Surface Tension Isotherms: Reconceptualizing Adsorption, Self-Assembly, and Micelle Formation via the Fluctuation Theory.

Given a surface tension isotherm (i.e., how interfacial free energy changes with the surfactant concentration), can we gain insight into how surfactant molecules interact at the interface and in the bulk solution? Historically, surfactants were modeled to bind onto a uniform interface, before aggregating stoichiometrically at the critical micelle concentration (CMC). However, this simple model contrasts with counterevidence, e.g., premicelles (smaller aggregates below CMC) and aggregate size distribution, necessitating a departure from stoichiometric aggregation models. To this end, a novel theory for surface tension will be established by synthesizing the statistical thermodynamic fluctuation theory for sorption and self-assembly in solution. This theory provides a link between the functional shape of a surface tension isotherm and the underlying interactions. We demonstrate that the gradient and curvature of a surface tension isotherm reveal a competition between surfactant sorption and bulk number fluctuation without employing any model assumptions. This novel theory proposes to (i) redefine the surfactant aggregation number using number fluctuations to replace the stoichiometric model, (ii) generalize the Szyszkowski-Langmuir isotherm (which implicitly assumes site-specific adsorption on uniformly distributed binding sites) with the novel ABC isotherm to capture the surface-bulk difference of surfactant number correlation, and (iii) replace the surfactant "area-per-molecule" with the projected area, by incorporating the thickness of the interface. This theory can be applied to surfactants and small molecules (e.g., alcohols) alike, eliminating the need for separate models over a spectrum of self-association propensities.