Spherical diffusion couple

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. 

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)... 

In single step voltammetry, the existence of chemical reactions coupled to the charge transfer can affect the half-wave potential Ey2 and the limiting current l. For an in-depth characterization of these processes, we will study them more extensively under planar diffusion and, then, under spherical diffusion and so their characteristic steady state current potential curves. These are applicable to any electrochemical technique as previously discussed (see Sect. 2.7). In order to distinguish the different behavior of catalytic, CE, and EC mechanisms (the ECE process will be analyzed later), the boundary conditions of the three processes will be given first in a comparative way to facilitate the understanding of their similarities and differences, and then they will be analyzed and solved one by one. The first-order catalytic mechanism will be described first, because its particular reaction scheme makes it easier to study. 

Impedance plots are modified at spherical electrodes. Examples of such plots, using an equivalent circuit in Eig. 4.1, are shown in Eig. 4.14. The high-frequency semicircle is related to the coupling of Ret and Cdepressed semicircle is related to spherical diffusion. 

This expression has been interpreted as the total resistance , being the sum of the diffusion (ro/Asi) and adsorption + internalisation (l/kKn) resistances [11,32] or a combination of permeabilities [19]. If the couple adsorption + internalisation is much more effective than diffusion (kKh > T>m/to), then r// 0 and we recover the steady-state maximum uptake flux for a spherical... 

Other results also confirm the important role of internal diffusion. Experimental activation energies (67—75 kJ mol"1) of the sucrose inversion catalysed by ion exchangers [506—509] were considerably lower than those of a homogeneously catalysed reaction (105—121 kJ mol"1) [505, 506,508] and were close to the arithmetic average of the activation energy for the chemical reaction and for the diffusion in pores. The dependence of the rate coefficient on the concentration in the resin of functional groups in the H+-form was found to be of an order lower than unity. A theoretical analysis based on the Wheeler—Thiele model for a reaction coupled with intraparticle diffusion in a spherical bead revealed [510,511] that the dependence of the experimental rate coefficient on acid group concentration should be close to those found experimentally (orders, 0.65 and 0.53 for neutralisation with Na+ and K+ ions respectively [511] or 0.5 with Na+ ions [510]). 

Because particles of different sizes are distributed throughout the bulk randomly, developing an exact model that couples diffusion to particle size evolution is daunting. However, a mean-field approximation is reasonable because diffusion near a spherical sink (see Section 13.4.2) has a short transient and a steady state characterized by steep concentration gradients near the surface. The particles act as independent sinks in contact with a mean-field as in Fig. 15.2. 

Diffusion coefficients can be related to molecular weight in three ways first by application of the Stokes-Einstein equation, second by combination with sedimentation data, and third by consideration of homologous polymer solutions. In the first method, an equivalent spherical size of the molecules is calculated from Dt, and an approximate molecular weight is found by combining these data with the appropriate density. In the second method, diffusion measurements are coupled with those of sedimentation velocity to give molecular weights, and in the third method, molecular weights may be determined directly from measurements of diffusion coefficients alone once a calibration has been... 

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/... 

This model approach, which produces curves much like those in Fig. 4.5, is applicable to the intraparticle diffusion of organic adsorptives in natural colloids that are approximately spherical.40 It can be coupled to an appropriate adsorption... 

The time required to establish such stationary states will depend on the diffusion coefficients of the ions in the solution and the size of the electrodes. For a small spherical electrode of radius ro the time for establishment of a quasi-stationary state will be of the order of f ro/ir D. For ro 0.1 cm and D 10 cm /sec, t is about 100 sec, BO that for large electrodes the times can become quite long. For ions, the diffusion of O and R is not independent of the speed of negative ions in the solution because the condition of electroneutrality requires a coupled diffusion mechanism. 

There is an inherent coupling of the behavior of the micro-scale variables to the behavior of macro-scale variables. This in itself presents complications when simrrlating these models. A few researchers have tried to address this problem of couphng of scales in these models. The solid state concentration term defined by the micro scale diffusion equation need to be coupled with the governing equations for the macro-scale to predict electrochemical behavior. Wang and co-workers used volume averaged equations and a parabolic profile approximation for solid-phase concentration. Subramanian et al. developed approximations assuming that the solid-state concentration inside the spherical electrode particle can be expressed as a polynomial in the spatial direction. 

One possibility that would lead to larger inferred porosities for the U-series was introduced by Qin (1992, 1993), who proposed that the retained melt was only in complete equilibrium with the surface of minerals and that solid-state diffusion limited the re-equilibration of the retained melt with the solid. In other respects, this model is identical to the ACM model of Williams and Gill (1989). Qin introduced a specific microscopic melting/diffusion model for spherical grains and coupled it to the larger-scale dynamic melting models. The net affect of this... 

The coupling of the flux and the time dependence of the counter-ion concentration is achieved through the material balance condition and Pick s Second Law (often termed the condition of continuity). For spherical particles and a constant interdifjusion coefficient, D, the foregoing considerations give the following partial differential equation for particle diffusion kinetics ... 

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