Bromate anion reduction: novel autocatalytic (EC″) mechanism of electrochemical processes. Its implication for redox flow batteries of high energy and power densities

Introduction

There is a growing need in fabrication of electrical energy sources of various types. Important place in this area is occupied by devices based on electrochemical cells which ensure a direct transformation of the chemical energy into electricity. Among commercially exploited systems of this kind, rechargeable batteries using lithium-ion intercalation materials are enjoying actually a vast popularity. Their weak features are a relatively slow recharging procedure, proportionality between the overall energy content of the device and its power, very high voltages (resulting inevitably in the inflammation danger), etc.

This situation creates a driving force to development of alternative directions based on fluidic media. The principal candidate of this kind is provided by famous fuel cells, in particular those of the hydrogen-air type where the electric energy is generated owing to the hydrogen oxidation reaction at anode and the oxygen reduction reaction at cathode, with the proton transport across the separating membrane. Under conditions where the air is directly available from the atmosphere such cell can provide extremely high energy density. Moreover such systems allow an independent scaling of the overall energy and total electric power, owing to external reservoir for hydrogen storage which volume may be varied independently of the parameters of the electricity-producing cell.

The key problem of fuel cells is related to the oxygen reaction which passes with acceptable overvoltages only at specially constructed catalytically active surfaces which include significant amounts of noble metals. Besides, intermediate products of the oxygen reduction reaction destroy rapidly these catalytic layers. As a result, such energy sources turn out to be relatively expensive and not sufficiently stable for long-term utilization.

These problems of fuel cells have led to development of systems which are called “redox flow batteries” (RFBs) [1], [2], [3], [4], [5], [6], [7]. In their initial variant both the anodic and cathodic reactions represent redox processes which proceed (quasi)reversibly at cheap electrode materials (e.g. various carbon-based ones) inside the discharge cell where the electricity is generated while the products of both reactions are regenerated outside the cell either chemically (e.g. with the use of oxygen from atmosphere as oxidizer) or electrochemically. The regeneration process(es) may be performed in separate device(s), e.g. at the station in the case of electromobile applications. Mostly inorganic reactants have been used in RFBs, e.g.

Fe 3 + + e − = Fe 2 + ,  V 3 + + e − = V 2 + ,  VO 2 + + 2 H + + e − = VO 2 + + H 2 O

Organic redox couples have also been proposed, e.g. quinones [7], [8].

RFBs pretend to resolve principal problems of fuel cells related to the oxygen reaction since their redox processes may be carried out at cheap electrode materials, with no use of noble metals or specially designed catalysts. Another important advantage of RFBs compared to fuel cells is the possibility to deal only with liquid media as both anolit and catholit, contrary to gas reactants in H₂–O₂ fuel cells which complicate highly the construction of the cell where one has to achieve a three-phase boundary around catalytic particles. At the same time RFBs share the principal positive feature of fuel cells, i.e. independent scaling of the overall energy and total power of the device.

An evident shortcoming of RFBs of the traditional type is the necessity to keep the redox oxidant and its reaction product inside reservoirs, compared to atmospheric air and water vapor for fuel cells. Moreover, the concentrations of reactants even in their saturated solutions are generally low. As a result, their energy density turns out to be quite insufficient for most mobile applications while this property is less important for stationary ones.

In some RFBs a pure redox reaction (with subsequent regeneration of the oxidant) at cathode is combined with hydrogen oxidation at anode. A recent example is given by the H₂–Br₂ cell where high values of the current and power density have been achieved [9], [10], [11]. A drawback of this system is the use of bromine as the oxidant so that this strongly toxic and corrosive agent has to be stored in reservoir.

Energy density estimates for bromate reaction and prospects of electric-energy sources

Quite a different type of the oxidant for RFBs, bromate anion (BrO₃⁻), has recently been proposed by our team [12]. One may expect that aqueous solutions of its salts should possess very high energy densities owing to the 6-electron transformation to bromide anion (Br⁻) in combination with very high concentrations of its salts (in particular, with Li⁺, Na⁺ and Ca²⁺ cations), detailed information on these saturated solutions within a broad temperature interval may be found in Fig. 2 of Ref. [12]. As for LiBrO₃, its solubility varies from 4.5 to 5.8 mole per 1 kg of solution for the temperatures from −40° to 100°C. What is also crucially important for this process, the solubility of the product of its reduction process, LiBr, is very high, too.

Theoretical estimate of the energy density of the lithium bromate is based on the reaction scheme:

Solubility of this salt [13], [14] is over 65%, i.e. 4.9 mole/kg (solution) at 25°C and 5.5 mole/kg (solution) at 60°C. Since the consumption of 1 mole of bromate is accompanied by passage of 6 F electrons, the redox-charge density of the saturated LiBrO₃ solution is 790 A h/kg at 25°C and 880 A h/kg at 60°C.

Standard potential of reaction (1C), E°, is 1.41 V (NHE) [15]. If the hydrogen reaction takes place at anode:

this value of E° gives about 1100 W h/kg at 25°C and 1300 W h/kg at 60°C for the energy density of the cathodic process.

Since the density of the saturated LiBrO₃ solution are about 1.8 g/cm³ [13], [16] the voluminous charge and energy densities of the cathodic process are 1400 A h/dm³ and 2000 W h/dm³ at 25°C while 1600 A h/dm³ and 2300 W h/dm³ at 60°C.

Both the charge and energy densities exceed strongly corresponding values for modern cathodes of lithium-ion batteries.

If the bromate reaction is used in the cathodic section of the H₂–BrO₃⁻ battery the global parameters of the cell will also depend on the fraction of the hydrogen mass inside its container. Further estimates are carried out for 5.7% (mass) [17] which corresponds to 1530 A h/kg for the charge density. It would give 520 A h/kg, 750 W h/kg at 25°C and 560 A h/kg, 810 W h/kg at 60°C for the overall charge and energy densities of the H₂–LiBrO₃ cell. These very high values represent obvious interest for application of such electricity-producing elements in both stationary and mobile devices.

