Rouse modulus

Fig. 12 Rouse modulus, Gr filled triangles), elastic modulus, G a, at = 250 rad s open triangles), and the lifetime of hydrophobic associations in HM PAAm hydrogels formed in 0.24 M CTAB-SDS solutions filled circles), plotted against the SDS contemt. Frran [37] with permission from the Royal Society of Chemistry...

Note a crucial difference between this series and the one for the Rouse modulus, eq. 2.12. The weighting q in the summand means that even as r — 0 the first term represents 81% of the series (as opposed to 1/AT of the series for the Rouse series) so the relaxation looks, for practical purposes, like a single exponential. 

Table 3.5 Rouse Theory Expressions for the Modulus (entries labeled 1) and Compliances (entries labeled 2) for Tension and Shear Under Different Conditions ...

We observed above that the Rouse expression for the shear modulus is the same function as that written for a set of Maxwell elements, except that the summations are over all modes of vibration and the parameters are characteristic of the polymers and not springs and dashpots. Table 3.5 shows that this parallel extends throughout the moduli and compliances that we have discussed in this chapter. In Table 3.5 we observe the following ... 

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... 

We can consider the friction coefficient to be independent of the molecular weight. At times less than this or at a frequency greater than its reciprocal we expect the elasticity to have a frequency dependence similar to that of a Rouse chain in the high frequency limit. So for example for the storage modulus we get... 

Therefore in this Rouse regime of unentangled semidilute solutions where hydrodynamic interaction is screened, both the reduced viscosity and reduced modulus decrease with increase in polymer concentration in salt free solutions... 

The terminal spectrum is furnished by cooperative motions which extend beyond slow points on chain in the equivalent system. The modulus associated with the terminal relaxations is vEkT, which is smaller by a factor of two than the value from a shifted Rouse spectrum. It is consistent with a front factor g = j given by some recent theories of rubber elasticity (Part 7). The terminal spectrum for E 1 has the Rouse spacings for all practical purposes, shifted along the time axis by an undetermined multiplying factor (essentially the slow point friction coefficient). Thus, the model does not predict the terminal spectrum narrowing which is observed experimentally. 

The front factor g as defined above5 is unity in all the earlier theories (17). Recently Duiser and Staverman (233) have obtained g = j and Imai and Gordon (259) g — 0.54 with Rouse model theories which make no a priori assumptions about the junction point locations after deformation. Edwards (260) also arrives at and Freed (261) deduces that g= 1 is an upper bound by similar approaches. The front factor usually assumed in the shifted relaxation theory of the plateau modulus is g = 1, although Chompff and Duiser (232) obtain g = j through their extension of the Duiser-Staverman result to entanglement networks. The physical reasons for the different values of g in different treatments are not clear at present. 

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... 

When reptation is used to develop a description of the linear viscoelasticity of polymer melts [5, 6], the same underlying hypothesis ismade, and the same phenomenological parameter Ng appears. Basically, to describe the relaxation after a step strain, for example, each chain is assumed to first reorganise inside its deformed tube, with a Rouse-like dynamics, and then to slowly return to isotropy, relaxing the deformed tube by reptation (see the paper by Montfort et al in this book). Along these lines, the plateau relaxation modulus, the steady state compliance and the zero shear viscosity should be respectively ... 

This function corresponds to a Rouse spacing of relaxation times and gives a better fit of the experimental data than Eq. 3-10. Hence the relaxation modulus of the (B) process may be written as a function of the entanglement density N/Ne ... 

But, for large polydispersities, the Rouse process (B) of the long chains overlaps the reptation process of the short chains. Consequently, the most general expression of the relaxation function (or relaxation modulus) must include all the relaxation processes described in part 3.2. 

As the Rouse dynamics is assmned to be linear with respect to the MWD and that the A and HF processes are mass independent, we define the relaxation modulus of a polydisperse linear polymer by ... 

From dynamic experiments and applying the time temperature superposition principle, the complex shear modulus is measured over about five decades and the Rouse model can be checked extensively [37]. 

Figure 27 Experimental complex shear modulus of unentangled polystyrene (M 8 500 g.mol-i) compared to the Rouse model [37].

For weakly entangled monodisperse and polydisperse polymer melts, J. des Cloizeavuc [26] proposed a theory based on time-dependent diffusion and double reptation. He combines reptation and Rouse modes in an expression of the relaxation modulus where a fraction of the relaxation spectrum is transferred from the Rouse to the reptation modes. Furthermore, he introduces an intermediate time Xj, proportional to M2, which can be considered as the Rouse time of an entangled polymer movii in its tube. But, in the cross-over region, the best fit of the experimental data is obtained by replaced Xj by an empirical combination of... 

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