Diffusion of Densely Entangled Polymer Chains by Reptation

3 Diffusion of Densely Entangled Polymer Chains by Reptation 

An approximate expression for the chain s self-diffusivity can be obtained using the theory of random walks. Let rrep be the average time required for the chain to move from one primitive tube to its successor. During an interval rrep, the chain s 

4This discussion of reptation is a simplified version of other rigorous treatments [36, 40]. 

An estimate for rrep may be found by considering the Brownian motion of the chain. At the transition from one isolated primitive tube to its successor, the chain must have executed enough excursions to travel the chain length Nb. If the distance associated with each excursion is u and the number of excursions is n, Eq. 7.47 also establishes the estimate by/n Nb. Therefore, the number of excursions, n, is approximately n = N2b2 ju2. If v is the excursion frequency, 

It is reasonable that v will be smaller for long chains. Therefore, as an approximation, let v = Vo/N, where u0 = constant. Therefore, 

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