Embed Size (px)
Citation preview
NORTHWESTERN UNIVERSITY
Evolving Faceted Surfaces: From Continuum Modeling, to Geometric
Simulation, to Mean-Field Theory
A DISSERTATION
SUBMITTED TO THE GRADUATE SCHOOL
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
for the degree
DOCTOR OF PHILOSOPHY
Field of Applied Mathematics
By
Scott A. Norris
EVANSTON, ILLINOIS
December 2006
2
c© Copyright by Scott A. Norris 2006
All Rights Reserved
3
ABSTRACT
Evolving Faceted Surfaces: From Continuum Modeling, to Geometric Simulation, to
Mean-Field Theory
Scott A. Norris
We first consider the directional solidification, in two dimensions, of a dilute binary alloy
having a large anisotropy of surface energy, where the sample is pulled in a high-energy
direction such that the planar state is thermodynamically prohibited. Analyses including
reduction of dynamics, matched asymptotic analysis, and energy minimization are used
to show that the interface assumes a faceted profile with small wavelength. Questions
on stability and other dynamic behavior lead to the derivation of a facet-velocity law.
This shows the that faceted steady solutions are stable in the absence of constitutional
supercooling, while in its presence, coarsening replaces cell formation as the mechanism
of instability.
We next proceed to introduce a computational-geometry tool which, given a facet-
velocity law, performs large-scale simulations of fully-faceted coarsening surfaces, first
in the special case with only three allowed facet orientations (threefold symmetry), and
then for arbitrary surfaces. Topological events including coarsening are comprehensively
4
considered, and are treated explicitly by our method using both a priori knowledge of
event outcomes and a novel graph-rewriting algorithm. While careful attention must
be paid to both non-unique topological events and the imposition of a discrete time-
stepping scheme, the resulting method allows rapid simulation of large surfaces and easy
extraction of statistical data. Example statistics are provided for the threefold case based
on simulations totaling one million facets.
Finally, a mean-field theory is developed for the scale-invariant length distributions
observed during the coarsening of one-dimensional faceted surfaces. This theory closely
follows the LSW theory of Ostwald ripening in two-phase systems, but the mechanism
of coarsening in faceted surfaces requires the derivation of additional terms to model the
coalescence of facets. The model is solved by the exponential distribution, but agree-
ment with experiment is limited by the assumption that neighboring facet lengths are
uncorrelated. However, the method concisely describes the essential processes operating
in the scaling state, illuminates a clear path for future refinement, and offers a generic
framework for the investigation of faceted surfaces evolving under arbitrary dynamics.
5
Acknowledgements
Soli Deo Gloria – To God alone be glory. With this great cry of the Reformation, the
composer J.S. Bach concluded each of his church Cantatas, and with that example I use
it to conclude this thesis. While fully accepting that evolution is the only well-reasoned
theory describing the mechanism of our origins, I still believe, on the basis of Christ’s
resurrection, that God is ultimately responsible in some way for everything that exists.
Thus, the credit for anything of worth in this work belongs ultimately and solely to Him.
I could thank God for many things here, but will limit myself to two. First, I am grateful
for Northwestern’s Graduate Christian Fellowship, where I found friends who love me,
colleagues who have shared my struggles, and a continual struggling conversation about
how to honor God amidst academic pursuit. Second, as I have struggled to justify the
pursuit of mathematics in a world with so much loneliness and suffering, I thank God for
giving me a vision of how my profession fits into the larger whole of the person He wants
me to become. My quest at this institution, and indeed my life as a whole, would be
much emptier without the presence of His truth and love.
Mom and Dad, thank you for your encouragement, love, and for teaching me all of
things growing up that have allowed me to succeed on my own. I am so grateful for all
the time and energy you have poured into me, and I hope I can do half as well with
my own children. Keith and Carla, thank you for being great in-laws – I’ll always look
forward to spending time with you guys. Finally, to my dear wife Tara, thank you for
6
your constant love and support – for celebrating with me in my successes, encouraging
me in my failures, and assuring me above all that you’d keep loving me no matter what
happened. It has been a long road, but sharing it with you has given me strength and
hope along the way.
I have been blessed to make some very good friends during my time here, and so I
thank: Michael and Bolu, for two fantastic years as roommates, much encouragement,
and the times I borrowed your cars; Matt, Edy, and Young Cheol, for a truly special year
of friendship and fellowship in small group, and many great eating expeditions; Kyle,
for many enjoyable lunches, and for teaching me to almost love the Mac; Gogi, for more
lunches and for laughing with me about the ups and downs of grad life; and Peter, for
all the wonderful conversations, reflections, and video game binges. You guys have truly
blessed my life, and it was you that made me sad to leave Chicago.
A good adviser makes graduate school easier, and I’ve been lucky to have two. Steve,
a huge thanks for your flexibility in allowing me to telecommute, and for always being
available for guidance and advice despite your busy schedule. Most of all, thank you for
your example of quiet humility and patience despite all you’ve achieved. Stephen, thank
you for our many conversations of all kinds, and for the inspiration of your sheer love of
doing mathematics. Also, thank you especially for sharing your life with me as well as
your ideas. To both of you, thanks for leading me to a topic of research that has turned
out to be a lot of fun to pursue. No matter the extent to which research is a part of the
rest of my life, I will look back fondly on this work.
Finally, despite rumors to the contrary propagated by PhD Comics, graduate students
must eat more than Ramen, and therefore I thank NASA for supporting me during the
7
majority of my time here. Though I have to wonder if my work will ever directly aid the
exploration of space, it is fun to be associated with a program that brought such wonder
to my childhood, and an honor to know they thought my work promising enough to fund.
In addition, I thank Northwestern and IGERT for support during my first and second
years, respectively, and NSF, via Stephen, for that extra bit of commuting money third
and fourth years.
8
Table of Contents
ABSTRACT 3
Acknowledgements 5
List of Tables 11
List of Figures 12
Chapter 1. Introduction 14
Chapter 2. Faceted Interfaces in Directional Solidification 20
2.1. Introduction 20
2.2. Background: Governing equations, basic state, and linear stability 27
2.3. Statics. Non-planar interface due to strong anisotropy, φ = 0 39
2.4. Energetics. Optimal Wavelength, Comparison with Planar State 51
2.5. Dynamics. Stability, Wavelength Trapping, and Coarsening 56
2.6. Conclusions and Comments 62
Chapter 3. Large-Scale Simulations of Coarsening Faceted Surfaces 65
3.1. Introduction 65
3.2. Faceted Surfaces: Description, Kinematics, and Dynamics 68
3.3. Topological Events 71
3.4. Demonstration 76
9
3.5. Conclusions 82
Chapter 4. The Kinematics of Faceted Surfaces with Arbitary Symmetry 84
4.1. Introduction 84
4.2. Data structures and simple motion: a 3D cellular network 87
4.3. Topological Events 90
4.4. Discretization and Performance of Topological Events 110
4.5. Demonstration and Discussion 117
4.6. Conclusions 122
Chapter 5. A Mean-Field Theory for Coarsening Faceted Surfaces 124
5.1. Introduction 124
5.2. Example Dynamics and Problem Formulation 127
5.3. Application: Our chosen facet dynamics 133
5.4. Solution and Comparison with Numerical-Experimental Data 134
5.5. Conclusions 136
References 140
Appendix A. Appendices for Chapter 1 149
A.1. Justification of the Quasi-steady state 149
A.2. Homogenized Linear Stability Analysis 151
Appendix B. Appendices to Chapter 3 154
B.1. Elaboration on Far-Field Reconnection 154
B.2. Non-Uniqueness 157
10
Appendix C. Appendix to Chapter 4 163
C.1. Numerical Simulation 163
11
List of Tables
3.1 Relevant coarsening phenomena. 78
12
List of Figures
2.1 Summary of linear stability results 36
2.2 Surface-energy minimizing slopes 40
2.3 Analysis of outer solutions 49
2.4 Sample faceted profiles 50
2.5 Representative behaviors of dynamic faceted interfaces 61
3.1 Facet Merge event 72
3.2 Merging Facet Pinch event 73
3.3 Vanishing Facet events 75
3.4 Example coarsening sequence 77
3.5 One-point geometric distributions 79
3.6 Correlational distributions 81
4.1 Diagram of neighbor relations 88
4.2 Normal diagrams for Vanishing Edge events 93
4.3 Neighbor Switch event 94
4.4 Irregular Neighbor Switch event 95
4.5 Facet Join event 96
13
4.6 Signatures of Constricted Facet events 98
4.7 Facet Pierce event 100
4.8 Irregular Facet Pierce event 101
4.9 Facet Pinch event 103
4.10 Joining Facet Pinch event 104
4.11 Example Vanishing Facet Event 106
4.12 Facets Vanishing in a group 108
4.13 Facets Vanishing as a step 109
4.14 Coarsening sequence with threefold symmetry 119
4.15 Coarsening sequence with fourfold symmetry 120
4.16 Coarsening sequence with sixfold symmetry 121
5.1 Survey of coarsening behavior 128
5.2 Representative facet configuration 129
5.3 Comparison between theory and experiment 138
5.4 Correlations in neighboring facet lengths 139
B.1 Method of listing binary trees 161
B.2 Saddle versions of Vanishing Edge events 162
14
CHAPTER 1
Introduction
This thesis, in the most general sense, is about what happens to materials with a large
anisotropy of surface energy when placed in dynamic situations. Thus, we begin with an
overview of the definition, causes, and effects of large surface energy anisotropy. A unique
property of crystals (in contrast to amorphous solids like glass) is that their material prop-
erties usually depend on orientation. This is due, in turn, to the packing symmetries of
the internal crystal lattice on which individual atoms reside. When such materials expose
a planar interface, the structure of that interface looks different at atomic scales depend-
ing on its orientation relative to the crystal lattice. These different surface configurations
cause the material to react differently with its surrounding environment. Thus, the sur-
face energy γ (and many other properties) depends on the surface orientation relative to
the lattice. This dependence is modeled by making these quantities functions of an angle
θ, which describes the deviation of the interface normal from some reference orientation
associated with the lattice. While many different functions describing anisotropy may be
considered, it will suffice for what follows to consider the particular form
γ(θ) = γ0[1 + α4cos(4θ)].
This surface energy for crystals in two dimensions exhibits fourfold symmetry, with mean
magnitude γ0 and relative anisotropy varying through the parameter α4.
15
The earliest studies of anisotropy focused on the equilibrium problem, which seeks the
surface-energy minimizing shape in two dimensions of a particle in equilibrium with a sur-
rounding melt, given the form of γ(θ). This problem was solved in 1901 by Wulff [1], who
devised a geometric method of constructing the correct shape, now known as the Wulff
construction. For the example surface energy γ considered here, increasing the value of
the anisotropy coefficient α4 from zero causes an initially round particle to change shape,
eventually developing missing orientations and corners at the critical value of α4 = 1/15.
A related problem inquires about the equilibrium shape of an initially planar interface
exposed to its melt at the high-energy orientation θ = 0. Here, Herring [2] also used
geometric considerations to show that the surface remains planar for small α4, but at the
identical critical value of α4 = 1/15, this interface becomes thermodynamically unstable
and is replaced by a completely faceted “sawtooth” interface. Herring also showed that
these two equilibrium results are linked, as it is precisely the missing orientations of the
former problem that are thermodynamically unstable in the latter. Furthermore, the
shared critical value is no accident – high-energy orientations will be unstable in any con-
text if the surface stiffness γ+γθθ of that orientation is negative [3]. This thermodynamic
instability is a key concept in the following chapters, and may be generally remembered
as follows: “an interface orientation is thermodynamically unstable if it can reduce its
surface energy by assuming a faceted sawtooth form.”
The modeling of thermodynamically-faceting surfaces in dynamic situations was ini-
tiated by Mullins [4] in 1961 and Cabrera [5] in 1963; these authors noted independently
the similarities between the process of surface faceting and the then-fledgling modeling
16
of evolving phase boundaries. However, even as understanding of phase boundaries pro-
gressed, including the seminal work of Cahn and Hilliard [6], the problem of faceting dy-
namics lay dormant for nearly thirty years, until it was picked up again and studied from
two perspectives – purely continuum thermodynamic models [7, 8, 9], and continuum
approximations of discrete step-flow models [10, 11, 12]. In each of these works, initially
planar interfaces are shown to undergo spinodal decomposition into faceted hill-and-valley
structures, just as one would expect from knowledge of the equilibrium problem. How-
ever, after this process is completed, coarsening phenomena are observed, where small
facets shrink to zero length and vanish, causing a corresponding continuous increase in
the average length of those that remain. Of even more interest is that these coarsening
faceted surfaces are often observed to obey dynamic scaling, in which the surface looks
the same at all sizes if first scaled by the average length. In such a state, the statistical
geometric properties of the surface remain constant even as the average lengthscale grows,
and this constant state can be considered to concisely summarize all of the salient infor-
mation present in the evolving system. This progression – from faceting, to coarsening,
to a dynamically scaling state – represents a typical surface behavior in models of many
systems, and is the pattern on which this thesis will focus.
In this historical context, then, we begin in Chapter 1 by applying mathematical
methods recently fruitful in other faceting contexts to the problem of the directional so-
lidification of a dilute binary alloy – a system which, though possessing a rich history of
its own, has been mostly neglected in the context of faceting. While the inclusion of a
solute field makes this system is slightly more complex than the pure-material systems
17
considered previously, the mathematical progression follows a similar course, with sim-
ilar results. Starting from a full free-boundary problem, thermodynamic instability of
an initially planar surface with negative surface stiffness leads to faceting. This occurs
initially on a small wavelength which, in turn, allows a simplification to a partial differ-
ential equation that governs just the interface position. Matched asymptotic analysis of
this equation reveals a family of steady faceted solutions, and energetic considerations
specify an optimal member. Turning to the unsteady problem, we follow Watson [13] to
obtain a key theoretical simplification – the further reduction of surface dynamics to a
facet-velocity law which describes the normal velocity of each facet as a function of its
geometric configuration. This law reveals that, below the critical pulling speed causing
supercooling, the steady faceted profiles are stable. However, above that critical speed,
coarsening occurs, which (a) replaces the formation of cells as the principal mechanism
of instability, and (b) appears at pulling speeds smaller than those at which cells would
otherwise appear.
Chapters 2 and 3 begin by observing that facet velocity laws of the type found in
Chapter 1 exist for many systems. Because they so efficiently describe the evolving
faceted surface, they allow correspondingly efficient computational methods; and since
the statistical study of dynamic scaling necessitates the simulation of large surfaces, the
development of a tool to exploit such theoretical simplification is urgently needed. While
easily implemented for 1+1D surfaces z = h(x, t), such tools are more difficult to construct
for the 2+1D surfaces z = h(x, y, t) commonly seen in crystal-growing experiments. This
is due to the occurrence of various topological events allowing surface re-organization and
coarsening. Existing simulation attempts must choose between speed [14, 15] and robust
18
topological handling [16, 17]; here we present a method that achieves both. The method
is outlined in Chapter 2 for threefold-symmetric surfaces, where it is used to study the
dynamic scaling exhibited by Watson’s annealing dynamics [13]. The method is then
more fully described and generalized to arbitrary symmetry in Chapter 3, where the same
phenomenon is illustrated by the dynamics derived in Chapter 1. In addition to speed
and topological accuracy, a further advantage of the method over previous approaches
is easy access to geometric data presented by the surface. This is achieved by choosing
a data structure which mirrors the natural structure of the surface, and demonstrated
by the exhibition of numerous statistical measures of surface geometry present in the
dynamically scaling states of the systems studied.
Finally, since dynamic scaling pushes complex evolving surfaces into a state which
can be effectively characterized by just a few statistics, it is natural to seek simplified
models which replicate this behavior. Chapter 4 illustrates just such a model, which
exploits the dynamics derived in Chapter 1 to describe the evolution of the facet length
distribution to the scale-invariant steady state found using the tool in Chapters 2 and
3. The model recalls the famous Lifshitz, Slyozov, and Wagner theory [18, 19, 20] of
Ostwald ripening in a two-phase system; however, due to the geometric consequences
of coarsening in faceted systems, it also includes a coagulation term reminiscent of that
in the work of Von Smoluchowski [21] and Schumann [22]. While certain simplifying
assumptions keep the resulting model from quantitatively matching numerically-collected
data, it qualitatively illustrates the essential forces at work in the dynamically scaling
state, while suggesting more general models that would increase its predictive value.
19
In summary, this thesis traces the particular faceting system of directional solidifica-
tion through a series of generally-applicable mathematical treatments. These include the
analytical reduction of a free-boundary problem to a facet-velocity law describing surface
evolution (Chapter 1), the numerical study of coarsening through an efficient computa-
tional tool (Chapters 2-3), and the development of a mean-field theory that attempts to
describe the resulting dynamically scaling state (Chapter 4). These three tools – con-
tinuum modeling, geometric simulation, and mean-field analysis – naturally parallel the
three stages of faceted-surface evolution – faceting, coarsening, and dynamic scaling. They
thus together form a framework for the comprehensive study and comparison of faceting
in many contexts.
20
CHAPTER 2
Faceted Interfaces in Directional Solidification
2.1. Introduction
In the process of directional solidification, a sample liquid is pulled through a tem-
perature gradient produced by the presence of a heating element and a cooling element,
set at temperatures above and below the freezing point of the liquid, respectively. At a
position in between these elements, the liquid freezes, forming a liquid/solid interface. If
the material being solidified is a binary alloy, then solute is rejected at the interface, and
must diffuse away into the bulk liquid, creating a solute gradient directed oppositely to
the thermal gradient. At high enough pulling speeds, the concentration gradient steepens
sufficiently to create an instability, replacing normally planar interface morphologies with
more complicated cellular or dendritic structures [23]. These phenomena may be easily
observed when transparent organic alloys are solidified within a narrow channel between
two glass planes, an environment known as a Hele-Shaw cell [24]. Because complex behav-
ior may thus be easily observed and measured, and because the nearly-two-dimensional
nature of the Hele-Shaw cell leads to analytical simplicity, this particular procedure has
been popular for the comparison of theory and experiment. Combined with the fact that
the instability described above is similar to important hydrodynamic instabilities [25],
21
the opportunities afforded by directional solidification have made it one of the most com-
monly studied forms of crystal growth, and placed it among the classical problems of
mathematical physics.
2.1.1. History 1: Directional Solidification
The first qualitative explanation of the morphological instability described above was
given in 1953 by Rutter and Chalmers [26]; these ideas were quantified the same year by
Tiller et. al. [27]. These authors argued that at high-enough pulling speeds, the steepen-
ing solute gradient at the interface eventually leads to a layer of liquid that is supercooled
for its chemical composition. In the presence of this consitutional supercooling, the pla-
nar interface was predicted to be unconditionally unstable. This idea was generalized in
1964 to include the effect of surace energy, when Mullins and Sekerka [23] performed a
linear stability analysis showing that surface energy could stabilize the interface against
small-wavelength perturbations even in the presence of constitutional supercooling1. This
accurately predicts the phenomenon of absolute stability, where the front restabilizes at
very high solidification rates as the effective surface energy regains dominance over the
solute gradient. Their analysis also predicted the critical wavelength at which instability
would occur, allowing for careful comparison with experimental results.
In 1970, Wollkind and Segel [28] extended this analysis into the weakly nonlinear
regime. For pulling velocities near the critical velocity, they derived ordinary differen-
tial equations (Landau equations) describing the post-instability amplitude of cellular
1If thermal conductivity is greater in the solid than in the liquid (common for metals), they also showedthat instability is possible without constitutional supercooling in the liquid. However, the central idea isthat a solute gradient large enough to overcome the thermal gradient is necessary for instability; thus,the idea of supercooling as a necessary condition is still instructive.
22
solutions having the critical wavelength predicted by Mullins and Sekerka. This analysis
predicted the conditions under which the instability is sub-critical or super-critical.
In limits where the critical wavelength is large compared to the solute boundary layer
thickness, the result of weakly nonlinear analysis is not an ODE governing the amplitude
of a single cellular mode, but rather a PDE governing the interface evolution as a whole. In
1983 Sivashinsky [29] performed the first such analysis of solidification, obtaining, in the
limit of small segregation coefficient, an equation that describes subcritical bifurcations.
In 1988, Brattkus and Davis [30] studied the near-absolute-stability limit, obtaining an
equation describing supercritical bifurcation. These cases were then generalized in 1990
into a single framework by Riley and Davis [31], who derived, in an intermediate limit,
equations able to capture the change from subcritical to supercritical bifurcation.
Beyond the Mullins-Sekerka instability, cellular interfaces are generically exhibited,
and as pulling speeds increase, these grow in amplitude, until a secondary instability
causes the formation of dendritic structures [32, 33]. In these regimes, numerical simu-
lation is a primary tool of investigation, and research has focused in several areas. In the
cellular regime, detailed examination of cell shape were carried out in [34, 35, 36], while
questions of wavelength selection have been investigated by looking for steady solutions
using an integral formulation [37, 38]. Finally, in the dendritic regime, phase-field meth-
ods have been developed to study the shape of solidifying structures in three dimensions
[39, 40].
The previous review covers only the behavior of the simplest possible solidification
model, including only the effects of solute diffusion and rejection and surface energy; a
23
more comprehensive review including many generalizations may be found in [41]. How-
ever, as we have seen, even this minimal model exhibits a wide variety of complex behavior.
The important general theme to extract from the above summary is the competition be-
tween a destabilizing solute gradient and a stabilizing surface energy. The progression
from planar, to cellular to dendritic solutions as pulling speed increases is driven by the
corresponding increase in solute gradients at the interfaces. However, the effective surface
energy is also dependent on pulling speed, and as it overtakes solute gradients in relative
strength it causes a reverse progression from dendritic, to cellular, and finally back to
planar states in the absolute stability limit. Since in what follows we consider the effect
of modifications to surface energy, it is important to keep this basic scenario firmly in
mind.
2.1.2. History 2: Anisotropy
The generalization to the above model of primary interest here is that we allow the
surface energy γ to be anisotropic. As discussed in the Introduction, we let γ depend
on the surface orientation θ, which measures the angle between the surface normal and a
reference orientation associated with the bulk crystal lattice. The anisotropy of γ may be
classified as either “small” or “large,” depending on whether or not the surface stiffness
γ + γθθ [42] is strictly positive or not. Large anisotropy, in which we are most interested,
has long been studied in equilibrium problems, where geometric considerations reveal
that cornering and faceting are generically present on energy-minimizing interfaces [1, 2,
3]. However, the study of large anisotropy in dynamic systems is problematic because,
in the Gibbs-Thompson-like equations describing interface motion, the surface stiffness
24
appears multiplying the curvature, which is ordinarily the highest derivative present in
the equation. Thus, for orientations with negative surface stiffness, the Gibbs-Thompson
equation is ill-posed mathematically, and some way to regularize the problem is needed.
The way forward is provided adding relevant physics in the form of a small corner energy,
which penalizes rapid changes in orientation [43, 42, 2]. Modeling this energy results
in the inclusion of higher-order derivative terms, which supply the needed mathematical
regularization [44, 45]. The strategy just described has been applied to enable the study
of large anisotropy in a variety of dynamic settings [7, 8, 9, 10, 11, 12]. The generic
behavior revealed by these studies is that initially planar surfaces rapidly decompose into
a faceted sawtooth, or “hill-and-valley” configuration with very small wavelength. Having
done so, such faceted surfaces then proceed to increase that wavelength via coarsening,
where small facets shrink and vanish, causing in increase in the average length of those
that remain.
2.1.3. History 3: Directional Solidification and Anisotropy
A natural step at this point is to inquire what happens when large anisotropy, with its ther-
modynamic instability leading to faceting, is added to the above model of solidification,
where solute gradients drive a morphological instability. However, despite the fascinating
behavior of faceted surfaces caused by large anisotropy in other dynamic contexts, most
work in directional solidification has considered only the case of small anisotropy2. In that
case, because all orientations are thermodynamically stable, no new instabilities appear,
2We note that [46, 47] considers large-anisotropy solidification, but in a low-energy direction whichis thermodynamically stable. Also, [48] considers faceted cellular solidification above the supercoolingspeed, but simply starts with a faceted array, and does not consider how such an array came to exist.
25
and so the effect on, for example, linear stability [49], amplitude equations [50], and
longwave reductions [51] is only modulatory. Such a treatment is valuable – part of the
appeal of the transparent organics used in Hele-Shaw cells is that they, like many met-
als, indeed possess only small anisotropies [52]. However, most non-metals have stronger
anisotropies, and when directionally solidified in thermodynamically unstable directions,
exhibit just the faceting behaviors predicted in other dynamic contexts (see, for example,
[53, 54]). Additionally, the solidification of these materials in such orientations is inter-
esting because of the possibility that the planar state, about which all subsequent analysis
is typically based, may cease to occur at all. Besides admitting qualitatively new system
behaviors, this possibility may require the development of novel analytical methods to
describe those behaviors. Developing such methods and exploring this behavior is the
aim of this work.
2.1.4. Kinetics
Finally, since directional solidification is a dynamic process, we also include in our model
the oft-neglected effect of attachment kinetics, which is also anisotropic. This property
effectively represents the mobility µ(θ) of a moving interface. Differences in mobility
correspond to differences in the amount of supercooling necessary to maintain a given
interface speed, with the end result that the inverse mobility µ−1(θ) also appears in
the Gibbs-Thompson equation. Anisotropy of attachment kinetics has no small/large
distinction as does that of surface energy; however, it can generate behaviors such as
traveling waves [49, 55], and can additionally cause faceting itself in some circumstances.
26
Because the relative importance of kinetic anisotropy and surface energy anisotropy is
unknown a priori, we include this effect in our analysis.
