Polymers local field

At this point it has been established that there are at least two basic mechanisms which contribute to the broad lines that are observed for the crystalline polymers. The residual zero frequency line broadening component can be analyzed in more detail. Specific attention can be given to factors which are a consequence of the chain-like character of the molecules. The local field at a given nucleus is the sum of the individual fields contributed by the neighboring magnetic nuclei. Segmental motions will induce a time dependence to the variables so that the individual contributions can be described by the equation (46)... 

Hooke s law relates stress (or strain) at a point to strain (or stress) at the same point and the structure of classical elasticity (see e.g. Love, Sokolnikoff) is built upon this linear relation. There are other relationships possible. One, as outlined above (see e.g. Green and Adkins) involves the large strain tensor Cjj which does not bear a simple relationship to the stress tensor, another involves the newer concepts of micropolar and micromorphic elasticity in which not only the stress but also the couple at a point must be related to the local variations of displacement and rotation. A third, which may prove to be very relevant to polymers, derives from non-local field theories in which not only the strain (or displacement) at a point but also that in the neighbourhood of the point needs to be taken into account. In polymers, where the chain is so much stiffer along its axis than any interchain stiffness (consequent upon the vastly different forces along and between chains) the displacement at any point is quite likely to be influenced by forces on chains some distance away. 

In the limit of the oriented gas model with a one-dimensional dipolar molecule and a two state model for the polarizability (30). the second order susceptibility X33(2) of a polymer film poled with field E is given by Equation 4 where N/V is the number density of dye molecules, the fs are the appropriate local field factors, i is the dipole moment, p is the molecular second order hyperpolarizability, and L3 is the third-order Langevin function describing the electric field induced polar order at poling temperature Tp - Tg. 

Equations (3) or (4), with refinements as necessary for "local field" effects, are an appropriate and useful basis for discussion of various models of non-conducting solutions of biological species considered in I. In many cases, however, solutions of interest have appreciable ionic concentrations in the natural solvent medium and the polymer or other solute species may also have net charges. Under these conditions, the electrical response is better considered in terms of the total current density Jfc(t) defined and expressed by linear response theory as... 

We have so far assumed that the nuclei are at rest in fixed positions with respect to each other. When molecular motion, — which in polymers will be chiefly group rotation —, takes place, the variables in equation (2) become functions of time1. If we assume for our present discussion that r is constant and only 6 varies with time (as would be true, for example, for the interaction of protons attached to the same carbon atom or to the same benzene ring), then the time averaged local field will be given... 

The situation is much more complicated in solids because the intermolecular effects can no longer be ignored, i.e. the approximation EM = 0 inherent in the simple formula for the local field (2.29) is not generally true. Consequently, although we can predict the molecular dipole moment from known group moments, it is not possible to calculate the molar polarisation and thereby the relative permittivity, without further elaboration of the dielectric model. In the case of a polymer there are further complications which arise from the flexibility of the long chains. 

For low naturally abundant nuclei, such as the nucleus, resonance lines reflecting the chemical shift anisotropy (CSA) can be observed for solid organic materials by eliminating the effects of local field with use of the so-called dipolar decoupling method. However, the CSA resonance lines are usually broad and, therefore, the respective CSA lines are superposed with each other for polymers composed of different C species. To separately measure these CSA lines, the following different methods were proposed ... 

The LHS of this equation differs from that of equation (9.17) because a more exact expression has been used for the internal field and the RHS now includes a factor g, called the correlation factor, which allows for the fact that the dipoles do not react independently to the local field. If /x, g, n and p are known for a polymer, it should therefore be possible to predict the value of s. Of these, g is the most difficult to calculate (see section 9.2.5). 

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