The asperities at the faying surfaces deform and the mating area grows so that a large fraction of the areas of the faces to be joined comes into contact with each other. In the second stage, atoms in the contact area diffuse via grain boundaries. This causes a rearrangement of grain boundaries and the elimination of pores in the bonded zone. In the third and final stage, volume diffusion dominates and the bond is formed. The success of the bonding process depends on three parameters: (1) bonding pressure, (2) bonding temperature and (3) holding time. The bonding pressure should be sufficient to make intimate contact at the faying interfaces, deform the surface asperities and fill the gaps between the mating surfaces. The applied pressure breaks the oxide film on the surfaces which, otherwise, hinders the diffusion process. It also suppresses the formation of voids due to the Kirkendall effect. The bonding temperature is generally kept at 0.5–0.7 times the melting temperature of the most fusible component in the assembly. The bonding temperature influences the interdiffusion across the weld interface and decides the extent of the diffusion zone and the nature of the reaction products formed in it. The optimum thickness of the bond region maximizes the bond strength. Diffusion bonding of similar materials results in the formation of a single phase in the diffusion zone. The width of the diffusion zone, w, follows the time dependence given by the expression w = Kp t1/n
(7.16)
where Kp is the penetration constant and n is the reaction index which is usually equal to 2 for the diffusion process. The temperature dependence of the penetration constant, Kp , follows an Arrhenius relationship given as Kp = Ko exp −Qp /RT
(7.17)
where Ko is a pre-exponential factor and Qp is the activation energy for diffusion controlled growth. The width of the diffusion zone formed in diffusion bonding can be estimated from these relations and the bonding parameters, viz., temperature and time, can be established. Diffusion bonding of dissimilar materials leads to the formation of intermediate phases in the diffusion zone. These phases are generally brittle in nature and form as bands. Their presence deteriorates the bond strength, and hence the formation of these intermediate phases in the diffusion zone should be avoided. The formation of intermediate phases can be avoided by using interlayers of different materials in between the two parent materials. If diffusion bonding between
586
Phase Transformations: Titanium and Zirconium Alloys
A and B leads to the formation of intermediate phases and if C is such that it does not form intermediate phases with either A or B in the diffusion zone, the formation of intermediate phases between A and B can be prevented by using C as an interlayer. Additionally, the interlayers used may help in (a) reducing the thermal expansion mismatch between the parent materials, (b) inhibiting the diffusion of undesired elements and (c) reducing the bonding temperature and/or pressure. The intermediate phase may have an incubation period for formation and hence its formation can be avoided if bonding is carried out at low temperatures or for small durations. As an illustrative example we consider the diffusion bonding of zircaloy-2 with stainless steel. Direct diffusion bonding between zircaloy-2 and stainless steel leads to the formation of Zr-based intermediate phases in the diffusion zone. Ni from stainless steel also diffuses into zircaloy-2. Diffusion bonding between zircaloy-2 and stainless steel has been studied by Kale et al. (1986) using Fe and Ti as interlayers. The Ti interlayer has been used in contact with zircaloy-2 as Zr–Ti is an isomorphous system. Similarly the Fe interlayer has been used in contact with Ti since in the Ti–Fe diffusion zone intermediate phases do not form, as shown by Kale et al. (1980). It also prevents the diffusion of Ni from stainless steel to zircaloy-2. Diffusion of Ni in zircaloy-2 is undesirable as it enhances hydride formation. Thus the whole assembly does not form intermediate phases in the diffusion zone and the bonding is good. Diffusion bonding between zircaloy-2 and stainless steel has also been studied by Bhanumurthy et al. (1994) using Nb, Cu and Ni interlayers. The assembly has the sequence: zircaloy-2/Nb/Cu/Ni/stainless steel. Zr–Nb, Cu–Ni and Ni–Fe, the major components of stainless steel, form solid solutions. The Nb–Cu phase diagram shows the occurrence of intermediate phases. But the temperature and time of diffusion bonding have been appropriately adjusted such that no intermediate phases form at any interface. Figure 7.24 shows the microstructure of the Zr-2/SS assembly indicating the absence of intermediate phases at various interfaces. The strength of the bond is very good. The use of interlayers in diffusion bonding may pose problems relating to corrosion as there are many interfaces in the assembly. In certain applications, the use of interlayers is not permitted due to detrimental physical/chemical service conditions. In such cases, direct diffusion bonding can be resorted to and the formation of brittle intermediate phases can be avoided by controlling the temperature and holding time in such a manner that the corresponding incubation period is not crossed. As an illustrative example, we consider the diffusion bonding of Ti and stainless steel, which has been studied by Kale et al. (1996). Ranzetta and Pearson (1969) have studied the diffusion reaction between Ti and stainless steel and have reported intermediate phase formation at 1223 K. Figure 7.25(a) and (b) show
Diffusional Transformations
587
Figure 7.24. Microstructure of the interface of a Zr-2/SS specimen diffusion bonded at 1173 K using several interlayers.
the backscattered electron (BSE) image and composition profile for the Ti/SS diffusion-bonded specimen at 1223 K for 2 h. At this temperature the diffusion zone shows several intermetallic compounds and the presence of layer structure of -phase shows inferior mechanical and corrosion properties. Kale et al. (1996) have bonded the samples at 1173 K for a short duration of 2 h to prevent the formation of intermediate phases in the diffusion zone. The bond has been found to exhibit a good strength of 150 MPa. The diffusion-bonded region has not shown any corrosion in a simulated solution. The joint has good integrity and the leak rate is less than 3 × 10−9 std cc per s of He.
7.3
PHASE SEPARATION
The tendency towards phase separation in the -phase of several Ti- and Zr-based alloys is reflected in the miscibility gap (1 + 2 -phase field) in the corresponding -isomorphous phase diagrams. The occurrence of the monotectoid reaction, 1 → +2 , is also a manifestation of the phase separation tendency in the -phase in a number of Ti and Zr alloys containing -stabilizing elements. Reference may be drawn to the classification of phase diagrams presented in Chapter 1, where
588
Phase Transformations: Titanium and Zirconium Alloys
12 3 σ-Phase
Titanium
Stainless steel 10 μm BSE1
20 kV 20 nA
(a)
60
Stainless steel
Concentration (at.%)
1
40
2
3
σ-Phase
Retained β(Ti, Fe, Cr, Ni)
80
Ti Cr Fe Ni
20
0 0
20
40
60
80
Distance (μm) (b)
Figure 7.25. (a) Backscattered electron image for the Ti-stainless steel diffusion-bonded specimen at 1223 K for 2 h. The micrograph shows -phase close to steel and also (1) Fe2 Ti(Cr), (2) TiCr2 + Fe2 Ti(Cr) and (3) FeTi (Cr,Ni) in the diffusion zone. (b) The composition profiles of Fe, Cr, Ni and Ti taken across the region marked at A–B in (a).
the -isomorphous and the -monotectoid systems are discussed. It is to be noted that the excess enthalpy of solution of the solid solutions of Ti and Zr alloy systems exhibiting the aforementioned classes of phase diagrams is invariably positive which is a necessary but not a sufficient condition for the existence of a miscibility gap within a single phase field. The mode of decomposition of a single phase into the equilibrium two-phase mixture within a miscibility gap is
Diffusional Transformations
589
primarily decided by the consideration of metastability and instability of the alloy system with respect to composition fluctuations. The decomposition can occur either through a discrete process in which second phase particles nucleate and grow or by an alternative process in which a concentration fluctuation of infinitesimal amplitude develops homogeneously and is gradually amplified. The former is known as a heterophase fluctuation or nucleation while the latter involves a homophase fluctuation or spinodal decomposition. These two processes are illustrated schematically in Figure 7.26. In the case of a heterophase fluctuation, nuclei or energetically stable solute-rich clusters form by thermal fluctuations. It will be shown in the next section that the formation of stable nuclei requires the development of localized concentration fluctuations of a sufficiently large amplitude. In contrast, when the solid solution becomes unstable, concentration fluctuations which are spatially extended but have small amplitudes develop through a process of spinodal decomposition. The thermodynamic conditions for these two distinct modes of phase separation are discussed in the following section. 7.3.1 Phase separation mechanisms Issues relating to metastability and instability of a solid solution can be best explained by considering a phase diagram showing a miscibility gap (Figure 7.27(a)) within which the high-temperature -phase decomposes into a mixture of two phases, 1 and 2 , which have different compositions but possess the same crystal structure as the -phase. Such a phase diagram corresponds to a solid solution with a positive excess enthalpy, HXS (positive deviation from the ideal solution behaviour). The free energy (G) versus composition (c) plots for such a binary A–B system at temperatures below Tm , the maximum temperature of immiscibility, exhibit a doubly inflected shape, as illustrated in Figure 7.27(b). The construction of common tangents to the G–c plots defines the compositions of the two phases which remain in equilibrium at different temperatures. The boundary of the two-phase field, also known as the coexistence curve, intersects the tie line for a given temperature at compositions given by the points of contact of the common tangent with the two depressions in the G–c plot. The condition of equilibrium is satisfied between the 1 - and the 2 -phases, having c1 and c2 atom fractions of B, respectively, as the chemical potentials (given by the intercepts of tangents on the G-axis) of either of the components in the 1 - and the 2 -phases are equal at the points where the common tangent touches the G–c plot, as has been illustrated in Figure 7.27(b) for the temperature T1 . It is also clear from this construction that the integral molar free energy G(c) of the single phase -solid solution between the points c1 and c2 is higher than that of the mixture of the two phases 1 and 2 , the free energy G(c) of the 1 + 2 mixture being given by the segment of the common tangent between c1 and c2 .
590
Phase Transformations: Titanium and Zirconium Alloys Stage
λ (t1)
2R* c6
Early
λ (t2)
Stable precipitate
Concentration, c
c3
→
c1
cS2 c4
→
Later
→
←
→
cS1
2R (t2) c2
→
→
→
→
c4
→
→
→
→
c3
cS2
Nucleus
←
→
cS1
→
2R (t3)
2R (t3)
c2 cS2 c4 Final c3 →
c1
cS1
→
→
→
→
Distance, x (a) “Heterophase” fluctuation (nucleation)
(b) “Homophase” fluctuation (spinodal)
Figure 7.26. Time evolution of phase separation by two mechanisms are illustrated (time, t1 < t2 < t3 ): (a) heterophase fluctuation (nucleation followed by growth) is shown by the composition profile at a nucleation site; (b) Homogeneous fluctuation (spinodal decomposition) is depicted by the development of a small amplitude composition fluctuation, subsequent growth of the amplitude and finally creation of sharp interfaces between the two product phases. Concentrations, ci , are indicated in Figure 7.27.
Let us now consider an alloy whose composition lies between c1 and c2 . If the alloy remains as a metastable single phase on quenching from the -phase field, its free energy at T1 will be given by the doubly inflected G–c plot shown in Figure 7.27(b). One can examine the decomposition modes of alloys of different
Diffusional Transformations
591
Single phase region
Temperature, T
β
Equilibrium phase diagram (coexistence curve)
Tm
T1
β1
β2
Two-phase region
β1+β2
(a)
Spinodal curve
c3 dG dG = dc c3 dc c6
C3+ΔC Identical slopes
C3
>0
Free energy, G
(b)
>0
ΔG<0 B A μβ1 (c1) = μβ2(c2)
β2 S2
>0
S1
ΔG
C3–ΔC
d2G dc2 <0
T = T1
C4+ΔC
β1 A A μβ1 (c1) =μβ2(c2)
C4–ΔC
A
c1c3 cS1 c4
c5 cS2
c2
C4
B
c6
c, concentration of solute, B
Figure 7.27. (a) Equilibrium phase diagram of a system showing miscibility gap ( → 1 + 2 ). Chemical spinodal curve (dotted line) is superimposed on the phase diagram. (b) The corresponding doubly inflected free energy (G)–concentration plot for a temperature T1 . Spinodes (where 2 G/c2 = 0) are indicated by S1 and S2 . Insets shown within dotted circles show that a small amplitude concentration fluctuation is metastable outside the spinodal region (e.g. at c3 ) whereas the alloy is unstable with respect to concentration fluctuation within the spinodal region (e.g. at c4 ).
compositions at the temperature T1 . The curvature of the G c curve changes sign at the points of inflection, S1 and S2 , defined by the following conditions:
2 G c2
= S1
2 G c2
=0 S2
(7.18)
592
Phase Transformations: Titanium and Zirconium Alloys
The miscibility gap can be divided into different regimes: those which correspond to 2 G/c2 > 0 and those which correspond to 2 G/c2 < 0. The concavity of the G c plot is upwards in the former case and downwards in the latter. Let us consider an alloy with a composition given by the point c3 for which 2 G/c2 > 0. Such an alloy is metastable with respect to the development of concentration modulations of very small amplitudes. This can be easily shown by constructing a straight line segment joining two points corresponding to alloy compositions given by c3 + c and c3 − c. The free energy of the mixture of two phases having compositions deviating from c3 by a small extent, c, will always remain above the G c plot in this regime where 2 G/c2 > 0. A tangent drawn at G c3 intersects the G c plot at the composition given by c5 . This implies that the nucleation of a -phase particle in the c3 ) alloy becomes thermodynamically possible on considerations of the chemical free energy change alone, only if the concentration of the element B in the nucleus exceeds c5 . The chemical free energy change within the volume of the nucleus is given by the vertical drop in the free energy from the tangent and is maximized at c6 where the tangent to the G c curve near the 2 -region is parallel to that drawn at c3 . In the regime where the G c curve is concave downwards 2 G/c2 < 0, any infinitesimal fluctuation in concentration will reduce the free energy of the alloy, as shown for the alloy of concentration c4 . This means that the alloy is unstable with respect to concentration fluctuations and, therefore, no concentration barrier exists in the phase separation process. It is also seen that an amplification of the concentration modulation, which can be represented by an increasing separation of the compositions of the solute-rich and the solute-lean regions, leads to a continuous decrease in the free energy. In the early stages of such a decomposition process, known as spinodal decomposition, the solute-rich and solute-lean regions remain coherent without creating any sharp interfaces between them as shown in Figure 7.26(b) and, spatially speaking, the concentration modulation extends homogeneously across the entire grain of the parent phase. The rate of spinodal decomposition is controlled by the interdiffusion coefficient, D, which is negative within the spinodal. The amplitude of the concentration modulation increases exponentially with time, with a characteristic time constant = − 2 /4 2 D, where is the wavelength of the modulation (if assumed to be one dimensional). The rate of transformation, therefore, may become very rapid by making as small as possible – a situation which corresponds to short wavelength modulation or chemical ordering. However, this would be an unrealistic effect as spinodal clustering occurs in systems exhibiting positive deviations from the ideal solution behaviour, i.e. in systems where like atoms have a tendency to cluster together. The opposing factors which do not allow the wavelength to decrease below a certain limit are the interfacial energy and the strain energy associated with the
Diffusional Transformations
593
composition modulation. Cahn and Hilliard (1959) and Cahn (1962) have included the contributions of the interfacial (gradient) and the strain energies along with the chemical driving force in working out the composition–temperature domain within which spinodal decomposition is possible and the time evolution of the composition modulation. Drawing reference to Figure 7.27, let us consider an alloy of composition c4 , which is within the spinodal regime. If this alloy decomposes into two parts with compositions, c4 + c and c4 − c, the chemical free energy change, Gc , is given by Gc =
1 2 G c 2 2 c2
(7.19)
As mentioned earlier, during the early stages of spinodal decomposition, sharp interfaces are not created between the A-rich and the B-rich regions. A composition gradient, which is produced at such diffuse interfaces, also causes an increment in the energy of the system due to an increase in the number of unlike atomic pairs at these interfaces when compared to the homogeneous solution. This energy component, known as the “gradient energy” (G ), can be expressed in terms of the amplitude, c, and the wavelength, , of a sinusoidal composition modulation as G
=K
c
2 (7.20)
where K is a proportionality constant dependent on the difference in the bond energies of like and unlike atom pairs. The strain energy built up in the system due to the development of a composition modulation arises essentially from the difference in the atomic sizes of the different components. The coherency strain energy, Gs , can be expressed as Gs = 2 c 2 E Vm
(7.21)
In this expression, is the fractional change in the lattice parameter, a, per unit composition change and is given by 1 a (7.22)
= a c E is an elastic constant given by E = E/ 1 − where E and are Young’s modulus and Poisson’s ratio, respectively, for a system with zero elastic anisotropy, and Vm is the molar volume. It is to be noted that Gs is independent of .
594
Phase Transformations: Titanium and Zirconium Alloys
When all the contributions to the total free energy change, G, accompanying the formation of a composition modulation are considered, one gets 2 G 2K 2 G = + 2 + 2 E Vm c 2 /2 (7.23) c2 The condition for a homogeneous solid solution to be unstable and to decompose spinodally is 2 G 2K > 2 + 2 2 E Vm c2
(7.24)
At the spinodal boundary, the system becomes just unstable, implying that the wavelength associated with the fluctuation will be approaching infinity. The coherent spinodal boundary, therefore, is given by the condition 2 G > 2 2 E Vm c2
(7.25)
The coherent spinodal line remains entirely within the chemical spinodal (2 G/c2 = 0). The maximum wavelength of the composition modulation that can develop within the coherent spinodal is given by the condition 2 G 2 2 = −2K (7.26) + 2
E V m c2 Let us now focus our attention on an alloy of composition c3 , which is located outside the spinodal but inside the miscibility gap at T1 . It is clear that a small amplitude fluctuation in composition will lead to an overall increase in the free energy as all the three components of free energy are positive (2 G/c2 is positive outside the spinodal). A critical fluctuation which will be thermodynamically stable will necessarily have a large deviation from the bulk composition. Cahn and Hilliard (1959), in their non-classical nucleation theory, have considered that the inhomogeneous solid solution in its metastable state contains homophase fluctuations with diffuse interfaces and a composition which varies throughout the cluster. A critical fluctuation is, therefore, characterized by its spatial extension or wavelength, , and its spatial concentration variation. The free energy change associated with the transformation from a homogeneous system to an inhomogeneous system can be expressed as G = g c − g co + K c 2 + 2 E c − co 2 dV (7.27) V
Diffusional Transformations
595
where g c and g co correspond to the free energy densities of the solid solutions and co is the composition of the homogeneous solid solution. For the case being discussed, co = c3 . Neglecting the elastic free energy term, 2 E c − co 2 , and assuming isotropy, the composition profile c r of a spherical fluctuation (r being the radial distance from the fluctuation centre) is obtained from a numerical integration (Cahn and Hilliard 1959) as K dc g g d2 c = − (7.28) 2K 2 +4 dr r dr c c c co with the boundary conditions dc/dr = 0 at the fluctuation centre (r = 0) and at a point far away from it (r → ) where c = co . The critical nucleus can then be defined as that fluctuation which remains in an unstable equilibrium with the matrix. The free energy change, G∗ , associated with the formation of a critical nucleus can be worked out by considering a minimum nucleation barrier (de Fontaine 1982) and is expressed as
2 dc g c∗ + K (7.29) r 2 dr G∗ = 4 dr o where c∗ corresponds to the composition of the centre of the fluctuation (nucleus). The free energy for nucleation, therefore, is given by the vertical distance from the tangent drawn on the G–c plot at co (co = c3 in Figure 7.27(b)). As the initial alloy composition, co , is chosen nearer to the spinodal point S1 , the solute concentration at the centre of the nucleus decreases and the composition profile becomes more diffuse. Furthermore, with the initial composition approaching the spinodal point, the spatial extent of the critical fluctuation, R∗ , increases rapidly and finally approaches infinity at co = cS1 . The non-classical nucleation theory, therefore, predicts a discontinuity in the decomposition mechanism at the spinodal line. The radius, R∗ , of the critical fluctuation outside the spinodal regime and the wavelength, c , of the spinodally decomposed structure have been plotted in Figure 7.28 for alloys of different initial compositions for illustrating the discontinuity at the spinodal line (Wagner and Kampman 1991). The generalized theory of phase separation proposed by Binder and co-workers takes into account nucleation in the metastable regime as well as spinodal decomposition in the unstable regime. The theory is essentially based on a cluster dynamics approach in which the attachment and splitting of clusters are considered. A homogeneous supersaturated solid solution, when aged at a temperature sufficiently high for solute diffusion to occur, will contain microclusters of solute atoms. A cluster containing i number of atoms is called an i-mer. The theory of the
596
Phase Transformations: Titanium and Zirconium Alloys Coexist.curve
↑ T To
Spinodal curve
A
B Unstable
Meta stable
λc (spinodal) R* (Cahn–Hilliard)
↑ R*,
R* (Binder)
λc
cαe
csβ
csα
cβe
Concentration, c
Figure 7.28. Phase diagram of a binary system with a miscibility gap. The coexistence curve and spinodal curves are also shown. The variation of characteristic length c and R∗ , according to different theories, with composition is shown.
nucleation of second phase particles in a solid solution has both static and dynamic components. The free energy of formation of an i-mer and the distribution, f(i), of i-mers are treated in the static part. In the dynamic part, the kinetics of the decomposition of the solid solution, which is described by the given distribution of non-interacting microclusters, are calculated in terms of the time evolution of f(i), which finally leads to the determination of the rate of formation of stable clusters, i.e. the nucleation rate. A detailed account of the kinetics of the early stages of decomposition and a comparison between the theoretical approaches and the predictions thereof for both classical and non-classical nucleation has been provided by Wagner and Kampman (1991). Binder (1977) has considered the time evolution of f i t , the cluster size distribution function, in terms of the rates of the following four processes:
Diffusional Transformations
(1) (2) (3) (4)
597
splitting of (i + i )-mers into i- and i -mers splitting of i-mers into (i − i )- and i -mers coagulation of (i − i )-mers and i -mers into i-mers coagulation of i- and i -mers into (i + i )-mers.
Thus the kinetics of the time evolution of f i t is given by i−1 d 1 f i t = i+i i f i + i t − f i t dt 2 i =1 ii i =1 i−1 1 + i−i i f i t f i − i t − ii f i t f i t 2 i =1 i =1
(7.30)
where and refer to the rate constants of the respective processes. These two rate constants can be replaced by a single rate constant, W, using the following expression which relates the cluster concentration, C i , in thermal equilibrium with the metastable matrix, and the rate constants W i i ≡ i+i i C i + i = ii C i C i
(7.31)
The time evolution of the cluster concentration can be obtained by numerical integration of Eqs. (7.30) and (7.31), provided inputs such as the initial cluster distribution, f i t = 0 , W i i and C i are available. In the absence of the necessary data for common alloys, solution of the kinetics equations has been obtained (Mirold and Binder 1977) with some plausible input values. These results have provided a profound, though qualitative, insight into the dynamics of cluster formation and growth. The most important feature of the nucleation theory of Binder and co-workers is that it addresses nucleation in the metastable regime as well as spinodal decomposition in the unstable regime of the miscibility gap. The size of the critical cluster, R∗ , predicted from this theory exhibits a monotonic variation when plotted against the composition, as shown in Figure 7.28. The divergency of both the critical radius, R∗ , of the nucleus and the wavelength, c , of the critical fluctuation, inherent in the Cahn–Hilliard theories in the vicinity of the spinodal line, is not seen in the critical cluster size. 7.3.2 Analysis of a phase diagram showing a miscibility gap The miscibility gap is the simplest system for a phase diagram analysis which involves the determination of thermodynamic quantities from a given phase diagram. The fact that a single phase decomposes into a mixture of two phases, all
598
Phase Transformations: Titanium and Zirconium Alloys
the three having the same crystal structure, is basically responsible for making the phase diagram analysis very simple. Let us consider the miscibility gap shown in Figure 7.27. The condition of equilibrium between the two phases, 1 and 2 , with compositions given by c1 and c2 , can be graphically represented by the construction of a common tangent. The numerical computation of thermodynamic quantities can be carried out using the method elaborated by Rudman (1970) and outlined in the following. The excess integral molar thermodynamic quantity, QXS , is expressed in terms of a power series of the form QXS = c 1 − c
N
aQi ci
(7.32)
i=0
where c, as defined earlier, is the atomic fraction of the solute and aQi are the coefficients of the series for the thermodynamic quantity Q. Assuming that the specific heat, Cp , is constant over a temperature range, the relative integral molar enthalpy, H, and entropy, S, are expressed by the following equations: H = H0 + Cp T − T0
(7.33)
S = S0 + Cp ln T/T0
(7.34)
where H0 and S0 are the integral molar enthalpy and entropy at the reference temperature, T0 . Using Eqs. (7.32)–(7.34), the composition and the temperature dependence of the excess integral molar free energy of the -phase is expressed as GXS = c 1 − c
N
i aG i c
(7.35)
i=0 C
C
p 0 0 0 0 G and ai p where aG i = ai − Tai + ai T − T0 − T ln T/T0 )] and ai , ai , ai represent the expansion coefficients of the excess integral molar free energy, enthalpy, entropy and specific heat, respectively. It is possible to express the free energy as a function of temperature and composition if one can find out the values of these coefficients. In a system showing a miscibility gap, the computation of these coefficients can be performed by using the following equilibrium condition which equates the partial molar free energies of a given component in the two solutions corresponding to the compositions c1 and c2 given by the phase boundary (coexistence curve) at a given temperature, T:
H
S
2 1 A c1 T1 − A c2 T1 = 0
H
S
(7.36)
Diffusional Transformations
599
2 1 B c1 T1 − B c2 T1 = 0
(7.37)
Substituting the values of c1 and c2 for different values of T, it is possible to evaluate aG i by a least square analysis. Menon et al. (1978) have presented the computed excess enthalpy, excess entropy and free energy as functions of compositions for the -phase in the Zr–Nb system. It has been shown by Rudman (1970) that the accuracy of the calculation of excess thermodynamic quantities does not significantly improve by the inclusion of more than two terms in the series expansion and in view of this each of the excess enthalpy and entropy of the -phase in the Zr–Nb system has been expressed in the form of a series containing two terms. The values of the coefficients obtained using the miscibility gap data from the phase diagram are aH0 = 335239 cal/mol aS0 = −06322 cal/mol K aH1 = 109674 cal/mol aS1 = −19094 cal/mol K and SXS for the -phase as functions of Figure 7.29 shows the quantities HXS composition. The asymmetrical miscibility gap in the -phase seen in the phase diagram is reflected in the HXS –c plot. The negative value of SXS is consistent with the clustering tendency which is clearly revealed from the positive deviation (positive value of HXS ) of the -phase from the ideal solution behaviour. Free
800
400
S xs (cal/mol k)
0
β
H xs (cal/mol) →
600
β
200
0
0
0.2
0.6
0.4
0.8
1.0
–0.2
–0.4 0
0.2
0.4
0.6
Concentration, c
Concentration, c
(a)
(b)
0.8
1.0
Figure 7.29. Computed (a) excess enthalpy and (b) excess entropy of the -phase of Zr–Nb alloys as a function of Nb concentration, c.
600
Phase Transformations: Titanium and Zirconium Alloys 200
650 K 750 K
0
–200
G (cal/mol) →
1050 K
–400
–600
1450 K
–800
–1000 0
0.2
0.4
0.6
0.8
1.0
Concentration, c
Figure 7.30. Computed free energy (G) versus Nb concentration (c) plot for the Zr–Nb system at various temperatures.
energy versus composition (G–c) plots (Figure 7.30) constructed at different temperatures show the doubly inflected shape at temperatures below about 1200 K, implying that the critical solution temperature of the -phase is nearly 1200 K. Flewitt (1974) has studied the isothermal decomposition behaviour of -Zr-Nb alloys and has experimentally determined the critical solution temperature to be close to 1250 K. Modelling of the Zr–Nb equilibrium phase diagram has been developed on a regular solution approximation by Kaufman (1959) and he has obtained a fairly good agreement with the experimental phase diagram by taking the values of the interaction parameters of the , and liquid phases to be 33472, 18410 and 6276 J/mol, respectively. The lattice stability terms used by Kaufman are given in Table 7.2.
Diffusional Transformations
601
Table 7.2. Lattice stability terms for Zr and Nb. Element Zr Nb
G→ (J/mol; T, in K) 3.766(1144 − T) −3.347(1875 + T)
G→L (J/mol; T, in K) 8.368(2125 − T) 8.368(2740 − T)
Abriata and Bolcich (1982) have made a critical assessment of experimental phase diagram data and results of thermodynamic modelling with regard to the Zr–Nb system. They have indicated that a reasonable agreement exists between the miscibility gap determined by different investigators and that phenomenological thermodynamic calculations are consistent with the experimental phase diagram. However, the ideal situation of coupling phase diagram and experimental thermodynamic data is still somewhat elusive for want of reliable thermodynamic measurements. The determination of the chemical spinodal line from the G –c plots (Figure 7.30) becomes possible by locating the points of inflection where the following condition is valid: 2 G =0 c2
(7.38)
The chemical spinodal (line 1) thus determined is shown superimposed on the Zr–Nb phase diagram in Figure 7.31. Cook and Hilliard (1965) have derived the spinodal line in a binary phase diagram in terms of the critical temperature, Tc , and the critical concentration, cc ,of the miscibility gap as T cs cc ce cc 1 − 0422 (7.39) Tc where cs and ce are the spinodal and the equilibrium concentrations at a temperature T. This formulation assumes a parabolic free energy function in the vicinity of cc . The spinodal curve (line 2) obtained from Eq. (7.39) is also plotted in Figure 7.31 to demonstrate the close agreement between the results obtained from the two approaches discussed. The coherent spinodal is defined by the equality 2 G + 2K 2 + 2 2 Y = 0 c2
(7.40)
where is the wave number ( = 2/) of the sinusoidal concentration fluctuation, K is the gradient energy coefficient and Y is a function of the crystallographic
602
Phase Transformations: Titanium and Zirconium Alloys
Temperature (K)
1400
1200
Chemical spinodal
β
1000
2 1
βI + βII
α + β1 800
α + βII
Coherent spinodal
600 0.0
0.2
0.4
0.6
0.8
1.0
Atomic fraction of niobium →
Figure 7.31. The phase diagram of the Zr–Nb system with superimposed chemical spinodal lines (line 1, determined from G–c plots; line 2, obtained from Eq. (7.39)) and the coherent spinodal line.
direction (associated direction cosines being l, m and n) along which the concentration modulation develops. Y is given by C11 + C12 1 (7.41) Y = C11 − 2C12 3 − 2 C11 + 2 C44 − C11 + C12 l2 m2 + m2 n2 + l2 n2 where Cij are elastic constants. The fractional changes in the lattice parameter, , is given by Eq. (7.22). Hilliard (1970) has shown that TS∗ − TS = 2 2 Y/S
(7.42)
where TS∗ and TS , respectively, indicate the temperatures of the coherent and the chemical spinodals and S = 2 s/c2 , where S is the entropy per unit volume. Using the values of entropy obtained from the phase diagram analysis of Menon et al. (1978) and from the reported values of elastic constants (Goasdoue et al. 1972) and lattice parameters (Pearson 1967) the coherent spinodal line has been determined from Eq. (7.40) and has been plotted in Figure 7.31 One of the advantages of the phase diagram analysis as elaborated here is that the spinodal line at temperatures lower than the monotectoid temperature, Tmono , can also be evaluated without using the extrapolated phase boundaries. An analysis of the phase diagram of the Zr–Ta system has also been performed by Menon et al. (1979) who have shown that the free energy hump between the
Diffusional Transformations
603
1 - and 2 -phases is much larger in the case of Zr–Ta than that in the case of Zr–Nb; this is consistent with the higher Tc associated with the former system. 7.3.3 Microstructural evolution during phase separation in the -phase Experimental investigations on spinodal decomposition in Ti- and Zr-based alloys are only a few in number. Flewitt (1974) has reported the results of a systematic study on the decomposition behaviour of the -phase within the miscibility gap in respect of the Zr–Nb system. A summary of these results is given in this section. The maximum of the miscibility gap and Tc for both chemical and strain spinodals lie at a composition of Zr–60 at.% Nb, which was selected by Flewitt for testing the occurrence of spinodal decomposition in the Zr–Nb system. On quenching this alloy in iced brine from a temperature above 1250 K, the homogeneous solid solution is retained in a metastable state, as demonstrated by the results of X-ray and electron diffraction. On subsequent ageing at temperatures above 883 K (the monotectoid temperature, Tmono ), fine modulations, as observed in TEM images obtained with the electron beam aligned along an <100> direction, develop and the contrast grows with increasing ageing time. The modulations are best observed when a g = h00 beam is excited and these are seen to be perpendicular to g, the excitation vector. The corresponding diffraction patterns show the presence of side bands, the spacing, r, of the side band from the main hkl reflection (at a distance r from the 000 position in the reciprocal space) giving the dominant wavelength, , of the concentration fluctuation (Daniel and Lispon, 1944): =
har h2 + k2 + l2 r
(7.43)
Wavelengths measured from bright- and dark-field micrographs have been shown to be consistent with those calculated from side band spacings. It is to be noted that the same alloy (Zr–60 at.% Nb), when subjected to a somewhat slower quench (such as water quench or oil quench), shows evidences of concentration modulations along <100> directions as revealed by the occurrence of mottling in diffraction contrast images and by the presence of side bands in diffraction patterns. This observation suggests that suppression of the spinodal decomposition, which is a homogeneous transformation, can be achieved only partially by kinetic means, i.e. by restricting the number of diffusive atomic jumps. The fact that the transformation is homogeneous is also testified by the observation that the modulation develops uniformly in the entire grain right up to the grain boundary. As has been pointed out in Section 7.3.1, the evolution of non-localized, spatially extended concentration fluctuations, the amplitude of which increases gradually
604
Phase Transformations: Titanium and Zirconium Alloys
with ageing time, can be defined as a true “spinodal decomposition” in the sense of the Cahn–Hilliard definition (1959). The time dependence of the concentration c(r,t) at a position r at an instant t is given by the linearized diffusion equation
c r t M 2 2 G + 2 2 Yc r t − 2K 4 c r t (7.44) = t
c2 c0 where M is the atomic mobility and is related to the interdiffusion coefficient D by the relation 2 G M = Dnv 2 (7.45) c c0 takes the sign of 2 G/c2 and is thus negative inside As M is always positive, D the spinodal regime, giving rise to an uphill diffusion of solute atoms. The reported values of the wavelength as a function of the ageing time (Flewitt 1974) show that the wavelength remains more or less constant at the very early stages of the decomposition process and at the subsequent stages it follows a coarsening law which can be expressed as = K1 t1/3 (Figure 7.32(a)). The rate constant (K1 ) in the Lifshiftz–Wagner equation is given by
8DcVm2 K1 = 9RT
1/3 (7.46)
where is the matrix/precipitate interfacial energy, D is the coefficient of diffusion of the solute in the matrix and Vm the molar volume of the precipitate. From the plot of log K 3 versus 1/T, the activation energy for the coarsening of modulations has been determined to be 64 ± 10 kcal/mol which is in good agreement with the value of 70 kcal/mol reported by Hartely et al. (1964) for self-diffusion in the -phase of Nb–Zr alloys. The morphology of the spinodally decomposed alloy is of interest and deserves a special mention, as it is through this distinctive criterion that a spinodal decomposition can be distinguished from a precipitation process. A recent review by Wagner and Kampman (1991) emphasizes that in view of the current theoretical developments, summarized by Binder (1991), it is rather difficult or even impossible to assess on a thermodynamic basis whether an alloy system is truly quenched into and aged within the spinodal regime of the miscibility gap. It is suggested that the morphological criterion can be used for identifying a spinodal decomposition process. The observed morphology (Flewitt 1974) of the spinodally decomposed Zr–Nb alloys of both symmetric (∼60 at.% Nb) and asymmetric compositions comprises
Diffusional Transformations
605
Rate constant K 13 (Å3/h)
108
107
106
105 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1/T × 103 K–1 (a) 0
T = 883 K
Gα
G (cal/mol)
–100
Gβ
N
cβ
2
–200
P cβ
1
cα
–300
0
0.2
0.4
0.6
0.8
1.0
Concentration, c (b)
Figure 7.32. (a) Rate constant for coarsening of the spinodally modulated structure as a function of reciprocal temperature in Zr–Nb alloys (Flewitt 1974); (b) free energy–concentration plots for the - and the -phase in the Zr–Nb system at the monotectoid temperature 883 K.
606
Phase Transformations: Titanium and Zirconium Alloys
a rod-like arrangement of solute-rich and solute-lean regions with the axes lying along the <100> directions. The irregularity of the initial morphology, as pointed out by Cahn (1962), arises from the range of wavelengths centred on a dominant wavelength at any instant. The diffracted intensity distribution around the side bands also indicates the presence of such a spectrum of wavelengths. Cahn (1962) has pointed out the importance of elastic anisotropy with regard to the final form of the periodic concentration fluctuations. The elastic anisotropy factor, 2C44 / C11 − C12 ), is less than unity for pure Nb but the addition of Zr to a level of about 40 at.% increases this factor to a value larger than unity. Under such a condition, concentration waves develop as plane waves on all the three {100} planes, so that an interlocking network of <100> rods, alternately enriched and depleted in solute content, appears in the spinodally decomposed microstructure. The structure becomes more and more interconnected with increasing reaction time. Lattice defects and grain boundaries play little role in the morphological evolution, indicating the homogeneous nature of the transformation. As the concentration waves get amplified, a situation is reached where the strain gradient at the interface between the solute depleted and enriched regions is adequate for nucleating misfit dislocations which are arranged in a hexagonal network built up from three sets of dislocations. Contrast analysis has shown that these two sets of dislocations are associated with a<100> Burgers vectors and have primarily edge characters while the third set comprises a<110> screw-type dislocations. For the Zr–60 at.% Nb alloy the critical wavelength for the loss of coherency has been found to be about 30 nm at 973 K. The wavelength at which coherency is lost is seen to decrease with the decomposition temperature, as is expected from the fact that the amplitude of the concentration wave (and correspondingly the mismatch between the regions lean and rich in the solute) increases with decreasing reaction temperature. 7.3.4 Monotectoid reaction – a consequence of -phase immiscibility The tendency of phase separation in the -phase, resulting from the positive deviation from the ideal solution behaviour, is manifested in the appearance of miscibility gaps in several binary Zr and Ti alloys. The occurrence of miscibility gaps in systems like Zr–Nb, Zr–Ta as reported quite sometime back in the phase diagram compilation by Hansen (1958). A number of binary Ti alloy phase diagrams such as Ti–V, Ti–Nb and Ti–Mo were classified some years back as -isomorphous systems. Recent assessments of many of these phase diagrams have identified the presence of miscibility gaps in the -phase field. The interplay of the two-phase transformation tendencies, namely, the / transformation of the terminal solid solutions, rich in either Ti or Zr, and the phase separation within the -phase field, is responsible for bringing about monotectoid
Diffusional Transformations
607
phase reactions in these systems. This point will be explained by taking the example of phase equilibria in the Zr–Nb system. In a later part of this section, other systems showing monotectoid reactions will be discussed. At temperatures between 883 K, the monotectoid temperature in the Zr–Nb phase diagram, and 1135 K, the temperature at which pure Zr undergoes the / phase transformation, the - and the -phases remain in equilibrium. The tie line within the ( + )-phase field gives the compositions of the two phases, c and c , in equilibrium, which is defined by equating the partial molar free energies of the two components in the - and the -phases: A c T = 1 A c1 T
(7.47)
B c T = 1 B c1 T
(7.48)
The numerical values of the right-hand sides of these equations at various temperatures can be obtained from the analysis of the miscibility gap in the phase diagram, described in Section 7.3.2. The partial molar free energies for the -phase for different compositions and temperatures can then be obtained using Eqs. (7.47) and (7.48). Thus G versus c plots for different temperatures, both above and below Tm , can be obtained. Using this formalism, Menon et al. (1978) have obtained G–c plots for both the - and -phases at different temperatures. The plots corresponding to the monotectoid temperature are given in Figure 7.32(b) as an illustrative example. Moffat and Kattner (1988) have considered not only the stable equilibria involving the -, 1 - and 2 -phases but also the metastable equilibria in respect of the -, 1 - and 2 -phases using the CALPHAD approach in which the free energy G c T of the given phase is expressed in terms of the lattice stability parameters of pure elements in the different phases and interaction parameters: G c T = 1 − c GA T + cGB T + RT c ln c + 1 − c ln 1 − c + l c 1 − c (7.49) where GA or GB is the lattice stability of element A or B, respectively, and l is the interaction parameter. These parameters, taken from Kaufmann and Bernstein (1970), are listed in Table 7.3. Figure 7.33(a) shows the calculated phase diagram of the Ti–Nb system. It may be noted that though the shape of the ( + )/ transus line is suggestive of a positive deviation from ideality, the extent of this deviation is not large enough to cause the appearance of the miscibility gap and the monotectoid reaction. In contrast, the calculated metastable phase diagram involving the -, 1 - and 2 -phases shows both the monotectoid reaction and the miscibility gap (Moffat and Kattner 1988). The regular solution parameters used by Moffat and Kattner (1988) for generating the metastable phase diagram are also listed in Table 7.3.
608
Phase Transformations: Titanium and Zirconium Alloys Table 7.3. Regular solution parameters for the Ti–Nb stable and metastable equilibria. Lattice stability parameters G→L T = 20 610 − 12134T Ti
G→L T = 19 337 − 12510T Ti
→L
→
GTi T = 16 259 − 8368T
GNb T = −6276 − 3347T
G→L Nb T = 16 669 − 11715T
G→ Nb T = −1273 − 0376T
→L GNb T
GNb T = −7549 − 3723T
→
= 22 945 − 8368T
G→L Nb T = 15 396 − 12091T Interaction parameters l = 13 075 l = 13 075
l = 2510
All values of G and l are in J/mol and that of temperature in Kelvin.
Weight per cent vanadium
Weight per cent niobium 0
0
10 20 30 40 50 60 70 80 90 100
10 20 30 40 50 60 70
80
90 100
1000
900
900 800
Temperature (°C)
Temperature (°C)
800 β
700
600 α
500
β1
700
676°C
18.51 α
500
β2 81.49
600 5.46 425°C
400
94.54
β→α
To
300 200 100
β→ω
To
0
400 0 Ti
10 20 30 40 50 60 70 80 90 100
Atomic per cent niobium (a)
Nb
0 Ti
10 20 30 40 50 60 70 80 90 100 V Atomic per cent vanadium
(b)
Figure 7.33. (a) The calculated phase diagram (Moffat and Kattner 1988) of the Ti–Nb system (solid line). Experimental points for locating the boundaries of + / phase fields are marked. (b) The calculated phase diagram (Moffat and Kattner 1988) of the Ti–V system showing the equilibrium 1 → + 2 reaction at 676 C and the metastable 1 → + 2 reaction at 425 C. The equilibrium phase diagram (solid line), the metastable phase diagram (dashed line) involving - and -phases, To lines (dotted lines) for the → and the → transformations are also shown.
Diffusional Transformations
609
The tendency of phase separation in the -phase has now been recognized in a number of systems which have earlier been designated as -isomorphous systems. The equilibrium and the metastable phase diagrams of the Ti–V system, calculated by Moffat and Kattner (1988), are superimposed in Figure 7.33(b) to illustrate that two monotectoid reactions, namely, the equilibrium 1 → + 2 reaction and the metastable 1 → + 2 reaction, can indeed occur in this system. Similar conclusions have also been arrived at for the Ti–Mo system. 7.3.5 Precipitation of -phase in supersaturated -phase during tempering of martensite Tempering of martensite is an important heat treatment often employed in heattreatable alloys of Ti and Zr. Unlike the steel martensites, supersaturation in Ti and Zr martensites is with respect to only substitutional alloying elements. Therefore during a tempering heat treatment the martensitic phase rejects those substitutional solute atoms which remain in excess of the solubility limit at the tempering temperature. Depending on the nature of the solute element, the precipitating phase which separates during tempering can be an intermetallic phase rich in the solute or the -phase. A great majority of heat-treatable alloys of Ti- and Zr-based alloys are + alloys in which a combination of - and -stabilizing elements are added. In the as-quenched state, such alloys consist of either a fully martensitic or a mixed primary + martensitic microstructure. On tempering, the martensitic -phase decomposes to produce a distribution of the -phase. The miscibility gap in the -phase plays an important role in deciding the phase transformation sequence during the tempering of such + alloys. This point will now be examined by taking the illustrative example of Zr–Nb alloys. The identification of the precipitating phases, the orientation relations between the precipitates and the matrix and the thermodynamics of phase evolution during tempering in the Zr–Nb system have been reported by Banerjee et al. (1976), Menon et al. (1978) and Luo and Weatherly (1988). It has been observed that the precipitation process is very sluggish at temperatures lower than 773 K (about 100 K below the monotectoid temperature of 883 K). The volume fraction of the precipitate phase has been found to be quite small, consistent with what is predicted from the equilibrium phase diagram. At temperatures below the monotectoid temperature, the equilibrium-precipitating phase is the Nb-rich 2 -phase (containing about 85 at.% Nb) which can be distinguished from the 1 -phase (Zr–20 at.% Nb) from an accurate determination of the lattice parameter of the precipitating phase. The most striking observations have been the following: (1) The phase which precipitates in the temperature interval of 773–883 K is the 1 -phase and not the equilibrium 2 -phase (Figure 7.34).
610
Phase Transformations: Titanium and Zirconium Alloys
m
m
(b)
(a)
m (c)
(d)
Figure 7.34. Precipitation along the twin boundaries in internally twinned martensite plates in Zr–2.5 Nb (a) 2 -phase precipitation at the tempering temperature of 773 K. (b) and (c) 1 -phase precipitation at the tempering temperature of 823 K, the dark-field image (c) is taken with a {110}1 reflection. (d) Diffraction pattern corresponding to (b) and (c) showing superimposition of reciprocal lattice sections of two twin-related - and the precipitate 1 -orientations.
(2) At temperatures below 773 K and above 883 K, precipitation of the equilibrium phases, 2 and 1 , respectively, takes place during tempering of the Zr–2.5 Nb martensite. (3) On tempering, the Zr–5.5% Nb martensite initially reverts back to the parent -phase via a composition-invariant process; subsequently the reverted -phase transforms into a structure consisting of Widmanstatten -plates in a 1 -matrix. These observations have been rationalized in terms of calculated free energy– composition G–c plots. The G–c plots in Figure 7.35 represent the free energy curves for the - and the -phases at 850 K, a temperature slightly below the monotectoid temperature. At such temperatures two common tangents can be constructed, one of which (line A in Figure 7.35) touches the - and the 1 -curves
Diffusional Transformations
611 T < T mono
K
Gα 0
D
G (cal/mol)
•P
(c p)
–50
B
E c1
K
–100
↓
Gβ
↓
–200
A
R •• S
–250 c1 0.0
β1
ΔG αβ (c 2, c n)
•Q
–150
–300
A
C
c3 0.05
c c2 c5
0.1
B (tangent to β2)
cn
c4 0.15
0.2
0.25
0.3
0.35
Composition, c
Figure 7.35. Free energy–concentration plots for the - and the -phases in the Zr–Nb system at 850 K showing that a metastable equilibrium (represented by the common tangent A) can be established between the - and the 1 -phases. See text for transformation sequences for alloys having compositions c and c5 .
while the other (line B) touches the free energy curves corresponding to the - and the 2 -phases. This implies that a metastable equilibrium between the - and the 1 -phases is feasible at such temperatures. The lines A and B merge at Tmono , the monotectoid temperature where a single line touches the free energy curves corresponding to the -, the 1 - and the 2 -phases. At a temperature slightly lower than Tmono the line A remains above the line B, implying that the structure consisting of and 1 is metastable (i.e. associated with a higher value of free energy) with respect to the equilibrium + 2 structure. On the basis of these G–c plots the sequences of transformation during tempering of alloys of different compositions can be rationalized. The common tangent between the G and G1 curves touches the former at a composition c1 while the G and the G curves intersect at a composition c2 . For an alloy with composition c, where c1 < c < c2 , the supersaturated martensite decomposes through a metastable step consisting of the + 1 structure before attaining the equilibrium + 2 structure. These two steps correspond to free energy changes shown by the free energy drop from C to D for the → + 1 decomposition followed by the drop from D to E corresponding to the + 1 → + 2 transformation (see inset of Figure 7.35). It is the evident that the driving forces for the → + 1 and
612
Phase Transformations: Titanium and Zirconium Alloys
the → + 2 reactions are not much different. The strong preference for the formation of 1 -precipitates during tempering at temperatures lower than but close to the monotectoid temperature can be understood in terms of the compositional barriers associated with the nucleation of the 1 - and the 2 -precipitates. As discussed earlier in the context of phase separation mechanisms, the chemical free energy change, G c2 , cn ), accompanying the nucleation process (where c2 and cn are the compositions of the supersaturated -martensite and the -nuclei, respectively), can be estimated by constructing a tangent to the G curve at c2 . The vertical distance between the tangent (marked CK in Figure 7.35) and the G curve gives the value of G (c2 , cn ) for a -nucleus composition, cn . The tangent CK intersects the G curve at a composition c3 ; therefore, the Nb concentration in the -nucleus must be higher than c3 . One can visualize the following two essential steps for the nucleation of a -particle in the -matrix: (a) clustering of Nb atoms enriching a local region and (b) crystallographic transformation of this Nb-rich region from hcp to bcc. The sharp rise in the free energy of the -phase with an increase in Nb content will preclude the formation of -regions substantially enriched with Nb. Alternatively the formation of -nuclei with Nb concentrations close to but higher than c3 is thermodynamically feasible. This route of nucleation will involve much smaller composition and free energy barriers. Such -nuclei will continue to get enriched with Nb and the free energy of the nuclei will follow the G curve. During the process of Nb enrichment the -nuclei composition will attain the level, c4 , where the common tangent between the G and G1 curves meets the G1 curve. A metastable equilibrium between the matrix - and the 1 -phase is established at this point. A further Nb enrichment of the -nuclei to approach the composition level corresponding to the 2 -phase is hindered by the presence of a large free energy barrier in the G curve. The magnitude of this barrier is given by the vertical distance NP (as shown in Figure 7.33) where the tangent to the G curve at N is parallel to the common tangent touching the 1 - and 2 -arms of the G curve. For a supersaturated martensite with a composition c > c2 (Figure 7.36), tempering at 8508 K can cause a reversion from to . The thermodynamic feasibility of a composition-invariant → transformation is shown for a -composition c5 by the vertical drop in the free energy from P to Q. In a subsequent step, the -phase decomposes into a mixture of + 1 phase mixture (metastable) and finally into the equilibrium + 2 structure. The overall transformation can then be described by the following scheme: c5 → c5 → + 1 → + 2 Drops in free energy for this transformation sequence are shown by the vertical segments, PQ, QR and RS. With decreasing temperature, the 1 -arm of the G
Diffusional Transformations
613
1 μm (a)
(b)
Figure 7.36. Light (a) and TEM (b) micrographs showing -grain boundary allotriomorphs at coarse -grain boundaries and fine -plates within the -grains in the Zr–5.5% Nb martensite tempered at 823 K. The formation of the coarse -grain structure indicates reversion of -martensitic into which subsequently decomposed into a structure consisting of -plates within -grains and -allotriomorphs at -grain boundaries.
curve goes up with respect to the G curve and below a certain temperature limit (T < (Tmono − 100 K)); it is not possible to construct a common tangent between the G curve and the 1 -portion of the G curve which implies that the establishment of a metastable equilibrium between the - and the 1 -phases is precluded. In such cases, direct nucleation of the 2 -phase takes place during tempering. Obviously this process involves the overcoming of a very large free energy barrier and hence the reaction is expected to be very sluggish as is indeed the case. Luo and Weatherly (1988) have studied the precipitation behaviour of the Zr–2.5% Nb alloy and have confirmed that both 1 - and 2 -precipitates form in this alloy during tempering. The rationale of the formation of bcc precipitates of two widely varying compositions (1 : Zr–20% Nb and 2 : Zr–85% Nb) at temperatures close to but below the monotectoid temperature has been corroborated by their studies. They have also pointed out that the precipitate phase forming at 773 K is exclusively 2 while both 1 - and 2 -precipitates form at 873 K. For the latter treatment, homogeneously nucleated precipitates within the martensite plates have been identified to be of the 2 -type and the twin boundary nucleated precipitates are invariably of the 1 -type. The crystallography of precipitation of 1 - and 2 -phases has been studied by Banerjee et al. (1976) and Luo and Weatherly (1988). Figure 7.37(a) and (b) depicts the key for the superimposed diffraction patterns of twin-related -crystals and of 1 -precipitates at twin boundaries. The latter study has convincingly demonstrated that the orientation relation followed by the needle-shaped 2 -precipitates (Figure 7.37(b)), which form within the martensite plates, is slightly different from the Burgers relation which is obeyed by the 1 -precipitates. In both the cases, the basal plane (0001) remains parallel to ¯ and the [110] ¯ directions {110} -type planes, but the angle, , between the [0110]
614
Phase Transformations: Titanium and Zirconium Alloys (0111)α (011)β T
1
(1010)α
M
(0002)α (011)β M
1
(1101)α
T
Matrix α Twin α Precipitate β1
(b)
(a)
Figure 7.37. (a) Key of the diffraction pattern shown in Figure 7.34(d). The 1 -precipitate bears Burgers orientation relationship with both the twin -orientations. (b) Twin boundary 1 -precipitates and needle-shaped 2 -precipitates within martensite plates in the tempered Zr–2.5 Nb alloy.
is different for 1 (Burgers relation, = 526 ) and for 2 ( = 38 –41 ) phases. They have also shown that the orientation relation obeyed by the needle-shaped 2 -precipitates is consistent with the invariant line strain (ILS) condition proposed by Dahmen (1982) and that the growth direction of these needles is very close to the ILS direction. Let us now consider the crystallography of twin boundary precipitation. It has been observed by Banerjee et al. (1976) and Luo and Weatherly (1988) that ¯ twin boundaries adhere to the Burgers orienta1 -precipitates forming at {1011} tion relation. The two 1 -variants that nucleate at the twin boundaries maintain ¯ and [1120] ¯ directions and the twin plane lies the parallelity between the [111]
within 1.5 of the (011) plane. The common close-packed direction in the two phases remains unrotated and gets only slightly contracted. Since a {110} -type ¯ -type twin plane, the mirror plane of the -phase is nearly parallel to a {1011} plate-shaped precipitates along the twin boundary can maintain nearly equivalent orientation relations with respect to both the twin-related crystals. The orientation of 1 -precipitates at the twin boundary is, therefore, the same as that of the parent -phase from which the twinned martensite plate has formed. The nucleation of such an orientation will require a very small surface energy, as the 1 -nucleus can maintain coherency on both sides of the twin plane. This low surface energy ¯ {110}1 plane will promote between the - and 1 -crystals along the {1011} spread of the 1 -crystal along the twin boundary. As a result, thin plate-shaped precipitates will form along the twin plane. 1 -precipitates of this specific variant,
Diffusional Transformations
615
0.5 μm (a)
α αT
(110)β1
α (110)β1 αT
→
→
α 0.5 μm
(b)
Figure 7.38. TEM morphology
(a) and schematic drawing (b) showing the morphological evolution ¯ of 1 -precipitates at 1011 twin boundaries of Zr–2.5 Nb martensite during tempering. The orientation of the twin boundary 1 -precipitates is the same as that of the parent -phase. and T
¯ twin related. are 1011
nucleated independently, will also join up to produce lamellae of 1 -precipitates along twins. Figure 7.38 illustrates the morphology of such 1 -precipitates. Few systematic investigations on microstructural changes occurring during tempering of Ti alloy martensites have been made. The close similarity between the crystallography of the / transformation in Zr and Ti alloys suggests that a similar mechanism of -phase precipitation along twin boundaries of the -martensite will control the morphological development of the two-phase structure during the tempering of internally twinned Ti martensites also. In several alloy systems of Ti and Zr, the precipitation of intermetallic phases occurs during tempering of martensites. A few examples of intermetallic phase precipitation have been discussed in Section 7.6.