2.1.5. Summary
The aim here, then, is to study the behavior of a binary alloy with large surface-energy
anisotropy that is directionally solidified in a high-energy orientation (with negative sur-
face stiffness). We are most interested in understanding the relationship between the
morphological instability expected due to solute gradients with the thermodynamic in-
stability expected due to negative surface stiffness. Specific questions we aim to answer
include the following. Does faceting occur as in other dynamic contexts? How does the
presence of solute gradients affect this behavior? Does a faceted steady state replace
the usual planar state below supercooling? If so, what are the characteristics of this
state, and what is the resulting effect on the solidified microstructure? What happens to
such an interface when the pulling speed is increased past its supercooling critical value?
Does the presence of this supercooled liquid layer cause further destabilization? Does
coarsening occur? Under what circumstances? Finally, since solidification is a dynamic
process, does anisotropy of attachment kinetics play a significant role in any of the above
considerations?
To briefly summarize our results, linear stability analysis reveals that, for large enough
anisotropy, the high-energy planar state is indeed thermodynamically unstable for all con-
centrations and pulling speeds. The search for the anticipated faceted steady solutions
reveals that concentration does not affect their shape, and leads to singularly perturbed
equation describing the steady interface. Matched asymptotic analysis of this equation
27
indeed yields a family of nearly faceted hill-and-valley structures. Members of this family
may be compared by considering an appropriate free energy, the minimization of which
yields an optimal wavelength that scales with the small corner energy. Questions con-
cerning interface dynamics lead to the derivation of a steepest-descent surface evolution,
and ultimately, an effective facet dynamics. This dynamics reveals that faceted interfaces
are stable below the supercooling speed, where coarsening is prohibited as a mechanism
for dynamic wavelength adjustment. In contrast, above the supercooling speed, but be-
low the usual Mullins-Sekerka limit, a reversal of the effective thermal gradient allows
coarsening, which replaces cellular growth as the mechanism of instability . Interestingly,
at no stage of this analysis does anisotropy of attachment kinetics play a significant role;
however, the presence of kinetic effects, often neglected elsewhere, is pivotal in deriving
the facet dynamics which so simplifies the later analysis.
2.2. Background: Governing equations, basic state, and linear stability
Governing equations. We consider the directional solidification of a dilute binary
alloy with anisotropic surface energy. We use the Frozen Temperature Approximation
(FTA) [25], in which one neglects latent-heat generation, assumes equal thermal conduc-
tivities in the solid and liquid3, and assumes that thermal diffusion is much faster than
solute diffusion, with the result that the temperature is linear over characteristic solute-
diffusion lengths. We also assume a one-sided model, which neglects diffusion in the solid
phase. Let x be the co-ordinate lateral to the initially planar surface, z be normal to that
surface, and shift to a frame of reference moving with the mean interface position. In this
3Note that this assumption precludes the presence of instability below the constitutional supercoolingvelocity, a possibility footnoted above and discussed in [23].
28
two-dimensional co-ordinate system, the temperature T (z), concentration C(x, z, t), and
interface position h(x, t) are then described by the following equations:
T = T0 + GT z all z(2.1a)
Ct = V Cz + D∇2C z > h(x)(2.1b)
C → C∞ z → ∞(2.1c)
Cn
= −[C(1 − k)]Vn/D z = h(x)(2.1d)
T I = Tm + mC +Tm
Lv
[
(γ + γθθ)K − ν(Kss + K3)]
− µ−1Vn
z = h(x).(2.1e)
Here, GT is the (imposed) temperature gradient, V is the speed at which the sample is
pulled through the temperature gradient, and C∞ is the original concentration of solute
in the bulk sample. These three parameters are the extrinsic parameters subject to
experimental control. The intrinsic parameters are more numerous. In the bulk, there
is a free parameter T0 which anchors the z-coordinate, the melting point Tm of the pure
solvent, the volumetric latent heat of fusion Lv, the diffusion coefficient D of the solute
in the liquid, and the (negative) liquidus slope m describing freezing-point depression.
At the interface, there is the local curvature K, the second derivative Kss of curvature
with respect to arc length, a modified normal velocity Vn
(described below), the normal
derivative Cn
of concentration, the interface temperature T I , the segregation coefficient
k describing solute rejection (the ratio, at an interface, of the concentration on the solid
side to that on the liquid side), a small parameter ν which is the magnitude of the
corner energy (typically near atomic scales), the anisotropic surface energy γ(θ), and the
29
anisotropic mobility µ(θ) describing attachment kinetics. Note the use of tilde to denote
a few dimensional quantities; this is for notational convenience later.
Here θ denotes the angle formed by the surface normal with the z-axis, so that hx =
− tan θ. Also note that Vn
describes the normal velocity in the (old) stationary coordinate
system. We’ll use Vn
to describe normal velocity in the (new) moving coordinate; the two
are related via the expression
(2.2) Vn
=V + ht
√
1 + h2x
= V cos θ + Vn.
To describe anisotropic surface energy and surface mobility, we use the identical
smooth, sinusoidal functions
γ(θ) = γ0γ(θ), γ(θ) = 1 + α4 cos[4(θ − φ)](2.3)
µ(θ) = µ0µ(θ), µ(θ) = 1 + β4 cos[4(θ − φ)],(2.4)
where γ0 and µ0 are dimensional constants giving the average magnitudes of γ and µ, and
γ and µ are nondimensional functions describing the variation in θ of the same. These
forms have fourfold symmetry, and allow the direction of solidification to vary from the
high-energy orientation by an angle φ. Thus, φ = π/4 describes solidification in a low-
energy direction (see [46]), while φ = 0 describes a high-energy direction. Equation (2.3)
yields a surface stiffness of γ + γθθ = 1 − 15α4 cos[4(θ − φ)], so that the cases of “small”
and “large” anisotropy are characterized respectively by α4 < 1/15 and α4 > 1/15.
30
Remark. It will be noted that surface energy and mobility have identical form, and
thus, orientations with lowest (highest) surface energy are also those with the lowest (high-
est) mobility. This is due, like anisotropy itself, to atomic packing patterns. Orientations
with low surface energy are typically atomically smooth, requiring energetically expensive
atomic layer nucleations for growth, and therefore exhibiting low mobility. By contrast,
orientations with high surface energy are typically atomically rough, offering a constant
supply of neighbor-rich “holes” for growth, and therefore exhibiting high mobility.
A recent overview of the derivation of these equations, and the associated assumptions,
may be found in Chapter 3 of [41]. The related equations there have been modified,
however, by the appropriate addition to Eqn. (2.1e) of regularization terms described in
[44, 45], to allow the possibility of “large” anisotropy.
Scaling and Non-Dimensionalization. For steady, x-independent solutions of
Eqns. (2.1), one finds that
C = C∞
[
1 +1 − k
kexp(−V
Dz)
]
(2.5a)
T0 = Tm +mC∞
k− µ−1(0)V(2.5b)
h = 0.(2.5c)
We pause to note several things. First, the point z = 0 was placed to ensure h = 0, which
specifies the constant T0. Second, the positive value Tm − T0 is the total freezing point
depression due to attachment kinetics and the presence of solute, which causes a displace-
ment of the interface from the T = Tm isotherm – this displacement will be important
later when we consider non-planar interfaces. Third, the concentration profile reacts to
31
this displacement by simply shifting with the interface – its shape is unchanged. Resum-
ing our argument, the form of the steady solution (2.5) informs the following scalings and
dimensionless variables:
[x, z, h] =D
V[x, z, h](2.6a)
t =D
V 2t(2.6b)
C =C∞
k
[
1 − (1 − k)C]
(2.6c)
T = Tm +mC∞
k− µ−1(0)V +
D
VGT T(2.6d)
T = z.(2.6e)
Here temperature, fixed by the FTA, is scaled so as to replace it by the variable z. Also,
the dimensionless concentration is scaled such that C = 0 at z = 0, which requires that
it scale with the negative physical concentration. After performing these scalings, and
eliminating all references to the fixed temperature T through (2.6e), we arrive at the
following non-dimensional equations describing the evolution of solute concentration and
interface shape (where bars have been dropped):
Ct = Cz + ∇2C for z > h(x)(2.7a)
C → 1 as z → ∞(2.7b)
Cz = hxCx + [1 − (1 − k)C](1 + ht) on z = h(x)(2.7c)
C = M−1h − Γs(θ)K + ν(
Kss + K3)
+ µ−1A(hx, ht) on z = h(x)(2.7d)
32
where
(2.8) s(θ; α4, φ) = γ + γθθ
has been introduced as a shorthand for the surface stiffness, and
(2.9) A(hx, ht) =(1 + ht)√
1 + h2x
µ−1(hx) − µ−1(0)
represents the effect of attachment kinetics on concentration at the interface.
The dimensionless constants (M−1, Γ, ν, µ) are respectively proportional to temper-
ature gradient, surface energy, corner energy, and mobility, and are given by
M−1 =D
V
k
|m(1 − k)|C∞GT(2.10a)
Γ =V
D
Tm
Lv
k
|m(1 − k)|C∞γ0(2.10b)
ν =V 3
D3
Tm
Lv
k
|m(1 − k)|C∞ν(2.10c)
µ−1 =k
|m(1 − k)|C∞V µ−1
0 .(2.10d)
Of particular interest is the parameter M−1, the reciprocal of the morphological number,
which can be written in the simpler form GT /|m|GC, where GC = (1 − k)C∞V/kD
is the solute gradient at the interface. Thus if M−1 < 1, the gradient of freezing point
depression is larger than the gradient of temperature, which precisely defines the existence
of constitutional supercooling. In addition, we also note that the parameter ν is typically
very small, and it will be important in later asymptotic analysis.
33
Basic State and linear stability analysis. The non-dimensional version of the
basic state (2.5) has the form:
(2.11) h0 = 0, C0 = 1 − exp(−z).
To determine the linear stability of this state, we introduce and study the evolution of a
small disturbance [h, C] as follows,
(2.12) h = h0 + h, C = C0 + C,
where [|h|, |C|] ≪ [|h0|, |C0|]. Linearizing in [h, C] we find that the disturbance obeys the
equations
Ct = Cz + Czz + Cxx for z > 0(2.13a)
C → 0 as z → ∞(2.13b)
Cz − h = −(1 − k)[C + h] + ht on z = 0(2.13c)
C + h = M−1h − s0Γhxx + νhxxxx + µ−1(AX hx + AT ht) on z = 0,(2.13d)
where Taylor series have been used to allow all boundary terms to be evaluated at z =
0. Here AX and AT describe the derivatives of A(hx, ht) with respect to hx and ht,
respectively, evaluated at hx = ht = 0. Also, it was first observed by Coriell and Sekerka
[49] that only the surface stiffness of the planar interface s0 is present;
(2.14) s0(α4, φ) = s(0; α4, φ) = (1 − 15α4) cos(4φ).
34
This term multiplies the nondimensional surface energy Γ; thus, anisotropy here simply
creates an effective surface energy parameter s0Γ compared to the isotropic case (where
s ≡ 1). From here, the introduction of normal modes [C, h] = [c(z), h] exp(σt) exp(iax)
leads, upon considering Eqns. (2.13a,b), to the selection of a decaying exponential solution
for c(z). Application of the boundary conditions (2.13c,d) then results a homogeneous sys-
tem of algebraic equations, the solution of which requires the following implicit dispersion
relation for σ:
(2.15)
M−1 = 1− s0Γa2 − νa4 + µ−1(iAXa+AT σ)+ µ−10 σ− σ + k
12
[
(1 +√
1 + 4(a2 + σ)]
− (1 − k).
We now investigate this relation in four cases of primary interest – small and large
surface energy anisotropy for high- and low-energy pulling orientations. We note that, for
each of these cases, anisotropic attachment kinetic terms vanish. For both high-energy
(φ = 0) and low-energy (φ = π/4) pulling directions, the term AX = 0, as can be seen by
considering the symmetries in equation (2.4). At intermediate pulling orientations, this
term causes traveling solutions [49], but that has already been investigated and will not
be considered further in this paper.
To study neutral stability boundaries, we set σ = 0 in expression (2.15), and note that,
since σ = AX = 0, kinetics plays no role in neutral stability for our choice of φ. Before
proceeding, however, we make an important observation. The resulting neutral stability
curves would live in (M−1, Γ, ν)-space, and measure the relative effects of morphological
number (i.e. constitutional supercooling) and surface energy. These are, indeed, the
theoretical effects at work, but they are awkward parameters to work with experimentally
35
– the directly controllable parameters of interest are pulling velocity and sample solute
concentration. Additionally, we see that ν is actually dependent on these parameters, and
so makes a poor description of corner energy. For these reasons, “natural parameters” also
introduced which scale independently on pulling speed, concentration, and corner energy
[56]. The relationships between these sets of parameters are given by the transformations
(2.16) V =
√
Γ
M−1C =
√
1
M−1ΓN =
M−1ν
Γ2.
With these parameter issues in mind, we now present neutral stability curves asso-
ciated with each of the four cases described above in both of the discussed parameter
spaces. Figure 2.1a shows curves in the experimental (log(C), log(V))-space with fixed N ,
while Figure 2.1b gives results in the theoretical (Γ, M−1)-space with fixed N (not fixed
ν). In each case, we fix k = 0.5, N = 10−4.
Before considering the effect of anisotropy, it is instructive to consider the stability
boundaries present in the isotropic problem (α4 = 0, s0 = 1), shown in blue in both figures.
In the theoretical space, the curve actually terminates on the axes; these terminal points
correspond to the linear asymptotes observed in the experimental space. Theoretically,
the interface is stable if M−1 > 1 or Γ > 1/k; the former is called the constitutional
supercooling boundary, while the latter is called the absolute stability boundary. These
correspond experimentally to the lower asymptote (shown) and the upper asymptote
(not shown). The succinct summary of behavior is that some amount of constitutional
supercooling M−1 < 1 is required for instability, but a large enough surface energy Γ can
stabilize no matter the supercooling; thus the name absolute stability [23].
36
−4 −2 0 2 4 6 8−6
−4
−2
0
2
4
6
log(C)
log(
V)
s0 < 0 s0 > 0
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
M−
1
Γ
s0 < 0
s0 > 0
Figure 2.1. Summary of linear stability results (color). Neutral stabilitycurves are shown in both (C, V)-space (a), and (Γ, M−1)-space (b). Blueshows isotropic stability boundary. Red shows solidification in high-energydirection (φ = 0). Green shows solidification in low-energy direction (φ =
π/4). Dashed lines represent α4 = 1/15 −√N (small anisotropy), while
dotted lines represent α4 = 1/15 +√N (large anisotropy). The black line
shows the constitutional supercooling boundary.
The effect of φ. Turning to the effect of anisotropy, we begin by considering the
effect of pulling direction. Recall that the only effect of anisotropy is the presence of the
constant s0 multiplying the non-dimensional surface energy Γ. For φ = π/4, s0 > 1, and
thus anisotropy enhances stabilizing effect of surface energy. However, for φ = π/4, s0 > 1,
which diminishes the surface energy stabilization. Thus, the φ = π/4 curves lie inside the
isotropic curves, while the φ = 0 curves lie outside of them. 4
The effect of α4. We next consider the effect of the anisotropy strength, expressed
in the constant α4. We see that the effect of α4 turns out to be φ-dependent. For both φ,
two values of α4 are considered, one just below the “strong anisotropy” value of 1/15, and
one just above. For the φ = π/4 case, crossing this boundary has no significant effect (s0
4The effect in (Γ, M−1)-space is simply to stretch or compress the neutral stability curve along the Γ-axisas compared to the isotropic case. The corresponding effect in (C, V)-space is a shift of the top branch.
37
simply crosses 2 – a quantitative change), while for the φ = 0 case, the effect is dramatic
(s0 crosses 0, which is a qualitative change).
The effect of s0. The above behavior is put into a single, simple framework if we
only consider s0. Whenever s0 > 0, surface energy is stabilizing. Thus, constitutional
supercooling is required for instability, causing the three positive-s0 curves to remain an-
chored to the constitutional supercooling boundary, while the absolute stability boundary
moves (s0 is, after all, simply a surface energy modifier). However, when s0 < 0, surface
energy becomes a destabilizing agent. Supercooling is no longer necessary for instabil-
ity, as the negative surface stiffness encourages bending; thus, the neutral stability curve
de-anchors from the supercooling boundary. Indeed, only the presence of corner energy
allows a stable region at all.
The de-anchored curves in Figure 2.1 have a very small negative value of s0, but as
s0 continues to decrease, the curves begin a singular migration upward (leftward) in the
theoretical (experimental) parameter space. Finally, for s0 < −2√N , the stable region
disappears entirely, signaling universal instability. Mathematically, this can be most easily
seen by converting to “natural” parameters, and finding the critical concentration Cc at
which instability occurs,
(2.17) Cc = mina
1V [1 + s0(aV)2 + N (aV)4]
1 − k12 [1+
√1+4a2]−(1−k)
.
It turns out that, for s0 < −2√N , Cc attains a negative value, implying instability for
all positive (physical) C. The form of (2.17) also allows us to describe the asymptotic
behavior of the neutral stability curve. Since the top and bottom branches represent
large- and small-V limits, respectively, and since C is largely a function of aV, large V
38
implies small a and vice-versa. If we let s0 = −2√N + ǫ, this allows an asymptotic
approximation of the singular migration of Cc, showing that
log(Ctop) ∼ − log(V) + log(ǫ) − log(√N )(2.18a)
log(Cbottom) ∼ log(V) + log(ǫ) − log(4N /k).(2.18b)
As ǫ → 0, the curve migrates singularly leftward, vanishing at ǫ = 0.
Finally, it is instructive to examine the change that occurs in the dimensional critical
wavelength of instability ac as s0 crosses zero. This wavelength, which is O(D) for much
of the visible part of the curves in Figure 2.1 for s0 > 0, becomes instead O(√
|s0|/N )
for s0 < 0. Thus, accompanying the transition to negative surface stiffness is a transition
of the instability wavelength from diffusional scales to corner scales. This occurs over the
same values as the migration of the NSC, and indicates that surface energy becomes the
primary cause of instability.
Summary. For large enough anisotropy s0 < −2√N , the typical conditional mor-
phological instability on diffusional scales is replaced by universal thermodynamic insta-
bility on capillary scales. This result is generally unsurprising; the thermodynamic insta-
bility of planar surfaces with negative surface stiffness was is a result that was obtained by
Herring [57]. However, that result is modified here by the presence of the regularization
term. In particular, besides stabilizing the planar solution for not-too-negative surface
stiffness, the corner energy coefficient provides a scale for the instability that occurs; no
such scale exists for Herring’s instability [58]. The obvious remaining question, then, is
what happens to the interface after instability?
39
2.3. Statics. Non-planar interface due to strong anisotropy, φ = 0
We now proceed to look for non-planar solutions present in the range of universal linear
instability, where s0 < −2√N . Specifically, we let α4 > 1/15 + 2
√N while pulling in
the thermodynamically unstable direction φ = 0. We begin by listing three expectations
based on heuristic reasoning; these will then be detailed.
Expectation 1: Shape. We have already stated Herring’s result that, in equi-
librium, planar surfaces with large anisotropy of surface energy are thermodynamically
unstable to lower-energy faceted surfaces. To explain this briefly, we consider the pro-
jected surface energy for a unit of length along an axis provided by the planar surface.
This energy is described by
(2.19) E =
∫
γ(θ)ds =
∫
γ(q)√
1 + q2dx,
where here and in what follows q = hx. In the presence of large anisotropy, this functional
is minimized by any5 faceted hill-and-valley structure with slopes of q∗(α4), where q∗
minimizes the integrand of (2.19). For the surface energy given in Eqn. (2.3), performing
this minimization yields an implicit relation for the value of q∗:
(2.20) α4 =(1 + q2)2
15 − 10q2 − q4.
The value q∗(α4) will be called the optimal slope, and is plotted in Figure 2.2. The same
result can also be obtained using the convexification argument found in [59], where q∗ is
found by locating double tangent points on the polar plot of inverse surface energy. Now,
5Note that all hill-and-valley structures have the same projected surface energy. Thus, while surfaceenergy anisotropy provides an energetically favored slope, it is not sufficient, by itself, to provide ascale[58].
40
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1
−0.5
0
0.5
1
α4
q*
Figure 2.2. Surface-energy-minimizing slope for various values of α4.
directional solidification includes many more effects than just surface energy anisotropy,
including, importantly, solute rejection and diffusion and anisotropy of attachment kinet-
ics. However, we have seen in the previous section that the thermodynamic instability
due to surface energy is preserved, and indeed dominant, in the linear theory. Thus, we
expect to see shapes similar to those found in the planar equilibrium problem – corners
separating facets with energetically favorable slopes near q∗.
Expectation 2: Size. Below the constitutional supercooling speed, the unstable
planar state exists within a positive effective thermal gradient (c.f. [60]). We expect the
interface to reduce surface energy by faceting, and the resulting change to the concen-
tration field will result in a nonlinear interaction between the two. However, because of
the thermal gradient, large interface deformations would result in large thermal energetic
penalties. Thus, we expect an interface with small amplitude. If the facet slopes are not
small, then this corresponds to a small wavelength λ as well, an expectation which will
be useful throughout the analysis that follows.
Expectation 3: Displacement. Equation (2.5) above shows how the planar state
is displaced from the T = Tm isotherm by the presence of attachment kinetics. Because
41
this effect depends on orientation and the normal velocity of the surface, any faceted
interface with slope q∗ 6= 0 will be displaced by a mean amount that is different from the
displacement of the planar interface. Turning to the solute concentration, we recall that
any such kinetic displacement does not affect the shape of the concentration field, which
is merely displaced along with the interface.
Formulation. In light of our expectations, then, we look for a small, faceted non-
planar interface h, having slopes near q∗ and with mean displacement Z from the z = 0
isotherm; this is accompanied by a corresponding small correction to the concentration
profile C. Just as in the above linear stability analysis, we write
(2.21) h = Z + h, C = C0(z − Z) + C,
where [|h|, |C|] ≪ [|h0|, |C0|]. In contrast to the linear theory, however, we do not assume
that the corrections are infinitesimal. Therefore, we expand C(h) and Cz(h) about z = Z
in the boundary conditions there, but we do not yet discard any nonlinear derivative
42
terms. The resulting semi-linearized equations governing h and C are then
Ct = Cz + Czz + Cxx for z > Z
(2.22a)
C → 0 as z → ∞
(2.22b)
Cz = hxCx + kh − (1 − k)C + ht on z = Z
(2.22c)
C = M−1Z + (M−1 − 1)h − Γs(θ)K + ν(
Kss + K3)
+ µ−1A(hx, ht) on z = Z.
(2.22d)
Looking for steady solutions (∂/∂t → 0), we first consider the interface h to be given,
and solve the system of equations (2.22a-c) governing the concentration C. This will
yield a steady solution for C which depends on the (fixed, but unknown) interface h. We
then evaluate the solution for C at Z, and insert this value into the left-hand side of the
Gibbs-Thompson condition (2.22d) to obtain a nonlinear equation describing h (c.f. [46]).
Now, the many nonlinear terms retained in equations (2.22) will prevent analytical
solutions at several points in the following analysis. This problem could be remedied
by considering a small-slope limit of these equations (c.f. [61]). Such a situation can
be created by choosing value of α4 that is only slightly above the critical value of 1/15,
introducing into the problem a small parameter ε = 15α4 − 1. The optimal slope q∗ can
be shown in this case to equal q∗ =√
3ε/8, suggesting a scaling x → √εx; if performed,
such a scaling would eliminate all the difficult nonlinearities. However, such a careful
43
selection of α4 is not very general, and so rather than applying this limit a priori, we
prefer to call upon it when analytical difficulty requires, and retain nonlinearities where
they can be handled. A postiore estimates well validate this approach.
2.3.1. Solution for C
Equations (2.22a-c) have a nonlinearity – the term hxCx in the boundary condition (2.22c)
describing solute balance. Recalling the small-slope approximation idea just discussed, let
us neglect it for a moment, and return to it later. Having done so, we perform a Fourier
transform in the x-coordinate, after which exponentials in z satisfying (2.22a) are easily
found. Application of the (transformed) boundary conditions (2.22b,c) (neglecting the
hxCx term) at z = 0 then yields the solution:
(2.23) C(x, z) =
∫ ∞
−∞−k
h(κ)
p(κ) − (1 − k)exp[−p(κ)(z − Z)] exp(iκx) dκ
where
h(κ) =1
2π
∫ ∞
−∞h(x) exp(−iκx)dx(2.24)
p(κ) =1 +
√1 + 4κ2
2;(2.25)
here bars indicate Fourier transforms (in the variable κ), and hats still indicate the small-
amplitude variables described above.
We now need to obtain a form for the value of (2.23) at z = Z, for insertion into Eqn.
(2.22d). Assuming periodic profiles h(x) with wavenumber κ0, then h(κ) is just a series of
δ-spikes occurring at wavenumbers nκ0. We can thus expand the expression for C(x, Z)
44
as
(2.26) C(x, 0) = −k
∞∑
n=1
ρ(nκ0)h(nκ0) cos(nκ0x),
where
(2.27) ρ(κ) =1
p(κ) − (1 − k)
(
≈ 1
κfor κ ≫ 1
)
.
We now see that this is roughly just h(x), but scaled by a factor ρ(κ0), and “rounded,”
with higher-frequency modes damped. We now use the first application of our expectation
of small wavelength. Given the asymptotic form ρ(κ) ∼ 1/κ, and the fact that the
wavelength satisfies λ = 2π/κ0, we see that C ∼ λh ∼ O(λ2). It turns out, then, that the
correction to the concentration caused by a small non-planar interface is very small, and
may usually be neglected in equations describing the interface h itself.
