Geometric partition functions of cellular systems: Explicit calculation of the entropy in two and three dimensions

A method is proposed for the characterisation of the entropy of cellular structures, based on the compactivity concept for granular packings. Hamiltonian-like volume functions are constructed both in two and in three dimensions, enabling the identification of a phase space and making it possible to take account of geometrical correlations systematically. Case studies are presented for which explicit calculations of the mean vertex density and porosity fluctuations are given as functions of compactivity. The formalism applies equally well to two- and three-dimensional granular assemblies.Comment: 14 pages, 4 figures, to appear in The European Physical Journal E - Soft Matte

Stress in planar cellular solids and isostatic granular assemblies: Coarse-graining the constitutive equation

A recent theory for stress transmission in isostatic granular and cellular systems predicts a constitutive equation that couples the stress field to the local microstructure. The theory could not be applied to macroscopic systems because the constitutive equation becomes trivial upon straightforward coarse-graining. This problem is resolved here for arbitrary planar structures. The solution is based on the observation that staggered order makes it possible to couple the stress to a reduced geometric tensor that can be coarse-grained. The method proposed here makes it possible to apply this idea to realistic systems whose staggered order is generally 'frustrated'. This is achieved by a renormalization procedure which removes the frustration and enables the use of the upscalable reduced tensor. As an example we calculate the stress due to a defect in a periodic solid foam

Modifying continuous-time random walks to model finite-size particle diffusion in granular porous media

The continuous-time random walk (CTRW) model is useful for alleviating the computational burden of simulating diffusion in actual media. In principle, isotropic CTRW only requires knowledge of the step-size, $P_l$, and waiting-time, $P_t$, distributions of the random walk in the medium and it then generates presumably equivalent walks in free space, which are much faster. Here we test the usefulness of CTRW to modelling diffusion of finite-size particles in porous medium generated by loose granular packs. This is done by first simulating the diffusion process in a model porous medium of mean coordination number, which corresponds to marginal rigidity (the loosest possible structure), computing the resulting distributions $P_l$ and $P_t$ as functions of the particle size, and then using these as input for a free space CTRW. The CTRW walks are then compared to the ones simulated in the actual media. In particular, we study the normal-to-anomalous transition of the diffusion as a function of increasing particle size. We find that, given the same $P_l$ and $P_t$ for the simulation and the CTRW, the latter predicts incorrectly the size at which the transition occurs. We show that the discrepancy is related to the dependence of the effective connectivity of the porous media on the diffusing particle size, which is not captured simply by these distributions. We propose a correcting modification to the CTRW model -- adding anisotropy -- and show that it yields good agreement with the simulated diffusion process. We also present a method to obtain $P_l$ and $P_t$ directly from the porous sample, without having to simulate an actual diffusion process. This extends the use of CTRW, with all its advantages, to modelling diffusion processes of finite-size particles in such confined geometries.Comment: 9 pages, 7 figure