Chain freely oriented

The extended or contour length of the freely oriented macromolecule will heal. Its rms end-to-end distance will be from the above equation. It can be seen that the ratio of the average end-to-end separation to the extended length is ct Since a will be of the order of a few hundred even for moderately sized macromolecules, d will be on the average much smaller than the chain end separation in the fully extended conformation. 

Figure 6-15. Radial distribution function of a chain of 104 freely orienting segments each of length 0.25 nm.

The effect of tetrahedral bond angles on the radial distribution function enters through the parameter b, the value of which decreased from 3/2nl2 in the freely orienting chain to 3/4nl2 in the chain with tetrahedral bond angles. 

The mean-square end-to-end distance of a freely orienting chain is deduced from equations (7) and (10) (73-76) as follows ... 

One common postulate or model used is that of the freely orienting chain where it is stipulated that two bonds may assume with equal probability any orientation whatsoever with respect to each other. Consequently the average projection of one bond on another will be zero and the average value of ai 82 or of any other pair will be zero so that eqn (3.4) reduces to ... 

There are two obvious objections to the application of the freely orienting chain molecule when applied to real polymer molecules. 

When a polymer network goes inside an FLC medium, we may consider the random networks of cross-Unk chains to be stretched between two smectic planes spaced L apart (Fig. 6.5). Gaussian like cross-links with orientationally uncorrelated chains, freely joined by spherical particles and connected by harmonic springs can be considered as shown in Fig. 6.6. r(z) = [x(z),y(z)], describes the path of cross links conformations where z stands for the direction of cross-link alignment without any loops and overhangs. Further, we assume the cross-link... 

In ethylene poljmerizations by Ni(II)-based a-diimine catalysts, the aryl groups are roughly perpendicular to the coordination plane so the bulky substituaits on the aryls are positioned at the axial directions to retard associative chain transfer ructions [6,7]. At elevated temperatures, the aryl groups may freely rotate away firm the perpoidicular orientation, resulting in increased associative chain transfes and a resulting decrease in MW of the PE. In addition such free rotation makes the sfructote of the cationic active species more unstable, resulting in fast decrease of activity. 

Exact Treatment for the Freely Jointed Chain (or Equivalent Chain).4 >5— Consider one of the bonds of a freely jointed chain acted upon by a tensile force r in the x direction. Letting xpi represent the angle between the bond and the o -axis, its component on the x-axis is Xi = l cos pi. The orientation energy of the bond is —rXi, and the probability that its x component has a value between Xi and Xi- -dxi therefore is proportional to... 

Figure 9. Orientational correlation function of the principal axis as a function of center-of-mass distance for freely jointed hard chains for N = 20.

Considering the large variation of / for the poly[2]catenand 51b, it is expected that little correlation will exist between the spatial orientation of neighboring monomer segments and that it will represent the closest synthetic equivalent of the freely jointed chain model [63]. In this model, a real polymer chain is replaced by an equivalent chain consisting of N rectilinear segments of length Z, the spatial orientations of which are mutually independent (Scheme 24) [63]. 

AMJhj is a function 6f a) the orientation of the inter-crosslink vector h. with the respect to the magnetic field B0 and b) the time-averaged value of the sum over k, which is the actual measure of the motional restrictions induced by crosslinking. In lightly crosslinked networks, presented by the freely-jointed model of the polymer chain 108), the residual part also can be described by the number of statistical segments in the chain section between crosslinks (Z) (Eq. (24)) ... 

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