616
Phase Transformations: Titanium and Zirconium Alloys
7.3.6 Decomposition of orthorhombic -martensite during tempering The tempering of hcp -martensites in alloys belonging to the -isomorphous and -monotectoid systems, as discussed in Section 7.3.5, results in the precipitation of the -phase. In contrast to this, tempering of the orthorhombic -phase leads to the following decomposition process:
→ enriched + → + Davis et al. (1979) have studied the decomposition process in detail in the Ti–Mo system and have shown that the -phase can decompose either by a discrete precipitation or by a continuous spinodal process. Some of the important results reported by them are summarized here. In the Ti–4% Mo alloy, the microstructural features of the martensite are suggestive of the presence of both hcp - and orthorhombic -phases. On tempering, six variants of precipitates appear in some of the plates while in some other plates large areas are covered by only two variants. The periodicity in the precipitate distribution in the latter gives rise to satellite reflections in the diffraction patterns. Such a periodic distribution of a pair of nearly orthogonal variants appears to originate as a consequence of elastic strain interactions between the variants. The elastic strain imposed by the pair introduces an orthorhombic distortion in the matrix which, in turn, suppresses the formation of four other variants locally. In Ti–4% Mo precipitate free zones are invariably present at the boundaries of martensite laths. Therefore a continuous mode of decomposition cannot be postulated in this alloy. In alloys containing 6 and 8% Mo, the tempering treatment results in a phase separation in the orthorhombic -phase (Figure 7.39). The observed composition modulation extends right up to the lath boundaries, suggesting the operation of a spinodal decomposition process within the -martensite. As the concentration
Figure 7.39. Microstructure of the tempered orthorhombic -martensite in Ti–8% Mo showing phase separation (after Davis et al. 1979).
Diffusional Transformations
617
modulation amplifies, the regions rich in Mo transform to the -phase while the regions depleted in Mo transform to the hcp -phase. The continuously changing compositions of the enriched and depleted regions of the -martensite with the progress of tempering are reflected in the rotation of the habit plane of modulation as recorded by Davis et al. (1979). The rotation of modulations continues in response to the changing elastic forces resulting from varying lattice parameters of the depleted and enriched regions of . Finally the appropriate orientation relation between the equilibrium - and -phases is established. The Ti–8% Mo martensite reverts, at least partially, to the parent -phase of the same composition when heated rapidly to temperatures above 798 K. Precipitation of -plates subsequently occurs within the matrix of reverted . Davis et al. (1979) have proposed a free energy–composition relationship in this system which can rationalize the aforementioned experimental observations. Since the transition from hcp to orthorhombic is observed in a continuous manner with increasing solute content, the free energy–composition plot for the - and the -phases can be drawn with a single line with two points of inflection located at c2 and c4 , as shown in Figure 7.40. The point (c3 ) of intersection of G and G curves defines the composition limit up to which martensite formation is feasible at this temperature. Below the composition c1 only the hcp -martensite can exist while between c1 and c3 both - and -martensites can form. At c2 there exists a point of inflection which marks the composition beyond which the orthorhombic -martensite undergoes spinodal decomposition during tempering if the tempering temperature is below the spinodal line.
G (β)
G (α″) Free energy
G (α′)
c1 Ti
c2
c3
c4
Mo concentration →
Figure 7.40. Free energy–Mo concentration diagram at T1 for Ti–Mo alloys showing doubly inflected (dashed line) plot for the orthorhombic -phase. Points of inflection of the G ) plot are located at c2 and c4 (after Davis et al. 1979).
618
Phase Transformations: Titanium and Zirconium Alloys
The tempering reaction in such a situation initiates by developing a homogeneous composition modulation, the amplitude of which increases with the duration of tempering. The regions depleted in Mo gradually enter the composition range where hcp is more stable and, therefore, these regions transform into that phase. The -regions enriched in Mo gradually reach compositions beyond c3 where the high stability of the -phase induces a → transformation. The martensite phase, , can revert back to the parent -phase when the tempering temperature exceeds T0 corresponding to the alloy compositions. 7.3.7 Phase separation in -phase as precursor to precipitation of - and -phases The precipitation of the metastable -phase and the equilibrium -phase takes place subsequent to phase separation in the -phase in those alloy systems in which the tendency for -phase separation is present. The Zr–Nb system can once again be used for illustrating this point. Flewitt (1974) has shown that the supersaturated Nb-rich -phase, having a composition within the coherent spinodal of the Zr–Nb system, on ageing at temperatures below the monotectoid temperature initially undergoes a spinodal decomposition following which a metastable hcp phase (designated as ) precipitates within the regions enriched in Zr. The strain field contrast associated with these -precipitates and the diffuse intensity in diffraction patterns reveal that these precipitates maintain at least partial coherency and a Burgers orientation relation with the matrix. Menon et al. (1978) have considered the possibility of the formation of -phase by a composition-invariant structural transition in the Zr-rich -regions present in a spinodally decomposed -alloy. The computed G −c and G −c plots corresponding to 673 K show that once the -phase is separated into the 1 - and 2 -phases, a composition-invariant transformation from 1 to is thermodynamically feasible, considering only the chemical free energy change. The drop in free energy, G (1 → ) shown in Figure 7.41 corresponds to the composition-invariant 1 → transformation. If a supersaturated -alloy is aged in a region outside the coherent spinodal, the decomposition into a mixture of 1 - and 2 -phases will still occur by a discrete nucleation process and the system will further reduce its energy by the monotectoid decomposition of the 1 -phase into the equilibrium + 2 mixture. The thermodynamic analysis of Menon et al. (1978) has pointed to the following two possible sequences of the transformation of the -phase in the Zr–Nb system: (1) → spinodal decomposition → 2 + 1 ; 1 → + 2 → + 2 . (2) → discrete precipitation → + 1 ; 1 monotectoid → + 2 .
Diffusional Transformations
619 T = 673 K
0
Gβ
β1
U
–100
ΔG β–α
G (cal/mol) →
–200
V
Gα
–300
–400
–500 0
0.05
c cs
0.10
0.15
0.20
c→
Figure 7.41. Free energy–concentration plots for the - and the -phases showing the possibility of the compositionally invariant transformation of the 1 -phase into the -phase, the 1 -phase resulting from a prior spinodal or discrete phase separation in the -phase. The line U represents the common tangent to the 1 - and 2 -arms of the G –c plot and the line V is the common tangent to G and G2 curves.
While rationalizing the relevant observations, Flewitt has suggested two separate G–c plots for the - and the -phases. The computed G–c plots obtained by phase diagram analysis show that the -phase is nothing but the -phase with Nb supersaturation and that its free energy can be expressed on the same G −c plot. At temperatures below 773 K, the formation of the metastable -phase may also intervene in the sequence of phase transformations. The thermodynamics of -precipitation has already been discussed in Chapter 6. The metastable equilibrium (represented by G–c plots in Figure 7.42) which can be established between the 1 -phase and the -phase leads to the appearance of the metastable (1 + ) phase field separating the single phase fields associated with the - and the 1 -phases (Figure 7.42). The G , G and G versus c plots demonstrate that at temperatures much below 773 K where a common tangent between the G curve
Phase Transformations: Titanium and Zirconium Alloys
Free energy
620
cb
Gω
Gβ cc
ca Gα
Concentration
Figure 7.42. Free energy–concentration plots showing metastable equilibrium between the - and the -phase.
and the 1 -arm of the G curve cannot be constructed, a metastable equilibrium represented by a common tangent between the G curve and the 1 -arm of the G curve can be drawn. In this context, it may be noted that three common tangents can be constructed between the G and the G curves, the tangents touching the G curve at three points denoted by ca , cb and cc (cc > cb > ca ), close to the three extrema of the G curve. It can be seen from this construction that there is no driving force for the nucleation of the -phase in the composition range ca < c < cb and, therefore, phase separation in the -phase must proceed to such an extent that the Zr concentration in some regions exceed the limit given by ca . The nucleation of -particles, therefore, can start only after the -phase separation process is nearly complete. The sequences of transformations of the -phase in a monotectoid system such as Zr–Nb, Ti–Nb can then be predicted from the computed G–c plots corresponding to the equilibrium and metastable phases as follows: (1) → 1 + 2 within the miscibility gap at temperatures above Tmono . The reaction initiates in a spinodal mode within the coherent spinodal regime and by a heterogeneous nucleation mode outside this. (2) → 1 +2 → +2 → +2 at temperatures where the -phase is unstable. (3) → 1 +2 → +2 → +2 at temperatures below Tmono where the -phase is metastable. Many of the Ti alloy systems which were earlier known as -isomorphous systems have now been redesignated as -monotectoid systems. Phase separation in the -phase, therefore, has a pronounced influence on the subsequent phase
Diffusional Transformations
621
transformations in these systems. The arguments which are used for predicting and explaining the phase transformation sequences in Zr–Nb alloys can be extended to a number of Ti-based alloys which exhibit a strong phase separation tendency in the -phase field. Two illustrative examples of the Ti–Mo and the Ti–Cr systems are discussed here to bring out the importance of the phase separation tendency of the -phase in dictating the subsequent phase reactions. The current version of the Ti–Mo phase diagram shows a monotectoid reaction (Figure 1.7) at 968 K and the critical temperature of -phase separation is 1117 K at 50 wt% Mo. The corresponding free energy–composition plots for the -phase at temperatures below the critical temperature will, therefore, be doubly inflected, the miscibility gap being defined by the compositions at which the common tangent touches the G plot with the spinodes defining the chemical spinodal line. The calculated coherent binodal (miscibility gap) and coherent spinodal lines (Figure 7.43) have recently been reported by Furuhara et al. (1998). They have observed that the retained -phase in a Ti–40 wt% Mo alloy when aged at 773 K produces uniformly distributed -precipitates within the -grains, while non-uniform grain boundary nucleated (allotriomorphs and side plates) -plates are formed when the ageing temperature is 873 K or higher. They have attributed the uniform nucleation of -plates within the grain to the occurrence of a phase 1200
Chemical spinodal (cal) β + β′
Temperature T (K)
1100
Chemical binodal (cal) β
1000
α
900
800
700 Ti
20
40
60
80
Mo
Mo (mass%)
Figure 7.43. Calculated phase diagram (Furuhara et al. 1998) of the Ti–Mo system showing a monotectoid transformation, phase separation within -phase and chemical spinodal lines.
622
Phase Transformations: Titanium and Zirconium Alloys
separation within the -phase prior to the nucleation of -plates. Though clear evidences in support of -phase separation, spinodal or otherwise, followed by the nucleation of either - or -precipitates within the phase-separated matrix have not been reported in -monotectoid systems of Ti, such a sequence is expected from thermodynamic considerations. The Ti–Cr system, which is known to belong to the class of -eutectoid systems, exhibits a strong tendency for phase separation. Analysis of thermodynamic data by Murray (1987), Kauffman and Nesor (1978), Menon and Aaronson (1986), Prasad and Greer (1993), Sluiter and Turchi (1991), Mebed and Miyazaki (1998) have indicated the presence of a miscibility gap, the location of which has been found to be somewhat different by different investigators. Figure 7.44 shows a superimposition of calculated binodal, chemical spinodal and coherent spinodal on the phase diagram of the binary Ti–Cr system. In case the formation of the equilibrium Laves phases, -, - and -TiCr2 , is suppressed during ageing of an alloy within the coherent spinodal, the alloy is expected to develop concentration modulations. Time-dependent evolution of such modulations by phase field modelling and by
1600
C14
Temperature (K)
1400
C15 β 1200
C36 1
1000
α
α+β
2 800
3 Ti
0.2
0.4
0.6
0.8
Cr
X Cr
Figure 7.44. Equilibrium phase diagram of the Ti–Cr system with superimposed metastable binodal and spinodal lines corresponding to -phase separation. Line 1 shows the metastable equilibrium boundary representing 1 + 2 phase separation when the formation of the intermetallic phases is suppressed. Line 2 indicates the calculated chemical spinodal. Line 3 is the calculated coherent spinodal for concentration modulations along <100> (after Menon and Aaronson 1986).
Diffusional Transformations
623
TEM experimental observations has been demonstrated by Mebed and Miyazaki (1998). The quenched metastable -phase, during ageing at 673 K, decomposes into solute-rich 2 - and solute-lean 1 -phases, with modulations occurring along the elastically soft <100> directions. In the early stages of decomposition, the 1 -regions in this compositionally modulated -matrix become amenable for a transformation either into the - or the -phase while the 2 -phase, which gets richer in the -stabilizing element Cr, remains untransformed. Fine scale modulation being responsible for the nucleation of - or -precipitates, a uniform distribution of these precipitate phases is generated unlike the grain boundary precipitation which occurs in homogenous -grains.
7.4
MASSIVE TRANSFORMATIONS
A massive transformation is a solid–solid phase transformation in which the product phase inherits the composition of the parent phase but unlike the case of a martensitic transformation, the growth of the product phase is controlled by diffusional jumps of atoms from the parent to the product lattice sites across the transformation front. The composition-invariant massive transformation in a two or a multicomponent system can, therefore, be compared with a diffusional polymorphic transformation of a single component system. Needless to say, a transformation of a two-component system in a composition-invariant process cannot take place under the equilibrium cooling condition (except in the case of a congruent transformation). A moderately rapid cooling from the high-temperature phase which provides the necessary supercooling is, therefore, a necessary prerequisite for initiating as well as propagating the transformation. Massive transformations have first been reported in Cu-based alloys by Massalski (1958, 1970). Systematic studies on massive transformation in Ti-based alloys have been carried out by Plichta and co-workers (1977, 1978, 1980) and it is through these studies that some of the special issues connected with massive transformations of Ti-based alloys have been identified. A brief account of these are given here. 7.4.1 Thermodynamics of massive transformations Thermodynamics of massive transformations can be conveniently discussed with reference to a typical eutectoid phase diagram in which the terminal solid solution has an extensive solubility of the solvent (Figure 7.45(a)) and the eutectoid composition, ce , is not far from the point showing the maximum solubility, cs , of the terminal solid solution. The corresponding free energy–composition (G–c) plots for the - and the -phases at a temperature T1 are shown in Figure 7.45(b). The
624
Phase Transformations: Titanium and Zirconium Alloys
To β β+γ
α+β T1
α Cs
T2
Ce α+γ
C1
(a) Gαc T1
Gβ Gβ
A Gα
B
Gα
I J
E C
T2
Gγ
α+γ
D
K
C 3 C 1C 4C o
C5
(b)
(c)
Figure 7.45. (a) Eutectoid phase diagram of a binary system which can exhibit a massive transformation. (b) G–c plots corresponding to Ti for the - and the -phase. Massive transformation at c3 and c4 will be driven by a change in free energy from A to C and B to D, respectively. Gc corresponds to the G–c plot for the -phase in the presence of the capillarity effect. (c) The driving force for the massive transformation at T2 for an alloy with concentration c5 is shown by the drop IJ. The equilibrium condition is denoted by the point K.
free energy curves for the two phases intersect at the composition co . This means that the To temperature for the composition-invariant → transformation for the alloy of composition co is located at T1 . The composition dependence of the To temperature has been schematically shown in Figure 7.45(a) by a dashed line. There are a number of Ti-based binary systems, such as Ti–Si, Ti–Au and Ti–Ag, which exhibit similar phase diagrams and in which massive transformation from the - to the -phase has been reported by Plichta et al. (1978, 1980). The change in free energy due to a composition-invariant → transformation is given by the vertical drop from the G curve to the G curve; this is shown in
Diffusional Transformations
625
Figure 7.45(b) by vertical segments AC and BD for two compositions c3 and c4 , respectively. From the consideration of the change in integral molar free energy, such a transformation from to is possible at T < To for the entire composition range 0 < c < co . This composition range can be divided into two parts, namely, 0 < c < c1 and c1 < c < co , the former conforming to the single phase -region, and the latter in the + region of the phase diagram at the selected transformation temperature T1 . Massive transformations are initiated by diffusional nucleation of the product phase at grain boundaries and/or other inhomogeneities in the parent phase. Homogeneous nucleation of massive product is rather difficult because the massive product, once it nucleates heterogeneously, grows at a very high speed towards complete transformation without allowing the creation of sufficient undercooling for homogeneous nucleation. Let us now consider the nucleation process in the single phase (0 < c < c1 ) and the two-phase (c1 < c < co ) regions of the phase diagram. As discussed earlier, the free energy change during nucleation is given by the vertical distance from the tangent drawn on the G–c curve of the parent phase. Application of this criterion, which can be easily shown by construction of parallel tangents on the G and G curves, demonstrates that the composition of the critical nucleus cannot be the same as that of the matrix. For example, free energy of the alloy having a solute concentration, c3 , in the -phase at the temperature T1 is given by the point A on the G curve. E is the point at which a tangent to the G curve can be drawn parallel to the tangent at point A on the G curve. Therefore the free energy change for nucleation is maximum when the -nuclei have a composition given by the point E which is different from c3 , the composition of the parent -phase. Therefore, nuclei of composition indicated by point E are expected to form predominantly. During the growth process due to the kinetic consideration the composition shifts from E to C to establish the composition invariance criterion. In case the free energy composition plot for the -phase is raised due to the capillarity effect to occupy the Gc curve (drawn with dashed line), it is possible to visualize a situation where a composition-invariant nucleation can occur. The product -phase exhibits a tendency for further transformation. It may be noted that in the composition range, c1 < c < co , the product -phase at T1 does not remain in equilibrium and exhibits a tendency for decomposition into a mixture of the - and -phases. At a temperature, T2 , → massive transformation can occur if the formation of the equilibrium -phase can be suppressed. The drop in free energy for an alloy with composition c5 is shown by the segment IJ in Figure 7.45(c); the free energy corresponding to the equilibrium + structure is indicated by the point K on the tangent common to G and G plots.
626
Phase Transformations: Titanium and Zirconium Alloys
7.4.2 Massive transformations in Ti alloys The study of a large number of binary Ti-based eutectoid systems (Plichta et al. 1978, 1980) has revealed that only in Ti–Ag, Ti–Au and Ti–Si the → m massive transformation occurs. In other systems, such as Ti–Fe, it is difficult to suppress the formation of the equilibrium -phase. A limited solubility of the solute elements in the -phase and a sharp drop in the To temperature with alloy addition are the factors responsible for the non-occurrence of massive transformation in such systems. Plichta et al. (1978, 1980) have shown that the morphology of the -quenched products in the composition ranges, Ti–4.7 to 13.5 at.% Ag, Ti–1.8 to 4.4 at.% Au and Ti–0.68 to 1.10 at.% Si can be easily recognized as the massive m -phase based on the following features: (1) Irregular grain shapes with jagged boundaries, as shown in Figure 7.46(a) characteristic of a massive transformation product, are observed. In cases where m is coexistent with martensitic (Figure 7.46(b)), they can be distinguished on the basis of their morphologies. (2) Energy dispersive analysis has established that the m -grains inherit the chemical composition of the parent -phase. The retention of the solutes in excess of their solubility limits has also been demonstrated by second phase precipitation within the m -regions during subsequent ageing treatments. (3) Unlike the martensitic , the m -phase does not exhibit any specific habit plane with respect to the parent -phase. The substructure, though containing a high density of dislocations, does not show any similarity with either dislocated lath on internally twinned plate martensites. The alloy composition ranges over which massive transformation has been observed are primarily in the hypoeutectoid regions of the respective phase diagrams, though in a few cases the massive transformation could be induced even in hypereutectoid compositions (e.g. Ti–13.5 at.% Ag). This is possible in systems in which the eutectoid composition is not far away from the maximum solubility limit of the equilibrium -phase (as shown in Figure 7.45(a). Thermal analysis experiments using continuous cooling have provided valuable information regarding the thermal arrest temperature as a function of the cooling rate, the enthalpy of transformation and the growth rate of massive m -phase. A typical cooling curve is shown in Figure 7.46(c) which shows the arrest temperature and the duration over which the massive transformation takes place. The growth rate, G, of a massive transformation product can be expressed approximately as G d/tg
(7.50)
Diffusional Transformations
50 μm
627
100 μm
(a)
(b) Ti – 17.5 w/o Ag 1 – 120°C/s 2 – 355°C/s 3 – 530°C/s 4 – 120°C/s 5 – 2420°C/s
Temperature (°C)
1000 900 tg
800
1 700 600 5 0
3
4 500
2
1000
1500
Time (ms)
(c) Figure 7.46. (a) Light micrograph showing irregular grain structure of the massive transformation product in the Ti–17.5 at.% Ag alloy. (b) Light micrograph showing a mixture of massive and martensitic product in the alloy. (c) Cooling curves exhibiting thermal arrest due to the massive transformation at different cooling rates in Ti–17.5 at.% Ag. (after Plichta et al. 1978, 1980).
where d is the grain diameter of the product m -grains and tg is the duration of thermal arrest, as shown in Figure 7.45(c). Although m -grains are very irregular in shape, they can be better approximated as equiaxed than as acicular for the measurement of d. The thermal arrest temperature versus cooling rate data obtained for some of the Ti–Au and Ti–Si alloys have shown an initial decrease in temperature with increasing cooling rate followed by a Ms plateau. Microstructural examination of the samples cooled at different rates has confirmed that martensitic transformation becomes operative when the critical cooling rate is exceeded.
628
Phase Transformations: Titanium and Zirconium Alloys
An estimate for enthalpy of transformation, HT ( → m ), for massive transformation can be made using the fact that the rate of heat generation, a = dq/dt, which matches the rate of heat extraction during the thermal arrest is proportional to the rate of transformation. This can be written as HT → m
dT df = Q = Cs s dt dt
(7.51)
where Cs and Ts are the heat capacity and temperature of the specimen, Cs being dependent on the volume fractions, X and Xm , of the and the m -phases at any instant of the transformation: (7.52) Cs = X C + Xm Cm = 1 − Xm C + Xm Cm Based on the thermal analysis experiments, Plichta et al. (1978) have determined thermodynamic data on massive transformations in Ti alloys. Table 7.4 lists the values of enthalpy and free energy changes and To temperatures for a few representative alloys. The experimentally measured growth rate, as a function of undercooling below To , for different alloys has yielded the activation enthalpy of the growth process. Burke and Turnbull (1952) have expressed the growth velocity, G, for an interfacecontrolled reaction as SDb F → m HDb kT exp − exp − (7.53) G= h R RT RT where is the width of the boundary, k is Boltzmann’s constant, h is Planck’s constant, SDb and HDb are the activation entropy and enthalpy for diffusion across the transformation front and F ( → m ) is the free energy change accompanying the → m transformation. A plot of log (−GT/F ( → m ))versus reciprocal of the absolute temperature therefore yields the value of HDb which has been found to lie between 50 and 93 kJ/mol. Table 7.4. Values of To , HT ( → m ) and G( → m ) for representative Ti alloys. Alloy composition (at.%) Ti–6.5% Ag Ti–2.6% Au Ti–1.1% Si Ti–47.55% Al
To
HT ( → m ) (J/mol)
G( → m
1136 1128 1140 1325
−2870 ± 460 −3035 ± 420 −2470 ± 500 −3712
−2870 + 255T 3035 + 272T 2470 + 218T −3712 + 231T
Diffusional Transformations
629
Massive transformations in alloys (except in cases of equilibrium congruent transformations) result in the formation of metastable phases. Therefore, for a massive transformation to occur, it must compete successfully with other reactions leading to equilibrium products. Usually the massive products nucleate at a much slower rate than the equilibrium precipitate phase(s). However, the growth kinetics of the massive transformation are usually several orders of magnitude faster than those of precipitation reactions involving solute partitioning. This is primarily due to the fact that the growth of a massive product occurs by diffusive atom movements only across the transformation front and does not involve long-range volume diffusion. Plichta et al. (1978) have reported the measured growth rates of m in Ti alloys which are in the range of 15 × 10−5 –2 × 10−4 m/s. These values are two to three orders of magnitude higher than the estimated growth velocities of the equilibrium -precipitates evolving from the parent -phase. Massive transformation has recently been studied in detail in titanium aluminides. Wang et al. (1992), Wang and Vasudevan (1992) and Veeraraghavan et al. (1999) have shown that in binary Ti–Al alloys with Al content between 46.5 and 48 at.% the massive – transformation can be induced by rapid cooling. Figure 7.47(a) and (b) shows the transformation products of the -phase of Ti–46.5 at.% Al alloy obtained by furnace cooling and water quenching. The continuous cooling transformation diagram of this alloy as shown in Figure 7.47(c) depicts that a cooling rate exceeding 300 C/s is required to suppress the lamellar transformation product completely. The experimentally measured S , has been found to be weakly dependent on the massive start temperature, M cooling rate. Some of the open questions regarding the mechanism of massive transformations have been addressed in the recent work on Ti–Al massive transformation. The first and foremost question is whether a strict orientation relationship exists between the parent and the product crystals during the nucleation and the growth stages of massive transformation. Recent results from TEM and orientation imaging microscopy experiments (Wang et al. 2002) have clearly shown that in a colony of the product m -crystals, though some of the m -crystals maintain the / orientation relation: (0001) (111) observed in lamellar product (discussed in Chapter 5), many of them do not have low index orientation relations with the parent . Presence of twin relation between m -crystals within a “colony”, however, is frequently encountered. By arresting the transformation at early stages it has been established that the nucleation step at grain boundaries and grain corners invariably involves formation of M -crystals which maintain orientation relationship with one of the contacting -grain but the growth of m -crystals in the orientation-related -grain remains considerably restricted. In contrast, the incoherent boundary rapidly propagates into the grain with which no low index
630
Phase Transformations: Titanium and Zirconium Alloys
(a)
(b)
1500 To
Temperature (°C)
1400 1300
To
γLs γMs
1200 1100
TE 100°C/s
1000
200°C/s 1000°C/s
900 800 0.1
300°C/s 500°C/s 400°C/s 1
10
Time (s) (c)
Figure 7.47. (a) Light and (b) TEM micrographs showing the massive product m in the 2 -matrix in the Ti–46.5 at.% Al alloy sample quenched from the single -phase field. (c) shows continuous cooling transformation diagram which depicts the onset of the → lamellar L (Ls ) and s of the → massive M (M ) transformations under different rates of cooling in the same alloy (after Wang et al. 1992, Veeraraghavan et al. 1999).
orientation relation exists. Structurally incoherent interphase boundaries which often exhibit faceting are the interfaces which propagate at the fastest rate in the parent -grain to accomplish → m massive transformation. The absence of /m orientation relation during the growth stage is further evidenced from the observation that an interphase boundary of a growing m -grain can advance into more than one parent grain without any change in orientation. Micrographs capturing several features of the nucleation, the growth and the interface structure associated with the → m transformation are shown in Figure 7.48.
Diffusional Transformations
(a)
631
(b)
(c)
(d)
Figure 7.48. (a) Light micrograph showing m -crystals formed at the grain boundary of (subsequently transformed into 2 ) in the Ti–46.5 at.% Al alloy sample in which the transformation is arrested by a special heat treatment. (b) A colony of m -crystals, mutually twin related (90 and 150 ), is nucleated at an -grain boundary in Ti–48 at.% Al–2 at.% Cr. (c) The growing m /2 interface boundaries are free of dislocations and other defects like misfit compensating ledges. (d) The m -crystal which bears usual orientation relation with 2 (I) but no low index orientation relation with 2 (II) grows into the 2 (II) grain (after Wang et al. 1992).
632
7.5
Phase Transformations: Titanium and Zirconium Alloys
PRECIPITATION OF -PHASE IN -MATRIX
A great majority of the + alloys of Ti and Zr are mechanically processed in the + phase field and are heat treated in such a manner that the -phase is precipitated in the -matrix. Such a solid state precipitation reaction often produces the -phase precipitates in the form of plates which maintain strict orientation relation and habit plane with the -matrix. A comparison can be drawn between these plate-shaped precipitates and the proeutectoid ferrite plates which form in steels. A considerable amount of attention has been paid to the mechanism of formation of plate-shaped precipitates of the product phase where the transformation takes place through a thermally activated diffusional process. The presence of features like orientation relationship, habit plane and sometimes surface relief suggests that a lattice shear mechanism takes part in the overall mechanism leading to the formation of such -plates. This viewpoint has been questioned and a controversy has persisted in the literature for several decades. In recent years, attempts have been made to resolve this controversy and, in the opinion of the present authors, evidences collected in this regard in Ti- and Zr-based alloys can indeed play a major role in clarifying many of the contentious issues. These alloys are particularly suitable for making a comparison between martensitic and diffusional transformations in view of the fact that the “lattice correspondence” and “atomic site correspondence” between the parent - and the product -phases are remarkably similar for both martensitic and diffusional transformations. Crystallographic features such as orientation relations, habit planes and interface structures associated with these transformations can be compared with an aim of examining whether the transformation mechanisms have characteristic imprints on these experimental observables. The feature which is often used to distinguish a martensitic from a diffusional transformation is the presence of a surface relief effect in the former. The origin of such surface relief is believed to be related to the invariant plane strain associated with the martensitic transformation (Bilby and Christian 1956). In contrast to this, Liu and Aaronson (1970) have presented experimental evidence that the formation of hcp (Ag2 Al) precipitate plates in Al–Ag alloys in a typical diffusional transformation is accompanied by the appearance of surface reliefs. In some recent articles, it has been recognized that diffusional transformations can exhibit surface relief despite the fact that lattice correspondence does not exist between the parent and the product lattices in such transformations. The presence of atomic site correspondence across the transformation front in diffusional transformations can preserve the shape deformation and produce a surface relief effect. The structure of the interface between the parent and product phases therefore attracts special attention for examining the presence of atomic site correspondence in diffusional
Diffusional Transformations
633
transformations. These aspects of -phase precipitation in the -phase matrix in Ti- and Zr-based alloys are discussed in this section. 7.5.1 Morphology The principal morphologies of -plates in the -matrix can be classified according to the Dube et al. (1958) morphological classification scheme, originally introduced with regard to the formation of proeutectoid ferrite plates in steels. Morphological descriptions, crystallographic and interfacial features and formation sequences of each of these morphological types are detailed in the following. Grain boundary allotriomorphs (GBA) are the plates which form along the highangle grain boundaries of the parent -phase. Usually these are the first plates to appear in the course of -precipitation (Figure 7.49). Allotriomorphs nucleate at and grow preferentially along grain boundaries in a manner similar to the “wetting” of the grain boundary surface by the emerging phase. The interfacial energies between the -plate and the two adjacent -grains must be low for making the nucleation kinetics of GBAs favourable. This is achieved by establishing the Burgers orientation relationship between the -plate with one of the -grains while the boundary with the other grain is usually irrational. Furuhara et al. (1988) have shown in a Ti–6.6 at.% Cr alloy that the Burgers-related interface maintains partial coherency as reflected in the presence of a periodic array of structural ledges with a uniform spacing of approximately 8 nm and a height of 2–3 nm. The schematic of a Burgers-related bcc/hcp interface, ¯ plane, contains one biatomic structural ledge. projected on to the (0001) (011) ¯ (21¯ 1) ¯ The terrace on which there is a close atomic fit is parallel to the (1100) plane and the structural ledge is associated with a Burgers vector of a/12 [111] (details discussed in Section 7.5.4). The interface of the -allotriomorph which is not related to the adjacent -grain by a Burgers relation shows a set of widely spaced ledges, relatively high (approximately 8 nm), following an irregular path and variably spaced. Two finer sets of linear defects, both uniformly spaced, are also observed and these have been identified as misfit dislocations. This observation suggests that the non-Burgersrelated / orientation also tends to maintain coherent facets which are separated by misfit correcting ledges. Furuhara and Maki (2001) have shown that morphologically indistinguishable -precipitates along a relatively straight prior -grain boundary belong to the same crystallographic variant. The selection of variant is made in such a manner that ¯ 111 direction remains nearly parallel to the grain boundary plane. the 1120 This is in agreement with the proposition that, in general, the low-energy facets ¯ 112 like the 0001 110 and 1010 make the least possible angle with the grain boundary plane.
634
Phase Transformations: Titanium and Zirconium Alloys
(A)
(B)
Figure 7.49. (A) Grain boundary allotriomorph of -phase along -grain boundaries. (B) Two orientations of -crystals appearing alternately along a -grain boundary. Orientations of the two -crystals, their basal planes being parallel are rotated with respect to one another by 10.5 around the normal to the basal plane (after Bhattacharyya et al. 2003).
All these conditions put quite stringent restrictions on the possible -variants that can be precipitated at -grain boundaries. The conclusions arrived at in these investigations are based on TEM observations made on a scale of a several micrometres. In a recent work, Banerjee et al. (2003) have extended the scale of observation by studying the orientation distribution of grain boundary allotriomorphs (-plates) along a grain boundary extending over a distance of about 10 mm in a compositionally graded Ti–8Al–XV sample prepared by the laser deposition technique. The strongly columnar growth morphology of -grains with a composition gradient has allowed in this case the study of -precipitation
Diffusional Transformations
635
along a long length of a grain boundary. Some of the important observations are summarized here: (1) Most of the grain boundary -precipitates exhibit Burgers orientation relation with one of the -grains (referred to as 1 -grain). (2) It is frequently observed that alternate -precipitates belonging to two different crystallographic variants which share the same (0001) plane are lying parallel to the same 011 plane of the I -grain. These two variants differ in their ¯ directions which are parallel to two different 111 directions on the 1120 same 011 plane of 1 . These two orientations shown in Figure 7.49 are related to each other by a rigid body rotation of nearly 11 around the axis perpendicular to (0001) 110 plane. (3) Out of the six possible 110 planes, the basal plane of the grain boundary -precipitate preferentially chooses the 110 plane which is closest to the grain boundary plane. (4) Grain boundary -precipitates exhibit a tendency towards maintaining the minimum possible misorientation from the Burgers orientation relation with the adjoining 2 -grain by the selection of a suitable orientation variant. In some rare instances, where the grain boundary plane permits, Burgers orientation relation is established with both the adjoining grains. Such a precipitate geometry can be compared with those of -precipitates along internal twins of -martensite plates (Section 7.3.5). (5) Occasionally -precipitates are encountered along the -grain boundaries which do not obey the Burgers orientation relation with either the 1 - or the 2 -grains. Figure 7.49 shows the distribution of -orientation along a -grain boundary as imaged by orientation imaging microscopy; the orientation of the two -grains, 1 and 2 , and that of the -precipitates are indicated by the pole figures placed alongside. Widmanstatten side plates form either by nucleating at grain boundaries or by branching out from GBAs as shown in Figure 7.50. The plates which nucleate at grain boundaries are designated as primary side plates while those created by branching of GBAs are known as secondary side plates. Side plates are often found to grow in a group resulting in the formation of a colony of parallel -plates. Unnikrishnan et al. (1978) have shown in a Ti–6 wt% Cr alloy that the growth of a group of -side plates can be treated in terms of the colony growth kinetics which is usually applicable to cellular transformations. However, important differences exist between the growth of a group of side plates and that of a colony of cellular reaction products (lamellar eutectoid or cellular precipitates) and these are listed below and are illustrated in Figure 7.51:
636
Phase Transformations: Titanium and Zirconium Alloys
(a)
(b)
Figure 7.50. Widmanstatten side plates: (a) bright- and dark-field TEM images of primary plates nucleating from the grain boundaries. (b) SEM images of secondary side plates formed by branching of grain boundary allotriomorphs.
βb
βb
βa
βa αb
αa
(a)
(b)
Figure 7.51. Schematic drawing for making a comparison between (a) growth of a colony of side plates and (b) growth of a nodule consisting of several plates in cellular precipitation.
Diffusional Transformations
637
(1) The growing tips of a group of side plates do not push the grain boundary as they propagate into a parent grain unlike in the case of a cellular transformation. Side plates maintain a fairly strict orientation relation with the grain in which they grow and the parent phase retained in the intervening space between adjacent -plates has the orientation of the untransformed -phase lying ahead of the transformation front. (2) The solute partitioning between the - and the -phases in the product takes place through a lattice diffusion process. A group of side plates which have the same orientation relation and habit plane grow by simultaneous movements of their tips as the diffusion field through the -phase ahead of the transformation front forces them to do so. The growth pattern is also consistent with the edgewise growth of plates by the movement of growth ledges at the tips of the plates. Intragranular plates Figure 7.52 constitute the third morphological variety of -plates and nucleate in the interior of the -grains. These plates also obey fairly strict orientation relations. Intragranular plates can form either in an isolated manner or in a group. The former continuously partition the prior -grain into smaller and smaller volumes with fresh generations of plates appearing. The selfsimilarity of the structure in decreasing scale is consistent with a fractal description (Figure 7.52). The formation and growth of a group of parallel intragranular plates which remain stacked in a parallel fashion within a given packet give rise to the “basket weave” morphology (Figure 7.53). A variation in the packet size of this structure, which can be induced by changing the alloy composition and the
10 μm
Figure 7.52. Fractal morphology of intragranular -plates which continuously partition the parent -grains causing a reduction in the size of the plates with every successive generation.
638
Phase Transformations: Titanium and Zirconium Alloys
Figure 7.53. Basket weave morphology of intragranular -plates which appear in a parallely stacked group within a colony.
extent of supercooling, is responsible for bringing about a significant change in the appearance of the microstructure in the optical microscope scale (typically for magnifications ranging from 50× to 500×). A detailed study of the morphology of the intragranular -plates in Ti–Cr alloys by Menon and Aaronson (1986) has shown that these plates can be classified into two types: “normal plates” which form at temperatures T > 873 K and “black plates” forming at temperatures T < 873 K. While normal plates exhibit not so perfect habit planes, the slender black plates are nearly perfect. The latter is named so for their dark etched appearance in optical microscopic investigations. The formation of two distinct morphologies of -plates in Ti–Cr alloys has been rationalized in terms of the G–c diagram of the system. Menon and Aaronson (1986) have shown that the -phase in the Ti–Cr system is associated with a strong clustering tendency which is reflected in the miscibility gap and a monotectoid reaction in the metastable phase diagram (Figure 7.44) when the formation of equilibrium intermetallic phases is suppressed. The precipitation of the -plates, far above the eutectoid temperature, is governed by the equilibrium set up between and 1 (Figure 7.54(a)), while at temperature below the eutectoid, -plates remain in equilibrium with the 2 matrix (Figure 7.54(c)). At temperatures close to the eutectoid, as shown in Figure 7.54(b), a metastable equilibrium between and 2 is possible in addition to the stable /1 equilibrium. The difference in morphology of normal plates and black plates, therefore, arises due to a significant difference in the lattice parameters of 1 - and 2 -phases which remain in contact with the growing -plates. As shown in the G–c plot (Figure 7.54(c)) corresponding to T = 850 K, a growing -plate can establish metastable equilibrium with the Cr-rich 2 -phase and not with the Ti-rich 1 -phase. It is, therefore, expected that during
Diffusional Transformations
639
Gα –500
T = 1000 K
G (J/mol)
α–β1
–1000 Gβ
–1500 Ti
0.2
0.4
0.6
0.8
Cr
Ccr
0
G (J/mol)
T = 950 K Gα
–500
Gβ α+β2
–1000
α+β1 Ti
0.2
0.4
0.6
0.8
G (J/mol)
Gβ
–500
–1000
Cr
Ccr
0
T = 850 K
α+β2 α+β 2
Gα Ti
0.2
0.4
0.6
0.8
Cr
Ccr
Figure 7.54. G–c plot for the Ti–Cr system at (a) 1000, (b) 950 and (c) 850 K.
the growth of -plates at T < 873 K a rim of Cr-rich -phase surrounds the growing plate even in alloys which are Ti-rich (as in the cited case for Ti-6.6 at.% Cr). Similar observations have been recorded in Zr–7 at.% Nb and Zr–10 at.% Nb (Banerjee et al. 1988) where -precipitates forming in the -matrix during isothermal decomposition establish local equilibrium with the 1 - and 2 -phases at 974 and 823 K, respectively. In the latter case, an enveloping rim of the 2 -phase has been observed around the -precipitates.
640
Phase Transformations: Titanium and Zirconium Alloys Table 7.5. Microstructure evolutions dictated by stable and metastable equilibrium in the Zr–Nb and the Ti–Cr systems. Transformation process
Sequence
Operating equilibrium
→ + 1 (800 K < T < 893 K) → + 2 (T < 800 K )
/1 Metastable
Phase separation– precipitation in -Zr-Nb
→ 1 + 2 → 1 + 2 → + 2
1 /2 1 / 2 /
-precipitation in -TiCr
→ + n (T > 873 K)
1 / Stable
→ + b T < 873 K
2 / Local metastable
Tempering of martensitic in Zr–Nb
/2 Stable
n : normal plates (); b : black plates ().
A comparison can be drawn between the precipitation of -phase from supersaturated Zr–Nb martensite (Section 7.3.5) and the precipitation of -plates from the -phase in the Zr–Nb and the Ti–Cr alloys. In all these cases, a competition between /1 and /2 equilibrium dictates the course of the transformation process. Table 7.5 summarizes key observations on the formation of equilibrium and metastable precipitates in the Zr–Nb and the Ti–Cr systems which can be rationalized in terms of the stable and metastable equilibrium between the -phase on one side and the 1 - and 2 -phases on the other. All the morphologies of -precipitates forming in the -matrix discussed so far are produced when the → transformation occurs directly and not mediated through the -phase. Unnikrishnan et al. (1978) have shown that two distinct morphologies of -precipitates appear in the alloy of the same composition (Ti–Cr) when it is subjected to the following heat treatment sequences: (1) -solutionization is followed by rapid cooling to the isothermal reaction temperature (973 K) where the -precipitation is allowed to proceed. (2) -solutionization is followed by water quenching to retain the -phase at room temperature and subsequently ageing at the same reaction temperature (973 K). Figure 7.55 (a) and (b) show the product -precipitates forming during the heat treatments (1) and (2), respectively. The difference in the morphologies of the -products has been rationalized (Unnikrishnan et al. 1978) in terms of heterogeneous nucleation of -precipitates on fine -particles present in the -quenched alloy subjected to the heat treatment (2). In a recent study by Ohmori et al. (1998),
Diffusional Transformations
60 μm
(a)
25 μm
(b)
60 μm
(c)
(d)
641
50 μm
(e)
50 μm
Figure 7.55. (a) and (b) Group of parallel Widmanstatten -plates growing from -grain boundaries during isothermal transformation of in Ti–6% Cr at 725 C. These -plates have nucleated directly from the -matrix (compare schematic drawing of Figure 7.51). (c)–(e) Group of -plates of different variants nucleated from -particles have formed within the -grain in Ti–6% Cr in the -quenched, followed by ageing at 725 C. (d) and (e) show -particles and -plates imaged with respective reflections in the dark field. (a), (b) and (c) are optical micrographs while (d) and (e) are TEM micrographs.
it has been demonstrated that -laths nucleate at / interfaces (Figure 7.56) and during their growth consume the -particles. The // orientation relationship have been established to be ¯ 211 ¯ 1120 ¯ ¯ ¯ ¯ ¯ ¯ 1010 1100 0001 011 0001 1210 111 ¯ (1010) ¯ , which is the The terrace plane between - and -phases is (1100) minimum misfit low index plane between the two phases.
642
Phase Transformations: Titanium and Zirconium Alloys
Figure 7.56. High-resolution micrograph showing the // orientation relation.
7.5.2 Orientation relation The orientation relation between the parent and the product -phases in a diffusional transformation has been studied in a number of Ti ( Furuhara et al. 1988) and Zr (Banerjee et al. 1988, Perovic and Weatherly 1989) alloys. The Burgers orientation relation has been found to be operative approximately in almost all cases. More accurate measurements reported in recent papers, however, have shown that a small deviation from the Burgers relation occurs and that this deviation is important in the context of the operation of the invariant line strain condition in diffusional transformations in these systems. As discussed in Chapter 4, the lattice correspondence operative in the → martensitic transformation of Ti- and Zr-based alloys is given by the following crystallographic relation: ¯ 21¯ 10 ¯ 110 0001 110 ¯ 0110 ¯ 001 This lattice correspondence, illustrated in Figure 4.20, is known as the Pitsch– Schrader relationship (Pitsch and Schrader 1958). The Burgers orientation relationship (Burgers 1934) can be obtained from this by rotating the hcp crystal by 5.26 about ¯ 10] ¯ the [0001] direction clockwise (or counterclockwise) in order to bring the [12
¯ direction (or the [1120] ¯ direction nearly in coincidence (within 1.5 ) with the [111]
Diffusional Transformations
643
direction to within 1.5 of the [111] direction). This rotation, as has been shown in Chapter 5, establishes the IPS condition when the 2% Bain strain along the direction perpendicular to the basal plane is neglected. The Burgers orientation relation which ¯ nearly (within 1.5 )[12 ¯ 10] ¯ also is usually described as (110) (0001) ; [111] ¯ plane to be appropriately parallel to (1010) ¯ plane. The importance brings the (112) of these two planes being parallel can be seen in Section 7.5.4 where the atomic matching between the two phases along these planes is considered. In some recent experiments on diffusional phase transformations in Ti- and Zr-based alloys, an accurate determination of the / orientation relation has yielded the following results:
¯ ¯ ¯ ¯ [101] [1011] ; [111] [1210]; [110] inclined to [0001] by ∼ 1.5 .
This Potter (1973) orientation relationship can be established only if the small angle between the [110] and [0001] directions is precisely determined. Since a careful determination of the angle between the directions [0001] and [110] has not been attempted in a great majority of earlier studies, it is difficult to assess in how many experimentally determined cases the reported orientation relationship truly corresponds to the Burgers or the Potter type, the latter being predicted from the condition of ILS. 7.5.3 Invariant line strain condition The morphological development of precipitate plates or laths in a diffusional transformation can be rationalized in terms of the hypothesis proposed by Dahmen and co-workers (1982, 1984, 1986). They have shown that the product phase in many diffusional transformations grows as a lath or a needle parallel to a vector, known as an invariant line, which remains unchanged in length and direction during the course of the phase transformation. If the overall transformation is described in terms of a linear homogeneous deformation, A, which is composed of a pure lattice (Bain) deformation, B, and a rigid body rotation, R , then A can be expressed as A = R · B Using the Pitsch–Schrader correspondence for the diffusional / transformation (which is the same as in the case of the martensitic transformation), B can be expressed as 1 1 1 B = 1 2 1 (7.54) 1 1 3
644
Phase Transformations: Titanium and Zirconium Alloys
where 1 = 3/2 1/2 a/a ; 2 = a/a ; and 3 = 21 c/a , the lattice parameters of the - and the -phases being given by a , c and a , respectively. The matrix representation of the rigid body rotation R , which brings back the undistorted vectors to their original positions, is given by 0997894 0002106 0064824 R = 0002106 0997894 −0064824 (7.55) −0064824 0064824 0995789 The condition for a vector, X, in the parent lattice to remain invariant on the application of the homogeneous strain, A, is AX = X Equation (7.56) is satisfied if det A − I = 0 and one of the eigen values is equal to 1. The predicted invariant line is extremely sensitive to the lattice parameters ratios, a /a and c /a . Substituting the values of the lattice parameters of the - and the -phases, Perovic and Weatherly (1989) have shown that the invariant line for the / transformation in the Zr–2.5 wt% Nb alloy lies very close ¯ direction, which is located at the intersection of the trace of (101) to the [212] ¯ (1011) and the cone of unextended vectors generated by the lattice strain, B. This is shown in the stereographic projection in Figure 7.57 in which the elliptical cone of unextended vectors is represented by a thick line. The rigid body rotation, R , can be decomposed into two rotations, the first one corresponding to that involved in rotating from the Pitsch–Schrader to the Burgers orientation while the second one rotates the Burgers to the Potter orientation. These two rotations are marked schematically in Figure 7.57. Dahmen and co-workers (1982, 1984, 1986) have suggested that laths or needles of the product phase in a diffusional transformation grow along the invariant line of the transformation with the orientation relationship being determined by the restrictions imposed by the invariant line criterion. In transformations where the product phase has a plate morphology with a well-developed habit plane, the habit plane must contain the invariant line while the selection of the other vector for defining this plane is made on the basis of minimization of the interfacial energy. It is attractive to consider a few interesting parallels that can be drawn between the habit planes of products of diffusion controlled and martensitic transformations. In the case of martensitic products, the habit plane is an invariant plane, i.e. all the directions lying on the plane are invariant as far as the macroscopic “average” habit plane and the total shape strain within the plate as a whole are concerned. In the microscopic scale, the habit plane of a martensitic plate will contain either
Diffusional Transformations
645
100
X2 [110]//[0110]
110 110 212 111
111
101
001 011 [001]//[2110]
010
101
011
010
111
111
110
110
100
X1 [110]//[0001]
Figure 7.57. Stereographic projection showing the direction of invariant line strain [2¯ 1 2] .
a row of dislocations marking the directions along which the lattice invariant slip planes intersect the habit plane in a dislocated martensite or a row of lines along which the twin planes meet the habit plane in a twinned martensite. In the latter case, the macroscopic habit is made up of zig zag segments which meet along the invariant line. In contrast, the habit plane in a diffusional transformation does not, in general, satisfy the invariant plane strain condition though at least one vector along the habit plane remains invariant. It is along this direction that laths or plates resulting from a diffusional transformation grow and this direction is often marked on the habit plane by a row of nearly equispaced dislocations, the line vectors of which lie parallel to the invariant line. The interface structure of -laths in the -matrix has been studied in Zr–Nb alloys by Perovic and Weatherly (1989) and Banerjee et al. (1997). In the Zr– 2.5 wt% Nb alloy, in which the volume fraction of the -phase is rather small, ¯ both studies have shown the presence of arrays of parallel 1/3 <1123> (or ) dislocations at the interfaces (Figure 7.58). These dislocations, which are exceptionally straight and maintain a spacing of about 6–8 nm depending on the ¯ alloy composition, have been found to lie along the common (101) (1011) plane.
