PHENOMENOLOGICAL APPROACH TO THE PROBLEM OF THE K13 SURFACE-LIKE ELASTIC TERM IN THE FREE-ENERGY OF A NEMATIC LIQUID-CRYSTAL

We consider the mathematical foundations of continuum theories of nema tic liquid crystals of the Frank-Oseen form which include, in addition , the surfacelike K13 term. Such theories present problems because (i) the free-energy functional F2, quadratic in the director derivatives, is unbounded from below and hence possesses no minima unless K13 is s trictly zero; and (ii) microscopic theories indicate that in the gener al case K13 does not vanish. The continuum theory presupposes the exis tence of weak director deformations. This is not consistent with the i dea, proposed by Oldano and Barbero, that there should be strong subsu rface director deformations, which are shown in the present paper to b e a formal consequence of (i). Instead we propose a resolution of the K13 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, R(infinity) bounds the total free energy F2 + R(infinity) f rom 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 F2, and which appear to give rise to a nonminim al family thereof, in fact give rise to a minimal family of director d istributions of the total free energy F2 + R(infinity). Moreover, no s pecific information on higher-order elastic terms enters the theory. T he theory further allows consideration of the derivative-dependent ter ms in the anchoring energy. Each such derivative is shown to be propor tional to a small parameter. As a result, all derivative-dependent anc horing terms are much smaller than the usual Rapini-Popoular term.