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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
June 2010
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UMI Number: 3402130
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c Copyright by Larry Kenneth Aagesen, Jr. 2010
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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 shapes 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 from a personal standpoint. 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. I’m also really happy to have had the pleasure of working with
the other members of the Voorhees group. You are an amazingly talented and unique
group of people. We still need to write that sitcom about the lab someday. People’s
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experiences in graduate school vary widely, so I am especially glad to have had such a
great experience here both professionally and personally.
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 17
Chapter 1. Introduction 18
Chapter 2. Background 22
2.1. Dendritic Microstructures 22
2.2. Coarsening 28
2.2.1. Coarsening of Spherical Particles 30
2.2.2. Coarsening of Dendritic Microstructures 31
2.3. Interfacial Curvature 33
2.3.1. Interface Shape Distributions 36
2.3.2. Rate of Change of Curvatures 37
2.4. Topology 39
2.5. Self-Similarity 40
2.6. Phase-Field Models 45
2.6.1. Phase-Field Modeling of Pure Material Solidification 48
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2.6.2. Phase-Field Modeling of Binary Alloy Solidification 49
Chapter 3. Methods 52
3.1. Three-Dimensional Microstructure Characterization 52
3.2. Interfacial Curvature 53
3.3. Topological Characterization 55
3.4. Calculation of Interfacial Velocities 55
3.5. Phase Field Models 57
3.5.1. Isothermal Solidification of Cu-Ni 57
3.5.2. Isothermal Coarsening of Al-Cu 58
Chapter 4. Phase-Field Simulation of Equiaxed Dendritic Solidification in a
Binary Alloy 61
4.1. Microstructure, Volume Fraction, and Surface Area 61
4.2. Interface Shape Distributions 664.3. Interfacial Topology 71
4.4. Comparison with Other Systems 79
4.5. Conclusions 81
Chapter 5. Rate of Change of Curvatures 83
5.1. Details of Algorithm 83
5.2. Examples 84
5.2.1. Cylinder Expanding in the Radial Direction 84
5.2.2. Translating Cylinder 86
5.3. Testing of Algorithm 87
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5.4. Strategies to Decrease Anisotropy of |∇H | 91
5.5. Conclusions 94
Chapter 6. Using Phase-Field Modeling to Predict Interfacial Velocities During
Coarsening 95
6.1. Methodology 95
6.2. Experimental Procedure 96
6.3. Validating the Phase-Field Model for Simulating Coarsening 97
6.4. Calculating Interfacial Velocities from Experimental Data 102
6.4.1. Choosing Simulation Parameters 103
6.5. Comparing Experiment to Theory 111
6.6. Analysis of Contributions to Error 112
6.6.1. Phase-Field Model at Large 113
6.6.2. Effect of Assumption DS = 0 115
6.6.3. Physical Parameters 117
6.7. Effect of Anti-trapping Current 119
6.8. Conclusions 120
Chapter 7. Self-Similar Pinch-off of Rods 121
7.1. Theoretical Formulation 122
7.1.1. Similarity Variables 123
7.2. Phase-Field Simulations 124
7.3. Solving for Interface Shape in Self-Similar Variables 128
7.3.1. Interface Shape After Pinching in 2D 129
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7.3.2. Interface Shape After Pinching in 3D 134
7.3.3. Interface Shape Before Pinching in 3D 137
7.3.4. Interface Shape Before Pinching in 3D for Solid Cylinders in a Liquid
Matrix 143
7.4. Comparison with Experimental Data 144
7.4.1. Conclusions 147
Chapter 8. Conclusions 150
References 153
Appendix A. Derivation of Perturbation Decay Constant σ 160
Appendix B. Determining Time Exponent of Similarity Solution for Rod
Pinching 165
Appendix C. Details of Boundary-Integral Equation After Pinching in Two
Dimensions 170
Appendix D. Details of Boundary-Integral Equation After Pinching in Three
Dimensions 173
Appendix E. Details of Boundary-Integral Equation Before Pinching in Three
Dimensions 179
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List of Figures
2.1 Formation of zones in a casting. 24
2.2 The conditions which give rise to constitutional undercooling. 26
2.3 δ /δ as a function of wavelength. 27
2.4 Four different models for isothermal coarsening. 33
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. 34
2.6 Interface Shape Distribution (ISD). 36
2.7 Particle radius distribution predicted by the LSW theory. 41
2.8 The pinch-off of a cylinder by Rayleigh instability as a function of
