Shape and Symmetry Determine Two-Dimensional Melting Transitions of Hard Regular Polygons

The melting transition of two-dimensional systems is a fundamental problem in condensed matter and statistical physics that has advanced significantly through the application of computational resources and algorithms. Two-dimensional systems present the opportunity for novel phases and phase transition scenarios not observed in 3D systems, but these phases depend sensitively on the system and, thus, predicting how any given 2D system will behave remains a challenge. Here, we report a comprehensive simulation study of the phase behavior near the melting transition of all hard regular polygons with $3\le n\le 14$ vertices using massively parallel Monte Carlo simulations of up to $1\times {10}^6$ particles. By investigating this family of shapes, we show that the melting transition depends upon both particle shape and symmetry considerations, which together can predict which of three different melting scenarios will occur for a given $n$. We show that systems of polygons with as few as seven edges behave like hard disks; they melt continuously from a solid to a hexatic fluid and then undergo a first-order transition from the hexatic phase to the isotropic fluid phase. We show that this behavior, which holds for all $7\le n\le 14$, arises from weak entropic forces among the particles. Strong directional entropic forces align polygons with fewer than seven edges and impose local order in the fluid. These forces can enhance or suppress the discontinuous character of the transition depending on whether the local order in the fluid is compatible with the local order in the solid. As a result, systems of triangles, squares, and hexagons exhibit a Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) predicted continuous transition between isotropic fluid and triatic, tetratic, and hexatic phases, respectively, and a continuous transition from the appropriate $x$-atic to the solid. In particular, we find that systems of hexagons display continuous two-step KTHNY melting. In contrast, due to symmetry incompatibility between the ordered fluid and solid, systems of pentagons and plane-filling fourfold pentilles display a one-step first-order melting of the solid to the isotropic fluid with no intermediate phase.

Popular Summary

Everyday materials have solid and fluid phases. Ice, for example, melts into liquid water at $0°\mathrm{C}$. Some materials have phases in between solid and fluid that give them useful properties, such as the liquid crystals in display monitors that can manipulate light. Two-dimensional materials can also exhibit a strange in-between state of matter called the $x$-atic phase, which lacks latticelike order (like a fluid) but has well-defined bonds (like a solid). While researchers have a good idea about what kinds of molecules can form liquid crystals (long rod-shaped ones), it is not clear what about a particle makes it form an $x$-actic phase. Using numerical simulations, we show that geometrical symmetry plays a large role in determining what kinds of particles exhibit $x$-atic phases and how the solid, $x$-atic, and fluid phases transition from one to the other.

Our analysis looks at the interactions among a large number (up to roughly one million) of identical polygons, from 3 to 14 sides. We find that ensembles of triangles, squares, and hexagons end up with large regions of $x$-atic order and smooth phase transitions because their body symmetry is compatible with that of the solid, maximizing entropy. Polygons with seven or more edges behave like hard disks with sharper transitions, while pentagons have no $x$-atic phase.

The way two-dimensional materials melt and form (or do not form) $x$-atic phases has been the subject of intense exploration since the 1970s. We have shown that just changing the number of sides in polygon-shaped particles can lead to diverse phase-transition behavior. Future studies might look at how rounding of polygons produces more phases.

Figure 1

The polygons studied in this work are the regular $n$-gons with $3\le n\le 14$ and the 4-fold pentille labelled $5^{*}$.

Figure 2

(a) Pictorial definitions of ${\overrightarrow{r}}_a$, ${\overrightarrow{r}}_b$, ${\theta }_a$, ${\alpha }_{ab}$, and ${\overrightarrow{q}}_0$. (b) We map order parameters to a color wheel for visualization. Each order parameter is rotated by its average in a given frame so that the average order parameter is colored green. The color wheel is the color part of the cubehelix [38] color map at constant apparent luminance ($v=0.5$, $\gamma =1$, $s=4.0$, $r=1$, $h=1$).

Figure 3

Example phase transitions from the three melting scenarios. We show data for (a)–(c) hexagons ($N=512^2$) following the continuous KTHNY scenario, (d)–(f) pentagons ($N=1024^2$) exhibiting a first-order isotropic fluid-to-solid transition, and (g)–(i) octagons ($N=1024^2$) following an intermediate scenario exhibiting a first-order fluid-to-hexatic transition. For each shape, the top left-hand panels show local density histograms [(a),(d),(g)]. The right-hand panels show bond and positional order parameters at selected densities [(b),(e),(h)]. The bottom left-hand panels show sub-block scaling analysis [(c),(f),(i)].

Figure 4

Defect counts as a function of density obtained with cell-edge counting. The particle count is the number of particles that belong to any defect cluster classified into neutral, Burgers-charged, and disclination-charged clusters. The panels show results of the defect analysis plots for polygons $n=5$ (a), $5^{*}$ (b), 6 (c), 7 (d), 8 (e), 9 (f), 10 (g), 11 (h), 12 (i), 13 (j), and 14 (k), averaged over 4–8 runs with 19–49 frames per run at the system size $N=128^2$. The defect count algorithm is expensive, so we do not run it on the largest size simulations we perform. We also show the phase diagram overlaid on the counts. Error bars at 1 standard deviation are smaller than the symbol size.

Figure 5

Representative snapshot of the hexagon system $N=128^2$ at density $\varphi =0.690$. (a) Particles are colored by the number of edges in the Voronoi cell. Red, blue, and green cells have seven, six, and five sides, respectively. (b) Particles are colored by $⟨\mathrm{IQ}⟩-\mathrm{IQ}$. Blue indicates large deviation from $⟨\mathrm{IQ}⟩$ and yellow indicates little deviation.

Figure 6

Potential of mean force and torque plots for polygons $n=3–13$ computed at the highest density isotropic fluid for each polygon. We choose the zero energy reference as the average over a large region $25\sigma$ from the center (not shown). The data are shown as a contour plot with selected free energy or entropy well contours. Computed from $N=256^2$ simulations.

Figure 7

Phase diagram of hard polygon melting behavior. Disk results ($n\to \infty$) are from Refs. [12, 25]. The label $5^{*}$ refers to the fourfold pentille. The $n=3$ solid is a honeycomb lattice with alternating triangle orientations, the $n=4$ solid is a square lattice, and the $n\ge 5$ solids are all hexagonal.

Figure 8

Latent heat (green diamonds) of the first-order phase transition and the two component factors plotted separately. $P_c^{*}$ (red circles) is the coexistence pressure, and the area difference per particle $\delta A$ is shown below (blue squares). Filled symbols are regular polygons, and the open symbols are the fourfold pentille. Error bars are estimated at 0.001 error in the transition densities. Data for disks ($n\to \infty$) are from Ref. [12].