Casimir interaction in smectic-A liquid crystals caused by coupled fluctuations of positional and orientational order

A theoretical study of the Casimir interaction in smectic-A systems, considering fluctuations of both types of smectic ordering—positional and orientational—including the coupling between them, is presented. Two model systems with plan-parallel geometry are studied: homeotropic cell and free-standing film. At large thicknesses of the system the behavior of the Casimir force is found to be primarily determined by positional fluctuations, whereas at small thicknesses also the orientational degrees of freedom greatly contribute to the interaction. The influence of different coupling strengths between orientational and positional order is presented. The dependence of the Casimir force on the director anchoring and surface-tension parameters is studied. The possibilities of experimental detection of the interaction are discussed.

Figure 1

Schematic presentation of a homeotropic cell (a) and a free-standing film (b).

Figure 2

Schematic presentation of fluctuation modes: (a) in-phase fluctuations of director and layers, and (b) out-of-phase fluctuations of director and layers. The dotted lines show local normal to the layers.

Figure 3

Casimir force ${\mathcal{F}}_{\mathrm{Cas}}$ in homeotropic smectic-A cell compared to approximate force ${\mathcal{F}}_{\mathrm{Cas}}^{\mathrm{lay}}$ for different director anchoring strengths: (a) $W\to \infty$, (b) $W={10}^{-3}\mathrm{J}∕{\mathrm{m}}^2$, (c) $W={10}^{-4}\mathrm{J}∕{\mathrm{m}}^2$, and (d) $W={10}^{-5}\mathrm{J}∕{\mathrm{m}}^2$.

Figure 4

Comparison between the coupled $\left({\mathcal{F}}_{\mathrm{Cas}}\right)$ and uncoupled $\left({\mathcal{F}}_{\mathrm{Cas}}^{\mathrm{unc}}\right)$ Casimir force in homeotropic cell for infinitely strong director anchoring $(W\to \infty )$ and various coupling constants $D$: (a) $D={10}^5\mathrm{N}∕{\mathrm{m}}^2$, (b) $D={10}^4\mathrm{N}∕{\mathrm{m}}^2$, and (c) $D={10}^6\mathrm{N}∕{\mathrm{m}}^2$.

Figure 5

Effect of director-layer coupling constant $D$ on the Casimir force: (a) $D={10}^5\mathrm{N}∕{\mathrm{m}}^2$, (b) $D={10}^4\mathrm{N}∕{\mathrm{m}}^2$, and (c) $D={10}^6\mathrm{N}∕{\mathrm{m}}^2$. Strong anchoring of the director $(W\to \infty )$ is assumed.

Figure 6

Casimir force in free-standing film compared to the force in homeotropic cell $\left[{\mathcal{F}}_{\mathrm{Cas}}(\gamma \to \infty )\right]$ for the director anchoring strength $W={10}^{-5}\mathrm{J}∕{\mathrm{m}}^2$, coupling constant $D={10}^5\mathrm{N}∕{\mathrm{m}}^2$, and different surface tensions: (a) $\gamma ={10}^{-2}\mathrm{J}∕{\mathrm{m}}^2$, (b) $\gamma =5\times {10}^{-2}\mathrm{J}∕{\mathrm{m}}^2$, and (c) $\gamma ={10}^{-1}\mathrm{J}∕{\mathrm{m}}^2$. The dashed lines represent ${\mathcal{F}}_{\mathrm{Cas}}^{\mathrm{lay}}\left(\gamma \right)$.

Figure 7

Comparison between the coupled $\left({\mathcal{F}}_{\mathrm{Cas}}\right)$ and uncoupled $\left({\mathcal{F}}_{\mathrm{Cas}}^{\mathrm{unc}}\right)$ Casimir force in a free-standing film for the director anchoring strength $W={10}^{-5}\mathrm{J}∕{\mathrm{m}}^2$, coupling constant $D={10}^5\mathrm{N}∕{\mathrm{m}}^2$, and different surface tensions: (a) $\gamma ={10}^{-2}\mathrm{J}∕{\mathrm{m}}^2$, (b) $\gamma =5\times {10}^{-2}\mathrm{J}∕{\mathrm{m}}^2$, and (c) $\gamma ={10}^{-1}\mathrm{J}∕{\mathrm{m}}^2$.