We now re-consider the neglected hxCx term. While we cannot solve the equations
on the concentration field with this term included, we can use a scaling argument to
show that the correction to the concentration field is still small. With λ again the small
wavelength, let the surface slope hx now be O(1). Also, let the value of C at the interface
be of unknown magnitude C. Now, we expect C at the interface to vary on the same
scale as h itself, suggesting that Cx ∼ O(C/λ). Finally, following the argument above,
equations (2.22a,b) can still be solved using Fourier transforms to show that the scale
of Cz at the interface is also O(C/λ). Replacing each term in equation (2.22c) with its
appropriate scale, we obtain the following dimensional form:
(2.28)Cλ∼ C
λ+ λ + C.
45
This “equation” is only for the purpose of identifying dominant terms, and is not an
equality (specifically, we cannot cancel the two C/λ terms). We observe first that, when
O(1) slopes are allowed, the hxCx term is of the same C/λ size as the Cz term, which
means (a) it cannot be neglected, and (b) probably has a qualitative effect on the shape
of the concentration field. However, looking for a balance of terms in equation (2.28), we
see that since C ≪ C/λ, we must balance C/λ and λ. This shows that again C ∼ λ2. We
thus conclude that the magnitude of C evaluated at h remains O(λh), and thus, is still
small enough to neglect in equations involving h itself.
2.3.2. Solution for h
Dropping the bars on h, a nonlinear equation governing the interface h is obtained by
inserting the just-derived value of C at the interface into the Gibbs-Thompson Equation
(2.22d). Since Equation (2.22d) contains terms h, and since we just saw that C ∼ O(λ2) ∼
O(λh), we conclude that this term is small enough to neglect. This leads to the equation:
(2.29) O(λ2) = M−1Z + Geffh − Γs(θ)K + δ2(Kss + K3) + µ−1A0(q)
where Geff = (M−1 − 1) is an effective thermal gradient, δ =√
ν is a small parameter
associated with corner energy, and A0(q) = A(hx, 0). The interface h is thus described
by a singularly perturbed nonlinear equation, which, like equations describing faceting in
other contexts, is similar to the Cahn-Hilliard equation describing phase separation [6].
The form of such equations suggests a matched asymptotic analysis; we follow [62] by
looking first for an inner solution describing corners, which will then provide boundary
conditions for an outer equation describing facets. This approach will yield a family of
46
composite solutions with varying wavelength, the comparison of which is the topic of
Section 2.4.
2.3.2.1. Inner Scale δ. Since the primary effect of strong anisotropy is expected to be
the presence of corners in solutions [1], we begin by looking at the inner scale where we
expect to find them. Using the small corner-energy parameter δ, an inner equation for
Eqn. (2.29) is found with the scaling [x, h] → (δ/√
Γ)[x, h]. To leading order, this gives
the equation
(2.30) Kss + K3 = s(θ)K.
Because this equation is strongly nonlinear, we limit ourselves to consideration of the
small-slope form. It will be seen later that only the existence of an inner solution is
necessary for further analysis, and the precise form of that solution is of little interest.
The small-slope form of (2.30) is [61]
(2.31) hxxxx = (8h2x − 1)hxx.
This equation, while still nonlinear, may be directly integrated as follows (letting q = hx):
qxxx = (8q2 − 1)qx(2.32a)
qxx =8
3q3 − q + A(2.32b)
1
2q2x =
2
3q4 − 1
2q2 + Aq + B(2.32c)
qx =√
2W(q).(2.32d)
47
In the final step, W(q) is a double-welled potential for the slope q, and is simply the small-
slope surface stiffness integrated twice. Solving this equation on (−∞,∞) and requiring
a bounded solution, we find that we must choose A and B according to the bitangent
construction, which lets both wells of W rest exactly on the q = 0 axis. This procedure
gives W(q) a final form of (2/3)(q2 − 3/8)2, and admits an exact solution for q:
(2.33) q = ±√
3
8tanh(
x√2).
This clearly represents a corner in h, and matches precisely the two energetically favored
slopes in the small-slope limit as described above. The inner, corner solution, in turn,
provides boundary conditions for the outer, facet equation – any outer solution must
connect to two corners, and must therefore twice achieve an optimal slope ±q∗.
Remark. While the corner solution (2.33) is valid only in the small-slope regime,
in general we require only the existence of some similar solution to provide boundary
conditions on the outer equations. Since the bitangent construction described here can be
viewed as the small-slope limit of the Wulff construction [61], this requirement is ensured.
2.3.2.2. Outer Scale. An outer equation describing facets is provided, as usual, by
simply neglecting the corner term because it is multiplied by δ2:
(2.34) Geffh = Γs(θ)K − [M−1Z + µ−1A0(q)].
Suitable solutions of Eqn. (2.34) must match our inner solutions (corners); i.e., they must
twice attain one of the preferred slopes q∗(α4). In addition, they must have a mean height
of zero, since mean displacement is described by the so-far-undetermined Z. To see what
solutions may exist satisfying these requirements, we numerically examine all solutions of
48
(2.34) by restating it as a dynamical system in (h, q)
h = q(2.35a)
q =
[
Geffh + M−1Z + µ−1A0(q)
Γs(θ)
]
(1 + q2)3/2,(2.35b)
where the dot represents differentiation in x. Now, to determine Z, we will first assume
that Z = 0. This will produce a solution h with some nonzero mean height H . Then, the
correct zero-mean solution is obtained by letting Z = (Geff/M−1)H .
A representative resulting phase plane is given in Figure 2.3a for Geff = Γ = µ−1 =
1, α4 = β4 = 0.5, q∗(α4) = 0.8908; only the upper half is shown since the system is
invariant under the transformation q → −q, x → −x. There, we see several interesting
families of solutions; however, only those enclosed in the red triangular regions meet the
boundary conditions just described. These solutions are displaced from the q-axis by the
A0(q) term, because the displacement due to attachment kinetics of the faceted state is
different from that of the planar state. Notably, if µ−1 = 0, then the system would be
invariant under the transformation q → −q, h → −h, and thus, symmetric about h = 0;
the corrective term Z would not be needed. Now, let these solutions be parametrized
by L, the total solution length in x; then for each parameter set (Geff, Γ, µ−1, α4, β4)
there exists an implicit relation qmin(L) describing the minimum slope attained by each
family member, and, following the above argument, a relation Z(L) describing the mean
displacement from zero. These functions are shownin Figs. (2.3b,c) for the parameter set
chosen above.
49
−1.5 −1 −0.5 0 0.5 1 1.5 2 2.50
0.2
0.4
0.6
0.8
1
h
q
0 1 2 3 4 5
0.4
0.5
0.6
0.7
0.8
0.9
1
L
q min
0 1 2 3 4 50.8
0.9
1
1.1
1.2
L
<h>
Figure 2.3. (a) The phase plane for outer equation solutions with positiveslopes (color). Admissible solutions live within the red triangular regions.Black solid (dashed) lines represent degenerate stable (unstable) regions.(b) The function qmin(L). (c) The function Z(L).
2.3.2.3. Composite Solution. Piecing together appropriate inner and outer solutions
(corners and facets), we plot in Figure 2.4 some example composite solutions [h, q](x) for
different L (note that L is the length of a single facet, and so the the total wavelength is
2L). These solutions are not precisely piecewise-planar (faceted), as would be expected
in the equilibrium problem. This is caused primarily by the presence of the thermal
50
gradient; Figure 2.4 shows how this gradient “pressures” the facets to bend in such a way
as to reduce the area of the solid which projects into the melt, and melt into the solid.
However, this effect is only strongly present at large wavelengths.
0 2 4 6 8 10−1
0
1
2
h
0 2 4 6 8 10
−1
0
1
q
0 0.2 0.4 0.6 0.8 1
1
1.2
1.4
h
0 0.2 0.4 0.6 0.8 1
−1
0
1
q
0 0.02 0.04 0.06 0.08 0.1
1.1
1.12
1.14
h
x0 0.02 0.04 0.06 0.08 0.1
−1
0
1
q
x
Figure 2.4. Sample height and slope profiles for various values of L, for theparameters given in the text. Note that the interface h(x) is very nearlyplanar even for O(1) wavelengths. As the wavelength decreases, it becomeseven more so – see figure (2.3b).
Since we have been anticipating a small-wavelength solution, we now inspect the small-
L limit, which reveals two important facts. First, it is seen in Figure 2.3b, and can be
shown analytically, that for small L, qmin(L) = q∗+O(L2). Since the slope of any solution
h(x; L) lies on [qmin(L), q∗], we have the important consequence that small wavelength
solutions are nearly linear. Second, if solutions are nearly linear, then A0(q) ≈ A0(q∗)
everywhere (not of course, in the corners, but its value is irrelevant there). In this limit,
51
then, the interface displacement is decoupled from the interface shape, and tends to the
value
(2.36) Z = −µ−1A0(q∗)M,
which is exactly the value at zero of the curve in Figure 2.3c. Meanwhile, the interface
shape tends to that which would occur if kinetics were neglected.
Summary. We have found a family of small, nearly-faceted steady interfaces which
replace the traditional planar state due to strong surface energy anisotropy. Accompany-
ing these interfaces is a correction to the concentration profile C which is smaller still, to
the point that the final effect on solid microstructure is negligible. Due to anisotropic at-
tachment kinetics, these solutions are displaced from the isotherm occupied by the planar
state. Questions of wavelength selection and stability will now be addressed in Sections
2.4 and 2.5.
2.4. Energetics. Optimal Wavelength, Comparison with Planar State
In Section 2.3 we used matched asymptotic methods to find a family of nearly-faceted
solutions, parametrized by their wavelength λ, that satisfy equation (2.29) describing the
steady interface. However, our matched asymptotic approach told us nothing about which
wavelengths λ, if any, are preferred, nor the mechanisms of such a preference. To inquire
about wavelength selection, we show that, in spite of the fact that the system is not in
equilibrium and is placed in a thermal gradient, the surface can still be characterized by
a free energy E(λ), which we then minimize to obtain an optimal λ. In the context of
faceted surfaces, this approach has been used previously by Voorhees et. al. [60], who
52
considered Eqn. (2.29) without corner energies. In addition, this approach has been used
on equations essentially identical to ours in the context of elastic bars (see [63, 64, 65, 66].
Here, we repeat the important points.
We begin by re-stating, for convenience, the equation describing h:
(2.37) O(λ2) = Geff(h + Z) − Γs(θ)K + δ2(Kss + K3) + µ−1A0(hx).
Now, we saw in Section 2.3.2.3 that, in the small-wavelength limit, solutions are nearly
perfectly faceted, allowing us to replace the function A0(q) with the constant A∗0 = A0(q
∗),
thus specifying the displacement Z. We then find that
(2.38) O(λ2) = Geffh − Γs(θ)K + δ2(Kss + K3) =δE
δh,
where
(2.39) E =
∫
R
1
2Geffh
2 dx +
∫
R
[
Γγ(θ) +1
2δ2K2
]
ds.
Thus, in the small-wavelength limit, the right-hand side of Eqn. (2.37) is the variational
derivative of a free energy functional given in Eqn. (2.39). The terms in this energy
represent, respectively, an effective thermal energy penalty ET due to supercooling, a
surface energy Es, and a corner energy Ec. Any solution to h(x), therefore, minimizes
this energy functional at least locally; the preferred solution minimizes it absolutely.
Since q is everywhere near q∗ (faceting), the anisotropic surface energy Es is nearly
constant over a solution period, and large-scale solution characteristics are determined by
a competition between ET and Ec. Then, since corner energy δ2 is expected to be small,
53
while the effective thermal gradient Geff may be O(1) below the supercooling boundary, we
expect an expensive supercooling penalty (projection into the melt) to be reduced by fre-
quent, inexpensive cornering, resulting in a small wavelength. This provides an energetic
rationale predicting for small wavelength which complements the geometric reasoning we
used earlier.
To obtain a precise value for the optimal wavelength, we calculate the average energy
〈E〉 (L) in terms of wavelength, and then minimize it. Since we expect small wavelength
solutions, and since even not-too-large solutions are almost perfectly linear, we simplify
our work by assuming perfectly faceted solutions with |hx| = q∗. The average surface
energy Es is then constant everywhere except in the corners, and we need only to consider
a balance between the supercooling energy on the facets, and the corner energies at the
corners. For a single solution period of wavelength λ, there are two facets and two corners,
giving total energies of
ET = 2
∫ λ/4
−λ/4
1
2Geffh
2 dx =Gq∗2
96λ3,(2.40a)
Ec ≈ 2
∫ ∞
−∞
1
2δ2q2
x dx ≈ δ
∫ q∗
−q∗
√
2W(q) dq = Iδ,(2.40b)
where I replaces the integral in (2.40b). The former quantity is exact and trivially
calculated, while the latter is actually a small-slope approximation of the corner energy
that substitutes h2xx for K2, dx for ds, and uses result (2.32d) to set up a change of
variables from x to q. Additionally, surface energy is higher in the corners, and so an exact
calculation of the total surplus energy at the corners would also integrate γ(q) − γ(q∗)
over the corner. However, both of these neglected contributions can be seen to be O(δ),
54
and thus merely change the form of I, and not its scale. Dividing by λ to get average
energies and differentiating in λ, we find that the minimum average energy satisfies
(2.41) 0 = Geffq∗2λ3 − 96Iδ.
The energy-minimizing wavelength is then
(2.42) λ = 2
(
12Iδ
Geffq∗2
)1/3
= O(δ1/3) = O(ν1/6).
Remark 1 (Optimal Wavelength near Supercooling). It will be noticed that this
wavelength blows up as the supercooling boundary Geff = 0 is approached. However, as
Geff approaches O(λ), we see that Geffh is O(λ2). Recalling from Section 2.3.1 that the
value of C at the interface is also O(λh) = O(λ2), this means that the concentration
correction due to the faceted interface is no longer small enough to neglect in the Gibbs-
Thompson equation, and thus the analysis leading to Eqn. (2.42) is no longer valid. Had
we specifically included the approximation C(x, 0) ≈ −χλh(x) in the Gibbs-Thompson
equation, we would have found that Geff → Geff + χλ, adding a term χλ4 to equation
(2.41). As Geff → 0, this new term would balance the corner energy term, giving
(2.43) λ = 2
(
2Iδ
χq∗2
)1/4
= O(δ1/4) = O(ν1/8).
Thus, while the concentration correction can be neglected for most purposes, it does
serve to limit the size of λ, protecting against blowup as the supercooling boundary is
approached.
55
Remark 2 (Comparison with the Energy of the Planar State). We also note that, in
reducing surface energy by faceting, a surface increases supercooling and corner energies.
Thus, faceting in this situation will be expected only if the total energy of the optimally-
faceted state is less than the surface energy of the planar interface. The condition for this
is, after some manipulation,
(2.44)Geffq
∗2
96λ3 − (E0 − Es
√
1 + q∗2)λ < −Iδ.
For the optimal solution wavelength just found, this translates to
(2.45) γ(0) − γ(q)√
1 + q2 >
(
Geffνq∗2I2
144
)1/3
.
Since ν ≪ 1, this condition is met for realistic solidification environments. However, the
relationship (2.44) will render some wavelengths in the family of possible wavelengths
unacceptable.
Summary. Our assumption of small wavelength, argued heuristically on geometric
grounds, has been shown to be verified on the basis of energetic arguments. Sufficiently
below the supercooling boundary, the optimal wavelength scales as δ1/3, or equivalently
ν1/6 [64, 65], where we recall that ν is the dimensional corner energy, having a scale on
atomic lengths. However, we note that it remains to be seen if this optimal wavelength
is, in fact, achieved. To speak to this question, we investigate the dynamics of a moving
interface in Section 2.5.
56
2.5. Dynamics. Stability, Wavelength Trapping, and Coarsening
Several of the remaining questions we wish to address concern dynamics; the evolution
of perturbations to our solution, the ability of the interface to change its wavelength, and
so on. To address these questions, we derive an evolution equation for h valid in the small-
wavelength limit. This equation may be obtained by using Laplace transforms on the fully
time-dependent problem (2.7), a process performed in Appendix A.1. However, for the
sake of clarity here, we merely state that we can justifiably make a quasi-static assumption
on the concentration field to neglect the Ct term in Eqn. (2.7a) (thus, the concentration
quickly equilibrates to a slowly-moving interface [49, 67, 68]). Then, following the same
analysis as is found in Section 2.3, we find a time-dependent value of C at the interface.
However, upon insertion into the Gibbs-Thompson relation, it may still be neglected due
to its small size. Thus, the appropriate evolution equation may be directly read from
the right-hand side of the Gibbs-Thompson equation; in the small-wavelength limit, that
equation is
(2.46)µ−1(q)
√
1 + q2ht +
[
A0(q) + µM−1Z]
= −µδE
δh,
with E defined in Eqn. (2.39).
Equation (2.46) turns out to be similar to a family of equations used to describe faceted
crystal growth in other contexts [69, 7, 8, 10, 9], and, like them, is a modification of the
Cahn-Hilliard equation describing phase separation. Of special importance to us in these
works is the observation that initially smooth surfaces evolving under Cahn-Hilliard-type
equations first rapidly decompose into faceted sawtooth surfaces, each facet of which then
slowly evolves as a unit. Since numerical simulations of (2.46) reveal a similar behavior,
57
we conclude that solution dynamics may be summarized by finding the facet-velocity law
which governs facet evolution [62].
To obtain this law, we first exploit the fact that small-wavelength solutions form near-
perfect facets to replace instances of the variable q with the constant q∗ as discussed above
in Section 2.3.2.3. 6 Then, the Z and A0(q∗) terms cancel, and we can write
ht = −[
µµ(q∗)√
1 + q∗2] δE
δh,(2.47)
which describes a steepest descent of the value of the energy E. We are now in a position
to follow Watson in [13], who considered faceted surfaces in three dimensions governed by
equations of precisely this form. Given a variational form like (2.47), and assuming that
individual facets each move as a unit on a slow timescale, he showed that facets evolve
according to the relation
(2.48)
[
dh
dt
]
i
= − 1
Ai
∂E
∂hi
,
where [dh/dt]i is the vertical velocity of the entire facet, Ai is its projected area, and
dE/dhi is the rate of change of the energy E resulting from vertical facet translation.
This result is independent of the form of the energy E, so we may apply it to our problem.
A quick calculation reveals that, for the energy (2.39), the value of dE/dhi is LiGeffhm,
6The variable q terms may not, of course, be so replaced in the corners. However, by writing out thevariational derivative in Eqn. (2.46) and making the space-time scaling [x, t] → [x, t]/δ, it can be seenthat corner evolution satisfies an equation with a much faster time scale than that which describes facetmotion. Thus, we make another quasi-static assumption that the corners remain in equilibrium with thefacets that form them, and proceed to consider only facet behavior.
58
with Li the width of the facet, and
(2.49) hm = 〈h〉
its mean height. Then, dividing by Li (instead of Ai), we arrive at the facet-velocity law
specific to our energy functional:
(2.50)
[
dh
dt
]
i
= −Geff
[
µµ(q∗)√
1 + q∗2]
hm
Remark. An argument might validly be raised at this point that the near-perfect
facet assumption is only valid for static surfaces, not evolving ones. However, if we make
the substitutions h → h + hm and ht → ht + V in equation (2.47), we see that the
overall facet velocity is governed by the discrete equation V = −AGeffhm which, since all
hm are small, has a characteristic slow time scale t/hm. The facet shape, on the other
hand, is governed again by equation (2.47), or at least the outer equation (2.34), which
evolves on the “regular” time scale t (also c.f. the previous footnote). Thus, a quasi-static
assumption on the shape of moving facets is justifiable, and shows that moving facets have
shapes which are identical to the static facet shapes found above.
2.5.1. Stability
From the above discussion, we infer that small-amplitude disturbances to a periodic
faceted profile will quickly settle into a disturbed faceted profile. Thus, stability may
be determined simply by examining the evolution of such profiles. Equation (2.50) shows
that, as long as the effective thermal gradient Geff > 0, facets are driven to the z = 0
isotherm. Since a sawtooth surface with all facets centered on z = 0 is easily seen to
59
be necessarily periodic, we see that initially non-periodic surfaces are driven to become
periodic ones, which are stable. If Geff < 0, the above argument no longer holds, facets are
driven away from the z = 0, and periodic surfaces are unstable. The instability criteria
for such surfaces is thus precisely Geff < 0, or M−1 > 1.
Remark 1. Interestingly, the instability requirement M−1 > 1 is just the super-
cooling criteria first hypothesized by Rutter and Chalmers and Tiller et. al. for the
isotropic problem, but which is only valid there in the absence of surface energy. The
difference between this behavior and the classical Mullins-Sekerka instability is illustrated
in Figure 2.1a where, for any bulk concentration C, a gap exists between the supercooling
boundary and the Mullins-Sekerka boundary. Inside this gap, isotropic surfaces are stable,
while faceted surfaces are not. Thus, for surfaces with strong surface energy anisotropy,
surface energy serves only to cause faceting – its usual role as a stabilizing agent is absent.
Remark 2. It will be noted that Equation (2.50), and thus the stability result just
given, are only valid for small-wavelength surfaces. Thus, one might inquire about the
stability of a periodic, small-wavelength faceted surface to disturbances that are large com-
pared to the surface wavelength. Such an analysis is carried out in Appendix A.2, where
it is shown that for large-wavelength disturbances, an effective homogenized surface en-
ergy does stabilize against supercooling. These disturbances thus first become unstable at
pulling speeds higher than the critical speed causing supercooling. Since small-wavelength
disturbances are already unstable at that point, large-wavelength disturbances need not
be considered.
60
2.5.2. Below supercooling: Wavelength Selection
While an energetically optimal solution wavelength was identified in Section 2.4, it is not
clear if, or how, a non-optimal solution would change its wavelength to become optimal.
Because surface energy keeps the surface faceted, the only way to increase wavelength
is through coarsening. However, coarsening of a periodic interface requires some kind of
facet motion, and a consideration of the free energy (2.39) reveals that any facet motion
on a periodic surface increases supercooling energy, and is therefore prohibited under the
dynamics (2.50). Energetically, smaller-than-optimal solutions are trapped in local energy
wells created by the thermal gradient. (Mechanisms of decreasing wavelength, such as
“facet shattering” [70] or “tip splitting” [71, 48], are not considered here.) However,
knowing that any one of a range of wavelengths is stable, we may still conclude that
the optimal wavelength may never be reached, and we are encouraged in this direction
by the fact that elastic materials modeled with similar equations [63, 64, 65, 66] are
experimentally observed to exhibit hysteresis in their equilibrium-pattern wavelength [72].
2.5.3. Above supercooling: Coarsening
Finally, when the solidification speed is increased beyond the constitutional supercooling
boundary, the sign of Geff changes, and the above-mentioned energy wells are replaced
by energy hills. Now, facet motion away from the z = 0 isotherm actually decreases
supercooling energy (since the thermal gradient is negative). As facets accelerate away
from z = 0, the surface coarsens as boundaries between facets meet and annihilate. An
example of this behavior is given in Figure 2.5, where it is also contrasted with below-
supercooling behavior. The coarsening process continues until the typical wavelength
61
is no longer small. At such point, the facet-dynamics model loses its validity, and the
full free-boundary problem must be considered. Consequently, the wavelength of a final
steady or unsteady state is selected not by near-instability analysis of competing cellular
modes, but by nonlinear dynamic interactions between the fully-faceted surface and its
associated concentration field. (See, for example, [48] for several facet-dynamics models
proposed to describe this late-time regime.)
1 1.2 1.4 1.6 1.8 2 2.2
1
2
3
4
5
6
7
8
x
t
(a)
16 17 18 19 20 21 22 23
1
2
3
4
5
6
7
x
t
(b)
Figure 2.5. Representative solution behavior below and above supercool-ing. The evolution of corners is shown; peaks are red, and valleys areblue. (a) Below the supercooling velocity, coarsening is prohibited, andnon-periodic interfaces are driven toward periodicity. (b) However, abovethe supercooling velocity, coarsening replaces cellular growth as the mech-anism of instability.
62
2.6. Conclusions and Comments
When materials with large anisotropy are solidified in high-energy orientations, neg-
ative surface stiffness renders the planar interface unstable for all solidification environ-
ments. Instead, the interface assumes one of a family of small, faceted profiles. Consid-
eration of attachment kinetics reveals a displacement of the faceted interface relative to
the planar one during solidification, whereas an associated correction to the concentra-
tion field results in a very slight variation in the composition of the final product. While
geometric considerations predict a small interface wavelength, the matched asymptotic
methods used to derive interface shapes do not reveal a wavelength selection mechanism.
Instead, the minimization of an appropriate surface free energy reveals that the optimal
solution wavelength scales as ν1/6 [64, 65], where ν is a very small corner energy parame-
ter. Questions of dynamics lead, in the small-wavelength limit, to the derivation of a facet
velocity law, which specifies the vertical velocity of each facet as a function of its mean
height. This approach result shows that (a) the stability boundary for faceted interfaces
is precisely the M−1 = 1 supercooling boundary; (b) wavelength change to reach the
optimal wavelength is inhibited below supercooling, leading to the prevalence of varied,
non-optimal solutions; and (c) above supercooling, coarsening replaces the usual cellular
growth as the mechanism of instability.
Perhaps the most interesting feature of the contrast between small and large anisotropy
is that the role of surface energy changes fundamentally. For isotropy and small anisotropy,
surface energy stabilizes against an instability driven by constitutional supercooling. The
presence of anisotropy in this regime simply quantitatively modifies a pre-existing mor-
phological instability. However, for large anisotropy, surface energy becomes destabilizing,
63
and drives its own, thermodynamic faceting instability before supercooling is reached. In
addition, the sole effect of surface energy is to drive the interface toward its optimal slope
via faceting. In the absence of its further stabilizing influence, the stability of the faceted
surface depends solely on the presence or absence of supercooling; interestingly, this is
exactly the qualitative instability criteria originally hypothesized by Rutter and Chalmers
and Tiller et. al. for the isotropic problem. Thus, where surface energy and supercooling
are competing effects for small or zero anisotropy, they are divorced for large anisotropy.
