Director orientation distribution

Fig. 4.6.1 Director orientation distribution of the K cell in the bend mode.

Fig. 4.6.2 Initial director orientation distribution of the Pi cell in the splay mode.

The anisotropy of the liquid crystal phases also means that the orientational distribution function for the intermolecular vector is of value in characterising the structure of the phase [22]. The distribution is clearly a function of both the angle, made by the intermolecular vector with the director and the separation, r, between the two molecules [23]. However, a simpler way in which to investigate the distribution of the intermolecular vector is via the distance dependent order parameters Pl+(J") defined as the averages of the even Legendre polynomials, PL(cosj r)- As with the molecular orientational order parameters those of low rank namely Pj(r) and P (r), prove to be the most useful for investigating the phase structure [22]. 

We begin, however, with the singlet orientational distribution function which is shown for the three liquid crystal phases in Fig. 6. In each phase the distribution is peaked at cos of 1 showing that the preferred molecular orientation is parallel to the director. The form of the distribution function is well represented by the relatively simple function... 

The order parameter S is the orientational average of the second-order Legendre polynomial P2(a n) (n = the director), and if the orientational distribution function is approximated by the Onsager trial function, it can be related to the degree of orientation parameter ot by... 

Orientational Distribution Function and Order Parameter. In a liquid crystal a snapshot of the molecules at any one lime reveals that they arc not randomly oriented. There is a preferred direction for alignment of the long molecular axes. This preferred direction is called the director, and it cun be used to define- an orienlalional distribution function, f W). where flH win Vilb is proportional to the fraction of molecules with their long axes within the solid angle sinbdw. 

Experiments by Muller et al. [17] on the lamellar phase of a lyotropic system (an LMW surfactant) under shear suggest that multilamellar vesicles develop via an intermediate state for which one finds a distribution of director orientations in the plane perpendicular to the flow direction. These results are compatible with an undulation instability of the type proposed here, since undulations lead to such a distribution of director orientations. Furthermore, Noirez [25] found in shear experiment on a smectic A liquid crystalline polymer in a cone-plate geometry that the layer thickness reduces slightly with increasing shear. This result is compatible with the model presented here as well. 

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) ... 

Doi molecular theory adds a probability density function of molecular orientation to model rigid rodlike polymer molecules. This model is capable of describing the local molecular orientation distribution and nonlinear viscoelastic phenomena. Doi theory successfully predicts director tumbling in the linear regime and two sign changes in the first normal stress difference,as will be discussed later. However, because this theory assumes a uniform spatial structure, it is unable to describe textured LCPs. 

Larson and Doi introduced a mesoscopic polydomain model based on LE theory. This model includes a domain orientation distribution function and incorporates director tumbling, distortional elasticity, and texture size. Larson-Doi model can qualitatively predict the steady flow behavior and transient behavior. However, discrepancies between the theoretical predictions and the experiments of model systems were observed, especially when the shear history includes rest periods. ° This model is restricted to low shear rates without perturbing the molecular orientation distribution function in each domain.f ... 

The negative Ni is the result of the coupling of molecular tumbling under flow and the local molecular-orientation distribution. At low shear rates, the director tumbles with the flow and Ni will be positive. At intermediate shear rates, nonlinear viscoelastic effects are important. The director tumbling competes with the steady director alignment along... 

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). 

The molecular orientation of the polymer matrix is described by means of a vector field n(x,y,z), with n being the unit vector (director) oriented along the mean molecular orientation direction. The molecular orientation of uniaxially drawn homogeneous polymers might be approximated, if one neglects the influence of the outer shape of the sample, to the cylindrical symmetry described by the orientation distribution function of the form ... 

As discussed in Ref. [46], the simulated trimers are complex entities with a wide distribution of molecular shapes. Since different shapes have to be oriented differently in the nematic phase, a complete description of the order in this phase can be given only by taking simultaneously into account orientation and conformation, which is not an easy task. A simplified, but less complete description can be given if orientational and conformational orders are individually considered. Figure 6 shows the orientational distribution of the rigid groups with respect to the nematic director (the z axis) for the nematic phases at temperatures close to the transition points. The plot for system Tf, refers to the metastable... 

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