The Polymer Stress and Birefringence Tensors

From the configuration distribution function, i/ o(R), the elasticity of flexible polymer molecules can be predicted. Suppose the ends of the polymer chain are held fixed so that the end-to-end vector of the chain is R. The number of internal configurations,, of the chain that satisfies this constraint is = ct fo(R), with c a constant. The entropy, S, is then (Wall 1942) 

If we now pull on a chain end to increase the end-to-end distance R, the force we need to exert to overcome the entropic spring force of the chain is 

To obtain the polymer contribution to the stress tensor for a fluid containing a large number Nc of springs, we define the position of one end of the spring i to be r, j and define the position of the other end to be r,-,2- We then define R, = r,2 — r, i. We note that the spring exerts a force —F. on point 2 and exerts a force F- on point 1. Now we use the Kirkwood Tbrmula for the stress tensor, Eq. (1-42). The summation must be carried out over the locations of the ends of all different springs. This summation can be expressed as 

We now define v = V to be the number Ns of springs per unit volume V. If this number is large, the sum in the above equation can be replaced by v times an ensemble average hence Bird et al. 1987b) 

Here ( without the subscript 0 is an average over the nonequilibrium distribution, 

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