Computational methods for a phase field model of grain growth kinetics

47 5.4 The average grain area and total time taken as function of time for FDM explicit.. 49 5.6 The average grain area and total time taken as function of time for FDM explicit... 535.9

COMPUTATIONAL METHODS FOR A

PHASE-FIELD MODEL OF GRAIN GROWTH

KINETICS

BIPIN KUMAR

(M.Sc., IIT Kanpur, India)

A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE

DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE

I warmly express heartfelt gratitude and sincere appreciations to my supervisor Dr.Lin Ping for his guidance full with patient, stimulating ideas and invaluable advicethroughout my study period in NUS Without his supervisions, even the very firststep was not possible It has been my pleasure being his research student

I would like to express my sincere thanks to Dr B.S.V Patnaik for all his help andguidance

Special thanks are very much due to Dr Shashi Bhushan (DNV), Venkateswarlu,Zacharry Harrish, Dr Ram Singh Rana (A*Star) and Dr Sanjiv Yadav (Chem-istry) On my mind, the proximity and moral support, I had from them, has putindelible print of my memorable stay in Singapore

I would next, like to thank to my officemates Jinghui, David, Chen Yidi, Wu Lei,and Xu Ying for their assistance Talks held with them will always remain in mythoughts Discussion held with Shapeev Alexander are duly acknowledged I wouldlike to express my sincere feelings to my house mates Yogesh, Raju, Mohan, Vedand my friends Sunil, Kaushal Pandey, Jan Frode Stene, Sulakshana, Khaing Ma-woo Toee and Mrs Desai

Finally, I would like to thank my parents and siblings Without their continuousencouragement and support, nothing would have been possible for me

i

iiLast but the most important feelings for Ms Preeti, who has accepted to knot with

me forever

I feel lack of words to acknowledge Him who has provided me with unlimited tential but with limited capabilities I bow my head to Him!!

1.1 Motivation and Objectives: 2

1.1.1 Motivation: 2

1.1.2 Objectives: 3

1.2 Organization of this thesis: 3

2 Material Science and Simulations 5 2.1 What are Materials? 5

2.2 Role of Computational Materials Science: 6

2.3 Relevance of Simulating Microstructural Evolutions: 7

2.4 Phase Field Model: 9

2.4.1 What is an Interface? 10

2.4.2 Sharp Interface: 10

2.4.3 Diffuse Interface: 11

2.5 Application of Phase Field Methods: 12

3 Theory and Model Description 14 3.1 Local Free Energy: 15

iii

CONTENTS iv

3.2 Need for advanced numerical approaches: 19

4 Computational Algorithms 21 4.1 Finite Difference Approach (Explicit): 22

4.2 Finite Difference Approach (Semi Implicit): 24

4.3 Operator Splitting Method: 26

4.3.1 Exact Solution of 3rd Degree Polynomial Equation: 27

4.3.2 Discussion of Solution: 27

4.4 Active Parameter Tracking Algorithm: 31

4.5 Finite Element Approach: 33

4.6 On the Implementation of Periodic Boundary Conditions: 38

4.7 On Solving Large-Scale System of Equations: 39

4.7.1 Format for Sparse Matrix: 41

4.7.2 Sparse Matrix-Vector Multiplication in CSR Format: 42

5 Results and Discussion 43 5.1 Finite Difference Method (Explicit): 43

5.2 FD with Operator Splitting Method: 48

5.2.1 Operator Splitting: Case 1 50

5.2.2 Operator Splitting: Case 2 52

5.2.3 Operator Splitting: Case 3 52

5.3 AIA-PT Method: 54

5.4 Result Comparison and Discussion: 54

5.4.1 Comparison of Result with other Researcher’s Results: 56

CONTENTS v

6.1 Conclusions: 656.2 Suggestions for Future Work: 66

Computational Materials science is fast catching up as an attractive and mentary approach to designing novel materials An apriori understanding of microstructures and its linkage to material properties is possible through the use of work-able models In this thesis, we explore a variety of computational algorithms forsolving partial differential equations that govern the kinetics of grain growth Tostart with, we employ the emerging phase-field approach to model the micro struc-tural evolutions

comple-To start with, we have to design the appropriate energy functional Then, solve theAllen-Cahn equations or the Complex Ginzhburg Landau equations (CGLE).Following computational schemes are devised for the phase-field equations:

