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Modeling Solid-State Precipitation

Book Description

Over recent decades, modeling and simulation of solid-state precipitation has attracted increased attention in academia and industry due to their important contributions in designing properties of advanced structural materials and in increasing productivity and decreasing costs for expensive alloying. In particular, precipitation of second phases is an important means for controlling the mechanical-technological properties of structural materials. However, profound physical modeling of precipitation is not a trivial task. This book introduces you to the classical methods of precipitation modeling and to recently-developed advanced, computationally-efficient techniques. If you’re a research professional, academic, or student, you’ll learn: • nucleation theory, precipitate growth, calculation of interfacial energies. • advanced techniques for technologically relevant multicomponent systems and complex thermo-mechanical treatments. • numerical approaches using evolution equations and discrete particle size distributions. • the major software products available for precipitation kinetics simulations.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright Page
  5. Contents
  6. List of Symbols
  7. List of Figures
  8. List of Tables
  9. Preface
  10. 1 Thermodynamic Basis of Phase Transformations
    1. 1.1 The Gibbs Energy
    2. 1.2 Molar Gibbs Energy and Chemical Potentials
    3. 1.3 Solution Thermodynamics
      1. 1.3.1 Mechanical Mixture and Ideal Solution
      2. 1.3.2 The Regular Solution
      3. 1.3.3 General Solutions—The CALPHAD Approach
    4. 1.4 Multiphase Systems and Driving Force for Precipitation
    5. 1.5 Curvature and Elastic Stress
      1. 1.5.1 The Gibbs–Thomson Equation
      2. 1.5.2 Elastic Misfit Stress
    6. 1.6 Equilibrium Structural Vacancies
  11. 2 Precipitate Nucleation
    1. 2.1 Paving the Way for Nucleation Theory
    2. 2.2 Nucleation of Liquid Droplets from Supersaturated Vapor
      1. 2.2.1 Thermodynamics of the Critical Nucleus
      2. 2.2.2 Overcoming the Nucleation Barrier
      3. 2.2.3 The Kinetics of Droplet Formation
      4. 2.2.4 The Zeldovich Factor
      5. 2.2.5 The Time Lag
      6. 2.2.6 Note on Thermodynamic Properties of Small Clusters
    3. 2.3 Solid-State Nucleation
      1. 2.3.1 The Precipitate–Matrix Interface
      2. 2.3.2 Free Energy of Nucleus Formation
      3. 2.3.3 Steady-State Nucleation Rate in Crystalline Solids
      4. 2.3.4 Time-Dependent Nucleation
      5. 2.3.5 The Volume Misfit Stress
      6. 2.3.6 Excess Structural Vacancies
    4. 2.4 Heterogeneous Nucleation
      1. 2.4.1 Heterogeneous Nucleation Sites
      2. 2.4.2 Potential Nucleation Sites in a Heterogeneous Microstructure
      3. 2.4.3 Nucleation Site Saturation
      4. 2.4.4 Effective Interfacial Energies in Heterogeneous Nucleation
      5. 2.4.5 Grain Boundary Energy
    5. 2.5 Nucleation in Multicomponent Environment
      1. 2.5.1 CNT in Multicomponent Environment
      2. 2.5.2 The Composition of the Critical Nucleus
    6. 2.6 Summary
  12. 3 Diffusion-Controlled Precipitate Growth and Coarsening
    1. 3.1 Problem Formulation
    2. 3.2 Diffusion-Controlled Growth with Local Thermodynamic Equilibrium
      1. 3.2.1 Local Equilibrium and Composition Profiles
      2. 3.2.2 Binary Diffusion-Controlled Growth—the Zener Model
      3. 3.2.3 The Quasi-Stationary Solution for Spherical Precipitates
      4. 3.2.4 Analytical Solution for High and Low Dimensionless Supersaturation
      5. 3.2.5 Influence of Capillarity on Precipitate Growth
    3. 3.3 Multicomponent Diffusion-Controlled Growth
      1. 3.3.1 The Multicomponent Local Equilibrium Tie-Lines
