Friction regime

Figure 3 Dynamic recrossmgs m the low and high friction regimes. Recrossmgs back to the reactive state lead to a lowering of the rate constant below the transition state theory value.

Fig. 10.15 Efficiency of cooling system (friction regime) (a) coefficient of efficiency vs. heat flux, (b) coefficient of efficiency vs. gravity, (c) coefficient of efficiency vs. capillary length, (d) coefficient of efficiency vs. capillary diameter. Reprinted from Yarin et al. (2002) with permission...

SMOLUCHOWSKI EQUATION—HIGH-FRICTION REGIME. At first We will restrict our attention to the one-dimensional case Eqs. (4.148) and... 

KRAMERS EQUATION—ARBITRARY FRICTION REGIME. In the presence Of inertial effects, the one-dimensional motion is determined by a bivariate Fokker-Planck equation. 

Fig. 4.4. Transition probability density p(x0 t, x) at different times [Eq. (4.155)] of a Brownian particle in the high-friction regime and in a force field deriving from an harmonic potential [initial condition p0(0, j 0) = 5(ac - )] x0 = -0.2 A, D = 0.1 A2 ns"1, r = 0.1 ns.

The multidimensional case has been discussed for the different friction regimes.168 176 177... 

Different attempts have been elaborated178-182 to find a general expression that accounts for the limiting forms (4.171) and (4.176) obtained in the high- and extreme-low-friction regimes. A new approach183 made conspicuous the connection between the diverse Kramers limits. 

The applicability of the reaction-rate formula in the intermediate-friction regime to the interpretation of conformational changes of bi-naphtyl in solution has been discussed.47 The intramolecular motion is governed by a bistable potential depending on a single variable, which is identified with the dihedral angle between the naphthalene moieties in binaphtyl. The influence of hydrodynamic interactions and coupling with the other coordinates of the molecules also are evoked. 

Fig. 4.6. Influence of the temperature on the Kramers reaction rate [intermediate friction regime Eq. (4.174)] for o>bt° = 5x10j (curves 1) or 104 (curves 2). The straight line represents e p(-QJkeT) with ( B = 2 kcal M. ...

Figure 11.12 Stribeck diagram Plot of friction coefficient /u versus ijkv/P,where rjk is the kinematic viscosity, v the velocity, and P the contact pressure. From left to right there are three different friction regimes Boundary lubrication with high friction and wear, mixed lubrication with medium friction and wear, and hydrodynamic lubrication with low friction and wear.

The active centers in silica are generated upon mechanical treatment not only due to the cleavage of the particles of material but also to their mutual contacts (so-called friction regime [11-13]). For this reason, the mechanical treatment can be used for generating defects at the surface of high-dispersed silica (aerosol). 

When the slip velocity is further increased, the Rouse friction [148] finally becomes dominant, for Vs>V ocN l. A linear friction regime is then recovered, with a constant extrapolation length, b much larger than b0 and comparable to what would be observed on an ideal surface without anchored chains [139]. 

Brinkman considered the bistable potential problem and showed that the diffusive, very-high-friction regime of Kramers was still correct. 

To sum up the current position of the experimental evidence on the viscosity effect on condensed phase reaction rates, we may say that the most conunonly observed effect is the inverse proportionality associated with the diffusive (high-friction) regime. In some cases, deviations are observed for lower viscosities which fit well with Kramers intermediate friction regime... 

As we have already discussed, further parameters of interest are the frequencies u>Q and U) deriving from the hi onic approximation at the bottom of the reactant well and the top of the barrier, respectively. Carmeli and Nitzan evaluated the reaction rate throughout the entire friction dominion ranging from the low-friction regime (w > y) to the high-friction regime. (This has also been commented on in S on II.B.) They also studied the dependence of the reaction rate on the correlation time... 

First of all we shall apply (101) to study the low-friction regime ... 

In other words, if we are exploring the low-friction regime, the interplay of non-Markovian statistics and external field renders the system still more inertial, thereby widening the range of validity of the formula provided by Kramers for the low-friction regime provided that y be replaced by... 

The Markovian description, inherent in Eqs. (2.1) and (2.2), is unrealistic for most chemical situations as it assumes that the relaxation time of the medium surrounding the molecule is faster than all molecular time scales. This assumption obviously breaks down in the low-friction regime where the escape rate is dominated (or at least influenced) by the well dynamics (that is, intramolecular motion), since typical molecular frequencies are of the same order or larger than intermolecular (solvent) frequencies. Even for higher friction, where the escape is dominated by the dynamics in the barrier regime, the Markovian assumption can fail because the characteristic barrier time... 

It is to be expiected that an external oscillating force will have a major effect in the low-friction regime where the motion is characterized by well-defined frequencies that can be pumped by the field. Recently treatments of such models were given by Carmeli and Nitzan, Sazonov and Zatsepin, Faetti et al., and Frouzoni et al. Here we follow the treatment of Carmeli and Nitzan. ... 

---