(i) Solve the equations by Finite difference method with a simple explicit timemarching scheme (This is the most popular approach)

(ii) Solve the Allen-Cahn equations using operator splitting method, where theequation is divided into a Poisson part and a cubic equation (Note that, the cubicequation has an exact solution)

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CONTENTS vii

(iii) A simple observation that only the interfaces have more than one activephase-field variables at every grid point was successfully exploited by devising anactive and implicitly adaptive parameter tracking (AIA-PT) approach

(iv) Solve the equations by Combing the two ideas of operator splitting andAIA-PT

List of Figures

2.1 Demonstration of multi-scale modeling for the design of radiation

resistant material : Ghoniem et al [26] 8

2.2 Typical image of grains [27] 9

3.1 Representation of Orientation field variable φ 14

4.1 Discretization of Laplacian 23

4.2 CSR format of a sparse matrix 42

5.1 Temporal evolution of microstructure for FDM(explicit) for Q=48 45

5.2 Temporal evolution of microstructure for FDM(explicit) for Q=60 46

5.3 Temporal evolution of microstructure for FDM (explicit) scheme for Q=36 47

5.4 The average grain area and total time taken as function of time for FDM (explicit) 48

5.5 Temporal evolution of microstructure for FDM (explicit) scheme for Q=48 49

5.6 The average grain area and total time taken as function of time for FDM (explicit) 50

viii

LIST OF FIGURES ix

5.7 Temporal evolution of microstructure for FDM (explicit) scheme forQ=48 515.8 Temporal evolution of microstructure for operator splitting case 1 forQ=36 535.9 The average grain area and total time taken as function of time foroperator splitting case 1 545.10 Temporal evolution of microstructure for operator splitting case 1 forQ=48 555.11 The average grain area and total time taken as function of time forFDM (explicit) 565.12 Temporal evolution of microstructure for operator splitting case 2 forQ=36 575.13 The average grain area and total time taken as function of time foroperator splitting case 2 585.14 Temporal evolution of microstructure for operator splitting case 3 forQ=36 595.15 The average grain area and total time taken as function of time foroperator splitting case 3 605.16 The average grain area and total time taken as function of time forAIA-PT method 605.17 Comparison of total time taken by FDM (Explicit),different operatorsplitting method and AIA-PI method for 10000 time steps 615.18 Comparison of average grain area as function of time for FDM (ex-plicit) case and operator splitting method case 1 62

LIST OF FIGURES x

5.19 Comparison of average grain area as function of time for FDM plicit) case and operator splitting method case 2 635.20 Comparison of average grain area as function of time for FDM (ex-plicit) case and operator splitting method case 3 635.21 Comparison of average grain area as function of time for FDM (ex-plicit) case and all three operator splitting method 64

(ex-Chapter 1

Introduction

Materials as a whole can be classified as either amorphous (glossy) or crystalline.For the latter the atoms are situated in a repeating (or) periodic array over largedistances, meaning a long range order exists If all such unit cells interlock inthe same way and have the same orientation, it is called a single crystal Singlecrystals exist in nature, but very difficult to grow (Example: Diamond) For agiven piece of material, if we find a collection of many small crystals or grains, it iscalled polycrystalline material, whose crystallographic orientation varies from grain

to grain Modeling such materials with a view to understand their properties isstill a great challenge Phase-field models developed in mid 90’s is a fertile groundfor both Mathematicians and Physicists alike In this thesis an attempt is made tomodel the microstructural evolution and kinetics of such materials