time. 43
4.1 Microstructures of Cu-Ni system undergoing dendritic solidification. 62
4.2 Microstructures of Cu-Ni system entering the coarsening regime. 63
4.3 V f and S v versus time. 64
4.4 S −1v vs. t1/3 during (a) solidification and coarsening, (b) coarsening
regime only. 64
4.5 V f and V f /S v versus time. 65
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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. 67
4.7 An isolated dendrite at t = 26.2 µs. 68
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. 70
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. 71
4.10 ISDs of Cu-Ni system undergoing coarsening at (a) 105 µs, (b) 131
µs, and (c) 157 µs. 72
4.11 A (1.31 × 1.83 × 1.44 µm) subset of the microstructure at 105 µs. 73
4.12 Scaled ISDs of Cu-Ni system undergoing coarsening at (a) 105 µs,
(b) 131 µs, and (c) 157 µs. 74
4.13 Portion of the simulation microstructure at (a) 105 µs, (b) 131 µs,
and (c) 157 µs. 75
4.14 Genus per unit volume versus time. 77
4.15 A handle (liquid tunnel) is formed as adjacent secondary arms join. 77
4.16 The breakup of liquid tubes due to capillarity-driven instability. 78
4.17 Number of independent bodies per unit volume for solid and liquid
phases. 79
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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. 85
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). 88
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. 90
6.1 Sample input array for perturbation decay simulation. 98
6.2 Decay of a perturbation from simulation results (red) compared
with analytical solution calculated from Equation 6.2 (blue). 99
6.3 (a) Decay constant error for perturbation with A = 10 with various
perturbation wavelengths, (b) decay constant error for constant
system size, varying . 100
6.4 (a) 100 × 80 × 150 voxel portion of microstructure used for
calculation of interfacial velocities, (b)-(c) solid-liquid interface
colored by normal interfacial velocities calculated from experimental
data. 104
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. 108
6.6 (a), (b) Two different views of the larger simulation volume selected
for velocity calculations. 109
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6.7 Comparison of experimental and simulated velocities. 111
6.8 Normalized histograms of experimentally measured and scaled
simulated interfacial velocites. 112
6.9 Difference between theoretical prediction and simulation results for
perturbation decay for high values of . 115
6.10 Interfacial velocity prediction using phase-field model without
anti-trapping current. 119
7.1 The microstructure of the Al-Cu alloy experimentally observed
during coarsening, with the liquid region capped at the boundary. 122
7.2 Phase-field simulations of liquid cylinder pinching off by bulk
diffusion. 125
7.3 From phase-field simulations, (a) minimum radius of a pinching
cylinder cubed, (b) distance between tips of cones after pinchingcubed versus time. 126
7.4 Cone angles measured as close as possible to the point of pinching,
(a) phase-field simulations, (b) experimental data. 128
7.5 Rod pinching due to Rayleigh instability, (a) before, (b) at, (c)
after, the time of pinching. 129
7.6 Shape of the 2D solid-liquid interface after pinching by interfacial-
energy driven bulk diffusion. 132
7.7 Convergence testing of 2D solid-liquid interface shape after
pinching. 133
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7.8 Shape of the 3D solid-liquid interface after pinching by interfacial-
energy driven bulk diffusion. 137
7.9 Convergence testing of 3D solid-liquid interface shape after
pinching. 138
7.10 Shape of the 3D solid-liquid interface before pinching by
interfacial-energy driven bulk diffusion. 141
7.11 Convergence testing of 3D solid-liquid interface shape before
pinching. 142
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. 143
7.13 The diameter of a pinching tube as a function of (t − ts)1/3. 145
7.14 Quantifying agreement between theoretically predicted and
experimentally measured interface shape. 148
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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. 73
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. 76
4.3 Computed genus versus time. 78
5.1 Summary of testing for expanding cylinder test scenario. 93
5.2 Summary of testing for translating cylinder test scenario. 94
6.1 Physical parameters used in phase-field simulations. 106
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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 [1].
