Rouse’s model

Fig. 16.9 shows the low frequency slopes of 2 and 1, respectively, as expected for viscoelastic liquids and the high frequency slopes Vi and 2/3 for Rouse s and Zimm s models, respectively. Experimentally it appears that in general Zimm s model is in agreement with very dilute polymer solutions, and Rouse s model at moderately concentrated polymer solutions to polymer melts. An example is presented in Fig. 16.10. The solution of the high molecular weight polystyrene (III) behaves Rouse-like (free-draining), whereas the low molecular weight polystyrene with approximately the same concentration behaves Zimm-like (non-draining). The higher concentrated solution of this polymer illustrates a transition from Zimm-like to Rouse-like behaviour (non-draining nor free-draining, hence with intermediate hydrodynamic interaction).

Both authors find that rm is equal to the relaxation time of a submolecule and should be quite independent of molecular weight. From this point of view the agreement between theory and practice is quite good, particularly for 1000. Unfortunately the calculations have only been carried out for the case of dipoles parallel to the chain, and Rouse s model does not apply when the dipole is in the side-groups, as is the case for all polymers that have been studied in dilute solutions. 

For relatively short polymer chains (DP <200), Rouse s model (15) for coiling polymers in lute solutions, where De 1/M, dictated that the Thiele modulus be... 

In order to compute theoretical values of x it is necessary to know a priori the corresponding value of x. Mooney developed an extension of Rouse s model for a chain with fixed ends [9]. Relaxation spectra resulted coincident with Rouse s original model, except for an additional contribution to the modulus with an infinite relaxation time. The maximum relaxation time for a chain of Ne monomers fixed in both ends became ... 

The product g = Cy performs the desired diagonalization yielding A = M in contrast to the old guess that M equals A of Rouse s model without hydrodynamic interaction. The values of intrinsic viscosity at zero gradient and zero frej uency are not affected by this result. But the values at finite y or O) are modified as soon as internal viscosity is considered. 

The extension of Rouse s approach from linear chains to other polymer systems is quite straightforward and leads eventually to the concept of generalized Gaussian structures (GGS), which are the subject of this review. In the framework of the GGS approach, a polymer system is modeled as a collection of beads (subject to viscous friction), coimected to each other by means of elastic springs in a system-spedfic way. Initially, the GGS concept was inspired by the study of cross-linked polymer networks however, its applications have turned out to cover large classes of substances, such as dendritic polymers, hybrid polymers, and hierarchically-built structmes. 

Zimm s model (1956) is also a chain of beads connected by ideal springs. The chain consists of N identical segments joining + 1 identical beads with complete flexibility at each bead. Each segment, which is similar to a submolecule, is supposed to have a Gaussian probability function. The major difference between the two models lies in the interaction between the individual beads. In the Rouse model, such interaction is ignored in Zimm s model, such interaction is taken into consideration. 

The slow mode s properties are substantially the properties expected of frustration-limited clusters. Changing the control variable primarily changes the number of clusters but not their size. The slow mode indeed scales as at small q, but at least in many systems as q at large q. The longest relaxation times in S q, t) and in the viscoelastic relaxation spectrum are indeed about the same(48). Comparison might also be made between slow mode properties and the Rouse cryptocrystallite model(91). 

Measurements of using Pecora s analysis " " have been performed on different systems using either polarized or depolarized light-scattering. The experimental values of are in the region of the values predicted by the Rouse-Zimm model, but no molecular weight dependences are reported. 

The velocity gradient leads to an altered distribution of configuration. This distortion is in opposition to the thermal motions of the segments, which cause the configuration of the coil to drift towards the most probable distribution, i.e. the equilibrium s configurational distribution. Rouse derivations confirm that the motions of the macromolecule can be divided into (N-l) different modes, each associated with a characteristic relaxation time, iR p. In this case, a generalised Maxwell model is obtained with a discrete relaxation time distribution. 