1

CHAPTER 1 INTRODUCTION 2

There is an extensive literature on the use of phase-field models These modelshave continuously varying field variables ψ(x) that lend themselves well to PDE’s.These models are often used as a way to approach difficult problems in interfacialdynamics such as phase-separation or diffusion limited aggregation For instance,Cahn-Hilliard equation describes phase separation from a uniform binary mixedstate to that of spatially separated multi-phase structure

1.1.1 Motivation:

The simulation and modeling route enables Mathematicians to contribute to theunderstanding of material processes and to achieve desirable properties and gaininsights Several unanswered, non-trivial questions exist in the field of material me-chanics, which are amenable to Mathematical modeling, albeit with simplifications.Even an iota (ǫ or δ ) progress would greatly aid the material technologists and theworld at large in our quest towards discovering novel materials Grain growth isone such process, whose kinetics can be described by suitable Mathematical mod-els Rich amount of literature exists on these coarsening processes, which is a goodground for probabilistic and/or deterministic models such as Monte Carlo and cel-lular automaton However, from Mathematical point of view, employing a set ofpartial differential equations is highly desirable For polycrystalline settings, where

a large number of orientations are likely, the governing coupled partial differentialequations at every grid point could lead to simulations that are highly memory andCPU intensive Assuming N ×N grid points in 2-D settings and Q order parameters(or) phase-field variables, in the system, at least N ×N×Q coupled equations need

phase-(i) Is it possible to employ efficient, stable numerical algorithms, that are rior and less CPU and memory intensive.

supe-(ii) Is it possible to compare the performance of existing algorithms vis-a-vis theproposed methods

(iii) Is it possible to view the kinetics of grain growth merely as evolution ofinterfaces and drastically reduce the number of phase-field variables to be solved atevery grid point

This thesis is organized as follows Chapter 2 starts with a rudimentary discussion ofwhat constitutes a material, the issues of interest in computational materials scienceand the relevance of the present simulations etc In chapter 3 we briefly introducethe theory behind the phase-field models of interest to Grain Growth kinetics andthe formation of mathematical models A repository of numerical methods which

Chapter 2

Material Science and Simulations

The Mathematical models that help us in understanding materials have far ing ramification in design and development of novel materials Materials not onlydirectly impact us, but advanced materials play a crucial, and enabling role under-lying virtually all technologies A simple and operative definition of a material is toview it as a particular arrangement of atoms In materials design and control thereare three main concerns

reach-(i) the atoms to be arranged,

(ii) kind of arrangement,

(iii) how to arrange these atoms

The last among the three, is more of a processing question and can be understood

by investigating the microstructural evolution at various scales A variety of toolsused at the human scale (say, macro scale), depend on materials for which some

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CHAPTER 2 MATERIAL SCIENCE AND SIMULATIONS 6

other length and time scales are critical All these scales can be broadly classified

as - micro, meso and macro

Mesoscale is in between the scales of engineering and atomistic science [1] This

is a convenient scale in materials design, since the material properties and behavior

is dominated by these mesoscale structures Simulations at these scales enable us toinvestigate the temporal evolutions of grain microstructures The various materialproperties (e.g mechanical properties) are thoroughly influenced by the grain topol-ogy and its evolution Such internal topological evolutions have been continuouslypossible mainly due to the following facts : (i) different lattices (ii) recrystallization(iii) transition of phases Furthermore, such evolutions are influenced by the con-nectivity of grains, which is a key microstructural feature All these different ways

of looking at the materials enable the scientists and engineers, to synthesize andprocess the materials, and control material structures at the molecular, nano, meso,and macro scales [2]

