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It has been shown that a cellular automaton model, based on digital images of continuum surfaces, can reporduce real sintering situations, using only local pixel-counitng rules, where curvature differences are the driving force for mass transport and surface relaxation. The curvature computation method has been shown to measure a function of the true curvature, in 2D or 3D. The algorithm is easily generalized to large random collections of particles in 2D, and to 3D problems as well. For three dimensional problems, one simply redefines curvature counting box to be a cube, or a sphere, and then the pixel air count will be a linear function of the total curvature at a surface pixel. Of course 3D pixels are now themselves cubes. It would also be possible to look at orthogonal planar slices within a curvature counting box and determine the curvature in each plane; their sum will be a measure of the 3D curvature.
Other physical features can also be built into the model. Grain boundary energies can be handled by checking whether a given pixel is at a boundary or not [13,14]. When measuring the curvature at such a pixel, a mismatch term can be put into the curvature box to bring in a grain boundary energy. It is important eventually to be able to build in elastic forces, so that stresses are taken into account as sintering proceeds, since grain boundary and lattice diffusion are driven by elastic stresses due to applied forces and surface tension. Simple methods for using elastic spring networks realistically mapped into digital images of model materials have been developed [26,27], and can possibly be applied with some further development to the densification and sintering problem.
Finally, we note that cellular automaton-type growth models have been advocated previously for enormously complicated problems like that of biological growth [28], where the microscopic "grain-level" growth mechanisms are unknown. The only alternative then becomes to develop growth simulations using essentially arbitrary rules, and then modify the rules until some agreement with reality is achieved. The rules are then closely examined to see if they can produce any insight into what the actual growth mechanisms might be. In this paper, we have not taken this approach, since there is much known about sintering mechanisms. Instead, we have tried to build in, as much as possible, the known physics of sintering. Computer simulation techniques were used not because of unknown microscopic mechanisms, but because the complexity of the problem renders it resistant to analytical methods.