      2. 3.3.2 Fast and Slow Local Equilibrium Transformation Regions
      3. 3.3.3 Local Equilibrium Controlled Precipitation in Multicomponent Systems
      4. 3.3.4 Approximate Treatment of Multinary Diffusional Transformations
    4. 3.4 Energy Dissipation at a Moving Phase Boundary—the Mixed-Mode Model
    5. 3.5 Mean-Field Evolution Equations for Precipitate Growth
      1. 3.5.1 The Thermodynamic Extremal Principle
      2. 3.5.2 Mean-Field Evolution Equations for Substitutional/Interstital Phases
      3. 3.5.3 Evolution Equations for General Sublattice Phases
      4. 3.5.4 Comparison with Local Equilibrium Based Growth Models
    6. 3.6 Precipitate Coarsening
      1. 3.6.1 The LSW-Theory of Precipitate Coarsening
      2. 3.6.2 Extensions of LSW Theory for Finite Phase Fraction Effects
        1. 3.6.2.1 The Modified LSW Theory of Ardell
        2. 3.6.2.2 The Brailsford and Wynblatt Theory
        3. 3.6.2.3 The Davies, Nash, and Stevens (LSEM) Theory
        4. 3.6.2.4 The Tsumuraya and Miyata Theory
        5. 3.6.2.5 The Marqusee and Ross Theory
        6. 3.6.2.6 The Tokuyama and Kawasaki Theory
        7. 3.6.2.7 The Voorhees and Glicksman Theory
        8. 3.6.2.8 The Enomoto, Tokuyama, and Kawasaki Theory
        9. 3.6.2.9 The Marder Theory
      3. 3.6.3 Comparison of Theories
      4. 3.6.4 Coarsening in Multicomponent Alloys
    7. 3.7 Summary
  13. 4 Interfacial Energy
    1. 4.1 The Nearest-Neighbor Broken-Bond Model
    2. 4.2 Composition Dependence of the Precipitate–Matrix Interfacial Energy
    3. 4.3 Generalization of the NNBB Approach—The GBB Model
      1. 4.3.1 Effective Bond Energies and Broken Bonds
      2. 4.3.2 Comparison Between Theory and Experiment
    4. 4.4 Interface Energy Correction for Small Precipitates
      1. 4.4.1 The Interface Energy Size Correction Function
      2. 4.4.2 Comparison with Size Correction in Vapor-Droplet Systems
    5. 4.5 Energy of Diffuse Interfaces
      1. 4.5.1 Free Energy of a Diffuse Interface
      2. 4.5.2 Regular Solution Approximation for Diffuse Interfaces
      3. 4.5.3 Comparison with Other Models
    6. 4.6 Summary
  14. 5 Numerical Modeling of Precipitation
    1. 5.1 Kolmogorov–Johnson–Mehl–Avrami (KJMA) model
      1. 5.1.1 Derivation of the KJMA Equation
      2. 5.1.2 Analysis of KJMA Parameters
      3. 5.1.3 Multiphase KJMA Kinetics
    2. 5.2 Langer–Schwartz Model
      1. 5.2.1 The original LS Model
      2. 5.2.2 Modified Langer–Schwartz Model
    3. 5.3 Kampmann–Wagner numerical Model
    4. 5.4 General Course of a Phase Decomposition
      1. 5.4.1 Heat Treatments for Precipitation
      2. 5.4.2 Stages in Precipitate Life
      3. 5.4.3 Evolution of Precipitation Parameters
      4. 5.4.4 Overlap of Nucleation, Growth, and Coarsening
    5. 5.5 Summary
  15. 6 Heterogeneous Precipitation
    1. 6.1 Precipitation at Grain Boundaries
      1. 6.1.1 Problem Formulation
      2. 6.1.2 Diffusive Processes
      3. 6.1.3 Evolution Equations for Precipitate Growth
      4. 6.1.4 Evolution Equations for Precipitate Coarsening
      5. 6.1.5 Growth Kinetics of Equisized Precipitates
      6. 6.1.6 Growth Kinetics of Nonequisized Precipitates
      7. 6.1.7 Coarsening Kinetics
    2. 6.2 Anisotropy and Precipitate Shape
      1. 6.2.1 Shape Parameter, h, and SFFK Evolution Equations
      2. 6.2.2 Determination of Shape Factors
      3. 6.2.3 Comparing Growth Kinetics
    3. 6.3 Particle Coalescence
      1. 6.3.1 Diffusion Kinetics of Clusters
      2. 6.3.2 Evolution of Precipitation Systems by Coalescence
      3. 6.3.3 Simultaneous Adsorption/Evaporation and Coalescence
      4. 6.3.4 Phenomenological Treatment of Particle Coalescence
      5. 6.3.5 Comparison with Experiment
    4. 6.4 Simultaneous Precipitation and Diffusion
