Walking analysis

The diffusion of an impurity atom in a crystal, say K in NaCl, involves other considerations that influence diffusion. In such cases, the probability that the impurity will exchange with the vacancy will depend on factors such as the relative sizes of the impurity compared to the host atoms. In the case of ionic movement, the charge on the diffusing species will also play a part. These factors can also be included in a random-walk analysis by including jump probabilities of the host and impurity atoms and vacancies, all of which are likely to vary from one impurity to another and from one crystal structure to another. All of these alterations can be... 

Berkowitz B, Emmanuel S, Scher H (2008) Non-Fickian transport and multiple rate mass transfer in porous media Water Resour Res 44, D01 10.1029/2007WR005906 Bijeljic B, Blunt MJ (2006) Pore-scale modeling and continuous time random walk analysis of dispersion in porous media. Water Resour Res 42, W01202, D01 10.1029/2005WR004578 Blunt MJ (2000) An empirical model for three-phase relative permeability. SPE Journal 5 435-445... 

Small elements lose their identity by the action of molecular diffusion, and the Einstein random walk analysis estimates this time as... 

In Sect. 2.1, the timescale over which the diffusion equation is not strictly valid was discussed. When using the molecular pair analysis with an expression for h(f) derived from a diffusion equation analysis or random walk approach, the same reservations must be borne in mind. These difficulties with the diffusion equation have been commented upon by Naqvi et al. [38], though their comments are largely within the framework of a random walk analysis and tend to miss the importance the solvent cage and velocity relaxation effects. 

Surface diffusion may be treated by a random-walk analysis. We assume that the molecular motion is completely random and that the jumps from site to site are of equal length, which is equal to the nearest-neighbor distance d. With these assumptions, the preexponential factor for diffusion is (23). 

The evidence for reflecting barriers at the edge of surface terraces derives from a number of different sources, one important example of which is field-ion microscopy. For example, it has been found that there is an enhancement in the likelihood of finding an adatom just adjacent to the top edge of a surface step. From the standpoint of the simple random walk analysis put forth earlier, one way to interpret the results of the types of experiments sketched above is via the random walker with reflecting walls. The basic idea in this case is the presumption that the random walker can exercise excursions within the confines of some finite region, but whenever the walker reaches the boimdary of the region it is reflected from the walls. 

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. 

It is in this region of time that the random walk or diffusion equation analysis is least adequate. The prime failure of the random walk analysis is to ignore the oscillatory motion of molecules. The encounter pair separates from encounter but gets reflected back towards each other by the solvent cage after a time about equal to the period of oscillations in a solvent cage. To incorporate this effect, Noyes [265] used the approximate form of h(t)... 

Figure 15.1. PhOLISE walking analysis/synthesis system architecture.

The same spin-walk analysis was also performed for the C3 and C6 side chain methylenes, on both a and P anomers. The chemical shifts of the sidechain methylene carbons determined in this fashion confirmed the assignments used to determine positional DS by 1-D NMR, as shown in the assigned spectrum in Figure 5. 

Peterson, S.C. and Noble, P.B., A two-dimensional random-walk analysis of human granulocyte movement. Biophys. J., 1972,12 1048-1055. 

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