Modelling the (Essential) Role of Proton Transport by Electrolyte Bases for Electrochemical Water Oxidation at Near-Neutral pH

Abstract

The oxygen-evolution reaction (OER) in the near-neutral pH-regime is of high interest, e.g., for coupling of OER and CO₂-reduction in the production of non-fossil fuels. A simple model is proposed that assumes equal proton activities in the catalyst film and the near-surface electrolyte. Equations are derived that describe the limitations relating to proton transport mediated by fluxes of molecular “buffer bases” in the electrolyte. The model explains (1) the need for buffer bases in near-neutral OER and (2) the pH dependence of the catalytic current at high overpotentials. The latter is determined by the concentration of unprotonated buffer bases times an effective diffusion constant, which can be estimated for simple cell geometries from tabulated diffusion coefficients. The model predicts (3) a macroscopic region of increased pH close to the OER electrode and at intermediate overpotentials, (4) a Tafel slope that depends on the reciprocal buffer capacity; both predictions are awaiting experimental verification. The suggested first-order model captures and predicts major trends of OER in the near-neutral pH, without accounting for proton-transport limitations at the catalyst–electrolyte interface and within the catalyst material, but the full quantitative agreement may require refinements. The suggested model also may be applicable to further electrocatalytic processes.

1. Introduction

The water oxidation reaction (or oxygen evolution reaction, OER) plays a key role in the global challenge towards the production of sustainable fuels [1,2,3]. In this framework, certain interesting applications require a catalyst that can be operated with reasonable efficiency in the neutral pH regime. Examples are the coupling of water oxidation with CO₂ reduction [4,5], the coupling of a photon absorber that can be corroded in extreme pH [6,7], and safety issues in home-localized devices [8,9]. However, operating the catalytic system at neutral pH can introduce mass-transport limitations, which are reported to prevent the development of efficient devices [10,11]. Local mass-transport limitation effects can be calculated based on the catalyst surface morphology [12]. The water oxidation reaction entails the release of four protons per each molecule of water produced. Thus, a high proton-transport capacity of the electrolyte is needed to prevent the formation of a local pH gradient. The choice of the electrolyte can strongly affect the catalytic performances [13] and has been the object of numerous literature studies [14,15,16,17,18,19]. In the Co oxide catalyst of the cobalt–phosphate type described first by Kanan and Nocera [20], we reported that the catalytic activity at high current density is proportional to the availability of unprotonated buffer molecules for a variety of tested buffer electrolytes [15]. The same has been observed for a Mn-based catalyst in phosphate buffer [21]. A related effect was observed for Mn-based catalysts where catalytic performances improved when the buffer pK_a was increased, which has been attributed to a coupled proton–electron transfer mechanism between the catalyst and the buffer molecules [22]. Other studies, mostly based on analysis of CV experiments, suggest that the catalytic activity is mostly limited by the diffusion of the buffer molecules [23] or the protons [19] inside the bulk of the catalyst.

One of the most commonly used diagnostic tools in electrocatalysis is the study of the current-potential relation at steady-state (so-called Tafel plot). The interpretation of the data is often based on the Butler–Volmer equation [24,25]. The Tafel slope represents the increment in overpotential needed to obtain an increment of one decade in the current density; it is a key feature for assessment of catalyst efficiencies. Theoretical Tafel slope values have been derived for different water oxidation mechanisms [26] and for different rate-limiting steps [27]. Shinagawa et al. reported a large collection of experimental values for the Tafel slope. Only some of them can be explained by existing models [27]. Bediako et al. developed a model for bulk active catalysts at neutral pH and proposed various Tafel-slope values for different rate-controlling phenomena [28], e.g., diffusion of protons or buffer components inside the film or at the film–solution interface. Corresponding experiments on a CoO_x catalyst resulted in a Tafel slope of 59 mV dec⁻¹ for high buffer concentrations and 118 mV dec⁻¹ for buffer-free solutions, in accordance with their model predictions. In the intermediate cases, they predicted a transition between the two behaviors controlled by the buffer diffusion in solution. Here we present a model to describe the proton concentration in the electrolyte close to the catalyst in different currents and pH regimes, and its effect on catalytic activity and Tafel slope. Unconventional values for the Tafel slope and a strong local pH increase are predicted, inter alia furnishing a theoretical background to previous experimental findings [15].

2. Results

2.1. Outline of the Model

We are aiming at a simple model that describes the influence of proton transport mediated by buffer molecules on the catalytic activity. Therefore, we consider three regions (illustrated in Figure 1):

• Region A: the (volume-active) OER catalyst film (Cat)—with its proton activity, X_Cat, and electric overpotential, ΔV.
• Region B: a solvent layer (L) at the outer catalyst-film surface, the “buffer layer”—with its proton activity, X_Layer.
• Region C: the bulk solvent (S)—with its proton concentration, c_H (pH = −log c_H).

