Surface-layer transitions in nematogenic materials characterized by a preferential planar surface interaction linear in the order parameter have been studied theoretically at temperatures above the bulk transition (T(NI)). The coupled Euler-Lagrange nonlinear differential equations obtained from the Landau-de Gennes free energy were solved exactly by numerical integration. This problem had been studied previously employing various limits and approximations with several differences in the phase diagram reported. The exact results allow one to determine which of these differences are artifacts of the approximations used and which are dependent upon the ratio of elastic constants. It is found, for physically relevant elastic constants, that there is always a uniaxially ordered surface layer at sufficiently high temperatures. For weak surface coupling, no surface phase transition occurs and the uniaxial layer remains the stable state until T(NI) is reached. When the surface coupling is increased, there is a single first-order (prewetting) transition from uniaxial to biaxial surface ordering as the temperature is reduced towards T(NI). This transition boundary becomes second order (by way of a tricritical point) when the surface coupling is further increased. We also find that the mean-field boundary is suppressed due to Berezinskii-Kosterlitz-Thouless (BKT)-type phase fluctuations. Also, these fluctuations can result in the re-entrant (with increasing surface coupling strength) uniaxial-biaxial phase boundary terminating on the bulk transition line rather than becoming asymptotic to it.
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