# Revisiting basic assumptions of the hydrogen evolution reaction (HER) for water electrolysis at near-neutral pH

Junzhe Sun, Luc Etcheverry, Alain Bergel

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Junzhe Sun, Luc Etcheverry, Alain Bergel. Revisiting basic assumptions of the hydrogen evolution reaction (HER) for water electrolysis at near-neutral pH. Chemical Engineering Journal, 2025, 507, pp.160490. 10.1016/j.cej.2025.160490. hal-04990908

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# Revisiting basic assumptions of the hydrogen evolution reaction (HER) for water electrolysis at near-neutral pH

Junzhe Sun, Luc Etcheverry, Alain Bergel*

Laboratoire de Génie Chimique, Université de Toulouse, CNRS, INPT, UPS, Toulouse, France

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## ARTICLE INFO

### Keywords:

- Modelling

- Mass transport

- Buffer

- Proton reduction

- Electrochemical deprotonation

- pH gradient

## ABSTRACT

Water electrolysis at near-neutral pH is attracting increasing attention due to its eco-compatibility, ease of use and lower capital expenditure in comparison to the water electrolysis processes currently on the market. However, the mechanism of hydrogen evolution reaction (HER) at near-neutral pH remains largely debated. The HER kinetics has been explained, in the absence of buffer, by the occurrence of successive proton and water reductions and, in the presence of buffer, by involving a direct electrochemical reaction of the buffer species. These hypotheses were mainly supported by considering the mass-transport limiting currents observed on the current-potential records. Here, a simple one-dimensional numerical model was designed using the sole water reduction reaction and focussing the mass transport of the species involved in pH control. The model perfectly recovered the three-phase shape of the intensity-potential curves. The experimental and numerical results matched well, whether in the absence or presence of buffer and in a large range of pH values. The model also succeeded in explaining experimental findings reported in the literature that compared the efficiency of phosphate and borate buffers for HER. It was thus shown that the single water reduction reaction was sufficient to explain HER at near neutral pH, provided that the impact of the local pH on its kinetics was taken into account. This drastic simplification of the basics will be a major help in simplifying the engineering involved in water electrolysis at near-neutral pH.

## 1. Introduction

For hydrogen to become a key energy carrier [1,2], it remains necessary to develop low-carbon production routes that could compete with production from the fossil sources [3,4]. Water electrolysis has long been seen as a promising technology [5-7], and the three main existing electrolysis technologies, namely alkaline [8,9], proton exchange membrane [10,11] and solid oxide electrolyte [12,13], have already been developed at industrial scale [14]. These three electrolysis technologies are characterized by the extreme conditions they implement: highly alkaline or acidic electrolytes or high temperature, respectively, which raise a number of issues, particularly in terms of durability, safety and environmental acceptability [15]. The cost of these technologies remains high because of the need of catalysts and membranes that remain stable in strong acidic or alkaline conditions. These conditions exacerbate the corrosion issues, and increase the cost of maintenance or disposing of the materials at end of life.

Water electrolysis at near-neutral pH is an emerging technology, which may overcome most of these issues [16]. This technology uses

electrolytes that contain a limited number of compounds, generally a weak acid and its salt, used as buffer compounds to control the pH. Phosphate buffer solution (PBS) is most often used [16-18], hydrogenocarbonates [19] and borates [20] have also been considered, and many other weak acids have been investigated as well, such as citric acid [21] and amino acids [22]. In milder conditions, corrosion is less severe, catalyst materials more stable and maintenance easier and safer, potentially making it possible for water electrolysis cells to be used as household devices [23]. Beyond the core area of direct application, the study of water electrolysis at near-neutral pH could also pave the way for direct seawater electrolysis, an alternative to avoid conflicts relating to a limited freshwater resource [24,25]. Natural water electrolysis has the great advantage of not needing additives but implies specific challenges relating to aggressive $Ca^{2+}/Mg^{2+}$ precipitation [26,27]. Similarly, as far as the cathode side is concerned, the hydrogen evolution reaction (HER) at near-neutral pH is also a key process for microbial electrolysis, an emerging process that could significantly reduce the electrical energy cost of hydrogen production [28-31].

A great number of recent studies have investigated water electrolysis

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: alain.bergel@toulouse-inp.fr (A. Bergel).</footnote>
at near-neutral pH, focusing on catalyst design for both HER [16,32,33] and oxygen evolution reactions [33–35]. Specific electrolyte engineering has also been developed, notably by the Takanabe research group [36–38]. Nevertheless, as noticed in a recent review article [16], the current understanding of HER in near-neutral electrolytes remains far behind that in acidic and alkaline conditions. In acidic electrolytes, the only primary reactant is the proton:

$$2H^+ + 2e^- \rightarrow H_2 \quad (1)$$

while in alkaline electrolytes, water is the sole primary reactant:

$$2H_2O + 2e^- \rightarrow H_2 + 2OH^- \quad (2)$$

In contrast, in near neutral pH conditions, even in the most simple electrolytes that contain only water and an inert salt, HER is more complex and shows a three-phase profile (Fig. 1A). This profile looks like a well-known diffusion-limited profile and has therefore been attributed for decades to the successive involvement of both reactants: first protons and then water [39–41]. It was postulated that, at low overpotential (phase I in Fig. 1A), H⁺ can HER (Reaction 1). When the overpotential increases, the depletion of H⁺ near the cathode surface becomes steeper, and the current plateau (phase II) was assumed to be a proton diffusion-limited current. When the overpotential is high enough, water molecules, in turn, are reduced directly at the cathode (Reaction 2) [16]. The three-phase behaviour is thus explained by the switch from proton reduction to water reduction controlled by mass transport of the protons. The switch is assumed to be located around pH 5 [39].

The objective in designing hydrogen production electrolysers is to sustain the highest possible HER current at the highest possible potential. Accordingly, the weak acids (AH) used around neutral pH mitigate the local proton depletion at the cathode surface by producing protons via the acid-base dissociation reaction [40,42–44]:

$$AH \rightleftharpoons A^- + H^+ \quad (3)$$

However, this mechanism has long been the subject of debate. In the 1980s [45] and 90s, a few pioneering studies observed that $H_2PO_4$ ions induce unexpected reduction current waves or current peaks on platinum electrodes. The proposed explanation was that phosphate species [45–49] and other weak acids [47,50] may undergo direct electrochemical reduction on platinum and silver electrodes:

$$AH + e^- \rightarrow 1/2H_2 + A^- \text{ on the electrode surface} \quad (4)$$

followed by acid-base dissociation:

$$A^- + H_2O \rightarrow AH + OH^- \text{ in solution} \quad (5)$$

which results in AH cycling. This hypothesis of direct cathodic reduction of phosphate species was later defended by our group using stainless steel electrodes [18,51,52]. We proposed the term “electrochemical deprotonation” as a more accurate description than the original name of “phosphate electroreduction” [45], as the understanding is that phosphate is not reduced but rather releases a hydrogen atom on the cathode surface. Actually, according to this scheme, the weak acid would act as a homogeneous catalyst of water reduction [53].

The occurrence of electrochemical deprotonation reactions has generally been supported by the observation of current waves, similar to that presented in Fig. 1.A, with the current plateau value proportional to the concentration of the weak acid. This plateau has been assumed to be controlled by the mass transport of the buffer species that reacts on the cathode surface [5,41,47]. Once again, as in the case for protons in the absence of buffer, rapid reasoning about the mass transport led to hypotheses about the species that react at the electrode. Recently, a comprehensive analysis of the reactions that might contribute to HER appears to confirm that all the proton-containing species, i.e. H⁺ (or $H_3O^+$), $H_2O$ and AH, react directly on the electrode surface [41]. Similarly, among studies comparing the capacity of different buffers to catalyse HER, some attribute different buffer efficiencies to the different rates of electrochemical deprotonation reaction [20,54].

However, other studies stick with the classic hypothesis on the sole reductions of protons and water, leaving out the hypothesis of electrochemical deprotonation of the buffer species [44,49,55]. The theoretical studies by the Mayrhofer group [40,56] also doubted the occurrence of electrochemical deprotonation of weak acids. By modelling mass transfer of the protons and the buffer species, they reproduced the current waves that were observed on voltammetric records due to the presence of weak acids (acetic acid, borate and phosphate). They explained these waves through the value of the interfacial pH and the switch proton/water, without any direct electrochemical deprotonation hypothesis. This approach involved the reductions of both protons and water, and transport of the charged species by migration was neglected because of the presence of the inert salt $NaClO_4$ at high concentration.

