Equilibrium Shapes for Grain Boundaries and Surfaces with Anisotropic Surface Tension Functions
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EQUILIBRIUM SHAPES FOR GRAIN BOUNDARIES AND SURFACES WITH ANISOTROPIC SURFACE TENSION FUNCTIONS
J.E. Taylor Department of Mathematics,
Rutgers University,
New Brunswick,
N.J.
08903
ABSTRACT The geometric configuration of grain boundaries and surfaces seems to play a significant role in phase transformations and surface phenomena. Determining the equilibrium configurations of such boundaries for a given surface tension function y is additionally an interesting mathematical problem; it reduces in the case of isotropic surface tension to the minimal surface problem. A framework is given here for determining such configurations in the other extreme case, where the equilibrium shape of a crystal of fixed volume (the Wulff shape) is a polyhedron. Some results obtained within this framework are outlined.
INTRODUCTION Given a surface tension (surface free energy) function y defined on the unit sphere, the space of unit normals, a construction for the surface Wy of least surface free energy surrounding a specified volume was devised by G. Wulff [11] and essentially proven to be correct by Dinghas [4]. (The methods of Dinghas were extended to more complete proofs in 151 and 16].) The result of this construction can be expressed as W = {x: x-v~y(v)
it
is
illustrated in
for every unit vector
v};
Figure 1.
\I
Fig. 1. The polar plot of a surface tension function which results from Wulff's construction.
Mat.
Res.
Soc.
Symp.
Proc.
Vol.
21 (1984)
Published by Elsevier
y
and the
Science Publishing Co.,
WY
Inc.
606
No such neat constructions are known for other variational problems; in particular, if gravity is introduced to the above problem, with for example a flat table for support, it is not even known if the region must be convex [2]. In case y is constant, so that the surface free energy is proportional to surface area and the Wulff construction yields a sphere, the related variational problems have been much studied mathematically. In particular, the minimal surface problem, that of determining the surface of least surface energy having a given curve as perimeter, has More recently, been the source of a great deal of mathematical activity. elliptic variational problems (ones involving a y for which WY has positive upper and lower bounds on its curvature) have also been Such y have been studied at least partly because extensively studied. "regularity" (smoothness) of their solutions can be proved. The opposite extreme is that the curvatures of WY are either zero The philosophy of the work outlined below is that or infinity everywhere. y where Wy is a polyhedron, one by considering surface tensions exchanges smooth surfaces for polyhedral ones, and differential equations More specifically, for a given problem, one for polynomial equations. first tries to show that the set of normal directions to a solution should One be only those of WY (and perhaps a controlled number of others). then determines all the possible local minimizing structures and how they Finally, the precise solution to the variatio
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