Kirkwood-Riseman theory

The details of the Kirkwood-Riseman theory are sufficiently involved that we shall not consider the derivation of this theory. We shall, however, examine in somewhat greater detail the cluster of variables we have designated by X as a measure of the permeability of the molecule to the flowing solvent. 

As discussed in connection with Eq. (9.47), the Kirkwood-Riseman theory predicts that a = 1 in the free-draining limit. This limit is expected for small values of n, however, and does not explain a > 0.5 for high molecular weight polymers. 

As in the case of the Rouse relaxation time, it is possible to express x2 in term of the intrinsic viscosity either by using Eq. (24) or the Kirkwood-Riseman theory [24, 47] ... 

The estimation of f from Stokes law when the bead is similar in size to a solvent molecule represents a dubious application of a classical equation derived for a continuous medium to a molecular phenomenon. The value used for f above could be considerably in error. Hence the real test of whether or not it is justifiable to neglect the second term in Eq. (19) is to be sought in experiment. It should be remarked also that the Kirkwood-Riseman theory, including their theory of viscosity to be discussed below, has been developed on the assumption that the hydrodynamics of the molecule, like its thermodynamic interactions, are equivalent to those of a cloud distribution of independent beads. A better approximation to the actual molecule would consist of a cylinder of roughly uniform cross section bent irregularly into a random, tortuous configuration. The accuracy with which the cloud model represents the behavior of the real polymer chain can be decided at present only from analysis of experimental data. 

Kirkwood and Riseman have developed a theory that allows for variable degrees of solvent drainage through the coil domain. We shall not go into this theory in any detail, except to note that it should reduce to Equation (87) in the free-draining limit and to the Einstein equation in the nondraining limit. The Kirkwood-Riseman theory can be written in the form... 

Using the nondraining limit of the Kirkwood-Riseman theory gives... 

Fig. 12. Viscosity-molecular weight relations in polystyrene. The three dotted curves show the effect of partial draining according to the Kirkwood-Riseman theory, Eqs. (50), (51) and (62), with three different choices of drainage parameter...

To investigate further the absence of the draining effect, three dotted curves are also given in Fig. 12. These correspond to the intrinsic viscosities in the theta solvent predicted by the Kirkwood-Riseman theory 139) according to Eqs. (50) and (51)1 with three different choices of the drainage parameter X. It is convenient to express the friction constant of a chain segment in the form of the Stokes law,... 

Auer, P. L., and C. S. Gardner J. Chem. Phys. 23, 1545, 1546 (1955). Application of Kirkwood-Riseman theory to various molecular models has been reported by Ullman, R. J. Chem. Phys. 40, 2193 (1964). 

Calculated from theories of Brinkman, and Debye-Bueche. Calculated from Kirkwood-Riseman theory. Calculated from Kuhn-Kuhn theory, using values in Ref. 155. Caleulated from theory of Peterlin. Calculated from Flory-Fox theory, using values in Ref. 155. (See pp. 381-385.)... 

Yamakawa (38) and Imai (83) have published an alternative description based on a random coil model and the Kirkwood-Riseman theory (62) and obtained for theta-solvent conditions an equation equivalent to ... 

The authors of Ref. [53] have shown, that frictional properties of fractal clusters can be different essentially for the usual results for compact (Euclidean) structures. It is known through Ref. [54], that the polymer melt structure can be presented as a macromolecular coils sets, which are fractal objects. Therefore, the authors [55] proposed general structural treatment of polymer melt viscosity within the framework of fractal analysis, using the model [53]. Within the framework of the indicated model the derivations for translational friction coefficient f(N) of clusters from N particles in three-dimensional Euclidean space were received, calculated according to Kirkwood-Riseman theory in the presence of hydrodynami-cal interaction between the cluster particles. The fundamental relationship of this theory is the following equation [53] ... 

Based on the Kirkwood-Riseman theory, if polymer chains are nondraining, it follows [Kirkwood and Riseman, 1948 Auer and Gardner, 1955] that intrinsic viscosity data can be related to the radius of gyration, Rg, of flexible polymers. This can be expressed in an equation of the form [Flory and Fox, 1951]... 

In the Kirkwood-Riseman theory, the perturbation of the rate of flow of the solvent by A — 1 chain elements is calculated for the Rth chain element and summed over all possible conformations. Suitable parameters that are used are the effective bond length b and the frictional coefficient of the monomeric unit. 

The Kirkwood-Riseman theory [93] and the Zimm theory [98] as well predict that n in eq 3.3 under the 0 condition decreases monotonically from 1 to 0.5 with increasing M, and thus fail to derive eq 3.4. Again, we have to invoke complete immobilization of solvent inside the polymer coil to explain this relation by these theories. The Yamakawa-Fujii theory shows that chain stiffness also counteracts the draining effect on Do, but a d/q value different from that needed for [q]ff has to be used in order to obtain a maximum suppression of the draining effect. 

As explained in Section 3.4.1, the observed M dependence of [77] and Doe can be deduced from the Kirkwood-Riseman theory (the Zimm theory as well) if it is postulated that the polymer coil is in the non-draining state at any chain length. Then we ask how accurately this theory can predict the prefactors Ke in eq 3.2 and Kg in eq 3.4. To discuss this problem it is usual to use the "hydrodynamic factors and pe defined as follows ... 

The intrinsic viscosity [//(r)] and the friction coefficient f r)g of an unperturbed ring polymer were first calculated by Bloomfield and Zimm [69] and by Fukatsu and Kurata [70] using the Kirkwood-Riseman theory with the preaveraged Oseen interaction tensor. In the non-draining limit their calculations yield... 

Still, the radius of gyration does not equal the effective radius of a polymer coil in solution in terms of the flow behavior. For the calculation of a so-called hydro-dynamic radius R that reflects this effective radius of the polymer coil in a flow field, several theoretical approaches have been made. For theta conditions for example, the modified Kirkwood-Riseman theory [47] predicts a ratio of radius of gyration to hydrodynamic radius of... 

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