Equilibrium structures and pretransitional fluctuations in a very thin hybrid nematic film

The stability of different structures of a nematic liquid crystal in a planar hybrid film is examined within the framework of a Gaussian description of order fluctuations. In a very thin film the director field is not bent smoothly but exhibits a steplike change if the anchorings at the confining substrates are strong (G approximately > or =10(-3) J/m(2)) and comparable in magnitude. A (dis)continuous structural transition to the bent-director state which occurs with increasing film thickness or decreasing temperature is governed by the lowest bending director fluctuation mode. Its relaxation rate exhibits a critical slowdown when the film thickness approaches the "supercooling" limit or transition point, respectively. The (dis)continuity of the structural transition depends on the temperature and film thickness. The upper limit for the corresponding tricritical point is determined. The lowest order parameter mode, which corresponds to fluctuations of the position of the central exchange region, is characterized by a nearly critical behavior of the relaxation rate. The spectra of the two noncritical biaxial fluctuation modes are degenerate, whereas the fluctuation profiles are just mirror images with respect to the middle plane of the film.