To sum up, in all cases, HER at near-neutral pH is routinely hypothesized to involve successively H⁺ and $H_2O$. In contrast, the hypothesis of electrochemical deprotonation of weak acids is, for many, considered speculative. A recent review article pointed out that the exact mechanism of HER in near-neutral electrolytes is still highly controversial because no evidence is available to support the assertion

<figure><img src="image_3.png" /><figcaption>Fig. 1. Schemes of the concentration profiles during potential scan. A) Common 3-phase shaped current-potential curves recorded at near-neutral pH. B) No buffer: H⁺ concentration profile for the « low j » zone. C) No buffer: H⁺ and $OH^-$ concentration profiles for the « high j » zone. D) With buffer: $AH^-$ and $A^{2-}$ concentration profiles for the « low j » zone. E) With buffer: $AH^-$ and $A^{2-}$ concentration profiles for the « high j » zone. The bulk concentrations $C^b_{AH}$ and $C^b_{A2}$ were 90 mM and 10 mM, respectively.</figcaption></figure>
that weak acid molecules can serve as direct reactant for HER [16]. As a consequence, it is often tacitly assumed that weak acids promote HER in one or both of these two ways [5,57]: mitigate the pH gradient on the electrode surface (Reactions 3 and 1), and/or directly deprotonate on the electrode surface (Reactions 4 and 5).

Which of species H⁺, H₂O and the buffer species are the actual reactant(s) on the cathode surface? This question has often been answered, more or less tacitly in the case of protons, based on a rapid analysis of current-potential curves (Fig. 1.A). Current plateaus appear, the intensity of which is proportional to the concentration of the protons or the buffer species. It has therefore been deduced that these transport-limited currents indicate that the corresponding species react at the electrode surface. Mass transport appears to be the stumbling block that needs to be considered if we are to make any progress in understanding HER mechanism in near-neutral conditions. In this aim, the present study designed a numerical approach to model mass transports in water electrolysis under near-neutral conditions. The model is based on the single reaction of water reduction at the cathode, i.e. only Reaction (2). The reaction was modelled using conventional Butler-Volmer kinetics, factoring in the value of the interfacial pH.

The model matched with several experimental findings, including some reported in the literature. Notably, it showed that there is no need to resort to the hypothesis of switching from proton reduction to water reduction, and it ruled out the need for an electrochemical deprotonation reaction from the buffer species. The model stated thus that water electrolysis under near-neutral conditions can be approached by using the sole water reduction reaction. The three phases observed on the current potential curves are driven by the impact of the interfacial pH on the water reduction kinetics. We are obviously not able to answer the question of which species is/are the actual reactant(s) at the cathode, but the model shows that mass transport can no longer be invoked to support that protons and buffer species react, contrary to what has often been more or less tacitly assumed. The model was elaborated with simplifying methods that are commonly used in the field of chemical engineering. This allowed us to devise an easy-to-use set of equations in the aim of convincing the broadest possible audience and prompting researchers to further develop this approach.

## 2. Numerical model

### 2.1. General framework

At the cathode surface, only the water reduction reaction (Reaction 2) was considered, using conventional Butler-Volmer kinetics:

$$j = j_0 \exp \left[ \frac{-\alpha n F}{RT} (E - E^0) \right] \quad (6)$$

where j is the current density (A m⁻²), j₀ is the exchange current density, α is the transfer coefficient, n = 2 is the number of electrons exchanged, F is the Faraday constant (96 485 C mol⁻¹), R is the gas constant (8.31 J mol⁻¹ K⁻¹), T is the temperature (K), E is the potential (V), and E⁰ is the formal potential (V).

The formal potential of Reaction 2 depends on the activity of the OH⁻ ions at the cathode surface as:

$$E^{0'} = E^{0} - \frac{2.3RT}{F} \log \left( \gamma_{OH^{-}} - \frac{C_{OH}^{0}}{C_{0}} \right) \quad (7)$$

where E⁰ is the standard water reduction potential, equal to -0.828 V/ SHE at 25 °C under a hydrogen pressure of 1 atm, γOH⁻ is the activity coefficient of the OH⁻ ions, C⁰ is the concentration of the OH⁻ ions at the cathode surface, C₀ = 1 M is the standard concentration, and the 2.3RT/F ratio equals 0.059 V at 25 °C. This equation can be transformed using the value of the pH at the cathode surface (pH⁰) (see Supplementary Data for details):

$$E^{0'} = -0.059 \, pH^{0} \quad (8)$$

The model was designed in steady state condition for galvanostatic operation, i.e. with an applied constant current density (j). The final objective was to plot current density-potential curves (E - j curves). For each given value of j, modelling mass transport provided the value of pH⁰, which was plugged into Equation (8) to get E⁰, from which the value of E was then calculated from Equation (6) transformed as:

$$E = E^{0'} - \frac{RT}{\alpha n F} \ln \left( \frac{j}{j_0} \right) \quad (9)$$

### 2.2. Modelling mass transport

Mass transport was modelled along the x-axis (one-dimensional model), perpendicular to the cathode surface, using the Nernst hypothesis. Accordingly, a stagnant diffusion layer of thickness δ was assumed to be present on the surface of the cathode, while the solution bulk beyond this layer was in well-stirred conditions. Inside the Nernst layer, the molar flux of a species i (φᵢ, mol s⁻¹ m⁻²) is expressed by the Nernst-Planck equation [58], in which the convective component is negated:

$$\phi i = -Di \frac{dC_i}{dx} - \frac{z_i F}{RT} DiC_i \frac{d\phi}{dx} \quad (10)$$

where Di is the diffusion coefficient (m² s⁻¹), Cᵢ is the concentration (mol m⁻³), zᵢ is the charge (positive, negative or nihil for cations, anions and non-charged species, respectively), and φ is the electrostatic potential (V). In the Nernst layer, neglecting the few molecular layers of the electrical double layer, the electrostatic potential varies linearly, and its gradient can be approximated by (see Supplementary Data for details):

$$\frac{d\phi}{dx} = \rho j \quad (11)$$

where ρ is the ionic resistivity of the electrolyte (Ω m). Equation (10) thus becomes:

$$\phi i = -Di \frac{dC_i}{dx} - ziDi \Lambda C_i \quad \text{with} \quad \Lambda = \frac{F}{RT} \rho j \quad (12)$$

At steady state, if the species is not involved in a chemical reaction, its molar flux is constant throughout the Nernst layer:

$$\frac{d\phi_i}{dx} = 0 \quad (13)$$

so that mass transport of the species i is described as:

$$\frac{d^2 C_i}{dx^2} + zi\Lambda \frac{dC_i}{dx} = 0 \quad (14)$$

### 2.3. Water reduction without buffer at low current density (Fig. 1B)

In the absence of a buffer species, it is assumed that the OH⁻ ions produced at the cathode surface react immediately with the protons that come from the bulk of the solution through the Nernst layer. Proton transport through the Nernst layer is described by Equation (14) with zH⁺ = 1, which gives:

$$\frac{d^2 C_{H+}}{dx^2} + \Lambda \frac{dC_{H+}}{dx} = 0 \quad (15)$$

At steady state, whatever the value of x, the molar flux of H⁺ through the Nernst layer is equal to its consumption rate at the cathode surface. At the cathode surface, OH⁻ ions are produced by the single Reaction (2). It was assumed that protons react immediately with the produced OH⁻ ions so that the molar flux of H⁺ is controlled by the current density j:
$$ \forall x, \phi_{H+} = -D_{H+} \frac{dC_{H+}}{dx} - D_{H+} \Lambda C_{H+} = \frac{-j}{F} \qquad (16) $$

At $x = \delta$, the $H^+$ concentration is equal to the concentration in the solution bulk:

$$ C_{H+}|_{x=\delta} = C_{H+}^{b} \qquad (17) $$

Solving the set of Equations (15–17) gives the profile of $H^+$ concentration in the Nernst layer (see Supplementary Data for details):

$$ C_{H+} = (C_{H+}^{b} - \Gamma_{H+}) \exp(\Lambda(\delta - x)) + \Gamma_{H+} \\ \text{where } \Gamma_{H+} = \frac{RT}{F^2 \rho D_{H+}} \qquad (18) $$

from which the expression of the interfacial $H^+$ concentration is extracted:

$$ C_{H+}^{0} = (C_{H+}^{b} - \Gamma_{H+}) \exp(\Lambda\delta) + \Gamma_{H+} \qquad (19) $$

leading to the value of the interfacial pH:

$$ pH0 = - \log \left( \gamma_{H+} \frac{C_{H+}^{0}}{C_{0}} \right) \qquad (20) $$

where $\gamma_{H+}$ is the activity coefficient of the proton, and $C_0 = 1$ M is the standard concentration. The value of $pH^0$ entered into Equation (8) gives $E^{0'}$, which is introduced into Equation (9) to get the value of the cathode potential $E$ that corresponds to the chosen value of the applied current density $j$.

