Diffusion Heuristic

Kinetic theories of adsorption, desorption, surface diffusion, and surface reactions can be grouped into three categories. (/) At the macroscopic level one proceeds to write down kinetic equations for macroscopic variables, in particular rate equations for the (local) coverage or for partial coverages. This can be done in a heuristic manner, much akin to procedures in gas-phase kinetics or, in a rigorous approach, using the framework of nonequihbrium thermodynamics. Such an approach can be used as long as... 

Heuristically, the dynamics proceed as follows the reaction term makes all active ((7 = 1) and refractory a = 2) sites cycle to their respective successor states. The diffusion term defines the manner in which activity (defined by sites with value (7 = 1) diffuses through the lattice. 

The time approach relies entirely on independent diffusion-reaction time without reference to distances. The reaction product inherits the time sequence of one of the parents chosen at random however, its residual time to react with another species is scaled inversely relative to its mutual diffusion confident. A heuristic correction is also made for the change of reaction radius (Clifford et al, 1986). 

Solution crystallization, general separation heuristics for, 22 319-320 Solution-diffusion (SD) models, 21 639-640, 660... 

The present analysis follows the approach taken by aU three of these authors, in which SDEs are constructed by choosing the drift and diffusivity coefficients so as to yield a desired diffusion equation. Peters [13] has pioneered an alternative approach, in which expressions for the drift and diffusivity are derived from a direct, but rather subtle, analysis of the underlying inertial equations of motion, in which (for rigid systems) he integrates the instantaneous equations of motion over time intervals much greater than the autocorrelation time of the particle velocities. Peters has expressed his results both as standard Ito SDEs and in a nonstandard interpretation that he describes heuristically as a mixture of Stratonovich and Ito interpretations. Peters mixed Ito—Stratonovich interpretation is equivalent to the kinetic interpretation discussed here. Here, we recover several of Peters results, but do not imitate his method. 

The top level of the electro-diffusion hierarchy is formed by the electroconvection phenomena, of which electro-osmosis is in several respects the simplest one. Certain aspects of electro-osmosis will be treated in Chapter 6. The higher we climb the hierarchy outlined the less rigorous our mathematics will become and the more vague heuristic statements will appear. 

The great advantage of this approach is the ease with which other effects and complications can be included and at a level of description appropriate rather than as heuristic additions to the diffusion equation. For instance, the corrections to the motion of the reactants to account for departures from purely diffusive motion can easily be incorporated. The rate kernel appropriate for pure diffusion and improved propagation are... 

It was pointed out in Chap. 8, Sect. 2.1 that there are primarily two reasons for the failure of the diffusion equation to describe molecular motion on short times. They are connected with each other. A molecule moving in a solvent does not forget entirely the direction it was travelling prior to a collision [271, 502]. The velocity after the collision is, to some degree, correlated with its velocity before the collision. In essence, the Boltzmann assumption of molecular chaos is unsatisfactory in liquids [453, 490, 511—513]. The second consideration relates to the structure of the solvent (discussed in Chap. 8, Sects. 2.5 and 2.6). Because the solvent molecules interact with each other, despite the motion of solvent molecules, some structure develops and persists over several molecular diameters [451,452a]. Furthermore, as two reactants approach each other, the solvent molecules between them have to be squeezed-out of the way before the reactants can collide [70, 456]. These effects have been considered in a rather heuristic fashion earlier. While the potential of mean force has little overall effect on the rate of reaction, its effect on the probability of recombination or escape is rather more significant (Chap. 8, Sect. 2.6). Hydrodynamic repulsion can lead to a reduction in the rate of reaction by as much as 30-40% under the most favourable circumstances (see Chap. 8, Sect. 2.5 and Chap. 9, Sect. 3) [70, 71]. 

As an example in which the second coefficient a2 is not constant consider diffusion in a medium whose diffusion coefficient varies in space. There are two plausible ways in which one can heuristically generalize the ordinary diffusion equation. In one dimension x, one may write either... 

Figure 39. The complete impedance spectrum of a mixed conductor contacted by ion-blocking electrodes.3,15 Regarding signal 3 The dashed curve corresponds to the heuristic approach (Eq. (64)), the straight line to the solution of the diffusion law with respect to the detailed behavior around the maximum see Figure 37. Figure 39 is the translation of Figure 38 into the frequency domain. Reprinted from J. Maier, Z. Phys. Chem. NF, (1984) 191-215. Copyright 1984 with permission from Oldenbourg Verlagsgruppe.

The first section of this chapter is devoted to the presentation of some diffusion models constructed on the basis of phenomenological considerations. These so-called heuristic models are often cited in the literature and used for the interpretation of experimental results. A special emphasis is to discuss how the mathematical formulae of these models can correlate with experimetal data and moreover to predict diffusion coefficients beyond the ranges experimentally investigated. This latter aspect is of great interest not only from a fundamental point of view but also in many practical fields where the possibility to predict a diffusion process might be a more economic alternative to its experimental investigation. 

It was shown in the above section that as a rule, at the base of the classical or microscopic diffusion models, there are ad hoc (heuristic) assumptions on a certain molecular behaviour of the polymer penetrant system. The fact that the mathematical formulae developed on such bases often lead to excellent correlations and even semipredictions of diffusion coefficients must be aknowledged. It is true that the classical models are not capable to predict diffusion coefficients only from first principles but this is often not an obstacle to hinder their use in certain types of investigations. Therefore we are quiet sure that this type of diffusion models will certainly be used in the future too for the interpretation of diffusion experiments. 

As announced above these findings are in astonishing agreement with the heuristic pictures of the diffusion mechanism discussed in the framework of some microscopic diffusion models. But, besides being free of the conceptual drawbacks (the ad hoc assumptions) of the classical diffusion models, the MD method of computer simulation of diffusion in polymers makes it possible to get an even closer look at the diffusion mechanism and explain from a true atomistic level well known experimental findings. For example the results reported in (119,120) on the hopping mechanism reveal the following additional features. 

In this respect one solution for the estimation of a Dp-value is to correlate the diffusion coefficient with the relative molecular mass, Mr, of the migrant and with matrix specific parameters at a given temperature T in Kelvin. This approach has already been successfully used (Piringer 1993,1994 Limm and Hollifield 1996). The estimation of the diffusion coefficient can be achieved for example using the following heuristic correlation (Piringer 1994 Baner et al. 1996) ... 

The prediction ofhuman oral absorption for diffusion rate-limited drugs based on heuristic method and support vector machine... 

Debye (1929) developed a model for the reorientation processes based on the assumption that collisions are so frequent in a liquid that a molecule can only rotate through a very small angle before suffering a reorienting collision (small-step diffusion). We give here a heuristic treatment of the Debye model. 

The stochastic approach to reaction-diffusion systems is not mathematically well-established. Though spatio-temporal stochastic phenomena ought to be associated with random fields, and not with stochastic processes, the usual investigation of such kinds of physicochemical problems starts from the master equation, and then it is extended by some heuristic procedure. From the physical point of view the role of spatial fluctuations is obviously important. It is well known that the density fluctuations are spatially correlated, and according to the modern theory of critical phenomena (e.g. Fisher, 1974 Wilson Kogut, 1974) small fluctuations are amplified owing to spatial interactions causing drastic macroscopic effects. 

Although there is no complete derivation of a generahzed CPE yet available which arises from nonuniform diffusion (NUD) in a finite-length region, one may heuristically modify the CPE and Warburg diffusion expressions in such a way as to generalize them both. The result is... 

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