Theory of anchoring on a two-dimensionally grooved surface

We investigate analytically the anchoring of a nematic liquid crystal on a two-dimensionally grooved surface of arbitrary shape, induced by the elastic distortions of a liquid crystal adjacent to the surface. Our theoretical framework applied to a surface with square grooves reveals that such a surface can exhibit bistable anchoring, while a direct extension of a well-known theory of Berreman [Phys. Rev. Lett. 28, 1683 (1972)] results in no azimuthal anchoring in the so-called one-constant case ($K_1=K_2=K_3$, with $K_1$, $K_2$, and $K_3$ being the splay, twist, and bend elastic constants, respectively). We show under the assumption of $K_1=K_2=K$ that the direction of the bistable easy axes and the anchoring strength crucially depend on the ratios $K_3/K$ and $K_{24}/K$, where $K_{24}$ is the saddle-splay surface elastic constant. To demonstrate the applicability of our theory to general cases and to elucidate the effect of surface shape and the elastic constants on the properties of surface anchoring, we also consider several specific cases of interest; one-dimensional grooves of arbitrary shape, rhombic grooves, and surfaces possessing $2N$-fold symmetry, including hexagonal grooves, and show the following: (i) The rescaled anchoring energy $f\left(\varphi \right)/f\left(\pi /2\right)$ of one-dimensional grooves, with $\varphi$ being the angle between the director $\mathit{n}$ and the groove direction, is independent of the groove shape. (ii) Whether two diagonal axes of rhombic grooves can become easy axes depends sensitively on $K_3/K$, $K_{24}/K$ and the angle $\alpha$ between the grooves. The angle $\alpha$ yielding the maximum anchoring strength for given groove pitch and amplitude depends again on $K_3/K$ and $K_{24}/K$; in some cases $\alpha =0$ (one-dimensional grooves), and in other cases $\alpha \ne 0$, gives the maximum anchoring strength. Square grooves $\left(\alpha =\pi /2\right)$ do not necessarily exhibit the largest anchoring strength. (iii) A surface possessing $2N$-fold symmetry can yield $N$-stable azimuthal anchoring. However, when $K_1=K_2=K_3$ and $N\ge 3$, azimuthal anchoring is totally absent irrespective of the value of $K_{24}$. The direction of the easy axes depends on $K_3/K$, $K_{24}/K$, and whether $N$ is even or odd.

Figure 1

(Color online) Plot of the reduced anchoring energies $f\left(\varphi \right)/f\left(\pi /2\right)$ for various $K_s$. Here we assume $K_1=K_2=K_3=K$.

Figure 2

(Color online) Illustration of the geometry of the square grooves. Thin green lines represent the direction of easy axes ($\varphi =\pi /4$, $3\pi /4$). The angle $\varphi$ can be understood as the angle between the director $\mathit{n}$ (or the $x$ axis) and the direction of one of the grooves (represented by thin solid black lines), as well as the angle between ${\mathit{q}}_1$ and the $y$ axis.

Figure 3

(Color online) Phase diagram of a surface with square grooves. Schematic illustrations of the easy axes are also given. At the point $\left(k_3=1,k_s=1\right)$ marked by a dot, no azimuthal anchoring is present.

Figure 4

(Color online) Plot of the rescaled anchoring energy $\tilde{f}\left(\varphi \right)$ for (a) $k_3=1.5$, (b) $k_3=0.7$, and (c) $k_3=0.1$ and $k_s=1.8$.

Figure 5

(Color online) Plot of the anchoring strengths $W\left(0\right)$ and $W\left(\pi /4\right)$ as a function of $k_3$ and $k_s$ for a surface with square grooves. Only the regions where $W\left(0\right)>0$ or $W\left(\pi /4\right)>0$ are shown.

Figure 6

(Color online) Illustration of the geometry of a surface with rhombic grooves. Thin green lines, diagonal lines of the rhombi, correspond to $\tilde{\varphi }=0$ or $\tilde{\varphi }=\pi /2$.

Figure 7

(Color online) Phase diagrams for $W\left(0\right)$ of rhombic grooves with (a) $\alpha =\pi /3$ and (b) $\alpha =5\pi /12$.