## 2.4. Water reduction without buffer at high current density (Fig. 1C)

The above approach is no longer valid for high values of current density, at which the molar flux of protons through the Nernst layer is no longer able to balance the fast production of $OH^-$ ions at the cathode. For instance, in Equation (19), high $j$ values result in negative values of the interfacial proton concentration. In this case, the production of $OH^-$ is no longer offset by the $H^+$ flux. Consequently, $OH^-$ moves away from the electrode surface until it meets $H^+$ and reacts with it. $OH^-$ and $H^+$ come into contact at a distance $\delta_R$ from the electrode surface ($\delta_R < \delta$) at a plane called the reaction plane.

The “reaction plane” concept has been commonly used in the field of chemical engineering to model fast reactions at gas-liquid interfaces [59]. The reactive species meet at a surface inside the diffusion layer, where they react together instantaneously. The concept has also been adapted to fast reactions involving electrochemically generated species [60]. Here, $OH^-$ and $H^+$ were assumed to react instantaneously when they meet at the reaction plane. For the sake of simplicity, it was assumed that their concentrations were nihil at the reaction plane:

$$ At \ x = \delta_R: $$

$$ C_{OH-}|_{x=\delta_R} = 0 \qquad (21) $$

$$ C_{H+}|_{x=\delta_R} = 0 \qquad (22) $$

While, strictly speaking, this hypothesis is not a physical possibility, it is used to express the instantaneous reaction between the two species. Using a more sophisticated hypotheses, such as the water ionic product ($\gamma_{OH-} \cdot C_{OH-} \cdot \gamma_{H+} \cdot C_{H+} = 10^{-14} \text{ mol}^2 \text{ L}^{-2}$), could be considered, but this would result in more cumbersome equations without adding any significant precision to the results.

According to this scheme, Equation (18) remained valid in the outer part of the Nernst layer, from $x = \delta_R$ to $x = 0$. Equation (18) was therefore used to calculate the value of $\delta_R$, assuming that the concentration of $H^+$ is nihil at the reaction plane:

$$ \delta_R = \delta + \frac{1}{\Lambda} \ln\left(1 - \frac{C_{H+}^b}{\Gamma_{H+}}\right) \qquad (23) $$

Then, the $OH^-$ concentration profile in the inner part of the Nernst layer, from $x = 0$ to $x = \delta_R$, was obtained by solving Equation (14) with $z_{OH-} = -1$:

$$ \frac{d^2 C_{OH-}}{dx^2} - \Lambda \frac{d C_{OH-}}{dx} = 0 \qquad (24) $$

considering the steady-state condition:

$$ \forall x, \phi_{OH-} = -D_{OH-} \frac{dC_{OH-}}{dx} - D_{OH-} \Lambda C_{OH-} = \frac{j}{F} \qquad (25) $$

and the boundary condition (Equation 21) at the reaction plane, leading to:

$$ C_{OH-} = \Gamma_{OH-} (1 - \exp(\Lambda(x - \delta_R))) \qquad (26) $$

The value of the $OH^-$ concentration at the cathode surface is then calculated as:

$$ C_{OH-}^{0} = \Gamma_{OH-} (1 - \exp(-\Lambda \delta_R)) \qquad (27) $$

which gives the interfacial pH value:

$$ pH^0 = 14 + \log \left( \gamma_{OH-} \frac{C_{OH-}^{0}}{C_{0}} \right) \qquad (28) $$

to be plugged into Equation (8) and then (9) to get the value of $E$.

## 2.5. Water reduction with $AH^-$/$A^{2-}$ buffer at low current density (Fig. 1D)

The buffer species was considered here as the charged form $AH^-$, which reacts with $OH^-$ by producing the di-anion $A^{2-}$:

$$ AH^- + OH^- \rightleftharpoons A^{2-} + H_2O \qquad (29) $$

This $AH^-$/$A^{2-}$ couple corresponds to the $H_2PO_4^-$/$HPO_4^{2-}$ and $HCO_3^-$/$CO_3^{2-}$ couples, which are the most commonly used in the framework of water electrolysis at near neutral pH. It is commonly assumed that the dissociation reaction (29) is very fast in comparison to the electrochemical reaction and mass transport, so that it is considered at equilibrium [55]. Accordingly, if the current density is not too high, the $AH^-$ species can reach the cathode surface, where it undergoes an immediate reaction with $OH^-$. In this scenario, the $AH^-$ species play the same role as the protons in the case above.

Through the Nernst layer, the mass transport of $AH^-$ and $A^{2-}$ are modelled by Equation (14) with $z_{AH-} = -1$ and $z_{A2-} = -2$, respectively:

$$ \frac{d^2 C_{AH-}}{dx^2} - \Lambda \frac{d C_{AH-}}{dx} = 0 \qquad (30) $$

$$ \frac{d^2 C_{A2-}}{dx^2} - 2\Lambda \frac{d C_{A2-}}{dx} = 0 \qquad (31) $$

At steady state, the molar flux of $AH^-$ to the electrode is constant in the Nernst layer and balances the electrochemical production of $OH^-$:

$$ \phi_{AH-} = -D_{AH-} \frac{dC_{AH-}}{dx} + \Lambda D_{AH-} C_{AH-} = - \frac{j}{F} \qquad (32) $$

and the molar flux of $A^{2-}$ is equal and opposite:

$$ \phi_{A2-} = -D_{A2-} \frac{dC_{A2-}}{dx} + 2\Lambda D_{A2-} C_{A2-} = \frac{j}{F} \qquad (33) $$

At $x = \delta$:

$$ C_{AH-}|_{x=\delta} = C_{AH-}^{b} \qquad (34) $$
$$C_{A2-}|_{x=\delta} = C_{A2-}^{b} \qquad (35)$$

Equations (30) and (31) solved with their respective boundary conditions give:

$$C_{AH-} = (C_{AH-}^{b} + \Gamma_{AH-}) \exp(\Lambda(x - \delta)) - \Gamma_{AH-} \\ \text{where } \Gamma_{AH-} = \frac{j}{\Lambda FD_{AH-}} = \frac{RT}{F^2 \rho D_{AH-}} \qquad (36)$$

$$C_{A2-} = (C_{A2-}^{b} - \Gamma_{A2-}) \exp(2\Lambda(x - \delta)) + \Gamma_{A2-} \\ \text{where } \Gamma_{A2-} = \frac{j}{2\Lambda FD_{A2-}} = \frac{RT}{F^2 \rho D_{A2-}} \qquad (27)$$

from which the interfacial AH⁻ and A²⁻ concentrations are extracted:

$$C_{AH-}^{0} = (C_{AH-}^{b} + \Gamma_{AH-}) \exp(-\Lambda\delta) - \Gamma_{AH-} \qquad (38)$$

$$C_{A2-}^{0} = (C_{A2-}^{b} - \Gamma_{A2-}) \exp(-2\Lambda\delta) + \Gamma_{A2-} \qquad (39)$$

Assuming that the dissociation equilibrium of AH⁻ is ensured at the electrode surface [55] gives the interfacial pH value:

$$pH^{0} = pKa - \log\left(\frac{\gamma_{AH-}C_{AH-}^{0}}{\gamma_{A2-}C_{A2-}^{0}}\right) \qquad (40)$$

where γAH⁻ and γA2⁻ are the activity coefficients of AH⁻ and A²⁻, respectively. Again, the value of pH⁰ used with Equations (8) and (9) gives the value of the cathode potential E that corresponds to the chosen value of j.

