Zimm model relaxation times

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]... 

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. 

The equations of motion (75) can also be solved for polymers in good solvents. Averaging the Oseen tensor over the equilibrium segment distribution then gives = l/ n — m Y t 1 = p3v/rz and Dz kBT/r sNY are obtained for the relaxation times and the diffusion constant. The same relations as (80) and (82) follow as a function of the end-to-end distance with slightly altered numerical factors. In the same way, a solution of equations of motion (75), without any orientational averaging of the hydrodynamic field, merely leads to slightly modified numerical factors [35], In conclusion, Table 4 summarizes the essential assertions for the Zimm and Rouse model and compares them. 

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... 

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... 

Fig. 5.2 Mode number dependence of the relaxation times and T2 solid lines) found for PIB in dilute solution at 327 K. The dashed-dotted line shows the relaxation time of the Rouse-Zimm model. The horizontal dashed line displays the value of (Reprinted with permission from [186]. Copyright 2001 American Chemical Society)...

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 ... 

Lodge, A. S., Wu, Y.-J. Exact relaxation times and dynamic functions for dilute polymer solutions from the bead/spring model of Rouse and Zimm, Report 16. Rheology Research Center, University of Wisconsin (July, 1972). 

Let us add here some remarks on the normal stress difference. According to the Rouse-Zimm model [132,133] the first normal stress difference may be related to the storage modulus G. Taking into account only the longest relaxation time x, one gets... 

In an earlier section, we have shown that the viscoelastic behavior of homogeneous block copolymers can be treated by the modified Rouse-Bueche-Zimm model. In addition, the Time-Temperature Superposition Principle has also been found to be valid for these systems. However, if the block copolymer shows microphase separation, these conclusions no longer apply. The basic tenet of the Time-Temperature Superposition Principle is valid only if all of the relaxation mechanisms are affected by temperature in the same manner. Materials obeying this Principle are said to be thermorheologically simple. In other words, relaxation times at one temperature are related to the corresponding relaxation times at a reference temperature by a constant ratio (the shift factor). For... 

In Zimm s model the expression for the relaxation times is more complex the relaxation times ti are proportional to p 3/z in the non-draining case instead of to p 2 in the free-draining case and S z = 2.369. This leads to slopes of 2 and 1 at sufficiently low frequencies, whereas for sufficiently high frequencies... 

EXTENSIONAL FLOW. In steady extensional flows, such as uniaxial extension, the single-relaxation-time Hookean dumbbell model and the multiple-relaxation-time Rouse and Zimm models predict that the steady-state extensional viscosity becomes infinite at a finite strain rate, s. With the dumbbell model, this occurs when the frictional drag force that stretches the dumbbell exceeds the contraction-producing force of the spring—that is, when the extension rate equals the critical value Sc. ... 

The value of the stress relaxation modulus at the relaxation time G(x) is of the order of kT per chain in either the Rouse or Zimm models, just as the strands of a network in Chapter 7 stored of order kT of elastic energy ... 

Substituting the prediction for the relaxation time of the Zimm model Tz %J / kT) [Eq. (8.25)] into the expression for intrinsic viscosity [Eq. (8.34)] leads to the Zimm prediction for intrinsic viscosity ... 

Similar scaling analysis of the mode structure can be applied to the Zimm model. The relaxation time of the pth mode is of the order of the Zimm relaxation time of the chain containing Njp monomers [Eq. (8.25)] ... 

In 0-solvents 1/2), the stress relaxation modulus decays as the - 2/3 power of time, while in good solvents (i/ 0.588) G(t) decays approximately as the - 0.57 power of time. Like the stress relaxation modulus of the Rouse model [Eq. (8.47)], Eq. (8.63) crosses over from kT per monomer at the monomer relaxation time tq to kT per chain at the relaxation time of the chain tz TqN [Eq. (8.25)]. Once again, an excellent approximation to the stress relaxation modulus predicted by the Zimm... 

Consistent with the fact that the longest relaxation time of the Zimm model is shorter than the Rouse model, the subdiffusive monomer motion of the Zimm model [(Eq. (8.70)] is always faster than in the Rouse model [Eq. (8.58)] with the same monomer relaxation time tq. This is demonstrated in Fig. 8.8, where the mean-square monomer displacements predicted by the Rouse and Zimm models are compared. Each model exhibits subdiffusive motion on length scales smaller than the size of the chain, but motion becomes diffusive on larger scales, corresponding to times longer than the longest relaxation time. ... 

This longest bending mode matches the shortest relaxation time in the Rouse and Zimm models [the relaxation time of a monomer ro, Eq. (8.15)],... 

Both the Rouse and Zimm models, as well as other molecular models to be discussed in Chapter 9, tacitly assume that the relaxation time associated with each mode has the same temperature dependence. Each mode s relaxation time is the product of temperature-independent factors and the... 

During their relaxation time r, polymers diffuse a distance of order their own size (r R /D). The relaxation times of the Rouse and Zimm models are then easily obtained from the diffusion coefficients ... 

The time-dependent viscoelastic response of polymers is broken down into individual modes that relax on the scale of subsections of the chain with Njp monomers. The Rouse and Zimm models have different structure of their mode spectra, which translates into different power law exponents for the stress relaxation modulus G t) ... 

The longest mode relaxes at time r (tz for the Zimm model, with exponent K = 1 /(3i/) and tr for the Rouse model, with exponent k — 1 /2). While the difference between these exponents is small, they can be measured quite precisely, allowing unambiguous identification of Rouse and Zimm motion. 

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. 

The relaxation time 1 and the pol3mer contribution to the viscosity rjp depend strongly on the polymer molecular weight, concentration, and equihbrium conformation. Kinetic theory can be used to obtain scaling behavior for these quantities. At dilute concentrations, for example, the Zimm bead-spring model predicts the relaxation time as a function of the drag on polymer chain segments. 

The other model is the Rodriguez, Zakin, Patterson (RZP) correlation (6) which is f/fpv = function where is the Zimm first-order relaxation time given as M(n-Tio)c (0.586RT) for n and tIq solution and solvent viscosity in poise and c as concentration in g/dl. The group (tJT/D 2) is a modified Deborah number in which D (pipe diameter) to the 0.2... 

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... 

---