Rhombic Preordering on a Square Substrate

A competition of incommensurate symmetries occurs whenever a system is forced to conform to an ordering that is different from the intrinsically preferred structure of the system itself. As a model system of such a competition, we study the rivalry between the triangular ordering of hard disks and the square symmetry induced by a periodic square substrate. By using density functional theory as well as Monte Carlo computer simulations, we determine the full phase behavior for the case of one particle per minimum. We observe a rhombic preordering structure preceding the hexagonal solid as a direct consequence of the competing symmetries. Furthermore, the square-rhombic transition is reentrant with increasing substrate interaction. Our predictions can be verified in experiments of colloids in laser fields.

Figure 1

Schematic view of the square substrate potential of lattice constant $a$. The disks have diameter $\sigma$ and on average there is only one particle located at each minimum of the substrate.

Figure 2

(a) Phase diagram for hard disks on a square substrate showing the three different phases, square modulated fluid, rhombic, and triangular crystal. The dashed line indicates the second order and the solid line the first order phase transition, where the dot denotes the triple point. Density contour plots of the three phases are included obtained for a fixed external potential with $V_0/k_{B}T=1$ and packing fractions (a’) $\eta =0.7069$, [(a”), (b)] $\eta =0.7257$, and (c) $\eta =0.7383$. The stars indicate the simulation parameters for Fig. 3.

Figure 3

Typical configurations from Monte Carlo simulations for a system with ${128}^2$ particles (only a small section of the system is shown) and the corresponding scattering function (insets) for a packing fraction $\eta =0.69$ and external field strength (a) $V_0/k_{B}T=1$, (b) 2, and (c) 4 (as indicated in Fig. 2). The particles are colored according to their local structure: black denotes no rhombic order, orange and blue denote perfect rhombic order with rows along the $x$ and $y$ directions, respectively. The scattering peak at ${\mathbf{k}}_R=(2\pi /a)$ (1, $1/2$) due to the rhombic preordering is indicated in the inset of (b).

Figure 4

Monte Carlo simulation results for packing fraction $\eta =0.69$ and for system sizes $N={32}^2$, ${64}^2$, ${96}^2$, and ${128}^2$. (a) The scattering function $S(\mathbf{k})$ at the $\mathbf{k}$ vector that is indicative of rhombic order, ${\mathbf{k}}_R\equiv 2\pi /a(1,1/2)$, as a function of the amplitude $V_0$ of the external field. $S^{\prime }({\mathbf{k}}_R)$ obtained from DFT in the inset. (b) The fluctuations of the energy divided by the system size $N$ as a function of $V_0$. The inset shows the average energy per particle, for which the error is of the order of the line thickness.