#### 2.2. Catalytic Electron Flux

The quantity µ

_{x} corresponds to the chemical potential of the protons within the catalyst film according to:

It is now assumed that the catalytic electrical current (J

_{e}) can be described by the following Butler–Vollmer type relation:

where ΔV, an electrical overpotential; J

_{e0}, formal exchange current density; T, the temperature in Kelvin e, elementary charge; k

_{B}, Boltzmann constant.

An alternative form of the above equation is used in the following (β = e/K

_{B}T):

Rearrangements lead to:

with

and X being a function of J

_{e} (X = X(J

_{e})). The value of s

_{ET} provides the so-called Tafel slope that is determined by α

_{ET}, which depends on electrocatalytic properties (catalytic mechanisms and other properties) of the material. For example, an α

_{ET} value of 1.0 corresponds to about 60 mV per decade (at room temperature); an α

_{ET} value of 0.5 corresponds to about 120 mV per decade. The Tafel slope describes a linear relation between the applied electric potential and the logarithm (log

_{10}) of the current, and it is often presented in units of mV per decade (of change in current density). The above value of s

_{ET} relates to the natural logarithm; the classical decadic Tafel slope is obtained by:

The generalized definition of the Tafel slope is (based on the natural logarithm):

The value of s_{ET} is the intrinsic Tafel slope of the catalyst material as it would be detectable in the absence of any limitations due to proton transport. Then X would be fully current-independent and equal to c_{H}; at room temperature the electric potential needed to maintain a given current would increase by about 60 mV per pH. Yet in the presence of proton-transport limitations, X is greater than c_{H}, and the magnitude of this difference increases with increasing current density (increasing J_{e}). Consequently, Equation (7) will provide Tafel slope values greater than s_{ET}.

#### 2.3. Proton Flux Mediated by Buffer Molecules

The following considerations aim at relations between the proton activity of catalyst material and the adjacent buffer layer, X, and the magnitude of the proton flux, J_{BH}.

(1) The value of X determines the equilibrium between the protonated and the unprotonated buffer base of the electrolyte in the buffer layer at the catalyst surface according to:

Here B_{L}^{−} and B_{L}^{H} are the effective concentration (i.e., activity) of the unprotonated and protonated buffer base in the buffer layer, respectively. The value of K_{aB} relates to the pK_{a} value of the buffer (−log K_{aB} = pK_{aB}).

With B

_{0} = B

_{L}^{−} + B

_{L}^{H} we obtain:

(2) In the bulk solvent (region C), the usual acid–base equilibrium prevails with a proton activity that corresponds to the bulk proton concentration (c_{H}, with pH = −log c_{H}). Thus,

B_{S}^{−} and B_{S}^{H} are the concentrations of the unprotonated base and protonated base in the bulk solution (i.e., far away from the catalyst surface), respectively. B_{0} provides the total buffer concentration, which is the same in all three regions.

(3) The protonated and the unprotonated buffer are exchanged between the solvent layer and the bulk by means of diffusion, both described by the same effective diffusion constant, k

_{D}. The flux of the proton-transporting base corresponds to a proton flux and, thus, a flux of positive charges, that is, a current. Thus, and for simplicity of the following considerations, the unit of choice for J

_{BH} is A cm

^{−2}. Accordingly, the B

^{H}-flux from the buffer layer to the bulk solvent is described by (F, Faraday constant):

The B^{H}-flux from the bulk solvent to the buffer layer is described by

For continuous operation of the catalyst, a net flux of protons (J_{BH}) from the buffer layer to the solvent is mediated by diffusion of the protonated form of the buffer molecules that equals the difference between J_{BH}^{S=>L} and J_{BH}^{L=>S}.

For continuous operation, the flux of unprotonated buffer molecules does not need to be considered explicitly because it matches the flux of the protonated buffer molecules. Moreover, it does not matter (for applicability of the above equations) how fast and where within the electrochemical cell the equilibrium distribution described by Equation (10) is reached. All details of the diffusion geometry and bulk-solvent equilibration are covered by the effective diffusion constant (k_{D}).

