(18l) A Classical Density Functional Theory of Entropic Colloidal Crystals

Hard particle crystals can self-assemble a wide variety of colloidal crystals through entropy alone ranging from crystals simple cubic crystals to more complicated structures like clathrates, quasicrystals, and even structures with up to 432 particles in the unit cell. Despite the absence of any explicit interactions beyond volume exclusion, hard particles will optimize their degrees of freedom to maximize system entropy when crowded, leading to emergent local and directional entropic forces, much like bonds resulting in ordered crystals isostructural to those found in atomic systems.

In this work, we present a classical density functional theory (cDFT) that predicts the relative thermodynamic stabilities of hard particle colloidal crystals. Unlike in the standard cDFT treatment of hard particles where hard particles are treated like a solvent, we the hard particles as fixed solute particles embedded in a solvent of fictitious pseudoparticles. We quantify the pseudoparticle and hard particle interactions as an external field and derive a free energy functional. By minimizing this functional, we can find the thermodynamically preferred pseudoparticle distribution for a given configuration of hard particle shapes. We show that this pseudoparticle distribution corresponds to the most probably crystal of the hard particle shape by comparing the energies between different crystal structures of a particular set of hard particle shape. We validate this by comparison with simulations. We show computational results for a variety of systems such as hard sphere systems, the stability of truncated tetrahedra as a function of truncation for the diamond and beta-tin crystal structures, and crystal structures of hard pentagons.