An additional interesting qualitative change is the mechanism by which solutions
evolve after becoming unstable. For materials with small or zero anisotropy, the late-
stage behavior of linearly-identified instabilities are well-described by the usual weakly
nonlinear cellular solutions, where a small band of admissible solution wavelengths is
apparent from onset, and destabilization to nearby wavelengths within that band can be
considered. In the large-anisotropy regime, by contrast, destabilization of periodic faceted
interfaces occurs when supercooling simply causes the interface to begin coarsening un-
der the appropriate facet dynamics. Here, no intrinsic wavelength is apparent at onset,
and the final wavelength will be selected only eventually, through nonlinear interactions
between interface shape and concentration profile.
While this work was motivated by the addition of the particular effect of large anisotropy
to the long-standing problem of directional solidification, its broader significance is best
seen by viewing it as a sample study of large-anisotropy surfaces in dynamic contexts.
In such systems, faceting is the generic outcome regardless of other environmental con-
ditions. This renders traditional analytical methods of limited value – linear stability
64
analysis simply returns the expected universal instability of the planar state, while fur-
ther destabilization occurs through coarsening rather than cellular instability. In the place
of these traditional methods, it was the derivation of the facet velocity law in Section 2.5
that allows real advancement of understanding. This approach would be of use in general
in the study of faceting interfaces.
65
CHAPTER 3
Large-Scale Simulations of Coarsening Faceted Surfaces
3.1. Introduction
When a crystalline material is cut along an energetically unfavorable direction and
then allowed to evolve by some mechanism, surface faceting may result. In this phe-
nomenon, an initially flat surface will decompose into a faceted, pyramidal hill-and-valley
configuration. Faceting is caused by the strong crystalline anisotropy of surface energy –
a faceted interface, while exposing more surface area than a flat one, may have a lower
surface energy if the facets have low-energy orientations. As a faceted surface continues
to evolve, it may also exhibit coarsening, whereby small facets continually vanish, and the
average length scale L of those that remain increases with time. Of primary interest is
whether such systems exhibit dynamic scaling, whereby the surface approaches a constant
statistical state which is preserved even as the length scale increases. This intriguing phe-
nomenon, observed in many coarsening systems, is studied because the system may be
described at all stages of evolution by a single set of statistical distributions.
The detailed statistical study of coarsening and dynamic scaling requires a method of
rapidly simulating large faceted surfaces, which may be developed as follows. Assuming
the orientation of each facet is prescribed and fixed, then individual facet motion is con-
strained to translation along its normal. Therefore, the evolution of a completely faceted
surface is concisely expressed by a discrete collection of individual normal facet velocities
66
(nucleation of new facets is not treated here). If the facet velocity law providing this
collection at each time can be determined, then the computational complexity of evolv-
ing the surface can be reduced to that of a system of ordinary differential equations; an
evolving surface which possesses such a known law is termed a Piecewise-Affine Dynamic
Surface (PADS) [13]. Such PADS are, in fact, known. In what are now known as van
der Drift models [73, 74], the facets of diamond grown under vapor deposition advance
according to a fixed velocity which depends only on orientation. Other configurational
rules have more recently been proposed or derived for systems as varied as the evolving
faceted interface between two elastic solids [75], the thermal annealing of a faceted crystal
with its melt [13], and several models of solidification [71, 48, 76, 62]. Because of the
variety of systems described by facet velocity laws, and the need for large simulations to
investigate coarsening and dynamic scaling phenomena, there is a strong incentive for the
development of a computational geometry tool to investigate evolving faceted surfaces.
Several such geometric methods have been considered in the past. For 1+1D surfaces
z = h(x, t) evolving in time t, many examples exist. Pfeiffer et. al. [71] and Shangguan
and Hunt [48] included, to our knowledge, the first such simulations in their proposals of
facet dynamics describing the solidification of pure silicon and binary alloys, respectively.
Later, Wild et. al. [77], Dammers and Radelaar [78], and Paritosh et. al. [79] all
used the same approach to study the evolution of diamond films under the previously-
known van der Drift evolution. Additionally, what are essentially 1+1D geometric surface
simulations are found in two simulations of the convective Cahn-Hilliard equation [76, 62]
– these authors actually develop explicit expressions for corner evolution. Finally, whereas
each of the above simulations were specifically implemented for the particular dynamics
67
being studied, Zhang and Adams [80, 81] have recently released a more general software
package which allows the selection of a variety of facet behaviors.
While their simplicity makes them efficient, the primary complication of direct geomet-
ric methods is the need to manually detect and resolve topological events. These changes
in the neighbor relations between facets occur when, as the surface evolves, facets merge,
split, or vanish. Trivial in 1+1D, difficulty associated with topology increases with the
number of dimensions considered. For 2+1D surfaces z = h(x, y, t), the only known geo-
metric simulations of faceted surfaces are due to Thijssen [14] and Barrat et. al. [15],
who studied the law for diamond films; a similar method was also applied to spiral-mode
growth of thin films [82]. These authors did not explicitly address topology, instead allow-
ing diamond grains to interpenetrate, and describing the actual surface as the envelope of
these grains. Indeed, the increased topological difficulty associated with high-dimensional
geometric methods is an oft-cited motivation for the development of “topology free” meth-
ods such as phase-field [83, 84] and level-set [16, 17] methods. However, these methods
sacrifice speed and ease of access to geometrical data. Furthermore, the appropriate reso-
lution of potentially non-unique events as described in [14] requires explicit intervention,
which is made difficult by topology free methods, and negates much of their benefit.
In light of these concerns, we have chosen an explicit resolution approach, and we find
that, for certain symmetries at least, the topological complexity has been overstated. In
particular, for 2+1D surfaces possessing only three facet normals (threefold symmetry),
only three kinds of topological event are possible, each of which recalls similar events
observed in the related 2D work of Roosen and Taylor et. al. [70, 85, 86]1. With
1The purely 2D work of these authors represents an important intermediate case between 1+1D and2+1D. Their method captured the kinematics of evolving completely faceted crystal domains in the
68
the abundance of systems for which facet-velocity rules are known, and the speed and
accessibility advantages offered by geometric methods, the challenges posed by topology
are worth tackling.
The aim here, then, is to present a general-purpose geometric method, which imple-
ments topological events, for the simulation of coarsening, threefold-symmetric faceted
surfaces in 2+1 dimensions. We shall begin with a description of the faceted surface and
associated data structure, and a discussion of the kinematics and dynamics which govern
its evolution. We then present illustrations and discussions of each class of topological
event, including detection and resolution procedures. Finally, we demonstrate our method
by simulating a total of one million facets under a sample dynamics describing thermal
annealing. The efficiency of this method allows such large collections of facets to be sim-
ulated rapidly, while the geometrical nature of the network allows the easy collection of
a rich variety of statistical data.
3.2. Faceted Surfaces: Description, Kinematics, and Dynamics
In this Section we present the basic elements of our method. We first give a mathemat-
ical description of completely faceted surfaces, and present a three-component structure
used to simulate them. We then discuss the kinematics of such surfaces; i.e., how facets
move and how their motion drives the evolution of other surface elements. Last, we
consider the imposition of a dynamics on the system, and we apply a sample dynamics
associated with thermal annealing.
plane, and was used to simulate dynamics associated with growth due to diffusion fields, attachmentkinetics, and surface diffusion. Of especial importance to us is that topological events were handledexplicitly, and indeed, each of the three events observed there exhibits aspects of a corresponding eventconsidered here.
69
3.2.1. Description
We consider evolving fully-faceted surfaces z = h(x, y, t) consisting of planar facets {Fi}
with prescribed normals {ni}. We consider surfaces formed by a single crystal with cubic
symmetry, and restrict our attention facets possessing one of three orthogonal normals,
given in spherical polar co-ordinates by
(3.1) ni ∈ (1, 2πi/3, α) , i = {0, 1, 2} , α = sin−1(1/√
2)
These normals represent the ([100],[010],[001]) orientations of a cubic crystal viewed from
the [111] direction (with a different choice of α, different symmetric orientations could be
modeled).
The facets F are bounded by and meet at straight edges, which in turn which in turn
meet at triple-junctions (while any number of edges greater than three could theoretically
meet at a junction, we assume that all junctions are formed by exactly three edges). This
facet-edge-junction structure is reminiscent of two-dimensional cellular networks [87, 88,
89, 90, 91]. Several other purely two-dimensional coarsening physical systems – such as
soap froths [92, 93] and polycrystalline grains [94, 95] – are also cellular in nature, and
have been effectively simulated using three-component models.
3.2.2. Kinematics
Since the orientation of each facet Fi is fixed and constrained by (3.1), its motion is
completely described by a displacement in the normal direction, parametrized by a local
distance parameter. The surface kinematics Vn are thereby captured by specifying the
instantaneous normal velocity Vi of each facet. The motion of edges and junctions, being
70
merely the intersections between the two and three facets that comprise them, are then
uniquely determined by the motion of those facets. In practice, we only use facet veloc-
ities indirectly, as a means to calculate junction velocities. When junctions are moved
correctly, edges (connections between two junctions) and thus facets (collections of edges)
are necessarily moved correctly as well.
3.2.3. Dynamics
With the kinematics of fully-faceted surfaces now set, we may consider imposing a surface
dynamics associated with some physical problem. To do so, we choose a facet-velocity law
which specifies Vi, yielding a piecewise-affine dynamic surface (PADS) [13]. (To connect
with mathematical theory, we note that this step amounts to imposing a vector field on
the manifold of fully-faceted surfaces just defined.)
For concreteness, we consider a PADS associated with the annealing of a faceted
surface. It has recently been shown [13] that, because the equations describing this system
are variational in nature, a principle of maximal dissipation may be applied, which shows
that the surface evolves so as to always maximally reduce its energy, which to leading
order is stored in the edges between facets. This approach allows the matched asymptotic
extraction of the facet velocity law, which is expressed by the equation
(3.2) Vi = − 1
A(Fi)
∂P
∂ni
.
Here A(Fi) represents the area of facet Fi, while ∂P∂ni
is the rate of change in total perimeter
P per unit displacement of Fi along its normal.
71
3.3. Topological Events
In presenting our data structure in Section 3.2.1, we described a set of neighbor re-
lations inherent to the surface. Surface elements of each class (facet, edge, junction)
neighbor members from each of the other classes. Together, this set of neighbor relations
comprises the topological state of the surface. As the surface evolves under a prescribed
dynamics (see Sec. 3.2.3), these relations may change as facets merge, split, or vanish.
Such changes to the topological state of the surface are called topological events. On
an actual faceted surface, these happen naturally as the system evolves; however, in an
approach like ours they must be performed manually. Each event is found by looking
for an appropriate “trigger” condition on the surface; once detected, the proper resolu-
tion follows from geometrical considerations. In this Section, we consider the occurrence,
detection and resolution of topological events, limiting our attention to those occurring
under the symmetry and dynamics already described.
3.3.1. Facet Merge
When an edge shrinks to zero length on an evolving surface with cubic symmetry, two
facets of like orientation meet, and merge to form a larger facet. This event is called
a Facet Merge. To see why this occurs, consider that each edge is composed of two
facets of which it is the intersection (its composite facets), and stretches between two
facets at which it terminates (its terminal facets). Because we only allow three distinct
facet orientations, the terminal facets of any edge necessarily have the same orientation.
Thus, when an edge shrinks to zero length in isolation (i.e. none of its neighbor facets are
vanishing), its terminal facets meet exactly. Having the same orientation, they merge into
72
a single facet, and the total number of facets on the surface is reduced by one. Thus, the
facet merge is one possible mechanism of coarsening for a PADS with cubic symmetry.
Figure 3.1 depicts a representative facet merge.
Figure 3.1. Example facet merge event, viewed from above. Arrows rep-resent gradients of the facets on which they appear. Dotted lines indicatepast edges no longer present.
Detection and Resolution. Edges in the data structure are directed, having an initial
and terminal junction. Since facets, and hence edges, have fixed orientation, each edge
thus has a unique orientation. The only way a tangent may change is to reverse direction
when an edge shrinks to zero, as just described. If such an “edge flip” occurs, it indicate
that a facet merge ought to have occurred during the preceding timestep. This serves as
our “trigger” condition. Having found the flipped edge, the like-oriented facets are first
adjusted to equal height in preparation for merging. Further resolution is then essentially
an exercise in labeling. All edges and faces which touch the shrinking edge are first
identified; then surface elements are created/deleted, and neighbor relations reassigned as
appropriate to effect the change shown in Figure 3.1. Complete numerical details will be
published elsewhere.
73
3.3.2. Merging Facet Pinch
In addition to merging together, facets may also be split apart if they are pinched by
non-adjacent neighbors. Under cubic symmetry, a facet may only be so pinched by two
of its neighbors with identical orientations. While the pinched facet splits in two, the
impinging neighbors (having the same orientation) merge to form a larger facet; the total
number of facets is thus conserved. This event is called a Merging Facet Pinch. It is
not a coarsening event, but rather a re-organization which allows further coarsening to
occur. In Figure 3.2, we see an example of this event.
Figure 3.2. Example merging facet pinch event, viewed from above. Arrowsrepresent normals of the facets on which they appear. Dotted lines representpast edges no longer present.
Detection and Resolution. To describe the trigger which indicates a merging facet
pinch, we note that the edges forming the boundary of a facet form a polygon in the
plane, A merging facet pinch is detected when, after a timestep, this polygon is found
to be self-intersecting. Such a polygon represents a geometric inconsistency, as surface
facets are necessarily simply-connected. This indicates that the facet in question was
pinched during the previous timestep, and a merging facet pinch should have occurred.
Having identified the impinging facets, we first ensure that they are of equal height to
74
allow proper merging. Then, as in the facet merge, further resolution of the merging facet
pinch is mostly an exercise in element labeling. Using neighbor relations inherent to the
network, we can collect all the affected edges and junctions. Elements are then created
and deleted as necessary to perform the change illustrated in Figure 3.2. Again, numerical
details will be presented elsewhere.
3.3.3. Removal of Vanishing Facets
The final class of topological events involves one or more contiguous facets shrinking
to zero area. When this occurs, these facets must be removed, and the surrounding
neighbor facets re-connected appropriately. The total number of facets clearly decreases,
making these events an additional mechanism of coarsening. A variety of these events
are possible kinematically; however, after many tests under both dynamics with random
initial conditions, we observe only three kinds: a “step removal,” a “ridge removal,” and
a “box removal.” We list these in Figure 3.3 for illustrative purposes. The appearance
and relative incidence of each configuration is a consequence of the dynamics.
Detection. In practice, we do not actually wait for a zero area facet to occur. Instead,
we seek to eliminate small facets whenever their area decreases below a certain threshold.
The gathering of two- and three-facet groups is accomplished by maintaining a second,
more liberal threshold. Whenever a facet is found to have decreased below the first
threshold, its neighbors are recursively examined to find those smaller than the second
threshold. This procedure is not foolproof, but since the area of vanishing facets tends
to zero, all facet areas must lie beneath the second threshold for at least some finite time
75
(a)
(b)
(c)
Figure 3.3. Three kinds of vanishing facet events. (a) One facet vanishingalone – a “step removal.” (b) Two facets vanishing simultaneously – a “ridgeremoval.” (c) Three facets vanishing simultaneously – a “box removal.”Arrows represent normals of facets on which they appear.
before the event. Therefore, if the procedure applied to a flagged facet does not yield a
recognized configuration, we abort and do nothing, to try again during a future timestep.
Resolution. After deleting the small facet or collection of facets, a “hole” is left in the
network. This must be repaired by reconnecting the surrounding facets, which we call
the “far field.” For the step removal, height averaging is again necessary before merging
the neighboring large facets together. For the ridge removal, the far field is O(4), and
76
the reconnection structure is a set of two points and a line, which must terminate at the
two (necessarily) identically-oriented far field faces. Finally, for the cube removal, the far
field is O(3), and a single point must result. In each case, the location of created points
is easily calculated from the positions of the surrounding facets.
3.4. Demonstration
We now apply our method to the annealing dynamics described above in Equation
(3.2). We begin in Figure 3.4 with a sequence of images from a relatively small test
run. There the reader may locate regions of the surface near to each of the topological
events described above. In the subsections which follow, we then present statistical data
averaged from 40 runs of 25,000 facets each. We first summarize the coarsening behavior
of annealing surfaces, including mechanism, power law, and convergence toward the dy-
namically scaling “steady state”. Next, we describe aspects of that state through some
easily-gathered morphometric data describing distributions of relative geometric quanti-
ties. Finally, we consider some topological and correlational statistics, which illustrate,
respectively, some properties of facets based on number of sides, and the degree to which
neighboring facets have similar geometric properties.
3.4.1. Rates and Mechanisms of coarsening, Convergence to dynamic scaling
We begin our statistical data with results concerning the rate and mechanism of coars-
ening, as well as the convergence to the scale invariant state (SIS – actual SIS data are
found below). First, coarsening is achieved primarily through the step-removal mecha-
nism shown in Figure 3.3a. As the system approaches the SIS, it exhibits the power-law
77
Figure 3.4. A top-down view of a small coarsening faceted surface. Spatialscale is constant, but irrelevant; time increases down the column.
coarsening seen in many dynamically scaling systems, with the characteristic morpho-
logical length scale LM growing as t 1/3. This has been observed in the past, and has
78
recently been explained by observing that the dynamics (3.2) are invariant under the
scaling x → αx, t → α3t [13].
Finally, we observe that the system converges to a scale-invariant state (SIS) at a
rate of E ∼ t−2/3, with E the 2-norm of the difference between any given statistical
distribution and its scale-invariant form. However, the mechanism by which convergence
to scale invariance occurs is coarsening, and so the convergence rate implicitly depends
on the coarsening rate. For this reason, we favor expressing E in terms of the fraction of
remaining facets N /N0 rather than time. This defines a function
(3.3) E(N ) ∼[ NN0
]p
for
[ NN0
]
≪ 1
which describes the coarsening efficiency of the dynamics being studied, where the
efficiency exponent p reflects the attractive strength of the scale-invariant state. This
representation allows the transparent comparison of coarsening phenomena in systems
with different coarsening rates; in particular, systems with p > 1 are expected to verifiably
achieve scale-invariance before running out of facets, while systems with p < 1 are not.
The dynamics studied here exhibit the intermediate value p = 1.
Coarsening Mechanism Step removalCoarsening Rate t 1/3
Convergence to SIS t−2/3
Efficiency Exponent p = 1
Table 3.1. Relevant coarsening phenomena.
79
3.4.2. Some 1D distributions
In Figure 3.5 we present a series of distributions of scaled, dimensionless geometric quan-
tities. That is, for a dimensional quantity q, we present a distribution of the dimensionless
q/ 〈q〉, where 〈q〉 denotes the system-wide average of that quantity. For a system in a state
of dynamic scaling, all such distributions are constant in time. In the following figures,
data have been time-averaged over the dynamic scaling regime to minimize noise.
0 2 4 6 8 10 12 14 16 18 200
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
nu
mb
er
of
fac
ets
number of sides
(a)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
ρ
L / <L>
(b)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
A / <A>
ρ
(c)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
P / <P>
ρ
(d)
Figure 3.5. (a) Distribution of facets by number of sides n. (b) Distributionof dimensionless edge lengths ρ(L/L), with contributions from concave andconvex edges. (c) Distribution of dimensionless facet areas ρ(A/A), withcontributions from 2n-sided facets. (d) Distribution of dimensionless facetperimeters ρ(P/P), with contributions from 2n-sided facets.
80
Figure 3.5a shows the probability of a facet having a given number of sides. Under cu-
bic symmetry, all facets under threefold symmetry have an even number of edges (because
only three facet orientations are available, and facets of like orientation cannot touch, the
neighbors of a facet with one orientation must facet alternate between the other two ori-
entations). In Figure 3.5b we show the distribution of edge lengths. The total distribution
is in dotted black, while the solid blue and green lines show the contributions from convex
and concave edges, respectively. The equality of these reflects the underlying up-down
symmetry of the dynamics. Finally, in Figures 3.5c,d we display distributions of facet area
and perimeter, respectively, relative to their global averages. These are broken down into
contributions from 2n-sided facets, illustrating the level of detail that may be extracted
using our method.
3.4.3. Topological results and neighbor relations
In our last set of data, we consider two scale-invariant topological properties, which de-
scribe average geometrical quantities as functions of the number of sides; and two corre-
lational properties, which are two-point distributions associated with neighbor pairs.
Figure 3.6a shows that average facet area grows linearly with the number of sides per
cell, a relationship known as Lewis’s law [96]. In Figure 3.6b, we show that the average
number of sides of the neighbors of n-sided cells, mn, obeys Aboav’s law [97, 98, 99]:
(3.4) mn = (6 − a) +6a + µ2
n,
with µ2 the second moment about the mean of the distribution of sides per cell, and a a
fitting parameter (we find µ2 = 7.07 and a = 5.345). These relationships are commonly
81
4 6 8 10 12 14 160.5
1
1.5
2
2.5
3
3.5
number of sides
A /
<A
>
(a)
4 6 8 10 12 14 16 185.5
6
6.5
7
7.5
8
number of sides
mn
(b)
0.5 1 1.5 2 2.5 3 3.5 4 4.5
0.5
1
1.5
2
2.5
3
3.5
4
4.5
A /
<A
>
A / <A>
(c)
0.5 1 1.5 2 2.5 3 3.5 4 4.5
0.5
1
1.5
2
2.5
3
3.5
4
4.5
P /
<P
>
P / <P>
(d)
Figure 3.6. (a) Lewis’s law, showing the average area of 2n-sided facets.Also shown is the average perimeter of the same. (b) The Aboav-Weaire law,showing the average number of sides of the neighbors of 2n − sided facets.(c) A two-variable distribution of the relative areas of neighboring facets.(d) A two-variable distribution of the relative perimeters of neighboringfacets.
observed in evolving 2D cellular network problems such as soap froth evolution and grain
growth.
Finally, in Figures 3.6c,d, we give a pair of distributions measuring the probability of
two neighboring facets having a given pair of areas and perimeters, respectively. These
data, together with the data in Figures 3.5c,d, can be used to determine whether the be-
haviors of neighboring facets are correlated. Understanding the extent of such correlations
will, in turn, inform the future pursuit of mean-field theories.
82
3.5. Conclusions
We have presented a computational geometry tool for the simulation of coarsening
faceted surfaces in 2+1 dimensions. Such surfaces are expressed geometrically as a 3D
cellular network consisting of facets, edges, junctions, and the connections between them.
Kinematic relationships between facet displacement and junction/edge motion have been
discussed, and an example dynamics has been imposed, resulting in a Piecewise-Affine
Dynamic Surface, or PADS. We considered a faceted surface with cubic symmetry, corre-
sponding to the growth of a cubic crystal in the [111] direction. For this symmetry group,
we identified and discussed the three classes of topological events:
• a facet merge in which two facets merge to form a larger facet
• a merging facet pinch in which one facet splits in two, and two others merge
• a facet removal in which one of more vanishing facets are removed.
The detection and implementation of these topological events is the main contribution of
our tool.
The primary benefits of our approach are its speed and easy access to a variety of
geometrical data, which are both highlighted through our demonstration on the facet
dynamics (3.2) associated with thermal annealing. Because the method is intrinsically
geometrical, we can easily extract statistics describing distributions of geometric quan-
tities, as well as those describing correlation among quantities and neighbors. On the
other hand, because the method efficiently handles topology, we can quickly measure not
only the dynamically scaling state itself, but also the convergence toward that state, as
described by the introduced quantity of coarsening efficiency.
83
It is our hope that this tool, generalized to arbitrary symmetry groups, will be widely
useful to anyone investigating fully-faceted surface evolution. The speed and design of
this tool will allow the rapid simulation of large faceted surfaces, and the consequent
collection of geometric statistics suitable for the comparison of different physical causes of
coarsening. Finally, neighbor relations inherent in the data structure will allow the search
for correlations in high-order statistics, the presence or absence of which should help to
inform the future pursuit of mean-field theories for coarsening faceted surfaces.
84
CHAPTER 4
The Kinematics of Faceted Surfaces with Arbitary Symmetry
4.1. Introduction
In many crystal-growing procedures of interest, a nano-scale faceted surface appears
and proceeds to evolve, often exhibiting coarsening and even dynamic scaling, whereby
characteristic statistics describing the surface remain constant even as the characteristic
lengthscale increases through the vanishing of small facets. For many evolving faceted
surfaces, a facet velocity law can be observed [74, 73], assumed [71, 48], or derived
[75, 62, 13] which specifies the normal velocity of each facet, often in configurational
form which depends on the geometry of the facet. In this way, the dynamics of a continu-
ous, two-dimensional surface can be concisely represented by a discrete collection of such
velocities, and overall computational complexity reduced to that of a system of ODE’s;
the resulting system is known as a Piecewise-Affine Dynamic Surface, or PADS. Such
theoretical simplification, in turn, enables the large-scale numerical simulations necessary
for the statistical investigation of coarsening and dynamic scaling.