One should also mention such merits of the bromate system as no inflammation or explosion danger, no need in noble/precious/rare elements, no corrosion activities or toxic-gas emissions during the bromate solution storage, no consumption of oxygen, no hazardous gas extinction, etc.

Bromate reduction via catalytic redox-mediator (EC′) mechanism

All these attractive features of the bromate anion as oxidant for cathode are counterbalanced by the lack of its electroactivity within the potential range positive with respect to the hydrogen electrode, for all tested electrode materials including noble metals. The conventional solution to overcome this problem of non-electroactivity of the principal reactant (denoted as A) is to apply the EC′ mechanism of electrochemical reactions [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28] which is based on addition of a reversible redox couple, C and B, into solution which triggers a combination of electrochemical and chemical steps. In the most general case the reaction scheme may be written down in the form [29]:

• (1-EC′) ( mostly reversible )  electrochemical step :  m C C + n e − ⇄ m B B

• (2-EC′) ( mostly irreversible )  chemical step :  m A A + m B B  →  m P P + m C C

This mechanism has been extensively studied for various regimes of the process, mostly assuming that all stoichiometric coefficients, m_i, are assumed to be equal to 1 while the general qualitative features of the mechanism are remaining the same for other values of the coefficients, in particular for the non-equal values of m_C and m_B, too. On the other hand, it is crucially important that the ratio of the coefficients in front of C and B, m_C/m_B, is the same in eqs. (1-EC′) and (2-EC′). Then, the passage of the cycle composed of reactions (1-EC′) and (2-EC′) does not change the total amount of the components of the redox couple, m_CC+m_BB, so that this couple plays the role of a redox-mediator catalyst.

For such systems the bulk solution contains (for the reduction process) both the electrochemically inactive oxidant, A, of a high concentration, A°, and the oxidized component of the reversible redox couple, C, the latter concentration, C°, being much lower than that of the former one, A°.

For illustration of predictions of such systems let us consider the EC′ process which takes place at a rotating disk electrode (RDE) under steady-state conditions. Then, the electrode surface is “equally available”, i.e. all concentration distributions depends on a single spatial coordinate, z, normal to the electrode surface (z=0) while the current density, j, is the same in all points of the surface.

To simplify the analysis the convective effects are simulated within the framework of the “stagnant Nernst layer model” [30], [31], [32], [33] where the concentrations of all species are assumed to be constant outside the (Nernst) diffusion layer (for z>z_d) while the convective terms are disregarded inside the layer (for z<z_d). The thickness of this diffusion layer is given by the Levich formula:

containing RDE revolution rate, Ω, and frequency, f, kinemati viscosity of solution, ν_kin, and diffusion coefficient of the transported species, D.

This definition of z_d is ambiguous for systems where one has to consider the transport of several species (e.g. A, B and C above) having different diffusion coefficients, D_A, D_B and D_C. This effect is taken into account automatically within the framework of the convective diffusion theory if this difference in diffusion coefficients is taken into account. An alternative (and much simpler) way to get correct quantitative results is to use our generalization of the Nernst layer model [29], [34], [35] which is outlined below in Section “Generalized Nernst layer model” for the bromate system.

It is assumed that this reaction (1-EC′), (2-EC′) is performed in the presence of background electrolyte excess. Then, effects of migrational transport may be disregarded for species under consideration, A, B and C. Then, the set of transport equations inside the diffusion layer has the form:

where the local rate, V(z), of the chemical step, eq. (2-EC′), is determined by the product of the local concentrations of species A and B:

while the stoichiometric coefficients, m_B and m_C, are determined by reaction scheme (2-EC′), m_A=1 to simplify the formulas.

At the outer boundary of the diffusion layer the concentration of each species should be equal to its value in the bulk solution:

while the balance relations determine the diffusional fluxes at the electrode surface:

where j is the cathodic current density, F, the Faraday constant.

One can get approximate analytical solutions for two regimes.

1. Weak current regime:

Passing current is so weak that the consumption of species A near the electrode surface does not lead to a significant gradient of its concentration so that the overall drop of its concentration across the diffusion layer is small compared to its bulk concentration:

which imposes a limitation to the current density to be small compared to the diffusion-limited one for species A, j_A^lim:

Then, one may replace approximately A(z) by A° in eq. (5) which leads to a linear differential equation for B(z). Its solution with the use of boundary conditions (6) and (7) allows one to get analytic expressions for other concentration distributions.

These formulas introduce an important parameter, “kinetic layer thickness (for weak currents)”, z_k:

The ratio of the diffusion and kinetic layer thicknesses, x_dk=z_d/z_k, determines the shape of the concentration profiles.

If x_dk≪1, i.e. the diffusion layer is much thinner than the kinetic one, then species B generated at the electrode via electrode reaction (1-EC′) leave the diffusion layer without reacting with species A. Thus, chemical step (2-EC′) does not affect the concentration profiles inside the diffusion layer and the whole process is reduced to the discharge of species C from the bulk solution at the electrode surface, with diffusion of generated B species towards the bulk solution.

Since the bulk concentration of species C is much lower than that of species A, C°≪A°, eq. (6), the current density, j, is limited by the diffusion-limited current for species C, j_C^lim:

so that condition of the “weak current regime”, eq. (8j), is satisfied automatically.

