Weak anchoring effects in smectic-A Fréedericksz transitions

Abstract

In smectic-A (SmA) liquid crystals, rod-like molecules are organized into layers with their optical axis oriented along the normal to these layers. When subject to an external field and for a certain critical threshold, these layers can buckle into a new different configuration. Here, we consider SmA liquid crystals which are confined between two parallel plates aligned in the so-called bookshelf geometry and subject to an external magnetic field. By assuming that instability is induced by a uniform field, we investigate the influence of the anchoring strength on the critical threshold field and on the layers shape by a perturbative analysis to the equilibrium equations. Main differences with respect to the standard Fréedericksz transition of nematics are highlighted. The behavior of this threshold effect suggests a new way to measure geometrical and constitutive parameters of a SmA sample.

Euler–Lagrange equations

Let $\mathcal {F}$ be a functional with fixed ends a and b:

$$\begin{aligned} \mathcal {F}[u] = \int \limits _a^b \psi (\zeta , u(\zeta ), u'(\zeta ),u''(\zeta ))\ \text {d}\zeta + \psi ^a(u(\zeta ),u'(\zeta ))|_{\zeta =a} + \psi ^b(u(\zeta ),u'(\zeta ))|_{\zeta =b}, \end{aligned}$$

(44)

and consider a perturbed argument function $u_\eta (\zeta ) = u(\zeta ) + \eta g(\zeta )$, where $\eta$ is a small parameter and g is a function which vanishes at the ends. Taylor expansion of $F(u_\eta ,u'_\eta , u''_\eta )$, up to $\mathcal {O}(\eta )$, yields

$$\begin{aligned} \mathcal {F}[u_\eta ]= & {} \int \limits _a^b\psi (u_\eta ,u'_\eta ,u''_\eta )\ \text {d}{\zeta } + \psi ^a(u_\eta ,u'_\eta )|_{{\zeta } =a} + \psi ^b(u_\eta ,u'_\eta ) |_{{\zeta } =b} \nonumber \\= & {} \int \limits _a^b \left[ \psi (u,u',u'') +\eta \left( \frac{\partial \psi }{\partial u} g + \frac{\partial \psi }{\partial u'} g' + \frac{\partial \psi }{\partial u''} g'' \right) \right] \text {d}\zeta \nonumber \\&+ \left[ \psi ^a(u,u') + \eta \frac{\partial \psi ^a}{\partial u'} g'\right] _{\zeta =a} + \left[ \psi ^b(u,u') + \eta \frac{\partial \psi ^b}{\partial u'} g'\right] _{\zeta =b} . \end{aligned}$$

(45)

From the product differentiation rule, we obtain

$$\begin{aligned} \frac{\partial \psi }{\partial u'} g'= & {} \left( g \frac{\partial \psi }{\partial u'}\right) ' - g \left( \frac{\partial \psi }{\partial u'}\right) ', \end{aligned}$$

(46a)

$$\begin{aligned} \frac{\partial \psi }{\partial u''} g''= & {} \left( g' \frac{\partial \psi }{\partial u''}\right) ' - \left[ g \left( \frac{\partial \psi }{\partial u''}\right) '\right] ' + g \left( \frac{\partial \psi }{\partial u''}\right) ''. \end{aligned}$$

(46b)

Thus, by substituting these two expressions into (45), integrating by parts, and applying the condition $g(a)=g(b) =0$, we obtain

$$\begin{aligned} \mathcal {F}[u_\eta ]= & {} \int \limits _a^b\psi (u,u',u'')\ \text {d}\zeta + \left[ g' \frac{\partial \psi }{\partial u''}\right] _a^b \eta +\int \limits _a^b\left[ \frac{\partial \psi }{\partial u} - \left( \frac{\partial \psi }{\partial u'}\right) ' + \left( \frac{\partial \psi }{\partial u''}\right) ''\right] \eta g\text {d}\zeta \nonumber \\&+ \left[ \psi ^a(u,u') + \eta \frac{\partial \psi ^a}{\partial u'} g'\right] _{\zeta =a} + \left[ \psi ^b(u,u') + \eta \frac{\partial \psi ^b}{\partial u'} g'\right] _{\zeta =b}. \end{aligned}$$

(47)

Stationary points of (44) are obtained by imposing

$$\begin{aligned} \delta \mathcal {F} = \lim _{\eta \rightarrow 0} \frac{\mathcal {F}[u_\eta ] -\mathcal {F}[u]}{\eta }=0. \end{aligned}$$

(48)

By the arbitrariness of g and $g'$, the Euler–Lagrange equation

$$\begin{aligned} \frac{\partial \psi }{\partial u} - \left( \frac{\partial \psi }{\partial u'}\right) ' + \left( \frac{\partial \psi }{\partial u''}\right) '' = 0 \quad \zeta \in (a,b) \end{aligned}$$

(49)

and the boundary conditions

$$\begin{aligned}&\frac{\partial \psi ^a}{\partial u'} - \frac{\partial \psi }{\partial u''} = 0 \quad \mathrm{at} \quad \zeta =a, \end{aligned}$$

(50a)

$$\begin{aligned}&\frac{\partial \psi ^b}{\partial u'} + \frac{\partial \psi }{\partial u''} = 0 \quad \mathrm{at} \quad \zeta =b \end{aligned}$$

(50b)

are derived.