Entropic Colloidal Crystal Prediction: A Quantum Density Functional Theory Inspired Approach

Abstract

In pursuit of a colloidal analogue to quantum density functional theory (DFT) predictions of atomic crystal structures, we report a new, classical DFT that predicts the relative thermodynamic stability of colloidal crystals of hard, convex particle shapes. In contrast to standard classical DFT approaches, our theory maps the hard particle system to an auxiliary system in which we treat the particles as fixed "nuclei" embedded in a fictitious, spatially varying density field that distributes throughout the auxiliary system. By minimizing the free energy of the auxiliary system, and through comparison with known equations of state and free energy calculations using thermodynamic integration, we show that the auxiliary system with the lowest free energy corresponds to the most probable crystal of hard shapes in the original system.

Figures (4)

• Figure 1: In an EB-DFT calculation, the original system (left) of hard particles (e.g., truncated tetrahedra assembled into a cubic diamond crystal structure) is mapped to an auxiliary system (right) in which the hard particles are fixed in place and surrounded by a fictitious, spatially varying density field. The density field in the auxiliary system is optimized to minimize the EB-DFT free energy functional. We assert that the arrangement of hard particles that corresponds to the minimum energy of the auxiliary system corresponds to the maximum entropy of the original system.
• Figure 2: Entropy difference between HCP and FCC in a hard sphere system calculated from EB-DFT (black filled circles) fit, using the corresponding values of $\gamma$ in the upper panel, to the "ground truth" revised Speedy equation of state (blue curve) from almarzaClusterAlgorithmMonte2009. The gray triangles show the EB-DFT prediction with fixed $\gamma$, using the smallest and largest values of $\gamma$ from the upper panel. Error bars, calculated as the standard deviation about the mean, are smaller than the symbols.
• Figure 3: Entropy difference between dimer crystal and quasicrystal approximant in a hard tetrahedron system calculated using EB-DFT, fit to FL data along with the best fitting exponent $\gamma$ shown in the upper panel. Errors were calculated using the standard deviations of the values obtained from the simulation replicas, and are smaller than the symbols.
• Figure 4: Entropy difference vs vertex truncation of a tetrahedron, where $t=0$ is a tetrahedron and $t=1$ is an octahedron. EB-DFT predicts the $\beta-Sn$ structure becomes more stable than diamond at a truncation value above $t\sim0.742$, in agreement with previous simulation results damascenoCrystallineAssembliesDensest2012. The values of $\gamma$ used in the calculations are shown in the upper panel. As in Fig. \ref{['fig:sphere']}, errors were calculated using the standard deviations of the free energies obtained from the simulation replicas.