On the random Interface models of microemulsions as the foundation of hydrophilic-lipophilic difference - net average curvature (HLD-NAC) equation of state

The HLD-NAC (Hydrophilic-Lipophilic Difference – Net Average Curvature) equation of state is a powerful semi-empirical framework used to predict the phase behavior, solubilization, and interfacial properties of surfactant-oil-water (SOW) systems it is based on the assumption of a normalized curvature for an idealized microemulsion, represented as the coexistence of two fictitious phases: water droplets dispersed in oil and oil droplets dispersed in water. The characteristic sizes of these droplets are combined into two quantities: the net-curvature H n and the average-curvature H a . The microemulsion structure is more accurately described as a random surface separating oil and water domains. This representation can be effectively framed in terms of Gaussian random fields, for which expressions for the surface-averaged mean and Gaussian curvatures, 〈H〉 and 〈K〉, are available. On this basis, we demonstrate that 〈H〉 has the same functional form as the net curvature H n in the NAC model. Furthermore, we show that the empirical HLD parameter corresponds to 〈H〉 normalized by the surfactant length, thereby enabling a direct prediction of the microemulsion composition at the emulsification failure from HLD. The resulting predictions show excellent agreement with experimental data. The possibility to evaluate the preferred mean and Gaussian curvature and compare it with the HLD equation of state opens the way to further in-depth analysis about the link between the film curvature and the parameters entering the HLD-NAC (viz. chemical structure of oil and surfactant, temperature and the ionic strength).