The Orientational Distribution Function

The rules of classical statistical mechanics give the orientational distribution function in terms of the potential function V as  

Expanding this equation to display its pertinent factors. 

We now see in Eq. [5] a self-consistent equation for the determination of the temperature dependence of P2 . The order parameter P2 appears on both the left and right hand sides of the equation. For every temperature T (or / ) we can use a computer to obtain the value (or values) of P2 that satisfies the self-consistency equation. This process has been accomplished and the results are depicted in Fig. 2. P2 = 0 is a solution at all temperatures this is the disordered phase, the normal isotropic liquid. For temperatures T below 0.22284 v/k, two other solutions to Eq. [5] appear. The upper branch tends 

As noted earlier, Qa/9 may be equally well-defined in terms of other macroscopic properties such as the refractive index or dielectric tensor. However, the simple relation [Eq. (3.6)] cannot be expected to hold for the dielectric anisotropy Ae and electric polarizability aij. This is due to complicated depolarization effects caused by the relatively large near-neighbor electrostatic interaction. The internal field corrections [3.3] are necessary in the electric case. It has been shown that Qa can be used to describe orientational order both in uniaxial and biaxial phases. Furthermore, measurement of Qa/3 is particularly useful when description of flexible molecules using microscopic order parameters becomes problematic. Experimentally, both magnetic resonance and Raman scattering techniques [3.3] may be employed to monitor the orientational order of individual molecules and to determine microscopic order parameters. 

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 

Now the orientational distribution function can be expanded in terms of Wigner rotation matrices of rank L 

Multiplying both sides by and integrating over the angles, it 

The averages are just the microscopic order parameters. There are a maximum of L independent order parameters in the principal axis system, but the principal axes may be defined only if symmetry allows. In particular, there are generally 25 order parameters for L = 2. In principle, the distribution function may be obtained from x-ray and neutron scattering studies. However, this is difficult in practice. If /(f2) is supposed to originate from an orientational pseudo-potential V(fi), then 

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Typical shapes of the orientation distribution function are shown in figure C2.2.10. In a liquid crystal phase, the more highly oriented the phase, the moreyp tends to be sharjDly peaked near p=0. However, in the isotropic phase, a molecule has an equal probability of taking on any orientation and then/P is constant. 

Here the bar indicates an average over the orientational distribution function.Here cos — 4)is the... 

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.

The quantitative texture investigations (based on the orientation distribution function) al-... 

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]. 

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... 

This rotation modifies the orientational distribution function f(a t) to f(a t) + 8f(a t) through Eq. (39) with Eq. (40b), the last term on the right hand side dominating over the others if the deformation is very rapid or the time 8t is very short. The argument r in f(r,a t) is omitted because we are concerned only with uniform solutions. 

Broad-line NMR derivative spectra were obtained using a Brucker HFX-90 spectrometer to record the resonance at 84.67 MHz. The specimens, made by compacting granular PTFE into preforms, sintering at 380°C, and cooling slowly at a rate of 0.02 deg/min had a specific gravity of 2.205. The second moment of tire NMR line shape is of interest because the fourth moment of the orientation distribution function is proportional to it. 

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. 

This is graphically depicted in Fig. 8.55. For simplicity, the x, y, and t from the orientation distribution function can be dropped. 

Since one end of a particle is indistinguishable from the other, the orientation distribution function must be periodic ... 

Since all particles are located between —7t/2 and ir/2, the orientation distribution function must be normalized such that... 

The averages in equation (5.7) represent averages over the orientation distribution function characterizing the angles 0, [Pg.83]

Evidently, Raman scattered light contains information about both the second and the fourth moments of the orientation distribution function. This is in contrast to birefringence and dichroism measurements, which respond only to anisotropies in the second moments. 

If the orientation distribution function is uniaxial, and symmetric about the x axis, the averages in equation (5.44) take on the following simple forms,... 

An exact solution for the orientation distribution function can be found in the absence of Brownian motion since the motion of the particles is deterministic and given by equation (7.107). In this case the orbit of each particle is uniquely determined from its initial conditions specified by the orbit constants, C and k. Indeed, the orientation distribu-... 

A consistent study of the linear and lowest nonlinear (quadratic) susceptibilities of a superparamagnetic system subjected to a constant (bias) field is presented. The particles forming the assembly are assumed to be uniaxial and identical. The method of study is mainly the numerical solution (which may be carried out with any given accuracy) of the Fokker-Planck equation for the orientational distribution function of the particle magnetic moment. Besides that, a simple heuristic expression for the quadratic response based on the effective relaxation... 

Taking thermal fluctuations into account, the motion of the particle magnetic moment is described by the orientational distribution function W(e,t) that obeys the Fokker-Planck equation (4.90). For the case considered here, the energy function is time-dependent ... 

In a fluid system, the rotational freedom of the particles affects the susceptibilities in two ways (1) the applied field (either AC or DC) deforms the orientational distribution function of the easy axes, which can never happen in a solid system with its fixed distribution of the particle axes and (2) if out of equilibrium, in a magnetic fluid the orientational diffusion of the particle axes works as an additional channel of magnetic relaxation that is, besides intrinsic processes, the magnetic moment can achieve equilibrium by rotating together with the particle in the suspended viscous liquid. Expressing the reference... 

The Fokker-Planck equation governing the evolution of the orientational distribution function W(e, n. t) for arbitrary i/ and i/j, that is, when the particle... 

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