Density Functional Theory of Model Systems with the Biaxial Nematic Phase

Present work is a theoretical study on the stability of the thermotropic biaxial nematic liquid crystal phase in model systems. Its main aim is to present the phase diagrams of spatially uniform liquid mesophases and to identify the molecular parameters that influence the stability of the biaxial nematic. The diagrams are obtained by means of the Local Density Functional Theory in low density approximation, and the relation between the molecular parameters of the models and macroscopic properties of the system close to the transition point are obtained by means of bifurcation analysis. We consider three model systems; the so-called L = 2 model (the lowest coupling model of the orientational part of pair potential), the biaxial GayBerne interaction, and the bent-core system. For the second one, we also briefly investigate the elastic constants and comment on the smectic phases. In every case we take into account rigid molecules. Firstly, we study a generalized version of Straley model, which should be considered as the simplest one giving rise to the biaxial nematic phase; it is to the biaxial nematic what Maier-Saupe model is to the uniaxial nematic. Its analysis allows to construct a generic, exact in mean field bifurcation diagram as function of potential parameters. By considering the symmetries of the L = 2 expansion of pair interaction, valid also for pair direct correlation function, we find the duality transformation which connects the states at different temperatures. The so-called self-dual points, i.e., the points in space of potential parameters left invariant under the action of the duality transformation, coincide with Landau points. The following Landau region is found as well as the tricritical points. Despite its simplicity, the model exhibits a rich behaviour. Many systems can be accurately approximated using this approach, and located on the presented diagram. The second model studied is a Gay-Berne potential generalized to the soft ellipsoids of three different axes. It showed the biaxial nematic in Monte Carlo study, and therefore it gives us the chance to confront our results with simulations and study the interaction parameter space in more detail. We also compare the bifurcation diagram following from Local Density Functional Theory with transition points acquired by the minimisation of the Helmholtz free energy. In comparison with the simulations, the approach used overestimates the transition temperatures. We study the interaction further and locate the Landau points (also called selfdual) induced by the increase of both the shape and energy biaxiality. In the former case, our results indicate that the introduction of attractive forces slightly shifts the position of the self-dual point from its location for hard biaxial ellipsoids. We find that the direct isotropic – biaxial nematic transition occurs at Landau points, and show how the concurring biaxialities influence their position. The results indicate that for this model self-dual region predicted for hard, biaxial ellipsoids by the so-called square root rule retains its significance, and provides qualitatively correct estimations of Landau points positions. Analysis of the bent-core system is aimed at studying the shape induced effects and the role of the dipole-dipole interaction. The model molecules are built using two and three Gay-Berne interacting, prolate, soft ellipsoids of uniaxial and biaxial symmetry. We study the influence of dipole-dipole interaction, in case of two and three uniaxial arms, by introducing a transverse dipole moment lying along the C2 symmetry axis, in the plane of an angle between the arms (opening angle). For non-polar case of two uniaxial arms, we find that by changing the opening angle we can obtain a Landau point at 107◦, in agreement with previous studies for hard bent-core molecules. Surprisingly, for this model the inclusion of attractive forces does not influence its position. Once the arms deviate from the uniaxial symmetry and become biaxial ellipsoids interacting via biaxial Gay-Berne potential, the self-dual point evolves to the line of Landau points ranging in the opening angle between 121◦ and 128◦. In both cases, the opening angles at which self-dual points occur, differ from the experimental estimates of 140◦. However, a study on system of bent-core molecules with opening angle of 90◦ was published recently, and those results remain in qualitative agreement with our analysis of the model of three uniaxial Gay-Berne parts where we find the bifurcation diagram with the isolated Landau point at the angle between the arms equal to 89◦. The introduction of weak dipoles (relative to the strength of the Gay-Berne energy) leads to the shifting of the Landau point towards lower angles. For three uniaxial parts model, moderate dipole magnitudes provide the diagram with a line of self-dual points, range of which in opening angle increases with dipole strength. The maximum length of the interval of the opening angle for which a direct isotropic – biaxial nematic transition occurs is 23◦ between 63◦ and 86◦. For the strongest dipoles studied, that line begins to shrink and shift towards higher angles. In bent-core models, again, the Landau points are the only places where isotropic – biaxial nematic transition takes place, the novel feature is the appearance of the self-dual line. Since the elastic constants are important quantities in the theory and applications of liquid mesophases, in final chapter we present the preliminary studies on temperature dependence of a set of bulk biaxial elastic constants for the biaxial Gay-Berne interaction in L = 2 model. Our results are in agreement with previous findings for the lattice biaxial model; we recover the splay-bend degeneracy, and find that the constants associated with one of three directors in the biaxial nematic are always negative. Also, in the rod-like molecular regime the relative values of the uniaxial elastic constants are higher than biaxial ones. This behaviour changes in vicinity of the Landau point, where the constants associated with one of the biaxial directors become dominant. Finally, we make a comment on the smectic-A phases. Although the thesis is devoted to the spatially uniform liquids, the competition between the smectic order and biaxial nematic should be considered since the former may gain stability earlier and prevent the formation of the spatially uniform biaxial ordering. In order to address this issue, and as an example, we limit ourselves to the orthogonal smectic-A phases. We present the behaviour of a complete set of leading order parameters for the biaxial Gay-Berne potential, based on the self–consistent equation for equilibrium one-particle distribution function in L = 2 model, including uniaxial and biaxial smectic-A states. The temperature dependence of order parameters is presented for two sets of interaction parameters. One of them, namely the one where molecules are strongest attracted to their sides (the lateral forces are strongest), was the only case for which Monte Carlo simulations discovered a stable biaxial nematic. We find that in this case the temperature range of the lower symmetric nematic state is significantly wider than for the model of strong face-to-face attractions. For fixed density and with increasing temperature, biaxial smectic is found to melt to biaxial nematic which in turn transforms to uniaxial nematic and for higher temperatures the system undergoes a phase transition to isotropic liquid. This phase sequence is in agreement with simulations results. In final section, we extend the approach used for spatially uniform states to include the orthogonal smectic phases of uniaxial and biaxial symmetry, and we present the additional bifurcation equations.