Rheological Constitutive Equation of the Rouse Model

In Chapter 6, we derived the stress tensor for the elastic dumbbell model. By following the derivation steps given there, one obtains the following result for the Rouse chain model with N beads per chain, which is equivalent to Eq. (6.50) for the elastic dumbbell model  

we have neglected the solvent contribution as we are mainly interested in the concentrated-solution or melt system. The first term of Eq. (7.47) arises from the tensile force on the bonds, and the second term from the momenta associated with the moving beads. Substituting Eqs. (7.34) and (7.35) into Eq. (7.47), we obtain 

Just as Eqs. (6.58) and (6.59) were obtained in Chapter 6, the integral forms of Eq. (7.53) can be obtained. Then, with Eq. (7.49), the integral form of the constitutive equation for the Rouse chain model is given by (with u replaced by p) 

Similar to the elastic dumbbell case, we can obtain the various viscoelastic properties from the constitutive equation of the Rouse model. The main difference between the two models is that the elastic dumbbell 

For the shear relaxation modulus, the Rouse model gives 

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The rheological constitutive equation of the Rouse model is that of an upper-convected Maxwell model, with the consequence that steady-state elongational flow only exists for strain rates lower than l/(2A,i). The steady-state elongational wscosity depends then on strain rate ... 

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