Theory of anchoring on a two-dimensionally grooved surface

We investigate analytically the anchoring of a nematic liquid crystal on a two-dimensionally grooved surface of arbitrary shape, induced by the elastic distortions of a liquid crystal adjacent to the surface. Our theoretical framework applied to a surface with square grooves reveals that such a surface can exhibit bistable anchoring, while a direct extension of a well-known theory of Berreman [Phys. Rev. Lett. 28, 1683 (1972)] results in no azimuthal anchoring in the so-called one-constant case (K1=K2=K3, with K1, K2, and K3 being the splay, twist, and bend elastic constants, respectively). We show under the assumption of K1=K2=K that the direction of the bistable easy axes and the anchoring strength crucially depend on the ratios K3/K and K24/K, where K24 is the saddle-splay surface elastic constant. To demonstrate the applicability of our theory to general cases and to elucidate the effect of surface shape and the elastic constants on the properties of surface anchoring, we also consider several specific cases of interest; one-dimensional grooves of arbitrary shape, rhombic grooves, and surfaces possessing 2N -fold symmetry, including hexagonal grooves, and show the following: (i) The rescaled anchoring energy f(phi)/f(pi/2) of one-dimensional grooves, with phi being the angle between the director n and the groove direction, is independent of the groove shape. (ii) Whether two diagonal axes of rhombic grooves can become easy axes depends sensitively on K3/K, K24/K and the angle alpha between the grooves. The angle alpha yielding the maximum anchoring strength for given groove pitch and amplitude depends again on K3/K and K24/K; in some cases alpha=0 (one-dimensional grooves), and in other cases alpha not equal 0, gives the maximum anchoring strength. Square grooves (alpha=pi/2) do not necessarily exhibit the largest anchoring strength. (iii) A surface possessing 2N -fold symmetry can yield N -stable azimuthal anchoring. However, when K1=K2=K3 and N>or=3, azimuthal anchoring is totally absent irrespective of the value of K24. The direction of the easy axes depends on K3/K, K24/K, and whether N is even or odd.