Weak-anchoring effects in a thin pinned ridge of nematic liquid crystal

A theoretical investigation of weak-anchoring effects in a thin two-dimensional pinned static ridge of nematic liquid crystal resting on a flat solid substrate in an atmosphere of passive gas is performed. Specifically, we solve a reduced version of the general system of governing equations recently derived by Cousins et al. [Proc. R. Soc. A 478, 20210849 (2022)] valid for a symmetric thin ridge under the one-constant approximation of the Frank-Oseen bulk elastic energy with pinned contact lines to determine the shape of the ridge and the behavior of the director within it. Numerical investigations covering a wide range of parameter values indicate that the energetically preferred solutions can be classified in terms of the Jenkins-Barratt-Barbero-Barberi critical thickness into five qualitatively different types of solution. In particular, the theoretical results suggest that anchoring breaking occurs close to the contact lines. The theoretical predictions are supported by the results of physical experiments for a ridge of the nematic $4^{\prime }$-pentyl-4-biphenylcarbonitrile (5CB). In particular, these experiments show that the homeotropic anchoring at the gas-nematic interface is broken close to the contact lines by the stronger rubbed planar anchoring at the nematic-substrate interface. A comparison between the experimental values of and the theoretical predictions for the effective refractive index of the ridge gives a first estimate of the anchoring strength of an interface between air and 5CB to be $(9.80\pm 1.12)\times {10}^{-6}{\mathrm{N}\mathrm{m}}^{-1}$ at a temperature of $(22\pm 1.5){}^{\circ }\mathrm{C}$.

Figure 1

A schematic of a two-dimensional pinned static ridge of nematic (N) with prescribed cross-sectional area $\tilde{A}$ resting on a flat solid substrate (S) in an atmosphere of passive gas (G), bounded by a gas-nematic interface at $\tilde{z}=\tilde{h}\left(\tilde{x}\right)$ and a nematic-substrate interface at $\tilde{z}=0$, with pinned contact lines at $\tilde{x}=\pm \tilde{d}$. The Cartesian coordinates $\tilde{x}$ and $\tilde{z}$ (with the $\tilde{y}$-axis out of the page), the contact angles ${\tilde{\beta }}^{-}$ and ${\tilde{\beta }}^{+}$, and the outward unit normals of the nematic-substrate interface and gas-nematic interface ${\mathit{\nu }}_{\mathrm{NS}}$ and ${\mathit{\nu }}_{\mathrm{GN}}$, respectively, are indicated. A typical director $\mathit{n}$ with director angle $\theta (\tilde{x},\tilde{z})$ and the height at the middle of the ridge ${\tilde{h}}_{\mathrm{m}}$ are also shown.

Figure 2

Sketches of (a) an $\mathcal{H}$ solution and (b) a $\mathcal{P}$ solution. The director field $\mathit{n}$ is indicated by the dark gray rods, with $\mathit{n}\equiv \widehat{\mathit{z}}$ in (a) and $\mathit{n}\equiv \widehat{\mathit{x}}$ in (b).

Figure 3

Sketches of (a) a ${\mathcal{D}}_{\mathcal{H}}$ solution, (b) a $\mathcal{D}$ solution, and (c) a ${\mathcal{D}}_{\mathcal{P}}$ solution. The director field $\mathit{n}$ is indicated by the dark gray rods, the critical thickness $z=h\left(x_{\mathrm{c}}\right)=\left|h_{\mathrm{c}}\right|$ is shown by a dashed line [note that $z=h\left(x_{\mathrm{c}}\right)=\left|h_{\mathrm{c}}\right|=0$ is not visible in (b)], the disclination line at $x=0$ is shown with a dotted line, the uniform regions $x_{\mathrm{c}}\le \left|x\right|\le 1$ in which $h\le |h_{\mathrm{c}}|$ are indicated in white, and the distorted region $\left|x\right|<x_{\mathrm{c}}$ in which $h>|h_{\mathrm{c}}|$ is indicated in gray.

Figure 4

The director angles on the nematic-substrate interface ${\theta }_{\mathrm{NS}}$ (solid line) and the gas-nematic interface ${\theta }_{\mathrm{GN}}$ (dashed line) for representative numerically calculated energetically preferred solutions when $K=1$ in the case of antagonistic anchoring with homeotropic anchoring on the gas-nematic interface and planar anchoring on the nematic-substrate interface with a constant difference between $C_{\mathrm{NS}}$ and $C_{\mathrm{GN}}=C_{\mathrm{NS}}+5.5>0$ (chosen to show a full range of behaviors in which elasticity and the anchoring of both interfaces are all comparable) for (a) $C_{\mathrm{NS}}=-4.5$, (b) $C_{\mathrm{NS}}=-4.25$, (c) $C_{\mathrm{NS}}=-3.5$, (d) $C_{\mathrm{NS}}=-3$, (e) $C_{\mathrm{NS}}=-2.75$, (f) $C_{\mathrm{NS}}=-2.5$, (g) $C_{\mathrm{NS}}=-2$, (h) $C_{\mathrm{NS}}=-1.25$, and (i) $C_{\mathrm{NS}}=-1$.

