A First-Principles Surface Reaction Kinetic Model for Hydrogen Evolution under Cathodic and Anodic Conditions on Magnesium

The surface reaction kinetic model derived in this paper is built upon the following set of assumptions:

• (1)  
  Water molecules dissociate upon adsorption on magnesium metal surfaces to produce adsorbed atomic hydrogen and adsorbed hydroxyl radical. (See Figure 3.) The exact charge states of these adsorbed species are not well defined. Quantum mechanical calculations for adsorption states indicate some localized charge transfer from the metal atoms on which they are adsorbed to the adsorbed states, but overall the reaction is charge neutral (no oxidation or reduction).¹³ This is Reaction 7.

  

  The reaction, therefore, consumes two available metal sites Mg^* and produces one surface site that has OH bound to it (Mg^*OH) and one surface site with H bound to it (Mg^*H).
• (2)  
  The Mg^*H sites may combine to produce hydrogen, H₂, and making available the two metal sites Mg^* (Figure 4). Note at this point that our mechanism has assumed that no charge transfer has yet taken place: it involves the recombination of neutral hydrogen atoms to make neutral H₂.

  
• (3)  
  If Reactions 7 and 8 were to continue, the surface would eventually be covered by Mg^*OH, allowing for no more hydrogen evolution. (Figure 5). However, there will be some rate at which the hydroxyl is removed from the surface by reduction to hydroxide (Figure 6a), making available once more the metal site Mg^*:

  

  It is during this step that the electron is consumed, making the total hydrogen evolution reaction through Equations 7–9 an electrochemical reduction reaction.At this point, we have been following the Tafel mechanism for hydrogen evolution reaction as elucidated in the theoretical work by Williams et al.¹³ This seems the most reasonable mechanism given the proclivity for Mg surfaces to reactively dissociate water and produce Mg^*H and Mg^*OH.¹² Collectively the reaction should be responsible for the cathodic evolution of hydrogen, the reaction overall being:

  

  with the states in [ ] representing the surface intermediates. The intermediate step, R3, is the one in which the electron transfer occurs, and will become slower and slower as potentials move in the anodic direction, thus leading to overall Mg^*OH poisoning of the HER at anodic potentials. However, this can only account for the cathodic branch of the hydrogen evolution observed in Figures 1 and 2, so the proposed mechanism in Equations 7–9 cannot be the whole story.To explain the anodic branch, we now propose that one more process must be occurring:
• (4)  
  Mg^*OH sites on the surface can alternatively be removed through anodic dissolution of Mg (Figure 6b). We assume here that sites where OH is adsorbed are the most vulnerable to dissolution, given that there is already some partial charge transfer from the Mg to the OH moiety to which it is attached, and the formation of a Mg-OH bond will tend to weaken adjacent metal-metal bonds.¹³ There are a number of possible representations for this anodic dissolution step; we choose here the simplest one, in which Mg²⁺ is liberated, along with the OH⁻ ion and the release of an electron (note that Mg²⁺ dissolution is a 2-electron process- here we assume the second electron is transmitted through the desorption of OH⁻):

  

  The final term, Mg^*, indicates that removing the Mg^*OH will leave a freshly exposed Mg^* site (initially Mg(m) where (m) stands for metal) which is available for further reaction according to Equation 7. Since dissolving surfaces will not be atomically smooth, but quite rough, we leave aside the details as to how the vacancy is dealt with, but rather, stick with the simplistic description for the sake of developing an analytical model (see further below). Also, the dissolution of Mg from sites that do not have OH adsorbed can occur in parallel with Reaction 11 and, as will be shown later on, does not have a mathematical effect on the details of the current model. This concept is essentially very similar to that proposed by D. Williams and R. Newman in a 2015 Faraday Discussion - "a fast-dissolving metal has more (albeit momentarily) bare metal sites which are the sites of hydrogen evolution" - although herein the detail is added that this route is necessary to overcome the otherwise difficulty of eliminating the adsorbed OH species that would prevent further reaction.¹⁴

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As will be shown, a reaction kinetic analysis – using reasonable activation energies and Arrhenius kinetics obtained from the theoretical literature and standard reference databases – that consists of Equations 7–9,11 and the law of conservation (the total number of surface sites must be conserved) can quantitatively account for the negative difference effect that is visible in both of Figures 1 and 2. Of omission from the mechanistic account is the formation of magnesium hydroxide films (distinct from metallic magnesium surface covered by a full or partial monolayer of adsorbed OH, Mg^*OH), suggested to also be relevant to hydrogen evolution by Salleh et al.¹⁵ We assume that this pathway is slower than the dissolution and HER kinetics, although certainly a treatment that considers the role of hydroxide in more detail could be made in a future work- especially given that theoretical data for the interactions of water and hydrogen with Mg(OH)₂ surfaces are also being newly made available.^12,16 Some preliminary discussion of this point based on theoretical data points for water dissociation and the formation of OH vacancies through OH⁻ dissolution from the Mg(OH)₂ surface will be made toward the end of this paper.

