The Zimm model

To describe the dynamics of polymers in dilute solution, we have to take into account the hydrodynamic interaction, which is expressed by the mobility matrix calculated in Chapter 3, 

In particular, for the O condition, eqns (4.4) and (4.42) give (in the continuous limit) 

Since H is a nonlinear function of R - R , eqn (4.43) is quite difficult to handle. To simplify the analysis, Zimm introduced the preaveraging approximation, which replaces H by its average, 

If we are considering problems near equilibrium, which is the case in the subsequent part of this chapter, we may use the equilibrium distribution function Veq( / ) in the average of eqn (4.44), 

In the 0 condition, the distribution of R - R is Gaussian with the variance n-m hence 

In the Zimm theory, the force on the ith bead has, besides the terms due to the hydrodynamic and restoring forces considered in the Rouse model, additional terms due to Brownian motion and hydrodynamic shielding. The Brownian motion force exerted on the beads is expressed by  

Now we must take into account the velocity disturbance v due to the motion of other beads in the chain to the solvent velocity at the ith bead that is, the solvent velocity consists of two components = v° + v. The velocity disturbance v  

Using normal coordinates, Eq. (4.67) can be rewritten such that the values of the relaxation times may be made to depend on the eigenvalues which are the elements of the diagonal matrix A that can be obtained from the following equation  

Zimm (1956) has shown that the solution of Eq. (4.72) can be obtained by solving the following integrodifferential equation  

The relaxation time x for the Zimm model is given by (Zimm 1956) 

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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 non-free draining character of flexible polymer chains was considered in the Zimm model [48], In this model, the effect of hydrodynamic interaction at the location of bead i is taken into account by an additional fluid velocity term vj ... 

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 ... 

In analogy with the Rouse model, the longest relaxation time (Xj) according to the Zimm model can again be put into a form which does not depend on N [44] ... 

The Zimm model predicts correctly the experimental scaling exponent xx ss M3/2 determined in dilute solutions under 0-conditions. In concentrated solution and melts, the hydrodynamic interaction between the polymer segments of the same chain is screened by the host molecules (Eq. 28) and a flexible polymer coil behaves much like a free-draining chain with a Rouse spectrum in the relaxation times. 

Since the hydrodynamic interaction decreases as the inverse distance between the beads (Eq. 27), it is expected that it should vary with the degree of polymer chain distortion. This is not considered in the Zimm model which assumes a constant hydrodynamic interaction given by the equilibrium averaging of the Oseen tensor (Eq. 34). 

Diffusion of flexible macromolecules in solutions and gel media has also been studied extensively [35,97]. The Zimm model for diffusion of flexible chains in polymer melts predicts that the diffusion coefficient of a flexible polymer in solution depends on polymer length to the 1/2 power, D N. This theoretical result has also been confirmed by experimental data [97,122]. The reptation theory for diffusion of flexible polymers in highly restricted environments predicts a dependence D [97,122,127]. Results of various... 

Zimm [34] extended the bead-spring model by additionally taking hydrodynamic interactions into account. These interactions lead to changes in the medium velocity in the surroundings of each bead, by beads of the same chain. It is worth noting that neither the Rouse nor the Zimm model predicts a shear rate dependency of rj. Moreover, it is assumed that the beads are jointed by an ideally Hookean spring, i.e. they obey a strictly linear force law. 

The dynamic structure factor is S(q, t) = (nq(r) q(0)), where nq(t) = Sam e q r is the Fourier transform of the total density of the polymer beads. The Zimm model predicts that this function should scale as S(q, t) = S(q, 0)J-(qat), where IF is a scaling function. The data in Fig. 12b confirm that this scaling form is satisfied. These results show that hydrodynamic effects for polymeric systems can be investigated using MPC dynamics. 

The coherent structure factor of the Zimm model can be calculated [34] following the lines outlined in detail in connection with the Rouse model. As the incoherent structure factor (83), it is also a function of the scaling variable ( fz(Q)t)2/3 and has the form... 

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

For the Zimm model, the approach and argumentation are quite similar ... 

Fig. 40a, b. NSE spectra of a dilute solution under 0-conditions (PDMS/ d-bromobenzene, T = = 357 K). a S(Q,t)/S(Q,0) vs time t b S(Q,t)/S(Q,0) as a function of the Zimm scaling variable ( t(Q)t)2/3. The solid lines result from fitting the dynamic structure factor of the Zimm model (s. Tablet) simultaneously to all experimental data using T/r s as adjustable parameter. 

Obviously, in the case of PS these discrepancies are more and more reduced if the probed dimensions, characterized by 2ti/Q, are enlarged from microscopic to macroscopic scales. Using extremely high molecular masses the internal modes can also be studied by photon correlation spectroscopy [111,112], Corresponding measurements show that - at two orders of magnitude smaller Q-values than those tested with NSE - the line shape of the spectra is also well described by the dynamic structure factor of the Zimm model (see Table 1). The characteristic frequencies QZ(Q) also vary with Q3. Flowever, their absolute values are only 10-15% below the prediction. 

In the case of dynamic mechanical relaxation the Zimm model leads to a specific frequency ( ) dependence of the storage [G ( )] and loss [G"(cd)] part of the intrinsic shear modulus [G ( )] [1]. The smallest relaxation rate l/xz [see Eq. (80)], which determines the position of the log G (oi) and log G"(o>) curves on the logarithmic -scale relates to 2Z(Q), if R3/xz is compared with Q(Q)/Q3. The experimental results from dilute PDMS and PS solutions under -conditions [113,114] fit perfectly to the theoretically predicted line shape of the components of the modulus. In addition l/xz is in complete agreement with the theoretical prediction based on the pre-averaged Oseen tensor. 

