Equilibrium Structures and Pretransitional Fluctuations in a Very Thin Hybrid Nematic Film A. ˇ Sarlah and S. ˇ Zumer Department of Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia (Dated: April 16, 1999) The stability of diﬀerent structures of a nematic liquid crystal in a planar hybrid ﬁlm is examined within the frame of Gaussian description of order ﬂuctuations. In a very thin ﬁlm the director ﬁeld is not bent smoothly but exhibits a step-like change if the anchorings at the conﬁning substrates are strong (G > ∼ 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 slow- down 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 non-critical biaxial ﬂuctu- ation modes are degenerate whereas the ﬂuctuation proﬁles are just mirror images with respect to the middle plane of the ﬁlm. I. INTRODUCTION Recently, the equilibrium ordering as well as dynamic properties of conﬁned liquid crystals have attracted a lot of attention of experimentalists and theorists. Studies devoted to the determination of the equilibrium order in diﬀerent conﬁning geometries with various constraining properties [1] have been lately followed by investigations of the pretransitional dynamics. The understanding of collective order ﬂuctuations gives a better insight in equi- librium ordering in the vicinity of phase and structural transitions as well as in the mechanism of the transition itself [2–9]. Possible technological applications have stimulated an increase of interest in hybrid nematic geometries [10]. Us- ing quasi elastic light scattering method Wittebrood et al. (see Ref. [11] and the references therein) have exper- imentally studied thickness dependence of the nematic– isotropic phase transition temperature and stability of ordered structures in a hybrid nematic ﬁlm obtained af- ter a spread of a liquid crystal droplet on a solid sub- strate. In their experimental setup with unequal anchor- ing strengths of the conﬁning substrates (solid substrate and a free liquid crystal surface) they were able to deter- mine the critical cell thickness for the hybridly aligned order which was in good agreement with the theoreti- cal expression obtained long ago by Barbero and Barberi [12]. In their study approximate director picture omitting positional dependence of the scalar order parameter and biaxiality was used. In the frame of Frank elastic theory an extensive study of pretransitional director dynamics in a hybrid cell has been done by Stallinga et al. [5]. Using the director description of the nematic liquid crys- talline ordering they have calculated relaxation times for tilt and twist ﬂuctuations in hybridly aligned structure and director ﬂuctuations in uniform director ﬁeld struc- ture. However, in their study they have neglected spatial dependence of the uniaxial and biaxial degrees of nematic order, which are quite important in the case of strong an- choring and thin cells. A couple of years ago, Palﬀy et al. [13] have shown that in highly constrained hybrid cells the nematic order can be either biaxial with the step-like proﬁle of director’s tilt angle or the director ﬁeld can be bent continuously. They have predicted a structural tran- sition between the two possible ordered conﬁgurations but did not probe the stability of both conﬁgurations. However, a more detailed description of the nematic or- der in planar hybrid geometry in relation to ﬁlm thickness and anchoring strength has been provided by Galabova et al. [14]. Another aspect of a nematic liquid crystal in a planar hybrid geometry are stripe domains studied by Pergamenshchik [15]. In his study, using Frank elastic theory with surface terms it has been shown that equi- librium modulated structures can appear. However, in that study only spatial dependence of the nematic direc- tor has been taken into account whereas other degrees of freedom of the nematic order have been neglected. In a cylindrical geometry Ziherl and ˇ Zumer [6] have stud- ied director ﬂuctuations in the vicinity of a disclination line of strength 1 whose structure is similar to structures in hybrid cells. They extended the approach based on Frank elastic theory by introducing spatially dependent rotational viscosity and elastic constants. This brief review shows that there is a lack of infor- mation on the dynamics related to the structural tran- sition between diﬀerent nematic conﬁgurations in highly constrained systems when non-director degrees of free- dom are crucial. This motivated us to start our analy- sis. In order to provide a simple but detailed description of a highly frustrated system we have examined a thin planar ﬁlm with hybrid surface conditions. In contrast to previous studies [5, 6, 12] we have focused our at- tention to highly constrained ﬁlms where biaxiality and non-homogeneous degree of nematic order play impor- tant role. Although the origins of highly frustrated sys- tem can be diﬀerent, i.e., speciﬁc conﬁning substrates in