Equations for the equilibrium surface pressure increase on the penetration of an insoluble monolayer by a soluble surfactant

This study addresses the penetration of a pre-existing insoluble monolayer by a bulk soluble surfactant. It is shown that if the adsorption isotherm is known describing the partitioning of the soluble component between the interface and the bulk, the equation for the resulting increase in surface pressure Delta pi can be derived using the Gibbs adsorption equation. The relationship is valid for any adsorption isotherm. A framework for studying monolayer penetration is presented based on these equations. Two example calculations are presented based on adsorption site models for the interface that allow closed form solutions for Delta pi(x(1),C-2), where C-2 is the molality of the soluble component and x(1) is the fractional coverage of the insoluble component. First, a Langmuir model is posed, where the insoluble surfactant reduces the adsorption of the soluble component. The Delta pi caused by the soluble surfactant at a given C-2 is exactly equal to the surface tension reduction caused by that surfactant in the absence of the insoluble monolayer. Second, a Frumkin model is posed, incorporating 1-2 interactions (between the insoluble (1) and soluble (2) components) and 2-2 self-interactions. The model predicts enhanced Delta pi with x(1) for 1-2 cohesion and 2-2 repulsion and diminishing Delta pi with x(1) for 1-2 repulsion and 2-2 cohesion when compared with the surface tension reduction achieved by the soluble component alone. The Frumkin model is discussed as a means for quantifying synergistic adsorption into the monolayer directly from a family of Delta pi curves for various, fixed x(1) as a function of C-2. A brief discussion of data available in the literature is presented; the trends observed are in qualitative agreement with the theoretical framework.