Minkowski Tensor Shape Analysis of Cellular, Granular and Porous Structures
Minkowski Tensor Shape Analysis of Cellular, Granular and Porous Structures
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DOI：
10.1002/adma.201100562
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发表时间：
2011-06-17
影响因子：
29.4
通讯作者：
Mecke, K.
中科院分区：
文献类型：
--
作者：
Schroeder-Turk, G. E.;Mickel, W.;Mecke, K.
预测具有空间复杂结构的材料的物理性质是材料科学中最具挑战性的问题之一。更好地理解这些材料的一个关键是它们的空间结构的几何特征。Minkowski张量是张量形状指数，允许定量表征复杂材料的各向异性，特别适合于开发张量值或取向相关物理性质的结构-性质关系。它们是基本的形状指数，在某种意义上是体积、曲面和积分曲率概念对张量值量的最简单推广。Minkowski张量建立在积分几何和随机几何的坚实数学基础上，具有很强的鲁棒性和完备性。闵可夫斯基张量的通用定义广泛适用于不同类型的形态，包括有序和无序结构。快速线性时间算法可用于其计算。本文提供了一个实际的概述不同用途的闵可夫斯基张量提取定量的物理相关的空间结构信息，从实验和模拟数据，无论是在2D和3D。应用程序，量化（a）对齐的共聚物膜由表面力显微镜成像的电场;（B）局部细胞各向异性的球形珠包模型的颗粒物质和闭孔液体泡沫模型;（c）表面取向的开孔固体泡沫研究的X射线断层扫描;和（d）缺陷密度和位置在分子动力学模拟结晶铜。
Predicting physical properties of materials with spatially complex structures is one of the most challenging problems in material science. One key to a better understanding of such materials is the geometric characterization of their spatial structure. Minkowski tensors are tensorial shape indices that allow quantitative characterization of the anisotropy of complex materials and are particularly well suited for developing structure-property relationships for tensor-valued or orientation-dependent physical properties. They are fundamental shape indices, in some sense being the simplest generalization of the concepts of volume, surface and integral curvatures to tensor-valued quantities. Minkowski tensors are based on a solid mathematical foundation provided by integral and stochastic geometry, and are endowed with strong robustness and completeness theorems. The versatile definition of Minkowski tensors applies widely to different types of morphologies, including ordered and disordered structures. Fast linear-time algorithms are available for their computation. This article provides a practical overview of the different uses of Minkowski tensors to extract quantitative physically-relevant spatial structure information from experimental and simulated data, both in 2D and 3D. Applications are presented that quantify (a) alignment of co-polymer films by an electric field imaged by surface force microscopy; (b) local cell anisotropy of spherical bead pack models for granular matter and of closed-cell liquid foam models; (c) surface orientation in open-cell solid foams studied by X-ray tomography; and (d) defect densities and locations in molecular dynamics simulations of crystalline copper.