## 2.6. Water reduction with AH⁻/A²⁻ buffer at high current density (Fig. 1E)

The above approach is no longer valid for high values of current density, at which the molar flux of AH⁻ is no longer able to balance the fast production of OH⁻. As explained above, the reaction plane concept can be used in this case. At high j values, the production of OH⁻ is no longer offset by the AH⁻ flux at the electrode surface and OH⁻ moves away from the electrode surface until it meets AH⁻ and reacts with it at the reaction plane. Considering that dissociation of the buffer species is very fast in comparison to the mass transport rates [55], OH⁻ and AH⁻ were assumed to react instantaneously at the reaction plane. For the sake of simplicity, their concentrations were taken to be nihil at the reaction plane:

At x = δR:

$$C_{OH-}|_{x=\delta R} = 0 \qquad (41)$$

$$C_{AH-}|_{x=\delta R} = 0 \qquad (42)$$

According to this scheme, Equations (38) and (39) were still valid in the outer part of the Nernst layer, from x = δR to x = δ. Equation (38) was thus used to calculate the value of δR, i.e. the value of x at which the concentration of AH⁻ is nihil (Equation 42), which gave:

$$\delta_R = \delta - \frac{1}{\Lambda} \ln\left(1 + \frac{C_{AH-}^b}{\Gamma_{AH-}}\right) \qquad (43)$$

A more sophisticated hypothesis, based on the dissociation reaction equilibrium, would result in more cumbersome equations without adding any significant precision to the results. The model's sensitivity to this assumption is discussed in the Supplementary Data file.

In the inner part of the Nernst layer, from x = 0 to x = δR, transport of the OH⁻ species is identical to the case in the absence of buffer (section 2.4). The value of δR resulting from Equation (43) is simply plugged into Equation (27) to obtain C₀OH⁻, which leads to pH⁰ with Equation (28) and

finally to E with Equations (8) and (9), as usual.

The same approach was developed for the AH/A⁻ couple, i.e. for a buffer species that is not charged, as is the case, for instance, with borate (B(H₂O)OH)₃/B(OH)₄. Calculations are detailed in the Supplementary Data file, and the resulting equations are gathered in Table S1.

# 3. Numerical data and experimental method

## 3.1. Model general information

The value of the physicochemical parameters used in the model are gathered in Table S2. The Butler-Volmer kinetics (Equation 6) is driven by the two parameters j₀ and α. α is sometimes expressed in terms of the Tafel slope (T_slope, see Supplementary Data file for details):

$$T_{\text{slope}} = \frac{2.3RT}{\alpha nF} \qquad (44)$$

The ionic resistivity of the electrolyte (ρ), to be used in Equation 16, was calculated as the ionic conductivity (K):

$$K = \sum_{i} \lambda_{i} C_{i} \qquad (45)$$

where λi is the molar ionic conductivity of each species i, and

$$\rho = \frac{1}{K} \qquad (46)$$

The activity coefficients were estimated through the Debye-Hückel equation:

$$\log(\gamma_i) = - \frac{0.51 z_i^2 \sqrt{I}}{1 + 3.3 r_i \sqrt{I}} \qquad (47)$$

where rᵢ is the radius of the solvated ion i, expressed in nm, and I the ionic strength of the electrolyte, expressed in mol L⁻¹:

$$I = \frac{1}{2} \sum_{i} z_{i}^{2} C_{i} \qquad (48)$$

The validity of this approach is discussed in the Supplementary Data file.

The thickness of the diffusion layer was calculated in the case of a rotating disk electrode with the Levich equation:

$$\delta = 1.61 D_i^{1/3} \omega^{-1/2} \nu^{1/6} \qquad (49)$$

where δ must be expressed in cm, Dᵢ (the diffusion coefficient of the species under consideration) in cm² s⁻¹, ω the angular rotation frequency (2π x rotation rate) in s⁻¹, and ν the kinematic viscosity of the electrolyte in cm² s⁻¹.

## 3.2. Model without buffer (Fig. 2)

Experimental voltammetries were recorded at 20 mV s⁻¹ in water containing 1 M KCl at pH values adjusted into the range of 2.5 to 4 by the addition of HCl. The model was implemented for identical electrolytes, factoring the addition of HCl into the calculation of the ionic resistivity, which was thus in the range of 6.60 10⁻² to 6.67 10⁻² Ω m for a pH of 2.5 to 4.

The exchange current density j₀ was taken equal to 8 A m⁻², the value applicable for hydrogen oxidation on platinum electrode in acidic condition [61]. The transfer coefficient α was taken equal to the theoretical value 0.5, corresponding to the current Tafel slope value of 59.1 mV/dec (Equation 44). The Nernst diffusion layer thickness calculated with the proton diffusion coefficient at infinite dilution (Dₕ⁺ = 9.31 10⁻⁵ cm² s⁻¹) was 33 µm for a rotating disk electrode (RDE) rotating at 1000 rpm (Equation 49).

In the second step, the proton diffusion coefficient was adjusted numerically to 6.9 10⁻⁵ cm² s⁻¹, resulting in a diffusion layer thickness
of 30 µm.

Finally, the kinetic parameters for the “high j” zone were adjusted to $α = 0.211$, corresponding to a $T_{slope}$ of 140 mV/dec, and $j_0 = 4.5, 7, 8, 9$, and $2.5 A/m^{-2}$ for the pH values of 4.0, 3.5, 3.0, 2.75, and 2.5, respectively.

### 3.3. Model with AH⁻/A²⁻ buffer (Figs. 3 and 4)

The $AH⁻/A²⁻$ equations were implemented with the physicochemical parameter values that correspond to the $NaH_2PO_4/Na_2HPO_4$ couple (Table S2). The Nernst diffusion layer thickness calculated with the diffusion coefficient of the $AH⁻$ species ($D_{AH} = 9.6 \times 10^{-6} cm^2 s^{-1}$) was 15.5 µm. For the “low j” zone, the kinetic parameters were taken equal to the theoretical values, i.e. the exchange current density $j_0$ was $8 A m^{-2}$ and the transfer coefficient $α$ was $0.5 (T_{slope} = 59.1 mV/dec)$.

For the “high j” zone, the kinetic parameters were adjusted. For the curves plotted at various buffer concentrations, the exchange current density $j_0$ in the “high j” zone was 8, 8, 2.5, and $0.3 A m^{-2}$ for 20, 10, 5, and 2 mM, respectively, and the transfer coefficient $α$ was $0.25 (T_{slope} = 118.13 mV/dec)$ for all concentrations.

For the curves plotted at 20 mM buffer and various pH values, only the transfer coefficient $α$ was adjusted in the “high j” zones to $0.180 (T_{slope} = 164.3 mV/dec)$ for the pH values of 4.35, 6.05, and 6.53, and to $0.154 (T_{slope} = 191.7 mV/dec)$ for the pH values of 7.09 and 8.7, while $j_0$ was kept at $8 A m^{-2}$ for all pH values.

### 3.4. Model for comparing HER in phosphate and borate (Fig. 5)

The $NaH_2PO_4/Na_2HPO_4$ couple was treated with the $AH⁻/A²⁻$ equations and the $B(OH)_3/NaB(OH)_4$ couple with the $AH/A⁻$ equations (grouped together in Table S1 of the Supplementary Data file). The Nernst layer thickness was taken equal to $10 µm$, corresponding to the value given by the Levich equation (see Supplementary Data file) for the rotating speed of 2500 rpm used in the experimental work reported by Jackson et al. [20]. In their experimental work, the authors determined Tafel slopes varying from 177 to 157 mV/dec for phosphate at concentrations from 19 to 500 mM. Accordingly, the values of $α$ used here in the model for phosphate corresponded to the linear decrease of the Tafel slopes with concentration reported by the authors ($T_{slope} = 177, 176, 174, 169, \text{and } 157 mV/dec$ for phosphate concentrations of 19, 43, 147, 250, and 500 mM, respectively). For borate, the Tafel slope remained constant and equal to 198 mV/dec for all concentrations. Accordingly, here $α$ was kept constant. The exchange current density $j_0$ was the unique parameter that was adjusted numerically. Its value was set to $0.12 A m^{-2}$ and kept constant for all concentrations of the two electrolytes.

