Thermodynamics, Diffusion and the Kirkendall Effect in Solids (eBook)

Aloke Paul is an associate professor in the Department of Materials Engineering, Indian Institute of Science (IISc), Bangalore, India. Professor Aloke Paul is currently running an active research group working on diffusion related problems with major areas of research including developing new phenomenological models in solid-state diffusion, materials in electronic packaging, bond coat in jet engine applications and growth of A15 intermetallic superconductors. He teaches a postgraduate level course on diffusion in solids and has guided several PhD and ME students. He has co-authored more than 80 articles in various international journals.Professor Tomi Laurila received his D.Sc. degree (with honours) in electronics production technology from the Helsinki University of Technology in 2001. He presently works as a teaching scientist in the group of Electronics Integration and Reliability and holds an adjunct professorship on Electronics Reliability and Manufacturing. His research currently involves the study of interfacial reactions between dissimilar materials used in micro and bioelectronics. He has published extensively, including around 70 scientific papers, book chapters and a text book, on topics such as the thermodynamic-kinetic analysis of interfacial reactions, electrochemical detection of biomolecules and issues related to reliability testing of electronic devices. Professor Laurila is also responsible for the teaching of material science, electronics reliability and bio-adaptive technology to undergraduate, postgraduate and postdoctoral students.Dr. Vesa Vuorinen received a D.Sc. degree in electronics production technology from the Helsinki University of Technology in 2006. Since then he has been working as a research scientist and full-time teacher in the group of Electronics Integration and Reliability at Aalto University. His research includes manufacturing and reliability of high-density electronics assemblies with emphasis on soldering metallurgy and thermodynamics of materials.Dr. Sergiy Divinski is a Privat-Docent at the Institute of Materials Physics, University of Münster, Germany, where he leads the radiotracer laboratory. He teaches graduate and postgraduate courses on diffusion in solids, numerical methods in material science and different aspects of materials science. He has co-authored more than 150 articles in various international journals and several book chapters in the field of diffusion in solids.

Preface 6
Acknowledgment 9
Contents 11
1 Thermodynamics, Phases, and Phase Diagrams 17
1.1…Thermodynamics System and Its State 18
1.2…The Laws of Thermodynamics 19
1.3…Heterogeneous Systems 21
1.4…Commonly Used Terms and First Glance at Phase Diagrams 23
1.5…Spontaneous Change 28
1.6…Free Energy and Phase Stability of Single-Component System 35
1.7…Pressure Effect of Single-Component Phase Diagram 40
1.8…Free Energy and Stability of Phases in a Binary System 42
1.8.1 Change in Free Energy in an Ideal System 43
1.8.2 Change in Free Energy in a System with Exothermic Transformation 46
1.8.3 Change in Free Energy in a System with Endothermic Transformation 46
1.9…Thermodynamics of Solutions and Phase Diagrams 48
1.9.1 The Chemical Potential and Activity in a Binary Solid Solution 48
1.9.2 Free Energy of Solutions 49
1.10…Lever Rule and the Common Tangent Construction 56
1.11…The Gibbs Phase Rule 59
1.12…Correlation of Free Energy and Phase Diagram in Binary Systems 61
1.13…Ternary Phase Diagrams 69
1.14…Stability Diagrams (Activity Diagrams, etc.) 77
1.15…The Use of Gibbs Energy Diagrams 80
1.15.1 Effect of Pressure on the Phase Equilibrium 89
1.15.2 Ternary Molar Gibbs Energy Diagrams 93
1.16…Interdependence of Chemical Potentials: Gibbs--Duhem Equation 96
1.17…Molar Volume of a Phase and Partial Molar Volumes of the Species 98
1.18…Few Standard Thermodynamic Relations 100
References 102
2 Structure of Materials 103
2.1…Hierarchical Structure of Materials 103
2.2…Atomic Bonding 103
2.3…Crystal Lattice 105
2.4…Grain Structure 107
2.5…Defects 109
2.5.1 Point Defects 109
