Friction 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 this nonadiabatic limit, the transmission coefficient is determined, via (2.8) by the ratio of the nonadiabatic and equilibrium barrier frequencies, and is in full agreement with the MD results [5a-5c]. (By contrast, the Kramers theory prediction based on the zero frequency friction constant is far too low. Recall that we emphasized for example the importance of the tail to the full time area of the SN2 (t). In the language of (3.14), the solvation time xs is not directly relevant in determining... 

Nonetheless, there can be nonnegligible atmosphere frictional reduction of the rate constant when the electrolyte concentration is sufficiently high that the coupling force magnitude is important [51] (this reduction requires GH theory for its description the zero frequency friction Kramers prediction can give an order of magnitude too large a reduction in the rate). 

The question is What is the appropriate friction factor for this zero frequency measurement If we take / = 6n 7]0a as in the RB development (a = a m — mass of gaussian segment) ... 

From the above discussion, it is obvious that the mode coupling theory calculations are quite involved and numerically formidable. Balucani et al. [16] have made some simple approximations to incorporate the self-consistency between the self-dynamic structure factor and the friction. This required the knowledge of only the zero frequency friction. The full self-consistent calculation is more elaborate and will be discussed later in this chapter. 

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. 

When the integration is performed, it is found that u%n/x oc 1 thus TC ffi2- From Eq. (115) the binary friction at zero frequency can be obtained as... 

The final expression for the zero-frequency friction for a large solute is given by... 

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. 

The solid line corresponds to the zero-frequency friction (Kramers result).57... 

Early treatments focused on a classical overdamped solvent model (zero-frequency friction) [13b, 14]. In the NA and WA cases, where initial- and final-state wells are sharply defined, separated by a narrow cusp-like barrier, one may adopt a steady-state model of the type displayed in Eq. 7,... 

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. 

In Section IV A the model is applied to diatomic dissociation on a Morse potential in the low-friction limit. This models diatomic dissociation in a low-density gas in which the collisional excitation is impulsive, being modeled by a zero-frequency friction, that is, the duration of the collision is assumed to be small relative to all other time scales. The objective of the section is to test whether the reduction to a one-dimensional effective potential P (l ) leads to an accurate formula for the dissociation rate. This is done by comparison... 

In the previous sections a model of the frequency-dependent collisional friction has been derived. Because the zero-frequency friction for a spherical particle in a dense fluid is well modeled by the Stokes-Einstein result, even for particles of similar size as the bath particles, there has been considerable interest in generalizing the hydrodynamic approach used to derive this result into the frequency domain in order to derive a frequency-dependent friction that takes into account collective bath motions. The theory of Zwanzig and Bixon, corrected by Metiu, Oxtoby, and Freed, has been invoked to explain deviation from the Kramers theory for unimolec-ular chemical reactions. The hydrodynamic friction can be used as input in the Grote-Hynes theory [Eq. (2.35)] to determine the reactive frequency and hence the barrier crossing rate of the molecular reaction. However, the use of sharp boundary conditions leads to an unphysical nonzero high-frequency limit to Ib(s). which compromises its utility. 

For a diatomic close to thermal equilibrium the assumption of impulsive collisions with the bath molecules is realistic only when the vibrational jjeriod is longer than the duration of a collision, but this is rarely the case for diatomics vibrating near the potential minimum. However, for realistic potentials the vibrational period lengthens near the dissociation threshold, and it is not so clear that an impulsive model will be quantitatively inaccurate in modeling dissociation, even though it may fail badly in describing vibrational relaxation deep with the well. A stochastic impulsive model of dissociation of a diatomic AB, which uses the zero-frequency frictions describ-... 

On the other hand is comparable to solvent structural relaxation times and is shorter than solvent reorientation and conformational relaxation times. In this case the effective friction for motion in the vicinity of Rj will be less than the zero frequency friction which describes diffusion for R > R. ... 

Thus the model predicts that the adiabatic effect is to reduce the energy relaxation rate by a factor of 4. Using the estimates of the zero-frequency friction discussed above and setting S= in Eq. (5.19) gives... 

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