Double-layer structure and interfacial tension at an ionic surfactant-laden interface

Hypothesis

The electrical double-layer structure at an ionic surfactant-laden interface is unique due to the nonlinear coupling of interfacial charging and adsorption kinetics. Consequently, the interfacial equation of state is nonideal even at low surfactant concentrations.

Analysis and computations

We analyze the equilibrium double-layer structure and interfacial tension of a planar interface that has a surface charge density derived from adsorbed ionic surfactants, as opposed to a surface whose charge or potential is specified a priori. Our analysis utilizes matched asymptotic expansions in the limit where the Debye length (${\kappa }^{-1}$) is much smaller than the surfactant depletion length (h), i.e. $\epsilon =1/\left(\kappa h\right)\ll 1$. The asymptotic analysis is verified against numerical computations.

Findings

The interfacial concentration of surfactant is asymptotically small due to electrostatic repulsion, scaling as ${\epsilon }^{2/3}{\Gamma }_{\infty }$, where ${\Gamma }_{\infty }$ is the maximum packing surface concentration. Moreover, the atypical double-layer structure consists of an inner layer of width $O\left({\epsilon }^{4/3}h\right)$ within a Debye layer of width $O\left(\epsilon h\right)$, outside of which is an electroneutral bulk solution of surfactant and counterions at uniform concentration $c_{\infty }$. The surfactant and counterion concentrations in the inner layer scale as $O\left({\epsilon }^{2/3}{c}_{\infty }\right)$ and $O\left({\epsilon }^{-2/3}{c}_{\infty }\right)$ respectively, whereas both are $O\left(c_{\infty }\right)$ in the Debye layer. The interfacial tension is predicted to decrease from its clean value (i.e. in the absence of surfactant) as $O\left({\epsilon }^{2/3}\right)$. This asymptotic prediction is in qualitative agreement with experimental data. Our elucidation of this unique double-layer structure could provide new approaches to analyze dynamic interfacial tension, dilatational surface rheology and emulsion stability.