It is furthermore assumed that the effective proton concentrations, that is, the proton activities in A and B are equal once fluxes (of electrons and protons) and concentrations have reached a steady-state. Thus, they can be described by a single quantity, which in the following is denoted as X (X_Cat = X_Layer = X). This approximation reduces the number of effective compartments from three to two. The assumption is exact only if proton transport inside the catalyst film and proton transfer from the film to the solvent are kinetically not limiting; these processes are assumed to be clearly faster than the proton transport over macroscopic distances within the solvent phase.

The value of X determines both the rate of water oxidation within the catalyst and the rate of proton transfer from the buffer layer towards the bulk solvent. The proton transfer is mediated by protonation of a buffer base, which diffuses to the bulk solvent.

2.2. Catalytic Electron Flux

The quantity µ_x corresponds to the chemical potential of the protons within the catalyst film according to:

µ_x = k_BT ln X

(1)

It is now assumed that the catalytic electrical current (J_e) can be described by the following Butler–Vollmer type relation:

J_e = J_e0 exp [α_ET(eΔV − µ_x)/k_BT]

(2)

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_BT):

ln J_e = ln J_e0 + α_ET β ΔV − α_ET ln X

(3)

Rearrangements lead to:

ΔV = s_ET ln J_e/J_e0 + β⁻¹ ln X

(4)

with

s_ET = (α_ET β)⁻¹

(5)

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 (log₁₀) 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:

s_10-ET = ln10 s_ET

(6)

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

s = d ΔV/d ln J_e = s_ET + β⁻¹ d ln X(J_e)/d ln J_e

(7)

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:

B_L⁻ X/B_L^H = K_aB

(8)

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 B₀ = B_L⁻ + B_L^H we obtain:

B_L^H = B₀ X/(K_aB + X)

(9)

(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^H = B₀ c_H/(K_aB + c_H)

(10)

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. B₀ 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):

J_BH^L=>S = F dB_L^H/dt = k_D B_L^H

(11)

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

J_BH^S=>L = F dB_S^H/dt = k_D B_S^H

(12)

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.

J_BH = J_BH^L=>S − J_BH^S=>L = k_D (B_L^H − B_L^S)

(13)

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 m²/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, B₀ − 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):

k_D = F D/(d_el/2)

(14)

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

J_BH = k_D B₀ [X/(K_aB + X) − c_H/(K_aB + c_H)]

(15)

Equation (15) implies that a maximal proton current density, J_max, exists for X >> K_aB.

J_max = k_D B₀ [1 − c_H/(K_aB + c_H)] = k_D B₀ [K_aB/(K_aB + c_H)]

(16)

We note that J_max is not only proportional to the concentration of buffer molecules (B₀) 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:

J_max = k_D B₀ [B_S⁻/(B_S⁻ + B_S^H)]

(17)

Thus, J_max scales with the fraction of unprotonated buffer molecules.

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

X = [c_H + (J_BH/J_max) K_aB]/[1 − (J_BH/J_max)]

(18)

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:

J_e = J_BH = J

(19)

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

ΔV = s_ET ln J/J_e0 + β⁻¹ ln {[c_H + (J/J_max) K_aB]/[1 − (J/J_max)]}

(20)

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):

ΔV = s_ET ln J/J_e0 − β⁻¹ pH

(21)

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:

s = s_ET + s_PT

(22)

with

s_PT = β⁻¹ d ln X/d ln J

(23)

We use the following identity:

d ln X/d ln J = (J/X) (dJ/dX)⁻¹

(24)

Substituting J by the expression derived in Section 2.3 for J_BH (Equation (15)) yields:

dJ/dX = K_DB₀ K_aB/(K_aB + X)²

(25)

and

s_PT = β⁻¹ J [K_DB₀ K_aB X/(K_aB + X)²]⁻¹

(26)

Using a highly approximative approach, we assume that X is close to c_H. For X = c_H, we obtain:

s_PT = β⁻¹ J K_D/β_B

(27)

with

β_B = B₀ K_aB c_H/(K_aB + c_H)²

(28)

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).

3. Discussion

In spite of its simplicity, the described model captures important aspects of the proton transport limitation in water oxidation at near-neutral pH.

In the absence of an explicit buffer base, either only water molecules or hydroxide ions can serve as proton-transporting buffer bases. Equation (15) shows that for H₂O molecules as proton acceptors, only minimal current densities can be reached in the neutral-pH region, because the local and bulk proton concentrations is, by many orders of magnitude, smaller than K_aB (for H₃O⁺/H₂O, formal pK_aB of −1.74 or K_aB of 55 M); the J_max of Equation (16) can never be reached (because always X << K_aB). For OH⁻ ions as proton acceptors, their low concentration in the near-neutral pH regime results in an extremely small value of J_max in Equation (16) (for H₂O/OH⁻, formal pK_aB of 15.74 or K_aB of 1.8 × 10⁻¹⁶). For water oxidation in the acidic and alkaline pH regime, however, Equations (15) and (16) predict sizeable current densities. We note that for water or OH⁻ ion as proton acceptors, Equation (15) is no longer exact. At very low and very high pH, this equation becomes increasingly imprecise.

