Average orientation function

For semi-crystalline polymers, the average orientation function 2 for the crystal c axes can be calculated from X-ray diffraction measurements (Chapter 3). Figure 8.15 shows how 2 increases linearly with the draw ratio, for polypropylene fibres and films, while the spherulitic microstructure survives. At 2 = 0.9, where the spherulites are destroyed and replaced by a microfibrillar structure, there is an increase in the slope of the 2 versus true strain relationship. It is impossible to achieve perfect c axis orientation... 

Figures 5.113-115 illustrates the need to parallelize the chains of poly(ethylene terephthalate) applying the three-phase structure, identified in Figs. 5.69-72 [25]. Figure 4.113 shows the model for the combination of the three phases [35,36]. The orientation in the mesophase matrix, measured by its average orientation function (OF) obtained from the X-ray diffraction pattern [24], is of most importance for the modulus and, surprisingly, also for the ultimate strength, as indicated by the left curves...

Samuels (6) expressed average orientation function (Xv) as average orientation of each phase (i.e., crystalline and amorphous) after weighing them by the amount of each phase present, which led to the following equation ... 

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]... 

The low electrical conductivity of PET fibers depends essentially on their chemical constituency, but also to the same extent on the fiber s fine structure. In one study [58], an attempt was made to elucidate the influence of some basic fine structure parameters on the electrical resistivity of PET fibers. The influence of crystallinity (jc) the average lateral crystallite size (A), the mean long period (L), and the overall orientation function (fo) have been considered. The results obtained are presented in the form of plots in Figs. 9-12. 

Information on molecular orientation can be useful in two primary ways. First, it is possible to use the orientation functions or averages to gain an understanding of the mechanisms of plastic deformation. Secondly the orientation averages can provide a basis for understanding the influence of molecular orientation on physical properties, especially mechanical properties. 

In this review the definition of orientation and orientation functions or orientation averages will be considered in detail. This will be followed by a comprehensive account of the information which can be obtained by three spectroscopic techniques, infra-red and Raman spectroscopy and broad line nuclear magnetic resonance. The use of polarized fluorescence will not be discussed here, but is the subject of a contemporary review article by the author and J. H. Nobbs 1. The present review will be completed by consideration of the information which has been obtained on the development of molecular orientation in polyethylene terephthalate and poly(tetramethylene terephthalate) where there are also clearly defined changes in the conformation of the molecule. In this paper, particular attention will be given to the characterization of biaxially oriented films. Previous reviews of this subject have been given by the author and his colleagues, but have been concerned with discussion of results for uniaxially oriented systems only2,3). 

Stochastic equation (A8.7) is linear over SP and contains the operators La and V.co of differentiation over time-independent variables Q and co. Therefore, if we assume that the time fluctuations of the liquid cage axis orientation Z(t) are Markovian, then the method used in Chapter 7 yields a kinetic equation for the partially averaged distribution function P(Q, co, t, E). The latter allows us to calculate the searched averaged distribution function... 

If orientation is assumed to occur only in one dimension (an oversimplification), birefringence and several related phenomena (infrared dichroism, etc.) measure the quantity , which is the average angle 8 between the molecular chain direction and that of the orienting force, such as the fiber stretch axis. It is convenient to introduce an orientation function [12]... 

The correlation between the heat offormation AHf and the work function WF. Table 5.11 is showing the heat of formation of AHf of the oxides, sulfides, chlorides and phosphorus compounds and the average work function (WF) of the elements. The work function is a complicated physical property related to the crystal orientation, surface composition, and chemisorption (Shpenkov, 1995). Taking an aluminum surface as an example (Huber and Kirk, 1966), the adsorption of dry oxygen to one monolayer coverage will lower the work function by 0.05... 

In order to avoid the use of adjustable parameters on rescaling the theoretical with the experimental orientation, and since we were interested in differences in the relaxation kinetics, we chose to study the orientation of the different blocks normalized by the average orientation, as a function of relaxation time. The relaxation times have been adimensionalized by the retraction time, Tg, which can be determined experimentally by applying the theoretical scaling law (see section 4.1). 

The orientation function (202) is highly sensitive with regard to the two extreme models of a liquid discussed by Prins and Prins, namely diatropism, where G is very large, and paratropism, where Gf,- tends to zero. For not too strong angular correlations, equation (202) can be averaged in a first approximation over all orientations, leading to ... 

The decay of the orientational correlation function is highly nonexponential and one needs at least four exponentials to fit it. The average orientational correlation time, t, is slower by about a factor of 20 than that of its bulk value. The orientational correlation function for the interfacial water molecules will, of course, decay in the very long time (of the order of tens of nanoseconds), either because of "evaporation" of the interfacial water molecules or rotation of the micelle. 

Of course, the above considerations may not be relevant to the problem at hand, since in solving the OZ equation, the important functions are y(l, 2), its Fourier transform and B(l,2). ° It is to be expected that y(l,2) will vary less quickly between different orientations and will be continuous even for hard core potentials. Thus, its expansion in spherical harmonics should be better behaved than that of gf(l,2). Computer simulation cannot be used to obtain y(l,2) but Lado has presented some evidence based upon his solution of the RHNC approximation for a hard diatomic fluid using a spherically averaged bridge function that the convergence is good. Nevertheless, the results he presents are, in our view, for a rather short diatomic bond length and may not be conclusive. 

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. 

It must be realized that these orientation functions are again only averages for the systems and in the case of fibers would indicate nothing about any distribution of orientation that might exist along the radius as was discussed in the birefringence section. Therefore Eq. (58) can be further generalized to be... 

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