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Fig. 4 shows how the model works for a simple 2-D scratch-smoothing example. All pixels shown are solid material, with the solid surface pixels highlighted in gray. The numbers in the surface pixels indicate the curvature at that point, determined using a b=3 box. There are two such pixels that have a curvature of 4, the highest value of the surface. From this point on, we use the terms "curvature" and "pixel count" interchangeably. On a given cycle, the model finds the curvatures of the entire solid surface, orders the resulting curvatures, and then randomly selects a prescribed number n of the highest curvature solid pixels, and removes them. In Fig. 4, n=1. The next step, shown in Fig. 4b, is to calculate the curvature of the air surface pixels, after the n solid pixels have been removed, and order them. The n lowest curvature air surface sites are then randomly filled with the n pixels selected previously. The next cycle is begun by recalculating the curvature of the solid surface, as shown in Fig. 4c, taking off the n highest curvature pixels at random, and then randomly filling the n lowest curvature air surface sites, as shown in Fig. 4d.
Figure 4: Illustrating the operation of the sintering model for a simple scratch example: (a) showing the solid surface in gray, labelled according to curvature values as measured by a b=3 box, (b) showing the air surface pixels (dashed), labelled with their b=3 curvature values, (c) same as (a) but for the second cycle and (d) same as (b) but for the second cycle.
This algorithm of moving the pixels models the physical situation of vapor phase transport with interface attachment as the rate-limiting step. In this case, the characteristic diffusion distance is infinite, which, as discussed in more detail in Sec. 5, allows densification to occur. Other pixel-moving-algorithms could be chosen for other cases. For example, a root mean square distance weighting could be used to simulate vapor-phase-transport-limited diffusion.
The model is run until an equilibrium shape has been reached. At this point, the solid surface curvatures are roughly equal, as measured within the resolution of the box being used to compute them. The effects of finite box resolution on the algorithm in general, and specifically on the equilibrium shape, will be discussed later.