2.5.1.1 Equilibrium Vacancy Concentration in Pure Elements 110
2.5.1.2 Equilibrium Concentration of Impurities in Pure Elements 113
2.5.2 Linear Defects 117
2.5.3 Two-Dimensional Defects 118
2.5.4 Volume Defects 119
2.6…Some Examples of Intermediate Phases and Their Crystal Structure 119
2.6.1 Defects in Intermediate Phases 121
2.6.2 Crystal Structures and Point Defects in Ordered Binary Intermetallics on an Example of Ni-, Ti-, and Fe-Aluminides 123
2.6.3 Calculation of Point Defect Formation Energies 124
2.7…Microstructure and Phase Structure 129
References 130
3 Fick’s Laws of Diffusion 131
3.1…Fick’s First and Second Laws of Diffusion 131
3.2…Solution of Fick’s Second Law to Estimate the Diffusion Coefficient 135
3.2.1 Solution for a Thin-Film Condition 135
3.2.1.1 Solution for a Semi-infinite Diffusion Couple (Error Function Analysis) 138
3.2.2 Solution for Homogenization (Separation of Variables) 151
References 155
4 Development of Interdiffusion Zone in Different Systems 156
4.1…Chemical Potential as the Driving Force for Diffusion and Phase Layer Growth in an Interdiffusion Zone 156
4.2…Few Practical Examples 164
4.3…Making Products by a Diffusion Process 175
References 180
5 Atomic Mechanism of Diffusion 182
5.1…Different Types of Diffusion 182
5.2…Interstitial Atomic Mechanism of Diffusion 188
5.2.1 Relation Between Jump Frequency and the Diffusion Coefficient 188
5.2.1.1 Jumps in the Presence of External Force 193
5.2.2 Random Walk of Atoms 196
5.2.3 Effect of Temperature on the Interstitial Diffusion Coefficient 199
5.2.4 Tracer Method of Measuring the Interstitial Diffusion Coefficient 202
5.2.5 Orientation Dependence of Interstitial Diffusion Coefficient 203
5.3…Diffusion in Substitutional Alloys 206
5.3.1 Measurement of Tracer Diffusion Coefficient 207
5.3.2 Concept of the Correlation Factor 208
5.3.3 Calculation of the Correlation Factor 211
5.3.4 The Relation Between the Jump Frequency and the Diffusion Coefficient in Substitutional Diffusion 213
5.3.5 Effect of Temperature on Substitutional Diffusion 216
5.3.6 Orientation Dependence in Substitutional Diffusion 221
5.3.7 Effect of Phase Transitions on Substitutional Diffusion 228
5.4…Diffusion Mechanisms in Intermetallics 228
5.4.1 Diffusion in Disordered Intermetallic Compounds 229
5.4.2 Diffusion Mechanisms in Ordered Intermetallic Compounds 231
5.4.3 Six-Jump Cycle Mechanism 231
5.4.4 Sublattice Diffusion Mechanism 236
5.4.5 Triple-Defect Diffusion Mechanism 238
5.4.6 Antistructure Bridge Mechanism 240
5.4.7 Interstitial Diffusion Mechanism 241
5.4.8 Other Diffusion Mechanisms 242
5.5…Correlation Factors of Diffusion in Intermetallic Compounds 243
5.5.1 Calculation of the Probabilities P 245
5.5.2 The Monte Carlo Calculation Scheme 248
References 252
6 Interdiffusion and the Kirkendall Effect in Binary Systems 254
6.1…Matano--Boltzmann Analysis 255
6.2…Limitation of the Matano--Boltzmann Analysis 263
6.3…Den Broeder Approach to Determine the Interdiffusion Coefficient 264
6.4…Wagner’s Approach to the Calculation of the Interdiffusion Coefficient 267
6.5…Change in Total Volume of the Diffusion Couple 272
6.6…The Kirkendall Effect 281
6.7…Darken Analysis: Relation Between Interdiffusion and Intrinsic Diffusion Coefficients 287
6.8…Relations for the Estimation of the Intrinsic Diffusion Coefficients 290
6.8.1 Heumann’s Method 291
6.8.2 Relations Developed with the Help of Wagner’s Treatment 294
6.8.3 Multifoil Technique to Estimate the Intrinsic Diffusion Coefficients 296
6.9…Different Ways to Detect the Kirkendall Marker Plane 301
6.10…Phenomenological Equations: Darken’s Analysis for the Relations Between the Interdiffusion, Intrinsic, and Tracer Diffusion Coefficients 303
6.11…Limitations of the Relations Developed by Darken and Manning’s Correction for the Vacancy Wind Effect 306
References 312
7 Growth of Phases with Narrow Homogeneity Range and Line Compounds by Interdiffusion 314
7.1…Time-Dependent Growth of the Phase Layer 314
7.2…Calculation of the Diffusion Parameters in Line Compounds or the Phases with Narrow Homogeneity Range: Concept of the Integrated Diffusion Coefficient 322
