Theoretical calculations of microstructural development are, in general, quite difficult, especially when considering materials consisting of an originally random collection of particles that are subsequently amalgamated by a processing step. Inorganic examples include random packings of ceramic  or metal particles , which are densified by heat treatment (sintering), and random dispersions of portland cement particles in water, which are solidified by hydraulic reactions  to form cement-based composites. The randomness of the original particle collection and the complicated physics and chemistry that take place during the processing to achieve a final microstructure in general preclude any analytical calculations except for extremely simple particle configurations, like a single pair of particles or a periodic array of particles. In order to consider collections of many particles, computer simulations become necessary, and in particular fundamental computer simulations .
For the types of materials considered above, computer simulation models of microstructural development are considered to be fundamental if they directly treat the material at the particle or grain level, which is the most relevant building block of the microstructure, and realistically incorporate as much of the known physics and chemistry as possible into the growth rules. Examples exist for cement-based materials [5,6,7], sedimentation in rocks , grain growth in powdered metals [9,10], and ceramics [11,12,13,14].
In this paper we concentrate on the problem of microstructural development during sintering, in the regime where surface mass transport driven by curvature differences dominates the densification process . Actually, "densification" of a powder compact during sintering also involves mass transport due to internal stresses, which we do not consider here. An example of surface evolution is the gradual smoothing of a surface scratch when temperatures are high enough for mass transport. A system of loose particles will also form inter-particle necks and sinter by surface or vapor diffusion. Such rearrangement is driven by gradients in chemical potential along the surface. When the surface tension is isotropic, the chemical potential is proportional to curvature. In this paper, we limit our discussion to isotropic surfaces, but note that anisotropic surfaces may be simulated in an analogous manner .
We use a digital-image approach, in which a particle is represented by a collection of neighboring pixels, that was developed previously for 2-D problems , and extend it to 3-D. A digital- image approach is used for the following reasons: 1) to handle random particle shapes, 2) to simulate the actual physics in a realistic way, which is only possible when the simulation can be controlled at the sub-particle pixel level, and 3) to compute physical quantities like transport and mechanical properties, once a microstructure is simulated. These physical property computations are quite difficult to do in a continuum representation, except for the simplest microstructures and physical properties .
The algorithm which we employ is a cellular automaton , consisting of: 1) curvature calculation via local neighbor-counting rules for each surface pixel, and 2) rules to rearrange the pixels. It is relatively easy to develop such schemes. Since each pixel only responds to its local environment, a simulation program is not limited to simple geometric shapes, but can operate on structures that are arbitrarily complicated at the continuum particle level.