Semi-infinite diffusion model

Reversible square-wave voltammogram — Figure. Theoretical dimensionless square-wave voltammogram of a reversible electrode reaction (see - reversible redox reaction) of a dissolved redox couple at a planar electrode assuming a semi-infinite diffusion model simulated with nEsw = 50 mV and a scan increment of the staircase ramp of 5 mV (see - square-wave voltammetry)... 

Equation (2.28) shows that any plane of constant concentration propagates proportionally to the square root of time. Based on the above considerations, Eq. 2.28 may serve the dual function (a) it is useful for preliminary validation of the value of the experimentally obtained DC and (b) can be used for the preliminary estimation whether limited or semi-infinite diffusion model should be applied. These estimates are using the thickness of the sample and the experiment duration, as well as the predetermined possible range of DC values. Eor example, when experiment duration equals 1 h, a nonporous electrode with the thickness of few millimeters behaves as semi-infinite media when diffusion coefficients are greater than 10 cm s ... 

In the case of non-steady-state diffusion, the diffusion layer becomes thicker with time because ions are transported from ever more distant areas. Then, the diffusion front gradually moves from the electrode surface to the bulk of the solution. Two main images are used in modern theories. According to semi-infinite diffusion model, the diffusion front is allowed to move from the electrode surface... 

Both the semi-infinite diffusion and Nernst s steady diffusion layer concept produce the same results when the diffusion front shift is not yet large as compared to In these conditions, the semi-infinite diffusion model, which is simpler from the mathematical point of view, is to be applied. However, the Nernst model is more general because the regularities of the first model follow from it at 5j.j oo. 

For all reactions, the mass transport regime is controlled by the diffusion of the reacting ligand only, as the mercury electrode serves as an inexhaustible source for mercury ions. Hence, with respect to the mathematical modeling, reactions (2.205) and (2.206) are identical. This also holds true for reactions (2.210) and (2.211). Furthermore, it is assumed that the electrode surface is covered by a sub-monomolecular film without interactions between the deposited particles. For reactions (2.207) and (2.209) the ligand adsorption obeys a linear adsorption isotherm. Assuming semi-infinite diffusion at a planar electrode, the general mathematical model is defined as follows ... 

The applicability of the foregoing procednre has been tested by modeling simple reaction under semi-infinite diffusion conditions (reaction 1.1) and EC mechanism coupled to adsorption of the redox couple (reaction (2.177)) [2]. The solutions derived by the original and modified step-function method have been compared in order to evaluate the error involved by the proposed modification. As expected, the precision of the modified step-function method depends solely on the value of p, i.e., the number of time subintervals. For instance, for the complex EC mechanism, the error was less than 2% for p>20. This slight modification of the mathematical procedure has opened the gate toward modeling of very complex electrode mechanisms such as those coupled to adsorption equilibria and regenerative catalytic reactions [2] and various mechanisms in thin-film voltammetry [5-7]. 

Figure 12. Modeling and measurement of oxygen surface diffusion on Pt. (a) Model I adsorbed oxygen remains in equilibrium with the gas along the gas-exposed Pt surface but must diffuse along the Pt/YSZ interface to reach an active site for reduction. Model II adsorbed oxygen is reduced at the TPB but must diffuse there from the gas-exposed Pt surface, which becomes depleted of oxygen near the TPB due to a finite rate of adsorption, (b) Cotrell plot of current at a porous Pt electrode at 600 °C and = 10 atm vs time. The linear dependence of current with at short times implies semi-infinite diffusion, which is shown by the authors to be consistent only with Model II. (Reprinted with permission from ref 63. Copyright 1990 Electrochemical Society, Inc.)...

In general, self-similar fractal surfaces do not exist in the real world. The fractal models may only approximate random surfaces. In addition, Eq. (183) for ( ) = 0.5 is formally identical with the semi-infinite porous model presented in the next section. The fractal model in the presence of diffusion is discussed in Refs. Ill and 118. Experimental verifications of the fractal model were also carried out for some electrodes. It was... 

