Rouse model intrinsic viscosity

The theoretical prediction of these properties for branched molecules has to take into account the peculiar aspects of these chains. It is possible to obtain these properties as the low gradient Hmits of non-equilibrium averages, calculated from dynamic models. The basic approach to the dynamics of flexible chains is given by the Rouse or the Rouse-Zimm theories [12,13,15,21]. How-ever,both the friction coefficient and the intrinsic viscosity can also be evaluated from equilibrium averages that involve the forces acting on each one of the units. This description is known as the Kirkwood-Riseman (KR) theory [15,71 ]. Thus, the translational friction coefficient, fl, relates the force applied to the center of masses of the molecule and its velocity... 

In 1959, Zimm and Kilb (34) made some calculations of the intrinsic viscosities of certain branched polymer molecules, taking into account the hydrodynamic interaction between portions of the polymer chain, using a modification of the Rouse procedure. They carried out these difficult calculations for a quite restricted range of models, obtaining numerical results for equalarmed stars with 3, 4, and 8 branches, and for one modified star with 2 long branches and 8 shorter branches. They found that their numerical results for this set of structures could be approximately represented by ... 

The Rouse model predicts that the intrinsic viscosity in a -solvent is proportional to molar mass. However, the Rouse model assumes no... 

Imai (56) constructed a theory fear the intrinsic viscosity and sedimentation constant of ring polymers using the Fixman method shown in 2.3.2 for a model similar to the Hearst-Harris model which will be described in 4.2.2. This model reduces to the Rouse model in a limiting case. The excluded volume potential is included in the form of Eq. (2.26) and the same type of calculation as described in 2.3.2 was performed for a steady shear flow. Dynamic mechanical properties were not treated, although the extension to include this case is only a matter of tedious calculations. 

Only recently has the theory of chain dynamics been extended by Peterlin (J [) and by Fixman (12) to encompass the known non-Newtonian intrinsic viscosity ofTlexible polymers. This theory, which is an extension of the Rouse-Zimm bead-and-spring model but which includes excluded volume effects, is much more complex than that for undeformable ellipsoids, and approximations are needed to make the problem tractable. Nevertheless, this theory agrees remarkably well (J2) with observations on polystyrene, which is surely a flexible chain. In particular, the theory predicts quite well the characteristic shear stress at which the intrinsic viscosity of polystyrene begins to drop from its low-shear Newtonian plateau. 

The product g = Cy performs the desired diagonalization yielding A = M in contrast to the old guess that M equals A of Rouse s model without hydrodynamic interaction. The values of intrinsic viscosity at zero gradient and zero frej uency are not affected by this result. But the values at finite y or O) are modified as soon as internal viscosity is considered. 

Rouse assumed the molecule to he freely draining, i.e., that the effect of the flow of solvent past one part of the molecule has no effect on another part. Another way of saying this is that he assumed no hydrodynamic interaction. As is noted in Section 2.5 on intrinsic viscosity, this led to predictions that were not in accord with observations for dilute polymer solutions. Zimm later developed a model that took into account hydrodynamic interaction, but it is not necessary to consider this here, as it is not relevant to our discussion of melt behavior where there is no solvent. 

Theoretical calculation of the intrinsic viscosity of regular stars goes back to 1959 [78]. In the free-draining case, the Rouse model for linear polymers applied to stars yields [79]... 

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