Kirkwood-Riseman model

According to perturbation calculations based on the Kirkwood-Riseman model [Kirkwood and Riseman (139) and Yamakawa and... 

Phillies, G. D. J., Low-shear viscosity of nondilute polymer solutions from a generalized Kirkwood-Riseman model, J. Chem. Phys., 116, 5857-5866 (2002b). 

The 3N -3 normal mode vectors are also called internal mode vectors, because they describe changes in the internal coordinates, the relative positions of the polymer beads. A set of internal coordinates - not to be confused with internal modes - that decompose changes in atomic coordinates into translations, rotations, and internal motions is given by Wilson, et a/. (42) and enhanced by McIntosh, et a/.(43). The Wilson, et al. coordinate decomposition is substantially the same as the Kirkwood-Riseman model, except that Kirkwood and Riseman focus on translation and rotation, while aggregating the internal modes into a fluctua-tional term(44). The Wilson decomposition differs from the Rouse and Zimm decompositions, which identify translations but focus on the internal modes. 

The hydrodynamic scaling model is an extension of the Kirkwood-Riseman model for polymer dynamics(l). The original model considered a single polymer molecule. It effectively treats a polymer coil as a bag of beads. For their collective coordinates, the beads have three center-of-mass translations, three rotations around the center of mass, and unspecified other coordinates. The use of rotation coordinates causes the Kirkwood-Riseman model to differ from the Rouse and Zimm models(2,3). The other collective coordinates of the Kirkwood-Riseman model are lumped as internal coordinates whose fluctuations are in first approximation ignored. The beads are linked end-to-end, the links serving to estabhsh and maintain the coil s bead density and radius of gyration. However, the spring constant of the finks only affects the time evolution of the internal coordinates it has no effect on translation or rotation of the coil as a whole. 

When a coil moves with respect to the solvent, each bead sets up a wake, a fluid flow described in first approximation by the Oseen tensor. The fluid flow velocity near each bead is perturbed by the wakes established by all the other beads, so the fluid flow created by all the beads must be computed in a self-consistent manner. To find concentration dependences, an extended Kirkwood-Riseman model is applied to several polymer chains. The extended model leads to a power series in polymer concentration. A process is then needed to take the power series to large concentration. The original calculation used a self-similarity argument to compute the concentration dependence of Dj(4). The retardation of motion of one polymer... 

The first indication that the preaveraging technique has a substantial effect on the calculation of the friction coefficients of star pol3miers is found in the calculation of the g-dependence of Dq, i.e., the contributions of internal modes [90,91]. Zimm showed that preaveraging of the hydrodynamic interactions in the Kirkwood-Riseman model was largely to blame for the discrepancy between ggp and experimental values of gjj [57]. Further refinements of the Monte Carlo simulations with thermodynamic interactions to represent 0-solvent conditions or good solvent conditions combined with non-preaveraged hydrodynamic interactions lead to close agreement between calculated and... 

It has been clearly shown that the hydrodynamic properties of star polymers cannot be compared with those of linear polymers within the Kirkwood-Riseman model with preaveraged hydrodynamic interactions. In order to match calculated with experimental results, non-preaveraging of hydrodynamic interactions becomes increasingly more important as /increases. It is interesting to note that non-preaveraging increases but decreases from the calculated preaveraged values. 

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. 

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 inhnitely dilute solutions where full hydrodynamic interaction between segments is present as in the Kirkwood-Riseman-Zimm model, the electrophoretic mobility is independent of N in both low (kR 1) and high (kR 1) salt concentrations ... 

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). 

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] ... 

In the Kirkwood-Riseman-Zimm (KRZ) model, unlike Rouse theory, the hydrodynamic interaction between the segments of a macromolecular chain is accounted for. In the limiting case of a tight macromolecular globe, the KRZ theory gives the expression for X,i that is similar to [7.2.27] ... 

Kirkwood-Riseman Theory (1948) This theory is based on a model in which the chain consists of a sequence of monomer units. When a polymer molecule is placed in a fluid of surrounding medium (solvent molecules), the flow is perturbed by the resistance offered by each polymer unit. This model is known as the pearl string (or pearl necklace) model, where each monomer unit is a bead (see Figure 8.5). The emphasis of the Kirkwood-Riseman theory is on the hydrodynamic resistance of the indivdual beads. When the individual resistance is summed, we obtain the resistance of the whole molecule. 

---