Gaussian orientational distribution

It is interesting to compare the results obtained with the numerical orientation distribution function with the results obtained with the Gaussian orientation distribution function (6.17). Substituting the expressions for a c and p[/g] given by (6.18) in (6.31) and minimizing the free energy with respect to k yields... 

Fig. 6.6 Isotropic-nematic phase coexistence for L/D = 10 and q = I in the reservoir representation. The Gaussian orientational distribution function result (solid) is compared to the coexistence computed using formal minimisation of the oriental distribution function (dashed curves). In the inset we plot the nematic order parameter S as a function of (f>2 °f the nematic phase that coexists with the isotropic phase...

As the depletant concentration becomes significant and attractions play a dominating role/ becomes sharply peaked. This is refiected in a strong increase of the nematic order parameter S, see the inset in Fig. 6.6. Hence, the Gaussian orientational distribution function is accurate at larger depletant concentrations. 

We have chosen Gaussian thickness distributions, because structure visualization by means of IDF or CDF exhibits thickness distributions that frequently look very similar to Gaussians97. The presented relations for the ID intensity and the IDF are the basic relations for many ID structure models, comprising the general analysis of materials made from layers, highly oriented microfibrillar materials, and the direction-dependent analysis of anisotropic materials. 

History. Wilke [129] considers the case that different orders of a reflection are observed and that the orientation distribution can be analytically described by a Gaussian on the orientation sphere. He shows how the apparent increase of the integral breadth with the order of the reflection can be used to separate misorientation effects from size effects. Ruland [30-34] generalizes this concept. He considers various analytical orientation distribution functions [9,84,124] and deduces that the method can be used if only a single reflection is sufficiently extended in radial direction, as is frequently the case with the streak-shaped reflections of the anisotropic... 

If a Gaussian can be used to describe the orientation distribution it follows... 

Problems arise, as the orientation distribution starts to split, but the split nature is not yet discernible. Thunemann [257] is discussing this problem in his thesis. He describes, how to determine the true tilt angle of the structural entities, and he determines the minimum tilt angle that is required for the split nature to become detectable (Fig. 9.8). We observe that, in practice, a split nature of Lorentzian orientation distributions (solid line) is detected earlier than a split nature of Gaussians - at least up to an apparent17 integral breadth of 70°. The reason is that Lorentzians are more pointed than Gaussians - in the vicinity of their maximum. 

Figure 9.8. Minimum average tilt angle, structural entities measured with respect to the fiber axis at which the split nature of the orientation distribution becomes observable -plotted as a function of the integral breadth Bg of the orientation distribution g (Solid line g (tp) is a Lorentzian. Dashed Gaussian... 

The Fitting Problem. In many studies in particular of natural fibers, orientation distributions are picked from spherical arcs in scattering patterns and then fitted by Gaussians or Lorentzians. The result is the finding of an isotropic background. At least part of this background is not related to structure, but to a fundamental misunderstanding. 

For a network of Gaussian chains having the same number n of links, uniaxially stretched by an amount L/Lo = A., the assumptions of affine displacement of jimction points and initial Gaussiein distribution of end-to-end vectors allows one to calculate the optical anisotropy of the network by integrating Eq.lO over the distribution of end-to-end vectors in the stretched state. By taking Treloar s expansion [11] for the inverse Langevin function, the orientation distribution function for the network can be put into the form of a power series of the number of Unks per chain ... 

A Gaussian probability distribution of fiber axis orientations was employed to account for the spectral broadening observed in both the parallel and... 

Assume that the degree of the ordering of liquid crystalline polymers is high and the orientational distribution function is simply Gaussian, Odijk (1986) developed the analytical formulae for elastic constants... 

In analog to the approach used by Odijk when dealing with elastic constants, Lee (1988) took the orientation distribution function approximately as Gaussian. When the system is highly ordered, the asymptotic expression can be deduced for viscosities of liquid crystalline polymers, e.g., the Miesowicz viscosities (in the unit of fj) are expressed by... 

V Vu w(a,p,y) w(N, r) W t) wq volume of a polymer segment. 6.1.1.3 scattering volume. 1.2.2 unit cell volume. 3.3.1 crystallite orientation distribution function. 3.6.3 end-to-end distribution of a Gaussian chain. 5.2.1 [5.12] slit-length weighting function. 5.6.1 constant value of W(t) with infinite slit approximation. 5.6.3... 

Even for the ideal case of completely random moment orientations, when there are more than three magnetic ions per hundred in a solid, the Lorentzian field distribution is unlikely to be even a rough approximation to the truth. Meanwhile, a Gaussian field distribution of local field is only to be expected when there are more than 30 magnetic ions per hundred. The local field distribution in the intermediate range has been simulated by Noakes (1991), but corresponding measmements are lacking. 

Comparison of these results with exact values (6.12) and (6.14) shows that the trial function chosen by Onsager works quite well. Odijk [15, 16] realized that for large values of k, Onsager s orientational distribution function can be approximated by a Gaussian distribution function... 

Instead of formal minimization of the free energy leading to an integral equation for the orientation distribution function / we will first use the Gaussian distribution function which simplifies the calculations considerably, while leading to reasonably good results. This is illustrated in Fig. 6.6, where we plot the isotropic-nematic phase coexistence curve for L/D = 10 and q= On the ordinate the relative reservoir concentration of penetrable hard spheres is plotted versus the... 

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