Phase-Field Methods in Materials Science and Engineering E-Book

Nikolas Provatas

• Herausgeber: Wiley-VCH
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 493

Veröffentlichungsjahr: 2011

Beschreibung

This comprehensive and self-contained, one-stop source discusses phase-field methodology in a fundamental way, explaining advanced numerical techniques for solving phase-field and related continuum-field models. It also presents numerical techniques used to simulate various phenomena in a detailed, step-by-step way, such that readers can carry out their own code developments.
Features many examples of how the methods explained can be used in materials science and engineering applications.

Leseprobe

Contents

Preface

1: Introduction

1.1 The Role of Microstructure Materials Science

1.2 Free Boundary Problems and Microstructure Evolution

1.3 Continuum versus Sharp Interface Descriptions

References

2: Mean Field Theory of Phase Transformations

2.1 Simple Lattice Models

2.2 Introduction to Landau Theory

References

3: Spatial Variations and Interfaces

3.1 The Ginzburg–Landau Free Energy Functional

3.2 Equilibrium Interfaces and Surface Tension

Reference

4: Nonequilibrium Dynamics

4.1 Driving Forces and Fluxes

4.2 The Diffusion Equation

4.3 Dynamics of Conserved Order Parameters: Model B

4.4 Dynamics of Nonconserved Order Parameters: Model A

4.5 Generic Features of Models A and B

4.6 Equilibrium Fluctuations of Order Parameters

4.7 Stability and the Formation of Second Phases

4.8 Interface Dynamics of Phase Field Models (Optional)

4.9 Numerical Methods

References

5: Introduction to Phase Field Modeling: Solidification of Pure Materials

5.1 Solid Order Parameters

5.2 Free Energy Functional for Solidification

5.3 Single Order Parameter Theory of Solidification

5.4 Solidification Dynamics

5.5 Sharp and Thin Interface Limits of Phase Field Models

5.6 Case Study: Thin Interface Analysis of Equation 5.30

5.7 Numerical Simulations of Model C

5.8 Properties of Dendritic Solidification in Pure Materials

References

6: Phase Field Modeling of Solidification in Binary Alloys

6.1 Alloys and Phase Diagrams: A Quick Review

6.2 Microstructure Evolution in Alloys

6.3 Phase Field Model of a Binary Alloy

6.4 Equilibrium Properties of Free Energy Functional

6.5 Phase Field Dynamics

6.6 Thin Interface Limits of Alloy Phase Field Models

6.7 Case Study: Analysis of a Dilute Binary Alloy Model

6.8 Numerical Simulations of Dilute Alloy Phase Field Model

6.9 Other Alloy Phase Field Formulations

6.10 Properties of Dendritic Solidification in Binary Alloys

References

7: Multiple Phase Fields and Order Parameters

7.1 Multiorder Parameter Models

7.2 Multiphase Field Models

7.3 Orientational Order Parameter for Polycrystalline Modeling

References

8: Phase Field Crystal Modeling of Pure Materials

8.1 Generic Properties of Periodic Systems

8.2 Periodic Free Energies and the Swift-Hohenberg Equation

8.3 Phase Field Crystal Modeling

8.4 Equilibrium Properties in a One-Mode Approximation

8.5 Elastic Constants of PFC Model

8.6 Multiscale Modeling: Amplitude Expansions (Optional)

References

9: Phase Field Crystal Modeling of Binary Alloys

9.1 A Two-Component PFC Model for Alloys

9.2 Simplification of Binary Model

9.3 PFC Alloy Dynamics

9.4 Applications of the Alloy PFC Model

References

Appendices

Appendix A Thin Interface Limit of a Binary Alloy Phase Field Model

A.1 Phase Field Model

A.2 Curvilinear Coordinate Transformations

A.3 Length and Timescales

A.4 Matching Conditions between Outer and Inner Solutions

A.5 Outer Equations Satisfied by Phase Field Model

A.6 Inner Expansion of Phase Field Equations

A.7 Analysis of Inner Equations and Matching to Outer Fields

A.8 Summary of Results of Sections A.2-A.7

A.9 Elimination of Thin Interface Correction Terms

References

Appendix B Basic Numerical Algorithms for Phase Field Equations

B.1 Explicit Finite Difference Method for Model A

B.2 Explicit Finite Volume Method for Model B

B.3 Stability of Time Marching Schemes

B.4 Semi-Implicit Fourier Space Method

B.5 Finite Element Method

References

Appendix C Miscellaneous Derivations

C.1 Structure Factor: Section 4.6.1

C.2 Transformations from Cartesian to Curvilinear coordinates: Section A.2

