Viscosity zero-frequency

In the Smoluchowski limit, one usually assumes that the Stokes-Einstein relation (Dq//r7)a = C holds, which fonns the basis of taking the solvent viscosity as a measure for the zero-frequency friction coefficient appearing in Kramers expressions. Here C is a constant whose exact value depends on the type of boundary conditions used in deriving Stokes law. It follows that the diffiision coefficient ratio is given by ... 

To conclude this section it should be pointed out again that the friction coefficient has been considered to be frequency independent as implied in assuming a Markov process, and that zero-frequency friction as represented by solvent viscosity is an adequate parameter to describe the effect of friction on observed reaction rates. 

In order to do so, let us first write down the expressions for the two heat mode contributions and the density mode contribution to the zero frequency and zero wavenumber viscosity, rjsI1H and rjspp respectively ... 

Thus the zero-frequency longitudinal viscosity in the first order in X is given by... 

When Eq. (233) is compared with Eq. (230), the zero-frequency value of the longitudinal viscosity in first order is found to be larger than its zeroth-order value. This suggests that in every loop of the self-consistent calculation the zero-frequency longitudinal viscosity will increase, which might lead to a divergence of the zero-frequency value of rfo(z) and [Pg.134]

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. 

Further simplification can be made by using a simple prescription for the wavenumber dependence of the structure factor, as shown by Balucani et al. [78]. The above prescription provides fairly accurate values for the zero-frequency friction and the viscosity. 

It is already found that the decay of the normalized viscosity is slightly slower than that of the friction and the ratio of the time constants is 160/ 124. Thus, the ratio of the contribution from the bare part to the zero-frequency friction to that of the viscosity is equal to 23.6966 x 124/160, which is equal to 18.364. Therefore, the ratio of the bare part of the zero-frequency friction to that of viscosity is nearly identical to 6n. 

An elegant explanation for the unusual viscosity dependence was provided by the non-Markovian rate theory (NMRT) of Grote and Hynes [149] which incorporates the idea of frequency dependence of the friction. According to this theory the friction experienced by the reactive motion is not the zero frequency macroscopic friction (related to viscosity) but the friction at a finite frequency which itself depends on the barrier curvature. The rate is obtained by a self-consistent calculation involving the frequency-dependent friction. 

Equation (320) predicts the TST result for very weak friction (Ar to ) and predicts the Kramers result for low barrier frequency (i.e., (ob —> 0) so that (2r) can be replaced by (0) in Eq. (322). If die barrier frequency is large (ia>b > 1013 s 1) and the friction is not negligible ( (0)/fi — cob), then the situation is not so straightforward. In this regime, which often turns out to be the relevant one experimentally, the effective friction (2r) can be quite small even if the zero frequency (i.e., the macroscopic) friction (proportional to viscosity) is very large. The non-Markovian effects can play a very important role in this regime. 

For a viscoelastic solid (like an organogel), any rheological description should give a constant finite elastic modulus and infinite viscosity at zero frequency or long times. The situation is somewhat comparable to that of a cross-linked network [2. The equilibrium shear modulus for small deformations is proportional... 

Experimental detection of the gel point is not always easy since the equilibrium shear modulus is technically zero at the gel point and any applied stress will eventually relax, but only at infinite time. From the classical theory, the attributes of the gel point are an infinite steady-shear viscosity and a zero equilibrium modulus at zero frequency limit (Figure 6-3) (Flory, 1953). These criteria have been widely employed to detect the gel point of chemical gels. However, because continuous shearing affects gel formation, accurate information from viscosity measurement is not possible in the close vicinity of the gel point. Further, information regarding the transition itself could only be obtained by extrapolation, thereby introducing uncertainties in the determination of the gelation moment. 

These coefficients, along with the shear viscosity rj = cyii/y, often approach constant values at low shear rates these are called the zero-shear values, rjo, 4>i,o, and 4 2,o- Figure 1-9 shows for a polyethylene melt that the zero-shear constant values of rj r]o and 4 1 — 4/1,0 are approached at low shear rates. For a viscoelastic simple liquid with fading memory, the zero-shear values of the viscosity and first normal stress coefficient are related to the zero-frequency values of the dynamic moduli by... 

This equation suggests a procedure to obtain the real component of the complex viscosity from the compliance functions at zero frequency. By taking Eq. (8.29) into account, Eq. (8.27) can be rewritten as... 

This expression is similar to the one obtained for viscoelastic liquids [Eq. (8.25)], the only difference being that r is the zero shear rate viscosity in liquids while q (0) for viscoelastic solids represents the real component of the complex viscosity at zero frequency. 

For a large particle in a fluid at liquid densities, there are collective hydro-dynamic contributions to the solvent viscosity r, such that the Stokes-Einstein friction at zero frequency is In Section III.E the model is extended to yield the frequency-dependent friction. At high bath densities the model gives the results in terms of the force power spectrum of two and three center interactions and the frequency-dependent flux across the transition state, and at low bath densities the binary collisional friction discussed in Section III C and D is recovered. However, at sufficiently high frequencies, the binary collisional friction term is recovered. In Section III G the mass dependence of diffusion is studied, and the encounter theory at high density exhibits the weak mass dependence. 

Other zero-frequency transport coefficients (thermal conductivity, viscosity, etc.) may also be expressed as areas under time-correlation functions by use of the methods described in Chapter 11. 

The Brillouin linewidth Td) depends on the dynamic shear and volume viscosities r s(w) and t v(w). If the hypersonic shear viscosity r s(Aa)(i)) is equal to the zero frequency shear viscosity and the small term caused by thermal conductivity is neglected, then measurements of Td) can be used to obtain the volume viscosity. Champion and Jackson (8) noticed that the volume viscosities determined in the above manner for the n-alkanes were essentially independent of temperature. The values of r(i) measured in the authors laboratory for n-hexadecane are plotted... 

The dynamic viscosity, = G"([Pg.291]

Link 8. Some Results We began our quest fully expecting to derive the Vogel-Fulcher (VF) equation (40) along the lines of the Adam-Gibbs (AG) approach (41) and then extend the results to frequencies other than zero. Instead, much to our surprise we found that our logical extension of the equilibrium theory (what we believe to be a logical extension) results in a much different functional form for the zero frequency viscosity. Whereas the VF equation... 

We note that althoirgh the location of the glass transition in temperatirre-ptessure space is determined solely by the conflgurational entropy (equation lb) the zero frequency viscosity is determined solely by the configurational free energy F, (equation 7). 

The relative viscosity of a colloidal solution (ratio of the solution viscosity to the solvent viscosity) at zero frequency limit m -> 0 due to the inter-particle interactions is given by a numerical integral in Q space (Q = kcr) of the following ex[ ression. 

Therefore the zero frequency viscosities, T o, are determined for resp>ective p values as shown in Figure 4. The characteristic frequency of of can be defined by the intersection of Tjo with the power law of the critical gel. of divides between the gel and the liquid behaviours. The value of of decreases with approaching to the gel point. 

---