Regensburg 2013 – scientific programme

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DY: Fachverband Dynamik und Statistische Physik

DY 17: Soft matter

DY 17.8: Talk

Wednesday, March 13, 2013, 11:45–12:00, H47

Quantifying shape in heterogeneous media by Minkowski-Tensors — •Michael A. Klatt, Gerd E. Schröder-Turk, and Klaus Mecke — Institut für Theoretische Physik, Universität Erlangen

We describe a novel approach to morphology and anisotropy analysis of complex spatial structure using so-called mixed volumes and Minkowski tensors, which are generalizations of the well-known scalar Minkowski functionals. The tensors are explicitly sensitive to anisotropic aspects of the structure and are relevant for example for elastic moduli or permeabilities of porous materials [1]. A theorem by Alesker (1999) ensures robustness and completeness of a morphological analysis based on Minkowski tensors.

To illustrate the technique we analyze analytically and numerically the spatial structure of the Boolean model of overlapping grains, which is among the most important models for porous and heterogeneous media, leading to good predictions of mechanical and transport properties, e.g., of rock [2]. The morphology of the Boolean model is usually quantified by the mean intercept length for which an analytic expression is presented. However, the commonly used MIL tensor is not well-defined.

An important geometric feature of heterogeneous media is percolation. An accurate estimation of the percolation threshold in Boolean models can be given in terms of mixed volumes and Minkowski tensors.

[1] G. Schröder-Turk et al., Adv. Mater. 23 2535-2553 (2011).

[2] C. H. Arns et al., Phys. Rev. Lett. 91 215506 (2003).