Rouse-Zimm description

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... 

Some possible approximations have been considered by Cates [56], who concentrated attention on macromolecular entanglements, which play an important role in the description of the behaviour of block polymers [86-89]. Cates believes that the fact that the concept of polymer fractal neglects the effects of macromolecular entanglements is the main drawback of this theory. Nevertheless, Cates [56] introduced several simplifications that make it possible to ignore these effects for dilute solutions and relatively low molecular masses. However, in the opinion of Cates, even in the case of predominant influence of entanglements, theoretical interpretation of this phenomenon is impossible without preliminary investigation of the properties of the system in terms of Rouse-Zimm dynamics, which can serve as the basis for a more complex theory. It was assumed [56] that the effects of entanglement can be due to the substantially enhanced local friction of macromolecules. 

Chapter 1 introduces basic elements of polymer physics (interactions and force fields for describing polymer systems, conformational statistics of polymer chains, Flory mixing thermodynamics. Rouse, Zimm, and reptation dynamics, glass transition, and crystallization). It provides a brief overview of equilibrium and nonequilibrium statistical mechanics (quantum and classical descriptions of material systems, dynamics, ergodicity, Liouville equation, equilibrium statistical ensembles and connections between them, calculation of pressure and chemical potential, fluctuation... 

All in all, the Rouse model provides a reasonable description of polymer dynamics when the hydrodynamic interactions, excluded volume effects and entanglement effects can be neglected a classical example of its applicability is short-chain polymer melts. Since the Rouse model is exactly solvable for polymer chains, it represents a basic reference frame for comparison with more involved models of polymer dynamics. In particular, the decouphng of the dynamics of the Rouse chain into a set of independently relaxing normal modes is fundamental and plays an important role in other cases, such as more complex objects of study, or in other models, such as the Zimm model. 

Two general classical bead-spring models have been developed for the description and analysis of the motions of flexible chains (see chapter Conformational and Dynamic Behavior of Polymer and Polyelectrolyte Chains in Dilute Solutions ). The Rouse model [54] is simpler (it does not take into account hydrodynamic correlations). The more advanced Zimm model accounts for hydrodynamic correlations and provides better description of the behavior [55]. In both cases, solution of the derived equations provides the so-called normal modes (relaxation times of different types of motions). The first mode describes the slowest motion of the... 

As the hydrodynamic interaction is screened in semidilute solutions, the molecular weight dependencies of the diffusion coefficient, the longest relaxation time, and the viscosity change in the semidilute solutions are exactly the same as in the Rouse model. However, since the Rouse model was originally designed for an isolated chain, the concentration dependencies of these quantities are not captured by the Rouse model. Nevertheless, we shall refer to the correct description of polymer dynamics in semidilute solutions as the Rouse regime. A summary of the main results for the Zimm model in dilute solutions... 

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