PhD thesis on Phase Field simulation and maths methods

8/11/2019 PhD thesis on Phase Field simulation and maths methods

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NORTHWESTERN UNIVERSITY

Phase-Field Simulation of Solidification

and Coarsening in Dendritic Microstructures

A DISSERTATION

SUBMITTED TO THE GRADUATE SCHOOL

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

for the degree

DOCTOR OF PHILOSOPHY

Field of Materials Science and Engineering

By

Larry Kenneth Aagesen, Jr.

EVANSTON, ILLINOIS

March 2010



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c Copyright by Larry Kenneth Aagesen, Jr. 2010

All Rights Reserved



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ABSTRACT

Phase-Field Simulation of Solidification and Coarsening in Dendritic Microstructures

Larry Kenneth Aagesen, Jr.

Phase-field models are used to simulate dendritic microstructures during solidifica-

tion and coarsening of metallic alloys. Equiaxed dendrites in a Cu-Ni alloy are sim-

ulated during rapid solidification and coarsening. The morphology of the solid-liquid

interface is characterized using Interface Shape Distributions. The interface’s topology

is quantified using genus and number of handles and voids. The system is found to

evolve non-self-similarly during solidification and the early stages of coarsening.

A method is developed to calculate the time rate of change of interfacial curvatures

for a diffuse interface representation of a microstructure, and an algorithm to calculate

the rate of change numerically is implemented. Accurate results are obtained, but

relatively large interface widths (> 10 points) are required.

A phase-field model to simulate isothermal coarsening in a binary alloy is imple-

mented and tested against the predictions of a linear stability analysis. Solid-liquid

interfacial velocities are calculated from experimental data for isothermal coarsening of

an Al-Cu alloy. The experimental data is used in the phase-field model, and interfacial



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velocities are calculated from simulations and compared to experiment. Qualitative

agreement between the velocity distributions is good, while quantitative agreement

differs by a factor of 2.9. The possible causes of disagreement are investigated, and the

diffusion coefficient of solute in the liquid, DL, is believed to be the greatest source of

the disagreement. This suggests the use of phase-field models as a means of determining

DL when other physical parameters are well known.

The shape of a liquid tube undergoing pinching by interfacial-energy driven bulk

diffusion is determined near the point of pinching. The characteristic length scale of

the process varies as t1/3. The shape is found in similarity variables using a boundary-

integral method in 2D after pinching, and in 3D before and after pinching. The shape of

a solid cylinder in a liquid matrix is also determined in 3D before pinching. The theory

is compared to experimental data for isothermal coarsening of an Al-Cu alloy. The

agreement between experiment and theory confirms that the interfacial morphology

near the singularity is universal, and that the dynamics is well described by the theorywell before pinch-off.



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Acknowledgements

First, I would like to thank my Ph.D. advisor, Peter Voorhees, from whom I learned

a great deal, and enjoyed working with very much. The things I learned from working

with him that I value most are the importance of checking yourself before drawing

conclusions and leaving no stone unturned in understanding and interpreting your

results. I am very grateful for the opportunity to have been a part his group.

I would also like to thank the members of my Ph.D. committee, Professors Monica

Olvera de la Cruz, Gregory Olson, and Katsuyo Thornton. I am also grateful to

Professor Mike Miksis from Northwestern’s Department of Engineering Sciences and

Applied Mathematics, for filling in on my committee for my Ph.D. defense, and for

his guidance in solving for the self-similar shape in Chapter 7. I also thank Professor

Alvin Bayliss of ESAM for support through the Research Training Group.

Returning to graduate school after seven years of working, I wasn’t sure what life

as a student again was going to be like. I am truly grateful to have made so many good

friends during my time at Northwestern that made being a graduate student here so

much fun. I must tip my hat to the Waldorf as the place where many of these great

times began, and of course to its residents. Man, that place was high-class. People’s

experiences in graduate school vary widely, so I am especially glad to have had such a

great experience here both professionally and personally.



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I would like to thank my family, especially my parents, who provided me the foun-

dation of a great education, the desire to work hard, and the values and beliefs to keep

it all in perspective. Above all I want to thank my wife Melissa. Your enthusiastic

support and encouragement were key in my decision to return to graduate school, and

your support never wavered no matter what challenges I encountered during our years

here. Finally, I want to dedicate this thesis to our son Nicholas. Your arrival was

undoubtedly the best part of my years in graduate school!

This work was financially supported by the Department of Energy, grant DE-FG02-

99ER45782 and by the National Science Foundation, RTG grant DMS-0636574.



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Contents

ABSTRACT 3

Acknowledgements 5

List of Figures 12

List of Tables 23

Chapter 1. Introduction 24

Chapter 2. Background 28

2.1. Dendritic Microstructures 28

2.2. Coarsening 34

2.2.1. Coarsening of Spherical Particles 36

2.2.2. Coarsening of Dendritic Microstructures 37

2.3. Interfacial Curvature 39

2.3.1. Interface Shape Distributions 42

2.3.2. Rate of Change of Curvatures 43

2.4. Topology 45

2.5. Self-Similarity 46

2.6. Phase-Field Models 51

2.6.1. Phase-Field Modeling of Pure Material Solidification 54



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2.6.2. Phase-Field Modeling of Binary Alloy Solidification 55

Chapter 3. Methods 58

3.1. Three-Dimensional Microstructure Characterization 58

3.2. Interfacial Curvature 59

3.3. Topological Characterization 61

3.4. Calculation of Interfacial Velocities 61

3.5. Phase Field Models 63

3.5.1. Isothermal Solidification of Cu-Ni 63

3.5.2. Isothermal Coarsening of Al-Cu 64

Chapter 4. Phase-Field Simulation of Equiaxed Dendritic Solidification in a

Binary Alloy 67

4.1. Microstructure, Volume Fraction, and Surface Area 67

4.2. Interface Shape Distributions 724.3. Interfacial Topology 77

4.4. Comparison with Other Systems 84

4.5. Conclusions 87

Chapter 5. Rate of Change of Curvatures 89

5.1. Details of Algorithm 89

5.2. Examples 90

5.2.1. Cylinder Expanding in the Radial Direction 90

5.2.2. Translating Cylinder 92

5.3. Testing of Algorithm 93



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5.4. Strategies to Decrease Anisotropy of |∇H | 97

5.5. Conclusions 100

Chapter 6. Using Phase-Field Modeling to Predict Interfacial Velocities During

Coarsening 101

6.1. Methodology 101

6.2. Experimental Procedure 102

6.3. Validating the Phase-Field Model for Simulating Coarsening 103

6.4. Calculating Interfacial Velocities from Experimental Data 108

6.4.1. Choosing Simulation Parameters 109

6.5. Comparing Experiment to Theory 117

6.6. Analysis of Contributions to Error 118

6.6.1. Phase-Field Model at Large 119

6.6.2. Effect of Assumption DS = 0 121

6.6.3. Physical Parameters 123

6.7. Effect of Anti-trapping Current 125

6.8. Conclusions 126

Chapter 7. Self-Similar Pinch-off of Rods 127

7.1. Theoretical Formulation 128

7.1.1. Similarity Variables 129

7.2. Phase-Field Simulations 130

7.3. Solving for Interface Shape in Self-Similar Variables 134

7.3.1. Interface Shape After Pinching in 2D 135



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7.3.2. Interface Shape After Pinching in 3D 140

7.3.3. Interface Shape Before Pinching in 3D 143

7.3.4. Interface Shape Before Pinching in 3D for Solid Cylinders in a Liquid

Matrix 149

7.4. Comparison with Experimental Data 150

7.4.1. Conclusions 153

Chapter 8. Conclusions 156

References 159

Appendix A. Derivation of Perturbation Decay Constant σ 166

Appendix B. Determining Time Exponent of Similarity Solution for Rod

Pinching 171

Appendix C. Details of Boundary-Integral Equation After Pinching in Two

Dimensions 176

Appendix D. Details of Boundary-Integral Equation After Pinching in Three

Dimensions 179

Appendix E. Details of Boundary-Integral Equation Before Pinching in Three

Dimensions 185



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List of Figures

2.1 Formation of zones in a casting [1]. Equiaxed dendrites close to

the mold walls form the outer equiaxed zone, then dendrites with

orientations closest to the direction of heat removal grow fastest

to form the columnar zone. Finally, equiaxed dendrites can form

at the center to form the inner equiaxed zone when the remaining

melt at the center of the casting becomes undercooled. 30

2.2 The conditions which give rise to constitutional undercooling. Top,

the solute concentration profile in the liquid during steady-state

directional solidification at velocity v. Assuming thermodynamic

equilibrium at the interface, the temperature in the liquid T L is

governed by the phase diagram, bottom right. T L is shown as a

function of z in the lower left. Constitutional undercooling occurs

if the temperature profile required for heat balance T q is less than

T L ahead of the interface. Figure from [1

]. 32

2.3 δ /δ as a function of wavelength for two cases: curve 1 unstable,

curve 2 stable [2]. 33



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2.4 Four different models for isothermal coarsening: (1) radial

remelting, (2) axial remelting, (3) arm detachment, (4) arm

coalescence [3]. 39

2.5 A interfacial patch with corresponding radii of curvature, R1 and

R2. n represents the unit normal vector, while p is the point of

interest. 40

2.6 Interface Shape Distribution (ISD). 42

2.7 Particle radius distribution predicted by the LSW theory [4]. The

horizontal axis is the radius of a particle divided by the average

particle radius for the entire collection of particles at time t. The

vertical axis is the probability of finding a particle with that radius.

Because the distribution is constant with time, the particle size

distribution is said to be self-similar. 47

2.8 The pinch-off of a cylinder by Rayleigh instability as a function

of time. For some systems, when the coordinates of the interface

position are scaled by B/(ts − t)α to put them into dimensionless

coordinates, the shape is independent of time. 49

4.1 Microstructures of Cu-Ni system undergoing dendritic solidification

at (a) 26.2 µs, (b) 41.9 µs, (c) 57.6 µs, and (d) 78.6 µs.

Visualizations are a 640 × 390 × 128 subset of the 640 × 640 × 640

simulation volume. The solid phase is capped at the boundary of

the visualization region. 68



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4.2 Microstructures of Cu-Ni system entering the coarsening phase at

(a),(b) 105 µs, (c),(d) 131 µs, and (e),(f) 157 µs. The solid phase

is capped at the boundary of the visualization region in (a),(c),(e)

and the liquid phase is capped in (b),(d),(f). Visualizations are a

640 × 390 × 128 subset of the 640 × 640 × 640 simulation volume. 69

4.3 V f and S v versus time. 70

4.4 S −1v vs. t

1/3

during (a) solidification and coarsening, (b) coarsening

regime only (last 3 data points of (a)). The straight line-fit to the

data is consistent with other observations of S −1v during coarsening,

but much longer simulation times would be required to prove this

scaling conclusively. 70

4.5 Volume fraction solid approaches the equilibrium value of 0.75.

V f /S v remains constant (±7%) during solidification, then begins to

increase as the system enters the coarsening regime. 71

4.6 ISDs for (a), (b) 26.2 µs, (c),(d) 157 µs. (a) and (c) were volume

smoothed with a width of 3, (b) and (d) with a width of 5.

Comparison shows no significant differences due to smoothing. 73

4.7 An isolated dendrite at t = 26.2 µs. Patches in Region 1 of the ISD

(κ1, κ2 both positive) are colored red. Patches in Regions 2 and 3

of the ISD (κ2 positive, κ1, negative) are colored grey. Patches in

Region 4 of the ISD (κ1, κ2 both negative) are colored blue. 74



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4.8 ISDs of Cu-Ni system undergoing dendritic solidification at (a)

26.2 µs, (b) 41.9 µs (c) 57.6 µs, (d) 78.6 µs. Color bar set to the

same scale for all plots. 76

4.9 A (1.31 × 1.83 × 1.44 µm) subset of the microstructure at 41.9 µs.

Patches with κ1 > 0, κ2 > 10 µm−1 are colored red. 77

4.10 ISDs of Cu-Ni system undergoing coarsening at (a) 105 µs, (b) 131

µs, and (c) 157 µs. 78

4.11 A (1.31 × 1.83 × 1.44 µm) subset of the microstructure at 105 µs.

(The liquid phase is capped at the boundary.) Patches with

κ1 < −8 µm−1, −4 µm−1 < κ2 < 12 µm−1 (liquid cylindrical

regions) are colored red. These patches show the origin of the

extended tail found mostly in Region 3 of in the ISDs of Figure 4.10.

These regions are associated with liquid tubes that are pinching off

and with the edges of interdendritic liquid regions that separate

solid dendrites. 79

4.12 ISDs of Cu-Ni system undergoing coarsening at (a) 105 µs, (b)

131 µs, and (c) 157 µs. ISDs scaled by S v and plotted with same

color bar. The changes in the ISDs with time mean that the

microstructure is not self-similar. 80

4.13 Portion of the simulation microstructure at (a) 105 µs, (b) 131

µs, and (c) 157 µs. Liquid phase is capped at the boundary. Red

arrow highlights a long liquid cylinder that pinches off by Rayleigh



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instability at two points simultaneously, leaving an isolated liquid

void in between after pinching. 81

4.14 Genus per unit volume versus time. 83

4.15 A handle (liquid tunnel) is formed as adjacent secondary arms

join, in red circle (t = 41.9 µs). The solid phase is capped at the

boundary. 83

4.16 The breakup of liquid tubes due to capillarity-driven instability.

The liquid phase is capped at the boundary. (a) 78.6 µs, (b) 157

µs. 85

4.17 Number of independent bodies per unit volume for solid and liquid

phases. The formation of either a liquid or solid body decreases

the genus by 1. The minimum value plotted for each series,

1.7

×106/mm3, represents a single continuous body divided by the

simulation volume. 85

5.1 Plot of φ = tanh

x2 + y2 − vt

/W

in the xy-plane for

vt = 25, W = 5, where black represents φ = −1, white represents

φ = 1. 91

5.2 Plot of mean curvature H for φ = tanh

x2 + y2 − vt

/W

along x for y = 0, z = 0. vt = 25, W = 2.5 (white), W = 5 (red). 94

5.3 |∇H | for all points in the xy plane of φ = tanh

x2 + y2 − vt

/W

.

Black represents |∇H | = 0 and white represents |∇H | = 1.5×10−3.

The red circle shows the position of the interface at vt = 25. 96



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6.1 Sample input array: White is solid, black is liquid. λ p = 120

gridpoints, perturbation amplitude A = 10, Ds

= 0. The gray

border is for visual clarity only and is not part of the simulation

region. 104

6.2 Decay of a perturbation from simulation results (red) compared

with analytical solution calculated from Equation 6.2 (blue). 105

6.3 (a) Decay constant error for perturbation with A = 10 with various

perturbation wavelengths, (b) decay constant error for constant

system size, varying . 106

6.4 (a) 100 × 80 × 150 voxel portion of microstructure used for

calculation of interfacial velocities (liquid phase is capped at

boundaries), (b)-(c) solid-liquid interface colored by normal

interfacial velocities calculated from experimental data (negative

velocities point into the liquid), experimental time steps 5 min 46

sec apart. In (b) the order parameter array was smoothed using

only a 5-voxel volume smooth, in (c), the order parameter array

was also smoothed by the phase-field model. 110

6.5 Interfacial velocities simulated using (a) 500,000, (b) 800,000, (c)

1,000,000, (d) 1,500,000 initial iterations of the phase-field model,

followed by composition field equilibration and 100,000 further

iterations for the velocity calculation. The liquid phase is capped

at the boundaries and surfaces are colored by interfacial velocity. 114



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6.6 (a), (b) Two different views of the larger simulation volume selected

for velocity calculations. 115

6.7 Comparing experimental and simulated velocities. The solid-liquid

interface is shown colored by normal velocities, with negative

velocities pointing into the liquid. Liquid is inside the bulb shape,

and the solid phase is transparent. Note color bars between left

and right images differ by a factor of 3. 117

6.8 Normalized histograms of experimentally measured and scaled

simulated interfacial velocites. The simulated velocities are scaled

by 1/2.9, the scale factor that produced the best least-squares fit

between the two normalized histograms. 118

6.9 Difference between theoretical prediction and simulation results for

perturbation decay for high values of . 121

6.10 Interfacial velocity prediction using phase-field model without

anti-trapping current. 125

7.1 The microstructure of the Al-Cu alloy experimentally observed

during coarsening, with the liquid region capped at the boundary.

The region boxed in red shows several liquid tubes at various stages

in the approach to pinching and one immediately after pinching. 128

7.2 Phase-field simulations of liquid cylinder pinching off by bulk

diffusion. Black represents liquid, white represents surrounding

solid. (a) Initial conditions, (b) shape just prior to pinching. 131



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7.3 From phase-field simulations, (a) minimum radius of a pinching

cylinder cubed, (b) distance between tips of cones after pinching

cubed versus time. The linear slope close to the point of pinching

supports t−1/3 dependence of the similarity variables close to the

point of pinching. 132

7.4 Cone angles measured as close as possible to the point of pinching,

(a) phase-field simulations, (b) experimental data. 134

7.5 Rod pinching due to Rayleigh instability, (a) before, (b) at, (c)

after, the time of pinching. The interface shape η = f (ξ ) which will

be solved numerically is highlighted in red. In (a), before pinching,

the independent variable ξ runs along the z and the dependent

variable η is along r. In (c), after pinching, the independent

variable ξ runs along r and the dependent variable η is along z .

Figure from [5]. 135

7.6 Shape of the 2D solid-liquid interface after pinching by interfacial-

energy driven bulk diffusion. For each curve, the solid lies to the

upper left, and the liquid to the lower right of the curve. 138

7.7 (a) Variation of 2D solid-liquid interface shape (after pinching,

α = 80) with truncation length of the numerical solution. The

shape changes little after ξ is increased above 5. (b) Error ratio

(defined in Equation 7.14) for an initial grid spacing of 0.5 (Error

ratio 1) and 0.25 (Error ratio 2). (c) Error ratio (as defined in

Equation 7.15) for initial truncation length ξ T . Grid spacing was



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0.2, initial truncation length was ξ T = 5 (Error ratio 1), and

ξ T = 10 (Error ratio 2). 139

7.8 Shape of the 3D solid-liquid interface after pinching by interfacial-

energy driven bulk diffusion. For each curve, the solid lies to the

upper left, and the liquid to the lower right of the curve. 143

7.9 (a) Variation of 3D solid-liquid interface shape (after pinching,

α = 80) with truncation length of the numerical solution. (b)

Error ratio (defined in Equation 7.14) for an initial grid spacing

of 0.2 (Error ratio 1) and 0.1 (Error ratio 2). (c) Error ratio (as

defined in Equation 7.15) for initial truncation length ξ T . Grid

spacing was 0.25, initial truncation length was ξ T = 5 (Error ratio

1), and ξ T = 10 (Error ratio 2). 144

7.10 Shape of the 3D solid-liquid interface before pinching by interfacial-

energy driven bulk diffusion. The solution at θ = 52 is expected to

be stable, while the solution at θ = 75 is expected to be unstable

(θ = 90 − α). 147

7.11 (a) Variation of 3D solid-liquid interface shape after pinching

(θ = 52) with truncation length of the numerical solution. (b)

Error ratio (defined in Equation 7.14) for an initial grid spacing

of 0.25 (Error ratio 1) and 0.125 (Error ratio 2). (c) Error ratio

(as defined in Equation 7.15) for initial truncation length ξ T . Grid

spacing was 0.25, initial truncation length was ξ T = 5 (Error ratio

1), and ξ T = 10 (Error ratio 2). 148



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7.12 Shape of the 3D solid-liquid interface before pinching by interfacial-

energy driven bulk diffusion for solid rods pinching off in a liquid

matrix. The stable solution is solution at θ = 50 (α = 40). 149

7.13 Tube diameter follows a power law prior to singularity. The

diameter of a pinching tube as a function of (t − ts)1/3 along with

the experimentally measured and theoretically predicted interface

shapes at various times and line showing least-squares fit to

experimental data. (Error bars show the maximum and minimum

measured diameter of tubes, which are not circular at all times.) It

is clear that even when the interfacial morphology is far from the

self-similar shape, the kinetics of the pinching process are still well

described by (t − ts)1/3. Figure courtesy of A.E. Johnson. 151

7.14 Quantifying agreement between theoretically predicted and

experimentally measured interface shape. The experimentally

determined microstructure is superimposed on the theoretically

predicted shape for the solution with a cone angle of 76 for several

times leading up to pinch-off (top). The theoretical shapes are

scaled to the same diameter as the experimental microstructures.

Bottom, the theoretical shape is shown colored by the distance from

each point to the closest point on the experimentally measured

shape. The mean deviation between experiment and theory

decreases as the system approaches the time of pinchoff. In the

case of the final time step, the mean deviation is less than the



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experimental resolution of the X-ray tomography scan (1.4 µm).

Times: (a) 142.5 min, (b) 155 min, (c) 165 min, (d) 167.5 min.

Figure courtesy of A.E. Johnson. 154



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List of Tables

4.1 Number of solid and liquid voids for each time. PBC= corrected

for effect of periodic boundary conditions. 79

4.2 Computed genus versus volume smoothing width for isolated

dendrites at t = 26.2 µs. The interface width is defined as the

number of interface points where the order parameter is between

10% - 90% of the equilibrium liquid value. 82

4.3 Computed genus versus time. 84

5.1 Summary of testing for expanding cylinder test scenario. 99

5.2 Summary of testing for translating cylinder test scenario. 100

6.1 Physical parameters used in phase-field simulations. 112



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CHAPTER 1

Introduction

Dendrites are tree-like structures that often form during solidification. They fre-

quently occur during technologically important processes such as casting, in which a

liquid metal or alloy is poured into a mold and solidifies to form a part in the shape

of the mold. It is important to understand what controls dendrites’ morphology be-

cause the dendritic microstructure formed during solidification has a strong effect on

the properties of the final part even after solidification is complete. For example, fine

dendritic microstructures are associated with improved mechanical properties such as

high ductility and tensile strength [6].

Solidification processes can be classified as directional or equiaxed, depending on

how heat is removed from the liquid. Dendrites can form in directional or equiaxial

solidification depending on whether a pure material or an alloy is solidifying. The

morphology of the dendrites, and therefore the properties of the part, are also strongly

affected by the process of coarsening, which occurs at the same time as solidification.

Coarsening occurs when a multiphase system decreases the surface area of interface

between phases, thereby decreasing its interfacial energy.

In solid-liquid mixtures in a metallic alloy, the coarsening process is kinetically

driven by the Gibbs-Thomson effect. Due to this effect, the equilibrium concentration

of solute at the solid-liquid interface differs from its equilibrium value. The solute

concentration at the interface is given by C = C 0 + ΓH , where C 0 is the equilibrium



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concentration, H is the mean interfacial curvature, and Γ is a material-dependent

constant. These variations in C at interfaces create solute concentration gradients,

which cause solute flux from regions of high to low concentration. Thus the variation

in interfacial curvature is the driving force for coarsening.

Due to the technological importance of understanding dendritic microstructures,

simplified models of dendrite coarsening have been developed in the past. These models

generally assume a simplified geometry for the dendrite primary and secondary arms,

calculate the solute concentrations at the solid-liquid interface based on the curvature of

the geometry, and finally calculate solute flux and therefore coarsening rate. Although

these models are useful for predicting basic trends, it is not possible to analytically solve

for the behavior of the complex microstructures that actually form in the mushy zone,

the region of mixed solid dendrites and surrounding liquid. Predicting the behavior of

real microstructures must rely on computer models where the real 3D microstructure

can be input.The past two decades have seen considerable advances in phase-field models, which

are particularly well-suited to simulating the solidification and coarsening of dendrites

in pure materials and alloys. Phase-field models are based on the basic thermodynamic

principle that the free energy of the system must decrease with time. The microstruc-

ture of the system being modeled is represented using one or more order parameters,

and a phenomenological free-energy functional of these order parameters is postulated.

Assuming the free energy must decrease linearly with time, the equations to calculate

the evolution of each order parameter with time can be derived. The microstructure



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to be simulated is represented digitally using the order parameters, and the equations

governing the order parameters are solved numerically to predict the system’s behavior.

Most phase-field models have been developed and tested on simplified 1D and 2D

test geometries. However, the considerable advances in computer processing power

and memory capacity during the same few decades is now opening the possibility to

simulate realistic microstructures in 3D using phase-field models. The development of

new techniques to capture real microstructures in 3D has unlocked another new pos-

sibility: to input the real microstructures into phase-field models and directly predict

the behavior of a real microstructure in 3D for systems of technological interest. The

goal of this thesis is to explore the capability of phase-field models to simulate the

evolution of dendritic microstructures in 3D, and to directly compare the predictions

of phase-field models and other theoretical techniques to 3D experimental data.

A 3D phase-field model is used to simulate the solidification and coarsening of a

Cu-Ni alloy during rapid solidification conditions, and the morphology and topologyare characterized throughout the process. These simulations demonstrate the limits of

what is currently possible in terms of system size and evolution time, and demonstrate

the capability of phase-field models to simulate systems which would be difficult or

impossible to observe experimentally.

Given the important role that curvature plays in determining the evolution of den-

dritic microstructures, a formal mathematical approach is developed to calculate the

time evolution of curvature using an order-parameter based representation of the 3D

microstructure. An algorithm to perform the calculation numerically is developed and

tested.