Figure 5

The height of the ridge $z=h$ (solid line) and the director field $\mathit{n}$ (dark gray rods) for representative numerically calculated energetically preferred solutions when $K=1$ in the case of antagonistic anchoring with homeotropic anchoring on the gas-nematic interface and planar anchoring on the nematic-substrate interface with a constant difference between $C_{\mathrm{NS}}$ and $C_{\mathrm{GN}}=C_{\mathrm{NS}}+5.5>0$ for the same parameter values as those used in Fig. 4, i.e., for (a) $C_{\mathrm{NS}}=-4.5$, (b) $C_{\mathrm{NS}}=-4.25$, (c) $C_{\mathrm{NS}}=-3.5$, (d) $C_{\mathrm{NS}}=-3$, (e) $C_{\mathrm{NS}}=-2.75$, (f) $C_{\mathrm{NS}}=-2.5$, (g) $C_{\mathrm{NS}}=-2$, (h) $C_{\mathrm{NS}}=-1.25$, and (i) $C_{\mathrm{NS}}=-1$. The critical thickness $z=h\left(x_{\mathrm{c}}\right)=\left|h_{\mathrm{c}}\right|$ is shown by a dashed line [note that $z=h\left(x_{\mathrm{c}}\right)=\left|h_{\mathrm{c}}\right|=0$ is not visible in (e)], the uniform region $x_{\mathrm{c}}\le x\le 1$ in which $h\le |h_{\mathrm{c}}|$ is indicated in white, and the distorted region $0\le x<x_{\mathrm{c}}$ in which $h>|h_{\mathrm{c}}|$ is indicated in gray.

Figure 6

The $C_{\mathrm{NS}}-C_{\mathrm{GN}}$ parameter plane for the representative value $K=1$. The regions of the parameter plane in which an $\mathcal{H}$ solution (shown in dark red), a ${\mathcal{D}}_{\mathcal{H}}$ solution (shown in light red), a ${\mathcal{D}}_{\mathcal{P}}$ solution (shown in light blue), and a $\mathcal{P}$ solution (shown in dark blue) is the energetically preferred solution are indicated. The $\mathcal{D}$ solutions are energetically preferred along the solid line corresponding to $h_{\mathrm{c}}=0$. The dashed lines correspond to $h_{\mathrm{c}}=-3/4$, the dotted lines correspond to $h_{\mathrm{c}}=3/4$, and the points corresponding to the representative solutions shown in Figs. 4 and 5 are indicated by dots ($•$) and labeled (a)–(i).

Figure 7

The effective refractive index $n_{\mathrm{eff}}$ given by (39) for the representative solutions labeled (a)–(i) shown in Figs. 4, 5, 6 for $n_{\mathrm{e}}=1.71$ and $n_{\mathrm{o}}=1.52$.

Figure 8

Schematic diagrams showing a plan view of (a) a transparent glass slide coated with a layer of PVA unidirectionally rubbed in the $\tilde{x}$-direction (shown by the gray arrows), (b) a unidirectionally rubbed PVA-glass substrate (the white rectangular region with the gray arrows) of length $\tilde{L}$ in the $\tilde{y}$-direction and width $2\tilde{d}$ in the $\tilde{x}$-direction with a Teflon surround (the dark gray region), and (c) a pinned static ridge of 5CB (the light gray rectangle). The Cartesian coordinates $\tilde{x}$ and $\tilde{y}$ (with the $\tilde{z}$-axis out of the page) are also indicated.

Figure 9

Polarizing optical micrograph of the small ridge of 5CB. The orientations of the polarizer and the analyzer are shown by the two crossed white arrows enclosed within the white circle. The Cartesian coordinates $\tilde{x}$ and $\tilde{y}$ (with the $\tilde{z}$-axis out of the page) and a $100\mu \mathrm{m}$ scale bar are also indicated.

Figure 10

The experimental values of the height of the ridge $\tilde{h}$ plotted as a function of $\tilde{x}$ for the right-hand edge (specifically, the region $400\le \tilde{x}\le 600\mu \mathrm{m}$) of the small ridge (shown by open circles) and the large ridge (shown by closed squares). The fitted theoretical predictions for a pinned static isotropic ridge ${\tilde{h}}_{\mathrm{I}}=3\epsilon ({\tilde{d}}^2-{\tilde{x}}^2)/\left(4\tilde{d}\right)$ as a function of $\tilde{x}$ with $\epsilon =0.071$ (dashed line) and $\epsilon =0.124$ (solid line), and $2\tilde{d}=1.2\times {10}^{-3}\mathrm{m}$ are also plotted.

Figure 11

The experimental values of the effective refractive index $n_{\mathrm{eff}}$ plotted as a function of $\tilde{x}$ for the right-hand edge (specifically, the region $400\le \tilde{x}\le 600\mu \mathrm{m}$) of the small ridge (open circles) and the large ridge (closed squares). The theoretical predictions for $n_{\mathrm{eff}}$ as a function of $\tilde{x}$ with $\epsilon =0.071$ (dashed line) and $\epsilon =0.124$ (solid line), ${\tilde{C}}_{\mathrm{NS}}=-5.0\times {10}^{-4}{\mathrm{N}\mathrm{m}}^{-1}, {\tilde{C}}_{\mathrm{GN}}=9.80\times {10}^{-6}{\mathrm{N}\mathrm{m}}^{-1}, \tilde{K}=7.2\times {10}^{-12}\mathrm{N}, n_{\mathrm{e}}=1.71$, and $n_{\mathrm{o}}=1.52$ are also plotted. The extraordinary and ordinary refractive indices of the nematic $n_{\mathrm{eff}}=n_{\mathrm{e}}$ (dashed gray line) and $n_{\mathrm{eff}}=n_{\mathrm{o}}$ (dashed gray line), respectively, are also plotted. For clarity, the error of $\pm 4\mu \mathrm{m}$ in the $\tilde{x}$-direction is omitted from the above plots.