Quantized Self-Assembly of Discotic Rings in a Liquid Crystal Confined in Nanopores

Disklike molecules with aromatic cores spontaneously stack up in linear columns with high, one-dimensional charge carrier mobilities along the columnar axes, making them prominent model systems for functional, self-organized matter. We show by high-resolution optical birefringence and synchrotron-based x-ray diffraction that confining a thermotropic discotic liquid crystal in cylindrical nanopores induces a quantized formation of annular layers consisting of concentric circular bent columns, unknown in the bulk state. Starting from the walls this ring self-assembly propagates layer by layer towards the pore center in the supercooled domain of the bulk isotropic-columnar transition and thus allows one to switch on and off reversibly single, nanosized rings through small temperature variations. By establishing a Gibbs free energy phase diagram we trace the phase transition quantization to the discreteness of the layers’ excess bend deformation energies in comparison to the thermal energy, even for this near room-temperature system. Monte Carlo simulations yielding spatially resolved nematic order parameters, density maps, and bond-orientational order parameters corroborate the universality and robustness of the confinement-induced columnar ring formation as well as its quantized nature.

Figure 1

(a) Birefringence experiment illustration and measured temperature evolution of the normalized retardation $R$ of HAT6 in the bulk and (b) confined state during cooling (blue) and heating (red). Insets: Normalized supercooling temperature for the isotropic-columnar transition of each annular layer with curvature radius $r$ along with a $r^{-2}$ fit. A Monte Carlo simulation snapshot illustrating the formation of concentric columnar rings. The colors represent the relative orientation of the molecules with respect to the horizontal axis, where blue means 0° and red 90° alignment. (c) Chemical potential $\mathrm{\Delta }\mu$-temperature $T$ phase diagram of HAT6 in the vicinity of the isotropic-columnar bulk transition. The excess energy $\mathrm{\Delta }{\mu }_n$ of a molecule in ring $n$ is indicated by arrows at the supercooling $T_n$ ($n=1\dots 5$), marked by red asterisks in panel (b).

Figure 2

(a) Circular columnar domains with $\{10\}$ (yellow) and $\{11\}$ (green) wall orientation in a cylindrical pore and reciprocal space maps ($\omega =90°$) assuming (b) perfect aligned and (c) a randomization of the domain orientations by maximal 15° with regard to the mean pore axis direction. (d) X-ray diffraction pattern of HAT6 ($T=340\mathrm{K}$, $\omega =75°$) confined in nanopores. (e) Temperature evolution of the (10) Bragg ring for a cooling-heating cycle. $R(T)$ from Fig. 1 serves as a guide to the eye. Insets: Enlarged reciprocal space and diffraction pattern focusing on the (10) Bragg ring.

Figure 3

(a) The temperature dependence of the average nematic order parameter $\overline{S}$ (blue circles) shows discontinuous jumps, marked with red crosses. Inset: Cross-sectional view on the molecular arrangement in the nanopore at a temperature $k_{B}T/{\epsilon }_0=4.94$. (b) Radial snapshots of molecular configurations at the $T$’s marked as ($1\dots 6$) in panel (a).

Figure 4

Radial dependence of the local density $\rho (r)$ and bond orientational order parameter ${\overline{q}}_6$ at different temperatures along the quantized phase transition.

Figure 5

(a) Dependence of the transition temperatures on the ring’s radial position. (b) The temperature evolution of the density profile. Upon cooling, five distinct regions of large density appear, corresponding to the circular concentric layers.