The numerics involved in the direct geometric simulation of an arbitrary PADS is
straightforward for one-dimensional surfaces, requiring nothing beyond traditional ODE
85
techniques except simple geometric translation between facet displacement and edge dis-
placement, and a small surface correction associated with each coarsening event. Conse-
quently, such simulations accompany many of the above facet treatments of facet dynam-
ics, and have also been independently repeated elsewhere [77, 100, 79, 80, 81]. However,
in two dimensions, the corrections due to coarsening events are much more involved, and
any code must be able to deal with a family of non-coarsening topological events that
alter the neighbor relations between nearby facets. Consequently, the fewer simulation
attempts use either fast but poentially imprecise envelope methods [14, 15, 82], or more
robust but slower phase-field [83, 84] or level-set [16, 17] methods to avoid explicitly
performing topological changes. Besides the speed/accuracy trade-off exhibited by these
approaches, both methods obscure the natural geometric simplicity of the native surface,
complicating the extraction of detailed surface statistics which, after all, motivates large
simulations in the first place. Additionally, as will be seen, the presence of non-unique
topological events requires explicit intervention regardless of topological scheme, which
negates much of the advantage of a “hands-free” treatment.
In the previous chapter, we introduced a direct-simulation method which explicitly
performs topological events along the way, thus preserving both simulation speed and
topological accuracy. In addition, by representing the surface as a collection of facets,
edges, and junctions, plus the neighbor relations between them, the method mirrors the
natural geometry of the surface being modeled, which allows easy extraction of geometric
statistics. There, however, the restricted case of threefold symmetry was chosen for ease of
topological implementation; under this symmetry, a limited number of topological events
were observed, and both vanishing facets and non-vanishing surface rearrangements could
86
be handled explicitly using a priori knowledge of the before and after surface states. While
many surfaces exhibit threefold symmetry, making the method useful even in this special
case, it could not handle other common crystal symmetries, notably fourfold and sixfold.
In this chapter, then, we generalize the previous model to allow the simulation of
surfaces with arbitrary symmetry groups. We begin in Section 4.2 with a brief summary
of the basic method, including surface representation, facet kinematics, and the applica-
tion of a dynamics. Next, in Section 4.3, we provide a careful enumeration of topological
events which may occur on surfaces of arbitrary symmetry; this includes discussion of the
Far-Field Reconnection algorithm, by which network holes left by vanishing facets may
be consistently repaired without knowledge of the post-event state. Then, we provide
in Section 4.4 a careful consideration of the consequences of using (necessarily discrete)
timesteps during the simulation of a surface whose evolution equations change qualita-
tively between steps (at topological events); the issues that arise are discussed in the
context of three sample strategies. The completed method is illustrated from three-, four-
, and six-fold symmetric surfaces in Section 4.5; these exhibit all of the topological events
likely to be encountered on a real surface, and demonstrate that the method is robust
enough to generically simulate faceted surfaces of any symmetry class for which a facet-
velocity law is uniquely specified. Finally, in addition to detailing the FFR algorithm,
the appendix includes a discussion of kinematically non-unique topological events, where
two resolutions are possible, and highlights the need to refer to the dynamics or even first
principles to decide how the surface should evolve in those cases.
87
4.2. Data structures and simple motion: a 3D cellular network
4.2.1. Characterization
We consider the evolution of a single-valued, fully-faceted surface z = h(x, y, t); this
definition explicitly forbids overhangs and inclusions. We assume that the surface bounds
a single crystal which exists on exactly one lattice; thus, we are not treating surfaces
with multiple grains. The surface is piecewise-affine, consisting of facets {Fi} with fixed
normals {ni}. These are bounded by and meet at edges E which are necessarily straight
line segments; edges in turn meet at triple-junctions J . This three-component structure
is reminiscent of two-dimensional cellular networks [101, 87, 89, 90, 91] and indeed,
while we consider three dimensional surfaces, the projection of the edge set onto the
plane z = 0 is a 2D cellular network. This structure and the neighbor relations inherent
within it suggest a doubly-linked object-oriented data structure, consisting of: (1) a set
of junctions, each having a location, pointing to three edges and three facets; (2) a set
of edges, each having a tangent, pointing to two junctions and two facets; and (3) a set
of facets, each having a normal, pointing to m edges and m junctions. These objects
and the associated neighbor relations are illustrated in Figure 4.2.1; this structure is the
natural structure of the surface, and uniquely and exactly describes it. We now consider
each element in more detail.
4.2.1.1. Junctions. A junction is a point in space formed where edges (and hence,
facets) intersect. The order n of a junction is simply the number of edges which meet
there. While junctions of any order n ≥ 3 are possible, we restrict ourselves here to
the case of order 3 junctions or “triple junctions.” This greatly simplifies analysis and
88
(a) (b) (c)
Figure 4.1. Neighbor relations for each kind of surface element. (a) Ajunction neighbors three edges and three faces. (b) An edge neighbors twojunctions and two faces. (c) A face neighbors m junctions and m edges(here m = 5).
code, as triple junctions are uniquely positioned by the three facets meeting there. The
intrinsic geometric information carried by a junction is its location. Junctions are stored
in a Junction class, which contains this location, as well as pointers to the three edges
and three facets which meet there.
4.2.1.2. Edges. An edge is a line segment formed by the intersection of exactly two
facets, and bounded by exactly two junctions. The intrinsic geometrical quantity of an
edge is its orientation, which is fixed since facets have fixed normals. Edges are stored
in the Edge class, which records the tangent, as well as pointers to the two neighboring
facets and two bounding junctions.
At creation, edges are “directed”: one junction is arbitrarily deemed the origin, and the
other the terminus, establishing a tangent. This has two important consequences. First,
if we imagine walking along the edge in the tangent direction, then one neighboring facet
may be labeled “left”, and the other “right.” This information allows us to distinguish
between convex and concave edges, and also to determine the clockwise direction around
a given facet, which is necessary for effective navigation of the network, as well as the
89
proper calculation of boundary integrals on facets. Second, the tangent allows us to detect
when an edge “flips” (see [70]); this will be discussed in more detail in Section 4.3.1.
4.2.1.3. Faces. A facet is a simply-connected planar polygonal region in space, which is
bounded by an equal number of edges and junctions. The intrinsic geometric information
carried by a facet is its normal, which is fixed. Our surface definition z = h(x, t) requires
that the normal of each facet is constrained to be on the hemispherical shell of unit-
length vectors with positive z component. The imposition of a particular symmetry on
the crystal may further restrict available normals, but no such restriction is here assumed.
Facets are stored in a Facet class, which contains the normal, as well as a list of bounding
edges and junctions, sorted in counter-clockwise order.
4.2.2. Kinematics
The intrinsic geometric means of characterizing surface evolution is by specifying the
normal velocity of each point on the surface. A piecewise-affine surface is composed
of a collection of planar, fixed-normal facets, whose motion is limited to displacement
along the normal. Therefore, the kinematics Vn of the entire surface may be expressed
by a discrete set of individual facet velocities Vi. As edges and junctions are merely
intersections between two and three facets, respectively, their motion is uniquely specified
by the motion of the facets that neighbor them. In particular, if p is the location in space
of a triple junction, then the velocity of that a triple junction may be calculated through
the expression
(4.1)dp
dt= A
−1v,
90
where the rows of A and entries of v are the unit normals and normal velocity, respec-
tively, of the three facets intersecting to form p. In practice, facet velocities are only
used indirectly to calculate junction velocities – if junctions are moved correctly, edges
(connections between two junctions) and thus faces (collections of edges) are necessarily
moved correctly as well.
4.2.3. Dynamics
All that remains now is to select a particular dynamics; that is, to specify an expression
for the normal velocity Vi of each facet. Having chosen one, we follow [13] and refer to the
resulting evolving structure as a Piecewise-Affine Dynamic Surface (PADS). Example
dynamics describing many different physical situations were listed in the introduction, and
the exact dynamics is not of special concern here (although we will select one for demon-
stration later). It is worth noting here, however, that most of the dynamics proposed
to date are configurational, depending on properties of the facet such as area, perimeter,
number of junctions, or mean height. Thus, sudden changes in the geometric properties
of a facet can lead to sudden changes in its velocity, an issue which will be explored in
more detail in Section 4.4.
4.3. Topological Events
We have just discussed how elements of each class (facet, edge, vertex) neighbor mem-
bers from each of the other classes. Taken together, the set of all of these neighbor
relations comprises the topological state of the surface. It is a complete record of every
neighbor relationship on the surface, and is unique for a given surface. As the system
91
evolves, these neighbor-relations may change as facets exchange neighbors, join together,
split apart, or vanish. Each of these cases is an example of a topological event, and rep-
resents a change to the topological state of the surface (topological events are a defining
feature of evolving cellular networks – again see [101, 87, 89, 90, 91]). To maintain an
accurate representation of the surface, a direct geometric method like that described here
must manually perform topological events as necessary. Because actual surface evolution
is fairly trivial, this is the main difficulty of our method.
A natural first question to ask at this point is “how many topological events are
possible?” To begin answering this question, we point out that on a physical surface,
topological events occur automatically, and by geometric necessity. If a detected event
signals the need to change neighbor relationships at some location on the surface, we may
therefore infer that failing to change them would produce a cellular network with “wrong”
relationships, that do not correspond to a physical surface. We call such erroneous con-
figurations geometrically inconsistent; examples include primarily edge networks that
intersect when viewed from above, since these correspond to overhangs and inclusions,
which are prohibited. Since topological events serve to avoid possible geometric inconsis-
tencies, we may discover what events are possible by considering how inconsistencies may
occur. This is most easily accomplished by considering each surface element in turn.
We first consider junctions, which are simply a location in space. A junction can, in
the course of surface evolution, leave the periodic domain, in which case it is wrapped to
the other side. However, this is only a bookkeeping operation, and does not represent a
real topological event. Turning to edges, we note that edges possess a directed length. As
already hinted in section 4.2.1.2, this length could become negative if the edge were to
92
“flip” [70] A flipped edge has no geometrical meaning on a single-valued surface, and so we
introduce a class of Vanishing Edge events which occur when edges reach zero length.
Finally, we consider facets. Since a facet has fixed orientation, its changing properties are
loosely its shape and size. Specifically, a facet is a simply connected planar region with
positive area. These two defining properties of facets lead, through consideration of their
potential violation, to two additional classes of topological event: Facet Constriction
events which prevent the formation of self-intersecting facets, and Vanishing Facet
events which remove facets from the network when they reach zero area.
4.3.1. Vanishing Edges
An adjacent point-point event occurs when an edge shrinks to zero length, and its junctions
meet. To consider what might happen to the faceted surface when this occurs, we first
label the immediate surroundings of an edge. Each edge is composed of two faces of which
it is the intersection, its composite faces, and stretches between two faces at which it
terminates, its terminal faces. In addition, we will also use the term emanating edges
to refer to those edges immediately neighboring the shrinking edge. Now, consider the
hemispherical shell of available facet normals (Section 4.2.1.3). The (necessarily distinct)
normals of the composite faces specify a great circle about this hemisphere, which divides
it into two parts. The normals of the terminal faces cannot lie on this boundary, and unless
they are identical (a special case), they form a second great circle around the hemisphere.
While terminal normals may not lie on the composite great circle, the reverse is not true,
and this fact effectively divides Vanishing Edge events into three sub-classes.
93
(a) (b) (c)
Figure 4.2. Normal Diagrams for different types of Vanishing Edge events.Blue dots represent the normals of composite faces, while red dots representthe normals of terminal faces. Dotted lines represent great circles betweentwo points. (a) Terminal great circle touches neither composite point. (b)Terminal great circle touches one composite point. (c) Terminal normalsoccupy the same point. Great circle undefined.
Figure 4.3.1 illustrates this idea, and gives an example of each of the three possible
cases. If the terminal great circle touches neither composite point, then the well-studied
Neighbor Switch occurs. If the terminal great circle touches one composite point, then
an Irregular Neighbor Switch results. Finally, if the terminal normals occupy the
same point, then no great circle is defined – the terminal facets have he same normal, and
when the edge between them shrinks to zero, they join into a single facet: a Facet Join.
4.3.1.1. Neighbor Switch. On a general surface, the most common Vanishing Edge
event is the neighbor switch, which is frequently encountered in other evolving cellular
networks. In this event, neither composite normal touches the terminal great circle, so
any three of the normals involved form a linearly independent set – this property is the
defining feature of the neighbor switch. When an edge with this configuration shrinks to
zero length, the surrounding facets simply exchange neighbors. Figure 4.3.1.1 gives an
example of this event.
Resolution. The neighbor switch is performed by the NS_repairman class. To resolve
this event, it simply deletes the old edge, and creates a new edge. The composite faces
94
Figure 4.3. Example of a Neighbor Switch Event. Arrows represent gradi-ents of regions in which they appear.
which formed the old edge become terminal faces of the new edge, and cease to neighbor
each other. Conversely, the terminal faces of the old edge become the composite faces of
the new edge, and thus become neighbors. This symmetric exchange in neighbor relations
is the cause of the name Neighbor Switch, which comes from the grain-growth literature
– the less-descriptive name “T1 process” in often used in the soap froth literature. In
addition to replacing the vanishing edge, the junctions on either side of this edge are
replaced. Each new junction is formed by the intersection of the deleted edge’s (formerly
non-adjacent) terminal faces with one of its composite faces.
Comments. Readers familiar with other cellular-network literature will note that the
example Neighbor Switch in Figure 4.3.1.1 lacks the typical “X” shape. This is due to
the constrained nature of facet normals, and hence, edge orientations. Additionally, we
note that the neighbor switch is a reversible event; in fact it is its own reversal. Finally, a
certain sub-class of neighbor switches posessing “saddle” structure are non-unique, as was
observed by Thijssen [14]. For a discussion of this non-uniqueness and its consequences,
see Appendix B.2.
4.3.1.2. Irregular Neighbor Switch. When the normal of one of the composite faces
lies on the great circle formed by the terminal normals, the neighbor switch cannot occur.
Here, the terminal faces cannot form a new junction with the offending composite face
95
because the three normals involved are not independent. Instead, when an edge with this
configuration shrinks to zero, two closely related events are possible, depending on the
configuration of the nearby edges. These events are collectively called Irregular Neigh-
bor Switches, with two varieties called a “gap opener” and “gap closer” that are exact
opposites. These are illustrated in figure 4.3.1.2.
Figure 4.4. An example of the Irregular Neighbor Switch event. From leftto right is the non-unique “gap opener.” From right to left is the unique“gap closer.” Arrows represent gradients of regions in which they appear,while circles indicate flat facets with zero gradient (vertical normal).
Resolution. The irregular neighbor switch is performed by the INS_repairman class.
Because one composite normal lies on the terminal great circle, exactly two of the emanat-
ing edges are parallel in R3. The gap opener occurs when these edges emanate from the
shrinking edge in opposite directions, while the gap closer occurs when the edges emanate
in the same direction. To resolve the gap opener, we select one of the parallel emanating
edges to be split apart (see below). The gap will go here, filled by the terminal face that
touches the other parallel edge, and will extend all the way to the far end of the split
edge, where a new edge is introduced to link the two edges resulting from the split edge.
This is all illustrated in Figure 4.3.1.2. To resolve the gap closer, simply reverse the steps.
96
Comments. Several comments on this pair of events are in order. First, while the gap
closer is uniquely resolved, the gap-opener is an inherently non-unique event, as either of
the parallel edges could be the one split (we will discuss this further in section B.2). Sec-
ond, both resolution options have the potentially dissatisfying property of being non-local
in effect, because the collision of two junctions causes an entire edge to split apart. What
is perhaps more likely is the nucleation of a new, tiny facet at the moment the junctions
collide; however, we have excluded that possibility from consideration here. Finally, while
common experimentally-encountered surfaces usually have either high symmetry (only a
few facet orientations) or no symmetry (as many orientations as facets), the irregular
neighbor switch with its three coplanar orientations requires what may be called “inter-
mediate symmetry,” where orientations are limited, but many are available. Because it
poses resolution difficulties, and because it is not encountered in any surfaces we wish to
study, we have not yet actually implemented this event.
4.3.1.3. Facet Join. Finally, we consider the special case where the terminal normals
are identical. When such an edge shrinks to zero length, the terminal faces meet exactly.
Having the same orientation, they then join to form a larger face. Figure 4.3.1.3 depicts
a representative facet join event.
Figure 4.5. An example facet join procedure. Arrows represent gradientsof regions in which they appear.
97
Resolution. Facet Joins are performed by the FJoin_Repairman class. To perform a
facet join, a new face is created to replace the joining faces, and all edges and junctions
that neighbored the old faces are re-assigned to this new face. Next, the vanishing edge
and its two junctions are deleted, leaving the four emanating edges to be considered.
These are most logically grouped into the (necessarily parallel) pairs of edges bordering,
respectively, the left and right composite faces of the vanished edge. In the example event
shown in Figure 4.3.1.3, these two pairs look different: one pair meets side-to-side, while
the other pair meets end-to-end. Computationally, however, this makes no difference;
each pair is replaced by a single edge connecting their remaining non-deleted junctions.
This behavior is generic for all face joins.
Comments. We note that the face join is, strictly speaking, non-reversible (though see
Section 4.3.2.1). The exact opposite of the face join would be a facet which spontaneously
“shatters,” as described in [70]; this behavior is certainly worth studying, but is not
currently implemented. Second, although this is a “special case” in general, for high-
symmetry crystal surfaces it may be very common – indeed, for the case of a cubic crystal
with only three available facet orientations considered in Chapter 2, Facet Joins are the
only Vanishing Edge event exhibited. Finally, we note that this event is the only Vanishing
Edge event which does not conserve the number of facets. It is, in fact, one mechanism
by which coarsening may occur, and may be the dominant mechanism for high-symmetry
surfaces.
98
4.3.2. Facet Constrictions
The second class of topological event occurs whenever a facet ceases to be simply-connected,
and results in that facet being split into two new facets. Remembering that the edges
of a facet trace out a polygon in the plane, we observe that the non-simply connected
polygon, if allowed to continue evolving, would become self-intersecting, which clearly has
no geometrical interpretation. So, how may an evolving polygon become self-intersecting?
Since the boundary consists of edges and junctions, there are three possible modes: (a)
two non-adjacent junctions meet, (b) a junction meets an edge, or (c) two edges meet.
Each case has a distinct “signature,” illustrated in Figure 4.3.2, which can be used to tell
them apart.
(b)(a) (c2)(c1)
Figure 4.6. Signatures of Constricted Facet events. (a) Non-AdjacentJunction-Junction collision signature. (b) Junction-Edge collision signa-ture. (c1),(c2) Asymmetric and Symmetric Edge-Edge collision signatures.
The Junction-Junction collision shown in Figure 4.3.2a represents the formation of a
perfect O(6) junction. While theoretically interesting, such events are not considered here;
we hypothesize that, given random initial data, two junctions not connected by an edge
will never exactly meet. Furthermore, by considering Figure 4.3.2, it can be seen that all
Junction-Junction collisions, if perturbed as we hypothesize, result in either junction-edge
or edge-edge collisions, and can therefore be resolved accordingly.
99
Junction-Edge collisions occur when a facet is pinched into two pieces by three of its
neighbors, depicted in Figure 4.3.2b. There, two adjacent neighbors of the facet, forming
a wedge, meet a third neighbor and pierce it. Two separate events are possible in this
class. In most cases, the wedge simply splits the central facet into two parts, in an event
called a Facet Pierce. However, if the normals of the wedge facets and the normal of the
central facet lie on the same great circle, then, as the central facet is split, the opposing
facet opens up a gap in the wedge: an Irregular Facet Pierce.
Edge-edge collisions occur when a facet is pinched by four neighbors, shown in Fig-
ure 4.3.2c. In these events, two non-adjacent, exactly parallel edges meet, which requires
that the normals of the impinging facets be coplanar with the normal of the pinched facet.
Again, two variations are possible. If the impinging faces have different normals, the event
is called a Facet Pinch. However, if they have the same normal, they join even as they
pinch the facet in question, in a process called a Joining Facet Pinch. In addition,
each event may occur in either symmetrical or asymmetrical flavors, which are shown in
Figure 4.3.2c1,c2 respectively. The meeting of two edges requires the involvement of two
junctions; these lie on the same edge for the symmetrical case, and on different edges for
the asymmetrical case, as seen in the figure.
4.3.2.1. Facet Pierce. The first self-intersection we will study is the simplest; the facet
pierce. It is a point-line event as described above; that is, a facet is split when a triangular
wedge formed by two adjacent neighboring facets intersects the edge formed with a third,
opposing neighbor. The facet pierce is functionally the opposite of a facet join, and is
illustrated in Figure 4.3.2.1.
100
Figure 4.7. Example of a facet pierce procedure. At the moment the eventoccurs, an O(5) junction is formed, which immediately breaks in one ofthree ways, depending on the dynamics. Arrows represent gradients of theregions in which they appear.
Resolution. Each Facet Pierce is performed by the FJoin_Repairman class. Given
the constricted facet, as well as the junction and edge which meet, it can label all of
the surrounding facet elements and deterministically reconnect them correctly. First, two
new facets are created to replace the constricted facet. The junctions and edges that
bordered the old facet can be reassigned to these based on the labels created initially.
The colliding junction and edge are deleted, to be replaced by three new junctions and
two new edges. The locations of the former and neighbor relationships of each can be
determined by considering Figure 4.3.2.1 and using the labels.
Comments. First, technically, at the moment of the event, an O(5) junction forms,
which as shown in Figure 4.3.2.1 may proceed to break in one of three ways. This does
not, however, constitute a non-uniqueness; rather, the dynamics governing the surface
101
evolution at the moment of topological change specify which exit pathway is chosen.
Second, while Thijssen [14] rightly objected to this resolution for the case of separate
grains, we find it satisfactory for the case of a single crystal considered here.
4.3.2.2. Irregular Facet Pierce. A special modification of the Facet Pierce just de-
scribed occurs when the normal of the opposing facet shares a great circle with the normals
of the facets forming the wedge. This event is called an Irregular Facet Pierce. Recall that
three new junctions were created during the facet pierce. However here, since the two
newly created facets have identical normals, and the remaining three have normals which
are not independent, those junctions cannot be created. Instead, as the wedge facets meet
the opposing facet, one of two things happen – either the center edge of the wedge is split
apart by the opposing facet (a “wedge split”), or the opposing facet is split apart by the
wedge (a “wedge extension”). We see an illustration of each possibility in Figure 4.3.2.2.
Figure 4.8. Example of an irregular facet pierce procedure. Arrows indicategradients; circles flat planes with no gradient.
Resolution. The Irregular Facet Pierce is performed by the IFP_Repairman class,
which at instantiation is given the constricted facet, as well as the junction and edge which
102
meet. This event is repaired quite similarly to the regular facet pierce, with modifications.
As is done there, two new facets are created to replace the constricted facet, and junctions
and edges bordering the old facet are reassigned to the new ones. The resolution differs
in how to replace the colliding junction and edge. If the “wedge split”resolution is chosen,
then the middle edge of the wedge and its far junction are also deleted – these are replaced
by two parallel edges and junctions. Finally, an edge is formed which links them and
borders the facet on the far side of the deleted edge. If the “wedge extension” resolution
is chosen, not only is the constricted facet split apart, but so is the one opposite the
edge split by the wedge. One must first determine which edge of this second split facet
the extended wedge will intersect. Having done so, that facet is deleted, to be replaced
by two new facets. The extension is formed by adding two edges parallel to the middle
edge of the wedge, and the edge it intersects is split in two. Two new edges and three
junctions must be created to link the extension with the edge it intersects. Finally, all
edges and junctions bordering the deleted facet, plus those created to form the extension,
are re-assigned appropriately to the new facets. Figure 4.3.2.2 is especially helpful here.
Comments. The event clearly recalls the “gap opener” described above. It shares
with that event three coplanar surface normals, and as a result, two possible resolutions.
Additionally, while the two options here are qualitatively different compared to the sym-
metric options of the gap opener, they are additionally both non-local effects due to a
local cause. Again, perhaps the best resolution is to nucleate a new facet, which we do
not yet consider. Finally, both events require “intermediate symmetry,” and for the same
reasons discussed above, we have not implemented this event.
103
4.3.2.3. Facet Pinch. We now turn to consider the case of Edge-Edge events, the first
of which is called a Face Pinch. Here the normals of the pinching facets are not identical,
and so junctions can be created as needed – an illustration of this event is shown in
Figure 4.3.2.3. This event is philosophically similar to the face split described above. In
each case, a facet is split into two by non-joining neighbors; the difference is just whether
the procedure is “sharp” or “blunt”; i.e., caused by parallel edges or a junction and an
edge.
Figure 4.9. Example of a symmetric face pinch. Arrows represent gradientsof regions in which they appear.
Resolution. Each Facet Pinch is performed by the FPinch_Repairman class. Because
of the similarities between the facet pierce and facet pinch, the associated Repairman
classes behave similarly as well. Here, the Repairman class constructor takes the con-
stricted facet and the two colliding edges. With this information, it can label all of
the surrounding facet elements and deterministically achieve the change shown in Fig-
ure 4.3.2.3. As with the Facet Pierce, two new facets are created to replace the con-
stricted facet, and the junctions and edges that bordered the old facet are reassigned
as required. The colliding edges are deleted, as are the associated junctions discussed
above. Five edges and four junctions are created to complete the reconnection, as shown
in Figure 4.3.2.3.
104
Comments. This event, like the Irregular Neighbor Switch and Irregular Facet Pierce,
requires a surface with “intermediate symmetry.” While it is uniquely resolved and poses
no great difficulty of implementation, we have not yet implemented it for this reason.
4.3.2.4. Joining Facet Pinch. Finally, a special modification of the face-pinch occurs
when the impinging facets have identical normals. The constricted is split in exactly
the same way as in a face pinch; however, since the two facets doing the “pinching” are
identically oriented, they join together to form a larger facet. We see an illustration of
this situation in Figure 4.3.2.4.
Figure 4.10. Example of an asymmetric face swap. Arrows represent gra-dients of regions in which they appear.