646
Phase Transformations: Titanium and Zirconium Alloys (a)
α
β
100 nm
Figure 7.58. Interfacial dislocations at the / boundaries in the Zr–2.5 Nb alloy.
Such dislocations can glide as the interface migrates. In addition to these, some ¯ 1/3 <1120> (or ) dislocations have been observed to lie both parallel to and across the dislocations. The density of the dislocations has been found to increase with increasing rotation of the / interface from the flat facets where only one set of dislocations is present. Because of the irregularity of the / surface in the Zr–2.5 wt% Nb alloy an accurate determination of the habit plane has not been possible. However, the line vectors of the dislocations have been identified to be the direction of the ILS. In a Zr–20 wt% Nb alloy, -laths distributed in the -matrix have been found to be quite amenable for the determination of the habit plane and also for the characterization of the dislocation structure at the / interface. Laths of the -phase formed in this alloy on isothermal treatment at 823 K are typically 100 × 200 nm in cross-section, with the length varying from 200 to 1000 nm. Several variants of -laths are often encountered in a single field of view; a typical example is depicted in Figure 7.59(a). Orientations of all the variants match the Burgers relation quite closely. The / interfaces of these laths have invariably been found to contain arrays of parallel, equispaced dislocations of type (Figure 7.59(b)). The spacing between adjacent dislocations varies from 8 to 10 nm. The habit plane of these laths, as defined by the plane containing the length and the width directions (the plane perpendicular to the thickness direction), has been found to lie between the {103} and {113} poles (Figure 7.58(c)) and the line vectors of the dislocations at the interface are along the <334> directions which match closely with the invariant line. The fact that the line vectors of such / interfacial dislocations are parallel to the long direction of the -laths
Diffusional Transformations
647
(b)
(a)
β
α 100 nm
100 nm
112
111
001 β
HC
110
(001) (0001)C HA
(211)//(1010) DA
122 LA
111
130
(110)//(0001)
131
(111) //(1120)
LC 212
HC trace
HC
LB
221
DC
HA trace
311
(121)//(1010)C β
HA DB
110
013 HB trace HC
001
113 111
(112)//(1100)
(c)
Figure 7.59. (a) Several variants of -precipitate laths in the -matrix. All variants exhibit arrays of equispaced dislocations lined up along the length of the precipitates. (b) Interfacial (/) dislocations, with line vectors parallel to the long direction of the precipitates. (c) Stereogram showing that three variants of orientation relationship are operative for three habit variants of precipitates shown in (a).
is demonstrated in the electron micrograph in Figure 7.60(a) in which two -laths, designated as A and B, are shown. The / interface at the side face of these laths is seen to be parallel to the foil plane examined in TEM. For the lath A this face is not retained within the sample sectioned, whereas this face is retained
648
Phase Transformations: Titanium and Zirconium Alloys (a)
(b)
B A
200 nm
100 nm
Figure 7.60. (a) Interfacial dislocations lying parallel to the foil plane on the / interface of plate B, the line vector of dislocation lines being parallel to the invariant line. (b) Equispaced dislocations at / interfaces lying along the long direction of this precipitate plate.
within the foil for the lath B. In the latter case dislocations are seen along the long direction of these laths and their line vector remains parallel to the invariant line. The foregoing observations on the morphology, orientation relation and crystallography of -laths forming from the -phase in a diffusional process can be summarized as follows: (1) The orientation relation between the - and the -crystals is very close to the Burgers relation. (2) The habit plane as defined by the plane containing the length, l, and the breadth, b, directions is an irrational plane. Habit plane poles of different variants are found to lie between the {103} and {131} poles. For all the variants of laths, ¯ {112} . habit planes remain close to the / conjugate pair {0110} (3) Interfacial dislocations with Burgers vector remain aligned along the invariant line and are arranged in a parallel array with a spacing of about 8–10 nm (Figure 7.60(b)). (4) The long direction of laths matches closely the invariant line direction <433> . 7.5.4 Interfacial structure and growth mechanisms Detailed studies have been made on the nature of the structure of the / interface in Ti–Cr alloys (Furuhara et al. 1991). Both diffraction contrast and phase contrast TEM experiments have been carried out for deciphering the interfacial structure. It is through these studies that the atomic movements necessary for the structural transformation from the bcc to the hcp phase during the diffusional process have been identified. As discussed earlier, two types of Widmanstatten -plates are encountered in the Ti–Cr system, namely, the normal -plates which maintain near equilibrium with the 1 -phase during growth and the black plates which form
Diffusional Transformations
649
in contact with the 2 -phase. Let us first discuss the -normal -interface structure which has the following characteristics: (1) As in the case of Zr–Nb alloys, normal -plates obey the Burgers relation almost exactly: ¯ [111] [1120] ¯ (0001) (011) ¯ ¯ (1100) (21¯ 1 ¯ (2) The broad face of the plate has an irrational habit plane close to (11, ¯ 11,13) . This habit plane results from the uniform arrangement of structural ledges which step down along the lattice invariant line (approximately [335] ) ¯ with (112) (1100) terraces. The concept of structural ledges introduced by Hall et al. (1972a,b) and Rigsbee and Aaronson (1978) envisages that these are steps, one to a few atom planes high which, when spaced regularly on an interphase interface, reduce the misfit along the rational interface direction normal to the ledge. How structural ledges reduce the misfit on a bcc/hcp interface is shown schematically in Figure 7.61. ¯ c/2 [0001] (3) A set of misfit dislocations with the Burgers vector a/2 [110] exists along the lattice invariant line on the broad face of -plates with a
[112]β
// [1100]α β (bcc)
[111]β // [1120]α
L
b = a /12[111]β
α (hcp)
Figure 7.61. Schematic representation of structural ledges showing the compensation of misfit at / boundaries.
650
Phase Transformations: Titanium and Zirconium Alloys
spacing of about 12 nm. These dislocations have a sessile character with respect to the migration of the broad face, and thus the growth of -plates must accompany diffusional jumps of atoms across the interface. The interfacial structure of the bcc/hcp interface as deduced from TEM observations and from modelling of the structure of the two phases across the plane ¯ contains two different kinds of linear defects at ¯ /(211) of good fit, i.e. (1100) the boundary: structural ledges and misfit compensating ledges. Aaronson and co-workers have distinguished between these two types of linear defects on the basis of the presence of extra half atomic layers in case of the misfit compensating ledges: such extra half layers are not present in the structural ledges. However, Christian (1994) has pointed out that steps, ledges and interface dislocations can all be described in terms of transformation dislocations of Burgers vector bt which in the case of structural ledges are smaller than the smallest lattice translation vector. It may be noted that both the direction and the magnitude of bt may be irrational. Furuhara et al. (1991) have shown that the broad face of an -lath contains a set of misfit dislocations, with c-type Burger vector. These dislocations, which have their line vectors parallel to the invariant line direction, 533 , loop around the -laths. Similar observations have been reported in Zr–Nb alloys by Zhang and Purdy (1994) and Banerjee et al. (1997). It has been hypothesized (Zhang and Purdy 1993a,b) that the optimum orientation relation is that which minimizes the interfacial misfit in the habit plane. The lines along which the two lattices match best are defined as O-lines (based on the O-lattice description of interfaces by Bollman 1970). The habit plane is, therefore, selected to be the plane containing an array of O-lines with the largest spacing between these lines. The comparison of experimental observations on and geometrical analysis of habit planes of -plates produced in a diffusional transformation from the -phase has established that the habit plane (broad face of -laths) is characterized by a single set of misfit dislocations whose line vectors are coincident with the invariant line vector and whose spacing is equal to the calculated distance between the O-lines. Experimental and theoretical work on the nature of / interfaces in diffusional transformation products of Ti- and Zr-based alloys has proved beyond doubt that there exists a good coherency between the bcc matrix and the hcp product across the interface. This is in agreement with diffusional transformations in other systems involving hcp/fcc (Howe et al. 1987) or bcc/fcc (Furuhara et al. 1995a,b) transformations. The presence of coherency at the transformation front is suggestive of an atomic site correspondence which should be distinguished from lattice correspondence, an essential feature of displacive transformations. During the growth of a plate or lath in a diffusional transformation, individual atoms cross
Diffusional Transformations
651
the transformation front by thermally activated diffusional jumps, maintaining the correspondence of atomic sites between the two phases. Such a growth process leads to a macroscopic shape change due to the transformation. This is reflected in the observation of tilting of originally flat surfaces and a change of direction of fiducial lines (scratches) inscribed on the surface of the parent phase sample. Such a change was earlier known to be the distinguishing feature of a martensitic transformation, but there is now good evidence that shape changes can occur in some diffusional transformations as well. It is worthwhile to draw a comparison between the interfaces/transformation fronts in martensitic transformations and those encountered in the / diffusional transformation. Both types are shown to be irrational and usually partially coherent. Fully coherent irrational interfaces are rare, but they are found in some martensitic transformations in Ti alloys, e.g. Ti–22 at.% Ta (Bywater and Christian 1972). In general, the partially coherent interfaces of martensite plates are usually irrational because of the invariant plane strain requirement and the fact that the lattice invariant shear is a simple shear. However, these interfaces should be necessarily glissile and should conserve the number of atoms during propagation. In contrast, partially coherent irrational interfaces in diffusional transformations are essentially epitaxial and are made up of terraces of rational interfaces, separated by steps or ledges. A new layer of the product phase forms by the migration of a growth ledge. This process does not conserve the number of atoms. In general, diffusional transformations involve non-conservative movements of transformation, dislocations at the transformation fronts. Ledges of any height that are not an intrinsic part of the interface structure may be regarded as growth ledges or transformation dislocations. Isolated steps on a coherent interface or steps on a faceted high-index interface are examples of such ledges. Aaronson and co-workers have distinguished between structural and growth ledges, the former being an intrinsic part of a high-index interface, as shown in Figure 7.61. However, the distinction between them is not always very sharp as the roles they play in misfit compensation and in the growth process may often merge. Superledges with large Burgers vectors, though unstable due to their high self-energies, are encountered on the transformation front in some cases of diffusional transformation. The growth of an -lath in Ti–Cr alloy has been schematically illustrated in Figure 7.62, which has been taken from the model proposed by Furuhara et al. (1995a,b). Growth ledges, which have been observed both on the broad face and the side facet of the lath, are responsible for thickening and widening of the lath. Each growth ledge on the side facet contains a misfit dislocation with the ¯ = c/2 [0001] on its riser. This dislocation is in a sessile Burgers vector, a/2, [110 orientation and, therefore, it must climb for the motion of the growth ledge riser.
652
Phase Transformations: Titanium and Zirconium Alloys
b = a /18[1120]α Broad face
Misfit-compensating c-type ledge [1100]α//[211]β
Side facet
Bi-atomic structural ledge Edge
[335]
Primary growth direction
Figure 7.62. Schematic illustration of the interfacial structures of -plates growing in ; both the broad face and the side facet are shown.
Since diffusion and partitioning of alloying elements accompany such a growth process, non-conservative motion of dislocations at the transformation front is facilitated. The thickening of a lath means addition of new layers on the broad face, which can be accomplished by a ledge growth mechanism as shown in Figure 7.62. The broad face of the lath is formed by the coalescence of growth ledges on this surface. This riser plane of growth ledges on the side facet contains the structural ledges and is thus semicoherent. The migration of the growth ledges should occur by the kink on ledge mechanism in which diffusing atoms get attached to the kink plane. Since these ledges do not coalesce to the extent necessary for forming a complete layer on the side facet, the broad face remains as a stepped interface with steps along the lattice invariant line. These steps are nothing but the misfit compensating c-type ledges. The ledge growth mechanism in a diffusional transformation requires (a) atomic diffusion to effect the partitioning of the alloying elements, (b) some mechanism by which new ledges are formed on the newly created layer and (c) maintenance of the semicoherent interfacial structure. In a steady state, these processes must
Diffusional Transformations
653
all conform to the overall rate that is determined by the process which consumes the major part of the driving force. The ledge growth mechanism described here can be compared with that of crystal growth from the vapour phase. In the latter case, atoms from the vapour phase condense on the terraces of a growing crystal, diffuse over the surface until they either evaporate back into the vapour or find a kink in a step where they ultimately join the crystal lattice. In the case of a solid–solid diffusional transformation, the growth, as described earlier, involves attachment of atoms at kinks in steps on solid–solid interfaces. The diffusion of atoms along the partially coherent interface, however, is not quite favourable. In martensitic transformations, such as the fcc → hcp transformation, the evidence suggests that a step can glide as a unit with such a high velocity that separate kink motion seems unlikely. In diffusional transformations, a kink on a superledge can undoubtedly provide a favourable site for attachment of freshly arriving atoms to join the lattice sites of the product crystal. The growth velocity of the interface by ledge growth mechanism can be modelled using the Jones and Trivedi (1971) treatment. Let us consider an interface with perfectly coherent terraces separated by an incoherent ledge at a spacing of as shown in Figure 7.63. The coherent areas can be considered to be entirely immobile with attachment of atoms to the growing phase occurring at the ledge. Cahn et al. (1964) expressed the growth rate of the boundary in the direction normal to the coherent areas as G=
av
(7.56)
where a is the height of the riser, is the average distance between the ledges and v is the lateral migration rate of the ledges. Jones and Trivedi have made a
a
G V
λ
V
Figure 7.63. Schematic representation of the ledge growth mechanism. a: height of the riser, : average spacing of ledges, V : velocity of ledge perpendicular to the growth direction, G: rate of interface movement along the direction normal to partially coherent areas.
654
Phase Transformations: Titanium and Zirconium Alloys
detailed treatment of v in which interface reaction kinetics are again taken into account. For reasons presented later, the interface kinetic coefficient, , will be assumed infinite, yielding the relationship. v=
D x − x
(7.57)
a x − x
where = a constant which can be likened to a diffusion distance and which is a complicated function of the Peclet number, p = va/ 2D, which in turn varies in a complicated manner with the supersaturation. Combining Eqs. (7.56) and (7.57): G=
D x − x
(7.58)
x − x
It may be noted that the ledge height, a, does not appear in the equation which is due to the fact that the velocity of a ledge is inversely proportional to its height. Jones and Trivedi (1975) have also investigated the course of the overlapping of the diffusion fields of the adjacent ledges. Figure 7.64 schematically shows the overlapping of two diffusion fields. It can be noticed that the diffusion field extends more on the broad face at the bottom of the riser than the face at the top of the riser. Therefore, the diffusion field of the trailing ledge affects the diffusion field of the leading ledge move than the latter affects the former. Modelling of this effect is rather complex because the overlapping diffusion fields cannot be simply superimposed, as that would alter the concentration gradient along the riser
–6.0
–5.0
–4.0
–3.0
–2.0
–1.0 0.0 1.0 Length/ledge height
2.0
3.0
4.0
5.0
6.0
Figure 7.64. Reduced isoconcentration curves about two ledges whose diffusion fields overlap. Numbers on the x-axis correspond to distances in multiples of the ledge height.
Diffusional Transformations
655
itself. The concentration gradient must be calculated carefully so that all portions of the riser migrate at the same rate. However, such calculations can be done using the confirmed mapping technique and it is possible to evaluate the concentration gradient on which the migration rate of each ledge depends. It is found that the concentration gradient for the trailing ledge is higher than the leading edge, which means that the trailing ledge will catch up and merge with the leading edge. 7.5.5 Morphological evolution in mesoscale The transformation of the -phase, either in an isothermal condition within the + phase field or during continuous cooling, produces a variety of microstructures. Since the properties of commercial Ti alloys are strongly influenced by the microstructure in mesoscale, a description of the microstructure evolution resulting from the / diffusional transformation is given here. During continuous cooling from the -phase field, the -phase appears first as GBAs. As mentioned earlier, -allotriomorphs are invariably related to one of the adjoining -grains by the Burgers orientation relation. The / interface of the allotriomorphs which tend to propagate more towards the Burgers-related -grains undergo a morphological instability, leading to the development of periodic protrusions (Figure 7.50) designated as saw tooth morphology. The invariant line direction corresponding to the operating variant of orientation relation is chosen to be the growth direction of the protrusions which develop into side plates. The combined influence of the stress field and concentration field ahead of the / interface is responsible for the growth of a group of -laths of identical orientation (same as that of the allotriomorph at a nearly equal spacing). The fully grown region of such parallel laths define a colony, a microstructural feature which has a strong bearing on mechanical properties such as yield strength and fracture toughness. The allotriomorphs cannot grow into the 2 grain with which it does not have a Burgers relation. However, a set of -laths with an orientation Burgers related to 2 can get sympathetically nucleated on the surface of the allotriomorph. They can grow in a similar manner into the 2 -grain forming a colony. With a somewhat slow cooling rate, the entire -grain volume is gradually filled up by a number of such colonies of -laths which nucleate from grain boundaries. The fact that -laths growing into the 1 -grain bear the same orientation of the allotriomorph is demonstrated in Figure 7.49 where two colonies are seen to grow from two allotriomorph orientations which share the same basal plane but are rotated by an angle of 10.5 (as described earlier in Figure 7.49). The invariant line directions for these two variants make an angle of 87 which is reflected in the large angle between their long axes (primary growth direction).
656
Phase Transformations: Titanium and Zirconium Alloys
With increasing cooling rate the volume fraction corresponding to the colony morphology decreases and the remaining part of the -grains transforms into a “basket weave” morphology which is characterized by the presence of laths of different orientations in the same region. As has been demonstrated through many examples in the book, phase transformation and microstructural evolution in commercial titanium and zirconium alloys are extremely complex. Traditional models that characterize microstructural features by their average values without capturing the anisotropy and spatial variation may not be sufficient to quantitatively define the microstructure and hence to establish a robust microstructure–property relationship. Recent progress in computer simulation of complex microstructures using the phase field method (Wang and Chen 2000) offers a unique opportunity to rigorously and realistically address the problem. Extensive efforts have been made in integrating thermodynamic modelling and phase field simulation to develop computational tools for quantitative prediction of phase equilibrium and spatiotemporal evolution of microstructures during thermal processing that account explicitly for precipitate morphology, spatial arrangement and anisotropy. Figure 7.65 shows an example of side plate formation in Ti–6Al–4V obtained from phase field simulations (Wang et al. 2005). Side plates, the -phase lamellae growing off grain boundary upon cooling, are the major microstructural constituents of many / Ti alloys. Two mechanisms have been proposed for the initiation of side plates from grain boundary : sym-
Figure 7.65. Simulated micrographs produced by the phase field simulation of formation of the -plates in the -matrix. A competition between formation of side plates and of basket weave structure in the 2D simulation is shown. (After Wang et al. 2005). (a) t = 5 s, (b) t = 10 s, (c) t = 15 s, (d) t = 20 s.
Diffusional Transformations
657
pathetic nucleation (Aaronson and Wells 1956) and interface instability (Mullins and Sekerka 1963). In the current example, the latter mechanism was assumed. Two-dimensional phase field simulations were carried out at 1123 K, with a starting microstructure of a thin layer of grain boundary that is in contact with supersaturated -grains. A random fluctuation of the / interface position was introduced by a random number generator (as shown in Figure 7.65(a)). The interfacial energy between and is assumed to be anisotropic, with the one of the interface parallel to the growth direction being 1/3 of that of the one perpendicular to the growth direction. The time evolution of the initial fluctuations in interface position into a colony of side plates is shown in Figure 7.65(b)–(d). It is readily seen that the spatial variation and shape anisotropy in precipitate microstructure are well captured by the simulation.
7.6
PRECIPITATION OF INTERMETALLIC PHASES
Precipitation of intermetallic phases can occur in Ti- and Zr-based alloys either in the supersaturated -matrix or in the matrix of a metastable intermetallic phase. The former situation is encountered in systems containing -stabilizing alloying elements, such as Al, Sn and Ga, and also a large number of Ti–X and Zr–X binary eutectoid systems in which the solubility of the element X in the -phase increases sharply with temperature. The precipitation of an ordered intermetallic phase from a disordered solid solution can go through several intermediate stages, depending on the clustering and/or ordering tendencies of the solid solution and the presence of coherent intermediate structures between the parent and the equilibrium precipitate phase. Section 7.6.1 is devoted to a discussion on some examples of such precipitation reactions. The formation of an ordered precipitate phase from a parent phase which is also chemically ordered has been exemplified in Ti aluminides. The crystallography of such transformations, which has been recently studied in detail, brings out some important aspects of the diffusional growth mechanism. This aspect has been elaborated in Section 7.6.2. The formation of intermetallic phases through cellular growth mechanisms is covered in Section 7.7. 7.6.1 Precipitation of intermetallic compounds from dilute solid solutions There are a number of Ti- and Zr-based alloys in which the equilibrium constitution is a mixture of the -phase and an intermetallic compound. Many of these systems exhibit an eutectoid reaction, essentially due to the fact that the solubility of the alloying element is higher in the -phase than in the -phase. Supersaturated -solid solution can be formed in such systems by quenching an alloy either from
658
Phase Transformations: Titanium and Zirconium Alloys
the -phase field or from the -phase field, the latter giving rise to a higher level of supersaturation in the martensitic -phase. On ageing, precipitation of intermetallic phases, usually the equilibrium phase richest in the solvent (Ti or Zr), occurs. The process of the precipitation of intermetallic phases in -solid solution has been studied in a number of binary and multicomponent alloys of Ti and Zr. As an illustrative example, we will first discuss the precipitation of ZrCr2 in the -phase dilute Zr–Cr alloys. Two structural variants of ZrCr2 have been reported. Both are topologically close-packed Laves phase structures: C14 (hexagonal, MgZr2 type) with a = 05079 nm and c = 08279 nm and C15 (cubic, MgCu2 type) with a = 0721 nm, the cubic form being the equilibrium structure at temperatures lower than 1000 C. Mukhopadhyay and Raman (1978) have reported that in an alloy having a composition of Zr–2 wt% Cr, -quenching cannot fully suppress precipitation of ZrCr2 which appears along closely spaced rows in the -matrix (Figure 7.66(a)). Such a microstructure suggests that these rows of precipitates trail the advancing transformation front at which the -phase rejected the and the ZrCr2 precipitate simultaneously. Such a process is thermodynamically possible under a condition where the extent of supercooling is sufficiently high to cause direct eutectoid reaction even in this hypereutectoid alloy. Ageing in the temperature range of 350–550 C has resulted in the formation of bimodal size distribution of precipitates, larger precipitates at -lath boundaries often being connected to form stringers. The presence of coarser precipitates either at lath boundaries or along dislocation is suggestive of enhanced solute diffusion along the defects being responsible for a faster growth of precipitates. The ZrCr2 precipitation reaction has also been studied in a heat treatment involving -solutionizing followed by isothermal holding at the reaction temperature in the range of 700–800 C. The resulting microstructures suggest that the ZrCr2 phase is the first phase to emerge from followed by the formation of -laths which remain supersaturated with solute (Figure 7.65(b)). Fine scale precipitation within the -laths occurs at a subsequent stage. The orientation relation between the and the ZrCr2 phase has been determined from superimposed diffraction patterns of and ZrCr2 (Figure 7.66(b)) to be the following: The most commonly used Zr alloys in nuclear industry are zircaloys which contain tin, iron, chromium and in some cases nickel (as shown in Table 7.6). Zircaloys are essentially -Zr-Sn solid solutions containing fine intermetallic precipitates. The distinct types of intermetallic particles are observed in zircaloy-2, namely, Zr2 (Fe,Ni) and Zr(Cr,Fe)2 . The former has a body-centred tetragonal structure (space group 14/mmm, D17 4n ) while the latter is a lower phase with a
Diffusional Transformations
659
m
m
(a)
(c)
(b)
(d)
Figure 7.66. Precipitation of ZrCr2 precipitates in the -matrix (a) shows rows of precipitate particles which appear to have formed along a trail following the advancing transformation front. (b) Coarse ZrCr2 particles forming at lath boundaries followed by fine scale precipitation within the matrix. (c) Diffraction pattern showing superimposed reciprocal lattice sections of -matrix (M) and ZrCr2 precipitates (P). (d) Key to the diffraction pattern.
C14 hexagonal structure. Usually precipitate particles forming at grain boundaries of zircaloy-2 are larger (typical diameter 3 m) and are of the Zr2 (Fe,Ni) type while intragranular precipitates (typical diameter ≤ 1 m) are of both Zr2 (Fe,Ni) and Zr(Cr,Fe)2 types. Presence of stacking faults within the latter is the distinguishing feature of the Zr(Cr,Fe)2 particles. Mechanical and corrosion properties of zircaloys-2 and 4 are strongly influenced by the size and the distribution of the ordered intermetallic particles which form during the thermal and mechanical processing of zircaloy components.
660
Table 7.6. Chemical composition of zirconium alloys for nuclear application. Zircaloy-2
Zircaloy-4
Zr–1Sn–1Nb
Zr–2.5Nb
Zr–1Nb
Excel
Zr–Nb–Cu
Ozhennite
Sn (wt%) Fe (wt%) Cr (wt%) Ni (wt%) O (ppm) N (ppm) Nb (wt%) Cu (wt%) Mo (wt%) H (ppm) C (ppm) C1 (ppm) P (ppm)
1.2–1.7 0.07–0.2 0.05–0.15 0.03–0.08 900–1300 <80 – – – <25 <270 <20 – (a)
1.2–1.7 0.18–0.24 0.07–0.13 – 900–1400 <65 – – – <25 150–400 – – (b)
0.9–1.1 0.1 – – 900–1300 <65 0.9–1.0 – – <20 – <0.5 – (c)
– <650 ppm <100 ppm <35 ppm 900–1300 <65 2.4–2.8 – – <5 <125 – <10 (d)
– <500 ppm – <200 ppm <1000 <60 1.0 – – <15 <200 – – (e)
3.5 – – – – <70 0.8 – 0.8 – – – – (f)
– – – – 900–1300 <65 2.4–2.8 0.3–0.7 – <25 <270 – – (g)
0.2 0.1 – 0.1 – – 0.1 – – – – – –
(a) Fuel tube and plug material for BWR and PHWR; (b) fuel tube and plug material for PWR and PHWR; (c) experimental fuel tube alloy; (d) pressure tube materials for PHWR; (e) fuel tube and end plug in VVER; (f) experimental pressure tube alloy; (g) garter spring material for PHWR. All the alloys contain A1 < 75 ppm, B and Cd < 0.5 ppm, Co and Mg < 20 ppm and Hf <50–150 ppm.
Phase Transformations: Titanium and Zirconium Alloys
Alloy element
Diffusional Transformations
661
The search for zirconium alloys suitable for use as a cladding material in high burn-up fuel has resulted in alloys in which both -forming Nb and intermetallicforming Sn and Fe are added. A typical composition of such an alloy is Zr–1Nb–1Sn–0.4Fe which exhibits better corrosion resistance and high radiation stability. These alloys are also suitable under conditions of partial boiling of coolant water as they do not show modular corrosion. Unlike zircaloys or binary Zr–1Nb alloys, these Zr-Nb-Sn-Fe alloys do not show a transition in the kinetics of irradiation-induced growth and creep up to a neutron fluence of 5 × 1026 nm−2 (neutron energy E > 0.1 MeV). The microstructure of these alloys are characterized by a distribution of intermetallic phase particles in the -matrix. The total volume fraction of precipitates in the Zr–1Nb–1Sn–0.4Fe alloy thermomechanically processed for cladding tube application is about 3–3.5. The majority of precipitates have a globular shape with diameters in the range of 0.1–0.2 m though a limited number of precipitates are as large as about 1 m. X-ray microanalysis of precipitates extracted on a carbon replica, as reported by Nikulina et al. (1996), has shown that the precipitates have an average composition of about 40Zr–40Nb–30Fe. Diffraction analysis, however, has shown that two types of intermetallic precipitates, namely, the (Zr,Nb)3 Fe phase having an orthorhombic structure and the Zr(Nb,Fe)2 phase having a hexagonal structure, are parent (Figure 7.67). It is interesting to note that the intermetallic precipitates in this alloy are susceptible for a change due to irradiation. When irradiated to a fluence of about 4×1025 nm−2 , the precipitate phases -Nb-Zr and Zr35 Fe appear at the expense of those present in the pre-irradiated condition. Intermetallic Ti5 Si3 precipitation in Ti–Si alloys has been studied in detail by Flower et al. (1971). Heterogeneous nucleation of precipitates on dislocations
Figure 7.67. Precipitation of Zr(Nb,Fe)2 and (Zr,Nb)3 Fe precipitates in the -matrix in the Zr–1Nb– 1Sn 115 M and 0.4 Fe alloy.
662
Phase Transformations: Titanium and Zirconium Alloys
has been reported. These precipitates being an important hardening agent, several commercial Ti alloys with Si addition have found extensive applications. The Ti–2% Cu alloy shows formation of Cu-rich Guinier Preston zones in the early stages of precipitation. During continued ageing, these zones lose coherency and undergo an in situ transformation into the Ti2 Cu phase (Williams et al. 1970). 7.6.2 Precipitation in ordered intermetallics: transformation of 2 -phase to O-phase Some cases of diffusional transformations have recently been studied systematically in the context of microstructure evolution in structural intermetallics. It is instructive to examine these transformations as examples of diffusional transformations which follow martensitic crystallography. Some key issues related to the crystallography of these transformations and to the mechanism of atomic movements are discussed in this section with reference to the specific example of the transformation of the 2 -phase to the O-phase in Ti–Al–Nb alloys. The transformation from the 2 -phase (DO19 structure) to the O-phase (orthorhombic structure) during isothermal ageing, investigated by Muralidharan et al. (1995), has revealed several special features as listed below: (1) The parent and the product phases are both ordered. (2) The transformation is associated with only a small lattice distortion, the three principal strains being positive, negative and zero, fulfiling the IPS condition without involving any lattice invariant strain. (3) The parent and the product phases differ in composition and the product grows with isothermal ageing; both these features indicate the involvement of diffusion and thermal activation in the transformation process. An alloy with the composition, Ti–28.5 at.% Al–13 at.% Nb, chosen for this study, on quenching from the -phase field, followed by annealing at 1198 K for 24 h, produces equiaxed grains of the 2 -phase. On subsequent ageing in the 2 +O field, plates of the O-phase are precipitated. The lattice correspondence between the parent 2 and the product O-phase structures is shown in Figure 7.68 which indicates that the O-phase structure has the same site occupancy for Ti, Al and Nb atoms as in the case of the parent 2 -structure but possesses an orthorhombic symmetry. The dimensions of the 2 and the O-phase unit cells along the principal strain directions, [100]O , [010]O and [001]O , are as follows: a2 = 0578 nm; b2 = 1001 nm; c2 = 0467 nm aO = 0595 nm; bO = 0970 nm; cO = 0467 nm
α2[0001]
[010]
C
1.001 nm
A
663
O [001]
A
C
0.970 nm
[0110]
Diffusional Transformations
B
[2110]
B
[100] 0.578 nm
Atom
Al
Layer A
0.595 nm Ti
Nb
Angle ABC
α2
O
60°
63°
Layer B. c /2 above
Figure 7.68. Structure of 2 - and O-phases projected along the <001> orientation. The small circles indicate atoms in a plane 21 above or below along the direction of projection. Note that the O-phase contains Nb atoms randomly distributed in Ti sites.
The 2 -structure is described here in terms of an orthohexagonal basis so that the correspondence between the parent and the product lattices is given by [100]2 [100]O ; [010]2 [010]O ; [001]2 [001]O The principal lattice distortions can then be evaluated to be along [100], 1 = 1029411 along [010], 2 = 0969031 and along [001], 3 = 1000000 The construction of a strain ellipsoid in accordance with the phenomenological theory of the martensitic transformation (Figure 7.69) shows the two invariant lines OA and OB which are generated by the lattice strain from OA and OB , respectively. A rigid body rotation by an angle, !, either clockwise or counterclockwise, brings back OA or OB to their original positions. Since the distortion along the 3 direction, perpendicular to the plane of the paper, is zero, the planes containing the
664
Phase Transformations: Titanium and Zirconium Alloys [0110]α2
[010]
B′
A′ A
B
+v Ha e bit so fo lu r tio n
r fo n it tio ab lu H so e -v
φ1 = 44.68°
θ = 1.73° φ2 = 42.95°
O
[100]
[2110]α2
Figure 7.69. Habit plane determination for the 2 to O transformation.
3 direction and OA or OB are the two habit plane solutions. The indices of the undistorted vectors on the basis of the parent lattice are given by <1K0> where
1 − 12 K=± 2 − 1 2
1/2 = ±0989
(7.59)
The rigid body rotation, ", is given by " = #1 − #2 where #1 and #2 are the angles made by the undistorted vectors OA and OB before and after the application of lattice strain; their values are #1 = tan−1 0989 = 4468 #2 = tan−1 0931 = 4295 " = 173
Diffusional Transformations
665
The indices of the habit planes which satisfy the IPS condition are determined by two vectors which remain invariant on the application of the combination of the lattice strain and the rigid body rotation. The indices of the plane containing [001] and [1 ± 0.9890] are {1 ± 1.7510} which work out, in the hexagonal four-index system, to be (83110) and (81130). This plane in the orthorhombic basis is {470}O . Three crystallographically equivalent O-phase variants, A, B and C, satisfy the lattice correspondence by making the [100]O axis parallel to one of the directions, ¯ [12 ¯ 10] ¯ or [1120] ¯ a . The small rigid body rotation of 1.73 in clockwise [21¯ 10] 2 2 2 and counterclockwise directions for each of the correspondence variants generates a total of six orientation variants designated as A+, A−, B+, B−, C+ and C−. Muralidharan et al. (1995) have experimentally determined the habit planes for each of these variants and the results obtained are presented in a stereogram in Figure 7.70. It is to be noted that the angles between the different habit variants are 30 , 60 and 90 , as indicated in Table 7.7. The excellent agreement between the experimentally determined and theoretically predicted habit planes clearly establishes the fact that the crystallography of the precipitation of the O-phase in the 2 -matrix satisfies the IPS condition.
(110) B+ (110) C– (0110) (1120) (130) A– (130) B+
(1210)
C+
B-
(1010) (110) A– (110) C+
(1100) A–
A+
B+ (2110)
(31180)
(0001)
(38110) (83110)
(81130) (1010)
(2110) (130) B– (130) C+
C–
(11830)
(1100) (110) A+ (110) B–
(11380)
(1210)
(1120)
(130) A+ (130)C– (0110)
Figure 7.70. Experimentally determined habit planes for 2 to O transformation for all the six variants.
666
Phase Transformations: Titanium and Zirconium Alloys Table 7.7. Angular relationship between O-phase variants. 1
Habit planes at 30
A−, B+; A−, C+;B−,C+; B−, A+; C−, A+; C−, B+
2
Habit plane at 60
A−, B−; A−, C−; B−, C−; A+, B+; A+, C+;B+, C+
3
Habit plane at 90
A−, A+; B−, B+, B+; C−, C+
The development of the morphology of intersecting O-phase plates and the grouping of O-plate variants during the thermally activated growth (both lengthening and thickening) process have been found to be controlled by the self-accommodation of the participating variants. The manner in which selfaccommodation is accomplished is described in the following by considering a few interesting examples. (a) Interactions of O-plate variants with habit planes at 30 : Figure 7.71(a) shows the interaction of pairs of variants, C + /A− and C + /B−. These variants, during ¯ 2 and (1210) ¯ 2 planes which the process of growth, come in contact along (1100) are nearly parallel to {110}O and {130}O planes. These plates do not terminate at the intersection point, and in such a situation the non-terminating plate tapers down in thickness. Combinations of three plates, each making a 30 angle with the neighbouring plates, are also encountered. The interfaces separating such plates are also of {110}O and {130}O types. Figure 7.71(a) shows how three plates meet at a point. Plate configurations like this suggest that the stress fields of the three plates interacted at the nucleation stage, leading to a low strain energy (high degree of self-accommodation) grouping of plates. (b) Plates with habit planes at 60 : Interactions of such plates lead to the propagation of one plate across the other. Plates are physically displaced in a manner analogous to that observed in the case of the intersection of twins or martensite plates, the offset created due to the displacement being related to the macroscopic shape strain associated with the intersecting plate. Such an interaction can also result in the nucleation of a third variant as shown in Figure 7.71(b) where A+ B+ and C+ variants meet. It is to be noted that the region where three variants overlap undergoes a retransformation into the 2 -phase. Selected area diffraction and microdiffraction analyses have established that the orientation of the retransformed 2 -region is not the same as that of the parent 2 -grain. The retransformed 2 -orientation is obtained by a 3.5 rotation of the parent 2 -orientation around the [0001] axis. As a consequence of this rotation, the interfaces created between the three O-phase variants and the retransformed 2 -region are again parallel to {470}O {83110}2 planes and are perpendicular to the original habit planes of the respective variants.
Diffusional Transformations
667
(a)
(b)
(c)
(d)
Figure 7.71. Interactions of O-plates: (a) at 30 , (b) at 60 , (c) at 90 , (d) combined features.
The 2 -phase is also observed at the centre of the coarsened star-shaped threeplate group containing, for example, the A+ B+ and C+ variants having their habit planes at 60 to one another. It remains unresolved whether the “retransformation” of the central region into the 2 -structure occurs during the early stages of plate formation or at the coarsening stage. (c) Plates with habit planes at 90 : Variants such as A+ and A− B+ and B− and C+ and C− meet each other at 90 (Figure 7.71(c)). A fresh nucleation of O-plates of other crystallographic variants inevitably occurs. The new variants which nucleate are at 30 to one of the intersecting plates. The freshly nucleated variants share common interfaces with those plates which are at 30 to them. The 60 star configuration with the retransformed 2 -phase is produced as a result of this interaction as shown in Figure 7.71(b).
668
Phase Transformations: Titanium and Zirconium Alloys
The strong elastic interactions among the different variants of the O-phase plates, as illustrated in these examples, can be rationalized in terms of self-accommodation of plates – a feature commonly encountered in martensitic structures. An examination of the average shape strain associated with a given group of variants can reveal the extent of self-accommodation that can be achieved in such a group. The groups in which the average shape strain is considerably reduced are those which appear more frequently. This analysis, however, assumes equal volume of each of the variants, neglects the presence of the matrix and, most importantly, ignores the anisotropy of strain around a plate-shaped precipitate and, therefore, the spatial arrangement of plates in a given group. Table 7.8 shows elements of shape deformation matrices for individual O-plate variants referred to a common set of principal axes, as defined in Figure 7.52. Six variants of O-plates can be grouped in 57 different ways taking two, three, four, five or six variants in a single group. A consideration of the symmetry of the transformation reduces the number of groups to 17. Average shape strains corresponding to these groups are presented in Table 7.9. It is seen that the group numbers 9, 10 and 17 yield an average shape strain which results in almost no distortion of the parent. Experimental observations have clearly pointed out that it is these groups that preferentially evolve during the nucleation and also during the growth processes. Stress coupling between the variants, therefore, is the motivating factor for the frequent association of these variants in a group. It may be noted from Table 7.9 that while the groups of variants whose habit planes lie at 30 and 60 , respectively, yield a negligible average shape strain, those whose habits lie at 90 are associated with large values of the dilatation as well as the shear component (group number 5). The interactions of plates in this group may lead to fresh nucleation of additional variants, resulting in the formation of groups with a more favourable average shape strain. Another important factor which controls the grouping of O-plate variants is the nature of the interfaces separating the plates in a group. A recent analysis by Bendersky and Boettenger (1994) shows that if the variants in a group are Table 7.8. Elements of shape deformation matrices for individual O variants. Variant A+ A− B+ B− C+ C−
11
12
13
21
22
23
31
32
33
1.0293 1.0293 0.9825 0.9857 0.9857 0.9825
−00018 +00018 −00252 −00270 0.0270 +00252
0 0 0 0 0 0
−00018 +00018 −00252 −00270 +00270 +00252
0.9691 0.9691 1.0159 1.0127 1.0127 1.0159
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
1 1 1 1 1 1
Diffusional Transformations
669
Table 7.9. Elements of average shape strain matrices associated with different groupings of O variants. Group number
Typical variant combinations
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
A− A+ A− C+ A− B+ A− B− A+ A− B− A+ A− C+ A+ C− C+ A+ A− C− A+ C+ B− A+ C+ B− A+ A− B+ C− A+ A− B+ C+ A+ A− B− C+ A+ A− B− C− A+ A− B− B+ Except B− All
11
12
13
21
22
23
31
32
33
1.0293 1.0075 1.0059 1.0075 1.0148 1.0148 0.9992 1.0137 0.9992 0.9992 1.0059 1.0067 1.0075 0.0067 1.0067 1.0019 0.9992
0 +00144 −00117 −00117 −00090 +00090 +00168 +00084 0 0 0 +00005 0 −00005 −00131 +00005 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 +00144 −00117 −00126 −00090 +00090 +00168 +00084 0 0 0 +00005 0 −00005 −00131 +00005 0
0.9691 0.9909 0.9925 0.9909 0.9836 0.9836 0.9992 0.9847 0.9992 1.000 0.9925 0.9917 0.9909 0.9917 0.9917 0.9965 0.9992
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
oriented with respect to each other in such a way that they come in contact along a common {110} or {130} twin plane, stress-free interfaces across which no discontinuities or displacements exist are created. Needless to mention, such low-energy interfaces readily form during the growth of plates. It may be noted that in group number 9 which consists of A+ B+ and C+ (or A− B− and C−) variants, such stress-free interfaces are not possible between the variants. In such a situation, retransformation into the 2 -phase occurs in the region where the plates come in contact. The retransformation is, therefore, motivated by two factors: first, self-accommodation of the variants in the group and second, the creation of low-energy interfaces between the retransformed 2 -grain and the three plates. Ternary Ti–Al–Nb intermetallics of compositions close to Ti2 Al–Nb and Ti4 AlNb3 remain in the two-phase field consisting of the B2 and the orthorhombic phases at a temperature of 973 K. Both the alloys, on rapidly cooling from 1673 K, produce an ordered B2 structure, the degree of ordering being very low in the case of the latter alloy. Ageing for an extended period (26 days) has been shown to result in the following sequences of transformations (Bendersky et al. 1991):
670
Phase Transformations: Titanium and Zirconium Alloys
(1) For Ti2 AlNb: Complete transformation of the retained high-temperature B2 phase into a highly faulted orthorhombic phase occurs. This is followed by recrystallization of the faulted grains into fault-free orthorhombic grains and finally precipitation of the bcc phase in the orthorhombic matrix. (2) For Ti4 AlNb3 : Precipitation of the equilibrium orthorhombic phase occurs in the weakly ordered B2 matrix. Bendersky et al. (1991) have proposed a tentative pseudobinary phase diagram for interpreting the sequence transitions which occur in these alloys.
7.7
EUTECTOID DECOMPOSITION
Eutectoid decomposition – a solid state phase reaction involving diffusion – is known to occur in a large number of alloy systems. In the context of a binary system, this reaction entails the decomposition, on cooling, of a high-temperature phase into a mixture of two low-temperature phases. The transformation mechanism as well as the morphology and the substructure of the decomposition products may vary from system to system and depend on the reaction temperature. For a given system, the appropriate reaction path is determined by the nucleation and growth processes that are kinetically favoured by the system. Eutectoid reactions can occur in higher component systems also (Rhines 1956). For example, in a ternary eutectoid reaction, the decomposition of a single phase may lead to the simultaneous evolution of three phases. In the present text, however, only binary eutectoids will be considered. Alloy eutectoids can be divided into two classes: those which originate through the allotropy of the base metal and those which are associated with the decomposition of an intermediate phase (Spencer and Mack 1962). Eutectoids in Fe-, Ti- and Zr-based alloys are examples that belong to the first category. Eutectoid reactions involving the decomposition of intermediate phase are quite numerous in Cu-based alloys and in many of these cases, the parent phase is an electron compound or nearly so (Spencer and Mack 1962). During an eutectoid transformation in a binary alloy, the two low-temperature equilibrium phases do not necessarily precipitate directly from the hightemperature phase; the formation of intermediate products comprising transition phases is not uncommon. Even when the equilibrium phases form, the extent of solute partitioning dictated by the equilibrium diagram is rarely achieved in the initial stages. Several modes of eutectoid decomposition, involving different reaction mechanisms and engendering different types of microstructure, have been observed.
Diffusional Transformations
671
Pearlite reactions, bainite reactions, massive reactions and proeutectoid precipitation reactions are the most important reactions in the context of eutectoid systems. In alloys of near-eutectoid compositions, the first two are the most commonly observed modes of eutectoid decomposition and give rise to characteristic microstructures: pearlite contains a lamellar distribution of the decomposition products while the state of aggregation of these product phases is essentially nonlamellar in bainite (Franti et al. 1978, Aaronson 1978). On the basis of Hillert’s (1962) suggestion pearlite may be regarded to form by a cooperative growth of the solute-lean and the solute-rich phases which grow side by side in a synchronous manner. In bainite, on the other hand, non-cooperative growth occurs and the two product phases grow alternatively rather than synchronously. As mentioned in Chapter 2, the eutectoid reaction occurs in a large number of binary Ti–X and Zr–X alloy systems, where X indicates a -stabilizing element. The decomposition in these cases can be represented as → + where is an intermetallic phase of the type Tim Xn or Zrm Xn . In many of these systems, the eutectoid compositions correspond to low levels of solute concentration so that it is possible to obtain structures with large volume fractions of the eutectoid product even in dilute alloys. Figure 7.72(a) shows a schematic binary eutectoid phase diagram in which metastable boundaries between different phase fields are extended. For example, the extension of the / + boundary, denoted by the broken line AA , essentially represents the composition of the -phase which remains in metastable equilibrium with the -phase at temperatures below the eutectoid temperature, Te . This point is illustrated in the schematic free energy–composition plot (Figure 7.72(b)) for the -, - and -phases at a temperature below Te . The temperature–composition regimes over which the parent -phase tends to separate either the - or the -phase or simultaneously both - and -phases can be identified from the phase diagram with the extrapolated metastable boundaries. This can be illustrated with the example of an alloy with the composition given by co . If this hypoeutectoid alloy is rapidly cooled down from the -phase field to a temperature T1 the -phase tends to separate the -phase, the reaction being driven towards the establishment of a metastable equilibrium between the - and -phases. It is to be noted that at T1 there is no thermodynamic driving force for the nucleation of the - and the -phases. In contrast to this situation, if the alloy is rapidly brought down to a temperature T2 which is below both the / + and the / + extrapolated boundaries, the parent -phase can nucleate both the - and the -phases simultaneously. This is the essential condition for the eutectoid decomposition. Since the - and the -phases which can simultaneously nucleate at T2 differ in composition, the nucleation of one of the phases facilitates the formation of the other phase in the vicinity of the former. Fine lamellae of
Phase Transformations: Titanium and Zirconium Alloys
Temperature →
672
β α+β
Te T1
β+γ
E
B
P1
γ
α B' T2
E'
P2
Co
E' Ce
Composition → (a) T1
Free energy, G
β α
γ
Composition, c (b)
Figure 7.72. (a) Phase diagram showing a eutectoid reaction. Metastable extensions of phase boundaries are drawn by dashed lines. (b) Free energy–concentration plots for and at T1 corresponding to the phase diagram shown in (a).
the two phases are nucleated one after another, resulting in the formation of a colony. Such a colony continues to grow by the movement of the colony boundary (advancing transformation front) and eventually boundaries of neighbouring colonies impinge on each other, marking the completion of the → + transformation. It is to be noted that while the equilibrium eutectoid decomposition occurs at a fixed composition, given by the point E in Figure 7.72(a), the reaction at lower temperatures can occur over a composition range shown by the shaded portion of the phase diagram.
