Kirkwood-Riseman-Zimm-Rouse model

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] ... 

These bead-spring models of Rouse and Kirkwood-Riseman-Zimm suffer from the artificiality of the beads and springs. The bead friction coefficient is an ad hoc phenomenological coefficient. This should arise naturally from the frictional forces coupling the polymer and solvent directly with the continuous version of the chain without beads and springs. 

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... 

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. 

The above results for the Rouse model are applicable to the experimental conditions where the hydrodynamic and excluded volume interactions and the entanglement effects can be completely ignored. We shall identify such an experimental regime later on. Now, we attempt to incorporate the hydrodynamic interaction in describing the chain dynamics in infinitely dilute solutions. The Rouse chain model incorporating the effect of hydrodynamic interaction is called the Kirkwood-Riseman modeF or Zimm model. These models differ from each other in certain subtle features and the numerical prefactors only the predicted molecular weight dependence of the longest relaxation time, viscosity of the solution, diffusion coefficient, etc. are the same. 

The general expectations embodied in Equations 7.12, 7.16, and 7.19 are borne out to be valid as shown by experiments in dilute solutions of uncharged polymers. Depending on the experimental conditions, the value of the size exponent changes and this change is directly manifest in D, rj, and t in terms of their dependencies on the molecular weight of the polymer and solvent conditions. In order to obtain the numerical prefactors for the above scaling laws and to understand the internal dynamics of the polymer molecules, it is necessary to build polymer models that explicitly account for the chain connectivity. The two basic models of polymer dynamics are the Rouse and Zimm models (Rouse 1953, Kirkwood and Riseman 1948, Zimm 1956), which are discussed next. 

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