Phase transitions universal behaviour

SmA phases, and SmA and SmC phases, meet tlie line of discontinuous transitions between tire N and SmC phase. The latter transition is first order due to fluctuations of SmC order, which are continuously degenerate, being concentrated on two rings in reciprocal space ratlier tlian two points in tire case of tire N-SmA transition [18,19 and 20], Because tire NAC point corresponds to the meeting of lines of continuous and discontinuous transitions it is an example of a Lifshitz point (a precise definition of tliis critical point is provided in [18,19 and 20]). The NAC point and associated transitions between tire tliree phases are described by tire Chen-Lubensky model [97], which is able to account for tire topology of tire experimental phase diagram. In tire vicinity of tire NAC point, universal behaviour is predicted and observed experimentally [20]. 

In view of the factors determining the behaviour of enormous ensembles of molecules, namely, intermolecular interactions and thermal motion, one can expect first-order phase transitions to be universal in one-component systems. In such cases, the isotherm enclosing three states of aggregation, has to have the form as in Figure 1.19. Four points corresponding to the thermodynamic stability boundary dFjdV = 0 A, B, C, D are marked on this isotherm. However, no critical point of the liquid-crystal transition has been discovered in spite of numerous attempts. 

Thus, the increase of the crosslinking density in the LCEs results in a shift of their phase-transition behaviour towards and beyond supercritical. As this trend is now confirmed for both the side- and main-chain nematic LCEs, i.e. two LCE families with drastically different crosslinking topologies, it can be anticipated that the link between the crosslinking concentration and the phase-transition criticality holds universally for most types of LCEs. 

Figure 5.22 Universal phase behaviour at the NAC point. Shifted phase transition temperature for five binary mixtures of mesogens are plotted as a function of the mole fraction, also shifted with respect to that at the NAC triple point...

Continuous phase transitions are characterized by universality thermodynamic observables that diverge with power-law exponents whose values are governed by symmetry considerations and are insensitive to other details of the materials [1, 2], The nematic-smectic-A (NA) transition, where an ordered liquid acquires additional one-dimensional periodicity is one of the outstanding unsolved problems in this field of study [3, 4], Here the critical behaviour appears non-universal. The order of the transition has been a matter of debate. The complexity of the NA transition arises from an intrinsic coupling between two order parameters. Indeed, even the direct determination of mean-field parameters has been a matter of recent study. Due to this complexity, there are still unresolved issues after more than three decades of research. The subtleties involved have been addressed theoretically via different approximations, leading to a rich addition to the phase transitions literature. Experimentally, these subtleties have inspired precise high-resolution experiments. This article focuses on experimental developments in the last decade that address aspects of the nature of the NA transition. 

In this Chapter we introduce some of the theoretical approaches for studying thin nematic liqnid crystalline systems, both on the microscopic and macroscopic level. In the former, one models the microscopic interactions between the constituing molecules, leaves the system to evolve, and then determines its macroscopic properties. If the obtained macroscopic behaviour is in agreement with the experimental evidence the modeled interaction is considered appropriate. On the other hand, the macroscopic description takes into account the universal properties of systems in the vicinity of phase and structural transitions. This means that they are based on the fact that in the vicinity of phase changes the macroscopic properties of the system do not depend on the details of the microscopic interactions but on the symmetry properties and dimensionality of the system in question. Most of our attention is focused on the effects of confinement on to liquid crystalline order. Finally, we will be interested in the resulting disjoining pressure. The evidences in experiments will be briefly mentioned. 

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