Rouse-Zimm theory

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 complex viscosity, i.e., the viscosity observed in the presence of an oscillatory shear rate, is a dynamic property that can be straightforwardly obtained from the Rouse, or Rouse-Zimm theory as the Fourier transform of the stress time-correlation function. Thus, these theories give [15]... 

Bixon M, Zwanzig R (1978) Optimised Rouse-Zimm theory for stiff polymers. J Chem Phys 68(4) 1896-1902... 

Note. R-Z, Rouse-Zimm theory. EN, electrical network analogy. 

Since molecular theories of viscoelasticity are available only to describe the behavior of isolated polymer molecules at infinite dilution, efforts have been made over the years for measurements at progressively lower concentrations and it has been finally possible to extrapolate data to zero concentration. The behavior of linear flexible macromolecules is well described by the Rouse-Zimm theory based on a bead-spring model, except at high frequencies . Effects of branching can be taken into account, at least for starshaped molecules. At low and intermediate frequencies, the molec-... 

Ferry also made seminal measurements on dilute solutions, using a precision shear wave apparatus. The Rouse-Zimm theory of chain dynamics was explored in great detail. 

In this chapter, we have presented the fundamentals of molecular theory for the viscoelasticity of flexible homogeneous polymers, namely the Rouse/Zimm theory for dilute polymer solutions and unentangled polymer melts, and the Doi-Edwards theory for concentrated polymer solutions and entangled polymer melts. In doing so, we have shown how the constitutive equations from each theory have been derived and then have compared theoretical prediction with experiment. The material presented in this chapter is very important for understanding how the molecular parameters of polymers are related to the rheological properties of homopolymers. 

The incorporation of non-Gaussian effects in the Rouse theory can only be accomplished in an approximate way. For instance, the optimized Rouse-Zimm local dynamics approach has been applied by Guenza et al. [55] for linear and star chains. They were able to obtain correlation times and results related to dynamic light scattering experiments as the dynamic structure factor and its first cumulant [88]. A similar approach has also been applied by Ganazzoli et al. [87] for viscosity calculations. They obtained the generalized ZK results for ratio g already discussed. 

There is an alternative and very direct way to generalize the Rouse-Zimm model for non-Gaussian chains. This approach takes advantage of the expression given by the original theory for the chain elastic potential energy in terms of normal coordinates ... 

Experimentally the overall size of the polymer chain can be studied by light scattering and neutron scattering. A great deal of theoretical work is present in the literature which tries to predict the properties of mixtures in terms of their components. The analytical model by Rouse-Zimm [85,86] is one of the earliest works to derive fundamental properties of polymer solutions. Advances were made subsequently in dilute and concentrated solutions using perturbation theory [87], self-consistent field theory [88], and scaling theory [89],... 

Figure 4-13 contains the predictions of the Rouse theory on the left and of the Zimm theory on the right. As is to be expected, the predictions of the Zimm theory that takes the hydrodynamic interactions into account predicts well experimental data. 

Freely Draining Gaussian Chain (Rouse Theory) Dominant HI Theta Solvent (Zimm Theory) Dominant HI Good Solvent... 

Figure 3.13 Linear viscoelastic data (symbols) for polystyrene in two theta solvents, decalin and diocty Iphthalate, compared to the predictions (lines) of the Zimm theory with dominant hydrodynamic interaction, h = oo. The reduced storage and loss moduli and G are defined by = [G ]M/NAksT and G s [G"]M/A /cbT, where the brackets denote intrinsic values extrapolated to zero concentration, [G jj] = limc o(G /c) and [G j ] = limc +o[(G" — cor)s]/c), and c is the mass of polymer per unit volume of solution. The characteristic relaxation time to is given by to = [rj]oMrjs/NAkBT. For frequencies ro

In a good solvent, where there are excluded-volume effects, G and G" can be fit to the Zimm theory simply by adjusting h downward, for finite Ns (Ferry 1980). Thus, as the solvent quality improves, the relaxation spectrum becomes more Rouse-like (since... 