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The predictions of a phase-field model are directly compared to experimental data

using data obtained from an in-situ X-ray tomography experiment. A solid-liquid

mixture in an Al-Cu alloy was coarsened isothermally, and the microstructure was

captured in real time using X-ray tomography. A portion of the microstructure is

input to a phase-field model for isothermal coarsening, and the time evolution predicted

from the phase-field model is compared to the experimental data. This is the most

direct comparison performed to date between the predictions of a phase-field model

and experimental data.

In this X-ray tomography experiment, many liquid tubes were observed during the

process of pinching off by interfacial-energy driven bulk diffusion. Close to the point of

pinching, the shapes appear to be universal, meaning they look identical as the effect

of initial conditions becomes less important and the shapes become locally determined.

They also appear to be self-similar, meaning that their shape is time-independent when

scaled by the appropriate time-dependent scale factor. Phase-field simulations are usedto show that the shape scales as t1/3 as the tube approaches the time of pinching. The

universal shape is solved in self-similar coordinates using a boundary integral method,

and the kinetics and shape predicted by theory are compared to experimental data.



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CHAPTER 2

Background

2.1. Dendritic Microstructures

Casting, the process of forming a part by pouring a liquid material (often a metallic

alloy) into a mold, was one of the most important technological developments of early

human history. An entire period of human history, the Bronze Age (3300-1200 BC),

is named for the advances in metallurgical technology during those years, including

casting. Casting remains one of the simplest, most convenient, and least expensive

ways to produce relatively complex metal shapes and is used in countless applications

today.

Dendrites are the dominant feature in the microstructure of castings. Dendrites are

tree-like branched structures formed by one phase during solidification. The properties

of the dendritic microstructure in a casting can have an important effect on the physical

properties of a cast part. Dendrites can form when either pure materials or alloys

solidify, but dendrite formation in alloys is the focus of this work because most metal

castings have some amount of alloying element present.

The main trunks of the growing dendrites are called the primary arms. The sec-

ondary arms grow out from perturbations on the primary arms, and tertiary arms can

even grow out from the secondary arms. Spacing between adjacent arms is often used as

a metric to characterize microstructures, especially secondary arm spacing (λ2). Past



29

work has shown that finer dendritic microstructures, as measured by a smaller λ2, have

improved mechanical properties, such as higher yield and ultimate tensile strength [6].

Figure 2.1 shows the types of dendritic microstructures that form in a casting [1].

When the liquid is poured into the mold, heat is extracted through the mold walls, and

the temperature of the liquid near the mold walls drops below the liquidus temperature

first. Solid particles begin to form near or on the mold walls when the temperature of

the liquid is undercooled enough to allow the free energy barrier for nucleation to be

overcome (often at the mold walls or at impurities located near the walls due to the

reduced energy barrier to nucleation). The nuclei form with random orientations and

the primary arms initially grow at equal rates in all directions, and are called equiaxed

dendrites. Soon after nucleation, the dendrite arms in the outer equiaxed zone which

grow parallel to and opposite the heat flow direction overgrow all other orientations,

forming the columnar zone which grows toward the center of the casting as heat as

extracted. During the final stages of solidification, the center of the casting has cooledto or slightly below the solidus temperature. Dendrite branches which break off from

the columnar zone can begin to grow in the center, and since the remaining liquid

is slightly undercooled with no preferred orientation, the growth is again equiaxial,

leading to the formation of the inner equiaxed zone.

Dendrites also form during directional solidification of metallic alloys. In a direc-

tional solidification process, the alloy to be solidified is contained in a crucible which

is passed through a temperature gradient at velocity v. As the liquid’s temperature

drops below the liquidus temperature, dendrites form in the solid as it grows into the

liquid. As in the columnar zone of a casting, the primary arms grow parallel to the



30

Figure 2.1. Formation of zones in a casting [1]. Equiaxed dendrites closeto the mold walls form the outer equiaxed zone, then dendrites withorientations closest to the direction of heat removal grow fastest to formthe columnar zone. Finally, equiaxed dendrites can form at the centerto form the inner equiaxed zone when the remaining melt at the centerof the casting becomes undercooled.

direction of heat extraction. By appropriate control of initial conditions, directional

solidification can be used to grow large, high-quality single crystals.

The type of dendritic microstructure formed, columnar or equiaxed, depends on how

heat is extracted. When the entire liquid is undercooled, as in the inner equiaxed zone

of a casting, heat is extracted nearly uniformly from a solid nucleus. The temperature

decreases radially outward from the nucleus in order for heat to be extracted through

the undercooled liquid. Any perturbation to the solid-liquid interface that grows further

out into the surrounding liquid will push further into the gradient. The perturbation

will therefore have an even steeper negative temperature gradient than the surrounding

unperturbed interface. This steeper gradient allows heat to be extracted even more



31

quickly from the perturbation, which makes it grow even faster preferentially outward.

Since perturbations are unstable in any direction, growth is equiaxial.

The reason a dendritic microstructure forms is more complicated for directional so-

lidification in an alloy (Figure 2.2). Consider an alloy with overall solute concentration

C 0 being directionally solidified, which has a phase diagram as shown in Figure 2.2.

When steady-state solidification conditions are reached (usually after a short initial

transient), the solid (shaded region at left, Figure 2.2) advances with velocity v into

the liquid (white region to right). The rejection of solute at the interface causes a jump

in solute concentration from C 0 in the solid to C 0/k in the liquid, where k = C S /C L

is the ratio of the solidus concentration to the liquidus concentration at a given tem-

perature (for the given phase diagram with straight solidus and liquidus, k is constant

with temperature and less than 1). The concentration drops off exponentially in the

liquid away from the interface:

(2.1) C L = C 0 + (C 0/k − C 0)exp

− vz DL

where z is the distance from the solid-liquid interface and DL is the diffusion coeffi-

cient of solute in the liquid. The solute concentration profile in the liquid causes the

liquidus temperature T L to increase away from the solid-liquid interface (curved lines

in Figure 2.2 show how solute concentration ∆C changes T L). The resulting liquidus

temperature profile is shown in the lower left, along with the the temperature profile

required for heat flow, T q. Without the effect of solute rejection, as in the case of a

pure material, any perturbation to the solid-liquid interface would result in the solid

protruding further and the temperature gradient becoming sharper, which would cause



32

the perturbation to re-melt and preserve the stability of the planar interface. However,

due to the variation of T L with z , under some conditions T q can be less than T L. Under

steady-state conditions the temperature is T q < T L, so solidification can occur. Any

perturbation increases the composition gradient, so the perturbation will tend to grow.

Because the constitution of the alloy determines the difference between T L and T q, this

effect is known as constitutional undercooling.

Figure 2.2. The conditions which give rise to constitutional undercool-ing. Top, the solute concentration profile in the liquid during steady-statedirectional solidification at velocity v. Assuming thermodynamic equi-librium at the interface, the temperature in the liquid T L is governed bythe phase diagram, bottom right. T L is shown as a function of z in thelower left. Constitutional undercooling occurs if the temperature profilerequired for heat balance T q is less than T L ahead of the interface. Figure

from [1].

A more rigorous approach to the criteria for growth of a perturbation to a solid-

liquid interface moving at velocity v , as in directional solidification, was developed by

Mullins and Sekerka [2]. Unlike the constitutional undercooling criteria, they took



33

capillarity into account. Assuming isotropic bulk and surface parameters and local

equilibrium at the interface, and no convection in the liquid, they calculated the time

rate of change δ of the amplitude of a sinusoidal perturbation of infinitesimal initial

amplitude δ superimposed on a planar solid-liquid interface as a function of perturba-

tion wavelength and solidification conditions. When the rate of change is positive, the

perturbation grows, leading to instability in the interface. In this model, the interface

is stable if there are no perturbation wavelengths which have positive growth rates

(Figure 2.3).

Figure 2.3. δ /δ as a function of wavelength for two cases: curve 1 unsta-ble, curve 2 stable [2].

The shape of a dendrite’s during steady-state growth has also been closely studied.

Ivantsov [7] showed that a dendrite with a parabolic tip (or a paraboloid in 3D) grows

with a constant shape under the constraint of constant interface temperature and

composition. The relationship between interface composition and velocity is:

(2.2) C t = C 0/ [1 − (1 − k)Iv(P )]



34

where C t is the composition at the tip of the dendrite, C 0 and k are as defined before,

and P is the solute Peclet number P = V R/2DL where V is the steady-state velocity,

R the tip radius, and DL the solute diffusion coefficient in the liquid. The Ivantsov

function is defined as Iv(P ) = P expP E 1(P ) and E 1(P ) is the first exponential integral.

Ivantsov’s solution, however, does not include the effect of capillarity, which can be

an important effect at high-curvature regions such as the tip of a dendrite. The more

recent development of solvability theory [8, 9] showed that anisotropy in the surface

energy perturbs the shape of the growing tip and selects the velocity of the tip. Four-

fold anisotropy in the surface energy is assumed for crystals with cubic symmetry. The

effects of this anisotropy lead to a perturbation to the paraboloid shape by a term of

the form [10]

(2.3) z = −r

2 + A4r4 cos4φ

where A4 = 1/88 and (r, φ) are polar coordinates in the plane normal to z , the growth

direction.

2.2. Coarsening

The process of coarsening also has an important effect on the properties of dendritic

microstructures. During casting or other processes where dendritic microstructures are

formed, coarsening occurs while the dendrites grow. Coarsening can be most easily un-

derstood as a consequence of thermodynamics, where the system seeks to reduce its

total energy by reducing its interfacial area and therefore its total interfacial energy.



35

Coarsening can be especially important at the center of a cast part, where the rela-

tively slow removal of heat can lead to a region of solid-liquid coexistence (called the

mushy zone) which can last long enough for coarsening to cause significant changes to

the microstructure. Coarsening in the mushy zone causes secondary arms to dissolve

away, leading to less fine dendritic microstructures and generally degrading mechanical

properties.

Kinetically, the process of coarsening is driven by differences in concentration at

the solid-liquid interface caused by the Gibbs-Thomson effect, in which the equilibrium

composition of an two-component system is modified by variations in curvature at

the solid-liquid interface. This composition difference is due to equilibrium pressure

changes caused by interfacial tension. The Gibbs-Thomson equation states

(2.4) P L − P S = γκ

where γ is the interfacial energy between the solid and liquid phases, and κ is the mean

curvature of the solid-liquid interface (to be explained in greater detail in Section 2.3).

Starting from the Gibbs-Thomson equation, it can be shown [11] that the equilibrium

composition of the liquid at the solid-liquid interface in a two-component system is

given by

(2.5) C L = C 0L + ΓH

C L is the composition of the liquid at a curved interface, C 0L is the equilibrium composi-

tion of the liquid at a flat interface, and Γ is the capillary length determined by material



36

parameters including interfacial energy, molar volume of the solid, and curvature of the

free energy function.

The differences in composition caused by the Gibbs-Thomson effect lead to solute

concentration gradients in the liquid. These gradients cause solute diffusion in the

liquid, leading to transport of solid from high-curvature regions to low-curvature re-

gions and therefore coarsening. The same process occurs in the solid, but since solute

diffusion through the solid is usually much slower than through the liquid, coarsening

in the liquid phase usually causes much greater changes to the microstructure.

2.2.1. Coarsening of Spherical Particles

Lifshitz and Slyozov [12] and Wagner [13] independently developed a theory of the

coarsening of spherical particles, generally referred to as the LSW theory. The model

assumes a system composed of A and B atoms containing a distribution of spherical

particles of the β phase composed entirely of B atoms in a matrix of the A-rich α

phase. Because of the Gibbs-Thomson effect, B atoms flow from the smaller particles

to the larger particles through the α phase, which causes the smaller particles to shrink

and the larger particles to grow. The mean particle size of the distribution grows as

(2.6) R3(t) − R3(0) = K LSW t

where R(t) is the mean particle radius at time t, and K LSW is a rate constant de-

pending on material parameters and temperature. The LSW theory also predicts a

distribution of particle radii about the mean radius that is time-independent when

scaled by R3(t). Because of this property, the particle size distribution is said to be



37

self-similar. Important limitations of this model are that it assumes that the spherical

particles are infinitely far apart and have a negligibly small volume fraction relative to

the matrix phase. Nonetheless, this model has proved useful in describing coarsening

in many binary alloys, and has also served as the basis for more advanced coarsening

models for spherical particles.

2.2.2. Coarsening of Dendritic Microstructures

Because dendritic microstructures are so much more complicated than a spherical par-

ticle distribution, developing a comprehensive theory of the coarsening of dendrites is

a much more difficult problem than a spherical particle distribution. However, due to

the important effects of dendritic microstructure on castings, directional solidification,

and other technologically important processes, coarsening in dendritic microstructures

has been studied extensively.

The change in spacing between the secondary arms λ2 has frequently been used

to quantify the time-dependence of the coarsening process. By measuring the time

evolution of the spacing between secondary arms during coarsening of dendritic Al-Cu,

Bower et. al. [14] showed that

(2.7) λ2 ∼ t1/3f

Thus λ2 evolves with time analogously to average particle radius in the LSW model.

Several different theoretical models have been used to describe the mechanism of

coarsening in dendrite secondary arms. Most have treated the arms of the dendrites as

either cylinders or teardrop-shaped protrusions. Kattamis et. al. [3] treated the arms



38

as cylinders, but assumed that one cylinder had a smaller radius than the others and

therefore dissolved with a shrinking radius, called radial remelting. Kahlweit et. al. [15]

treated the dendrite arms as cylinders with spherical tips. Since the tips have a smaller

radius of curvature, dissolution primarily occurs at the tip in this model and the arms

shrink from their tips down to the roots while the radii remain practically unchanged.

This process is known as axial remelting. The first attempt to model dendrite arms

as teardrop-shaped objects considered individual dendrites which would detach from

the primary arm as the thin neck at the arm shrank due to solute diffusion from the

neck to the wider portion of the teardrop. [16, 17] The next such model considered

two adjacent teardrop-shaped secondary arms which eventually coalesce into a single

arm. [18] These models are summarized in Figure 2.4.

λ2 is a useful metric to describe a dendritic microstructure because it has been

correlated with physical properties, and because it is easy to measure given a physical

specimen that has been properly prepared. However, the use of λ2 has disadvantages.It is highly dependent on which 2D plane is visible after sample preparation, and

because it is based on a 2D section, cannot fully describe the complexity of a 3D

microstructure. Also, it has been observed that after extremely long coarsening times,

an initially dendritic microstructure can break up into spherical particles [19], in which

case λ2 loses its meaning.

In order to adequately characterize a dendritic microstructure in 3D, the inverse

of surface area per unit volume, S −1v , has proven useful. It also scales with t1/3 [19],

is independent of the morphology of the system, and does not depend on stereological

assumptions necessary for 2D images. The relationship S −1v ∼ t1/3 has been shown to



39

Figure 2.4. Four different models for isothermal coarsening: (1) radialremelting, (2) axial remelting, (3) arm detachment, (4) arm coales-cence [3].

hold during the coarsening of directionally solidified dendritic Sn-Bi [19], Al-Cu [20],

and Pb-Sn [21] and equiaxially solidified dendritic Al-Cu [22].

2.3. Interfacial Curvature

Because curvature drives coarsening as shown by Equation 2.5, it is important to

have a rigorous mathematical description to characterize the curvature at every point

of an interface. For any given patch of surface area on an interface, two orthogonal

planes can be chosen whose intersections c1 and c2 with the interface are curves with

radius of curvature R1 and R2. Both these planes also contain the normal vector to the

interface n. The principal curvatures κ1 and κ2 are the inverse of the radii of curvature

of the two curves c1 and c2, as in Figure 2.5. The mean curvature H is the average of



40

Figure 2.5. A interfacial patch with corresponding radii of curvature, R1

and R2. n represents the unit normal vector, while p is the point of interest.

the curvatures

(2.8) H = 1

2 (κ1 + κ2)

Through linear algebra, it can be shown that the mean curvature H is independent

of the choice of curves c1 and c2. Another invariant of the choice of curves is the

Gaussian curvature K :

(2.9) K = κ1κ2 = 1

R1R2

Although the principal curvatures are easier to understand physically, H and K are

easier to calculate for digital representations of 3D microstructures. This can be done



41

using either a surface-mesh based description of the interface, or using a diffuse inter-

face representation. For a diffuse interface representation, the mean curvature can be

calculated using

(2.10) H = 1

2 (∇ · n)

where n is the normal vector to the interface calculated from gradients of the order

parameter

(2.11) n = ∇φ

|∇φ|

The Gaussian curvature can be calculated using

(2.12) K = n · adj(He(φ))n

where He(φ) is the 3

×3 Hessian matrix of second partial derivatives of the order

parameter, and adj(He(φ)) is the adjoint of the Hessian matrix [23]. More detail on

the numerical implementation of these methods will be provided in Chapter 3. The

principal curvatures can then be calculated from H and K using

κ1 = H −√

H 2 − K (2.13)

κ2 = H +√

H 2 − K (2.14)



42

2.3.1. Interface Shape Distributions

In order to summarize the distribution of the principal curvatures found throughout the

interface of a multiphase microstructure, the Interface Shape Distribution, or ISD, has

been developed [24, 25]. An ISD is a two-dimensional contour plot of the probability

P (κ1, κ2) of finding a patch with curvatures κ1 and κ2 between κ1 + dκ1 and κ2 + dκ2.

(κ2 is always defined to be greater than or equal to κ1.)

Figure 2.6. Interface Shape Distribution (ISD).

The ISD can be broken down into four regions (Figure 2.6):



43

• Region 1: solid on the concave side of the interface.

• Region 2: saddle-shaped with H > 0, K < 0.

• Region 3: saddle-shaped with H < 0, K < 0.

• Region 4: liquid on the concave side of the interface.

Other features of interest occur along the axes and the line κ1 = κ2:

• The interface is planar along the line κ1 = κ2 = 0.

• No patches appear below the line κ1 = κ2 since κ2 ≥ κ1 by definition.

• Interfacial patches on the κ1 = κ2 > 0 line correspond to solid spherical shapes,

and patches along κ1 = κ2 < 0 are liquid spherical shapes.

• For κ1 = 0, the interface is cylindrical with solid inside, and for κ2 = 0, the

interface is cylindrical with liquid on the inside.

2.3.2. Rate of Change of Curvatures

Because coarsening changes the morphology of interfaces, we are concerned with not

just a static description, but also in how curvatures change as the interface itself

changes. Mendoza et. al. calculated the change in curvatures of a complex microstruc-

ture by computing the time rate of change of H and K using an order parameter

description as in Equation 2.10 and 2.12 [26]. However, this approach did not consider

the effect of the motion of the interface itself on curvatures. To account for this effect,

another term must be added which has the form of a convective derivative. Following

is a more detailed exposition of this effect.

Let S (t) be a surface evolving in time with v(x, t) defined to be the velocity field of

S (t), which gives the velocities for evolving particles constrained to S . Given a scalar



44

field Φ(x, t), smooth in a 3-dimensional neighborhood of S (t) for all t, then by the

chain rule [27],

(2.15)

Φ = ∂ Φ

∂t + v · ∇Φ

Therefore the time derivative of mean curvature for a point on the interface moving

through space with velocity v can be found in the same way:

(2.16)

H =

∂H

∂t + v · ∇H

Similarly, the time derivative of Gaussian curvature for a point on the interface moving

through space with velocity v is

(2.17)

K = ∂K

∂t + v · ∇K

Using a diffuse interface model,

H and

K can be found for all points in the 3D grid,

and their values on the interface can be found by interpolating values from the grid to

points on the interface.

Taken together with the local velocity of the interface v and its derivatives with

respect to principal directions 1, 2, the time rate of change of the mean and Gaussian

curvatures H and K are given by [28]:

(2.18)

H = −(2H 2 − K )v − 12

(v11 + v22)

(2.19)

K = −2HKv − H (v11 + v22) +√

H 2 − K (v11 + v22)



45

(where we have used the notation

H and

K rather than ∂H ∂t and ∂K

∂t as in [28] to be

clear that we are considering a point moving with the interface).

2.4. Topology

Topology is the study of solid bodies which are not changed by elastic deformations

such as stretching and bending. In materials science, methods from topology are used

to characterize the connectivity of multiphase microstructures. The use of topological

measurements as a means of characterizing microstructures was pioneered by DeHoff

et. al. [29]

One of the most convenient measures of the topology of a microstructure is the

genus, which is defined as the maximum number of cuts along closed simple curves

that can be made in a body without breaking it up into smaller bodies. Genus is a

topological invariant of a surface, meaning that it does not change if the surface is

elastically deformed. Genus can also be related to the number of loops or handles in

the structure minus the number of independent bodies or voids:

(2.20) g = h − v + 1

To illustrate genus, consider a few simple examples. For a simple sphere, g = 0,

since there is no cut that can be made along a simple curve that will render the body

still connected. Using Equation 2.20, a sphere has no handles and is itself a single void,

giving g = 0. A sphere, cube, pyramid, and cylinder are all topologically equivalent

and therefore all have g = 0. For two isolated spheres, g = −1 because you would need

to join the two surfaces (a “negative cut”) in order to produce a connected body. A



46

torus or doughnut-shape will remain connected after a single cut, so it has a genus of

1.

Genus can also be related to other geometric properties of a microstructure. Us-

ing the Gauss-Bonnet theorem, genus can be related to the integral of the Gaussian

curvature K of the interface [30]:

(2.21) g = 1 − 1

4π

S

KdS

where the integral is over the surface area of the interface.

Genus can also be related to the Euler characteristic, another topological invariant.

For a surface of polyhedra, the Euler characteristic χ is calculated from the number of

vertices V , edges E , and faces F :

(2.22) χ = V − E + F

The genus can be calculated from the Euler characteristic using

(2.23) g = 1 − χ

2

2.5. Self-Similarity

In the context of a system evolving in time, a microstructure that is self-similar is

time-independent when scaled by the appropriate time-dependent characteristic length.For example, in the LSW theory [12, 13], the particle size distribution is self-similar

when scaled by the average particle radius R(t), which is proportional to t1/3 (Fig-

ure 2.7).



47

Figure 2.7. Particle radius distribution predicted by the LSW theory [4].The horizontal axis is the radius of a particle divided by the averageparticle radius for the entire collection of particles at time t. The verticalaxis is the probability of finding a particle with that radius. Because the

distribution is constant with time, the particle size distribution is saidto be self-similar.

Since this first use of the concept of self-similarity in materials science, other mate-

rials systems have been investigated to determine if they exhibit self-similar evolution

in the late stages of coarsening. Mendoza [20] and Kammer [21] investigated the

evolution of directionally solidified dendritic Al-Cu and Pb-Sn, respectively, coarsened

just above the eutectic to form a solid-liquid mixture. They found that S −1v ∝ t1/3,

and thus that S −1v is an appropriate characteristic length scale for coarsening in this

system, analogous to R(t) for a spherical particle distribution. Since in each case the

morphology is a complicated interconnected structure, ISDs were used to characterize

the morphologies, and were scaled by S −1v . They found that for these directionally

solidified systems, the scaled ISDs were not self-similar, even for extremely long coars-

ening times. Instead, the final morphology was liquid cylinders whose axis was aligned

with the solidification direction.

Fife conducted a similar study of dendritic Al-Cu during coarsening [22], but in

this case the dendrites were produced by equiaxial solidification. She found that again,



48

S −1v ∝ t1/3, but that in this case, the ISDs were self-similar for sufficiently long coarsen-

ing times. When compared with [20] and [21], this shows how the initial morphology

of the dendritic structure can have a significant impact on its morphological evolution.

Self-similarity has also been found in the shape of the interface between phases

before and after pinch-off by Rayleigh instability. As the pinch-off is approached, a

finite-time singularity occurs as the curvature at the point of pinch-off becomes infinite.

The length and time scales near the singularity become much smaller than the scale of

the initial conditions that caused the singularity to form.

The following example is presented to clarify this concept. Consider a rod of one

phase embedded in another, with surface energy σ at the interface between the two

phases. To reduce its surface energy, the rod will pinch off if a perturbations to the

interface to the interface position forms with a wavelength of 2πR or greater, where

R is the radius of the perturbation. This is the well-known Rayleigh instability, which

causes a stream of liquid flowing from a faucet to break up into drops.In Figure 2.8, the time evolution of the rod is shown. As the cylinder approaches

the time of pinch-off, its shape approaches that of two opposing cones with angle 2α.

Depending on the physical laws that govern the system as it approaches pinching, the

radial and axial positions of the interface may be made time-independent by scaling by

the appropriate time-dependent characteristic length. Scaling by this time-dependent

factor is referred to as a similarity transformation, and is of the form:

(2.24) η = rB

(ts − t)α, ξ =

zB

(ts − t)α



49

Figure 2.8. The pinch-off of a cylinder by Rayleigh instability as a func-tion of time. For some systems, when the coordinates of the interface

position are scaled by B/(ts − t)α to put them into dimensionless coor-dinates, the shape is independent of time.

where r is the radial coordinate of the interface position, z is the axial coordinate, η

and ξ are the radial and axial coordinates in the time-independent similarity variables,

B is a dimensional constant, ts is the time of pinching, and α is the exponent specific to

the physical process, such as α = 2/3 for pinching of an inviscid fluid drop or α = 1/4

for pinching of solid rods by surface diffusion. Figure 2.8 shows that the interface shape

becomes time-independent after scaling by the factor B/(ts − t)α.

