Liquid fraction profile in a liquid foam under an applied voltage

A liquid foam, a dense assembly of gas bubbles in a surfactant solution, is a deformable porous material. As classically observed in divided systems, electrokinetic transport can be induced, for example, when an electric field is applied from either side of the foam sample. We determine here the liquid fraction profile obtained when a foam is submitted to an electro-osmotic flow, when the surfactant induces so-called rigid or mobile interfaces. We show that the main governing equation for the liquid fraction repartition in space and time is diffusivelike and similar to the one describing pressure-induced drainage only. The electric field however significantly affects the general profile by modifying the boundary conditions. A capillary number that compares electro-osmotic stress and capillary pressure in the foam geometry is introduced and characterizes the magnitude of electrically induced flow in a macroscopic foam. In particular, the ability of a foam to capture liquid from a reservoir is quantitatively estimated.

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Figure 1

Shown on the left is the scheme of the experiments performed by Reuss, which shows an electro-osmotic flow [1]. On the right is a picture of a reproduction of the Volta battery.

Figure 2

Image of a liquid foam to identify its structural elements. The Plateau borders are small liquid channels that link three films, as represented in the cross section, and the nodes are at the junction of four PBs. Here $R$ is the typical radius of a bubble.

Figure 3

Scheme of the unit cell (one bubble) of a Kelvin structure [8].

Figure 4

Scheme of the two cases considered. In case A the foam is in contact with a liquid reservoir. One electrode is in the reservoir, the other one is at the top of the foam. In case B the foam is squeezed between two electrodes and isolated from the outside. Here $L$ is the height of the foam along the $x$ direction.

Figure 5

Stationary liquid fraction profile for rigid interfaces in (a) case A from Eq. (30) with $\overline{{\varphi }_r}=1.1$ and (b) case B from Eq. (32).

Figure 6

Linearized transient liquid fraction profile for rigid interfaces plotted at different times $T$ with ${\mathrm{Ca}}_r=0.1$ in (a) case A (with ${\overline{\varphi }}_r=1$) and (b) case B. The black line corresponds to the steady-state profile obtained from Eqs. (30) and (32) for cases A and B, respectively.

Figure 7

Liquid fraction profile obtained numerically for rigid interfaces in (a) case A (with ${\overline{\varphi }}_r=1.1$) and (b) case B with ${\mathrm{Ca}}_r=1$.

Figure 8

Stationary liquid fraction profile for mobile interfaces in (a) case A (with ${\overline{\varphi }}_r=1.1$) and (b) case B.

Figure 9

Linearized liquid fraction profile as a function of the position for different times $T$ for (a) case A and (b) case B. We set ${\mathrm{Ca}}_m=0.1$ and ${\overline{\varphi }}_r=1$ to remain in the linear regime.

Figure 10

Numerical solutions obtained in (a) case A and (b) case B for liquid fraction evolution, at different times with ${\mathrm{Ca}}_m=1$ and ${\varphi }_r=1.1$. The black lines correspond to the stationary profiles determined from Eq. (50) for case A and Eq. (51) for case B.