Diffusional Transformations
673
Not much work appears to have been carried out on the mechanisms of eutectoid decomposition in Zr alloys although this reaction is a very common feature of Zr–X systems where X is a -stabilizing element. The Ti–X eutectoids, which are nearly as numerous, are relatively better explored. Aaronson et al. (1957, 1958, 1960) have studied the Ti–Cr eutectoid in detail and have found that in hypoeutectoid alloys eutectoid decomposition occurs by a bainite or non-lamellar mode while in hypereutectoid alloys a lamellar structure does form, though only at large undercoolings. TEM investigations of Williams (1973) have shown that in the Ti–Cu system, the decomposition of alloys of near-eutectoid compositions, during continuous cooling, leads to the formation of a lamellar structure. Franti et al. (1978) have carried out a survey of eutectoid decomposition in a large number of Ti–X systems. Microstructural studies, using mainly light microscopy and supplemented by the determination of T –T –T curves, undertaken in this work have generated a large volume of useful information. They have found that in hypoeutectoid Ti–Bi, Ti–Co, Ti–Fe, Ti–Mn, Ti–Ni, Ti–Pb and Ti–Pd alloys a bainitic reaction leading to the formation of a non-lamellar dispersion of particles of the intermetallic phase, (Tim Xn ), in proeutectoid occurs. However, in hypoeutectoid Ti–Cu alloys both bainitic and pearlitic (giving rise to a lamellar structure) modes are observed. The occurrence of the pearlitic mode has been noted to be more frequent in near-eutectoid Ti–X alloys: pearlite is the only eutectoid decomposition structure in these cases when X stands for Co, Cr, Cu and Fe. But in the Ti–Ni, Ti–Pd and Ti–Pt systems, only the bainitic mode has been observed even in alloys of near-eutectoid compositions. It has been found that the T –T –T curve for the initiation of the pearlite reaction is, in general, displaced to longer times with decreasing eutectoid temperature. However, the corresponding curve for bainite reaction initiation does not show such a regular, straightforward dependence. It is well known that in Fe–C alloys of near-eutectoid as well as hypoeutectoid compositions, pearlite is the primary product of the eutectoid decomposition of austenite when the reaction occurs at temperatures not far below the eutectoid temperature. At lower reaction temperatures, the bainite reaction predominates. There is overlap of pearlite and bainite over quite a wide range of intermediate temperatures. The results of Franti et al. (1978) clearly indicate that the situation is quite different in a large number of Ti–X alloys of hypoeutectoid compositions – the bainite reaction is the only mode of eutectoid decomposition. This difference has been attributed (Franti et al. 1978) to differences in the morphology and the distribution of the proeutectoid phase in the two cases. It has been pointed out that pearlite evolves through cooperative growth and the principal barrier to pearlite formation pertains to the “initiation process” involving the formation of viable nuclei of the two constituents phases and the evolution of their growth patterns
674
Phase Transformations: Titanium and Zirconium Alloys
into the cooperative mode required for the development of the pearlite colony. Appreciable areas of disordered interphase boundary between the parent and the proeutectoid phases are needed for the evolution of pearlite because such disordered interface regions have “sufficient flexibility of shape” to permit the initiation of cooperative growth. Once the initiation barrier is overcome, pearlite growth can occur rapidly because of the short diffusion paths involved. In hypoeutectoid Fe–C alloys, over wide ranges of C content and reaction temperature, the predominant morphology of the proeutectoid ferrite phase is of the grain boundary allotriomorph type. An appreciable fraction of the interfacial area of these grain boundary allotriomorphs has a disordered structure – a situation that facilitates pearlite formation and makes pearlite the major product of eutectoid decomposition. The pearlite reaction occurs so rapidly that the evolution of side plates from the grain boundary allotriomorphs and the formation of intragranular plates are suppressed. Only at very large undercoolings does an appreciable population of Widmanstatten plates with partially coherent interfaces form. When this happens, pearlite formation becomes difficult because of lack of disorder at the interfaces and the bainitic mode of decomposition becomes dominant. In hypoeutectoid Ti–X alloys, however, the situation is quite different in that side plates and intragranular plates constitute the predominant proeutectoid morphology even at small undercoolings. The partially coherent boundaries of these plates are unsuitable for initiating pearlite formation. However, the ledges on these boundaries may act as nucleation sites for isolated precipitates of the intermetallic phase, . The bainite reaction is, therefore, preferred. In the singular case of the Ti–Cu system, where pearlite does form in alloys of hypoeutectoid composition, it has been suggested (Franti et al. 1978) that the kinetics of pearlite nucleation are so rapid that the reaction can start at some grain boundary allotriomorphs in the very short period available before the possibility of pearlite formation is foreclosed by the evolution of closely spaced side plates from the allotriomorphs. The observation that the occurrence of a pearlite structure is more common in near-eutectoid alloys than in their hypoeutectoid counterparts has been rationalized along the following lines (Franti et al. 1978). In alloys of near-eutectoid compositions, the driving forces for the precipitation of the - and the -phases are more nearly equal than in the hypoeutectoid case. Therefore, even if -allotriomorphs precipitate first, there is a reasonable chance that at suitable undercoolings, intermetallic phase precipitation can compete with side plate formation at the allotriomorph boundaries. Precipitation of the -phase at such locations can provide an opportunity for the initiation of the process of pearlite formation. Thus systems in which the pearlite reaction predominates near the eutectoid composition are those where the nucleation of -precipitates at / interfaces can occur rapidly. In fact, it has been seen (Franti et al. 1978) that in near-eutectoid Ti–Cu, Ti–Co and Ti–Fe
Diffusional Transformations
675
alloys, the T –T –T curves for the initiation of the proeutectoid and the pearlite reactions almost coincide. It is likely that many of the features of the Ti–X eutectoids are pertinent to Zr–X eutectoids also. However, detailed experimental confirmation in this regard is yet to come. 7.7.1 Active eutectoid systems In Ti–X and Zr–X eutectoid systems, the relative “activity” of the eutectoid decomposition process varies significantly from system to system. “Active” eutectoid systems (Jaffee 1958) are those in which the -phase decomposes rapidly into the - and the -phases; the latter is not necessarily the equilibrium intermetallic phase pertinent to the system. In alloys of near-eutectoid compositions, this -phase decomposition occurs so fast that it cannot be suppressed even by a rapid -quenching. The state of aggregation of the decomposition products in such alloys is lamellar, so that a fine, pearlite-like microstructure is obtained on rapid cooling. This structure may be too fine to be resolved by light microscopy and its details can be seen only by techniques like transmission electron microscopy. On subsequent ageing at temperatures below the eutectoid temperature, discontinuous coarsening may occur and the fine lamellar structure may be replaced by a coarse lamellar structure. The coarsening reaction is driven by (a) the chemical free energy change associated with the replacement of the metastable phases resulting from incomplete partitioning of the alloying elements in the rapidly formed fine lamellae by the equilibrium phases and (b) the reduction in the surface energy due to a significant annihilation of interfaces separating the product phase lamellae. In a “sluggish” eutectoid system, the eutectoid decomposition of the -phase is slow and it is possible to retain this phase metastably at low temperatures even on slow cooling. On the basis of observations made so far on Ti–X and Zr–X eutectoid systems, it has been suggested in a qualitative manner (Jaffee 1958, Mukhopadhyay et al. 1979) that active eutectoid decomposition occurs in those systems where the eutectoid temperature is high, the eutectoid composition is solute lean and the intermetallic phase resulting from the reaction is rich in the base metal. The formation of a fine lamellar structure, resolvable by TEM, on rapidly quenching alloys of near-eutectoid compositions from the -phase field, has been observed in the systems Zr–Cu, Zr–Fe, Zr–Ni (Mukhopadhyay 1985) and Ti–Cu (Williams et al. 1970). In all these cases, the aforementioned criteria for the occurrence of active eutectoid decomposition hold good. A survey of the reported eutectoid temperatures and compositions corresponding to binary Ti–X and Zr–X eutectoid systems also reveals that, in general, the latter are associated with higher eutectoid temperatures and eutectoid compositions
676
Phase Transformations: Titanium and Zirconium Alloys
leaner in the solute as compared to the former (Massalski et al. 1992). Whether these facts imply that active eutectoid decomposition is more common in Zr–X than in Ti–X eutectoids is a question that is yet to be resolved. In the following sections, the observations made on the active eutectoid decomposition in near-eutectoid Zr–Cu and Zr–Fe alloys will be reviewed. 7.7.2 Active eutectoid decomposition in Zr–Cu and Zr–Fe system Active eutectoid decomposition in a Zr–Cu alloy of near-eutectoid composition has been studied by Mukhopadhyay et al. (1979) and their observations and inferences will be discussed at some length here. These workers have attempted to provide a morphological and crystallographic description of the decomposition product and have also proposed a possible mechanism of the transformation. It has already been mentioned that a similar decomposition reaction has been observed in neareutectoid Ti–Cu alloys (Williams et al. 1970) The eutectoid reactions in these systems are → +Zr2 Cu and → +Ti2 Cu, respectively. Both the intermetallic phases are isostructural with MoSi2 (Pearson 1967). The lattice parameters are a = 032204 nm c = 111832 nm for Zr2 Cu and a = 02944 nm c = 10782 nm for Ti2 Cu so that the values of the axial ratio are 3.47 and 3.66, respectively. Pearson (1972) has pointed out that the values of the c/a ratio for MoSi2 -type phases are clustered in two ranges: 2.4–2.5 and 3.2–3.7. In the former, each atom has a CN 10 environment while in the latter (e.g. in Zr2 Cu and Ti2 Cu) the CN 10 polyhedron is distorted so as to produce an approximately CN 8 environment. It can be mentioned here that a CN 8 environment also exists in a phase with the bcc structure (e.g. -Zr or -Ti). A comparison of the -Zr and the Zr2 Cu structure is shown in Figure 7.73. The microstructural observations made on a -quenched, near-eutectoid (Zr–1.6 wt% Cu) alloy could be summarized as follows (Mukhopadhyay et al. 1979). The quenched material consists of several colonies with lamellar structure, consecutive lamellae of the -phase being separated by ribbon-like features. In a given cell, the -lamellae have a single crystallographic orientation and so have ¯ -type the ribbons. In some regions, the latter are straight and parallel to the {1011} planes while in some other regions in the same colony these have a “wavy” appearance (Figure 7.74(e) and (f)). In the transition region, a one to one correspondence between the straight and the wavy segments exists. The average colony size and the average interlamellar spacing are about 2.0 and 0.1 m, respectively. Some of these features are illustrated in Figure 7.60. The ribbons give rise to extra spots in the selected area diffraction patterns and these spots could be indexed in terms of the Zr2 Cu structure in an approximate manner in the sense that an exact matching between the observed and the calculated interplanar angles is not obtained. This suggests that the structure of the observed second phase is similar
Diffusional Transformations
677
β unit cell a = 3.609 Å
a
C
a
β' unit cell a = 3.40 Å
a
Zr Cu
Zr2Cu unit cell a = 3.22 Å; c = 11.18 Å
Figure 7.73. Unit cells of , metastable and equilibrium Zr2 Cu phase.
but not identical to that of Zr2 Cu. The matching is better in regions with wavy ribbons than in regions with straight ribbons. Assuming the phase constituting the ribbons to be the Zr2 Cu phase, the following planar and directional matching ¯ ¯ with the -phase could be obtained: (0001) (013)Zr2 Cu ; [1100] [031]Zr2 Cu . This orientation relationship (Figure 7.75) is consistent with that observed between the - and the Ti2 Cu phases in the Ti–Cu system (Williams 1973) It is also found that the flat faces of the ribbons correspond to one of the following Zr2 Cu ¯ ¯ planes: (110), (110) (103), and (103). It could be seen that habit plane variants such as (110) and (103) are quite similar with respect to atomic arrangement on these planes and interplanar spacing though small differences in interatomic and interplanar distances do exist because of the difference in the values of a and c/3 for the Zr2 Cu unit cell. No interfacial dislocations are observed on the flat faces of the ribbons, implying that the coherency strains are not large enough to warrant the generation of misfit dislocations. However, considerations of the ¯ planes – the habit planes of the ribbons in terms mismatch between the {1011} of the -structure – on the one hand and (110) and (103) planes of Zr2 Cu on the other indicate that interfaces separating and Zr2 Cu crystals should, in all likelihood, be semicoherent and, therefore, misfit dislocations should be present
678
Phase Transformations: Titanium and Zirconium Alloys
0.5 μm
0.5 μm (a)
0.5 μm (b)
(c)
0.2 μm
0.2 μm
0.2 μm (d)
(e)
(f)
Figure 7.74. Microstructure of active eutectoid product in Zr–1.6 wt% Cu (a) period and straight lamellae of + structure resembling internally twinned martensite structure. (b) Straight lamellae degenerating into wavy lamellae. (c) and (d) Bright and dark ( -reflection) field image of + structure. (e) and (f) Bright- and dark-field images show one to one correspondence between straight and wavy lamellae.
114
112
006
004
002
116
114
112
116
1211
0111
1013
110
112
114
1102
1102
1211
116
000
110
002
004
112
114
0111
006
116 1013
α-Reflections Zr2Cu Reflections
Figure 7.75. Diffraction pattern showing orientation relation between the - and the -phases.
Diffusional Transformations
679
at such interfaces. It appears that the observed second phase in the -quenched alloy has neither the exact structure nor the exact stoichiometry of the equilibrium Zr2 Cu phase, presumably because the partitioning of the Cu atoms into the two constituent phases of the decomposition product remains incomplete during the rapid cooling process. While attempting to work out a possible mechanism of the observed active eutectoid decomposition, Mukhopadhyay et al. (1979) have pointed out that the Zr2 Cu unit cell could be constructed by stacking three bct cells, one above the other (Figure 7.62). In fact, the Zr2 Cu structure could be built by introducing a state of order in a disordered bcc structure (designated as in such a manner that there would be approximately a threefold increase in the unit cell dimensions along the c-axis. The lattice parameters of the bcc phases and are not much different and it could be considered very likely that a one to one correspondence, implying the parallelism of the corresponding orthogonal axes, exists between the - and -lattices. Making use of such a correspondence, the indices of planes and directions in either of the two bcc structures could be expressed in terms of indices referred to in the Zr2 Cu structure. Using such relationships and the observed lattice correspondence between the and the Zr2 Cu phases, it has been possible to obtain the orientation relation between the parent - and the -phases. It is seen that the lattice correspondence between these phases is the Burgers correspondence. It is ¯ -type habit plane variants of the second also noticed that all the observed {1011} phase lamellae are derived from {110}-type mirror planes of the -phase. Further, ¯ the (1102) plane is derived from the (001) plane. It could be seen in suitable ¯ selected area diffraction patterns that the (1102) spot of the -phase and the (002) spot of the -phase are near coincident and that superlattice spots subdivide the line joining the central spot and the (002) spot into three equal segments. This indicates that the metastable phase constituting the ribbons has attained a certain extent of long-range order and this has resulted in an approximately threefold increase in the dimension of the unit cell in the [001] direction. In the light of the aforementioned considerations, it has been suggested that during -quenching, the -phase decomposes into a mixture of the -phase and a partially ordered phase derived from . Regions containing straight and wavy lamellae in the same colony possibly correspond to different levels of Cu enrichment and, therefore, to different degrees of long-range order. The crystallographic description of the lamellar product structure with respect to the parent -structure is illustrated schematically in Figure 7.76. The presence of a lattice correspondence suggests that it might be possible for the lamellar, two-phase decomposition product to form from the -phase essentially by cooperative atom movements. However, the development of partial order within the second phase lamellae would necessitate solute transport to these regions by diffusion.
680
Phase Transformations: Titanium and Zirconium Alloys
(110)β
Tra ns for m
ati on f
ron t
(0001)α
(0001)α
β′(Zr2Cu)
(110)β′
α
Figure 7.76. Lattice correspondence between the parent - and the product - and -structures across the transformation front.
The interfaces between the parent -phase and the -phases lamellae would be similar to martensite interfaces and would thus be glissile. The boundaries separating the -phase and the second phase lamellae would also be expected to be glissile in view of the one to one lattice correspondence between the - and the -phases. The propagation of these second type of interfaces across the parent -region would make the -lattice shrink by about 5% to yield the -lattice and would, at the same time, cause a certain degree of long-range order to be established in this newly formed -lattice. This ordering would require the long-range migration of Cu atoms within the untransformed phase, ahead of the transformation front. Since the solubility of Cu in -Zr is much smaller than that in -Zr (Massalski et al. 1992) such a process would not be unlikely. The release of Cu atoms from an advancing -lamella would render the untransformed -phase region ahead of
Diffusional Transformations
681
it enriched in Cu. The composition gradient built up in this manner would result in a continuous flow of Cu atoms towards the -regions ahead of the adjacent lamellae of the metastable second phase. Thus morphology wise and also from the point of view of the mechanism of the distribution of Cu atoms in the two product phases, the mode of decomposition would be akin to a pearlite reaction. However, in a typical pearlite reaction, the transformation front is incoherent and no lattice correspondence exists across this interface (Hornbogen 1972). In the Zr–Cu case, the presence of wavy lamellae suggests that at a certain stage in the progress of the reaction, the crystallographic requirements are relaxed and the transformation front ceases to be coherent, making the reaction tend to resemble a normal pearlite reaction. A simplified analysis has been carried out with a view to compare the observed reaction rates with those estimated. It has been found that the observed and the estimated growth rates are more or less of the same order at temperatures not far below the eutectoid temperature. This analysis emphasizes the fact that solute partitioning between the two product phases would be quite significant in spite of the extremely fast reaction rates. This is possible due to rapid anomalous diffusion in the -phase and the exceedingly small interlamellar spacing in the decomposition product. To summarize, Mukhopadhyay et al. (1979) suggest that the rapid eutectoid decomposition in near-eutectoid Zr–Cu alloys, resulting in the formation of a lamellar aggregate of the -phase and a partially ordered, bcc, metastable phase (structurally similar but not identical to Zr2 Cu), occurs through the following steps. (1) Partitioning of solute atoms takes place ahead of the transformation front by bulk diffusion. (2) Cooperative atom transport occurs across this front, leading to the formation of lamellae of the -phase and a metastable second phase which has an ordered bcc structure. During this process, the front remains coherent and the ¯ habits. (3) Subsequently the transformation front lamellae strictly adhere to {1011} becomes incoherent, facilitating the partitioning of Cu atoms and the attainment of the stoichiometry and state of order characteristic of the equilibrium Zr2 Cu phase. It has been observed that an active eutectoid decomposition, leading to the formation of a lamellar structure on rapid -quenching occurs in near-eutectoid Zr–Fe alloys also (Mukhopadhyay 1985). The microstructural features associated with this decomposition reaction are quite similar to those pertaining to the Zr–Cu case and will be described briefly here. Rhines and Gould (1962) have suggested that in the Zr–Fe system the intermetallic phase richest in Zr is Zr4 Fe and that the eutectoid reaction involves this phase: → + Zr4 Fe. However, Buschow (1981) has found that in conventionally melted Zr–Fe alloys, the Zr4 Fe phase does not occur and that Zr3 Fe is the first intermetallic phase to form on the Zr-rich side.
682
Phase Transformations: Titanium and Zirconium Alloys
Thus the eutectoid reaction in the Zr–Fe system is → +Zr3 Fe. The Zr3 Fe phase has a base-centred orthorhombic structure (Re3 B type) with a = 03326 nm b = 10988 nm and c = 08807 nm. A Zr–4 wt% Fe alloy, on being rapidly cooled from the -phase field, exhibits a pearlite-like lamellar structure. In each colony, -lamellae are separated by ribbon-like features which are straight in some regions and wavy in some others. As in the Zr–Cu case, a one to one correspondence exists between the straight and the wavy segments. In some instances, the second phase lamellae have a fragmented appearance. This phase (constituting the ribbons) gives rise to spots in selected area diffraction patterns and can be imaged in the dark field by using these reflections. A limited amount of analysis has shown that these extra diffraction spots can be indexed in terms of the Zr3 Fe structure. Similar features have been observed in a rapidly quenched Zr–3 wt% Fe alloy, except that in this case the colonies have a lath-like, and not equiaxed, shape. Moreover, some of these laths do not have a lamellar structure and appear to contain only the -phase. In either alloy, ageing at temperature below the eutectoid temperature causes the lamellar structure to be replaced by a coarse distribution of globular Zr3 Fe particles in the -matrix. Some of these features are illustrated in Figure 7.77. As discussed earlier, the mechanism proposed for the active eutectoid decomposition in near-eutectoid Zr–Cu alloys makes use of the fact that the Zr2 Cu structure is related to and can be derived from the bcc structure in a simple manner. A similar situation exists in respect of the Ti–Cu system. In the case of Zr3 Fe, however, such a simple structural relationship is not apparent. Nevertheless, from considerations of reaction kinetics and post-reaction microstructure, the decomposition process does appear to be very similar in all these three cases.
(a)
(b)
Figure 7.77. Microstructures of active eutectoid product in Zr–3 wt% Fe.
Diffusional Transformations
7.8
683
MICROSTRUCTURAL EVOLUTION DURING THERMO-MECHANICAL PROCESSING OF Ti- AND Zr-BASED ALLOYS
Bulk metal working of Zr and Ti alloys comprises a series of steps, each of which has important roles to play in determining the microstructure of the finished product. It generally involves two stages of hot working following the melting step and subsequently a few steps of secondary working operations for achieving suitable microstructure, specified shapes and desired mechanical properties (McQueen and Bourell 1988, Weiss and Semiatin 1999). Amongst these steps, not working in the form of forging, extrusion or rolling is very important, as in addition to breaking the cast microstructure and shaping the ingot, this step is responsible for providing a chemically homogeneous product with a suitable microstructure for subsequent processing. In industry, Ti and Zr alloys are hot worked in the , or in the ( + ) phase regime, depending on alloy composition and specific microstructural/mechanical property requirements. A difference in crystal structure, coupled with a very high diffusivity in the -phase (2–3 orders of magnitude higher than that in the -phase), promotes the processes of recovery, recrystallization and diffusional deformation at lower temperatures and higher strain rates in the -phase as compared to the -phase. Consequently, the constitutive deformation behaviour of these phases under hot working conditions are different and this dictates the evolution of microstructure in the product. In general, the important structural changes occurring during hot working are recovery and recrystallization, leading to the annihilation of various crystal defects and softening of the material which dynamically balance strain hardening. In addition to the dynamic recrystallization (DRX) and dynamic recovery(DRV) processes, superplastic deformation, in which large-scale structural modifications are not involved, has also been reported in a number of Ti alloys and in a few Zr alloys (McQueen and Bourell 1988). During DRV, structural changes result primarily from the annihilation and rearrangement of dislocations into low-energy arrays. These mechanisms lead to the formation of low-angle sub-boundaries, dislocation arrays which have a limited capability for migration. The nucleation process in DRX involves dynamic creation and propagation of interfaces which separate highly deformed areas from newly formed, relatively strain-free grains. The growth begins with migration of this interface when it assumes the configuration of a large angle boundary. The nucleation mechanism may be different, depending on the strain rate. At low strain rates nucleation is by boundary bulging and at high strain rates the nucleation results from conversion of subgrain walls into high misorientation boundaries as a result of the accumulation of glide dislocations. During superplastic deformation, the major contributors to superplastic deformation are grain boundary or interface
684
Phase Transformations: Titanium and Zirconium Alloys
sliding, grain rotation and grain switching, and consequently large-scale structural modifications, as seen in DRX, are not possible. Although the dislocations are not involved in a major way during plastic deformation, they play a significant role in the accommodation process. However, minor structural modifications such as agglomeration of second phase particles is possible during the grain switching process. Structural changes can also occur during the intervals of hot working in the absence of stress (static) as well as during cooling after hot working (McQueen and Bourell 1988). These static modifications in structure brought about by recovery and recrystallization are important in deciding the final mechanical properties of both the alloy systems. Because of these structural changes during hot deformation and different types of phase transformations occurring in Ti alloys as has been mentioned earlier, there is extensive opportunity for thermomechanical processing (TMP). A large volume of research has been carried out to understand various aspects of TMP of Ti alloys including the recently developed Ti aluminides. It has been found that besides processing parameters (temperature, deformation strain rate and amount of deformation), factors like initial microstructure, alloy composition and physical properties such as stacking fault energy and diffusivity have a profound effect on the evolution of the microstructure (McQueen and Jonas 1975). 7.8.1 Identification of hot deformation mechanisms through processing maps In order to understand the microstructural evolution, the underlying hot deformation mechanisms must be identified by studying the constitutive behaviour of the work piece (relationship between stress, strain rate and strain) under hot working conditions. This is necessary as each of the hot deformation mechanisms is distinctly different from the others in terms of the permitted ranges of strain rate and temperature for its operation, its dependence on initial microstructure and associated strain rate sensitivity (m = ln / ln $˙ ) of stress. In addition to these hot deformation mechanisms, Zr and Ti alloy systems are known to exhibit strain localization leading to non-uniform microstructure during deformation processing. Thus control of microstructure and avoidance of non-uniform microstructure are of paramount importance in optimizing the final mechanical properties. There are several approaches of materials modelling to study the constitutive behaviour as well as for predicting hot deformation mechanisms, namely, study of the shape of the stress–strain curve, kinetic analysis of the deformation process and construction of processing maps (Prasad and Seshacharyulu 1998a,b) with identification of different processes operating in different domains. Amongst these approaches mentioned, the construction of processing maps has been found to be quite useful in various alloys (Prasad and Seshacharyulu 1998a,b, Prasad 1990). The processing maps are developed using the principles of the dynamic materials model which
Diffusional Transformations
685
considers the work piece being subjected to hot deformation essentially as a dissipator of power (Prasad and Seshacharyulu 1998a, Prasad 1990). The constitutive behaviour describes the manner in which the power is converted at any given instant into two forms: most of it through a temperature rise (G content) and a smaller part through a microstructural change (J co-content). The factor that partitions power between these two processes is the strain rate sensitivity of flow stress which is unity for an ideal linear dissipator. At a given temperature, strain J is given by (Prasad and Seshacharyulu 1998a,b, Prasad 1990) J = $˙ m/ m + 1
(7.60)
where $˙ is the strain rate. The efficiency of power dissipation ( ) of a work piece can be obtained by comparing its power dissipation through microstructural changes with that occurring in an ideal linear dissipator (m = 1) and is given by = 2m/ m + 1
(7.61)
The variation of with temperature and strain rate gives the power dissipation map. Over this frame is superimposed a continuum instability map in which a dimensionless instability parameter, %(˙$) defined as (Kalyanakumar 1987), % ˙$ =
ln m/m + 1 +m ≤ 0 ln $˙
(7.62)
is plotted as a function of strain rate and temperature. The instability map is developed on the basis of extremum principles of irreversible thermodynamics applied to large plastic flow as given by Zeigler (1963). Flow instabilities are predicted to occur when %(˙$) assumes negative values. The processing maps exhibit domains in which shows local maxima where specific microstructural mechanisms operate as well as regimes where flow instabilities like adiabatic shear bands or flow localization occur. To illustrate the concept, a contour map (isoefficiency contours on a strain rate temperature frame) for commercial pure Zr is shown in Figure 7.78. The processing map shows two domains, one corresponding to DRV and the other related to the process of DRX (as marked). These domains have been identified on the basis of the examination of the deformed microstructure (conditions $˙ , T ; Figure 7.79) of samples quenched from the processing temperature (Chakravartty 1992). In the following, some efforts made so far to understand the microstructural evolution in Ti- and Zr-based alloys using various methodologies, especially construction of processing maps, are highlighted.
686
Phase Transformations: Titanium and Zirconium Alloys Zirconium- 2 (β-quenched)
Strain = 0.4
2
2 8 1
11
Log strain rate
15 0
19 41
24 –1
DRX
28
–2
37 32 37 41
–3 650
DRV
15 24
690
730
770
810
850
Temperature (°C)
Figure 7.78. Processing map for zirconium for a strain of 0.4 showing domains of dynamic recrystallization (DRX) and dynamic recovery (DRV). The number against each contour indicates percentage efficiency.
Figure 7.79. The optical micrograph of -Zr deformed at 800 C and 0.1 s−1 within the DRX domain to a strain of 0.7 showing dynamically recrystallized grains.
Diffusional Transformations
687
7.8.2 Development of microstructure during hot working of Ti alloys 7.8.2.1 -alloys Prasad and Seshacharyulu (1998b) have studied the hot deformation characteristics of commercially pure -Ti over wide ranges of strain rate (0.001–100 s−1 ) and temperature 873–1123 K by constructing processing maps. DRX has been observed in the temperature range of 873–1123 K and strain rate range of 0.001–0.1 s−1 . The occurrence of DRX has been found to be sensitive to the O content, a lowering of which results in increasing the strain rate and temperature of the DRX domain. In addition to DRX, DRV has also been reported in higher purity Ti. Weiss and Semiatin (1999) have recently reviewed the deformation behaviour of -Ti alloys and have observed that during deformation below the -transus temperature, -alloys undergo DRV in which the rate of hardening by generation of dislocations is balanced by the rate of softening due to dislocation annihilation, resulting in a steady-state flow curve. The deformed microstructure shows both elongated and equiaxed morphology of the -phase with substructure within. Under similar deformation conditions a near--Ti alloy IMI 685 with equiaxed (+) morphology has been reported to exhibit superplasticity (Gopalkrishna et al. 1997). The deformed microstructure of this micro-duplex alloys has been found to contain partially recrystallized and plate-like in a transformed -matrix. However, the same alloy, with a lower O content and with an acicular morphology has been found to show a domain of – spheroidization at strain rates lower than 0.001 s−1 in the temperature range 1123–1233 K. The constitutive behaviour of these alloy systems has been studied above the -transus temperature and it has been observed that the shape of the stress–strain diagrams are of steady-state type and the apparent activation energy associated with the process of deformation is in the range 180–220 kJ/mol which is close to the activation energy for selfdiffusion. Thus it appears that deformation in the -phase is controlled by dynamic recovery (Weiss and Semiatin 1999). Large-grain superplasticity and DRX have also been reported in -alloys during deformation at temperatures in the -phase field, respectively, at lower and higher strain rates (Prasad and Seshacharyulu 1998b). It has been noted that some Ti alloys like IMI 685 have very narrow processing window because of the occurrence of a flow instability at strain rates as low as 0.1 s−1 and higher over a wide range of temperatures. Therefore, in practice processing steps are carefully designed for successful forging and microstructural control. 7.8.2.2 + alloys These alloys offer considerable scope for obtaining a variety of microstructures by TMP (McQueen and Bourell 1988). Depending on the detailed thermo-mechanical
688
Phase Transformations: Titanium and Zirconium Alloys
treatment, lamellar, equiaxed or bimodal microstructures can be obtained. The deformation characteristics of the two-phase alloys have been studied in great detail over wide ranges of temperature and strain rate, for different initial microstructures. In an early work (Shakanova et al. 1980) on Ti–6Al–4V and Ti–6Al–2Mo– 2Cr alloys, the evolution of microstructure during hot deformation was studied over the temperature range of 1123–1373 K and strain rate range of 0.0001–1 s−1 for two different initial microstructures, namely, the ( + ) equiaxed and the transformed -structures. The -phase was observed to have undergone polygonization during deformation over the entire range of temperatures studied, but as the deformation temperature was increased (T > 1193 K), DRX became the dominant deformation mode at lower strain rates. Increasing the temperature of deformation enhanced the process of DRX. Both polygonization and DRX were also observed in the -phase in which subgrains were formed. DRX and polygonization was recorded on deformation in both the initial microstructures but the DRX process was dominant in the case of equiaxed morphology. These alloys also exhibited superplasticity with equiaxed morphology on deformation at lower strain rates (≈0.0001 s−1 ) and at temperatures in the range 1198–1223 K. Sastry et al. (1980) carried out detailed TEM investigation of hot deformed Ti–6Al–4V material at a strain rate of 0.05 s−1 and at temperatures above 850 C and reported that both DRX and DRV occurred as evidenced by a hexagonal network of dislocations in the -phase, formation of small equiaxed -grains and absence of shearing in -phase. Recently, Seshacharyulu et al. (2000, 2002) studied the effect of initial microstructures (equiaxed and -transformed) and impurity content on the hot working behaviour of Ti–6Al–4V by constructing processing maps over a wide range of strain rate and temperature. The processing map for the equiaxed morphology exhibited two domains, viz., a domain of fine grain superplasticity in the ( + ) region and another domain of DRX in the -phase field. The ratecontrolling accommodation process during superplastic deformation was identified to be cross slip in -phase which was located at the triple junction of -grains. The activation energy for the process of DRX in the -phase was close to that for self-diffusion in . During deformation in the ( + ) phase field , the -plates of the transformed -structure assumed equiaxed morphology by the process of globularization. The primary -grain size varied linearly with the Z parameter (Zener Hollomon parameter Z = $˙ exp [Q/RT], where $˙ is the strain rate, T is the temperature in K and Q is the apparent activation energy) in a manner similar to that observed during DRX. The deformation in the -phase field resulted in DRX with characteristics similar to that observed in the initial equiaxed microstructure. Chen and Coyne (1976) carried out isothermal forging experiments on Ti–6Al–4V and identified DRX as the rate-controlling deformation process on the basis of the apparent activation energy obtained by kinetic analysis. Weiss et al. (1986)
Diffusional Transformations
689
carried out detailed TEM investigation of the hot deformed microstructure of a Ti–6Al–4V alloy with Widmanstatten morphology and reported that -lamellae break up during deformation either heterogeneously by forming intense shear bands or homogeneously by formation of a dislocation substructure. Depending on the extent of localized shear strain, the -lamellae may sever completely or partially. These sheared regions, during the course of deformation, will either form / grain boundaries or will get converted to a structure in which -regions are separated by -phase by its complete severance by percolation of -phase. During the homogeneous deformation, however, dislocations rearrange to form recovered substructure leading to the formation of subgrains which divide the lamellae into smaller units. The -phase then penetrates along the / sub-boundaries. 7.8.2.3 -alloys Although the -alloys are cold workable and can be recrystallized at temperatures as low as 1073 K to obtain fine grains, the initial bulk working of the ingot is done at temperatures above their -transus followed by secondary working either above or below the transus (Weiss and Semiatin 1998). Numerous hot working studies have been conducted on -alloys both in the single phase as well as two-phase ( + ) regime. The various aspects of thermo-mechanical processing of -Ti alloys have been reviewed by Weiss and Semiatin (1998). These alloys were found to exhibit discontinuous yielding followed by almost a steady-state flow behaviour, formation of serrated -grain boundaries on deformation and the presence of subboundaries ranging from medium- to high-angle boundaries during deformation in the -phase. The deformation in the -field appeared to be controlled by dynamic recovery with an apparent activation energy which matched with that of self-diffusion. Similar conclusion was drawn by Robertson and McShane (1998) who examined the effect of two initial microstructures, viz., -transformed and equiaxed ( + ), on the deformation behaviour of Ti–10V–2Fe–3Al alloy at lower strain rates. Balasubrahmanyam and Prasad (2001, 2002) have characterized the hot deformation behaviour of two -alloys, Ti–10V–2Fe–3Al and Ti–10V–4.5Fe– 1.5Al over a wider range of strain rates and temperatures using processing maps. They found that the processing of -alloy (Ti–10V–4.5Fe–1.5Al) showed a single domain in the temperature range 1023–1173 K and strain rate range 0.01–0.1 s−1 . On the basis of the microstructural features, the variation of grain size of the deformed grain structure (predominantly equiaxed) with temperature and tensile ductility variations, it was concluded that the prevailing deformation mechanism in the domain was DRX. The processing maps of the metastable -alloy (Ti–10V– 2Fe–3Al) on the other hand showed two domains during deformation in the similar temperature and strain rate ranges. In the temperature range 923–1023 K and at strain rates lower than 0.1 s−1 , the material deformed superplastically exhibiting
690
Phase Transformations: Titanium and Zirconium Alloys
large ductility. The accommodation process was identified to be dynamic recovery from the nature of stress-strain diagram, which was of steady state type, and from the apparent activation energy value, which matched with the activation energy for self-diffusion in -Ti. At temperatures higher than 1073 K and strain rates lower than ∼0.1 s−1 , the alloy exhibited large grain superplasticity (LGSP) with a deformed microstructure containing stable subgrain structure within large -grains. LGSP was also reported to occur during hot deformation of -Ti alloys by Griffith and Hammond (1972). 7.8.2.4 Ti-aluminides Titanium aluminides offer a combination of properties such as low density, high strength to weight ratio, high modulus and good elevated temperature strength properties which make them excellent candidate material for gas turbine engine and airframe structure in advanced aircrafts. The aim during primary ingot breaking of these materials is to prevent catastrophic fracture during shape change as well as to provide suitable homogeneous microstructure to enhance hot workability during secondary operations (Semiatin et al. 1992, 1998). One of the main limitations of these materials is their limited workability during conventional metal working processes such as forging, rolling and extrusion because of the generation of various internal defects. The importance of primary hot working can be emphasized by taking a typical example of a near--TiAl alloy. This variety of aluminides, with aluminium content more than 46%, does not have single phase -field as in the case of conventional titanium alloys. This coupled with significant coring of the dendritic arms that result from the peritectic solidification and presence of ordered phases even at very high temperatures (> 1573 K) make primary ingot breakdown a technological challenge. The hot deformation mechanisms of these materials during secondary processing have been found to be a strong function of the starting microstructure, deformation temperature and strain rate in a manner similar to that observed with conventional Ti alloys. The major issues of thermo-mechanical processing of Ti-aluminides have been reviewed by Semiatin et al. (1998). A considerable amount of effort has been directed to identify the suitable hot working conditions and the associated deformation mechanisms to obtain defect-free products for a variety of single phase 2 , single phase , twophase (2 + ) and near- alloys by developing workability maps as well as by microstructural observation of hot deformed materials. Semiatin et al. (1998) and Huang et al. (1995, 1998) studied the deformation behaviour of cast Ti–24Al–11Nb over a range of temperature and strain rate and identified three different regimes of deformation: a regime of warm working where deformation led to distortion of -laths, wedge cracking and cavity formation, a region of globularization of -laths in the two-phase (2 +) and a domain of hot working in the single phase .
Diffusional Transformations
691
Sagar et al. (1994, 1996) studied the hot deformation behaviour of a similar alloy in two different initial microstructural states employing dynamic materials model. They identified a domain of DRX of 2 -phase in both the starting microstructures. In addition, they have also observed a second domain of DRX and a domain of superplastic deformation of -phase for different combinations of strain rate and temperature. The flow behaviour and microstructural development during forging of super 2 (Ti-25Al-10Nb-3V-1Mo) have been studied by Huang and co-workers (1995, 1998). It was found that while both DRV and deformation heating were involved in microstructural evolution during hot deformation of (2 + ) alloys at rates lower than 10 s−1 , the dominant deformation mechanism was DRX at rates higher than 10 s−1 . However, both DRX and DRV coupled with certain amount of grain boundary sliding were responsible for flow softening during deformation in the -phase field at both high and low rates of deformation. In view of very restricted workability of large lamellar microstructure of aluminides, several TMP methods are being increasingly employed to break down the initial microstructure to smaller units (Martin 1998, Sun et al. 2002). For microstructural refinement of orthorhombic Ti–22Al–27Nb, Martin (1998) used near-isothermal processes involving rolling at temperatures just below the -transus followed by a thermal treatment to obtain a fine grain metastable -phase. The lamellar microstructure of near- Ti–Al, which does not have an overlying -phase, is broken down by extrusion in the 2 + phase field by dynamic recrystallization. Subsequently further microstructural refinement is achieved by multiple isothermal forgings. Recently the technique of hot-die incremental deformation has been used with intermittent recovery to breakdown the large 2 + lamellae into fine 2 + microstructure (Pan et al. 2001). Although the hot deformation mechanisms that are involved in conventional Ti alloys are also present in the titanium aluminide alloys, the window of hot working of Ti-aluminides in the as-cast condition is very restricted because of the ease of intergranular fracture, cavitation and flow localization even at higher temperatures. An initial thermo-mechanical treatment to break up the lamellae into fine microstructure is therefore essential for good workability, with the added capability of achieving even superplasticity. 7.8.3 Hot working of Zr alloys The number of commercial alloys of Zr is considerably smaller than that of Ti. Zr-based alloys are chosen primarily for the manufacture of some critical structural components in nuclear reactors. As has been indicated earlier, these alloys are fabricated by processing routes, involving a melting stage, hot working in two stages and cold working with or without annealing. It has been observed by Chedale et al. (1972) that the scale of the final microstructure and texture are determined primarily in the second hot working stage which is extrusion. It has
692
Phase Transformations: Titanium and Zirconium Alloys
been shown that the extrusion speed and temperature are two important parameters which have significant effect on the development of texture, microstructure and mechanical properties of the finished products (Chedale et al. 1972, Konishi et al. 1972). It is therefore essential to optimize them in order to arrive at the desired final properties. 7.8.3.1 and near--Zr alloys Jonas and co-workers (Lutton and Jonas 1972, Abson and Jonas 1973) studied the high-temperature deformation behaviour of -Zr and -Zr-tin alloys in compression in detail and examined the microstructural development by TEM. They concluded dynamic recovery to be occurring in the temperature range 898–1098 K and strain rate range 10−4 to 3 × 10−3 s−1 on the basis of the observed shape of stress–strain diagram, kinetic analysis performed and detail TEM investigation of deformed microstructure. Although the deformation was carried out at or above the recrystallization temperature (static), no recrystallized grain has been observed in the deformed material. The grain shape in general was elongated and was found to contain sub-boundaries. The mean size and perfection of these boundaries decreased with decreasing temperature, increasing strain rate and increasing tin content. On the other hand, Ostberg and Attermo (1962) carried out forging of Zircaloy-2 (a near--alloy) at 1073 K and at a strain rate of 0.4 s−1 and observed that the original Widmanstatten -plates were broken up into new grains which were equiaxed. They concluded that DRX was involved in the modification of the microstructure. It may be noted that the deformation of Zircaloy-2 at 1073 K is in the -phase and deformation at temperatures higher than 1083 K and up to -transus will be essentially in two-phase ( + ). Garde et al. (1978) studied hot tensile deformation behaviour of zircaloy-2 in the two-phase ( + ) field at various strain rates. The microstructure evolved during deformation at temperatures around 1123 K and lower strain rates (< 10−3 s−1 ) was characterized by equiaxed -grains separated by -phase. The phenomenon of grain boundary sliding has been found to be responsible for the occurrence of equiaxed grain structure. When the deformation rates are increased (≈0.1 s−1 and at the same temperature), elongated -grain structure separated by a thin film of is usually seen. Ostberg and Attermo (1962) have observed similar elongated morphology of separated by -films on forging at 1173 K and 0.4 s−1 (estimated strain rate). A few -plates have been found to change from single crystals to fragmented structure of new grains/subgrains as evidenced by difference in contrast under polarized light. The formation of new grains from the original -plates into smaller units is suggestive of the occurrence of dynamic recrystallization process in the -phase of the two-phase ( + ) mixture. Transgranular deformation mechanisms involving dislocations are important at these rates of deformation to produce the elongated
Diffusional Transformations
693
morphology. These observations clearly indicate the effect of strain rates on the mode of deformation and evolution of microstructure. As regards the effect of initial microstructure, an equiaxed grain structure has been found to exhibit better hot ductility than a Widmanstatten microstructure. In ( + ) material of zircaloy-4, the nickel-free variety of zircaloy-2, a hot working activation energy of 155 kJ/mol was reported in the temperature range 840–970 C and at strain rates greater than 3 × 10−3 s−1 (Rosinger et al. 1979). This value of activation energy was similar to that obtained in -phase. Zr and zircaloy-2, when deformed in the -phase field at low rates ≈ 10−3 s−1 , exhibit superplasticity and high ductility (Garde et al. 1978, Bocek et al. 1976). Chakravartty et al. (1991, 1992a,b,c) studied the hot deformation characteristics of commercial pure Zr (containing approximately 1000 ppm of oxygen) with a -quenched starting microstructure in both - and -phase fields using processing maps generated by carrying out compression testing in the strain rate range of 10−3 –102 s−1 and over the temperature range of 650–1050 C. The processing map revealed (Figure 7.78) two safe domains and a regime of instability: (a) a domain of DRX in the temperature range of 730–850 C and strain rate range of 10−2 –0.1 s−1 with its peak efficiency of 40% at 800 C and 0.1 s−1 which was considered as optimum hot working parameters, (b) a domain of dynamic recovery at temperatures lower than 700 C and strain rates lower than 10−2 s−1 and (c) a regime of flow instability at strain rates higher than 1 s−1 and temperatures above 670 C. The characteristics of dynamic recrystallization were found to be similar to those of static recrystallization regarding the sigmoidal variation of grain size (or hardness) with temperature, although the DRX temperature was much higher than the static recrystallization temperature 873 K. Within the domain of DRX, the grain size increased with increase of temperature and decrease of strain rate (Chakravartty 1992, Chakravartty et al. 1992b). The manifestation of instability was in the form of localized shear band (Figure 7.80). During hot working, the regime of instability should be avoided. The processing map of Zr in -phase showed only a single domain in the temperature range 925–1050 C and at strain rates lower than 0.1 s−1 with the following features: (a) the efficiency of power dissipation is high (≈ 60%) with a corresponding high strain rate sensitivity of flow stress (0.45), (b) the stress strain curves show steady-state behaviour with no flow softening and (c) flow stress values are generally low. On the basis of these informations, the domain has been interpreted to represent LGSP in -Zr (Chakravartty et al. 1992a,b,c). Hot working characteristics of zircaloy-2, with -transformed initial microstructure, have been investigated by Chakravartty et al. (1992b) using processing maps in the temperature range of 650–950 C. In this range, zircaloy-2 exists either as predominantly -phase or as a mixture of and . The processing maps
694
Phase Transformations: Titanium and Zirconium Alloys
125 μm
Figure 7.80. The optical micrograph of Zirconium samples deformed at 700 C and 100 s−1 showing adiabatic shear band across which a variation of microstructure can be seen. The instability parameter assumes negative value under the conditions of deformation.
(Figure 7.81) exhibited three distinct safe domains in the temperature range studied with a discontinuity at around 850 C. The domains occur at 800 C and 0.1 s−1 , 650 C and 10−3 s−1 and 950 C and 10−3 s−1 which have been characterized to be the domain of DRX, the domain of DRV and the domain of superplasticity (marked SPD), respectively. It has been reported that the initial -transformed microstructure is converted to equiaxed grains of during the process of DRX. Besides the domains of stable flow, at strain rates higher than 1 s−1 and temperature of 700–850 C, zircaloy-2 exhibits microstructural instability in the form of adiabatic shear band. A comparison of the processing maps of zircaloy-2 and commercially pure Zr in -quenched condition reveals that the maps are strikingly similar. The DRX domain of both the maps extend over similar strain rate and temperature ranges, although the peak efficiency (38%) for DRX for zircaloy is slightly lower than that of Zr (42%). The addition of tin to Zr as in zircaloy-2 lowers the stacking fault energy of Zr (240 mJ/m2 ) significantly (Sastry et al. 1974) while the diffusion coefficient is not greatly altered (16 × 10−16 m2 /s in -Zr while 3 × 10−17 m2 /s in Zr–1.03Sn alloy) (Chakravartty 1992, Chakravartty et al. 1992b). A simple model was used to explain the DRX characteristics of both Zr and zircaloy-2 (Chakravartty 1992). Of the two competitive processes involved in DRX, namely formation of interfaces (the nucleation step which involves dislocation generation and their simultaneous recovery) and migration of a large-angle boundary, it was found that the process of formation of interface is the rate-controlling step in both these materials. The thermal recovery by diffusion-controlled climb plays
Diffusional Transformations Zircaloy- 2 (β-quenched)
695 Strain = 0.4
2 2 1
Log strain rate
18
0
7
11 21
25
30
11 38 DRX
–1
38 35
44
–2 30
25 30 DRV 35 –3 650 710
SPD 49 56 64
25 770
830
890
950
Temperature (°C)
Figure 7.81. Processing map of -quenched zircaloy-2 for a strain of 0.4 showing various domains: (a) domain of dynamic recrystallization (marked DRX), (b) domain of dynamic recovery (marked DRV) and (c) domain of superplasticity (marked SPD). The number against each contour indicates percentage efficiency.
an important role in both these materials (Chakravartty 1992, Chakravartty et al. 1991). The characteristics of dynamic recovery and flow instability in both these materials were also similar (Chakravartty et al. 1991). The variation of grain size with temperature and strain rate, observed within DRX domain of zircaloy2, would permit microstructural control. In the domain of superplasticity, the efficiency value was high and the overall deformed grain structure was equiaxed. In order to validate the processing maps of zircaloy-2, extrusion trials have been carried out on full size billets and ingots used for making various zircaloy-2 products for nuclear reactor use. Two different temperatures (720 and 800 C) and various extrusion ratios and ram speeds were chosen (Chakravartty et al. 1991). The experimental extrusion conditions were located on both the processing map (Figure 7.82) and instability map (Figure 7.83). Most of the extrusions carried out at 800 C are within the DRX domain and the extrusions carried out at 720 C fall in the instability regime. The extrusion carried out at 800 C showed dynamically recrystallized grains while the microstructure of extruded material depicted signs of flow localization (Chakravartty et al. 1992a,b,c).
696
Phase Transformations: Titanium and Zirconium Alloys Zircaloy- 2 (β-quenched) 2
Extrusion at 720°C Extrusion at 800°C 5
9
Log strain rate
1
13 13
0 39
–1
20 24
DRX 28
–2
32 28
32
35 39 DRV –3 650 680
24 730
20
770
810
850
Temperature (°C)
Figure 7.82. Extrusion conditions superimposed on processing map of -quenched zircaloy-2 for a strain of 0.4. The extrusions carried out at 800 C are within the DRX domain while those carried out at 720 C are outside the DRX domain.
7.8.3.2 + alloys Jonas and co-workers (Jonas et al. 1979, Choubey and Jonas 1981, Rodriguez et al. 1985) studied flow properties of Zr–2.5Nb alloys by constant true strain rate compression testing in the strain rate range 10−4 to 1 s−1 and at temperatures from 750 to 1000 C. The alloys showed considerable flow softening in the ( + ) phase region and steady-state flow behaviour in single phase -region. The strain rate sensitivity was found to increase from 0.18 during deformation in the (+) phase field (800–875 C) to 0.22 in the -phase field (900–1000 C). The mechanism involved during hot deformation, however, has not been identified. Zr–2.5Nb alloys have been found to exhibit superplasticity in the temperature range 720–850 C and at strain rates in the range 10−2 –10−4 s−1 (Nutall 1976, Holm et al. 1977, Rosinger et al. 1979). Superplasticity has also been reported in cold worked and stress relieved Zr–2.5Nb pressure tube material during tensile deformation in the temperature range of 650–800 C at strain rates lower than 10−3 s−1 (Singh et al. 1993). The development of microstructure during high-temperature tensile deformation of Zr–2.5Nb pressure tube (hot extruded and cold worked) has been studied by Perovic et al. (1985) using TEM. At deformation temperatures lower than 600 C, the starting lamellar morphology is retained but the original dislocation structure is destroyed and is replaced by sharp sub-boundaries which fragment
Diffusional Transformations Zircaloy- 2 (β-quenched)
697 Strain = 0.4
2
1 –25
–63 –38
–88
–13
Log strain rate
0.00 0
–1
SAFE
–13 0.00
Extrusion at 720°C Extrusion at 800°C
–2
–3 650
680
730
770
810
850
Temperature (°C)
Figure 7.83. Extrusion conditions superimposed on instability map of -quenched Zircalloy-2 for a strain of 0.4. The extrusion carried out at 800 C are in the stable flow regime while those carried out at 720 C are within the instability regime.
the elongated -plates. When deformation is carried out at temperatures lower than -transus, the initial lamellar structure changes over to an approximately equiaxed structure of - and -grains. In this temperature region (>700 C and < -transus) during tensile deformation of Zr–2.5Nb in equiaxed ( + ) microstructure, Nutall (1976) observed that initial equiaxed morphology is maintained even after large deformation (>500%) and suggested superplasticity to be the deformation mechanism. The development of microstructure during hot extrusion of Zr–2.5Nb has been studied by Chedale et al. (1972) and Perovic et al. (1985). The temperature and extrusion ratio have been found to be the two major factors that influence the microstructural evolution in the billets. The extrusion of the -quenched billet at 780 C results in a microstructure in which ( + ) grain structure has been aligned by working. The -constituent is present in the form of lenticular plates which are joined together by sub-boundaries. The hot deformation behaviour of two-phase ( + ) alloys, Zr–2.5Nb and Zr– 2.5Nb–0.5Cu in the temperature range 650–1050 C and at strain rates in the range 0.001–100 s−1 have been studied by Chakravartty et al. (1995, 1996) using processing maps generated by carrying out compression testing. Two different initial
698
Phase Transformations: Titanium and Zirconium Alloys
microstructures of Zr–2.5Nb alloy, namely equiaxed ( + ) and -transformed (Chakravartty et al. 1992a,b,c), were studied. In both these materials, a domain of DRX has been identified in the processing maps by noting variation of the efficiency values with temperature and strain rate, nature of flow behaviour and by carrying out detailed metallographic investigations using both optical and transmission electron microscopy (Figure 7.84). While a steady-state flow behaviour has been observed in the equiaxed ( + ) material, with a peak efficiency of 45% at 850 C and 0.001 s−1 , the -transformed material exhibits stress–strain curves with continuous flow softening and higher peak efficiency (>50%) under similar deformation conditions. Deformation of equiaxed ( + ) structure in the DRX domain results in the development of equiaxed -grains distributed in a matrix of . A majority of these grains have been found to be further subdivided into subgrains separated by twist or tilt boundaries. In the DRX domain of -transformed material, speroidization of deformed microstructure occurred as a result of the shearing of -platelets followed by globularization (Figure 7.85). This observation is similar to that observed by Weiss and co-workers (1986) during two-phase deformation of ( + ) titanium alloys. A simple model has been used to analyse the characteristics of DRX by considering the rates of nucleation and growth of recrystallized grains (Prasad and Ravichandran 1991). Calculations of these two rates show that they are nearly equal and that the nucleation of DRX is essentially controlled by mechanical recovery involving cross slip of dislocations (Chakravartty et al. 1996). Tensile tests carried out over a range of temperatures and at a strain rate of 0.001 s−1 showed a peak in ductility within the DRX domain. The optimum hot working temperatures for equiaxed ( + ) and -transformed preform microstructures are 850 and 750 C, respectively, while the optimum strain rate is 0.001 s−1 for both. It may be noted that the DRX temperature 850 C, of Zr–2.5Nb, is higher than in -Zr (800 C) and this increase is expected in view of the back stress generated by the long-range elastic interactions of the solute. Further, in comparison with the processing map for -Zr, the map for Zr–2.5Nb showed that the DRX domain has shifted to lower strain rates by two orders of magnitudes (from 0.1 s−1 in -Zr to 0.001 s−1 in Zr–2.5Nb). The lowering in DRX strain rate has been attributed to an increase in the probability of recovery or a decrease in link length of dislocation or both. In view of likely increase of stacking fault energy and solid solution strengthening caused by niobium, it is possible that both these contributions are responsible for the lowering of the strain rate for DRX (Chakravartty et al. 1996). Figure 7.86 shows a TEM photograph of the dynamically recrystallized Zr–2.5Nb which exhibits the extensive formation of twist boundaries due to enhanced cross slip of screw dislocations and its role in forming the interfaces. Kapoor and Chakravartty (2002) studied
Diffusional Transformations
699
Zr–2.5Nb (E)
Strain = 0.4
2 3 8 1
14
Log strain rate
19
24
0
29
–1
35 –2
40 45
–3 650
710
770
830
890
950
Temperature (°C)
(a) Zr–2.5Nb (Q)
Strain = 0.4
2 7 10 17 1
21
Log strain rate
24 0 28 31 –1 35 38 42
–2
45 49 52 –3 650
710
770
830
890
950
Temperature (°C)
(b)
Figure 7.84. Processing maps for (a) + equiaxed starting microstructure and (b) -transformed starting microstructure of Zr–2.5Nb for a strain of 0.4. That the maps show a single domain in the ( + ) phase filled for a strain rate lower than 5 × 10−1 s−1 . The number against each contour indicates percentage efficiency.