Although this example has focused on the time prior to pinch-off, self-similarity can

also occur after pinching in the shapes of the cones as they draw away from each other.



50

Self-similarity was first found in the shape of a tapered liquid sheet after breaking

and a liquid wedge just after contact with a flat solid surface [31]. An axisymmetric

film bridge of inviscid liquid was found to exhibit self-similar behavior near breakup

into two droplets with an even smaller satellite droplet in between [32]. An important

result of this work was that a unique cone angle for the liquid droplets’ shape was

found prior to film breakup, regardless of initial conditions. A similar result was found

for the splitting of an inviscid liquid droplet in air when a thin bridge between the two

larger drops forms [33]. Wong et. al. investigated the pinch-off of rods via capillarity-

driven surface diffusion [5]. They found that self-similar solutions to the shape existed

both before and after pinch-off, but the solutions before and after were of a different

nature. After pinch-off, a solution was found for any cone angle 2α. However, prior

to pinching, a solution was found only for a unique angle α = 46.04. This implies

that the pinching event introduces a universality in the morphology, which persists

after pinching, even though solutions are found at any angle after pinching. Zhanget. al. investigated a similar geometry as [33] for a fluid of viscosity λη (where λ is a

constant) surrounded by another fluid of viscosity η [34]. They found the self-similar

shapes and asymptotic angles of the liquid bridges, which were in good agreement with

the experimental results of Cohen et. al. [35]. In summary, self-similarity is found in

a variety of systems undergoing topological singularities driven by interfacial energy

reduction, and the singularities cause universality in the interface shapes. Although

some experiments have verified these theoretical predictions, self-similarity has not yet

been observed experimentally in a system of interest to materials scientists.



51

2.6. Phase-Field Models

In the past few decades, great progress has been made in simulating various solidifi-

cation processes through the use of phase-field models. Phase-field models use an order

parameter to represent the phase of a system at each point. For instance, in models

of solidification, the order parameter may be assigned a value of φ = 0 for the solid

phase and φ = 1 for a liquid. The interface between the two phases is represented by

a continuous variation of the order parameter between the equilibrium values, with a

width that is typically much larger than the actual width of an interface between two

phases in a real system. Because of the continuous variation of the order parameter,

this type of model is referred to as a diffuse-interface model, as opposed to a sharp-

interface model, in which the boundaries between different phases must be explicitly

tracked.

To model the dynamics of the system, it is assumed that a free-energy functional

(function of a function) exists that depends on the order parameter φ, its gradient ∇φ,

and other variables such as temperature and composition. For instance, in the case of

an isothermal, constant volume process such as coarsening, the Helmholtz free energy

is the appropriate thermodynamic quantity to be represented by a functional:

(2.25) F =

V

f (φ , c , T , . . . ) +

2φ2 |∇φ|2 +

2c2 |∇c|2

dV

This functional is minimized with respect to changes of the appropriate variables

using the calculus of variations. In order to ensure that the free energy decreases



52

monotonically with time, it is assumed that

(2.26) ∂φ

∂t = −M φ

δ F δφ

= −M φ

∂f

∂φ − 2φ∇2φ

This is known as the Allen-Cahn equation [36]. The free energy density f (φ , c , T , . . . )

is typically a double-well potential function of the order parameter with local minima

at the equilibrium values of solid and liquid. The width of the interface is a balance

between the local free energy and the gradient energy; the system wants to minimize

the interface width to minimize its free energy, but then order parameter cannot change

too quickly or the gradient energy term will become excessively large.

In a single-component system, evolution is also determined by temperature varia-

tion and the net rate of latent heat production at the solid-liquid interface, which is

proportional to ∂φ∂t :

(2.27) C P ∂T

∂t + L

∂φ

∂t =

∇ ·(k

∇T )

where T is the local temperature, C P is the constant pressure heat capacity, L is the

latent heat of solidification, and k is the thermal conductivity.

The model can easily be extended to a binary alloy by modifying the free energy

density using ideal or regular solution models. For regular solution model of compo-

nents A and B,

f (φ,c,T ) =(1 − c)f A(φ, T ) + cf B(φ, T )

+ RT [c ln c + (1 − c)ln(1 − c)] + c(1 − c) [ΩS (1 − p(φ)) + ΩL p(φ)]

(2.28)



53

where f A and f B are the (double-well) free energy density functions for component A

and B alone, c is the mole fraction of component B, ΩS and ΩL are the regular solution

parameters of the solid and liquid, and p(φ) is an interpolating function with values

p(0) = 0, p(1) = 1. In the case of a multicomponent system, it is usually assumed

that heat diffusion through the system occurs much more quickly than solute diffusion,

so that system can be considered isothermal or having a fixed temperature gradient

with conditions set by external constraints. Using this assumption, the evolution of

the system is determined by the Allen-Cahn equation coupled with the Cahn-Hilliard

equation [37]

(2.29) ∂c

∂t = ∇ ·

M cc(1 − c)∇

δ F δc

= ∇ ·

M cc(1 − c)∇

∂f

∂c − 2c∇2c

For a more general, thermodynamically rigorous formulation, an entropy functional

S is postulated [38, 39]

(2.30) S =

V

s(e,c,φ) − 2e

2|∇e|2 − 2c

2|∇c|2 − 2φ

2|∇φ|2

dV

where s is the entropy density and e is the internal energy density. Using this

functional, the system is evolved such that entropy is monotonically increasing as a

function of time. By making the appropriate substitutions and simplifications, the

equations for heat diffusion, solute diffusion, and phase field evolution can be derived,

and all the previously discussed models can be recovered.

To prove that their models are consistent with physical laws, most authors perform

an asymptotic or thin-interface analysis. In this procedure, it is shown that when



54

the interface between phases is made sufficiently thin, the equations which define the

analytical solution to the problem are recovered from the equations that define the

phase-field model. For example, when an asymptotic analysis is applied to a phase-

field model of alloy solidification, the equations for solute conservation, solute diffusion,

and the Gibbs-Thomson equation are recovered.

2.6.1. Phase-Field Modeling of Pure Material Solidification

Phase-field models have been especially popular for modeling solidification problems.

For pure materials, many authors have modeled dendrite growth in an undercooled

liquid. [40–48] To simulate dendrite growth, some form of anisotropy in the system is

required, either in the gradient energy coefficient φ or the kinetic coefficient M φ. The

concentration of the liquid is set to be supersaturated for the given temperature, and

a seed of solid or thermal noise is introduced to provide a nucleation site.

McFadden et. al. [42] helped validate some of the original models of 2D dendritic

growth in a pure material [40, 41] by using an asymptotic analysis to show that their

model for anisotropy of the gradient energy term reduced to the anisotropic form of

the Gibbs-Thomson equation in the sharp-interface limit. Karma and Rappel [43]

introduced a new method for performing a sharp-interface analysis which allowed the

use of much wider interface thicknesses (on the same size scale as the capillary length)

and therefore facilitated modeling much larger physical systems. It also allowed the

selection of model parameters such that modifications to the interface kinetic coefficient

M φ could be made much smaller than modifications to the gradient energy coefficient,

which could be useful for modeling systems under a variety of undercoolings. They then



55

applied the model to 3D dendritic growth and showed that their results for dendrite

tip velocity and radius were comparable to the analytical solutions of solvability theory

and to experimental results in succinonitrile [44]. Karma and Rappel also used a phase

field model with intentionally introduced thermal noise to allow the system to grow

secondary arms [45]. Bragard et. al. modeled dendritic solidification in undercooled

nickel, using the results of molecular dynamics simulations to obtain realistic values

for the kinetic and gradient energy coefficients [48].

Although most models have been numerically simulated on a uniform rectangular

grid using standard finite-differencing techniques, some models have been implemented

with more efficient computational techniques. Provatas et. al. modeled dendrite growth

using an adaptive grid whose spacing was set progressively larger farther away from

the solid-liquid interface. [49] That way, the resolution was fine near the interface

where ∇φ and ∇c were large, and resolution was much coarser far from the interface

where φ and c were not spatially varying. Plapp and Karma introduced a model whichused an ensemble of random walkers to solve the diffusion equation for heat far from

the interface, which took progressively larger steps with increasing distance from the

growing interface [46].

2.6.2. Phase-Field Modeling of Binary Alloy Solidification

By modifying the free energy density to include corrections for alloy composition,

phase-field solidification models can be extended to binary alloys [41,50–57]. However,

to avoid the additional computation necessary to solve for coupled heat and solute

diffusion, it is usually assumed that heat diffuses much more quickly than solute, so



56

the system is either isothermal or held at a fixed temperature gradient by external

constraints.

Wheeler et. al. introduced a model for isothermal binary alloys in which the compo-

sition gradient energy coefficient c = 0 [50]. They used an asymptotic analysis to show

that the solute diffusion equation and solute conservation at interfaces were recovered

in the sharp-interface limit. Wheeler et. al. then expanded this model to include a

nonzero c in order to reproduce the experimentally observed effect of solute trapping,

in which the composition of solid growing rapidly into a liquid region is higher than

the equilibrium solid composition [41]. Elder et. al. created a model of an isother-

mal eutectic system and included a stochastic noise term in both the Allen-Cahn and

Cahn-Hilliard equations to stimulate nucleation [52]. Depending on initial conditions,

they reproduced in 2D diffusion-limited growth, lamellar growth, and spinodal decom-

position (or Ostwald ripening for off-eutectic compositions). Warren and Boettinger

created a model for an ideal binary alloy in 2D based on an entropy functional withanisotropic interfacial energy (φ) [54].

Diepers et. al. added the capability to model convective flow in the melt of an

isothermal binary system [55]. They simulated coarsening of an array of particles

without and with convective flow and found that particle radii evolved with t1/3 without

flow, in agreement with the LSW theory, and changed to t1/2 with flow, in agreement

with theoretical predictions.

A major advance in phase-field modeling of binary alloys came with the realiza-

tion by Karma that unequal diffusivities in the solid and liquid phases would lead

to three important nonequilibrium effects at the solid-liquid interface, (i) a chemical



57

potential jump across the interface leading to solute trapping, (ii) an interface stretch-

ing correction to solute conservation, and (iii) a surface diffusion correction to solute

conservation [56]. These effects scale with the thickness of the interface, meaning that

models with relatively large values of interface thickness (for computational efficiency in

modeling physically large systems) would see solute trapping and other nonequilibrium

effects appear at a much lower interface velocity than would be expected physically.

Karma then employed several clever tricks in order to eliminate these effects. He intro-

duced an antitrapping current into the Cahn-Hilliard equation which would eliminate

the effects of solute trapping:

(2.31) ∂c

∂t = ∇ ·

M c∇δ F

δc − jat

He then used an asymptotic analysis to show how to choose the parameters of

the phase-field model correctly in order to eliminate all three nonequilibrium effects.

Although this approach is less appealing from a thermodynamic standpoint since the

system’s evolution is no longer determined from an energy functional alone, the as-

ymptotic analysis shows that it accurately reproduces the sharp-interface equations

and therefore should accurately model the evolution of the system. It even has the

additional benefit of allowing corrections to the kinetic coefficient to the interface to

be made arbitrarily small, as in [43].



58

CHAPTER 3

Methods

3.1. Three-Dimensional Microstructure Characterization

In a diffuse-interface or order parameter-based representation of a microstructure,

the phase of the system at each point is represented by the value of the order parameter

in a 3D array. This array is made up of voxels, or volume elements, the 3D equivalent of

the 2D pixels that make up a digital image. Much useful information can be obtained

from analysis of the microstructure using this 3D array as a starting point. The volume

fraction V f of each phase can be calculated by counting voxels of each phase.

In many cases we are interested in the interface between the two phases. A digital

representation of the interface is needed to calculate surface area, interfacial curvatures,

and topological characteristics. This digital representation is created starting from the

3D order parameter-based array. A threshold value of the order parameter is chosen,

usually halfway between the equilibrium values of the order parameter in each phase.

The surface dividing the phases at the threshold value is then calculated from the 3D

voxel array using a marching cubes algorithm [58]. This surface is represented digitally

as a set of small polygons in space connected at their vertices. The positions of the

vertices are specified by their (x,y,z ) coordinates. This surface mesh calculation is

performed in the IDL software package using the SHADE VOLUME command.



59

3.2. Interfacial Curvature

Once the digital representation of the interface is calculated, the curvatures at each

vertex of the interface can be computed. This has also been accomplished in the IDL

software package’s programming language, using the algorithm described in [59]. The

mean and Gaussian curvatures H and K are calculated first since they are invariants

of the curvature tensor for each vertex. Then, principal curvatures κ1 and κ2 are

calculated using Equation 2.13 and 2.14. The surface area and surface area per unit

volume S v can also be calculated from the surface areas of the polygons that make

up the interface. Interface Shape Distributions are made by summing the area around

each vertex (the Voronoi area) in a bin for the appropriate combination of κ1 and κ2

and plotting the bins as a contour plot.

The curvatures of the interface can also be calculated using a different algorithm.

Using a diffuse-interface representation of the microstructure, the mean and Gauss-

ian curvatures are calculated at each grid point of the 3D volume based array using

Equation 2.10 and 2.12:

H = 1

2 (∇ · n)

K = n · adj(He(φ))n

Rewriting these equations in terms of the order parameter φ,

(3.1)

H =φxx

φ2

y + φ2z

+ φyy (φ2

x + φ2z) + φzz

φ2

x + φ2y

− 2 (φxφyφxy + φyφzφyz + φxφzφxz)

2

φ2x + φ2

y + φ2z

3/2



60

K =φ2

xφyyφzz

−φ2

yz+ φ2

y (φxxφzz

−φ2

xz) + φ2

zφxxφyy

−φ2

xy

φ2x + φ2

y + φ2z

2− 2 [φxφy (φxyφyz − φxyφzz) + φyφz (φxyφxz − φyzφxx) + φxφz (φxyφyz − φxzφyy)]

φ2x + φ2

y + φ2z

2

(3.2)

where subscripts denote derivatives in the x, y,z directions. First and second deriva-

tives of the order parameter are calculated using central differences:

φx = φi+1,j,k − φi−1,j,k

2∆x

(3.3)

φxx = φi+1,j,k − 2φi,j,k + φi−1,j,k

(∆x)2(3.4)

where φi,j,k represents the value of the order parameter at grid coordinates i, j, k, and

similar for derivatives in the y, z directions. These stencils are accurate to second-

order, meaning they have an error of order (∆x)2. A four-point stencil is used for

mixed partial derivatives:

(3.5) φxy = φi+1,j+1,k − φi+1,j−1,k + φi−1,j−1,k − φi−1,j+1,k

∆x∆y

which is also of second-order accuracy. Once H and K are determined at all grid

locations i, j, k, their values on the interface can then be determined by interpolating

the values at the fixed gridpoints onto the interface using the IDL INTERPOLATE

function.



61

3.3. Topological Characterization

The topological characteristics of the microstructure are quantified using the genus.

The genus can be calculated using the Gauss-Bonnet theorem:

(3.6) g = 1 − 1

4π

S

KdS

In this case, the Gaussian curvatures K are calculated using the order parameter

representation of the microstructure using Equation 3.2. The values of K at the fixed

3D grid points are then interpolated onto the vertices of the digital representation of the

interface described in Section 3.1. This is performed in IDL using the INTERPOLATE

function. Equation 3.6 is then numerically integrated using the areas of the polygons

making up the interface.

3.4. Calculation of Interfacial Velocities

Using an order-parameter based representation of a 3D microstructure, the veloci-

ties of the interface can be calculated using:

(3.7) v = φt

|∇φ|

where φt is the time derivative of the order parameter, which is calculated using a finite

difference approximation between times t and t + ∆t:

(3.8) φt = φi,j,k(t + ∆t) − φi,j,k(t)

∆t



63

3.5. Phase Field Models

3.5.1. Isothermal Solidification of Cu-Ni

The isothermal solidification of a Cu-Ni alloy (58 at.% Ni) was simulated using the

phase-field method as described in [60]. The solidification temperature is 1574 K. At

this temperature and for this alloy composition, the equilibrium solid volume fraction

is 75%. The free energy is given by the functional

(3.12) F =

V

2φT

2 |∇φ|2 + f (φ,c,T ) + f ori

dV

where φ is an order parameter representing the state of the system at each point

(φ = 0 for solid, and φ = 1 for liquid), T is the temperature, φ is the gradient energy

coefficient, which determines the strength of the gradient term, f is the local free

energy density, and f ori accounts for the interfacial energy of grain boundaries. The

gradient term ensures a diffuse interface between solid and liquid phases. The free

energy density is given by

(3.13) f (φ,c,T ) = wT g(φ) + [1 − p(φ)] f S (c) + p(φ)f L(c)

where g is a double well function with minima at 0 and 1, w is the free energy scale,

p(φ) interpolates smoothly between 0 and 1, and f S (c) and f L(c) are the local free

energy densities in the solid and liquid phases, respectively. The orientation energy

f ori is assumed to be proportional to the misorientation angle between adjacent grains.

The system is allowed to evolve by relaxational dynamics, with random supercritical

solid seeds added to stimulate nucleation. The simulations are carried out on a 640 ×



64

640 × 640 simulation volume using periodic boundary conditions, with grid spacing of

13.125 nm per gridpoint and time step of 1.31 ns. The diffusion coefficient in the liquid

is DL = 10−9 m2/s, and DS = 0 is assumed in the solid. The interfacial energy is taken

to be isotropic and equal to 5.25 × 10−2 J/m2. The 10-90% thickness of the interface

was 64.6 nm, resulting in approximately 5 points through the interface. Due to the

rapid solidification velocity that results from the high supersaturation at the start of

the simulation, growth is expected to be limited by interface kinetics. Thus, to produce

a dendritic morphology, the mobility M φ is anisotropic. Nucleation and growth of the

dendrites causes the solid volume fraction to grow with decreasing Avrami-Kolmogrov

exponent p as the equilibrium volume fraction is approached.

3.5.2. Isothermal Coarsening of Al-Cu

The phase field model used to simulate coarsening is based on one developed by

Echebarria, Karma, et. al. for a dilute binary alloy. [61] The original work includes

a fixed temperature gradient, but our model assumes the system is isothermal (adding

the temperature gradient to our implementation is possible if necessary). The free-

energy functional is

(3.14) F = V

f (φ, T m) + f AB(φ,c,T ) +

σ2φ

2 |∇φ|2

dV

where f (φ, T m) is the double well potential for the order parameter φ

(3.15) f (φ, T m) = H (−φ2/2 + φ4/4)



65

with a solid represented by φ = 1 and liquid by φ = −1. f AB interpolates between the

two bulk free energies

(3.16) f AB(φ,c,T ) = f A(T m) − (T − T m)s(φ) + RT m

v0(c ln c − c) + (φ)c

where s(φ) and (φ) interpolate independently between entropy and internal energy

between the phases, and the expression for the mixing entropy has been simplified by

assuming that c is small so that (1 − c) ≈ 1 and ln(1 − c) ≈ −c. The dynamics of

the system are controlled by the Allen-Cahn equation, derived as before from the first

variation of the order parameter, and the Cahn-Hilliard equation modified to include a

phenomenological antitrapping current as in Equation 2.31. This approach allows the

use of unequal diffusivities in the liquid and solid phases while simultaneously canceling

spurious effects at the interface caused by these unequal diffusivities, including, as

mentioned before, (i) a chemical potential jump across the interface leading to solute

trapping, (ii) an interface stretching correction to solute conservation, and (iii) a surface

diffusion correction to solute conservation.

The model is implemented on a uniform rectangular grid in three dimensions us-

ing FORTRAN 90, with explicit time steps and central differences used to evaluate

derivatives. The time step

(3.17) ∆t = 0.6(∆x)2

4D

(where ∆x is the grid spacing in units of interfacial width W , and D is the nondimen-

sional diffusion coefficient as described in [61]) is chosen to ensure numerical stability

in 3D. The model is run on parallel computing systems using MPICH-MX message



66

passing to pass data at interfaces between nodes. Since the model is intended for use

with experimental data represented as a 3D binary array, where the interfaces between

phases are initially sharp in the raw data, it is necessary to have a procedure to allow

the interfaces to reach their equilibrium shape (a hyperbolic tangent profile in 1D)

without allowing the interface positions to move significantly while equilibrating. The

procedure is to first allow both the order parameter and composition to simultaneously

evolve for only a short time to allow the order parameter to equilibrate. The order pa-

rameter is then held fixed and composition is allowed to equilibrate. Once composition

has equilibrated, both variables are then allowed to evolve simultaneously again.



67

CHAPTER 4

Phase-Field Simulation of Equiaxed Dendritic Solidification in

a Binary Alloy

4.1. Microstructure, Volume Fraction, and Surface Area

The isothermal solidification of a Cu-Ni alloy (58 at.% Ni) was simulated using

the phase-field method as described in Section 3.5.1. As shown in Figure 4.1, isolated

dendrites grow from the supercritical seeds with random orientations relative to one

another. As time progresses, the dendrites continue to grow and begin to impinge on

one another, and the solid phase gradually becomes interconnected. Following 78.6

µs, (Fig. 4.1(d)) the system has essentially reached the equilibrium volume fraction

solid of 75%. Afterwards, coarsening begins to dominate the changes to the system’s

microstructure. Figure 4.2 shows the microstructure after the system enters the coars-

ening regime. Due to the high V f , it is helpful to also visualize the system with the

liquid phase capped at the boundary.

The system’s volume fraction solid V f initially grows very rapidly, then slows as

the system approaches its equilibrium V f , as determined by the system’s position on

the phase diagram (Figure 4.3). S v also grows rapidly in the beginning as V f grows.However, as the system approaches its equilibrium V f , S v begins to decrease due to the

effect of coarsening.



68

(a) (b)

(c) (d)

Figure 4.1. Microstructures of Cu-Ni system undergoing dendritic solid-ification at (a) 26.2 µs, (b) 41.9 µs, (c) 57.6 µs, and (d) 78.6 µs. Visual-izations are a 640 × 390 × 128 subset of the 640 × 640 × 640 simulationvolume. The solid phase is capped at the boundary of the visualizationregion.

As previously discussed, S −1v has been recognized as a useful quantity to describe the

length scale of the system during coarsening. However, its behavior during solidification

has not been previously explored. As S v is rapidly increasing during solidification,

S −1v rapidly decreases, then begins to grow again as the system begins coarsening

(Figure 4.4(a)). Soon after the system begins coarsening, it is found that for the

limited amount of time simulated, S −1v increases linearly with t1/3 (Figure 4.4(b)).



69

(a) (b)

(c) (d)

(e) (f)

Figure 4.2. Microstructures of Cu-Ni system entering the coarseningphase at (a),(b) 105 µs, (c),(d) 131 µs, and (e),(f) 157 µs. The solid phaseis capped at the boundary of the visualization region in (a),(c),(e) and theliquid phase is capped in (b),(d),(f). Visualizations are a 640 ×390×128subset of the 640 × 640 × 640 simulation volume.



70

!

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!#)

!#*

!#+

!

$

%

&

'

(

)

! %! '! )! +! $!! $%! $'! $)! $+!

V o l u m e F r a c t i o n S o l i d

S v

( µ m - 1 )

Time (µs)

,-

./

Figure 4.3. V f and S v versus time.

!

!#$

!#%

!#&

!#'

(

(#$

(#%

(#&

(#'

$

$ ) % * &

S v

- 1 ( µ m )

t 1/3 (µs1/3)

(a)

! # $%$&'() * $%$+,-

./ # $%---

$

$%$0

$%+

$%+0

$%1

$%10

$%2

& &%0 0 0%0 ,

S v

- 1 ( µ m )

t 1/3 (µs1/3)

(b)

Figure 4.4. S −1v vs. t1/3 during (a) solidification and coarsening, (b)coarsening regime only (last 3 data points of (a)). The straight line-fit tothe data is consistent with other observations of S −1v during coarsening,but much longer simulation times would be required to prove this scalingconclusively.

This is consistent with other observations of coarsening behavior, but data over several

decades of time is necessary confirm this scaling behavior conclusively, which was not

practical given the time required to perform these simulations.



71

During solidification, S −1v is not a useful length scale because surface area increases

rapidly as dendrites nucleate and grow. However, as shown in Figure 4.5, for this

system, V f /S v remains constant to within ±7% during solidification, and thus this

quantity is a useful length scale to characterize this microstructure. As the system

approaches the equilibrium solid volume fraction and coarsening begins to determine

the changes in the microstructure, S −1v begins to increase with the characteristic t1/3

dependence of coarsening, while V f remains constant, making V f /S v a characteristic

length useful throughout the solidification and coarsening process.

!

!#$

!#%

!#&

!#'

!#(

!#)

!#*

!#+

!#,

$

!

!#!(

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! %! '! )! +! $!! $%! $'! $)! $+!

V o l u m e F r a c t i o n S o l i d ( V f )

V f

/ S v

( µ m )

Time (µs)

-./01

-.

Figure 4.5. Volume fraction solid approaches the equilibrium value of 0.75. V f /S v remains constant (±7%) during solidification, then beginsto increase as the system enters the coarsening regime.