If x_dk≫1, i.e. the diffusion layer is much thicker than the kinetic one, then species B generated at the electrode are consumed by reaction (2-EC′) inside the kinetic layer, i.e. deeply inside the diffusion layer, the kinetic layer thickness being about z_k, i.e. independent of the diffusion layer thickness, z_d (Fig. 1b). This chemical step also consumes species A resulting in diminution of their concentration inside the kinetic layer, thus inducing their diffusional transport from the bulk solution (Fig. 1a). The distribution of the reaction product, species C, which are generated from species A and B inside the diffusion layer diffuse towards the electrode surface to regenerate species B (Fig. 1c). It implies that species B and C participate in a redox cycle circulating inside the kinetic layer. As a result, the current density, j, is limited by the diffusion-limited current across the kinetic layer which will be called “the catalytic current density”, j^cat:

Fig. 1:

Concentration profiles: A(z)/A°, D_BB(z)/(D_A A°) and D_CC(z)/(D_A A°) as functions of z/z_k for x_dk=z_d/z_k=3 (lines 2, 2′) corresponding to j_A^lim=8.8 A cm⁻² and x_dk=z_d/z_k=8 (lines 1, 1′) corresponding to j_A^lim=3.3 A cm⁻². Current density: j=j^max=j^cat=26 mA cm⁻² (solid lines 1, 2) or j=0.5 j^max=13 mA cm⁻² (dashed lines 1′,2′). Stoichiometric coefficients: m_A=1, m_B=6, m_C=3, n=6. Other parameters: A°=1 M, J_CA=j_C^lim/j_A^lim=10⁻³, D_A=1.48 10⁻⁵ cm² s⁻¹, D_B=2.08 10⁻⁵ cm² s⁻¹ [13], D_C=1.18 10⁻⁵ cm² s⁻¹ [36], k=33 M⁻¹c⁻¹ (for k°=1.2 M⁻³ s⁻¹ and proton activity a_H=5.3 M for 2 M H₂SO₄) [13], [37], [38], z_k=3.6 μm (these values are identical to those for the bromate reaction in Section “Bromate system without added redox catalysts. Novel autocatalytic redox-mediator (EC″) mechanism” below, to underline both similarities and differences in predictions for these two systems).

Since the diffusion layer is much thicker than the kinetic layer one, z_d≫z_k, for plots in Fig. 1 the concentration profiles for B and C species are practically independent of their ratio, x_dk=z_d/z_k (except for the most external areas of the kinetic layer), as one can see from comparison of plots for x_dk=3 and x_dk=8. Since the maximal current density, j^max, is defined as the current where C(0)=0, its values for for x_dk=3 and x_dk=8 are practically identical, in conformity with eq. (11) where j^max≅j^cat, i.e. it is independent of the diffusion layer thickness, z_d. For a fixed x_dk value both B(z) and C°−C(z) are proportional to the passing current density, j (solid and dash lines in Fig. 1b,c).

1. Thin kinetic layer regime:

If the kinetic layer is much thinner than the diffusion one which implies the condition:

one can consider both weak and strong currents. Since kinetic layer is relatively thin and the derivative, dA/dz, is equal to 0 at the surface, eq. (7), one can approximately consider its value as being constant inside the kinetic layer, i.e. A(z)≅A(0). Then, one can solve again the resulting equation for B(z): D_B d²B/dz²≅m_B k A(0) B(z), in combination with boundary conditions (6) and (7) as well as with a relation between B(0) and A(0), to find their values as a function of the imposed current density, j.

If the condition of the weak current regime, eq. (8j), is also satisfied for this current, then thus found profile of B(z) is practically identical to the one found in case 1, corresponding to the kinetic layer thickness equal to z_k. It means that the calculation of the concentration profiles for x_dk=3 or for x_dk=8 with the use of expressions for the thin kinetic layer regime will result practically in the same plots as those derived for the weak current regime (Fig. 1).

On the contrary, if the current density, j, becomes comparable with the diffusion-limited one for species A, j~j_A^lim, the difference of the concentrations of species A across the diffusion layer, A°−A(0), becomes comparable with its bulk concentration, A°. Then, the kinetic layer thickness, z_kj=(D_B/m_B k A(0))^1/2, increases as a function of j in view of the diminution of A(0).

According to eq. (11) the current density may become comparable with the diffusion-limited one for species A, j~j_A^lim, only for very large values of the parameter, x_dk=z_d/z_k ~ j_A^lim/j_C^lim=1/J_CA, or even larger values. The variation of z_kj as a function of the current density, j, is illustrated in Fig. 2 for J_CA=0.001 and correspondingly x_dk=z_d/z_k=1000. One may note that the condition for the kinetic layer to be much thinner than the diffusion layer one, z_d/z_kj≫1, is perfectly fulfilled.

Fig. 2:

Dependence of the kinetic layer thickness, z_kj (divided by its value for weak currents, z_k) on the current density, j, for x_dk=z_d/z_k=1000. Other parameters are given in Fig. 1.

The corresponding concentration profiles for x_dk=1000 and J_CA=0.001 are given in Fig. 3 for various values of the current density, j. One can see that in conformity with Fig. 2 for values of the current-dependent kinetic layer thickness, z_kj, the variation of the concentrations of species B and C is localized near the electrode surface. It implies that all species B generated at the electrode by reaction (1-EC′) are consumed inside the kinetic layer by the chemical step, eq. (2-EC′), while all species C produced by the latter diffuse to the electrode, to participate again in reaction (1-EC′).

Fig. 3:

Concentration profiles: (a) A(z)/A°, (b) D_B B(z)/(D_A A°) and (c) D_C C(z)/(D_A A°) as functions of z/z_k for x_dk=z_d/z_k=1000 (corresponding to j_A^lim=26 mA cm⁻²) and various values of the current density: j=j^max=16 mA cm⁻² (lines 1), j=8 mA cm⁻² (lines 2) or j=4 mA cm⁻² (lines 3). Other parameters are given in Fig. 1.

Thus, these reactions of species B and C form a catalytic redox cycle which transforms species A into their product, species P. It reduces the concentration of species A inside the kinetic layer, thus inducing its diffusional transport across the diffusion layer from the bulk solution. Intensity of these processes is determined by the passing current, j (Fig. 3): its increase leads to a greater amplitudes of variation of the B and C concentrations inside the kinetic layer (Fig. 3b,c) as well as to a greater slope of A(z) outside the kinetic layer (Fig. 3a).

