Phase Boundaries of Bulk 2D Rhombi

Abstract

We conducted replica exchange Monte Carlo simulations to investigate the phase diagram of identical hard rhombi systems in two dimensions. The rhombi shape varies from nearly square-like, as their minor angle a approaches 90 degrees, to needle-like, as it approaches 0 degrees. For angles near 90 degrees, we observe an isotropic fluid, a rhombatic fluid, a rotator phase, and a columnar space-filling structure with increasing density. Conversely, as a approaches 0 degrees, the results resemble the needle limit. Even for angles as small as a = 20 degrees, we still obtain isotropic, nematic, and rhombatic fluids before reaching a rhombic solid, but the nematic phase gains importance with decreasing a. At a approximately 60 degrees, aperiodic space-filling structures with long-range six-fold orientational symmetry dominate over periodic candidates such as the rhombic and rhombille. This aperiodic solid undergoes a melting process leading to a phase with quasi-long-range six-fold orientational symmetry, a hexatic fluid, before reaching the isotropic phase.

Figures (9)

• Figure 1: a) Rhombus with side $\sigma$ and minor angle $a$. As $\sigma$ is taken as the unit of length, the only relevant parameter defining the hard rhombi system is $a$. For all panels in this figure we have set $a=60^\circ$. b) Periodic rhombic tiling (also known as centered rectangular tiling). The rectangular unit cell is outlined with black lines. c) Another possible periodic rectangular tiling. d) Poly-rhombic pattern (locally rhombic but with several different rhombic-like regions). e) Columnar arrangement. f) Rhombille conway2008 (also called tumbling blocks smith2002) periodic pattern. In panels c) and d), rhombi are colored according to their orientation, whereas in panel f), they are colored according to their local pattern (blue for cube-like shapes and red for stars).
• Figure 2: a) Probability density functions, b) compressibility factor, c) dimensionless isothermal compressibility, and d) order parameters as a function of the packing fraction for rhombi with a minor angle $a=85^\circ$. Black and red dashed lines in panels b) and c) correspond to angles $a=90^\circ$ and $80^\circ$, respectively. In panel d), black and red symbols depict orientational order parameters $P_2$ and $P_4$, respectively, while blue and cyan symbols correspond to bond-orientational order parameters $\Psi_4$ and $\Psi_6$, respectively. The vertical dotted lines signal phase transitions for the $a=85^\circ$ case. The red and cyan lines denote isotropic-rhombatic fluid-fluid and rotator-columnar solid-solid transitions, respectively. The green vertical line corresponds to the fluid-solid transition obtained from the building of quasi-long-range positional correlations and long-range bond-orientational correlations.
• Figure 3: a) $\log_{10}(gdr-1)$, with $gdr$ being the peaks of the radial distribution function, and b) $\log_{10}(g_4)$ as a function of distance. Both panels correspond to systems with $N=5000$, and for $a= 90^{\circ}$ (black), $85^{\circ}$ (red), and $80^{\circ}$ (cyan). Squares in panel a) and solid lines in b) correspond to $\eta=0.848$, 0.850, and 0.872 for decreasing $a$. Circles and short dashed lines correspond to $\eta=0.800$, and diamonds and dash-dotted lines correspond to $\eta=0.760$. The straight lines in the plots serve as visual guides. The upper line in panel b) is horizontal.
• Figure 4: Snapshots and their corresponding SSFs (computed by using freud ramasubramani2020freud) for rhombi with $a=85^\circ$ at increasing packing fraction. In the third snapshot, dashed lines form angles of $85^\circ$, following the columns and rows formed by the particles. In the fourth snapshot, dashed lines form angles of $90^\circ$ (left) and $85^\circ$ (right), with only the horizontal lines aligning along the columns (horizontal in this case) formed by the particles.
• Figure 5: a) Probability density functions, b) compressibility factor, c) dimensionless isothermal compressibility, and d) order parameters as a function of the packing fraction for rhombi with a minor angle $a=40^\circ$. Black and red dashed lines in panels b) and c) correspond to angles $a=45^\circ$ and $35^\circ$, respectively. In panel d), black and red symbols depict orientational order parameters $P_2$ and $P_4$, respectively, while blue and cyan symbols correspond to bond-orientational order parameters $\Psi_4$ and $\Psi_6$, respectively. The vertical dotted lines signal phase transitions for the $a=40^\circ$ case. The black and red lines denote isotropic-nematic and nematic-rhombatic fluid-fluid transitions, respectively. The green vertical line corresponds to the fluid-solid transition obtained from the melting of a rhombic solid. The black and red arrows point out $\chi$ peaks for the $a=45^\circ$ and $35^\circ$ curves, respectively.