      1. 6.4.1 Numerical Treatment in the Local-Equilibrium Limit
      2. 6.4.2 Comparison of Local-Equilibrium Simulations with Experiment
      3. 6.4.3 Coupled Diffusion and Precipitation Kinetics
  16. 7 Diffusion
    1. 7.1 Mechanisms of Diffusion
      1. 7.1.1 Diffusion in Crystalline Materials
      2. 7.1.2 The Principle of Microscopic Time Reversal
      3. 7.1.3 Random Walk Treatment of Diffusion
      4. 7.1.4 The Einstein–Smoluchowski Equation
    2. 7.2 Macroscopic Models of Diffusion
      1. 7.2.1 Phenomenological Laws of Diffusion
      2. 7.2.2 Special Solutions of Fick’s Second Law
        1. 7.2.2.1 Spreading of a Diffusant From a Point Source
        2. 7.2.2.2 Diffusion into a Semi-Infinite Sample
      3. 7.2.3 Numerical Solution
      4. 7.2.4 Diffusion Forces and Atomic Mobility
      5. 7.2.5 Multicomponent Diffusion
    3. 7.3 Activation Energy for Diffusion
      1. 7.3.1 Temperature Dependence of the Diffusion Coefficient
      2. 7.3.2 Diffusion Along Dislocations and Grain Boundaries
    4. 7.4 Excess Structural Vacancies
      1. 7.4.1 Vacancy Generation and Annihilation
      2. 7.4.2 Modeling Excess Vacancy Evolution
        1. 7.4.2.1 Annihilation at Dislocation Jogs
        2. 7.4.2.2 Annihilation at Frank Loops
        3. 7.4.2.3 Annihilation at Grain Boundaries
      3. 7.4.3 Vacancy Evolution in Polycrystalline Microstructure
    5. 7.5 Summary
  17. 8 Design of Simulation
    1. 8.1 General considerations
    2. 8.2 How to design and interpret a solid-state precipitation simulation
  18. 9 Software for Precipitation Kinetics Simulation
    1. 9.1 DICTRA—Diffusion-controlled transformation
      1. 9.1.1 General information
      2. 9.1.2 Basic Concepts
        1. 9.1.2.1 Sharp Interface
        2. 9.1.2.2 Local Equilibrium
        3. 9.1.2.3 Diffusion
        4. 9.1.2.4 Microstructure
        5. 9.1.2.5 Nucleation and Surface Energy
      3. 9.1.3 DICTRA Precipitation Simulation
        1. 9.1.3.1 Interactive formulation of a problem in DICTRA
        2. 9.1.3.2 Results of the simulation
      4. 9.1.4 Further Modules
        1. 9.1.4.1 Para-equilibrium Model
        2. 9.1.4.2 Pearlite Module
    2. 9.2 PrecipiCalc—Software for 3D multiphase precipitation evolution
      1. 9.2.1 General Information
      2. 9.2.2 Software Implementation
      3. 9.2.3 Example of PrecipiCalc Simulations
      4. 9.2.4 Summary
    3. 9.3 MatCalc—The Materials Calculator
      1. 9.3.1 General Information
      2. 9.3.2 The Kinetic Model
      3. 9.3.3 MatCalc Precipitation Simulation in the GUI Version
      4. 9.3.4 MatCalc Precipitation Simulation Using Scripting
      5. 9.3.5 Using MatCalc with External Software
      6. 9.3.6 Software-relevant Literature and Web Sources
        1. 9.3.6.1 Modeling
        2. 9.3.6.2 Application
        3. 9.3.6.3 Examples
    4. 9.4 PanPrecipitation—An Integrated Computational Tool for Precipitation Simulation of MultiComponent Alloys
      1. 9.4.1 Introduction
      2. 9.4.2 Kinetic Models
      3. 9.4.3 Software Design and Data Structure
      4. 9.4.4 Examples
        1. 9.4.4.1 Example 1: Precipitation behavior of a Model Ni- 14 at% Al alloy
        2. 9.4.4.2 Example 2: Coarsening of Rene88DT
        3. 9.4.4.3 Example 3: Precipitation hardening behavior of Al-Mg-Si alloys
      5. 9.4.5 Discussion
    5. 9.5 TC-Prisma
      1. 9.5.1 General Information
      2. 9.5.2 Kinetic Model
      3. 9.5.3 Performing TC-Prisma Simulations “From Scratch”
        1. 9.5.3.1 Define System
        2. 9.5.3.2 Define Simulation Conditions
        3. 9.5.3.3 Start Simulation
        4. 9.5.3.4 Plot Results
      4. 9.5.4 Performing Simulations Using Scripts
    6. 9.6 Comparison of Software Codes
  19. Appendix
  20. References
  21. Index
  22. Last Page
  23. Backcover