Rounded hard squares confined in a circle

Abstract

Packing under confinement could generate rich ordered structures through entropic effects, which is a fundamental problem in condensed matter, biophysics and material science. The influence of confinement to the anisotropic hard particles--particularly regarding the emergence of topological defect structures--remains poorly understood. Recent studies have shown that granular rods confined within circular boundaries can cluster into square like super-particles, forming four disclinations. In this study, we employ Monte Carlo simulations in the NPT ensemble to investigate how circular confinement influences the ordered structures of rounded-corner hard-squares with varying roundness. At low roundness, the system forms an integrated cross-shaped domain with tetratic order and four +1/4 disclinations in the corners, along with some column shifts. As roundness increases, we found a new partition structure, where particles self-assemble into six domains separated by six +1/4 disclinations and a central -1/2 disclination. Our findings reveal that the interplay between confinement geometry and colloid shape can drive entropy governed structural transitions, offering new insights for the design of topological metamaterials.

Figures (6)

• Figure 1: Average of local bond angular orders ($\bar{\Psi}_{4}$ and $\bar{\Psi}_{6}$) as functions of roundness $\zeta$. The state diagram is segmented into four distinct regions, with representative snapshots illustrating the corresponding structures at $\zeta=0.0,0.41,0.6$, and $0.8$, respectively. The particles sketched above the diagram depict rounded‑corner hard-squares with roundness values of $\zeta=0.1,0.4,0.6$, and $0.8$.
• Figure 2: Topological defects under square and partition structures. The snapshot corresponding to the partition structure is taken at $\zeta=0.4$, and that for the square structure at $\zeta=0.1$. In each snapshot, red particles indicate crystalline domains, blue triangles denote $+1/4$ disclinations, and the purple hexagon represents a $-1/2$ disclination. Enlarged views of the defect regions are shown in the center. Orientation changes are computed using the orientation whose angular difference from the neighboring particle is smaller than $\pi/4$, consistent with the particle symmetry and indicated by the arrows.
• Figure 3: Column shifts in square structure. Snapshots for $\zeta=$ (a) $0.0$, (b) $0.1$, (c) $0.2$, with Color coding based on bond angular order $\text{$\Psi$}_{4}$. The shifted columns are highlighted by red rectangles.
• Figure 4: Detailed structures of the square and partition states in the confined RCHS system. Each panel contains two snapshots with distinct color code. In both frames, the left image visualizes the crystalline domain structure: particles within the same domain are assigned the same color, while gray particles are excluded from any domain classification. The right image displays the corresponding radial order. Panel (a) shows the square phase at $\zeta=0.1$, and panel (b) illustrates the partition phase at $\zeta=0.4$.
• Figure 5: Order parameters as functions of roundness $\zeta$ from $0$ to $0.9$. (a) Domain orientational order $P_{4}$(orange line) and $P_{6}$(blue line), with a schematic illustrating the definition of domain orientation. (b) The standard deviation of $P_{n}$, representing fluctuations of domain orientational order. (c) The average local four-fold orientational order $\bar{\Phi}_{4}$ (blue line); the global four-fold orientational order $\Phi_{{\rm G}}$ (green line); radial order $\bar{\Phi}_{{\rm r}}$ (orange line).