Nematic distribution functions

Here we have used the zero-field nematic distribution function PQ( ) for convenience of notation. The degree of net polar alignment can be seen to be enhanced in the liquid crystal over the isotropic case. The limiting cases are isotropic distributions and the Ising model (in which only 6=0 and 6=n are allowed orientations). By retaining only the leading terms in the last equation one sees that in the high temperature limit... 

X p Zxxp is diminished from that calculated for an isotropic distribution and approaches zero as the nematic distribution function becomes more tightly squeezed around 0=0 and 8=ir. This is related to the fact that the projection of DANS z-axes onto the laboratory X-axis is reduced. 

The nematic mean-field U, the molecule-field interaction potential, WE, and the induced dipole moment, ju d, are evaluated at different orientations using Equation (2.263), and then the coefficients of their expansion on a basis of Wigner rotation matrices can be calculated, according to Equation (2.268). The permittivity is obtained by a self-consistency procedure, because the energy WE and the induced dipole moment / md, as well as the reaction field contribution to the nematic distribution function p( l), themselves depend on the dielectric permittivity. 

This can be inserted in equation (02.2.3) to give tlie orientational distribution function, and tlius into equation (02.2.6) to deteniiine the orientational order parameters. These are deteniiined self-consistently by variation of tlie interaction strength iin equation (c2.2.7). As pointed out by de Gemies and Frost [20] it is possible to obtain tlie Maier-Saupe potential from a simple variational, maximum entropy metliod based on tlie lowest-order anisotropic distribution function consistent witli a nematic phase. 

One of the primary features of the Gay-Berne potential is the presence of anisotropic attractive forces which should allow the observation of thermally driven phase transitions and this has proved to be the case. Thus using the parametrisation proposed by Gay and Berne, Adams et al. [9] showed that GB(3.0, 5.0, 2, 1) exhibits both nematic and isotropic phases on varying the temperature at constant density. This was chosen to be close to the transitional density for hard ellipsoids with the same ellipticity indeed it is generally the case that to observe a nematic-isotropic transition for Gay-Berne mesogens the density should be set in this way. The long range orientational order of the phase was established from the non-zero values of the orientational correlation coefficient, G2(r), at large separations and the translational disorder was apparent from the radial distribution function. 

Figure I. Schematic representation of statistical orientational distribution functions in isotropic (I) and nematic (N) potentials. Dashed curves depict adjusted distributions under electric field poling.

Note 4 The extent of the positional correlations for the molecules in a nematic phase is comparable to that of an isotropic phase although the distribution function is necessarily anisotropic. 

With increasing flow rate, the orientational state in the nematic solution should change. Larson [154] solved numerically Eqs. (39) and (40b) with Vscf(a) given by Eq. (41) for a homogeneous system (T[f ] = 0) in the simple shear flow to obtain the time-dependent orientational distribution function f(a t) as a function of k. The non-steady orientational state in the nematic solution can be described in terms of the time-dependent (dynamic) scalar order parameter S[Pg.149]

Here Wgn are the coefficients of the expansion of the electrostatic free energy, which can be obtained from the free energy Wfl(li), according to Equation (2.269). T2n are the irreducible spherical components of the (second rank) surface tensor, which describe the anisometry of the molecular shape, and can be calculated in the form of integrals over the molecular surface [25]. Given the nematic potential the distribution function... 

Nematic phases are characterised by a uniaxial symmetry of the molecular orientation distribution function f(6), describing the probability density of finding a rod with its orientation between 6 and 6 + d0 around a preferred direction, called the director n (see Fig. 15.49). An important characteristic of the nematic phase is the order parameter (P2), also called the Hermans orientation function (see also the discussion of oriented fibres in Sect. 13.6) ... 

From slow-shear-rate solutions of the Smoluchowski equation, Eq. (11-3), with the Onsager potential, Semenov (1987) and Kuzuu and Doi (1983, 1984) computed the theoretical Leslie-Ericksen viscosities. They predicted that ai/a2 < 0 (i.e., tumbling behavior) for all concentrations in the nematic state. The ratio jai is directly related to the tumbling parameter X by X = (1 -h a3/a2)/(l — aj/aa). Note the tumbling parameter X is not to be confused with the persistence length Xp.) Thus, X < I whenever ai/a2 < 0. As discussed in Section 10.2.4.1, an approximate solution of Eq. (11-3) predicts that for long, thin, stiff molecules, X is related to the second and fourth moments Sa and S4 of the molecular orientational distribution function (Stepanov 1983 Kroger and Sellers 1995 Archer and Larson 1995) ... 

The Maier-Saupe theory of nematic liquid crystals is founded on a mean field treatment of long-range contributions to the intermolecular potential and ignores the short-range forces [88, 89]. With the assumption of a cylindrically symmetrical distribution function for the description of orientation of the molecules and a nonpolar preferred axis of orientation, an appropriate order parameter for a system of cylindrically symmetrical molecules is... 

The nematic phase differs from the usual isotropic liquid phase by the existence of a long-range orientational order [13]. The strength of this order may be quantified by the first non-zero moment S = (3 cos jS — 1) of the orientational distribution function f(jS). jS is here the angle between the axis of the rod-like moiety with the average orientation direction called the director n (Fig. 3). 

Therefore, a diffuse ring of scattering centered at the origin of reciprocal space is attached to each particle. Thus, the nematic order parameter which characterizes the distribution function of a single particle is derived from the interferences among a cluster of particles. This assumption is somewhat similar to a mean-field treatment and tends to overestimate S. 

The fairly good quality of the fits validates both Leadbetter s assumptions and the Maier-Saupe distribution function. However, the values of S obtained and even the quality of the fits obviously depend on the odd or even number of (CH2) groups in the flexible spacer. This odd-even effect is widespread and well known in the field of main-chain LCPs and will be discussed later in this article. The nematic order parameter of main-chain LCPs may reach values as high as 0.85 which demonstrates the very high orientation of the nematic phase of these polymers. Such a large orientation is undoubtedly responsible for the good mechanical properties of this type of materials. The treatment described above therefore provides a very easy way of characterizing the orientational order of a nematic phase. It has also been tested for thermotropic side-chain LCPs and found to be satisfactory as well [15]. Unfortunately, it has not been used yet in the case of lyotropic LCPs except for some aqueous suspensions of mineral ribbons (Sect. 5) which are not quite typical of this family of materials. 

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