The time evolution for 5 different sets of initial data, including comparisons between surface diffusion (SD) and surface attachment limited kinetics (SALK) are illustrated in Figures 3-7. Computations of SD are displayed in a time sequence of light gray images, while images for SALK are in dark gray. The kinetic coefficients in each case have been scaled so that evolution occurs roughly on the same time scale for each.
In Figure 3, 
is a square. The initial data
is an elongated rectangle. Figure 4 begins with the
same initial data, but uses a 16-gon as its Wulff shape. Note that in
both cases, the particle flowing by SALK remains convex. In contrast,
particle flowing by SD becomes non-convex, forming ``bulbs'' on the
ends of the initial rectangle. These examples show an important
difference between SD and SALK. In SD, motion is so as to reduce local
gradients in chemical potential, and this ``proximity effect'' can
produce non-convex regions from previously convex ones. In SALK, by
contrast, convex regions always will remain convex because there is no
local force driving them to become concave.
In Figure 5 the analog
of the rectangle has been rotated by 
. is a square, and so
what was a rectangle is now a pair of staircases. Here, not only is
the flow different for SALK and SD, but the end configuration as well.
For SALK, volume
diffuses uniformly to the zero 
stairsteps from the positive
ends causing all the steps not adjacent to the ends to move
diagonally as a unit, since equations 16 and
17 imply that all
facets with 
move with the same velocity.
In motion by SD, on the other hand, the middle region of the zero
steps hardly moves at all. The steps near the end are removed
one by one as volume is pumped from the ends towards the center. In
fact, the small edges are rapidly moving off down the staircase as
described in Section 3.1.1. Eventually,
the particle breaks into two particles each of which asymptotically
approach . The ability to detect and handle such topological
changes is one of the strengths of the general computational scheme.
This example further demonstrates how SALK cannot take non-convex
curves to convex ones. The staircase in Figure 5
is the crystalline analog to a convex curve in the surface energy
sense: it contains no points of negative curvature.
One might wonder if it is possible that stepping might only occur at
``topological times'', as it does in Figures 3
and 4, in which all stepping occurs only at the
start of the computation. Figure 6 shows that this is not
the case, and that stepping may occur in SD even at regular times. The
Wulff shape is again a square, and the initial condition is L-shaped,
having one of the branches very elongated. In SD flow, bulbs form on
the ends of each branch at time 
via the proximity effect.
Interestingly, another step forms on the inside long edge of the L so
as to turn the single long edge into a long zero edge and an
edge with negative curvature at the base of the shorter leg of the L.
The two legs of the L shrink at approximately the same speed, and so
the shorter leg becomes a single bulb before the bulb at the other end
of the long leg has time to travel down the leg. Note that as the
short leg shrinks, the edge which initially formed the outside of the
short leg becomes very short, and runs up the long leg, eventually
disappearing. At even later times, the material that was contained in
the short leg eventually reforms itself into a large bulb and then
motion continues as for the elongated rectangle. It is important to
note, however, that a new step formed as the blob on the short leg
turned the corner to begin moving down the long leg.
Topological changes may occur in a surprising and complicated manner.
The spiral in Figure 7 is an illustrative example for
a square . Initially the ends of the spiral have large positive
, and the `interior' facets have zero or small 
. Most of the
action takes place at the inner end where 
and the gradient in
is very large. As the inner end unwraps and forms bulbs (a
manifestation of the proximity effect), it collides with another part
of the crystal forming a hole. The two surfaces then flow
independently, until they collide again, re-forming a simply connected
domain. The process repeats once: the system approaches an equilibrium
with a Wulff-shaped hole in it. Note that the flow does not scale
uniformly with 
. Each of the topological changes
produces large discontinuous changes in the chemical potential which
reestablish large potential gradients in regions which had previously
nearly exhausted them. Thus, the overall effect is that the evolution
starts fast and slows until a topological change occurs, and then
proceeds rapidly in those regions proximate to the event which caused
the topological change.
Although the spiral of Figure 7 did not undergo a
topological change under SALK, it is possible to have initial data
which does produce topological changes for that type of flow.
Figure 8, which uses an octagonal Wulff shape,
starts as a large particle with a deep notch cut
out of it. Under both flow by SALK and by SD, the notch closes off and
the system equilibrates to a Wulff-shaped crystal with a Wulff-shaped
hole in it. The reason the flow by SALK is able to produce the
topological change is because of the two opposing zero-
facets.
Because the initial shape is a simply connected solid, the average
is positive. Thus all zero curvature facets have the same
positive velocity, and so they can be placed so as to collide.