C.3 Newton’s Method for Nonlinear Algebraic Equations: Section 6.9.5

Index

Related Titles

Raabe, D., Roters, F., Barlat, F., Chen, L.-Q. (eds.)Continuum Scale Simulation of Engineering MaterialsFundamentals - Microstructures - Process Applications885 pages with 410 figures and 12 tables2004HardcoverISBN: 978-3-527-30760-9

Holzapfel, G. A.Nonlinear Solid MechanicsA Continuum Approach for Engineering470 pages2000SoftcoverISBN: 978-0-471-82319-3

Hubert I. Aaronson (Editor)Lectures on the Theory of Phase Transformations, 2nd Edition294 pagesApril 2010PaperbackISBN: 978-0-87339-476-5

Vitaly V. SlezovKinetics of First Order Phase Transitions429 pagesHardcoverSeptermber 2009ISBN: 978-3-527-40775-0

Linda E. ReichlA Modern Course in Statistical Physics427 pagesPaperbackAugust 2009ISBN: 978-3-527-40782-8

The Authors

Prof. Nikolas ProvatasMcMaster UniversityMat. Science & Engineering1280, Main Street WestCA, Hamilton L8S-4L7USA

Prof. Ken ElderOakland UniversityDepartment of PhysicsMI, Rochester 48309-4487USA

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Cover DesignBackground – 2D adaptine meshTop right – PFC, Model Simulation of a 3D PolycrystalBottom right – Phase Field Simulation of a 3D dendrite, courtesy of Laiszlo Granasy

ISBN: 978-3-527-40747-7

Preface

The idea for this book grew out of a series of workshops that took place at the McMaster University from 2002 to 2005 in which a couple of dozen researchers and students (called the Canadian Network for Computational Materials Science or CNCMS) were invited to discuss their research and their visions for the future of computational materials science. One serious concern that surfaced from the discussions and the proceedings regarded the gaping hole that existed in the standard pedagogical literature for teaching students – and professors – about computational and theoretical methods in phase field modeling. Indeed, unlike many other fields of applied physics and theoretical materials science, there is a dearth of easy-to-read books on phase field modeling that would allow students to come up to speed with the details of this topic in a short period of time. After sitting on the fence for a while, we decided to add our contribution by writing an introductory text about phase field modeling.

The aim of this book is to provide a graduate-level introduction of phase field modeling for students of materials science who wish to delve deeper into the underlying physics of the theory. The book begins with the basic principles of condensed matter physics to motivate and develop the phase field method. This methodology is then used to model various classes of nonequilibrium phase transformations that serve as paradigms of microstructure development in materials science phenomena. The workings of the various phase field models studied are presented in sufficient detail for students to be able to follow the reasoning and reproduce all calculations. The book also includes some basic numerical algorithms – accompanied by corresponding Fortran codes that come on a CD with this book – that students can use as templates with which to practice and develop their own phase field codes. A basic undergraduate-level knowledge of statistical thermodynamics and phase transformations is assumed throughout this book. Most long-winded mathematical derivations and numerical details that can serve as references but would otherwise detract from the flow of the main theme of the text are relegated to appendices.

It should be specified at the outset that this book is not intended to be an exhaustive survey of all the work conducted throughout the years with phase field modeling. There are plenty of reviews that cover this angle and many of these works are cited in this book. Instead, we focus on what we feel is missing from much of the literature: a fast track to understanding some of the “dirty” details of deriving and analyzing various phase field models, and their numerical implementation. That is precisely what we have observed new students wishing to study phase field modeling are starving for as they get started in their research. As such, this book is intended to be a kind of “phase field modeling for dummies,” and so while the number of topics is limited, as many of the details as possible are provided for those topics that are covered.

