Entropic crystal–crystal transitions of Brownian squares

Abstract

When a monolayer of hard microscale square platelets, produced lithographically, is osmotically concentrated in a flat plane to raise the particle area fraction ϕ A , an order-order transition occurs between a hexagonal rotator crystal and a rhombic crystal. Strikingly, phases having fourfold symmetry are not observed at any ϕ A . The rhombic lattice angle α increases continuously with ϕ A , as the system maximizes its total rotational and translational entropy. A cage model, based on packing rotationally swept squares, or "squaroids," reasonably predicts the measured αðϕ A Þ, indicating that rotational entropy and the square particle shape combine to produce the rhombic unit cell.colloid | phase behavior | structure | two dimensions | thermal fluctuations T hermodynamic structures of liquid crystalline mesophases depend in an important way on the geometrical shapes of their constituent molecules. This shape dependence is particularly clear for hard-core repulsive interactions, because the state having minimum free energy is determined by maximizing the entropy through the available free volume per molecule. For solutions of long hard rods, Onsager first showed that the onset of nematic liquid crystalline order depends on molecular geometry through the excluded volume (1). Molecules in two dimensions (2D) are particularly interesting because their shapes and symmetries can determine whether or not a first-order freezing phase transition is replaced by an intervening mesophase, which exhibits a spatial power-law decay in orientational order but only short-range translational order. For hard disks, a sixfold hexatic phase can interpolate between the isotropic liquid and hexagonal 2D crystalline phases (2-9). For hard squares, Monte Carlo (MC) simulations (10) predict that a high-density crystal phase having square symmetry will melt into a tetratic mesophase having a fourfold orientational order at lower densities: The fourfold symmetries of the square crystal and the tetratic mesophase reflect the underlying symmetry of the constituent square particle. However, no experiments have yet been made on systems of Brownian squares that test these predictions.Lithographic methods have facilitated the synthesis of uniform particulate dispersions containing shape-designed colloids that can serve as model systems for molecular liquid crystals (11-16). When properly controlled, these customized dispersions can be used to study interesting and fundamental problems of statistical mechanics of dense many-particle systems. Although the 2D phase behavior and jammed states of a thermal system of hard pentagons, which cannot fully tile a plane, have been examined (17), a very different set of phases and phase transitions could be observed by investigating a system of Brownian squares which can fully tile a plane.To investigate this, we have explored the phase behavior of a model aqueous dispersion of lithographic square platelets (i.e., "squares") that diffuse in a plane. We form a monolayer of microscale squares near the bottom s...

Shape-designed frustration by local polymorphism in a near-equilibrium colloidal glass

“…We tilt the cell by about 5°from horizontal, causing kites to slowly move toward the lower portion of the cell by weak gravitational sedimentation as they also diffuse. We allow equilibration to occur over 7 months, which is more than 5× longer than that used for other 2D systems of shapes which have formed liquid crystals or crystals after only a few weeks of similarly slow compression (30)(31)(32). As ϕ A is increased quasi-statically, the average rotational and translational diffusion of individual kites becomes strongly restricted (i.e., bound), whereas the system of kites retains a disordered spatial structure and uniform distribution of pointing directions, as shown in Fig.…”

“…or a crystal phase possessing at least quasi-long-range order (29)(30)(31)(32). Thus far, no empirical observations of 2D Brownian systems exist that show a disordered liquid phase being slowly compressed into a disordered glassy solid-like state.…”

Phase diagram of octapod-shaped nanocrystals in a quasi-two-dimensional planar geometry

“…34,36 Contrary to the simulation results, a hexagonal plastic-crystal (rotator) phase appeared in experiments of square colloidal particles under confinement. 37 This finding could be explained by the roundness of particles. 38 Despite this strong interest in (quasi-)2D systems, there are still many unanswered questions, even for a relatively simple system consisting of monodisperse hard disks.…”

“…Our results for L/D < 1.7 are reminiscent of the results obtained in experiments and simulations of rounded square particles. 37,38 This correspondence can be explained by the shape of the octapods for these values of L/D, since the particles have an interaction cross section that is roughly a rounded square with concave edges. For octapod-shaped nanoparticles there are also experimental indications that a rhombic monolayer can form when the pod length-to-diameter ratio is small.…”

“…The mesophase behavior of convex anisotropic particles in quasi-2D, such as rods and polygonal (e.g., square and pentagonal) particles, has for instance been studied in experiments and by simulations. [29][30][31][32][33][34][35][36][37][38] a) Electronic mail: W.Qi@uu.nl b) Electronic mail: M.Dijkstra1@uu.nl For rods, theoretical investigations 29,31 suggested a possible Kosterlitz-Thouless (KT) isotropic-nematic phase transition, when the aspect ratio exceeded 7.0, and its existence was confirmed by simulations and experiments. [30][31][32] For hard squares and rectangles, a tetratic phase with quasi-long-range orientational order was observed in simulations.…”