Rouse model relaxation modes

The relaxation time of a monomer, tq [Eq. (8.15)] is the shortest relaxation time of the Rouse model, with mode index p = N, making xjyi = TQ. The mode with index p = 1 is the longest relaxation mode of the chain with relaxation time equal to the Rouse time ti tr, and corresponds... 

The purpose of these comparisons is simply to point out how complete the parallel is between the Rouse molecular model and the mechanical models we discussed earlier. While the summations in the stress relaxation and creep expressions were included to give better agreement with experiment, the summations in the Rouse theory arise naturally from a consideration of different modes of vibration. It should be noted that all of these modes are overtones of the same fundamental and do not arise from considering different relaxation processes. As we have noted before, different types of encumbrance have different effects on the displacement of the molecules. The mechanical models correct for this in a way the simple Rouse model does not. Allowing for more than one value of f, along the lines of Example 3.7, is one of the ways the Rouse theory has been modified to generate two sets of Tp values. The results of this development are comparable to summing multiple effects in the mechanical models. In all cases the more elaborate expressions describe experimental results better. 

The earliest and simplest approach in this direction starts from Langevin equations with solutions comprising a spectrum of relaxation modes [1-4], Special features are the incorporation of entropic forces (Rouse model, [6]) which relax fluctuations of reduced entropy, and of hydrodynamic interactions (Zimm model, [7]) which couple segmental motions via long-range backflow fields in polymer solutions, and the inclusion of topological constraints or entanglements (reptation or tube model, [8-10]) which are mutually imposed within a dense ensemble of chains. 

Fig. 5.1 Mode number dependence of the relaxation times Tj and T2 (solid lines). The dashed-dotted line shows the relaxation time ip in the Rouse model (Eq. 3.12). The horizontal dashed line displays the value of r. The dashed and the dotted lines represent the relaxation time when the influence of the chain stiffness is considered mode description of the chain statistics iq (dashed, Eq. 5.11) and bending force model tp (dotted, Eq. 5.7). The behaviour of the relaxation time used in the phenomenological description is also shown for the lowest modes (see text). (Reprinted with permission from [217]. Copyright 1999 American Institute of Physics)...

Note that another origin of the slowing down of the chain relaxation compared to the Rouse prediction could be a reduction of the weights of the higher modes, which in the Rouse model are proportional top (see Eq. 3.19). 

Thus, the remaining difference from the Rouse model is a mode-dependent friction coefficient p=(Hpp) for (p>0), which leads to a relaxation mode spectrum with a different mode munber p-dependence. The second term in Hpq is the bead friction with the surrounding me um (solvent), which is the only term present in the Rouse model. The ratio of the diagonal (Rouse-like) friction and the solvent-mediated interaction strength may be expressed by the draining parameter The Rouse model has B=0, whereas the assumption... 

Furthermore, it may be seen that for all the normal modes of relaxation, including the most rapid, the freely jointed chain model and the Rouse model are identical if we set n = N + 1 that is, the relaxation time xp of the pth normal mode of a freely-jointed chain is the same as that of a Rouse marcromolecule composed of N + 1 subchains, each of mean square end-to-end length b2. Moreover, for the special choice a = 0, Eq. (10) is true for arbitrarily large departures from equilibrium. We thus seem to have confirmed analytically the discovery of Verdier24 that quite short chains executing a stochastic process described by Eqs. (1) and (3) on a simple cubic lattice display Rouse relaxation behavior. Of course, Verdier s Monte Carlo technique permits study of excluded volume effects, quite beyond the range of our present efforts. 

Adachi K, Kotaka T (1993) Dielectric normal mode relaxation. Prog Polym Sci 18 585—622 Adelman SA, Freed KF (1977) Microscopic theory of polymer internal viscosity Mode coupling approximation for the Rouse model. J Chem Phys 67(4) 1380-1393 Aharoni SM (1983) On entanglements of flexible and rodlike polymers. Macromolecules 16(11) 1722-1728... 

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... 