Simulations through Mathematical modeling have played a constructive role in thedevelopment of new materials, in enhancing our fundamental understanding of ma-terial behavior Indeed, computational Materials science is an exciting synergybetween materials and computing power, each feeding back into the developments

of the other in a tightly coupled fashion For instance the CPU time is expanding

at an exponential rate (thanks to Moor’s law), which is only possible by delvingdeep into the world of materials and technology The time scales and length scales

at which one can probe through simulation is getting more and more realistic and

CHAPTER 2 MATERIAL SCIENCE AND SIMULATIONS 7

close to predictable

The advances in understanding material microstructures can bring valuable sights in the field of material science [3] Thus, simulation and modeling of materialprocesses through micro, meso and macroscopic modeling is gaining fast acceptance

in-as a cost effective and complementary tool to experimentation However, achievingrelevant time scales and bridging the gap among different scales is a real chal-lenge and is in the realm of multiscale materials modeling The following simulationmodes are prerequisite in building capability for the reliable and accurate prediction

of phenomena and properties in a wide range of materials [1]

• discovering novel relations and paradigms for complex behavior,

• benchmarking test forms of the mathematical models,

• direct calculation of parameter input for mathematical models,

• validation of models by comparison the result with experiments, and

• predicting the complexities involved in materials processes and phenomenon.Combining these five modes, simulation becomes a powerful, revolutionary tool toaccelerate conceptual advances

Evo-lutions:

All engineering materials contain certain type of microstructure, virtually in everymaterial processing phase The microstructural evolutions are common in manyfields including biology, hydrodynamics, chemical reactions, and phase transforma-tions

CHAPTER 2 MATERIAL SCIENCE AND SIMULATIONS 8

Figure 2.1: Demonstration of multi-scale modeling for the design of radiation tant material : Ghoniem et al [26]

resis-Microstructure is a broader term, which refers to spatial distribution of structuralfeatures which can be phases of different compositions and/or crystal structures, orgrains of different orientations, domains of different structural variants, domains ofdifferent electrical and magnetic polarization, and structural defects The lengthscales of these structures is typically in the range of nanometers to a few tens ofmicrons Microstructural evolution takes place to reduce the total free energy thatmay include the bulk chemical free energy, interfacial energy, elastic strain energy,magnetic energy, electrostatic energy, and/or under applied external fields such asapplied stress, electrical, temperature, and magnetic fields[25] Indeed, the simu-lation of microstructures is a hunt for key mechanisms associated with that scale.The current trend in materials modeling is to feed on to higher scales as depicted

in Fig.2.1 In Fig.2.2, a typical micrograph from [27] depicts material

microstruc-CHAPTER 2 MATERIAL SCIENCE AND SIMULATIONS 9

Figure 2.2: Typical image of grains [27]

ture, where a few grains can be visually identified The capability of predictingequilibrium and non-equilibrium phase transformation phenomena at a microstruc-tural scale is among the most challenging topics in materials science Therefore, it

is highly desirable that we are able to understand the kinetics of microstructuralevolutions

A phase field implicitly describes a microstructure, both the compositional/ tural domains and the interfaces, as a whole by using a set of field variables Thefield variables are continuous across the interfacial regions and hence the interfaces

struc-in a phase-field model are diffuse There are two types of field variables, conservedand non-conserved Conserved variables have to satisfy the local conservation con-dition Interfaces between phases is central to understanding the elegant features ofphase-field models

CHAPTER 2 MATERIAL SCIENCE AND SIMULATIONS 10

2.4.1 What is an Interface?

An interface refers to a boundary between two different phases From a ical point of view these problems constitute a class of so-called moving boundaryproblems Usually analytical treatment of these type of problems is very restricted.Thus, it seems natural to search for adequate numerical treatment In numericaltreatment the key part is to simulate the effect of the boundary by treating it ex-plicitly, i.e with no smearing of information at the interface resulting in numericaldiffusion For moving boundaries, techniques for such an explicit treatment includeblock-structured domain decomposition, overset meshes or unstructured boundary-conforming curvilinear grids to discretize the domain are employed, in which thecomputations are performed on the fixed grid while at the same time the interface

mathemat-is tracked explicitly as independent curve

The phase boundaries are more than the interfaces known from materials science.Talking of interface in a broader sense one could include evolving boundaries from,e.g combustion, image processing, computer vision, control theory, seismology andcomputer aided design, as well