### 3.5. Model for calculation of the optimal values of current density (Fig. 6)

The $AH⁻/A²⁻$ equations were used. The Nernst diffusion layer thickness calculated with the $AH⁻$ species diffusion coefficient ($D_{AH} = 9.6 \times 10^{-6} cm^2 s^{-1}$) was 15.5 µm for an RDE rotating at 1000 rpm. Then, arbitrary values of 10 and 50 µm were used for comparison. Only the “low j” zone was useful for calculation of the $j_{optimal}$ values. The kinetics parameters were taken equal to the theoretical values, i.e. the exchange current density $j_0$ was $8 A m^{-2}$ and the transfer coefficient $α$ was $0.5 (T_{slope} = 59.1 mV/dec)$.

Calculations were run with equimolar solutions of $AH⁻/A²⁻$. In this case, the pH was in the range of 6.64 to 6.35 for buffer concentrations of 0.25 to 2 M. The solutions prepared with $AH⁻$ alone were approached numerically with 99.35 % of $AH⁻$ and 0.65 % $A²⁻$. This composition corresponds to the values that give a pH of 4.35 at 20 mM with 1 M KCl, i.e. the value measured experimentally. In such conditions, the calculated pH value of the “$AH⁻$ alone” solutions ranged from 4.56 to 4.25 for buffer concentrations of 0.25 to 2 M.

### 3.6. Experimental

Phosphate buffer solution (PBS) was used for experiments because this is the most used in the context of water electrolysis at near-neutral pH [16–18]. The various concentrations of KCl and phosphate buffer were obtained by adding the required volumes of 2 M KCl, 1 M $NaH_2PO_4$, $Na_2HPO_4$, and 0.3 M $Na_2HPO_4 \cdot 12H_2O$ in distilled water. Cyclic voltammetries (CV) were recorded with a platinum rotating disk electrode (RDE, Radiometer Copenhagen CTV101) of 5 mm diameter ($0.196 cm^2$) rotated at 1000 rpm. This rotation speed was chosen as the result of a compromise: on the one hand, the speed must be high enough to ensure that the bubbles, which formed at the lower potentials, were rapidly expelled from the surface of the electrode so as not to significantly disturb the Nernst layer; on the other hand, rotation speeds that were too high resulted in high current densities, and therefore very large releases of hydrogen bubbles at the lowest potentials, which could affect mass transport and invalidate the Nernst hypothesis. CV were recorded, using a Biologic VSP1 potentiostat, at 20 mV/s from the open circuit potential to the lower potential limit ranging from −1.0 to −1.3 V/SCE. The lower limit was adjusted to avoid too high a reduction current, which would disturb the current record due to excessive hydrogen bubbling. The upper potential limit was always 0.3 V/SCE to avoid chloride oxidation, which could modify the chemical state of the platinum surface. All potentials were measured with respect to a saturated calomel reference electrode (SCE, potential of 0.241 V/SHE), and a large platinum grid (surface area around $3 cm^2$) was used as the auxiliary electrode. It was placed opposite the RDE at a distance of about 3 cm, at the bottom of the reactor. Around 3 min of electrolysis at 0.3 V/SCE under strong nitrogen bubbling was done before each CV record to ensure reproducibility of the CV curves. Nitrogen was flushed above the solution during CV records. Three CVs were recorded successively. The first was generally slightly different from the second and third, which were similar. The third cycles were used in this study.

## 4. Results and discussion

### 4.1. HER without buffer species

Voltammetries were recorded in water containing 1 M KCl at pH values ranging from 2.5 to 4. The curves showed the expected three-phase shape [39–41] (Fig. 2A) with and intermediate current plateau (phase II). The reduction current at the plateau increased when the pH decreased, i.e. the greater the proton concentration. The concentration of the protons in solution ($C_{H+}^b$) was calculated with the pH of the solution ($pH^b$) using:

$$pHb = - \log \left( \gamma_{H+} \frac{C_{H+}^b}{C_0} \right) \qquad (50)$$

with $γ_{H+} = 0.74$ and $C_0 = 1 M$. The current density at the plateau was exactly proportional to the concentration of protons in the bulk of the solution (Fig. 2E).

The model was implemented with similar conditions using common values of the two parameters involved in the water reduction kinetics, i. e. $j_0 = 8 A m^{-2}$ and $α = 0.5$ (with $n = 2$). The values of the $H⁺$ and $OH⁻$ diffusion coefficients were taken equal to their values at infinite dilution. The model produced $j-E$ curves (Fig. 2B) with exactly the same general shape as the experimental ones. Nevertheless, the values of the reduction current densities at the plateau were slightly higher than experimental ones (Fig. 2E). To align the theoretical curves with the experimental ones, the value of the proton diffusion coefficient was adjusted numerically: by bringing the $D_{H+}$ value from $9.31 \times 10^{-9}$ (infinite dilution) to $6.9 \times 10^{-9} m^2 s^{-1}$ (Fig. 2C), the theoretical reduction current densities at the plateau perfectly matched the experimental values (Fig. 2E). The adjusted value of $6.9 \times 10^{-9} m^2 s^{-1}$ gave a ratio of 0.74 with respect to the infinite dilution value. This order of magnitude matched
<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Water electrolysis without buffer species at various pH. A) Experimental *j*-E curves recorded at 20 mV s⁻¹ in KCl 1 M, at the pH values indicated on the curves. B) Theoretical *j*-E curves with *D*<sub>H⁺</sub> = 9.31 10<sup>-9</sup> m² s<sup>-1</sup>. C) Theoretical *j*-E curves with *D*<sub>H⁺</sub> = 6.9 10<sup>-9</sup> m² s<sup>-1</sup>. D) Theoretical *j*-E curves with *D*<sub>H⁺</sub> = 6.9 10<sup>-9</sup> m² s<sup>-1</sup> and adjusted values of the kinetics on the “high *j*” zones. For all the theoretical curves, dashed and solid lines correspond to the “low *j*” and “high *j*” zones, respectively. E) Experimental (circles) and theoretical (squares) current density at -0.75 V/SCE as a function of the proton concentration in the bulk; black and red squares were obtained with *D*<sub>H⁺</sub> = 9.31 10<sup>-9</sup> and 6.9 10<sup>-9</sup> m² s<sup>-1</sup>, respectively. F) pH at the cathode surface (pH<sup>0</sup>) at -0.50 and -0.90 V/SCE as a function of pH in the bulk (pH<sup>b</sup>).</figcaption></figure>

the ratio values of 0.82 and 0.75 reported for proton diffusion in electrolytes that contained KCl 1 or 2 M, respectively [62].

First, it was observed that the two zones of the *j*-E curves, at low (dashed lines) and high (solid lines) current density, were perfectly connected to each other in each case. This confirmed the relevance of the approach proposed here, based on the combination of the “low *j*” and “high *j*” mass transport schemes, with the reaction plane hypothesis in the “high *j*” case.

The values of the current density at the plateau were not sensitive to the value of the kinetic parameters (*j*₀ and *α*), but were highly sensitive to the value of the H⁺ diffusion coefficient. This confirmed that phase II was fully controlled by the rate of proton transport. Actually, the model predicted that the current density at the plateau, here measured at -0.75 V/SCE, was strictly proportional to the proton concentration in the bulk, as observed experimentally (Fig. 2E).

The model perfectly matched the experimental curves in the phase I and phase II zones. It should be noted that for both the theoretical and

experimental curves, the reduction started at around -0.4 V/SCE, without any numerical adjustment to improve this part of the curves.

Fig. 2F gives the value of the pH at the cathode surface (pH⁰) as a function of the pH of the solution at potentials of -0.50 and -0.90 V/SCE, which roughly corresponded to the transition from phase I to phase II and from phase II to phase III. At -0.50 V/SCE, at the beginning of the current plateau (phase I to phase II), pH⁰ varied proportionally to pHᵇ, as pH⁰ = pHᵇ + 0.9 (equation of the dotted straight line on the figure). On the other hand, phase II consistently switched to phase III at pH⁰ value of around 10.8 regardless of the pH in the bulk. The model showed that the occurrence of the three phases corresponded to considerable changes in pH⁰, and that the cathode can thus be exposed to pH values much higher than the value of the solution.

