A — Contour Length Fluctuations of the Primitive Chain

Appendix 9.A — Contour Length Fluctuations of the Primitive Chain 

The time-correlation function 5L 0)5L t)) of Eq. (9.3) will be derived by considering the polymer chain as a Gaussian chain consisting of No segments each with the root mean square length b. Let 5 (t) be the contour position of the nth bead relative to a certain reference point on the primitive path. Then the contour length of the primitive chain at time t is given by 

The force constant, 3kT/b, used in Eq. (9.A.2) originates from the entropy related to the number of microstructural configurations in a Rouse segment. The microstructural configurations have to be considered in three-dimensional space. Involving three-dimensional space is related to the idea that the size of the Rouse segment, b, is much smaller than the tube 

The fluctuation forces also cause the curvilinear diffusion along the primitive path. However, because of the constraint effect of the tube, the positive and negative fluctuation forces perpendicular to the primitive path cancel each other out when averaged over a time period longer than required for a segment to travel over an entanglement distance, and thus make no net contribution to the curvihnear diffusion and the translational diffusion of the center of mass. Thus, in the study of the cmvilinear diffusion, Eq. (9.A.3 ) should be used, instead of Eq. (9.A.3). 

As discussed in the Doi-Edwards theory (Chapter 8), a fictitious tensile force Feq given by Eq. (8.21) must be assumed to apply at both chain ends. Hence the boundary condition for Eq. (9.A.2) is 

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