Several authors have suggested possible relationships between V f and S v during

solidification. Cahn [62] suggested that S v ∝ V 2/3

f or(1 − V f )2/3 as V f approaches 0 or

1, respectively. Rath [63] suggested that S v ∝ V mf (1 − V f )n, where m, n are param-

eters fit to the experimental data. Limodin et.al. [64] found that experimental data

from solidification of Al-10 wt.% Cu during continuous cooling at -3C/min was fit

well by Rath’s equation. However, the present simulations differ from the conditions

employed in Limodin’s experiments since we are considering isothermal solidification



72

and thus a volume fraction of solid of 1 can never be reached. Li and Beckermann [65]

analyzed data from an isothermal dendritic growth experiment performed in micro-

gravity and found scaling relationships between dendrite tip radii, surface area, and

volume fraction, but the results are only applicable to a single dendrite without making

assumptions about the density of nuclei.

4.2. Interface Shape Distributions

Interface shape distributions were created for the microstructures at each time step

as described in Section 3.1. It was necessary to smooth the microstructures in order to

eliminate noise from the ISDs. Volume smoothing with a width of 3 and 5 voxels was

tested as described in Section 3.4. Figure 4.6 shows that resulting ISDs were consistent

at both the early and late stages of simulation. 5 voxel smoothing was chosen for the

remaining ISDs for consistency with the smoothing necessary in Section 4.3.

Figure 4.8 shows how the ISDs of the simulation change during the initial stages

of the transformation when solidification dominates. At 26.2 µs, where the structure

consists mostly of free-growing dendrites, approximately 50% of interfacial area has

both principal curvatures positive. These patches are primarily located at the tips of the

still-forming primary and secondary dendrite arms. 45% of interfacial area has K < 0

or κ1 < 0, κ2 > 0 (saddle-shaped patches), and is mostly found between adjacent tips

of the growing secondary arms. The 5% remaining area has both curvatures negative

and is found in the regions between adjacent primary and secondary arms. These

regions are shown highlighted in color for a single isolated dendrite in Figure 4.7.



73

-15 -10 -5 0 5 10

-10

-5

0

5

10

15

0 .0 00 E+ 00 6 .1 06 E- 03 1 .2 21 E- 02 1 .8 32 E- 02 2 .4 42 E- 02 3 .0 53 E- 02

!![1/"m]

! #

[ 1 / " m ]

(a)

-15 -10 -5 0 5 10

-10

-5

0

5

10

15

0 .0 00 E+ 00 6 .3 51 E- 03 1 .2 70 E- 02 1 .9 05 E- 02 2 .5 40 E- 02 3 .1 75 E- 02

!![1/"m]

! #

[ 1 / " m ]

(b)

-15 -10 -5 0 5 10

-10

-5

0

5

10

15

0 .0 00 E+ 00 6 .3 50 E- 03 1 .2 70 E- 02 1 .9 05 E- 02 2 .5 40 E- 02 3 .1 75 E- 02

!![1/"m]

! #

[ 1 / " m ]

(c)

-15 -10 -5 0 5 10

-10

-5

0

5

10

15

0 .0 00 E+ 00 6 .3 50 E- 03 1 .2 70 E- 02 1 .9 05 E- 02 2 .5 40 E- 02 3 .1 75 E- 02

!![1/"m]

! #

[ 1 / " m ]

(d)

Figure 4.6. ISDs for (a), (b) 26.2 µs, (c),(d) 157 µs. (a) and (c) werevolume smoothed with a width of 3, (b) and (d) with a width of 5.Comparison shows no significant differences due to smoothing.



74

Figure 4.7. An isolated dendrite at t = 26.2 µs. Patches in Region 1of the ISD (κ1, κ2 both positive) are colored red. Patches in Regions 2and 3 of the ISD (κ2 positive, κ1, negative) are colored grey. Patches inRegion 4 of the ISD (κ1, κ2 both negative) are colored blue.

As volume fraction increases, there is a general shift of the distribution to saddle-

shaped interface patches and a decrease in the intensity of the peak for solid cylindrical

patches (κ1 = 0). After 78.6 µs, where the system has essentially reached its equilibrium

volume fraction, only 35% of patches have both principal curvatures positive, while 59%

are saddle-shaped, and the peak of the distribution is almost completely contained in

the saddle-shaped region. This shift is due to the dendrites impinging on one another

and therefore a slower increase in concave interfacial area relative to saddle-shaped

interfacial area. However, during the process, there is actually an increase in the region

where κ2 > 10 µm−1 between 41.9 µs and 57.6 µs, which starts to shrink back down

by 78.6 µs. Figure 4.9 shows that this region is dominated by the tips of secondary



75

arms which form as the system’s volume fraction increases. As the equilibrium volume

fraction is approached, these high-curvature regions are the first to be eliminated by

coarsening, which is evident in the ISDs from later times (see Figure 4.10). Despite

the significant change in the volume fraction from 26.2 to 78.2 µs, there is no major

change in the breadth or peak location of the ISDs. This implies that there is no major

change in the scale of the curvatures in the structures during solidification, which is

consistent with the constant length scale, V f /S v, noted above.

Figure 4.10 shows the evolution of the ISDs at later times. Saddle-shaped regions

with relatively low curvature are the dominant feature of the microstructure. The

peak is mostly comprised of patches that are located on the gently curved solid-liquid

interfaces. These gently curved interfaces are frequently separated by a thin liquid

layer between adjacent solid dendrites which approach each other, but the remaining

liquid is too enriched in solute to freeze. The high-κ2 regions corresponding to dendrite

tips are gone. However, regions with κ1 < −8 µm

−1

, κ2 ≈ 0, also relatively high inthe magnitude of the curvature, persist longer. Many of these liquid cylindrical shapes

are caused by liquid tubes pinching off then retracting, as shown in Figure 4.11. Many

liquid cylindrical shapes are also found at the edges of the interdendritic liquid. This

tail remains relatively constant throughout the ISDs shown in Figure 4.10, indicating

that the interdendritic liquid remains and that pinching events continue to occur.

As discussed in Section 2.5, during coarsening, microstructures may become self-

similar, in which case the ISD becomes time-independent when scaled by S v. For the

early stages of coarsening simulated in this system, the ISDs are not self-similar after

scaling (Figure 4.12). Therefore, the system has not yet begun to exhibit self-similar



76

-15 -10 -5 0 5 10

-10

-5

0

5

10

15

0 .0 00 E+ 00 6 .3 51 E- 03 1 .2 70 E- 02 1 .9 05 E- 02 2 .5 40 E- 02 3 .1 75 E- 02

!![1/"m]

! #

[ 1 / " m ]

(a)

-15 -10 -5 0 5 10

-10

-5

0

5

10

15

0 .0 00 E+ 00 6 .3 50 E- 03 1 .2 70 E- 02 1 .9 05 E- 02 2 .5 40 E- 02 3 .1 75 E- 02

!![1/"m]

! #

[ 1 / " m ]

(b)

-15 -10 -5 0 5 10

-10

-5

0

5

10

15

0 .0 00 E+ 00 6 .3 50 E- 03 1 .2 70 E- 02 1 .9 05 E- 02 2 .5 40 E- 02 3 .1 75 E- 02

!![1/"m]

! #

[ 1 / " m ]

(c)

-15 -10 -5 0 5 10

-10

-5

0

5

10

15

0 .0 00 E+ 00 6 .3 50 E- 03 1 .2 70 E- 02 1 .9 05 E- 02 2 .5 40 E- 02 3 .1 75 E- 02

!![1/"m]

! #

[ 1 / " m ]

(d)

Figure 4.8. ISDs of Cu-Ni system undergoing dendritic solidification at(a) 26.2 µs, (b) 41.9 µs (c) 57.6 µs, (d) 78.6 µs. Color bar set to thesame scale for all plots.



77

Figure 4.9. A (1.31 × 1.83 × 1.44 µm) subset of the microstructure at41.9 µs. Patches with κ1 > 0, κ2 > 10 µm−1 are colored red.

coarsening. However, the system may eventually exhibit self-similar coarsening after

sufficiently long coarsening time, as in [66].

4.3. Interfacial Topology

The topological characteristics of the simulated Cu-Ni alloy were also investigated.

The genus, g, was the primary metric used to quantify the system’s topology. Since g =

h−v +1, the number of voids v contributes to genus. To quantify this contribution, the

number of voids of each phase was counted using the IDL LABEL REGION function.

Since the simulations use periodic boundary conditions, single voids which extend over

the boundaries cause an over-count of the number of voids by one. To correct for this,

an IDL program was written to scan along the x = 0, y = 0, and z = 0 planes and

subtract one from the void count for every void which extended across the periodic

boundary.

Table 4.1 shows that periodic boundary conditions have an important effect on

the count of bodies. By visualizing the edges of the simulation region, it was found

that the majority of the excess voids removed were small tips which barely extended



78

-15 -10 -5 0 5 10

-10

-5

0

5

10

15

0 .0 00 E+ 00 6 .3 50 E- 03 1 .2 70 E- 02 1 .9 05 E- 02 2 .5 40 E- 02 3 .1 75 E- 02

!![1/"m]

! #

[ 1 / " m ]

(a)

-15 -10 -5 0 5 10

-10

-5

0

5

10

15

0 .0 00 E+ 00 6 .3 50 E- 03 1 .2 70 E- 02 1 .9 05 E- 02 2 .5 40 E- 02 3 .1 75 E- 02

!![1/"m]

! #

[ 1 / " m ]

(b)

-15 -10 -5 0 5 10

-10

-5

0

5

10

15

0 .0 00 E+ 00 6 .3 50 E- 03 1 .2 70 E- 02 1 .9 05 E- 02 2 .5 40 E- 02 3 .1 75 E- 02

!![1/"m]

! #

[ 1 / " m ]

(c)

Figure 4.10. ISDs of Cu-Ni system undergoing coarsening at (a) 105 µs,(b) 131 µs, and (c) 157 µs.



79

Figure 4.11. A (1.31 × 1.83 × 1.44 µm) subset of the microstructure at105 µs. (The liquid phase is capped at the boundary.) Patches withκ1 < −8 µm−1, −4 µm−1 < κ2 < 12 µm−1 (liquid cylindrical regions)are colored red. These patches show the origin of the extended tailfound mostly in Region 3 of in the ISDs of Figure 4.10. These regionsare associated with liquid tubes that are pinching off and with the edgesof interdendritic liquid regions that separate solid dendrites.

Time (µs) V f (% solid) Solid voids Solid voids (PBC) Liquid voids Liquid voids (PBC)

26.2 7.69 82 58 1 141.9 26.0 84 11 2 157.6 49.6 93 1 11 178.6 69.2 57 1 18 2105 72.7 28 1 25 8131 73.1 15 1 35 15157 73.3 12 1 35 20

Table 4.1. Number of solid and liquid voids for each time. PBC= cor-rected for effect of periodic boundary conditions.

over the periodic boundary. The solid phase becomes continuous at 41.9 µs, when

V f = 50%. The system is bicontinuous at that time as well. Following 41.9 µs, the

solid phase remains continuous, while the liquid phase begins to develop isolated voids.



80

-5 -4 -3 -2 -1 0 1 2

-2

-1

0

1

2

3

4

5

0 .0 00 E+ 00 8 .4 46 E- 02 1 .6 89 E- 01 2 .5 34 E- 01 3 .3 78 E- 01 4 .2 23 E- 01

!!/S

v

! "

/ S

v

(a)

-5 -4 -3 -2 -1 0 1 2

-2

-1

0

1

2

3

4

5

0 .0 00 E+ 00 8 .4 46 E- 02 1 .6 89 E- 01 2 .5 34 E- 01 3 .3 78 E- 01 4 .2 23 E- 01

!!/S

v

! "

/ S

v

(b)

-5 -4 -3 -2 -1 0 1 2

-2

-1

0

1

2

3

4

5

0 .0 00 E+ 00 8 .4 46 E- 02 1 .6 89 E- 01 2 .5 34 E- 01 3 .3 78 E- 01 4 .2 23 E- 01

!!

/Sv

! "

/ S

v

(c)

Figure 4.12. ISDs of Cu-Ni system undergoing coarsening at (a) 105 µs,(b) 131 µs, and (c) 157 µs. ISDs scaled by S v and plotted with same colorbar. The changes in the ISDs with time mean that the microstructure isnot self-similar.



81

(a) (b)

(c)

Figure 4.13. Portion of the simulation microstructure at (a) 105 µs, (b)131 µs, and (c) 157 µs. Liquid phase is capped at the boundary. Redarrow highlights a long liquid cylinder that pinches off by Rayleigh in-stability at two points simultaneously, leaving an isolated liquid void inbetween after pinching.

These isolated liquid voids form when a long liquid tunnel pinches off by a Rayleigh

instability at two points simultaneously, leaving a liquid void in between (Figure 4.13).

The calculated number of voids v was also used to test the accuracy of the genus

calculation. Genus was calculated as described in Section 3.3. The original phase-field

data at time 26.2 µs was compared to data volume-smoothed with widths of 3, 5, 7,

and 9 (Table 4.2).



82

Smoothing Interface width (points) GenusNone 5 -13703 6 -92.95 8 -54.47 9 -44.59 11 -32.9

Table 4.2. Computed genus versus volume smoothing width for isolateddendrites at t = 26.2 µs. The interface width is defined as the numberof interface points where the order parameter is between 10% - 90% of the equilibrium liquid value.

Since there are 58 solid voids at t = 26.2 µs, each of which has 0 handles, the

expected genus is -57. The computed value with a smoothing width of 5 gives the best

agreement with the computed value (within 4.6%). Smoothing with a width of 3 does

not produce wide enough interfaces for an accurate Gaussian curvature calculation,

while smoothing with a width of 7 or more begins to weld adjacent solid dendrites

together.

Genus was calculated for each time step after the order parameter was volume

smoothed with width 5. Results are shown in Table 4.3 and plotted on a per unit

volume basis in Figure 4.14. Genus is initially negative as isolated solid dendrites

without handles nucleate throughout the simulation volume. Handles in the solid

structure (or equivalently, tunnels of liquid) form as adjacent dendrite arms coalesce,

as shown in Figure 4.15. From 26.2 to 105 µs, handles form much more quickly than

new solid particles nucleate, causing genus to increase.

As the system approaches its equilibrium solid volume fraction and coarsening

begins to dominate (105 µs to 157 µs), genus decreases due to the breakup of liquid

tubes that result from a capillary-driven instability, as shown in Figure 4.16. Each



83

!"

$

"

%

&

'

(

)

$ %$ '$ )$ *$ "$$ "%$ "'$ ")$ "*$

G e n u s / V o l [ ! 1 0 9 / m m 3 ]

Time (µs)

Figure 4.14. Genus per unit volume versus time.

Figure 4.15. A handle (liquid tunnel) is formed as adjacent secondaryarms join, in red circle (t = 41.9 µs). The solid phase is capped at theboundary.



84

Time (µs) Genus26.2 -54.4041.9 213.857.6 102278.6 2292105 2836131 2666157 2492

Table 4.3. Computed genus versus time.

pinching event will cause genus to decrease by one. At the same time, although the

solid phase remains continuous, isolated liquid droplets begin to form (Figure 4.13),

which also causes genus to decrease. However, it can be seen by comparing Figure 4.14

and 4.17 that gv decreases by 5.8 × 108/mm3 while the number of liquid bodies per

unit volume increases by only 3.2 × 107/mm3 from 105 µs to 157 µs, meaning that the

pinch-off of handles is a much more important contributor to the change in topology

than the formation of liquid voids at this early stage of coarsening.

During coarsening, quantitative comparison between systems is facilitated by con-

verting the genus to a scaled genus, gvS −3v . This is a dimensionless number that can

most easily be understood as the genus per characteristic volume, and is expected

to be a constant for systems undergoing self-similar coarsening. From 105 µs to 157

µs, gvS −3v increased from 0.068 to 0.087, meaning that the system is not topologically

self-similar during the early stages of coarsening.

4.4. Comparison with Other Systems

The morphology and topology of other dendritic microstructures produced under

different conditions have also been characterized [20, 21, 66]. In this section, we will



85

(a) (b)

Figure 4.16. The breakup of liquid tubes due to capillarity-driven insta-bility. The liquid phase is capped at the boundary. (a) 78.6 µs, (b) 157µs.

!

!#!$

!#!%

!#!&

!#!'

!#(

!#($

! $! %! &! '! (!! ($! (%! (&! ('!

B o d i e s / V o l [ ! 1 0 9 / m m

3 ]

Time (µs)

Solid Bodies/Vol

Liquid Bodies/Vol

Figure 4.17. Number of independent bodies per unit volume for solid and

liquid phases. The formation of either a liquid or solid body decreases thegenus by 1. The minimum value plotted for each series, 1.7 × 106/mm3,represents a single continuous body divided by the simulation volume.



86

compare these experiments to the results of Section 4.2 and 4.3. The use of simulations

allows direct characterization of the solidification process that was not available in

these experiments, and shows that the morphology and topology of the system during

solidification were not self-similar.

In [66], the morphology and topology of dendritic microstructures produced by

equiaxial solidification in Al-Cu were characterized using ISDs and genus per unit vol-

ume. In these experiments, the initial morphology of the dendrites is controlled by the

interfacial energy anisotropy, not the kinetic coefficient anisotropy as in these simula-

tions. In both cases, the morphology and topology of the initial microstructures evolve

non-self-similarly during the early stages of coarsening. In [66], after long coarsening

times (18 hours and longer), the morphology and topology did become self-similar for

both solid volume fractions investigated (46% and 72%). Such long coarsening times

were not accessible for these simulations due to the extremely long computation times

that would be required. The scaled genus gvS

−3

v of the 74% solid sample was 0.02for the liquid phase genus and 0.06 for the solid phase genus, both of which are in the

same order of magnitude as the Cu-Ni simulations, but the several orders of magnitude

difference in time scales makes any conclusions about the universality of scaled genus

difficult.

The morphology of dendritic microstructures after directional solidification in Al-

Cu [20] and Pb-Sn [21] were also characterized using ISDs. In these systems, a strong

directionality was observed as coarsening proceeded due to the directional solidification

process, and self-similarity was not observed in the morphology. Like the present



87

simulations, the initial morphology introduced by the solidification process lacks self-

similarity, and in the case of the experiments, this non-self-similar evolution persists

long into the coarsening regime.

The topology of the Al-Cu system after directional solidification was also investi-

gated [67]. The system’s solid volume fraction was 74%, very close to the 75% of the

simulated Cu-Ni system being studied. In spite of the very different time scales (the

earliest coarsening time investigated in [67] was 10 min), the scaled genus gvS −3v = 0.11

at 10 min was quite close to the value found at t = 157 µs of 0.087 for the simulations.

The topology of the Al-Cu system continued to evolve non-self-similarly into the late

stages of coarsening, as gvS −3v decreased to 0.45 after 964 min of coarsening. In both

systems, in the early stages of coarsening there were many more handles than voids,

so handles contributed much more to genus. In the later stages of [ 67], the number of

voids becomes comparable to the number of handles, causing the scaled genus gvS −3v to

decrease. The large number of liquid droplets formed in [67] result from the pinchingof liquid cylinders, similar to the process described in Section 4. Again, since very long

coarsening times were not accessible for these simulations, it is not known whether

the gvS −3v for these simulated dendrites will achieve a steady-state value or begin to

decrease due to the formation of liquid voids.

4.5. Conclusions

A rapidly solidifying Cu-Ni alloy was simulated using a phase-field model, and

the morphology and topology of the dendritic microstructure were characterized. V f

increased rapidly until it reached the equilibrium value of 75%. S v rapidly increased



88

during solidification, then began to decrease as the microstructure began to coarsen.

V f /S v remained relatively constant during solidification and then began to increase

during coarsening, suggesting its possible use as a characteristic length throughout the

solidification and coarsening process.

The morphology of the microstructure was characterized using ISDs. Initially the

majority of interfacial area had both principal curvatures positive, then shifted to a

majority of saddle-shaped patches as the equilibrium V f was reached. During the

coarsening regime, the peak of the ISD remained in the saddle-shaped region. There

was also an extended tail in the liquid-cylindrical region caused by liquid cylinders

pinching off and by the edges of interdendritic liquid regions. The system’s morphology

evolved in a non-self-similar manner early in the coarsening regime.

The microstructure’s topology was characterized using the genus. g was initially

negative as isolated dendrites without handles nucleated, then increased rapidly as

handles formed by adjacent dendrite arms grew together. Genus then decreased asthe system entered the coarsening regime, due to the pinch-off of liquid tubes and the

formation of liquid voids by simultaneous Rayleigh instabilities in long tubes. Pinching

of liquid tubes has a much larger effect on the changes in topology than formation of

liquid voids. The system’s topology also evolved in a non-self-similar manner early in

the coarsening regime.



89

CHAPTER 5

Rate of Change of Curvatures

5.1. Details of Algorithm

As discussed in Section 2.3.2, the time evolution of mean and Gaussian curvatures

can be found using Equation 2.16 and 2.17:

H = ∂H

∂t + v · ∇H

K = ∂K

∂t + v · ∇K

An algorithm to calculate

H and

K was developed. The algorithm takes as input

the microstructures of a two-phase system at two subsequent time steps separated by

time ∆t. The overall approach is to first calculate mean and Gaussian curvatures

using Equations 2.10 and 2.12 at all points on the grid, using centered finite-difference

approximations for first and second derivatives. The partial derivatives of H and K

with respect to time are found using a finite difference approximation at each point in

the 3D grid:

(5.1)

∂H

∂t =

H (t + ∆t)

−H (t)

∆t

and similarly for K . The components of the term v · ∇H and v · ∇K are calculated

next. The x, y,z components of velocity are calculated using Equations 3.9-3.11. The



90

components of ∇H and ∇K are calculated from H and K using upwind differenc-

ing [68], which uses the direction of the velocity field at each point in the 3D grid to

determine which direction to take a one-sided finite difference to find ∂H ∂x etc.

Once all components of Equation 2.16 and 2.17 are calculated, they are summed

at all points on the 3D grid. To find the values of

H and

K on the interface between

phases, the INTERPOLATE function in the IDL programming language is used.

5.2. Examples

To show the importance of the term v · ∇H and to provide test cases for the

algorithm, following are a few examples using simple functions which allow analytical

solutions to be found.

5.2.1. Cylinder Expanding in the Radial Direction

Let a two-phase system be represented on a three-dimensional Cartesian grid by an

order parameter φ with equilibrium values φ+ = 1 and φ− = −1 and a diffuse interface

of varying width between phases. Let the order parameter be given by the function

(5.2) φ = tanh

x2 + y2 − vt

/W

where v is the velocity of the interface, t is time, and W is a constant controlling the

interface width. This forms a cylinder whose central axis runs along the z -direction.

The cylindrical interface is given by the level curve φ = 0 and moves outward with

constant velocity v , with φ− inside the cylinder and φ+ outside.



91

Figure 5.1. Plot of φ = tanh

x2 + y2 − vt

/W

in the xy-plane for

vt = 25, W = 5, where black represents φ = −1, white represents φ = 1.

The normal vector n is given by

(5.3) n = ∇φ

|∇φ| =

x

x2

+ y2

, y

x2

+ y2

, 0and the mean curvature is

(5.4) H = 1

2 (∇ · n) =

∂

∂x

x

2

x2 + y2

+

∂

∂y

y

2

x2 + y2

=

1

2

x2 + y2

This shows that the curvature of a patch of surface on the cylinder is just one-half of the

curvature of the circle in the xy plane, as expected from the equation H = (κ1 + κ2)/2

since one of the principal curvatures for a cylinder is zero.

This equation also shows the importance of the term v · ∇H when calculating the

time derivative of curvature. Intuitively the curvature of a cylinder expanding in time

must be changing. Since there is no explicit time dependence in H , ∂H ∂t

= 0. Therefore,



92

in the level set representation, the motion of the interface (outward at constant velocity

v) must be accounted for by the term v · ∇H to find the time rate of change of H .

Since the cylindrical interface is moving outward in the normal direction at velocity

v,

(5.5) v = vn =

x x2 + y2

v, y x2 + y2

v, 0

and the time rate of change of mean curvature

H for a point on the cylindrical interface

is given by

H = ∂H

∂t + v · ∇H (5.6)

H = 0 +

x x2 + y2

v, y x2 + y2

v, 0

·

∂H

∂x ,

∂H

∂y , 0

(5.7)

H = −v

2 (x2 + y2)(5.8)

which shows that

H is nonzero, as expected.

5.2.2. Translating Cylinder

Another example which will demonstrate this method is a translating cylinder with

constant radius moving at constant velocity vx in the x direction. In this case, we

intuitively expect that since the cylinder’s radius is not changing, H = 0. The cylinder

can be represented mathematically by the function

(5.9) φ = tanh

(x − vxt)2 + y2 − k

/W



93

The normal is given by

(5.10) n = ∇φ

|∇φ| =

x − vxt (x − vxt)2 + y2

, y

(x − vxt)2 + y2, 0

and mean curvature is

(5.11) H = 1

2 (∇ · n) =

1

2

(x − vxt)2 + y2

In this case since there is explicit time dependence in the mean curvature,

∂H

∂t =

∂

∂t

1

2

(x − vxt)2 + y2

(5.12)

= vx(x − vxt)

2 [(x − vxt)2 + y2]3/2(5.13)

which is nonzero as expected. To find the time derivative of curvature at a point

moving with the interface, since v = vxx,

H = ∂H

∂t + vx

∂H

∂x

= vx(x − vxt)

2 [(x − vxt)2 + y2]3/2 − vx

(x − vxt)

2 [(x − vxt)2 + y2]3/2

= 0

as expected.

