Entropically stabilized growth of a two-dimensional random tiling

The assembly of molecular networks into structures such as random tilings and glasses has recently been demonstrated for a number of two-dimensional systems. These structures are dynamically arrested on experimental time scales, so the critical regime in their formation is that of initial growth. Here, we identify a transition from energetic to entropic stabilization in the nucleation and growth of a molecular rhombus tiling. Calculations based on a lattice-gas model show that clustering of topological defects and the formation of faceted boundaries followed by a slow relaxation to equilibrium occur under conditions of energetic stabilization. We also identify an entropically stabilized regime in which the system grows directly into an equilibrium configuration without the need for further relaxation. Our results provide a methodology for identifying equilibrium and nonequilibrium randomness in the growth of molecular tilings, and we demonstrate that equilibrium spatial statistics are compatible with exponentially slow dynamical behavior.

Figure 1

(Color online) Schematic of (a) lattice and tiles; (b) defect diffusion mediated by tile detachment and reattachment; (c) annihilation of a defect pair (left to right) or generation of defect pair (right to left). (d) The fraction of empty sites as a function of time (in units of Monte Carlo sweeps) ($k_{B}T=0.3$ and lattice size $N={10}^6$). (e) Dependence of $c_g$, $c_{eq}$, and $\Delta c$ [defined in text and in (d)] on $\mu$.

Figure 2

(Color online) (a)–(c) Simulated growth of tilings for varying $\mu$ with $k_{B}T=0.2$: (a) $\mu =0.5$, high nucleation density, and irregular islands $\left[c\left(t\right)=0.7\right]$; (b) $\mu =1.5$, reduced nucleation density, and facetted islands $\left[c\left(t\right)=0.7\right]$; (c) $\mu =1.8$, example of an inhomogeneous tile distribution within strongly facetted island; (d) schematic of growth along a straight interface for $1<\mu <2$.

Figure 3

(Color online) (a) An inhomogeneous tiling resulting from strongly faceted growth of multiple islands, $k_{B}T=0.2$, $\mu =1.8$, and $N=1.6\times {10}^5$ ($\sim 4\times {10}^4$ tiles shown). (b) Clustering of topological charge shown in a charge density map. The value at a point corresponds to the number of upward-pointing minus downward-pointing defects within a range of three times the average defect separation. (c) The tiling after relaxing to an entropically maximized equilibrium state ($\sim 4\times {10}^4$ tiles shown), (d) the corresponding $c\left(t\right)$ behavior, and (e) height correlation functions during relaxation, indicating convergence to maximum randomness (top line: ${10}^3$ MCSs; second top line: ${10}^4$ MCSs; middle line: ${10}^5$ MCSs; second bottom line: ${10}^6$ MCSs; bottom line: ${10}^7$ MCSs).

Figure 4

(Color online) (a) Height correlation functions calculated for tilings immediately after the initial growth stage [where $c\left(t\right)=c_g$] at $k_{B}T=0.3$ with varying $\mu$ (top line: $\mu =1.90$; second top line: $\mu =2.00$; middle line: $\mu =2.02$; second bottom line: $\mu =2.04$; bottom line: $\mu =2.05$). (b) Calculations of stationary interfaces marking the tiled-untiled phase equilibrium. (c) Illustrative representation of the $\mu \text{−}T$ parameter space, indicating the region where tilings are entropically stabilized.