Reptation model stress relaxation modulus

The stress relaxation modulus in the reptation model is proportional to the fraction of original tube remaining at time t (see Fig. 9.1). As time goes on, sections of the original tube are abandoned when the chain end first visits them. Such a problem is called a first-passage time problem. 

The stress relaxation modulus G t) for the reptation model was calculated by Doi and Edwards in 1978 by solving the first-passage problem for the diffusion of a chain in a tube (see Problem 9.6) ... 

The stress relaxation modulus then decays exponentially at the reptation time [Eq. (9.22)]. The terminal relaxation time can be measured quite precisely in linear viscoelastic experiments. Hence, Eq. (9.82) provides the simplest direct means of testing the Doi fluctuation model and evaluating... 

These tube length fluctuation modes (see Section 9.4.5) of the neighbouring chains affect the constraint release modes of a given chain. If entanglements between chains are assumed to be binary, there should be a duality between constraint release events and chain in a tube relaxation events. A release of an entanglement by reptation or tube length fluctuation of one chain in its tube leads to a release of the constraint on the second chain. If this duality is accepted, the distribution of constraint release rates can be determined self-consistently from the stress relaxation modulus of the tube model. 

What value of a corresponds to the double reptation model In Section 9.3, we have presented theoretical arguments and experimental data supporting the value a =4/3 in -solvents (and melts with ideal chain statistics). What is the expression of the stress relaxation modulus of tube dilation models corresponding to a = 4/3 ... 

Graessley and Stmglinski (1986) developed a theory, that predicts the linear dynamic viscoelastic properties of binary blends of monodisperse flexible homopolymers, based on the tube model, by incorporating constraint release and path fluctuations into the reptation motion (see Chapter 4). They assumed that the stress relaxation modulus of a binary blend, Gb(0, may be represented by... 

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