The broad organization of the material in following chapters is as follows. The first half of the book begins by establishing the basic phase field phenomenology, from its basic origins in mean field theory of phase transformations to its basic form now in common use as the base of most modern phase field models in computational materials science and engineering. Phase field theory is applied to several examples, with a special emphasis placed on the paradigms of solidification and solid-state transformations. An appendix is also dedicated to the important issue of mapping the phase field model onto specific sharp interface limits. The last two chapters of this book deal with the development of more complex class of phase field models coined “phase field crystal” models. These are an extension of the original phase field formalism that makes it possible to incorporate elastic and plastic effects along side the usual kinetics that governs phase transformations. We will see that these models constitute a hybrid between traditional phase field theory and atomistic dynamics. After motivating the derivation of phase field crystal models from classical density functional theory, these models are then applied to various types of phase transformation phenomena that inherently involve elastic and plastic effects. It is noted that some sections of the book are marked as “Optional.” These are sections that can be skipped at first reading without losing the main flow of the text and without detracting from the minimum path of topics comprising the basic principles of phase field theory.

Writing this book involved the valued help of many people. We would like to thank all the graduate students in the Department of Materials Science and Engineering at the McMaster University who took MATLS 791 in the Fall of 2009. Their help and advice in editing and proofing the first edition of this book is greatly appreciated. I (NP) would like to thank my wife Photini and sons Spiro and Aristotle for their love and patience during the writing of this book; doing science for a living is fun but their love is what living is actually about. I also suppose thanks are in order to Starbucks Coffee for providing me – at the cost of lots of overpriced bitter coffee – many hours of escape from the mundane administrative environment of a modern university in order that I could work on this book in peace and talk politics with other patrons. I would also like to thank the Technical Research Centre of Finland (VTT) and the Helsinki University of Technology for hosting me during my sabbatical leave in 2009 and for flipping the bill for some of my travels to Helsinki where I also worked on this manuscript and other cool stuff.

As with anything in print, this book very likely contains typos and oversights. We would be delighted to hear from readers about any such errors or omissions. Please do not hesitate to contact us at [email protected] or [email protected].

I (KE) would like to thank my wife Nancy, daughter Kate and parents Fay and Stan for the tremendous support they have given me over many years and throughout the writing of this text. In addition I would like to thank Tapio Ala-Nissila and the Helsinki University of Technology (now Aalto University) for providing me the opportunity to give several short courses on phase field and phase field crystal modelling. Some of the material developed for those courses has found its way into the text.

Nikolas ProvatasKen Elder

2

Mean Field Theory of Phase Transformations

The origins of the phase field methodology – the focus of this book – have been considerably influenced by mean field theory of first- and second-order phase transformations. It is thus instructive to begin first with a discussion of some simple phase transformations and their description via mean field theory. Using this as a framework will better allow the concept of an order parameter to be defined and generalized to include spatial variations. This will thus set the stage for the later development of phase field models of solidification and solid-state transformation phenomenon. Before proceeding, the reader should have a basic background of statistical thermodynamics. For a quick review, the reader may refer to Refs [1–3].

Common first-order transformations include solidification of liquids and condensation of vapor. They are defined by a release of latent heat and discontinuous first derivative of the free energy. Moreover, just below a first-order transformation, nucleation of the metastable phase is required to initiate the transformation. Finally, in first-order transformations, two phases can typically coexist over a wide range of temperatures, densities (pure materials), or impurity concentrations (alloys). In contrast, second-order transformations occur at well-defined temperature, density, or concentration. There is no release of latent heat and the transformation begins spontaneously due to thermal fluctuations. A paradigm example is phase separation of a binary mixture or spinodal decomposition in metal alloys. Another is the spontaneous ferromagnetic magnetization of iron below its Currie temperature.

An important concept used again and again in the description of phase transformations is that of the order parameter. This is a quantity that parameterizes the change of symmetry from the parent (disordered) phase to the daughter (ordered) phase appearing after the transformation. For example, a crystal phase has fewer rotational and translational symmetries compared to a liquid. The order parameter typically takes on a finite value in the ordered state and vanishes in the disordered state. First-and second-order phase transitions are distinguished by the way the order parameter appears below the transition temperature. In a first-order transformation, the order parameter of the ordered state emerges discontinuously from that of the disordered phase, below the transformation temperature. In the second-order transformation, the disordered state gives way continuously to two ordered phases with nonzero order parameter. Another example of a change of symmetry characterized by changes in the order parameter includes the average magnetization. For some phase changes, such as vapor → vapor + liquid, there is no change in the structural symmetry groups of the parent and daughter phases. In such case, effective order parameters can often be defined in terms of density differences relative to the parent phase.