Rouse-model viscosity

Within the Rouse model for polymer dynamics the viscosity of a melt can be calculated from the diffusion constant of the chains using the relation [22,29,30] ... 

In summary, the chain dynamics for short times, where entanglement effects do not yet play a role, are excellently described by the picture of Langevin dynamics with entropic restoring forces. The Rouse model quantitatively describes (1) the Q-dependence of the characteristic relaxation rate, (2) the spectral form of both the self- and the pair correlation, and (3) it establishes the correct relation to the macroscopic viscosity. 

In addition to the Rouse model, the Hess theory contains two further parameters the critical monomer number Nc and the relative strength of the entanglement friction A (0)/ . Furthermore, the change in the monomeric friction coefficient with molecular mass has to be taken into account. Using results for (M) from viscosity data [47], Fig. 16 displays the results of the data fitting, varying only the two model parameters Nc and A (0)/ for the samples with the molecular masses Mw = 3600 and Mw = 6500 g/mol. 

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... 

The description of the chain dynamics in terms of the Rouse model is not only limited by local stiffness effects but also by local dissipative relaxation processes like jumps over the barrier in the rotational potential. Thus, in order to extend the range of description, a combination of the modified Rouse model with a simple description of the rotational jump processes is asked for. Allegra et al. [213,214] introduced an internal viscosity as a force which arises due to a transient departure from configurational equilibrium, that relaxes by reorientational jumps. Thereby, the rotational relaxation processes are described by one single relaxation rate Tj. From an expression for the difference in free energy due to small excursions from equilibrium an explicit expression for the internal viscosity force in terms of a memory function is derived. The internal viscosity force acting on the k-th backbone atom becomes ... 

In Fig. 6.22 the results of a viscosity scahng by f— fxT/rj (T) of the relaxation data are shown. Such a scaling is motivated by the Rouse model and should hold for the a-relaxation. The pure PPO data (right) behave according to this expectation in contrast the PP0-IiC104 curves deviate considerably. This indicates that the coupling factor between microscopic friction and viscosity depends on temperature, possibly due to transient cross-linking via Li-ions. 

The recoverable compliance Je° is very sensitive to molecular weight distribution, especially to the tail of the distribution at high molecular weights. According to the Rouse model [Eq.(4.28)], when samples with the same Mw are compared, their compliances should vary as MZMZ+ JM. Based on the success of the Rouse model mixing law for viscosity, one might hope for correlations of the form ... 

The viscosity is thus directly proportional to chain length, as in the Rouse model. The authors argue however in favor of an additional factor proportional to 1/3 because the distance between molecular centers in the direction of the velocity gradient, k2, increases by this factor as n increases. Their result is therefore... 

It may be surprising that the effect of the nearest-neighbor bond correlations on the one-dimensional chain depends on the sign of P i.e., that the spectrum broadens when extended conformations are favored and narrows when compact conformations are favored. No simple qualitative explanation of this result has occurred to us. The usual internal viscosity always produces a narrowing of the spectrum. This effect is easily introduced into a one-dimensional Rouse model an internal viscous force is... 

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... 

A more quantitative interpretation of the scattering data involves molecular models. As an example, we turned to the Rouse model [20] developed for solutions and extended to melts [21]. As this model is unable to account for the molecular weight dependence of zero-shear viscosity (t o M ) above the critical molecular weight (Mc 35 000 for PS), the analysis wall be extended as a next step to other models which are more realistic for entangled systems. A basic result of the Rouse model relates the monomeric friction coefiBcient Co and the zero-shear viscosity t o ... 

Because of hydrodynamic interactions, the Rouse model fails in its prediction of the scaling of the low-shear viscosity, rjo, with molecular weight. Note from Eqs. (3-41) and (3-42a) that for the Rouse model we obtain... 

Fig. 6. Comparisons of viscosity and recoverable compliance predictions by the Doi-Edwards theory with experimental observations. The predicted tio h too large, but its chmn length dependence is slightly weaker than observed. The predicted J is too small, but independent of chain length as observed. The dashed lines indicate predictions of the Rouse model...

The monomer relaxation time Tq and the chain relaxation time of the Rouse model tr can be rewritten in terms of the solvent viscosity 7] ... 

The Rouse model predicts that the intrinsic viscosity in a -solvent is proportional to molar mass. However, the Rouse model assumes no... 

Compare this Zimm diffusion coefficient Dz with the Rouse diffusion coefficient Dr of part (ii). Hint. The viscosity of an unentangled melt of shorter /Vg-chains is predicted by the Rouse model [Eq. (8.53)]. 

The viscosity of a polymer melt is predicted to be proportional to molar mass for unentangled melts (the Rouse model) and proportional to the cube of molar mass for entangled melts (the reptation model). 

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

Imai (56) constructed a theory fear the intrinsic viscosity and sedimentation constant of ring polymers using the Fixman method shown in 2.3.2 for a model similar to the Hearst-Harris model which will be described in 4.2.2. This model reduces to the Rouse model in a limiting case. The excluded volume potential is included in the form of Eq. (2.26) and the same type of calculation as described in 2.3.2 was performed for a steady shear flow. Dynamic mechanical properties were not treated, although the extension to include this case is only a matter of tedious calculations. 

The shear viscosity of the Rouse model can be obtained by substituting Eq. (7.58) into Eq. (4.30) or, as Eq. (6.70) was obtained, it can be calculated by substituting the steady-state shear deformation into Eq. (7.55). These two different calculations give the same result ... 

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