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Entropically-stabilised 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 timescales so the critical regime in their formation is that of initial growth. Here we identify a transition from energetic to entropic stabilisation 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 occurs under conditions of energetic stabilisation. We also identify an entropically-stabilised 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 non-equilibrium randomness in the growth of molecular tilings, and we demonstrate that equilibrium spatial statistics are compatible with exponentially slow dynamical behaviour.