2.4.2 Sharp Interface:

The interface between two distinct domains is infinetly thin In the sense of ics, if a grid is assumed to have been superimposed on the domain, at every gridpoint either we have phase 1 or phase 2 but not a combination of these two This isprecisely the reason for the popularity of the diffuse interface models The boundarycondition typically yields a normal velocity at which the interface is moving This

numer-is called sharp interface approach

CHAPTER 2 MATERIAL SCIENCE AND SIMULATIONS 11

2.4.3 Diffuse Interface:

In the case of diffuse interface any density of an extensive quantity (e.g the massdensity, number density, orientation etc.), between two co-existing phases variessmoothly from its value in one phase to its values in the other Essentially thediffuse interface is connected to such an additional order parameter Clearly suchmodels have advance numerical treatment as well as understanding of interfacialgrowth phenomena

The Phase field method to model interfacial growth is to understand it as anumerical technique which helps to overcome the necessity of solving for the preciselocation of the interfacial surface explicitly in each time step of numerical simula-tion This can be achieved by the introduction of one or several additional phasefield variables They are the key elements of the resulting phase-field modeling ap-proach for studying systems out of equilibrium In such an approach the phase-fieldvariables are continuous fields which are functions of x, and time t They are intro-duced to describe the different relevant phases Typically these fields vary slowly

in bulk regions and rapidly, on length scales of the order of the correlation length

ξ, near interface ξ is also a measure for the finite thickness of the interface Thefree energy functional F determine the phase behavior Together with equations

of motion this yields a complete description of the evolution of the system Forexample, in a binary alloy the local concentration or sub-lattice concentration can

be described by such fields

With the contribution of Cahn and collaborators, the phase-field models are morethan just trick to overcome numerical difficulties Rather they are rigorously de-rived based on the variational principles of irreversible thermodynamics In another

CHAPTER 2 MATERIAL SCIENCE AND SIMULATIONS 12

approach, a given sharp interface formulation of the growth problem is the correctdescription of the physics under consideration On the basis of this assumption, aphase-field model can be justified by simply showing that it is asymptotic to thecorrect sharp interface description, i.e the latter arises as the sharp interface limit

of the phase-field model when the interface width is taken to be zero Thereforeemployed in this way phase-field models do not seem to be of much help to eluci-date the physics of the interfacial region beyond what is captured within the sharpinterface model equations One can assume a phase-field model to be thermody-namically consistent and to describe a physical situation, for which an establishedsharp interface formulation exists, as well, certainly, in the sharp interface limit thephase-field model should correspond precisely to that sharp interface formulation

A variety of material processing applications have been comfortably handled byPhase-field models The application of phase-field method have been focused onthe three major materials processes: solidification, solid-state phase transformation,and grain growth and coarsening Examples of existing phase-field applications are:Solidification

Solid-State Phase Transformations

• Spinodal phase separation

CHAPTER 2 MATERIAL SCIENCE AND SIMULATIONS 13

• Precipitation of cubic ordered intermetallic precipitates from a disordered trix

ma-• Cubic-tetragonal transformations

• Hexagonal to orthorhombic transformations

• Ferroelectric transformations

• Phase transformations under an applied stress

• Martensitic transformation in single and polycrystals

Coarsening and Grain Growth

• Coarsening

• Grain growth in single-phase solid

• Grain growth in two-phase solid

• Anisotropic grain growth

Chapter 3

Theory and Model Description

Basic to phase field theories are continuous field variables φ1(x, t), φ2(x, t), , φQ(x, t)(referred to as order parameters) which are functions of material points x and time

t In the context of polycrystalline materials, the order parameters represent thevolume fraction of grains of a particular orientation A schematics microstructurerepresented by the orientation fields in 2-D is shown as in fig (3.1)

