Dynamics Rouse-Zimm model

Fig. 5.10 Chain dynamic structure factor of PDMS (a) and PIB (b) in toluene solution at 327 K at the Q-values 0.04 A" (empty circle), 0.06 A (filled circle), 0.08 A" (empty square), 0.10 A (filled square), 0,15 A (empty diamond), 0,20 A (filled diamond), 0,25 A (empty triangle), 0.30 A" (filled triangle), 0.40 A" (plus). Solid lines correspond to fitting curves Rouse-Zimm model for PDMS and Rouse-Zimm with intrachain viscosity for PIB (see the text). (Reprinted with permission from [186]. Copyright 2001 American Chemical Society)...

The depolarized scattering for the Rouse-Zimm dynamical model of flexible polymer chains (cf. Section 8.8) may also be calculated. Ono and Okano (1971) have performed this calculation for q = 0 (zero scattering angle) and find that the scattered light spectral density is a series of Lorentzians each with a relaxation time characteristic of one of the Rouse-Zimm model modes. However the contribution of each mode to the spectrum is equal. This behavior should be contrasted with that of the isotropic spectrum where the scattering spectrum is dominated by contributions from the longest wavelength modes. 

Just as the Gaussian chain is the basic paradigm of the statistics of polymer solutions, so is its extension to the bead-spring model still basic to current work in the held of polymer dynamics. The two limiting cases of free draining (no hydrodynamic interaction between beads, characterized by the draining parameter A = 0) and non-free draining (dominant hydrodynamic interaction, A= CO, due to Rouse and Zimm, respectively, are sufficiently familiar that the approach is often known as the Rouse-Zimm model. ... 

The Rouse-Zimm Model Describes the Dynamics of Viscoelastic Fluids... 

Before turning to dynamics, we should hke to point out that, because no solvent is explicitly included, the Rouse model [37,38] (rather than the Zimm model [39]) results in the dilute limit, as there is no hydrodynamic interaction. The rate of reorientation of monomers per unit time is W, and the relaxation time of a chain scales as [26,38]... 

The Rouse model, as given by the system of Eq, (21), describes the dynamics of a connected body displaying local interactions. In the Zimm model, on the other hand, the interactions among the segments are delocalized due to the inclusion of long range hydrodynamic effects. For this reason, the solution of the system of coupled equations and its transformation into normal mode coordinates are much more laborious than with the Rouse model. In order to uncouple the system of matrix equations, Zimm replaced S2U by its average over the equilibrium distribution function ... 

The Rouse and Zimm models are valid only under 0-conditions. To extend their range of applicability into good solvent conditions, several improvements have been proposed to include excluded volume effects. Dynamical scaling, however, provides probably the simplest approach to the problem [30],... 

As in the case of the Rouse dynamics (see Sect. 3.1.1), the intermediate incoherent scattering law for dominant hydrodynamic interaction (Zimm model) can be... 

Fig. 35. Dynamic structure factors S(Q,t)/S(Q,0) as dependent on Qt for the Rouse and the Zimm model...

In contrast to -conditions a large number of NSE results have been published for polymers in dilute good solvents [16,110,115-120]. For this case the theoretical coherent dynamic structure factor of the Zimm model is not available. However, the experimental spectra are quite well described by that derived for -conditions. For example, see Fig. 42a and 42b, where the spectra S(Q, t)/S(Q,0) for the system PS/d-toluene at 373 K are shown as a function of time t and of the scaling variable (Oz(Q)t)2/3. As in Fig. 40a, the solid lines in Fig. 42a result from a common fit with a single adjustable parameter. No contribution of Rouse dynamics, leading to a dynamic structure factor of combined Rouse-Zimm relaxation (see Table 1), can be detected in the spectra. Obviously, the line shape of the spectra is not influenced by the quality of the solvent. As before, the characteristic frequencies 2(Q) follow the Q3-power law, which is... 

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 culmination of this trend is illustrated in Fig. 5.2 by dynamic data on undiluted polystyrene of low molecular weight (124). Agreement with the Rouse model here is by no means as good as that seen in Fig. 5.1 with the Zimm model for a high molecular weight polystyrene at infinite dilution. Indeed, the value of Je° deduced from G (to) for the sample in Fig. 5.2 exceeds the value from... 

Explain the length scales over which the reptation, Rouse, and Zimm models describe dynamics in semidilute entangled solutions of linear polymers. 

For flexible polymers the structural change due to intramolecular motions must be large enough for the light wave to detect the difference between the various molecular shapes. Only under these circumstances will intramolecular interference affect the lightscattering spectral distributions. An extreme example of this case, the Rouse-Zimm dynamic model of the Gaussian coil, is discussed in detail in Section 8.8. 

Only recently has the theory of chain dynamics been extended by Peterlin (J [) and by Fixman (12) to encompass the known non-Newtonian intrinsic viscosity ofTlexible polymers. This theory, which is an extension of the Rouse-Zimm bead-and-spring model but which includes excluded volume effects, is much more complex than that for undeformable ellipsoids, and approximations are needed to make the problem tractable. Nevertheless, this theory agrees remarkably well (J2) with observations on polystyrene, which is surely a flexible chain. In particular, the theory predicts quite well the characteristic shear stress at which the intrinsic viscosity of polystyrene begins to drop from its low-shear Newtonian plateau. 

Within this framework, one can draw a rather simplified picture about the dynamics of the polymers in the entangled state. If the characteristic length scale of a motion is smaller than a, the entanglement effect is not important, and the dynamics is well described by the Rouse model (or the Zimm model if the hydrodynamic interaction is not screened). On the other hand, if the length-scale of the motion becomes larger than a, the dynamics is governed by reptation. 

In the Rouse-Zimm bead spring model of polymer solution dynamics, the long-range global motions are associated with a broad spectrum of relaxation times given by equation (10) and where tj is the relaxation time of the h such normal mode of the chain... 

Another important aspect of unentangled models not covered in this chapter is hydrodynamic interactions, that is, the behavior of a single chain in a solvent of small molecules. This situation is believed to be well described by the Zimm model however, critical consideration of all observables will surely reveal some differences. It is also interesting to see the transition between Zimm and Rouse dynamics as the solvent chains become longer. 

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. 

The Rouse model has been extended to deal with the dynamics of chains in dilute solution [18], In solution a moving bead perturbs the solvent flow around another bead, leading to effective, so-called hydrodynamic, interactions. The Zimm model generalizes the Rouse model by taking hydrodynamic interactions into account. 

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... 

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. 

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... 

However, the inclusion of the long-range correlations places the Zimm model into a different dynamic universality class than the Rouse model. Although there exist sophisticated renormalization- oup... 

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. 

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