Diffusion and friction theory

The pair kinetic theory equation given in Section VII.D can be used to extend the Smoluchowski results outlined earlier. In this section, we present the microscopic derivation of the Smoluchlowski equation from the kinetic theory and also obtain expressions for the space and time nonlocal diffusion and friction tensors, which appear in this theory. 

In liquid solution, Brownian motion theory provides the relation between diffusion and friction coefficient... 

Thus one must rely on macroscopic theories and empirical adjustments for the determination of potentials of mean force. Such empirical adjustments use free energy data as solubilities, partition coefficients, virial coefficients, phase diagrams, etc., while the frictional terms are derived from diffusion coefficients and macroscopic theories for hydrodynamic interactions. In this whole field of enquiry progress is slow and much work (and thought ) will be needed in the future. 

The whole problem of computing pressure distributions in particulate packings is one of great complexity. In addition to the fact that we are unable to deal with a material whose apparent density is not uniform, we must consider added difficulties such as diffusion, sliding friction, deformation of individual particles, cohesive forces, and perhaps others. The quantitative relationships of these factors to particle size must remain empirical for the time being. In the paragraphs to follow we shall be concerned only with a limited theory of the problem of particles under pressure. 

The values of A, S, d and X obtained from experimental data on the kinetics of the Kerr effect agree qualitatively with those determined for the same polymers by translational diffusion and sedimentation (Table 3). The agreement between geometrical molecular characteristics obtained from the phenomena of rotattonal and translational friction indicates that the hydrodynamic and the oinformational models on which this theory is based are valid. This is an evidence of the kinetic r idity of the investigated chains. 

In chemical kinetics the concept of the order of a reaction forms the basis of a kinematics which constitutes a frame for most of the molecular theories of chemical reactions. The fundamental magnitudes of this kinematics are the concentrations and the specific rate constants. In simple cases only the time enters as an independent variable, whereas in a diffusion process both time and space are involved. Diffusion processes are generally described in terms of diffusion coefficients, volume concentrations and thermodynamic potential or activity factors. Partial volume factors and friction coefficients associated with the components of the diffusing mixture are also essential in the description. A feature of the macro-dynamical theory is that it covers any region of concentration. Especially simple equations are connected with the differential diffusion process (diffusion with small concentration differences), for which the different coefficients or factors mentioned above are practically constant. 

Ah8olute-rate4heory Approach, Eyring and others (6, 8, 14, 23, 29) have extended the theory of absolute rates to the problem of liquid diffusion and viscosity with considerable success. Viscosity is a measure of the force per unit area required to overcome the frictional resistance between two layers of molecules of a liquid in maintaining unit relative velocity of the two layers. In diffusion, molecules of the diffusing solute move past those... 

KAPRAL - My comment concerns the effect on the rate coefficient of including spatial dependence in the diffusion or friction coefficient. We investigated this problem some time ago (M. Schell, R. Kapral and R.I. Cukier, Chem. Phys. Lett.) for a model bimolecular reaction using a space-dependent friction coefficient obtained from a kinetic theory calculation (R.I. Cukier, 3.R. Mehaffey and R. Kapral, J. Chem. Phys.). The effects are rather small, amounting to no more than about forty per cent corrections to the rate coefficient. The small size of the effect arises from averaging over a range of spatial configurations. 

Ben-Amotz, D., and Drake, J.M. (1988). The solute size effect in rotational diffusion experiments A test of microscopic friction theories. J. Chem. Phys. 89 1019-1029. 

This result comes from the idea of a variational rate theory for a diffusive dynamics. If the dynamics of the reactive system is overdamped and the effective friction is spatially isotropic, the time required to pass from the reactant to the product state is expected to be proportional to the integral over the path of the inverse Boltzmann probability. 

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