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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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
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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 [1].
Figure 2.1 shows the types of dendritic microstructures that form in a casting [2].
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
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Figure 2.1. Formation of zones in a casting [2]. 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
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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
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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 [2].
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 [3]. Unlike the constitutional undercooling criteria, they took
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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 [3].
The shape of a dendrite’s during steady-state growth has also been closely studied.
Ivantsov [4] 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 )]
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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 [5, 6] 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 [7]
(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.
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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 [8] 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
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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 [9] and Wagner [10] 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
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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. [11] 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. [12] treated the arms
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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. [13]
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. [14, 15] The next such model considered
two adjacent teardrop-shaped secondary arms which eventually coalesce into a single
arm. [16] 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 [17], 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 [17],
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
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Figure 2.4. Four different models for isothermal coarsening: (1) radialremelting, (2) axial remelting, (3) arm detachment, (4) arm coales-cence [12].
hold during the coarsening of directionally solidified dendritic Sn-Bi [17], Al-Cu [18],
and Pb-Sn [19] and equiaxially solidified dendritic Al-Cu [20].
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
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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
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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 [21]. 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)
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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 [22, 23]. 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):
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• 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 [24]. 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
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field Φ(x, t), smooth in a 3-dimensional neighborhood of S (t) for all t, then by the
chain rule [25],
(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 [26]:
(2.18)
H = −(2H 2 − K )v − 12
(v11 + v22)
(2.19)
K = −2HKv − H (v11 + v22) +√
H 2 − K (v11 + v22)
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(where we have used the notation
H and
K rather than ∂H ∂t and ∂K
∂t as in [26] 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. [27]
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
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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 [28]:
(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 [9,10], the particle size distribution is self-similar when
scaled by the average particle radius R(t), which is proportional to t1/3 (Figure 2.7).
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Figure 2.7. Particle radius distribution predicted by the LSW the-ory [29]. The horizontal axis is the radius of a particle divided by theaverage particle radius for the entire collection of particles at time t. Thevertical axis is the probability of finding a particle with that radius. Be-
cause the distribution is constant with time, the particle size distributionis said to 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 [18] and Kammer [19] 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 [20], but in
this case the dendrites were produced by equiaxial solidification. She found that again,
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S −1v ∝ t1/3, but that in this case, the ISDs were self-similar for sufficiently long coarsen-
ing times. When compared with [18] and [19], 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
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-5
-4
-3
-2
-1
0
1
2
3
4
5
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
1
2
3
4
5
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
1
2
3
4
5
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
1
2
3
4
5
-5 -4 -3 -2 -1 0 1 2 3 4 5
Solid
Liquid
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.
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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 [30]. 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 [31]. 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 [32]. Wong et. al. investigated the pinch-off of rods via capillarity-
driven surface diffusion [33]. 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 [32] 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.
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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
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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)
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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
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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
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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
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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
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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].
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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.
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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
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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.
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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
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The components of |∇φ| are calculated using central differences as in Equation 3.3. The
magnitudes of velocities on the fixed 3D grid are then interpolated onto the polygonized
interface using the IDL INTERPOLATE function. When the components of velocity
vx, vy, and vz are needed, they can be calculated from the magnitude of velocity and
derivatives of the order parameter:
(3.9) vx =
−∂φ
∂t∂φ1/∂x
|∇φ1|2
(3.10) vy = −
∂φ
∂t
∂φ1/∂y
|∇φ1|2
(3.11) vz = −
∂φ
∂t
∂φ1/∂z
|∇φ1|2
When calculating velocities from an experimentally measured 3D microstructure,
the order parameter data is initially binary. Before the velocity calculation, the order
parameter data is smoothed to obtain a smoother velocity profile when interpolated
onto the interface. The binary data at each point is averaged with surrounding data
points using the IDL SMOOTH function with a width of 5 voxels. This means that
each point is averaged with the nearest two points in the +x direction and the nearest
two points in the −x direction for a total of five points being averaged, and similarly
for the y, z directions.
