Unequal diffusion coefficients

For some biological systems, the species that eventually crosses the cell membrane has travelled through different media, each one with its own mass transfer characteristics. As an example, we deal with the case where the two media are the bulk solution and the cell wall (with the separation surface parallel to the cell membrane) with diffusion as the only relevant mass transfer phenomenon in each medium. Apart from having different parameters in the differential equations in each medium (due to the unequal diffusion coefficients), we need to impose two new boundary conditions at the separating plane which we denote as a. The first boundary condition follows from the continuity of the material flux ... 

It should be noted that it is not always possible to solve the problem of unequal diffusion coefficients in such a rigorous way [147]. 

For a spherical electrode, by making rs electrolytic solution (upper sign), one obtains... 

In the case of unequal diffusion coefficients, when spherical electrodes are considered, the currents of the first and second potential pulses under the above conditions are (see also Eq. (2.147) of Sect. 2.5.2 and Appendix F),... 

The use of double potential pulse chronoamperometry is of great interest in electrochemistry for an accurate determination of both diffusion coefficients DQ and Dr, and this interest is enhanced when this technique is applied to small size spherical electrodes like the SMDE or gold microhemispheres or microspheres. There is a great number of redox couples for which highly unequal diffusion coefficients appear such as room temperature ionic liquids [23], ferrocene/... 

To use Eq. (5.50) under CV conditions, values of AE < 0.01 mV must be used. Unequal Diffusion Coefficients... 

This is the only case in which a rigorous explicit expression of the cyclic voltammogram has been deduced by considering unequal diffusion coefficients for species O and R. When the electrode geometry is considered, it has not been possible to obtain an explicit solution and Fick s second law differential equations need to be solved using numerical procedures. 

Martin and Unwin simulated chronoamperometric feedback allowing for unequal diffusion coefficients of the oxidized and reduced forms of the redox mediator [63]. Unlike steady-state SECM response, the shape of the tip current transients is sensitive to the ratio of the diffusion coefficients, y = Do /Dr (Fig. 8). When Do = DR, the tip current attains a steady-state value much faster... 

Often, for convenience, diffusion coefficients of all species in a mechanism are assumed equal, however unrealistic this probably is. If the reader wants to assume this, the d s in what follows can simply be set equal to unity. A recent paper [504] underlines the point, finding significant effects of unequal diffusion coefficients, and Pedersen et al. [438] in fact measured differences of about 20% between the diffusion coefficients of some organic compounds and the radical anions formed by their reduction. Such differences may well be significant in a simulation. 

This search for the atomistic basis of the diffusion coefficient will commence from the picture ofrandom-walking ions (see Sections 4.2.4 to 4.2.6). It will be recalled that a net diffusive transport of ions occurs in spite of the completely random zigzag dance of individual ions, because of unequal numbers of ions in different regions. 

Equation [4.8.10] is Pick s second law, which we have seen before, see [4.6.34b] and [4.8.11] is equivalent to [4.6.14]. In passing It is mentioned that for symmetrical electrolytes of unequal diffusion coefficients, this pair of equations has to be replaced by... 

An interesting observation that has been pointed out is that the difference in rate between forward and reverse exchange of two counterions may arise not only from unequal diffusion coefficients but also from unequal dissociation constants of complexes formed with the fixed groups. 

Even if there are equal driving forces for the two components to diffuse across the membrane, they will have unequal concentrations one the downstream side because of their different diffusion coefficient. This is the basis for separation in a porous membrane. 

For example, for the special case tiiOf Cf = n Oy Cf, = 3ti. Thus, while in controlled-potential voltammetric methods two substances at equal concentration with equal diffusion coefficients show two waves of equal height, in chronopotentiometry unequal transition times arise. The long second transition results from the continued diffusion of Oi to the electrode after ti, so that only a fraction of the applied current is available for reduction of O2 (Figure 8.5.1). 

These equations hold for polarography as well (within the expanding plane approximation) with t = /max di op time) and the area A given by (7.1.3). The approach of Section 7.2.2 applies. Treatments taking account of spherical diffusion and unequal diffusion coefficients have also been presented (27, 28). Note how the limiting current in po-... 

In binary substitutional alloys, it is possible for one atomic species to diffuse faster than the other. This unequal diffusion (the Kirkendall effect [83-85]) is consistent with the vacancy mechanism of diffusion. Each component in a binary alloy (e.g., A and B) has its own diffusion coefficient, and the Kirkendall effect is due to the fact that D for A is different from D for B. An overall chemical diffusion coefficient D (interdiffusion coefficient) can be defined as... 

A mathematical analysis (171) of the processes occurring in the network presented in Figure 32, has shown that regardless of the intrinsic product selectivity, the observed isomer distribution is altered as a result of unequal diffusion rates and isomerization rate of the isomers. The emichment of p-xylene is possible by the much greater diffusion coefficient of p-xylene. Then when the rate of diffusion is much slower than isomerization, at low conversion, the xylene mixture will be p-xylene rich. At higher conversions (longer residence times), the primary products will reenter in the zeolite, and will reisomerize to form secondary products, and the final composition will depend on the conversion level. It can be then concluded from the work of Wei (171) that the relation between para selectivity and conversion will be dependent on the square root of the ratio between the rate of disproportionation and the rate of isomerization [(Ko/Ki) "]. 

Following the procedure detailed in Chapter 3 for the case of unequal diffusion coefficients, with a non-uniform spatial grid, central difference scheme for the spatial derivative and the backward imphcit time integration, the coefficients for the Thomas algorithm for species y = A, B are given by... 

O. V. Klymenko, R. G. Evans, C. Hardacre, I. B. Svir, and R. G. Compton. Double potential step chronoamperometry at microdisk electrodes Simulating the case of unequal diffusion coefficients, J. Electroaruil. Chem. 571, 211-221 (2004). 

The deviation expected as a result of (i) unequal diffusion coefficients of the oxidized and reduced species, and (ii) electrode sphericity can be described as follows [3] ... 

Electrophoresis due to the macroscopic electric field generated by the concentration gradient of the electrolyte whose anion and cation have unequal diffusion coefficients... 

The ECcat reaction scheme [10] in which A is recycled via the first order homogeneous reaction of B with a suitable reagent in excess leads to an identical expression for to the ECjn v case. If diffusion coefficients are equal I2 also replaces Ij in the expression for Ca. This system in which the simple electron transfer expressions have I2 replacing I is simple to deal with via the ideas discussed. When the diffusion coefficients are unequal Ca depends on a more complex convolution described in outline below. 

Inequality in diffusion coefficients of electroactive species m ely causes matching of fluxes to occur at unequal shifts of concentrations from initial conditions at the electrode. The effect is to shift the axis to be centred on the En2 potential rather than E. The situation for species engaged on both sides of chemical reactions is more complex in some cases demands rather elaborate forms of convolution to be made. Even the ER y equal D case replaces the Ij of equation (17) say with (Ii+K.l2)/(1+K) thus increasing the complexity of parameter extraction. 

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