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Theoretical calculations of microstructural development are, in general, quite difficult, especially when considering materials that are made up of an originally random collection of particles that are subsequently amalgamated by a processing step. Inorganic examples include random packings of ceramic [1] or metal particles [2], which are densified by heat treatment (sintering), and random dispersions of portland cement particles in water, which are solidified by hydraulic reactions [3] 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 realistic collections of particles, fundamental computer simulations become necessary [4].
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 [8], 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 [1]. We use a digital-image approach [15], for three reasons. The first is to be able to handle random particle shapes. The second is to be able to simulate the actual physics in a realistic way, which only becomes possible when the simulation can be controlled at the sub-particle pixel level. The third reason for a digital-image approach is to be able, once a microstructure is simulated, to subsequently compute physical quantities like transport and mechanical properties. These kind of computations are quite difficult to do in a continuum representation, except for the simplest microstructures and physical properties [16,17].
The specific problem discussed in this paper is that of the evolution of a surface due to its curvature, which we generically call sintering. 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. Strictly speaking, such rearrangement is driven by gradients in chemical potential along the surface. When the surface tension is isotropic, the potential is proportional to curvature. For anisotropic surfaces, the potential at a particular orientation is somewhat more complicated, but has been treated [18]. In this paper, we limit our discussion to isotropic surfaces, but note that anisotropic surfaces may be simulated in an analogous manner.
Mathematically tracking a surface under the influence of its own dynamics is notoriously difficult. For the case at hand, the velocity of the surface in the direction of its normal is proportional to the divergence of the gradient (in the coordinates of the surface) of the curvature: a nonlinear fourth-order PDE with moving boundary conditions results. Calculations have either employed useful approximations [19] or numerical techniques [20] and are always limited to simple cases. A simulation of sintering of many particles will require 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 above atomic length scales. In this paper, we illustrate a useful algorithm which calculates an approximation to the curvature of a discrete representation of a surface and then updates the surface according to simple rearrangement rules.
The algorithm which we employ is a cellular automaton [21], 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. It is anticipated that cellular automaton techniques will be useful in investigating the behavior of many-particle sintering compacts and may lead to a new understanding of these processes. The purpose of this paper is to establish and demonstrate the local rules of counting which simulate curvature and to show pixel-moving-rules which simulate transport via a vapor phase. We compare our results to known scratch-smoothing calculations and demonstrate the technique for sintering of one or more particles and for pore closure due to vapor diffusion, all in two dimensions.