Diffusion, segregation and solid-state phase transformations, livre ebook

This book contains a set of solved problems on the thermodynamics of phase transformations, diffusion and kinetics in alloys. These problems, preceded by a reminder of the course, are completed by answers detailing each stage of the calculations. The topics covered in the book (precipitation, ordering, segregation on crystalline defects, growth of a thin film, oxidation, creep, doping of semiconductors, etc.) are essential for the design of materials and the optimisation of heat treatments and properties, whether for structural materials (steels, Ni, Al, Cu, Ti-based alloys, etc.) or functional materials (semiconductors, alloys and magnetic multilayers, etc.). They often lead to an industrial problem (aeronautics, nuclear power plants, microelectronics, etc.). This book is intended for teachers and students of bachelor’s and master’s degrees, but also for engineering students, PhD students and researchers.

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III

CHAPTER 1

Thermodynamics of Equilibria and Phase Diagrams . . . . . . . . . . . . . . . . . . . 1

1.1 Course Reminders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Precipitation and Chemical Ordering . . . . . . . . . . . . . . . . . . . . 1

1.1.2 Thermodynamic Functions and Potentials . . . . . . . . . . . . . . . . 9

1.1.3 Binary Equilibria: Ideal and Regular Solid Solutions. . . . . . . . . 13

1.1.4 The Critical Transition Theory and the Landau Theory . . . . . . 19

1.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.2.1 Equilibrium Vacancy Concentration . . . . . . . . . . . . . . . . . . . . . 23

1.2.2 Intergranular Segregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.2.3 Phase Diagrams and Regular Solution . . . . . . . . . . . . . . . . . . . 32

1.2.4 Order–Disorder Transformations and Bragg–Williams Theory . 40

1.2.5 Ordering in FCCs and Landau’s Theory . . . . . . . . . . . . . . . . . . 49

1.2.6 Ordering in BCCs and Landau’s Theory. . . . . . . . . . . . . . . . . . 55

CHAPTER 2

Diffusion and Transport in Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.1 Course Reminders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.1.1 Diffusion at the Atomic Scale . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.1.2 Fick’s Two Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2.1.3 Hetero-Diffusion, Kirkendall Effect . . . . . . . . . . . . . . . . . . . . . . 72

2.1.4 Diffusion Short Circuits: Influence of Defects . . . . . . . . . . . . . . 74

2.1.5 Driven Diffusion in a Force Field . . . . . . . . . . . . . . . . . . . . . . . 75

2.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

2.2.1 Fundamental Mechanisms of Diffusion – Einstein’s Equation . . 78

2.2.2 Elementary Mechanisms and Self-Diffusion . . . . . . . . . . . . . . . . 82

2.2.3 Diffusion in CuAl Alloys. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

2.2.4 Surface Hardening by Cementation . . . . . . . . . . . . . . . . . . . . . 89

2.2.5 Diffusion in Thin Sandwich Films . . . . . . . . . . . . . . . . . . . . . . . 94

2.2.6 Kirkendall Effect and Diffusion in CuZn . . . . . . . . . . . . . . . . . . 100

2.2.7 Short-Circuits of Diffusion: Grain Boundaries and Dislocations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

2.2.8 Driven Diffusion in a Force Field and Spinodal Decomposition . 108

2.2.9 A New Fick Law According to Howe . . . . . . . . . . . . . . . . . . . . 115

2.2.10 Driven Diffusion in a Force Field: Application to Nabarro-Herring Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

2.2.11 Driven Diffusion of Impurities in a Force Field, the Formation of Cottrell Atmospheres . . . . . . . . . . . . . . . . . . 126

2.2.12 Diffusion in Semiconductors: The Generation of an Internal Nernst Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

2.2.13 Driven Diffusion in a Force Field: Oxidation of Nickel. . . . . . . 139

CHAPTER 3

Kinetics of Formation of a New Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

3.1 Course Reminder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

3.1.1 Continuous and Discontinuous Precipitation. . . . . . . . . . . . . . . 147

3.1.2 Instability and Metastability . . . . . . . . . . . . . . . . . . . . . . . . . . 150

3.1.3 Homogeneous Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

3.1.4 Heterogeneous Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

3.1.5 Nucleation of Metastable Phases . . . . . . . . . . . . . . . . . . . . . . . 164

3.1.6 Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

3.1.7 The Spinodal Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 168

3.1.8 Coarsening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

3.1.9 Overall Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

3.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

3.2.1 Eutectoid Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

3.2.2 Nucleation of Precipitates in Ni-Al Alloys . . . . . . . . . . . . . . . . 185

3.2.3 Nucleation and Growth of Precipitates in Dilute Cu-Co Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

3.2.4 Heterogeneous Nucleation of Precipitates in AlCu . . . . . . . . . . 195

3.2.5 Deal and Grove’s Growth Law of a Spherical Precipitate . . . . . 200

3.2.6 Growth of an Oxide Film on Surface, Deal and Grove’s law . . . 205

3.2.7 Growth and Coarsening of Spherical Precipitates . . . . . . . . . . . 211

3.2.8 Coarsening in NiAl Superalloys . . . . . . . . . . . . . . . . . . . . . . . . 220

3.2.9 Zener’s Model for the Growth of a Precipitate with Incoherent Planar Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

3.2.10 The Spinodal Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 230

3.2.11 Zeldovich’s Kinetic Theory of Nucleation and Incubation Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

3.2.12 Capture Coefficient, Nucleation Flux and Incubation Time . . . 244

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257