Rodlike molecules in extreme confinement

A unique feature of colloid particles and biopolymers is the molecule's intrinsic rigidity characterized by a molecular-level length scale. Under extreme confinement conditions at cellular scales or in nanodevices, these molecules can display orientational ordering accompanied by severe density depletion. Conventional liquid-crystal theories, such as the Oseen-Frank or Landau–de Gennes theories, cannot capture the essential molecular-level properties: the boundary effects, which extend to a distance of the rigidity length scale, and the drastic variations of the inhomogeneous molecular density. Here we show, based on a simple interpretation of the Onsager model, that rodlike molecules in extreme annular confinement produce unusual liquid-crystal defect structures that are independent phases from the patterns usually seen in a weaker confinement environment.

Figure 1

Rodlike molecules of length $L$ confined in concentric annular circles of radii $R_1$ and $R_2$. (a),(b) Typical allowed (yellow) and disallowed (blue) configurations, realized in our theory. (c) Two phase regimes which exist when $R_2/L\gg 1$: a twofold defect state ($D_2$) and a uniform defect-free state ($D_{\infty }$), divided by a phase boundary. Multiple states exist in the extreme confinement region of $R_2/L\sim 1$, as illustrated in phase diagram (d), based on the numerical solution of the extended Onsager free energy for interacting, rodlike molecules. An overall number density of rods, ${\rho }_0=20/L^2$, is assumed here, where ${\rho }_0$ is the number of rods per unit area on a two-dimensional surface. The phase boundaries are determined by a comparison of the free energies. The color of the rods illustrated in (c) and (d) represents the angles that the rods make with respect to the horizontal axis, specified in the inset of (c). The shaded area is defined in Appendix pp2.

Figure 2

Defect structures under the extreme confinement condition. Three illustration methods are used, all based on the numerical solution of ${\rho }_c(\mathbf{r},\mathbf{u})$. The first row in each plot is a reconstructed, real-space illustration; the second row is the scalar orientational order parameter, calculated from the distribution function; and the third row is the center-of-mass distribution function, after averaging over all orientations. The first-order phase boundaries separating $D_m$ from $D_m^{\prime }$ are represented by the change of the background shades: from white to light gray, where $m=2,3$, and 4 in (a) $R_1/R_2=0.1$, (b) $R_1/R_2=0.3$, and (c) $R_1/R_2=0.45$, respectively. The color used in the first row of each plot has the same meaning as in the inset of Fig. 1. The intensities of both $\mathrm{\Lambda }$ and ${\mathrm{\Phi }}_c$ are represented by the blue color bar. The light-blue and green circles indicate the defect locations on $D_m$, of $-1/2$ and $1/2$ winding numbers, respectively.

Figure 3

Phase transitions between $D_m^{\prime }$ and $D_m$. For each case of (a) $m=2$, (b) $m=3$, and (c) $m=4$, two branches of the reduced free energy per rod, triangles for $D_m^{\prime }$ and circles for $D_m$, are plotted in (a)–(c). The variance of the nematic direction at $r=R_2, {\sigma }^2$, is used as the order parameter of the $D_m^{\prime }\text{−}{D}_m$ transitions and displayed in (d)–(f) for $m=2, m=3$, and $m=4$, respectively. Jumps of ${\sigma }^2$ can be seen at the transition points.

Figure 4

All topologically different states found from our solutions to the extended Onsager model for the current geometry. Some of them show up in the phase diagrams, shown in Figs. 1 and 1. The snapshots in the first column illustrate the positions of the rods and their orientations, where the color represents the angle of a rod with respect to the horizontal axis specified by the color bar. For easy reference, we produce simulated cross-polarizer images of these states, $I_{\alpha } (\alpha =\pi /4,5\pi /16$, and $3\pi /8)$, according to (13), where $\alpha$ is the angle between the first polarizer direction and the horizontal axis of these plots.

Figure 5

Illustration of the limit of the rod length, $L^{*}$, which allows the rod to just fit into the annular confinement.