Dipolar Crystals: The Crucial Role of the Clinohexagonal Prism Phase

We report a new phase called clinohexagonal prism (CHP) that accounts for all the ground states of dipolar hard spheres prepared at any density. This phase merely consists of an oblique prismatic lattice with a hexagonal base. Our calculations show that at intermediate densities, a special close packed body-centered orthorhombic phase coincides with the CHP phase in the ground state for a wide density window. In the high packing regime, i.e., in the vicinity of the density of the hexagonal close packed phase, it is a limiting case of the CHP phase with vanishing obliquity that emerges. These findings provide a unified and clarified view of the solid-solid transitions occurring at zero temperature in dipolar systems and should be relevant in other related molecular or soft matter systems governed by anisotropic (and possibly isotropic) soft potentials.

Figure 1

Clinohexagonal prism (CHP) crystal structure and some special relevant limits or cases body-centered orthorhombic (BCO), BCT, HCP, face-centered cubic (FCC), and hexagonal primitive (HP). Corresponding values of the reduced cohesive energy $u$ are provided too. Standard unit cells are in gray and equivalent (special) CHP cells for the phases BCO, BCT, and FCC are represented by blue dashed lines. The characteristic angle of 60° is indicated as well. $F$ and AF stands for ferromagnetic and antiferromagnetic arrangements, respectively. Lengths $a$, $b$, $c$ are given in units of the bead diameter $d$. The packing fractions ${\eta }_{\mathrm{BCT}}$, ${\eta }_{\mathrm{HCP}}$, ${\eta }_{\mathrm{FCC}}$, ${\eta }_{\mathrm{HP}}$ of BCT, HCP, FCC, HP structures, respectively, are reported too. The constitutive beads are omitted for the sake of clarity.

Figure 2

Reduced crystal cohesive energy $u$ as a function of the rescaled packing fraction ${\eta }^{*}$. The two (CHP) special/limit structures BCT and HCP are illustrated with obliquity $\vartheta =30°,0°$, respectively, see also text. The inset is a magnification around the CHP/right hexagonal prism (RHP) transition occurring at around ${\eta }^{*}=0.951$.

Figure 3

(a) Angular order parameters $\vartheta$ and $\phi$ as a function of the relative packing fraction ${\eta }^{*}$ in the CHP representation, see Eqs. (4) and (5). The inset illustrates $\vartheta$ and $\phi$ in the CHP unit cell. (b) Sidelengths $a,b\ge a,c=1$ of the BCO unit cell, see also the inset, as a function of ${\eta }^{*}$. The shaded region corresponds to ${\eta }^{*}>{\eta }_{\mathrm{tr}}^{*}$ where the BCO phase is beaten by the RHP. The profile $a({\eta }^{*})$ follows Eq. (7) and those of $b({\eta }^{*})$ and $a/b$ are obtained by combining it with Eq. (6).

Figure 4

Sidelength behavior of $a$ (left) and $c$ (right), around CHP/RHP transition, as a function of ${\eta }^{*}$. The inset illustrates the corresponding geometric order parameters of the CHP cell. Notice that $c=1$ when ${\eta }^{*}<{\eta }_{\mathrm{tr}}^{*}$, not shown due to the severe difference in length scale variation involved there.