Proposing a mechanism as in Reactions 7–9, 11 provides a qualitative rationale that might explain the negative difference effect, but some reasonable, quantitative model needs to be developed in order to see if the proposed mechanism is truly viable. To begin constructing the quantitative model we note that, under the proposed mechanism, the surface sites will consist of available Mg sites Mg^*, hydroxylated sites Mg^*OH, and hydrogenated sites Mg^*H. The model in its current form only considers the actively dissolving part of the magnesium surface, and so does not consider those surfaces that have been passivated by thin films of Mg(OH)₂. This neglect is not meant to assume that these surface films are not important, but that they have not yet been included in the current model, and, we are assuming implicitly that the passivation to form the Mg(OH)₂ film occurs at a time-scale much slower than the Reactions 7–9,11. Another type of surface site not considered herein are those that may be occupied by anions, like chloride. The role of chloride in affecting surface properties will be treated in a separate publication which uses density functional theory calculations for adsorption energies of different species on Mg surfaces to construct a surface phase diagram for magnesium in aqueous media.^3,17,18

With these points in mind, we will continue for now to only focus on Mg^*, Mg^*OH and Mg^*H, and the conservation law:

where θ_M, θ_H, and θ_OH stand for the surface fractional coverage of metal sites, hydrogenated sites, and hydroxylated sites, respectively.

In addition, we have the kinetic rate equations (K₁–K₄) for Reactions 7–9,11:

where the upper case "K_i"s are the reaction rates, and the lower case "k_i"s are rate constants. The rate constants for the electrochemical reactions k₃ and k₄ will be potential dependent, following Tafel type relations:

The sign of the exponential differs from k₃ to k₄ due to the switch from a cathodic elimination of OH to an anodic elimination of OH conjoint with Mg dissolution. Under standard conditions (i.e. concentrations of 1 M for OH⁻ and Mg²⁺ and at standard temperature and pressure), U₀ for k₃ will be −0.83 V NHE, the equilibrium potential for hydrogen evolution, and U₀ for k₄ will be −2.38 V NHE, the equilibrium potential for magnesium dissolution.¹⁹ To simulate the effect of different pH and background Mg²⁺ concentrations we can shift the potentials using the Nernst equation. Within the treatment that follows we use [Mg²⁺] = 2×10⁻⁵ M and pH 11 based on the solubility product of Mg(OH)₂ and typical pH resulting from magnesium dissolution.²⁰ We will assume a typical β value of 0.5, although, at some point, it should be feasible to determine β entirely from first-principles.

The kinetic rate equations (K₁–K₄, i.e. Equations 13–16) are expressions for the rate of change in the consumption and production of the reaction sites, quantitatively expressed by the fractional surface coverages θ_M, θ_H, and θ_OH. Thus, they contain the differential equations that control the evolution of θ_M, θ_H, and θ_OH over time. For example, according to Equation 7, θ_M will decrease at twice the rate at which θ_H and θ_OH increase. However, the differential equations for these variables will be tightly coupled. The differential equations resulting from the simultaneous occurrence of Reactions 7–9,11 are:

Since the adsorbed surface species are intermediates to the hydrogen evolution reaction, as illustrated in Equation 10, the steady state approximation can be applied- that is, that the rates of change captured in these three differential equations should, on average, go to zero. Here, again, it is appropriate to remember that corroding Mg surfaces are eventually protected by the formation of a Mg(OH)₂ film, however, this film is observed to form behind the evolving active corrosion front (see work by G. Williams),²¹ hence we assume this is at a time scale several orders of magnitude away from the events captured in Reactions 7–9,11. In fact, it seems likely that this film might form following the reaction sequence 7–9,11, as a consequence of the re-precipitation of Mg(OH)₂ once a critical pH and magnesium concentration are reached in the near-surface environment.

