Statistics of a Polymer Chain

The special structure of polymer molecules that distinguishes them from other species is their long, flexible chain structure. To describe this situation, let us first consider an isolated polymer chain and then extend the results to ensembles of chains, that is, to the bulk polymer. An isolated linear polymer chain is capable of assuming many different conformations. Because of 

We shall first derive the average properties of an ideal polymer chain that is infinitely long, possesses negligibly small volume, and has freely jointed links. Next we shall examine the influence of fixed bond angles between adjacent links. The concept of the statistically equivalent random chain will then be introduced to rationalize the validity of using these model chains to represent the behavior of real polymer chains. Finally, the equation of state for a single polymer chain will be discussed. This equation is the starting point for equations (6-32) and (6-33) 

Suppose our ideal polymer chain has n links, each of length / then the fully extended length of the chain would be  

However, the fully extended conformation is only one of a great many it would be more meaningful to consider an average size of the macromolecule such as the mean square end-to-end distance, r2. As the name implies, the end-to-end distance is just the length of the vector connecting the two ends of the ideal chain. This average can be that for a given molecule at a number of times or that of an ensemble of identical molecules at the same time.5 Thus, for p chains that do not interact with one another, 

The square of the end-to-end distance of a particular chain is obtained from equation (c). 

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The conformational statistics of a polymer chain will change if the chain is placed in a random environment. We can model the effect of the random environment by introducing an interaction energy... 

As iioLt d above, G [7 (a)] contains all the, inronnation related to the conformational statistics of a polymer chain. 

As illustrated in Figure 2, elastomeric networks consist of chains joined by multifunctional junctions. As early as 1934, it was suggested by Guth and Mark and by Kuhn that the elastic retractive force exhibited by rubber upon deformation arises from the entropy decrease associated with the diminished number of conformations available to deformed polymer chains. It is, therefore, of primary interest to study the statistics of a polymer chain and to establish the elastic equation of state for a single chain. 

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