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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 ×
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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)
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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
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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.
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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.
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(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)).
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(a) (b)
(c) (d)
(e) (f)
Figure 4.2. Microstructures of Cu-Ni system entering the coarseningregime at (a),(b) 105 µs, (c),(d) 131 µs, and (e),(f) 157 µs. Thesolid phase is capped at the boundary of the visualization region in(a),(c),(e) and the liquid phase is capped in (b),(d),(f). Visualizationsare a 640 × 390 × 128 subset of the 640 × 640 × 640 simulation volume.
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!
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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.
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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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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
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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.
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-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.
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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
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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
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-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.
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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
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-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.
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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)
Liquidvoids
Liquidvoids
(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.
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-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.
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(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).
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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
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!"
$
"
%
&
'
(
)
$ %$ '$ )$ *$ "$$ "%$ "'$ ")$ "*$
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.
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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.
(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.
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!
!#!$
!#!%
!#!&
!#!'
!#(
!#($
! $! %! &! '! (!! ($! (%! (&! ('!
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.
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 [18, 19, 66]. In this section, we will
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.
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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 −3v of the 74% solid sample was 0.02
for 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 [18] and Pb-Sn [19] 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
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
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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 −3
v to
decrease. The large number of liquid droplets formed in [67] result from the pinching
of 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
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.
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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 as
the 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 inthe coarsening regime.
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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
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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.
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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,
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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
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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
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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
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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.
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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
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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.
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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)
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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.
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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.
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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
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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 [23]. 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.
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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.
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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
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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.
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-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 with
various 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.
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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.
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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
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∆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))
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(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.
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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
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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
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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
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(a) (b)
(c) (d)
Figure 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 iterationsfor the velocity calculation. The liquid phase is capped at the boundariesand surfaces are colored by interfacial velocity.
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(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:
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• 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.
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(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. Comparison of experimental and simulated velocities. Thesolid-liquid interface is shown colored by normal velocities, with negativevelocities pointing into the liquid. Liquid is inside the bulb shape, andthe solid phase is transparent. Note color bars between left and rightimages differ 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
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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
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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
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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,
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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
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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
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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
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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.
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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.
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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.
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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.
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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
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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
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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
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(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
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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 [32]. However, those simulations used an adaptive grid
spacing and time stepping algorithm that allowed the interface shape to be determined
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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.
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(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
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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 [33].
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,
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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.
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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.
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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,
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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)
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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 [33], 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)
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= −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
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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
Equation 7.19 is substituted into Equation 7.18 and a set of integrodifferential
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.
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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
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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
R a t i o o f E r r o r s
!
Error ratio 1
Error ratio 2
(b)
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5
R a t i o o f E r r o r s
!
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
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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
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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
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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
R a t i o o f E r r o r s
!
Error ratio 1
Error ratio 2
(c)
Figure 7.11. (a) Variation of 3D solid-liquid interface shape before 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.
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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.
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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
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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.
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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
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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
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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
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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.
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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
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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.
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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.
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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
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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
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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
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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)
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−DsΓsk2 σ + Dsk2
Ds exp−
σ + Dsk2
Ds z
exp(i k · r + σt)(A.13)
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.
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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
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(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)
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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
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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
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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.
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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)
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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)
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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)
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APPENDIX D
Details of Boundary-Integral Equation After Pinching in
Three Dimensions
The integrodifferential equation is
(D.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 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)
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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
The normal vector to the interface is given by
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)
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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
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(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)
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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)
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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)
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APPENDIX E
Details of Boundary-Integral Equation Before Pinching in
Three Dimensions
The integrodifferential equation is
(E.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 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)
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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.
The normal vector to the interface is given by
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)
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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
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(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)
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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)
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