Three types of rules are used: random arrangement of noninteracting spheres (Boolean model, this model is not restricted to spherical shape of pores), dense-random packing of hard spheres (DRP), and random packing of partially penetrable spheres each consisting of a hard core and a soft shell (cherry-pit model). The considered models are based on rules for arranging spheres in the three-dimensional space. There is no restriction regarding the volume fraction of pores. The pore surfaces have, at least piecewise, spherical curvature with arbitrary distribution of radii of curvature (for simplicity, called pore size distribution below). In this review we consider models for random isotropic porous media of both closed-pore and open-pore type. Activities of bridging ideas developed in physics and materials science on the one hand and of methods established by mathematicians on the other hand can be found, for example, in. Parallel to this progress in physics, mathematical methods for the description of random systems have been improved as documented in textbooks on stochastic geometry and its applications. Beginning with Ziman’s famous book on models of disorder treatises like that on the physics of structurally disordered matter, the physics of foam, mechanical properties of heterogeneous materials, effective medium theory for disordered microstructures, structure-property relations of random heterogeneous materials, transport, and flow in porous media followed. The development of models for random porous media and the setting-up of structure-property relationships have benefitted substantially from the rich body of theoretical work on disordered matter. The methods presented are exemplified by applications: small-angle scattering of systems showing fractal-like behavior in limited ranges of linear dimension, optimization of nanoporous insulating materials, and improvement of properties of open-pore systems by atomic layer deposition of a second phase on the internal surface. Effective medium theory is applied to calculate physical properties for the models such as isotropic elastic moduli, thermal and electrical conductivity, and static dielectric constant.
Volume fraction, surface area, and correlation functions are given explicitly where applicable otherwise numerical methods for determination are described. Besides systems built up by a single solid phase, models for porous media with the internal surface coated by a second phase are treated. A parameter is introduced which controls the degree of open-porosity. The Poisson grain model, the model of hard spheres packing, and the penetrable sphere model are used variable size distribution of the pores is included.
Both closed-pore and open-pore systems are discussed.
The models are isotropic both from the local and the macroscopic point of view that is, the pores have spherical shape or their surface shows piecewise spherical curvature, and there is no macroscopic gradient of any geometrical feature. Models for random porous media are considered.
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