Figure 8

(Color online) Phase diagrams for $W\left(\pi /2\right)$ of rhombic grooves with (a) $\alpha =5\pi /12$, (b) $\alpha =3\pi /8$, (c) $\alpha =\pi /8$, and (d) $\alpha =0$. Notice that in (c), there exist a very narrow region around $k_3\simeq 0$, in which $W\left(\pi /2\right)<0$.

Figure 9

(Color online) Implicit plot of $c\left(\alpha ,k3,ks\right)=0$ or $\partial W\left(0\right)/\partial \alpha =0$ for $0<\alpha \le \pi /2$.

Figure 10

(Color online) Plots of $W\left(0\right)$ as a function of $\alpha$ for (a) $k_3=1$ and (b) $k_3=2$.

Figure 11

(Color online) Illustration of the geometry of grooved surfaces with $2N$-fold symmetry: (a) $N=3$ (hexagonal) and (b) $N=4$. The angle $\varphi$ can be understood as the angle between the director $\mathit{n}$ (or the $x$ axis) and the direction of one of the grooves (represented by thin solid black lines), as well as the angle between ${\mathit{q}}_1$ and the $y$ axis.

Figure 12

(Color online) Phase diagram of a surface with hexagonal grooves. Schematic illustrations of the easy axes are also given. On the vertical line $k_3=1$, no azimuthal anchoring exists.

Figure 13

(Color online) Plot of the rescaled anchoring energy $\tilde{f}\left(\varphi \right)$ for (a) $k_3=1.5$, (b) $k_3=0.7$, and (c) $k_3=0.02$ and $k_s=1.8$.

Figure 14

(Color online) Plot of the anchoring strengths $W\left(0\right)$ and $W\left(\pi /6\right)$ as a function of $k_3$ and $k_s$ for a surface with hexagonal grooves. Only the regions where $W\left(0\right)>0$ or $W\left(\pi /6\right)>0$ are shown.

Figure 15

(Color online) Plot of the rescaled anchoring energy $\tilde{f}\left(\varphi \right)$ for $N=4$ and (a) $k_s=1$, and (b) $k_s=2$.

Figure 16

(Color online) Plot of the rescaled anchoring energy $\tilde{f}\left(\varphi \right)$ for $N=5$ and (a) $k_s=1$, and (b) $k_s=2$.

Figure 17

(Color online) (a) Plot of $h_0^{\left(0\right)}/6$, $h_1^{\left(0\right)}$, $h_2^{\left(0\right)}$, $h_1^{\left(1\right)}$, and $h_2^{\left(1\right)}$ as a function of $\alpha$ to show their non-negativeness in the range $0\le \alpha \le \pi /2$. (b) Plot of $h_0^{\left(1\right)}$ as a function of $\alpha$. Note that $h_0^{\left(1\right)}\left(\alpha \right)=0$ at $\alpha =2\text{arcsin}\left(\sqrt{2}/3\right)\simeq 0.982$.

Figure 18

(Color online) Plots of (a) $h_0^{\left(0\right)}\left(\pi -\alpha \right)$ and (b) $h_1^{\left(0\right)}\left(\pi -\alpha \right)$ and $h_2^{\left(0\right)}\left(\pi -\alpha \right)$ as a function of $\alpha$. Note that $h_2^{\left(0\right)}\left(\pi -\alpha \right)$ becomes zero at $\alpha =2\text{arccos}\left(\sqrt{6}/3\right)$.

Figure 19

(Color online) Plots of (a) $h_0^{\left(1\right)}\left(\pi -\alpha \right)$ and (b) $h_1^{\left(1\right)}\left(\pi -\alpha \right)$, $h_2^{\left(0\right)}\left(\pi -\alpha \right)$, and $d\left(\alpha \right)$ as a function of $\alpha$. Note that $h_2^{\left(0\right)}\left(\pi -\alpha \right)$ becomes zero at $\alpha =2\text{arccos}\left(\sqrt{6}/3\right)$ and that $d\left(\alpha \right)=0$ at $\alpha =2\text{arccos}\left(\sqrt{546\pm 52\sqrt{3}}/26\right)$.