Resolution. Each Joining Facet Pinch is performed by the JFPinch_Repairman class,
which operates similarly to the Repairman classes associated with the Facet Pierce and
Facet Pinch. This class is again instantiated with the constricted facet and the two
meeting edges, which allows the necessary labeling. Again, two new facets are created
to replace the constricted facet, but in this case the two facets which meet must join,
and so another new facet must be created to replace them – necessary junction and edge
reassignments are again easily carried out. Finally, rather than deleting the edges which
meet and the associated junctions involved, the meeting edges are simply re-connected as
shown in Figure 4.3.2.4.
105
Comments. Note that the final configuration is similar to the original configuration;
in fact, with suitable facet motion, the surface could return to its original configuration
via another face swap; the event is thus self-reversible in a sense. Also, since both a facet
pinch and a facet join occur simultaneously, the total numbers of each surface element
remain unchanged during this event.
4.3.3. Vanishing Facets
The final class of topological event occurs when a facet shrinks to zero area and is removed.
However, as has been noted numerous times previously in the context of cellular networks,
very small facets can result in stiff dynamics that are difficult to numerically simulate
accurately. For this reason, we follow previous authors by pre-emptively removing facets
with areas below some small threshold (but see Section 4.4.1). This process is summarized
in Figure 4.3.3. There, we see a single small flat facet vanishing into a pentagonal well
(4.3.3a). Being smaller than the allowed threshold, it is removed, leaving a “hole” in the
network (4.3.3b). The facets and edges bordering this hole we call the far field, and
they need to be reconnected correctly to patch the hole. The correct reconnection for this
particular well is shown in Figure 4.3.3c.
The principal difficulty in this process occurs during the reconnection step (Fig-
ure 4.3.3c). Here, we are assigning new neighbor relationships to the far-field facets, which
also involves the creation of new edges and junctions to form boundaries between them.
In other cellular-network problems, these neighbor relationships (and hence the reconnec-
tion) is usually chosen randomly, under the reasoning that any error introduced is small
106
(a) (b) (c)
Figure 4.11. (a) a single flat facet vanishing into a pentagonal well. (b)removing the facet leaves a far field outside the dotted circle, which mustbe reconnected. (c) the unique reconnection shown inside the dotted circle.Arrows indicate gradients.
enough to neglect and quickly corrected1. However, because the faceted-surface network
represents a piecewise-planar geometrical surface, we are not free to choose randomly.
Since each facet in the far field has a normal and a local height, neighbor relationships
determine junction locations and thus edge placement. However, the final reconnection
must be geometrically consistent – all facets must be simply connected, and thus no edges
may intersect. If we were to randomly choose our neighbor relationships, the resulting
reconnection would likely fail this test, and would thus represent a non-physical “surface.”
To guarantee a geometrically consistent reconnection, we must search through all virtual
reconnections until we find one that does not result in any self-intersecting facets.
Several questions immediately arise:
1: How can we effectively characterize a “reconnection”?
2: How many virtual reconnections are there to search?
3: How can we efficiently list all these choices?
4: Can we be sure a good reconnection exists?
1See, however, [102, 103], where the effects of this random choice in soap froths is investigated and foundto be significant. A deterministic method of re-connection is proposed, based on the assumption that acell loses sides as it shrinks until it has only three.
107
5: Is this reconnection unique?
For our method to be effective, all but the last of these questions must be answered
satisfactorily. The detailed answers to (1-3) are found in the appendix, but we will
summarize them here. The edges and junctions created during an O(n) reconnection may
be effectively characterized as a binary tree with n − 2 nodes. The number of m−noded
binary trees is given by the Catalan number Cm = 2n!n!(n+1)!
. Finally, these trees may be
efficiently listed using a greedy recursive algorithm in O(Cm) time. For the fourth question
regarding existence, we argue heuristically that a facet reaching zero area proves the
existence of its own reconnection, since a surface with a zero-area facet is functionally the
same as the surface with that facet removed. We then assume the existence of that same
reconnection for some window of time before the facet reaches zero. A fuller proof would
appeal to manifold theory. Finally, the fifth question regarding uniqueness is addressed
in Section B.2.
Having established these facts, we have a robust method for reconnecting an arbitrary
far field of facets. Before considering some special cases of this method, let us summarize
the general process so far: Whenever facets smaller than a threshold area are detected,
we:
a: remove them, leaving a hole in the mesh.
b: list all virtual reconnections (VR’s) as n-node binary trees.
c: use associated neighbor relationships to find edge locations.
d: test each VR until one with no intersecting edges is found.
108
We note that this approach represents a comprehensive reconnection method for any
cellular network problem. Though it is necessary for the faceted surface problem, it may
be useful in any situation where a verifiably optimal reconnection is sought.
4.3.3.1. Special Case: Facets Disappearing in Groups. It is possible for groups of
facets to shrink together, in such a way that they cannot be removed sequentially. For an
example, consider the configurations in Figure 4.3.3.1. In such a case, it is necessary to
identify and collect a contiguous group of small facets for simultaneous deletion – we call
this a near field. Any facet neighboring the near field is assigned to the far field, which
may be reconnected as described previously after the near field is deleted.
Figure 4.12. Example of a group of disappearing facets. Reconstructionshown in dotted lines.
To gather the near-field facets, we maintain a second, more liberal threshold. When-
ever a face shrinks below the first threshold, as described above, its neighbors are re-
cursively examined to collect those smaller than the second threshold. This method is
rather simplistic, and, in cases of oddly-shaped pyramids, may not return the entire near
field. This, in turn, will result in an incorrect far field, which will most likely be non-
reconnectable. However, a group of facets vanishing together eventually all head to zero
area, and for some window of time before they would physically vanish, all are small
109
enough to be detected in this way. Thus, we allow the code to “skip over” small facet
combinations that it cannot remove successfully, and try again during a future timestep.
4.3.3.2. Special Case: Facets Disappearing as Steps. It is also possible, on high-
symmetry crystal surfaces, that the small facet or group of facets forms a “step” between
two much larger facets of identical orientation, but different height. Figure 4.3.3.2 illus-
trates this situation, in which the near field is bounded by exactly four facets, two of
which have identical orientations. In such a case, the final fate of the surface is that the
small facets vanish as the large facets join together. The method described above contains
no provision for joining far-field facets during reconnection, and so there is no way to
reconnect the far field produced in this case.
Figure 4.13. Example of a step removal. Left: A chain of small facetsseparates two large facets of identical orientation. Right: The small facetshave been removed, and the large facets joined.
Having identified a near field as forming a step, one solution is to delete the small
facets, then move the two large faces to the same height and join them. This results in two
pairs of unconnected edges, which are each deleted and replaced with an appropriate single
edge. Since facet groups forming steps are, in fact, bordered by four facets generically,
a separate Repairman class could be written to handle this case. However, the small
110
adjustment to the positions of the large facets can lead to subtle problems, as will be seen
in Section 4.4. Therefore, a more robust if less elegant approach is to simply add one of
the large parallel facets to the (step-forming) near field; a good choice is the one with
fewer edges. Since the far field surrounding this modified near field requires no joins, it
can be repaired using the FFR method. Again, failures are possible as described in the
above section, but resolution is always possible near enough to the time the event would
physically occur.
4.4. Discretization and Performance of Topological Events
We have now discussed the general kinematics of a PADS, and surveyed all topo-
logical events which may occur as the surface evolves. Before our treatment is complete,
however, we must consider with care the application of a time-stepping scheme. The
accurate performance of topological events under such a scheme is problematic because,
while events on a continuously evolving surface happen at precise times (Ei at ti), any
time-stepping method invariably skips over these times. This has three consequences,
concerning detection, consistency, and accuracy. After discussing them briefly, we will
present three possible timestepping methods which illustrate them in more detail.
Detection. Because timestepping will always skip over moments of topological
change, we must abandon hope of simply finding topological events ready to perform.
Instead, we must either look ahead before each timestep and anticipate when events will
occur (a predictive method); or step before looking, and then by examining the network
infer where events should have occurred (a corrective method). Class A events can be
111
easily be detected either way, while class B events are easier to correct, and class C are
easier to predict.
Consistency. Once the occurrence of an event has been detected by either means,
it must be performed in a way that preserves geometric consistency – i.e., the network
always corresponds to a physical surface s = h(x). For example, two joining facets can
only be mechanically fused if they exhibit the same local height. If, in addition to the
occurrence of an event, a detection scheme can determine the exact time at which it
occurred, then one strategy is to move the network to the precise event time, at which
resolution is trivial. However, one may wish to attempt resolutions at other times, and
the geometric consequences of doing so must be weighed.
Accuracy Finally, we must consider the possibility of error that is produced during
topological change. This error is most easily understood if we view the evolving surface
in its abstract form as a highly nonlinear system of ODE’s. The (usually configurational)
evolution function is moderated by the topological state; thus, topological events can
represent sudden, qualitative changes in the evolution function. A naive time-stepping
scheme which steps over these without appropriate measures will produce large localized
errors at moments of topological change.
4.4.1. Method 1. Predict Events, Travel Exactly to Each Event
Assume that, at all times, we accurately predict the time and location of the next topo-
logical event2. Then, a straightforward timestepping strategy which avoids consistency
2An example of this approach may be found in the early soap froth simulations of [92], where edges andcells shrinking to zero are anticipated. A similar predictive approach could be developed for facets whichbecome non-simply connected, by anticipating the possibility of junctions crossing edges.
112
and accuracy concerns is to continually calculate the time of the next topological event
(accurate to the the order of the time-stepping method), and then step from event to
event. Under this approach, time is divided into slices with constant equations of motion,
guaranteeing that that the system always evolves under the correct equations, and accu-
rately representing the continuously evolving surface. In addition, high-order single-step
methods such as Runga-Kutta methods may be used to obtain high accuracy.
Though neatly eliminating consistency and accuracy concerns, this method has a
serious disadvantage. The frequency of topological events scales with the system size,
and since we can never step farther than the next event, we effectively make the timestep
dependent on system size – ∆t ∼ O(N−1). Since moving the system through a single
timestep is itself an O(N) operation, then advancing the system through any O(1) period
of time takes O(N2) time. While acceptable for the detailed study of a small surface, it
is obviously undesirable for the statistical study of large surfaces. This is chiefly because,
consistency concerns aside, it makes little sense to halt the entire surface at every single
topological event, when each of these involves only a few facets. Thus, our next method
has as its chief objective the use of timesteps which are independent of system size.
4.4.2. Method 2. Use Fixed timestep – Late Correction of Observed Events
A second strategy is to take fixed timesteps, use a corrective method of topological detec-
tion, and attempt to perform topological corrections late. Since timestep is independent
of system size, many events will now occur per timestep, the size of which is chosen
to produce a fixed small percentage of facets undergoing topological change each step.
While this approach theoretically eliminates the O(N2) contribution to running time, it
113
introduces hurdles to event detection, as well as geometrically consistent and accurate
resolution.
Detection. We just stated that, in this corrective detection scheme, more than one
event occurs per timestep. Whether or not this is a problem depends on the Domain of
Influence of each event, defined to be the set of network elements that event affects. If
these sets contain no common elements, then the associated events occur too far apart in
space to affect each other – they are independent. Consequently, a detection routine can
hand them in arbitrary order to the repair routines, there to be confidently performed
in isolation. However, occasionally two or more domains of influence overlap. In this
situation, called a Discrete Compound Event, the associated topological events are
no longer independent, and a detection routine can no longer ensure a priori their correct,
consistent resolution when handed off. Even worse, the very signatures used to identify
separate events may be obscured in the resulting “tangle,” such that the routine does not
even recognize what has happened. Given the variety of event signatures described in
Section 4.3, and the many combinations in which they might occur, creating a complete
list of all DCE’s would be prohibitive if not impossible. Instead, we reason that, on a
random surface, no two events will ever occur at exactly the same moment (It is possible to
artificially construct faceted surfaces such two or more events must occur simultaneously
– we do not consider this case). Thus, if we simply refine our timestep when necessary,
formerly overlapping events can be sorted out, and detected in sequence. A robust strategy
for handling compound events is thus to (a) retrace the problematic timestep, (b) refine
it into smaller slices, and (c) repeat steps (a) and (b) recursively, until only single events
are detected.
114
Consistency. Since the surface is allowed to evolve unrepaired past numerous topo-
logical events per timestep, surface regions near these events will be geometrically in-
consistent after the step. To say the same thing, facets involved in the bypassed events
will have incorrect neighbor relationships. However, we have already classified all possible
events, so having identified which event occurred, and which facets were involved, we know
a priori what the correct neighbor relationships should be after the event. This knowl-
edge, along with knowledge of the position of each facet involved, allows us to reconstruct
the consistent surface that should have emerged during the event3.
Unfortunately, not all events can be consistently corrected at a late time in this way.
In particular, Facet Joins and Joining Facet Pinches involve the joining of two facets
that meet each other at a single local height. Since this condition exists for only a
single instant, such events cannot be performed in a geometrically consistent way at any
time other than the “correct” one. To accommodate this requirement while preserving
a topology-independent timestep, we are forced to manually adjust the height of the
joining facets before the event is performed. Besides the error induced by this strategy
(discussed next), this need illustrates a second problem that can arise. In a Repair-
Induced Inconsistency, the very act of performing one event, because it is done late,
triggers a second event that was not detected originally. An example is when the just-
described height adjustment required for the delayed repair of a facet join triggers, say,
a neighbor-switching event. Since this newly-triggered event was not originally detected,
the system is left in an inconsistent state after all repairs are made. An ad-hoc strategy
3This strategy is similar to the Far-Field Reconnection algorithm described above, except that exceptthat the correct neighbor relationships are already known. However, FFR is a general algorithm forfinding correct relationships between neighbors. Thus, many of the above topological events describedabove may be performed “lazily,” by simply identifying the involved facets, and applying FFR.
115
to find such RII’s is infeasible for the same reason as is a complete listing of all possible
Discrete Compound Events (indeed, an RII may produce a DCE, which rules out a simple
multiple-rechecking strategy). Thus, a similar retrace/refine/repeat strategy is required,
with the added requirement that all events performed prior to detecting the RII must first
be undone.
Accuracy. Finally, as alluded above, repairing topological events after they occur
can introduce large isolated errors. This can be due to the “fudging” required for the
delayed repair of facet joins and swaps, but more generally is caused by facets involved
in (uncorrected) topological events having been evolved under the wrong equations of
motion for part of the relevant timestep. Consider an event Ei : tj < ti < tj+1, with
domain of influence Di. Since topological events likely correspond to a change in the
surface’s evolution equation, the facets in Di are moved using the wrong equations for
the time interval [ti, tj+1]. Since the equations guiding Di are wrong by as much as O(1)
for a time of order O(∆t), facets in Di may accumulate O(∆t) location errors during
the timestep in which the event occurs. Since the quantity of topological events does
not depend on ∆t, the method retains first order accuracy globally. However, this error
introduces a barrier to achieving higher-order accuracy later on.
4.4.3. Method 3. Localized Adaptive Replay
The previous method, alas, contains one subtle problem that keeps it from being a true
O(N) method. This problem is that the frequency of DCEs and RIIs, though small,
still scales with the system size, and these necessitate timestep refinement. So although
the late method does not have to explicitly step according to the O(1/N) time between
116
topological events, yet to accurately detect and resolve those events it is still implicitly
driven by the refinement strategy to step along a time associated with DCEs and RIIs.
While this characteristic time is longer than that between individual topological events,
and does not greatly slow the simulation of tens of thousands of facets, it still results in
a method that is formally O(N2), which becomes prohibitive when considering systems
of millions of facets. Thus, we now sketch a third method, not yet implemented, which
eliminates this effect all together. In addition, the method allows us to perform topological
events in a way which confers all the accuracy benefits of the first, predictive method.
We first re-state that, on any given timestep, most facets are not involved in any
topological changes. While it was therefore obviously wasteful to move the entire sur-
face from event to event in the first method, it is also conceptually wasteful to perform
a global retrace/refine/repeat step to DCEs and RIIs in the second method. Instead,
after every timestep, we should identify for each DCE/RII the Topological Subdomain
containing all facets involved in the event. The few facets within these subdomains would
be retraced/refined/repeated as required, while the rest of the (unaffected) facets would
be left undisturbed in their post-timestep state. Since operating on a given, constant
number of facets takes O(1) time, and since the number of events per timestep scales only
like O(N), we see that a single timestep and all associated corrections – including DCEs
and RIIs – can now be performed in O(N) time, with a final state that is guaranteed to
be consistent. This produces a true O(N) method. In addition, this “Localized Replay”
strategy has an accuracy benefit. Regular, recognized topological events also have easily
identifiable topological subdomains. If the facets within these domains are retraced, then
117
the predictive detection mechanism of the first method can be applied within the domain
to eliminate consistency and accuracy problems associated with late removal.
One difficulty remains, however. Facets involved in topological events may, under
configurational facet-velocity laws, exhibit abrupt changes in velocity a result of the event.
During the remaining segment of timestep, these facets may “break out” of the subdomain
initially created to contain them, and begin interacting with facets outside of it. Thus,
we would need a mechanism to detect this, and start over with a larger subdomain if it
occurs. Finally, if subdomains can change size, then there is the possibility that two nearby
subdomains will come to overlap as the algorithm progresses. Therefore, we must include
the ability to merge them if necessary, start over with the new, larger sub-domain, and
repeat adaptively until everything can be sorted. This adaptivity ensures the robustness
of the method, as highlighted by the method’s formal name of Adaptive Localized
Replay. The reader may note that the pattern of adaptive repetition is similar to that
used to resolve DCEs and RIIs above, and worry that another, even smaller O(N2) effect
lurks in the shadows. However, in both of the previous methods, such effects were due to
the global response to a local problem. Since this latter method is designed to be localized,
there is no longer any mechanism to generate such effects.
4.5. Demonstration and Discussion
We demonstrate our method using the sample dynamics derived in Chapter 1, associ-
ated with the directional solidification of a strongly anisotropic dilute binary alloy. When
a sample is solidified at a pulling velocity which is greater than some critical value, solute
gradients caused by solute rejection at the interface create a solute gradient which opposes
118
and overcomes the thermal gradient, resulting in a negative effective thermal gradient. In
this environment, facets move away from the freezing isotherm at a rate proportional to
their mean distance from the isotherm, as given by the dynamics
(4.2) Vi = 〈h〉i .
In Figures 4.5, 4.15, and 4.16, this dynamics is applied to surfaces with common
three-, four-, and six-fold symmetries to illustrate the flexibility of our topology-handling
approach. A series of snapshots from the coarsening surface are presented, in which
surface configurations near to many of the topological events described above may be
observed. (However, neither the Irregular Neighbor Switch, Irregular Facet Pierce, nor
Facet Pinch occur because no three facet normals are coplanar in these symmetries; indeed,
these events are not expected to occur on most physical surfaces, and were included for
theoretical completeness.)
119
Figure 4.14. A top-down view of a small coarsening faceted surface withthreefold symmetry. Spatial scale is constant, but irrelevant; time increasesdown the column.
120
Figure 4.15. A top-down view of a small coarsening faceted surface withfourfold symmetry. Spatial scale is constant, but irrelevant; time increasesdown the column.
121
Figure 4.16. A top-down view of a small coarsening faceted surface withsixfold symmetry. Spatial scale is constant, but irrelevant; time increasesdown the column.
122
About half of computational time is spent looking for topological changes, which is
significant but not prohibitive. With appropriate timestep choice, using even the still-
inefficient timestepping method 2 above, a surface of 25, 000 facets may be simulated to a
99 percent coarsened state in about an hour on currently available workstations. With the
implementation of method 3 above, this time should be cut in half, and since method 3
is truly O(N), a single million-facet simulation should take about a day. Looking further
ahead, since facet velocity calculations and topology checks require only local information,
the method should be easily parallelizable, making possible even larger speed gains.
4.6. Conclusions
We have presented a complete method for the simulation of fully-faceted interfaces
of a single bulk crystal, with arbitary symmetry, where an effective facet velocity law
is known. The surface, which is reminiscent of two-dimensional cellular networks, is
encoded numerically in a geometric three-component structure consisting of facets, edges,
and junctions, and the neighbor relationships between them. Consistent surface evolution
specified by the facet velocity law is accomplished via a simple relationship between facet
motion and junction motion. Although requiring the explicit handling of topological
events, the method is efficient, using the natural structure of the surface, and accessible,
allowing easy extraction of geometrical data. This combination makes it ideal for the
statistical study of extremely large surfaces necessary for the investigation of dynamic
scaling phenomena.
A comprehensive listing of all topological events has been presented. These allow
single-crystal surface with arbitrary symmetry (or no symmetry at all) to be simulated.
123
Events are classified into three categories, representing three ways that surface elements
can become geometrically inconsistent. These are: edges which approach zero length,
facets which become constricted, and facets which approach zero area. Resolution strate-
gies for the former two classes can be determined a priori, while repairing surface “holes”
left by vanishing facets requires a novel Far-Field Reconnection algorithm, which itera-
tively searches through all virtual reconnections to find one which produces a consistent
surface. Finally, intrinsic non-uniqueness of several events is discussed; since ours is a
purely kinematic method, decisions regarding resolution of these events must be made
ahead of time through consideration of the dynamics or other physics.
In addition, a detailed discussion of the issues associated with a discrete time-stepping
scheme has been presented. The core issue is that topological events, which occur at
discrete times throughout surface evolution, invariably fall between timesteps, with con-
sequences for the detection of events, as well as their geometrically consistent and numer-
ically accurate resolution. Since topological change corresponds (under configurational
facet velocity laws at least) to qualitative changes in the local evolution function, some
way to reach these in-between times must be introduced, while recognizing that only a
few facets are involved in topological change during each timestep. A comparison of three
approaches showed that the optimal solution is one of Localized Adaptive Replay, where
large timesteps are taken to improve speed, but local surface subdomains associated with
topological change are reverted, and then replayed in a way that re-visits events with the
necessary precision as necessary. While further work remains to implement this approach,
the method as presented is capable of comparing million-facet datasets via averaged runs.
124
CHAPTER 5
A Mean-Field Theory for Coarsening Faceted Surfaces
5.1. Introduction
In many examples of faceted surface evolution, a facet velocity law giving the normal
velocity of each facet can be observed, assumed, or derived. Examples of such dynamic
laws describe growth of polycrystalline diamond films from the vapor [74, 73], evolution
of faceted boundaries between two elastic solids [75], the evaporation/condensation mech-
anism of thermal annealing [13], and various solidification systems [71, 48, 76, 62]. Such
velocity laws are typically configurational, depending on surface properties of the facet
such as area, perimeter, orientation, or position, and reduce the computational complex-
ity of evolving a continuous surface to the level of a finite-dimensional system of ordinary
differential equations. This theoretical simplification enables and invites large numerical
simulations for the study of statistical behavior. This has been done frequently for one-
dimensional surfaces [71, 48, 76, 62, 77, 100, 79, 80, 81], while less frequently for
two-dimensional surfaces due to the necessity of handling complicated topological events
[13, 14, 15, 16, 17]. Such inquiries reveal that many of the systems listed above exhibit
coarsening – the continual vanishing of small facets and the increase in the average length
of those that remain. Notably, these systems also display dynamic scaling, in which com-
mon geometric surface properties approach a constant statistical state, which is preserved
even as the length scale increases.
125
The dynamical scaling behavior of coarsening faceted surfaces recalls the process of
Ostwald ripening [104], in which small solid-phase grains in a liquid matrix dissolve, while
larger grains accrete the resulting solute and grow. As this proceeds, the distribution of
relative particle sizes tends to a constant state1. In fact, general 2D faceted surfaces
fall conceptually into the same class of phase-ordering systems, except that the a vector
order parameter reflecting surface normal replaces the scalar order parameter reflecting
phase [105]; other systems exhibiting similar behavior include coarsening cellular networks
describing soap froths and polycrystalline films [87, 88, 89, 90, 91], and films growing
via spiral defect [82]. Each of these generalizations is characterized by a network of
evolving boundaries which separate domains of possibly differing composition, and exhibit
coarsening and convergence toward scale-invariant steady states.
Since dynamic scaling pushes complex systems into a state which can be effectively
characterized by just a few statistics, it is natural to seek simplified models which repli-
cate this behavior. The canonical example of this approach is the celebrated theory of
Lifshitz, Slyozov, and Wagner describing Ostwald ripening [18, 19, 20]. Generically,
such an approach selects a distribution of some quantity, and includes just enough of the
total system behavior to specify the effective behavior of that quantity – for example, the
original LSW theory first identifies the average behavior of particles as a function of size,
and uses that result to identify a continuity equation describing distribution evolution.
Ideas of this kind have been applied to several of the higher-order cellular systems intro-
duced above, notably froths [106, 107] and spiral-growth films [82]. To the extent that
1Indeed, it was observed some time ago that facets of alternating orientations on a one-dimensionalsurface are analogous to alternating phases of a separating two-phase alloy [4, 5], and the Cahn-Hilliardequation [6] which models phase separation has been used, in modified form, to describe several differentkinds of faceted surface evolution [8, 9].
126
such approaches mirror experimental data, they can yield valuable physical insight which
cannot be gained by considering single particles, nor even by direct numerical simulation
of larger ensembles. However, to date no similar attempt has been made for evolving
faceted interfaces.
Given the wide variety of examples of purely faceted motion, the membership of this
problem class in the wider class of phase ordering systems, and the past success in apply-
ing mean-field analyses to these systems, it is somewhat surprising that no such attempt
has been made to describe the mean-field evolution of faceted surfaces. In this chapter,
therefore, we take a first step in that direction by introducing a framework for describing
the distribution of facet lengths in 1D faceted surface evolution. Our approach closely
resembles the LSW theory of Ostwald ripening, in that a facet-velocity law allows the
effective behavior of facets by length, and thus the specification of a continuity equation
governing the evolution of the length distribution. However, our model differs in that
facets do not vanish in isolation as do grains in Ostwald ripening – instead each vanishing
facet causes its two immediate neighbors to join together. This process of merging is not
treated in LSW theory, and requires the introduction of a convolution integral reminis-
cent of equations due to Smoluchowski [21] and Schumann [22] describing coagulation.
