Thermodynamically Stable Colloidal Solids: Interfacial Thermodynamics from the Particle Size Distribution

True thermodynamic stability of a solid colloidal dispersion is generally unexpected, so much that thorough experimental validation of proposed stable systems remains incomplete. Such dispersions are under investigated and would be of interest due to their long-term stability and insensitivity to preparation pathway. We apply classical nucleation theory (CNT) to such colloidal systems, providing a relationship which links the size-dependent interfacial free energy density of the particles to their size distribution, and use this expression in the fitting of previously reported size distributions for putatively thermodynamically stable nanoparticles. Experimental data from a gold-thiol system exhibiting inverse coarsening or "digestive ripening" can be well-described in terms of a power-law dependence of the interfacial free energy $\gamma$ on radius based on capacitive charging of the nanoparticles, going as $r^{-3}$ , as suggested by prior authors. Data from magnetite nanoparticles in highly basic solutions also can be well-fit using the CNT relation, but with $\gamma$ going as $r^{-2}$ . Slightly better fits are possible if the power of the radius is non-integral, but we stress that more complex models of $\gamma$ will require richer data sets to avoid the problem of overfitting. Some parameters of the fits are still robustly at odds with earlier models that implicitly assumed absolute thermodynamic stability: first, the extrapolated free energy density of the flat surface in these systems is small and positive, rather than strongly negative; second, the shape of the distributions indicates the solution phase to be supersaturated in monomer relative to the bulk, and thus that these two systems may only be metastable. For future work, we derive expressions for the important statistical thermodynamic and chemical parameters of the interface energy in terms of 1) the surfactant concentration, 2) the temperature dependence, and 3) the concentrations of particles in the tail of the distribution.