A geometric formulation of the law of Aboav–Weaire in two and three dimensions
A geometric formulation of the law of Aboav–Weaire in two and three dimensions
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DOI：
10.1088/1751-8113/45/6/065001
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发表时间：
2011-11
期刊：
影响因子：
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通讯作者：
J. Mason;R. Ehrenborg;E. Lazar
中科院分区：
文献类型：
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作者：
J. Mason;R. Ehrenborg;E. Lazar
阿博阿夫-韦尔定律是一个简单的数学表达式，源自经验观察，即颗粒的边数与相邻颗粒的平均边数相关，并且通常仅限于自然的二维微观结构。人们已经进行了许多尝试来从理论上证明这种关系的合理性，或者在三个维度上推导出类似的关系。本文提供了几个精确的几何结果，其表达式与通常的 Aboav-Weaire 定律相似，但附加了可用于确定 Aboav-Weaire 定律何时是合适的近似值的术语。具体来说，我们推导了适用于各个颗粒簇的几个局部关系，以及除了单个参数 z 之外在二维和三维上相同的相应全局关系。推导需要定义和研究平均超曲率，这是一个以前未考虑的物理量。我们将精确结果的近似值与二维和三维的广泛模拟结果进行比较，并且我们提供了一个紧凑的表达式，在复杂性和准确性之间取得了平衡。
The law of Aboav–Weaire is a simple mathematical expression deriving from empirical observations that the number of sides of a grain is related to the average number of sides of the neighboring grains, and is usually restricted to natural two-dimensional microstructures. Numerous attempts have been made to justify this relationship theoretically, or to derive an analogous relation in three dimensions. This paper provides several exact geometric results with expressions similar to that of the usual law of Aboav–Weaire, though with additional terms that may be used to establish when the law of Abaov–Weaire is a suitable approximation. Specifically, we derive several local relations that apply to individual grain clusters, and a corresponding global relation that is identical in two and three dimensions except for a single parameter ζ. The derivation requires the definition and investigation of the average excess curvature, a previously unconsidered physical quantity. An approximation to our exact result is compared to the results of extensive simulations in two and three dimensions, and we provide a compact expression that strikes a balance between complexity and accuracy.