When we apply the steady state assumption, it becomes possible to solve for the steady state coverages θ_M, θ_H, and θ_OH and hence the reaction rates K₁–K₃, Equations 13–15. The reaction rate K₂ is the rate corresponding to hydrogen evolution reaction (i.e. the observable volumetric evolution rate shown in Figure 2). The algebraic expression for K₂ resolves to:

The variables α and γ are introduced for simplicity, and the somewhat complex form for the rate equation arises due to the necessity of solving a quadratic equation when the steady state approximation is applied to the differential equations. The variable α can be related to the relative rate of elimination of the Mg^*OH sites relative to hydrogen recombination, and the variable γ can be related to the relative rates of hydrogen recombination compared to water dissociation.

Having now presented the basis for, and the results of, the derivation of the hydrogen evolution rate, K₂, as a function of the fundamental rate constants k_1-4, we will now use a first-principles approach to quantify the rate constants, and hence see if the proposed combination of mechanisms can account for the negative difference effect of hydrogen evolution on pure magnesium.

While it is not straightforward to estimate the rate constants, k₁–k₄, we can in fact make some reasonable guesses by applying the Arrhenius equation which resolves to the requirement of estimating a prefactor A and an activation energy E_a:

Reaction 7 is the dissociation of water over the magnesium surface. According to the density functional investigations by Dr. Williams, the barrier to water dissociation over a pair of vacant magnesium sites is 1.06 eV calculated with explicit solvent molecules.^c,12 The value for the prefactor A is taken to be 1.0 × 10¹³, i.e. the standard "attempt frequency" based on the typical vibrational frequency for atoms and molecules in the condensed phase.²²

Similarly, the barrier to hydrogen recombination (Reaction 8) was computed to be 1.04 eV.¹² Other theoretical estimates for this reaction have a similar barrier.^23,24 According to Pozzo, the attempt frequency for hydrogen recombination is around 1.0 × 10¹⁴.²³

Reaction 9 is the cathodic desorption of hydroxide. This reaction will be electrochemical in nature, and so the Arrhenius term should be coupled with the Tafel equation. The activation barrier for desorbing hydroxide can be estimated using some reasonable physical arguments. The initial state- adsorbed hydroxyl- is likely to have a semblance of the solvation shell, so we assume it has approximately 1/2 of the solvation energy in the initial state. During the transition to a completely solvated hydroxide ion, the hydroxyl must first desorb, which has an estimated energy of 5.0 eV.^d The hydroxyl species must then receive an electron to become OH-; the energy associated with this is 3.7 eV, from experimentally determined electron affinity of OH.²⁵ At the equilibrium potential for hydrogen evolution this value should be modified to the value 3.7−4.44 = −0.74 eV, since the absolute potential of the hydrogen evolution reaction is 4.44 V.²⁶ The hydroxide ion then receives the rest of its solvation shell; half of the solvation enthalpy of OH- is 2.4 eV, using the enthalpies tabulated by Marcus.²⁷ Put altogether, this gives an estimated barrier to the cathodic desorption of OH of 1.9 eV. The standard prefactor of 1.0 × 10¹³ is used here, and the Tafel equation with a β value of 0.5 applied.

The final reaction step considered in the scheme is the dissolution of the magnesium ion. We estimate the barrier using the cohesive energy of Mg, 1.51 eV.²⁸ Density functional theory studies of the dissolution pathway for a copper adatom on copper showed that the early stage of dissolution was associated with the breaking of the metal bonds,²⁹ and hence, the cohesive energy should serve as a reasonable estimate for the activation barrier. Alternative estimates and/or values derived from detailed studies made using density functional theory could be made in future work. The standard prefactor of 1.0 × 10¹³ is used. The Tafel relation is combined with this equation, using a β value of 0.5, as discussed earlier in the derivation.

The rate of hydrogen evolution, K₂ (Equation 14), can now be determined by using the rate constants estimated in the preceding section to determine the constants α and γ. α is a function of potential, and so we calculate α for potentials ranging between −2.8 and −1.2 V NHE. Consequently, we arrive at a HER that has two branches: a cathodic branch, and an anodic branch. see Figure 7.

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The model data plotted in Figure 7a has a strong resemblance in structure and a local minimum at similar potential to the experimental data shown in Figure 1² highlighted here in the side by side plot shown in Figure 7b. Since the pre-factor is estimated to within an order of magnitude in the rate constants, we only show the relative rate scaled to the asymptotic HER values. As seen, there is a difference in the absolute potential at which the current drops to a minimum, as well as some differences in the steepness of the curve. These differences could be due to a multitude of factors, some of which may include (a) errors in the assumed rate constants, (b) neglecting the effect of other species (chloride, for instance), (c) ignorance of any microstructure effects, such as microgalvanic coupling within the model developed herein, (d) neglect of solid-state (rather than adsorption only) films of MgO and Mg(OH)₂ as well as mixed oxy-hydroxides, and last but not least (e) faulty assumptions in the expressions used to derive the model of equations 7–9,11.