We note that Equation (13) can be understood in terms of Fick’s first diffusion law: The diffusion flux is proportional to the concentration gradient; the diffusion coefficient, D (in m

^{2}/s), is the proportionality constant. To illustrate the relation to Fick’s law, we consider a particular simple electrode geometry: two parallel electrodes of identical area, the OER anode and the cathodic counter electrode, are located at a fixed distance, d

_{el}, from each other; the unstirred electrolyte fills the volume exclusively in between of the two electrodes. Maintenance of a constant proton current mediated by diffusion of the protonated buffer base, B

^{H}, requires a constant gradient of the B

^{H} concentration; the B

^{H} concentration equals B

_{L}^{H} at the anode, B

_{0} − B

_{L}^{H} at the cathode, and B

_{L}^{S} at an intermediate position between anode and cathode, as illustrated by

Figure 1. Then the concentration gradient is given by (B

_{L}^{H} − B

_{L}^{S})/(d

_{el}/2), and multiplication by D yields the proton flux in mol m

^{−2} s

^{−1}.

For the idealized electrode geometry, we obtain (F, Faraday constant):

Using Equations (9)–(13), eventually we obtain:

Equation (15) implies that a maximal proton current density, J_{max}, exists for X >> K_{aB}.

We note that J

_{max} is not only proportional to the concentration of buffer molecules (B

_{0}) and the effective diffusion constant (k

_{D}) but is also pH-dependent. Since the last factor in Equation (16) provides exactly the fraction of unprotonated buffer molecules in the bulk solution, we can write Equation (16) also in the following form:

Thus, J_{max} scales with the fraction of unprotonated buffer molecules.

Using Equation (16), rearrangement of Equation (15) yields:

For very small J_{BH} (J_{BH} << J_{max}), X equals c_{H}. Then the proton activity in buffer layer and catalyst (X) corresponds to the pH in the bulk electrolyte so that the local pH is very close to the bulk pH.

#### 2.4. Current–Voltage Relation of OER Catalysis with Proton-Transport Limitations

For continuous operation of the electrocatalytic system (i.e., under steady-state conditions), the electric current density (J

_{e},

Section 2.2) needs to be equal to the proton current density (J

_{BH},

Section 2.3), which in the following is denoted by J:

By means of Equation (4) and Equation (17) we obtain:

with J

_{max} according to Equation (16).

Figure 2 shows current–voltage relations for a buffer with a pK

_{a} of 7 simulated based on Equation (20). At high potentials, the maximal current density, J

_{max}, is reached.

At low potentials and current densities, for J << J

_{max}, we obtain from Equation (20):

In this current regime, the slopes are equal to the intrinsic Tafel slope of the electrocatalyst material (s_{ET}, here assumed being 60 mV per decade), and the potential (vs. NHE) has a Nernstian dependence, i.e., decreases by 60 mV per pH unit.

At intermediate current densities, the slope is generally increased. This increase is minimal at pH 7 (for pH = pK

_{a}) and more pronounced at higher and lower pH. This is seen more clearly in

Figure 3, which shows the Tafel slope calculated by Equation (7).

The maximal current density is determined by the concentration of the unprotonated buffer molecules and, thus, increases with increasing electrolyte pH. In clear contrast, the pH dependence of the Tafel slope at intermediate current densities is characterized by a minimum at the pK_{a} of the buffer molecule. In the following, a rough approximation is presented that relates the Tafel slope to the capacity of the buffer molecules to stabilize the pH, i.e., the buffer capacity β_{B}.

Equation (7) implies for the total Tafel slope:

with

We use the following identity:

Substituting J by the expression derived in

Section 2.3 for J

_{BH} (Equation (15)) yields:

and

Using a highly approximative approach, we assume that X is close to c

_{H}. For X = c

_{H}, we obtain:

with

β

_{B} is the pH-buffering capacity of the B

^{H}/B

^{−} buffer system of the electrolyte, which is maximal for c

_{H} = K

_{B} (pH = pK

_{aB}). Equation (27) correctly predicts the minimum of the Tafel slope at pH values close to the pK

_{a} of the buffer molecules (

Figure 4).