Orientation distribution moments

Figure A2.4.10. Orientational distribution of the water dipole moment in the adsorbate layer for tlu-ee...

FIG. 3 Left density profile, p z), from a 500 ps simulation of a thin film consisting of 200 TIP4P water molecules at room temperature. Right orientational distribution, p cos d), with 3 the angle between the molecular dipole moment p and the surface normal z. The vertical lines in the left plot indicate the boundary z-ranges,... 

The left side of Fig. 7 shows the orientational distribution of the molecular dipole moment relative to the surface normal in various distance... 

These moment studies have been performed on polymer systems such as polyethylene (or on penetrants in polymer systems) in which the interacting spins (protons or fluorines) reside on the same or on adjacent atoms. This allows essentially no freedom of variation in the internuclear vectors upon deformation of the network. The primary informational content therefore relates to independent segmental orientation distributions. By placing single spins on alternate segments, there should be much greater sensitivity to changes in the chain extension upon bulk deformation. 

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. 

In such instances no knowledge of the orientation distribution or of / is required to compute 6V Once 6X is obtained, / can be calculated from equation (49), and using this / the 6 values for other moments can be obtained directly from the observed dichroic ratios. These relations demonstrate the potentialities of polarized radiation studies in providing structural information on high polymers. 

In addition to describing the conformation of the hydrocarbon chains for amphiphilic molecules at the A/W interface, external reflectance infrared spectroscopy is also capable of describing the orientation of the acyl chains in these monolayers as a function of the monolayer surface pressure. The analysis of the orientation distribution for an infrared dipole moment at the A/W interface proceeds based on classical electromagnetic theory of stratified layers (2). In particular, when parallel polarized radiation interacts with the A/W interface, the resultant standing electric field has contributions from both the z component of the p-polarized radiation normal to the interface, as well as the x component of the p-polarized radiation in the plane of the interface. The E field distribution for these two... 

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... 

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... 

When measuring absorption spectra, one records a signal that is related to the wavelength-dependent probability of making a spectroscopic transition. From the molecular point of view, this probability is proportional to the dot product jl p where (L is the molecular transition moment and p is the photon polarization direction. When the orientational distribution of the molecules is isotropic (not crystalline, liquid crystalline, or bound to a surface), its absorption spectrum represents the orientationally averaged probability of making a spectroscopic transition and the measured spectrum is independent of polarization direction. When the orientational distribution of the molecules is anisotropic, the probability of making a spectroscopic transition depends on the polarization direction, and that dependence can be exploited to deduce the direction of the transition moment relative to the laboratory frame. Because transition moments are often trivially related to the orientation of the molecule, structural information can be deduced from polarized absorption measurements on anisotropic samples. 

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... 

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