Surface Pressure of Charged Colloids at the Air/Water Interface

Charged colloidal monolayers at the interface between water and air (or oil) are used in a large number of chemical, physical and biological applications. Although a considerable experimental and theoretical effort has been devoted in the past few decades to investigate such monolayers, some of their fundamental properties are not yet fully understood. In this paper, we model charged colloidal monolayers as a continuum layer of finite thickness, with separate charge distribution on the water and air sides. The electrostatic surface free-energy and surface pressure are calculated via the charging method and within the Debye-H{\"u}ckel approximation. We obtain the dependence of surface pressure on several system parameters: the monolayer thickness, its distinct dielectric permittivity, and the ionic strength of the aqueous subphase. The surface pressure scaling with the area per particle, ${a}$, is found to be between ${a}^{-2}$ in the close-packing limit, and ${a}^{-5/2}$ in the loose-packing limit. In general, it is found that the surface-pressure is strongly influenced by charges on the air-side of the colloids. However, when the larger charge resides on the water-side, a more subtle dependence on salt concentration emerges. This corrects a common assumption that the charges on the water-side can \textit{always} be neglected due to screening. Finally, using a single fit parameter, our theory is found to fit well the experimental data for strong to intermediate strength electrolytes. We postulate that an anomalous scaling of $a^{-3/2}$, recently observed in low ionic concentrations, cannot be accounted for within a linear theory, and its explanation requires a fully-nonlinear analysis.