Competition between shape anisotropy and deformation in the ordering and close packing properties of quasi-one-dimensional hard superellipse fluids

TL;DR

The paper probes how particle shape controls ordering and close packing in quasi-one-dimensional fluids of hard superellipses with tunable deformation parameter $n$ and aspect ratio $k$, using an exact transfer-operator framework. It finds a competition between shape anisotropy: increasing $k$ promotes nematic order while increasing $n$ promotes tetratic order, with a characteristic maximum in the pressure ratio $P/P_{\parallel}$ signaling the quasi-isotropic-to-nematic crossover. In the close-packing limit, the exponents obey $\alpha=2-1/n$, $\beta=1/n-3/2$, and $\gamma=1/2-1/n$, independent of $k$, and satisfy universal relations $\alpha+\beta=1/2$, $\beta+\gamma=-1$, and $\alpha+2\beta+\gamma=-1/2$; these results extend previously observed universality for hard superdisks to the broader class of superellipses. The work highlights a robust, shape-driven mechanism governing ordering and packing in confined anisotropic fluids and suggests broader applicability of the universal close-packing laws to other quasi-1D geometries and dimensions.

Abstract

We investigate the orientational ordering and close-packing behavior of a quasi-one-dimensional (q1D) system of hard superellipses, where the centers of the particles are confined to a line, but they can rotate freely within a two-dimensional plane. The particle shape is tuned between an ellipse and a rectangle by varying the deformation parameter (n). The elongation of the particle is changed using the aspect ratio (k). The pressure ratio between freely rotating and parallel hard superellipses, which displays a single peak, serves as an effective marker for the continuous structural change from quasi-isotropic to nematic ordering. Our findings reveal a competition between the parameters k and n, with k promoting nematic alignment and n favoring tetratic ordering. Notably, in the close-packing regime, the packing properties become independent of k, as the relevant exponents depend solely on n. Furthermore, certain combinations of these exponents exhibit universality, remaining invariant with respect to particle shape

Figures (8)

• Figure 1: (a) The effect of the deformation parameter n on the shape of hard superellipses with aspect ratio $k=2$ and (b) contact distance between two neighboring hard superellipses with orientations $\varphi_{1}$ and $\varphi_{2}$ in the q1D geometry in which the centers of the superellipses are constrained to move along the x-axis but can rotate freely within the $x$-$y$ plane.
• Figure 2: Effect of deformation parameter ($n$) and aspect ratio ($k$) on the orientation distribution function. The upper panel shows the results at $P^*=0.1$, while the lower one at $P^*=10$. The horizontal dashed line corresponds to the isotropic distribution ($f=1/\pi$).
• Figure 3: The pressure ratio $P/P_{\parallel}$ as function of $\rho^*=\rho d$ for (a) hard ellipses ($n = 2$) and varying aspect ratio $k$, and for varying deformation parameter $n$ for (b) $k=4$ and (c) $k=10$. Solid lines are exact results from TOM, dashed lines show $P_\text{iso}/P_{\parallel}$ . Insets show the two approximations for the density of quasi-isotropic--nematic structural change (full lines: $\rho_\text{max}$, dashed lines $\rho_\text{c,iso}$, the latter is also marked by the dashed vertical lines in the main graphs).
• Figure 4: The nematic order parameter $S$ as a function of $\rho^*=\rho d$ for various values of n (solid lines: $k=10$, dashed lines: $k=4$). Symbols indicate $S(\rho_\text{max})$.
• Figure 5: $P/P_{\parallel}$ as a function of density for two values of $k$ ($k=8.5$ and 10) and $n$ ($n=2$ and $n=20$). The vertical dashed lines mark the maximum of $P/P_{\parallel}$ which approximately separates the quasi-isotropic and nematic structures.