Onsager Model for the Structure of Rigid Rods Confined on a Spherical Surface

Recent studies have suggested that a monolayer of self-avoiding hard rods confined on a spherical surface may display a distribution texture corresponding to splay, tennis-ball, rectangle, or cut-and-rotate splay symmetries. We investigate the system on the basis of a generalized Onsager model which includes both excluded-volume and entropic effects. The numerical solution indicates that the splay state, where on average rods line up in parallel to the longitudinal lines on the spherical surface, is the only stable state.

Figure 1

Idealized illustration of four possible structures considered in this work: splay (a), tennis ball (b), rectangle (c), and cut-and-rotate splay (d).

Figure 2

Minimized Onsager free-energy difference, Eq. (4), plotted as a function of reduced number density $\rho L^2$ [(a)–(c)] and as a function of the cut-and-rotate angle $\alpha$, for $L/R=0.5$. The solid curve (black) in (a)–(c) is the splay branch of the free energy. Dashed lines were produced by fixing the leading symmetry-breaking term in the density distribution expansion (2): ${\varphi }_{3,2,0}$ fixed at $-0.1$ (brown), $-0.08$ (yellow), $-0.06$ (blue), $-0.04$ (green), and $-0.02$ (red) from the top to bottom curves for tennis ball (a), ${\varphi }_{2,2,0}$ at 0.1 (brown), 0.08 (yellow), 0.06 (blue), 0.04 (green), and 0.02 (red) for rectangle (b), and ${\varphi }_{3,-2,0}$ at 0.1 (brown), 0.08 (yellow), 0.06 (blue), 0.04 (green), and 0.02 (red) for cut-and-rotate splay (c). In (d), circles, squares, diamonds, and triangles represent $\rho L^2=7.96$, 11.94, 15.92, and 19.90, respectively.

Figure 3

Orientational and positional order parameters, (a) $\Omega$ and (b) $\Sigma$, plotted as functions of reduced density, $\rho L^2$, for $L/R=0.5$. Data points associated with nonzero $\Omega$ and $\Sigma$ were produced from splay configurations. The area inside the two vertical lines indicates a narrow transition region.

Figure 4

(a) Transition densities ${\rho }_{\mathrm{iso}}{L}^2$ (down triangles) and ${\rho }_{\mathrm{spl}}{L}^2$ (up triangles), and (b) orientational and spatial order parameters, $\Omega$ and $\Sigma$, at ${\rho }_{\mathrm{spl}}{L}^2$ plotted as functions of $L/R$.