Fluids in porous media: a morphometric approach - Mecke, Klaus; et al. (2005)
Also the thermodynamic behaviour of fluids in porous media, i.e., the shape dependence of the grand canonical potential and of surface energies of a fluid bounded by an arbitrarily shaped convex pore, can be calculated in the thermodynamic limit fully from the knowledge of the Minkowski functionals, i.e., of only four morphometric measures. This remarkable result is based on Hadwiger's theorem on the completeness of the additive Minkowski functionals and the assumption that a thermodynamic potential is an 'additive' functional which can be understood as a more precise definition for the conventional term 'extensive'. As a consequence, the surface energy and other thermodynamic quantities contain in the thermodynamic limit, beside a constant term, only contributions linear in the mean and Gaussian curvature of the pore and not an infinite number of curvature terms. Finally, starting from a microscopic density functional for an inhomogeneous fluid in a porous medium the phase coexistence (capillary condensation) and the critical point of the fluid is determined in terms of structure functions and morphological measures of the pore space and calculated explicitly for specific random porous structures using results from integral geometry.