A difference between theory and experiments was observed in phase III, where the model (Fig. 2C) showed shorter current plateaus than the experimental curves (Fig. 2A). The explanation can be found in Fig. 2F, which shows that, whether the solution was neutral or acidic, the
cathode surface was always exposed to a pH of around 10.8 when phase III started. The value of the kinetic parameters used in the model ($j_0 = 8 \text{ A m}^{-2}$ and $\alpha = 0.5$), which apply to efficient platinum cathodes in acidic condition [61], were consequently no longer valid in the alkaline conditions to which the cathode was exposed during phase III. The kinetic parameters were consequently adjusted for the “high j” zone, in particular with $\alpha = 0.211$ that corresponded to a Tafel slope of 140 mV/dec. The final result (Fig. 2D) nicely matched the experimental data (Fig. 2A).

The presence of three phases has been attributed to the switch from proton reduction to water reduction. Here, by contrast, the model took into account only one reaction, the reduction of water. Using this single reaction, the model reproduced exactly the same three successive phases as observed experimentally. As observed experimentally, the current densities at the plateau were also proportional to the concentration of protons in solution. It should be mentioned that phase I, phase II and the current plateaus were shown by the model without any other parameter adjustment than the correction of the proton diffusion coefficient, in accordance with values reported in the literature. Adjustment of the Tafel slope, due to the high local pH increase at the lower overpotentials, was only needed to match the phase III. These results demonstrated the validity of this simpler approach to modelling HER at near neutral pH, factoring in only one reaction instead of the two different reactions. The water reduction reaction alone is sufficient, provided that the effect of the interfacial pH on its equilibrium potential is taken into account. The key is to design a correct ion transport scheme in order to account for the impact of the interfacial pH on the water reduction reaction kinetics.

## 4.2. HER with phosphate buffer

Voltammetries were recorded in water containing 1 M KCl with different concentrations of equimolar $NaH_2PO_4$ and $Na_2HPO_4$ (Fig. 3A). Hereafter, the term “buffer concentration” refers to the total

concentration of phosphate in the bulk, i.e. the sum of the $H_2PO_4^-$ and $HPO_4^{2-}$ concentrations. Three-phase type curves were obtained as expected, with the current density at the plateau proportional to the buffer concentration (Fig. 3B).

The pH of the solution was equal to 6.53, while the theory would appear to predict, for an equimolar mixture of $NaH_2PO_4$ and $Na_2HPO_4$, a pH value equal to the pKa of the acidic equilibrium, i.e. 7.21. This discrepancy owes to the effect of the activity coefficients, which were far from being equal to 1 (see Supplementary Data file). It is therefore important to recall the need to use activities instead of concentrations in the acidic equilibrium and for pH calculations. The model was run in similar conditions. With the theoretical values of the kinetic parameter ($j_0 = 8 \text{ A m}^{-2}$ and $\alpha = 0.5$), the model produced *j-E* curves identical to the experimental curves in the “low j” zones. As observed in condition without a buffer, the model showed shorter plateaus, in terms of potential range, than the experimental curves and a steeper increase in the reduction current densities in the “high j” zone. The kinetic parameters for the “high j” zone were consequently adjusted to match the experimental data. $j_0$ was decreased with decreasing buffer concentrations in the range of 8 to 0.25 $\text{A m}^{-2}$, while $\alpha$ was kept constant at 0.75 ($T_{\text{slope}} = 118.13 \text{ mV/dec}$). This numerical adjustment of the “high j” zones resulted in numerical curves that nicely matched the experimental data (Fig. 3B and C).

The switch from phase I to phase II was evaluated as the point where the current density was equal to 90 % of the plateau current density, and the switch from phase II to phase III as the point where the current density was equal to 110 % of the plateau current density. With these criteria, the phase I/II switch occurred at potentials of around -0.70 to -0.76 V/SCE for buffer concentrations ranging from 2 to 20 mM, and the switch phase II/III at potentials of around -0.95 V/SCE (Fig. 3D). Whatever the buffer concentration, phase II started at $pH_c$ around 7.9, and phase III started at $pH^0$ above 9, while the pH of the bulk was 6.53.

Similar experiments were run with the buffer concentration kept at

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** Water electrolysis with $AH^- / A^2-$ buffer species (pH 6.53) at various concentrations. A) Experimental *j-E* curves recorded at 20 mV s⁻¹ in KCl 1 M with the phosphate concentrations ($NaH_2PO_4 / Na_2HPO_4$ equimolar) indicated on the curves. B) Experimental (circles) and theoretical (squares) current density at -0.85 V/SCE as a function of the buffer concentration. C) Theoretical *j-E* curves with adjusted values of the kinetics on the “high j” zones; dashed and solid lines correspond to the “low j” and “high j” zones, respectively; the blue lines indicate the switch from one phase to the other. D) Values of $pH^0$ for current densities equal to 90 and 110 % of the plateau current density as a function of the buffer concentration.</figcaption></figure>
20 mM at a pH ranging from 4.35 to 8.70 (Fig. 4A). As expected, the reduction current densities increased with decreasing pH. The model was run in similar condition, always using the theoretical values of the kinetic parameters ($j_0 = 8 A m^{-2}$ and $\alpha = 0.5$) for the “low j” zones. Only the transfer coefficient $\alpha$ was adjusted numerically in the “high j” zones (Fig. 4B). Once again, the model nicely matched the experimental data.

The model produced the three-phase shaped j-E curves fully similar to those often reported in the literature with various weak acids, notably phosphate in the ranges 0.1 to 10 mM [40,56], 25 to 100 mM [51], 10 to 500 mM [18,53], and other weak acids such as acetate [40,44], borate [40], lactic acid [55], aspartic, and glutamic acids [22].

Experimentally, it has also been observed that the value of the current density at the plateau (phase II) is proportional to the buffer concentration in the bulk [53,55]. The model recovered the same proportional correlation. The occurrence of current waves due to the buffer species, with a plateau current density that is proportional to the buffer concentration, has been advanced as a strong evidence supporting the hypothesis of the direct electrochemical deprotonation of the AH species (Equation 4).

By contrast, here, the model did not assume any direct electrochemical deprotonation reaction. It used only the water reduction reaction (Reaction 2), taking care to carefully assess the value of the interfacial pH and to factor in the impact of this interfacial pH on the water reduction kinetics.

Moreover, the model did not take into account the proton reduction reaction and consequently did not use the switch from proton reduction to water reduction in order to explain the three-phase shape of the j-E curves. In conclusion, the model shows that using the basic water reduction reaction alone is sufficient to describe the action of the buffer species on HER at near neutral pH. Variation of the pH on the electrode surface has a key impact on the kinetics of this reaction, and this impact

<figure><img src="image_6.png" /><figcaption>Fig. 4. Water electrolysis with buffer species $AH^- / A^2-$ 20 mM at various pH. A) Experimental j-E curves recorded at 20 mV s$^{-1}$ in KCl 1 M with at the pH values indicated on the curves. B) Theoretical j-E curves with adjusted values of the kinetics on the “high j” zones; dashed and solid lines correspond to the “low j” and “high j” zones, respectively.</figcaption></figure>

is sufficient to explain the three-phase shape of the j-E curves.

It should be mentioned that the first part of the curves, called the “low j” zones, were always modelled with the theoretical values of the kinetic parameters, without numerical adjustments. This means that phase I, phase II and the current density values at the plateau were predicted by the model without any parameter adjustment. Only the “high j” zones had to be numerically adjusted, by reducing the values of the kinetic parameters. This need to modify the kinetic parameters in the “high j” zones is perfectly consistent with the alkalinisation of the medium indicated by the model. HER is known to become a sluggish reaction in alkaline conditions [63], even with electrode materials such as platinum that are very efficient HER electro-catalysts in acidic conditions [64,65]. This has commonly been attributed to the change in the hydrogen binding energy with the pH, while some authors note that the underlying reasons for the sluggish kinetics of HER in alkaline conditions is still controversial [66].

