Classical density functional theory, unconstrained crystallization, and polymorphic behavior

While in principle, classical density functional theory (cDFT) should be a powerful tool for the study of crystallization, in practice this has not so far been the case. Progress has been hampered by technical problems which have plagued the study of the crystalline systems using the most sophisticated fundamental measure theory models. In this paper, the reasons for the difficulties are examined and it is proposed that the tensor functionals currently favored are in fact numerically unstable. By reverting to an older, more heuristic model it is shown that all of the technical difficulties are eliminated. Application to a Lennard-Jones fluid results in a demonstration of power of cDFT to describe crystallization in a highly inhomogeneous system. First, we show that droplets attached to a slightly hydrophobic wall crystallize spontaneously upon being quenched. The resulting crystallites are clearly faceted structures and are predominantly HCP structures. In contrast, droplets in a fully periodic calculational cell remain stable to lower temperatures and eventually show the same spontaneous localization of the density into “atoms” but in an amorphous structure having many of the structural characteristics of a glass. A small change of the protocol leads, at the same temperature, to the formation of crystals, this time with the fcc structure typical of bulk Lennard-Jones solids. The fcc crystals have lower free energy than the amorphous structures which in turn are more stable than the liquid droplets. It is demonstrated that as the temperature is raised, the free energy differences between the structures decrease until the solid clusters become less stable than the liquid droplets and spontaneously melt. The presence of energy barriers separating the various structures is therefore clearly demonstrated.

Figure 1

Phase diagram for the Lennard-Jones potential cutoff at $r_c=3\sigma$ as determined from the model DFT functional minimized at constant chemical potential. The dashed lines are for visual purposes only and indicate the critical point and the triple point.

Figure 2

Cooling droplet resulting in crystallization. The images show the density (as a contour plot) with both the intensity and the opacity of the color increasing from very light-blue and completely transparent at zero density to dark blue and 70% opaque at $n{\sigma }^3=1.5$ for the liquid droplets. For the crystal, the maximum density reaches 216 and the contours used in the plot are equally spaced at $0.8\le n{\sigma }^3\le 10$. The top three images from left to right are at reduced temperatures of $T^{*}=0.8,0.7$, and 0.6 and $T^{*}=0.5$ for the bottom left image. The last two images are different views of the same crystal that spontaneously formed at $T^{*}=0.45$.

Figure 3

The local density of droplets at temperatures $k_{B}T/\epsilon =0.8,0.7,0.6$ from left to right in the first row and $0.5,0.4,0.3$ in the second row: the last figure shows the amorphous cluster that forms spontaneously. For the droplets, the figure displays a planar slice through the center of the droplet with the same color scale, ranging from zero to density $\rho {\sigma }^3=1.5$. For the amorphous cluster, the representation is a contour plot with the same characteristics as in Fig. 2 and with the actual maximum density being $n{\sigma }^3=148$.

Figure 4

(a) Amorphous structure obtained with DFT shown as a contour plot as in Fig. 3. (b) Corresponding coordinates identified using particle tracking. Blue particles are identified as defective icosahedron and others (red) are deliberately shown smaller.

Figure 5

(a) Pair distribution function of amorphous structures as obtained at different temperatures. (b) Population of face-centered-cubic, hexagonal compact, defective icosahedron, and icosahedron clusters.

Figure 6

A comparison of the amorphous (left) and crystalline (right) clusters obtained in the homogeneous case, shown as contour plots. Note the faceting of the latter structure.

Figure 7

The (canonical, Helmholtz) free energy of the liquid, amorphous, and crystalline clusters as functions of the temperature. For the amorphous and crystalline clusters, the curves were obtained by starting with the lowest temperature structures, increasing the temperature, and minimizing. The inset shows the free energy of the amorphous and solid clusters relative to the liquid droplet.

Figure 8

Vacancy concentration as a function of average density determined using the mRSLT model both by minimizing the free energy at constant particle number and at constant chemical potential with subsequent minimization with respect to the lattice spacing in both cases. Simulations results due to Kwak et al. [74] and Bennett and Alder [75] are shown as are the WBII results of Oettel et al. [16].