For each value of the principal parameter, x_dk=z_d/z_k, the interval of possible values of the current density, j, is limited from above by the value which is called “the maximal current density”, j^max. This dependence in the form of j^max/j_A^lim vs. x_xdk, is shown as line 1′ in Fig. 4a.

Fig. 4:

Plots for the maximal current density, j^max, in bilogarithmic coordinates: (a) dimensionless value, j^max/j_A^lim, as a function of x_dk=z_d/z_k (line 1′); (b) j^max vs. RDE frequency, f (line 1). Dashed straight lines: asymptotic behavior of plots 1′ and 1 in various ranges: j^max=j_C^lim (lines 2′ and 2), j^max=j_A^lim+j_C^lim (lines 3′ and 3), j^max=j^cat (lines 4′ and 4). Values of parameters are given in Fig. 1.

In view of relation (3) between the RDE revolution rate, f, and the diffusion layer thickness, z_d, the above results for the concentration profiles and for the maximal current allows one to predict their dependence on f. In particular, line 1 in Fig. 4b represents a plot for j^max vs. the RDE revolution rate, f. This figure also shows the relationship between j^max and x_dk.

As one can see from eq. (10), the maximal current is close to the diffusion-limited one for species C (added redox couple component): j^max≅j_C^lim (line 2 in Fig. 4b) i.e. its value is inversely proportional to z_d, i.e. to x_dk for very small x_dk values, x_dk≪1. Since the diffusion-limited current density for species A, j_A^lim, is also inversely proportional to z_d their ratio, j^max/j_A^lim, is close to a constant, J_CA, within this range (line 2′ in Fig. 4a).

For a very broad range of moderate x_dk values, 1≪x_dk≪j_A^lim/j_C^lim=1/J_CA, maximal current is close to the “catalytic one”: j^max≅j^cat, eq. (11), i.e. its value is independent of the diffusion layer thickness, i.e. of x_dk or f (line 4 in Fig. 4b), while the ratio shown as line 4′ in Fig. 4a (j^max/j_A^lim) is proportional to x_dk.

At last for even larger x_dk values:

the maximal current density, j^max, is approaching to the sum of the diffusion-limited current densities of species A and C:

As a result, j^max varies again as z_d⁻¹ or f^1/2, i.e. x_dk⁻¹ (line 3 in Fig. 4b) while the ratio, j^max/j_A^lim, is close to a constant: j^max/j_A^lim≅1+J_CA (line 3′ in Fig. 4a).

As a whole, the plots for the maximal current density (lines 1′ and 1 in Fig. 4a,b) demonstrates the existence of three characteristic ranges described by eqs. (10), (11) and (13j) separated by two transition zones where x_dk ~ 1 and x_dk~j_A^lim/j_C^lim. The latter value which is proportional to the ratio of the bulk concentrations of species A and C, A°/C°, is usually very large, i.e. J_CA=j_C^lim/j_A^lim≪1, to minimize the amount of the added redox couple. Then, the medium interval of frequencies, f, or x_dk where the maximal current is independent of the diffusion layer thickness and it is close to the “catalytic one”, j^max≅j^cat, eq. (11), is very extended: the ratio of their values for transitions (line 2 to line 4 and line 4 to line 3) is J_CA⁻¹ (i.e. 1000 in Fig. 4a) for the x_dk axis and J_CA⁻² (i.e. 10⁶ in Fig. 4b).

Since the interval of the rotation frequency values, f, is always restricted by technical reasons it is practically impossible to observe both the transition between the thick and thin kinetic layer regimes (x_dk~1) and the approach to the diffusion-limited current for species A, j^max~j_A^lim, for the same system. Moreover, as one can see from Fig. 4a,b this approach of j^max to j_A^lim takes place for such enormous thicknesses of the diffusion layer that the value of this current, j_A^lim, is even lower than j^cat.

As a result, the maximal current, j^max, can only become comparable with j_A^lim within the range where the thicknesses of the diffusion and kinetic layers are of the same order of magnitude if the amount of the added redox couple is comparable with the one of the principal oxidant, C°~A°, which contradicts to the principle of catalysis. In the “normal” case where the amount of the added catalytic component, C°, is very small compared to that of the principal oxidant, A°, i.e. J_CA≪1, the maximal current within the practically interesting range of rotation frequencies is proportional to C° so that it is very small compared to diffusion-limited current for species A.

Besides, the use of such externally added redox components inside the H₂–BrO₃⁻ flow cell is problematic, either, because of the necessity to avoid a strong crossover of the components of the redox couple, B and C, across the proton-exchange membrane. This condition excludes the dominant majority of reversible redox couples which are usually used in flow cells.

As a whole the approaches based on the conventional EC′ mechanism have probably to be taken out from the consideration as non-prospective.

Bromate system without added redox catalysts. Novel autocatalytic redox-mediator (EC″) mechanism.

In the first paper on the use of the bromate system for electrical energy production [12] Tolmachev et al. proposed a different approach based on the presence of Br₂ inside the bromate solution. The bromine concentration is determined by the rate of the bromate-anion decomposition under strongly acidic conditions but even for 3–4 M H₂SO₄ it remains to be within a tracer level, i.e. well below millimolar range, as it is evidenced by the absence of the absorption band near 400 nm in the spectrum of such a solution.

Unlike bromate anion, bromine is reduced rapidly at numerous substrates, including those of various carbon materials:

The reaction product, bromide anion, reacts irreversibly with BrO₃⁻ via the comproportionation reaction:

with regeneration of bromine which can participate again in the electrochemical step, eq. (14E). Thus, the reactions (14E) and (14C) form a cycle where the components of the redox couple, Br₂ and Br⁻, play a mediating role in transformation of the principal oxidant, BrO₃⁻.

