Spontaneous deformations of the uniform director ground state induced by the surfacelike elastic terms in a thin planar nematic layer

We study the effect of the divergence (surfacelike) $K_{24}$ and $K_{13}$ terms on stability of the uniform ground state of a nematic phase. It is shown that the $K_{13}$ term can effectively boost the action of the $K_{24}$ term. As a result, even if the two Ericksen stability conditions are satisfied, spontaneous deformations can occur in geometries with a sufficiently small volume-to-surface ratio. For a specific example, we show that this mechanism can destabilize the uniform planar director field in a thin nematic layer with planar anchoring and produce spontaneous periodic director modulation. The critical thickness below which the predicted modulated phase occurs is found to be $h_c=-{2L}_a\left[1-\right(1\mathrm{}-{2K}_{24}{/K}_{11}{)}^2{+K}_{13}^{*}{/K}_{11}],$ where $L_a$ is the polar anchoring extrapolation length, $K_{11}$ and $K_{24}$ are the standard Nehring-Saupe elastic constants, and $K_{13}^{*}$ is the effective elastic constant of the layer.