In this paper we have set up a problem of shape evolution, driven entirely by reduction in an interfacial energy that is so anisotropic that only shapes with a limited set of facets orientations need be considered. This is called crystalline anisotropy. In this extreme of anisotropy, problems of equilibrium and shape evolution become much simpler than with lesser anisotropy, and in many cases simpler even than with isotropy. This paper is an exploration the crystalline formulations for this type of problem specialized to 2-D. Three dimensional problems are generally more difficult, but there are many reasons to believe that the advantages of the crystalline formulation will be even greater.
We considered two kinds of material flows, surface diffusion (SD) and surface attachment limited kinetics (SALK). For SD we postulated a chemical potential field in a surrounding transport medium that varies continuously with position along the surface and around corners. Fluxes, proportional to gradients in chemical potential, are required to be continuous at corners, and to have a constant divergence on each facet proportional to the instantaneous growth rate of that facet. Furthermore the chemical potential at each facet is required to have an average value so that there is no free energy change when the facet moves to grow or dissolve the crystal-an averaged local equilibrium. For SALK we assumed that the kinetic process of attachment or detachment is so slow compared to diffusion in the transport medium, that the chemical potential in that medium is constant, and that the interface motion of each facet was a function, here taken to be linear, of the local deviation from equilibrium. The surrounding transport medium could be the surface as in surface diffusion or, with special restrictions, a surrounding vapor or solution phase.
An immediate advantage of the crystalline formulation is that we are able to show that solutions exist for all times for SD and SALK. At present only short time solutions have been shown to exist for the isotropic SD problem.
We have set up an algorithm for solving the evolution equations and solved
them for specific anisotropies as parameterized by various Wulff shapes 
,
and for various initial shapes. We identified the conditions for
which stepping will occur by stability and variational arguments;
proof that these are the proper conditions will be published in the
mathematics literature.
The computational algorithm handles both stepping and merging correctly.
Topological changes do occur in the examples and are readily made.
Topological changes generally occur when surfaces contact:
In crystalline problems with the restriction that,
if n is in so is -n,
the contact is along parallel plane segments; sometimes the adjacancy
conditions are not preserved, and infinite weighted curvatures occur for the
missing facets, but these infinities are easily handled by introducing
a fan of orientations at the corners.
This restriction on limits the applicability to crystals
with some symmetry;
in 2-D, a two-fold or an inversion axis is sufficient, and, in 3-D,
intersecting 2-fold axes, 2/m or an inversion center is sufficient, but
crystals with lower symmetries could satisfy this condition,
since there is no requirement that 
or even that
.
Let us compare the results of the calculations for the square and the
regular 16-gon.
There is an increase in computational difficulties with increasing 
, and
this provides a practical limit for the magnitude of . As long as we stay
with regular -gons, we expect the calculations to converge to that of an
isotropic limit, but in this case little is known about this isotropic
limit.
There are numerical results for the axisymmetric flow in 3D by
Nichols and Mullins[12].
Proof of convergence to
isotropic flow (
, where 
denotes mean curvature)
as the number
of sides in increases is still an open question, but computer
experimental evidence indicates that it does.
Although we used a regular 16-gon for , we could just well have taken any
irregular convex -gon, with parallel sides and with small
enough to make calculation feasible, to approximate many anisotropies
with low symmetries or combined forms of high symmetry crystals. For example,
a combined form with square symmetry with four faces each of direction
(10) and (11) would be represented by an octagon often described as a truncated
square. If we add eight facets from some (hk), is a 16-gon that can be
used to approximate the anisotropy of many crystals whose lattices have
the symmetry of a square.
The comparison of SD vs. SALK shows a number of salient phenomenae which will be discussed below; they are:
and initial shapes, under some conditions they occur in both; sometimes they
occur in one but not the other.
Although we feel they are more likely to occur in SD than in SALK, we have
displayed an example where it occurs in SALK but not in SD.
can develop facets which have the opposite
sign during SD but not during SALK.
Let us discuss these in more detail, beginning with proximity and stepping
effects. When the initial shapes are long rectangles steps form immediately
in SD and growing bulbs form at the ends of the
rectangles.
The growth of the bulbs has the effect of diminishing local gradients
of the chemical potential and drives the
system to look locally like the underlying .
The local removal of gradients produces the proximity effect.
In SALK, the stepping does not occur; if incipient bulbs are
placed in the initial data by forcing non-zero lengths steps onto
the long sides, they disappear on a short time scale.
These calculations suggest that, because of the proximity effect, it should
be possible to distinguish between the two extreme types of area-conserving
flow by observing how shapes evolve.
