Position-dependent friction

B. Carmeli and A. Nitzan, Theory of activated rate processes position dependent friction, Chem. Phys. Lett. 102, 517 (1983). 

The first is a variational approach that maps the position-dependent problem to an effective parabolic barrier transfer problem, with an effective friction that is position-independent. This approach leads to a result for the rate that can be interpreted as a Grote-Hynes coefficient with a position-dependent friction. 

The bilinear coupling case (i.e. position-independent friction) corresponds to G (s) = 1, or equivalently, to Y s=s . The position-dependent friction Eq. (39) can then be rewritten as... 

In an earlier work we performed a quantum calculation using the exponential resummation technique and found results that agreed qualitatively with those of Azzouz and Borgis. When we allowed for a position-dependent friction, we obtained a function g(s) that is plotted in Fig. 2. The results for the quantum rate are presented in Tables II and 111. The column g(s) = s refers to the position-independent case, as calculated in our earlier work on this system. 

Since DFT calculations are in principle only applicable for the electronic ground state, they cannot be used in order to describe electronic excitations. Still it is possible to treat electronic exciations from first principles by either using quantum chemistry methods [114] or time-dependent density-functional theory (TDDFT) [115,116], First attempts have been done in order to calculate the chemicurrent created by an atom incident on a metal surface based on time-dependent density functional theory [117, 118]. In this approach, three independent steps are preformed. First, a conventional Kohn-Sham DFT calculation is performed in order to evaluate the ground state potential energy surface. Then, the resulting Kohn-Sham states are used in the framework of time-dependent DFT in order to obtain a position dependent friction coefficient. Finally, this friction coefficient is used in a forced oscillator model in which the probability density of electron-hole pair excitations caused by the classical motion of the incident atom is estimated. 

Electron-hole pairs have already been treated on the Hartree-Fock level in otherwise classical high-dimensional molecular dynamics simulation using the molecular dynamics with electronic friction method [120]. In this approach, the energy transfer between nuclear degrees of freedom and the electron bath of the surface is also modelled with a position-dependent friction term, but additionally temperature-dependent fluctuating forces are included. 

The friction (more generally, molecule-solvent interaction) is taken in the Kramers model to be a constant, independent of the position along the reaction coordinate. As seen, generalization to position-dependent friction is trivial in the Smoluchowski limit. In many systems position-dependent friction should be considered also in the underdamped case. An obvious example is desorption where the dissociating particle ceases to feel the thermal bath as it draws further away from the surface. 

The position-dependent part of the friction is manifest in the spatial dependence of the coupling function g (s). The usual quantum Kramers problem is recovered when g(s) = s. An implicit assumption in Eq. (36) is that the functional form of the coupling g(s) is the same for all modes k. 

Some narrowing of the spectrum does occur when there is a strong positional dependence in the frictional coefficient. For example, the reduced compliance is ... 

For the perfectly mixed continuous reactor, the CSTR, the ratio VT/ Fy only represents the mean residence time, /p,av however, it is still possible to compare the performance of the CSTR with the performance of the BR by letting the mean residence time fp av = t. Interestingly, when the reaction rate shows a positive dependence on reactants concentration, the BR is more effective than the CSTR. This is because the batch reactor experiences all the system compositions between initial and final values, whereas the CSTR operates at the final composition, where the reaction rate is smaller (under the above hypotheses). Finally, one can compare the two continuous reactors under steady-state conditions. The CSTR allows a more stable operation because of back-mixing, which however reduces the chemical performance, whereas the PFR is suitable for large heat transfer but suffers from larger friction losses. 

The rotational friction coefficient of a spherical molecule in solution is calculated applying the Navier-Stokes equation for a continuum solvent with a position-dependent viscosity as a model of "microscopic viscosity." The rotational friction coefficient decreases with decreasing surface viscosity. The results are compared with the translational fnction and viscosity B coefficients which are previously obtained from the same model. The B coefficient is most sensitive to a local viscosity change The Gierer-Wirtz model overestimates the effect of the "microscopic viscosity" on the translational friction coefficient comparing with the present results... 

The first result agrees with what solution chemists expect for the effect of the "microscopic viscosity " The second result tells us that the sensitivity of the friction coefficients on a local viscosity change largely depends on the mode of solvent motions. The shear mode (the viscosity B coefficient) is the most sensitive of the three It is to be noted that these results do not depend on the particular choice of the functional form of the position-dependent viscosity as expected. 

In the small damping limit it is also possible to obtain an energy diffusion equation for the case where the friction kernel (and the associated random noise) are position dependent. A convenient model with such property is given by ... 

A similar conclusion has been reached by Ciccotti et al. s-iao jj, their studies of the model ion association reaction. Their system consisted of two equally massive ions, modeled as Lennard-Jones spheres with a positive or negative charge, in a solvent of dipolar molecules. Each solvent molecule was modeled as a Lennard-Jones sphere with a dipole moment of either 2.4 or 3.0 D and with a mass equal to that of the ionic mass. As with the simulations of Karim and McCammon, Ciccotti et al. started the dynamics at the transition state, as determined from the free energy calculations, and ran 104-144 trajectories to determine the transmission coefficient. The values of the transmission coefficient they found were 0.18 in the 2.4 D solvent and 0.16 in the 3.0 D solvent (which are surprisingly, and perhaps coincidentally, close to the results of Karim and McCammon e). Ciccotti et al. also calculated the frequency-dependent friction that the solvent exerted on the reaction coordinate in order to compare the simulation results with Grote-Hynes theory for the rates. The comparison with Grote-Hynes theory was quite close, although within the outer reaches of the calculated uncertainties in the molecular dynamics transmission coefficients. 

Aside from the applied force Fa of the voice coil current, the piston experiences a restoring force Fj from the suspension whenever the piston is displaced from its equilibrium position, a frictional force Ff resulting from flexure of the spider and surround whenever the piston is in motion, a restoring force Fj, resulting from compression or rarefaction of the air trapped in the sealed cabinet behind the loudspeaker cone whenever the piston is displaced from its equilibrium position, and a force of radiation F, exerted by the air in front of the piston whenever the piston is moving so as to generate sound waves in the air. Table 1.2 collects aU of these forces and indicates their relationship to the piston velocity which is to be considered the dependent variable in the following analysis. 

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