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

## TL;DR

This paper introduces a new theory that explains how surfactants behave at interfaces and in solutions using surface tension isotherms and fluctuation theory.

## Contribution

A novel fluctuation-based theory is proposed to redefine surfactant aggregation and adsorption without relying on traditional stoichiometric models.

## Key findings

- The gradient and curvature of surface tension isotherms reveal interactions between surfactant sorption and bulk fluctuations.
- The new ABC isotherm generalizes the Szyszkowski–Langmuir model by capturing surface–bulk differences in surfactant correlations.
- The theory replaces the 'area-per-molecule' concept with a projected area that includes interface thickness.

## Abstract

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.