Institut für Theoretische Physik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstr. 7, 91058 Erlangen, Germany.
Adv Mater. 2011 Jun 17;23(22-23):2535-53. doi: 10.1002/adma.201100562.
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.
预测具有复杂空间结构的材料物理性质是材料科学中最具挑战性的问题之一。更好地理解此类材料的关键之一是对其空间结构进行几何特征描述。Minkowski 张量是张量形状指标，可用于定量描述复杂材料的各向异性，特别适合于开发张量值或与方向有关的物理性质的结构-性质关系。它们是基本形状指标，在某种意义上是对体积、表面和积分曲率等概念的最简单推广张量值数量。Minkowski 张量基于积分和随机几何提供的坚实数学基础，并具有强大的稳健性和完备性定理。Minkowski 张量的多功能定义广泛适用于不同类型的形态，包括有序和无序结构。它们的计算有快速的线性时间算法。本文提供了 Minkowski 张量的不同用途的实用概述，用于从实验和模拟数据中提取定量的物理相关空间结构信息，包括 2D 和 3D 数据。展示的应用包括：(a) 通过表面力显微镜成像电场对共聚物薄膜的取向进行量化；(b) 对颗粒物质的球形珠粒堆积模型和封闭液泡沫模型的局部细胞各向异性进行量化；(c) 通过 X 射线断层扫描研究开孔固体泡沫的表面取向；以及(d) 对结晶铜的分子动力学模拟中的缺陷密度和位置进行量化。
