Phenomenological approach to the problem of theK13surfacelike elastic term in the free energy of a nematic liquid crystal

We consider the mathematical foundations of continuum theories of nematic liquid crystals of the Frank-Oseen form which include, in addition, the surfacelike ${\mathit{K}}_{13}$ term. Such theories present problems because (i) the free-energy functional ${\mathit{F}}_2$, quadratic in the director derivatives, is unbounded from below and hence possesses no minima unless ${\mathit{K}}_{13}$ is strictly zero; and (ii) microscopic theories indicate that in the general case ${\mathit{K}}_{13}$ does not vanish. The continuum theory presupposes the existence of weak director deformations. This is not consistent with the idea, proposed by Oldano and Barbero, that there should be strong subsurface director deformations, which are shown in the present paper to be a formal consequence of (i). Instead we propose a resolution of the ${\mathit{K}}_{13}$ problem which is consistent with weak director distortions alone. The resolution involves a formal consideration of all the terms in the total free energy containing high-order derivatives, the infinite sum of which, ${\mathit{R}}_{\mathrm{\infty }}$, bounds the total free energy ${\mathit{F}}_2$+${\mathit{R}}_{\mathrm{\infty }}$ from below. A consequence of this resolution is that the Euler-Lagrange equations which follow from a naive consideration of the Oseen-Frank free-energy functional ${\mathit{F}}_2$, and which appear to give rise to a nonminimal family thereof, in fact give rise to a minimal family of director distributions of the total free energy ${\mathit{F}}_2$+${\mathit{R}}_{\mathrm{\infty }}$. Moreover, no specific information on higher-order elastic terms enters the theory. The theory further allows consideration of the derivative-dependent terms in the anchoring energy. Each such derivative is shown to be proportional to a small parameter. As a result, all derivative-dependent anchoring terms are much smaller than the usual Rapini-Popoular term.