We feel that such reasoning from
qualitative shape changes is a simpler and more reliable hallmark of
mechanism than methods based on scaling laws applied to some set of
stereological measures, in which values are plotted on a log-log plot and
exponents are used to determine which is the active diffusive mechanism in
some materials process.
Such averaging is open to misinterpretation and conclusions
are often equivocal.
Of course, there are mixed cases where neither process is dominant
and calculations of such cases will be presented elsewhere.
However, scaling laws would do no better when no mechanism is
dominant: interpolation of scaling exponents is suspect.
The whole motion is very stable, except for near some of the topological
changes.
The final state is independent of the number of infinitesimal
steps if more than required are inserted,
and is also insensitive to using ad
hoc step positions in the initial data.
This is an indication that the stepping algorithm by which
a stable potential is obtained effectively finds a unique
potential.
A theorem about such uniqueness will appear elsewhere [8].
Computer experiments where steps where injected at random onto
the surface 
at each forward difference iteration and
did not produce behavior different from that with implementation of
the stepping algorithm alone.
While the breaking into separate particles of the staircase in
Figure 5, the
the bulb formation in
Figures 3-4,
and the formations of inclusions in 7 is
reminiscent of Rayleigh-like
instabilities such as that calculated in 3-D by Nichols and Mullins[12], they are
not really related to it at all since this is a two dimensional model.
The Rayleigh instability is is based on a path of continuously decreasing
surface area in 3-D; the existence of instability is independent of the
particular kind of transport
mechanism, although the wavelength depends on it. It does not occur in
isotropic two dimensional rods.
Rather, the occurance of a Rayleigh-like break-up of the long rectangles in
our calculations is a combination of the crystallinity and the transport
mechanism since it does not occur in
Figures 3-4
for SALK and since
the staircase in figure 5 will always break into only
two pieces
independent of the length of stairs.
It is currently unknown whether the analogies to the staircase's break
of figure 5 for -gonal will disappear
as is increased, but it is suspected that it will as long as it is
a staircase - i.e., the axis of the staircase is not parallele to a facet.
On an atomic scale in 3D, a facet may contain steps, ledges, and surface vacancies which can move about the surface while it is growing. In the crystalline model, any facet is represented a being mathematically flat; so, interpretation of the results can only be made at scales where the averaging out of defect nature of the surface is sensible. Vicinal surfaces will always contain ledges and in the crystalline model are represented as a combination of many flat surfaces; the size of such surfaces assumed to be large enough so that averaging is again sensible. An extrapolation from an atomistic model to the continuized equations of motions presented in this paper will appear in a subsequent publication.
Experimental observations of cubes of MgO particles ( is a cube)
which are just barely overlapping along their edges break into two isolated
cubes have been made by Rankin[13].
Experimentally, the breaking occurs if the overlapped region, which
is a rectangular strip, is small and does not occur if it is large.
While our calculations are only two dimensional, there is similarity of her
micrographs to our Fig. 5.
The examples repudiate some common misconceptions about microstructural evolution.
It is sometimes stated that a system evolves so as to reduce its total free energy the fastest possible way with no regard to the actual diffusive mechanisms. The contrast between the two types of diffusion in the Figures contradicts this statement. Actually, both mechanisms each decrease the free energy as quickly as possible: but in different metrics, as discussed in [10].
It is sometimes suggested that the free energies of the end-states will determine the equilibrium end-state; this is contradicted by the case of surface diffusion in Figures 5 and 7, in which the system does not have the lowest possible energy for its area. Rather, equilibrium occurs when all gradients are extinguished and there is no means for the system to, for instance, remove the metastable equilibrium hole in Figure 7.
The ambient chemical potential 
can increase continuously in
time for
the case of SALK for any set of positive crystals (and decreases
for any set of negative crystals).
This is apparent from Eqs. 17-the numerator is
a constant if all the mobilities are while the denominator can only
decrease.
Mass would tend to flow from a system of unequilibrated surfaces
to a system which contains only Wulff shapes.
The average chemical potential can change discontinuously when
a topological change occurs such as when a particle disappears.
Such considerations apply to the standard isotropic mean-field
coarsening models as well.
In this paper we have described two different types of area-preserving motion for completely faceted two-dimensional interfaces and have established rigorous rules for their motion. The crystalline method has now been used to examine four different types of flows: motion by mean curvature [3] and crystal growth in a diffusion field [2] as well as surface diffusion and surface attachment limited kinetics. The calculated results seem to be qualitatively consistent with experimental observations, although three dimensional flow is a topic for future research. Open questions about two-dimensional behavior remain, and will be addressed in a forthcoming publication.