Continuous diffusion

As a reactant molecule from the fluid phase surrounding the particle enters the pore stmcture, it can either react on the surface or continue diffusing toward the center of the particle. A quantitative model of the process is developed by writing a differential equation for the conservation of mass of the reactant diffusing into the particle. At steady state, the rate of diffusion of the reactant into a shell of infinitesimal thickness minus the rate of diffusion out of the shell is equal to the rate of consumption of the reactant in the shell by chemical reaction. Solving the equation leads to a result that shows how the rate of the catalytic reaction is influenced by the interplay of the transport, which is characterized by the effective diffusion coefficient of the reactant in the pores, and the reaction, which is characterized by the first-order reaction rate constant. 

F = Function of the molecular volume of the solute. Correlations for this parameter are given in Figure 7 as a function of the parameter (j), which is an empirical constant that depends on the solvent characteristics. As points of reference for water, (j) = 1.0 for methanol, (j) = 0.82 and for benzene, (j) = 0.70. The two-film theory is convenient for describing gas-liquid mass transfer where the pollutant solute is considered to be continuously diffusing through the gas and liquid films. 

The use of equipment close to the temperature at wliich the material was diffusion treated will result in continuing diffusion of chromium, aluminum etc., into the substrate, thus depleting chromium with consequent loss in oxidation and corrosion resistance. For aluminum, this effect is noticeable above 700°C in steels, and above 900°C in nickel alloys. For chromium, the effect is pronounced above 850°C for steels and above 950°C for nickel alloys. 

Wiley, R.C. et al.. Efficiency studies of a continuous diffusion apparatus for the recovery of betalaines from the red table beet, J. Food Sci., 44, 208, 1979. 

In all of the structures based upon hexagonal close-packed anions illustrated, continuous diffusion paths through empty sites can be traced, and a population of point defects is not mandatory to facilitate atom transport. 

At high frequencies diffusion of the reactants to and from the electrode is not so important, because the currents are small and change sign continuously. Diffusion does, however, contribute significantly at lower frequencies solving the diffusion equation with appropriate boundary conditions shows that the resulting impedance takes the form of the Warburg impedance ... 

Figure 11,26 Concordia diagram. (A) Interpretation of straight discordia path as a result of a single episode of Pb loss. (B) Continuous diffusion model of Tilton (1960) applied to world minerals of a = 2800 Ma common age. Reproduced with modifications from G. Faure (1986), Principles of Isotope Geology, 2nd edition, copyright 1986 by John Wiley and Sons, by permission of John Wiley Sons and from Tilton (1960) by permission of the American Geophysical Union.

Fig. 4. Simplified version of Digby s semiconductor theory of biomineralization. In the arthropod (top) ions are continually diffusing out of the animal across the cuticle at different rates setting up a potential with the outer surface positive. This causes a flow of electrons leaving the inner surface rich in proteins and the outer surface with hydroxyl ions. The alkaline outer surface favors CaC03 formation. In molluscs (bottom) muscular movements cause salt flow through the periostracum followed by an alkaline reaction on the inside inducing CaC03 deposition. (After Simkiss445 )...

Figure 23-16 Longitudinal diffusion gives rise to B/u, in the van Deemter equation. Solute continuously diffuses away from the concentrated center of its zone. The faster the flow, the less time is spent on the column and the less longitudinal diffusion occurs.

At small radiation doses (the number of radiation-produced defects), the mean distance l between components of such geminate pairs (the vacancy and an interstitial atom) is much less than the mean distance between different pairs Iq = n-1/3, where n is defect concentration. The initial defect distribution is described by the distribution function f(r). Below a certain temperature (typically < 30 K for interstitial atoms and 200 K for vacancies in alkali halides), defects are immobile. With a temperature increase, the defects perform thermally activated random hops between the nearest lattice sites. This is usually considered to be continuous diffusion. 