Let us assume for the moment that the bulk solution contains a great excess of proton, compared to bromate, speaking nothing of bromine. Then, kinetics of reactions (14E) and (14C) is determined by the above Br containing species while the proton concentration is close to its bulk-solution value everywhere in solution.

Formally, eqs. (14E) and (14C) represent a particular case of the general scheme for the EC′ mechanism given by eqs. (1-EC′), (2-EC′) if C=Br₂, B=Br⁻, A=BrO₃⁻+6H⁺, P=Br₂, m_C=2.5, m_B=5, m_A=1, m_P=0.5. If this conclusion were correct, then intensive currents could only be expected for high concentrations of Br₂ in the bulk solution, i.e. for a significant chemical decomposition of BrO₃⁻. However, the analysis carried out in a detailed theoretical study [39] has given quite different results.

The reason why the principles of the EC′ mechanism are inapplicable with the system described by eqs. (14E), (14C) is the product(s) of the chemical step, P in eq. (2-EC′), are assumed to be non-reactive while in reaction (14C) the principal reaction product, Br₂, is electroactive, i.e. it can participate in electrode reaction (14E). This circumstance changes radically the predictions.

The theory proposed in Ref. [39] is based on the same set of transport-kinetic eqs. (4), (5) and boundary conditions (6), (7) as the one given in Section “Bromate reduction via catalytic redox-mediator (EC′) mechanism” above for the EC′ mechanism where the stoichiometric coefficients are determined by those in eqs. (14E) and (14C), in particular:

Their comparison with eqs. (4–7) shows that almost all coefficients in these relations remain the same (m_A=1, m_B=6, m_C=3, n=6), except a different coefficient, m_B′=5, in eq. (15V) for B(z).

According to Refs [37], [38], [40], [41], the rate of the comproportionation reaction, eq. (14C), is also given by formula (5), i.e. it is proportional to the product of the local concentrations of species A and B while the rate constant, k, depends on pH [14], [42], [43].

The set of non-linear differential eqs. (15V) combined with relation (15k) and boundary conditions (15zd) and (15j) was solved [39] with the use of the same analytical procedure described above in Section “Bromate reduction via catalytic redox-mediator (EC′) mechanism” for the EC′ mechanism.

The solution for the weak current regime corresponds to the approximate constancy of the concentration of species A (BrO₃⁻), eq. (8A), which implies an upper restriction to the current density, eq. (8j). Then, eqs. (15) and (5) provide a closed linear differential equation for B(z) with the corresponding boundary conditions for this function, eqs. (6) and (7). Its solution gives expressions for A(z) and C(z).

Similar to the EC′ mechanism the key parameter is given by the ratio of the diffusion and kinetic layer thicknesses, x_dk=z_d/z_k, with the use of the same definitions, eqs. (3) and (9), for z_d and z_k.

1. Weak current regime:

For small values of this parameter: x_dk≪1 the results for the concentration distributions and for the maximal current are quite identical to those for the EC′ mechanism, Section “Bromate reduction via catalytic redox-mediator (EC′) mechanism”. In particular, the presence of the principal oxidant, A (BrO₃⁻), does not play any role under these conditions and the process consists in the electroreduction of C (Br₂) at the electrode after its diffusion from the bulk solution. As a result, the maximal current density is close to the diffusion-limited one for species C:

so that it is much smaller than the diffusion-limited current density for species A (BrO₃⁻):

For greater values of the parameter: x_dk≫1, i.e. if the diffusion layer is much thicker than the kinetic one, z_d≫z_k, species B (Br⁻) generated at the electrode are consumed by reaction (14C) inside the kinetic layer, i.e. deeply inside the diffusion layer, the kinetic layer thickness being about z_k, i.e. independent of the diffusion layer thickness, z_d. This chemical step also consumes species A (BrO₃⁻) resulting in diminution of their concentration inside the kinetic layer, thus inducing their diffusional transport from the bulk solution. The concentration profiles for species A and B for this regime (Fig. 5a,b) look qualitatively very similar to those in Fig. 1 for the EC′-mechanism, with slightly increased amplitudes of their variation due to greater values of the maximal current, j^max, for the same values of x_dk, see below.

Fig. 5:

Concentration profiles: A(z)/A°, D_BB(z)/(D_A A°) and D_CC(z)/(D_A A°) as functions of z/z_k for x_dk=z_d/z_k=3 (lines 2,2′) corresponding to j_A^lim=7.4 A cm⁻², j^max=37 mA cm⁻² and x_dk=z_d/z_k=8 (lines 1, 1′) corresponding to j_A^lim=2.8 A cm⁻², j^max=1.5 A cm⁻². Current density: j=37 mA cm⁻² (solid lines) or j=18 mA cm⁻² (dashed lines). Unlike for plots in Fig. 1 for the EC′ mechanism the values of j in Fig. 6 for x_dk=8 are much lower than that of the maximal current, j^max, since the latter is out of the weak current range. Other parameters are given in Fig. 1.

On the contrary, quite a new feature appears in the distributions of species C (Br₂) for various current densities in Fig. 5c, compared to those in Fig. 1c. Besides a rapid increase of C(z) inside the kinetic layer one can see its linear diminution outside this layer. As a result, the concentration of species C passes through a maximum within an outer area of the kinetic layer, with their diffusion in both directions, towards the electrode surface (z=0) where they are subject to electrochemical step (14E), as well as to the bulk solution.

