Diffusion spherical

Macdonald and coworkers [158-164] obtained an exact solution of the finite-length diffusion impedance in unsupported conditions where the Nemst-Planck and continuity equations for both negative and positive mobile charges were solved and involved full satisfaction of Poisson s equation. One can expect such conditions in diluted electrolytic solutions and in poorly conductive solids. 

In the case of spherical diffusion, there is an additional term in the differential equation indicating an increase of the mass transfer to a sphere [144]. These equations may be written for the concentration phasors as 

These equations can be solved for semi-infinite external diffusion, where both Red and Ox forms are in the solution outside the sphere (diffusion to a spherical or hemispherical hanging mercury electrode, metallic solid spherical electrode), or they may diffuse inside the sphere (amalgam formation at mercury electrode, intercalation of Li into particles, hydrogen absorption into spherical hydrogenabsorbing particles). 

This case arises, for example, when working with dropping or hanging mercury electrodes. Let us consider semi-infinite diffusion to a sphere of radius with both oxidized and reduced forms soluble in the solution. In this case Eq. (47) should be substituted by 

These equations may be rearranged into a simpler form [Eq. (51)] substituting it = rCp and 7 = tCr  

Taking into account that dU/dr= Cp + rdC /dr and that at r = Tq (at the electrode surface), dCp/dr = 7/nFDp, the following solutions are obtained  

The mass-transfer impedance may be obtained from Eq. (64). Assuming a reversible dc process, one obtains, similar to the case of linear diffusion  

The influence of the nonlinearity of diffusion on the observed complex plane plots is shown in Fig. 13. Spherical mass transfer causes the formation of a depressed semicircle at low frequencies instead of the linear behavior observed for linear semi-infinite diffusion. For very small electrodes (ultramicroelectrodes) or low frequencies, the mass-transfer impedances become negligible and the dc current becomes stationary. On the Bode phase-angle graph, a maximum is observed at low frequencies. 

4 Concentration Profiles. Cottrell Equation. As previously mentioned, the region close to the electrode surface where the concentrations 

During an electrode reaction in an unstirred solution, the thickness of the diffusion layer grows with time up to a limiting value of about 10- 4 m, beyond which, because of the Brownian motion, the charges become uniformely distributed. At ambient temperature the diffusion layer reaches such a limiting value in about 10 s. This implies that in an electrochemical experiment, the variation of concentration of a species close to the electrode surface can be attributed to diffusion only for about 10 s, then convection takes place. 

The graphs that show the dependence of the concentration of a species on distance from the electrode surface and how it evolves with time are called concentration profiles. 

To obtain such diagrams, one must mathematically solve Fick s second law  

The resolution makes use of non-elementary mathematical treatments (Laplace transformation). Neglecting such treatments, one obtains  

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PasquiU Atmo.spheric Diffusion, Van Nostrand, 1962) recast Eq, (26-60) in terms of the dispersion coefficients and developed a number of useful solutions based on either continuous (plume) or instantaneous (puff) releases, Gifford Nuclear Safety, vol, 2, no, 4, 1961, p, 47) developed a set of correlations for the dispersion coefficients based on available data (see Table 26-29 and Figs, 26-54 to 26-57), The resulting model has become known as the Pasquill-Gifford model. 

Most electrochemical studies at the micro-ITIES were focused on ion transfer processes. Simple ion transfer reactions at the micropipette are characterized by an asymmetrical diffusion field. The transfer of ions out of the pipette (ejection) is controlled by essentially linear diffusion inside its narrow shaft, whereas the transfer into the pipette (injection) produces a spherical diffusion field in the external solution. In contrast, the diffusion field at a microhole-supported ITIES is approximately symmetrical. Thus, the theoretical descriptions for these two types of micro-ITIES are somewhat different. 

FIG. 2 Schematic representation of different microhole geometries, (a) Recessed microdisk interface, spherical-linear, linear-spherical diffusion, (b) quasi-inlaid microdisk interface, spherical-spherical diffusion, (c) Long microhole with quasi-inlaid interface, spherical-linear diffusion. (Reprinted with permission from Ref. 13. Copyright 1999 Elsevier Science S.A.)... 

The mathematical formulations of the diffusion problems for a micropippette and metal microdisk electrodes are quite similar when the CT process is governed by essentially spherical diffusion in the outer solution. The voltammograms in this case follow the well-known equation of the reversible steady-state wave [Eq. (2)]. However, the peakshaped, non-steady-state voltammograms are obtained when the overall CT rate is controlled by linear diffusion inside the pipette (Fig. 4) [3]. 

For the HMDE and for a solution that contains only ox of a reversible redox couple, Reinmuth102, on the basis of Fick s second law for spherical diffusion and its initial and boundary conditions, derived the quantitative relationship (at 25° C)... 

As a last example in this section, let us consider a sphere situated in a solution extending to infinity in all directions. If the concentration at the surface of the sphere is maintained constant (for example c — 0) while the initial concentration of the solution is different (for example c = c°), then this represents a model of spherical diffusion. It is preferable to express the Laplace operator in the diffusion equation (2.5.1) in spherical coordinates for the centro-symmetrical case.t The resulting partial differential equation... 

Diffusion of electroactive species to the surface of conventional disk (macro-) electrodes is mainly planar. When the electrode diameter is decreased the edge effects of hemi-spherical diffusion become significant. In 1964 Lingane derived the corrective term bearing in mind the edge effects for the Cotrell equation [129, 130], confirmed later on analytically and by numerical calculation [131,132], In the case of ultramicroelectrodes this term becomes dominant, which makes steady-state current proportional to the electrode radius [133-135], Since capacitive and other diffusion-unrelated currents are proportional to the square of electrode radius, the signal-to-noise ratio is increased as the electrode radius is decreased. 

