Rotational isomeric state model of polymers

Paul J. Flory addressed the problem of predicting the shape of a polymer molecule and so provided an understanding of the way in which chemical structure influences conformational changes in polymers. His approach was essentially an extension of that presented above for pentane and the substituted hexanes. He considered that the range of the important interactions was very short. This approximation is surprisingly accurate and has allowed theoretical prediction of the equilibrium structures of many polymers, at least when the molecules are present as separate entities in dilute solution. 

The conformation of the polymer can be described in terms of the position vectors of the backbone. For each bond there will be a value of (p which indicates whether that bond is in a gauche or trans conformation. If we have N + 1 monomers then we have N + 1 position vectors R, R, R,  

Polymers are large molecules and the degree of polymerisation, i.e. the number of monomers incorporated in the chain, wUl typically range from 100 to greater than 1000. The number of possible conformations for such a molecule would be around 2.0 X 10, a very large number indeed. Yet, despite these complications, we can stiU use this analysis to obtain reasonable estimates of the potential energy sur-frces and thence predictions of the conformational dynamics of polymer chains. 

If it is assumed that the chain is flexible, that is that the barrier between conformational states is sufficiently low that rapid interchange can occur, then the time averaged shape of the polymer will be described by the distribution between the available conformations. This allows a statistical mechanical analysis, called the rotational isomeric model. This indicates that the population of the gauche states causes the chain to adopt a random quasi-spherical shape and not the often pictured zigzag extended aU trans form. The analysis allows calculation of two fundamental parameters the radius of gyration, which is the size of the polymer molecule if it were to undergo free internal rotation, and the related end-to-end distance. These quantities define the shape of the isolated polymer 

The success of the rotational isomeric model lies in predicting the time averaged shape and size of polymer molecules in dilute solution. However, we have to look further to find the way in which the molecule moves between the various states that are averaged in the model. 

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