700
Phase Transformations: Titanium and Zirconium Alloys
α
β
α 1.2 μm
Figure 7.85. TEM micrograph of -transformed microstructure of Zr–2.5Nb deformed at 800 C and 0.001 s−1 to a strain of 0.3, showing initial stages of dynamic recrystallization involving fragmentation and shearing of -plates followed by globularization.
α
α
β
0.4 μm
Figure 7.86. TEM micrograph of equiaxed ( + ) microstructure of Zr–2.5Nb on deformation at 800 C and 0.001 s−1 . The dynamically recrystallized -grains are found to contain twist boundaries in the form of hexagonal network of screw dislocations. Enhanced cross slip of screw dislocations results in the formation of interface required for dynamic recrystallization.
deformation behaviour in the -phase field of Zr–2.5Nb alloy and reported LGSP at temperatures greater than 950 C and at strain rates lower than 0.01 s−1 . The instability map generated revealed that Zr–2.5Nb alloy undergoes microstructural instability at temperatures lower than 700 C and strain rates higher than 1 s−1 and the manifestation of the instability has been in the form of flow
Diffusional Transformations
701
localization in both the initial microstructures and the severity of flow localization has been found to increase with rate of deformation (Chakravartty et al. 1996)). In Zr–2.5Nb–0.5Cu, copper lowers the to transition temperature and partitions entirely to -phase. It is expected therefore that copper modify the deformation characteristics of -phase alone in the alloy. The deformation behaviour has been studied in the temperature range of 650–1050 C encompassing both the ( + )and -phase fields (transformation temperature for this alloy being 870 C). The hot deformation characteristics of this material is similar to that of Zr–2.5Nb in the ( + ) phase deformation in terms of flow behaviour and occurrence of DRX domain. The domain of DRX is seen with its peak efficiency at about 750 C and 0.001 s−1 (Chakravartty et al. 1995). In addition to the DRX domain, another domain was found in the -phase field of this material at 1050 C and 0.001 s−1 which has been characterized to be that of LGSP (Chakravartty et al. 1995). The instability map reveals that microstructure instabilities will occur at temperatures greater than 800 C and strain rates higher than 30 s−1 . 7.8.3.3 -alloys The deformation characteristics of -Zr-Nb alloys (containing 10–20% Nb) have been investigated by Jonas and co-workers (1979) in the temperature range from 725 to 1025 C and with strain rate range 10−5 –10−1 s−1 to assess flow properties of the metastable -phase in the two-phase ( + ) field of Zr–2.5Nb alloy. The flow curves obtained on Zr–Nb alloys exhibited flow softening, and magnitude of this effect decreased as the test temperature is increased. The occurrence of flow softening has been attributed to the disintegration or over-ageing of Nb clusters during straining. 7.8.4 Development of texture during cold working of Zr alloys The overall fabrication schedule which consists of one or two hot working steps followed by a series of cold working steps determine the microstructure of the finished product. The development of crystallographic texture during the process of TMP can again be divided in two parts – the first being produced during hot working operation while the second resulting from the cold working and intermediate annealing steps. The anisotropic crystal structure of -phase makes it quite sensitive to texture development which contributes significantly towards anisotropy of mechanical and other physical properties of the finished product. Although similar deformation characteristics of Ti and Zr result in similar textural development during TMP in their alloys, the texture development in Zr alloys has attracted greater attention owing to the fact that the texture plays a significant role in-service performance of Zr alloys in irradiation environment. A brief account of texture development of zirconium alloys during deformation processing
702
Phase Transformations: Titanium and Zirconium Alloys
and its importance in mitigating some life-limiting degradation mechanisms in actual components is given here. One of the prime concerns in deformation processing is the development of a favourable crystallographic texture which is usually represented in terms of the distribution frequency of important crystallographic planes (e.g. basal or prismatic planes) with reference to the axial, radial and circumferential directions for tubular products and to the rolling, normal and transverse directions for plate products. As both slip and twinning contribute to plastic deformation of Zr, the deformation texture is strongly dependent on the process variables such as rate of deformation, temperature, extent of deformation and state of stress (Tenckhoff 1988). These variables in turn dictate available slip/twin systems and their respective critical resolved shear strength. The development of texture with increasing degree of cold work is due to the lattice rotations caused by a number of deformation systems. Twinning causes spontaneous lattice rotations through large angles though the resulting strain is small. Consequently at low values of plastic strain, the contribution of twinning is more significant in texture development. After large plastic strains, texture development is mainly controlled by slip deformation which tends to rotate the basal pole by ≈ 20 –40 from the radial normal direction towards the circumferential/transverse direction, provided the major compressive deformation is along the former (Tenckhoff 1988). In the production of zircaloy tubing, tube reduction processes, such as pilger milling involving triaxial stresses and strains, are commonly employed. The reduction in the cross-section (RA) is achieved by reduction in the wall thickness (RW) and the diameter (RD). Depending on the value of the ratio Q (= RW/RD), different crystallographic textures are developed in pilgered tubes. A high wall thickness to diameter reduction (Q > 1) produces a texture with the basal pole aligned parallel to the radial direction while a deformation with Q < 1 leads to a texture with the basal poles aligned parallel to the circumferential direction. Cold rolling of sheets also produces a similar texture with the basal poles being about 20 –40 away from the normal direction. Texture development in different types of metal-forming operations in zirconium alloys is schematically illustrated in Figure 7.87. The crystallographic texture of zirconium alloys influences the physical, mechanical and corrosion properties both out-of-pile and in-pile. Under neutron irradiation, single crystals of -zirconium shrink along the c-axis and dilate on the basal plane with the volume remaining more or less constant. This means a tube with a predominant circumferential basal pole texture will grow along the axial direction under irradiation. In order to minimize such growth, the fraction of basal poles along the axial direction needs to be increased. Similarly the creep strength along the hoop and axial directions of the cladding can be tailored by appropriately
Diffusional Transformations
Type of deformation
Body dimension initial–final
703
Schematic (0002) pole figure
Element
AD RD
AD
T
TD
u b
TD RW >1 RD
e RD
r
AD
AD
e d u c
TD
TD RW =1 RD AD
t
RD
AD
i o
TD
TD
n RW <1 RD
RD Sheet rolling
AD
RD
TD TD
AD R Wire drawing
T
A TD
Figure 7.87. Various textures developed during cold working by pilgering, sheet rolling and wire drawing operation involved in fabrication of zirconium alloy components.
modifying the texture. Anisotropy of plastic deformation and resistance to stress corrosion cracking also depend on the texture. By far the most important application of texture control is in achieving a favourable orientation distribution of brittle hydride platelets in cladding tubings with a strong radial basal pole texture.
704
Phase Transformations: Titanium and Zirconium Alloys
The development of texture in a two-phase Zr–2.5Nb alloy has recently been studied in detail by Kiran Kumar et al. (2003). The main findings of this work are summarized below to emphasize the role of -phase (though small in volume fraction yet present as continuous matrix phase) in controlling the development/evolution of texture. Two different microstructures of this alloy, namely, martensitic single phase hcp structure obtained by rapid quenching from the -phase and two-phase + structure obtained by solutionizing in the -phase followed by annealing in the + phase field, were considered. The initial bulk crystallographic textures were similar in these structures. On cold rolling, while the single phase -structure showed considerable textural changes, the bulk texture of two-phase structure remained virtually unchanged. The Taylor-type deformation model which considers different combinations of slip-twin systems and constraints could explain adequately the texture evolved in single phase material. However, it could not explain the lack of textural development in two-phase + structure. In the two-phase structure, microtexture observations showed that -plates remained approximately single crystalline even after cold working, while the continuous -matrix underwent significant orientational changes as evidenced by relative hardening estimated by X-ray peak broadening. In addition to these, the aspect ratio of -plates remained unchanged with cold rolling suggesting the absence of effective macroscopic strain in hcp -phase. On the basis of these observations, a simple model, which considered that the plastic deformation was solely concentrated in the matrix -phase within which the embedded -plates were subjected to in plane rigid body rotation, was proposed. The salient feature of this model is depicted in the Figure 7.88. The schematics of Figure 7.88(a) explain the alignment of -plates floating in continuous -phase matrix along RD and on the rolling plane (i.e. in plane rigid body rotation) on deformation. The nature of this alignment of -plates on deformation has been substantiated by microstructural observations which showed -plates did not align in any specific direction on the other two cross-sections LT and ST and there was no dimensional changes of -plates in any cross-section (particularly in the rolling plane). In order to test this model, random in-plane rigid body rotation(s) were simulated on the starting texture. This was accomplished by imparting random rotations commensurate with the range of angular realignment necessary for -plates (Figure 7.88(b)). An orientation distribution function (ODF) of two-phase material clearly shows the absence of texture development on deformation involving in-plane rigid body rotation. However, a simultaneous combination of in-plane rigid body rotation and deformation (Figure 7.88(c)) could not reproduce the texture and thus was found unsuitable for explaining the lack of textural development in -plates.
Diffusional Transformations (0 2 2 1) <3 2 1 2>
705 (0 2 2 1) <2 1 1 0>
β-Matrix Cold rolling
α-Plates
RD
(0 1 1 4) <2 3 4 4>
(0 1 1 2) <1 4 4 4>
(0 1 1 4) <2 1 1 0>
TD (0 1 1 4) <2 1 1 0>
(a) 0
90
90
90
0
90
90 0
90 0
90 0
90
0
0
90 PHI2=0
PHI2=10
PHI2=20
90 PHI2=40
PHI2=50
PHI2=60
(b)
PHI2=30
90 PHI2=0
PHI2=10
PHI2=20
90 PHI2=40
PHI2=50
PHI2=60
PHI2=30
(c)
Figure 7.88. The in-plane rigid body rotation model: (a) schematic diagram illustrating rotation(s) on the rolling plane and along RD, of -plates in a continuous -matrix. An imaginary unit cell represented few of the -plates, both before and after deformation. It is to be noted that the shades shown in these unit cells are just to give a sense of necessary rotation and do not correspond to the indicated crystallographic planes. The schematic overestimates -fraction, but otherwise dimension tolerances were kept in consideration for 60% reduction. (b) The simulated, with random in-plane rigid body rotation, ODF of the two-phase material. Absence of texture development in (b) corresponds to the experimental observation in two-phase structure. (c) Simulated ODF for 20% deformation (by unidirectional rolling) simultaneously with in-plane rigid body rotation or random ND rotation. Random ND rotations were applied after small incremental strain (2%). The ODF fails to capture lack of texture development typical of deformed -plates in two-phase material.
706
Phase Transformations: Titanium and Zirconium Alloys
7.8.5 Evolution of microstructure during fabrication of Zr–2.5 wt% Nb alloy tubes Although the hot working step imparts large deformation and provides a suitable microstructure by operation of one or more of the dynamic softening processes identified in titanium and zirconium alloys, the final product properties are considerably influenced by several other factors involved in TMP subsequent to hot working. These are rate of cooling and associated phase transformation after hot working, ageing, cold working and annealing. A variety of TMP techniques have been employed in several titanium alloys to tailor microstructure for obtaining specific mechanical properties, the details of which are well documented and are understood (Fleck et al. 1984). From these investigations, it appears that a knowledge of phase transformation and an understanding of the softening (recovery and recrystallization) processes that occur statically are crucial in imparting desired mechanical properties to the finished product. In order to highlight this, it is now proposed to give an account of a study on the evolution of microstructure during fabrication of Zr–2.5 wt% Nb alloy pressure tubes for application in PHWRs. The fabrication schedule of Zr–2.5Nb pressure tubes, as largely practiced, involves hot working of the ingot in several stages either by forging or by extrusion followed by a single cold working step like tube drawing which introduces the required 25% cold work and shapes the final product within the dimensional tolerances specified. The last hot working step, hot extrusion, is carried out in the ( + ) phase field with an accurate control of the temperature, the deformation rate (ram speed) and the extrusion ratio. The volume fractions of the two phases, and consequently their compositions, the aspect ratios of the - and -grains in the ( + ) fibrous microstructure and the crystallographic texture of the product are essentially determined by the process parameters of the last hot extrusion step. Figure 7.89 shows some typical microstructure of the hot extruded tube in both longitudinal and transverse directions as revealed by SEM and TEM. SEM photograph shows that the two-phase ( + ) structure is elongated in the direction of extrusion. The -phase (bright constituent) stringers are discontinuous and separate the elongated -phase units. The details of -grain structure, however, are not clearly revealed in SEM. TEM micrographs of both longitudinal and transverse direction reveal that the elongated -units are essentially made of cluster of nearly equiaxed fine -grains separated by thin -phase film. This observation suggested the operation of DRX in Zr–2.5Nb alloy during hot extrusion. In addition to high-angle boundaries in the structure, some of the equiaxed grains have also been found to contain sub-boundaries. The combination of high strength with a good ductility and toughness of Zr–2.5Nb pressure tubes is essentially derived from the fine ( + ) fibrous microstructure consisting of elongated -grains containing about 0.5Nb and the -phase (15–20Nb) stringers primarily located at -grain boundaries. Such
Diffusional Transformations
707
Figure 7.89. SEM and TEM micrographs of two-phase ( + ) microstructure of hot extruded tube: (a) elongated morphology in the longitudinal section (SEM), (b) serrated morphology in the transverse section (SEM), (c) and (d) dynamically recrystallized equiaxed -grains within -stringers in longitudinal and transverse sections, respectively (TEM).
a structure cannot remain stable if it is subjected to a cold working–full annealing cycle which tends to coarsen the structure into a distribution of equiaxed - and -grains, resulting in a substantial drop in the ultimate tensile strength. In recent years, there has been a sustained effort towards improving the resistance to irradiation-induced growth (one of the life-limiting factors of pressure tubes in PHWRs) through the modification of microstructure of the pressure tube material (Srivastava et al. 1995). However, the challenge was to qualify the product in terms of microstructure, dislocation density, crystallographic texture and short-time mechanical properties as specified for the conventionally processed material. This has been achieved by incorporating cold working by drawing or pilgering in two steps (instead of one in the conventional process) with an optimized intermediate annealing treatment (Figure 7.90). Srivastava et al. (1995) studied the evolution of microstructure and of tensile properties during each fabrication step of the modified route and compared it with that of the conventional route. The development of microstructure in both the routes was identical during extrusion in the two-phase field as the - and I -phases dynamically recrystallized and the I -phase layers were sandwiched between elongated -phase (stringers).
708
Phase Transformations: Titanium and Zirconium Alloys Conventional route
Modified route
β-Quenching 1000°C/30 min
β-Quenching 1000°C/30 min
Hot extrusion Ext. ratio 11:1 800°C
Hot extrusion Ext. ratio 8:1 800°C
Vacuum stress relieving 480°C/3 h
Cold drawing 25%
I – Pilgering 50–55% reduction
Vacuum annealing 550°C/6 h
II – Pilgering 25% reduction
Autoclaving 400°C/24 h
Autoclaving 400°C/72 h
Figure 7.90. A comparison of fabrication routes of Zr–2.5Nb pressure tube material.
The elongated -units were essentially made up of a number of nearly equiaxed -grains. The -phase precipitated within the I during cooling after hot extrusion following either route. However, in the modified route, the aspect ratio of both the phases was lower, presumably because of lower extrusion ratio employed and the I volume fraction was higher (with lower niobium content) owing to faster cooling rate subsequent to extrusion. A typical microstructure in radial–axial section is shown in Figure 7.91(a) which shows elongated -grains separated by -phase. Subsequent to hot working, in the conventional route, 20–25% cold work is given to produce a dislocation density of the order of 1014 m−2 which results in optimum combination of tensile strength and in reactor creep behaviour (Figure 7.91(b)).
Diffusional Transformations
0.7 μm
(a)
0.3 μm
(b)
0.2 μm
(c)
(d)
709
0.8 μm
(e)
0.4 μm
Figure 7.91. Transmission electron micrograph of Zr–2.5Nb alloy: (a) as-extruded microstructure showing elongated morphology of -phase separated by -phase stringers. (b) First pilgered microstructure illustrating the very high dislocation density. The dislocations are concentrated primarily at the / interface. (c) Incomplete recrystallization of the -stringers as evidenced by the presence of a substantial number of dislocations after annealing at 500 C for 6 h. (d) Coarsening of the -grains caused by redistribution and agglomeration of -phase. The -phase is located at the triple junctions of -grains after annealing at 600 C for 1 h and (e) completely recrystallized -lamellae obtained after annealing at 550 C for 3 h. The lamellar morphology of the two phases is not altered by this annealing treatment.
710
Phase Transformations: Titanium and Zirconium Alloys
The purpose of intermediate annealing treatment prior to second pilgering in the modified route is to annihilate all the cold work introduced in the first pilgering step so as to get a microstructure and flow properties similar to that obtained after extrusion. Retention of this structure is necessary for obtaining optimum tensile properties at service temperature of 310 C. In this investigation, the annealing temperature was varied between 500 and 650 C and annealing duration varied between 1 and 6 h. It was observed that the elongated two-phase structure could be retained only if the annealing temperature is kept below 575 C. Annealing at lower temperature, for example, at 500 C for 6 h resulted in recovery of -stringers as evidenced by the presence of sub-grain boundaries. In addition to these, a high density of unrecovered dislocations was also noticed (Figure 7.91(c)). The tensile strength of the resultant structure was higher than the as-extruded material. In contrast to this, annealing at higher temperature, e.g. 600 C, even for 1 h resulted in complete recrystallization and morphological modifications of both - and phases (Figure 7.91(d)). The annealed microstructure comprised essentially of nearly equiaxed -grains, and I -phase was noticed primarily at the boundaries and trijunctions of -grains. The significant coarsening of the structure accompanied by morphological changes resulted in lowering of tensile strength to a level below that of the hot extruded material. An annealing treatment at 550 C for 6 h caused nearly complete recrystallization of phase without altering the elongated morphology of the + I microstructure (Figure 7.91(e)). A comparison of tensile flow behaviour of this annealed material with that of as-extruded material revealed that the flow stresses at different levels of plastic strains are nearly the same. On the basis of the observed similar tensile flow properties and morphology of the two-phase structure, it was suggested that annealing at 550 C for 6 h would be the optimum condition for achieving the desired results.
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Chapter 8
Interstitial Ordering 8.1 Introduction 8.2 Hydrogen in Metals 8.2.1 Ti–H and Zr–H phase diagrams 8.2.2 Terminal solid solubility 8.3 Crystallography and Mechanism of Hydride Formation 8.3.1 Formation of -hydride in the - and -phases 8.3.2 Lattice correspondence of -, - and -phases 8.3.3 Crystallography of → transformation 8.3.4 Crystallography of → transformation 8.3.5 Mechanism of the formation of -hydrides 8.3.6 Hydride precipitation in the / interface 8.3.7 Formation of -hydride 8.4 Hydrogen-Related Degradation Processes 8.4.1 Uniform hydride preciptiation 8.4.2 Hydrogen migration 8.4.3 Stress reorientation of hydride precipitates 8.4.4 Delayed hydride cracking 8.4.5 Formation of hydride blisters 8.5 Thermochemical Processing of Ti Alloys by Temporary Alloying with Hydrogen 8.6 Hydrogen Storage in Intermetallic Phases 8.6.1 Laves phase compounds 8.6.2 Thermodynamics 8.6.3 Ti and Zr-based hydrogen storage materials 8.6.4 Applications 8.7 Oxygen Ordering In -Alloys 8.7.1 Interstitial ordering of oxygen in Ti–O and Zr–O 8.7.2 Oxidation kinetics and mechanism 8.8 Phase Transformations in Ti-Rich End of the Ti–N System References
720 721 722 725 728 728 729 730 735 737 737 739 741 742 743 745 746 747 753 754 754 756 756 761 764 764 769 772 780
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Chapter 8
Interstitial Ordering List of Symbols ija : External stress eijT : Stress-free transformation strain ¯ ¯ VH /Vhyd : Partial molar volume of hydrogen/hydride B Correspondence matrix relating chosen coordinate system (xi ) and xR : that of the -bcc (B) F Correspondence matrix relating chosen coordinate system (xi ) and xR : that of the -fct (F) S SB : Strain in x and B coordinate system P PB : Lattice invariant shear in x and B coordinate system FB : Macroscopic shape strain in B coordinate system HB : Habit plane in B-coordinate system SH : Lattice strain in orthohexagonal coordinate system J : Flux of hydrogen atoms D: Diffusivity of hydrogen in metal cx : Hydrogen concentration at point x Q∗ V ∗ : Heat and volume of transparent of hydrogen in metal P: Hydrostatic pressure CBFT : A threshold hydrogen concentration for blister formation ∗ : Hydrogen concentration corresponding to the terminal solid cTSS solubility Te : Eutectoid temperature in the Ti–H system Ec : Configurational internal energy Pij ij : Number of ij pairs and corresponding interaction energies
H M : Concentration of dissolved hydrogen in metal pH2 : Partial pressure of hydrogen Hs : Enthalpy of solution Gs : Strain energy density associated with hydride precipitates Accommodation strain energy Gacc s : : Strain energy due to applied stress field, residual stress and stress Gint s field due to other inclusions CoT : Equilibrium concentration of hydrogen in solution at T growth diss /Co : Hydrogen concentration during growth/dissolution of hydride Co acc W el : Strain energy due to elastic accommodation acc W p : Strain energy due to plastic accommodation 719
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rp : Plastic zone radius p: Hydrostatic pressure within the hydride precipitate K: Bulk modulus : Ratio of bulk moduli of hydride and matrix : Poisson’s ratio
: Shear modulus
8.1
INTRODUCTION
The formation of interstitial compounds in a metal matrix can be treated as a process of interstitial ordering whereby interstitial solutes, which remain distributed randomly in the interstitial sites, assume an ordered array with a lowering of the temperature of the system. An excellent example of interstitial ordering is the formation of Fe4 N in the fcc lattice of dilute Fe–N alloys (Jack 1948). In the structure of Fe4 N, the N atoms occupy the octahedral holes at the body-centred positions of the fcc lattice in such a way that each N atom is separated from its neighbours by the maximum possible distance. However, the stoichiometric Fe4 N alloy does not exhibit an order-to-disorder transformation at an elevated temperature as other phase reactions come into play. The bct martensite forming in the Fe–C system can also be interpreted as having an interstitially ordered structure. The occupation of the octahedral interstices in the bcc lattice causes a distortion of the lattice to produce a tetragonal symmetry. Zener (1946) has pointed out that such an ordered configuration of C atoms is produced below a critical temperature. Khachaturyan (1978, 1983) has presented a theory of interstitial ordering by considering the interstitial solid solution as a system of interacting particles in the periodic field of the host lattice, and has discussed the formation of various types of superlattices. The formation of interstitial compounds, Tix I and Zrx I, where I stands for H, N and O, in the metallic matrix of titanium- and zirconium-based alloys can be treated as interstitial ordering. In this chapter, we will examine the mechanism and to a limited extent, the kinetics of the formation of interstitial compounds such as hydrides, nitrides and oxides of Ti and Zr. Emphasis will be laid on those transformations in which the interstitial compound phase maintains a lattice relationship with the parent phase. As we proceed, we will encounter situations in which interstitial ordering is accompanied by changes in the dimension and stacking sequence of the parent lattice. The crystallography of such transformations has been discussed in detail to examine whether the phenomenological theory of martensite crystallography remains valid in these cases.
Interstitial Ordering
721
The formation of hydrides in Ti and Zr is discussed in the first part of this chapter. As will be elaborated later, the transformation of the (hcp) phase into the -hydride (fct) phase is an excellent example of a hybrid transformation in which the lattice of the metal atoms undergoes a displacive transformation while the movement of H atoms is by diffusion. As in the case of shear transformations, the hydride precipitation process is strongly influenced by external stress–a phenomenon which has serious implications in respect of the in-service degradation mechanisms of alloys based on these metals. Oxygen atoms dissolved in the -phase of Ti and Zr occupy the octahedral interstitial sites of the host hcp lattice. Ordering of O atoms has been reported at compositions near the O/metal ratios of 13 and 21 . The structures and transformation behaviours of interstitial alloys of such compositions are discussed in a subsequent part of this chapter. The corrosion rate of zirconium alloys in service in nuclear reactors is essentially controlled by the growth of oxide layer on zirconium alloy components. The mechanism and the kinetics of oxide growth are also outlined in this chapter. The case of the formation of Ti2 N and TiN phases in titanium alloys is chosen as examples of nitrogen ordering. The crystallography of the phase transformations involved in the formation of titanium nitride coatings is discussed in the last section of this chapter.
8.2
HYDROGEN IN METALS
Hydrogen is soluble to some extent in almost all metals. When present in excess of a certain limit, H can have a damaging effect on the mechanical properties of a metal, particularly its fracture properties. These effects have been studied extensively in common metals such as Fe, Cu and Ni in which H is present either in solution or is precipitated as gaseous bubbles. In another class of metals such as Ti, V, Zr and Nb which have a strong chemical affinity for H, metal hydrides are precipitated when the H content exceeds the solubility limit. In these metals H solubility is quite high at elevated temperatures, but decreases sharply with lowering of temperature leading to hydride precipitation (Dutton 1976). Dissolution of H gas in a metal involves ionization of H atoms, with the electrons entering the conduction band of the host metal and the ions being positioned at interstitial sites. The equation of dissolution of hydrogen in metals can be represented as 1 H → HM 2 2
(8.1)
722
Phase Transformations: Titanium and Zirconium Alloys
Table 8.1. Selected hydrogen solubility parameters for Ti and Zr. Phase -Ti (hcp) -Ti (bcc) -Zr (hcp) -Zr (bcc)
Enthalpy of solution Hs (kcal/mole)
Maximum concentration (at.%)
H solubility temperature (K)
−108 −139 −122 −154
6.72 >50 5.9 >50
573 >873 823 >1023
where the subscript ‘M’ refers to H dissolved in metals. Sievert’s law for the concentration of dissolved H, H M is
H M = pH2 1/2 exp−G/RT
(8.2)
where G is the standard free energy change for the reaction, pH2 is the partial pressure of H and R and T have their usual meanings. The corresponding enthalpy change, H, has two components: (a) the enthalpy of dissociation of the H molecule (about 52 kcal/mole) and (b) the enthalpy of solution Hs of H atoms in the metal lattice. Depending on the sign of Hs , metals are classified as endothermic and exothermic occluders of hydrogen (Dutton 1976). This classification serves to distinguish the stable phase formed when the solid solubility limit is exceeded. The endothermic occluders are in equilibrium with the H gas, often in the form of internal gas bubbles, and the exothermic occluders are in equilibrium with a precipitated solid hydride phase. Ti and Zr are exothermic H occluders, the enthalpy of solution and the maximum solid solubility limit of H in the (hcp) and in the (bcc) phases being indicated in Table 8.1. 8.2.1 Ti–H and Zr–H phase diagrams The equilibrium phase diagrams of the Ti–H and Zr–H systems are shown in Figures 8.1(a) and (b) in each of which the metastable -hydride phase field is also marked. The two equilibrium hydride phases are the -hydride phase and the -hydride phase. In both the systems the -hydride phase results from an eutectoid decomposition of the -phase, as indicated in the phase reaction = +
(8.3)
The equilibrium -hydride (fcc) phase is stable in the composition range of 56.7– 66.0 at.% H (Zuzek 2000). In this phase H atoms occupy some of the tetrahedral interstices at ( 41 41 41 ) and equivalent positions. Only a very small fraction of H atoms are located in octahedral holes. The volume misfit of -hydride with respect to -Ti and -Zr is about 24% and 17.1%, respectively.
Interstitial Ordering
723
Hydrogen (at.%) 0 10 20
30
40
50
Hydrogen (at.%)
60
0 10 20
1173
40
50
60
1273
1073
1173
β-Ti
973
1073
873
Temperature (K)
Temperature (K)
30
773 673 573
573 K
δ 1.33
0.15
473
2.16
α-Ti
373 273
ε
173
β-Zr
δ
973 873
823 K
773
0.07
1.43
~0.659
ε
α-Zr
673 573 473 373
73
273
–27 0 Ti
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Hydrogen (Wt%) (a)
0 Zr
0.2 0.4
0.6 0.8
1
1.2
1.4 1.6 1.8
2
Hydrogen (Wt%) (b)
Figure 8.1. (a) Ti–H and (b) Zr–H phase diagrams.
The tetragonal -hydride phase has a c/a ratio less than unity and a stoichiometry close to TiH2 and ZrH2 . H atoms occupy the interstitial sites of tetragonal symmetry at ( 41 41 41 ) or equivalent points. There are two such sites per Zr/Ti atom. While the radius of the interstitial site in the -hydride structure is 0.0378 nm that in the -hydride structure is 0.043 nm (in the case of Zr hydrides). The atomic radius of H is about 0.040 nm which matches closely with the sizes of the interstitial sites in both and hydrides. The metastable -hydride phase has a fct structure with a c/a ratio greater than unity and a stoichiometry close to ZrH and TiH. The lattice parameters of this phase depend on the alloy composition and the thermal treatment. Typically, the -hydride phase in a dilute Zr–H alloy has lattice parameter values of a = 04596 nm and c = 04969. H atoms occupy ordered tetrahedral interstitial sites in a fct Zr lattice. The hcp phase of Ti and Zr can take a certain amount of H in solid solution. There are two types of interstitial sites in the hcp structure – tetrahedral and octahedral (Figure 8.2); their radii are 0.0364 and 0.068 nm, respectively, in -Zr. For titanium the sizes are 0.0343 and 0.0619 nm, respectively. Neutron diffraction experiments have shown that H atoms occupy tetrahedral interstitial sites in the phase of Ti and Zr (Narang et al. 1977). There are four such sites of tetrahedral coordination at ( 23 13 61 ), ( 23 13 65 )(00 13 ) and (00 23 ). Darby et al. (1978) in a theoretical study, have calculated the relative energy of H in tetrahedral and octahedral sites in the hcp -phase assuming that the energy terms most strongly dependent on
724
Phase Transformations: Titanium and Zirconium Alloys a√2
a√3
a
2√2
a
Metal atoms Tetahedral interstices
Metal atoms Octahedral interstices (a)
(b)
A
B
C a0
3√
a0 (c)
Figure 8.2. (a) Octahedral, (b) tetrahedral interstitial sites in hexagonal close packed structures and √ (c) a single interstitial layer described in terms of a hexagonal net with transalation of 3ao .
site symmetry are those pertaining to the electronic and protonic electrostatic contributions. Their calculations have shown that the tetrahedral site is preferred over the octahedral site in agreement with experimental observations. There have been several attempts to predict H solubility in the -phase. Using a strain energy-based model, Sinha et al. (1970) have obtained a saturation solubility of 62 at.% H in -Zr at room temperature. However, the experimentally observed maximum solubility limit is about 6 at.% at the eutectoid temperature (823 K) in the Zr–H system. In a thermodynamic study (Northwood and Kosasih 1983), it has been assumed that hydrogen atoms occupy only alternate sites in the -phase and that they oscillate between the two interstitial sites with an estimated vibration frequency of 121 × 1013 Hz. Sinha et al. (1970) have used this concept in modifying their strain energy model but the results have still shown a solubility limit much higher than the experimentally observed value. Considering H as a quantized oscillator, a solubility limit of 0.492 at.% at room temperature has been obtained, which is in good agreement with the observed value (Sinha et al. 1970).
Interstitial Ordering
725
8.2.2 Terminal solid solubility The maximum hydrogen concentration, which can be retained in solid soltuion without forming hydride precipitate, is called terminal solid solubility (TSS) (Pan et al. 1996). The amount of hydrogen in excess of solid solubility precipitates as one of the hydride phases depending on the cooling rate and the hydrogen concentration (Northwood and Kosasih 1983). Similar to any solid state phase transformation process, the free energy of hydride precipitation has three components, viz. chemical free energy, interfacial energy and strain energy. The driving force of chemical free energy change associated with the transformation of matrix to hydride is primarily opposed by the strain energy arising from the density difference between the matrix and hydride precipitate. The interfacial energy is negligibly small when hydride precipitates are fine (<1 m in diameter with a aspect ratio of ∼20). The strain energy, Gs , can again be expressed as the sum int acc Gs = Gacc represents strain energy density s + Gs (Puls 1990). While Gs associated with the accommodation of individual hydride platelets, Gint s is due to the change in the accommodation strain energy in the presence of an externally applied stress or residual stress or stress field due to other inclusions. Experimentally determiend values of TSS of hydrogen in zircaloy-2 are shown in Figure 8.3 (Singh et al. 2005). TSS obtained during heating corresponds to the dissoltuion of hydrides and hence termed as terminal solid solubility for dissoltuion (TSSD). On the other hand, TSS determined during cooling corresponds to the
TSSD (Heating) 700
Temperature (K)
α-Zr
TSSP (Cooling)
600
500 α-Zr + hydride 400
300 0
100
200
300
CH (ppm)
Figure 8.3. Terminal solid solubility of hydrogen in Zircaloy-2 obtained during heating (TSSD) and cooling (TSSP) using dialatometry.
726
Phase Transformations: Titanium and Zirconium Alloys
precipitation of hydrides and is termed as terminal solid solubility for precipitation (TSSP). As can be seen in the Figure 8.3, TSSP is higher than TSSD. The experimental results on TSS of hydrogen in hydride forming metals are analysed in terms of a hypothetical equilibrium solvus coT , the concentration of hydrogen in solution in equilibrium with hydride, coT = A exp −Hs /RT
(8.4)
where A is a constant and Hs is the enthalpy of solution with respect to the hydride (Puls 1990). The equilibrium TSS defined in Eq. (8.4) cannot be determiend experimentally as hydride precipitation is always associated with a large value of strain energy of accommodation. The accommodation strain energy is reported to be dependent on the size of the precipitate and its tendency to grow or dissolve leading to hysteresis behaviour between the TSS obtained during heating and cooling. During nucleation the accommodation energy is fully elastic, whereas for a growing precipitate the accommodation energy is partly due to elastic and partly due to plastic deformation of the matrix and precipitate. Usually interfacial free energy is neglected and in that case chemical free energy released due to phase transformation must provide for the strain energy associated with elastic and plastic deformation. The strain energy stored in the material due to elastic deformation can be recovered but the strain energy spent in plastic deformation is partly locked in and around the defects and is partly dissipated in the form of sound, heat, etc. Thus, the TSS expression for the precipitate growth and dissolution during cooling down and heating up, respectively, in terms of the stress-free solvus defined by Eq. (8.4) is given as (Puls 1990): acc acc coGrowth = coT exp W el + W p /RT
(8.5)
acc coDiss = coT exp W el /RT
(8.6)
and
acc
acc
The expression for strain energy associated with elastic, W el , and plastic, W p , accommodation have been obtained for the case of spherical, misfitting inclusions having isotropic positive or negative misfit strains (Puls 1990):
Interstitial Ordering
acc y2 1 rp 3 1 W el p2 = − 1 − + 2K 3 a 6 V hyd acc
Wp
V hyd
=
3 rp rp 1 rp 3 1 − ln + a a 3 a 3 y2
727
(8.7)
(8.8)
where rp = plastic zone radius a = radius of the hydride precipitate V hyd = partial molar volume of hydride p = Kmatrix eT 1 − = hydrostatic pressure within the hydride precipitate K = bulk modulus = Khyd /Kmatrix = 1 + /41 − , where is the Poisson’s ratio = E/21 + and is solution to KeT 2eT + − = exp 1 − − 1 y 2y
(8.9)
Influence of stress on TSS: The partial molar volume of hydrogen in solution in -zirconium is positive and hence, in the presence of a tensile hydrostatic stress, the solid solubility is enhanced. Also, the hydride precipitation in these alloys is associated with positive misfit strains. Thus, in the presence of tensile stress, the tendency to hydride precipitation is also enhanced. The equilibrium concentration of dissolved hydrogen cPT can be obtained corresponding to the aforementioned solvus by multiplying the right-hand side of Eqs. (8.5) and (8.6) by the following factor (Puls 1990). exp pV H − V hyd ija eijT /RT
(8.10)
which represents the change in strain energy of the system, when one mole of hydrogen in solution under the hydrostatic pressure p, is precipitated under an external stress of ija with the stress free transformation strain eijT . While V H represents partial molar volume of hydrogen atoms in solution, V hyd is the specific volume of hydride precipitate per mole of hydrogen. It turns out that for zirconium alloys, the two interaction energies are nearly equal (pV H ∼ V hyd ija eijT ) and under uniform applied stress, TSS changes negligibly. Thus, TSSP and TSSD do not change significantly in the presence of stress.
728
8.3
Phase Transformations: Titanium and Zirconium Alloys
CRYSTALLOGRAPHY AND MECHANISM OF HYDRIDE FORMATION
As mentioned earlier, three types of hydride phases, namely, the -phase (fct, nearly equiatomic, ZrH, TiH), the -phase (fcc, ZrH15 TiH15 ) and the -phase (tetragonal with c/a < 1, ZrH2 , TiH2 ) are encountered in Ti–H and Zr–H systems. The kinetics of the formation of these hydride phases are considerably different. While the → and the → transformations are so rapid that they cannot be suppressed by quenching, the other two hydride phases can form only on ageing. Increasing H contents and slower cooling rates promote the formation of and hydrides. 8.3.1 Formation of -hydride in the - and -phases The metastable -hydride phase, which has an fct structure (c/a >1.0 ) can coexist with the as well as with -hydride phases. The formation of the -hydride phase is favoured by a fast cooling rate from a temperature where H is taken into the solid solution (typically, above about 625 K for Zr containing about 40 ppm H). Northwood and Kosasih (1983) have suggested a shear mechanism for the → transformation. The temperatures at which -hydride can form are too low for any significant self-diffusion of Zr (or Ti) atoms to occur. Interstitial diffusion of H atoms within the time scale of the transformation is still possible. The overall → transformation process can, therefore, be viewed as a shear transformation of the hcp lattice of -Zr/Ti with an accompanying H redistribution and interstitial ordering. The morphological features (Figure 8.4(a); Dey et al. (1982) Dey and Banerjee (1984)) of -hydride precipitates in the -matrix include plate shape with specific habit plane and internal twinning which suggest an invariant plane strain
200 nm
(a)
Figure 8.4. Micrographs showing -hydride in (a) - and (b) -phases.
300 nm
(b)
Interstitial Ordering
729
transformation. The crystallography of the transformation is, therefore, analysed in terms of the phenomenological theory of martensitic crystallography. The -hydride phase can also form in the matrix of -stabilized alloys (Flewitt et al. 1976, Dey et al. 1984). The → transformation also exhibits several features of shear transformations such as low transformation temperatures (at which self-diffusion of Zr or Ti is negligible), a plate morphology and internal twinning of hydride precipitates. In + alloys, H is partitioned between the two phases. Since the solubility of H in the -phase is much more than that in the -phase, precipitation of hydrides in + alloys is encountered more frequently in the -phase and along / boundaries than in the -phase. 8.3.2 Lattice correspondence of -, - and -phases Based on the observed orientation relations between the -hydride and the - and the -phase matrix, Dey et al. (1984) have proposed the following lattice correspondence in respect of these phases (as shown in Figure 8.5). ¯ 1120 ¯ 101 ¯ X1 001 [110]β [1100]α ½[121]γ X2
[111]β [1210]α [110]γ
X1 X3
[001]β [1120]α ½[101]γ
[110]β [0001]α ½[111]γ
Figure 8.5. Lattice correspondences between -, - and -hydrides.
730
Phase Transformations: Titanium and Zirconium Alloys
¯ 1100 ¯ ¯ X2 110 121 X3 110 0001 111 It may be noted here that while studying the precipitation of -hydride from the -phase, Weatherly (1981) has noted -hydride precipitates of two distinct classes:
¯ habit plane and the orientation (a) Type I -hydride precipitates with 1017 ¯ [12 ¯ 10] ¯ . relationship (111) (0001) ; [110]
¯ (b) Type II -hydride precipitates with 1010 habit plane and the orientation ¯ [1210] ¯ . relationship (001) (0001) ; [110] The lattice correspondences for both type I and type II -plates could be regarded as crystallographically equivalent if one considers the to transition to have two components: the first being the to transition in accordance with the Burgers correspondence and the second being responsible for converting the structure to the -hydride structure. The next two sections have, therefore, been devoted to crystallographic descriptions of the → and the → transformations, respectively. 8.3.3 Crystallography of → transformation In the crystallographic analysis presented here the orthogonal basis vectors ¯ , [110] ¯ [110] , respecx1 , x2 , x3 are taken to be parallel to the directions [001] tively. These vectors are shown in Figure 8.5 which depicts how the (110) plane can be distorted to the (111) plane. The correspondence matrices relating the chosen axial system (xi ) on one hand and the bcc () and fct () axial systems on the other, represented by x RB and x RF , respectively, are given by ⎡
0 B ⎣0 xR = 1¯
⎤ 1¯ 1 1 1⎦ 0 0
⎡
and
1 F ⎣ 0 xR = 1¯
⎤ 1 1 2¯ 1 ⎦ 1 1
(8.11)
The distortion associated with the → transformation can be conceptually divided into two components, S1 and S2 . While S1 is responsible for straining the (110) plane to match the dimensions of the (111) plane and for adjusting the interplanar spacing of the former phase to be equal to that of the latter, the second component, S2 , is necessary for effecting the necessary change in the stacking sequence. The principal distortions, 1 , corresponding to S1 , can be worked out from the lattice correspondence shown in Figure 8.5, to be the following:
Interstitial Ordering
[110]β
(110)β
731
B
B
A
A
A
C
B
B
B
A
A
A
[001]β
(0001)α
(111)γ
BCC Distorted by S1
HCP Distorted by S2
FCT
(a)
(b)
(c)
Figure 8.6. Simple shear acting homogeneously on every atomic plane.
√ Along x1 1 = a /2 2a √ Along x2 2 = a /2 3a √ √ Along x3 3 = 2a / 3a The shear, S2 , which changes the AB AB type stacking of the distorted (110) planes (distorted by S1 ) to the ABC ABC type stacking of the (111) planes, is a simple shear acting homogeneously on every atomic plane (Figure 8.6). The shear S2 , being a pure shear, can be expressed as: ⎡ ⎤ 1 0 0 S2 = ⎣ 0 1 03535 ⎦ (8.12) 0 0 1 The total lattice distortion matrix can, therefore, be expressed as S = S1 · S2 where both S1 and S2 are on the basis of the axial system, xi as indicated in Figure 8.5. In order to consider a specific example, the lattice parameters of the -hydride phase and of the -phase in a Zr-20% Nb alloy are substituted for obtaining the lattice strain matrix, S1 , which can be expressed as: ⎡ ⎤ 09734 0 0 S1 = ⎣ 0 11322 0 ⎦ (8.13) 0 0 10856
732
Phase Transformations: Titanium and Zirconium Alloys 110
Rigid body rotation Habit plane trace B′
After S1
110
B
Y
Before S1
Undistorted cones
(110)trace
Direction of S1
001
A A′
∧
∧
110
ab = A′B′ For S2 = tan 11°
311 301 Habit plane pole
a,b before S2 and S1 A,B after S2 A,B after S2 and S1
100
110
Figure 8.7. Stereographic projection showing lattice distortion, S1 , and the shear, S2 involved in the to transformation.
When S1 is applied to a unit sphere, the set of vectors which remain undistorted generate an elliptical cone. The positions of this cone before and after the application of S1 are represented in the stereographic projection shown in Figure 8.7. It can be seen that the superimposition of a certain fraction of the shear S2 can lead to the fulfilment of the invariant plane strain (IPS) condition for a plane defined by the undistorted vectors a and b as shown in Figure 8.7. The pole of this irrational plane is located within the stereographic triangle defined by the ¯ and (301) . This habit plane prediction matches well with poles (311) , (100) that experimentally determined for the -hydride precipitates forming within the -phase matrix in a Zr-20% Nb alloy. The Wechsler, Liebermann and Read (WLR) methodology of martensite crystallography is applied for the → transformation for the precise determination of the habit plane and the magnitude of the lattice invariant shear. The lattice strain S = S1 · S2 represented with reference to the axial system, xi , can be transformed into the fct axial system by employing a similarity transformation:
Interstitial Ordering
733
⎡
⎤ 09088 02234 0 0 ⎦ SB = x RB S B Rx −1 = ⎣ 01768 13090 0 0 09734
(8.14)
As mentioned earlier, a superposition of the S2 shear on S1 can generate the -hydride lattice. If the direction of S2 is reversed, a second variant of the -hydride crystal is created. These two -orientations have a twin relation between them. These two shears, denoted as S+2 and S−2 are along the following two directions: ¯ 111 121 ¯ S2+ 110 110 ¯ 111 12 ¯ 1 ¯ S2− 110 110 Consistent with this fact, the lattice-invariant shear (LIS), has been selected to ¯ , the lattice-invariant shear, P with reference to the axial system be (110) [110] defined by the shear direction, d, the shear plane normal, n, and the vector, dxn, can be expressed as: ⎡ ⎤ 1 0 g P = ⎣0 1 0⎦ (8.15) 0 0 1 where, g is the magnitude of LIS. The LIS matrix, PB for a shear along (110) ¯ represented in the bcc axial system, will be [110] ⎡ ⎤ ⎤ 1 1 0 0 0 0 g PB = ⎣ 0 1 0 ⎦ + ⎣ 1¯ 1¯ 0 ⎦ 2 0 0 0 0 0 1 ⎡
(8.16)
The macroscopic shape strain, FB , is then given by F B = PB · S B ⎤ ⎤ ⎡ 05427 05427 0 0 0904 02137 = ⎣ 01814 12991 0 ⎦ + g ⎣ 05427 05427 0 ⎦ 0 0 09734 0 0 0 ⎡
(8.17)
The IPS condition can be met by adjusting the value of g. In the present case, the magnitude of the lattice invariant shear, g, has been found to be 0.1699 and
734
Phase Transformations: Titanium and Zirconium Alloys
substituting this in Eq. (8.17) the average macroscopic shape strain matrix can be determined to be ⎤ ⎡ 09962 01215 00 (8.18) FB = ⎣ 00892 12069 00 ⎦ 00 00 09734 The strains in three principal directions could be determined by finding the eigen values of the macroscopic strain matrix FB and these are given by: 1 = 09734
2 = 10
and
3 = 12131
where 1 , 2 and 3 are the strains along the three principal axes. It could be thus seen that 1 < 1, 2 = 1 and 3 > 1 which is the necessary and sufficient condition for IPS (Lieberman et al. 1957, Wayman 1964). Therefore, an invariant plane strain condition in → hydride transformation could be achieved when the magnitude of the shear is 0.1699 for the LIS system given by shear S+2 . Furthermore, applying the WLR matrix methodology (Lieberman et al. 1957, Wayman 1964), the habit plane or the interface plane (HB ) for the correspondence variant given in Figure 8.5 and for the LIS given by shear S+2 corresponding to two opposite shear directions are found to be the same: HB + = −00850 29968 10000 and HB − = −00850 29968 10000 Following the method of comptuation described in WLR theory (Lieberman et al. 1957), the other relevant crystallographic parameters for the given shear system were calcualted and the results are presented in Table 8.2. Table 8.2. The computed crystallographic parameters of → transformation, for correspondence variants shown in Figure 8 and first shear system. Crystallographic parameters
Computed values Positive solution
Negative solution ⎤ ⎡
⎡
090376 10001 −00059 −00020 ⎣−00059 12098 00700 ⎦ ⎣ 007084 Shape strain matrix −005749 00025 −00873 09709 00089 −03163 09486 00089 Shape strain shear direction −00262 09230 −03839 00262 Shape strain direction Magnitude of shape strain
0.2398
⎤ −007302 006930 105375 −005101⎦ −004362 104140 −03163 −09486 −09230 −03839 0.2398
Interstitial Ordering
735
It is to be noted that the LIS chosen here are the same as S+2 and S−2 and that the LIS magnitude of (0.1699) corresponds to nearly half of the shear S2 . This means that a -plate which consists of two twin related -variants (one corresponding to S2+ and the other to S−2 shear) with the ratio of their thicknesses equal to 1:3 will satisfy the IPS condition. Experimentally, a twin thickness ratio of 1:3 is often observed in -plates containing (111) twins, the twin plane being derived from the parent (110) plane. The habit plane solutions obtained from the WLR analysis have been found to be of the (130) type which also match closely with the experimentally determined habit plane indices. From these observations it is clear that the phenomenological theory of martensite crystallography can be used for the prediction of crystallographic observables such as the habit plane and the magnitude of lattice-invariant shear (or ratio of twin thicknesses) in the context of the precipitation of -hydride plates in the -phase matrix (Srivastava et al. 2005). 8.3.4 Crystallography of → transformation The crystallography of the → transformation and the internal structure of -hydride precipitates in the -matrix have been studied in detail by Weatherly (1981), who has pointed out that there are two distinct orientation relationships and habit planes exhibited by -hydride precipitates in the -matrix. These are grouped as type I and type II, the crystallographic details of which are given in Table 8.3. The hcp -structure can be transformed into the fcc -structure by altering the stacking sequence from the ABABAB to the ABCABCABC type . The lattice strain required for transforming the hcp -lattice to an fcc lattice (an imaginary precursor to the -hydride phase) consists of a small dilation of the (0001) planes (to increase the c/a ratio from 1.593 to the ideal value of 1.633) and a simple shear to change the stacking sequence. This fcc structure (with a lattice parameter of 0.4575 nm can be transformed into the fct -hydride structure (a = 04596 nm and c = 04969 nm by a second homogeneous deformation. The overall ¯ , [1010] ¯ and [0001] is lattice strain, referred to the axes [1210]
Table 8.3. Orientation relationship and habit planes for Type I and Type II -Hydride precipitation in -Zr. -Hydride Type I Type II
Habit plane
Orientation relationship
¯ } {1010 ¯ } {1017
¯ 0] ; (001) (0001) [110] [121 ¯ [1210] ¯ ; (111) (0001) [110]
736
Phase Transformations: Titanium and Zirconium Alloys
⎡
⎤ 10048 0 0 10592 03356 ⎦ SH = ⎣ 0 0 −00471 10308
(8.19)
It may be noted here that the / correspondence observed for the type II -hydride precipitates is crystallographically equivalent to that depicted in Figure 8.5. The fact that the lattice strain along one of the principal axes is ¯ direction for less than 0.5% allows one to neglect the strain along the [1210] an approximate analysis which can be reduced to two dimensions. Weatherly (1981) has shown from the construction of the unit circle and of the deformed ellipse that the IPS condition is satisfied for habit planes which are indicated by the undistorted vectors (AB) and after rigid body rotation (A B ) in Figure 8.7 ¯ zone axis. As seen in this construction, two which corresponds to the [1210] ¯ possible habit planes are predicted, both having a normal lying along the [1210] direction. The habit plane normal for the first solution lies approximately 12 from the [0001] pole and a rigid body rotation of 3 is required to bring the undistorted plane back to its original orientation. For the second solution, the ¯ pole and a rigid body rotation habit plane normal lies 16 away from a (1010) ¯ makes an angle of 20 is required. The experimentally observed habit of (1017) of about 14 with (0001) , indicating the validity of martensitic crystallography in the case of -hydride precipitation in -Zr.