5.3. Testing of Algorithm

The algorithm was then tested on the examples of Section 5.2. The algorithm to

calculate H and K was tested first using a cylinder as represented using a hyperbolic



94

tangent function as in Equation 5.2 (as shown in Figure 5.1) with vt = 25, and varying

interface width W (units of length are dimensionless gridpoints). At the interface, the

curvature calculation should give H = (1/25+0)/2 = 0.02. After calculating H at the

fixed gridpoints and interpolating H to the interface, a minimum of H = 0.0199 and a

maximum of H = 0.0203 were found for the W = 5 case, in good agreement with the

expected value.

Figure 5.2. Plot of mean curvature H for φ =

tanh

x2 + y2 − vt

/W

along x for y = 0, z = 0. vt = 25,

W = 2.5 (white), W = 5 (red).

For an accurate calculation of the term ∇H , H should vary smoothly through the

interface. Figure 5.2 shows the effect of varying the number of points through the

interface (controlled by W ) on the profile of H . As W is increased and the number

of points through the interface increases, H varies more smoothly. To quantify the



95

number of points through the interface, we define the interface as the region where

−0.9 < φ < 0.9. Using this definition, W = 2.5 gives 5 points through the interface,

and W = 5 gives 15 points. As Figure 5.2 shows, the magnitude of H is close to 0.02 at

the interface (plotted as x = 25, 75) for either value of W , but the derivative is much

smoother for W = 5, which will be required for later calculation of ∇H .

The example case of the expanding cylinder was also used to test the calculation

of interfacial velocities and v · ∇H . Equation 5.2 for an expanding cylinder was again

used, with vt = 25 for the array at time t and vt = 25.01 used for time t + ∆t, and

W = 5. In the case of the expanding cylinder, the expected value of

H on the interface

is given by Equation 5.8 as

(5.14)

H = −v

2 (x2 + y2) =

∆x

2∆t (x2 + y2) = 1.38 × 10−4/(gridpoints · τ 0)

where v = ∆x/∆t = 0.173 gridpoints/τ 0 (which is determined by is the arbitrary

dimensionless time step ∆t = 0.0577τ 0).

Initial testing showed the calculation of

H was inaccurate, showing a four-fold

anisotropy with accurate values along the x, y axes and incorrect values along y =

x, y = −x. To determine the source of error, each component was calculated separately.

The values of ∂H ∂t

were close to zero, as expected. The components of velocity were

calculated using Equation 3.9-3.11. The quantity v2 = v2x +v2y +v2z was calculated for all

points in the 3D Cartesian grid and then interpolated to the interface. The values of v2

on the interface ranged between 0.0308 and 0.0304 (gridpoints/τ 0)2, in good agreement

with the expected value of 0.0299 (gridpoints/τ 0)2. Therefore the components ∂H ∂t

and

v were eliminated as the sources of error.



96

Figure 5.3. |∇H | for all points in the xy plane of φ =

tanh

x2 + y2 − vt

/W

. Black represents |∇H | = 0 and white

represents |∇H | = 1.5 × 10−3. The red circle shows the position of theinterface at vt = 25.

|∇H | was also calculated for all points on the 3D grid for time t and interpolated to

the interface. The expected value from the analytical solution was 8.00 × 10−4. Along

the x and y axes, the calculated value was 8.28 × 10−4, in reasonably good agreement

with the expected value; however, there was significant grid anisotropy in the calculated

value of |∇H |, with the maximum values found along the lines y = x and y = −x.

At these positions on the interface the calculated value of |∇H | was 1.5 × 10−3, nearly

twice the expected value.

Figure 5.3 shows that there is four-fold anisotropy present in |∇H |. Although the

calculated value of H was accurate, the additional derivative needed to calculate |∇H |increases the amount of anisotropy. To test the effect of increasing the interface width

on the anisotropy found in |∇H |, the parameter W in Equation 5.2 was increased to



97

10 and |∇H | was interpolated onto the interface. The anisotropy was considerably im-

proved, with |∇H | ranging from 8.36×10−4

to 8.58×10−4

, but at the cost of increasing

the width of the interfacial region to 29 points, prohibitively large for practical use.

5.4. Strategies to Decrease Anisotropy of |∇H |

Having isolated the inaccuracy in

H to grid anisotropy in the calculation of |∇H |,

several modifications to the algorithm were attempted to decrease the anisotropy of

|∇H | when interpolated onto the interface. The calculation of the components of

∇H and ∇K was changed from upwind differencing to isotropic finite differences as

described in [69]. For the first derivative, a six-point stencil without directional bias

in the lowest-order error term can be found:

H x = 1

2∆x 1

6 (H i+1,j+1,k − H i−1,j+1,k) +

4

6 (H i+1,j,k − H i−1,j,k)

+1

6 (H i+1,j−1,k − H i−1,j−1,k)

+ O(∆x2)

(5.15)

(Additional terms are needed to remove directional bias in the z -direction, but these

were not needed for this test since the cylinder is symmetric in the z -direction.)

However, the use of these isotropic finite differences resulted in little change in the

anisotropy of |∇H |. Fourth-order accurate stencils were also tried for the components

of ∇H and ∇K :

(5.16) H x = 1

12∆x2 (−H i−2,j,k + 8H i−1,j,k − 8H i−1,j,k + H i+2,j,k) + O(∆x4)

but again little improvement was found.



98

With changes in the algorithm to calculate ∇H and ∇K showing little benefit,

upwind differencing was restored, and changes were instead made to the algorithm for

calculating H and K . Isotropic stencils for first, second, and mixed partial derivatives

from [69] were used. The isotropic second-derivative stencil is

φxx = 1

∆x2

1

12 (φi+1,j+1,k − 2φi,j+1,k + φi−1,j+1,k)

+

10

12 (φi+1,j,k − 2φi,j,k + φi−1,j,k) +

1

12 (φi+1,j−1,k − 2φi,j−1,k + φi−1,j−1,k)

(5.17)

and the stencil for mixed partial derivatives remains as in Equation 3.5. Using isotropic

derivative stencils resulted in the maximum value of |∇H | decreasing to 1.27 × 10−3,

meaning error decreased from 89% to 59%. Since this error is still unacceptably high,

the code for calculating H and K was modified to use fourth-order accurate stencils

for first, second, and mixed partial derivatives. The first derivative stencil remains as

in Equation 5.16. The second derivative stencil is

(5.18) φxx = 1

12∆x2 (−φi−2,j,k + 16φi−1,j,k − 30φi,j,k + 16φi+1,j,k − φi+2,j,k) + O(∆x4)

and the mixed partial derivative stencil is

φxy = 1

144∆x∆y [φi−2,j−2,k − 8φi−2,j−1,k + 8φi−2,j+1,k − φi−2,j+2,k

− 8φi−1,j−2,k + 64φi−1,j−1,k − 64φi−1,j+1,k + 8φi−1,j+2,k

+ 8φi+1,j−2,k − 64φi+1,j−1,k + 64φi+1,j+1,k − 8φi+1,j+2,k

−φi+2,j−2,k + 8φi+2,j−1,k − 8φi+2,j+1,k + φi+2,j+2,k]

(5.19)



99

W vt at (t + ∆t) N. points −0.9 < φ < 0.9 Max. Error in

H

5 25.01 15 1.50%5 25.1 15 1.62%5 25.5 15 2.10%4 25.01 11 3.27%3 25.01 9 33.0%

Table 5.1. Summary of testing for expanding cylinder test scenario.

Using fourth-order accurate stencils, the accuracy of the calculation of

|∇H

|improved

considerably, with variation on the interface only from 8.02 × 10−4 to 8.5 × 10−4.

With confidence in the calculation of its components,

H was calculated for an

expanding cylinder using fourth-order accurate stencils for H and upwind differencing

for ∇H , and with vt = 25 for the array at time t, vt = 25.01 for time t + ∆t, and

W = 5. The values of

H interpolated on the interface deviated by only 1.50% from

the expected value of 1.38 × 10−4 calculated from the analytical solution.

Table 5.1 summarizes the results of testing the calculation of H for different test

conditions. The interface can move over a wide range of velocities with little impact

on the accuracy of the calculation of

H , as long as the interface is relatively wide.

Decreasing the width of the interface below 10 points quickly results in increasing

error due to grid anisotropy.

The code was also tested on the translating cylinder described by Equation 5.9,

in which case

H = 0 is expected. Results are shown in Table 5.2. The calculated

minimum and maximum values of

H on the interface are near zero as expected, and

consistent with Table 5.1, error increases with decreasing interface width and increasing

vxt.



100

W vxt at (t + ∆t) N. points −0.9 < φ < 0.9 Max.

H Min.

H

5 25.01 15 6.59 × 10−5

−6.01 × 10−5

5 25.1 15 9.37 × 10−5 −8.56 × 10−5

4 25.01 11 6.66 × 10−5 −6.17 × 10−5

3 25.01 9 1.08 × 10−4 −8.57 × 10−5

Table 5.2. Summary of testing for translating cylinder test scenario.

5.5. Conclusions

An algorithm to numerically calculate

H and

K was developed for an order pa-

rameter based representation of a microstructure. The algorithm was tested using

expanding and translating cylinders represented using a hyperbolic tangent function,

and the numerical results were compared to the analytical solutions. Fourth-order ac-

curate stencils for derivatives were needed in the calculation for H and K to obtain

accurate results. Due the the additional points sampled by the fourth-order accurate

stencils, relatively wide interfaces (> 10 points) were needed for accurate results.



101

CHAPTER 6

Using Phase-Field Modeling to Predict Interfacial Velocities

During Coarsening

6.1. Methodology

Phase-field models are especially useful for predicting the evolution of complex mi-

crostructures because they avoid the need to explicitly track the boundaries between

phases. However, due to the difficulty of experimentally observing microstructural

evolution in real time, there have been few attempts to directly compare phase-field

predictions of microstructural evolution with experimental data. Past work has com-

pared the evolution of parameters such as average secondary dendrite arm spacing [70]

or volume fraction during a phase transformation [71].

The development of synchrotron-based X-ray tomography has allowed, for the first

time, the details of microstructural evolution to be captured in 3D in real time. This

technique is capable of providing data that will allow the most direct comparison to

date of the predictions of a phase-field model to experimental data. In this chapter, we

will quantitatively compare 3D phase-field predictions of interfacial velocities with real-

time X-ray tomography data from the isothermal coarsening of a Al-Cu alloy, held justabove the eutectic temperature to form a solid-liquid mixture. This work thus serves

as a validation of the phase-field method and also shows how the results of phase-field



102

simulations can be used to determine physical parameters which are difficult to measure

independently, in this case the diffusion coefficient of solute in the liquid phase.

6.2. Experimental Procedure

Experimental observations of the Al-Cu alloy were provided by Dr. Julie L. Fife

of Northwestern University. Samples of Al-15 wt.%Cu (>99.99% purity, provided by

Ames Laboratory) were directionally solidified as in [25]. These samples were then

cut into 2 mm diameter specimens in preparation for in-situ X-ray tomography. The

sample was placed inside a thin-walled boron nitride holder, glued atop an alumina

rod, and then mounted on a rotating stage. A custom made furnace, developed at

Risø National Lab, was used to coarsen the samples at 553 C, 5 C above the eutectic

temperature. The sample was held at temperature while tomography is performed.

X-ray tomography was conducted on the TOMCAT beamline located at the X02DA

port of the Swiss Light Source (SLS) at the Paul Scherrer Institut (Villigen, Switzer-

land). X-ray photon energy of 30 keV was used, with an exposure time of 400 ms,

and 721 projections were captured over 180 of rotation. A total of 1024 slices were

collected every 5 min. 46 sec. Continuous scans were completed for up to 12 hours

for each sample. The reconstructed, two-dimensional grayscale images were segmented

using filters to convert them to binary. These images were then combined to create a

three-dimensional binary array of the order parameter φ with a resolution of 1.4 µm

in each direction. Further information about the TOMCAT beamline and the details

of the experiment can be found in [72] and [73], respectively.



103

6.3. Validating the Phase-Field Model for Simulating Coarsening

The phase-field model for predicting the interfacial velocities of the Al-Cu alloy

during coarsening was described in Section 3.5.2. Before using the Al-Cu microstruc-

ture in the model, the model was tested to validate its performance against a known

analytical solution. The decay of a sinusoidal perturbation at a solid-liquid interface

was simulated in the phase-field model, and the decay rate of the perturbation was

compared to the predictions of a linear stability analysis. The problem is defined

by the diffusion equation in the solid and the liquid, the Gibbs-Thomson condition

for equilibrium composition of the solid and liquid at a fixed interface, the far-field

condition, and the flux condition for conservation of solute at the interface:

(6.1)

C 0l − C 0s

V = −Dl∇C l · n + Ds∇C s · n

where the interfacial velocity V is in the +z direction. It is assumed that the compo-

sition decays exponentially from the value at the interface (determined by the Gibbs-

Thomson condition) to its equilibrium value, and that the amplitude A of the sinu-

soidal perturbation decays away in time with decay constant σ . It can be shown that

for λ p A W (where W is the width of the interface)

(6.2)

C 0

l − C 0

s

σ = −DlΓlk2 σ + Dlk2

Dl − DsΓsk2 σ + Dsk2

Ds

where Dl, Ds are the equilibrium liquid and solid diffusivities, Γl, Γs are the capillary

lengths in liquid and solid, λ p is the wavelength of the perturbation, and k = 2π/λ p.



104

Figure 6.1. Sample input array: White is solid, black is liquid. λ p = 120gridpoints, perturbation amplitude A = 10, Ds = 0. The gray border isfor visual clarity only and is not part of the simulation region.

(See Appendix A for details of the linear stability analysis.) This equation can be

solved numerically for σ .

To simulate the perturbation decay, a 2D array with a sinusoidal perturbation at

the solid-liquid interface was used as input to the phase-field model. Wavelength was

set to a fixed number of gridpoints, and grid spacing ∆x = ∆y = ∆z = 0.8W where

W is the dimensionless interface width. No-flux boundary conditions were used. Other

parameters used were for the Al-Cu system, where C 0S = 5.65%, C 0L = 33.0%, and

each timestep ∆t = 0.6(∆x)2/4DL where DL is the dimensionless diffusion coefficient

in the liquid. The diffusion coefficient in the solid DS is assumed to be zero. Figure

6.1 shows the input array with solid (φ = 1) represented by white and liquid (φ = −1)

represented by black.

The size of the liquid region is much larger than the solid region in this example to

allow the diffusion field to reach its equilibrium value at the boundary of the simulation



105

Figure 6.2. Decay of a perturbation from simulation results (red) com-pared with analytical solution calculated from Equation 6.2 (blue).

(at the top of the y-direction in this plot). Since Ds was set to 0 this was not necessary

on the solid side.

Once initial conditions were established as described in Section 3.5.2, each simula-

tion was allowed to evolve so the perturbation could decay, and the composition and

order parameter fields were periodically output to disk. For the initial (equilibrated)

array and for each output, the position of the interface was determined along the x = 0

edge of the boundary by linearly interpolating to find the point where φ = 0. A plot of

the perturbation amplitude as a function of time is shown in Figure 6.2 along with the

expected perturbation amplitude from the linear stability analysis. The decay constant

σ was calculated using least-squares fitting to an exponential decay function.



106

-45%

-40%

-35%

-30%

-25%

-20%

-15%

-10%

-5%

0%

5%

0 20 40 60 80 100 120 140 160 180 200

D e c a y C o n s t a n t E r r o r

Perturbation Wavelength (gridpoints)

% Error

(a)

0%

5%

10%

15%

20%

25%

0 20 40 60 80 100 120

D e c a y C o n s t a n t E r r o r

Epsilon

% Error

(b)

Figure 6.3. (a) Decay constant error for perturbation with A = 10 withvarious perturbation wavelengths, (b) decay constant error for constantsystem size, varying .

Simulation results were first compared to theory by varying the wavelength of the

perturbation λ p and comparing the decay constant predicted by the linear stability

analysis to the simulated decay constant. In the simulations, the perturbation ampli-

tude A = 10 gridpoints and the parameter = W/d0 (which sets the capillary length

d0 and therefore the strength of capillary forces) was set to 30. Agreement between

the linear stability analysis and simulation (Figure 6.3(a)) was good except for smaller

wavelengths where the assumption λ p A is no longer valid.

The parameter = W/d0 is expected to have an impact on the accuracy of the

phase-field model. In [61], the model was tested against the predictions of a linear

stability analysis for the amplification of a perturbation to a solid-liquid interface during

directional solidification. Agreement between simulations and experiment was worse as

was increased to a value of 100. The effect of was also tested in our implementation

of the model. The system’s physical size was kept constant by maintaining the product

of λ (in grid points) and constant, so that only the effect of changing was tested.



107

The system’s physical size is set by the product of the perturbation wavelength in

grid points and the grid spacing ∆x which converts from grid spacing to physical

dimensions:

(6.3) λ = λgrid∆x

and since ∆x = ∆y = ∆z = 0.8W (as set in the simulation parameters) and W = d0,

(6.4) λ = 0.8λgridW = 0.8λgridd0

Since d0 is a physical parameter of the system, Equation 6.4 shows that the system’s

physical size is maintained constant by keeping λgrid constant. The same scaling was

applied to the perturbation’s initial amplitude. Similar to [61], agreement became

worse as increased toward 100 (Figure 6.3(b)).

The effect of the antitrapping correction (as described in Section 3.5.2) on the decay

of the perturbation was investigated by repeating the calculation for a perturbation

of λ = 120, A = 10, = 30 with the antitrapping current set to zero. The decay

constant found from this simulation was 1.46 × 10−4, meaning that deviation from the

analytical prediction increased from 1.00% to 6.12% when the antitrapping current was

not included. This shows that the antitrapping current does minimize errors caused by

unequal diffusivities between the two phases, although agreement between simulation

and theory is still relatively good without antitrapping current. This is likely due to the

low velocity of the interface in this coarsening simulation relative to the much larger

velocity in directional solidification, which the model of [61] was created to simulate.



108

6.4. Calculating Interfacial Velocities from Experimental Data

Interfacial velocities in the normal direction were calculated from the experimental

microstructure as described in Section 3.4. The order parameter arrays were from two

subsequent tomographic scans, the first after 2.6 hours of isothermal coarsening, the

second 5 min 46 s later. Because the 3D arrays created by tomographic scans were large

(451 × 451 × 349 voxels), the full arrays are too large to simulate using phase-field

modeling in a reasonable amount of time. For this reason, a 100 × 80 × 150 voxel

subset of the array was selected for the velocity calculation and input into the phase-

field model. The arrays were smoothed using a 5-voxel volume smooth. To determine

the velocities on the solid-liquid interface, the velocities calculated at each point of the

order parameter array were linearly interpolated onto the level set representation of

the interface calculated from the first tomographic scan.

Figure 6.4 shows the subset of the microstructure used for the velocity calcula-

tion at the initial time step, and the velocities calculated using two different methods

of smoothing the order parameter array. In Figure 6.4(b), the order parameter was

smoothed only using the 5-voxel volume smoothing. Although the velocity profile is

fairly smooth across the surface, there are subtle variations caused by variations in

surface curvature in the experimental microstructure after only one smoothing oper-

ation. To determine whether these variations cause significant error in the calculated

velocities, the volume-smoothed order parameter arrays were input into the phase-field

model. The phase-field model quickly reduces small-scale surface curvature variations

in order to reduce the total interfacial energy of the system. The phase-field model

evolved both order parameter arrays for 60,000 iterations, with a interface width of



109

∆x = 0.4W and = 1000. Figure 6.4(c) shows that the interfacial velocities calcu-

lated using phase-field smoothing of the experimental microstructures does not differ

significantly from those calculated using only volume-smoothing. Therefore, the ex-

perimental velocities of Figure 6.4(b) will be used as the basis of comparison with

simulation results.

Interfacial velocities were also calculated from experimental time steps near those

shown in Figure 6.4. The purpose of these tests was to verify that there were no

anomalies in the experimental microstructures which could lead to errors in the velocity

calculation, and that the microstructure was not changing too fast to allow accurate

velocity calculation using a 5 min 46 sec interval. Velocities were calculated from the

2.6 hour microstructure and the microstructure two time steps (11 min 32 seconds)

before, and also one four time steps (23 min 4 sec) after. Velocities calculated were

consistent with the initial calculation.

6.4.1. Choosing Simulation Parameters

Many factors were considered in selecting parameters for the phase-field simulations

used to predict interface evolution. Previous testing showed that at least 10 points

through the interface (−0.9 < φ < 0.9) were needed for accurate calculation of

H

and

K from phase-field simulations (not covered in this work but an eventual goal of

the sponsoring project). Therefore, ∆x = 0.4W was selected, which gave an interface

width of 10 points, as opposed to ∆x = 0.8W , which gave an interface 6 points wide.

Initial testing of the velocity calculation with ∆x = 0.4W showed that the liquid

particle in the upper portion of the simulation volume (at z = 200 µm, Figure 6.4(a))



110

(a)

(b) (c)

Figure 6.4. (a) 100 × 80 × 150 voxel portion of microstructure used forcalculation of interfacial velocities (liquid phase is capped at boundaries),(b)-(c) solid-liquid interface colored by normal interfacial velocities cal-culated from experimental data (negative velocities point into the liquid),experimental time steps 5 min 46 sec apart. In (b) the order parameterarray was smoothed using only a 5-voxel volume smooth, in (c), the orderparameter array was also smoothed by the phase-field model.



111

had a significant negative velocity over the entire surface, in contrast with experimental

data and the simulation’s assumption of DS = 0. It was found that due to the wide

interface in comparison with the particle’s size, the order parameter never reached the

equilibrium value for liquid in the particle; the minimum value of the order parameter

in the liquid particle was φ = −0.74 instead of -1. Therefore, the particle was shrinking

in an effort to reach the equilibrium value to minimize the system’s energy. Because

there are many features in the system with similar sizes as this particle, it was necessary

to increase the grid point density of the simulation volume to allow the interfaces to

be adequately resolved. This was accomplished by using the REBIN command in IDL

to produce a simulation volume of 200 × 160 × 300 and changing the parameter ,

which sets the physical size scale of the system as follows. Since by the definition of ,

W = d0,

(6.5) ∆x = 0.4W = 0.4d0

(6.6) = ∆x

0.4d0

Since the grid spacing ∆x is given by the resolution of the X-ray experiment, prior

to the rebin command, ∆x = 1.4 µm. After rebinning, ∆x = 0.7 µm. d0 and other

physical parameters used are shown in Table 6.1.

(6.7) = 0.7 µm

0.4 × 2.27 × 10−3 µm = 771



112

Parameter Value ReferenceDL (liquid diffusion coefficient) 2.4

×10−9 m2/s [74]

mL (liquidus slope) −7.25 K/at. % [75]k (partition coefficient) 0.145 [75]Γ (Gibbs-Thomson coefficient) 2.41 × 10−7 m·K [76]d0 (Capillary length) 2.27 nm Γ

mL(c0

L−c0

S) [61]

Table 6.1. Physical parameters used in phase-field simulations.

The parameters of Table 6.1 are used to convert simulation time to real time. To

convert the dimensionless time step per iteration to physical time, the dimensionless

time step per iteration (Equation 3.17) is multiplied by the characteristic time τ as

described in [61]:

(6.8) τ = a1a23d20

DL

where a1, a2 are dimensionless constants and DL is the diffusion coefficient of solute in

the liquid.

As discussed in Section 3.5.2, before the velocity simulations were begun, the or-

der parameter and composition fields were allowed to equilibrate in a 2-step process.

The first step is to allow the order parameter and composition fields to simultaneously

evolve. The number of iterations must be long enough to fully equilibrate the order

parameter, but short enough that the microstructure does not begin to change signifi-

cantly. Testing was performed on a flat interface to estimate the number of iterations

required to equilibrate the order parameter at the solid-liquid interface. A 10 × 10

× 30 array was created with a sharp solid-liquid interface at z = 15. This array was

smoothed with a 5-voxel volume smooth and resized to 20 × 20 × 60 to match the



113

interface shapes of the experimental microstructure as closely as possible. The phase-

field code was modified so that every 104

iterations it subtracted the value of the order

parameter at each point from its value 104 iterations ago, and continued to run until

the maximum change in the order parameter at any point was less than 10−8. This

provided an estimate of the number of iterations required for the interfaces to equi-

librate. The simulation was run with simulation parameters ∆x = 0.4W , k = 0.145,

DS = 0, and = 700 (based on a rough estimate of the capillary length). 8.5 × 105

iterations were required for the interface to reach its equilibrium shape.

The resized subset of the experimental data was then input in to the phase-field

model. To further test the number of iterations required for the first step in the in-

terface equilibration process, the number of iterations in the first step was varied, and

the composition field was then equilibrated holding the order parameter fixed. The

resulting φ, c fields were evolved together for another 100,000 iterations, and interfa-

cial velocities were calculated between the beginning and end order parameter arrays(Figure 6.5). The simulation parameters were ∆x = 0.4W , k = 0.145, DS = 0, and

= 700, and no-flux boundary conditions. The large positive and negative velocities

near the boundaries of the simulation volume are due to the no-flux boundary condi-

tions imposed. The velocities on the isolated liquid droplets and the bulb-shaped liquid

region in the center of the simulation volume were chosen as the basis for comparison

with the experimental data, as they were best isolated from anomalous effects at the

boundaries due to the assumption DS = 0.

