Orientational Distribution Functions for Molecules

The translational and orientational degrees of freedom can be treated separately (this follows from fundamentals of group theory which states that groups of translations and rotations are subgroups of the crystalline space groups P r, Q) = P(r) x P(0). Here x is a symbol of the group product. In particular case of the isotropic liquid or nematic phase (no positional order) f (r, Q) = pf (Q) where p = constant is density. 

The frame r], is attached to a molecule. Then Euler angles correspond to 

In this chapter we consider only an orientational distribution function [10, 

Why do we need it Because it is a kind of a bridge between the microscopic and macroscopic descriptions of the nematic phase. We define a value 

We can use this normalization condition to find they(Q) function, at least, for the isotropic liquid or isotropic liquid crystal phase. Indeed, in this case there is no angular dependence of f(Q.) i.e. /(d ,i ,4 ) = const. After integrating we find /( l ,1 , P)iso=l/87t  

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The equatorial spots are extended in the vertical direction and have the form of arcs the intensity decreases with increasing i -angle as shown in Fig. 5.21b. This is a result of a non-ideal orientational order the higher the order parameter S, the shorter the arcs. From the diffractogram one can find the distribution of intensity and calculate S [7]. In some cases, even the orientational distribution function for molecules /(i ) can be calculated from experimental data as schematically shown in Fig. 5.22. Generally, the shape of the function is determined by different Legendre polynomials P2(cosi ), P4(cosi ), etc. (see Section 3.3) and, in principle, different order parameters P2, P4 etc. can be found from experiment. 

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]. 

Now the initial orientational distribution function for an anisotropic molecule in an equilibrium ensemble is12... 

For molecules, the orientation must be taken into account if the true nature of the distribution is to be determined. The radial distribution function for molecules is usually measured between two fixed points, such as between the centres of mass. This may then be supplemented by an orientational distribution function. For linear molecules, the orientational distribution function may be calculated as the angle between the axes of the molecule, with values ranging from —180° to +180°. For more complex molecules it is usual to calculate a number of site-site distribution functions. For example, for a three-site model of water, three functions can be defined (g(0—O), g(0—FI) and g(H—H)) An advantage of the site-site models is that they can be directly related to information obtained from the X-ray scattering experiments. The 0—0, O—IT and H-H radial distribution functions have been particularly useful for refining the various potential models for simulating liquid water. 

The molecular origin of these relaxations has been established for dipolar molecular liquids by Debye [122] who has shown that the applied electric field perturbs the orientational distribution function for the dipolar molecules, leading to a static relative permittivity So greater than n, where n is the optical refractive index, and a dispersion for c (/) accompanied by a peak in c" f). 

Fig. 3.19 Orientational distribution function of molecules for two different values of order parameter S2...

Now Eq. (3.23) is an approximation since, in reality, the fi2 has cylindrical symmetry in nematics. This point is deferred [see Eq. (3.57)], but note that additional terms [3.10] are needed in the above equation for (U12). The average over the orientations of molecule 2 only influences and requires an orientation distribution function /( 2) molecule 2. Since there is no (f) dependence in nematics because of uniaxial symmetry, the integral over (f) vanishes unless m is zero and L is even. Thus,... 

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... 

Fig. 2. Schematic representation of the orientational distribution function f 6) for three classes of condensed media that are composed of elongated molecules A, soHd phase, where /(0) is highly peaked about an angle (here, 0 = 0°) which is restricted by the lattice B, isotropic fluid, where aU. orientations are equally probable and C, Hquid crystal, where orientational order of the soHd has not melted completely.

For a removal attempt a molecule is selected irrespective of its orientation. To enhance the efficiency of addition attempts in cases where the system possesses a high degree of orientational order, the orientation of the molecule to be added is selected in a biased way from a distribution function. For a system of linear molecules this distribution, say, g u n ), depends on the unit vector u parallel to the molecule s symmetry axis (the so-called microscopic director [70,71]) and on the macroscopic director h which is a measure of the average orientation in the entire sample [72]. The distribution g can be chosen in various ways, depending on the physical nature of the fluid (see below). However, g u n ) must be normalized to one [73,74]. In other words, an addition is attempted with a preferred orientation of the molecule determined by the macroscopic director n of the entire simulation cell. The position of the center of mass of the molecule is again chosen randomly. According to the principle of detailed balance the probability for a realization of an addition attempt is given by [73]... 

