More realistic chains - the excluded-volume effect

4 More realistic chains - the excluded-volume effect 

The fact that real chains cannot cross over themselves and therefore do interact at distances greater than the Kuhn link length is often called the excluded-volume effect and it leads to an expected increase in the RMS chain length and to a change in the way that it depends on n. Later it will be seen that the effect is not important in changing / nns or its dependence on n in homopolymer melts or in solids, but the fact that the chains cannot interpenetrate each other nevertheless has important effects on their mechanical properties. 

Two segments of a chain cannot interpenetrate because there is a repulsive force between them when they approach closer than a certain 

According to Boltzmann, the entropy of a system is given by S = k nW, where W is the number of configurations available to the system. The word configurations is used in the statistical-mechanics sense and corresponds here to the conformations of the chain. Consider a Gaussian chain with a particular end-to-end vector r. The probability of finding a chain with this value of r is proportional to the number of conformations that it can take up and equation (3.6) shows that W = C exp(—b r ), where C is independent of r. The entropy is then given hy S = k nW = k nC — kb P.  

Setting AAjAr = 0 and assuming that the resulting value of r is equal to r ns leads to = iat v/ 2b c) and substituting for b from equation (3.7) leads to 

---