A Generalized Theoretical Model for the Relationship Between Critical Micelle Concentrations, Pressure, and Temperature for Surfactants

Directional solvent extraction (DSE) has been gaining interest as a water treatment technology in recent years. DSE utilizes the process of micellization for the purposes of species separation between water and complex chemical systems. In this article, we develop a conformal geometric algebra-based formulation that models surfactants, their solubilities, and critical micelle concentration (CMC), with relation to temperature and pressure. Molecules are represented as spatially distributed networks embedded in R-4,R-1 space, and the mathematical characterizations of these molecules are shown to be effective in modelling CMC as a function of temperature and pressure. One of the contributions of this work is the utilization of this formulation to develop a governing expression, in the form of a three-dimensional relationship, between CMC, pressure, and temperature for a general surfactant. In prior works, the CMC-temperature plane and CMC-pressure plane expressions have been extensively documented for sodium alkyl sulfates. In this work, we extend the formulation to model the CMC of decanoic acid, sodium octyl sulfate, sodium decyl sulfate, sodium dodecyl sulfate, and sodium tetradecyl sulfate. Using this theoretical model, a relationship between CMC and the directional solubility of water in a surfactant is determined. Directional solubility is related to temperature and pressure, and on this basis, we devise a directional solubility-pressure-temperature expression for an arbitrary surfactant to improve the state of the art for DSE. From this expression, we propose a novel isothermal DSE process for water treatment.