Phase behaviour and defect structure of soft rods on a sphere

TL;DR

The paper addresses how spherical topology influences phase behavior and defect structures of soft, repulsive spherocylinders confined to a spherical surface. It employs particle-resolved molecular dynamics with a three-step expansion protocol to map the phase diagram as a function of aspect ratio $A$ and packing fraction $\eta$, identifying crystal (K), smectic (Sm), nematic (N), and isotropic (I) phases and their characteristic defects. A nematic phase emerges only above a critical aspect ratio $A_c$ in the range $(6,7)$, with four $+\tfrac{1}{2}$ defects arranged on a great circle; lower-$A$ systems melt sequentially from K to Sm to I, with Sm–I and N–I transitions appearing at larger $A$. The results show robust phase behavior against finite-size effects and provide experimentally testable predictions for Pickering emulsions and biological morphogenesis, while highlighting how curvature and topology govern defect structures on curved substrates.

Abstract

Using particle-resolved molecular-dynamics simulations, we compute the phase diagram for soft repulsive spherocylinders confined on the surface of a sphere. While crystal (K), smectic (Sm), and isotropic (I) phases exhibit a stability region for any aspect ratio of the spherocylinders, a nematic phase emerges only beyond a critical aspect ratio lying between 6.0 and 7.0. As required by the topology of the confining sphere, the ordered phases exhibit a total orientational defect charge of +2. In detail, the crystal and smectic phases exhibit two +1 defects at the poles, whereas the nematic phase features four +1/2 defects which are connected along a great circle. For aspect ratios above the critical value, lowering the packing fraction drives a sequence of transitions: the crystal melts into a smectic phase, which then transforms into a nematic through the splitting of the +1 defects into pairs of +1/2 defects that progressively move apart, thereby increasing their angular separation. Eventually, at very low densities, orientational fluctuations stabilize an isotropic phase. Our simulations data can be experimentally verified in Pickering emulsions and are relevant to understand the morphogenesis in epithelial tissues.

Figures (7)

• Figure 1: The schematics of the system: A) A soft repulsive spherocylinder (SRS) with body length $L$ and diameter $D$. B) The SRSs anchored tangentially on the surface of a sphere. The positions of the particles are denoted by the coordinates of their center of masses $\vec{\bm{r_i}}$, with the origin of the coordinate system at the centre of the sphere. The polar and azimuthal unit vectors $\hat{\theta_j},\hat{\phi_j}$ are shown at the position of the $j-th$ particle. The direction of the long axis of the $j-th$ spherocylinder or the unit orientation vector is denoted as $\hat{\bm{s_j}}$. The distance of closest approach between two spherocylinders is given by $d_m$. C) One particular face of the sphere between two latitudinal lines and two longitudinal lines is shown. For a layered smectic structure, the center of masses of the particles have an angular periodicity $\theta_0$ along the longitude. For a crystal phase, an additional periodicity $\phi_0$ appears along the latitude.
• Figure 2: Snapshot of states obtained during expansion of a spherical surface of soft repulsive spherocylinders for the aspect ratio $A = 4.0$, with $N=10000$ at $T^* = 1.0$ - A) The crystal phase with both orientational and positional ordering, obtained at a packing fraction of $\eta = 0.84$. The orientations of the spherocylinders are ordered along the longitude of the sphere. The positional order is two-fold: it consists of multiple latitudinal layers that are stacked with inter-layer periodicity along the longitude, and it also features intra-layer periodicity along the latitude. B) At $\eta = 0.689$, we observe a smectic phase which retains the layered structure with longitudinal periodicity, while the intra-layer ordering in latitudinal direction is lost. For both the states shown in A) and B), there is a $+1$ defect at each of the poles. At a lower packing fraction of $\eta = 0.387$, there is no positional or orientational ordering of the particles on the sphere (C). We observe no nematic phase in this particular value of the aspect ratio of the spherocylinders. The visualisations are done using VMD humphrey1996vmd.
• Figure 3: Determination of transitional packing fractions at phase transitions for $A = 2.0$ for a spherical shell of soft repulsive spherocylinders. A) The nematic ($S$) and smectic ($\zeta$) order parameters plotted as a function of packing fraction $\eta$. The nematic order parameter indicates the orientational ordering in the system, whereas the smectic order parameter indicates the presence of a layered structure. Both order parameters show a sharp rise at a packing fraction $\eta_I \approx 0.627$, which demonstrates that the orientational ordering in the system appears with the layered structure and thus there is no nematic phase for the system. B) The radial distribution function of the $\phi$ co-ordinates of the particles, plotted at different packing fractions. The appearance of a larger number of peaks in the plot at higher $\eta$ shows the emergence of periodicity in the latitude direction, indicating a crystal-smectic transition.
• Figure 4: The existence of nematic phase for $A = 8.0$. In the range $0.470 \leq \eta \leq 0.535$, the structure shows a high value of $S$, but a low value of $\zeta$, indicating a nematic phase. The snapshot of the molecular configuration for a nematic phase at a packing fraction $\eta = 0.5$ is also shown. $\eta \leq 0.437$ shows an isotropic phase and $\eta \geq 0.567$ exhibits smectic phase. The pink shaded areas denote the transition packing fractions.
• Figure 5: Phase diagram for a system of SRSs on the surface of a sphere at $T^* = 1.0$. The crystal (K), smectic (Sm) and isotropic (I) phases are observed for all aspect ratios considered. The points indicate the packing fractions delimiting the phase transition. The black lines are the guides to the eye. The colored shaded regions represent the uncertainty in the transition packing fractions. The nematic phase (N) appears only above a critical aspect ratio of the SRSs ($A_c$), the value of which lies between 6 and 7, Above the critical aspect ratio $A_c$, as the SRSs become longer, the transition packing fraction for the nematic phase decreases. The packing fraction for the K-Sm transition of the system is approximately around $0.7$, which is similar to the freezing packing fraction for hard discs on a 2d plane li2022hardbernard2011twokapfer2015two or sphere giarritta1992statistical.