Nematic liquid crystal in a tube: The Fréedericksz transition

We study the configurations of a nematic liquid crystal confined between two concentric cylinders, of radii ${\mathit{r}}_1$ and ρ${\mathit{r}}_1$>${\mathit{r}}_1$, with homeotropic anchoring conditions at the walls. For ρ<${\mathrm{\rho }}_{\mathit{c}}$=exp[π √(${\mathit{K}}_{\mathit{b}}$/${\mathit{K}}_{\mathit{s}}$) ] the director is purely radial, while for ρ>${\mathrm{\rho }}_{\mathit{c}}$ the director ‘‘escapes into the third dimension’’ and gains an axial component. This represents a geometrically induced Fréedericksz-like transition. When combined with a radial field and a nematic liquid crystal with a dielectric anisotropy ${\mathrm{\epsilon }}_{\mathrm{\perp }}$-${\mathrm{\epsilon }}_{\mathrm{\parallel }}$>0, the critical voltage of the Fréedericksz transition can be made arbitrarily small. This is of interest for liquid-crystal displays, since in the planar geometry the critical voltage is a material constant. A related temperature-induced Fréedericksz-like transition is expected in semiflexible nematic polymers.