VOLUME 87, NUMBER 20 PHYSICAL REVIEW LETTERS 12 NOVEMBER 2001
Shocks Preempt Continuous Curvature Divergence in Interface Motion
V. Tsemekhman
1
and J. S. Wettlaufer
1,2
1
Applied Physics Laboratory, University of Washington, Seattle, Washington 98105-5640
2
Department of Physics, University of Washington, Seattle, Washington 98105-1560
(Received 18 June 2001; published 24 October 2001)
The dichotomy between two approaches to interface motion is illustrated in the context of two-
dimensional crystal growth. Analyzing singularity formation based on the curvature of the interface
predicts a continuous divergence of curvature in contrast to the discrete loss of orientations predicted
when the evolution is described by an equation for the two-vector of the interface. We prove that the
formation of a shock in the latter approach preempts continuous curvature divergence predicted in the
former approach. The results are broadly applicable to kinematic interface motion problems, and we
connect them with experiments reported by Maruyama et al. [Phys. Rev. Lett. 85, 2545 (2000)].
DOI: 10.1103/PhysRevLett.87.205701 PACS numbers: 64.70.Dv, 68.08.Bc, 68.15.+e, 81.30.Fb
An understanding of the motion of interfaces constitutes
a broad enterprise in physical and mathematical science;
in the former case practitioners are often motivated by a
particular setting, such as pattern formation in condensed
matter systems, and in the latter case, a principal goal
deals with the development of methodologies that apply
in many settings. Although an interface may separate, for
example, two immiscible fluids, phases, or magnetic states
of a single phase, it is predicting the space-time evolu-
tion of that interface that captures interest. For example,
in a Hele-Shaw cell, a flow drives the interface separating
two immiscible fluids, and in crystal growth, disequilib-
rium drives a solidification front, but possible topological
changes in the interface, in particular the formation of fi-
nite time singularities, are central to the dynamics of these
and other systems (e.g., [1,2]). Here, we describe several
basic features of interface motion during crystal growth in
which a dual description of singularity formation leads to a
paradox which we resolve using mathematical proof. The
relevance of the principal results may span many interface
motion problems.
A broadly useful illustration of interface motion arises
in the context of geometric models for the evolution of a
two-dimensional phase boundary wherein the local normal
growth velocity does not depend on curvature and is only a
function of the local orientation of the surface normal for a
given thermodynamic growth drive [3]. Two approaches,
one based on the global shape dynamics of the two-
dimensional curve, the other describing the local curvature
evolution, have been suggested [3]. While shocks are as-
sociated with the first approach, a continuously divergent
curvature is a dramatic aspect of the second. The global
shape dynamics is determined by the properties of the
solutions to the evolution equation
≠
C
≠t
2V
N , (1)
where
C
Cu, t is an evolving plane curve parame-
trized by a variable u,
N is the (inward pointing) unit nor-
mal vector, and V u, Dm is the normal growth velocity.
The angle between the positive x axis and the unit tangent
vector
T is defined by u, and the (constant) thermodynamic
growth drive Dm is the departure of the chemical potential
of the solid phase from that at bulk coexistence.
Exact solutions for this type of evolution equation can
be expressed in terms of characteristics which represent
straight rays along which the orientation of the normal is
preserved [4]. The characteristic for a given orientation u
is given by [5]
r u, t r
0
u 1 2V u
N 1 V
0
u
T t . (2)
In contrast, if we parametrize Eq. (1) by the angle u, we
obtain for the evolution of curvature k along the trajectory
of the location on the boundary with fixed orientation of
the normal
ku, t
k
i
u
1 1
˜
V uk
i
ut
, (3)
where k
i
u is the curvature at the location on the initial
seed crystal’s boundary with orientation u,
˜
V V 1 V
00
,
and the primes denote differentiation with respect to u
[3,6]. One observes that the curvature decreases with time
for all orientations with
˜
V . 0. The curvature grows con-
tinuously and diverges at finite time for all orientations
with
˜
V , 0, and is stationary at the zeros of
˜
V .
Growing shapes resulting from Eqs. (1) and (3) have
been analyzed in detail [3,4,7] for various functional forms
of the growth function V u. The analysis in [3] predicted
that for a general class of V u the growth shape contains
areas of decreasing curvature both at the roughest and vici-
nal orientations. The region between these flattening areas
grows with increasing curvature and develops a corner that
eventually absorbs all rough orientations. The prediction
has since been confirmed by experiment [6].
We describe the contrast in these approaches by focusing
on the problem of diverging curvature. Previously, numeri-
cal evidence led to a conjecture that curvature divergence
is preempted by the formation of a shock [3]. According to
Eq. (3) the divergence of curvature is a continuous limiting
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