Rouse model theoretical results

Over the entire Q-range within experimental error the data points fall on the line and thus exhibit the predicted Q4 dependence. The insert in Fig. 7 demonstrates the scaling behavior of the experimental spectra which, according to the Rouse model, are required to collapse to one master curve if they are plotted in terms of the Rouse variable u = QV2 /wt. The solid line displays the result of a joint fit to the Rouse structure factor with the only parameter fit being the Rouse rate W 4. Excellent agreement with the theoretical prediction is observed. The resulting value is W/4 = 2.0 + 0.1 x 1013 A4s 1. 

The elastic dumbbell model studied iu Chapter 6 is both structurally and djmamicaUy too simple for a poljmier. However, the derivation of its constitutive equation illustrates the main theoretical steps involved. In this chapter we shall apply these theoretical results to a Gaussian chain (or Rouse chain) containing many bead-spring segments (Rouse segments). First we obtain the Smoluchowski equation for the bond vectors. After transforming to the normal coordinates, the Smoluchowski equation for each normal mode is equivalent in form to the equation for the elastic dumbbell. 

Based on the fluctuation-dissipation theorem, the equilibrium-simulated Gs t) is predicted to be equivalent to the step strain-simulated Gs t) in the linear region. In Fig. 16.2, the equilibrium-simulated Gs(t) curves for two-bead, five-bead and ten-bead Rouse chains are also shown. These equilibrium-simulated Gs(t) results are in perfect agreement with the step strain-simulated results and the Rouse theoretical curves, illustrating the fluctuation-dissipation theorem as applied to the Rouse model and confirming the validity of the Monte Carlo simulations. 

Analysis of the Rouse Modes. In order to test the applicability of the Rouse model we calculated the basic quantities, the Rouse modes, and compared the simulation results with the theoretical predictions. The Rouse modes are defined as the cosine transforms of the position vectors, Vn, to the monomers. For the discrete polymer model under consideration they can be written as (66)... 

The KSR and Rouse models were subj ected to numerous experimental tests. A reasonably good agreement between the theoretical predictions and experimental data was demonstrated for a variety of dilute polymeric solutions. Further advance in the molecular-kinetic approach to description of relaxation processes in polymeric systems have brought about more sophisticated models. They improve the classical results by taking into account additional factors and/or considering diverse frequency, temperature, and concentration ranges, etc. For the aims of computer simulation of the polymeric liquid dynamics in hydrodynamic problems, either simple approximations of the spectrum, Fi(A), or the model of subchains are usually used. Spriggs law is the most used approximation... 

The exact results of Equations 7.28, 7.31, and 7.32 for the Rouse model are in disagreement with the general expectations of Section 7.2.1 and are never observed experimentally in dilute solutions of polymers, where the model was intended. The reason for this discrepancy is the absence of hydrodynamic coupling between the segments in the Rouse model. Nevertheless, due to the general complexity of the problem of polymer dynamics, there have been considerable activities in simulating the dynamics of polymer chains in silico, usually without hydrodynamic interactions. For this artificial situation of a Rouse chain (i.e., without hydrodynamic interaction) with the size exponent v, the various theoretical results can be summarized as... 

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... 

Thus we see that the N dependence of Dg changes from the Rouse type Dg 1/iV to the reptation type Dg 1/iV as the chain length increases and crosses over rie. However, it is too early to conclude with this result that the premise of the reptation idea has been justified theoretically, because Skolnick et al. [ 14] have shown that an equation similar to eq 2.39 can be obtained from a different dynamic model, as explained below. 

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 tube model forms the basis for detailed theoretical approaches to the dynamics of an entangled polymer melt pioneered by Doi and Edwards [13, 14]. The present discussion will be restricted to a simple description of some of its basic features. Consider, for example, the relaxation behavior of a chain subjected to a step shear strain at t = 0. As shown in Figure 14.11, the applied strain results in a deformation of the tube, and hence of the chain trapped inside it. The first relaxation occurs within the tube at times t < Te, where is the Rouse relaxation time for a chain with Ne = ajVf statistical segments, and hence a mean square end-to-end distance equal to a. At t > t, the constraints due to the tube begin to dominate and the only way the stress can relax further relax is for the chain to escape the deformed tube and re-establish a random (unperturbed) conformation. As sketched in Figure 14.11, it achieves this through Brownian motion backward and... 

Before embarking on a discussion of the results of these studies let us add one historical note. The difficulty with swinging the polymer tails in a conformational transition has been recognized for many years. A means of circumventing was proposed by Schatzki. Verdier and Stockmayer had earlier invoked a similar principle but used it only to produce Rouse modes. We know now that slow Rouse modes are insensitive to the details of the faster time-scale dynamics. The proposed motions are completely local, and involve going from one equilibrium rotational isomeric state to another by moving only a finite, small number of atoms. Mechanisms of this class have come to be known as crankshaft motions (a term applicable in the strictest sense only to the Schatzki proposal). Because of the limited amount of motion and the simplicity of the dynamics these models are easy to understand, analyze, and simulate. This probably contributes to the continued attention devoted to them. The crankshaft idea has helped to focus attention on the necessity to localize the motion associated with conformational transitions, but complete localization is too restrictive. There are theoretical objections that can be raised to the crankshaft mechanism, but the bottom line is that no signs of it are found in our simulations. 

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