A thermodynamic model and related equations were developed to improve our understanding of the size selective fractionation of ligand stabilized nanoparticles using a CO₂ gas expanded liquid precipitation process. This size selective precipitation process is based on controlled reduction of the solvent strength of the organic solution through increases in the antisolvent CO₂ concentration via pressurization. A decrease in the solvent strength reduces the repulsive Osmotic contributions to the total interaction energy of the dispersion by weakening the interactions between the solvent and the ligand tails. Eventually, these repulsive forces in the dispersion are lowered to the point where they can no longer overcome the van der Waals forces of attraction between particles, thereby causing the particles to precipitate. This results in an ability to tune the size of the nanoparticles that are dispersed or precipitated simply by adjusting the CO₂ pressure. In this chapter, Osmotic energy and Hamaker constant expressions were developed for CO₂ gas expanded solvent mixtures consisting of an organic solvent (e.g. hexane) and antisolvent (e.g. CO₂). These expressions, coupled with appropriate thermodynamic data, were used to estimate the maximum size of dodecanethiol capped Ag nanoparticles that could be dispersed at a given CO₂ pressure by equating the total interaction energy to the Boltzman threshold stabilization energy (-3/2 k_BT). A parameter sensitivity analysis was performed to determine which parameters most affect the predicted maximum particle size that could be dispersed at a given CO₂ pressure. It was found that the length of the ligand tail that interacts with the solvent medium had the most influence on these total interaction energy calculations. Assuming complete solvation of the entire ligand tail, the model always overpredicted the size of the particles that would be precipitated at a given CO₂ pressure compared to the experimental results. Consequently, we have proposed that the entire length of the dodecane chain is not accessible to the solvent medium. As such, three phenomenological model variations were developed that provide for differing abilities of the ligand tails to interact with the solvent including an Extended Ligand Length Solvation Model, a Condensed Phase Model, and a Limited Ligand Length Solvation Model with the best fit of the experimental size separation data coming from the Limited Ligand Length Solvation Model. Equations for a surface fraction model based on UNIQUAC parameters and its viability for this process is also discussed.

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