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 zeta potential at free interfaces, and for electrokinetic transport in liquid foams.

💡 Research Summary

The authors investigate electrokinetic phenomena at liquid‑air interfaces by means of atomistic molecular dynamics simulations of aqueous electrolyte films coated with the ionic surfactant sodium dodecyl sulfate (SDS). Two complementary numerical experiments are performed: (i) a streaming‑current (SC) setup in which a Poiseuille‑type pressure gradient is imposed by a body force on the fluid, and the resulting electric current is measured; (ii) an electro‑osmotic (EO) setup in which an external electric field drives flow and the fluid velocity is recorded. Both approaches yield the zeta potential ζ through the standard linear relations I_e = (εζ/η)(–∇p) for SC and v_eo = (εζ/η)E for EO, with ε the permittivity and η the viscosity of SPC/E water.

Simulations are carried out for two salt concentrations (0.26 M and 0.068 M, corresponding to Debye lengths λ_D = 0.57 nm and 1.1 nm) and for a wide range of surfactant surface densities c = 0.047–3.0 nm⁻², i.e. surface charges Σ = –e c ranging from –7.6 to –480 mC m⁻². The film thickness is set to ten Debye lengths to avoid overlap of the two interfacial double layers.

The key result is a non‑monotonic dependence of ζ on surfactant coverage. At high coverage the measured ζ is much smaller than would be expected from the large surface charge, whereas decreasing the surfactant density leads to a rapid increase of ζ. Remarkably, even in the limit of vanishing coverage ζ does not tend to zero but saturates at a finite negative value of about –75 mV.

To rationalize these observations the authors combine three physical ingredients that are specific to liquid‑air interfaces: (1) a partial slip boundary condition. The velocity jump (slip velocity v_s) between the liquid and the surfactant layer is related to the shear rate by v_s = b γ̇, where the slip length b scales inversely with surfactant density (b ∝ c⁻¹). This scaling follows from a simple friction model F = α η R c v_s, with R the effective hydrodynamic radius of a surfactant head and α a geometric factor (α = 3π for a hemispherical head). (2) ion‑binding to the surfactant heads. At high surfactant density almost all counter‑ions are bound (binding fraction θ → 1), effectively reducing the net surface charge to Σ_eff = –e c (1 – θ). (3) a shift of the shear plane position z_s with coverage, which modifies the electrostatic potential V(z_s) felt by the fluid.

Using these ingredients the total zeta potential is expressed as ζ = ζ_slip + ζ_noslip = Σ b/ε + V(z_s). In the dilute limit the no‑slip term vanishes (V(z_s) → 0) while the slip term becomes independent of c because Σ b = –e / (3πR) is constant, yielding the finite saturation value. The authors verify this prediction analytically (Eq. 3) and find excellent agreement with the simulation data. For intermediate coverages they compute V(z) from the exact Poisson‑Boltzmann solution for a single charged wall, insert the measured b(c), z_s(c) and θ(c) into the formula, and obtain theoretical curves that capture the main trends of the simulated ζ values.

The study demonstrates that, unlike solid‑liquid interfaces where ζ scales roughly with surface charge and vanishes at zero charge, free liquid‑air interfaces can exhibit substantial zeta potentials even with minute impurity concentrations. This has direct implications for the long‑standing controversy over the sign and magnitude of the surface potential of pure water, as well as for practical applications such as electro‑driven foam stability, bubble manipulation in microfluidic channels, and water purification using gas‑phase carriers. The work highlights the necessity of accounting for interfacial slip and ion binding when interpreting electrokinetic measurements at free interfaces.