Statistical theory of two-dimensional grain growth—I. The topological foundation

Abstract

For designing a statistical theory of 2-dimensional grain growth the two-parametric topological distribution function ϕin describing the number of grains of a grain size class i (equivalent radius Ri) and of the number of sides n was investigated. Of special importance are the (statistical) relationships between Ri and n. It could be shown that many of such relationships proposed in literature did not satisfy the generally valid topological law n̄ = 6 and that the form of this relationship is not unique but depends on the nature of the microstructure. In the present paper mainly two special cases, the random (“Voronoi”) and, in particular, the equiaxed (“quasicircular”) microstructure, are considered. For the latter case which is the case being approximated during grain growth the here called “special linear relationship” n̄i = 3 + 3Ri/R̄ was found to be valid by measurements of Ri and n of a great number of grains. The “physical meaning” of this relationship was displayed by deriving it with the help of a simplified model of circular grains applying the principles of complete and of random surface covering. This relationship which will be very important for the theory of grain growth (Part II) was shown to be valid in the case of random distribution of equiaxed grains of different size. It was further shown that with the circular grain model and the above mentioned two principles also other topological relationships (e.g. the Aboav-Weaire equation) could be interpreted.