Molecular modeling Fokker-Planck equation

The starting point of a molecular constitutive theory is a simple mechanical model for the molecule that captures its most salient traits. Thus, flexible polymer molecules have been represented by elastic dumbbells and bead-spring chains, and rigid polymers by rigid dumbbells and rigid rods. For its simplicity, the evolution of the model molecule is easily described by a convection-diffusion equation. Then a Fokker-Planck equation is written for the probability distribution function of an ensemble of these molecules. Finally, the macroscopic stress tensor is derived in terms of the distribution function. This kinetic theory framework was pioneered by Kirkwood (see, for example, Ref. ). 

The construction of Cooper and Mann (7) for the surface viscosity includes the substrate effect by a model that represents the result of very frequent molecular collisions between the small substrate molecules and the larger molecules of the monolayer. This was done by adding a term to the Boltzmann equation for the 2D singlet distribution function that is equivalent to the friction coefficient term of the Fokker-Planck equation from which Equations 24 and 25 can be constructed. Thus a Brownian motion aspect was introduced into the kinetic theory of surface viscosity. It would be interesting to derive the collision frequency of Equation 19 using the better model (7) and observe how the T/rj variable of Equation 26 emerges. 

In Sect. 3, the Noyes approach to analysing reaction rates based on the molecular pair approach is discussed [5]. Both this and the diffusion equation analysis are identical under conditions where the diffusion equation is valid and when the appropriate recombination reaction rate for a molecular pair is based on the diffusion equation. Some comments by Naqvi et al. [38] and Stevens [455] have obscured this identity. The diffusion equation is a valid approximation to molecular motion when the details of motion in a cage are no longer of importance. This time is typically a few picoseconds in a mobile liquid. When extrapolating the diffusion equation back to such times, it should be recalled that the diffusion is a continuum form of random walk [271]. While random walks can be described with both a distribution of jump frequencies and distances, nevertheless, the diffusion equation would not describe a random walk satisfactorily over times less than about five jump periods (typically 10 ps in mobile liquids). Even with a distribution of jump distances and frequencies, the random walk model of molecular motion does not represent such motion adequately well as these times (nor will the telegrapher s or Fokker-Planck equation be much better). It is therefore inappropriate to compare either the diffusion equation or random walk analysis with that of the molecular pair over such times. Finally, because of the inherent complexity of molecular motion, it is doubtful whether it can be described adequately in terms of average jump distances and frequencies. These jump characteristics are only operational terms for very complex quantities which derive from the detailed molecular motion of the liquid. For this very reason, the identification of the diffusion coefficient with a specific jump formula (e.g. D = has been avoided. 

By contrast, when both the reactive solute molecules are of a size similar to or smaller than the solvent molecules, reaction cannot be described satisfactorily by Langevin, Fokker—Planck or diffusion equation analysis. Recently, theories of chemical reaction in solution have been developed by several groups. Those of Kapral and co-workers [37, 285, 286] use the kinetic theory of liquids to treat solute and solvent molecules as hard spheres, but on an equal basis (see Chap. 12). While this approach in its simplest approximation leads to an identical result to that of Smoluchowski, it is relatively straightforward to include more details of molecular motion. Furthermore, re-encounter events can be discussed very much more satisfactorily because the motion of both reactants and also the surrounding solvent is followed. An unreactive collision between reactant molecules necessarily leads to a correlation in the motion of both reactants. Even after collision with solvent molecules, some correlation of motion between reactants remains. Subsequent encounters between reactants are more or less probable than predicted by a random walk model (loss of correlation on each jump) and so reaction rates may be expected to depart from those predicted by the Smoluchowski analysis. Furthermore, such analysis based on the kinetic theory of liquids leads to both an easy incorporation of competitive effects (see Sect. 2.3 and Chap. 9, Sect. 5) and back reaction (see Sect. 3.3). Cukier et al. have found that to include hydrodynamic repulsion in a kinetic theory analysis is a much more difficult task [454]. 

The Cooper-Mann theory of monolayer transport was based on the model of a sharply localized interfacial region in which ellipsoidal molecules were constrained to move. The surfactant molecules were assumed to be massive compared with the solvent molecules that made up the substrate and a proportionate part of the interfacial region. It was assumed that the surfactant molecules had many collisions with solvent molecules for each collision between surfactant molecules. A Boltzmann equation for the singlet distribution function of the surfactant molecules was proposed in which the interactions between the massive surfactant molecules and the substrate molecules were included in a Fokker-Planck term that involved a friction coefficient. This two-dimensional Boltzmann equation was solved using the documented techniques of kinetic theory. Surface viscosities were then calculated as a function of the relevant molecular parameters of the surfactant and the friction coefficient. Clearly the formalism considers the effect of collisions on the momentum transport of the surfactant molecules. 

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