Orientation distribution parameter

The X-ray diffraction measurements result in a numerical parameter (sin2 angle between the chain axis, being parallel to the symmetry axis of the crystallite, and the fibre axis, (sin2 [Pg.479]

Here 533 is the fibre compliance, 3 the modulus of elasticity of the chain and > the orientation distribution parameter of the crystallites with respect to the fibre axis, which is zero for perfect orientation and 2/3 for random orientation. Figure 4 shows the observations which confirm expression (1). The constant A represents a measure of the mechanical anisotropy of the crystallite ... 

Other possible models for fibers are the uniform strain and the elastic unwrinkling model. The first does not describe the observed linear relation between the dynamic compliance and the orientation distribution parameter of the chains in the polymer fibers the second is not essentially different from the classical series model. 

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. 

The anisotropy of the liquid crystal phases also means that the orientational distribution function for the intermolecular vector is of value in characterising the structure of the phase [22]. The distribution is clearly a function of both the angle, made by the intermolecular vector with the director and the separation, r, between the two molecules [23]. However, a simpler way in which to investigate the distribution of the intermolecular vector is via the distance dependent order parameters Pl+(J") defined as the averages of the even Legendre polynomials, PL(cosj r)- As with the molecular orientational order parameters those of low rank namely Pj(r) and P (r), prove to be the most useful for investigating the phase structure [22]. 

The distribution of the intermolecular vector is also of value in distinguishing between smectic A and smectic B phases with the latter having long range bond orientational order [23, 24]. At the local level we can define a bond orientational order parameter, PeCn) for molecule i at position q by [25]... 

In order to simplify the discussion and to keep the derivation of the formulae tractable, the major part of this analysis is limited to a polymer fibre with a single orientation angle 0. This angle is assumed to be a kind of average angle and a characteristic parameter of the orientation distribution of the chain axes. 

In the previous sections the theoretical relations describing the strength of a polymer fibre as a function of the intrinsic parameters, such as the chain modulus, the modulus for shear between adjacent chains, the orientation distribution and the chain length distribution, have been discussed. In this section the dependence of the strength on the time and the temperature will be investigated. 

According to the simple Eq. 115 and the full Eq. 140, the lifetime of a fibre measured at a constant load decreases with increasing orientation parameter. The dependence of the slope of the curve, log( b) vs ob, on the initial orientation distribution has been calculated for PpPTA fibres using Eq. 140. Figure 70 shows that at constant load for increasing orientation angle the lifetime curves become steeper, while at the same time the lifetime decreases. This effect has been observed for nylon 66 yarns as shown in Fig. 71, where the lifetime data... 

The presented derivations of the load rate and the lifetime relationships applying the shear failure criterion are based on a single orientation angle for the characterisation of the orientation distribution. Therefore these relations give only an approximation of the lifetime of polymer fibres. Yet, they demonstrate quite accurately the effect of the intrinsic structural parameters on the time and the temperature dependence of the fibre strength. 

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In order to check further the correctness of this procedure, we used the deduced values for the distribution parameters and the residual linewidth to simulate the experimental spectra at X-band and S-band, using an expression which specifically includes the linewidths dependence on the distribution parameters. An orientation dependent linewidth was used, eq. 5, with Hj,= H. ... 

These expressions contain two orientation-dependent parameters p and ct(N), which can be calculated from the equilibrium orientational distribution function T(a) minimizing AF. When the Onsager trial function (Eq. (17)) is used for f(a), p and ct(N) are given as functions of the unknown parameter a the functional form of ct(N) has already appeared in Eq. (18), while p is expressed asymptotically as [2]... 

The order parameter S is the orientational average of the second-order Legendre polynomial P2(a n) (n = the director), and if the orientational distribution function is approximated by the Onsager trial function, it can be related to the degree of orientation parameter ot by... 

With increasing flow rate, the orientational state in the nematic solution should change. Larson [154] solved numerically Eqs. (39) and (40b) with Vscf(a) given by Eq. (41) for a homogeneous system (T[f ] = 0) in the simple shear flow to obtain the time-dependent orientational distribution function f(a t) as a function of k. The non-steady orientational state in the nematic solution can be described in terms of the time-dependent (dynamic) scalar order parameter S[Pg.149]

Fuhs et al.m investigated P p0 Aj in multilayers of Synechocystis PCC 6803 oriented on mylar sheets by transient W-band EPR. They could show an enhanced resolution of structural parameters of the RP in this model system. A problem is the uncertainty of the orientation distribution (width 30 10°). Limitations and possibilities of the method are discussed in this work. The technique is interesting for all systems for which no single crystals are available. 

Orientational Distribution Function and Order Parameter. In a liquid crystal a snapshot of the molecules at any one lime reveals that they arc not randomly oriented. There is a preferred direction for alignment of the long molecular axes. This preferred direction is called the director, and it cun be used to define- an orienlalional distribution function, f W). where flH win Vilb is proportional to the fraction of molecules with their long axes within the solid angle sinbdw. 

The microdomain orientation as a function of the electric field strength was monitored by a series of scanning force microscopy (SFM) images taken in the center between the electrodes. The entire electrode length of 6 mm was screened in steps of a few tens of microns. From the azimuthal intensity distribution of the 2D Fourier transformations of the SFM images, the orientational order parameter P2 was calculated according to ... 

The discovery of confinement resonances in the photoelectron angular distribution parameters from encaged atoms may shed light [36] on the origin of anomalously high values of the nondipole asymmetry parameters observed in diatomic molecules [62]. Following [36], consider photoionization of an inner subshell of the atom A in a diatomic molecule AB in the gas phase, i.e., with random orientation of the molecular axis relative to the polarization vector of the radiation. The atom B remains neutral in this process and is arbitrarily located on the sphere with its center at the nucleus of the atom A with radius equal to the interatomic distance in this molecule. To the lowest order, the effect of the atom B on the photoionization parameters can be approximated by the introduction of a spherically symmetric potential that represents the atom B smeared over... 

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