¯ -hydride plates of type I which exhibit the 1010 type habit plane have been found to follow one of the two orientation relations listed in Table 8.3. For -plates corresponding to the first orientation relation, (001) (0001) , internal twins are seen along two sets of 110 planes which are obliquely inclined to the habit plane. In contrast, for -plates following the second orientation relation, (111) (0001) , the twin planes are parallel and perpendicular to the habit
plane. ¯ The transformation mechanism associated with -plates exhibiting 10 17
¯ habit planes (type II), involves a simple shear of 30 on a 1010 plane in ¯ 10> ¯ direction. It is, however, interesting to note that the lattice correa < 12 spondences for the -plates of type I and type II could be regarded as crystallographically equivalent if one considered the to transition to occur in two conceptual steps: the first step being the to transition in accordance with the Burgers correspondence followed by the second step comprising the to transition as discussed in the preceeding section. A stereographic projection showing the approximate orientation relations between and through an intermediate orientation brings out the equivalence of two lattice correspondences (Figure 8.7). It could be seen from this stereographic projection that for the type I plates, the (110) plane which is derived from the (0001) plane becomes the
Interstitial Ordering
737
(111) plane, whereas for the type II plates, other 011 planes are converted to 111 planes. 8.3.5 Mechanism of the formation of -hydrides The -hydride phase precipitates form in both the - and the -alloys at high quenching rates. Alloys containing hydrogen to a level exceeding the terminal solubility, will invariably produce -hydride precipitates. The charcteristic features of -hydride precipitates, namely, surface relief on polished surfaces, strict orientation relationship and habit plane, operation of the invariant plane strain condition, periodic internal twinning and rapid transformation kinetics suggest that the transformation involves a lattice shear. The fact that the transformation occurs at sufficiently low temperatures where self-diffusion of zirconium (or titanium) atoms is negligible also suggests that the transformation of the - or the -lattice into that of -hydride does not involve diffusive jumps of the metal atoms. However, hydrogen migration which is known to be very fast even at such low temperatures (<550 K) is necessary for the attainment of the right stoichiometry and interstitial ordering for the formation of the -hydride structure. Considering all these factors, the precipitation of -hydride in either the - or the -matrix can be approximately described as a hybrid transformation involving a shear transformation of the parent Zr-lattice with the accompanying process of hydrogen partitioning and interstitial ordering. In this way, the -hydride precipitation can be grouped in the class of Bainitic transformation. In view of the close lattice correspondence of the -, the - and the -structures, the fact that hydrogen is a strong -stabilizer and the crystallographic observations reported by Dey et al. (1982), Dey and Banerjee (1984) and Srivastava (1996, 2005), it is attractive to envisage that the -hydride formation in the -phase occur through an intermediate -step. It is possible that as hydrogen segregation occurs in certain localized regions within the -phase, these microscopic regions first transform into the -structure which finally goes over to the -hydride structure to accommodate the large concentration of hydrogen atoms. Since direct experimental evidence in support of this step wise transformation, → → , has not been obtained, the size of the embryonic -phase appearing in the path of the → transformation cannot be ascertained. 8.3.6 Hydride precipitation in the / interface Transmission electron microscopic examiantion of electropolished thin foils of + titanium alloys has revealed the presence of an interface phase at / interfaces (Rhodes and Williams 1975, Unnikrishnan et al. 1978). This interface phase has been found to possess fcc structure with a lattice parameter of 0.44 nm. It has also been observed that this fcc phase is present only when the -phase has evolved
738
Phase Transformations: Titanium and Zirconium Alloys
(a)
0.3 μ
(b)
0.3 μ
Figure 8.8. -hydride precipitates enveloping precipitates in the -matrix.
through a diffusional transformation from the parent -phase and not along the martensitic interface. The observation of the fcc phase in a wide variety of / titanium alloys suggests that its origin is fundamentally connected with the nature of the bcc → hcp transformation. Later experiments (Banerjee and Arunachalm 1981, Banerjee and Williams 1983) have conclusively proved that the / interface phase is -hydride which forms during the electropolishing operation (Figure 8.8). The detailed crystallographic analysis reported by Banerjee and Arunachalm (1981) has shown that the -Hydride phase bears two distinct types of orientaiton relationship with the -phase hydride plates at the / interface are designated as type I and type II, based on the following orientation relationship they follow with the -phase: ¯ 1100 ¯ 001 0001 Type I 110 ¯ 1120 ¯ Type II 111 0001 110 Morphologically these two types of / interface hydride can be distinguished as the type I hydride forms a monolithic layer along the interface while the type II precipitates are usually finely divided and are stacked along the faults in the -phase. Let us examine the possible causes for the formation of a -hydride envelope around the -plates in a -matrix. The large difference in the solubility of hydrogen in the - and the -phases is one of the contributing factors. During electropolishing of samples, hydrogen ingress occurs to a considerable extent and once the limit of solubility is exceeded, hydride precipitation commences. A higher solubility of hydrogen in the -phase is expected to provide a preponential path for hydrogen
Interstitial Ordering
739
entry into the sample through the -phase region. The -phase region in contact with the -phase, therefore, attains the limit of solubility at the earliest instance. The other factor which is responsible for the preferential -hydride formation at the / interface is the reduced nucleation barrier for -hydride at such locations. This point is explained considering the crystallographic mismatch associated with the type I and the type II -hydride at the / interfaces. Let us consider first type II hydrides which form along the basal planes of the -phase. The mismatch along the habit plane is a uniform dilation of 3.3% and a large dilation of 8.3% perpendicular to the habit plane. The presence of c-dislocations along the / interface, necessary for compensating misfits on
¯ 112 1100 interface planes, facilitate accommodation of the large dilation perpendicular to the habit plane of hydrides which nucleate at a periodic intervals in the -side of / interfaces. The accommodation of the type I hydride at the interface is not immediately obvious. Banerjee and Arunachalam (1981) have argued that the 17% dilation in the direction normal to the habit, which offers the main constraint against the nucleation of type I hydride, can be accommodated if the / interface is atomistically rough. The presence of ledges on the / interface appear to be responsible for inducing nucleation of type I hydride phase. The formation of hydrides at the / interfaces in samples undergoing a → diffusional transforamtion can be visualized as follows. As the → transformation proceeds, partitioning of alloying elements occurs. Since the difference in solubility of hydrogen in the - and the -phase is large, the -regions at the interface becomes supersaturated with hydrogen resulting in hydride precipitation. Since hydrogen solubility drops down drastically at temperatures below 573 K, hydride precipitation occurs only at such low temperatures in samples containing very low hydrogen level (<50 ppm). 8.3.7 Formation of -hydride The -hydride of Zirconium is the equilibrium phase with fcc structure having lattice parameter a = 04775 nm. As mentioned earlier, hydrogen atoms occupy some of the tetrahedral interstitial sites 41 41 41 and equivalent positions in the fcc lattice.The radius of the site is about 0.0378 nm and there are two such sites per zirconium atom (Northwood and Kosasih 1983). A number of studies involving electron microscopy, dilatometry and X-ray diffraction have shown that the metastable -hydride phase is favoured under a fast quenching rate while with ageing at temperatures in the range of 150–250 C -hydride precipitates transform into -hydride precipitates in an irreversible manner. During continuous cooling from an elevated temperature, the formation of the equilibrium -hydride precipitates occur preferentially when the cooling rates
740
Phase Transformations: Titanium and Zirconium Alloys
are slow, particularly during the nucleation stage. The volume change associated with the formation of the -hydride from the Zr phase is about 17.2%. This is substantially larger than that for the -phase formation in Zr (∼12.3%). The formation of the -hydride is also promoted as the hydrogen level in the alloy is increased. This is reflected in the time–temperature–transformation diagrams for the formation of - and -hydrides in Zr–H alloys of different hydrogen concentrations (Figure 8.9).
αZr
673
(a)
cH = 640 ppm
δ
γ
473
Temperature (K)
0
cH = 500 ppm
αZr 673
(b)
δ γ
473
0 673
αZr
(c)
cH = 50 ppm
473
δ γ
0
10–2
10–1
10
10
102
103
104
Time (sec)
Figure 8.9. Time–Temperature–Transformation diagram for - and -hydride formation in Zr alloys for hydrogen concentrations of (a) 640 ppm, (b) 500 ppm and (c) 50 ppm.
Interstitial Ordering
741
The orientation relationship between the - and the -hydride phase has been found by Northwood and Kosasih (1983) and Une et al. (2004) to be the following: ¯ 1120 ¯ 111 0001 110
8.4
HYDROGEN-RELATED DEGRADATION PROCESSES
Zirconium alloys which are extensively used in structural components in nulcear reactors absorb hydrogen during service. A stringent control of hydrogen level is maintained in these alloys during the entire processing route starting from the production of zirconium sponge followed by alloy melting and fabrication. In fact, recent specifications of some components such as pressure tubes allow a maximum of 5 ppm of hydrogen in the finished products. It is the in-service absorption and accumulation of hydrogen which cause serious degradation of mechanical properties of these components. The problems resulting from hydrogen ingress into zirconium alloys get further aggrevated due to migration of hydrogen leading to localization at points of tensile stress concentration and at relatively cooler regions where the terminal solubility is substantially lower. The severity of the hydrogen-related degradation processes can be recognized from the fact that they are responsible for limiting the service life of several zirconium alloy components in nuclear reactors. Deuterium and/or hydrogen is picked up by zirconium alloy components in a nuclear reactor from the following sources: (1) Deuterium/hydrogen evolved in the corrosion reaction Zr + 2D2 O → ZrO2 + 4D (2) Radiolytic decomposition of D2 O/H2 O D2 O → D + OD (3) Moisture present in the fuel – UO2 pellet which is encapsulated in the cladding tubes made of zirconium alloys; (4) Hydrogen added in the coolant water for scavenging nascent oxygen evolved due to radiolytic decomposition 2OD → D2 O + O (5) Hydrogen/deuterium produced in the corrosion reaction between zirconium alloy components and the moisture present in the annulus gap between calandria tubes and pressure tubes in pressurized heavy water reactors.
742
Phase Transformations: Titanium and Zirconium Alloys
Hydrogen ingress in zirconium alloy components occurs along the entire surface area of these components which remain in contact with coolant or fuel. The mobility of hydrogen at operating temperatures of nuclear reactors is quite high and therefore, apart from embrittlement due to uniform hydride precipitation several other dynamic processes, such as stress reorientation of hydrides, localized hydride blister formation at cold spots and delayed hydride cracking can result in serious degradation of the material. 8.4.1 Uniform hydride precipitation As long as hydrogen remains in solid solution, it does not cause any deleterious effect on mechanical properties of zirconium alloys. As the hydrogen concentration exceeds the TSS limit, precipitation of hydride plates occurs in the matrix. Because of the strain associated with the hydride formation, hydride precipitates acquire a plate shape. The bainite-like transformation mechanism is responsible for sheafs of contiguous hydride plates as shown in Figure 8.10 (Singh et al. 2006). A uniform distribution of the embrittling hydride phase manifests in the form of reduction in tensile ductility, impact and fracture toughness. However, for a significant reduction in these properties, certain minimum volume fraction of the brittle hydride phase is required. The degree of embrittlement strongly depends on the hydride plate orientation with respect to the direction of applied tensile stress. The control of crystallographic
Figure 8.10. Sheafs of contiguous hydride plates formed in Zr–2.5 Nb alloy.
Interstitial Ordering
743
texture of zirconium alloy components is, therefore, of utmost improtance in combating hydride embrittlement. The factors which control the orientation of hydride plates are as follows: (a) Crystallographic texture of the major phase in the fabricated components dictates of
the orientation of hydride plates which exhibit a microscopic habit ¯ (Northwood and Gilbert 1978) making an angle of about 14 with 1017 the basal plane of the hcp Zr and a submicroscopic habit of (0001) (Une et al. 2004). (b) Presence of applied or residual stress determine the plane of hydride precipitation which occurs on the plane normal to the tensile stress direction. By suitable control of fabrication parameters of tubular products, particularly the Q-factor which is defined as the ratio of the wall thickness reduction to the diameteral reduction, the desired crystallographic texture can be introduced. Since a high Q-factor orients the basal poles in the radial direction, thin walled tubes are pilgeried with a high wall thickness reduction. Hydrides forming in such tubes remain primarily circumferential which are least deleterious as far as supporting a hoop stress in the tube is concerned. In tube burst test, the total circumferential elongation, which is a measure of ductility, has been found to be significantly reduced with an increase in the volume fraction of hydrides. In view of the importance of unfavourably oriented hydride plates in thin walled tubes, the ratio of ‘radial’ hydride lengths and total hydride lengths given by a parameter called Fn is monitored as a part of technical specifications. The value of Fn is always kept low for the acceptance of such tubes for fuel cladding applications. In tubular products of larger wall thickness, such as pressure tubes, the orientation of uniformly distributed hydride precipitates changes across the thickness of the wall. This variation is primarily due to the variation in microtexture and microstructure and residual stress across the wall thickness. Though microstructure and texture of pressure tubes are carefully controlled for pressure tubes giving due considerations of hydride precipitate orientations, acceptance criteria usually do not include the Fn number specification. 8.4.2 Hydrogen migration Hydrogen migration at operating temperatures (∼ 300 C) of nuclear reactors is responsible for restructuring of hydride precipitates. Phenomena such as stress reorientation of hydrides, formation of hydride blisters and delayed hydride cracking which are serious causes of material degradation are all outcome of in-service hydrogen migration. Studies on hydrogen migration have revealed that hydrogen migrates down the concentration and temperature gradient and up the tensile stress
744
Phase Transformations: Titanium and Zirconium Alloys
gradient in zirconium (and also in titanium) alloys. The general diffusion equation, in one dimension, considering all the three gradients is given by (Northwood and Kosasih 1983): d ln cx Q∗ dT −Dcx ∗ dP RT + −V J= RT dx T dx dx
(8.20)
where J is hydrogen flux, D hydrogen diffusivity in metal, (= 0217 exp − 8380/RT mm2 /s), cx hydrogen concentration at point x, Q∗ and V ∗ , heat and volume of transport of hydrogen in metal and P is the hydrostatic stress. Two distinct cases, namely, thermal migration and stress directed migration can be considered using the generalized diffusion equation for hydrogen migration. In absence of a stress gradient, thermal migration, governed by the concentration and the temperature gradients, occurs. Let us consider a case where the bulk hydrogen concentration is more than the TSS of hydrogen at Tmax , the maximum temperature of the sample. Imposition of a temperature gradient on such a sample will introduce a concentration gradient. This is due to the fact that hydrogen atoms participating in the diffusion process are those which remain in solid solution and not those which are locked up in the hydride phase. At the low temperature end, more of hydride phase will be precipitated and level of hydrogen in solid solution will be given by TSS at that temperature. This will result in setting up of a hydrogen flux from the high temperature to the low temperature end of the sample. There exists, however, a critical hydrogen concentration called thermal migration threshold for a given thermal boundary condition, below which there is no net migration of hydrogen. At this concentration of hydrogen, a dynamic equilibrium is established in which the hydrogen flux due to concentration gradient exactly balances the opposing flux due to thermal gradient. This condition can be described as: d ln cx Q∗ dT −Dcx RT + =0 (8.21) J= RT dx T dx Solving this equation, one can find out the threshold hydrogen concentration, which is also known as blister formation threshold, as the formation of hydride blisters at cold spots can occur when this threshold is exceeded. This phenomenon is discussed in detail in Section 8.4.5. The migration of hydrogen can also occur when only a stress gradient is present. Hydrogen has a tendency to migrate up the tensile stress gradient. Such stressdirected migration has serious implications in the processes like stress induced hydride reorientation and delayed hydride cracking.
Interstitial Ordering
745
As in the case of thermal migration, stress-directed migration of hydrogen is also associated with a threshold hydrogen concentration, cxth , which is determined by the dynamic equilibrium given by the following equation: −Dcx d ln cx ∗ dP J= RT +V =0 RT dx dx
(8.22)
The stress-directed migration is manifested into processes such as stress-induced reorientaion of hydride precipitates and migration of hydrogen to locations of high tensile stress concentration leading to delayed hydride cracking. These processes often encountered during service are sometimes responsible for limiting the service life of a zirconium alloy component in a nuclear reactor. 8.4.3 Stress reorientation of hydride precipitates As mentioned earlier, the crystallographic texture of zirconium alloy components is carefully controlled so that the hydride precipitates are oriented favourably with respect to the major stress direction. The initial distribution of hydride precipitates may undergo appreciable changes due to the dissolution and reprecipitation of hydride plates during service. Since the TSS of hydrogen increases exponentially as the temperature is raised from ambient to the reactor operating temperature, a large fraction of hydride precipitates dissolve on heating. Subsequently, they may precipitate under stress during cooling in grains which have basal planes nearly perpendicular to the major tensile stress direction. A threshold value of stress must be exceeded for the reorientation to occur. The threshold stress increases with the yield strength of the material but decreases with increasing solutionizing temperature (Northwood and Kasasih 1983). Experimental evaluation of threshold stress can be made using specially designed tapered tensile specimens (Figure 8.11(a)), which when strained along its axis produces a gradient in tensile stress along its length (Singh et al. 2004). The stress level at which reorientation of hydrides can be observed gives the threshold stress value (Figure 8.11b). It has been reported that the degree of reorientation is insensitive to time under stress and to small variation in texture, grain size and dislocation density (Northwood and Kosasih 1983). The threshold stresses at the typical reactor operating temperature of 300 C for hydride reorientation are reported to be about 180–250 MPa and 90–120 MPa for cold worked and stress relieved Zr–2.5 Nb and cold worked and stress relieved zircaloy-2, respectively (Lietch and Puls 1992). For the pressure tubes, if the hoop stress in service exceeds the threshold stress value, reorientation of hydride plates can cause a shift of preference from circumferenial to radial orientation of hydrides.
22
Phase Transformations: Titanium and Zirconium Alloys
37
746
22 38 63 91 All dimensions in mm
(a) 134.7
142.3
152.6 157.5 Inside diameter of the tube
160.7
168.7
600 μm
(b)
Figure 8.11. (a) Tapered gage tensile specimen. (b) Montage showing hydride orientation as a function of stress (MPa).
8.4.4 Delayed hydride cracking Delayed hydride cracking is a form of localized hydride-embrittlement phenomenon, which in the presence of a tensile stress field manifests itself as a subcritical crack growth process. It is caused due to hydrogen migration up the tensile stress gradient to the region of stress concentration. Once the solid solubility limit is exceeded in the region of localized stress concentration, brittle hydride plates precipitate normal to the operating tensile stress and these plates continue to grow till a critical size is formed. At this stage, hydride plates of critical size cracks under concentrated stress leading to the growth of the crack. This crack growth is delayed by the time required for hydrogen to reach the crack tip and form hydride platelet of critical size. Hence this phenomenon is called delayed hydride cracking (DHC), which is characterized by the crack growth rate called DHC velocity (VDHC ).
Interstitial Ordering
747
Based on the important processes occurring, the region ahead of a crack tip for a sample or a component of a zirconium alloy experiencing DHC can be divided into three zones (Singh et al. 2002a). These are process zones where the stress is larger than the yield stress, o , of the material, a reorientation zone where o > > th , the threshold stress for reorientation of hydride being th and a migration zone where < th . The processes which operate in these zones are schematically illustrated in Figure 8.12. Experimental observations of the DHC phenomenon have revealed the following important features (Singh et al. 2002a): (1) DHC crack initiation is associated with an incubation period and fracture surface is usually marked with striations (Figure 8.12(b)). (2) The initiation of DHC is associated with a threshold stress intensity factor, KIH , below which the DHC crack growth velocity, VDHC , is too small to be detected; (3) KIH is practically independent of material strength; (4) For a given material and for KIH < KI < KIC , VDHC is independent of the applied stress intensity factor, KI (where KIC is the fracture toughness of the material); (5) VDHC increases with an increase in the strength of the material; (6) VDHC increases with temperature due to increased hydrogen diffusivity and solubility at higher temperatures. Though softening of material causes an increase in the process zone size, the increased hydrogen diffusivity has a dominating effect on VDHC . The variation of VDHC with temperature is reported to exhibit an Arrhenius type relationship, the activation energy for DHC being in the range of 40–70 KJ/mole which corresponds to the sum of the activation energy of hydrogen diffusion (30 kJ/mole) and the enthalpy of mixing of hydrogen in zirconium (38 kJ/mole). 8.4.5 Formation of hydride blisters In the presence of a thermal gradient, hydrogen migrates down the temperature gradient. Once the solid solubility at the localized low temperature spot (hereforth called cold spot) is exceeded, hydride precipitation starts. A sustained thermal gradient allows continuation of hydrogen migration from the high temperature to the low temperature region as a steady state concentration gradient persists under such a condition. The volume fraction of the hydride phase at the cold spot increases to near 100% and a semiellipsoidal module of hydride forms at the cold spot. Since the transformation of zirconium metal into hydride is associated
748
Phase Transformations: Titanium and Zirconium Alloys
σy = K
σy
√2πr
Stress (σ)
σo
Modified stress distribution elastic + plastic
σth
Distance (r)
rp
Process zone
Reorientation Migration zone zone (a)
80 μm (b)
Figure 8.12. (a) Schematic representation of three zones ahead of a sharp crack. (b) Fracture surface showing striations resulting from stepwise growth of delayed hydride cracks.
Interstitial Ordering
749
with an increase in volume, a bulge appears on the surface of the cold spot. This hydride bulge, due to its appearance, is called a hydride blister. The phenomenon of blister formation attracted special attention of the nuclear engineering community after the failure of a zircaloy-2 pressure tube in a pressurized heavy water reactor, PHWR (Pickering - Unit 2) in Canada. In the PHWR design, several hundred pressure tubes are arranged in a lattice in a large vessel called calandria (Figure 8.13). Each of these pressure tubes which contains a series of fuel bundles pass through calandria tubes, the gap between a pressure tube and the surrounding calandria tube is maintained by a set of spacers known as garter springs. Pressure tube carries coolant water at around 300 C whereas the calandria tube remains in contact with moderator water maintained at a temperature of 60–80 C. Pressure tubes have a tendency of sagging because of the deflection due to the fuel load and due to the in-reactor creep. When the sagging of the pressure tube results in the establishment of a contact between a pressure tube and a calandria tube, the contact points become cold spots (as shown in Figure 8.14(a) and (b)) which are susceptible to hydride blister formation.
Figure 8.13. Lattice arrangement of pressure tubes and calandria tubes in pressurized heavy water reactors.
750
Phase Transformations: Titanium and Zirconium Alloys Presure tube
Calendria tube 80°C 300°C
Garter spring (a) Before creep deformation 80°C 300°C
Contact point (b) After creep sagging OD of the pressure tube
Circumferential direction
380 μm
Region III
Radial direction
O
Region II
Region I
(c)
SYY - STRESSES
VIEW : –54.10852 RANGE : 110.4911
110.5
98.73
86.98
75.22
63.46
51.71
39.95
28.19
16.43
4.677
–7.080
–18.84
–30.59
–42.35
–54.11
ROTX 0.0 ROTY 0.0 ROTZ 0.0
EMRC–NISA/DISPLAY
Y X
MAR/05/01 15:46:26
(d)
Figure 8.14. (a) and (b) show pressure tube/calandria tube contact creating cold spot on the power. (c) A section of hydride blister showing both the radial and circumferential hydrides. (d) Computed stress field inside the blister and the matrix surrouding it.
Interstitial Ordering
751
The parameters which influence the growth rate of hydride blisters are the difference in temperature between the bulk of the material and the cold spot, thermal conductivity at the contact point and the hydrogen content of the material. Larger the temperature difference and sharper the thermal gradient, higher the blister growth rate. A higher hydrogen level in the bulk of the material no doubt promotes the growth of blisters. Metallographic examination of the section of a hydride blister along the radial-circumferential plane of a pressure tube shows three distinct regions (Figure 8.14(c)), viz. the core region where the hydride volume fraction is nearly 100%, the reoriented hydride region where a large fraction of hydride plates are reoriented from their original habit to directions radial to the semiellipsoidal blister and the region where hydride precipitates retain their original habit. The stress field present around a growing blister is responsible for reorienting hydrides in the region adjacent to the blister (Figure 8.14(d)) (Singh et al. 2003). Two distinct types of blister morphology have been encoutnered, namely (a) a single convex hump; and (b) a nearby circular ring which is made up by joining a series of independently nucleated blisters (Singh et al. 2002b). The former type is formed when all the hydrogen present in the alloy is first brought into solid solution and the cold spot is created subsequently. Under this condition, blister nucleation occurs right at the cold spot. In any case, these blsiters cannot grow beyond a size where their periphery touches the temperature corresponding to the TSSP of hydrides. The latter type (ring morphology) is formed when the cold spot is maintained during the heating-up operation. As a consequence, the hydride precipitates which remain undissolved in the vicinity of the cold spot act as nucleating centres for blister formation. As the hydrogen flux is directed towards the cold spot, hydrogen atoms are arrested at the point where undissolved hydride precipitates are present. The ring of blisters, therefore, mark the temperature contour corresponding to the TSS limit for hydride dissolution under the imposed thermal gradient. In this context it may be emphasized that the TSS limit for precipitation and for dissolution of hydrides in -zirconium are different and are denoted by TSSP and TSSD, respectively. While TSSP is the controlling factor for the single blister morphology, the ring type blister formation is governed by TSSD (Singh et al. 2002b). For the nucleation of blisters, a minimum hydrogen concentration, cBFT , called blister formation threshold is required. However, cBFT is dependent on the TSS, ∗ , at the cold spot temperature. The relationship of these concentrations with the cTSS bulk and the cold spot temperatures can be derived from the steady state condition in absence of stress, where the net flux J of hydrogen is given by Eq. 8.21.
752
Phase Transformations: Titanium and Zirconium Alloys
This partial differential equation can be reduced to Q∗ 1 dcx + =0 cx dT RT 2
(8.23)
Separating the variables and on integrating between the bulk dissolved hydrogen ∗ at the cold spot temperature, T ∗ we get concentration, cb and cTSS
∗ cTSS
cb
dcx Q∗ T dT =− cx R Tb T 2 ∗
(8.24)
which reduces to Q∗ Tb − T ∗ cBFT = exp − ∗ cTSS R Tb · T ∗
(8.25)
∗ The ratio of the two concentrations cBFT /cTSS , increases with an increase in the ∗ cold spot temperature, T , as is shown in Figure 8.15 where the bulk temperature, Tb is maintained at 573 K (Singh et al. 2002b). Since cTSS depends on the direction of approach of temperature (i.e. TSSP and TSSD), cBFT will also vary depending on whether it is measured during a heating or a cooling operation.
50
0.7
TSSD TSSP c BFT /c TSS
0.6 0.5
30
0.4
20
0.3 0.2
c BFT/c TSS
c TSS (ppm)
40
10 0.1 0
250
0.0
300
350
400
450
500
550
Tcs (K)
Figure 8.15. Hydrogen concentration corresponding to blister formation threshold (cBFT ) as a function of the cold spot temperature, Tcs .
Interstitial Ordering
8.5
753
THERMOCHEMICAL PROCESSING OF Ti ALLOYS BY TEMPORARY ALLOYING WITH HYDROGEN
Titanium and most of its commercial alloys have a high solubility for hydrogen, being capable of absorbing upto about 6 at.% hydrogen at 600 C. Since hydrogen entry in titanium alloys is reversible, temporary alloying with hydrogen can be very effectively used in stabilizing the -phase which in turn assists in refinement of the microstructure and in improvement of fabricability of these alloys. Since the alloy chemistry is changed during the processing operation, the technique is known as thermochemical processing. The basic principle of this processing technique is discussed here as an example where hydrogen-induced phase transformations have beneficial effects. After the processing is completed, hydrogen can be extracted from the alloy by vacuum annealing. The concept of constitutional solution treatment can be explained using a pseudo-binary phase diagram for Ti–6Al–4V with hydrogen (Figure 8.16). Since hydrogen stabilizes the -phase, it is possible to solutionize Ti–6Al–4V into a single phase -solid solution by hydriding at a temperature above the eutectoid temperature (Te ) of 800 C. This hydrogenated alloy (containing 0.5–1 wt% H) on cooling transforms into either + TiH or Orthorhombic martensite ( ) depending on the cooling rate. The latter on subsequent ageing decomposes into a mixture of + TiH. The large volume expansion associated with hydride formation results in an accumulation of high density of dislocations in the matrix. When this material is dehydrogenerated, a fine -phase forms by the decomposition of the hydride to a mixture of and spherodized -phase. The low transformation and dehydrogenation temperatures and strain in the matrix are responsible in substantial refinement of the microstructure and consequent improvement in mechanical properties. Hydrogen (at.%)
Temperature (K)
10
20
30
40
β
1273
β+X
α+β 1073 α
X
873
α+X 0.2
0.4
0.6
0.8
1.0
Hydrogen (wt%)
Figure 8.16. Ti-6V-4A1 + H pseudo binary phase diagram.
1.2
754
Phase Transformations: Titanium and Zirconium Alloys Table 8.4. Thermochemical treatment cycles. Designation
Treatment sequences
Constitutional solution treatment -quench followed by hydride-dehydride Hot isostatic pressing or vacuum hot pressing High Temperature hydrogenation
Hydrogenate at T > Te → cool to T < Te → dehydrogenate -solution treatment → water quench → hydrogenate at T < Te and dehydrogenate at T < Te Hydrogenate to increase -volume fraction → hot deformation → dehydrogenate Hydrogenate in → cool to room temperature → dehydrogenate at T < Te
A variety of thermochemical processing cycles, as listed in Table 8.4, have been tried. Froes and Eylon (1990) have summarized the effects of these thermochemical treatments on the microstructure and mechanical properties of titanium alloys.
8.6
HYDROGEN STORAGE IN INTERMETALLIC PHASES
8.6.1 Laves phase compounds Hydrogen storage materials (HSMs) are usually intermetallics containing interstices with a suitable binding energy for hydrogen which allows its absorption or desorption near room temperature and atmospheric pressure. A promising candidate for hydrogen storage is the class of Laves phase compounds having the formula unit of AB2 . Laves phase include fcc (MgCu2 ), hexagonal C14 (MgZn2 ) and di-hexagonal C36 (MgNi2 ) structures. Since they transform from one to another during heating and cooling (typically C14 at high temperatures and C15 at low temperatures), hydrogen absorption and desorption can essentially be considered as a phase transformation process. The C15 structure is an fcc-based structure containing six atoms (two formula units) in the primitive unit cell, while C14 and C36 structures are hexagonal structures containing 12 and 24 atoms in the primitive unit cell, respectively. The interstitial sites occupied by hydrogen in AB2 Laves phases are tetrahedral sites formed by two A and two B atoms (2A2B site) or by one A atom and 3 B atoms (1A3B site) or four B atoms (4B site). There are 17 tetrahedral sites per formula units: twelve 2A2B sites, four 1A3B and one 4B site for both C14 and C15 structures. While in C15 structures, A or B atoms within each type of sites (2A2B, 1A3B and 4B) are locally equivalent, this is not the case for C14 structure. Depending on the local environment of the tetrahedral interstice, 12 2A2B sites and 4 1A3B sites per formula units in the
Interstitial Ordering
755 Y
Active
X
Z
(a)
5
4 3 7
2
8 6 9 1 (b)
(c)
Figure 8.17. (a) B2 crystal structure, (b) Crystal structure of Laves phase compounds C14 (hexagonal) open circles stand for A atoms and solid circles for B atoms and (c) the C15 Laves structure, where the larger and the smaller circles represent A (Zr) and B (X = V, Cr, Mn, Fe, Co, Ni) atoms, respectively. Here, for example, a 2A2B site is formed by the atoms labeled 2, 4, 7 and 8; a 1A3B site by atoms 1, 6, 8 and 9; and a 4B site by atoms 6, 7, 8 and 9.
C14 structures are further sub-grouped into six 2A2B (I) sites, three 2A2B(k2 ) sites, 1.5 2A2B(h1 ) sites, 1.5 2A2B(h2 ), one 1A3B(f) site and three 1A3B (k1 ) sites. To depict the interstitial sites formed by A and/or B atoms, a ball stick model of the fcc structure is given in Figure 8.17. The 2A2B sites has the largest interstitial hole size and the 4B site has the smallest hole size (Hong and Fu 2002). The volume change associated with hydrogen absorption could be as large as +20%. Such a large volume change during hydrogen absorption and desorption results in cyclic loading and unloading of the matrix which may lead to disintegration of the host lattice.
756
Phase Transformations: Titanium and Zirconium Alloys
8.6.2 Thermodynamics Most of the thermodynamic properties of a HSM can be obtained from pressure– composition isotherms (PCI). Figure 8.18(a) illustrates a typical PCI diagram (Sinha et al. 1985). The points of intersections of mildly sloping plateau with the steeply sloping portion of an isotherm define the phase boundaries / + and ( + /, where is the solid solution of hydrogen in the host alloy and is the hydride of the alloy. The plateau (mildly sloping region) pressure for formation of hydrides (Pf ) is higher than that corresponding to dissociation of hydrides (Pd ) due to hysteresis effect. The degree of hysteresis is expressed by the logarithmic function (1/2) RT 1n Pf /Pd which is the free energy loss per mole of atomic hydrogen in completing a hysteresis loop. With increase in temperature, plateau pressure increases. Equilibrium pressures in the plateau region and its slope are important in deciding the application of HSM. Using Vant Hoff equations: ln Pi =
Hi Si − RT T
where i stands for f (formation) or d (dissociation) of hydrides, the enthalpy of formation and dissociation can be obtained from PCI by plotting plateau pressure against the inverse of absolute temperature. The negative of the slope of such a plot will yield enthalpy change value and the intercept will yield the change in entropy values. 8.6.3 Ti- and Zr-based hydrogen storage materials In general, a good HSM for automobile applications should have dissociation temperature and corresponding equilibrium vapour pressure comparable to that of the ambient. Most of the developed hydrogen storage alloys can be classified in two groups, viz. AB5 and AB2 types. LaNi5 is a typical example of the AB5 type hydrogen storage alloys, which is mainly used in Ni-metal hydride batteries. Hydrogen storage capacity of AB5 alloys is about 1.5 wt% (Taizhong et al. 2004). In the AB2 type alloy, A = Ti or Zr and B = Ni, V, Mn Cr etc. These alloys usually possess the structure of Laves phase. The AB2 type Laves phase alloys have been regarded as promising HSMs because of high hydrogen storage capacity. Both Ti- and Zr-based AB2 have been explored for applications as HSM and are briefly discussed in the next sections. Zr-based Laves phase alloy hydrides are very stable with very low dissociation pressure for hydrogen and hence are not suitable for automobile application. On the other hand, Ti-based Laves phase alloy hydrides are unstable but their plateau characteristics are not suitable for automobile applications. Thus for both Ti- and Zr-based AB2 type alloy, substitution of both A- and B-type element are being used to modify the plateau pressure, its slope and hysteresis to suitable values.
Interstitial Ordering
757
30°C ZrCr1 – γ Fe1 + γ
Pressure (Nm)
0.5 0.3 0.2
0
1
2
3
(a)
0.6
Composition (HM)
Ti0.99 Zr0.01 Fe
Ti0.8 Zr0.2 Fe
0.5
Ti0.9 Zr0.1 Fe
Ti0.9 Zr0.1 Fe (No H.T.)
0.4 0.3
0.2
0.1
0.01
TiFe 0.1
1
10
100
1000
Time (min) (b)
Figure 8.18. (a) Typical pressure–composition isotherms for ternary Zr–Cr–Fe alloys. (b) Influence of Zr addition on the hydrogen absorption curves for TiFe alloys.
758
Phase Transformations: Titanium and Zirconium Alloys
8.6.3.1 Ti-based hydrogen storage materials Among the Ti-based systems, TiFe, TiNi2 , TiNi, TiMn2 , TiCr and TiV intermetallic compounds were investigated for hydrogen storage and purification. One of the significant problems associated with TiFe was the difficulty in activation. Substitution of Nb or Ta or V for Fe and substitution of Nb or Cu for Ti in TiFe is reported to accelerate activation. The TiFe alloys, to which Fe2 O3 with Cu were added for Ti, comprised of TiFe matrix and precipitates of Fe2 Ti and the oxide (Fe7 Ti10 O3 ) phases. The interface between the matrix and the precipitates were suggested to be active sites for the hydriding reaction and as entrance sites for hydrogen to diffuse into the alloy. Addition of small quantities of Zr and Nb enhances the ease of activation (Figure 8.18(b)). TiMn alloys have been investigated due to their easy activation, good hydriding– dehydriding kinetics, high hydrogen storage capacity and relatively low cost. TiMnx (with x = 1 to 2) have C14 type hexagonal crystal structure. The hydrogen storage capacity of Ti–Mn hydrogen storage alloy was reported to be 1.5–1.8 wt%. There are two major categories of Ti–Mn HSM, i.e. TiMn2 and TiMn15 . The main problem with the Ti–Mn alloys is their high equilibrium plateau pressure, which limits their practical application. Another shortcoming of these alloys is that hydrides in these alloys are characterized by large hysteresis effects. V is reported to reduce the hysteresis. Homogenization treatment helps in obtaining clear plateau pressure. Several studies have focused on the roles of solid–gas and electrochemical reactions. The activation of the alloys plays a key role in hydrogen absorption process, since it defines the reaction rate of the hydrogen with the metal and its incorporation in its structure. The hydrogen storage capacity of HSM is a function of the crystal structure–lattice parameter and void size. By suitable alloying addition, void size can be manipulated resulting in increase in hydrogen storage capacity. In late nineties, it was reported that TiV with a bcc structure absorbs more hydrogen than conventional intermetallic compounds. Zr substitution in Ti–Cr–V system decreases the equilibrium pressure and the hysteresis of the PCI. However, it increased the slope of PC isotherm and reduced the hydrogen storage capacity by forming ZrCr2 . By selecting suitable composition range in the quarternary Ti–Zr–Cr–V system, formation of this phase can be avoided (Figure 8.19). The atomic radius of zirconium (1.62 Å) is larger than that of Ti (1.47 Å) and its electronegativity (1.4) is smaller than that of Ti (1.6). Generally, a metal hydride becomes stable with increasing difference between the electronegativities of hydrogen (2.1) and the metal atom. By adding Zr, therefore, a decrease of the hysteresis and the plateau pressure, and an increase in the width of the plateau and the hydrogen storage capacity due to an increase in lattice volume is expected. Under heat treated condition, the hydrogen storage capacity of Ti016 Zr005 Cr022 V057 is reported to be 3.55 wt% (Table 8.5).
Interstitial Ordering
759
Ti
at.%
Cr
at.%
at.%
V
Figure 8.19. Composition range showing the largest effective hydrogen storage capacity in the Ti–Cr–V alloy system.
Table 8.5. Hydrogen storage capacities of the composition controlled and heat treated Ti–Zr–Cr–V alloys. Composition
Ti012 Zr005 Cr045 V026 Ti012 Zr005 Cr045 V026 Ti012 Zr005 Cr045 V026 Ti012 Zr005 Cr045 V026 Ti012 Zr005 Cr045 V026 Ti012 Zr005 Cr045 V026 Ti012 Zr005 Cr045 V026 Ti012 Zr005 Cr045 V026
H-storage capacity
Effective H-storage capacity
H/M
wt%
H/M
wt%
1.50 1.44 1.50 1.73 1.67 1.57 1.71 1.87
2.90 2.74 2.87 3.34 3.18 2.99 3.24 3.35
0.85 0.77 0.79 1.07 0.93 0.86 1.02 1.13
1.65 1.47 1.51 2.07 1.77 1.63 1.92 2.14
8.6.3.2 Zr-based hydrogen storage materials Various studies performed so far on Zr-based AB2 type HSM have been directed to increasing the equilibrium hydrogen vapour pressure in the Zr-based Laves phase alloys without markedly reducing the absorption capacity by partial substitution of A or B type elements by other elements. The A element can be substituted by Ti while the B element by Cr, Mn, Fe, Co, Ni and Cu. The reduction in hydride stability is possible due to any or all of the following effects: the reduction in the size of the hydrogen occupation site, the change in chemical affinity for hydrogen and the electronic factor. The reduction in binding energy of hydrogen in C15 Zr X2 ( X = V, Cr, Mn, Fe, Co, Ni) is shown in Figure 8.20. Most
760
Phase Transformations: Titanium and Zirconium Alloys
Binding energy of H (kJ/mol H)
30
0
–30
2A2B site 1A3B site
–60
4B site
–90
V
Cr
Mn
Fe
Co
Ni
Figure 8.20. Binding energy of hydrogen in C15 ZrX2 (X = V, Cr, Mn, Co, Ni).
research has been concentrated on the ZrMn2 type and ZrCr2 type pseudobinaries. Elemental substitution in pseudo-binaries can be classified according to the following: (1) A element substitution, primarily with Ti: e.g. Zr1−x Tix Cr2 and Zr1−x Tix Mn2 . (2) B element substitution with a different transition element: e.g. Zr(Mn1−x (Fex )2 , Zr(Cr1−x Fex )2 . (3) Hyperstoichiometric substitution of B element: e.g. ZrMn2 + x, ZrMn2 Fex , ZrMn1 + xFe1+y (4) Any combination of the above: e.g. Zr1−x Tix MnFe and Zr1−x Tix (Fe1−y Mny )z alloys. The hydrogen absorption characteristics of Zr-based stoichiometric AB2 HSM such as ZrMn2 , ZrV2 , ZrFe2 , ZrCo2 and ZrCr2 were first studied by Shaltiel’s group (Shaltiel et al. 1977). The hydrides of the first two alloys were observed to be too stable to be of practical significance. The stoichiometric ZrMn2 hydride, for example, exhibits a dissociation pressure of 0.007 atm at 323 K. ZrMn2 crystallizes in the C14 (MgZn2 ) structure. The same structure is observed for ZrMn2 alloys in which Zr has been partially substituted by Ti and Mn by Fe. Since the hydride in ZrMn2 is stabilized primarily by interaction of hydrogen with Zr, it was suggested that the vapour pressure of hydride might be increased by having a system that was
Interstitial Ordering
761
substoichiometric in Zr. Investigation of ZrMn2+x binaries, ZrMn2 Fex ternaries and a series of related systems have established that the dissociation pressure and the hydrogen capacity of these alloys depend on the Zr:Mn:Fe ratio in the lattice and the dissociation pressure was manifold higher than ZrMn2 without significantly reducing the hydrogen storage capacity (Sinha et al. 1985). Though these alloys were found to absorb large quantities of hydrogen, the hydrides formed were too stable to be of practical significance. ZrMn1+x Fe1+y , where Zr is partially substituted by Mn and/or Fe and Zr1−x Tix MnFe, where Zr is partially substituted by Ti are reported to be more attractive than LaNi5 because of favourable combination of properties such as lower cost, rapid kinetics of hydrogen sorption and lower endothermal nature of dehydrogenation. In comparison with Mn and Fe, the hyperstoichiometric Ni and Co is reported to be more effective in augmenting the decomposition pressure of ZrMn2 hydride, although their presence reduced hydrogen storage capacity of the alloy. Sinha et al. (1985) reported that the multicomponent Zr-based alloy having C14 structure appear to be very promising HSM. The ZrCrFe1+x alloys, in particular, have many attractive properties. The vapour pressure of ZrCr2 hydride is augmented by several orders of magnitude without significantly reducing its hydrogen storage capacity when 50% Cr is replaced by Fe and in addition, hyperstoichiometric Fe is present in the lattice. The ZrCrFe15 alloy, for example, exhibits a dissociation pressure of its hydride of about 2 atm at 23 C. The hydrogen storage capacity of this alloy is comparable to that of LaNi5 . The effectiveness of alloying elements in destabilizing ZrMn2 hydride is reported to be in the following order: Cr < Mn < Fe < Ni < Co. Sinha et al. (1982) reported that 20% substitution of Zr by Ti raises the equilibrium pressure five fold in (Zr1−x Tix )Mn2 . The change in PCIs for hydrogen absorption and desorption in ternary ZrCr1−y Fe1+y alloy is shown in Figure 8.21 with increasing Fe content (Park and Lee 1990). Substitution of iron by chromium raises the two phase plateau pressures for hydride formation and dissociation. The extent of hysteresis was also observed to increase. The overall trend in PCIs of Zr09 Ti01 Cr1−y Fe1+y for the ranges 0 < y < 0.4 is shown in Figure 8.22 (Park and Lee 1990). Unlike Ti-free ZrCr1−y Fe1+y alloy, well-defined plateaux with low slopes are formed in the two phase region for the whole composition range y. It is reported that 10% substitution of Zr by Ti in ZrCr1−y Fe1+y alloy yield the best of the Laves phase alloy for various applications due to their low hysteresis and low slopes. 8.6.4 Applications In the recent years, hydrogen has received worldwide attention as an alternative energy carrier which can substitute petroleum in internal combustion engines. The possible applications of HSMs are in automobile industry, moderators in nuclear
762
Phase Transformations: Titanium and Zirconium Alloys 30°C Zr0.8 Ti0.2 Cr1 – y Fe1 + y
30°C Zr0.9 Ti0.1 Cr1 – y Fe1 + y Y
Y
0.4 0.3 0.2 0.1 0
10
PH2 (atm)
PH2 (atm)
10
0.4 0.2 0
1
1
0.1
0.1 0
2
1
H/AB2 (a)
3
0
2
1
3
H/AB2 (b)
Figure 8.21. Pressure–composition isotherms for Zr–Cr–Fe–H2 systems in which Zr has been partially replaced by (a) 10% Ti and (b) 20% Ti at 303 K.
reactors, heat pumps, refrigerators, getters, etc. With the development of fuel cell technology and the widening of hydrogen application fields, methodology of hydrogen storage has attracted more and more attention. Three systems of interest are glass microspheres, liquid hydrogen and metal hydrides. The disadvantages for glass microspheres are that charging requires high hydrogen pressure and temperatures (573 K) while for discharging the microspheres have to be heated to 473 K. On the other hand, liquid hydrogen requires complex handling techniques, a safe cryogenic container and involves cost of liquefaction. For a refrigeration cycle having 33% efficiency for refrigeration, the refrigeration will require 10 KWh/kg of hydrogen, which accounts for nearly 25% of heat of combustion of hydrogen. In addition to several advantages, the unique feature of metal hydride is the equilibrium existing between the hydrogen and the metal. On cooling, hydrogen is reabsorbed into the metal, in contrast to two other systems in which, hydrogen once librated, remains in the gas phase. However, several of their properties are insufficient to preclude any large scale engineering applications, e.g. hydrides can be contaminated by oxygen, nitrogen, water vapour and carbon-dioxide, all of which lead to loss of hydrogen storage capacity. The driving force for current
Interstitial Ordering
0
10
20
763
Oxygen (at.%) 40 50
30
60
70
{
2473
Tin O2n-1
2273 ~2158 K
2073
2143 K
2115 K
1993 1943 (β-Ti)
γ-TiO
β-Ti2O3
1873
~1523 K
1473 β-TiO
(α-Ti) 1273 1155 K
β-Ti1–xO
α-TiO
Ti3O2
1073 873
TiO2 (rutile)
1673
higher magneli phases
Temperature (K)
L
α-Ti1–xO
Ti2O Ti3O
673 5
0
10
15
Ti
25
20
30
35
40
45
Oxygen (wt%)
(a) Oxygen (at.%) 0
10
20
30
40
50
60
70
L+G
3073 2873
L
~2650 K
2673
γZrO2–x 5.5%, 2403 K
2473
Temperature (K)
2273 2073
~2983 K
~2338 K 1.9 2243 4.1 2.0 2128
22
8.6 10.5
β(Zr) 1873
~1798 K 7.4
1673
23.5 βZrO2–x
~1478 K 1473 (α'Zr)
6.9 ~1243 K
1273 6.7
1136 K (α3"Zr)
1073 873
αZrO2–x
(α2"Zr)
~773 K 26.0
6.6 (α4"Zr )
(α1"Zr)
673 473 0
5
10
15
20
Oxygen (wt%)
(b)
Figure 8.22. (a) Ti–O and (b) Zr–O binary alloy phase diagrams.
25
30
35
764
Phase Transformations: Titanium and Zirconium Alloys
research on HSM is to attain weight reduction, cost reduction, optimizing temperatures, heat transfer to and from hydride bed, kinetics of hydrogen sorbtion and other parameters. As an example, the requirement for the fuel storage system of the hydrogen automobile are (Sinha et al. 1982) listed below: (1) A minimum delivery rate of 1.25 kg/h to meet the suburban driving cycle defined by the Society of Automobile Engineers; (2) A high energy density (energy per unit mass); (3) A low storage volume such that it can be accommodated in a car; (4) A high degree of safety during charging and discharging; (5) A reasonable charging time; (6) No necessity of an external energy source to start-up; and (7) An energy as small as possible for hydrogen retrieval. Hydrogen storage in metals is of special interest, since certain metals, alloys and intermetallic compounds react rapidly with hydrogen in a reversible manner at nearly ambient temperatures and pressures. The safety of the hydride tanks are comparable to that of petrol tanks, which is widely accepted in the community. The energy to weight ratio of the metal hydrides are about five to ten times greater than that of lead-acid battery, although they are still 10–20 times less than the energy to weight ratio of petrol. The important properties of hydrides for aforementioned applications are hydrogen storage capacity, dissociation temperature, vapour pressure of dissociation, ease of activation (is a function of temperature and pressure at which absorption and desorption begins), hysteresis and enthalpy of absorption and desorption.