The value of the C concentration at this maximum, C^max, compared to its bulk-solution one, C°, reflects the intensity of the catalytic process owing to the redox cycle of steps (14E) and (14C). They are practically identical for small values of x_dk while the effect increases rapidly for higher x_dk values: it is still weak for x_dk=3, even for j=j^max (Fig. 5c), while already for relatively weak currents (as it follows from the value of the maximal current, j^max, compared to the values of j in Fig. 5c) at x_dk=8 the value of C^max is much greater than that of C° and it increases rapidly with j. Since the maximal current density, j^max, at x_dk=8 is comparable to the diffusion-limited one for species A, j_A^lim, the further analysis for higher intensities of the current (see below) cannot be carried out on the basis of formulas for the weak current regime.

In conformity with these results for the concentration profiles the dependence of the maximal current, j^max, on x_dk or f (lines 2′ and 2 in Fig. 6) for the case of a thin diffusion layer, x_dk≪1, is similar to that for the EC′ mechanism (lines 2′ and 2 in Fig. 4):

Fig. 6:

Plots for the maximal current density, j^max, in bilogarithmic coordinates: (a) dimensionless value, j^max/j_A^lim, as a function of x_dk=z_d/z_k (line 1′); (b) j^max vs. RDE frequency, f (line 1). Dashed straight lines: asymptotic behavior of plots 1′ and 1 in various ranges: j^max=j_C^lim (lines 2′ and 2), j^max=1.2 j_A^lim+j_C^lim (lines 3′ and 3). Dotted lines EC′: predictions for the EC′ mechanism (identical to lines 1′ and 1 in Fig. 4). Values of parameters are given in Fig. 1.

so that its intensity is low compared to j_A^lim, in conformity with the condition of the weak current regime.

Change of the shape of the C(z) distribution (Fig. 5c) for moderate x_dk values, x_dk≫1, results in a difference of predictions of the weak-current approximation for the maximal current (lines 1′ and 1 in Fig. 6), compared to those for the EC′ mechanism (lines 1′ and 1 in Fig. 4). In the latter case the maximal current density was close to a constant value independent of x_dk or f, j^max≅j^cat, eq. (11) (lines 4′ and 4 in Fig. 4, lines EC′ in Fig. 6). On the contrary, the maximal current for the bromate system at x_dk≫1 is a growing function of x_dk (lines 1′ and 1 in Fig. 6), its slope increasing rapidly so that in the vicinity of a critical value, x_dk=6, the weak current approximation becomes inapplicable for predictions of the maximal current. Therefore, for its greater values, x_dk>6, this approximation is only valid for currents densities satisfying to condition (8j) while even stronger currents can pass through the system.

1. Thin kinetic layer regime:

Similar to Section “Bromate reduction via catalytic redox-mediator (EC′) mechanism” for the EC′ mechanism, useful analytical results are also available for the case where the diffusion layer thickness is much greater than the kinetic layer one, x_dk=z_d/z_k≫1, eq. (12) so that one can neglect the spatial variation of A(z) inside the kinetic layer: A(z)≅A(0). Then, one can derive again approximate analytical expressions for all concentration distributions (in particular, for the current-dependent kinetic layer thickness, z_kj) as well as a transcendental equation for the maximal current density, j^max.

Comparison of lines 1′ and 4′ in Fig. 6a as well as lines 1 and 4 in Fig. 6b for the maximal current density, j^max, demonstrates a radical difference in predictions for the systems considered in Sections “Bromate reduction via catalytic redox-mediator (EC′) mechanism” (EC′ mechanism) and “Bromate system without added redox catalysts. Novel autocatalytic redox-mediator (EC″) mechanism” (bromate system). A rapid growth of the maximal current density, j^max, in the range of moderate x_dk values (between 1 and 6) makes its values comparable with the diffusion-limited current for species A (BrO₃⁻), j_A^lim. Despite a slower increase of j^max in the range of greater x_dk values (between 6 and 10) the maximal current density reaches the level of j_A^lim and even exceeds it, their ratio approaching to a constant value equal to 1.2+J_CA for greater x_dk values (Fig. 6a), in deviation from its limiting value equal to 1+J_CA for the EC′ mechanism (Fig. 4a). The behavior of the plot in Fig. 6b is slightly different because of the dependence of the diffusion-limited current for species A, j_A^lim, on the diffusion layer thickness, z_d, i.e. on x_dk or rotation frequency, f. As a result, after a rapid growth of j^max at moderate x_dk values the maximal current passes through a maximum (Fig. 6b), with its decrease for even greater x_dk values.

Thus, both limits of these dependences are similar for the EC′ mechanism and the bromate system: j^max≅j_C^lim, eq. (10), for x_dk≪1 as well as j^max≅j_A^lim+j_C^lim, eq. (13j), for the EC′ mechanism or j^max≅1.2 j_A^lim+j_C^lim for the bromate system in the limit of x_dk→∞. However, the behavior within the intermediate range is absolutely different, as it is evident from comparison of lines 1 and 4 in Fig. 6b, or lines 1′ and 4′ in Fig. 6a. Line 4 demonstrates a usual behavior of the maximal current as a function of the RDE frequency: a weaker agitation of the solution by RDE (lower rotation frequency, f) does never result in a higher current, in the best case the current is nearly constant, as it is in the intermediate range of x_dk values: j^max≅j^cat. On the contrary, the bromate system demonstrates an anomalous behavior in Fig. 6b: diminution of the RDE frequency leads within a certain range to explosive increase of the maximal current. Within a relatively narrow interval the maximal current increases by several orders of magnitude, from a diffusion-limited one for species C (Br₂), j_C^lim, to the current exceeding the diffusion-limited one for species A (BrO₃⁻), j_A^lim. Thus, a tracer amount of Br₂ in the bulk solution may induce currents of an enormous intensity, strongly exceeding 1 A/cm² for the BrO₃⁻ concentration in the molar range.

