Surfacelike-elasticity-induced spontaneous twist deformations and long-wavelength stripe domains in a hybrid nematic layer

We consider the effect of the divergence ${\mathit{K}}_{24}$ term ∇[n(∇n)+n×∇×n] in the nematic free energy, which has been ignored for a long time. The ${\mathit{K}}_{24}$ term is shown to be able to cause spontaneous twist deformations. This mechanism is irrespective of the bulk Frank constant anisotropy in contrast to the well-known mechanism associated with the smallness of the twist elastic constant ${\mathit{K}}_{22}$. For geometries with sufficiently large surface-to-volume ratios, it can also be effective in other condensed media described by similar free-energy functionals, but with considerably less anisotropic constants than liquid crystals (e.g., the B phase of liquid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ and ferromagnets). As a specific model we consider the formation of long-wavelength stripe domains in a usual but rather thin hybrid nematic layer. Degenerate boundary conditions are imposed on the surfaces of the latter. Recent experiments show the existence of such domains in a hybrid nematic layer whose state had always been regarded as homogeneous in the layer plane. The theory worked out in the paper allows one to incorporate all the harmonics of the periodic domain structure in the vicinity of the critical point. The dependence of the domain period on the layer thickness, obtained in the paper, makes it possible to find the value of the elastic constant ${\mathit{K}}_{24}$ from the experimental data.