Skeletal bond vectors

The purpose of this Appendix is to review the derivation of the relaxation spectrum (Eq. (15)) for the GRM. We begin by considering a polymer configuration under ideal or theta conditions. Letting [r n)] represent the set of N skeletal bond vectors, we write the end-to-end vector, R, of the chain as... 

Allegra has shown that the probability distribution for more general linear combinations of the skeletal bond vectors of an unperturbed chain is also Gaussianly distributed. In particular, if we decompose the conformation into the Fourier modes, 7 (p), defined by... 

The vector r joining the two ends of the chain takes different values resulting from rotations about the individual bonds. For chains with more than about 50 skeletal bonds, the probability W(r)dxdydz that one end of r is at the origin and the other end is in an infinitesimal volume dV — dxdydz is satisfactorily represented by the Gaussian function [31,32]... 

The end-to-end vector r of any non-crystallized chain portion comprising n skeletal bonds with a length l will be taken as Gaussian-distributed, i.e.,... 

A method is developed for calculating even moments of the end-to-end distance r of polymeric chains, on the basis of the RIS approximation for rotations about skeletal bonds. Expressions are obtained in a form which is applicable in principle to arbitrary k, but practical applications are limited by a tremendous increase in the order of the matrices to be treated, with increasing k. An application is made to the PE chain by using the familiar three-state model. Approximate values of the distribution function Wn (r) of the end-to-end vector r, Wn (0), and , are calculated from these even moments. 

A general theory of dichroism induced by strain in polymeric networks Is developed by adaptation of methods developed earlier for treating strain birefringence. It is generally applicable to dichrolc bands associated with any specified conformation involving sequences of one or more consecutive bonds. The transition dipole moment is introduced in the local framework of the skeletal bonds associated therewith. Possible differences in transition moments for various conformations and repeat units are taken into account. Numerical calculations for PE chains show gauche bonds, rather than trans, to be more favorably oriented with respect to the chain vector r. 

Response of the mean square dipole moment, < J2>, to excluded volume is evaluated for several chains via Monte-Carlo methods. The chains differ in the manner in which dipolar moment vectors are attached to the local coordinate systems for the skeletal bonds. In the unperturbed state, configurational statistics are those specified by the usual RIS model for linear PE chains. Excluded volume is introduced by requiring chain atoms participating in long-range interactions to behave as hard spheres. 

Considering one chain segment characterized by a normahzed p stretching vector, and a number n of skeletal bonds, the transverse magnetic relaxation function of nuclear spins, attached to this segment, is generally expressed as a product... 

Chains of the usual length between junctions in a rubber network consist of several hundred skeletal bonds. The distribution function W(r) for the vector r connecting the ends of a chain of this length is satisfactorily approximated by the Gaussian function i.e.,... 

Fig. 1.1. A two-dimensional projection of an n-alkane chain having 200 skeletal bonds [3]. The end-to-end vector starts at the origin of the coordinate system and ends at carbon atom number 200.

In order to facilitate the task of transforming every bond vector to the reference frame affiliated with the first bond, it is helpful to define a reference frame for each skeletal bond of... 

Figure 2.2 (a) Backbone structure of a polyethylene chain, (b) A typical conformation of a skeletal chain. R, is the distance vector of the /th united atom from the center of mass (CM) of the conformation, and a, is the bond vector connecting (/— l)th and th united atoms. 

For the calculation of Hory and Semiyen started out from the mass action Eqs. (5.20) and (5.21) like J + S, but then introduced a probability (IT(0)) for chain conformations bringing the chain ends in close neighborhood to each other with vectors of the functional groups allowing for immediate ring closure. Finally, they arrived at Eqs. (5.22) and (5.23) which were, in turn, used for the calculation of The calculated values were in good agreement with the experimental results for cyclosiloxanes with DP >15 (30 skeletal bonds). 

Recently, an elegant approach to the bonding in clusters has been developed by Stone [35], whose Tensor Surface Harmonic Theory derives cluster skeletal molecular orbitals as expansions of vector surface harmonic functions. The skeletal molecular orbitals are generated from basis sets with 0(o) and l(jt) nodes with respect to a radial vector passing through the atoms. 

The mathematical aspects developed above can be summarised as follows. Whenever a nido polyhedron with C y (n > 3) symmetry is generated from a closo deltahedron it will have a non-bonding e set of molecular orbitals. This e set is localised predominately on the open face of the polyhedron and the n and components conform to the vector diagrams shown in Figs. 19 and 22. If this e set is filled, then the nido polyhedron is characterised by n -I- 2 skeletal electron pairs. 

Figure 3.29 Model potential surface for isomerization of a cyanine dye. The branching space is shown in Figure 3.28(c). The x-axis corresponds to the reaction coordinate X3), in this case a complex mixture of conrotatory and disrotatory C=C torsion. Notice that the reaction path is quasi-parallel to the seam itself. From Figure 3.28(c), the branching-space vectors correspond to skeletal deformation rather than torsion. The reaction path encounters the seam at the point MEP-CI where one C=C bond is rotated 90°. (See the color plate.) Adapted from Hunt and Robb. ...

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