Topological defects in two-dimensional liquid crystals confined by a box

When a spatially uniform system that displays a liquid-crystal ordering on a two-dimensional surface is confined inside a rectangular box, the liquid crystal direction field develops inhomogeneous textures accompanied by topological defects because of the geometric frustrations. We show that the rich variety of nematic textures and defect patterns found in recent experimental and theoretical studies can be classified by the solutions of the rather fundamental, extended Onsager model. This is critically examined based on the determined free energies of different defect states, as functions of a few relevant, dimensionless geometric parameters.

Figure 1

Typical nematic defect states (top) and their free-energy (per molecule) branches as functions of $a/L$ for fixed (a) $[b/a,L^2\rho ]=[1,10]$ and (b) $[1.2,10]$. Defect points of winding numbers $-1/2$ (blue) and $-1$ (yellow) are seen in the diagonal, long-axis, and $\mathsf{X}$-shaped states. The crossing of these branches yields a phase boundary for the transition between the two involved states.

Figure 2

Phase diagrams in terms of $a/L$ and $L^2\rho$ for four given aspect ratios, $b/a=1,1.2,2$, and 3. A second-order phase transition curve (dashed) separates the parameter space into ordered and isotropic (ISO) states.

Figure 3

Nematic defect structures found from the solutions to the extended Onsager model for $b/a=1$. Every structure is visualized by six methods: nematic-director field, fluid density variation $\varphi$, main-axis orientational order parameter $\mathrm{\Lambda }$, and three gray-scale crossed-polarizer images calculated according to (10) for $\alpha =\pi /4$, $5\pi /16$, and $3\pi /8$. The reduced parameters used to produce these structures are $[L^2\rho ,a/L]=[10.0,8.0]$ for (a) and (f) and $[10.0,10.0]$ for (b)–(e) and (j)–(l). The color scale used for columns 2 and 3 is indicated by the white-blue color bar and the grayscale used for all $I_{\alpha }$ plots by the black-white gray bar. The blue, yellow, and green circles label the defect locations of $-1/2$, $-1$, and $+1/2$ winding numbers, respectively. The three gray-scale images of D and U states closely match the real crossed-polarizer images of a liquid crystal confined in 80-$\mu \mathrm{m}$ square cells, reported in Ref. [1].

Figure 4

Nematic defect structures found from the solutions to the extended Onsager model for a rectangular box of aspect ratio $b/a=2$. Every structure is visualized by six methods: nematic director field, nematic fluid density variation $\varphi$, main-axis orientational order parameter $\mathrm{\Lambda }$, and three gray-scale crossed-polarizer images calculated according to (10) for $\alpha =\pi /4$, $5\pi /16$, and $3\pi /8$. The system parameters used to produce these structures are $[L^2\rho ,a/L]=[10.0,7.5]$ for (a), $[8.0,6.0]$ for (e), and $[10.0,10.0]$ for (b)–(d) and (f)–(l). The color and grayscales and the meaning of colored circles are the same as in Fig. 3.

Figure 5

Phase diagram and the probabilities for appearance of metastable states, $P$, in the nematic phase for a given $L^2\rho =10$. The solid phase boundary in all plots divides D, L, and X states. The shaded areas are metastable regions. The color bars specify the relative probabilities, in comparison with those in states D and L.

Figure 6

Comparison to nematic textures found experimentally and theoretically. These plots are adapted from (a) Ref. [5] (experiment on granular rods); (b) Ref. [2] (experiment on actin filaments); (c), (k), and (n) Ref. [30] (solution to LdG model); (d) Ref. [24] (solution to the Onsager model); (e) Ref. [28] (MC simulations); (f) and (o) Ref. [3] (solution to the Oseen-Frank model); (g), (h), (l), (m), and (p) Ref. [3] (experiment on Y21M and fd-wt viruses); as well as (j) Ref. [4] (experiment on colloidal rods). All plots were reproduced with consents from the original authors. Systems in (a)–(g) are in the D state, (h)–(k) L state, (l)–(o) U state, and (p) T state. More images have been published in Ref. [3]; the most characteristically representative ones are reproduced here.

Figure 7

Additional nematic defect structures found from the solutions to the extended Onsager model for $b/a=1$ and $b/a=2$. See the caption of Fig. 3 for an explanation of the physical quantities illustrated. The reduced parameters used to produce these structures are $[L^2\rho ,a/L]=[6.0,12.5]$. The color and grayscales, as well as the meaning of the colored circles, are the same as in Fig. 3. In addition, the orange circle labels the defect location of a defect of the $+1$ winding number.

Figure 8

Phase diagram and the probabilities for appearance of metastable states, $P$, in the nematic phase (for a given $L^2\rho =6$). The solid phase boundary in all plots divides the D and L states. The shaded areas in plots (a)–(k) represent the metastable regions of the structures listed in Fig. 7. The color bars specify the relative probabilities, in comparison with those in states D and L.

Figure 9

(a) Winding angle of a sharp angle boundary (left) and a smooth boundary (right), (b)–(f) the total winding numbers of different types of boundary geometries, and (g) the types of defect points labeled by their winding numbers. Every type of frustrations is represented by the total number of corners along the boundary, $n$. Two example defect patterns of the same total winding number are shown for each case.