Viscosity frequency dependence

In the case of electron transfer reactions, besides data on the dynamic Stokes shift and ultrafast laser spectroscopy, data on the dielectric dispersion (w) of the solvent can provide invaluable supplementary information. In the case of other reactions, such as isomerizations, it appears that the analogous data, for example, on a solvent viscosity frequency dependence 17 ( ), or on a dynamic Stokes fluorescence shift may presently be absent. Its absence probably provides one main source of the differences in opinion [5, 40-43] on solvent dynamics treatments of isomerization. 

During dynamic measurements frequency dependences of the components of a complex modulus G or dynamic viscosity T (r = G"/es) are determined. Due to the existence of a well-known analogy between the functions r(y) or G"(co) as well as between G and normal stresses at shear flow a, seemingly, we may expect that dynamic measurements in principle will give the same information as measurements of the flow curve [1],... 

Moreover, if for pure polymer melts the correlation of the behavior of the functions ri (co) andrify) under the condition of comparing as y takes place, as a general rule, but for filled polymers such correlation vanishes. Therefore the results of measuring frequency dependences of a dynamic modulus or dynamic viscosity should not be compared with the behavior of the material during steady flow. 

When bounding walls exist, the particles confined within them not only collide with each other, but also collide with the walls. With the decrease of wall spacing, the frequency of particle-particle collisions will decrease, while the particle-wall collision frequency will increase. This can be demonstrated by calculation of collisions of particles in two parallel plates with the DSMC method. In Fig. 5 the result of such a simulation is shown. In the simulation [18], 2,000 representative nitrogen gas molecules with 50 cells were employed. Other parameters used here were viscosity /r= 1.656 X 10 Pa-s, molecular mass m =4.65 X 10 kg, and the ambient temperature 7 ref=273 K. Instead of the hard-sphere (HS) model, the variable hard-sphere (VHS) model was adopted in the simulation, which gives a better prediction of the viscosity-temperature dependence than the HS model. For the VHS model, the mean free path becomes ... 

The dramatic slowing down of molecular motions is seen explicitly in a vast area of different probes of liquid local structures. Slow motion is evident in viscosity, dielectric relaxation, frequency-dependent ionic conductance, and in the speed of crystallization itself. In all cases, the temperature dependence of the generic relaxation time obeys to a reasonable, but not perfect, approximation the empirical Vogel-Fulcher law ... 

Some information concerning the intramolecular relaxation of the hyperbranched polymers can be obtained from an analysis of the viscoelastic characteristics within the range between the segmental and the terminal relaxation times. In contrast to the behavior of melts with linear chains, in the case of hyperbranched polymers, the range between the distinguished local and terminal relaxations can be characterized by the values of G and G" changing nearly in parallel and by the viscosity variation having a frequency with a considerably different exponent 0. This can be considered as an indication of the extremely broad spectrum of internal relaxations in these macromolecules. To illustrate this effect, the frequency dependences of the complex viscosities for both linear... 

Fig. 13. Frequency dependences of the complex viscosity, q, for melts of linear and branched PMM A with different M values. (Reproduced with permission from [88]. Copyright 2001 American Chemical Society.)...

Fig. 9.9 Reduced frequency dependence of storage modulus, loss modulus and complex viscosity of neat PLA and various nanocomposites (PLANCs). Reprinted from [40], 2003, Elsevier Science.

The term S represents the strength of the network. The power law exponent m was found to depend on the stochiometric ratio r of crosslinker to sites. When they were in balance, i.e. r = 1, then m - 1/2. From Equations (5.140) and (5.141) this is the only condition where G (co) = G (cd) over all frequencies where the power law equation applies. If the stochiometry was varied the gel point was frequency dependent. This was also found to be the case for poly(urethane) networks. A microstructural origin has been suggested by both Cates and Muthumkumar38 in terms of a fractal cluster with dimension D (Section 6.3.5). The complex viscosity was found to depend as ... 

A particular question of interest is whether the DNA torsional motions observed on the nanosecond time scale are overdamped, as predicted by simple Langevin theory, and as observed for Brownian motions on longer time scales, or instead are underdamped, so that damped oscillations appear in the observed correlation functions. A related question is whether the solvent water around the DNA exhibits a normal constant viscosity on the nanosecond time scale, or instead begins to exhibit viscoelastic behavior with a time-, or frequency-, dependent complex viscosity. In brief, are the predictions for... 

Frequency dependent complex impedance measurements made over many decades of frequency provide a sensitive and convenient means for monitoring the cure process in thermosets and thermoplastics [1-4]. They are of particular importance for quality control monitoring of cure in complex resin systems because the measurement of dielectric relaxation is one of only a few instrumental techniques available for studying molecular properties in both the liquid and solid states. Furthermore, It is one of the few experimental techniques available for studying the poljfmerization process of going from a monomeric liquid of varying viscosity to a crosslinked. Insoluble, high temperature solid. 

The magnitude of the ionic mobility c and the rotational mobility of the dipole t depends on the extent of the reaction and the physical state of the material (5). As such, c and t determined from the frequency dependence of (u), provide two molecular probes for monitoring the reaction advancement and the viscosity during cure. 

The correlation during cufe of -log a with log 17 (viscosity) measured at 10 radians/sec and the ability to use frequency dependent (<<)) measurements to determine a, thereby accurately... 

Figure 4.13 The viscosity of the thick laminate as determined from the frequency dependence of the FDEMS sensors at the surface, thirty-second, sixty-fourth, and center plies...

Here pg and p f are the mass densities of the gel and the solvent, respectively, K is a bulk modulus, c0 is the speed of sound, and i s is the solvent shear viscosity. The solvent bulk viscosity has been neglected. The terms proportional to / arise from an elastic coupling in the free energy between the density deviation of gel and that of solvent The p in Eq. (6.1) coincides with the shear modulus of gels treated so far. We neglect the frequency-dependence of the elastic moduli. It can be important in dynamic light scattering, however, as will be discussed in the next section. 

Fig. 2.17 Frequency dependence of dynamic shear moduli computed using a model for the linear viscoelasticity of a cubic phase based on slip planes, introduced by Jones and McLeish (1995). Dashed line G, solid line G".The bulk modulus is chosen to be G — 105 (arb. units). The calculation is for a slip plane density AT1 - 10 5 and a viscosity ratio rh = = 1, where rjs is the slip plane viscosity and t] is the bulk viscosity. The strain...

B. Frequency Dependence of Viscosity Comparison with Maxwell Relation... 

The relation between friction and viscosity goes beyond the Stokes relation. The Navier-Stokes hydrodynamics has been generalized by Zwanzig and Bixon [23] to include the viscoelastic response of the medium. This generalization provides an elegant expression for the frequency-dependent friction which depends among other things on the frequency-dependent bulk and shear viscosities and sound velocity. 

A further motivation of this study comes from the following observations. Many chemical dynamic processes, such as nonpolar solvation dynamics [70], can be described in terms of the frequency-dependent viscosity because... 

An attempt has been made to answer the following questions. What is the relation between r)s(t) and (r) at short times Does the ratio between the two retain a Stokes-like value at all times And how does the relation behave as a function of frequency The analysis seems to suggest that if one includes only the binary interaction in the calculation of the time scale of the short-time dynamics, both viscosity and friction exhibits nearly the same time scale. When the triplet dynamics is included, both the responses become slower with the viscosity being affected more than the friction. The time scale of both the responses axe of the order of 100 fs. It is shown that both the frequency-dependent viscosity and the friction exhibit a clear bimodal dynamics. 

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