Distribution orientation

Orientaticm of a spheroid is determined by the balance of hydrodynamic forces and rotary Brownian motion. Hydrodynamic forces tend to align the major axis with the flow, while Brownian motion tends to randomize the orientation. The relative importance of each is expressed in terms of the Peclet number Pe, the ratio of the time scales for Brownian motion (l/D ) to that for convective motion (1//0- 

For rigid spheres the rotary Brownian diffusion coefficient is 

If the particles are small, shear rate and viscosity of the suspending fluid are low, so Brownian motion randomizes orientations completely and Pe = 0. With high viscosity and shear rates or with large particles, the disperse phase will orient with the flow as Pe 00. 

Prolate spheroid with orientation vector e along its major axis. The spherical coordinate system to describe e is shown with the Cartesian system of the shear flow of surrounding fluid. Both coordinate systems are fixed to the center of the particle. 

The probability that e will lie within some solid angle d l = sin is /(0i, The probability that e will lie on the 

Polymer molecules in fabricated items rarely adopt a true random walk. Manufacturing processes stretch out molecules and then freeze them in an extended configuration before they have time to relax to the random state. Manufacturers exploit orientation in order to control physical properties. 

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

FigureBl.5.16 Rotational relaxation of Coumarin 314 molecules at the air/water interface. The change in the SFI signal is recorded as a fimction of the time delay between the pump and probe pulses. Anisotropy in the orientational distribution is created by linearly polarized pump radiation in two orthogonal directions in the surface. (After [90].)...

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... 

Figure C2.2.10. Orientational distribution functions for (a) a highly oriented liquid crystal phase, (b) a less well...

This can be inserted in equation (02.2.3) to give tlie orientational distribution function, and tlius into equation (02.2.6) to deteniiine the orientational order parameters. These are deteniiined self-consistently by variation of tlie interaction strength iin equation (c2.2.7). As pointed out by de Gemies and Frost [20] it is possible to obtain tlie Maier-Saupe potential from a simple variational, maximum entropy metliod based on tlie lowest-order anisotropic distribution function consistent witli a nematic phase. 

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.

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... 

A discontinuous fiber composite is one that contains a relatively short length of fibers dispersed within the matrix. When an external load is applied to the composite, the fibers are loaded as a result of stress transfer from the matrix to the fiber across the fiber-matrix interface. The degree of reinforcement that may be attained is a function of fiber fraction (V/), the fiber orientation distribution, the fiber length distribution, and efficiency of... 

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

While thin polymer films may be very smooth and homogeneous, the chain conformation may be largely distorted due to the influence of the interfaces. Since the size of the polymer molecules is comparable to the film thickness those effects may play a significant role with ultra-thin polymer films. Several recent theoretical treatments are available [136-144,127,128] based on Monte Carlo [137-141,127, 128], molecular dynamics [142], variable density [143], cooperative motion [144], and bond fluctuation [136] model calculations. The distortion of the chain conformation near the interface, the segment orientation distribution, end distribution etc. are calculated as a function of film thickness and distance from the surface. In the limit of two-dimensional systems chains segregate and specific power laws are predicted [136, 137]. In 2D-blends of polymers a particular microdomain morphology may be expected [139]. Experiments on polymers in this area are presently, however, not available on a molecular level. Indications of order on an... 

Pulsed deuteron NMR is described, which has recently been developed to become a powerftd tool for studying molectdar order and dynamics in solid polymers. In drawn fibres the complete orientational distribution function for the polymer chains can be determined from the analysis of deuteron NMR line shapes. By analyzing the line shapes of 2H absorption spectra and spectra obtained via solid echo and spin alignment, respectively, both type and timescale of rotational motions can be determined over an extraordinary wide range of characteristic frequencies, approximately 10 MHz to 1 Hz. In addition, motional heterogeneities can be detected and the resulting distribution of correlation times can directly be determined. 

As a prelude to the discussion it is necessary to consider the definition of orientation in terms of the Euler angles, and the definition ofan orientation distribution function in terms ofan expansion ofLegendre functions. These definitions set the scene for examining the information which can be obtained from different spectroscopic techniques. In this review, infra-red and Raman spectroscopy and nuclear magnetic resonance, will be considered. 

Deformation Schemes and Theoretical Consideration of Orientation Distribution... 

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