Periodic Fréedericksz transition for nematic-liquid-crystal cells with weak anchoring

Lonberg and Meyer [Phys. Rev. Lett. 55, 718 (1985)] have recently discovered in polymer nematics the existence of a critical value $r_c$=0.303 for the ratio r=$K_2$/$K_1$ between the twist and splay Frank elastic constants, below which a periodic splay-twist distortion is energetically favored with respect to the well-known aperiodic splay distortion which appears in a Fréedericksz transition. In this paper the effect of weak-anchoring conditions on the critical r value and on the other parameters of interest is theoretically investigated. Analytical expressions between these parameters are given. It is shown that the periodic transition is not allowed in the range 0.5rr can be controlled in a wide range.