Electrical conductivity and tortuosity of solid foam: Effect of pore connections

Numerical and analytical methods at both micro- and mesoscales are used to study how the electrical resistivity and the high-frequency tortuosity of solid foam are modified by the presence of membranes that partially or totally close the cell windows connecting neighbor pores. Finite-element-method simulations are performed on two pores connected by a single-holed membrane and on well-ordered Kelvin foam. For two pores connected by a single-holed membrane, we show that the equation for pore access resistance obtained by Sahu and Zwolak [Phys. Rev. E 98, 012404 (2018)2470-004510.1103/PhysRevE.98.012404] can predict, after a few modifications, the electrical resistivity at the membrane scale for a large range of membrane apertures. Considering these analytical results, we build a pore-network model by using two kinds of conductances at the pore scale: interpore conductance and intrapore conductance. Local interpore resistances govern foam electrical conductivity at small membrane aperture size, but when the membrane aperture has the same order of magnitude as the pore size, the intrapore resistances are no longer negligible. An important success of this pore-network model is that it can be used to study the effects of percolation on the foam electrical conductivity by using pore-network simulations on larger samples containing a few thousand pores and having different proportions of closed membrane randomly distributed over the sample. The tortuosity is found to be drastically larger than one in foam containing membranes with small apertures or a significant fraction of closed membranes.