The dynamics of flexible molecules and macromolecules

All the above theory also applies to flexible molecules, except for the assumption that G(t) decays exponentially. Let us now consider a somewhat more sophisticated motional model in which the relevant internuclear vector, e.g. the C-H bond, jumps freely within a cone [6] of semi-angle X, but is prevented from further movement either by other parts of the molecule, or by, for example, neighbouring polymer chains. This will turn out to be a surprisingly good approximation for local motions in a dissolved or molten polymer. The added motion fairly rapidly reduces G(t) from 1 to 

One may note that varies from unity at X = 0 to zero at X = njl. It is one kind of order parameter for the final state. The decay of G(t) by random angular jumps is once again exponential, as in the previous case where X was equal to n. However, its time constant Tq may well be far shorter than 

If the molecule also undergoes slow isotropic rotation as before, then this will also tend to reduce G(t) towards zero, but much more slowly than the timescale for libration. (The same will often turn out to be true for the looping motions of a polymer chain). Thus, 

This model can also be extended to further levels of librational motion, each with its own order parameter and correlation time, and has been underpinned more formally in the three-r case of two librational levels plus rotation [7]. It is clearly of a general nature. One may indeed choose to abandon relationship (4.23) and to re-interpret in (4.24) as a more general order parameter. One thereby achieves a model-free theory, which has advantages over alternative, more specific models for polymer motion (see chapter 6), when the motional details are not clear. 

However, the nature of the motion that corresponds to the overall tumbling of an isolated, polymer molecule remains unclear. It clearly has a relatively slow component, because the linewidths are considerably greater than one would deduce from equations (4,22) and (4.25). They may be approximated by the more complex model invoking three levels of motion, mentioned above, provided that the slowest motion has Tr as long as 18 ns. One also notes empirically that the slowest level of motion, in the bulk state, becomes more dominant when, for example, the rubber is blended with particles of an inert, solid filler. This results in broader resonances, beyond what would 

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