Uniaxial nematic transitions

Figure 13.13 Phase transition temperatures from isotropic phase to various structures as a function of chirahty. (a) isotropic-uniaxial nematic transition, (b) isotropic-uniaxial cholesteric transition, (c) isotropic-biaxial cholesteric transition, (d) isotropic-cubic transition.

Camp P J, Mason C P, Allen M P, Khare A A and Kofke D A 1996 The isotropic-nematic transition in uniaxial hard ellipsoid fluids coexistence data and the approach to the Onsager limit J. Chem. Phys. 105 2837-49... 

The above spectral densities can be modified for the occurence of chain flexibility, and for the director being oriented at dLD w.r.t. the external BQ field in the L frame. For CD bonds located in the flexible chain, the effect of DF is reduced due to an additional averaging of the time dependent factor (/f g) by conformational transitions in the chain. Consequently, the spectral densities given in Eqs. (60)-(62) are modified by replacing Soc%0(Pm,q) by the segmental order parameter YCD of the C-D bond at a particular carbon site on the chain.146,147 As observed experimentally,148,149 the spectral densities in a flexible chain show a SqD dependence when DF dominate the relaxation rates. The general expression of Jm(co 0LD) due to DF in uniaxial nematic phases is given by... 

The degree of orientational order in a uniaxial nematic is given by the order parameter S, defined by Eq. (2-3). S is zero in the isotropic state, and it approaches unity for hypothetically perfect molecular alignment (i.e., all molecules pointing in the same direction). In single-component small-molecule nematics, such as MBBA, S varies with temperature from 5 0.3 at Tni, the nematic-isotropic transition temperature, to S 0.7 or so at lower... 

At mesoscopic, continuous scale, the nematic order is characterized by the local orientation n (r) and the quality of the order. We restrict ourselves to uniaxial nematics. The parameters are introduced in the frame of the Landau-de Gennes analysis, which is convenient close to the N1 transition, using the pedagogical presentation by Sheng and Priestley. In the local coordinate system where n(r) coincides with the z-axis, the local order parameter is given by ... 

The majority of the existing molecular theories of nematic liquid crystals are based on simple uniaxial molecular models like sphe-rocylinders. At the same time typical mes-ogenic molecules are obviously biaxial. (For example, the biaxiality of the phenyl ring is determined by its breadth-to-thick-ness ratio which is of the order of two.) If this biaxiality is important, even a very good statistical theory may result in a poor agreement with experiment when the biaxiality is ignored. Several authors have suggested that even a small deviation from uniaxial symmetry can account for important features of the N-I transition [29, 42, 47, 48], In the uniaxial nematic phase composed of biaxial molecules the orientational distri-... 

Finally, the transition lines from uniaxial nematic to biaxial are determined Eq. (8) with the constraint a2 = d -. At lowest order in b, the slope of the two lines at their meeting point a = Z7 = 0 is given by . 

The complete phase diagram is obtained with the selection rule (tI (7 and reproduced in Fig. 1 [1]. Note that a direct transition I-Nb occurs at a single point a = b = 0 on the line (Eq. 10) [7]. The biaxial nematic region separates two uniaxial nematics N+ and N of opposite sign. N -Nb transitions are second order since the condition 2 = (7 can be approached continuously from the biaxial phase. 

The only difference between the nematic phase and the isotropic phase is the orientational order. A proper description of this orientational order requires the introduction of a tensor of the second rank [7, 8]. This tensor can be diagonalized and for anisotropic liquids with uniaxial symmetry, the nematic phase can be described by only one scalar order parameter. The thermodynamic behavior in the vicinity of the N-I transition is usually described in terms of the mean-field Landau-de Gennes theory [7]. For the uniaxial nematic phase one can obtain the expansion of the free energy G in terms of the modulus of an order parameter Q. 

