Nematic wetting and filling of crenellated surfaces

We investigate nematic wetting and filling transitions of crenellated surfaces (rectangular gratings) by numerical minimization of the Landau–de Gennes free energy as a function of the anchoring strength, for a wide range of the surface geometrical parameters: depth, width, and separation of the crenels. We have found a rich phase behavior that depends in detail on the combination of the surface parameters. By comparison to simple fluids, which undergo a continuous filling or unbending transition, where the surface changes from a dry to a filled state, followed by a wetting or unbinding transition, where the thickness of the adsorbed fluid becomes macroscopic and the interface unbinds from the surface, nematics at crenellated surfaces reveal an intriguingly rich behavior: in shallow crenels only wetting is observed, while in deep crenels, only filling transitions occur; for intermediate surface geometrical parameters, a new class of filled states is found, characterized by bent isotropic-nematic interfaces, which persist for surfaces structured on large scales, compared to the nematic correlation length. The global phase diagram displays two wet and four filled states, all separated by first-order transitions. For crenels in the intermediate regime re-entrant filling transitions driven by the anchoring strength are observed.

Figure 1

Schematic representation of a surface cusp. The thick continuous line depicts the surface close to a cusp, characterized by an opening angle $\Delta \phi$. Strong anchoring conditions are enforced at the surface, with ${\theta }_1$ and ${\theta }_2$ their asymptotic values as the cusp is approached from either side. The integration contour $C$ is shown as a thin curved line, with $\mathbf{\eta }$ its outward normal. See text for details.

Figure 2

Crenellated surfaces. $l_1$ is the separation between crenels, and $l_2$ and $h$ are the width and depth of the crenels. The structure is periodic with wavelength $\lambda =l_1+l_2$.

Figure 3

Interfacial states of crenellated surfaces for nematics with positive elastic anisotropy. ($D$) dry state, ($W_u$) uniform wet state,($W_d^s$) symmetric wet state, ($W_d^a$) asymmetric wet state, ($F_b$) bent filled state, $\left(F_b^{+}\right)$ bent filled state with droplets, ($F_u$) unbent filled state, $\left(F_u^{+}\right)$ unbent filled with droplets.

Figure 4

Phase diagram for a crenellated surface with the same crenel width and separation, $l1=l2=10\xi$. The lines correspond to first-order phase transitions that separate dry $\left(D\right)$, wet $\left(W\right)$, and filled $\left(F\right)$ states. Two distinct wet states and four distinct filled states are observed. At the wetting transition, the nearly uniform wet state $\left(W_u\right)$ at low roughness changes continuously to a symmetric distorted wet state $(W_d^s$) as the anchoring increases. There is a bistable asymmetric nematic wet state $\left(W_d^a\right)$ at intermediate roughness. The wet states are separated by a weak first-order transition indicated by the grey line. There are also filled states, characterized by a bent $\left(F_b\right)$ or unbent NI interface $\left(F_u\right)$. The unbent filled states are bistable. At strong anchoring the top surfaces promote the nucleation of nematic droplets or films for bent $\left(F_b^{+}\right)$ and unbent $\left(F_u^{+}\right)$ filled states. See Fig. 3 for a sketch of the corresponding states.

Figure 5

Scaling of the free energy, $\mathcal{F}-{\sigma }_{nw}^{\perp }(l_1+l_2+2h)-(2\pi /3)\mathcal{K}\text{ln}(h/\xi )$ of distorted wet states as a function of the anchoring strength $w$ with the system size, for roughness $r=1.2$ and $r=1.5$. $\mathcal{K}\left(k\right)$ is an effective elastic constant. The curves for equal roughness collapse at moderate to high values of the anchoring strength. The collapse occurs at lower values of the anchoring as the depth of the crenels $h$ increases.

Figure 6

Bent filled state approximated by two circular domains, of radius $h$, of nematic bent around the bottom corners connected by a flat layer of uniform homeotropic nematic, of height $h_f$.

Figure 7

Filling $F_{u\left(b\right)}$ to filling $F_{u\left(b\right)}^{+}$ transition. Anchoring strength $w_t$ as a function of the crenel separation $l_1$. The full-black line corresponds to the anchoring strength $w_t$ obtained numerically. The dashed-red line is the analytical estimate $w_t={\sigma }_{ni}^{\perp }R\left(l_1\right)\xi /l_1$, where $R\left(l_1\right)=l_1/\xi +4+O(1/l_1)$. The inset depicts the nematic droplet on the top surface. The nematic director is strongly anchored to the surface and is homeotropic everywhere.

Figure 8

Phase diagrams for surfaces with $l_1=l_2=10\xi$ (full-black line) and $20\xi$ (dashed-red line). Re-entrant filling is enhanced. The grey lines represent the weak first-order transition between the two wet states (see Fig. 4).

Figure 9

Phase diagrams for systems with $l_1=l_2=25\xi ,30\xi$. At low and high roughness the phase diagram is similar to the previous ones. At intermediate roughness, there is a clear enhancement of the re-entrant regions, as the wet state is stable between the filled states $F_b$ and $F_b^{+}$. The grey lines represent the weak first-order transition between the two wet states (see Fig. 4). Blue (vertical) and green (horizontal) arrows indicate re-entrant paths by varying the anchoring and the roughness, respectively.

Figure 10

Phase diagrams for systems with $l_1=10\xi ,l_2=20\xi$ and $l_1=20\xi ,l_2=40\xi$. The stability of the bent filled, $F_b$ and $F_b^{+}$, is enhanced. The grey lines represent the weak first-order transition between the two wet states (see Fig. 4).

Figure 11

Phase diagrams for $l_1=20\xi ,l_2=10\xi$ and $l_1=40\xi ,l_2=20\xi$. The grey lines represent the weak first-order transition between the two distorted wet states (see Fig. 4).