Such a cardinal difference in predictions for systems in Sections “Bromate reduction via catalytic redox-mediator (EC′) mechanism” and “Bromate system without added redox catalysts. Novel autocatalytic redox-mediator (EC″) mechanism” originates from a positive disbalance in the cycle composed by reactions (14E) and (14C). One can see that 5 Br⁻ anions (B species) consumed by reaction (14C) produce 6 Br⁻ anions after passage of the cycle, owing to the added Br atom of the BrO₃⁻ anion. In other words, the product of the transformation of the principal reactant, BrO₃⁻, represents a component of the mediator redox couple, Br₂/Br⁻. It means that the cycle of reactions (14E) and (14C) is an autocatalytic one since passage of cycles increases the amount of the redox catalyst.

The different limit of the ratio, j^max/j_A^lim, at x_dk→∞ for the bromate system (Fig. 6a) equal to 1.2+J_CA originates from the non-electroactivity of species A (BrO₃⁻ anion) at the electrode surface so that it is consumed by the comproportionation step, eq. (14C), inside the kinetic layer. As a result, its diffusion-limited flux, N_A≅ D_A A°/(z_d−z_kj), under conditions of a very high rate of this reaction is determined by the thickness of the outer part of the diffusion layer, z_d−z_kj, where the current-dependent kinetic layer thickness, z_kj, approaches to z_d/6 so that z_d−z_kj≅(5/6) z_d and N_A≅(6/5) D_A A°/z_d. At the same time the second contribution to the maximal current is given by the discharge of species C (Br₂) from the bulk solution at the electrode surface so that their diffusion-limited flux is equal to N_C≅D_C C°/z_d. The superposition of their currents gives for the maximal current the dependence: j^max≅1.2 j_A^lim+j_C^lim shown as line 3′ in Fig. 6a and line 3 in Fig. 6b.

This all allows us to conclude that the bromate process under study corresponds to a novel mechanism of electrochemical processes (EC″) which may be called “redox-mediator autocatalysis”.

Figure 7 provides an illustration of these autocatalytic features of the process in terms of the concentration profiles. Even though the concentration profiles for species A (BrO₃⁻) and B (Br⁻) look similar to those in Fig. 3a,b for the EC′ mechanism one should note that a similar concentration drop of species A across the diffusion layer requires an extremely large thickness of the diffusion layer (z_d=1000 z_k), compared to its moderate value: z_d=8 z_k in Fig. 7a. It is the reason why the maximal current density, j^max, may reach such enormous values under such conditions, see Fig. 6b. Such intensities of the current are directly related with accumulation of very high concentrations of species B and C inside the kinetic layer (Fig. 7b,c), as a direct consequence of the autocatalytic character of the process, while these concentrations for the EC′ mechanism are always limited by the added bulk concentration of the catalytic component, C°, see Fig. 3b,c.

Fig. 7:

Concentration profiles: (a) A(z)/A°, (b) D_B B(z)/(D_A A°) and (c) D_C C(z)/(D_A A°) as functions of z/z_k for x_dk=z_d/z_k=8 (corresponding to j_A^lim=3.3 A cm⁻²) and various values of the current density: j=j^max=1.8 A cm⁻² (lines 1), j=0.5 j^max=0.9 A cm⁻² (lines 2) or j=0.25 j^max=0.45 A cm⁻² (lines 3). Other parameters are given in Fig. 1.

Increase of the passing current, j (at a fixed value of x_dk equal to 8, Fig. 7), both the drop of the A concentration and the amplitudes of the B and C concentrations grow. At the same time the concentration of C (Br₂) species at the electrode surface passes through a maximum, with its decrease for the further increase of the current. Its zero value determines in all cases the value of the maximal current (line 1 in Fig. 7c).

As for the current-dependent kinetic layer thickness, z_kj, it is directly related to the relative value of the A concentration at the surface: z_kj≅z_k [A(0)/A°]^−1/2 so that its value increases for higher current densities, as it is visible from Fig. 7b.

Its approximate behavior at very large x_dk values: z_kj≅z_d/6 is explained by the disbalance of the redox cycle which generate one extra B species per 5 ones entering the cycle. Therefore, for the steady-state character of the process, in order to keep the accumulated amounts of B (Br⁻) and C (Br₂) species inside the kinetic layer constant in time, one sixth of all C species generated by chemical step (14C) has to diffuse to the bulk solution while five sixth of this amount are to go to the electrode, to participate in the next cycle. It implies that the ratio of the diffusion distances for the former and the latter transport processes, z_d−z_kj and z_kj, should be about 5 so that z_d≅z_kj/6.

All this analysis was based on approximate analytical expressions for the concentration distributions and for the maximal current density. The same set of equations and boundary conditions has been solved numerically [43]. The calculated concentration distributions are in good agreement with analytic predictions (Figs. 5 and 7) for small and moderate x_dk values (up to about 20). Very close proximity of the analytical and numerical results (within 2%) was also found for the dependence of the maximal current density on x_dk or on f (Fig. 6) within the whole, i.e. extremely large range of the corresponding parameter.

Thus, one may conclude that all astonishing predictions of our theory follow directly from the above model of the ion transport accompanied by the chemical and electrochemical transformations.

Effects of pH

The above analysis of the bromate process in Section “Bromate system without added redox catalysts. Novel autocatalytic redox-mediator (EC″) mechanism” is carried out under the conditions of great excess of protons so that its concentration, H(z), may be considered as being practically constant everywhere in solution: H(z)≅H°, H° being its bulk solution concentration. It should be kept in mind that protons are consumed by the same comproportionation reaction, eq. (14C), in parallel to species A (BrO₃⁻). It is evident from the above analysis that the concentration gradients of species A and H (protons) are negligible under conditions of the weak current regime. Therefore, only the thin kinetic layer regime should be analyzed.

