Normal Mode Theories Based on Motion of Isolated Flexible Chains

5 Normal Mode Theories Based on Motion of Isolated Flexible Chains 

We have so far discussed two types of theories, those based on the site model, and thosebased on the WLF equation and its ramifications, which deal with time-temperature equivalence. The site model theories predict constant activation energies and are more applicable to relaxation transitions originating from localised chain motions, whereas the WLF equation theories deal with the glass transition behaviour in amorphous polymers. 

In the introductory section on amorphous polymers (Section 7.1.1), we considered the relaxation spectrum of amorphous polymers and noted that it was quite complex. The normal mode theories, now to be discussed, attempt to predict the relaxation spectrum for amorphous polymers, as well as the time-temperature equivalence. 

These theories are associated with the names of Rouse, Bueche and Zimm [26-28] and are based on the idea of representing the motion of polymer chains in a viscous liquid by a series of linear differential equations. They are essentially dilute solution theories, but we shall see that, rather unexpectedly perhaps, they can be extended to predict the behaviour of the pure polymer. Because of its simplicity, we will give an account of the theory due to Rouse [26]. 

Consider the motion of the bead situated at the point xiyiii) between the rth and (i + l)th submolecules. The origin of coordinates is the bead between the (i — l)th and ith submolecules, that is at the other end of the /th submolecule. 

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