8.7
OXYGEN ORDERING IN -ALLOYS
8.7.1 Interstitial ordering of oxygen in Ti–O and Zr–O The oxygen atoms dissolved in -Ti and Zr occupy the octahedral interstitial sites of the hcp host lattice. These interstitial sites are surrounded by six metal atoms as shown in Figure 8.2. Dagerhamn (1961) have first reported that the incorporation of oxygen atoms in -Ti and Zr causes a somewhat peculiar variation of the lattice parameters with oxygen content and oxygen ordering occuring at and near by the stoichiometric compositions of O/Ti = 13 (Holmbeog 1962) and O/Zr = 21 (Hirabayashi et al. 1974). Taking into account the formation of interstitially ordered superstructures, the partial phase diagrams of the Ti–O and the Zr–O systems have been constructed (Jostons and McDougall 1970, Hirabayashi et al. 1974) which
Interstitial Ordering
765
have been incorporated in the recently compiled binary phase diagrams (Massalski 1992) as shown in Figure 8.22. The ordered hcp phases in Ti–O and Zr–O systems are present in wide composition ranges and are produced by second-order ordering of interstitial oxygen atoms. The octahedral holes in the hcp lattice form a simple hexagonal lattice with a = ao and c = co /2 where ao and co are the lattice parameters of the hcp metal lattice which remains unchanged through the ordering transformation. The ordered structures so produced can be described as a stacking of interstice layer normal to the c-axis on which oxygen atoms are distributed. A single interstitial layer √ can be conveniently described in terms of a hexagonal net with translation of 3ao , as shown in Figure 8.2(c). Oxygen atoms can occupy A, B and C positions which have their coordinates at (0,0), ( 23 , 13 ) and ( 13 , 23 ), respectively. If only A positions are filled by oxygen atoms, the plane is designated as A plane, If A and B positions are filled, the plane is (AB) and when all A, B and C positions are filled, the interstitial plane is indicated as (ABC). Vacant oxygen layers are denoted by (V ). The arrangement of oxygen occupation is not identical in the Ti–O and the Zr–O systems. While in the Zr–O system oxygen atoms in fully ordered state are distributed regularly in every layer of interstice plane with the spacing of co /2, those in the Ti–O system are positioned in every second layer with the spacing co . This seems to be reasonable due to a larger spacing for Zr (co = 0515 nm) compared to that of Ti (co = 0468 nm). As the oxygen concentration changes, the layerwise distribution of oxygen interstitials alters and a series of closely related structures are produced. For the Ti–O system, the oxygen arrangements in Ti6 O, Ti3 O and Ti2 O can be described by the following sequence of interstital and vacant layers: Ti6 O : AVBV, Ti3 O : ACVBCV, Ti2 O : ABCVABCV where V denotes a vacant interstice plane and A, B, C are oxygen interstice planes as described earlier. These structures, as schematically illustrated in Figure 8.23, are closely related to each other and the structural change can be viewed to occur continuously as a function of composition. This point can be illustrated by plotting the occupation probabilities, p, q and r as a function of composition, where p, q and r are defined for the sites A at Z = 0 and B at Z = 21 , the sites C at Z = 0 and 21 , and the sites A at Z = 21 and Z = 0, respectively (Yamaguchi et al. 1970). This shows that the oxygen distribution in the compoO sition range, 16 ≤ Ti ≤ 13 , the structure changes by continuous filling of C sites by oxygen atoms in the ACVBCV layered sequence. The oxygen ordering in the Zr–O system is somewhat complicated. The stacking O sequences of the oxygen layers are basically expressed as ABC at Zr ≤ 13 O 1 and ACBACB at Zr ≤ 2 . The former is isomorphous with -Ni3 C, being
766
Phase Transformations: Titanium and Zirconium Alloys
√3 ao C
Co
B
α'
2Co
B
(d)
A
A (a) Ti6O
(b) Ti3O
(c) Ti2O
Figure 8.23. Schematic illustration of packing sequence of various Ti-oxides.
called Zr3 O1−x or ZrOx type. The latter is denoted as the ZrO2 type, being a derivative structure of -Fe2 N in which the occupation probabilities, for the sites A, B and C are given as 1 ≥ pA pB > pC . Another type of superstructures O ≥ 13 . This structure known as ZrO with the sequence ABAB is found at Zr is isomorphic with Ni3 N. In addition to the three superstructures, a series of long period structures with various types of stacking sequences are encountered near the O = 13 . Hirabayashi et al. (1974) have listed the long stoichiometric composition Zr period stacking structures of interstially ordered Zr–O alloys at compositions near O = 13 . It is observed that with an increase in the oxygen content, the probability Zr of hexagonal stacking increases continuously. The preference for ‘hexagonal’ O ≥ 13 can be explained in terms stacking in hyperstoichiometric compositions, Zr of strain ordering. If the ‘cubic’ stacking of the oxygen layers were extended over the hyperstoichiometric region, the nearest neighbour pairs of oxygen atoms along the c-axis with the distance c2o would inevitably be created by introduction of excess oxygen atoms in any vacant site. On the other hand, the hexagonal stacking can contain the excess atoms without forming the closest pairs upto the O = 21 , since the same kind of octahedral holes remain unoccupied composition Zr O at the stoichiometric composition Zr = 13 (in the (A) (B) (A) (B) stacking, C sites remaining vacant everywhere. The nearest neighbour distances of the oxygen atom pairs along the c-axis are 3co and co for cubic and hexagonal stackings, respectively, in the Zr–O system. 2 The probability of finding oxygen pairs with interatomic distance of co increases O ratio as shown in Figure 8.24. with the Zr
Interstitial Ordering
Probability of O–O pairs
1/3
1/2 1.0
0.8
0.8 Co
0.6
0.6
Zr–O
0.4
0.4
0.2
0.2
0.0
0.1
0.2
0.3
0.4
Probability of h stacking
1/6
1.0
767
0.5
Composition O/Zr
Figure 8.24. Plot of probability of finding Oxygen atom pairs with interatomic distance of co versus O ratio. Zr
The order–disorder transformation in the Ti–O and the Zr–O systems occur in two steps, the disordered -phase first orders into the -phase in which oxygen atoms are distributed in a regular array of interstitial planes (interplanar ordering), but oxygen atoms within the plane remain randomly distributed. Since oxygen interstitial planes are placed on every second layer in the Ti–O system and on every layer in the Zr–O system, the -phase in these two cases are structurally not the same. The second ordering steps results in a periodic distribution of oxygen atoms within these planes (intraplanar ordering). The sequence of transformations, i.e. → can be shown in the order parameter versus temperature plot for a O Ti–O alloy ( Ti = 032), (Figure 8.25) which shows that the interplanar disordering, → occurs at T2 while the intraplanar disordering is complete at T1 . O The disordering process of the Zr–O alloy ( Zr = 13 ) involves changing of stacking sequences of long period superstructures (diminishing periodicity with increasing temperature) and randomization of the ordered arrangement of oxygen atoms on interstitial planes. The overall transformation, → remains to be of the secondorder type, the order parameter continuously dropping as the transition temperature (Tc ) is approached. The domain structure formed due to oxygen ordering has been found to be similar to the B2 ordering (Hirabayashi et al. 1974). This is because only two antiphase domain configurations as shown in Figure 8.26 are possible in the Ti–O system depending on whether oxygen atoms occupy either the (000) plane or the (00 21 ) plane. At the antiphase boundaries which have been drawn parallel and normal to the c-axis (in Figure 8.26) for simplicity, the interstitial oxygen
768
Phase Transformations: Titanium and Zirconium Alloys
Degree of order
1.0
0.8
SΙΙ
0.6
SΙ
0.4
0.2
0
T1
T2
∨
∨
O/Ti = 0.32
200
400
600
700
Temperature (K)
Figure 8.25. Order parameters versus temperature plot for
↓
APB
↓
O Ti
= 032.
← Ti ←O
↑ C
Figure 8.26. The domain structure formed due to oxygen ordering in Ti–O system.
layers are displaced by a vector c/2 keeping the metal lattice identical across the boundaries. The phenomenon of oxygen ordering is known to have strong influences on several properties such as hardness, electrical resistivity, thermoelectromotive force and magnetic susceptibility, an account of which is summarized by Hirabayashi et al. (1974) As the oxygen level increases towards the equiatomic TiO composition, the -phase appears in the microstructure. The -phase refers to the NaCl type structure which exists over a wide range of compositons including the stoichiometric
Interstitial Ordering
769
TiO. While in the - and the -structures titanium atoms are placed in a hcp lattice, the -structure can be regarded as an fcc arrangement of titanium atoms in which oxygen atoms occupy octahedral interstitial positions with high equilibrium concentrations of vacancies in each sublattice. The → phase transformation, studied by Jostsons and McDougall (1970), has shown several interesting features such as strict adherence to the orientation relation and habit plane, shape change and close relation between the two structures. The usual hcp/fcc rela¯ [110] ¯ has been observed between the and the tion, (0001) (111) ; [1120] -phase. Chill cast samples of Ti–O (35–44.5 at.% O) show grains of parallel plates in the -matrix. The → transformation produces martensitic type surface relief effect. The observed transformation characteristics are consistent with the fact that the two lattices are fully coherent at their interfaces. Specimen of Ti–O alloys ( 31–34 at.% O) can be heated from the -phase to the + phase field at temperatures above 1450 C. A heating and cooling cycle in which the / + transus is crossed results in the formation of -plates in the -matrix during heating and subsequent → → transformation during cooling. 8.7.2 Oxidation kinetics and mechanism Two forms of oxidation have been recognized for zirconium alloys. These are uniform and localized forms of oxidation. Under most conditions of temperature and environment, oxidation of zirconium alloys by coolant water or steam results in growth of uniform oxide films, especially in early stages. However, in some isolated regimes of temperature and environment, viz. in 300 C boiling or oxygenated water and in high temperature (>450 C) and high pressure (>5 MPa), a local form of corrosion occurs. The growth of oxide layer as a result of the corrosion reaction between zirconium alloys and water can be described in terms of two distinct stages, usually known as pre- and post-transition stages. The pre-transition stage is characterized by a decreasing rate of weight gain which corresponds to cubic or quadratic kinetics relation between the weight gain and time (Figure 8.27) expressed in terms of effective full power days (EFPD) of operation of a nuclear reactor. The deviation from usual parabolic growth kinetics in this stage is attributed to the fact that the rate controlling process of the diffusion of oxygen ions through the oxide layer is not through lattice diffusion but through grain-boundary diffusion. The oxide layer grows in the early stages maintaining epitaxy. The ratio of the oxide volume to that of the parent (Pilling–Bedworth ratio) is 1.56 for the Zr/ZrO2 system. Thus, as the oxide grows, the stress build-up due to the volume expansion accompanying oxidation induces a fibrous texture of oxide crystal, which minimizes the compressive stress in the plane of the oxide layer. The presence of the compressive
770
Phase Transformations: Titanium and Zirconium Alloys
70
Zr-2 M pick-up
50
60 50
40 40 30
Break away
30 Zr–2.5 Nb oxidation
20
20
Max. H pickup (mg dm–2)
Maximum oxide thickness (μm)
Zr-2 oxidation 60
10
10 H pick-up Zr–2.5 Nb 1
2
3
4
5
Time (EFPD) × 1000
Figure 8.27. Long term in-reactor, oxidation and hydrogen pick-up behaviour of zircaloy-2 and Zr–2.5 Nb pressure tubes, showing parabolic and then accelerated linear oxidation and hydrogen kinetics in zircaloy-2. A low and uniform rate of corrosion and hydrogen pick-up is seen in Zr–2.5 Nb alloy.
stress is also a factor responsible for the stabilization of the tetragonal phase in the ZrO2 layer. An examination of the metal/oxide interface reveals that a thin layer (∼10 nm thick) of amorphous oxide is present right on the metal surface. Fine crystallites of zirconia are distributed in the amorphous matrix. The volume fraction and size of the crystallites increase with the distance from the metal surface. The growth of the oxide crystals finally results in the formation of a columnar structure. The intercrystalline boundaries provide the diffusion path of O−2 ions from the environment to the reaction front at the metal/oxide boundary. The oxidation process and the nature of the oxide layer on a zirconium alloy sample are schematically shown in Figure 8.28 As the oxide layer grows, the compressive stress at the outer layer of oxide is not sustained and consequently the tetragonal phase becomes unstable and transforms into the monoclinic phase. Such a transformation causes the formation of a fine interconnected porosity in the oxide film which allows the oxidizing water to come in contact with the metal surface. With the development of an equilibrium pore and crack structure in the oxide layer, the oxidation rate becomes effectively linear, a characteristic feature of post-transition oxidation behaviour. Alloying elements, particularly tin, niobium, and iron present in the -solid solution strongly influence both the kinetics and the mechanism of oxide growth in zirconium alloys.
Interstitial Ordering
771
Electrons travel by surface conduction Iron oxide
O2 + 4e– → 2O2–
Monoclinic ZrO2
Intermetallic precipitate (Zr – Fe – Ni)
{
2-Diffusion O along
Columnar oxide
Amorphous oxide
→
≈2 μm
Thin ZrO2 layer
crystallite boundaries
ZrO2 crystals Zr
+202–
→ ZrO2 + 4e
Metallic conduction
Zircaloy - 2
Figure 8.28. Schematic diagram showing the mechanism of the oxidation process and the oxide film structure on zircaloy.
Non-uniform oxide formation, usually referred to as nodular corrosion, is the limiting design consideration in boiling water reactors (BWR). Several mechanisms have been proposed for nodule nucleation which can occur in various sites such as -Zr grain-boundaries and in areas at which the continuous dense oxide layer ruptures in the initial stages due to the presence of large intermetallic precipitates. The best microstructure for resisting nodular corrosion in the operating condition of a BWR consists of a distribution of fine intermetallic precipitates (dia <0.15 micron). It may be noted here that precipitate size has an opposite impact on corrosion rate in the pressurised water reactor (PWR) conditions where fine precipitates (<0.1 micron) increase the uniform corrosion rate. In addition to metallurgical factors, water chemistry has a strong influence on the corrosion process in zirconium alloys. The control of water chemistry required in different reactor systems is achieved by suitable additions of lithium hydroxide, boric acid, hydrogen/deuterium, oxygen, iron and zinc. In PWR, boric acid is added for reactivity control. The pH of the coolant is adjusted by addition of lithium hydroxide
772
Phase Transformations: Titanium and Zirconium Alloys
which renders the coolant slightly alkaline, in order to reduce the corrosion rates of structural materials (stainless steels and Inconels) in the primary heat transport circuit and thereby inhibit deposition of corrosion products on the fuel cladding. Radiolysis of water produces oxidizing species which enhances oxidation rate of zirconium alloys in radiation environment. The build-up of these can be suppressed by adding hydrogen in the coolant water. Dissolved hydrogen concentration in PWR is maintained at a level of 2.2–4.5 ppm with a view to enhancing recombination with oxygen radicals formed by radiolysis. Dissolved oxygen in pressurised heavy water reactor (PHWR) is kept between 10–50 ppb and it has been observed that the corrosion of zirconium alloys rises to an exceptionally high value at high oxygen concentrations. The BWR coolant contains higher level of oxygen, typically 200–400 ppb. Hydrogen addition in BWR condition is not very effective due to the segregation of hydrogen in the steam phase.
8.8
PHASE TRANSFORMATIONS IN TI-RICH END OF THE TI–N SYSTEM
The Ti-rich end of the binary Ti–N phase diagram (Figure 8.29) shows the -stabilizing nature of N leading to an extended phase field of the -hcp Ti–N solid solution, the -Ti2 N phase and the -Tix N1−x phase. While the -phase is present only in a narrow composition range, the -phase is found in a wide range of compositions from 30 to about 53% N. The -Ti2 N phase possesses a tetragonal (antirutile) structure and the -phase has a cubic NaCl structure. The equilibrium phase diagram shows several invariant phase reactions, the eutectoid, → + being important in the phase selection using reactive evaporation deposition process to form titanium nitride coatings. The other solid state invariant reaction involves the (Ti2 N) and the (TiN) phases which react to form the -phase (Si2 Th structure) through possibly a peritectoid reaction. The congruent transformation, → at 1373 K is an interesting example of a polymorphic transformation between two interstitially ordered structures. A metastable tetragonal phase , with ordered N atoms, forms during ageing of supersaturated dilute Ti–N alloy (N content well below the stoichiometry of Ti2 N). The sequence of transformation processes that occur during the evolution of the -phase, studied in detail by (Sundararaman et al. 1989), serves as a good example of an interstitial ordering process in which interstitial sites in the parent metal lattice are progressively filled up with interstitial solute atoms leading to the generation of several intermediate structures. Nitrogen occupies octahedral sites in hcp -titanium. The size of the octahedral interstice in the Ti lattice is not sufficient to accomodate a solute atom without
Interstitial Ordering
0
10
773
Nitrogen (at.%) 30
20
40
50
3773 3563 K
L
3273
Temperature (K)
2773 2623 K 7.0 2293 K 1.9
2273
1943
10
(δ-TiN)
4.0
(β-Ti)
1773 (α-Ti)
1373 K
1273
8
11 1073 K
10
(ε-Ti2N)
δ'
773 0 Ti
10
5
15
20
Nitrogen (wt%) (a) Nitrogen (wt%)
0
10
20
3373 3173
L
2973
Temperature (K)
2773 2573 2373
~2258 K 2128 K
2173
0.8
~2153 K 3.0
5
9
ZrN
1973
(β-Zr)
1773 1573
(α-Zr)
1373 1173 1136 K
973 0 Zr
10
20
30
40
Nitrogen (at%) (b)
Figure 8.29. (a) Ti–N and (b) Zr–N binary alloys phase diagrams.
50
60
70
774
Phase Transformations: Titanium and Zirconium Alloys
dispalcing neighbouring atoms. The stress field created by introduction of N atoms in the interstitial sites of hcp -Ti tends to get minimized by establihsing an ordered arrangement of interstitial sites. The octahedral interstitial sites in an hcp structure are shown by small circles while the positions of the host Ti atoms by large open circles (Figure 8.30(a)), a and c being the lattice parameters of Ti. Sundararaman et al. (1989) proposed the following scheme of filling these interstitial positions by N atoms in the Ti–N system: (a) The distance between the two adjacent basal plane layers of the interstitial lattice (shown by dashed lines in Figure 8.30) being small (= 2c ), only alternate interstitial layers are progressively filled by N atoms with increasing concentration of N.
Y
X C' B' Y
c
A'
X C B A
a
√3
√3a
a Ti N
Figure 8.30. (a) An interstitial sublattice is outlined inside the metal atom lattice. (b) Ordered configuration of nitrogen interstitials in the interstitial sublattice. For clarity, the host lattice is not outlined. A, A, B, B and C, C refer to three types of interstitial sites in alternate layers. Note the nitrogen occupancy in alternate layers. Open circles represent nitrogen interstitials and shaded ones the forbidden sites.
Interstitial Ordering
775
(b) A single layer of the basal plane of the interstitial site, as schematically shown in Figure 8.30(b), can be considered to have three types of sites designated as A, B and C. The nearest √ distance between two sites with same designation, A-A, B-B or C-C, is 3a. (c) The complete filling of A sites generates the Ti6 N structure, of A and B (or A and C) sites generates Ti3 N and of A, B and C sites produces Ti2 N structure. (d) Such progressive filling of interstitial sites on alternate basal planes can account for the dilute interstitial TiNx alloys (x < 21 ). The configurational internal energy, Ec , of TiNx alloys can be expressed as Ec = PNV NV + PNN NN + PVV VV
(8.26)
where Pij and ij represent the total number of pairs and interaction energies involving species, i and j, respectively. The subscripts N and V stand for nitrogen atoms and vacant interstitial sites, respectively. The interaction energy entering the Eq. 8.26 can be considered as effective bond energies between the species concerned, mediated by the host lattice. The coordination numbers for first (Z1 ) and second (Z2 ) near neighbours, their respective distances and the interaction energies are presented in Table 8.6. Using a number of simplifying assumptions such as (a) vv = 0, (b) ij is not concentration dependent and substituting physicochemical data from Balasubramanian and Kirkaldy (1985), Sundararaman et al. (1989) have estimated the configurational internal energies, E R and E O , of the random Table 8.6. Number (Zi ) and distance (di ) of first and second nearest-neighbours in a hcp-based interstitial solid solution for the proposed ordering scheme. Ordering
All A sites completely filled X = 1/6 A and B sites completely filled X = 1/3 A, B and C sites completely filled X = 1/2
First-(subscript 1) and second-(subscript 2) nearest neighbours Z1
d1
N-N
Z2
2
c
NN
3
a
6
a
√ Remarks: c = 4.68, a = 2.959 , a 3 = 51094.
N-N
6
d2 √ a 3
NN
2
c
2NN
NN
2
c
2NN
2NN
776
Phase Transformations: Titanium and Zirconium Alloys
1.5
8.0
1.0
4.0
0.5
0.0
0.0
–04.0
–0.5
–08.0
–1.5
–ER
–12.0
–2.0
–Eo
–16.0
Eo, ER (KJ/mol) →
ΔE (KJ/mol) →
12.0
–20.0
–2.5 –3.0 –3.5 –4.0 –4.5
–ΔE 0.1
0.2
0.3
0.4
0.5
X→
Figure 8.31. Variation of internal energy with composition for random (E R ) and ordered (E O ) solid solutions.
and ordered state of N in Ti–N interstitial solid solution as a function of x (Figure 8.31). It is shown that both configurations reaches a minimum energy at x = 02. The effect of ordering of N atoms in the intersitial sites on the lowering of the configurational energy, as revealed from E versus x plot is maximum at x = 1/6 corresponding to the ordered stochiometric superstructure Ti6 N. Sundararaman et al. (1989) have also shown that the activity ratio, aoN /aRN where aoN and aRN corresponds to the activities of N in ordered and random configuration in Ti–N interstial solution (dilute), shows two minima when plotted against x. These two minima corresponds with x = 16 and 13 and are separated by a maximum. Such plots with double minima suggest a phase separation tendency which results in the formation of Ti6 N and Ti3 N regions. For 0 < x < 16 , the solution prefers to be ordered as dictated by the repulsive interactions along the c-axis, and at x = 16 , the solution being fully ordered, a minimum at the Ti6 N composition is exhibited. For x values marginally exceeding 16 , one can visualize the alloy as the dilute solution of nitrogen in Ti6 N. The strong and replusive first neighbour interaction
Interstitial Ordering
777
is reported to be responsible for interstitial ordering and the stability of Ti6 N superstructure (Sundararaman et al. 1989). For x > 16 , second-nearest neighbour interaction comes into play rendering Ti6 N phase unstable. With increasing N concentration, filling of alternate B and C sites in alternate layers bring back the dominant first neighbour interactions and the nitrogen activity in the ordered solution is lowered once again. Thus we get a minimum at x = 13 in which case the order is again partially restored. Nitrogen atoms in a hexagonal titanium lattice have been found to distort the lattice as shown in Figure 8.32. The distortion of the host lattice can be resolved into two components, one along the c-direction and the other one in direction perpendicular to it. The c-component of the distortion results in a monotonic increase of the lattice parameter c (Figure 8.33) whereas the distortion component along the basal plane periodically distorts the titanium atoms towards the central nitrogen atom (Figure 8.32). In other words, a nitrogen atom serves as a contraction centre as far as the basal plane is concerned; while it serves as an expansion centre along the c-direction. This is brone out by the lattice parameters “a” and “c” variation with composition, as depicted in Figure 8.33. The c-axis exhibits a monotonic increase with nitrogen content while the lattice parameter “a” initially increases and then reaches a saturation value. Associated with such a local distortion, there is an elastic stress field around each interstitial atom which can drive a martensitic transformation of the ordered hexagonal structure to a tetragonal one. The volume change associated with the structural change in the case of the Ti–N system is small enough to be accommodated by a lattice shear. It is also clear that any such C
A
B
C
Ti C
N
Figure 8.32. Schematic illustration of the second neighbour active interaction. Note that it is mediated by the intervening metal atoms and that an interstitial atom serves as a contraction centre in the basal plane. The arrows point to the direction of displacement of the metal atoms.
778
Phase Transformations: Titanium and Zirconium Alloys N, wt. % 5
10
15
C /a
1.62 C/a 1.58
C,Å
4.80 4.76 C 4.72 4.68
a,Å
2.96
a
2.95 2.94 0.0
0.05
0.11
0.176
x
Figure 8.33. Lattice parameter variation with composition for Ti–N system.
homogeneous crystal lattice distortion must preserve the ordered configuraiton of nitrogen atoms inherited from the parent lattice. Sundararaman et al. (1980) have experimentally shown the existence of a martensitic transformation as one of the precursor steps in the precipitation process of Ti2 N in Ti–N alloys. The salient features of the precipitation reaction as experimentally observed by Sundararaman et al. (1980, 1983) are briefly presented with a view to augment the theoretical support concerning the martensitic transformation: (a) There exists a specific orientational relation between the -phase and the TiNx ¯ 011TiN 1 < x < 1 ; 1210 ¯ 011 ¯ TiN ; phase. (1010 2 x3 x (b) The TiNx precipitates always from on a definite habit plane; (c) The transformation is very rapid; it initiates during quenching; (d) The volume change associated with the transforamtion TiNx is small; = (e) The principal strains along three mutually perpendicular directions are: 1210 ¯ 488%; 0001 = 522%; 1010 ¯ = 128%
Interstitial Ordering
779
The above values nearly satisfy the requirement of an invariant plane strain condition for a martensitic transformation. The magnitude of lattice invariant shear is, therefore, quite small. The above mentioned lattice correspondence is consistent with the observed orientation relation between the and the TiNx phase (Figure 8.34). The interaction of strain fields in the -matrix, results in the formation of the observed mottled contrast (Figure 8.35(a)). The induction of the lattice transformation of the host lattice (hcp Ti) and the interstitial ordering of N atoms provides yet another example of a coupled displacive–diffusional (interstitial) transformation. (0001)α (100)Ti2N (100)Ti2N (1010)α ||(011)Ti2N
(010.378)Ti2N [1210]α [011]Ti2N
Figure 8.34. Schematic representation of orientation relation between -(hcp) ordered solid solution ¯ 011Ti N , [1210 ¯ 011 Ti N . and tetragonal Ti2 N. The orientational relation is (1010 2 2
3]
[01
0]
[10 0.2 μm
(a)
2.5 nm
(b)
Figure 8.35. (a) Quenched microstructure exhibiting mottled or tweed contrast typical of the initial decomposition stage of supersaturated Ti–N solid solution. (b) High resolution structure image showing the presence of and (isostructural with the Ti2 N phase) regions separated by coherent interfaces.
780
Phase Transformations: Titanium and Zirconium Alloys
REFERENCES Balasubramanian, K. and Kirkaldy, J.S. (1985) Calphad, 9, 103. Banerjee, D. and Arunachalam, V.S. (1981) Acta Metall., 29, 1685. Banerjee, D. and Williams, J.C. (1983) Scr. Metall., 17, 1125. Dagerhamn, T. (1961) Acta Chem. Scand., 15, 214. Darby, M.I., Read, M.N. and Taylor, K.N.R. (1978) Phys. Status Solidi (a), 50, 203. Dey, G.K. and Banerjee, S. (1984) J. Nucl. Mater., 124, 219. Dey, G.K., Banerjee, S. and Mukhopadhyay, P. (1982) J. Phys. C4, 43. Dutton, R. (1976) Metall. Soc. CIM, 16, 16. Flewitt, P.E., Ash, P.J. and Crocker, A.G. (1976) Acta Metall., 24, 669. Froes, F.H. and Eylon, D. (1990) Hydrogen Effects in Material Behaviour (eds N.R. Moody and A.W. Thompson) TMS, Warrendale, PA, p. 261. Hirabayashi, M., Yamaguchi, S., Asano, H. and Hiraga, K. (1974) Proc. Int. Conf. on Order–Disorder Transformations in Alloys (ed. H. Warlimont) p. 266. Holmberg, B. (1962) Acta Chem. Scand., 16, 1245. Hong S. and Fu C.L. (2002) Phys. Rev.(B), 66, 99. Jack, K.H. (1948) Proc. Roy. Soc., A195, 34. Jostsons, A. and McDougall (1970) Phys. Status Solidi, 29, 873. Khachaturyan, A.G. (1978) Prog. Mater. Sci. 22, 1. Khachaturyan, A.G. (1983) Theory of Structural Transformations in Solids, Wiley, New York. Lieberman, D.S., Read, T.A. and Wechsler, M.S. (1957) J. Appl. Phys., 28, 532. Leitch, B.W. and Puls, M.P. (1992) Metall. Trans., 23A, 797. Massalski, T.B. (1992) Binary Alloys Phase Diagram, 2nd edition, ASM International Materials Park, OH, pp. 2066–2067, 1652–1655, 1792–1793, 2078–2079, 1659–1660, 1799–1800. Narang, P.P., Paul, G.L. and Taylor, K.N.R. (1977) J. Less-Common Metals, 56, 125. Norwood, D.O. and Gilbert, R.W. (1978) J. Nucl. Mater., 78, 112. Norwood, D.O. and Kosasih, U. (1983) Int. Metals Rev., 28(2), 92. Pan, Z.L., Ritchie, I.G. and Puls, M.P. (1996) J. Nucl. Mater., 228(2), 227. Park, J.M. and Lee, J.Y. (1990) J. Less-Common Metals, 160, 259. Puls, M.P. (1990) Metall. Trans. A, 21A, 2905. Rhodes, C.G. and Williams, J.C. (1975) Metall. Trans. 6A, 1071, 2103. Shaltiel, D., Jacon, I. and Davidov, D. (1977) J. Less-Common Metals, 53, 117. Sidhu, S.S., Heaton, Le Roy, Campos, F.P. and Zauberis, D.D. (1963) Am. Chem. Soc., 39 Singh, R.N., Kumar, N., Kishore, R., Roychaudhary, S. and Sinha, T.K. (2002a) J. Nucl. Mater., 304, 189. Singh, R.N., Kishore, R., Sinha, T.K. and Kashyap, B.P. (2002b) J. Nucl. Mater., 301, 153. Singh, R.N., Kishore, R., Sinha, T.K., Banerjee, S. and Kashyap, B.P. (2003) Mater. Sci. Eng. A, 339, 17. Singh, R.N. Kishore, R. Singh, S.S., Sinha, T.K. and Kashyap, B.P. (2004) J. Nucl. Mater., 325, 26.
Interstitial Ordering
781
Singh, R.N., Mukherjee, S., Gupta, A. and Banerjee, S. (2005) J. Alloys Compd., 389, 102. Singh, R.N., Lala Mikin, R., Dey, G.K., Sah, D.N., Batra, I.S. and Stahle, P. (2006) J. Nucl. Mater., 359, 208. Sinha, V.K., Shetty, M.N. and Singh, K.P. (1970) Trans. Faraday Soc., 66, 1981. Sinha, V.K., Pourarian, F. and Wallace, W.E. (1982) J. Less-Common Metals, 87, 283. Sinha, V.K., Yu, G.Y. and Wallace, W.E. (1985) J. Less-Common Metals, 106, 67. Srivastava, D. (1996) Ph.D Thesis, Indian Institute of Science, Bangalore, India. Srivastava, D., Neogy, S., Dey, G.K., Banerjee, S. and Ranganathan, S. (2005) Mater. Sci. Eng. A, 397, 138. Sundararaman, D., Terrance, A.L.E., Seetharaman, V. and Raghunathan, V.S. (1980) Titanium 80, Science and Technology, Proc. Fourth Int. Conf. on Titanium, Vol. 2, The Met. Society of AIME, New York, p. 1521. Sundararaman, D., Terrance, A.L.E., Van Tendeloo, G. and Van Landuyt, J. (1983), J. Phys. Status Solidi (A), 76, K-109. Sundararaman, D., Raju, S. and Raghunathan, V.S. (1989) J. Phys. Chem. Solids, 50, 1101. Une, K., Nogita, K., Ishimoto, S. and Ogata, K. (2004) J. Nucl. Sci. Technol. 41, 731. Taizhong, H., Zhu, W., Jinzhou, C., Xuebin, Y., Baojia X., Naixin, X. (2004) Mat. Sci. Eng. (A), 385, 17. Unnikrishnan, M., Menon, E.S.K. and Banerjee, S. (1978) J. Mater. Sci., 13, 1401. Wayman, C.M. (1964) Crystallography of Martensitic Transformation, Mcmillan, New York. Weatherly, G.C. (1981) Acta Metall., 29, 501. Yamaguchi, S., Hiraga, K. and Hirabayashi, M. (1970) J. Phys. Soc. Japan 28, 1014. Zener, C. (1946) Trans. AIME, 167, 550. Zuzek, E. and Abriata, A. (2000) Phase Diagrams of Binary Hydrogen Alloys, Monograph Series on Alloy Phase Diagrams, Vol. 13, ASM International, Materials Park, OH.
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Chapter 9
Epilogue
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Chapter 9
Epilogue
The preceding eight chapters have illustrated the wide variety of phase transformations encountered in Ti- and Zr-based systems and have demonstrated that the whole subject of phase transformations in condensed matter can be studied using examples taken from these systems. It has also been shown that the various concepts and formalisms introduced in the study of phase transformations in alloys can be applied to intermetallics and ceramics as well. In this concluding chapter, let us examine some general trends in the phase transformations in Ti- and Zr-based alloys. As has been discussed in earlier chapters, solid–solid phase transformations in these systems are essentially governed by the competition between the different allotropes, , and that are structurally related through unique lattice correspondences, the Burgers relation for / and that involving the collapse of a set of adjacent {222} planes for /. In fact, the structural relationships between these phases are so strong that the same relationships remain valid for both displacive and diffusional transformations. An inspection of phase diagrams of binary and ternary alloys reveals that in a fairly large number of Ti, Zr–transition metal systems, the liquidus temperature drops down considerably with alloy additions. This enhanced stability of the liquid phase makes it possible to amorphize these alloys under non-equilibrium processing conditions. The chemical short-range order present in some of these liquid phases which are stabilized to sufficiently low melting temperatures tends to form clusters with icosahedral symmetry. Such tendencies are reflected in the formation of quasicrystalline phases on crystallization of some of the amorphous alloys in these systems. Another strong tendency which is responsible for several phase reactions in these systems is the clustering tendency in the -phase. Spinodal decomposition, phase separation, monotectoid decomposition, metastable phase reactions during tempering of some martensites are all consequences of the clustering tendency in the -phase in a number of alloy systems. Tendencies for chemical ordering in the -, - and -phases, present in several binary and multicomponent systems, are responsible for the formation of ordered derivations of these phases. Important examples of such chemical ordering are hcp → D019 , bcc → B2 and → B82 . Symmetry relationships between the disordered parent structures and their ordered derivatives play a dominant role in dictating the path of the transformation sequence in several alloy systems. 785
786
Phase Transformations: Titanium and Zirconium Alloys
The hcp structure is a special case √ of the orthorhombic structure in which the ratio of lattice parameters b/a = 3. With the introduction of some alloying elements beyond certain limits, the hcp structure is distorted to an orthorhombic √ symmetry (b/a = 3). This tendency of orthorhombic distortion is also present in some ordered intermetallics having the D019 structure which undergoes a transition to the O-phase (orthorhombic) following the same lattice correspondence as that prevailing between hcp and orthorhombic structures. We will briefly discuss these tendencies which determine the phase transformation sequences in Ti- and Zr-based systems. Pressure–temperature phase diagrams of single component systems comprising Group IVA metals, Ti, Zr and Hf show the three phases, hcp, bcc and (hexagonal), in the solid state. The most interesting aspects of these phases are their unique lattice correspondences and the relationship of their stability regimes with their electronic density of state. Introduction of alloying elements and/or change in external variables such as pressure and temperature can induce a change in the electronic structure which in turn is responsible for bringing about the phase transition. Out of the three allotropes, the -phase with the bcc structure is associated with the highest symmetry. This is also the first phase to appear from the liquid phase on cooling. Let us, therefore, consider the -phase as the reference state and examine how this bcc structure transforms into the hcp - and the hexagonal -structures. As has been elaborated in Chapters 1 and 4, the -phase experiences a tendency to undergo a transition into the -phase as the / transformation temperature is approached. The lattice correspondence between and is such that a distortion of a {110} plane can introduce a sixfold symmetry, and a rather small distortion along the <110> direction can establish the right c/a ratio of the product hcp structure. In addition to this lattice strain, an atomic shuffle is necessary for bringing the atoms in every alternate basal planes in the right position. The -phase in some alloys cannot maintain the hexagonal symmetry and distorts into an orthorhombic structure. In addition to the tendency for the bcc to orthohexagonal transition, the -phase experiences another type of lattice instability which can be described as a longitudinal displacement wave of wave vector k = 2/3 <111>. The introduction of this periodic displacement in the -lattice generates the -structure as has been discussed in Chapters 1 and 6. Variations in external conditions such as pressure and temperature and chemical composition shift the relative stabilities of these phases and the corresponding transformations can be induced. These two transformation tendencies lead to either an orthohexagonal structure or to the -structure. These tendencies are present not only in dilute alloys but also in several intermetallic compounds as has been elaborated in Chapters 4–6.
Epilogue
787
The -phase in several Ti and Zr alloys also exhibits a tendency for phase separation as reflected in the presence of a miscibility gap in the -phase field and a monotectoid reaction of the type 1 → + 2 . This phase separation or clustering tendency is instrumental in giving rise to a number of metastable steps in the phase transformation sequence of both - and -alloys. Chemical ordering of -, - and -phases can produce a host of ordered intermetallic structures in binary and ternary alloys of Ti and Zr. A comparison between several equilibrium and metastable intermetallic structures in these systems vis-àvis the -, - and -structures reveals that the former can be construed as ordered derivatives of the latter. The mechanism of formation of these intermetallic phases from the parent -phase, therefore, involves displacive atom movements (either from bcc to orthohexagonal or from bcc to ) in conjunction with replacive (chemical) ordering. The path associated with such transformations can be determined considering the symmetry changes associated with the displacive and replacive ordering processes. In spite of the diversity of solid-state phase transformation processes in Ti- and Zr-based alloys, an understanding of the mechanism involved can often be gained by taking into the account the fact that the structures are invariably closely related and that the transformations are often guided by the following general tendencies: (1) bcc → orthohexagonal lattice distortions, (2) bcc → transformation by the shuffle dominated lattice collapse process, (3) phase separation tendency of the -phase in several systems containing -stabilizing elements and (4) chemical ordering tendencies of the -, - and -phases leading to the formation of their respective superlattice structures. Coming to the issue of the formation of amorphous and quasicrystalline structures, the stabilization of the liquid phase by alloying Ti and Zr with various transition metals is reflected in the plunging of the liquidus temperature in the respective phase diagrams. In case the formation of the equilibrium intermetallic phases is suppressed by a non-equilibrium processing technique at a temperature below the glass transition temperature, the amorphous phase becomes the next best alternative for the system to adopt. The presence of the chemical short-range order in the liquid phase in compositions close to those of relevant intermetallic compounds is a prerequisite for bringing about vitrification in these systems. The nature of clusters is undoubtedly very important in promoting the stability of the amorphous phase as is seen from the formation of bulk glasses in a host of ternary and quaternary Zr–Al–transition metal alloys. These clusters, some of which have icosahedral or decagonal symmetries, have a tendency to retard the
788
Phase Transformations: Titanium and Zirconium Alloys
kinetics of crystallization, thereby making the corresponding alloys amenable for glass formation even at a relatively slow cooling rate. In order to identify different general tendencies of phase transformations, some recent experimental techniques have been found to be extremely convenient. To illustrate this point let us cite the examples of studies carried out on miniature compositionally graded samples (Banerjee et al. 2003). Phase transformation studies on a given binary or multicomponent alloy system invariably necessitate the preparation of a series of alloys of different compositions and thermal analysis, phase analysis and microstructural characterization of different microconstituents after suitable heat treatments. This has been the general practice till recently; now it is possible to produce compositionally graded samples with desired gradients by laser melting of controlled powder feeds emanating from multiple nozzles. Figure 9.1 illustrates the operating principle of such a laserprocessing unit. In this process, either prealloyed powder or elemental powders can be taken as the powder feed. When powders of different compositions are fed from separate feeders with a precise control of feed rates at a small size molten puddle, the possibility of making samples with graded composition opens up. The process control computer continuously controls the position (x y z) of the hot spot (molten puddle) with respect to the component/sample being processed and simultaneously controls the feed rates of powders from different nozzles. Such a manufacturing process (which is essentially an extension of rapid prototyping process) has the potential not only of producing components of intricate shapes, but also of preparing samples of desired compositional gradients. Glove box CO2 laser system Powder feed system
Stage control
Figure 9.1. Experimental set-up for production of compositionally graded samples.
Epilogue
789
Suitable slices of one such graded sample can be prepared and given appropriate heat treatments for inducing different phase transformations. Microscopic examination of graded samples by a scanning electron microscope (SEM) attached with microanalysis and orientation imaging facilities can yield morphological, crystallographic and compositional information on the microconstituents at a level of about 0.1 m resolution. Such studies can be complemented by TEM studies by picking up samples from related areas by focussed ion beam sectionizing (FIBS) technique. Nanoindentation experiments on small sample areas can further supplement the investigation by providing mechanical properties data from each of the microconstituents. Recent work reported on Ti–V and Ti–Mo alloys have shown how these techniques can be gainfully employed for phase transformation studies on a system from a single compositionally graded sample (Banerjee et al. 2002, 2003). The composition profile of a graded Ti–V sample of 25 mm length produced by laser melting of prealloyed powders of different compositions is shown in Figure 9.2. Several longitudinal slices of such a sample after subjecting to different heat treatments such as (a) as cast, (b) -quenched and (c) -solutionized followed by air cooled show a variety of microstructures as described in the following. The -quenched sample shows the presence of lath martensite in regions corresponding to a V level of 0–3 at.%, plate martensite in the region of 3–11 at.% V
Ti–25 at.% V
Ti
∼ 25 mm 25
#14 #13
Composition (at.% V)
20 #12 15 #10
#11
#9 10
#8 #7 #6
5 #4
#3 0
#5
#1 #2 0
5
10
15
20
25
Distance (mm)
Figure 9.2. Variation of composition of a Ti–V compositionally graded sample.
30
790
Phase Transformations: Titanium and Zirconium Alloys
and near complete -stabilization in regions containing more than 12% V. TEM investigations have revealed that a distribution of fine -plates in the composition range of 12–15% V. Microstructures of the as-cast sample match closely with those corresponding to -solutionizing and air cooling. This points out that the laser-processing technique on a sample size, like the present one, essentially produces a normalized structure. Microstructures of the compositionally graded samples (both as-cast and -solutionized followed by air cooled) show a gradual change in the morphology of Widmanstatten -plates and monoatomic increase in the volume fraction of the retained -phase with increasing V content (Figure 9.3). While the -plates are long and slender in alloys containing up to about 5 at.%, they are short and stout in alloys more enriched with V. Though the as-cast condition is not expected to yield equilibrium microstructure, the volume fractions of the - and the -phases and their respective compositions are seen to approach equilibrium values particularly in samples lean in V. With an increase in the V level, as the / + transus temperature comes down, the departure from the equilibrium values of volume fraction and composition is larger, suggesting a more incomplete partitioning of alloying elements in the - and the -phases. The reduction in the length of the plates is also consistent with a slower growth kinetics of the plates and an increased gap in the composition of and . In regions corresponding to V level of about 5–8 at.%, the presence of a bimodal size distribution of -plates can be attributed to a two-stage decomposition process – primary plate forming when the temperature of deposited alloy drops below the / + transus and the secondary plates subsequently appearing in the retained -phase during later heating cycles. Crystallographic information from these samples at a mesoscopic level (typically at a magnification of 2000×) obtained from orientation imaging microscopy (01 m) has yielded an orientation distribution map for the matrix and the different variants of the -phase (Figure 9.4). As per the Burgers relationship, six equivalent variants of planar matching exist, where the basal (0001) plane is parallel to one of the six variants of the 110 plane of the -phase. Again for a given variant of planar matching (say (0001) (110) ), there are two distinct options of directional matching, namely [1120] [111] and [1210] [111] . OIM has identified all the 12 orientation variants and provided quantitatively the area fractions occupied by each of them. The orientation distribution information presented in pole figures (Figure 9.4) corresponding to {110} , {111} , (0001) and 1120 poles have oneto-one correspondence with the 110 poles and the 111 poles matched with the 1120 poles. Observations like this not only validate the Burgers relationship, but also show special disposition of orientation variants in specific locations. It has been shown recently (Bhattacharyya et al. 2003) that -phase forming at grain boundaries (described in Chapter 7 as “Grain Boundary Allotriomorphs”)
Epilogue
791
10 μm
Ti–1.8 at.% V
Ti–5 at.% V
Ti–3 at.% V
Ti–6.8 at.% V
10 μm
Ti–8 at.% V
Ti–10 at.% V
Figure 9.3. Laser-processed (as-cast) sample with composition gradient shows variation in the aspect ratio of -plates and in the volume fraction of the -matrix (lighter shade) with increasing V contents.
often correspond to a stack of two alternate orientation variants which have the same planar matching (say (0001) (110) ) with directional matching alternating between [1120] [111] and [1210] [111] . According to the nomenclature introduced in describing martensite variants (Chapter 4) these are designated as ( − +) and ( + +), respectively.
792
Phase Transformations: Titanium and Zirconium Alloys
Figure 9.4. Distribution of different orientation variants of -plates in -matrix as revealed from orientation imaging microscopy.
Microstructural examinations of a second slice of the same compositionally graded Ti–V bar – which has been given the -quenching treatment – reveal the presence of lath martensite in the region corresponding to 0–3% V, plate martensite in the region having 3–11% V and nearly complete -phase stabilization in the region of V > 12%. Microstructural observations on a finer scale from selected regions have become feasible due to the availability of FIBS technique. TEM examination of samples picked up from regions of interest reveals the following:
Epilogue
793
g = 1012
2 μm
500 nm
Figure 9.5. Parallel stacking of -laths with small angle boundaries laths.
(1) Regions of the laser-processed sample in the composition of 1–2% V show Widmanstatten -laths, contiguously stacked with small angle boundaries between the adjacent laths. In addition to the interfacial dislocations, an array of discrete fine -plates are seen (Figure 9.5) along these interlath boundaries. These fine plates again exhibit an internal substructure akin to martensitic substructure (Figure 9.6). Based on this observation, Banerjee et al. (2003) have proposed that the interlath plates indeed form by a martensitic process. This is possible because under the non-equilibrium cooling condition prevailing during laser processing, partitioning of V in the - and the -phases is not complete. Under such a situation, the metastable + / transus can intersect the Ms line making the martensitic transformation possible within the -layer at the interlath boundary. (2) In regions having average V content above 5%, primary and secondary -plates are seen to be distributed in the -matrix (Figure 9.7) which exhibits a mottled structure. Diffraction patterns corresponding to the -matrix clearly indicate the presence of both - and -reflections. For the region corresponding to an average composition of 5% V, the -matrix (containing 15% V) shows a distribution of fine - and -particles. With further V enrichment of the -matrix (1˜ 7.5% V), only -particles are seen to be present. This is consistent with the general trend of enhancement of the stability of in comparison to that of as the -phase is enriched with V. (3) In regions containing average V content above 20%, the -phase is retained. However, the -phase with 20% V has been found to be amenable to stressinduced → transformation as revealed from nanoindentation experiments. Experimental results on the microstructure of laser-processed compositionally graded sample of about 1 in. length are presented here to illustrate that some of the major trends of phase transformations of titanium (and also zirconium) alloys can all be identified from such a small sample. Some of the recent experimental
794
Phase Transformations: Titanium and Zirconium Alloys
g = 0002
g = 1010
100 nm
g = 1011
g = 1011
200 nm
Figure 9.6. Vanadium-enriched interlath regions exhibiting martensite-like substructure.
techniques have made it possible to study a whole range of phenomena in a single sample. Metallurgists for many years have been using Jominy test bars for achieving graded quenching rates in a single sample. The present trend of combinatorial materials science can explore, in a similar manner, a wide range of phase transformations in a single compositionally graded sample. Such experimental techniques will no doubt complement the recent theoretical development of first principle predictions of phase stability of alloys. In this concluding chapter, it is worthwhile to draw a comparison between the observed crystallographic features of diffusional, martensitic and mixed-mode transformations encountered in titanium and zirconium alloy. As mentioned earlier, an approximate − Burgers orientation relationship is a dominant feature in all types of transformations discussed, namely, martensitic transformation, Widmanstatten -precipitation, -hydride precipitation and active eutectoid decomposition. This statement, however, is being made by not distinguishing the
Epilogue
Figure 9.7. -plates in V-stabilized -matrix containing dispersion of -particles.
795
796
Phase Transformations: Titanium and Zirconium Alloys
Burgers and the Potter relations, which differ by a rotation of 1 5 . The range of transformations discussed involves shear, diffusional and mixed-mode mechanisms, and it is attractive to examine the applicability of the invariant line strain (ILS) and the invariant plane strain (IPS) criteria in determining the habit planes of advancing transformation fronts associated with these transformations. The macroscopic habit plane for the martensitic plates and laths can be precisely predicted on the basis of the IPS criterion. The interfaces of a martensite lath consist ¯ 1123 ¯ of arrays of c + a dislocations, suggesting the operation of the 1011 slip mode as the lattice invariant shear (LIS). Since the LIS is accomplished by the passage of these dislocations, it is necessary for the line vectors of these dislocations to be along the invariant (IL) vector. The lath boundary structure, however, is more complex, as in the lath martensites, the parent -phase is fully consumed and adjacent plates come in contact, resulting in a superimposition of their interface structures. In twinned martensitic plates for both ( − +) and ( + +) solutions, for the two twin-related variants to remain in coherence with the parent , it is geometrically necessary for them to meet along the IL vector. The same argument is also valid for a group of three martensite crystals that form the indentation morphology. These self-accommodating groups of crystals meet along a line that is the IL. The line of intersection of habit plane segments corresponding to two adjacent twin-related martensite variants within a twinned plate following the ( − +) solution of the Bowles–Mackenzie analysis matches the ILS direction. Martensite plates that obey the ( + +) solution contain a stack of thin twins, which again intersect the habit planes along the ILS direction. The geometries of the habit planes in these cases are schematically illustrated in Figure 9.8(a) and (b). The habit planes of Widmanstatten laths/plates have also been found to be irrational. However, habit plane predictions based on the IPS criterion do not match those experimentally observed. Two types of Widmanstatten products, namely, single crystal laths and internally twinned plates, have been encountered. The lath interface exhibits an array of dislocations with c + a Burgers vectors, where dislocation line vectors are parallel to the ILS direction. In fact, the growth direction of the laths is marked by the line vector of these interfacial dislocations, which lie along all four surfaces parallel to the long axis of the laths. It is also seen that the ¯ habit plane poles are located in the vicinity of the pole, where [112] and 1010 come in coincidence, consequent to the operation of the Burgers relation. This observation is consistent with that of Furuhara and Aaronson (1991) who reported that the habit of (hcp) plates in the matrix of (bcc) Ti–Cr alloys lies close to ¯ directions. 130 and 131 poles, which are located near the [112] 1010 They also invoked a periodic occurrence of structural ledges, which accounts for
Epilogue IL
IL
Habit (301)β-(301)β
)β
33
(4
)β 33 (4
797
(1011)α
(a)
(d)
α Precipitate in β
1
(21
IL
Average habit (301)β-(311)β
(1011)α//(110)β
(1011)α
β )β
2) β
(b)
(11
Martensite Class A (α−ω+) thick 1011 twins IL
(e)
α Precipitate in β
Martensite Class (α+ω+) thin 1011 twins
IL
IL
Transformation front
α
α
α
β′
γ Hydrie in β
(101
1)α/
α
/(11
)γ
(f)
(111
(c)
0)β
Habit (301)β-(311)β
Active Eutectoid Decomposition β→α+β′
Figure 9.8. A schematic illustration of habit planes/transformation fronts in various types of diffusional, martensitic and mixed-mode processes. The role of the IL vector in all processes is revealed.
¯ the rotation of the average habit interface from the terrace plane (112) 1010 on which patches of good atomic fit across the / interface exist. The criterion of the selection of the habit plane in a diffusional transformation has been discussed in Banerjee et al. (1997) and references therein. There is a general agreement that one of the vectors that defines the habit plane is the ILS direction. The second vector fulfils one of the following conditions: • • • •
a vector that remains unrotated, a direction of low strain, a direction of easy slip in the parent or product phase, a direction favoured due to the elastic anisotropy of the matrix–precipitate assembly and • a direction along which structural ledges are aligned.
