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Mathematically tracking a surface under the influence of its own dynamics is notoriously difficult. Calculations have either employed approximations [18] or numerical techniques [19] and are always limited to simple cases. A simulation of sintering of many particles requires a new approach. Analysis begins by replacing an atomically stepped surface with a continuous representation, which is then represented digitally, by pixels. The size of a pixel is a small fraction of a particle, but is far larger than atomic length scales.
The many-particle structure being considered is a 3-D box filled with uniform spheres. The spheres are dropped from the top of the box, and are allowed to move laterally to try to work their way lower down in the box, after contact with a previously deposited sphere. This is a modified digital version of the well-known Visscher-Bolsterli algorithm [20], although we have not tried to fully optimize the packing density. Using monosize digital spheres, we can easily obtain packing densities of about 50%. Impenetrable walls are placed at the bottom and sides of the box, and free boundary conditions are maintained at the top.
Once an initial structure is defined, with solid and pores labelled and defined pixel by pixel, the mass rearrangement or sintering algorithm is applied. The first step is to measure the curvature at every solid surface pixel and every air surface pixel (pore pixels touching a solid surface), and make histograms for each of these two results. Then a number n of the highest curvature pixels are chosen at random, removed, and placed at random at n of the lowest curvature air pixel sites. This operation is iterated until all curvature differences are gone and an equilibrium shape is reached, as measured by the current diameter counting sphere being used [15].
This algorithm of moving the pixels models the physical situation of curvature difference driven vapor phase transport with interface attachment as the rate-limiting step. In this case, the characteristic diffusion distance is infinite, so that densification can occur via this mechanism alone. There are no grain boundaries or elastic forces in the algorithm at present. 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 algorithm we have developed to measure curvature is motivated by the following exact result. Consider a sphere 1 of radius R, whose curvature, at some point P on the surface, we wish to measure. Center another sphere 2 at P, with radius b, b << R. Define the volume that is inside 2 but outside 1 by V. It is obvious that V is independent of the position of P on the surface. The volume V can be easily derived using standard formulae [21], or by direct integration, and is given by:
The volume V is exactly linear in K=2/R, the true curvature of the sphere. The equivalent formula in 2-D has correction terms that are higher order in b/R. In a digital image mode, to measure the curvature at a given surface pixel, we center a digital counting sphere at the pixel of interest, and count how many pixels lie in the counting sphere, but outside the surface being considered. This number, by eq. (1), gives a linear measure of curvature.
To demonstrate the algorithm, we have taken digital spheres, of various radii R, and run the algorithm using two different values of b for the counting sphere. Fig. 1 shows the results, plotted vs. K=2/R. The fitted straight lines agree well with the data points. The error bars are the result of averaging the pixel count over the digitally rough surface of the digital spheres whose curvature is being measured. The fourth root of the ratio of the 2b=13 slope to the 2b=9 slope is 1.45, while 13/9 = 1.44, which agrees well with the b4 scaling of the slopes predicted by eq. (1). The 2b=9 slope is larger than that predicted by eq. (1) by only 7%, and the 2b=13 slope is larger than the exact result by 9.0%.
Figure 1: Showing the pixel count from the curvature algorithm applied to spheres with a range of radii R, plotted vs. the true curvature K=2/R. Two values of 2b (9 and 13), the counting sphere diameter, were used.