Butler-Volmer boundary conditions

The two successive electron transfer reactions are assumed to obey the Butler-Volmer law with the values of standard potentials, transfer coefficient, and standard rate constants indicated in Scheme 6.1. It is also assumed, matching the examples dealt with in Sections 2.5.2 and 2.6.1, that the reduction product, D, of the intermediate C, is converted rapidly into other products at such a rate that the reduction of B is irreversible. With the same dimensionless variables and parameters as in Section 6.2.4, the following system of partial derivative equations, and initial and boundary conditions, is obtained ... 

Fick s first and second laws (Equations 6.15 and 6.18), together with Equation 6.17, the Nernst equation (Equation 6.7) and the Butler-Volmer equation (Equation 6.12), constitute the basis for the mathematical description of a simple electron transfer process, such as that in Equation 6.6, under conditions where the mass transport is limited to linear semi-infinite diffusion, i.e. diffusion to and from a planar working electrode. The term semi-infinite indicates that the electrode is considered to be a non-permeable boundary and that the distance between the electrode surface and the wall of the cell is larger than the thickness, 5, of the diffusion layer defined as Equation 6.19 [1, 33] ... 

Since the electrochemical reactions are supposed to take place at the electrodeelectrolyte interface, then the Butler-Volmer equation, regulating the electrochemical kinetics, sets the boundary condition, whilst j (production rate) in Equation (3.37) is replaced with J (current density produced), as explained in detail in Section 3.7.2. 

The boundary condition for Equations (3.68) and (3.69) is given by the Butler-Volmer equation ... 

For the quasireversible case, two species A and B must again be considered and the two boundary conditions are the flux condition (2.49) and the dimensionlesss form of the Butler-Volmer equation. The forward and backward heterogeneous rate constants kj and kb are normalised ... 

Additional parameters specified in the numerical model include the electrode exchange current densities and several gap electrical contact resistances. These quantities were determined empirically by comparing FLUENT predictions with stack performance data. The FLUENT model uses the electrode exchange current densities to quantify the magnitude of the activation overpotentials via a Butler-Volmer equation [1], A radiation heat transfer boundary condition was applied around the periphery of the model to simulate the thermal conditions of our experimental stack, situated in a high-temperature electrically heated radiant furnace. The edges ofthe numerical model are treated as a small surface in a large enclosure with an effective emissivity of 1.0, subjected to a radiant temperature of 1 103 K, equal to the gas-inlet temperatures. 

Note that this does not mean that the concentration profile of species A is equivalent to that of the E mechanism since it will be influenced by the chemical reaction through the surface boundary conditions. Thus, the chemical reaction affects the surface concentration of species B, which is related to that of species A through the Nernst equation (for reversible systems), or more generally, through the Butler-Volmer or Marcus-Hush relationships. Therefore, the surface concentration of species A, and as a consequence the whole concentration profile, will reflect the presence of the chemical process. 

For the electrode plane ( = 0), two separate boundary conditions are required as the plane is composed of two different materials the electroactive microdisc and the insulating supporting surface. As the microdisc surface is electroactive, a potential-dependent boundary condition is applied the Nernst equation, Butler-Volmer or Marcus-Hush models may be used as appropriate. 

Lastly, the system described by Reaction (5) might be quasireversible or even irreversible, in which case the boundary condition is given by the Butler-Volmer equation. It is preferable to express it in dimensionless form, using potential p, and the heterogeneous rate constant ko... 

In Table 4, the boundary conditions at the electrode surface involve the Butler-Volmer law describing the kinetics of electron transfer between the electrode and the cosubstrate rather than the Nernst law describing its thermodynamics (as in Table 2), because it is interesting... 

The current density at the pore wall, j, depends of the local overvoltage, t], according to some Butler-Volmer kinetics, which are not given here explicitly. The first boundary condition [Eq. (28.72)] is equivalent to the definition of (< —

electrode pore, so it defines the electric potential there. The second boundary condition [Eq. (28.73)] demands that no charge flux exits the electrolyte at the top of the pore. This differential equation can be solved in combination with a reaction rate expression, for example, Butier-Volmer kinetics. 

The concentration and surface reactions are declared as MATLAB variables and then passed to COMSOL, similarly to declaring initial conditions. The reaction rates rbv Mo and rbv Mr were defined as COMSOL global expressions and are thus written in single quotes. The boundary types are defined by COMSOL as follows NO for no flux, N for Neumann (Butler-Volmer flux af the electrode surface), and C for concentration. Only the concentration boundary condition applies at the biofilm/bulk liquid interface thus, the first index value for bnd. c 0 is zero. Likewise, the reaction of the mediators occurs only at the electrode boundary thus, the second index value for bnd. N is zero. 

Near the electrode the concentrations will be determined in part by the potential of the electrode (through the Butler-Volmer equation). At a distance beyond the diffusion layer, the concentrations will be at bulk values. These are the boundary conditions for Equation (1). 

In the world of numerical analysis, one distinguishes formally between three kinds of boundary conditions [1, 2] the Dirichlet, Neumann (derivative) and Robin (mixed) conditions they are also sometimes called [1, 3] the first, second and third kind, respectively. In electrochemistry, we normally have to do with derivative boundary conditions, except in the case of the Cottrell experiment, that is, a jump to a potential where the concentration is forced to zero at the electrode (or, formally, to a constant value different from the initial bulk value). This is Dirichlet only for a single species simulation. If the simulation involves two species (e.g. the reduced and oxidised form) and the surface kinetics obeys the Butler-Volmer equation, flux conditions must apply, i.e. derivatives are involved, see Sect. 5.5.1. If species do not undergo electrode reactions, zero-flux conditions prevail at the location of the electrode surface, involving also derivatives. In what follows below, we briefly treat the single species case, which includes the Cottrell (Dirichlet) condition as well as derivative conditions, and then the two-species case. In a later section in this chapter, a mathematical formalism is described that includes all possible boundary conditions for a single species and can be useful in some more fundamental investigations. 

Two boundary conditions are required for each species one in bulk and one at the electrode. Conventionally the concentrations are set to their initial values in bulk, and at the electrode either the Nernst equation or Butler-Volmer equation is applied to describe the electrode kinetics. These equations have the general form/(c,o E) = 0, where the applied potential is a linear function of time. Conservation of mass also requires that the fluxes of reactant A and product B are equal and opposite at the electrode surface. 

The bulk concentration (Co and Cr) of each species is set to the one used during the experiment and the adsorbed species A is not present in this subdomain. At the tip boundary, z=0,0 < r < a, the inward flux condition following Butler-Volmer kinetics (Equations 16.13 and 16.14) is written and in the case of a linear potential sweep. Equation 16.15 is added ... 

On the cathodic surfaces, the Butler-Volmer equation prevails. The associated boundary condition is... 

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