### 4.3. Phosphate vs. Borate

A recent paper by Jackson et al. compared the capacity of phosphoric and boric acids to promote HER [20]. A gold rotating disk electrode was used to compare various concentrations of sodium phosphate pH 6.1 and sodium borate at pH 8.4. The same comparison was reproduced here with the model, using the appropriate equations, the $AH^- / A^2-$ equations for the $NaH_2PO_4/Na_2HPO_4$ couple, the $AH^- / A^-$ equations for the B (OH)$_3/NaB(OH)_4$ couple (Table S1), and the related physico-chemical values for both species (Table S2). The values of $\alpha$ were extracted from the experimental values of the Tafel slopes given in the paper [20].

Fig. 5 shows that the model was in agreement with the experimental data reported by Jackson et al. The model well predicted the general shape of the current density-potential curves and the value of current density for the highest buffer concentration. It kept a correct match for the lower concentrations. In any case, the clear difference between phosphate and borate buffers was well matched. The model confirmed that phosphate at pH 6.1 was more efficient in promoting HER than borate at pH 8.4. It should be mentioned that the kinetics parameter $j_0$ was the only parameter that was adjusted numerically, and its value was the same ($j_0 = 0.12 A m^{-2}$) for all the concentrations and for the two buffers. The difference in efficiency between the two buffers was therefore not due to an artificial numerical adjustment. Furthermore, this value of $j_0$ was weaker than the values used in the previous sections, which makes sense since this work used a gold electrode whereas a platinum electrode had been used in previous sections.

The higher efficiency of phosphate may be attributed to the lower pH value (6.1 vs. 8.4 for borate), which benefits HER. To account for this issue, the authors of the experimental study [20] attempted to compare the intrinsic effect of the two buffers by suppressing the direct effect of pH by plotting the j-E curves with respect to the reversible hydrogen electrode (RHE) rather than the standard hydrogen electrode (SHE). The potential expressed with respect to RHE ($E_{RHE}$) depends on the pH of the solution as:

$$E_{RHE} = E_{SHE} + 0.059pH \qquad (51)$$

Plotting the curves with respect to RHE shows that the difference in efficiency was due not only to the direct effect of pH (6.1 vs. 8.4), but also to the difference in the kinetics. The authors explained this difference by assuming the occurrence of direct deprotonation reactions of phosphate and borate with different reaction orders. A reaction order of 0.6 and close to 0 were found for phosphate and borate, respectively [20]. The present model showed that introducing these deprotonation reactions was not necessary. It makes it clear that the difference in efficiency observed between the two buffers can be fully explained by the different values of the Tafel slopes on the kinetics of the water reduction reaction.

In conclusion, based on this comparison, the model succeeds in matching the different behaviour of phosphate and borate with regard to
<figure><img src="image_7.png" /><figcaption>Fig. 5. Experimental and theoretical j-E curves of sodium phosphate pH 6.1 and sodium borate pH 8.4. Experimental j-E curves from Jackson et al. [20] for A) sodium phosphate and B) sodium borate at concentrations of 19 (black), 43 (red), 83 (blue), 147 (green), 250 (purple) and 500 mM (orange); Theoretical j-E curves with the experimental values of the Tafel slopes for C) sodium phosphate and D) sodium borate, at 19, 43, 147, 250 and 500 mM with the same colour chart.</figcaption></figure>

HER, still using only the water reduction reaction (Reaction 2). Assuming direct deprotonation reactions with different orders was not required to recover the experimental data.

## 4.4. Discussion and outlooks

The main finding of the present study is that the model based solely on the water reduction reaction is sufficient to explain HER at near neutral pH. Do protons and/or the buffer species actually react at the cathode surface or not? The model cannot answer this basic question. In contrast, it definitively shows that the shape of the current-potential curves can no longer be used to support that protons or buffer species directly react at the cathode surface, as commonly done to date. The shape of the current-potential curves, with current density at the plateaus proportional to the concentrations of the protons and buffer species, was matched by the model based on the single water reduction reaction.

It should be mentioned that the first part of the current-potential curves, i.e. phases I and II, which include the current plateaus, were perfectly matched by using the theoretical values of the kinetic and mass transport parameters. The exchange current density was extracted from the literature and the Tafel slope was calculated with $α = 0.5$. Diffusion coefficients were extracted from the literature and only the diffusion coefficient of protons was adjusted numerically. This adjustment was justified by the high concentration of inert salt, in accordance with the literature. Then, for the high values of the current density, Tafel plots were adjusted numerically, in accordance with the qualitative observations reported in the literature. These adjustments were relating to the phase III only, which was less important for the key message provided by the model.

This study therefore introduces a much simpler approach than that currently established in the literature. This makes it possible to progress in electrolyte engineering without having to factor in additional reactions, including the switch from proton reduction to water reduction and speculative direct electrochemical deprotonation of buffer compounds. In contrast, the model shed light on the essential role of the local pH at the cathode surface. This means that modelling electrolysers for water electrolysis at near-neutral pH requires the tertiary current

distribution to be resolved. The secondary current distribution, which is currently chosen even in the most recent studies [67–69], should be used with care, limited to the low current densities, when pH gradient are not too steep.

In addition, this work has also shown that the reaction plane hypothesis was appropriate for modelling the j-E curves in the absence of buffer, leading to theoretical curves very close to the experimental ones (Fig. 2). In the presence of buffer, the overall shape of the theoretical curves was similar to the experimental ones but the current plateaus were more stable and longer in terms of potential range for the model than for the experimental curves (Figs. 3 and 4). This may suggest that acid dissociation, which was assumed to be immediate in the model, may be not fast enough to be at equilibrium at the reaction plane.

The model can be used in the context of electrolyte engineering in order to define the optimal current density that could be expected for a given buffer concentration or, conversely, to determine the most appropriate buffer concentration to be used in order to reach a given current density value. In HER context, the general target is to reach the highest possible current density at the highest possible potential. Both the model and experimental curves (Figs. 3 and 4) show that, at the current plateaus, a small change in the current can correspond to a large change in potential. Monitoring an HER process in the zone of the current plateau would therefore be too risky because a small increase in current could result in a drastic decrease in the potential. We therefore suggest selecting 90 % of the current density at the plateau as the optimal current density value. Examining the modelled curves reveals that the switch from the “low j” zone to the “high j” zone (frontier between dashed and continuous lines on the graphics) offers a good estimate of the current density at the plateau. This switch corresponds to the exact balance of the $OH^−$ production by the transport of the $AH^−$ species, so that the $AH^−$ concentration is equal to zero at the cathode surface ($C_{AH−}^0 = 0$). This is calculated using Equation (38):

$$0 = (C_{AH-}^b + \Gamma_{AH-}) \exp(-\Lambda\delta) - \Gamma_{AH-} \quad (52)$$

leading to the value of j at the plateau:
$$j_{\text{plateau}} = \frac{RT}{F\rho\delta} \ln\left(1 + \frac{C_{\text{AH-}}^{b}}{\Gamma_{\text{AH-}}}\right) \quad (53)$$

Taking 90 % of this value as the optimal current density ($j_{optimal}$) gives:

$$j_{\text{optimal}} = 0.9 \frac{RT}{F\rho\delta} \ln\left(1 + \frac{C_{\text{AH-}}^{b}}{\Gamma_{\text{AH-}}}\right) \quad (54)$$

This equation correlates $j_{optimal}$ with the concentration of the buffer ($C_{AH-}^b$) and the thickness of the Nernst diffusion layer ($\delta$), as illustrated in Fig. 6. In any case, $j_{optimal}$ is proportional to the buffer concentration (Fig. 6A). The values of $j_{optimal}$ plotted for $\delta = 15.5 \ \mu\text{m}$ showed that using the buffer in the form of $AH^-$ alone, instead of an equimolar mixture of $AH^-$ and $A^{2-}$, increases $j_{optimal}$ by 76 %. This is logical because $AH^-$ is the species that balances the production of the $OH^-$ ions at the electrode surface. The situation is reversed at the anode, where acidification must be mitigated:

$$H_2O \rightarrow 1/2O_2 + 2H^+ + 2e^- \quad (55)$$

The same model could now be designed for the anode side in order to determine the optimal buffer composition.