Figure 6.5 shows that after 800,000 initial iterations, the large anisotropy in ve-

locity seen in the bulb and liquid droplets (most clearly visible in Figure 6.5(a)) has



115

(a) (b)

Figure 6.6. (a), (b) Two different views of the larger simulation volumeselected for velocity calculations.

decreased significantly. Because of this result and the results obtained on the flat

interface described previously, 800,000 iterations was chosen as the number of initial

iterations to equilibrate the order parameter, followed by composition field equilibra-

tion as described previously. Again, this was chosen to equilibrate the order parameter

while maintaining the system microstructure as close to the original as possible.

Figure 6.5 shows the strong effect the no-flux boundary conditions imposed on the

simulation have on the calculated velocities at the boundaries. To further investigate

the effects of boundary conditions, a larger system volume was selected for input into

the phase-field model (Figure 6.6). The principles used in choosing this simulation

volume were:



116

• The liquid bulb at the center of the simulation volume will be the basis for

quantitative comparison of theory and experiment.

• The simulation volume is larger to provide a buffer for the bulb shape from the

effects of the no-flux boundary conditions. Additionally, the volume is chosen

so that the solid-liquid interfaces intersect the simulation boundaries as close

to perpendicular as possible, to minimize solute flux created by artificially

large curvatures due to no-flux boundary conditions.

• The height in the z -direction is large enough that the order parameter field

surrounding the liquid bulb reaches its equilibrium value before intersecting

the boundary.

• A large liquid volume remains connected to the liquid bulb to act as a reservoir

for solute in the simulation volume.

This 150 × 120 × 115 voxel simulation volume was rebinned in IDL as before to

a size of 300 × 240 × 230 voxels and loaded in to the phase-field model. Simulation

parameters were ∆x = 0.4W , = 771, k = 0.145, and no-flux boundary conditions

were used. The initial microstructure was equilibrated 800,000 iterations, followed by

holding the order parameter fixed and equilibrating the composition field. After the

order parameter and composition fields were equilibrated, the system was allowed to

evolve for 0.5, 1, and 5 seconds of real time and interfacial velocities were calculated.

The interfacial velocities were approximately the same for all three simulation times.



117

(a) Interfacial velocities calculated from experi-mental data. 5 minutes 46 seconds between X-rayscans.

(b) Interfacial velocities calculated from phase-fieldsimulations. 5 seconds simulated time elapsed forvelocity calculation.

Figure 6.7. Comparing experimental and simulated velocities. The solid-liquid interface is shown colored by normal velocities, with negative ve-locities pointing into the liquid. Liquid is inside the bulb shape, and thesolid phase is transparent. Note color bars between left and right imagesdiffer by a factor of 3.

6.5. Comparing Experiment to Theory

Experimental results are compared with simulation data in Figure 6.7. Interfacial

velocities are shown for a 150×160×180 voxel (105×112×126 µm) subset of the mi-

crostructure. The qualitative features of the velocity distribution on the liquid bulb

are similar, but it is important to note that the color bars of Figure 6.7(a) and 6.7(b)

differ by a factor of 3.

Figure 6.7 shows that qualitative agreement between experimental and simulated

velocities is good, with the model accurately reproducing most of the finer details of

the velocity distribution and their relative magnitudes correctly. However, as shown



118

Figure 6.8. Normalized histograms of experimentally measured andscaled simulated interfacial velocites. The simulated velocities are scaledby 1/2.9, the scale factor that produced the best least-squares fit betweenthe two normalized histograms.

by the color bars, the overall magnitude of the velocities estimated by the phase-field

simulations is higher by approximately a factor of three.

To better quantify how much the simulations overestimate the interfacial velocities,

the distributions of velocities from experiment and simulation (Figure 6.7) were plotted

as normalized histograms. A least-squares fit was used to determine the scale factor

between the normalized histograms (Figure 6.8). The simulated interfacial velocities

were a factor of 2.9 ± 0.1 greater than experimentally measured velocities.

6.6. Analysis of Contributions to Error

To understand the causes of the differences between experimentally measured and

simulated velocities, the following potential causes were considered: accuracy of the

phase-field model at high , use of a dilute solution approximation in the model, the



119

assumption DS = 0, and the physical parameters used to convert from dimensionless

simulation results to physical units.

6.6.1. Phase-Field Model at Large

To quantify the contribution of error from the phase-field model, it is necessary to

understand the limits of the model’s validity. Echebarria et. al. achieved well-converged

results using in the range of 10 to 100, even though their sharp-interface analysis

assumes that is of order unity. They showed that in the limit of vanishing interface

kinetics, results would still be well-converged for larger as long as W/R 1, where

W is the interface width and R is the local radius of curvature of the solid-liquid

interface, or equivalently, W H 1, where H is the local interfacial mean curvature.

For future work, it was desired to maintain the ratio of grid spacing to interface width

at ∆x/W = 0.4 in order to maintain approximately 10 points through the interface for

accurate calculations of mean and Gaussian curvatures and their rates of change. This

means well-converged results could be obtained as long as 2.5∆xH 1. This clearly

illustrates the tradeoff between accuracy, which requires ∆x small, and computation

time, memory and storage requirements, which increase as (1/∆x)3.

The mean curvature of the solid-liquid interface was calculated for the experimental

data using H = 12(∇ · n). The highest mean curvature was found at the tip of the

bulb-shaped region where H = 0.07 µm−1. To test the accuracy of the model at

this curvature and for an interface width of W = 0.4∆x, the decay of a sinusoidal

perturbation at a stationary solid-liquid interface with the same curvature and interface

width was simulated in two dimensions and compared with the results of the linear



120

stability analysis. The maximum curvature of the perturbation is found at its point of

maximum amplitude and is given by

(6.9) H = 4π2A

λ2

where A is the perturbation amplitude and λ is the wavelength. Assuming no diffu-

sivity in the solid, the linear stability analysis predicts that the perturbation decays

exponentially with decay constant σ :

(6.10) σ = DLd20k4

2

1 −

1 +

4

d20k4

where k is the magnitude of the wave vector of the perturbation.

To evaluate the effect of and grid spacing ∆x, perturbation decay simulations were

run maintaining a constant physical system size by simultaneously varying and grid

spacing as in Section 6.3. The decay constants of the perturbations were determined at

the time when the maximum curvature of the perturbation was equal to the curvature

at the tip of the bulb, and compared with the predictions of decay constants from the

linear stability analysis. The difference between the linear stability analysis predictions

and simulation results is shown in Figure 6.9.

The error for = 1400 and grid spacing of ∆x = 1.4 µm (the native resolution of

the experimental data) was 26%. At this grid spacing, W H = 0.23, which compares

reasonably with the results of [61], which reported significant error when W H 0.2.

Since = 771 was used in the simulations for velocity calculations, the phase-field

model is not a major contributor to error in the velocity predictions. Additionally,



121

0%

5%

10%

15%

20%

25%

30%

0 200 400 600 800 1000 1200 1400 1600

D e c a y c o n s t a n t E r r o r

Epsilon

Error

Figure 6.9. Difference between theoretical prediction and simulation re-sults for perturbation decay for high values of .

using this much larger value of than previously tested allows this model to be used

to simulate much larger physical systems for studies of isothermal coarsening.

6.6.2. Effect of Assumption DS = 0

The phase-field model used for these calculations assumes zero solute diffusivity in

the solid phase. Using DS = 1.5 × 10−13 m2/s [77], DS is more than four orders of

magnitude less than DL. Solute flux balance at the interface requires that

(6.11) (C 0L − C 0S )vn = DL∂C L∂n

− DS ∂C S

∂n

where vn is the velocity of the interface in the normal direction, derivatives with respect

to n are spatial derivatives taken in the normal direction, C 0L and C 0S are the equilibrium

liquid and solid compositions, and we assume that (C L − C S ) ≈ (C 0L − C 0S ). By

estimating the magnitude of ∂C S∂n , we will show that the contribution from the term

DS ∂C S∂n

is three orders of magnitude smaller than (C L − C S )vn. This will show that

the assumption DS = 0 used in the phase-field model is valid for calculating interfacial



122

velocities, and that this assumption is not a significant contribution to the disagreement

between simulations and experiment.

To estimate the amount of diffusion through the solid, we focus on the spherical tip

of the bulb-shaped liquid region. First we show that solute gradients remaining from

solidification should be eliminated by the time of this scan. The distance through the

solid from this spherical tip to the nearest interfacial regions (not shown) is 40 µm.

The system had been isothermally coarsened for approximately 2.6 hours at the time of

this scan. The characteristic length for diffusion through the solid at this time is given

by√

Dst = 37 µm, meaning that solute concentration gradients in the solid remaining

from the solidification process should be relatively smooth on the scale of the distance

between the spherical cap and nearest interfaces. Therefore, the solute flux through

the solid is driven primarily by the Gibbs-Thomson effect.

To estimate solute flux through the solid, it is assumed that since the spherical tip is

moving with velocity vn = 0.005 µm/s, vn/DS = 3.4 × 10

4

m

−1

, and the diffusion fieldsurrounding the spherical tip is approximately Laplacian. The solute concentration

field surrounding the spherical cap can be approximated by the concentration field

surrounding a spherical particle of radius R [78, 79]:

(6.12) C S (r) = C 0S + R

r

C S (r = R) − C 0S

where C S (r = R) is found from the Gibbs-Thomson equation. Using this assumption,

the flux of solute through the solid is given by

(6.13) − Ds∂C S

∂r = DS

[C S (r = R) − C 0S ]

R



123

By the Gibbs-Thomson equation,

(6.14)

C S (r = R) − C 0S

= (C 0L − C 0S )d0H

and using R = 1/H = 1/0.07 µm−1 at the tip of the spherical cap, the flux of so-

lute through the solid is [1.7 × 10−6 µm/s × (C 0L − C 0S )]. This is more than three

orders of magnitude less than the left-hand side of Equation 6.11, (C 0L − C 0S )vn =

[5 × 10−3 µm/s × (C 0L − C 0S )]. This shows that the flux through the solid is a much

smaller contribution to interface motion than flux through the liquid, and that DS = 0

is a valid assumption for these simulations.

6.6.3. Physical Parameters

Another possible source of disagreement between theory and experiment is the physical

parameters used in converting the dimensionless model results to dimensional units,

shown in Table 6.1. The phase-field model uses a dilute solution approximation which

assumes straight liquidus and solidus lines. Since the values of k and mL used were

taken from the Al-Cu phase diagram [75] at the coarsening temperature, the dilute

solution approximation should not cause error since the system’s position on the phase

diagram does not change.

The capillary length d0 is calculated from the Gibbs-Thomson coefficient and phase

diagram parameters as in [61]. The Gibbs-Thomson coefficient was measured by

Gunduz and Hunt in their grain boundary cusp experiments [76]. This measurement

was taken at very nearly eutectic temperature and liquid composition, and so should



124

be accurate for this experiment. The estimated error in the measured Gibbs-Thomson

coefficient is ±5% [76]).

Of the necessary parameters, the most difficult to measure is the diffusion coefficient

of solute in the liquid. By Equation 6.8, DL affects the units of dimensionless time

used in the phase-field equations:

τ = a1a23d20

DL

The effect of a lower DL is to reduce the simulated interfacial velocity. The value of

DL = 2.4 × 10−9 m2/s (at the eutectic point) used was determined by Lee et. al. [74].

Their measurement was lower than any previously measured values because they care-

fully controlled experimental conditions to try to eliminate the effect of convection in

the liquid, and showed how the presence of convection increased the effective diffu-

sion coefficient. DL is believed to be the largest possible contribution to disagreement

between the present experiments and simulation results, and the trend in DL with in-

creasingly accurate experimental measurements points toward better agreement. The

least-squares fit of simulated to experimental velocities can even be used to obtain a

better estimate of DL, due to the inverse proportionality of the time step to DL:

(6.15) DL = 2.4 × 10−9 m2/s

2.9 = 8.3 × 10−10 m2/s

The use of this estimate of DL will be further discussed in Chapter 7.



125

Figure 6.10. Interfacial velocity prediction using phase-field model with-out anti-trapping current.

6.7. Effect of Anti-trapping Current

To evaluate the effect of the anti-trapping current in the phase-field model on

interfacial velocity prediction and computation time, the simulation was repeated with

the anti-trapping term removed from the phase-field model.

As Figure 6.10 shows, there is very little difference between the model predictions

with and without the anti-trapping current. This is due to the relatively low interfacial

velocities found under conditions of isothermal coarsening relative to directional solid-

ification. By removing this term from the model, the computation time per iteration

was reduced by a factor of 1.6 without penalty in accuracy.



126

6.8. Conclusions

The interfacial velocities of a portion of a dendritic Al-Cu alloy undergoing coars-

ening were determined experimentally using X-ray tomography and simulated using

phase-field modeling. The features of the experimental and simulated velocity distri-

butions agreed well qualitatively, while the magnitudes of the velocity distributions

differed by a factor of 2.9. The diffusion coefficient of solute in the liquid is believed to

be the largest contribution to the discrepancy. These results suggest that the phase-

field method could be used to validate or even measure experimental parameters such

as DL. In validating the phase-field model, it was shown that much larger values of

the length scale parameter than previously tested were valid, allowing much larger

physical systems to be simulated for isothermal coarsening. Finally, it was shown that

for the present isothermal coarsening simulation, the antitrapping current could be

neglecting, increasing computational efficiency by a factor of 1.6.



127

CHAPTER 7

Self-Similar Pinch-off of Rods

As discussed in Section 2.5, self-similar pinch-off has been observed experimentally

in fluid dynamics systems. Although it has been shown theoretically that self-similar

pinch-off should occur in pinching by surface diffusion, the possibility of self-similarity

during pinching by bulk diffusion has not been explored theoretically or experimentally.

In the experiments described in Section 6.2, during the coarsening of the solid-liquid

mixture in the Al-Cu alloy, many liquid tubes were observed undergoing the pinching

process following Rayleigh instabilities. The mechanism of pinching is bulk diffusion

inside the liquid tube, with negligible contribution from diffusion through the solid.

Hundreds of tubes pinching were observed throughout the experiment. Because the

evolution of the tubes was monitored as a function of time, the shape of the tubes could

be observed for several time steps before the time of pinching. In most of the pinching

events, a cone shape with a similar angle was asymptotically approached sufficiently

close in space and time to the point of pinching. Figure 7.1 shows numerous liquid

tubes at various stages of the pinching process.

Motivated by experimental data as shown in Figure 7.1, we began to investigate

the possibility of a self-similar solution for the interface shape of a rod pinching by

bulk diffusion.



128

Figure 7.1. The microstructure of the Al-Cu alloy experimentally ob-served during coarsening, with the liquid region capped at the boundary.The region boxed in red shows several liquid tubes at various stages in

the approach to pinching and one immediately after pinching.

7.1. Theoretical Formulation

The pinching process is caused by the diffusional motion of Cu solute atoms through

the liquid driven by the interfacial energy between the solid and liquid phases. The

diffusion field in the liquid is given by

(7.1) ∂C L

∂t = DL∇2C L

where C L is the concentration of solute and DL is the diffusion coefficient of solute in

the liquid. We assume that DS = 0 in the surrounding solid. We seek a solution to



129

Equation 7.1 consistent with the boundary conditions. The first is the Gibbs-Thomson

equation

(7.2) C L = C 0L +

C 0L − C 0S

d0H

where C L is the solute concentration on the liquid side of the interface, C 0L and C 0S are

the equilibrium solute concentrations at a flat interface, d0 is the capillary length, and

H is the mean interfacial curvature. The other boundary condition is the interfacial

mass balance,

(7.3)

C 0L − C 0S

v = −DL

∂C L∂n

where v is the interfacial velocity in the direction normal to the interface, and ∂C L/∂n

is the derivative of the concentration field in the normal direction.

7.1.1. Similarity Variables

In cylindrical coordinates, the position of the solid-liquid interface as a function of

time is given by r = f (z, t). If a valid similarity coordinate system can be found,

the shape of the interface can be written in a time-independent form as η = f (ξ ), or

defined parametrically F = η − f (ξ ) = 0. Prior to pinching, the similarity variables

are expected to be of the form

(7.4) η = rB

(ts − t)α, ξ =

zB

(ts − t)α



130

where r is the radial coordinate of the interface position, z is the axial coordinate, η

and ξ are the radial and axial coordinates in the time-independent similarity variables,

and B is a dimensional constant. (The dependent and independent variables will be

reversed when solving for the post-pinching shape, see Section 7.3.)

To determine whether a valid similarity coordinate system exists, we define the

difference between the solute concentration in the liquid and the equilibrium concen-

tration in terms of a scaled concentration in similarity variables, K (ξ, η):

(7.5) C L (r,z,t) − C 0L = tβ AK (ξ, η)

Substituting (7.5) into (7.2) and (7.3) and using (7.4), we find that α = −β = 1/3

is required for a self-similar solution that is consistent with the boundary conditions

(see Appendix B for details.) The diffusion equation allows a similarity solution only

for α = 1/2. However, sufficiently far from the time of pinching, when the interfacial

velocity is moving slowly in comparison to the time required for the diffusion field to

relax, the quasi-stationary approximation ∂C L/∂t ≈ 0 can be used. In this regime,

the diffusion field is governed by Laplace’s equation, ∇2C L = 0, and α = 1/3 is a

valid temporal exponent. From (7.2) and (7.3), a similarity solution also requires that

B = (DLd0)−1/3 and A = (C 0L − C 0S ) (d20/DL)1/3

.

7.2. Phase-Field Simulations

Before determining the shape of the interfaces in the self-similar coordinate system,

a pinching event was simulated to determine whether the quasi-steady approximation

held long enough for self-similar behavior to be observed, and whether the cone angles



131

(a) (b)

Figure 7.2. Phase-field simulations of liquid cylinder pinching off by bulkdiffusion. Black represents liquid, white represents surrounding solid. (a)Initial conditions, (b) shape just prior to pinching.

observed in modeling were similar to the angles measured in the experimental data.

A liquid cylinder with a sinusoidal perturbation in interface position along its axis,

surrounded by solid, was created as a 3D array in IDL and loaded into the binary

alloy phase field model described in Section 3.5.2. The diameter of the cylinder was 50

gridpoints, the wavelength of the perturbation was 176 gridpoints, and perturbation

amplitude was 3 gridpoints. The size of the simulation volume was 176 × 100 × 100,

= 2, and the interface width was ∆x = 0.8W resulting in an interface width of 5

points. Periodic boundary conditions were used. Figure 7.3 shows the initial conditions

of the system and the shape just prior to the time of pinching.

A plot of r3 (where r is the minimum radius of the cylinder) versus t should be

linear if t1/3 is a valid choice of the similarity variable. Early in the pinching process,

r3 versus t does not follow a linear slope (Figure 7.3(a)), because initial conditions still

exert a strong effect on the dynamics of the pinching process. However, as the time of

pinching is approached, r3 versus t begins to follow a linear slope as the effects of initial

conditions become less important and the shape becomes locally determined. The slope



132

y = -0.0123x + 24603

R ! = 0.99913

0

1000

2000

3000

4000

5000

6000

7000

1.0E+06 1.2E+06 1.4E+06 1.6E+06 1.8E+06 2.0E+06

R 3 (

g r i d p o i n t s

3 )

Time (iterations)

080721a

080804a

(a)

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

50000

1.85E+06 1.90E+06 1.95E+06 2.00E+06 2.05E+06 2.10E+06 2.15E+06 2.20E+06 2.25E+06

( D i s t a n c e f r o m x

= 0 t o c u s p ) 3 ( g

r i d p o i n t s 3 )

Time (iterations)

080804a

(b)

Figure 7.3. From phase-field simulations, (a) minimum radius of a pinch-ing cylinder cubed, (b) distance between tips of cones after pinchingcubed versus time. The linear slope close to the point of pinching sup-ports t−1/3 dependence of the similarity variables close to the point of pinching.

becomes linear when r ≈ 10. Since the original radius of the cylinder is 25, the slope is

actually linear for a considerable fraction of the pinching process. Since the similarity

variables should also be valid after pinching is complete, the distance between the tips

of the cones formed after pinching should also follow the same scaling, and d3 versus

t should also be linear. Although this behavior is difficult to observe experimentally

because of the much faster speed of the receding tips, the simulation time step can be

made adequately small to observe this behavior. Figure 7.3(b) shows that d3 versus t

is also linear sufficiently close to the point of pinching.

These results suggest that the use of similarity variables with t−1/3 dependence is

justified. In some past work, the universal shape of the interface was determined from

time-dependent simulations [33]. However, those simulations used an adaptive grid

spacing and time stepping algorithm that allowed the interface shape to be determined



133

arbitrarily close to the point of pinching. As Figure 7.3(a) shows, the time step of

these phase-field simulations was changed as the time of pinching was approached, but

the lack of adaptive grid spacing in our phase-field model means that as the point of

pinching is approached, the interface width becomes the same order of magnitude as

the size scale of the pinching tips, so there is insufficient resolution to determine the

shape. Additionally, due to the use of the quasi-steady approximation, the scaling is

expected to break down sufficiently close to the point of pinching. For these reasons,

the problem was transformed to similarity variables and shape of the interface was

solved in this coordinate system.

The shapes of the cones from phase-field simulations and experimental data were

also compared to see if the possibility of a universal shape is suggested (Figure 7.4).

To measure the cone angles, lines were drawn approximating the shape of the cone on

a 2D cross-section of the pinching cylinders as close as possible to the time of pinching.

The cone angles were measured using the Measurement tool in Adobe Photoshop.The angle measured from the phase-field simulations was 68 while the angle from the

experimental data (which was taken from two different angles on three pinching events)

ranged from 67 to 71. Thus, the presence of a universal shape is suggested, but it

must be emphasized that due to the limited resolution of the experimental data and

the phase-field simulations, the cone angles are only approximate.



134

(a) (b)

Figure 7.4. Cone angles measured as close as possible to the point of pinching, (a) phase-field simulations, (b) experimental data.

7.3. Solving for Interface Shape in Self-Similar Variables

After transforming the problem to the self-similar coordinate system, the shape of

the interface was found numerically using a boundary-integral method. In the quasi-

steady approximation, the diffusion equation becomes Laplace’s equation in the simi-

larity coordinate system ∇2ξ,ηK = 0. Since this is a harmonic function, Green’s Third

Identity can be used along with the free-space Green’s function for Laplace’s equa-

tion to yield an integrodifferential equation for the scaled concentration K (ξ, η) on the

interface:

(7.6) 1

2

K (ξ 0, η0) = S K (ξ, η)

∂g

∂n −g

∂K

∂n dS

where g is the free-space Green’s function for Laplace’s equation. The following sub-

sections explain the details of how the interface shape η = f (ξ ) is solved in 2D and

3D, before and after the time of pinching. Figure 7.5 is included to clarify where the



135

Figure 7.5. Rod pinching due to Rayleigh instability, (a) before, (b) at,

(c) after, the time of pinching. The interface shape η = f (ξ ) which willbe solved numerically is highlighted in red. In (a), before pinching, theindependent variable ξ runs along the z and the dependent variable η isalong r. In (c), after pinching, the independent variable ξ runs along rand the dependent variable η is along z . Figure from [5].

solid and liquid regions, and dependent and independent variables, are located in the

plots of the following subsections.

7.3.1. Interface Shape After Pinching in 2D

The interface was first determined in 2D after the time of pinching, since the simpler

mathematics in 2D would allow testing of the boundary integral code, which was

modified from a FORTRAN program to solve for the 2D shape of a solidifying wedge,



136

provided by Professor M. J. Miksis of Northwestern University. The coordinate system

is given by

(7.7) ξ = rB

t1/3, η =

zB

t1/3

The integrodifferential equation is given by Equation 7.6. In 2D the free-space

Green’s function for Laplace’s equation is

g =

1

2π ln |ri

(ξ, η) −r

(ξ 0, η0)|(7.8)

= 1

2π ln

(ξ − ξ 0)2 + (η − η0)2(7.9)

= 1

2π ln

(ξ − ξ 0)2 + [f (ξ ) − f (ξ 0)]2(7.10)

since the interface is defined by η = f (ξ ). The integration of Equation 7.6 is performed

along the ξ -axis. Substituting for the Green’s function and using the solute conservation

boundary condition to substitute for ∂K

∂n the following integrodifferential equation forK [ξ, f (ξ )] is obtained (see Appendix C for details):

K [ξ 0, f (ξ 0)] = 1

π

∞0

K [ξ, f (ξ )]

[−f (ξ ) + f (ξ 0) + f (ξ )(ξ − ξ 0)]

(ξ − ξ 0)2 + [f (ξ ) − f (ξ 0)]2

− 1

3 [f (ξ ) − f (ξ )ξ ] ln

(ξ − ξ 0)2 + [f (ξ ) − f (ξ 0)]2

dξ

+1

π

∞0

K [ξ, f (ξ )]

[−f (ξ ) + f (ξ 0) + f (ξ )(ξ + ξ 0)]

(ξ + ξ 0)2 + [f (ξ ) − f (ξ 0)]2

− 1

3 [f (ξ ) − f (ξ )ξ ] ln

(ξ + ξ 0)2 + [f (ξ ) − f (ξ 0)]2

dξ

(7.11)

where primes denote derivatives in the ξ -direction.