798
Phase Transformations: Titanium and Zirconium Alloys
In a recent article, Zhang and Purdy (1993a,b) have shown that using an extension of the concept of the O-lattice to a system containing invariant lines, a complete Burgers vector balance can be accomplished at the / interface of the Zr–2.5Nb alloy by a single set of dislocations with the Burgers vector [010] and dislocation spacing of 10 nm. This prediction remains valid for Zr–20Nb also, as the lattice parameters of the Zr–20Nb alloy are not significantly different than those for the Zr–2.5Nb alloy. The observed interfacial structure (a single set of parallel dislocations along the ILS direction) (see Figure 7.91) and the experimentally determined habit plane indices are consistent with the predictions of Zhang and Purdy (1993a,b) who have shown that the habit plane is given by an O-lattice plane associated with a plane of the reference lattice, which contains the Burgers vector of the dislocations. The habit plane envisaged in this case differs from that predicted from the atomistic model (see Section 7.5.4) proposed in a similar system of Ti–Cr alloys, though the indices of the macroscopic habit plane and the orientation relation are quite similar in the two cases. Matching of the atomic planes across the / boundary in Ti–Cr alloys has been achieved by the introduction of biatomic structural ledges with Burgers vector 16 a and of misfit compensating c-type ledges. It appears that in the case of the / interface in Zr–Nb alloys, the misfit along the c and a components is compensated by a single set of c + a dislocations lying along the ILS direction (Figure 9.8(c) and 7.91). Examples of internally twinned Widmanstatten -plates have been reported in ¯ twinning and a habit plane pole Banerjee et al. (1997). These plates exhibit 1011 lying between 130 and 131 . This habit does not match that predicted from the IPS consideration. A detailed analysis of the macroscopic facets that constitute the average habit plane could not be performed with sufficient accuracy. However, ¯ planes. It is, therefore, the major facet appeared to be close to the [112] [1010] quite attractive to consider that the macroscopic habit plane is made up of steps of ¯ planes on which good atomic fit can be established. The direction [112] [1010] along which two adjacent twin variants meet on the habit plane has been found to be the ILS direction (Figure 9.8(d)). The crystallography of precipitation of the -hydride plates in either the - or matrix obeys the IPS criterion. As mentioned earlier, this transformation involves a shear transformation of the or zirconium lattice, with the accompanying process of hydrogen partitioning. For such a transformation, the phenomenological theory of martensite crystallography provides an accurate prediction of the habit plane. The habit plane retains the direction of the IL, which is defined by the line of intersection of the habit plane and the twin plane (Figure 9.8(e)). Active eutectoid decomposition of the parent -phase occurs through a cooperative growth of the - and metastable -phases. Based on crystallographic
Epilogue
799
observations, it can be inferred that the Burgers relation exists between and , while the / relationship is based on the parallelism between their respective ¯ ) cube axes. The line along which the / interface (which is parallel to 1011 intersects the transformation front is the IL vector (Figure 9.8(f)). Crystallographic descriptions of different types of transformation products, as schematically illustrated in Figure 9.8, bring out the importance of the IL vector in defining the habit plane (or the transformation front). The tendency of maintaining at least one undistorted vector on the advancing transformation is not unexpected. It is increasingly being realized that either partial or full coherency is maintained in a great majority of solid-state phase transformations, both martensitic and diffusional. This point has been amply demonstrated in the case of transformations in titanium- and zirconium-based alloys. The operation of Burgers or near Burgers orientation relations has an overwhelming influence on all types of transformations in titanium- and zirconium-based alloy systems. The diffusional transformations are, however, distinguishable from displacive transformations from the nature of the habit plane. While the habit plane fulfilling the IPS criterion is a necessity in displacive or diffusional-displace (e.g. formation of -hydride) processes, the habit planes in pure diffusional transformations are determined by the strong tendency of maintaining regions of good fit between the parent and the product phases. Since in the latter case interfaces often deviate from being planar, a rotation of the boundary is achieved by rotating the interface around the invariant line vector. A typical -lath formed through a diffusional process possesses an interface characterized by an array of c + a dislocations, which are all aligned along the IL vector. Even for the active eutectoid decomposition, which involves partitioning of substitutional alloying elements, the two product phases appear to grow in such a manner that the crystallographic registry of the parent is maintained simultaneously with two product crystals. The velocity of interface motion is, therefore, limited by the requirements of chemical diffusion for attaining the appropriate extent of alloy partitioning and for the ordering processes within one of the product phases. In the preface of the book, we have highlighted the diversity of phase transformations in titanium- and zirconium-based alloys and subsequently have justified that the whole subject of phase transformations can be covered using examples taken from these alloys. Finally, we are only emphasizing that this wide range of transformations are driven by only a few trends of structural and chemical instabilities. It is also noted that different transformation mechanisms leave their imprints on the crystallographic and morphological features of the transformation products and it is through these experimental observables one deciphers the relevant operating mechanisms.
800
Phase Transformations: Titanium and Zirconium Alloys
REFERENCES Banerjee, S., Dey, G.K., Srivastava, D. and Ranganathan, S. (1997) Metall. Trans., 28A, 2201. Banerjee, R., Collins, P.C., and Fraser, H.L. (2002) Metall. Mater. Trans. A, 33, 2129. Banerjee, R., Collins, P.C., Bhattacharayya, D., Banerjee, S. and Fraser, H.L. (2003) Acta Mater., 51, 3277. Bhattacharyya, D., Viswanathan, G.B., Denkuberger, R., Furrer, D. and Fraser, H.L. (2003) Acta Mater., 51, 4679. Furuhara, T. and Aaronson, H.I. (1991) Acta Metall. Mater., 39, 2857. Zhang, W.Z. and Purdy, G.R. (1993a) Acta Metall. Mater., 41, 543. Zhang, W.Z. and Purdy, G.R. (1993b), Philos. Mag., 68, 291.
Index -TiCr2 -TiCr2
alloys 21, 22, 284, 687 alloys 21, 284, 504–506, 539, 567, 689–690, 701 -hydride 120, 721, 722, 728, 729, 730, 733, 736, 737, 738, 794, 798 -hydride (fct) phase 721 -hydride 722, 723 -hydride 722, 723, 728, 739 -phase immiscibility 606–609 -phase 5, 10, 21, 26, 49, 189, 217, 223, 447, 529, 656, 683 -phase 5, 9, 13, 21, 22, 25, 43, 48, 50, 52–53, 134, 145, 283, 310, 384, 421, 447, 475–484, 486, 488, 512, 523, 559, 599, 609–615, 632, 655, 688, 739, 753, 787 -phase 6, 7, 9, 10, 12, 13, 26, 35, 36, 43, 44, 50, 52, 251, 284, 490, 492–494, 504, 512, 516, 529, 536, 544 -precipitation 492, 536, 537, 539–540 -stabilizers 21, 22, 154 -stabilizers 21, 25, 53, 174, 325, 383, 417 -stabilizing elements 21–23, 148, 175, 251, 281, 303, 310, 417, 475, 516, 587, 609 -transformed 688, 689, 693, 694, 698, 699, 700 -transition 9, 12, 106, 419, 474, 481–484, 491–495, 520 + alloys 21, 22, 284, 327, 609, 632, 687–689, 696, 729 – lattice correspondence 485 – lattice correspondence 488 –-isomorphous systems 27 / interface 559, 646, 647, 650, 657, 737, 738, 799 2 -phase 382, 383, 418, 420, 421, 423, 434, 443, 445, 447, 448, 451, 453, 458, 662–670 -eutectoid systems 27, 622 -isomorphous systems 27, 29, 30, 606, 609, 620 -TiCr2 33, 62
33, 62 33, 62, 622
A15 (cP8, Cr3 Si type) 54–62 ab initio phase diagram calculations 16 Accommodation strain energy 719, 725, 726 Accommodation stress 264 “Active” eutectoid decomposition 560 Affine transformation 103, 267 Aged 45, 50, 475, 476, 492, 538, 547 Ageing 44, 50, 53, 421, 435, 448, 453, 475, 529, 533, 539, 546, 603, 658, 706, 728, 772 Allotrope 5, 286 Amorphous alloys 162, 163, 165, 176–179, 183, 204, 205, 207, 209, 210, 250 Amorphous phase 160, 161, 164, 165, 175, 176, 178, 182, 183, 185–187, 191, 192, 203, 207, 208, 210, 217, 220–229, 232, 233 Anomalous diffusion behaviour 13, 568 Antiferroelectric 109 Antiferromagnetic 109 Applied stress 263–265, 324, 331, 340, 350, 354, 355, 370, 725, 727, 747 Athermal 43, 44, 49–52, 101, 105, 107, 109, 110, 264, 265, 277, 278, 280–283, 338, 339, 362, 474–477, 486, 490–492, 494, 495, 500, 501, 508–510, 512, 517, 529, 539, 540, 546, 560 Athermal → transition 50, 475, 486, 491, 501, 512 Athermal martensitic 43, 278 Athermal nucleation 110 Atomic layer stacking 55 Atomic shuffles 111, 316, 320, 504, 505 Atomic site correspondence 632, 650 Austenite 47, 121, 153, 260–262, 267, 269, 270, 279, 324, 325, 354, 355, 358, 359, 371, 673
801
802 Autocatalytic 305, 322 Avrami exponent 193, 197, 202, 204, 211 B2 (cP2, CsCl type) 54, 60, 61 B82 141, 156, 210, 441, 518, 519, 521, 525, 527, 528, 529, 530, 533, 535, 785 Bain distortion 269, 272, 292–296, 298 Bain strain 45, 269, 273, 274, 276, 293, 294, 314, 316, 321, 431, 432, 504, 643 Bainite-like transformation 742 Bainitic transformation 121, 122, 737 Ball milling 226, 228, 229 Basket weave morphology 421, 637, 656 BaTiO3 111, 113, 114, 260 BCC special points 404 Bf (oC8, CrB type) 54, 60 Black plates 638, 648 Blister formation threshold 744, 751 Boiling water reactors 771 Bridge 351 Bulk Metallic Glasses 157, 158, 171, 205 Burgers 45, 46, 276, 289, 293, 294, 302–304, 310, 312, 314–316, 332, 334, 336, 339, 420, 421, 424, 538, 539, 547, 558, 606, 613, 614, 618, 633, 635, 642–644, 646, 648–651, 655, 679, 730, 736 Burgers correspondence 294, 421, 679, 730, 736 Burst 278, 362, 547, 550, 743 Burst-like behaviour 362 C14 (hP12, MgZn2 type) 54, 59, 61 C15 (cF24, Cu2 Mg type) 54, 59, 61 C16 (tI12, CuAl2 type) 60 C32 (hP3, AlB2 type) 60 Cellular precipitates 635 Ceramics 55, 62, 63, 90, 92, 106, 107, 261, 369, 371, 372, 558, 559 Chemical bonding 12 Chemical diffusion 555, 570, 578, 799 Chemical free energy 202, 238, 261, 263, 264, 278, 287, 324, 371, 456, 580, 592, 593, 612, 618, 675, 725, 726 Chemical potential 98, 129, 130, 132, 152, 285, 395, 530, 571, 577, 589
Index Chemical spinodal 558, 594, 601, 602, 621, 622 Classification 21, 27, 28, 57, 90–93, 101, 105–107, 111, 115, 283, 284, 540, 587, 633, 722 Clausius–Clapeyron 8, 359 Cluster approximation 394, 395, 396, 398, 399, 412 Cluster variation method 392 Coherency 65, 99, 240, 264, 265, 350, 370, 504, 593, 606, 614, 618, 633, 650, 662, 677 Coherent phase transformations 386 Coherent spinodal 99, 525, 558, 594, 601, 618, 620, 621, 622 Cold rolling 702 Composition-invariant transformations 91, 110, 435 Composition modulation 525, 593, 594, 616, 618 Composition-induced destabilization of crystalline phases 220 Compressive 290, 702, 769, 770 Concentration modulation 99, 381, 438, 557, 592, 602, 603, 622 Concentration wave 99, 117, 120, 183, 379, 390, 391, 392, 393, 406, 416, 437, 438, 459, 460, 524, 525, 527, 528, 606 Concomitant clustering and ordering 407 Configurational energy 393, 395, 776 Constitutional solution treatment 753 Continuous or homogeneous transitions 97 Cooling rate 48, 128, 141, 142, 144–151, 153, 158–160, 165, 171–173, 175, 180, 182, 205, 281, 383, 420, 421, 433, 434, 446, 448, 450, 452, 626, 627, 629, 655, 656, 708, 725, 728, 739, 753 Coordination factor 17 Coordination numbers 59, 775 Correlation functions 393, 395, 397, 398, 399 Correspondence matrices 487, 489, 730 Correspondence variant 294, 343, 348, 350, 665, 734 Corrosion 157, 586, 587, 659, 661, 702, 721, 741, 769, 771, 772
Index Coupled transformation 122 Cross slip 332, 333, 688, 698 Crystallization 92, 110, 111, 145, 157, 158, 160, 170, 172, 175–178, 181, 182, 184–189, 191–197, 200–205, 207, 209–211, 213, 220, 250 Crystallization kinetics 193, 200, 210, 211 Crystallographic texture 701, 702, 704, 706, 707, 743, 745 Crystallography 44, 50, 266, 269, 270, 282, 290, 292, 303, 320, 340, 342, 345, 366, 372, 418, 424, 474, 484, 534, 613–615, 648, 657, 662, 665, 720, 721, 728–730, 732, 735, 736 Cu-Zn-Al 362 d-band occupation 12 d-band 13, 18–20, 508, 516–518 D019 54, 55, 59, 61, 62, 227, 228, 229, 377, 382, 383, 407, 412, 413, 416, 417, 418, 420, 421, 423, 427, 428, 430, 431, 436, 437, 438, 439, 440, 441, 442, 444, 447, 451, 785, 786 D88 54, 60, 520, 521, 525, 527, 528, 529, 530, 535 Deformation twins 331, 334, 546 Degradation Processes 741 Degree of self-accommodation 308, 666 Delayed hydride cracking 742, 743, 744, 745, 746 Dendritic 137, 141, 145, 146, 150, 173, 175, 189, 203, 212, 690 Density functional theory 14, 15, 385 Density of states 18, 19, 384, 516 DHC 746, 747 DHC velocity 746 Diad 296, 346, 351 Differential scanning calorimetry 340 Diffraction effects 51 Diffuse Scattering 51, 112, 249, 471, 474, 495, 499, 533 Diffusion anisotropy difference 565 Diffusion bonding 555, 584, 585, 586 Diffusion mechanisms 157, 555, 560 Diffusion zone 559, 573, 574, 578, 579, 584, 585, 586, 587
803 Diffusional 43, 48, 50, 65, 101, 103–107, 109–111, 120–122, 129, 135, 148, 149, 185, 204, 266, 281, 293, 419, 453, 500, 504, 520, 558–560, 623, 625, 632, 642–645, 648, 650–653, 655, 657, 662, 683, 738, 739, 779 Diffusional transformation 103, 120, 558, 632, 643, 650, 785, 799 Dilation parameter 276 Discontinuous coarsening 450, 456, 457, 675 Discrete transformations 97, 109, 110 Dislocations 49, 121, 264, 276, 279, 280, 310, 312, 315, 316, 327, 329, 331–335, 337–339, 342, 387, 441, 451–455, 537–540, 546–548, 550, 562, 606, 626, 633, 645, 646, 648–652, 661, 677, 683, 684, 687–689, 692, 698, 710, 739, 753 Displacement ordering 98, 156, 419, 501–503, 509, 518, 550 Displacement 43, 101, 111, 114, 120, 431, 451, 506–508, 786 Displacement vector 316, 317, 319, 431, 432, 451, 454, 473 Displacement wave 43, 101, 120, 419, 427, 485, 486, 499, 500, 506, 510, 513, 527, 528, 531, 532, 533, 786 Displacive 43, 101, 109, 113, 120–122, 386, 389, 419, 425, 426, 429, 430, 492, 495, 518, 519, 525, 534 Displacive transformation 43, 100, 101, 103–107, 111, 120, 154, 261, 492, 518, 559, 650, 721 Distorted hexagonal 46, 281 Divacancies 561, 562 Domain structure 319, 419, 428, 767 Dynamic recovery (DRV) 683, 685, 686, 687, 691, 694 Dynamic recrystallization (DRX) 683, 687, 694, 698, 701, 706 Dynamic strain ageing 539, 540, 542, 546 Effective full power days (EFPD) 769 Elastic energy 263, 323 Electron concentration (e/a) factor 17 Electron diffraction 51, 52, 118, 174, 175, 341, 362, 495, 533, 603
804 Electron to atom (e/a) ratio 24 Electronegativity factor 17 Electronic structure 13–15, 24, 384, 385, 386, 393, 412, 413, 507, 510 Ellipsoidal inclusion 306 Embrittlement 181, 182, 471, 536, 537, 742, 746 Enantiotropic 92 Euler’s formula 300 Eutectic crystallization 111, 185, 187, 192, 194–197, 202–204 Eutectoid Decomposition 22, 27, 28, 65, 91, 111, 153, 186, 447, 448, 556, 560, 670, 671, 672, 673, 674, 675, 676, 681, 682, 722, 794, 798, 799 Exchange mechanisms 561 Exothermic 182, 184, 187, 189, 194, 207, 210, 340, 360, 361, 722 Extended defects 562 FCC special points 407 Fe24 Zr76 191 Fe30 Zr70 191 Fe40 Zr60 191 Fe–B 157, 162, 177, 187, 320, 356 Fe-Ni-Co-B 162 Ferroelectric 100, 106, 107, 111, 113, 114 Ferromagnetic 94, 107, 116, 162 Ferrous 47, 260, 263, 277, 279, 287, 320, 322, 326, 332 FeZr2 191 FeZr3 191 Fick’s laws 555, 562 First principles 11, 14, 17, 18, 117, 386, 393, 412, 503, 507, 510 First-order transformation 94, 97, 101, 105, 128, 262 First-order transitions 5, 93, 98, 107, 115 Flow localization 685, 691, 695, 701 Flow softening 691, 693, 696, 698, 701 Flow stresses 260, 328, 710 Fractal morphology 321, 637 Fractal 308, 313, 637 Fracture 64, 145, 226, 327, 352, 353, 370, 371, 537, 539, 547, 655, 690, 691, 721, 742, 747
Index Gas atomization 150 Gd–Co40–50 163 Gd–Fe32–50 163 Geometrically close packed (GCP) structures 57 Glass formation in diffusion couples 215, 220 Glass formation 134, 157, 158, 160, 161, 165, 168, 171, 205, 206, 213, 215, 220, 226 Glass transition 158, 161, 181–185, 193, 194, 200, 205, 209, 210, 218 Glass-forming abilities (GFAs) 157, 158, 160, 171 Glissile 105, 120, 121, 277, 278, 350, 651, 680 Gliding 332, 337, 504 Glissile interface 105, 120, 121, 278 Grain boundary allotriomorphs 633, 664, 674, 690 Grain-boundary diffusion 769 Ground states 377, 381, 397, 398, 403 Group number 18, 24, 26, 668, 669 Group/subgroup relations 5, 377, 424 Growth 49, 97, 98, 101, 105, 109, 110, 113, 116, 120, 122, 128, 129, 135–137, 139, 140, 144–146, 155, 158, 162, 164, 165, 172, 175, 177, 185, 188, 189, 191–194, 196, 197, 199, 200, 202–206, 208, 211, 212, 223, 225, 264, 265, 277–280, 308, 314, 315, 322, 355, 358, 362, 381, 421, 442, 451–453, 455, 457, 492, 494, 499, 504, 510,527, 539, 547, 559, 581, 582, 585, 597, 614, 623, 625, 626, 628–630, 634, 635, 637, 639, 641, 648, 650–653, 655, 657, 658, 661, 666, 668, 669–671, 673, 674, 681, 683, 698, 702, 707, 721, 726, 746, 747, 751, 769 Growth ledges 637, 651, 652 Habit plane 45, 109, 110, 270, 293, 303, 314, 348, 366, 368, 666, 732, 799 Habit plane variants 277, 313, 314, 348, 350, 352, 677, 679 Hafnium (Hf) 4 Hall–Petch constant 337, 338 Hall–Petch relation 337, 338 Hardening 326, 327, 329, 335, 336, 337, 339, 341, 536, 538, 604, 662, 683, 687 HCP special points 406
Index Heterogeneous 165, 169, 171, 175, 183, 213, 221, 222, 279, 381, 426, 451, 455, 533, 620, 625, 640, 661, 689 Heterogeneous nucleation 165, 169, 171, 175, 213, 222, 279, 455, 553, 620, 640, 661 Heterophase fluctuation 101, 279, 589 Hexagonal, H net 55 Homogeneous deformation 102, 103, 267, 268, 343, 345, 505, 643, 689, 735 Homogeneous nucleation 165, 170–173, 206, 278, 279, 527, 625 Homogenization treatment 758 Homophase fluctuation 589, 594 Hume-Rothery phases 384 Hydride blister 742,743, 744, 747, 749, 751 Hydride blister formation 742, 749 Hydride precipitation 717, 721, 726, 727, 736, 742, 747, 794 Hydrogen absorption 754, 755, 758, 760, 761 Hydrogen charging 225 Hydrogen ingress 738, 741, 742 Hydrogen migration 737, 743, 744, 746, 747 Hydrogen storage capacity 756, 758, 761, 762, 764 Hydrogen storage materials 55, 62, 754, 756, 758, 759, 760, 761, 764 Hydrostatic 215, 479, 494, 499, 508, 719, 720, 727, 744 Hysteresis 9, 10, 50, 353, 354, 356, 362, 475, 479–481, 726, 756, 758, 761, 764 Icosahedral phases 248–251 Incommensurate -structures 509, 515 Inhomogeneous shear 45, 47, 49, 275 Instability diagrams 410 Instability temperature 98, 107, 115, 528 Interaction energies 14, 380, 403, 427, 465, 727, 775 Interdiffusion 127, 128, 204, 224, 225, 555, 557, 559, 570, 572, 573, 574, 576, 577, 578, 585, 592, 604 Interface instability 146, 657 Interface phase 737, 738 Interfacial energy 206, 238, 263, 264, 280, 456, 458, 504, 580, 592, 604, 644, 657, 725 Interfacial equilibrium 129
805 Interfacial structure 555, 648, 650, 652, 798 Interlayers 225, 585, 586 Intermetallics 17, 54, 55, 57, 61, 90, 92, 106, 107, 111, 231, 261, 356, 382, 474, 558, 662, 669, 754 Intermetallic compounds 22, 153, 218, 225, 231, 414, 458, 559, 587, 657, 758, 787 Intermetallic Phases 26, 33–35, 38, 43, 53–55, 57, 59, 61, 62, 160, 162, 164, 175, 188, 284, 384, 414, 443, 518, 525, 560, 615, 638, 657, 658, 676, 754 Internal twinning 728, 729, 737 Internal twins 267, 276, 293, 304, 312, 316, 347, 635, 736 Interstitial ordering 719, 720, 728, 737, 764, 772, 779 Interstitially ordered superstructures 764 Interstitials 231, 232, 236, 339, 382, 561, 765 Intraplanar ordering 767 Intrinsic diffusivities 571 Invariant line strain 276, 555, 559, 614, 642, 643, 796 Invariant line 276, 302, 342, 504, 555, 559, 614, 642, 643, 644, 645, 646, 648, 650, 655, 663, 679, 796, 798 Invariant plane strain (IPS) 45, 47, 110, 260, 266, 504, 632, 728, 737, 796 Invariant plane strain condition 45, 110, 504, 645, 734, 737, 779 Invariant plane strain transformation 45, 47, 276, 632, 651, 732, 734, 779 Invariant plane 45, 47, 110, 260, 267, 276, 314, 504, 532, 644, 651, 654, 728, 732, 734 IPS condition 270, 272, 275, 291, 294, 304, 315, 643, 662, 665, 733, 735, 736 Irrational 244, 246, 267, 333, 347, 633, 648–651, 732 Isomorphous 22, 26–30, 282, 285, 287, 327, 418, 475, 586–588, 606, 609, 616, 620, 765 Isothermal kinetics 194, 196, 202, 264 Kagome net or K net 55 Kirkendall effect 572, 585
806 L12
54, 55, 59, 61, 201, 228, 382, 390, 391, 392,402, 408, 409, 413, 436, 439, 440, 441, 442, 529, 530, 533 L1o (tP4, AuCu type) 54, 59, 61 La78 Ni22 163 Lamellar 111, 382, 420, 440, 441, 443, 444, 448, 450, 451, 452, 453, 454, 456, 457, 629, 635, 671, 673, 675, 676, 679, 681, 682, 688, 691, 696, 697 Lamellar eutectoid 635 Lamellar microstructure 450, 451, 691 Lamellar structure 443, 450, 456, 457, 673, 675, 676, 681, 682, 697 Lath martensite 48, 308, 310, 320, 328, 329, 331, 332, 437 Lath morphology 48, 277, 310 Lattice collapse mechanism 471, 474, 492, 498, 499, 504, 509, 518, 527 Lattice correspondence 45, 103, 289, 305, 342, 363, 368, 431, 485, 488, 515, 786 Lattice distortion matrix 731 Lattice-invariant shear (LIS) 45, 260, 269, 271, 293, 295, 296, 312, 314, 315, 346, 347, 366, 368, 504, 651, 732, 733, 779, 796 Lattice parameters 7, 46, 51, 227, 290, 345, 427, 443, 512, 520, 602, 613, 644, 723, 786, 798 Lattice shear 44, 46, 120, 270, 271, 345, 532, 632, 737, 777 Lattice-site correspondence 560 Lattice strain dominated 111 Lattice strains 45, 46, 100, 106, 111, 269, 270, 271, 431, 663, 731, 786 Laves phase 55, 59, 60, 62, 235, 250, 251, 622, 658, 754, 756, 759, 761 Linear muffin tin orbital (LMTO) 9, 11, 15, 19, 385, 412 Liquid/solid interface 129, 134, 135, 146 Local density approximation (LDA) 11, 14, 15, 16, 385, 386 Localized hydride-embrittlement 746 Long-range order (LRO) 387 Mackay cluster 251, 252 Macroscopic flow behaviour 335 Macroscopic shape deformation 109
Index Macroscopic shape distortion 300, 301 Macroscopic shear 44, 266, 270, 274, 298, 303, 372, 504 Macroscopic strain 264, 270, 275, 504, 547, 704, 734 Macrosegregation 141, 143–145 Martensite interfaces 264, 280, 680 Martensite 26, 30–32, 35, 43–49, 106, 110–112, 153, 155, 260, 261, 263–267, 269–271, 273–281, 283, 287, 290, 293, 295, 296, 302–310, 313–317, 319–329, 331–340, 347–352, 354, 355, 358–362, 365, 368, 370, 372, 418, 421, 437, 504, 532, 546, 559, 609–618, 626, 635, 640, 645, 651, 666, 680, 720, 732, 735, 753 Martensite phase 26, 43, 44, 48, 279, 280, 305, 354, 355, 357, 421, 618 Martensitic transformation 25, 26, 43–45, 47, 49, 100, 101, 105, 106, 109–111, 113, 114, 155, 260–266, 270, 275–279, 281–285, 287, 288, 290, 292, 293, 306, 312, 324, 326, 327, 339, 342, 348, 352, 354–356, 370–372, 418, 419, 447, 475, 492, 494, 500, 504, 530, 623, 627, 632, 642–644, 646, 651, 653, 663, 777–779 Martensitic 25, 26, 34, 35, 38, 43–47, 49, 64, 100, 101, 105, 106, 109–114, 141, 148, 149, 149, 153–155, 260–266, 270, 275–285, 287, 288, 290, 292, 293, 297, 302, 304, 306, 312, 319, 320, 322, 324, 326–328, 331, 337–340, 342, 348, 350, 352–356, 361–363, 368, 370–372, 418, 421, 439, 447, 475, 477, 482, 492, 494, 500, 504, 530, 546, 559, 609, 623, 626, 627, 632, 642, 643, 644, 651, 653, 658, 662, 663, 668, 704, 729, 736, 738, 769, 777–779 Massive 48, 106, 109, 110, 185, 205, 217, 221, 223, 281, 421, 447, 448, 450, 453–456, 559, 623–626, 628–630, 671 Massive transformation 106, 109, 110, 185, 205, 421, 447, 448, 450, 453–456, 559, 623–626, 628–630 Matano interface 574 Maximum resolved shear stress 303 Mean field 94, 385, 388
Index Mean free slip length 332, 334, 335, 337 Mechanically driven systems 226 Mechanism 101–104, 228, 451–453, 499–518, 560–562, 589, 648, 737 Melt extraction 150 Melt spinning 150, 151, 173, 191 Metallic glasses 110, 134, 157, 158, 162, 164, 165, 171, 176–179, 181, 183–185, 187, 193, 205, 207, 210 Metal–metal glasses 181, 187, 188, 204 Metal–metalloid glasses 187, 204 Metastable 23, 35, 53, 92, 129, 215, 403, 410, 502, 611, 671, 793 Metastable Zr3 Al 337, 382, 437, 441, 530 Mf 47, 48, 110, 278, 283, 350, 353–356, 360, 361, 363 Microsegregation 141–144 Microstructural evolution 235, 555, 603, 656, 683, 684, 685, 691, 697 Minor orientations 314 Mirror plane 272, 274, 275, 293, 296, 314, 315, 342, 346, 351, 356, 614, 679 Miscibility gap 29, 31, 35, 52, 54, 98, 99, 409, 410, 555,558, 588, 589, 596, 597, 598, 599, 601, 609, 621, 622, 787 Misfit compensating ledges 650 Mixed diffusive/displacive transformation 120 Mixed Mode Transformations 115 Modes of crystallization 185, 200 Molar free energy 131, 285, 435, 571, 589, 598, 625 Molecular dynamics (MD) 16, 235, 385 Monoclinic 14, 41, 54, 62–67, 113, 340, 345, 347, 348, 362–366, 369–372, 770 Monotectoid reaction 29, 31, 32, 35, 53, 187, 555, 559, 587, 606, 607, 609, 621, 787 Monotropic transitions 92 Monovacancies 561, 562 Monte Carlo (MC) 14, 16 Morphological stability 135, 138, 139 Morphologies 48, 51, 172, 173, 277, 304, 309, 343, 500, 626, 633, 638, 640 Morphology and substructure 48, 304
807 Ms
44, 47, 48, 101, 110, 112, 113, 140, 154, 155, 262, 263, 265, 277, 278, 281–283, 287, 288, 303, 310, 312, 313, 320, 322, 324, 325, 331, 361, 363, 418, 627 Muffin Tin (MT) 15 Multilayered structures 237 m-ZrO2 63
Nearly-free-electron (NFE) 18 Neutron irradiation 235, 237, 702 Ni10 Zr7 578 Ni5 Zr 578 Ni5 Zr2 579 Ni–B 162, 187 NiZr 578 NiZr2 578 Nodular corrosion 771 Non-ferrous 260, 326 Non-collinear 292 Non-Equilibrium Phases 23, 26, 43 Nucleation 97, 98, 101, 105, 109, 110, 113, 116, 120, 122, 146, 158, 162, 164, 165, 167–172, 175, 185, 188, 192–194, 196, 197, 199, 200, 202, 206, 208, 211–213, 217, 222, 223, 225, 226, 229, 265, 277–279, 305, 322, 324, 325, 348, 349, 381, 439, 451, 452, 455, 479, 494, 527, 528, 533, 539, 547, 580, 582, 589, 592, 594–597, 612–614, 618, 620–623, 625, 629, 630, 633, 640, 657, 661, 666–668, 670, 671, 674, 683, 694, 698, 726, 739, 740, 751, 771 O-phase 417, 418, 420, 421, 422, 423, 427, 428, 431, 433, 434, 435, 555, 556, 662, 665, 666, 786 Octahedral and tetrahedral voids 57 Omega phase 49 Order–disorder 235, 380–382, 384, 386, 388, 389, 401, 414, 459, 460, 767 Order–disorder transformation(s) 235, 384, 386, 767, Ordered -Structures 421, 518–536 Ordering tie lines 458
808 Orientation relations 45, 50, 103, 175, 192, 267, 293, 302, 303, 310, 313, 314, 363, 443, 454, 488, 613, 635, 642, 679, 798, 799 Orientation relationship 45–47, 50, 51, 103, 175, 192, 274, 276, 293, 302, 303, 310, 362, 364, 365, 368, 383, 420, 424, 440, 441, 444, 457, 485, 530, 629, 632, 633, 641–644, 677, 730, 735, 737, 738, 741 Orientational relation 778 Orientational variant 506 Orthohexagonal 281, 290, 296, 532, 663 Orthorhombic 46, 54, 66, 111, 113, 200, 282, 345, 418, 427 Orthorhombic -martensite 281, 283, 418, 421, 616–618 Oxidation 383, 443, 769, 770, 772 Paraelectric 113 Partial dislocations 279, 316, 452, 453, 455 Partial molar volume of hydrogen 719, 727 Partially crystalline alloys 171 Partially Stabilized Zirconia (PSZ) 65, 369–371, 373 Partitionless polymorphic solidification 161 Partitionless solidification 110, 131, 133–135, 156, 160, 161, 185, 205 Peierls–Nabarro 339 Peritectoid reaction 27, 141, 187, 382, 433, 434, 437, 441, 442, 529, 772 Perovskite structure 113 Phase diagrams 7, 8, 11, 13–15, 17, 18, 24, 26–29, 43, 53, 54, 65, 98, 129, 133, 134, 160, 161, 163, 218, 232, 282, 286–288, 369, 387, 388, 410, 413, 414, 418, 475, 479, 516, 587, 588, 606, 609, 624, 626, 722, 764, 765 Phase rule 7, 92 Phase separation 43, 52, 53, 99, 115, 185, 210, 327, 409, 411, 500, 523, 555, 559, 587, 603, 606, 609, 612, 616, 618, 620, 621, 787 Phase separation in -phase 43, 52, 53, 559, 587, 589, 603, 606, 609, 618, 620–622
Index Phase stability 13, 14, 16–18, 24, 27, 52, 62, 148, 237, 240, 284, 383, 384, 433, 508 Phase transformation (transition) 5, 8, 21, 62, 89–92, 94, 105–107, 109, 115, 120, 128, 140, 145, 147, 148, 152, 153, 157, 176, 186, 195, 271, 285, 352, 361, 369, 382, 383, 386, 409, 417–419, 430, 443, 450, 474, 495, 512, 529, 530, 535, 558, 560, 606, 607, 609, 619, 621, 623, 643, 656, 684, 706, 721, 725, 726, 753, 754, 769, 772 Phenomenological theory of martensite crystallography 266, 270, 504, 720, 735, 729 Phonon dispersion curves 498, 507 PHWR 382, 706, 707, 749, 772 Pilger milling 702 Plasma rotating electrode 150 Plastic accommodation 313, 331, 347, 719 Plastic deformation 178, 263, 265, 324, 325, 334, 354–356, 561, 683, 684, 688, 691, 702–704, 726 Plate morphology 49, 277, 309, 313, 328, 644, 729 Plate-shaped 504–506 Plateau pressures 761 Point defects 231, 232, 233, 235, 533, 561 Polarized light microscopy 302 Polydomain 277, 305, 323, 350 Polymorphic crystallization 110, 175, 178, 185, 187, 189, 191, 194, 197, 202, 204, 205 Polymorphic transformations 65, 92, 110, 223, 362 Polymorphism 4, 5 Polymorphous transformation 5 Portevin–Le Chatelier (PLC) effect 539 Post-solidification transformations 140 precipitate particles 370, 659 Premartensitic 279, 361 Pressure-induced → 494 Pressure–composition isotherms (PCI) 756, 758, 761 Primary crystallization 177, 187–189, 192–194, 196, 202–204 Primary plates 304, 313, 348, 349 Principal axes of deformation 268 Principal strains 46, 268, 301, 366, 662, 778 Processing maps 699
Index Promotion energy factor 17 Pseudoelasticity 339, 340 Pseudoplastic 353–355, 359 Pt–Sb 162 Quantum Monte Carlo (QMC) 14, 16 Quantum structural diagrams (QSD) 17 Quasicrystalline structures 241, 243 Quenched-in nuclei 165, 172, 192, 197, 199, 203 R phases 103, 342 Radiation-induced amorphization 229, 235 Rapid or nonquenchable 105 Rapid solidification 36, 128, 139, 150, 152, 153, 155, 159, 160, 169–171, 228, 231, 250, 522 Rapid solidification processing 36, 150–153 Recovery 346, 353, 684, 692, 710 Reconstructive transformations 104 Recrystallization 145, 199, 421, 670, 683, 684, 692, 710 Resistivity 181, 279, 340, 341, 362, 479, 482, 768 Reversion stress 260, 356, 359 Rigid body rotation 45, 103, 271, 274, 276, 292, 314, 635, 643, 644, 663, 704, 705, 736 Rupture 371, 382, 771 Schiebung 279 Schmidt factors 334 S–d electron transfer 7, 11, 12, 20 Second (or higher order) 5, 502 Second-order transitions 93, 115, 528 Self-diffusion 176, 564, 568, 604, 688, 689, 690, 728, 729 Self-similarity 241, 313 Self-accommodation 277, 305, 306, 308, 320–323, 340, 342, 345, 347, 349, 350, 355, 361, 362, 666, 668, 669 Serrated flow 540, 542, 545, 546, 550 Shape memory effect 339, 340, 352, 353, 355, 356 Shape strain 263, 265, 269–271, 303, 306, 308, 316, 320, 322, 366, 373, 504, 644, 666, 668,733, 734
809 Shear band 693, 694 Shear direction 103, 295, 296, 302, 303, 506, 733, 734 Shear instability 213, 215 Shear moduli 337 Shear modulus 25, 47, 112, 215, 260, 331, 337, 538, 720 Shear plane 295, 296, 504, 733 Shear transformation 111, 120, 721, 728, 729, 737, 798 Shock pressure 471, 481, 489, 499, 500, 504 Shock wave 8–10, 506 Shuffle dominated 111 Similarity transformation 272, 294, 732 Single surface trace analysis 310 Sintering 370 Site Occupancies 458, 520 Size factor 17, 23, 29, 338 Slip mode 322, 334 Sluggish or quenchable 105 Softmode 111 Solid solubility 27, 28, 65, 150, 152, 285, 573, 722, 725–727, 746, 747 Solid State Amorphization 212, 215, 226, 228, 229 Solidification 36, 110, 128–142, 144–148, 150–156, 159–161, 163, 169–171, 175, 185, 205, 206, 228, 231, 250, 522, 690 Solubility limit 66, 152, 327, 609, 629, 721, 722, 746 Solute partitioning 121, 131, 175, 492, 629, 637, 670, 681 Sp-band 18 Special point ordering 401 Specific volume of hydride precipitate 727 Spinodal clustering 98, 99, 115, 116, 409, 438, 592 Spinodal decomposition 99, 156, 524, 525, 559, 589, 592, 593, 595, 597, 603, 604, 616, 617, 618, 785 Spinodal ordering 98, 99, 403 Splat quenching 150, 151, 175 Square, S net 55 Stacking domains 319 Static concentration wave 389 Static concentration wave model 389 Stereographic projection 316, 644, 732, 736
810 Stiffness moduli, C11 , C44 and C12 25 Strain coupling 305, 306, 355 Strain energy density 166, 306, 307, 308, 349, 719, 725 Strain energy 47, 99, 166, 238, 265, 277, 305, 347, 370, 479, 504, 592, 666, 719, 724, 727 Strain hardening 683 Strain-induced martensitic 324 Strain rate sensitivity 540, 548, 684, 685, 693, 696 Strain spinodal 279, 603 Strain-induced plates 265 Strengthening mechanisms 327 Strengthening 22, 326–328, 331, 338, 698 Stress-assisted 48, 265, 322, 324–326, 354, 546 Stress-assisted martensite 260, 324–326 Stress directed migration 744, 745 Stress free transformation strain 259, 306, 719, 727 Stress reorientation of hydrides 742, 743 Stress–strain 325, 335, 337, 352–354, 356, 358, 537, 540, 547, 548, 550, 684, 687, 690, 692, 698 Structural ledges 633, 649, 650, 652, 796, 796, 798 Structural relaxation 158, 176, 180, 181, 183, 184, 193, 412 Structure maps 17 Substitutional 23, 27, 30, 177, 277, 327, 338, 339, 387, 415, 533, 539, 546, 559, 561, 562, 609 Substructure 45, 48, 49, 293, 303–305, 313, 316, 317, 319–322, 327–329, 332, 421, 626, 670, 687, 689 Supercooled liquid 137, 140, 160, 163, 164, 182, 207, 208, 211–213 Supercooling 47, 99, 129, 133, 137, 138, 154, 161, 213, 214, 262, 263, 265, 281, 287, 324, 361, 403, 416, 438, 479, 527–529, 623, 638, 658 Superlattice 99, 109, 117, 227, 320, 380, 381, 419, 437, 451, 454, 458, 518, 527, 679, 720 Superplasticity 685, 691, 695, 701
Index Supersaturation 120–122, 236, 282, 283, 327, 438, 441, 447, 453, 457, 609, 619, 654, 658 Surface relief 266, 269, 270, 310, 362, 364, 492, 632, 737, 769 Symmetry 5, 6, 15, 51, 52, 55, 58, 99–101, 114, 117, 176, 185, 214, 241, 243–245, 251, 271, 272, 305, 316, 370, 386, 396, 401, 403, 406, 416, 421, 422, 424–429, 431, 450, 485, 495, 496, 500, 502, 519, 520 Symmetry tree 471, 534, 535 Sympathetic nucleation 655 Syncretist Classification 105 TEM 45, 49, 173, 189, 191, 196, 220, 250, 310, 331, 451, 648, 675, 692, 789 Tempered 284 Tempering 284, 327, 421, 559, 609–613, 615–618 Tempering of martensite 555, 559, 609, 615 Temporary Alloying 753 Tensile 145, 157, 270, 290, 291, 335, 352, 356, 370, 372, 540, 689, 692, 696–698, 707, 708, 710, 727, 741–746 Terminal solid solubility 717, 719, 725, 726 Terminal solid solubility for precipitation (TSSP) 725, 726 Terminal solubility 737, 741 Tetragonal 12, 54, 55, 63, 65, 66, 113, 362–368, 372 Tetragonal phase (t-ZrO2 63 Tetragonal shear constant 12 Texture 691, 701, 702, 704, 707, 743, 745, 769 The C11b (tI6, MoSi2 type) structure 60 Thermal arrest memory effect 340, 360, 361 Thermal cycles 354 Thermal expansion coefficient 369 Thermal migration 744, 745 Thermal shock resistance 369 Thermal stability 176, 181, 208, 210 Thermo-Mechanical Processing 683, 689, 690 Thermochemical processing 753, 754 Thermodynamic interaction parameter 26
Index Thermodynamics 8, 92, 93, 128, 129, 133, 215, 220, 261, 288, 386, 387, 393, 408, 609, 619, 623, 685, 756 Thermoelastic equilibrium 265, 340, 354, 356, 358, 359 Thermoelastic 265, 340, 350, 354–356, 358, 359 Thermomechanical processing (TMP) 684 Thin Film Multilayers 237 Thin twins 314, 315 Threshold stress 356, 548, 550, 745, 747 Threshold value of stress 745 Ti-V 11, 27, 29, 30, 32, 33, 53, 288, 325, 482, 483, 489, 606, 609 Ti2 AlNb 421, 422, 423, 432, 433, 670 Ti2 Al–Nb 669 Ti2 Cu 662, 676, 677 Ti2 N 39, 721, 772 Ti45 Zr38 Ni17 251 Ti4 AlNb3 669, 670 Ti5 Al3 520 Ti–Ag 576, 578, 624, 626 Ti–Al 27, 29, 30, 34, 35, 284, 377, 381, 383, 412, 413, 414, 415, 416, 430, 433, 446, 453, 464 Ti–Al–Nb 383, 417, 421, 430, 432, 433, 519, 662, 669 Ti-Al-V 284 Ti–Au 383, 624, 626, 627 Ti–Be 160, 161, 188 Ti–Bi 673 Ti–Co 673, 674 Ti–Cr 27, 29, 30, 33, 34, 250, 316, 474, 621, 622, 638, 639, 640, 648, 651, 673, 758, 798 Ti-Cr-Si 250 Ti–Cu 27, 157, 160, 163, 179, 232, 310, 673, 674, 675, 676, 677, 682 Ti–Fe 161, 582, 583, 586, 626, 673, 674 Tight-binding (TB) 18, 19, 215 Ti–H 27, 717, 719, 722, 723, 728 Ti–Mn 27, 250, 314, 316, 673, 758 Ti-Mn-Si 250 Ti–Mo phase diagram 621 Ti–Mo 27, 29–32, 53, 288, 536, 537, 546, 606, 609, 616, 621
811 Ti–N 27, 29, 30, 39, 220, 232, 339, 340, 458, 464, 477, 482, 606, 607, 620, 673, 772, 774, 776–778 TiN 39, 40, 721, 772, 778 Ti–Ni Shape Memory Alloys 339 Ti–Ni 27, 220, 232, 339, 340, 673 Ti–O 27, 764, 765, 767, 769 Ti–Pb 673 Ti–Pd 356, 673 Ti–Pt 673 Ti–Si 624, 626, 627, 661 Titanium (Ti) 4, 7, 9, 21, 24, 626, 629, 656, 687, 690, 691, 698, 706, 720, 721, 723, 737, 738, 744, 753, 754, 769, 772, 777 Titanium nitride 721, 772 Ti–V 27, 29, 30, 32, 33, 53, 288, 325, 482, 483, 489, 606, 609 Ti–X 24, 26–29, 43–46, 48, 53–55, 57, 59, 61, 62, 148, 418, 657, 671, 673–676 Ti-Zr 27, 29, 30, 249–252, 285–287, 327, 758 Ti-Zr-Fe 249–251 Ti-Zr-Ni 249, 251, 252 Topologically close packed (TCP) structures 57 Toughening 369–372 Toughness 22, 65, 145, 327, 369–371, 655, 706, 742, 747 Tracer-diffusion 555, 564, 566 Transformation-induced plasticity 371 Transformation sequences 141, 257, 340, 377, 409, 428, 429, 430, 432, 447, 530, 621, 786 Transformation temperatures 47, 323, 340, 729 Transformation twins 49, 334 Transient nucleation 168–170 Transition metals 4, 7, 8, 15, 18, 20–22, 24–26, 29, 157, 162, 163, 177, 225, 231, 249, 480, 507, 516, 518 Transmission electron microscopic 737 Triangular, T net 55 TRIP steels 371 Triple point 8–10, 218, 219, 480 True plastic strain 260, 328 True strain 328, 696 True stress 328, 337
812
Index
TSS 719, 720, 725, 742, 745, 751 TSSD 725, 726, 727, 751, 752 TSSP 726, 727, 751, 752 Twinning 270, 271, 274, 275, 277, 295, 296, 300, 313, 322, 333, 342, 346, 355, 445, 484, 505, 546, 702, 728, 729, 737, 798 Two-way shape memory 354 Type I twins 296, 345, 346, 347, 350, 351, 730, 735, 736, 738, 739 Type II twins 346, 347, 351 Umklapp 279 Undistorted 45, 46, 267–271, 274, 292, 293, 295, 296, 301, 485, 505, 509, 644, 664, 732, 736 Unrotated 45, 241, 267, 271, 274, 300, 614 Vacancy mechanism 561, 562, 564, 565, 566, 569, 571 Vapour pressure 756, 759, 760, 761, 764 Variant 146, 351, 527 Viscosity 157, 158, 161, 165, 168, 170, 171, 175, 177, 184, 206, 210 Vitrification 128, 157, 158, 160, 164, 165, 171, 172, 175, 192–194, 208, 215, 221–223 Wechsler, Liebermann and Read (WLR) methodology 732, 734, 735 Weldments 145 Widmanstatten 49, 141, 148, 149, 327, 328, 337, 338, 421, 435, 447, 450, 610, 635, 648, 674, 689, 692, 693 Widmanstatten side plates 635 WLR 732, 734, 735 WLR theory 734 X-ray diffraction 739 Yield strength
165, 362, 363, 474, 482,
324, 339, 537, 655, 745
Zener anisotropy ratio 25–26 Zigzag habit 314 Zircaloys 23, 30, 658, 659, 661 Zirconia 62, 63, 65, 227, 362, 365, 366, 368–371, 770
Zirconium (Zr) 4, 7, 10, 23, 24, 656, 661, 701, 702, 706, 720, 721, 727, 737, 739, 741–745, 747, 751, 758, 769–772 Zr alloys 4, 23, 24, 30, 47, 111, 120, 128, 145, 150, 152, 171, 172, 283, 284, 287, 313, 320, 327, 338, 381, 475, 501, 525, 587, 604, 658, 673, 683, 691, 692, 701 Zr-Nb-Sn-Fe 661 Zr(Cr,Fe)2 658, 659 Zr(Nb,Fe)2 661 Zr2 (Fe,Ni) 658, 659 Zr–2.5% Nb 24, 294, 312, 613 Zr2 Al 38, 39, 141, 156, 210, 382, 437, 441, 442, 518, 519, 525, 527–530, 584 Zr2 Al, Ti2 Al 518 Zr2 Cu 211, 212, 222, 676, 677, 679, 681, 682 Zr2 Ni 174, 175, 188, 189, 200, 202, 204 Zr3 (Fe, Ni) 190 Zr3 Al 30, 38, 39, 55, 61, 227, 228, 382, 436,439, 440, 441, 442, 523, 525, 529, 530, 531 Zr3 Fe 35, 191, 192, 200, 202, 681, 682 Zr4 Fe 681 Zr5 Al3 38, 141, 520, 521, 525, 527, 528, 529 Zr5 Al4 521, 522 Zr76 Fe12 Ni12 200 Zr76 Fe16 Ni8 170–172, 200 Zr76 Fe20 Ni4 200, 204 Zr76 Fe24 189, 191, 194, 200, 202 Zr–Al 29, 30, 38, 39, 116, 140, 156, 207, 208, 210, 381, 382, 437, 522, 523, 530, 574, 582, 584 Zr–Co 163, 220 Zr–Cr 23, 658, 758 ZrCr2 62, 658, 758, 760, 761 Zr–Cu 157, 160, 163, 207, 218, 220, 222, 223, 232, 556, 675, 676, 681, 682 Zr–Fe 29, 30, 35, 36, 163, 172, 187, 195, 200, 220, 232, 249, 250, 251, 676, 681, 682 Zr–H 29, 30, 39, 41, 717, 722, 723, 724, 728, 740 Zr–Nb 11, 23, 29, 30, 35, 52, 53, 153, 154, 155, 310, 477, 481, 492, 515, 586, 599, 600, 601, 603, 606, 607, 609, 618, 620, 621, 640, 645, 649, 650, 661, 701, 798
Index (Zr,Nb)3 Fe 661 Zr–Nb system 30, 35, 599, 601, 603, 607, 609, 618 Zr–Ni 157, 160, 163, 164, 172, 200, 204, 218, 220, 221, 225, 232, 249, 251, 252, 578, 579, 675 Zr–Ni–Cu 160 Zr–Ni–Fe 163 Zr–O 29, 30, 41, 63, 764–767 ZrO2 polymorphs 62, 63
813 ZrO2 62, 63, 65, 66, 363, 770 ZrO2 –CaO system 66 ZrO2 –MgO system 66 ZrO2 –Y2 O3 system 67 Zr–Sn 29, 30, 36, 38, 658 Zr–Ta system 52, 53, 602 Zr–Ti 310, 328, 334–338, 586, 692 Zr–X 24, 26, 27, 29, 43–45, 48, 53–55, 57, 59, 61, 62, 148, 657, 673, 675, 676
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