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由晶粒长大过程的曲率驱动本质出发, 以不同于MacPherson和Srolovitz的方法, 推导出凸型多面体晶粒的三维von Neumann关系式, 无任何其他形状假设及晶粒尺寸分布或拓扑分布要求. 在应用于凸型多面体晶粒时, 本文结果与MacPherson和Srolovitz给出的结果完全一致. 对于凸型多面体晶粒, 三维个体晶粒长大速率是晶粒平均切直径和晶粒棱总长度的函数, 符合Kinderlehrer指出的$n$维体积的变化速率仅与胞的(n-2)维特征量有关的规律.

Since J.von Neumann derived an exact formula for the growth rate of a cell in a two-dimensional cellular structures in 1952, people has attempted to find an exact extension of this result into three dimensions for a half century. In 2007, an exact three-dimensional von Neumann relation was reported in Nature (Nature, 2007, 446: 1053) by MacPherson and Srolovitz, which was regarded as a great progress of the long-time intense effort. However, the derivation of the exact three-dimensional von Neumann relation was complex and the quantity of “the mean width” of a real grain was difficult to measure. In this paper, based on the capillarity-driven nature of the grain growth, we derived the exact three-dimensional von Neumann-Mullins relation for a convex polyhedron grain in a simple method, which is independent of any additional assumptions concerning any grain size distribution, topology distribution or grain shape. It is shown in this paper that the three-dimensional growth rate of a convex polyhedron grain is related to two one-dimensional quantities: the grain’s mean caliper diameter and the sum of the length of its edges, which agrees with the property pointed by D. Kinderlehrer (Nature, 2007, 446: 995): the rate of change of n-dimensional volume is related to (n-2)-dimensional features of the cell and no others.