We apply our method to one particular facet dynamics, associated with the directional
solidification of faceting binary alloys. However, the method is general and can be applied
to any dynamics where effective facet behavior is accessible.
127
5.2. Example Dynamics and Problem Formulation
To exhibit our method, we consider the dynamics derived in Chapter 1. During the
directional solidification of a strongly anisotropic binary alloy, small-wavelength faceted
surfaces develop. If the alloy is solidified above a critical velocity, a layer of supercooled
liquid is created at the interface, which drives a coarsening instability governed by the
facet dynamics
(5.1)
[
dh
dt
]
i
= 〈h〉i .
Figure 5.1a displays representative surface evolution during coarsening. There, the loca-
tions of corners are plotted over time. We see that this system exhibits binary coarsening,
whereby a single facet shrinks to zero length, causing two corners meet and annihilate. As
coarsening proceeds, the average facet length increases (Figure 5.1b), and a scale-invariant
distribution of relative facet lengths is reached (Figure 5.1c).
To describe the evolution of a scale-invariant length distribution such as that shown
in Figure 5.1c, we will derive equations governing the evolution of the facet distribution
ρ(x), where the value of ρ at x represents the number of facets with length L = x. This
distribution can be used to obtain the total number of facets N(t), the average facet
length L(t), and the (constant) total surface area A, via the relations
N =
∫ ∞
0
ρ(x)dx(5.2)
A =
∫ ∞
0
xρ(x)dx(5.3)
L = A/N.(5.4)
128
82 84 86 88 90 92 94
2
4
6
8
x
t(a)
0 1 2 3 40
0.5
1
1.5
2
2.5
3
t
log
<L>
(b)
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
L / <L>
P
(c)
Figure 5.1. Survey of coarsening behavior. (a) A representative example ofthe kink/anti-kink evolution (red/blue). (b) Facet lengthscale growth withtime. (c) Scale-invariant distribution of relative facet lengths (the tail isgaussian).
Additionally, we will refer in what follows to the normalized probability distribution of
facet lengths P (x), and the probability distribution of relative facet lengths P (x), given
respectively by
P (x) = P (x)/N(5.5)
P (x) = LP (x/L),(5.6)
with x = x/L.
129
To simplify the calculations, we assume that all facets have slopes of ±1, and neigh-
boring facet lengths are uncorrelated. Additionally, in what follows, it will be helpful to
consider Figure 5.2, which shows a representative facet F , and its two neighbors F (−) and
F (+).
L L(+)L(−)
h + L/2
h − L/2
h + L/2 − L(−)
h − L/2 + L(+)
Figure 5.2. A diagram illustrating a representative facet and its two neigh-bors. Here h is an arbitrary reference height that occurs at the midpoint ofthe center facet.
5.2.1. Flux Law
We begin by observing that, because all slopes are fixed with alternating values ±1, then
the rate of length change for any single facet is independent of its own vertical velocity,
and is instead completely determined by the vertical velocity of its immediate neighbors.
This general, geometric property can be obtained by inspection of Figure 5.2, and may
be written
(5.7)dLi
dt=
1
2(−1)i
[
dhi+1
dt− dhi−1
dt
]
.
130
where odd (even) facets have negative (positive) slopes. Assume for now that a consid-
eration of the facet dynamics (5.1), the existing facet distribution ρ, and Equation (5.7)
allow the derivation of an effective mean facet behavior
(5.8) φ(L) =
⟨
dL
dt
⟩
(L),
which gives the average rate of length change as a function of facet length. Then ρφ gives
the total flux of facets in the space of facet lengths parameterized by x. This allows us to
use the divergence theorem to write a simple continuity equation for ρ(x), as follows:
(5.9)∂ρ
∂t+
∂
∂x[ρ(x)φ(x)] = 0.
5.2.2. Coarsening Terms
Since equation (5.9) aims to describe a coarsening faceted surface, it must accurately
address the primary feature of coarsening – the shrinking and vanishing of facets over
time. To see if it does, we consider Figure 5.2 again, and imagine that the facet F goes
to length 0. We see that three things occur: first, the facet F itself vanishes; second, the
neighbors of F also vanish; and third, a new facet is created which is the merging of the
neighbors of F . If these processes are not captured by Eqn. (5.9), then we must add
terms to it so that it does.
The first process of facet vanishing is indeed captured by the continuity equation
(5.9). In our framework, vanishing facets shrink to zero length and flow through the
domain boundary at x = 0. Equation (5.9) naturally exhibits this behavior, and allows
the easy extraction of the rate R at which coarsening occurs. This is simply the flux at
131
the origin, given by
(5.10) R = −ρ(0)φ(0).
The second process, neighbor loss, is not captured by (5.9). To include it, we model it
as a sink S, which eliminates facets in a way that is probabilistically accurate. Recalling
the assumption that adjacent facet lengths are not correlated, we can assume that Li−1
and Li+1 are each described by the (normalized) distribution P (x). Thus, the appropriate
sink is
(5.11) S(x) = −2P (x).
The final process of facet creation is also not captured by (5.9). We model it, in
contrast to neighbor loss, as a source Ψ, which creates facets of length Li−1 + Li+1. Since
these variables are independent, and each described by P , then the sum Li−1 + Li+1 is
described by the joint probability function
(5.12) Ψ(x) = P2(x) =
∫ x
0
P (s)P (x− s) ds,
obtained by integrating a two-point (probability) distribution P (x)P (y) along lines of
constant x + y.
The net modifications required by coarsening may now be summed into a single term
(5.13) C(x) ≡ S(x) + Ψ(x) = −2P (x) +
∫ x
0
P (s)P (x − s) ds.
132
Since C describes the additional effects of facets coarsening (i.e. leaving the domain), it
must be multiplied by R, the rate at which this occurs, and then added to Eqn. (5.9).
This gives the result
(5.14)∂ρ
∂t= − ∂
∂x(ρφ) + RC(x).
5.2.3. Assuming a Scaling Viewpoint
Equation (5.14), with C defined as in (5.13), now correctly describes the distribution, by
length, of facets on a coarsening surface. In particular, it models the two primary features
of coarsening – the decrease in the number N of facets over time, and the corresponding
increase in the average length L of those that remain. Indeed, by performing the same
integrals used to obtain N and L above to the entire equation (5.14), we can estabilish
that
∂N
∂t= −2R(5.15a)
∂L
∂t=
∫ ∞
0
Pφdx.(5.15b)
These rates of change in N and L accurately describe the relative change in the number of
facets and average length scale of any initial faceted surface. However, the ultimate aim
of this chapter is to obtain a description of the (normalized) distribution of relative facet
lengths P (x), which reaches a steady state during dynamic scaling. Since P is defined in
terms of P , which in turn is defined in terms of ρ, and since since we know that ρ evolves
133
according to Equation (5.14), we can use several iterations of the chain rule to show that
(5.16)∂P
∂t= − ∂
∂x
[
P (x)φ(x)]
+ R[
∫ ∞
0
P (s)P (x − s)ds + 2(P + x∂P
∂x)
]
,
where φ(x) = φ(Lx)/L, and R = −P (0)φ(0).
5.3. Application: Our chosen facet dynamics
The framework just derived was completely general, describing the coarsening of any
faceted surface, and indeed any binary system exhibiting binary coarsening. We here apply
that general framework to the specific facet dynamics (5.1), by deriving the appropriate
effective flux function φ(x). Using the dynamics itself, the general kinematic form of φ
given in Equation (5.7), and recalling the diagram in Figure 5.2, we perform the following
calculation:
dL
dt=
1
2
(
dh(−)
dt− dh(+)
dt
)
(5.17a)
=1
2
(⟨
h(−)⟩
−⟨
h(+)⟩)
(5.17b)
=1
2
(
1
2[2h + L − L(−)] − 1
2[2h − L + L(+)]
)
(5.17c)
=1
2
(
L − 1
2[L(−) + L(+)]
)
(5.17d)
=1
2
(
L − 1
2
∫ ∞
0
xP2(x) dx
)
(5.17e)
φ(x) =1
2(x − L(t))(5.17f)
φ(hatx) =1
2(x − 1) .(5.17g)
134
In step (5.17e), since both L(−) and L(+) obey the probability distribution P , the sum L(−)
and L(+) is again modelled by the joint probability function P2. The effective contribution
to φ is then obtained by performing a weighted integral of possible sums x multiplied by
their relative prevelance P2. Since the total mass of P2 equals unity, that integration
describes the center of mass of P2, which by considering the form of Eqn. (5.12) can be
shown to equal 2L(t). This result, in turn, informs the final nondimensionalized result in
step (5.17g). We see from this that facets smaller than the average shrink, while facets
larger than the average grow. Broadly speaking, this is how coarsening works, so our
approximation at least has the right form.
Having calculated φ, we now write the complete evolution equation for the distribution
of relative facet lengths under the specific facet dynamics (5.1). Dropping all hats, we
have
(5.18)∂P
∂t= −1
2
∂
∂x[(x − 1)P ] + R
[∫ x
0
P (s)P (x− s) ds + 2∂
∂x(xP (x))
]
.
5.4. Solution and Comparison with Numerical-Experimental Data
Solution of Equation (5.18) is currently performed numerically, by letting arbitrary
initial conditions relax to a steady state; our numerical method is given in Appendix C.1.
This state is unsurprisingly independent of the initial condition chosen, but surprisingly
simple in form – it is simply the exponential distribution P (x) = exp(−x), which is
easily shown (after the fact) to satisfy Equation (5.18). We now proceed to compare the
characteristics of this predicted steady state with those of the actual steady state found
by direct simulation of the dynamics (5.1); our main results are shown in Figure 5.3.
135
We begin in Figure 5.3a by comparing the distribution P itself. The predicted expo-
nential distribution is shown in blue; comparison with the green actual distribution reveals
qualitative but not quantitative agreement. In particular, while the tail of the predicted
distribution is (obviously) exponential, the tail of the actual distribution is gaussian. As a
consequence, our mean-field steady state exhibits far too great an incidence of extremely
long facets.
Seeking the cause of this discrepancy, we next test the accuracy of our effective flux
function φ(x). Figure 5.3b shows the contour plot of the distribution of length/velocity
pairs ρ(x, φ). Finding the mean velocity for each length gives the statistical φ (dashes),
which turns out to compare favorably with the predicted φ (solid). Both are linear with
form φ = α(x − 1), and while the actual slope of 0.39 differs from the predicted slope
of 0.5, they can be made to agree by simply scaling time, since every term in Equation
(5.18) contains either φ or φ(0). So this approximation seems to be valid.
Finally, we examine the coarsening terms: the sink S(x) in Figure 5.3c and the source
Ψ(x) in Figure 5.3d. These are functionals of P , and so we are not surprised that the
predicted values (blue) are different from the values calculated from the actual steady state
(green). However, for both S and Ψ, we assumed that neighboring facet lengths were
uncorrelated. If we instead calculate statistically the S and Ψ generated by vanishing
facets (that is, the neighbors of vanishing facets), we get the curves in red, which are
different not only from the predicted quantities, but also from the quantites we would
have gotten from using the actual distribution P and assuming no correlations.
This suggests that the ultimate culprit is the assumption that neighboring facet lengths
are uncorrelated. Going back to the simulation, we now measure the correlation of the
136
lengths of nearby facets as a function of neighbor distance, in Figure 5.4. There we see
a small but significant correlation for at least the first two neighbors. This produces
the discrepancy between the green and red curves in Figures 5.3c,d above, and may be
responsible for the discrepancy in the tail as well. This result is not surprising, as the
main weakness of the original LSW theory which inspires our approach was also a failure
to address correlations; later generalizations which corrected this deficiency agreed well
with experimental data [108].
5.5. Conclusions
We have presented a mean-field theory for the evolution of length distributions as-
sociated with coarsening faceted surfaces. In the spirit of LSW theory, a facet-velocity
law governing surface evolution is used to establish a characteristic length-change law;
this in turn leads to a simple continuity equation governing the evolution of the facet
length distribution ρ(x). However, because the vanishing of any facet forces the joining
of its two neighbors, this equation must be modified by the addition of appropriate terms
describing coarsening, including a convolution term recalling models of coagulation. Our
model therefore serves, apart from the direct application to facet dynamics, as a study in
the union of these two mechanisms of steady statistical behavior.
The scale-invariant distribution is tracked by studying the evolution of the normalized
probability distribution of relative facet lengths P (x), which preserves both zeroeth and
first moments. The resulting equation is solved by the exponential distribution, and
numerical simulation reveals that any initial condition converges to this solution. This
result unfortunately does not agree quantitatively with the more gaussian distribution
137
obtained by sampling a large surface simulated directly under the facet dynamics. Further
investigation reveals that the likely culprit is the assumption that neighboring facet lengths
are uncorrelated. Indeed, a similar assumption plauged the original LSW theory, and
relaxing that assumption resulted in much better agreement with experiment.
However, even with this deficiency, the model captures the essential feature of the
dynamically scaling state – an effective facet behavior law which grows large facets and
shrinks small ones, moderated by competing terms describing coarsening and continuous
change of viewpoint, which respectively redistribute probability density toward infinity
and zero, respectively. While later improvements to our model addressing neighbor cor-
relation will undoubtedly increase its predictive capabilities, these same forces will still
balance in the steady state. The model as presented thus serves as a qualitative explana-
tion of the essential features of the scaling state, as well as a guide to further reasearch
efforts.
138
0 1 2 3 4 5 6 7 80
0.2
0.4
0.6
0.8
1
L / <L>
P
(a)
L / <L>
φ
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−1
0
1
2
3
(b)
0 2 4 6
−12
−10
−8
−6
−4
−2
0
L / <L>
log|S|
(c)
0 2 4 6 8−12
−10
−8
−6
−4
−2
0
L / <L>
log|Ψ
|
(d)
Figure 5.3. (a) Comparison between the theoretically-predicted (blue) andstatistically-gathered (green) steady states. The former exhibits exponen-tial decay, while the latter is gaussian. (b) Contour of the statistical dis-tribution of length/velocity pairs ρ(x, φ). For each length, mean statisticalvelocity is plotted as dotted line, while the predicted velocity φ(x) is a solidline. (c,d) Comparison of log(−S/2), log(Ψ) as obtained by various means:from the predicted steady state (blue), from the actual steady state (green),and from measuring the neighbors of vanishing facets in the actual steadystate (red).
139
0 2 4 6 8 10 12 14 16 18 20−0.5
0
0.5
1
1.5
neighbor distance
corr
elat
ion
coef
ficie
nt
Figure 5.4. Statistically-sampled correlation of facet lengths as a functionof neighbor distance.
140
References
[1] G. Wulff, Zur frage der Ge-. schwindigkeit des wachstums und der auflsung von.krystallflchen, Z. Kristallogr. 34 (1901) 449, For a recent review, see M. Wortis,in Chemistry and Physics of Solid Surfaces, Edited by R. Vanselow and R. Howe(Springer, Berlin, 1988), vol. 7.
[2] C. Herring, Some theorems on the free energies of crystal surfaces, Physical Review82 (1) (1951) 87–93.
[3] A. Chernov, The spiral growth of crystals, Soviet Physics - Uspekhi 4 (1961) 116–148.
[4] W. Mullins, Theory of linear facet growth during thermal etching, PhilosophicalMagazine 6 (71) (1961) 1313–1341.
[5] N. Cabrera, On stability of structure of crystal surfaces, in: Symposium on Proper-ties of Surfaces, ASTM Materials Science Series – 4, American Society for Testingand Materials, American Society for Testing and Materials, 1916 Race St., Philadel-phia, PA, 1963, pp. 24–31.
[6] J. Cahn, J. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy,J. Chem Phys 28 (1958) 258.
[7] J. Stewart, N. Goldenfeld, Spinodal decomposition of a crystal surface, PhysicalReview A 46 (10) (1992) 6505–6512.
[8] F. Liu, H. Metiu, Dynamics of phase separation of crystal surfaces, Physical ReviewB 48 (9) (1993) 5808–5818.
[9] A. Golovin, S. Davis, A. Nepomnyashchy, A convective Cahn-Hilliard model for theformation of facets and corners in crystal growth, Physica D 122 (1998) 202–230.
[10] M. Siegert, M. Plischke, Slope selection and coarsening in molecular beam epitaxy,Physical Review Letters 73 (11) (1994) 1517–1522.
141
[11] M. Seigert, Coarsening dynamics of crystalline thin films, Physical Review Letters81 (25) (1998) 5481–5484.
[12] P. Smilauer, M. Rost, J. Krug, Fast coarsening in unstable epitaxy with desorption,Physical Review E 59 (6) (1999) R6263–R6266.
[13] S. Watson, S. Norris, Scaling theory and morphometrics for a coarsening multiscalesurface, via a principle of maximal dissipation, Physical Review Letters 96 (2006)176103.
[14] J. Thijssen, Simulations of polycrystalline growth in 2+1 dimensions, Physical Re-view B 51 (3) (1995) 1985–1988.
[15] S. Barrat, P. Pigeat, E. Bauer-Grosse, Three-dimensional simulation of CVD dia-mond film growth, Diamond and Related Materials 5 (1996) 276–280.
[16] G. Russo, P. Smereka, A level-set method for the evolution of faceted crystals, SIAMJournal of Scientific Computing 21 (6) (2000) 2073–2095.
[17] P. Smereka, X. Li, G. Russo, D. Srolovitz, Simulation of faceted film growth inthree dimensions: Microstructure, morphology, and texture, Acta Materialia 53(2005) 1191–1204.
[18] I. Lifshitz, V. Slezov, Kinetics of diffusive decomposition of supersaturated solidsolutions, Soviet Physics JETP 38 (1959) 331–339.
[19] I. Lifshitz, V. Slyozov, The kinetics of pecipitation from supersaturated solid solu-tions, J. Phys. Chem. Solids 19 (1961) 35–50.
[20] C. Wagner, Theorie der alterung von niederschlagen durch umlosen, Zeitschrift furElektrochemie 65 (1961) 581–591.
[21] M. von Smoluchowski, Drei vortrage uber diffusion, brownsche molekularbewegungund koagulation von kolloidteilchen, Physikalische Zeitschrift 17 (1916) 557–571.
[22] T. Schumann, Theoretical aspects of the size distribution of fog particles, QuarterlyJournal of the Royal Meteorological Society 66 (1940) 195–207.
[23] W. Mullins, R. Sekerka, Stability of a planar interface during solidification of adilute binary alloy, Journal of Applied Physics 35 (2) (1964) 444–451.
[24] R. Trivedi, Interdendritic spacing: Part II. A comparison of theory and experiment,Metallurgical Transactions A 15 (6) (1984) 977–982.
142
[25] J. Langer, Instabilities and pattern formation in crystal growth, Reviews of MoternPhysics 52 (1) (1980) 1–28.
[26] J. Rutter, B. Chalmers, A prismatic substructure formed during solidification ofmetals, Canadian Journal of Physics 31 (1953) 15.
[27] W. Tiller, J. Rutter, K. Jackson, B. Chalmers, The redistribution of solute atomsduring the solidification of metals, Acta Metallurgica 1 (1953) 428–437.
[28] D. Wollkind, L. Segel, A nonlinear stability analysis of the freezing of a dilute binaryalloy, Philos. Trans. Roy. Soc. London A 268 (1970) 351–380.
[29] G. Sivashinsky, On cellular instability in the solidification of a dilute binary alloy,Physica D 8 (1983) 243–248.
[30] K. Brattkus, S. Davis, Cellular growth near absolute stability, Physical Review B38 (16) (1988) 11452–11460.
[31] D. Riley, S. Davis, Long-wave morphological instabilities in the directional solidi-fication of a dilute binary mixture, SIAM Journal of Applied Mathematics 50 (2)(1990) 420–436, connects [29] and [30].
[32] S. Coriell, G. McFadden, Morphological stability, in: D. Hurle (Ed.), Handbook ofCrystal Growth, Vol. 1b, Elsevier Science Publishers, 1993, Ch. 12, pp. 785–857.
[33] B. Billia, R. Trivedi, Pattern formation in crystal growth, in: D. Hurle (Ed.), Hand-book of Crystal Growth, Vol. 1b, Elsevier Science Publishers, 1993, Ch. 14, pp.899–1073.
[34] G. McFadden, S. Coriell, Nonplanar interface morphologies during unidirectionalsolidification of a binary alloy, Physica D 12 (1984) 253–261.
[35] L. Ungar, M. Bennett, R. Brown, Cellular interface morphologies in directionalsolidification. III. The effects of heat transfer and solid diffusivity, Physical ReviewB 31 (9) (1985) 5923–5930.
[36] L. Ungar, R. Brown, Cellular interface morphologies in directional solidification. IV.The formation of deep cells, Physical Review B 31 (9) (1985) 5931–5940.
[37] A. Karma, Wavelength selection in directional solidification, Physical Review Let-ters 57 (1986) 858–861.
143
[38] D. Kessler, H. Levine, steady-state cellular growth during directional solidification,Physical Review A 39 (1988) 3041–3052.
[39] W. Boettinger, J.A.Warren, C. Beckermann, A. Karma, Phase-field simulation ofsolidification, Annual Review of Materials Science 32 (2002) 163–194.
[40] B. Echebarria, R. Folch, A. Karma, M. Plapp, Quantitative phase-field model ofalloy solidification, Physical Review E 70 (2004) 061604.
[41] S. Davis, Theory of Solidification, Cambridge University Press, 2001.
[42] C. Herring, Surface tension as a motivation for sintering, in: W. Kingston (Ed.),The physics of powder metallurgy, McGraw-Hill, 1951, p. Chap. 8, proceedings of asymposium held at Bayside, L. I., New York, August 24-26, 1949.
[43] J. Gibbs, On the equilibrium of heterogeneous substances, Vol. 1, Longmans, Green,and Co., 1928, Ch. 3, p. 219.
[44] S. Angenent, M. Gurtin, Multiphase thermomechanics with interfacial structure. 2.Evolution of an isothermal interface, Archive for Rational Mechanics and Analysis108 (1989) 323–391.
[45] A. Di Carlo, M. Gurtin, P. Podio-Guidugli, A regularized equation for anisotropicmotion-by-curvature, SIAM Journal of Applied Mathematics 52 (4) (1992) 1111–1119.
[46] R. Bowley, B. Caroli, C. Caroli, F. Graner, P. Nozi‘eres, B. Roulet, On directionalsolidification of a faceted crystal, Journal de Physique (France) 50 (1989) 1377–1391.
[47] B. Caroli, C. Caroli, B. Roulet, Directional solidification of a faceted crystal. II.phase dynamics of crenellated front patterns, Journal de Physique (France) 50(1989) 3075–3087.
[48] D. Shangguan, J. Hunt, Dynamical study of the pattern formation of faceted cellulararray growth, Journal of Crystal Growth 96 (1989) 856–870.
[49] S. Coriell, R. Sekerka, The effect of the anisotropy of surface tension and interfacekinetics on morphological stability, Journal of Crystal Growth 34 (1976) 157–163.
[50] G. McFadden, S. Coriell, R. Sekerka, The effect of surface tension anisotropy oncellular morphologies, Journal of Crystal Growth 91 (1988) 180–198.
144
[51] R. Hoyle, G. McFadden, S. Davis, Pattern selection with anisotropy during direc-tional solidification, Phil. Trans. Roy. Soc. Lond. A 354 (1996) 2915–2949.
[52] K. Jackson, J. Hunt, Transparent compounds that freeze like metals, Acta Metal-lurgica 13 (1965) 1212–1215.
[53] F. Melo, P. Oswald, Destabilization of a faceted smectic-A – smectic-B interface,Physical Review Letters 64 (12) (1990) 1381–1384.
[54] D. Shangguan, J. Hunt, In situ observation of faceted cellular array growth, Metal-lurgical Transactions A 22A (1991) 941–945.
[55] G. Young, S. Davis, K. Brattkus, Anisotropic interface kinetics and tilted cells inunidirectional solidification, Journal of Crystal Growth 83 (1987) 560–571.
[56] G. Merchant, S. Davis, Morphological instability in rapid directional solidification,Acta Metallurgica et Materialia 38 (1990) 2683–2693.
[57] C. Herring, The use of classical macroscopic concepts in surface energy problems, in:R. Gomer, C. Smith (Eds.), Structure and Properties of Solid Surfaces, Universityof Chicago Press, 1953, p. 5.
[58] W. Mullins, Solid surface morphologies governed by capillarity, in: W. Robertson,N. Gjostein (Eds.), Metal Surfaces: Structure, Energetics, and Kinetics, AmericanSociety for Metals, Metals Park, OH, 1963, Ch. 2, pp. 17 – 66.
[59] F. Frank, The geometrical thermodynamics of surfaces, in: W. Robertson,N. Gjostein (Eds.), Metal Surfaces: Structure, Energetics, and Kinetics, AmericanSociety for Metals, Metals Park, OH, 1963, Ch. 1, pp. 1–16.
[60] P. Voorhees, S. Coriell, R. Sekerka, G. McFadden, The effect of anisotropic crystal-melt surface tension on grain boundary groove morphology, Journal of CrystalGrowth 67 (1984) 425–440.
[61] S. J. Watson, Crystal growth, coarsening and the convective Cahn-Hilliard equation,in: P. Colli, C. Verdi, A. Visintin (Eds.), Free Boundary Problems: Theory andApplications, International Series of Numerical Mathematics, Vol. 147, Birkhaeuser,2003.
