Thermodynamic Model of Liquid−Liquid Phase Equilibrium in Solutions of Alkanethiol-Coated Nanoparticles

A thermodynamic model for a mixture of alkanethiol-coated nanoparticles (NPs) and low-molecular-weight (nonpolymeric) solvent is developed, and calculations of liquid−liquid phase equilibria for different values of NP core radius, alkanethiol chain length, solvent molar volume, and alkanethiol−solvent interaction parameter, are presented. The model takes into account the swelling of the organic coronas and the dispersion of particles with swollen coronas in the solvent. The energetic interaction between alkyl chains and solvent is considered, both within the corona and between the outer alkyl segments and free solvent. Swelling involves mixing of alkanethiol chains and solvent in the corona and stretching of the organic chains. Dispersion considers an entropic contribution based on the Carnahan−Starling equation of state and an enthalpic term calculated considering the surface contacts between alkyl segments placed in the external boundary of the corona and the molecules of free solvent. Two different kinds of phase equilibrium are found. One of them, observed at high values of the interaction parameter, is the typical liquid−liquid equilibrium for compact NPs in a poor solvent where a complete phase separation is observed when cooling (increasing the interaction parameter). The second liquid−liquid equilibrium is observed at low values of the interaction parameter, where swelling of coronas is favored. In this region two different phases coexist: one more concentrated in NPs that exhibit relatively compact coronas and the other one more diluted in NPs with extended coronas. In diluted solutions of NPs the deswelling of the fully extended coronas takes place abruptly in a very small temperature range, leading to a solution of compact NPs. This critical transition might find practical applications similar to those found for the abrupt shrinkage of hydrogels at a critical temperature.