The model exhibits the expected Rouse relaxation modes at high temperature. At lower temperature, the chain motions can still be decomposed into Rouse normal modes, but the normal modes no longer relax via single exponential decays. Instead, the decay of each mode is described by a stretched exponential, and the stretching increases decreases) as the mode number increases. In addition, the temperature-dependence of the relaxation rates is described by the VFTH expression. The emergence of these features of real glasses in such a simple model suggests that such features are insensitive to molecular details. 

This expression effectively interpolates between a modulus level of order kT per monomer at the shortest Rouse mode t ro) to a modulus level of order kT per chain at the longest Rouse mode (/ = tr xqN ) using a power law. We already know that the stress relaxation modulus has an exponential decay beyond its longest relaxation time [Eq. (7.112)]. Therefore, an approximate description of the stress relaxation modulus of the Rouse model is the product of [Eq. (8.47)] and an exponential cutoff ... 

For high frequencies uj > 1 /tq, there are no relaxation modes in the Rouse model. The storage modulus becomes independent of frequency, and equal to the short time stress relaxation modulus, which is kT per monomer G uj) (pkTjb. This high-frequency saturation is not included in Eqs (8.49) and (8.50). At low frequencies a < 1/tr, the storage modulus is proportional to the square of frequency and the loss modulus is pro-portional to frequency, as is the case for the terminal response of any viscoelastic liquid. 

The stress relaxation times Xp of the Rouse model are half of the correlation times of normal modes (see Problem 8.36). 

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. 

Calculate the stress relaxation modulus of the Rouse model (Eq. 8.55) by showing that after a small step shear strain 7 at time t 0 the correlation function of normal modes decays as Xpx t)X y(t)) — i kTjkp) exp (- tjxp). 

The Rouse model (Rouse, 1953) extends these theories to multiple beads and springs (or multiple-relaxation modes). Here the expression for the viscosity becomes... 

Hermans and Van Beek626 have recently used the new model of polymer molecules suggested by Rouse.46 At high frequencies the whole molecule cannot follow the field so it is divided into a number of submolecules small enough to follow the field and yet sufficiently large to have a Gaussian distribution. Dielectric relaxation for the case of dipoles parallel to the chain has been calculated by Founder sum transforms. The distribution of relaxation modes appears though the multiplicity of the mathematical solution for the diffusion equation. 

As has already been said and can be observed in Figures 12.5 and 12.7, results deviate from Maxwellian behavior at high frequencies. An upturn is observed in G", which has been ahributed in other systems to a transihon of the relaxation mode from reptation-scission (Cates model) to breathing or Rouse modes [28, 31]. In the systems presented in this chapter, if the relaxation mechanisms were just living reptation at long times -I- Rouse relaxations at short ones, results should be fitted by Equations 12.3 and 12.4, where a Rouse relaxation mode has been added to the Maxwell model, subscripts M and R referring to Maxwell and Rouse relaxations, respectively ... 

Fig. 6. The vector mode relaxation rates as a function of k/(N f 1). Results shown are from BD simulations of a 16 carbon alkane chain with the indicated torsional barrier height. The line indicates the prediction of the Rouse model. (Reproduced from [27] with permission)...

Results of calculations for the relaxation rates v are shown by the empty circles in Fig. 8 as a function of 1/n [10]. Unlike the predictions of the Rouse model, a linear dependence between the relaxation rates and the size of the kinetic unit is observed. The filled circles are from the BD simulations of Fixman [99,100] for a polyethylene-like chain. Dependence of Vkk on the mode number for a 16 bond polyethylene chain are shown by the empty circles in Fig. 9. The filled circles are from the BD simulations of Fixman [99, 100] for a similar chain. The curve is the best fitting curve through the empty circles. It is interesting to note that both the DRIS approach and Fixman s simulations yield a plateau value for the rate of most of the relaxational modes, while a few slowest modes exhibit distinct lower values. 

The classical picture is based on the notion of relaxation modes for one chain. It first appeared in a 1953 paper by P. E. Rous and was based on the following model ... 

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