Viscosity dynamic structure factor

Comparing Eqs. (83), (84) and Eqs. (21), (22) it follows immediately that Rouse and Zimm relaxation result in completely different incoherent quasielastic scattering. These differences are revealed in the line shape of the dynamic structure factor or in the (3-parameter if Eq. (23) is applied, as well as in the structure and Q-dependence of the characteristic frequency. In the case of dominant hydrodynamic interaction, Q(Q) depends on the viscosity of the pure solvent, but on no molecular parameters and varies with the third power of Q, whereas with failing hydrodynamic interaction it is determined by the inverse of the friction per mean square segment length and varies with the fourth power of Q. 

The dynamics of highly diluted star polymers on the scale of segmental diffusion was first calculated by Zimm and Kilb [143] who presented the spectrum of eigenmodes as it is known for linear homopolymers in dilute solutions [see Eq. (77)]. This spectrum was used to calculate macroscopic transport properties, e.g. the intrinsic viscosity [145], However, explicit theoretical calculations of the dynamic structure factor [S(Q, t)] are still missing at present. Instead of this the method of first cumulant was applied to analyze the dynamic properties of such diluted star systems on microscopic scales. 

The incorporation of non-Gaussian effects in the Rouse theory can only be accomplished in an approximate way. For instance, the optimized Rouse-Zimm local dynamics approach has been applied by Guenza et al. [55] for linear and star chains. They were able to obtain correlation times and results related to dynamic light scattering experiments as the dynamic structure factor and its first cumulant [88]. A similar approach has also been applied by Ganazzoli et al. [87] for viscosity calculations. They obtained the generalized ZK results for ratio g already discussed. 

It is noteworthy that the neutron work in the merging region, which demonstrated the statistical independence of a- and j8-relaxations, also opened a new approach for a better understanding of results from dielectric spectroscopy on polymers. For the dielectric response such an approach was in fact proposed by G. Wilhams a long time ago [200] and only recently has been quantitatively tested [133,201-203]. As for the density fluctuations that are seen by the neutrons, it is assumed that the polarization is partially relaxed via local motions, which conform to the jS-relaxation. While the dipoles are participating in these motions, they are surrounded by temporary local environments. The decaying from these local environments is what we call the a-process. This causes the subsequent total relaxation of the polarization. Note that as the atoms in the density fluctuations, all dipoles participate at the same time in both relaxation processes. An important success of this attempt was its application to PB dielectric results [133] allowing the isolation of the a-relaxation contribution from that of the j0-processes in the dielectric response. Only in this way could the universality of the a-process be proven for dielectric results - the deduced temperature dependence of the timescale for the a-relaxation follows that observed for the structural relaxation (dynamic structure factor at Q ax) and also for the timescale associated with the viscosity (see Fig. 4.8). This feature remains masked if one identifies the main peak of the dielectric susceptibility with the a-relaxation. 

Fig. 5.10 Chain dynamic structure factor of PDMS (a) and PIB (b) in toluene solution at 327 K at the Q-values 0.04 A" (empty circle), 0.06 A (filled circle), 0.08 A" (empty square), 0.10 A (filled square), 0,15 A (empty diamond), 0,20 A (filled diamond), 0,25 A (empty triangle), 0.30 A" (filled triangle), 0.40 A" (plus). Solid lines correspond to fitting curves Rouse-Zimm model for PDMS and Rouse-Zimm with intrachain viscosity for PIB (see the text). (Reprinted with permission from [186]. Copyright 2001 American Chemical Society)...

RoLe has to be calculated for a chain with one end fixed to a surface). Fig-ure 6.15 displays a fit of the measured spectra at both temperatures with the complete dynamic structure factor where the Rouse relaxation rate was taken from an earlier experiment. Fit parameters were the surface tension and the effective local viscosity of the short labelled PEP segments. The data are well... 

B. Effect of the Dynamic Structure Factor on Friction and Viscosity... 

The above expression has been used by Leutheusser [34] and Kirkpatrick [30] in the study of liquid-glass transition. Leutheusser [34] has derived the expression of the dynamic structure factor from the nonlinear equation of motion for a damped oscillator. In their expression they refer to the memory kernel as the dynamic longitudinal viscosity. 

Mode coupling theory provides the following rationale for the known validity of the Stokes relation between the zero frequency friction and the viscosity. According to MCT, both these quantities are primarily determined by the static and dynamic structure factors of the solvent. Hence both vary similarly with density and temperature. This calls into question the justification of the use of the generalized hydrodynamics for molecular processes. The question gathers further relevance from the fact that the time (t) correlation function determining friction (the force-force) and that determining viscosity (the stress-stress) are microscopically different. 

An analysis of the relevant integrals shows that the dominant contribution of the density mode to the viscosity and the friction comes from intermediate length scale (8 > qa > 3). That is, more than 90% of the contribution comes from a region surrounding the sharp first peak of the static structure factor— that is, around qa = 2n. At these values of the wavenumber, the dynamic structure factor is well-determined by the following simple mean field expression ... 

The long-time tail in the solvent dynamic structure factor is responsible for the large value of the viscosity in supercooled liquid. The solute dynamics for smaller solutes are faster than that of the solvent. Thus, the solute dynamics is decoupled from this long-time tail of the solvent dynamic structure factor. 

When the bead-and-spring chain is not in the ideal state, the intramolecular force is given in Eqn. (3.1.3). As it may be seen, in general, the force is not simply transmitted by first-neighboring atoms, but it has a long-range character. The relaxation times are given by Eqn. (3.1.11) after they are known, the dynamic viscosity i (cu) and the atomic correlation function B(k, t) are obtained from Eqs. (3.1.15) and (3.1.18) (for the periodic chain), and the complex modulus and dynamic structure factors are easily constructed. 

MCT is a popular liquid viscosity theory (Gee, 1970 Gotze et al., 1981 Leutheuser, 1984, Jackie, 1989). It is based on the description of the dynamical properties of density fluctuations in terms of a dynamical structure factor. There are inherent density fluctuations in liquids, which decay with characteristic relaxation times. The decay becomes slower as the temperature is lowered due to increase of viscosity. It is controlled by dynamically correlated collisions. The equations governing the decay are non-linear. Analysis of the non-linear equation of motion of the density fluctuations gives a density correlation function of the type... 

Borsali et al. [73] started with the general Eq. (2.24) derived by Hess and Klein and evaluated the shear viscosity by replacing the dynamic structure factor by the mean fidd expression S(q,t) = exp[ — Dq t/S(q)] and assuming Gaussian chain statistics for the calculation of the form factor. Numerical calculations of the resulting integral show the peak position to vary as... 

Finally the book reaches properties that are determined by the collective properties of the dissolved polymers, including the dynamic structure factor, the polymer slow mode, the zero-shear viscosity, and linear and nonlinear viscoelasticity. Chapter 11 treats the dynamic structure factor S(q,t) of polymer solutions as... 

Simulation results for the structure factor in two-dimensions with A/a = 1.0 and collision angle a = 120°, and A/a = 0.1 with collision angle a = 60° are shown in Figs. 2a and 2b, respectively. The solid lines are the theoretical prediction for the dynamic structure factor (see (36) of [57]) using c = s/lk T/m and values for the transport coefficients obtained using the expressions in Table 1, assuming that the bulk viscosity 7 = 0. As can be seen, the agreement is excellent. 

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