Historically, the Warburg impedance, which models semi-infinite diffusion of electroactive species, was the first distributed circuit element introduced to describe the behavior of an electrochemical cell. As described above (see Sect. 2.6.3.1), the Warburg impedance (Eq. 38) is also analogous to a uniform, semi-infinite transmission line. In order to take account of the finite character of a real electrochemical cell, which causes deviations from the Warburg impedance at low frequencies. 

FIGURE 4.6 Schematic illustration of the transient semi-infinite diffusion of a species i from the surface into the hulk of a medium. The concentration of species i at the surface of the medium is assumed to he held fixed at c while the initial concentration of species i within the bulk of the medium is assumed to be c°. As time elapses, species i diffuses deeper and deeper into the medium from the surface. Since the medium is semi-infinitely thick, this process can proceed indefinitely and the concentration of species i never reaches c anywhere inside the medium except at the surface. This figure assumes that c > c° however, the reverse situation (which would involve out-diffusion of i from the bulk) could be similarly modeled. 

Though in the general case, mathematical expressions of the Nernst model are more complicated than of those semi-infinite diffusion, stationary mass transport is described by a rather simple Eq. (3.12). In this connection, there occurs an interesting possibility to use superposition of both models, which is convenient to apply when i is the periodic time function. Perturbation signals of this type are considered in the theory of electrochemical impedance spectroscopy. In this case, i(t)... 

It should be noted that the aforementioned model deals with the semi-infinite diffusion, which is presented by Warburg impedance. As the diffusion layer of finite thickness 5 is formed at the RDE surface, it is necessary to meet certain relations between 5, specified by Eq. (3.5), and the depth of penetration of the concentration wave Xq = y/2Dfco. Investigations in this field show [25,26] that the two aforementioned diffusion models are in harmony when < 0.15. At Schmidt number Sc = v/D = 2000, this condition is satisfied when co > 6 2 (or/ > 0.1m, where m is rotation velocity in rpm). 

As already noted, the Warburg conductivity Yg = 0.34S2 cm s°, established at pH 2, is consistent with the theoretical model of semi-infinite diffusion. However, it can be seen from Table 8.7 that Tg falls with pH and becomes... 

For a semi-infinite diffusion process at cathode represented by Warburg impedance, the Nyquist plot appears as a straight line with a slope of 45°. The impedance increases linearly with decreasing frequency. The infinite diffusion model is only valid for infinitely thick diffusion layer. For finite diffusion layer thickness, the finite Warburg impedance converges to infinite Warburg impedance at high frequency. At low frequencies or for small... 

This way of determining the cation transference number involves some underlying assumptions, too binary electrolyte with the cation as active species, no convection, semi-infinite diffusion, and one-dimensional cell geometry. Furthermore, the method combines the results of three different measurements, which is very time-consuming. Nevertheless, the calculation of transference numbers does not assume ideality or diluted solutions, making it more appHcable for modelling transport parameters of hthium-ion batteries. 

Diffusion resistance Zp,yy to current flow carried by electroactive species can create impedance, frequently known as the Warburg element [23, p. 376]. If the diffusion layer Lp is assumed to have an unlimited thickness within the experimental AC frequency range, than a "semi-infinite" diffusion may become the rate-determining step in the Faradaic kinetic process. In the "semiinfinite" diffusion model the diffusion layer thickness Lp is assumed to be always much smaller than the total thickness of the sample d (Lp d. The equation for the "semi-infinite" Warburg impedance Z m) is a function of concentration-driven potential gradient dV/rfC. The "semi-infinite" diffusion limitation is modeled by characteristic resistance and a Warburg infinite diffusion component Z that can be derived [8] as ... 

As in the previous chapter, the semi-infinite diffusion at a planar electrode is considered, where the adsorption is described by a linear adsorption isotherm. The modeling of reaction (2.173) does not require a particular mathematical procedure. The model comprises equation (1.2) and the boundary conditions (2.148) to (2.152) that describe the mass transport and adsorption of the R form. In addition, the diffusion of the O form, affected by an irreversible follow-up chemical reaction, is described by the following equation ... 

The time-dependent diffusion equations for Red appropriate to the axisymmetrical geometry, shown in Fig. 10, are identical to Eqs. (9) and (10), given earlier. Although phase 2 is assumed to be semi-infinite in the z-direction, the model can readily be modified for the situation where phase 2 has a finite thickness [61]. 

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