7.3…Calculation of the Average Interdiffusion Coefficient 327
7.4…Comments on the Relations Between the Parabolic Growth Constants, Integrated and Average Interdiffusion Coefficients 330
7.5…Calculation of the Ratio of the Intrinsic Diffusion Coefficients 335
7.6…Calculation of the Tracer Diffusion Coefficients 338
7.7…The Kirkendall Marker Velocity in a Line Compound 340
7.8…Case Studies 343
7.8.1 Calculation of the Integrated and the Ratio of the Tracer Diffusion Coefficients 343
7.8.2 Calculation of the Absolute Values of the Tracer Diffusion Coefficients 345
7.8.3 Diffusion Studies in the Ti-Si System and the Significance of the Parabolic Growth Constant 348
References 351
8 Microstructural Evolution of the Interdiffusion Zone 352
8.1…Stable, Unstable, and Multiple Kirkendall Marker Planes 353
8.2…A Physicochemical Approach to Explain the Morphological Evolution in an Interdiffusion Zone 363
8.3…The Application of the Physicochemical Approach in an Incremental Diffusion Couple with a Single Product Phase 375
8.4…The Application of the Physicochemical Approach to Explain the Multiphase Growth 381
8.5…Effect of Electrical Current on the Microstructural Evolution of the Diffusion Zone 392
References 399
9 Interdiffusion in Multicomponent Systems 401
9.1…Interdiffusion and Intrinsic Diffusion Coefficients in Multicomponent Systems 401
9.2…Average Effective and Integrated Diffusion Coefficients in Multicomponent System 413
9.3…A Pseudobinary Approach 419
9.3.1 Estimation of Diffusion Parameters in a Binary System 420
9.3.2 A Pseudobinary Approach in a Ternary System 423
9.3.3 A Pseudobinary Approach in a Multicomponent System 427
9.3.4 Estimation of Diffusion Parameters in Line Compounds Following the Pseudobinary Approach 428
9.4…Estimation of Tracer Diffusion Coefficients in a Ternary System 431
9.5…Determination of Phase Diagram Following Diffusion Couple Technique 434
References 440
10 Short-Circuit Diffusion 443
10.1…Fisher Model of GB Diffusion 445
10.1.1 Approximate Solution of the Fisher Model 448
10.1.2 Exact Solutions of the Fisher Model 452
10.1.3 Comparison of the Solutions of GB Diffusion Problem 456
10.2…Kinetic Regimes of GB Diffusion 457
10.2.1 C Regime of GB Diffusion 460
10.2.2 B Regime of GB Diffusion 461
10.2.3 A Regime of GB Diffusion 462
10.2.4 BC Transition Regime of GB Diffusion 464
10.2.5 AB Transition Regime of GB Diffusion 466
10.3…Determination of the Segregation Factor s 473
10.4…Nonlinear GB Segregation and GB Diffusion 478
10.5…Microstructures with Hierarchy of Short-Circuit Diffusion Paths 481
10.5.1 Kinetic Regimes of GB Diffusion in a Material with a Hierarchic Microstructure 481
10.5.1.1 C--C Regime 486
10.5.1.2 C--B Regime 486
10.5.1.3 B--B Regime 487
10.5.1.4 AB--B Regime 491
10.5.1.5 A--B Regime 492
10.5.1.6 A Regime 493
10.5.2 Temperature Dependence of Interface Diffusion in Material with a Hierarchic Microstructure 494
10.6…Dependence of GB Diffusion on GB Parameters 497
10.7…Effect of Purity on GB Diffusion 498
10.8…Grain Boundary Interdiffusion 499
10.8.1 Coupling of Diffusion and Strain for GB Interdiffusion 499
10.8.2 Kirkendall Effect in GB Interdiffusion 501
10.8.3 Morphology of Growing Phases Affected by GB Diffusion 503
References 504
11 Reactive Phase Formation in Thin Films 506
11.1…Role of Nucleation 506
11.1.1 Activation Energy Delta g* 510
11.1.2 Interfacial Free Energy sigma 510
11.1.3 (Elastic) Strain Energy Delta hd 511
11.1.4 The Chemical Driving Force 511
11.1.5 Nucleation Issues in Solid-State Amorphization 513
11.2…Metastable Structures and Nucleation on Concentration Gradient 516
11.3…Role of the Interfaces 523
11.4…The Role of Grain Boundaries 524
11.5…Role of the Impurities 526
11.6…Phase Formation in Thin-Film Structures 527
11.6.1 Linear-Parabolic Treatment 527
11.6.1.1 Growth of One Phase Between Pure A and B 528
11.6.1.2 Multiphase Growth Between Pure A and B 529
11.6.1.3 ‘‘Pure’’ Diffusional Approach to Multiphase Growth 531
11.6.1.4 Linear-Parabolic Approach to Multiphase Growth 532
11.6.2 Interfacial Reaction Barrier Approach 534
11.6.3 Similarities Between the Growth Models 538
References 539
Author Biography 542