137

In the similarity coordinate system, K (ξ, η) = ∇ · n. This can be written in terms

of the interface shape f (ξ ) as

(7.12) K [ξ, f (ξ )] = f (ξ )

[1 + f (ξ )]3/2

Equation 7.12 is substituted into Equation 7.11 and a set of integrodifferential

equations for f (ξ ) is obtained. Boundary conditions are that f (ξ ) approaches a straight

line with angle α from the ξ -axis (where 2α is the cone angle, see Figure 7.5) as ξ → ∞

and f (0) = 0, which ensures finite curvature at the origin.

The integrodifferential equations are discretized. The integration is performed nu-

merically using the trapezoidal rule on a truncated domain along ξ . A Taylor expan-

sion is used to approximate integrable singularities which occur in terms containing

ln

(ξ − ξ 0)2 + [f (ξ ) − f (ξ 0)]2

when ξ = ξ 0. L’Hopital’s rule is used on the term in the

first line of 7.11 to show that

(7.13) limξ→ξ0

[−f (ξ ) + f (ξ 0) + f (ξ )(ξ − ξ 0)]

(ξ − ξ 0)2 + [f (ξ ) − f (ξ 0)]2 =

f (ξ )

2[1 + f (ξ )2]

which is used to numerically evaluate this term when ξ = ξ 0. Newton’s method is then

used to iteratively solve for f (ξ ), starting with a straight line with angle α as an initial

guess for the shape.

The interface shapes found are shown in Figure 7.6, and correspond to the red line

in Figure 7.5(c). As expected from comparing to the results of the surface diffusion

case, after pinching, a solution was found for any angle α except for very sharp angles

α < 20, when the straight-line initial guess is a poor approximation for the final shape.



138

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

!

f (!)

30°

40°

50°

60°

70°

80°

Figure 7.6. Shape of the 2D solid-liquid interface after pinching byinterfacial-energy driven bulk diffusion. For each curve, the solid liesto the upper left, and the liquid to the lower right of the curve.

The solution for the shape should converge to the exact solution as the grid point

spacing along the ξ -axis goes to zero, and the truncation length in the ξ -direction goes

to infinity. Convergence is therefore tested by varying the grid spacing and by varying

truncation length in the ξ -direction. Varying the grid spacing produced no visible

change in a plot of the interface shape. Varying truncation length with fixed spacing,

the shape did change slightly as ξ was increased from 5 to 10, but much less as ξ was

increased further (Figure 7.7(a) shows results for α = 80).

Let f (ξ i) be the exact solution at ξ i, and f ∆ξ(ξ i) be the estimate from numerical

approximation with a grid spacing ∆ξ . Since Newton’s Method is a second-order

convergent approximation scheme, once the grid spacing becomes sufficiently small,



139

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

!

f (!)

Length=5

Length=10

Length=20

Length=40

(a)

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5 6 7 8 9 10

f ( ! )

!

Error ratio 1

Error Ratio 2

(b)

0

1

2

3

4

5

6

7

0.0 1.0 2.0 3.0 4.0 5.0

R a t i o o f E r r o r s

!

Error ratio 1

Error ratio 2

(c)

Figure 7.7. (a) Variation of 2D solid-liquid interface shape (after pinch-ing, α = 80) with truncation length of the numerical solution. Theshape changes little after ξ is increased above 5. (b) Error ratio (definedin Equation 7.14) for an initial grid spacing of 0.5 (Error ratio 1) and 0.25(Error ratio 2). (c) Error ratio (as defined in Equation 7.15) for initialtruncation length ξ T . Grid spacing was 0.2, initial truncation length wasξ T = 5 (Error ratio 1), and ξ T = 10 (Error ratio 2).

the error ratio

(7.14) Error ratio =

f ∆ξ(ξ i) − f ∆ξ/2(ξ i)f ∆ξ/2(ξ i) − f ∆ξ/4(ξ i)



140

should approach 4.0 at all points ξ i. Figure 7.7(b) shows that for the solution with

α = 80

, as the initial grid spacing ∆ξ is decreased from 0.5 to 0.25, the expected

quadratic convergence with grid spacing is observed.

Similar testing for truncation length is shown in Figure 7.7(c). In this case, the

error ratio is defined as

(7.15) Error ratio = |f ξT (ξ i) − f 2ξT (ξ i)||f 2ξT (ξ i) − f 4ξT (ξ i)|

where ξ T is the truncation length in the ξ -direction, and the number of grid points is

simultaneously varied to keep the grid spacing constant. Quadratic convergence is also

observed for variation of truncation length when grid spacing is 0.25.

7.3.2. Interface Shape After Pinching in 3D

After verifying correct operation and convergence of the code in 2D, the code was

modified to solve for the shape of the interface after pinching in 3D. Since by analogy

with the surface diffusion case, a solution is expected for any cone angle after the time

of pinching [5], the 3D version of the code could be debugged without simultaneously

needing to search for the correct cone angle. The coordinate system is again given in

Equation 7.7, and the integrodifferential equation is given by Equation 7.6. The system

is assumed to be axially symmetric about the η-axis. In 3D, the free-space Green’s

function for Laplace’s equation is

g = −1

4π |ri(ξ, η) − r(ξ 0, η0)|(7.16)



141

= −1

4π [(ξ cos θ

−ξ 0 cos θ0)2 + (ξ sin θ

−ξ 0 sin θ0)2 + (η

−η0)2]1/2

(7.17)

where θ is the angular coordinate in the radial coordinate system. The integration is

performed along the ξ -axis, and the angular integral is rewritten as a complete elliptic

integral of the first kind and approximated using the polynomial approximation of [80].

Substituting for the Green’s function and using the solute conservation boundary con-

dition to substitute for ∂K ∂n

the following integrodifferential equation for K [ξ, f (ξ )] is

obtained (see Appendix D for details):

2πK [ξ 0, f (ξ 0)] =

∞0

ξ d ξ

K [ξ, f (ξ )]

×

f (ξ )4(ξ + ξ 0)

(l2− + k2)3/2 − 4[f (ξ ) − f (ξ 0)]

(l2− + k2)3/2

×

4

n=0 an − bn ln

1 − k2

l2− + k21 − k2

l2− + k2n

+

−16ξ 0f (ξ ) [ξ 2 − ξ 20 − [f (ξ ) − f (ξ 0)]2]

(l2− + k2)5/2 +

32ξξ 0[f (ξ ) − f (ξ 0)]

(l2− + k2)5/2

× 4

n=1

n

an − bn ln

1 − k2

l2− + k2

1 − k2

l2− + k2

n−1

−4

n=0

bn

1 − k2

l2− + k2

n−1

− 4[f (ξ )ξ − f (ξ )]

3(l2− + k2)1/2

4

n=0

an −

bn

ln1−

k2

l2− + k21

− k2

l2− + k2

n

(7.18)

where an, bn are numerical constants defined in [80], l− ≡ (ξ − ξ 0)2 + [f (ξ ) − f (ξ 0)]2,

and k2 ≡ 4ξξ 0. In this coordinate system, the curvature can be written in terms of the



142

interface shape f (ξ ) as

(7.19) K [ξ, f (ξ )] = f (ξ )

[1 + f (ξ )2]3/2 +

f (ξ )

ξ [1 + f (ξ )2]3/2


equations for f (ξ ) is obtained. Having reduced the 3D problem to a 1D integral, the

system of equations is solved numerically using the boundary conditions and numerical

methods as described in Section 7.3.1. L’Hopital’s rule and Taylor expansions were used

as necessary for terms where ξ = ξ 0.

The interface shapes found are shown in Figure 7.8. As in the 2D case, after

pinching, a solution was found for any angle α except for very sharp angles α < 20,

when the straight-line initial guess is a poor approximation for the final shape.

Convergence was again tested by varying the grid spacing and by varying truncation

length in the ξ -direction. Varying the grid spacing produced no visible change in a plot

of the interface shape. The interface shape changed much less with changing truncation

length than the 2D case (Figure 7.9(a) shows results for α = 80).

The error ratios for varying grid spacing and truncation length are shown in Fig-

ure 7.9(b) and 7.9(c). The error ratios to begin to approach 4 but not as closely as the

results of the 2D solution. This may be due to the use of a numerical approximation

for the angular integral.



143

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

!

f (!)

30°

40°

50°

60°

70°

80°

Figure 7.8. Shape of the 3D solid-liquid interface after pinching byinterfacial-energy driven bulk diffusion. For each curve, the solid liesto the upper left, and the liquid to the lower right of the curve.

7.3.3. Interface Shape Before Pinching in 3D

To solve for the shape prior to pinching, the dependent and independent variables are

reversed, and the coordinate system is

(7.20) η = rB

(ts − t)1/3, ξ =

zB

(ts − t)1/3

The system is assumed to be axially symmetric about the ξ -axis in this case. The ge-

ometry and portion of the interface shape being determined are shown in Figure 7.5(a).

In 3D, the free-space Green’s function for Laplace’s equation is again

(7.21) g = −1

4π [(ξ cos θ − ξ 0 cos θ0)2 + (ξ sin θ − ξ 0 sin θ0)2 + (η − η0)2]1/2



144

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.2 0.4 0.6 0.8 1

!

f (!)

Length=5

Length=10

Length=20

(a)

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5


!

Error ratio 1

Error ratio 2

(b)

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5


!

Error ratio 1

Error ratio 2

(c)

Figure 7.9. (a) Variation of 3D solid-liquid interface shape (after pinch-ing, α = 80) with truncation length of the numerical solution. (b) Errorratio (defined in Equation 7.14) for an initial grid spacing of 0.2 (Errorratio 1) and 0.1 (Error ratio 2). (c) Error ratio (as defined in Equa-tion 7.15) for initial truncation length ξ T . Grid spacing was 0.25, initialtruncation length was ξ T = 5 (Error ratio 1), and ξ T = 10 (Error ratio2).

The integration is performed along the ξ -axis, with limits from −∞ to +∞, and the

angular integral is again rewritten as a complete elliptic integral of the first kind and

approximated using the polynomial approximation of [80]. Substituting for the Green’s



145

function and using the solute conservation boundary condition to substitute for ∂K ∂n the

following integrodifferential equation for K [ξ, f (ξ )] is obtained (see Appendix E for

details):

2πK [ξ 0, f (ξ 0)] =

∞0

dξf (ξ )

K [ξ, f (ξ )]

×

−4f (ξ )(ξ − ξ 0)

(l2− + k2)3/2 +

4(η + η0)

(l2− + k2)3/2

M

k2

l2− + k2

+

32ηη0(ξ − ξ 0)f

(ξ )(l2− + k2)5/2

− 16η0[η2

− η20 − (ξ − ξ 0)

2

](l2− + k2)5/2

M d

k

2

l2− + k2

− 4

3

[f (ξ )ξ − f (ξ )] l2− + k2

M

k2

l2− + k2

+ K [ξ, f (ξ )]

−4f (ξ )(ξ + ξ 0)

(l2+ + k2)3/2 +

4(η + η0)

(l2+ + k2)3/2

M

k2

l2+ + k2

+

32ηη0(ξ + ξ 0)f (ξ )

(l2+ + k2)5/2 − 16η0[η2 − η2

0 − (ξ + ξ 0)2]

(l2+ + k2)5/2

M d

k2

l2+ + k2

− 4

3

[f (ξ )ξ −

f (ξ )] l2+ + k2 M

k2

l2+ + k2

(7.22)

where M is the polynomial approximation for a complete elliptical integral of the first

kind defined in [80], M ξ is the series approximation differentiated with respect to ξ ,

l− ≡ (ξ − ξ 0)2 + [f (ξ ) − f (ξ 0)]2, l+ ≡ (ξ − ξ 0)2 + [f (ξ ) + f (ξ 0)]2, and k2 ≡ 4ξξ 0. In this

coordinate system, the curvature can be written in terms of the interface shape f (ξ )

as

(7.23) K [ξ, f (ξ )] = 1

η

1 + f (ξ )2− f (ξ )

[1 + f (ξ )2]3/2



146


equations for f (ξ ) is obtained. Having reduced the 3D problem to a 1D integral, the

system of equations is solved numerically using the boundary conditions and numer-

ical methods as described in Section 7.3.1. L’Hopital’s rule and Taylor expansion to

approximate integrable singularities were used as necessary for terms where ξ = ξ 0.

Since by analogy with the surface diffusion case, solutions are not expected at

arbitrary cone angles [5], an iterative process was used to search for the solutions.

A coarse grid spacing along ξ was used to search for solutions as α was varied using

the boundary condition f (ξ ) = 0. α was initially varied through the region where

a solution is expected based on the experimental data and phase-field simulations.

Solutions were obtained for some values of α in this range, and these solutions were

used as an initial guess for the solution with a finer grid spacing. As the grid spacing

was made finer, the boundary condition at ξ = 0 was changed, fixing f (0) rather than

f

(ξ ) = 0. The angle and f (0) were iteratively refined until a smooth solution withf (0) = 0 was found. The angle was converged to θ = 90−α = 52±1. The interface

shape found is shown in Figure 7.10.

As shown in [81], in the similarity coordinate system a countably infinite number

of solutions is expected at various angles prior to pinching, but only the solution

with the largest cone half-angle α is expected to stable in time-dependent variables.

The presence of at least one additional solution prior to pinching was verified in the

similarity variables (see Figure 7.10).



147

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

f ( ! )

!

52°

75°

Figure 7.10. Shape of the 3D solid-liquid interface before pinching byinterfacial-energy driven bulk diffusion. The solution at θ = 52 is ex-pected to be stable, while the solution at θ = 75 is expected to beunstable (θ = 90 − α).

Varying the grid spacing produced no visible change in a plot of the interface shape.

The interface shape changed much less with changing truncation length than the 2D

case (Figure 7.11(a) shows results for θ = 52).

Convergence was again tested by varying the grid spacing and by varying truncation

length in the ξ -direction. Changing truncation length produced a smoother f (ξ ) when

ξ was increased above 5. The error ratios for varying grid spacing and truncation length

are shown in Figure 7.11(b) and 7.11(c). In this case the solution does converge, but not

quadratically. This is likely due to the use of a numerical approximation for the angular

integral, a different boundary condition at ξ = 0, and the difficulty of identifying the

exact angle of the solution. However at this angle the solution could be resolved to an



148

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

f ( ! )

!

Length=5

Length=10

Length=20

(a)

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5 6 7 8 9 10

R a t i o o f e r r o r s

!

Error ratio 1

Error ratio 2

(b)

0

2

4

6

8

10

12

14

16

18

20

0.0 1.0 2.0 3.0 4.0 5.0


!

Error ratio 1

Error ratio 2

(c)

Figure 7.11. (a) Variation of 3D solid-liquid interface shape after pinch-ing (θ = 52) with truncation length of the numerical solution. (b) Errorratio (defined in Equation 7.14) for an initial grid spacing of 0.25 (Errorratio 1) and 0.125 (Error ratio 2). (c) Error ratio (as defined in Equa-tion 7.15) for initial truncation length ξ T . Grid spacing was 0.25, initialtruncation length was ξ T = 5 (Error ratio 1), and ξ T = 10 (Error ratio2).

extremely fine grid spacing of 0.03125 and the solution converged, so this is believed

to be the correct angle in spite of the fact that the error does not converge as quickly

as in the 2D after pinching case.



149

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

f ( ! )

!

50°

Figure 7.12. Shape of the 3D solid-liquid interface before pinching byinterfacial-energy driven bulk diffusion for solid rods pinching off in aliquid matrix. The stable solution is solution at θ = 50 (α = 40).

7.3.4. Interface Shape Before Pinching in 3D for Solid Cylinders in a Liquid

Matrix

To broaden the applicability of these techniques, the problem of solid cylinders pinching

off in a liquid matrix was also considered. The formulation of Section 7.3.3 is again

used, the only differences being that the sign of K and the direction of the normal

vector change. A solution was found using the solution search method as described in

Section 7.3.3 at α = 40.



150

7.4. Comparison with Experimental Data

The results of Section 7.3.3 were compared with experimental data for pinching

liquid cylinders. Quantitative analysis of the experimental microstructures was per-

formed by A.E. Johnson of Northwestern University. Since the data is obtained in

three dimensions, it is possible to view the tube along its axis to measure accurately

its smallest cross section or diameter. If the pinching process proceeds in a self-similar

fashion, then the diameter of the tube should decrease linearly as (t−ts)1/3. Figure 7.13

shows that the diameter does decrease linearly as (t − ts)1/3. This behavior was also

observed in each of the 19 other pinching events investigated.

To test the validity of the quasi-stationary approximation, we compare the kinetics

predicted by the similarity solution to the experimental data of Figure 7.13. The error

bars give the minimum and maximum diameter of the liquid tube at a given time. They

are nonzero because the diameter is not circular at all times. Figure 7.13 shows that as

the singularity is approached, the cross section becomes increasingly circular, consistent

with the theoretical prediction. The excellent fit to a straight line indicates that the

temporal exponent of 1/3 holds, showing that the quasi-stationary approximation is

valid over nearly the entire time interval of the experiment.

The prefactor of the temporal power law of the theoretical prediction for time

dependence of diameter can be determined from theory and compared to experimental

data. Using Equation 7.20, the radius at the center (ξ = 0) of a pinching liquid tube,

R(t), evolves as

(7.24) R(t) = f (0)(DLd0)1/3 (ts − t)1/3



151

0

5

10

15

20

25

-25 -20 -15 -10 -5 0

P

i n c h - o f f D i a m e t e r ( µ m )

(t -t s)1/3 (s1/3)

Figure 7.13. Tube diameter follows a power law prior to singularity. Thediameter of a pinching tube as a function of (t

−ts)1/3 along with the

experimentally measured and theoretically predicted interface shapes atvarious times and line showing least-squares fit to experimental data.(Error bars show the maximum and minimum measured diameter of tubes, which are not circular at all times.) It is clear that even when theinterfacial morphology is far from the self-similar shape, the kinetics of the pinching process are still well described by (t−ts)1/3. Figure courtesyof A.E. Johnson.

where f (0) = 0.60 and 0.37 for the solutions with cone angles (2α) of 76, and 30,

respectively, see Figure 7.10. Thus, sufficiently close to the singularity, the time-

dependence of the pinch-off process is independent of initial conditions and is a function

of only f (0) and materials parameters.



152

From Equation 7.24, the slope of the line shown in Figure 7.13 is related to the

materials parameters and f (0), the interface location in similarity variables at ξ = 0.

Linear slopes were determined for 19 separate pinching events based on diameters

measured for at least 3 time steps prior to ts. The average slope from these was

1.28 µm/s1/3 with a standard deviation of 0.178 µm/s1/3. From Equation 7.24, the

predicted slope, or prefactor of the power law, is given by 2f (0)(DLd0)1/3 (the factor

of 2 converts radius to diameter). Using DL = 8.3× 10−10 m2/s (Chapter 6), d0 = 2.27

nm [76], and f (0) = 0.60 , the theoretically predicted rate constant is 1.47 µm/s1/3

.

The 16% difference between the theoretically predicted and experimentally measured

rate constants indicates an excellent agreement between theory and experiment, given

the uncertainties in the materials parameters. If we had used, instead, f (0) for the

solution with a cone angle of 30, the predicted rate constant would be 0.91 µm/s1/3,

or a difference of 30% from that measured experimentally. This suggests that only the

solution with a cone angle of 76

is observable experimentally, in agreement with [81].Moreover, the agreement between theory and experiment is excellent from very early in

the pinching process, indicating that theory can be used to determine the time required

for a rod-like structure to break up. The agreement between theory and experiment

also shows that phase-field modeling can be used in combination with experimental

techniques to obtain accurate measurements of DL as described in Chapter 6.

As a further test of the theory, a 3D representation of the theoretically predicted in-

terface shape was created. Using an affine transformation, we aligned the experimental

data to the theoretically predicted shape for the solution with a cone angle of 76, and

linearly scaled the predicted shape such that the diameter at the center of the pinching



153

tube was the same as that measured experimentally. This process was repeated for a

series of times leading to the singularity. The distance from the theoretically predicted

shape to the closest point on the experimentally measured shape was calculated, see

Figure 7.14. As expected, the agreement between theory and experiment improved as

the singularity is approached, since initial conditions have less influence and the shape

becomes locally determined. Just before pinch-off, the mean deviation between shapes

is less than the 1.4 µm resolution of the X-ray tomography scan, indicating excellent

agreement between theory and experiment. Very similar diagrams result if we plot

the distance between the experimentally measured and theoretically predicted shapes

in similarity variables, confirming that the improved agreement as ts is approached

is not just a result of the changing length scale. In some cases, the initial shapes of

the rods were asymmetric about the pinching axis and thus the singularity did not

achieve high degrees of self-similarity under the time and space resolution constraints

of the experiment. However, in all 19 cases examined the agreement between theoryand experiment improved as the singularity was approached. Thus to the resolution of

the tomography, the experiments confirm that some distance from the pinching point,

the shape approaches a cone with the predicted cone angle 2α = 76.

7.4.1. Conclusions

The pinching of liquid cylinders caused by interfacial-energy driven bulk diffusion was

shown to be a self-similar process within the quasi-steady approximation. Sufficiently

close to the point of pinching, the interface approaches a universal shape. The shape is

self-similar and follows a t1/3 power law. The problem was transformed into similarity



154

25 um 25 um 25 um 10 um

25 um 25 um 25 um 10 um

0 8!m+

(a) (b) (c) (d)

Figure 7.14. Quantifying agreement between theoretically predicted andexperimentally measured interface shape. The experimentally deter-mined microstructure is superimposed on the theoretically predictedshape for the solution with a cone angle of 76 for several times lead-ing up to pinch-off (top). The theoretical shapes are scaled to the samediameter as the experimental microstructures. Bottom, the theoretical

shape is shown colored by the distance from each point to the closestpoint on the experimentally measured shape. The mean deviation be-tween experiment and theory decreases as the system approaches thetime of pinchoff. In the case of the final time step, the mean deviation isless than the experimental resolution of the X-ray tomography scan (1.4µm). Times: (a) 142.5 min, (b) 155 min, (c) 165 min, (d) 167.5 min.Figure courtesy of A.E. Johnson.

variables, and self-similar interface shapes were found numerically for any cone angle

after pinching for both 2D and 3D. In 3D, before pinching, solutions to the self-similar

shape were not found at all cone angles, as expected. The stable solution and the

first unstable solution were found numerically. The stable solution was also found for

solid cylinders in a liquid matrix. Both the interfacial morphology and kinetics of the



155

pinching process measured experimentally agree with the predictions of theory for the

solution with a cone angle of 76

. This implies that it is possible to observe self-similar

behavior before the quasi-stationary approximation fails and that that the shape with

the highest cone angle is the only stable shape.



156

CHAPTER 8

Conclusions

A Cu-Ni alloy undergoing equiaxial dendritic solidification was simulated using the

phase-field method. Due to the rapid solidification velocity, interface kinetics controlled

the microstructure’s formation. As the system solidified, S v increased, then began to

decrease as the system approached the equilibrium volume fraction and coarsening

began to dominate changes to the microstructure. V f /S v remained approximately

constant during solidification then began to increase as S v decreased due to coarsening.

ISDs were calculated for the solid-liquid interface of the Cu-Ni system. During

solidification, ISDs show a shift from a majority of positive principal curvatures to a

majority of saddle-shaped patches. The proportion of high solid-cylindrical patches

increases then decreases, due to the presence of secondary arms. Overall, the changes

in the ISDs during solidification were relatively small considering the large changes in

the microstructure, consistent with the constant length scale V f /S v. During coarsen-

ing, ISDs were dominated by saddle-shaped interfacial patches on liquid walls. Liquid

cylindrical regions also remain during coarsening as pinching due to Rayleigh instabil-

ities occurs. Evolution was not morphologically self-similar during the early stages of

coarsening.

The topology of the Cu-Ni system was quantified using the genus. Genus was

initially negative as isolated dendrites nucleated, then increased rapidly as adjacent

dendrite arms coalesced. Genus then decreased in the coarsening regime as liquid tubes



157

pinch off during coarsening. Liquid voids were formed from long liquid tubes as two

Rayleigh instabilities happened simultaneously. Pinch-off of handles had approximately

twenty times greater effect on topology than liquid void formation. The system is not

topologically self-similar during the early stages of coarsening.

A method to calculate the rate of change of curvatures for a moving interface

was developed that properly accounts for the motion of the interface itself. A nu-

merical algorithm was developed to calculate these quantities numerically from an

order-parameter based representation of a microstructure. The algorithm was tested

and numerical methods were modified to calculate quantities accurately. ∇H was the

most sensitive parameter in the calculation, and needed 4th-order accurate stencils for

derivatives for accurate results. This necessitated interfaces at least 10 points wide.

A phase-field model was used to predict interfacial velocities during coarsening.

Interfacial velocities were calculated from experimental data taken from an isothermal

coarsening experiment of an Al-Cu alloy. The experimentally observed microstructurewas input in to the phase-field model and evolved. Good qualitative agreement was

observed between simulation and experimental results, while the simulated velocity

distribution was greater than the experiment by a factor of 2.9. The diffusion coefficient

of solute in the liquid is believed to be the main cause of the disagreement. This method

can be used to estimate DL when other physical parameters of the system are well

known. The effect of the anti-trapping current was tested. For this experiment, the

anti-trapping current caused negligible change in the simulated velocities, and setting

it equal to zero allowed the calculation to run 1.6 times faster.



158

The self-similar pinching of liquid tubes by interfacial energy driven-bulk diffusion

was investigated. The problem was transformed to similarity variables, and t−1/3

scal-

ing of the similarity variables was found to be consistent with the boundary conditions.