Figure 3.1: Representation of Orientation field variable φ

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CHAPTER 3 THEORY AND MODEL DESCRIPTION 15

The order parameters φi are non-conserved quantities therefore their local lution rates are linearly proportional to the variational derivative of the total freeenergy with respect to local order parameter Thus the evolution of the order pa-rameters is specified by the time dependent Ginzburg-Landau equations for each ofthe Q order parameters

In the phase-field model one of the key component is local free-energy density tion The main difference among different phase field models lies in the construction

func-of the energy function f as a function func-of field variables In many func-of phase-field els, particularly in solidification modeling a double-well form function is used as

CHAPTER 3 THEORY AND MODEL DESCRIPTION 16

φ = −1 and φ = +1 represent the liquid and solid states, respectively ∆f is the tential height between the two states with the minimum free energy If φ represents

po-a conserved composition field, the two minimpo-a represent the two equilibrium phpo-aseswith different compositions and ∆f is the driving force for the transformation of asingle homogeneous phase (φ = 0) to a hetrogenous mixture of two phases repre-sented by φ = −1 and φ = +1 For some processes, it may be more desirable tohave two minima of free energy located at φ = 0 and φ = 1, then following function

where I(φ) = ∞ for |φ| > 1 and 0 otherwise

On the similar way Chen et al.[30], we choose a specific form of the free energyfunction as

¶+ γ

· · · (0, 0, · · · , 1) respectively” as described in [32]

CHAPTER 3 THEORY AND MODEL DESCRIPTION 17

If we assume α = β = 1, then the first summation from right hand side ofequation (3.6) becomes

It shows that each term in the first summation in the right hand side is a well potential with well located at φi = −1 and φi = 1 However, the first sum-mation alone cannot satisfy the requirement [32] since it will have total number of

double-2Q minima located at the positions where each φi is either equal to -1 or 1, such as(φ1, φ2, · · · , φQ) = (1, 1, · · · , 1) Therefore, the cross terms were added to equation(3.6)

By choosing proper value of γ, the potential equation can fulfill that requirement.One can rewrite the equation (3.6), for choosing a proper γ, as follows

CHAPTER 3 THEORY AND MODEL DESCRIPTION 18

CHAPTER 3 THEORY AND MODEL DESCRIPTION 19

which are a set of Q coupled, non-linear, parabolic equations

In the diffuse interface description, the free energy of an inhomogeneous system,such as microstructure depends on the gradients of the field variables α in theequation is the gradient energy coefficient, which characterizes the energy penaltydue to the field inhomogeneties at the interfaces i.e the interfacial energy contri-bution to the total free energy For a given free energy model and a given set ofgradient energy coefficients, the specific interfacial energy can be calculated for anequilibrium interface It is important to realize that the integral of the gradientenergy term only counts part of the interfacial energy

In real materials, there are infinetly many grain orientations that are likely i.e

Q = ∞ However for the purpose of computer simulations, the number of field ables (ordered parameters) has to be a finite quantity But there are two drawbacks

vari-in usvari-ing a limited number of order parameters Svari-ince each order parameter sents a particular grain orientation, intermediate values of orientations are excluded.This may be viewed as restricting the likely grain orientations to a finite discreteset of angles (with loss of rotational invariance of the free energy) or each orderparameter as representing a wider range of orientations In the context of solidifi-cation of a polycrystalline material, this prevents the potential nucleation of a newrandomly oriented grain Obviously, a larger number of order parameters is requiredfor a more continuous description of frequency distribution of grains of different ori-entations Furthermore, another unfortunate consequence of limiting the number

repre-of order parameters is the coalescence repre-of grains during grain growth Coalescence