An even more serious problem concerns the corresponding time scales on the most microscopic level, vibrations of bond lengths and bond angles have characteristic times of approx. rvib 10-13 s somewhat slower are the jumps over the barriers of the torsional potential (Fig. 1.3), which take place with a time constant of typically cj-1 10-11 s. On the semi-microscopic level, the time that a polymer coil needs to equilibrate its configuration is at least a factor of the order larger, where Np is the degree of polymerization, t = cj 1Np. This formula applies for the Rouse model [21,22], i. e., for non-... 

In the case of coherent scattering, which observes the pair-correlation function, interference from scattering waves emanating from various segments complicates the scattering function. Here, we shall explicitly calculate S(Q,t) for the Rouse model for the limiting cases (1) QRe -4 1 and (2) QRe > 1 where R2 = /2N is the end-to-end distance of the polymer chain. 

Figure 6 shows the measured dynamic structure factors for different momentum transfers. The solid lines display a fit with the dynamic structure factor of the Rouse model, where the time regime of the fit was restricted to the initial part. At short times the data are well represented by the solid lines, while at longer times deviations towards slower relaxations are obvious. As it will be pointed out later, this retardation results from the presence of entanglement constraints. Here, we focus on the initial decay of S(Q,t). The quality of the Rouse description of the initial decay is demonstrated in Fig. 7 where the Q-dependence of the characteristic decay rate R is displayed in a double logarithmic plot. The solid line displays the R Q4 law as given by Eq. (29). 

Like the dynamic structure factor for local reptation it develops a plateau region, the height of which depends on Qd. Figure 20 displays S(Q,t) as a function of the Rouse variable Q2/ 2X/Wt for different values of Qd. Clear deviations from the dynamic structure factor of the Rouse model can be seen even for Qd = 7. This aspect agrees well with computer simulations by Kremer et al. [54, 55] who found such deviations in the Q-regime 2.9 V Qd < 6.7. 

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 classical theory predicts values for the dynamic exponents of s = 0 and z = 3. Since s = 0, the viscosity diverges at most logarithmically at the gel point. Using Eq. 1-14, a relaxation exponent of n = 1 can be attributed to classical theory [34], Dynamic scaling based on percolation theory [34,40] does not yield unique results for the dynamic exponents as it does for the static exponents. Several models can be found that result in different values for n, s and z. These models use either Rouse and Zimm limits of hydrodynamic interactions or Electrical Network analogies. The following values were reported [34,39] (Rouse, no hydrodynamic interactions) n = 0.66, s = 1.35, and z = 2.7, (Zimm, hydrodynamic interactions accounted for) n = 1, s = 0, and z = 2.7, and (Electrical Network) n = 0.71, s = 0.75 and z = 1.94. 

Early-time motion, for segments s such that UgM(s)activated exploration of the original tube by the free end. In the absence of topological constraints along the contour, the end monomer moves by the classical non-Fickian diffusion of a Rouse chain, with spatial displacement f, but confined to the single dimension of the chain contour variable s. We therefore expect the early-time result for r(s) to scale as s. When all prefactors are calculated from the Rouse model [2] for Gaussian chains with local friction we find the form... 

Qualitatively the rubber-like and the generahzed Rouse model of Ronca arrive at similar results for S(.ham(Q>0 in the transition region. Quantitatively the Q-dis-persion of the plateau heights, however, is more pronounced in the Ronca model. 

Moreover, from Fig. 3.18 it is apparent that the model of des Cloizeaux also suffers from an incorrect Q-dependence of S(Q,f) in the plateau region, which is most apparent at the highest Q measured. It is important to note that the fits with the reptation model were done with only one free parameter, the entanglement distance d. The Rouse rate was determined earlier through NSE data taken for Kr. With this one free parameter, quantitative agreement over the whole range of Q and t using the reptation model with d=46.0 1.0 A was found. 

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