While both side-on side-chain liquid crystal polymers [16] and low molecular weight liquid crystals [17] have been reported to have biaxial nematic phases, only in a lyotropic liquid crystal system [18], has the uniaxial-biaxial nematic transition been studied in detail from the point of view of critical phenomena. This transition is found to be second order. So far, it is the one liquid crystal system where earlier theoretical expectations [19] of fluctuation dominated phase transitions and later experimental results are most fully in agreement with respect to both static [20] and dynamic [21] aspects of critical phenomena. In particular, static critical phenomena predicts 3D-XY exponents which have been observed (with irrelevant corrections to scaling) by Saupe et al. [19]. Transport parameters were not expected [19] to show any singularities at the transition [19] as later verified by Roy et al. [21] because the dynamics of the biaxial order parameter is nonconserved (Model A) [19]. 

In general, liquid crystal molecules do not have the D200 symmetry of the uniaxial nematic phase. Since an interface acts as a field, its presence can provide polar order. Such surface polar ordering, confined to a single molecular layer, has been observed [99]. Surface SHG can also be used to probe the orientational distribution at the surface, and anchoring transitions [100]. 

Phase biaxiality in lyotropic system was detected by NMR a long time ago [55, 56]. For the ternary system, sodium dodecylsul-phate/decanol/water, the transitions from the uniaxial nematic N and N phases to... 

Based on the interaction employed in the Maier-Saupe theory of the nematic state, Preiser [4] was the first to predict the possible existence of an N, phase as an intermediate between two uniaxial nematic phases. Later, a number of other theoretical investigations were carried out [5-7] using various models to predict the possibility of obtaining an liquid crystal. In all these models, a system consisting of hard rectangular plates was considered. These approaches gave the same result, an Nb phase should be obtained between two uniaxial nematics of opposite sign, i.e., those made up of rod-like and plate-like molecules. They also predicted that a transition from a uniaxial nematic to a biaxial nematic would be second order. 

The values of the scaled transition temperature, l TNi/e2oo, together with the transitional values of the ord parameters ni, ni and ni are listed in TABLE 1 for several values of the biaxiality parameter, X. As the molecular biaxiality increases so the major order parameter at the transition decreases indeed when X is 0.3 ni has decreased to about half that for a uniaxial molecule. In contrast although as expected the biaxial ordo- parameter ni increases with X the value is extremely small. It would seem, therefore, that the molecular biaxiality has the greatest effect on Ni and that ni is essentially negligible. The influence of X on the second rank order parameter is mimicked by its fourth rank counterpart ni indeed when X is 0.3 this order parameter is four times smaller than that for uniaxial molecules. For X of 0.4 all of the transitional order parameters are extremely small, hinting at the approach of a second ord transition. This occurs when X is 1 / >/6 but now the transition from the isotropic phase is directly to a biaxial and not a uniaxial nematic phase, as we shall see. 

For molecules of this shape, the isotropic phase is found to undergo a transition directly into the biaxial phase on compression. A biaxial phase has also been observed in simulations of mixtures of rods and platelets, in which the different species are modelled ellipsoids with aspect ratios of x = 20 and X = 1/20, respectively [19]. However, here the range of the biaxial phase is severely limited by demixing into two coexisting uniaxial nematics, one rich in rods, the other rich in discs. 

The non-collective motions include the rotational and translational self-diffusion of molecules as in normal liquids. Molecular reorientations under the influence of a potential of mean torque set up by the neighbours have been described by the small step rotational diffusion model.118 124 The roto-translational diffusion of molecules in uniaxial smectic phases has also been theoretically treated.125,126 This theory has only been tested by a spin relaxation study of a solute in a smectic phase.127 Translational self-diffusion (TD)29 is an intermolecular relaxation mechanism, and is important when proton is used to probe spin relaxation in LC. TD also enters indirectly in the treatment of spin relaxation by DF. Theories for TD in isotropic liquids and cubic solids128 130 have been extended to LC in the nematic (N),131 smectic A (SmA),132 and smectic B (SmB)133 phases. In addition to the overall motion of the molecule, internal bond rotations within the flexible chain(s) of a meso-genic molecule can also cause spin relaxation. The conformational transitions in the side chain are usually much faster than the rotational diffusive motion of the molecular core. 

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