At high overpotentials, the catalytic current approaches a maximum value (J_max according to Equation (16)). Its pH dependence is determined by the concentration of unprotonated buffer molecules in the bulk electrolyte. This prediction has been verified by Klingan et al., where it was shown for nine buffer bases that the pH dependence of the current density at high overpotentials is determined by the protonation state of the respective buffer base [15].

At very low overpotentials, the catalytic current is not affected by proton transfer to the bulk electrolyte, but exclusively by the intrinsic electrocatalytic properties of the material. It is perfectly well described by a Tafel slope behavior with an increase in current density by a factor of 10 per pH unit at constant electric potential. At constant current density, a decrease in the required electric overpotential by ln(10) e/kT (about 60 mV at room temperature) per pH unit is predicted, as repeatedly observed in experimental studies [29,30,31].

At intermediate overpotentials, the Tafel slope increases in comparison to its low-overpotential value, which has been assigned before to mass transport limitations and largely can be avoided by the employment of a rotating-disc electrode [28,32,33,34]. In the described model, the mass transport limitations result from diffusion of protonated and unprotonated buffer bases in the electrolyte, without involving buffer base diffusion within the catalyst film. The diffusion process is described by an effective diffusion constant, k_D, per mol of buffer molecules. The use of a rotating disc electrode (or just stirring of the solution) would increase the value of k_D. The model predicts a macroscopic acidification in the electrolyte close to the catalyst surface, a prediction that still requires experimental verification. Moreover, the model predicts that at intermediate overpotentials the Tafel slope does not reflect the concentration of unprotonated buffer molecules, but rather the pH-buffering capacity of the B_H/B⁻ pH-buffer system (Equation (28), Figure 4). This dependence on an effective buffering capacity has not been reported before (to our best knowledge), and it, thus, represents another testable prediction of the presented model.

Application of the model in the extreme alkaline or acidic pH region (pH < 2 or pH > 12) also is possible, by substituting the unprotonated/protonated buffer base with the relevant water species: OH⁻/H₂O at alkaline pH, H₂O/H₃O⁺ at acidic pH. Then Equation (15) predicts sizeable current densities, inter alia because the diffusion coefficient (D) is about 10 times higher for H₃O⁺ and five times higher for OH⁻ than for the commonly used buffer bases (e.g., HPO₄²⁻/H₂PO₄⁻). We note that for water or OH⁻ ion as proton acceptors, Equation (15) is no longer exact; at very low pH (<−1) and very high pH (>15) the imprecision will be sizable. At high current densities, the influence of Ohmic electrolyte conductivity and the associated electric field gradient is likely to result in further deviations from Equation (15). In any event, the here presented model is qualitatively in line with the long-standing finding that at elevated concentrations of OH⁻ or H₃O⁺ ions, high current densities can also be ensured in the absence of an additional proton-accepting base, in clear contrast to the near-neutral pH regime,

4. Conclusions

The following phenomena are explained or predicted by the here developed model:

• For pH values ranging from about 2 to 12, the OER reaction essentially requires buffer molecules that facilitate proton transport. In the absence of an explicit buffer base, either only water molecules or hydroxide ions could serve as proton-transporting buffer bases; both cannot support significant current densities.
• In the presence of a buffer base, the maximal current density is determined by the concentration of unprotonated buffer molecules and, thus, by the solution pH and the pK_a value of the used buffer, in line with experimental findings [15].
• At intermediate current densities, the Tafel slope is increased in comparison to the intrinsic Tafel slope measured at low current densities. The Tafel slope increase is avoided only when macroscopic mass transport limitations are negligible, which is achievable experimentally, e.g., by employing catalyst films deposited on rotating disc electrodes. This behavior has been observed repeatedly and is, at least, qualitatively well explained by the presented model. Whether quantitative agreement with the here described first-order model can be reached is still unclear; extension of the model to include also (e.g.,) limitations of proton transport within the catalyst or at the catalyst–solvent interface may be required.
• At intermediate current densities, the Tafel slopes reflect the buffering capacity of the B^H/B⁻ couple, a prediction that still requires experimental verification.
• The limitations by proton transport are assigned to proton transport in the electrolyte phase. This assumption predicts (macroscopic) acidification of the electrolyte near the electrode surface, a prediction that still awaits experimental verification. Additional factors may contribute to proton transport limitations, specifically rate limitations by proton transfer at the catalyst–electrolyte interface or within the catalyst film. In the present model, these are excluded by assuming equal proton activities within the catalyst film and the near-surface electrolyte.