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 crossover from 0- to good solvent conditions in the internal relaxation of dilute solutions was investigated by NSE on PS/d-cyclohexane (0 = 311 K) [115] and on PDMS/d-bromobenzene(0 = 357K) [110]. In Fig. 45 the characteristic frequencies Qred(Q,x) (113) are shown as a function of t = (T — 0)/0. The QZ(Q, t) were determined by fitting the theoretical dynamic structure factor S(Q, t)/S(Q,0) of the Zimm model (see Table 1) to the experimental data. This procedure is justified since the line shape of the calculated coherent dynamic structure factor provides a good description of the measured NSE-spectra under 0- as well as under good solvent conditions. 

The line shape of the spectra is independent from the solvent conditions and in complete agreement with the calculations based on the Zimm model... 

The theoretical approach described before dealt with the short-time dynamic response of the star molecules. However, in the case of completely labelled stars [148] it was found that the line shape of the Zimm model provides a good description of the NSE spectra not only in the short-time regime (t < 5 ns), but also on longer time scales. 

The NSE spectra from full stars are well described by the scattering law of the Zimm model in the whole experimental time window (20 ns)... 

The hydrodynamic interaction is introduced in the Zimm model as a pure intrachain effect. The molecular treatment of its screening owing to presence of other chains requires the solution of a complicated many-body problem [11, 160-164], In some cases, this problem can be overcome by a phenomenological approach [40,117], based on the Zimm model and on the additional assumption that the average hydrodynamic interaction in semi-dilute solutions is still of the same form as in the dilute case. 

Figure 61 presents the Q(Q)/Q2 relaxation rates, obtained from a fit with the dynamic structure factor of the Zimm model, as a function of Q. For both dilute solutions (c = 0.02 and c = 0.05) Q(Q) Q3 is found in the whole Q-range of the experiment. With increasing concentrations a transition from Q3 to... 

Fig. 70. NSE spectra for 2% linear h-PI in deuterated n-decane at Q/A 1 values of 0.064 ( ), 0.089 ( ) and 0.115 ( ). The solid lines represent a common fit with the dynamic structure factor of the Zimm model (see Table 1) neglecting possible effects of translational diffusion. (Reprinted with permission from [174]. Copyright 1993 The American Physical Society, Maryland)...

The Zimm model is a little more complex to evaluate but is essentially a sum of Maxwell models with a dependence on the sum of the modes as p-i/2 Thjs applies to all but the first few modes. A comparison between the two models is shown in Figure 5.24a and b. 

At low frequencies the loss modulus is linear in frequency and the storage modulus is quadratic for both models. As the frequency exceeds the reciprocal of the relaxation time ii the Rouse model approaches a square root dependence on frequency. The Zimm model varies as the 2/3rd power in frequency. At high frequencies there is some experimental evidence that suggests the storage modulus reaches a plateau value. The loss modulus has a linear dependence on frequency with a slope controlled by the solvent viscosity. Hearst and Tschoegl32 have both illustrated how a parameter h can be introduced into a bead spring... 

But p decreases with salt concentration with an apparent exponent of k which changes from 0 at low salt concentration to — at high salt concentrations. The N-independence of p arises from a cancellation between hydrodynamic interaction and electrostatic coupling between the polyelectrolyte and other ions in the solution. It is to be noted that the self-translational diffusion coefficient D is proportional to as in the Zimm model with full... 

The PDMS and the PIB chains consist of around 78 monomers rendering the usually applied long wavelength approximation of the Zimm model... 

Using the same Ansatz as for the Rouse model the scattering function S(Q,t) for the Zimm model simply emerges by replacing the above expressions for D and Tp(for 0-solvents) into the summation, analogous to (Eq. 3.19). In the limit... 

Note that h is proportional to n1/2 in 0-solvents, and thus to N112. For 0 = 0 the flow disturbance is zero, the chain is said to be free draining, and the original Rouse model is recovered. For hP, flow in the coil interior is presumed to be substantially reduced, the chain is frequently said to behave as an impenetrable coil, and the Zimm model is obtained. Equations (4.10-4.12) continue to apply for all values of h, although the distribution of relaxation times depends on h. Some results for the two limiting cases and large N are ... 

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... 

Although shear rate effects are more pronounced in good solvents, the intrinsic viscosity decreases with shear rate even in 0-solvents, where excluded volume is zero (317,318). The Zimm model employs the hydrodynamic interaction coefficients in the mean equilibrium configuration for all shear rates, despite the fact that the mean segment spacings change with coil deformation. Fixman has allowed the interaction matrix to vary in an appropriate way with coil deformation (334). The initial departure from [ ]0 was calculated by a perturbation scheme, and a decrease with increasing shear rate in 0-systems was predicted to take place in the vicinity of / = 1. 

In dilute solutions, hydrodynamic interactions between the monomers in the polymer chain are strong. These hydrodynamic interactions also are strong between the monomers and the solvent within the pervaded volume of the chain. When the polymer moves, it effectively drags the solvent within its pervaded volume with it. For this reason, the best model of polymer dynamics in a dilute solution is the Zimm model, which effectively treats the pervaded volume of the chain as a solid object moving through the surrounding solvent. 

From the Einstein relation [Eq. (8.4)] the diffusion coefficient of a chain in the Zimm model is reciprocally proportional to its size R ... 

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