This time-dependence is not consistent with the diffusion equation, but holds in the

quasi-steady approximation. Phase-field simulations showed that t−1/3 scaling held

sufficiently close to the time of pinching.

The interface shape of a pinching liquid tube was solved in similarity variables

using a boundary integral method. The shape was found after the time of pinching in

2D, and before and after pinching in 3D. The shape was also found for solid cylinders

surrounded by liquid prior to pinching in 3D. Experimental data was compared to

the pre-pinch-off solution for liquid cylinders. Good agreement was found between

experiment and theory for the exponent, prefactor of time dependence, and interface

shape, meaning that a universal, self-similar interface shape exists sufficiently close to

the time of pinching.



159

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166

APPENDIX A

Derivation of Perturbation Decay Constant σ

Assume a sinusoidal perturbation exists at a solid-liquid interface. The position of

the interface is given by h(x,y,t) = exp(i k · r + σt). The evolution of the interface

will be determined by the diffusion equation in solid and liquid:

∂C s

∂t = Ds∇2C s(A.1)

∂C l

∂t = Ds∇2C l(A.2)

and the value of concentration at the interface will be given by the Gibbs-Thomson

equation:

C s = C se + ΓsH (A.3)

C l = C le + ΓlH (A.4)

The solute conservation boundary condition at the interface is

(A.5)

C le − C se

V = −Dl∇C l · n + Ds∇C s · n



167

Where the interfacial velocity V is in the normal direction; and the flux in the solid

has opposite sign because the normal into the solid is equal and opposite to the normal

into the liquid. Assuming concentration is of the form

C l = B l0 + Bl

1(z )exp(i k · r + σt)(A.6)

C s = Bs0 + Bs

1(z )exp(i k · r + σt)(A.7)

Substituting into the diffusion equation:

Bl1(z )σ exp(i k · r + σt) = −DlBl

1(z )k2 exp(i k · r + σt) + Dl∂ 2Bl

1

∂z 2 exp(i k · r + σt)

and cancelling factors of exp (i k · r + σt) and :

Bl1(z )σ = −DlBl

1(z )k2 + Dl ∂ 2Bl1

∂z 2

Assume the solution is of the form B l1(z ) = E exp (−αz ) for z > 0

E exp (−αz )(σ + Dlk2) = D lα2E exp (−αz )

so

α =

σ + Dlk2

Dl



168

Choose the exp+βz solution for the solid so concentration does not go to infinity

for −z → ∞. So the solutions are:

Bl1(z ) = E exp

−

σ + Dlk2

Dl z

Bs1(z ) = F exp

σ + Dsk2

Ds z

To solve for B l,s0 , use the fact that far from the interface, the concentration will be

at its equilibrium value as determined from the phase diagram.

C le = B l0 + E exp (−

σ + Dlk2

Dl z )exp(i k · r + σt)

C se = Bs0 + F exp

σ + Dsk2

Ds z

exp(i k · r + σt)

So as z

→ ±∞, the exponential terms go to zero, and C le = B l

0, C se = Bs0. And we

find

(A.8) C l = C le + E exp

−

σ + Dlk2

Dl z

exp(i k · r + σt)

(A.9) C s = C se + F exp σ + Dsk2

Ds z exp(i k · r + σt)

Equating to the linearized Gibbs-Thompson equation:

(A.10) C le + E exp

−

σ + Dlk2

Dl z

exp(i k · r + σt) = C le + Γl

∂ 2h

∂x2 +

∂ 2h

∂y2



169

The C le terms drop out, and

∂ 2h

∂x2 +

∂ 2h

∂y2 = −k2 exp(i k · r + σt)

So canceling factors of exp (i k · r + σt) and substituting in to Equation A.10,

E exp

−

σ + Dlk2

Dl z

= −Γlk2

This must be evaluated at the interface, where z = exp(i k

·r + σt). Using the

approximation exp (−x) ≈ 1 − x, since is small and therefore z is small:

E

1 −

σ + Dlk2

Dl exp(i k · r + σt)

= −Γlk2

The term with exp(i k · r + σt) can be dropped since will be of order 2.This leaves

E = −Γlk2 and F = −Γsk2, so

(A.11) C l = C le − Γlk2 exp−

σ + Dlk2

Dl z

exp(i k · r + σt)

(A.12) C s = C se − Γsk2 exp

σ + Dsk2

Ds z

exp(i k · r + σt)

Using the linearized form of the flux condition (for small perturbations, V = ht)

C le − C se

σ exp(i k · r + σt) = −DlΓlk2

σ + Dlk2

Dl exp

−

σ + Dlk2

Dl z

exp(i k · r + σt)

−DsΓsk2

σ + Dsk2

Ds exp

−

σ + Dsk2

Ds z

exp(i k · r + σt)(A.13)



170

Cancelling exp(i k · r + σt) and again using exp (±x) ≈ 1±x, and dropping terms

proportional to 2:

(A.14)

C le − C se

σ = −DlΓlk2

σ + Dlk2

Dl − DsΓsk2

σ + Dsk2

Ds

which can be solved numerically for σ.



171

APPENDIX B

Determining Time Exponent of Similarity Solution for Rod

Pinching

To determine the time exponents α and β required for a similarity solution, the

assumed similarity solution for the solute concentration

(B.1) C L (r,z,t) − C 0L = (ts − t)β AK (ξ, η)

is substituted into the boundary conditions for the rod pinching problem. First is the

Gibbs-Thomson equation:

(B.2) C L = C 0L + C 0L

−C 0S d0H

Substituting we obtain

(B.3) (ts − t)β AK (ξ, η) =

C 0L − C 0S

d0H

H = ∇ · n in real-space coordinates. To transform this derivative into similarity

variables, we use the chain rule to show

(B.4) ∂F

∂z =

∂F

∂ξ

∂ξ

∂z = F ξ

∂

∂z

zB

(ts − t)α

=

B

(ts − t)αF ξ



172

(F = η − f (ξ ) = 0 is the function that parametrically defines the interface, and F ξ

denotes partial differentiation of F with respect to ξ ). In other words, real-space

derivatives simply give B(ts−t)α times derivatives in the similarity variables. Using this

result we obtain

(B.5) H = ∇ · n = B

(ts − t)α∇ξ,η · n

where the subscripts ξ, η to ∇ denote that the divergence operator is applied in the

similarity coordinate system. Substituting back into Equation B.3

(B.6) (ts − t)β AK (ξ, η) =

C 0L − C 0S

d0

B

(ts − t)α (∇ξ,η · n)

In order for a valid similarity solution for the concentration and interface shape to

exist, the time-dependence on the left and right hand sides of the equation must cancel,

meaning that α = −β is required for a similarity solution.

The solute conservation boundary condition is next used so that α and β can be

determined.

(B.7)

C 0L − C 0S

v · n = −DL

∂C L∂n

This boundary condition must also be transformed into similarity coordinates.

(B.8) ∂C L∂n = n · ∇C L = n · ∇ C 0L + (ts − t)β AK (ξ, η)



173

Again using the fact that real-space derivatives give B(ts−t)α

times derivatives in the

similarity variables,

(B.9) ∂C L

∂n = AB(ts − t)β −α (n · ∇ξ,ηK )

The material derivative at the interface can be used to find the normal velocity of the

interface in similarity variables.

(B.10)

DF

Dt =

∂F

∂t + v · ∇F = 0

Dividing the above equation by |∇F | and using the fact that n = ∇F |∇F |

,

(B.11) v · n = −∂F ∂t

|∇F |

Using the chain rule,

(B.12) ∂F ∂t

= ∂F ∂ξ

∂ξ ∂t

+ ∂ F ∂η

∂η∂t

= α(ts − t)

(F ξξ + F ηη)

and that real-space derivatives give B(ts−t)α

times derivatives in the similarity variables,

(B.13) |∇F | = B

(ts − t)α|∇ξ,ηF |

gives

(B.14) v · n = αtα−1

B

(F ξξ + F ηη) F 2ξ + F 2η



174

Substituting Equation B.9 and B.14 into B.7 yields

(B.15) (C 0L − C 0S )α(ts − t)α−1

B

F ξξ + F ηη F 2ξ + F 2η

= DL(ts − t)β

(ts − t)αAB (n · ∇ξ,ηK )

For a valid similarity solution to exist, the time dependence on the left and right hand

sides must cancel, yielding α−1 = β −α. Substituting α = −β from Equation B.6, the

required exponents for a similarity solution consistent with the boundary conditions

are

(B.16) α = 1

3, β = −1

3

However, these exponents must be substituted in to the diffusion equation in self-similar

coordinates to determine in a self-similar solution exists. Substituting the assumed self-

similar solution for solute concentration (Equation B.1) in to the diffusion equation and

applying the chain rule

(B.17) (ts − t)β −1 [AK − αA (K ξξ + K ηη)] = (ts − t)β

(ts − t)2αAB2∇2

ξ,ηK

Therefore, for the time dependence to cancel on the left and right sides, β − 1 =

β − 2α or α = 1/2 must hold, meaning that α = 1/3 is not a valid solution for the

diffusion equation in similarity coordinates. However, in the quasisteady approximation

where ∂C L

∂t ≈0,

(B.18) ∇2C L = (ts − t)β −2αAB2∇2ξ,ηK = 0



175

and the time dependence will always disappear, leaving α = 1/3 a valid exponent, and

reducing the problem of solving for the shape to solving Laplace’s equation in similarity

variables on the boundary.



176

APPENDIX C

Details of Boundary-Integral Equation After Pinching in Two

Dimensions

The integrodifferential equation is

(C.1) 12

K (ξ 0, η0) =

S

K (ξ, η) ∂g

∂n − g ∂K

∂n

dS

which applies on the boundary defined by η = f (ξ ). The free-space Green’s function

for two dimensions is

(C.2) g = 1

2π ln

(ξ − ξ 0)2 + [f (ξ ) − f (ξ 0)]2

The surface integral of Equation C.1 must also include the arbitrarily shaped cap

enclosing the area inside the curve η = f (ξ ). However, as ξ, η → ∞, C L → C 0L so

K → 0 and therefore ∂K ∂n

→ 0. Therefore this boundary makes no contribution to the

integral.

The integral along the surface can be performed along the ξ -axis. The surface area

element is

d s =

d ξ 2 + d η2(C.3)

= d ξ

1 + f (ξ )2(C.4)



177

The normal vector to the interface is given by

n = ∇F

|∇F | = F ξξ + F ηη

F 2ξ + F 2η

(C.5)

= f (ξ )ξ − η

1 + f (ξ )2(C.6)

where F = η − f (ξ ) = 0 defines the interface, and we have chosen the normal to the

interface pointing outward from the liquid region. To find ∂g∂n ,

∂g

∂n = n · ∇g =

f (ξ )ξ − η

1 + f (ξ )2

·

∂g

∂ξ ξ +

∂ g

∂ηη

(C.7)

= 1

2π

1 + f (ξ )2

f (ξ )(ξ − ξ 0) − (η − η0)

(ξ − ξ 0)2 + (η − η0)2

(C.8)

Using η = f (ξ ),

(C.9) ∂g

∂n

= 1

2π

1 + f

(ξ )2

−[f (ξ ) − f (ξ 0)] + f (ξ )(ξ − ξ 0)

(ξ − ξ 0)

2

+ [f (ξ ) − f (ξ 0)]

2 From the solute conservation boundary condition in similarity coordinates, Equa-

tion B.15, once the materials constants and time dependence are cancelled,

∂K

∂n = n · ∇K = −α

(F ξξ + F ηη) F 2ξ + F 2η

(C.10)

= f (ξ ) − f (ξ )ξ

3

1 + f

(ξ )2

(C.11)



178

Substituting g , ∂g∂n , and ∂K

∂n in to Equation C.1,

K [ξ 0, f (ξ 0)] = 1

π

∞

−∞

K [ξ, f (ξ )]

[−f (ξ ) + f (ξ 0) + f (ξ )(ξ − ξ 0)]

(ξ − ξ 0)2 + [f (ξ ) − f (ξ 0)]2

− 1

3 [f (ξ ) − f (ξ )ξ ] ln

(ξ − ξ 0)2 + [f (ξ ) − f (ξ 0)]2

dξ

(C.12)

So that the integration can be performed over +ξ values only, the integral can be split

into positive and negative regions. The integral over negative ξ can be rewritten using

a change of variable and the properties that with respect to ξ , f (ξ ) is even, f (ξ ) is

odd, and K [ξ, f (ξ )] is even. This change of variable results in Equation 7.11,

K [ξ 0, f (ξ 0)] = 1

π

∞0

K [ξ, f (ξ )]

[−f (ξ ) + f (ξ 0) + f (ξ )(ξ − ξ 0)]

(ξ − ξ 0)2 + [f (ξ ) − f (ξ 0)]2

− 1

3 [f (ξ ) − f (ξ )ξ ] ln

(ξ − ξ 0)2 + [f (ξ ) − f (ξ 0)]2

dξ

+1

π

∞0

K [ξ, f (ξ )]

[−f (ξ ) + f (ξ 0) + f (ξ )(ξ + ξ 0)]

(ξ + ξ 0)2 + [f (ξ ) − f (ξ 0)]2

− 1

3 [f (ξ ) − f (ξ )ξ ] ln

(ξ + ξ 0)2 + [f (ξ ) − f (ξ 0)]2

dξ

(C.13)



179

APPENDIX D

Details of Boundary-Integral Equation After Pinching in

Three Dimensions


(D.1) 12

K (ξ 0, η0) =

S

K (ξ, η) ∂g

∂n − g ∂K

∂n

dS


for three dimensions is

(D.2) g = −1

4π [(ξ cos θ − ξ 0 cos θ0)2 + (ξ sin θ − ξ 0 sin θ0)2 + (η − η0)2]1/2

The surface integral of Equation D.1 must also include the arbitrarily shaped cap

enclosing the area inside the curve η = f (ξ ). However, as ξ, η → ∞, C L → C 0L so

K → 0 and therefore ∂K ∂n

→ 0. Therefore this boundary makes no contribution to the

integral.

The integral along the surface is performed in radial coordinates. The surface area

element is

d s = ξ d θ

d ξ 2 + d η2(D.3)

= d θξ d ξ

1 + f (ξ )2(D.4)



180

So the integrodifferential equation is

(D.5) 1

2K (ξ 0, η0) =

2π

0

∞

0

d θξ d ξ

1 + f (ξ )2

K (ξ, η)∂g

∂n − g

∂K

∂n

The solute conservation boundary condition can again be used to write ∂K ∂n in terms of

ξ and f (ξ ) as

(D.6) ∂K

∂n =

f (ξ ) − f (ξ )ξ

3

1 + f (ξ )2


n = ∇F

|∇F | = F ξξ + F ηη

F 2ξ + F 2η

(D.7)

= f (ξ )ξ − η

1 + f (ξ )2(D.8)

where F = f (ξ )

−η = 0 defines the interface, and we have chosen the normal to the

interface pointing outward from the liquid region. To find ∂g∂n

,

∂g

∂n = n · ∇g =

f (ξ )ξ − η

1 + f (ξ )2

·

∂g

∂ξ ξ +

∂ g

∂ηη

(D.9)

= 1

1 + f (ξ )2

f (ξ )

∂g

∂ξ − ∂g

∂η

(D.10)

Since we are trying to find the surface η = f (ξ ), let θ0 = 0 and η0 = f (ξ 0).

g = −1

4π

(ξ cos θ − ξ 0)2 + (ξ sin θ)2 + (f (ξ ) − f (ξ 0))21/2(D.11)

= −1

4π

ξ 2 + ξ 20 − 2ξξ 0 cos θ + (f (ξ ) − f (ξ 0))21/2(D.12)



181

Let

(D.13) g− ≡ 2π

0

d θ 1

ξ 2 + ξ 20 − 2ξξ 0 cos θ + (f (ξ ) − f (ξ 0))21/2

Using the trigonometric identity 1 − cos θ = 2sin2 θ2

and performing a change of

variables θ = 2φ,

(D.14) g− =

π0

2dφ

ξ 2 + ξ 20 − 2ξξ 0 + 4ξξ 0 sin2 φ + (f (ξ ) − f (ξ 0))2

−1/2

Let

(D.15) l− ≡ (ξ − ξ 0)2 + [f (ξ ) − f (ξ 0)]2

(D.16) k2 ≡ 4ξξ 0

Therefore

g− = 4

l−

π/2

0

dφ 1 + k2

l2−

sin2 φ(D.17)

g− = 4

l−M

−k2

l2−

(D.18)

where M is a complete elliptic integral of the first kind. Using the identity

(D.19) M (−m) = 1√ 1 + m

M

m

1 + m



182

(D.20) g− = 4

l2− + k2M

k2

l2− + k2Using the polynomial approximation for M (m) of [80],

(D.21) M (m) =4

i=0

[ai − bi ln(1 − m)](1 − m)i + ε(m)

where ai and bi are known constants, and |ε(m)| ≤ 2 × 10−8,

(D.22) g− = 4 l2− + k2

4

i=0

ai −

bi ln1

− k2

l2− + k21

− k2

l2− + k2

i

Substituting back in to the integrodifferential equation,

2πK [ξ 0, η0] =

∞0

ξ dξ

K [ξ, f (ξ )]

− f (ξ )

∂g−

∂ξ +

∂ g−

∂η

+ g−

3 [−f (ξ ) + f (ξ )ξ ]

(D.23)



183

and differentiating Equation D.22 with respect to ξ and η , we obtain

2πK [ξ 0, f (ξ 0)] =

∞0

ξ d ξ

K [ξ, f (ξ )]

− f (ξ )

× −4(ξ + ξ 0)

(l2− + k2)3/2

4n=0

an − bn ln

1 − k2

l2− + k2

1 − k2

l2− + k2

n

+ 4

(l2− + k2)1/2

4n=1

n

an − bn ln

1 − k2

l2− + k2

1 − k2

l2− + k2

n−1

−4

n=0

bn

1 − k2

l2− + k2

n−1

4ξ 0 [ξ 2 − ξ 20 − [f (ξ ) − f (ξ 0)]2]

[l2− + k2]2

− 4[f (ξ ) − f (ξ 0)]

(l2− + k2)3/2

4n=0

an − bn ln

1 − k2

l2− + k2

1 − k2

l2− + k2

n

+ 4

(l2− + k2)1/2

4n=1

n

an − bn ln

1 − k2

l2− + k2

1 − k2

l2− + k2

n−1

−

4

n=0

bn1

−

k2

l2

− + k2

n−1

8ξξ 0 [f (ξ ) − f (ξ 0)]

[l2− + k

2

]2

− 4

3

[f (ξ )ξ − f (ξ )]

(l2− + k2)1/2

4n=0

an − bn ln

1 − k2

l2− + k2

1 − k2

l2− + k2

n

(D.24)



184

Grouping coefficients of the series, we obtain Equation 7.18:

2πK [ξ 0, f (ξ 0)] =

∞0

ξ d ξ

K [ξ, f (ξ )]

×

f (ξ )4(ξ + ξ 0)

(l2− + k2)3/2 − 4[f (ξ ) − f (ξ 0)]

(l2− + k2)3/2

×

4n=0

an − bn ln

1 − k2

l2− + k2

1 − k2

l2− + k2

n

+−

16ξ 0f (ξ ) [ξ 2

−ξ 20

−[f (ξ )

−f (ξ 0)]2]

(l2− + k2)5/2 +

32ξξ 0[f (ξ )

−f (ξ 0)]

(l2− + k2)5/2

× 4

n=1

n

an − bn ln

1 − k2

l2− + k2

1 − k2

l2− + k2

n−1

−4

n=0

bn

1 − k2

l2− + k2

n−1

− 4[f (ξ )ξ − f (ξ )]

3(l2− + k2)1/2

4

n=0

an − bn ln

1 − k2

l2− + k2

1 − k2

l2− + k2

n

(D.25)



185

APPENDIX E

Details of Boundary-Integral Equation Before Pinching in

Three Dimensions


(E.1) 12

K (ξ 0, η0) =

S

K (ξ, η) ∂g

∂n − g ∂K

∂n

dS


for this geometry is

(E.2) g = −1

4π [(η cos θ − η0 cos θ0)2 + (η sin θ − η0 sin θ0)2 + (ξ − ξ 0)2]1/2

The surface integral of Equation E.1 must also include the arbitrarily shaped caps at

ξ = ±∞ that enclose the liquid volume. However, as ξ → ±∞, C L → C 0L so K → 0 and

therefore ∂K ∂n

→ 0. Therefore these boundaries make no contribution to the integral.

The integral along the surface is performed in radial coordinates. The surface area

element is

ds = ηdθ d ξ 2 + d η2(E.3)

= dθf (ξ )d ξ

1 + f (ξ )2(E.4)



186

So the integrodifferential equation is

(E.5) 1

2K (ξ 0, η0) =

2π

0

∞

−∞

dθf (ξ )dξ

1 + f (ξ )2

K (ξ, η)∂g

∂n − g

∂K

∂n

The solute conservation boundary condition can again be used to write ∂K ∂n in terms of

ξ and f (ξ ) as

(E.6) ∂K

∂n =

f (ξ ) − f (ξ )ξ

3

1 + f (ξ )2

In this case we choose the normal vector to the interface pointing out of the liquid.


n = ∇F

|∇F | = F ξξ + F ηη

F 2ξ + F 2η

(E.7)

= −f (ξ )ξ + η

1 + f (ξ )2(E.8)

where F = η − f (ξ ) = 0 defines the interface. To find ∂g∂n ,

∂g

∂n = n · ∇g =

−f (ξ )ξ + η

1 + f (ξ )2

·

∂g

∂ξ ξ +

∂ g

∂ηη

(E.9)

= 1

1 + f (ξ )2

−f (ξ )

∂g

∂ξ +

∂g

∂η

(E.10)

Since we are trying to find the surface η = f (ξ ), let θ0 = 0.

g = −1

4π

(η cos θ − η0)2 + (η sin θ)2 + (ξ ∓ ξ 0))21/2(E.11)

= −1

4π

η2 + η20 − 2ξξ 0 cos θ + (ξ ∓ ξ 0)2

1/2(E.12)



187

where the (ξ −ξ 0)2 term applies for positive ξ and the (ξ +ξ 0)2 term applies for negative

ξ . Let

(E.13) g∓ ≡ 2π0

dθ 1

η2 + η20 − 2ηη0 cos θ + (ξ ∓ ξ 0)2

1/2

Using the trigonometric identity 1 − cos θ = 2sin2 θ2

and performing a change of

variables θ = 2φ,

(E.14) g∓

= π0 2dφ

η2

+ η2

0 − 2ηη0 + 4ηη0 sin2

φ + (ξ ∓ ξ 0)2−1/2

Let

(E.15) l∓ ≡ (η − η0)2 + [ξ ∓ ξ 0)]2

(E.16) k2 ≡ 4ηη0

Therefore

g∓ = 4

l∓

π/2

0

dφ 1 + k2

l2∓

sin2 φ(E.17)

g∓ = 4

l∓M

−k2

l2∓

(E.18)

where M is a complete elliptic integral of the first kind. Using the identity

(E.19) M (−m) = 1√ 1 + m

M

m

1 + m



188

(E.20) g∓ = 4

l2∓ + k2M

k2

l2∓ + k2Using the polynomial approximation for M (m) of [80],

(E.21) M (m) =4

i=0

[ai − bi ln(1 − m)](1 − m)i + ε(m)

where ai and bi are known constants, and |ε(m)| ≤ 2 × 10−8,

(E.22) g∓ = 4

l2∓ + k2

4

i=0

ai −

bi ln1

− k2

l2∓ + k21

− k2

l2− + k2

i

Substituting back in to the integrodifferential equation,

2πK [ξ 0, η0] =

∞−∞

f (ξ )dξ

K [ξ, f (ξ )]

f (ξ )

∂g∓

∂ξ − ∂ g∓

∂η

− g∓

3 [−f (ξ ) + f (ξ )ξ ]

(E.23)

The integral is split into positive and negative domains, then the negative portion is

changed to an integral over +ξ by performing a change of variables and using the

properties that ξ is odd, f (ξ ) is even, f (ξ ) is odd, and K is even. In this manner the

integral can be rewritten as

2πK [ξ 0, η0] =

∞0

f (ξ )dξ

K [ξ, f (ξ )]

f (ξ )

∂g−

∂ξ − ∂ g−

∂η

−

g−

3 [

−f (ξ ) + f (ξ )ξ ]

+

∞0

f (ξ )dξ

K [ξ, f (ξ )]

− f (ξ )

∂g+

∂ξ − ∂ g+

∂η

− g+

3 [−f (ξ ) + f (ξ )ξ ]

(E.24)



189

and differentiating Equation E.22 with respect to ξ and η, and grouping coefficients of

the series, we obtain Equation 7.22:

2πK [ξ 0, f (ξ 0)] =

∞0

dξf (ξ )

K [ξ, f (ξ )]

×

−4f (ξ )(ξ − ξ 0)

(l2− + k2)3/2 +

4(η + η0)

(l2− + k2)3/2

M

k2

l2− + k2

+32ηη0(ξ − ξ 0)f (ξ )

(l2− + k2)5/2 − 16η0[η2 − η2

0 − (ξ − ξ 0)2]

(l2− + k2)5/2 M d k2

l2− + k2− 4

3

[f (ξ )ξ − f (ξ )] l2 k2

M

k2

l2 + k2

(E.25)

PhD thesis on Phase Field simulation and maths methods

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