Formation of Polymorphic Cluster Phases for a Class of Models of Purely Repulsive Soft Spheres

We present results from density functional theory and computer simulations that unambiguously predict the occurrence of first-order freezing transitions for a large class of ultrasoft model systems into cluster crystals. The clusters consist of fully overlapping particles and arise without the existence of attractive forces. The number of particles participating in a cluster scales linearly with density, therefore the crystals feature density-independent lattice constants. Clustering is accompanied by polymorphic bcc-fcc transitions, with fcc being the stable phase at high densities.

Figure 1

The phase diagram of the GEM-4 model, as obtained by DFT. The shaded area represents the coexistence region of the liquid and the cluster bcc phase. The dash-dotted curve is the $\lambda$ line of the system, calculated in the MFA.

Figure 2

Two simulation snapshots of a GEM-4 system for $T^{*}=0.4$ and ${\rho }^{*}=2.5$ and 7 (left and right). The middle panel shows a close-up of one cluster. Particle diameters are not drawn to scale but are chosen to optimize the visibility of the structures.

Figure 3

DFT and MC results for the modified free energy density $\beta \tilde{F}{\rho }^{*}/N\equiv \beta F{\rho }^{*}/N+K_1{\rho }^{*}$ of the GEM-4 system against ${\rho }^{*}$ at $T^{*}=1$. A thermodynamically irrelevant term $K_1{\rho }^{*}$ has been added for clarity of presentation. The error bars of the MC results are smaller than the symbol size. Inset: a close-up on the region of stability of the cluster bcc lattice. Here, a different linear term $K_2{\rho }^{*}$ has been added.

Figure 4

Cluster size $n_c$ and localization parameter $\alpha {\sigma }^2$, for the GEM-4 system at $T^{*}=1$, plotted against the density ${\rho }^{*}$. Lines: DFT results, points: MC simulations. There are small discontinuities at the density of the bcc-fcc transition. The left inset shows DFT results that corroborate the $n_c\propto {\rho }^{*}$ relation. The dashed part of the line is an extrapolation to low densities, for which the crystal is unstable. The right inset shows the probability distribution of cluster population $N_c$ from simulations at ${\rho }^{*}=6.5$ (left curve) and ${\rho }^{*}=9$ (right curve).

Figure 5

Cluster density profile ${\rho }_{\mathrm{cl}}(r)$ of the GEM-4 system at $T^{*}=0.1$ and ${\rho }^{*}=9$. Inset: Semilogarithmic plot of ${\rho }_{\mathrm{cl}}(r)$ against $r^2$, demonstrating the Gaussian shape of the former.