Spherical diffusion has peculiar properties, which can be utilized to measure fast reaction rates. The diffusion current density of a species i to a spherical electrode of radius ro is given by ... 

Figure 1. Outline of the uptake model showing the spherical diffusion of species M through the medium towards two different sites where adsorption is followed by internalisation. The radius of the organism is taken as ro...

P Ratio of calculated maximal cellular flux by spherical diffusion to nutrient uptake flux ... 

It is assumed i) that the concentration c remains constant and ii) that transport by diffusion is rate controlling, i.e., the adsorbate arriving at the interface is adsorbed fast (intrinsic adsorption). This intrinsic adsorption, i.e., the transfer from the solution to the adsorption layer is not rate determining or in other words, the concentration of the adsorbate at the interface is zero iii) furthermore, the radius of the adsorbing particle is relatively large (no spherical diffusion). 

Spherical Diffusion. If, as it might happen, the electrode is spherical rather than planar (e.g. using a hanging drop mercury electrode), See Figure 19, Fick s second law should be integrated by corrective terms accounting for the sphericity, or the radius r, of the electrode ... 

Figure 19 (a) Typical electrode for spherical diffusion (b) parameters of the spherical... 

Some of these stability issues can be addressed by the use of protective barrier membranes, at the risk of aggravating another fundamental challenge reactant mass transfer. Typical reactants present in vivo are available only at low concentrations (glucose, 5 mM oxygen, 0.1 mM lactate, 1 mM). Maximum current density is therefore limited by the ability of such reactants to diffuse to and within bioelectrodes. In the case of glucose, flux to cylindrical electrodes embedded in the walls of blood vessels, where mass transfer is enhanced by blood flow of 1—10 cm/s, is expected to be 1—2 mA/cm. ° Mass-transfer rates are even lower in tissues, where such convection is absent. However, microscale electrodes with fiber or microdot geometries benefit from cylindrical or spherical diffusion fields and can achieve current densities up to 1 mA/cm at the expense of decreased electrode area. ... 

Eor spherical diffusion, 1 = 2nFDQd, with parameters as defined in ref 30 and electrode diameter d = 10 gm. 

The above equation is important and very useful because many solutions obtained for the simple one-dimensional diffusion equation can be applied to the spherical diffusion problem. Below is an example. 

Often it is necessary to treat diffusion between different layers as three dimensional diffusion. For isotropic minerals such as garnet and spinel (including magnetite), diffusion across different layers may be considered as between spherical shells, here referred to as "spherical diffusion couple." Oxygen diffusion in zircon may also be treated as isotropic because diffusivity c and that Tc are roughly the same (Watson and Cherniak, 1997). If each shell can be treated as a semi-infinite diffusion medium, the problem can be solved (Zhang and Chen, 2007) as follows ... 

Figure 5-26 The concentration evolution for a "spherical diffusion couple." The radius of the initial core is a. The initial concentration is Cl = 0.2 in the core and C2 = 0.4 in the mantle. Note that the position for the midconcentration between the two halves moves toward smaller radius, which is due to the much larger volume per unit thickness in the outer shell. From Zhang and Chen (2007).

Although the shape of the profile of a "spherical diffusion couple" is similar to that of a one-dimensional diffusion couple, one difference is that, whereas the midconcentration position stays mathematically at the initial interface for the normal diffusion couple, the midconcentration position moves with time in the "spherical diffusion couple." Initially, the concentration at the initial interface (r = a) is the mid-concentration Cmid = (Ci + C2)/2. However, as diffusion progresses, the concentration at r = a is no longer the mid-concentration. Rather, the location of the mid-concentration moves to a smaller r. Define the mid-concentration location as Tq. Then Tq x a(l — z /2) for small times. If layer 1 is the solid core (meaning r extends to 0), the concentration at the center begins... 

The following table gives measured Fe concentrations in garnet as a function of distance from the center. Treat the diffusion profile as a spherical diffusion couple. Fit the data to find jDdt... 

Figure A3-3-4 Diffusion profile evolution in a "spherical diffusion couple."...

Zhang Y. and Chen N.S. (2007) Analytical solution for a spherical diffusion couple, with applications to closure conditions and geospeedometry. Geochim. Cosmachim. Acta submitted. 

Fig. 4.2.1 Spherical diffusion model for the growth of a tabular grain. (From Ref. 6.)...

J- Barton and J. O M. Bockris, Proc. Roy. Soc. London A268 485 (1962). Spherical diffusion control experimentally established. 

For example, the treatment of diffusion that is to follow is solely restricted to semi-infinite linear diffusion, i.e., diffusion that occurs in the region between x = 0 and x —> +oo, to a plane of infinite area. Thus, diffusion to a point sink—called spherical diffusion—is not treated, though it has been shown to be relevant to the particular problem of the electrolytic growth of dendritic crystals from ionic melts. 

To avoid the difficulties associated with the spherical diffusion equation, a useful hypothesis is the linear-driving-force concept. This arises when a parabolic concentration profile within the spherical particles is supposed - which is a good approximation in cases where there is a Thiele modulus of a maximum volume of 2-5 (that is, with some intra--particle resistance [50]). In these conditions, the volume-averaged intra-particle concentration is defined as ... 

While this equation is thought to overestimate the diffusion-limited rate constant slightly, it is a good approximation. If the diffusing particles are approximately spherical, diffusion constants DA and DB can be calculated from Eq. 9-25, and Eq. 9-28 becomes Eq. 9-29. 

At a spherical electrode, one must consider a spherical diffusion field as discussed in Sect. 2.4. Fick s second law is then written... 

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