One key feature to note in Figure 7a is the plateau in the hydrogen evolution rate at the cathodic and anodic extremes plotted. This plateau is a consequence of the reaction limitation regarding the kinetics of water dissociation and hydrogen recombination. That is to say, when the potential is sufficiently negative, the OH no longer poisons the surface due to cathodic desorption of OH⁻, and so the reaction can proceed as fast as the surface can dissociate water, and/or as fast as the hydrogen atoms can recombine. At the positive potentials, the OH poisoning is remedied by the anodic desorption of MgOH⁺ (or some other synergistic combination of Mg²⁺ + OH⁻, the quantum chemical calculations have not yet made that clear) and hence the same scenario also applies. The graph is also symmetric because the two reactions (cathodic desorption of OH⁻ and anodic desorption/dissolution of Mg-OH) are both one-electron steps and we have assumed the same Tafel constants in our model. That symmetry may break as further first-principles or well-resolved experimental data can be obtained. In reality, the precipitation or solid-state formation of Mg(OH)₂ films on the surface will significantly perturb the shape of these curves as anodic potentials are encountered; that is beyond the scope of this interpretive, analytical model. It should also be noted that the curves are scaled to these plateau values due to the difficulty in calculating pre-factors for rate constants using first-principles.

To highlight the importance of surface adsorption effects on controlling the reaction behavior, in Figure 8 are shown the relative surface coverages of free metal (θ_M), hydroxide (θ_OH) and hydrogen (θ_H) that can be extracted from the steady state analysis performed using the rate equations derived in this paper. The coverage is shown as a function of the electrochemical potential. It is seen that the hydroxide coverage reaches a maximum at the open circuit potential, and then decreases as anodic or cathodic polarization occurs. The free metal sites (θ_M) maintain around 15% coverage, except for in the vicinity of the OCP, and it can be presumed that the conventional 2-electron dissolution of Mg can proceed at these sites following the usual Butler-Volmer kinetics. It could be that a combination of this 2-electron dissolution from free metal sites alongside the 1-electron dissolution at the OH-adsorbed sites is responsible for the non-stoichiometric net valence for Mg reported by Petty et al.,⁴ and further advanced by Atrens.³⁴

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The hydrogen evolution rate can also be plotted against the applied current density, as was done for the experimental data shown in Figure 2. The theoretical version, obtained using the first-principles rate constants in this work, is reproduced in Figure 9.

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Once again, the phenomenology between Figures 2 (from experimental data) and Figure 9 (the current model) is in considerably good agreement, indicative that the proposed model, grounded in the latest density functional theory studies of fundamental magnesium surface chemical properties, provides a potentially useful model for accounting for the negative difference effect.

Further exploration of the phenomenology surrounding the role of surface orientation, defects (i.e. vacancies, steps, kinks, ...), the dependence of adsorption isotherms with changes in the coverage of OH, O or H, and alloying elements should shed further light on the details of this mechanism, and could be made using a first-principles approach such as density functional theory. As more details are provided through either theory or experimental characterization, the simplistic analytical model provided in this paper would need to give way to 3-dimensional simulations using techniques such as kinetic Monte Carlo that can take into account the evolving shape of the surface, improving the realism of the model, but at the same time taking away from the intuitive benefits the analytical model derived herein provides.^30,31

At present the estimated rate constants do not allow for precise prediction of the absolute rates of hydrogen evolution or the anodic/cathodic current densities associated with the change in potential. Pre-factors can be more accurately determined from first-principles when certain kinds of accelerated molecular dynamic simulations are employed, for example.³²

An additional area of concern is that the formation of hydroxide films is not taken into account in this model- rather, we assume that the hydroxide films take a longer time scale to form compared to the events that lead to the fast moving corrosion front in magnesium alloys. It might be the case that, even when the system is cathodically polarized, a nanoscale hydroxide film is present on the material, and, as shown recently, this hydroxide film might have considerable activity for HER.¹⁵

There are some density functional theory studies for hydrogen recombination and water dissociation on magnesium oxide surfaces with and without defects that might provide some useful starting points for estimating rate constants.^16,33 Those estimated rate constants could then potentially be used in a surface reaction kinetic model that uses a model similar to that derived in this work to provide analogous predictions for the activity of a surface that has the Mg(OH)₂ structure, with defect sites providing the place of Mg^* as potential hosts for H^* and OH^* obtained from water dissociation and serving as the catalytic points for hydrogen recombination. The mechanisms may be slightly different due to the alternative ways in which water, hydrogen and hydroxyl/hydroxide can interact with ideal, stepped and defective surfaces of the oxides and hydroxides. Example steps to add to the proposed mechanism include the formation of defects due to Mg²⁺ or Mg(OH)⁺ dissolution, and the healing of defects via interactions with water, possibly involving the concomitant H₂ evolution, see, for example Ref. 34.