This function is the integrally normalized probability for each water molecule being oriented such that it makes an angle B between its OH bond vectors and the vector from the water oxygen to the carbon atom. This function is calculated for those molecules within 4.9 A of the carbon atom (nearest neighbors), as this distance marks the first minimum in the pair distribution function for that atom. The curve in Figure 10 is typical for hydrophobic hydration (22). 

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... 

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. 

The applied electric field perturbs the orientational distribution function of the dipolar molecules. Dielectric relaxation due to classical molecular reorientational motions is a form of pure absorption spectroscopy whose frequency range of interest for materials, including polymers, is between 10 6 and 1011 Hz. 

Extension of this theory can also be used for treating concentrated polymer solution response. In this case, the motion of, and drag on, a single bead is determined by the mean intermolecular force field. In either the dilute or concentrated solution cases, orientation distribution functions can be obtained that allow for the specification of the stress tensor field involved. For the concentrated spring-bead model, Bird et al. (46) point out that because of the proximity of the surrounding molecules (i.e., spring-beads), it is easier for the model molecule to move in the direction of the polymer chain backbone rather than perpendicular to it. In other words, the polymer finds itself executing a sort of a snake-like motion, called reptation (47), as shown in Fig. 3.8(b). 

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) ... 

An axially symmetric molecule is characterized by its linear polarizability in the principal axes a x and a y = a" and a" = af/. It is a good approximation to assume that its second- and third-order polarizability tensors each have only one component and respectively, which is parallel to the z principal axis of the molecule. For linear and nonlinear optical processes, the macroscopic polarization is defined as the dipole moment per unit volume, and it is obtained by the linear sum of the molecular poiarizabilities averaged over the statistical orientational distribution function G(Q). This is done by projecting the optical fields on the molecular axis the obtained dipole is projected on the laboratory axes and orientational averaging is performed. The components of the linear and nonlinear macroscopic polarizabilies are then given by ... 

For organized lamellar structures, like monolayers, Langmuir-Blodgett films and bulk lamellar phases of amphiphiles, the orientation distribution of the molecules is usually described using an axially symmetric orientation distribution function, N(0, with 0 the azimuthal tilt angle. This distribution... 

The Onsager potential is based on the excluded-volume effect among rod-like molecules. Consider an ensemble of rigid rods of uniform length L and diameter b, with a number density u and an orientation distribution function T (m) for the molecular orientation u. For a test molecule oriented in u, the effect of all the other molecules can be represented by a mean-field Onsager potential ... 

If the orientation dependence of the resonance frequency of a spin 5 is determined by just one interaction, it can be exploited for use as a protractor to measure angles of molecular orientation. In powders and materials with partial molecular orientation, the orientation angles and, therefore, the resonance frequencies are distributed over a range of values. This leads to the so-called wideline spectra. From the lineshape, the orientational distribution function of the molecules can be obtained. These lineshapes need to be discriminated from temperature-dependent changes of the lineshape which result from slow molecular reorientation on the timescale of the inverse width of the wideline spectrum. The lineshapes of wideline spectra, therefore, provide information about molecular order as well as about the type and the timescale of slow molecular motion in solids [Sch9, Spil]. 

Molecular order is descnhedhy the orientational distribution function P(0) [Mcbl]. This is the probability density of finding a preferential direction n in the sample under an angle 0 in a molecule-fixed coordinate frame (Fig. 3.2.2(a)). For simplicity, macroscopically uniaxial samples with cylindrically symmetric molecules are considered. Then, one angle is sufficient to characterize the orientational distribution function. In practice, not the angle 0 itself but its cosine is used as the variable and for weak order the distribution function is expanded into Legendre polynomials P/(cos 0),... 

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