Figure 1

The geometry of the nematic cell with cross section $A$ and volume $V=Ad$. The upper boundary is characterized by strong homeotropic anchoring. The lower substrate is patterned. The pattern consists of periodic stripes of anchoring energies per area $W_a$ and $W_b$ with widths ${\zeta }_a$ and ${\zeta }_b$, respectively, so that $\zeta ={\zeta }_a+{\zeta }_b$. The easy directions [see Eq. (2)] at both boundaries are normal to the interfaces and we consider such values of $W_a$ and $W_b$ for which the thermal average of the director field ${\mathbf{n}}_0$ is homogeneous.Reuse & Permissions

Figure 2

The fluctuation-induced force $\mathcal{F}$ as a function of the reduced separation $d∕{\lambda }_{\mathrm{eff}}$ in the case in which the patterned substrate can be described by an effective homeotropic anchoring [$d⪢\zeta$, see Eq. (26)]. A crossover from repulsion to attraction occurs for a separation $d∕{\lambda }_{\mathrm{eff}}\simeq 1.19$ followed by a minimum at $(d∕{\lambda }_{\mathrm{eff}},\mathcal{F}∕k_{B}TA∕{\lambda }_{\mathrm{eff}}^3)\simeq (1.65,-0.003)$.Reuse & Permissions

Figure 3

Amplitude of the fluctuation-induced force $\mathcal{F}$ relative to asymptotic behavior $d^{-3}$ as a function of the reduced separation $d∕{\lambda }_{\mathrm{eff}}$ in the case in which the patterned substrate can be described by an effective homeotropic anchoring [$d⪢\zeta$, see Eq. (26)]. At $d∕{\lambda }_{\mathrm{eff}}=0$ this amplitude attains the value $3\zeta \left(3\right)∕\left(16\pi \right)\simeq 0.072$ linearly [Eq. (29)] and approaches the value $-\zeta \left(3\right)∕\left(4\pi \right)\simeq -0.096$ for $d∕{\lambda }_{\mathrm{eff}}\to \infty$ [Eq. (30)].Reuse & Permissions

Figure 4

The fluctuation-induced force $\mathcal{F}$ as a function of the patterning ratio ${\zeta }_a∕\zeta$ for a fixed reduced separation $d∕\zeta$ (fulfilling the requirement $d⪢\zeta$, see Fig. 2) and fixed reduced extrapolation lengths ${\lambda }_{a\left(b\right)}∕\zeta$ [Eq. (26)]. The crossover from repulsion to attraction indicates that upon increasing ${\zeta }_a∕\zeta$ the system transforms from the effective strong-weak to the effective strong-strong anchoring regime. For the given set of the parameters (${\lambda }_a∕\zeta =5,{\lambda }_b∕\zeta =12$, and $d∕\zeta =10$), the force becomes attractive when more than 31% of the substrate consists of strong anchoring parts with $W_a>W_b$, i.e., ${\lambda }_a=K∕W_a<{\lambda }_b=K∕W_b$. Note that here $\mathcal{F}$ is measured in units of ${\zeta }^3$ instead of ${\lambda }_{\mathrm{eff}}^3$ as in Fig. 2.Reuse & Permissions

Figure 5

Rescaled relative decrease of the fluctuation-induced force in the case in which the pattern is characterized by alternating anchoring strengths as a function of the reduced separation $d∕{\lambda }_b$ compared to the repulsive force ${\mathcal{F}}_{\mathrm{eff}}=3k_{B}TA\zeta \left(3\right)∕\left(16\pi d^3\right)$ between two homogeneous boundaries characterized by infinitely strong and zero anchoring energy, respectively. For $d⪡\zeta$ Eq. (31) holds. Both integrals in Eq. (31) happen to have the same form as in Eq. (26) (which holds, however, for $d⪢\zeta$) with ${\lambda }_{\mathrm{eff}}$ replaced by ${\lambda }_a$ and ${\lambda }_b$, respectively. According to the discussion in the second paragraph following Eq. (26), for the parameter values ${\zeta }_a={\zeta }_b$ and ${\lambda }_a∕{\lambda }_b=0.4$ chosen here these integrals are positive for $d∕{\lambda }_b<0.48$ for the contribution associated with ${\lambda }_a$ and $d∕{\lambda }_b<1.19$ for the contribution associated with ${\lambda }_b$. Thus one expects that on the basis of Eq. (31) $\mathcal{F}$ reduces to ${\mathcal{F}}_{\mathrm{rep}}$ as given by Eq. (29) for $d∕\zeta \to 0$. The solid curve corresponds to $\mathcal{F}$ given by Eq. (31) and the other curves correspond to the full numerical results obtained from Eq. (25) for the indicated values of $\zeta ∕{\lambda }_b$. All curves appear to vanish for $d∕\zeta \to 0$ and thus confirm the above expectation. Moreover, by increasing the periodicity the difference between the solid curve and the numerical results vanishes and thus confirms Eq. (31) as a reliable approximation in the limit $d∕\zeta \to 0$.Reuse & Permissions

Figure 6

Rescaled relative decrease of the fluctuation-induced force in the case in which the pattern is characterized by alternating anchoring strengths as a function of the reduced separation $d∕{\lambda }_b$ compared to the attractive force ${\mathcal{F}}_{\mathrm{attr}}=-k_{B}TA\zeta \left(3\right)∕\left(4\pi d^3\right)$ between two homogeneous substrates characterized by infinitely strong anchoring energies. For $d⪢\zeta$ Eq. (26) holds. According to the discussion in the second paragraph following Eq. (26) this integral is negative for $d∕{\lambda }_b>0.68$ for the parameter values ${\zeta }_a={\zeta }_b$ and ${\lambda }_a∕{\lambda }_b=0.4$ chosen here. Thus one expects that $\mathcal{F}$ reduces to ${\mathcal{F}}_{\mathrm{attr}}$ given by Eq. (30) for $\zeta ∕d\to 0$. The solid curve corresponds to $\mathcal{F}$ given by Eq. (26) and the other curves correspond to the full numerical results obtained from Eq. (25) for the indicated values of $\zeta ∕{\lambda }_b$. All curves appear to vanish for $\zeta ∕d\to 0$ and thus confirm the above expectation. Moreover, by decreasing the periodicity $\zeta$ the difference between the solid curve and the full numerical results vanishes and thus confirms Eq. (26) as a reliable approximation in the limit $\zeta ∕d\to 0$.Reuse & Permissions

Figure 7

The fluctuation-induced force as a function of the reduced separation $d∕{\lambda }_b$ in the case in which the pattern is characterized by competing anchoring energies. The solid curve shows the result for small periodicity $(\zeta ⪡d)$ where the pattern can be described by an effective anchoring energy per area $W_{\mathrm{eff}}={\zeta }_a{W}_a∕\zeta +{\zeta }_b{W}_b∕\zeta <0$ [Eq. (32)]. The dashed curve shows the result for large periodicity $(\zeta ⪢d)$ for which the proximity force approximation is valid [Eq. (36)]. The data points $(+)$ show our full numerical results [Eq. (25)] for $\zeta ∕{\lambda }_b=1.5$. This illustrates that as expected $\mathcal{F}$ approaches the proximity force solution (dashed curve) for $d∕\zeta \to 0$ and the effective anchoring solution (solid curve) for $d∕\zeta \to \infty$.Reuse & Permissions