Finite Electrode Kinetics

So far we have considered that the electrode reaction is very fast (reversible) such that equihbrimn conditions apply for the surface concentrations of the electroactive species A and B Ca,o = Nevertheless, electron 

There are different formalisms for the modelling of the potential dependence of the rate constants. The empirical Butler-Volmer (BV) model has been the most widely used over a number of years in electrochemistry due to its simplicity and successful quantitative description of a vast number of electrochemical systems (in the absence of bonds being broken or formed). According to the BV model, the rate constants show a simple exponential dependence with the applied potential according to the following expressions  

we are going to deal with the implementation of the new surface boimdary condition (Eq. (4.20)) in our simulation. As mentioned before, when the diffusion coefficients of both electroactive species are the same the total concentration of the species at any point of the solution and at any time is constant ca x, t) - - CQ x,t) = c + Cg. Along with Eq. (4.21), this enables us to rewrite Eq. (4.20) as a function of only the concentration of species A  

Note that a new dimensionless variable appears that accounts for the kinetics of the electrode reaction  

After discretisation of the problem using a two-point approximation for the surface gradient we obtain 

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Equation (92) suggests that the current density should approach infinity near the disc edges in practice, effects associated with finite electrode kinetics preclude this. 

In practice, this is physically unrealistic since finite electrode kinetics must restrict the flux to a finite value in the vicinity of the disc edge. 

Thus the current density (flux) is predicted to be infinite at the upstream edge of the electrode. In reahty, this would not be attained, since finite electrode kinetics preclude the passage of an infinite current. Nevertheless, the flux would be very large at x = 0. 

The solution of the Laplace equation is not trivial even for relatively simple geometries and analytical solutions are usually not possible. Series solutions have been obtained for simple geometries assuming linear polarisation kinetics "" . More complex electrode kinetics and/or geometries have been dealt with by various numerical methods of solution such as finite differencefinite elementand boundary element. ... 

The first limitation is related to interference of the anode and the cathode. The finite permeability of the Nation membrane to fuel and oxygen results in crossover of fuel from the anode to the cathode, and oxygen crossover in the opposite direction. This may have a significant influence on electrode kinetics. 

Current and potential distributions are affected by the geometry of the system and by mass transfer, both of which have been discussed. They are also affected by the electrode kinetics, which will tend to make the current distribution uniform, if it is not so already. Finally, in solutions with a finite resistance, there is an ohmic potential drop (the iR drop) which we minimise by addition of an excess of inert electrolyte. The electrolyte also concentrates the potential difference between the electrode and the solution in the Helmholtz layer, which is important for electrode kinetic studies. Nevertheless, it is not always possible to increase the solution conductivity sufficiently, for example in corrosion studies. It is therefore useful to know how much electrolyte is necessary to be excess and how the double layer affects the electrode kinetics. Additionally, in non-steady-state techniques, the instantaneous current can be large, causing the iR term to be significant. An excellent overview of the problem may be found in Newman s monograph [87]. 

Tertiary current distribution. This method of analysis applies to those systems where there is significant mass transport and electrode polarization effects. Electrode kinetics is considered, with electrode surface concentrations of reactant and/or products that are no longer equal to those in the bulk electrolyte due to finite mass transfer resistance. The analysis of tertiary current distributions is complex, involving the solution of coupled... 

In addition to the overvoltage losses due to electrode kinetics, one has to take into account the voltage loss in the ion-conducting electrolyte, due to the finite electrolyte conductivity, and (minimized) losses due to the electric resistance of electrode and cell materials, including contact resistances. 

Bond AM, Mahon PJ (1997) Linear and non-linear analysis using the Gldham- Zoski steady-state equation for determining heterogeneous electrode kinetics at microdisk electrodes and digital simulation of the microdisk geometry with the fast quasi-expUdt finite difference method. J Electroanal Chem 439 37-53... 

Safford LK, Weaver MJ (1991) Cyclic voltammetric wave-shapes for microdisk-electrodes coupled effects of solution resistance, double-layer capacitance, and finite electrochemical kinetics. J Electroanal Chem 312 69-96... 

Fig. 9.4 Potential difference between the working and reference electrodes (a) and relative error in the determination of WE polarization resistance (b) as functions of misalignment between the WE and CE normalized by the solid-electrolyte thickness (s/d), as calculated by the finite-element analysis assuming linear electrode kinetics [8, 25]. At high s/d ratios, the experimentally measured value stabilizes at a small Nemst potential due to gas-phase polarization of the working electrode,...

