Approximate constitutive equation for reptation model

We shall now derive the constitutive equation for reptation dynamics. To simplify the analysis, we assume that tiie contour length of the primitive chain remains at the equilibrium value L under macroscopic deformation (inextensible primitive chain). This assumption is valid if the characteristic magnitude of the velocity gradient is much less than 1/Tj, i.e. 

Since nonlinear behaviour starts to be observed when the characteristic magnitude of the velocity gradient becomes of the order of I/t, the assumption is not restrictive for a polymer of 1, i.e., M M . In practice, the condition (7.169) is not always satisfied, and the elongation of the contour length can be important, but this will not be considered here. 

For the sake of simplicity, we shall use a slightly different notation in this and the following two sections the equilibrium contour length will be denoted by L (because L and L need not be distinguished for the inextensible model), and the segments of the primitive chain are labelled from -L/2 to L/2. (Thus the segment 0 corresponds to the middle of the chain.) 

First we express the transformation rule of the inextensible primitive chain in mathematical terms. Let R s) and 6. s) be the conformations of the primitive chain before and after the deformation. The transformation rule is explained in Fig. 7.21, i.e., 

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