The Stress Tensor and Rheological Constitutive Equations

Once the singlet distribution function has been found, we are in a position to evaluate the various contributions to the fluxes that depend on (see Table 1). In this section we discuss the contnbutions to the stress tensor, and in the next two sections the contnbutions to the mass and heat flux vectors. In these sections, for illustrative purposes, we restrict ourselves to the Rouse bead-spring chain and the Hookean dumbbell models, for which we can use the singlet distribution functions , given in Eqs. (13.5) and (13.8). 

We confine our attention here to dilute solutions of several polymer speaes m a solvent. According to Sect. 7, the stress tensor is a sum of four contnbutions, the first three of which involve the singlet distribution function (s), whereas the fourth involves the doublet distribution fiinction(4)  

As shown here, we can also regard the stress tensor as being a sum of contributions over the individual chemical species plus a double sum over all pairs of speaes. If the solvent species (subscript is now treated separately, and primes indicate sums over all solute speaes, then Eq. (14.1) can be written as  

It has been common practice in the kinetic theory of dilute solutions to lump the first and third terms together and refer to this combination as it the solvent contribution, and to neglect the fourth and fifth terms. Omitting the fifth term is appropriate for dilute solutions in which polymer-polymer interactions can be Ignored. Neglect of the fourth term, however, may not always be appropriate omission of this term disregards solvent effects, such as the phenomenon of solvent modification that has been observed in recent years [23a, 23b, 23c] This term refers to the tendency for solvent molecules to become preferentially oriented in the neighborhood of the polymer molecules. 

In this section we will then use the simphfied version of Eq. (14.2) as has been customary in dilute solution theories [1,2,3,4] and wnte for a solution with a single solute a N  

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