A significant step forward has been achieved in computer modeling of time-evolution of three-dimensional grain aggregates, by projecting the equation of motion for the grain boundaries with continuous degrees of freedom into that of grain vertices, thus leaving a finite set of equations of motion for vertices. This model is referred to as the vertex model. It is noted that such a coarse-graining can be carried out not in ad-hoc manner but in such a way that the grain edges are approximated by straight lines in the course of projecting out unimportant continuous degrees of freedom. However, we are immediately confronted with the fact that the grain faces bounded by these straight edges are not always planar in general which is characteristic of three-dimensional cellular systems.
To circumvent this difficulty supplementary vertices called virtual vertices are introduced on grain faces by which a unique triangulation of the faces can be achieved. The actual introduction of virtual vertices, which should still be simple enough to be practically implemented, has been performed in the following two ways; (i) Take the positions of virtual vertices as the centers of gravity of each grain face (center-of-gravity model). (i) Take the initial positions of virtual vertices as the centers of gravity as in (i), but their subsequent motions are determined selfconsistently by the equations of motion (virtual-vertex model).
Equations of motions of vertices are then readily derived for each case. Note here that some lower cutoff resolution below which the movement of vertices cannot be described by the equations of motion does exit because of the coarse-grained character of the vertex models. Such movements occurring in very limited spatial regions can be adequately described by local topology changes which can always be resolved into recombination processes between the two grains and/or tetrahedron annihilation process. In this way, dynamics of grain aggregates can be unambiguously described by the set of equations of motion for all the vertices together with the elementary topological processes. The model permits further refinements of arbitrary degree by increasing the number of virtual vertices in appropriate manner.
Grain growth kinetics is discussed on the basis of the vertex model by comparing the results obtained by the vertex model simulations with those obtained by the Potts model simulations.