When p > Pc, one can define P p) to be the fraction of bonds belonging to the infinite cluster. The percolation predictions of the modulus G, the longest relaxation time r, and the viscosity rj depend on whether one uses the Rouse-Zimm (R-Z) theory, or the analogy to an electrical network (EN). The exponent for the modulus G is predicted to be greater than either of these (i.e., around 3.7) if bond-bending dominates (Arbabi and Sahimi 1988). Further details about these exponents can be found in Chapter 5 of Drinker and Scherer (1990), as well as in Martin and Adolf (1991). 

The effect of excluded volume as predicted by the Tschoegl equation (39) is to shift the dynamic properties to more Rouse-like behavior. If one keeps h at infinity and takes = 1/3, then one obtains curves for [G ju and [G"]k which lie between the corresponding quantities for h = 1 and 25 in Fig. 2.2. Thus the effect of increasing e is qualitatively equivalent to decreasing h in the original Zimm theory (29). This effect of g diminishes as h decreases, and disappears at h = 0. 

Fig. 2.3. Relaxation spectrum of Ogasa-lmai theory (47) compared with those erf Rouse theory (27) and Zimm theory (29). Strength Gp of p-th relaxation mode is plotted against logarithm erf reduced relaxation time t,/tp. Strength of mode of the longest relaxation time at left side is nkTon each panel...

Non-0-SoIutions. A number of data have been published for the intrinsic complex modulus of flexible polymers in good solvents. Tanaka, Sakanishi and coworkers studied the system of PIB in cyclohexane (2,92,98), polymethyl methacrylate (PMMA) in chloroform (2,9/), PMS in benzene and in toluene (94,99) and a few copolymers of styrene-butadiene in toluene and in cyclohexane (100). Schrag, Ferry et al. reported the results on PS in a-chloronaphthalene and in a chlorinated diphenyl (3), polybutadiene (PBD) in a-chloronaphthalene and in Decalin (101) and PMS in Decalin and in a-chloronaphthalene (102). In all these cases, the frequency dependence of the intrinsic complex modulus is qualitatively described as more Rouse-like as compared with the results for -solutions. Apart from the physical appropriateness, any of the three versions of the Zimm theory can be applied to the experimental results the original Zimm integrodifferential equation... 

Figure 3-16. Predictions of the Rouse and Zimm theories. [After J. D. Ferry et al., J. Phys. Chem., 66, 536 (1962). Reprinted by permission of the American Chemical Society.]...

In Figure 3-16 we have compared the prediction of the Rouse and Zimm theories.1 The limiting values of the slopes of plots of log G versus log co at large co are one-half for the Rouse treatment and two-thirds for the Zimm treatment. This arises from the different distributions of relaxation times in the Rouse and Zimm theories. In Figure 3-17 the results of DeMallie9 for the... 

McKinney (1965) for rigid dumbbells as a function of random coil theories of Rouse and Zimm are shown in Fig. 19. At high frequencies, the reduced moduli of the Rouse theory become equal and increase together with a slope of 1/2, while those in the Zimm theory remain unequal and increase in a parallel manner with a slope of 2/3. 

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. 

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. 

Experimentally, most dilute polymer solutions in 0-solvents fit the Zimm theory best, whereas as the concentration and/or the solvent quality is increased the behaviour becomes more Rouse-like (A is decreased). Several theories exist to describe physical properties for intermediate values of A. 

We leave out small numerical factors such as 3 for this dimensional analysis. As another simpliheation, here we consider only the slowest relaxation time (the Rouse and Zimm theories treat all the relaxation modes of polymers). The solution to Equation (33.30) is... 

This treatment has become conveniently but imprecisely known as the Zimm theory, so that the names of Zimm and Rouse are labels for dominant hydrodynamic interaction and free draining, respectively. 

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