Empty smectic liquid crystals of hard nanorings: Insights from a second-virial theory

Inspired by recent simulations on highly open liquid crystalline structures formed by rigid planar nanorings, we present a simple theoretical framework explaining the prevalence of smectic over nematic ordering in systems of ring-shaped objects. The key part of our study is a calculation of the excluded volume of such nonconvex particles in the limit of vanishing thickness to diameter ratio. Using a simple stability analysis we then show that dilute systems of ring-shaped particles have a strong propensity to order into smectic structures with an unusual antinematic order while solid disks of the same dimensions exhibit nematic order. Since our model rings have zero internal volume, these smectic structures are essentially empty, resembling the strongly porous structures found in simulation. We argue that the antinematic intralamellar order of the rings plays an essential role in stabilizing these smectic structures.

Figure 1

Representative configuration of the smectic-A (SmA) phase formed by a system of $N=500$ rigid rings obtained by molecular dynamics simulations (see Ref. [19]). This system is modeled as a collection of $n_b=56$ tangent beads of diameter $\sigma$ interacting via a soft WCA potential. The radius of the rings (from the center of the particles to the center of the peripheral beads) is $r=8.92\sigma$. The particle density is $\rho r^3=N{r}^3/V=1.5$, which corresponds to a very-low-packing fraction $\varphi =\pi N{n}_b{\sigma }^3/\left(6V\right)=0.062$. The system has been replicated along the direction of the layers to aid the visualization of the SmA phase. The typical lamellar spacing is indicated by ${\lambda }^{*}$. The rings are aligned antinematically with their normals pointing perpendicular to the lamellar director $\widehat{\mathbf{n}}$. Traces of interlayer rings with transverse, nematic order are visible in the center of the image.

Figure 2

Visualization of the excluded volume of a pair of infinitely thin hard rings (a) and disks (b) with radius $r=D/2$ at fixed mutual orientation defined within a particle-based frame spanned by the three unit vectors. The overlap figure of rings is strongly nonconvex and contains sharp inward cusps.

Figure 3

Overlap between a ring and a disk; rolling the disk around the circle in the $\{{\widehat{\mathbf{w}}}_{\mathbf{1}},\widehat{\mathbf{v}}\}$ plane at fixed mutual orientation projects a typical dimer composed of two overlapping circles with radius $r=D/2$. Its surface can be calculated by decomposing the area into a circular section (I, checkerboard), a circle segment (II, stripes), and a triangular section (III, waves). The dotted lines in the top-right sketch denotes the convex hull of the fused circles.

Figure 4

(a) Bifurcations in a binary isotropic fluid mixture of hard disks and rings. Shown are the normalized particle density ${\rho }_0^{*}{r}^3$ plotted versus the mole fraction $x$ of rings. The emergent type of liquid crystalline order is given by the curve with the lowest density. Pure rings ($x=1$) exhibit a direct transition from isotropic to smectic order, whereas pure systems of disks ($x=0$) form a nematic phase. (b) Characteristic lamellar distance ${\lambda }^{*}$ corresponding to the smectic phase expressed in units of the particle diameter $D$.

Figure 5

Characteristic eigenfunctions $f^{*}$ [from Eq. (5)], indicating the preferred orientational order in the new phase: (a) nematic phases of pure disks show regular uniaxial nematic order, whereas smectic phases of pure disks exhibit typical antinematic order (b). The orientational distribution of the rings in the smectic phase deviates from the standard second-order Legendre polynomial form (black curve).