Modeling interface growth and shape changes is fundamental to the study of microstructural evolution, but tracking of the evolution of an interface during diffusive transport is a notoriously difficult problem, especially for anisotropic surface energies. Use of the crystalline formulation, in which the interfaces are assumed to be completely faceted, can avoid many of these difficulties. In this paper, we apply this crystalline formulation to the special cases of sintering, coarsening, and shape equilibration where the only driving force comes from the reduction of the energy associated with surfaces and where volume is conserved. No other thermodynamic driving forces, such as those which accompany phase changes, are assumed present here, and flows which do not conserve volume, such as motion by mean curvature (e.g. grain growth) are treated elsewhere. (See [1] for a review.)
Volume conserving surface motion typically arises with heterophase surfaces in which long range diffusion of a conserved quantity is required for motion. We will assume that surface velocities are entirely determined by surface processes; either the diffusion is confined to be along the surface (SD) or with more general diffusion in a surrounding transport medium that the motion is limited by surface attachment limited kinetics (SALK). To have volume conservation we have to assume no change in the amount of matter stored in the transport medium. With these restrictions we obtain geometric laws for surface motion, which allow us to apply recently developed methods for calculating the time evolution of shapes. Part of these developments is the realization that although these laws become very difficult with mild surface energy anisotropies, they become simpler if the anisotropy in surface energy is so extreme that all microstructures are composed of plane segments of a finite number of low energy orientations, for some problems even simpler than the isotropic cases. We therefore explore in this paper this extreme case of anisotropy. Last we restrict ourselves to evolution of shapes on the plane, two-dimensional polygonal ``particles.''
The surface energy function, 
, which underlies the
driving force, is the work at constant pressure and temperature
which
must be supplied to create a unit area (length in two-dimensions) of surface
with orientation 
.
It is useful to represent by the shape with the lowest
surface energy for a fixed volume, the
Wulff shape
.
For the special case that is a constant,
is a sphere (circle in two dimensions).
When is not constant, the shape differs from a sphere
and in extreme cases becomes polyhedral.
For a general in any dimension, is given by the Wulff
construction:
.
Note that in this construction the length scales of
have units of 
. The distance that a facet is from the origin of
is for that , but the length 
of that facet
in depends not only on the of that facet but also those of the
two adjacent facets. The Wulff construction can be easily inverted to
recover
for those that appear in . If has missing orientations, the
values of for those
not in cannot in general be recovered, but usually do not appear in
physical problems.
Recently developed
methods [3][2]
have shown that computations of surface evolution can often be
greatly simplified when is ``crystalline'', that is
completely
faceted (composed of only flat surfaces and their
intersections).
In this case, the underlying surface
energy density 
can be given as a list of
magnitudes for all orientations n which are the normal
of a facet in .
Any orientations not in will break
up into facets that do appear in while preserving
the average orientation of the surface.
We are interested in motions which decrease surface energy. The limit of
change
in surface energy with volume under small deformations is by definition the
weighted mean curvature (or weighted
curvature in two dimensions), 
[4].
``Small" usually means local, but in the case of facets should
mean motion of an entire facet by a small distance.
is thus defined on a facet-by-facet basis by considering energy
changes involved in moving the entire facet keeping its normal direction fixed. While
the chemical
potential may vary along a facet,
the average chemical potential across the facet equals on that
facet.
For polygonal ,
has a particularly
simple form, based on , given in Section 2.
The methods developed for anisotropic problems can be
useful for approximating isotropic problems. Motion by crystalline
curvature, where the normal velocity of each facet
is proportional to its crystalline curvature and volume is not conserved,
has been proven to converge to motion by (isotropic)
curvature for two dimensional problems, as approaches a circle
(though a sequence of regular polygons with the
number of sides becoming infinite)
[7][6][5].
It remains to be shown whether other types of motion will likewise converge
to the isotropic case. In the case of surface diffusion, there does
not now exist a good way of computing isotropic surface diffusion, so
its approximation by this crystalline method is currently the best way
of computing it, assuming that these approximations do in fact
converge to it.