Orientational distribution functions singlet

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... 

Note 5 Even for molecules with cylindrical symmetry, does not provide a complete description of the orientational order. Such a description requires the singlet orientational distribution function which can be represented as an expansion in a basis of Legendre polynomials with L an even integer. The expansion coefficients are proportional to... 

The form an infinite set of order parameters which are necessary, in principle, to provide a complete description of the orientational order of a nematic. The limiting values of all of these are particularly convenient thus for perfect order when the symmetry axes of the molecules are parallel to the director they all take the value of unity. At the other extreme when the system is disordered, that is in the isotropic phase, the order parameters all vanish. Another limiting case occurs when the molecular syimnetry axes are perpendicular to the director so that the nematic phase remains uniaxial. Now the limiting values of the order param ers dqrend on L and for the first three = -V2, = % and

= -5/16. Und the same limiting conditions the singlet orientational distribution functions are given by... 

To see the nature of the orientational order parameters needed to characterise the biaxial nematic we start with the singlet orientational distribution function. Since the molecules are taken to be rigid the distribution is a function of the three Euler angles, a 3y, which we denote by Q connecting the director... 

The singlet orientational distribution function is obtained directly from the potoitial of mean torque as... 

Fig. 7. The potential of mean torque U p)jkT for a rod-like molecule as a function of the angle between the molecular symmetry axis and the director the corresponding singlet orientational distribution function f(P) is also shown.

Here the bar indicates an ensemble average and /(cu) is the normalized singlet orientational distribution function for finding the director at an orientation co in a molecular frame. Of this infinite set only the second rank quantities 2, can readily be measured and these five components, corresponding to m =0, 1, 2, form the ordering tensor which is the irreducible analogue of the more familiar Saupe ordering matrix. This matrix is defined by... 

Here is the probability of finding a molecule in the th conformation irrespective of its orientation and / ((u) is the singlet orientational distribution function for this conformer. Such a factorization is equivalent to assuming that the intramolecular energy is independent of the molecular orientation—an assumption which is eminently reasonable for the interactions governing the conformations of mesogenic molecules. Equation (11) can be rewritten using the factorized joint distribution function as... 

For smectic phases the defining characteristic is their layer structure with its one dimensional translational order parallel to the layer normal. At the single molecule level this order is completely defined by the singlet translational distribution function, p(z), which gives the probability of finding a molecule with its centre of mass at a distance, z, from the centre of one of the layers irrespective of its orientation [19]. Just as we have seen for the orientational order it is more convenient to characterise the translational order in terms of translational order parameters t which are the averages of the Chebychev polynomials, T (cos 2nzld)-, for example... 

The two singlet distribution functions are not in themselves sufficient to characterise the order in a smectic A phase because there is, in general, a correlation between the position of a molecule in a smectic layer and its orientation. We need, therefore, the mixed singlet distribution function P(z,cos ) which gives the probability of finding a particle at position z and at an orientation P with respect to the director [18,19]. At the level of description provided by the order parameters it is necessary to introduce the mixed order parameter... 

The correlation between the translational and orientational order is reflected by the mixed singlet orientational and translational distribution function P(z, cos ). The results for this are shown in Fig. 7 for the smectic A... 

In a similar fashion, we can define the singlet distribution function for location and orientation, which by analogy to (2.14) is defined as... 

Watanabe and Klein have reported MD simulations of the hexagonal mesophase of sodium octanoate in water with hexagonal symmetry. The singlet (i.e., one atom) probability distribution functions of the carbon atoms on the hydrocarbon chains show close similarity to those in the micelle. The dynamics of water molecules close to the head groups shows lower mean square displacements, and their orientational correlation function decays more slowly than those of waters farther from the head groups, as was seen in a recent bilayer simulation.6 ... 

In this section we generalize the concept of molecular distribution to include properties other than the locations and orientations of the particles. We shall mainly focus on the singlet generalized molecular distribution function (MDF), which provides a firm basis for the so-called mixture model approach to liquids. The latter has been used extensively for complex liquids such as water and aqueous solutions. 

The orientation of molecules in a mesophase can be specified by a singlet distribution function /(fi), where Q, denotes the Eulerian angles (0,0,-0) that transform between the molecular frame and the director frame. The average of any single-molecule property X(n) over the orientations of all molecules is defined by... 

The orientational order for the nematic is now defined in terms of the orientational ordering tensors for each conformational state. The conformational order is defined by the singlet distribution function, fconi( (Pi X which as we have seen can be expanded in a basis of Fourior functions, but we shall not pursue this aspect here. 

When the molecule is non-rigid we need to define an ordering tensor for each rigid sub-unit if we are to begin to characterize the orientational order of the mesophase. The orientation of the director in an axis system set in a sub-unit changes because of fluctuations in both the conformational state of the molecule and its orientation. The ordering tensor for the sub-unit is therefore calculated from the joint singlet distribution function as... 

Macroscopic properties of amorphous polymers are known to depend heavily on the local spatial and orientational distribution of chain segments, belonging to the same or to a different chain. We theref( e have to discuss these distributions in detail. We will use a statistical description for this purpose, concentrating on the singlet and pair distribution functions. These fimctions are introduced in the following sections, as well as the means by which they can be determined experimentally. We also discuss the results obtained so far for amorphous polymers. 

We know from statistical mechanics that a detailed understanding of the condensed noncrystalline state requires, in principle at least, the knowledge of a hierarchy of n particle distribution functions, which give die probability of finding clusters of n particles with particular positions and orientations within a system composed of N particles. It has been demonstrated in die past that it is sufficient in most cases to consider just the two lowest distribution functions, namely the singlet and the pair distribution functions. 

The singlet distribution function, which is schematically depicted in Figure 3, gives the probability of observing a particle at location with the orientation IIj (where represents the three Euler angles aj, Pi.Yi) relative to some frame of reference. This function contains information on the variation of particle density as a fimction of the location within the sample as well as on long range orientational order. 

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