Properties of Alternative Models

Curtiss et al. (102a) have recently developed general formulas for t]0 — t]s and Je° for free-draining bead-connector models with arbitrary numbers of beads, connecting arrangements and force-distance laws for the connectors. The expressions depend on averages over the unperturbed distribution of configurations for the model  

The number of beads in the model macromolecule is n, and is the Stokes law friction coefficient of each bead. The are to be evaluated for each macromolecule in its own internal coordinate system, with origin at the molecular center of gravity and axes (k = 1,2,3) lying along the principal axes of the macromolecule. The coordinates of the ith bead in this frame of reference are (x ]),-, (x2)i, and (x3)f. The averaging indicated by is performed over all macromolecules in the system. Thus, i + 2 + 3) is simply S2 for the macromolecules. The viscosity is therefore identical, for all free-draining models with the same molecular frictional coefficient n and the same radius of gyration, to the expression from the Rouse theory  

In rigid models each macromolecule has the same set of 4 so 4 = 4 and For flexible models the averages are calculated from the 

These results make it clear that the forms of t]0 — rjs and Je° are completely independent of model details. Only the numerical coefficient of Je° contains information on the properties of the model, and even then the result depends on both molecular asymmetry and flexibility. Furthermore, polydispersity effects are the same in all such free-draining models. The forms from the Rouse theory cany over directly, so that t]0 - t]s, translated to macroscopic terms, is proportional to Mw and Je° is proportional to the factor A/2M2+, /A/w. Unfortunately, no such general analysis has been made for models with intramolecular hydrodynamic interaction, and of course these results apply in principle only to cases where intermolecular interactions are negligible. 

Completely rigid models appear to provide rather peculiar short time response. The stress relaxation modulus for rigid dumbbells is (102)  

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