The practical polarization-resistance technique involves one more assumption than those listed in Section 11.1. It is assumed that the polarization equation can be approximated by a linear relation between the measuring current density and the polarization for small polarization values. Equation (7) can be obtained by expanding the exponentials in Eq. (1) into Taylor series and ignoring all terms higher than first power. There has been a certain amount of controversy about the criteria of validity for this approach, both for the corrosion case and for the electrode kinetics case (see Ref. 116 and references therein). By now, this controversy seems to have been cleared up with the following conclusions. While the polarization curve is never absolutely linear at finite polarization values, the assumption of linearity often introduces only a small error. Furthermore, this error approaches zero, by mathematical definition, as A approaches zero, but the rate of approach depends strongly on the values of and b. The error... 

The traditional approach to understanding both the steady-state and transient behavior of battery systems is based on the porous electrode models of Newman and Tobias (22), and Newman and Tiedermann (23). This is a macroscopic approach, in that no attempt is made to describe the microscopic details of the geometry. Volume-averaged properties are used to describe the electrode kinetics, species concentrations, etc. One-dimensional expressions are written for the fluxes of electroactive species in terms of concentration gradients, preferably using the concentrated solution theory of Newman (24). Expressions are also written for the species continuity conditions, which relate the time dependence of concentrations to interfacial current density and the spatial variation of the flux. These equations are combined with expressions for the interfacial current density (heterogeneous rate equation), electroneutrality condition, potential drop in the electrode, and potential drop in the electrolyte (which includes spatial variation of the electrolyte concentration). These coupled equations are linearized using finite-difference techniques and then solved numerically. 

As evidenced at the beginning of the treatment of the electrode kinetics, the Butler-Volmer equation only accounts for the kinetics of the charge transfer, once supposing that the supply of electroactive species at the electrode occurs at infinite rate, i.e., assuming that infinite rate of mass transfer is operative. This is out of the reality a finite rate characterizes the mass transport. As a consequence of the charge transfer, in fact, mass transfer of reactant and product to and from the electrode surface, respectively, is induced. 

Here F is the Faraday constant C = concentration of dissolved O2, in air-saturated water C = 2.7 x 10-7 mol cm 3 (C will be appreciably less in relatively concentrated heated solutions) the diffusion coefficient D = 2 x 10-5 cm2/s t is the time (s) r is the radius (cm). Figure 16 shows various plots of zm(02) vs. log t for various values of the microdisk electrode radius r. For large values of r, the transport of O2 to the surface follows a linear type of profile for finite times in the absence of stirring. In the case of small values of r, however, steady-state type diffusion conditions apply at shorter times due to the nonplanar nature of the diffusion process involved. Thus, the partial current density for O2 reduction in electroless deposition will tend to be more governed by kinetic factors at small features, while it will tend to be determined by the diffusion layer thickness in the case of large features. 

Linearizing the kinetic term as before, a set of three unknown linear equations is obtained, which is completed by the finite difference expression of the initial and boundary conditions. Inversion of the ensuing matrix allows the calculation of C at each node of the calculation grid and finally, of the current flowing through the electrode, or of the corresponding dimensionless function, by means of its finite difference expression. Calculation inside thin reaction layers may thus be more efficiently carried out than with explicit methods. The combination of the Crank-Nicholson... 

In the real world, the simple redox couple may be perturbed by finite ET rates, by adsorption of O and/or R on the electrode surface, and by homogeneous (i.e., in solution) chemical kinetics involving O and/or R. Various combinations of heterogeneous ET steps (E) with homogeneous chemical steps (C) are encountered. It should be clear that if one or more species in equilibrium in solution are electroactive, electrochemistry can be used to perturb the equilibrium and study the solution chemistry. 

Two impedance arcs, which correspond to two relaxation times (i.e., charge transfer plus mass transfer) often occur when the cell is operated at high current densities or overpotentials. The medium-frequency feature (kinetic arc) reflects the combination of an effective charge-transfer resistance associated with the ORR and a double-layer capacitance within the catalyst layer, and the low-fiequency arc (mass transfer arc), which mainly reflects the mass-transport limitations in the gas phase within the backing and the catalyst layer. Due to its appearance at low frequencies, it is often attributed to a hindrance by finite diffusion. However, other effects, such as constant dispersion due to inhomogeneities in the electrode surface and the adsorption, can also contribute to this second arc, complicating the analysis. Normally, the lower-frequency loop can be eliminated if the fuel cell cathode is operated on pure oxygen, as stated above [18],... 

In order to have theoretical relationships with which experimental data can be compared for analysis it is necessary to obtain solutions to the partial differential equations describing the diffusion-kinetic behaviour of the electrode process. Only a very brief account f the theoretical methods is given here and this is done merely to provide a basis for an appreciation of the problems involved and to point out where detailed treatments can be found. A very lucid introduction to the theoretical methods of dealing with transient electrochemical response has appeared (MacDonald, 1977) which is highly recommended in addition to the classic detailed treatment (Delahay, 1954). Analytical solutions of the partial differential equations are possible only in the most simple cases. In more complex cases either numerical methods are used to solve the equations or they are transformed into finite difference forms and solved by digital simulation. 

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