Elliptical soft colloids in smectic-$C$ films

We investigate theoretically the elliptical shapes of soft colloids in freely standing smectic-$C$ films, that have been reported recently. The colloids favor parallel alignment of the liquid crystal molecules at their surfaces and, for sufficiently strong anchoring, will generate a pair of defects at the poles of the colloidal particles. The elastic free energy of the liquid crystal matrix will, in turn, affect the shape of the colloids. In this study we will focus on elliptical soft colloids and determine how their equilibrium shapes depend on the elastic constants of the liquid crystal, the anchoring strength, the surface tension, and the size of the colloids. A shape diagram is obtained analytically, by minimizing the Frank elastic free energy, in the limit of small eccentricities. The analytical results are verified, and generalized to arbitrary eccentricities, by numerical minimization of an appropriate Landau free energy. The latter is required for an adequate description of the topological defects when the liquid crystal correlation length is comparable to the size of the colloidal particles.

Figure 1

(Color online) Soft colloids in a smectic-$C$ layer with correlation length $\xi =0.1\mathcal{R}$ and $\omega =\mathcal{WR}∕\mathcal{K}=10$. The vectors represent the $\mathbf{c}$ director, while the colored regions correspond to strong variations of the order parameter and signal the surface defects. (a) $\sigma =\gamma \mathcal{R}∕\mathcal{K}=40$ and optimal eccentricity $e^{*}=0.2$; (b) $\sigma =13.4$ and $e^{*}=0.4$; (c) $\sigma =7.8$ and $e^{*}=0.6$; (d) $\sigma =5.8$ and $e^{*}=0.8$.

Figure 2

${\mathcal{U}}_2^{\left(2\right)}$ and ${\mathcal{U}}_4^{\left(2\right)}$ as functions of the reduced anchoring strength $\omega =\mathcal{WR}∕\mathcal{K}$. Higher order coefficients lead to small corrections (less than $0.6\%$) to the optimal eccentricity $e^{*}$.

Figure 3

Optimal eccentricity $e^{*}$ as a function of $\sigma =\gamma \mathcal{R}∕\mathcal{K}$, at fixed values of $\omega =\mathcal{WR}∕\mathcal{K}=0.1$, $1.0$, $10$, $\infty$.

Figure 4

Shape diagram: lines of constant eccentricity in the $\omega$-$\sigma$ plane. The circle and the ellipse schematically shown occur for $e=0$ and $e=0.9$, respectively.

Figure 5

Reduced free energy $\tilde{\mathcal{F}}$ as a function of the eccentricity for $\epsilon ={10}^{-4}$, $\omega =10$, and $\sigma =6$.

Figure 6

Optimal eccentricities as a function of $\epsilon$ for (a) $\omega =1$ and (b) $\omega =10$ and $\sigma =10$.

Figure 7

Optimal eccentricity $e^{*}$ as a function of $\sigma$ at constant $\omega =1$, $10$, $100$ and $\epsilon ={10}^{-2}$, ${10}^{-4}$. Lines and symbols represent the analytical and numerical results, respectively.

Figure 8

Shape diagram. Lines and symbols denote equilibrium shapes at constant eccentricity, in the $\omega \text{−}\sigma$ plane. The lines were obtained analytically while the symbols are the result of a numerical minimization, for systems with $\epsilon ={10}^{-2}$.