Explicitly stable fundamental-measure-theory models for classical density functional theory

A microscopic approach to crystallization: Challenging the classical/non-classical dichotomy

We present a fundamental framework for the study of crystallization based on a combination of classical density functional theory and fluctuating hydrodynamics that is free of any assumptions regarding order parameters and that requires no input other than molecular interaction potentials. We use it to study the nucleation of both droplets and crystalline solids from a low-concentration solution of colloidal particles using two different interaction potentials. We find that the nucleation pathways of both droplets and crystals are remarkably similar at the early stages of nucleation until they diverge due to a rapid ordering along the solid pathways in line with the paradigm of "non-classical" crystallization. We compute the unstable modes at the critical clusters and find that despite the non-classical nature of solid nucleation, the size of the nucleating clusters remains the principle order parameter in all cases, supporting a "classical" description of the dynamics of crystallization. We show that nucleation rates can be extracted from our formalism in a systematic way. Our results suggest that in some cases, despite the non-classical nature of the nucleation pathways, classical nucleation theory can give reasonable results for solids but that there are circumstances where it may fail. This contributes a nuanced perspective to recent experimental and simulation work, suggesting that important aspects of crystal nucleation can be described within a classical framework.

Crystal Polymorphism Induced by Surface Tension

We use classical density functional theory (cDFT) to calculate fluid-solid surface tensions for fcc and bcc crystals formed by hard spheres and Lennard-Jones (LJ) particles. For hard spheres, our results show that the recently introduced "explicitly stable" functionals perform as well as the state of the art, and for both interaction potentials, our results compare well to simulation. We use the resulting bulk and interfacial energies for LJ to parametrize a capillary model for the free energy of small solid clusters and thereby determine the relative stability of bcc and fcc LJ clusters. We show a crossover from bcc to fcc stability as cluster size increases, thus providing insight into long-standing tension between simulation results and theoretical expectations. We also confirm that the bcc phase in contact with a vapor is unstable, thus extending earlier zero-temperature results. Our Letter demonstrates the potential of cDFT as an important tool in understanding crystallization and polymorphism.

Using classical density functional theory to determine crystal-fluid surface tensions

Classical density functional theory is used to determine the fluid-solid surface tensions for low-index faces of crystals of hard spheres and Lennard-Jones particles. The calculations make use of the recently introduced explicitly stable fundamental measure theory model for hard spheres, and we show that this gives state-of-the-art accuracy compared to simulation. For the Lennard-Jones system, results are presented for both solid-liquid and solid-vapor interfaces, and in both cases the FCC results compare favorably with existing results from the literature. We find that the BCC crystal has significantly lower solid-liquid surface tension than the FCC structure. For the solid-vapor interface, our results indicate that the BCC phase is unstable with respect to transition to the HCP structure, in agreement with various zero-temperature results in the literature.

Classical density functional theory in the canonical ensemble

Classical density functional theory for finite temperatures is usually formulated in the grand-canonical ensemble where arbitrary variations of the local density are possible. However, in many cases the systems of interest are closed with respect to mass, e.g., canonical systems with fixed temperature and particle number. Although the tools of standard, grand-canonical density functional theory are often used in an ad hoc manner to study closed systems, their formulation directly in the canonical ensemble has so far not been known. In this work, the fundamental theorems underlying classical DFT are revisited and carefully compared in the two ensembles showing that there are only trivial formal differences. The practicality of DFT in the canonical ensemble is then illustrated by deriving the exact Helmholtz functional for several systems: the ideal gas, certain restricted geometries in arbitrary numbers of dimensions, and, finally, a system of two hard spheres in one dimension (hard rods) in a small cavity. Some remarkable similarities between the ensembles are apparent even for small systems with the latter showing strong echoes of the famous exact of result of Percus in the grand-canonical ensemble.

Direct Correlation Function of a Crystalline Solid

Direct correlation functions (DCFs), linked to the second functional derivative of the free energy with respect to the one-particle density, play a fundamental role in a statistical mechanics description of matter. This holds, in particular, for the ordered phases: DCFs contain information about the local structure including defects and encode the thermodynamic properties of crystalline solids; they open a route to the elastic constants beyond low temperature expansions. Via a demanding numerical approach, we have explicitly calculated for the first time the DCF of a solid: based on the fundamental measure concept, we provide results for the DCF of a hard sphere crystal. We demonstrate that this function differs at coexistence significantly from its liquid counterpart-both in shape as well as in its order of magnitude-because it is dominated by vacancies. We provide evidence that the traditional use of liquid DCFs in functional Taylor expansions of the free energy is conceptually wrong and show that the emergent elastic constants are in good agreement with simulation-based results.

Self-consistent construction of bridge functional based on the weighted density approximation

A parameter-free bridge functional is presented using a weighted density approximation (WDA). The key point of this scheme is the utilization of Baxter's relation connecting the second-order direct correlation function (DCF) to the higher-order DCF with the density derivative. The free energy density required for the WDA is determined in a self-consistent manner using Baxter's relation and Percus's test particle method. This self-consistent scheme enables us to employ any type of potential model for simple liquids. The new functional is applied to calculate density distribution functions for the inhomogeneous fluids interacting via the hard-sphere, Lennard-Jones, and hard-core Yukawa potentials under an external field from a planar wall and a slit pore.

Classical density-functional theory applied to the solid state

The standard model of classical density-functional theory (cDFT) for pair potentials consists of a hard-sphere functional plus a mean-field term accounting for long ranged attraction. However, most implementations using sophisticated fundamental measure hard-sphere functionals suffer from potential numerical instabilities either due to possible instabilities in the functionals themselves or due to implementations that mix real- and Fourier-space components inconsistently. Here we present a new implementation based on a demonstrably stable hard-sphere functional that is implemented in a completely consistent manner. The present work does not depend on approximate spherical integration schemes and so is much more robust than previous algorithms. The methods are illustrated by calculating phase diagrams for the solid state using the standard Lennard-Jones potential as well as a new class of potentials recently proposed by Wang et al. [Phys. Chem. Chem. Phys. 22, 10624 (2020)PPCPFQ1463-907610.1039/C9CP05445F]. The latter span the range from potentials for small molecules to those appropriate to colloidal systems simply by varying a parameter. We verify that cDFT is able to semiquantitatively reproduce the phase diagram in all cases. We also show that for these problems computationally cheap Gaussian approximations are nearly as good as full minimization based on finite differences.