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)] 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.

Figure 1

Two pores connected by a single perforated membrane. Depending on the physical problem under study, the boundary conditions are different. For the fluid flow case, a pressure drop is applied between the top and the bottom faces, a no-slip condition is applied over the membrane, and a slip condition is applied on the lateral faces. For the electrical conductivity case, an electrical potential difference is applied between the top and bottom faces and the electrical field is parallel to all the others faces (membrane and lateral faces).

Figure 2

Fluid flow conductance $G_{fl}$ of two interconnected pores as a function of the membrane aperture size for various aspect ratios $H/D_b$. Dots are the FEM results. The solid line is plotted by using Sampson's equation.

Figure 3

Cylindrical pores: electrical conductance $G_e=1/R_e$ of two interconnected pores normalized by the electrical conductance without membrane ($D_b^2{\sigma }_f/H$) as a function of the membrane aperture size for various aspect ratios $H/D_b$. (a) Comparison between FEM results (symbols) and predictions of Eq. (6) from Sahu and Zwolak (dashed lines) and Eq. (7) (solid lines). (b) Comparison between FEM results (symbols) and predictions of Eq. (5) from Rosenfeld and Timsit (dashed lines) and Eq. (7) (solid lines). The same data are used in (b) and in its inset.

Figure 4

Rectangular pores: normalized electrical conductance $G_e=1/R_e$ of two interconnected pores as a function of the membrane aperture size for various aspect ratios $H/D_b$. Symbols are the FEM results. Dashed and solid lines are plotted by using Eqs. (6) and (7), respectively.

Figure 5

Shape of current streamlines in a pore associated with four regions: I, bulklike; II, transition region; III, access region; and IV, membrane thickness channel. Note that the first region is observed only if the pore is long enough ($H\gtrsim D_b$).

Figure 6

Periodic unit cell based on the Kelvin structure ($\varphi =0.995$). Depending on the physical problem considered, the boundary conditions are different.

Figure 7

Top view from the median plane of the tested configurations showing the positions of the closed membranes: (a) $K_0$, (b) $K_1$, (c) $K_2$, and (d) $K_3$. Note that the horizontal median plane is a plane symmetry.

Figure 8

Permeability $K$ of PUCs having different configurations of closed membranes for various membrane apertures. Symbols are the FEM results. Solid and dashed lines are plotted by using Eq. (8).

Figure 9

(a) Flow pattern and (b) its equivalent pore-network scheme. (c) Pore-network scheme to solve the electrical conduction problem by using two kinds of conductances: interpore conductance and intrapore conductances. (d) Intrapore conductances for square and hexagonal windows of a Kelvin cell.

Figure 10

Fluid flow passing through the $K_0$ configuration of the PUC [Fig. 7]. Colored lines correspond to the active bond in the pore-network model for foam permeability [Fig. 9].

Figure 11

(a) Electrical conductivity ${\sigma }_e$ and (b) tortuosity ${\alpha }_{\infty }$ of the PUCs as a function of the membrane aperture size for various configurations of closed membranes ($K_0, K_1, K_2$, and $K_3$) with $\varphi =0.995$. The same data are used to the inset graph of (a). Symbols correspond to the FEM results. Dashed lines are plotted by using Eq. (8), taking into account only interpore electrical conductances [Fig. 9]. Solid lines are plotted by using pore-network calculations, taking into account both intra- and interpore electrical conductances [Fig. 9]. Due to the symmetries of the PUC configurations, analytical expressions for effective conductivity can be found for each configuration (see Appendix pp2).

Figure 12

Viscoinertial characteristic frequency $f_v$ of the $K_i$ configuration as a function of the aperture size calculated from FEM results. On the left abscissa, the frequency is normalized, and on the right abscissa, the frequency corresponds to the case of foam filled by air with $D_b=1\text{mm}$ and $\varphi \approx 1$.

Figure 13

Electrical conductivity ${\sigma }_e$ and tortuosity ${\alpha }_{\infty }$ of the $K_0$ configuration as a function of the porosity.

Figure 14

Pore-network simulations on random Kelvin foam with various fractions of open windows and identical membrane aperture size ($\varphi =0.995$): (a) electrical conductance ${\sigma }_e$ and fraction of open porosity ${\varphi }_o/\varphi$ as a function of the open window fraction $x_{ow}$ and ratio ${\sigma }_e\left(1\right)/{\sigma }_e\left(x_{ow}\right)$ as a function of the open window fraction $x_{ow}$, and (b) ratio ${\sigma }_e\left(1\right){\varphi }_o/{\sigma }_e\left(x_{ow}\right)\varphi$ as a function of the open window fraction $x_{ow}$. Note that (b) leads to the ratio ${\alpha }_{\infty }\left(x_{ow}\right)/{\alpha }_{\infty }\left(1\right)$ in considering the pore space volume to use in the definition of the tortuosity, the open-pore space. Moreover, ${\sigma }_e\left(1\right)={\sigma }_{e,K_0}$ can be calculated by using Eq. (9).

Figure 15

Conductance pore network associated with each PUC configuration: (a) $K_0$, (b) $K_1$, (c) $K_2$, and (d) $K_3$.