Anomalous $\zeta$ Potential in Foam Films

Electrokinetic effects offer a method of choice to control flows in micro- and nanofluidic systems. While a rather clear picture of these phenomena exists now for the liquid-solid interfaces, the case of liquid-air interfaces remains largely unexplored. Here, we investigate at the molecular level electrokinetic transport in a liquid film covered with ionic surfactants. We find that the $\zeta$ potential, quantifying the amplitude of electrokinetic effects, depends on the surfactant coverage in an unexpected way. First, it increases upon lowering surfactant coverage from saturation. Second, it does not vanish in the limit of low coverage but instead approaches a finite value. This behavior is rationalized by taking into account the key role of interfacial hydrodynamics, together with an ion-binding mechanism. We point out implications of these results for the strongly debated measurements of the $\zeta$ potential at free interfaces and for electrokinetic transport in liquid foams.

Figure 1

(a) Snapshot of a typical system (${\lambda }_D=0.57\mathrm{nm}$, $c=3.0{\mathrm{nm}}^{-2}$); water molecules are not represented. (b),(c) Sketches of electro-osmosis (EO) and streaming current (SC) numerical experiments in foam films.

Figure 2

$\zeta$ potential as a function of surfactant coverage $c$, measured from EO and SC simulations, for two Debye lengths ${\lambda }_D$. The solid and dashed lines are predictions from Eq. (2), for ${\lambda }_D=0.57$ and 1.1 nm, respectively. The dot-dashed line is the dilute limit given by Eq. (3).

Figure 3

Top: (a) Poiseuille velocity profiles in the SC configuration (${\lambda }_D=0.57\mathrm{nm}$), for increasing surfactant coverage $c$ from top to bottom. The velocity is measured in the surfactant reference frame and divided by the normalized applied force $F^{*}=F/F(c=0.047{\mathrm{nm}}^{-2})$ for comparison purposes. The origin of the $z$ axis is taken at the average position of the surfactant sulfur atoms. (b) Cartoon illustrating the characterization of the hydrodynamic boundary condition. Bottom: (c) Shear plane position $z_s$ and (d) slip length $b$ as a function of the surfactant coverage, for two Debye lengths ${\lambda }_D$. The slip length is fitted using Eq. (1) (dashed line).

Figure 4

Typical charge density profiles of surfactants and ions, for ${\lambda }_D=1.1\mathrm{nm}$ and two surfactant coverages $c$. The solid red lines represent the absolute surfactant charge $|{\rho }_f|=-{\rho }_f$. The dashed green lines represent the ionic charge ${\rho }_e$. The dash-dotted blue lines represent the water density profile. Each data set is plotted for both the SC and EO simulations. The origin of the $z$ axis is taken at the average position of the surfactant sulfur atoms. Inset: Fraction of bound ions $\theta$ versus surfactant coverage $c$, for ${\lambda }_D=1.1$ and 0.57 nm (violet circles and orange squares, respectively).