Rouse theory relaxation times

Thus the relaxation spectrum resulting from the average coordinates equation11 of our model has the same form as that of Rouse, of Kargin and Slonimiskii, or of Bueche. In order to relate the parameters of the model to those of the Rouse theory, the time scale factor a must somehow be connected to the frictional coefficient for a single subchain of a Rouse molecule. To achieve this comparison, we may23 study the translational diffusion coefficients as computed for the two models. 

It is convenient to use the time scale r, which is called the Rouse characteristic relaxation time and is a combination of parameters of the theory... 

According to the Rouse terminal relaxation time theory [14], a polymer solution always has a corresponding characteristic relaxation frequency Ochar)- Here, the codur is the (0.79 corresponding to the point of maximum curvature of the flow curve. Therefore the calculated M is believed to be the peak MW (Mp). Mp indicates the maximum probability of... 

In tube theories, strain hardening is produced by chain stretch, which is discussed in Section 11.3.2. For linear polymers in extensional flow, chain stretch requires that the strain rate exceed the inverse Rouse stretch relaxation time given by Eq. 11.1, i.e., / 5 k T) =... 

As in the case of the Rouse relaxation time, it is possible to express x2 in term of the intrinsic viscosity either by using Eq. (24) or the Kirkwood-Riseman theory [24, 47] ... 

In semi-dilute solutions, the Rouse theory fails to predict the relaxation time behaviour of the polymeric fluids. This fact is shown in Fig. 11 where the reduced viscosity is plotted against the product (y-AR). For correctly calculated values of A0 a satisfactory standardisation should be obtained independently of the molar mass and concentration of the sample. 

Sikorsky and Romiszowski [172,173] have recently presented a dynamic MC study of a three-arm star chain on a simple cubic lattice. The quadratic displacement of single beads was analyzed in this investigation. It essentially agrees with the predictions of the Rouse theory [21], with an initial t scale, followed by a broad crossover and a subsequent t dependence. The center of masses displacement yields the self-diffusion coefficient, compatible with the Rouse behavior, Eqs. (27) and (36). The time-correlation function of the end-to-end vector follows the expected dependence with chain length in the EV regime without HI consistent with the simulation model, i.e., the relaxation time is proportional to l i+2v The same scaling law is obtained for the correlation of the angle formed by two arms. Therefore, the model seems to reproduce adequately the main features for the dynamics of star chains, as expected from the Rouse theory. A sim-... 

Chompff and Duiser (232) analyzed the viscoelastic properties of an entanglement network somewhat similar to that envisioned by Parry et al. Theirs is the only molecular theory which predicts a spectrum for the plateau as well as the transition and terminal regions. Earlier Duiser and Staverman (233) had examined a system of four identical Rouse chains, each fixed in space at one end and joined together at the other. They showed that the relaxation times of this system are the same as if two of the chains were fixed in space at both ends and the remaining two were joined to form a single chain with fixed ends of twice the original size. 

In this respect, another insufficiency of Lodge s treatment is more serious, viz. the lack of specification of the relaxation times, which occur in his equations. In this connection, it is hoped that the present paper can contribute to a proper valuation of the ideas of Bueche (13), Ferry (14), and Peticolas (13). These authors adapted the dilute solution theory of Rouse (16) by introducing effective parameters, viz. an effective friction factor or an effective friction coefficient. The advantage of such a treatment is evident The set of relaxation times, explicitly given for the normal modes of motion of separate molecules in dilute solution, is also used for concentrated systems after the application of some modification. Experimental evidence for the validity of this procedure can, in principle, be obtained by comparing dynamic measurements, as obtained on dilute and concentrated systems. In the present report, flow birefringence measurements are used for the same purpose. 

In this case the theory, apart from the characteristic Rouse relaxation time r, contains three more parameters, namely the relaxation time r of the medium, the measure B of the increase in the resistance of the particle when it moves among the chains, and the measure of internal viscosity E associated with resistance to the deformation of the coil due to the present of ambient macromolecules. 

These are exactly the known results (Doi and Edwards 1986, p. 196). The time behaviour of the equilibrium correlation function is described by a formula which is identical to formula for a chain in viscous liquid (equation (4.34)), while the Rouse relaxation times are replaced by the reptation relaxation times. In fact, the chain in the Doi-Edwards theory is considered as a flexible rod, so that the distribution of relaxation times naturally can differ from that given by equation (4.36) the relaxation times can be close to the only disentanglement relaxation time r[ep. 

The Rouse model is the earliest and simplest molecular model that predicts a nontrivial distribution of polymer relaxation times. As described below, real polymeric liquids do in fact show many relaxation modes. However, in most polymer liquids, the relaxation modes observed do not correspond very well to the mode distribution predicted by the Rouse theory. For polymer solutions that are dilute, there are hydrodynamic interactions that affect the viscoelastic properties of the solution and that are unaccounted for in the Rouse theory. These are discussed below in Section 3.6.1.2. In most concentrated solutions or melts, entanglements between long polymer molecules greatly slow polymer relaxation, and, again, this is not accounted for in the Rouse theory. Reptation theories for entangled... 

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

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 power laws for viscoelastic spectra near the gel point presumably arise from the fractal scaling properties of gel clusters. Adolf and Martin (1990) have attempted to derive a value for the scaling exponent n from the universal scaling properties of percolation fractal aggregates near the gel point. Using Rouse theory for the dependence of the relaxation time on cluster molecular weight, they obtain n = D/ 2- - Df ) = 2/3, where Df = 2.5 is the fractal dimensionality of the clusters (see Table 5-1), and D = 3 is the dimensionality of... 

The relaxation time associated with the p mode (p > 1) is related to the largest relaxation time by the expression Xp = x /p. Thus the second mode, i2 = Xr/4, describes changes over distances of one-half the molecule, and so forth. Equation (11.13) suggests that is strongly dependent on molecular weight and temperature. The dependence of this latter parameter on temperature arises from the factor l/T and, especially, the enhancement caused in the molecular mobility (1 / o) by increase in temperature. Accordingly, the Rouse theory is in qualitative agreement with the experimental results. 

As shown in Figure 27c, the rotational relaxation time varies as a power of length, zr N with /u = 2.6 0.4. The experimental result is in remarkable agreement with the scaling behavior of the rotational relaxation zr°c A/1 l2v with theory = 2.5, which follows from the Rouse model and the Flory exponent v = 3/4.189... 

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. 

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...

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