Orientation distribution second moment

However, case (ii) above, where there is biaxial symmetry of the distribution function, but no preferred orientation of the structural units about their Ox3 axes is a feasible proposition. Kashiwagi et al.10) and later Cunningham et al. n) have given expressions for the second moment... 

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. 

Specimens were elongated the indicated amount, then released from the grips of the tensile machine, cut parallel to the draw direction, and placed in NMR tubes. From the derivative NMR spectra, the second moments of die orientation distribution were measured at temperatures of 158 and 345°K. 

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. 

Equation (7.24) predicts that the refractive index tensor is proportional to the second-moment tensor of the orientation distribution of the end-to-end vector. This expression was developed using a number of assumptions, however, and is strictly valid only for small... 

The optical apparatus used in this work was described in section 8.6 and has the capability of providing both Raman scattering and birefringence measurements simultaneously. The Fourier expansion of the overall Raman scattering signal is given by equation (8.51), and the coefficients are given by equations (8.52) to (8.54). In these expression, a simple, uniaxial form for the Raman tensor was assumed. From these coefficients, the anisotropies in the second and fourth moments of the orientation distribution can be solved as... 

Figure 7.7 Fast but anisotropic segmental motion results in a solid-like contribution to the NMR signal. This contribution is expressed in terms of a fractional contribution q of the second moment M2 of the rigid lattice line of a single chain or residual dipolar interactions between protons. The line splitting caused by the dipole-dipole interaction depends on the orientation angle q of the internuclear vector of the coupling protons in the magnetic field B(). The distribution of orientation angles changes with the...

Pseudo-affine model, the deformation process of polymers in cold drawing is very different from that in the rubbery state. Elements of the structure, such as crystallites, may retain their identity during deformation. In this case, a rather simple deformation scheme [12] can be used to calculate the orientation distribution function. The material is assumed to consist of transversely isotropic units whose symmetry axes rotate on stretching in the same way as lines joining pairs of points in the bulk material. The model is similar to the affine model but ignores changes in length of the units that would be required. The second moment of the orientation function is simply shown to be ... 

Figure 2. Second moment of the orientation distribution as a function of uniaxial strain.

Ordinary methods of polarimetry, such as birefringence and dichroism, provide information about the second moment of the orientation distribution of the microstructural elements... 

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) ... 

For all Azo-PURs, the quantum yields of the forth, i.e., trans—>cis, are small compared to those of the back, i.e., cis—>trans, isomerization—a feature that shows that the azo-chromophore is often in the trans form during trans<->cis cycling. For PUR-1, trans isomerizes to cis about 4 times for every 1000 photons absorbed, and once in the cis, it isomerizes back to the trans for about 2 absorbed photons. In addition, the rate of cis—>trans thermal isomerization is quite high 0.45 s Q 1 shows that upon isomerization, the azo-chromophore rotates in a manner that maximizes molecular nonpolar orientation during isomerization in other words, it maximizes the second-order Legendre polynomial, i.e., the second moment, of the distribution of the isomeric reorientation. Q 1 also shows that the chromophore retains full memory of its orientation before isomerization and does not shake indiscriminately before it relaxes otherwise, it would be Q 0. The fact that the azo-chromophore moves, i.e., rotates, and retains full orientational memory after isomerization dictates that it reorients only by a well-defined, discrete angle upon isomerization. Next, I discuss photo-orientation processes in chromophores that isomerize by cyclization, a process that differs from the isomeric shape change of azobenzene derivatives. 

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 ... 

Hence we see that the stress depends on only the second and the fourth moments of the orientation distribution, (pp) and (pppp). However, in general, it is necessary to know the complete orientation distribution function Mp, t) to calculate exact results for these moments by means of (2 104). 

It should be remembered that birefringence does not separate the second moment of the orientation distribution for each component but rather only gives the average second moment which contains orientation distributions of all components. 

The measurement of orientation by sonic techniques has received relatively little attention. This method along with that of birefringence and dichroism measures only the second moment of the orientation distribution function. It does offer, however, some advantages, probably the most important being that it can be easily used for measuring the average orientation in fibers. 

The technique itself simply amounts to measuring the sound velocity along a fiber or in a long thin film. This velocity compared with the velocity measured in the same material that is randomly oriented (isotropic) leads to the second moment of the orientation distribution function. 

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