Inextensible primitive chain

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. 

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.,... 

This is the time-evolution equation for the tangent vector 0(5, t) of the inextensible primitive chain. Thus the equation for/(0, s, t) is... 

First, consider a simpliflcation in which it is assumed that the contour length of the primitive chain remains at the equilibrium length L under the imposed deformation. This assumes an inextensible primitive chain and is seen as a reasonable approximation for slow flows or long times. Then, the deformation of the primitive chain is given by considering that the segment in the middle of the chain changes position affinely as... 

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.)... 

We consider the inextensible chain model. Figure 7.24 eiqilains the change of polymer conformation under the double step strain. Figure 7.24a shows the undeformed state just before the first deformation. Figure 7.24h represents the state immediately after the deformation the primitive chain is deformed by the shear Yi. Figure 7.24c indicates the state just before the second deformation the inner part AB still remains in the deformed tube, while the outer parts are in the undeformed tube. Now when the second deformation is applied, the inner part AB is deformed by the shear Yi + Yi from the equilibrium state, while the outer part is deformed by the shear Yi- It is important to note that the second shear stretches the contour length of the outer part by the factor < (72), but that of the inner part by the factor... 

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