Groupe de Physique Statistique

Spatially complex disordered matter such as foams, gels or polymer phases are of increasing technological importance due to their shape-dependent material properties. But the shape of disordered structures is a remarkably incoherent concept and cannot be captured by correlation functions alone which were almost a synonym for structural analysis in Statistical Physics since the very first X-ray scattering experiments. However, in the last 20 years numerous methods such as AFM and computer tomography have been developed which allow quantitative measurements of complex structures directly in real space. Integral geometry furnishes a suitable family of morphological descriptors, known as Minkowski functionals, which are related to curvature integrals and do not only characterize connectivity (topology) but also size and shape of disordered structures. Furthermore, Minkowski functionals are related to the spectrum of the Laplace operator, so that structure-property relations can be derived for complex materials. Percolation thresholds and fluid flow in porous media, for instance, can be predicted by measuring the Minkowski functionals of the pore space alone. Also, evidence was found in hard sphere fluids that the shape dependence of thermodynamic potentials in finite systems can be expressed solely in terms of Minkowski functionals.