They have calculated the continuous diffusion equation (3.2.30) with U(r) = -a/r3 for several kinds of nn F, H centres in the crystalline lattice. Figure 3.9 demonstrates well that both defect initial separation and an elastic interaction are of primary importance for geminate pair recombination kinetics. The 3nn defects are only expected to have noticeable survival probability. Its magnitude agrees well with equation (3.2.60). 

Up to now we have been considering defect diffusion in continuous approximation, despite the fact that crystalline lattice discreteness was explicitly taken into account defining the initial distribution for geminate pairs. Note, however, that such continuous diffusion approximation is valid only asymptotically when defects (particles) before recombination made large number of hops (see Kotomin and Doktorov [50]). This condition could be violated for recombination of very close defects which can happen in several hops. The lattice statement of the annihilation kinetics has been discussed in detail by Schroder et al. [3, 4, 83], Dederichs and Deutz [34]. Let us consider here just the most important points of this problem. 

These results yield an impressive justification for the use of the continuous diffusion approximation even for short distances and small recombination regions. 

In this Section we describe briefly the principal effects arising during replacement of the continuous diffusion, equation (4.1.23), by its more general analog... 

This trend continues as A decreases further (Fig. 4.13). At last, for small A < 0.05ro the reaction profile practically coincides with the limiting case of continuous diffusion, equation (4.3.11). It is seen in Fig. 4.14 how fastly the transient time required to reach the steady-state increases with decreasing A. For A > ro this happens very rapidly, whereas for small A (continuous diffusion) it goes very slowly. 

As it follows from equation (4.3.29), in another extreme case, A — 00, ftr tends to r, as it is indicated by a broken line in the insert of Fig. 4.14. It is also demonstrated in [85] that the continuous diffusion approximation, equation (4.1.63), gives quite reasonable reaction rates up to A 5ro. This comes from the fact that K(t) is a convolution of the correlation function and the exponentially decaying reaction probability o(r), that is, the essential deviation of the reaction profile from the diffusion limit does not affect the reaction rate considerably. 

The results obtained in these computer simulations of the hopping reactions were applied in [86] to the centre recombination in KC1-T10 stimulated by step-like temperature increase. As it is clearly seen from Fig. 4.15, taking directly into account finite hop lengths (A = ao/(2 /2) 2.2 A to be compared with ro < 1 A of the electron Tl° centre) permits us to obtain a much better agreement with the experimental data than the standard continuous diffusion approximation (curves 1 and 3, respectively). 

Fig. 4.15. A comparison of experimental delayed kinetics of an increase of tunnelling luminescence intensity after sudden change of their mobility (temperature increase from 175 to 180 K) in KC1 with theory [86], 1 - hopping kinetics for A = 2n> obtained by means of equation (4.4.1), 2 - experimental curve, 3 - results of continuous diffusion approximation...

Fig. 5. A donor impurity is diffused into the silicon from a gaseous phase, (a) A shallow n region lias been created, (b) Continued diffusion, longer times, or higher temperatures increase the extent of the n region, (c) Surface diffusion has caused spreading of the n region along the SiCVsilicon interface, (d) A crystal defect, such as a dislocation, has provided a path for anomalously high diffusion and led to penetration of the junction to unanticipated distance from the surface. (See Fig. I for legend)...

PCBs that have historically accumulated in the sediments may return to the water through resuspension-desorption and through continuous diffusion from sediment porewaters. 

It was assumed that the chlorine atoms could be removed from the film if a continuous diffusion path connecting the chlorine atom with the surface through oxygen vacancies existed. Here, when the diffusion path was determined, rule (b) was taken into account a chlorine atom can occupy the site in the anion sublattice if there is at least one cation vacancy in its coordination shell. 

There are also two essential simplifications of the reaction mechanism accepted in almost all the theories. They imply that the reactivity of the partners is spherically isotropic and their encounter motion can be considered as a continuous diffusion at any distance. Both these assumptions are simply the approximations that have been overcome already with DET under the pressure of experimental facts. 

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