[62] S. Watson, F. Otto, B. Rubinstein, S. Davis, Coarsening dynamics of the convectiveCahn-Hilliard equation, Physica D 178 (3-4) (2003) 127–148.
[63] J. Ericksen, Equilibrium of bars, Journal of Elasticity 5 (1975) 191–202.
145
[64] S. Muller, Minimizing sequences for nonconvex functionals, phase transitions andsingular perturbations, in: Kirchgassner (Ed.), Problems involving change of type,Vol. 359 of Lecture Notes in Physics, Springer-Verlag, Berlin, 1990, Ch. 1, pp. 31–44.
[65] S. Mller, Singular perturbation as a selection criterion for periodic minimizing se-quences, Calculus of Variations 1 (1993) 169–204.
[66] L. Truskinovsky, G. Zanzotto, Ericksen’s bar revisited: Energy wiggles, J. Mech.Phys. Solids 44 (8) (1996) 1371–1408.
[67] R. Sekerka, A time-dependent theory of stability of a planar interface during dilutebinary alloy solidification, in: H. Peiser (Ed.), Crystal Growth, Pergamon, Oxford,1967, pp. 691–702.
[68] R. Sekerka, Application of the time-dependent theory of interface stability to anisothermal phase transformation, Journal of Physics and Chemistry of Solids 28(1967) 983–994.
[69] J. Villain, Continuum models of crystal growth from atomic beams with and withoutdesorption, Journal de Physique I (France) 1 (1991) 19–42.
[70] A. Roosen, J. Taylor, Modeling crystal growth in a diffusion field using fully facetedinterfaces, Journal of Computational Physics 114 (1) (1994) 113–128.
[71] L. Pfeiffer, S. Paine, G. Gilmer, W. van Saarloos, K. West, Pattern formationresulting from faceted growth in zone-melted thin films, Physical Review Letters54 (17) (1985) 1944–1947.
[72] S. Fu, I. Muller, H. Xu, The interior of the pseudoelastic hysteresis, Mater. Res.Soc. Symp. Proc. 246 (1992) 39–42.
[73] A. van der Drift, Evolutionary selection, a principle governing growth orientationin vapor-deposited layers, Philips Research Reports 22 (1967) 267–288.
[74] A. Kolmogorov, To the ”geometric selection” of crystals, Dokl. Acad. Nauk. USSR65 (1940) 681–684.
[75] M. Gurtin, P. Voorhees, On the effects of elastic stress on the motion of fully facetedinterfaces, Acta Materialia 46 (6) (1998) 2103–2112.
[76] C. Emmot, A. Bray, Coarsening dynamics of a one-dimensional driven Cahn-Hilliardsystem, Physical Review E 54 (5) (1996) 4568–4575.
146
[77] C. Wild, N. Herres, P. Koidl, Texture formation in polycrystalline diamond films,Journal of Applied Physics 68 (3) (1990) 973–978.
[78] A. Dammers, S. Radelaar, 2-dimensional computer modeling of polycrystallinegrowth, Textures and Microstructures 14 (1991) 757–762.
[79] Paritosh, D. Srolovitz, C. Battaile, J. Butler, Simulation of faceted film growth intwo dimensions: Microstructure, morphology, and texture, Acta Materialia 47 (7)(1999) 2269–2281.
[80] J. Zhang, J. Adams, FACET: a novel model of simulation and visualization ofpolycrystalline thin film growth, Modelling Simul. Mater. Sci. Eng 10 (2002) 381–401.
[81] J. Zhang, J. Adams, Modeling and visualization of polycrystalline thin film growth,Computational Materials Science 31 (3-4) (2004) 317–328.
[82] T. Schulze, R. Kohn, A geometric model for coarsening during spiral-mode growthof thin films, Physica D 132 (1999) 520–542.
[83] J. Taylor, J. Cahn, Diffuse interfaces with sharp comers and facets: Phase fieldmodels with strongly anisotropic surfaces, Physica D 112 (1998) 381–411.
[84] J. Eggleston, G. McFadden, P. Voorhees, A phase-field model for highly anisotropicinterfacial energy, Physica D 150 (1-2) (2001) 91–103.
[85] W. Carter, A. Roosen, J. Cahn, J. Taylor, Shape evolution by surface diffusion andsurface attachment limited kinetics on completely faceted surfaces, Acta Metallur-gica et Materialia 43 (12) (1995) 4309–4323.
[86] A. Roosen, W. Carter, Simulations of microstructural evolution: Anisotropic growthand coarsening, Physica A 261 (1998) 232–247.
[87] J. Stavans, The evolution of cellular structures, Reports on Progress in Physics56 (6) (1993) 733–789.
[88] V. Fradkov, D. Udler, Two-dimensional normal grain growth: Topological aspects,Advances in Physics 43 (6) (1994) 739–789.
[89] H. Frost, C. Thompson, Computer simulation of grain growth, Current Opinion inSolid State and Materials Science 1 (3) (1996) 361–368.
147
[90] C. Thompson, Grain growth and evolution of other cellular structures, Solid StatePhysics 55 (2001) 269–314.
[91] G. Schliecker, Structure and dynamics of cellular systems, Advances in Physics51 (5) (2002) 1319–1378.
[92] D. Weaire, J. Kermode, Computer simulation of a two-dimensional soap froth I.Method and motivation, Philosophical Magazine B 48 (3) (1983) 245–259.
[93] D. Weaire, J. Kermode, Computer simulation of a two-dimensional soap froth II.Analysis of results, Philosophical Magazine B 50 (3) (1984) 379–395.
[94] M. Anderson, D. Srolovitz, G. Grest, P. Sahni, Computer-simulation of grain-growth.1. Kinetics, Acta Metallurgica et Materialia 32 (5) (1984) 783–791.
[95] D. Srolovitz, M. Anderson, P. Sahni, G. Grest, Computer-simulation of grain-growth.2. Grain-size distribution, topology, and local dynamics, Acta Metallurgica et Ma-terialia 32 (5) (1984) 793–802.
[96] F. Lewis, The correlation between cell division and the shapes and sizes of prismaticcells in the epidermis of cucumis, Anatomical Record 38 (3) (1928) 341–376.
[97] D. Aboav, The arrangement of grains in a polycrystal, Metallography 3 (1970)383–390.
[98] D. Weaire, Some remarks on the arrangement of grains in a polycrystal, Metallog-raphy 7 (1974) 157–160.
[99] D. Aboav, Arrangement of cells in a net, Metallography 13 (1) (1980) 43–58.
[100] J. Thijssen, H. Knops, A. Dammers, Dynamic scaling in polycrystalline growth,Physical Review B 45 (15) (1992) 8650–8656.
[101] J. Glazier, D. Weaire, The kinetics of cellular-patterns, Journal of Physics-Condensed Matter 4 (8) (1992) 1867–1894.
[102] B. Levitan, E. Domany, Topological model of soap froth evolution with deterministicT2 processes, Europhysics Letters 32 (7) (1995) 543–548.
[103] B. Levitan, E. Domany, Dynamical features in coarsening soap froth: Topologicalapproach, International Journal of Modern Physics B 10 (28) (1996) 3765–3805.
[104] W. Ostwald, Z. Phys. Chem. 34 (1900) 495.
148
[105] A. Bray, Theory of phase-ordering kinetics, Advances in Physics 43 (1994) 357–459.
[106] C. Beenakker, Evolution of two-dimensional soap-film networks, Physical ReviewLetters 57 (19) (1986) 2454–2457.
[107] H. Flyvbjerg, Model for coarsening froths and foams, Physical Review E 47 (6)(1993) 4037–4054.
[108] M. Marder, Correlations and ostwald ripening, Physical Review A 36 (2) (1987)858–874.
[109] M. Miksis, L. Ting, Effective equations for multiphase flows – waves in a bubblyliquid, in: Advances in Applied Mechanics Vol. 28, Vol. 28 of Advances in AppliedMathematics, Academic Press, 1992, pp. 141–260.
149
APPENDIX A
Appendices for Chapter 1
A.1. Justification of the Quasi-steady state
The quasi-steady approximation is often justified by identifying a small Peclet number,
which measures the ratio of typical structural lengths to diffusional lengths. No Peclet
number is directly obtained from the non-dimensionalization performed above; however,
a typical definition of the Peclet number looks like
(A.1) P =V L
D,
where V , L, and D are characteristic velocities, lengths, and the diffusion coefficient.
With this definition, a consideration of the facet-velocity (2.50) shows that, for non-
periodic evolving faceted surfaces, V ∼ λ, L ∼ λ, and D ∼ 1. This gives an O(λ2) Peclet
number, in agreement with the original quasi-steady assumption.
However, we can more rigorously arrive at the steepest-descent form (2.46), at least
in the small-slope sense. Beginning again with Equations (2.7), but neglecting the hxCx
term, we now keep all the time derivatives, and use Laplace and Fourier transforms to
solve the equations. In Fourier-Laplace space, we obtain the following solution:
LF [ht + kh]
m(κ, s) − (1 − k)= −LF [GR](A.2)
m(κ, s) =1
2(1 +
√
1 + 4(κ2 + s))(A.3)
150
where LF indicates the Fourier-Laplace transform, GR represents the right hand side
of the Gibbs-Thompson equation, and κ and s are the Fourier and Laplace variables,
respectively.
Now, the right-hand side can be directly inverted, which simply returns GR. The
question is what to do about the left-hand side. The effect of the quasi-steady approxi-
mation Ct → 0 is to replace m(κ, s) with m(κ, 0), but we’ll attempt to directly perform
the inverse Laplace transform to see if this is justified. We begin with the convolution
theorem for Laplace transforms, which says that
(A.4) L−1 [F (s)G(s)] =
∫ t
0
f(τ)g(t− τ)dτ
and we let
(A.5) F (s) = L[kh + ht], G(s) =1
m(κ, s) − (1 − k).
Our task is thus to find the inverse Laplace transform of G(s), and examine the resulting
integral. While this expression exhibits no easily-invertible form, we can approximate
the Bromwich integration to leading order for small t, and arrive at the leading-order
approximation (in κ) of
(A.6) g(t) ≈ 1√π
exp(−κ2t)√t
.
(We want the small t behavior because of the form of Eqn. (A.4) – the values function
g(τ) with small τ multiply the values of h and ht with τ near t, which are expected to
matter the most.) Then, in the small-wavelength limit κ → ∞, we can extract the leading
151
order behavior of the convolution as
L−1F (s)G(s) ≈ [kh + ht]
∫ ∞
0
2√π
exp[−(κ2 + 1/4)t]√t
dτ(A.7a)
≈ 1
κ0[kh + ht](A.7b)
which is exactly what we would get from the the quasi-steady approximation in the same
limit (see the argument leading to Eqn. (2.26)).
A.2. Homogenized Linear Stability Analysis
In Section 2.5.1, we developed a stability criteria for small-wavelength periodic faceted
surfaces. However, that criteria was only valid for small-wavelength disturbances – namely,
perturbations to the facet heights of the periodic surface. Here, we consider the stability
of the micro-faceted solution to perturbations with wavelength much larger than the
solution wavelength. We will show that long-wavelength disturbances are categorically
less destabilizing than the small disturbances considered in the main text.
We again consider disturbed solutions of the form
(A.8) h = h0 + h, C = C0 + C,
but where [h0, C0] now include the faceted corrections derived in the text. We then insert
forms (A.8) into the original non-dimensional governing equations (2.7). The result of
this is a system of equations similar to those describing the linearization about the planar
state (2.13). However, since [h0, C0] are no longer planar, there are some additional terms
152
in the boundary conditions at z = h having variable coefficients:
Ct = Cz + Czz + Cxx for z > 0(A.9a)
C → 0 as z → ∞(A.9b)
Cz = kh − (1 − k)C + ht +[
C0xhx + h0
xCx + h0xC
0xzh
]
on z = 0(A.9c)
C = Geffh on z = 0(A.9d)
− Γ[
SX hx + SXX hxx
]
+ ν[
KX hx + KXX hxx + KXXX hxxx + KXXXX hxxxx
]
,
where S = s(hx)K, K = (Kss + K3), subscripts indicate derivatives with respect to the
proper derivative of h (i.e. SX indicates the derivative of S with respect to hx), and each
quantity is evaluated at h0. Now, for disturbances [hn, Cn] with wavelength much greater
than λ (the wavelength of [h0, C0]), we may use the technique of homogenization. In
this approach, we “average out” the effects of the (relatively) rapidly varying coefficients
above. Following [109], we define a fast-space variable ξ = x/λ (where λ is the small
wavelength of the steady solution). Then we let
[h, C] = [h, C](x, ξ, t), where ξ =x
λ(A.10)
[h, C] =
∞∑
0
λn[hn, Cn](A.11)
and collect like powers of λ. With the assumption that all φ are bounded and four-times
differentiable, the large orders λ−4 through λ−1 serve to establish that h0 through h3 are
ξ-independent, as are C0 and C1. Then we take the average of the order λ0 equations,
153
and find as a solvability condition, equations (A.9), but with all coefficients averaged over
one period.
Now, considerations of the form of h, as well as symmetries in S and K, reveal that
many of the variable coefficients vanish (h0x, C
0x, SX , KX , KXXX). Of those that remain,
〈h0xC
0xz〉 is small compared to k, 〈KXX〉 is small compared to 〈SXX〉, and 〈KXXXX〉 merely
modifies ν; this leaves SXX for our consideration. Recalling the definition of S, and
comparing Eqns. (2.13) and (A.9), we see that this term is an effective, period-averaged
surface stiffness which replaces the surface stiffness of the planar state s0. Critically, this
turns out to be positive for all values of the anisotropy coefficient α4, and thus all solution
slopes q∗. Thus, whereas the planar state had a negative surface stiffness leading to
universal instability, the nonplanar solutions under consideration have a positive effective
homogenized surface stiffness. The homogenized linear stability analysis then is identical
to the original Mullins-Sekerka stability analysis for positive surface stiffness, leading to a
Mullins-Sekerka-like region of instability similar to that in Figure 2.1. However, this region
of instability lies entirely within the supercooled region M−1 > 1, indicating that facet-
wise instability and coarsening as described in Section 2.5.3 will always occur at lower
pulling velocities than the long-wave instability discussed here. Therefore, this analysis
is needed only for completeness, and serves functionally only to rule out behaviors not
already considered in the main text.
154
APPENDIX B
Appendices to Chapter 3
B.1. Elaboration on Far-Field Reconnection
Our method of removing facets and facet groups requires patching a “hole” in the
network left by the deleted facets. This requires selecting a geometrically consistent
reconnection from a list of potential, or virtual reconnections. As outlined in the text,
this involves searching through a complete list of virtual reconnections and testing each
for geometric consistency. In this Appendix, we address in more detail questions (1-3)
posed in Section 4.3.3 regarding the details of this method. For convenience, we repeat
them here:
1: How can we effectively characterize a “reconnection”?
2: How many potential reconnections are there to search?
3: How can we efficiently list all potential choices?
We show here that an effective means of answer these questions is to think of reconnec-
tions as extended binary trees. This characterization enables us to easily count potential
reconnections, distinguish them through naming, and suggests an algorithm for efficiently
listing them for testing. An exhaustive illustration of the process is given for the case
of an O(5) far field in Figure B.1. It will be useful to refer to that diagram during the
following discussion.
155
B.1.1. Characterization: Binary Trees
Patching network holes always involves finding unknown neighbor relations between a
given number of adjacent facets – that is, no facets are ever created, only edges and
junctions. These are always connected into a single graph. In fact, the edges and junctions
created during reconnection (the “reconnection set”) form a binary tree1. In Figure B.1,
the trees associated with each possible reconnection are shown in thick blue lines. The
far-field edges touching the reconnection set shown in gray represent the completion of
this tree. That is, they take the interior tree, and add leaves to it so that every node
of the interior tree is a triple-node. Each virtual reconnection corresponds to a unique
interior tree and completed tree in this manner.
B.1.2. Enumeration: The Catalan Number
The counting of binary trees is, fortunately, a solved problem of graph theory. Given
n nodes, they may be arranged in Cn distinct binary trees, where Cn is the nth Catalan
Number ;
(B.1) Cn =2n!
n!(n + 1)!.
Now, re-connecting an O(n) far field requires the creation of n−3 edges and n−2 nodes;
this may be visually confirmed for the case n = 5 in Figure B.1. This creates an n − 2
noded binary tree, and so an O(n) far field has Cn−2 virtual reconnections to search.
1A binary tree is a graph consisting of edges and nodes such that: (1) each node connects to 1,2 or 3other nodes, (2) no cycles exist. Condition (1) is met because we consider only triple junctions, whilecondition (2) is met because the creation of new facets is excluded.
156
B.1.3. Naming
If take the completed binary tree and arbitrarily select a root node, then from each virtual
interior tree we can generate a unique sequence of letters which identify it and encode its
construction. This can be formed in one of two ways. The first way is to specify the tree
by a set of recursive function calls. Each branch point has left and right branches, each
of which may terminate in either a leaf, or another branch point. The middle column of
Figure B.1 gives such a function for each tree shown in the left column. The second way
is to walk around the tree in a counter-clockwise manner, recording each branch point or
leaf as it is encountered. Either method produces a series of letters that uniquely identify
the tree. Since the beginning ’LB’ and terminal ’LL’ are guaranteed, we may use only an
abbreviated version consisting of n − 3 of each letter.
B.1.4. Listing
The problem is now reduced to generating all possible letter combinations. We can recur-
sively build these combinations letter by letter using a greedy algorithm which chooses
’B’ over ’L’ if possible. This approach is subject to three restrictions which must be true
of a “legal” word. At each step, (a) L ≤ B + 1, (b) B ≤ n − 3, (c) L ≤ n − 3. The
function we use is sufficiently short that we simply reproduce it here:
list_trees(n) {
rec_list_trees(n, 0, 0, ’’) ;
}
rec_list_trees(n, leaves, branches, word) {
157
max = n-3 ;
if (branches < max)
rec_list_trees(n, leaves, branches+1, word+’B’) ;
if (leaves < max and leaves <= branches)
rec_list_trees(n, leaves+1, branches, word+’L’) ;
if (leaves == max and branches == max)
print(word) ;
}
Thus, using this algorithm to generate a complete list of reconnection labels, we perform
the reconnection associated with each one, and test it for geometric consistency. This
allows us to efficiently find the correct reconnection.
B.2. Non-Uniqueness
We have already mentioned that certain topological events are ambiguous in their
resolution. Here we review the instances of non-uniqueness, discuss their cause and im-
plications, and suggest a method of treating them in a numerical scheme.
B.2.1. Review. Saddles.
We begin our list with an important ambiguity not discussed in the main text. As noted by
Thijssen [14], the lowly Neighbor Switch can be non-unique if the four facets involved have
a “saddle” configuration – where the edges neighboring the vanishing edge (the emanating
edges) form an alternating sequence of two valleys and two ridges. This is illustrated in
Figure B.2.1a. There, from either starting position on the left, two topological resolutions
158
are possible. One involves changing neighbor-relations, while the other does not. This
latter resolution, which we term a “neutral pass,” represents a Vanishing Edge event
which needs no resolution. However, it still results in a (prohibited) flipped edge, so if
it is to actually occur, a bookkeeping operation must take place to correct this. Moving
on, we recall the non-unique “gap opener” flavor of Irregular Neighbor Switch, which is
not reproduced here. However, as a Vanishing Edge event, it may also come in the saddle
variety, and admits a neutral-pass resolution option which is displayed in Figure B.2.1b.
Finally, the Irregular Facet Pierce event is also not reproduced.
B.2.2. Resolution Strategy: Analytical Backtracking
None of these cases of ambiguity of resolution can be resolved at the kinematic level.
Since the particular evolution pathway leading to such ambiguities is the product of a
long chain of mathematical reasoning, we therefore look backward in this chain to resolve
them. The kinematics are most directly provided by the dynamics, which is a good first
place to start. One option is to see which resolution is more strongly self-reinforcing; if
one choice under the given dynamics immediately works to reverse itself, for example, the
other option should probably be chosen. Moving farther back the chain of reasoning, one
may consider the argument used to derive the given dynamics. For example, the dynamics
(4.2) comes from the constant reduction of an undercooling energy by moving away from
the maximally-expensive z = 0 isotherm. Therefore, an energy-informed choice is to
choose the resolution which minimizes this energy, i.e., maximizes the total integrated
distance from z = 0. Taking another step back, perhaps such arguments come, as this
one does, from a partial differential equation describing surface evolution. Such equations
159
can be simulated directly, and the resulting resolution choices studied. Ultimately, the
original physical model may have to be considered, perhaps at the atomic level. For each
dynamics we wish to study, one must apply this chain of reasoning to find the correct
ambiguity-resolution strategy. Ideally, considerations at all levels should produce identical
results.
B.2.3. Implications for Far-Field Reconnections
The non-uniqueness of particular topological events reflects a deeper, underlying prob-
lem. For each non-unique event, facet heights of competing resolutions are identical; only
neighbor relationships between these facets differ. Thus, non-uniqueness in topological
events indicates the presence of, and indeed is caused by, multiple possible FFR-style
reconnections of a given group of far-field facets. What are the implications for the FFR
algorithm of early facet removal? While it may seem helpful at first to list prohibited
resolution configurations and feed these to the FFR algorithm, this approach grows in-
creasingly brittle as the number of prohibited configurations grows. Instead, a more robust
approach is to cause an FFR yielding multiple valid results to fail. Having discovered res-
olution strategies for each individual event as described above, we simply let these rules
apply until the far field is small enough to be unique (similar to [102, 103]). A theory
predicting when far fields will have multiple resolutions would aid in this process.
B.2.4. Comments
Finally, we note that the benefits of phase-field and level-set methods, which automati-
cally capture topological change, are only clear if topological changes are unique. These
160
methods are ultimately only kinematic, and offer no clear criteria for resolving kinematic
non-uniqueness. Thus, a dynamic selection criteria would seem to be required no matter
the method used, and indeed, the implicit handling of topology offered by these methods
may actually hinder the dynamic selection of kinematically ambiguous events.
161
LB( L, B(B(L,L),L))
LB( L, B(L,B(L,L))) LBLB
Calling Function Permutation CodeReconnection/
B(B(L,L),L),
L)
LB(
BBLL
LBBLBLLL
LBBBLLLL
BLBL
LBBLLBLL
BLLB
LBLBBLLL
LBBL
LBLBLBLL
LB( B(L,B(L,L)), L)
LB( B(L,L), B(L,L))
Binary Tree Essential Code
Figure B.1. Our method of listing binary trees. The five possibilities foran O(5) far field are listed in the order given by our algorithm. Left: Thereconnection with “reconnection set” binary tree in blue. Middle: A recur-sive function describing the binary tree. Right: The letter code and reducedletter code describing the binary tree. The red lines in the top figure illus-trate the “counter-clockwise walk” which also produces the letter code. Asimple algorithm can enumerate all abbreviated letter codes, which can beused to reconstruct the binary trees for testing.
162
(a)
(b)
Figure B.2. Saddle versions of (a) the Neighbor Switch, and (b) the Irregu-lar Neighbor Switch. The saddle versions of these events produce additionalnon-uniqueness beyond that described in the main text.
163
APPENDIX C
Appendix to Chapter 4
C.1. Numerical Simulation
This section describes the simulation of Equation (5.18), which is repeated here for
convenience:
∂P
∂t= −1
2
∂
∂x[(x − 1)P ] + R
[∫ x
0
P (s)P (x− s) ds + 2∂
∂x(xP (x))
]
.
Since the spatial derivatives here are purely advective, we use an upwinding finite differ-
ence scheme to approximate them, which easily captures the evolution of discontinuous
initial conditions, and is accurate to second order in space. Standard Simpson’s quadra-
ture methods are used for the convolution integral. For time-stepping, a Crank-Nicholson
integration on the linear terms combined with a second-order Adams-Bashforth integra-
tion of the integral is quite stable, allwing the use of large timesteps to quickly reach the
relaxed steady state. However, two numerical issues arise that require special mention.
First, we note that the correct boundary condition on Eqn. (5.18) is that P vanishes
at infinity. To be able to specify this condition, and also to include the whole domain
using a finite number of grid points, we perform the change of variables ξ = exp(−x),
which maps the domain [0, ∞) to [1, 0]. The resulting equation, with like terms collected,
164
is:
(C.1)∂P
∂t= [(2R− 0.5) ln(x) − 0.5]x
∂P
∂ξ+ (2R− 0.5)P + R
∫ 1
ξ
1
sP (s)P (ξ/s)ds.
Now the entire domain of P is included, and the boundary condition at ∞ can be easily
implemented; additionally, gridpoints are distributed unevenly, with a bias toward zero
to increase resolution of the tail of P . Note, however, that the convolution describing
P2 now contains an integrable singularity requiring custom quadrature, and additionally
requires an interpolation of P to obtain its values at the points ξ/s.
Second, while Eqn. (5.16) was reached by carefully including terms to conserve the
mass M(0) and center of mass M(1), it turns out that neither is a numerically stable
quantity. In fact, by performing appropriate multiplications and integrations, we can
write down a system of ODEs governing the evolution of M(0) and M(1):
∂M(0)
∂t= P (0)
(
M(0)2 − 1)
(C.2a)
∂M(1)
∂t= M(1)/M(0) − 1(C.2b)
From this dynamical systems perspective, the point (M(0), M(1)) = (1, 1) is indeed a
fixed point, but an unstable one. This poses no problems for the theory per se, as
equation (5.16) contains no mechanism for perturbing the solution away from this fixed
point. However, inevitable numerical error will so perturb any calculation, after which
such perturbation grows. To avoid this pitfall, it is thus necessary to periodically re-scale
the solution manually back to this fixed point by interpolation.