Fig. 6A also shows that $j_{optimal}$ is very sensitive to the hydrodynamic conditions. Increasing $\delta$ from 15.5 to 50 $\mu\text{m}$ decreased the current density by 69 %, while decreasing $\delta$ to 10 $\mu\text{m}$ increased the current by 55 %.

Plotting $j_{optimal}$ as a function of the potential at which it is reached gives another piece of interesting information (Fig. 6B). Lowering $\delta$ can drastically increase $j_{optimal}$ but it also results in decreasing the potential

<figure><img src="image_8.png" /><figcaption>Fig. 6. Water electrolysis with buffer species $AH^- / A^{2-}$ under various operating conditions. A) Optimal $j$ as a function of the buffer concentration. B) Optimal $j$ as a function of the related potential; each curve involves 8 points corresponding to buffer concentrations ranging from 0.25 to 2.0 M by 0.25 M steps. For both graphics, black circles: equimolar $AH^-$ and $A^{2-}$, $\delta = 15.5 \ \mu\text{m}$; empty circles: $AH^-$ alone, $\delta = 15.5 \ \mu\text{m}$; squares: $AH^-$ alone, $\delta = 50 \ \mu\text{m}$; triangles: $AH^-$ alone, $\delta = 10 \ \mu\text{m}$; orange triangles: $AH^-$ alone, $\delta = 10 \ \mu\text{m}$, addition of KCl 1 M.</figcaption></figure>

that must be applied. For example, a $\delta$ value of 10 $\mu\text{m}$ (triangles on Fig. 6) allows 173 $\text{A m}^{-2}$ to be produced at -0.590 V/SHE with a buffer concentration of 0.25 M, while $\delta$ of 50 $\mu\text{m}$ (squares on Fig. 6) can lead to the same current density, produced at -0.576 V/SHE at a buffer concentration of 1.25 M. Stirring the solution increases the current density that can be reached, but can also drive down the potential that must be applied. The model can therefore provide invaluable help in choosing the optimal current density-potential-concentration triptych.

The model also shed light on the possible benefit of adding an inert salt. Adding KCl 1 M resulted in a 10 % increase in $j_{optimal}$ whatever was the buffer concentration (Fig. 6A). This effect is not due to the decrease in the solution ohmic drop, because ohmic drop was not factored into the model. In fact, adding an inert salt tends to decrease the detrimental effect of the electric field, which drives $AH^-$ away from the cathode surface. This last example highlights the importance of taking into account the effect of migration on the transport of ionic species, which is not yet commonly done in the theoretical approaches to HER.

The simple model presented here can be used to analyse and compare the data reported by different studies. In particular, the burgeoning field of new catalyst development [16,33] presents results that can be obtained under very different experimental conditions, in terms of the nature and concentration of the buffer, whether or not salts are added, hydrodynamic conditions, etc. The model can be a helpful tool in harmonising and comparing these various data.

Building on the foundation developed here, the model can be improved in several ways. Characterizing hydrodynamics will be a key area for improvement, given its high impact (Fig. 6). In this objective, factoring in the natural convection driven by the buoyancy effect will be essential, in order to bring us closer to modelling industrial water electrolysis processes [70]. Here, the Nernst layer thickness was controlled by the RDE rotation speed. In contrast, the Levich equation and Nernst layer hypothesis are no longer valid in an industrial electrolyser run at high current density. Bubble evolving and release coupled with the forced convection of the electrolyte control the local hydrodynamic on the electrode surface [71–73]. The hypothesis developed here should therefore be rewritten in the appropriate hydrodynamic framework. Mass transport is governed by migration and diffusion, which both depend on the composition of the solution and notably its viscosity. For instance, a study that examined the diffusion of phosphate species at pH 5, identified the concentration of 2 M as the optimum [57]. The HER performance decreased at higher concentration due to the decrease in the diffusion coefficient at too high an electrolyte viscosity. Factoring in the impact of electrolyte composition on ion mobility would thus be a relevant avenue for improvement. If necessary, including the effect of temperature may also be considered in order to get closer to industrial water electrolysis conditions [17].

Finally, it should be kept in mind that the numerical results were highly dependent upon the kinetic parameters, here implemented in the form of $j_0$ and $\alpha$, which was determined from the experimental Tafel slopes. This underscores the vital need for fundamental studies aimed at deciphering the mechanisms of water reduction on the cathode surface, if the objective is to understand the meaning and to predict the value of these parameters. For instance, identifying the differences in mechanisms between acid and alkaline environments [74] is of prime importance, given the wide variation in local pH shown by the model. Pursuing in-depth mechanistic studies would also be worthwhile to identify the effect of the various compounds that can adsorb on the electrode surface and impact the water reduction kinetics, notably buffer compounds and the inert salts that may be added into the electrolyte to increase its ionic conductivity. For instance, it was shown that adsorbed phosphate groups can facilitate HER [75]. By way of illustration, it is of note that, in the context of $H_2O_2$ oxidation, it has been stated that surface binding of $H_2O_2$ at platinum electrodes was strongly affected by the presence of $H_2PO_4^-$ ions [76]. The kinetics of water reduction in the specific electrolytes used under near neutral pH conditions should be a key subject of further study. This type of study is
essential for understanding the variations in kinetic parameters and possibly predicting them. However, such studies of the kinetic mechanisms will not alter the simplification in terms of engineering which is introduced here by demonstrating that a model with a single reaction is sufficient to develop the engineering of water electrolysis at near neutral pH.

## 5. Conclusions

A basic approach to mass transport modelling in the context of HER at near-neutral pH was developed. The close correlation between the theoretical and experimental data leads us to propose revisiting some fundamental bases. The switch from proton reduction to water reduction and the additional electrochemical deprotonation reaction of the buffer species are not required to explain the three-phase shape of the j-E curves recorded, whether in the absence or presence of a buffer. The water reduction reaction alone is sufficient, provided that the impact of the local pH at the cathode surface is taken into account.

The model cannot settle the question of the nature of the real reactants on the cathode surface: protons, buffer species, or water molecules only, but it states that mass transport can no longer be invoked to support that protons and buffer species are actual reactants. Unravelling the reduction pathway and the effects on HER kinetics of the various species that can adsorb on the cathode surface, notably the buffer species or the inert ions, remains a key focus for further study. These possible impacts were factored in here as a whole, through the two parameters of current density exchange and Tafel slope, but a better basic understanding would be of great help to spur progress in developing water electrolysis at near-neutral pH.

This work leads to a drastic simplification of the models to be developed for the engineering of electrolysers working at near-neutral pH. It is no longer necessary to work on two or three reactions at the cathode, but only water reduction reaction is required, taking into account local pH variations. The model also shows that the local pH is the key parameter for controlling the kinetics. The change in local pH can be extreme and lead to changes in the Tafel slope, which no longer corresponds to that of the pH of the bulk at large overpotential. This also means that modelling electrolysers for water electrolysis at near-neutral pH requires the tertiary current distribution to be resolved. Finally, as a secondary result, this study also reminds of the need to consider activities rather than concentrations, which is still rare in papers in this field.

The model has been kept simple enough to remain user-friendly, and we hope that this will make it useful to a large community, serving as a basis for further development towards electrolyser modelling, and also in harmonising the numerous findings reported in the context of catalyst design.

### CRediT authorship contribution statement

**Junzhe Sun:** Writing – original draft, Methodology, Investigation, Formal analysis, Conceptualization.
**Luc Etcheverry:** Investigation, Methodology.
**Alain Bergel:** Writing – review & editing, Writing – original draft, Validation, Supervision, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Conceptualization.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgements

This research was conducted within the framework of the “HyTEAM” joint laboratory (Technologies Electrochimiques Alternatives Multi-

échelle pour la production d'hydrogène décarboné) co-operated by the Laboratoire de Génie Chimique (LGC, Université de Toulouse, CNRS, INPT, UPS, Toulouse, France) and the company 6TMIC (www.6t-mic.com, Castanet-Tolosan, France), with financial support from the French Région Occitanie in the framework of the programme "Groupe-ment pour la Recherche Appliquée Innovante avec les Entreprises (GRAINE)".

## Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.cej.2025.160490.

### Data availability

Data will be made available on request.

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