Curvilinear polygons, finite circle packings, and normal grain growth

A key element of analytical models for curvature-driven grain growth is the formulation of a growth law for individual grains. Grain size distributions can be derived through the use of such growth laws. The present paper reviews some recent progress in developing a mean-field statistical theory of normal grain growth. A paradox related to curvilinear polygons is shown to support the expectation that the grain size distribution has a finite cutoff. The circumference of the circumscribing circle and the perimeter of the curvilinear polygon become exactly equal when the number of sides is 12. The boundary of an average 12-sided grain has an integral curvature that is equal in magnitude to the integral boundary of a completely circular grain. For this reason, grains with more than 12 sides are suggested to be energetically unfavourable, a result which is in excellent agreement with simulation data. In the concluding part, finite circle packings are employed to interpret a topological relationship found for Potts-model-type simulations of 2D grain growth. It is briefly indicated how spherical packings may help to develop some intuition about topological relationships found for normal grain growth in three dimensions.