We can make already some rough estimates for how processes may be different for the hydroxide film by looking at theoretical calculations that have already been performed on Mg(OH)₂ and MgO surfaces. Water dissociation on Mg(OH)₂ has been shown to have a barrier of 27.6 kcal/mol (1.2 eV, similar to the value calculated by Dr. K. S. Williams for Mg(0001), as used in this work) and OH⁻ dissolution of 43.2 kcal/mol (1.9 eV, essentially equal to the value used here for Mg surface).¹⁶ The hydrogen recombination barrier on a defective MgO surface is 0.61 eV,³³ suggesting that the overall rate at which these processes occur might be faster on an oxidized surface. Using these rough numbers for oxidized surfaces gives outputs that are entirely analogous to those shown in Figures 7 and 9.

The theory concerning the formation of Mg⁺ as a precursor to Mg²⁺ evolution³⁴ and its tentative placement on the traditional Pourbaix diagram³⁵ may have some fundamental origin in the proposed reaction step 11, which we recount here:

This reaction has the "appearance" of being a one-electron dissolution process, but, in reality, there is a second electron that we consider to have transferred to OH⁻ during the dissolution event. One could also write a possible pathway for Mg dissolution as:

which is effectively the same, but we consider the OH⁻ to be a ligand that supports the dissolution of Mg (similarly to Bender et al.¹⁰ and Casey et al.⁷). This microkinetic reaction step could be occurring, and would give the "appearance" of a univalent Mg as an intermediate to Mg dissolution. Since it is conjectured that 2-electron electrochemical steps are less likely (and therefore kinetically much slower) than 1-electron steps, it is possible that the dissolution of Mg occurs in this way rather than as the direct 2-electron generation of Mg²⁺.

The model can also be adapted to include a second route for dissolution of Mg²⁺ from the sites that do not have OH adsorbed. These are the free sites represented in the surface kinetic model as θ_M. Including the rate of dissolution for Mg²⁺ from these sites could be coupled into the kinetic model, in which case Mg²⁺ dissolution will occur in parallel with the processes occurring here, but mediated according to the extent of relevant surface sites, see Figure 8. Many of the macroscopic observations occurring in experimental tests will be a combination of the mechanisms occurring in this work, alongside features such as the precipitation of Mg(OH)₂, and the formation of distinct anodic and cathodic sites on the surface, that cannot be well-represented in the simplified mathematical model of this work. However, incorporation of all these effects in a wider multiscale model could well be achieved in a well-coordinated and collaborative effort as future work. As suggested above, one route could be to use kinetic Monte Carlo. Another route could be to use cellular automata.³⁶

A reaction scheme has been presented whereby H₂ can be generated on Mg via Tafel recombination of hydrogens obtained by the chemical (i.e. not electrochemical) dissociation of water on Mg, and the "cathodic consumption" of electrons is attributed to the desorption of OH⁻. Under anodic potentials, this cathodic consumption is disallowed and so HER would otherwise stop due to OH poisoning of the surface. We propose that, in the case of Mg, the OH can be removed anodically through the dissolution of the Mg atom to which the OH is adsorbed – assuming that OH might be facilitating the dissolution, leading to a one-electron dissolution pathway. This effectively creates fresh surface whereby more water can react with Mg producing H and OH. Using first-principles data from density functional theory and reference tables to estimate rate constants, we show that the proposed pathway can account for both cathodic and anodic HER, and the theoretically predicted phenomenology is in remarkable agreement with the measured relations between HER and potential and HER and applied current density. While there are many theoretical questions that remain to be answered, it appears that the proposed model might reasonably hold the key to understand the negative difference effect on Mg.

The author gratefully acknowledges several exceptionally beneficial and stimulating discussions that led to the development of this model and the improvement of this work with Dr. Kristen S. Williams (Boeing), Michael F. Francis (EPFL) and Gerald S. Frankel (Ohio State University). The recommendations and input from peer-reviewers was also gratefully received and benefited the paper. The Ohio State University is gratefully acknowledged for supporting this work.