Free-energy model for nanoparticle self-assembly by liquid crystal sorting.

We modeled the experimentally observed self-assembly of nanoparticles (NPs) into shells with diameters up to 10 μm, via segregation from growing nematic domains. Using field-based Monte Carlo simulations, we found the equilibrium configurations of the system by minimizing a free-energy functional that includes effects of excluded-volume interactions among NPs, orientational elasticity, and the isotropic-nematic phase-transition energy. We developed a Gaussian-profile approximation for the liquid crystal (LC) order-parameter field that provides accurate analytical values for the free energy of LC droplets and the associated microshells. This analytical model reveals a first-order transition between equilibrium states with and without microshells, governed mainly by the competition of excluded-volume and phase-transition energies. By contrast, the LC elasticity effects are much smaller and mostly confined to setting the size of the activation barrier for the transition. In conclusion, field-based thermodynamic methods provide a theoretical framework for the self-assembly of NP shells in liquid crystal hosts and suggest that field-based kinetic methods could be useful to simulate and model the time evolution of NP self-assembly coupled to phase separation.