Then, species B (Br⁻) do not spread into the outer part of the diffusion layer (being consumed inside the kinetic layer one) while the other components, A, C (Br₂) and H diffuse there without chemical transformations. In view of the stoichiometric coefficients for species A and H in the chemical step, eq. (14C), the assumption of the proton excess means that the diffusion-limited fluxes for these components, N_A and N_H, should satisfy to the condition: N_H≫6 N_A. Since the diffusion distances are identical for species A and H (equal to z_d−z_kj, see above) condition (17N) takes the form: H°≫6 A° D_A/D_H. Since D_H/D_A≅6.5 [13] the condition of the proton excess is reduced to the inequality:

Even though in this case the proton concentration, H(z), is close to its bulk value, H°, everywhere in solution and it plays the role of a background electrolyte (thus allowing one to neglect the migrational component of the ion fluxes), pH of the solution may affect strongly the rate of the bromate process under conditions of a strong current via the dependence of the rate constant of the comproportionation reaction, eq. (14C) [14], [42], [44]:

where the proton activity is related to pH: a_H=10^−pH. Increase of the solution acidity, i.e. of a_H and consequently of k leads to diminution of the weak-current kinetic layer thickness, z_k. Therefore, the range of the anomalous increase of the maximal current which takes place in the range where 3<x_dk=z_d/z_k<10 (Fig. 6b) is shifted to smaller values of the diffusion layer thickness, i.e. to greater values of the rotation frequency, f. As a result, the maximal value of j^max increases rapidly, as one can see from comparison of lines 1 (a_H=3 M) and 2 (a_H=5 M) in Fig. 8a.

Fig. 8:

Relation between the maximal current, j^max, and the RDE frequency, f: (a) Proton excess, A°=1 M, a_H=3 M, k=11 M⁻¹c⁻¹, z_k=6.1 mkm (line 1), A°=1 M, a_H=5 M, k=30 M⁻¹c⁻¹, z_k=3.7 mkm (line 2), A°=2 M, a_H=5 M, k=30 M⁻¹c⁻¹, z_k=2.6 mkm (line 3); (b) Bromate excess, D_H=9.31 10⁻⁵ cm²/s [13], A°=5 M, H°=1 M, k=1.2 M⁻¹c⁻¹, z_k=8.3 mkm (line 1), A°=5 M, H°=2 M, k=4.8 M⁻¹c⁻¹, z_k=4.1 mkm (line 2). Values of other parameters are identical to those in Fig. 1.

Effect of the bulk-solution concentration of bromate ion, A°, is shown by lines 2 (A°=1 M) and 3 (A°=2 M) in Fig. 8a.

For applications of the bromate process in RFBs one needs to use a high concentration of bromate, i.e. A° should be on the level of several moles per liter. Addition of an extra acid to satisfy the condition of proton excess for the bulk concentrations, eq. (17HA), would mean to diminish strong the charge and energy density of the bromate solution. Besides, it may lead to chemical decomposition of BrO₃⁻ ions. In this context it is prospective to deal with a concentrated solution of bromate where the added acid concentration satisfies to the opposite condition: H°≪A° [45], [46] Then, it is the bromate concentration. A(z) which is practically constant across the diffusion layer: A(z)≅A° while the diffusional transport of protons will limit the current density.

The set of equations for concentrations and boundary conditions within the framework of the Nernst layer model has the form analogous to eqs. (15V), (15zd), (15j), (15k) where the rate constant of step (14C)t, k, depends on the local pH value, eq. (17k0):

Similar to Sections “Bromate reduction via catalytic redox-mediator (EC′) mechanism” and “Bromate system without added redox catalysts. Novel autocatalytic redox-mediator (EC″) mechanism”, one can get approximate analytical solutions of these eqs. [45], [46] for two ranges of parameters corresponding to

• weak current regime:

Its criterion replacing eq. (8j) includes now the diffusion-limited current density of protons, j_H^lim:

Then, not only the concentration of species A (BrO₃⁻) but also that of species H (protons) are practically constant across the diffusion and kinetic layers: H(z)≅H°. Therefore, the same results are valid for the concentration distributions and for the maximal current (assuming that it satisfies to condition (17w), i.e. for x_dk<6, see Section “Bromate system without added redox catalysts. Novel autocatalytic redox-mediator (EC″) mechanism”).

• thin kinetic layer regime:

Its thickness should be much smaller than the diffusion layer one, z_d, which implies the inequality: x_dk= z_d/z_k≫1 where the thickness of the kinetic layer for weak currents is defined by modified eq. (9):

Then, the proton concentration, H(z), is approximately constant inside the kinetic layer: H(z)≅H(0). It allows one to get an analytical solution first for B(z), then for the other concentration profiles. The only difference is in another form of the transcendental equation for the current-dependent kinetic layer thickness, z_kj.

These solutions provide expressions for all characteristics of the process for any possible values of parameters: x_dk, j and J_CH=j_C^lim/j_H^lim. Qualitative results are similar to those for the bromate system in excess of protons, Section “Bromate system without added redox catalysts. Novel autocatalytic redox-mediator (EC″) mechanism”. Fig. 8b exposes plots for the maximal current density, j^max, as a function of the RDE frequency, f, for two bulk-solution concentrations of protons, H°. Unlike the case of the proton excess where the maximal current tends to the same BrO₃⁻ diffusion-limited one, j^max≅j_A^lim, for sufficiently low frequencies (lines 1 and 2 in Fig. 8a) similar plots in Fig. 8b approach to the corresponding H⁺ diffusion-limited one, j^max≅j_H^lim, which depend on the pH value of the bulk solution, i.e. on H°.

As a whole, the maximal value of j^max (Fig. 8a,b) increases rapidly for higher bulk-solution concentrations of both bromate anions and protons, owing to the combination of two effects:

• increase of the corresponding diffusion-limited current density, j_A^lim (Section “Bromate system without added redox catalysts. Novel autocatalytic redox-mediator (EC″) mechanism”) or j_H^lim (Section “Effects of pH”),

• diminution of the weak-current kinetic layer thickness, z_k=[D_B/5 k° A° (a_H)²]^1/2, eqs. (9) and (17k0), which shifts the maximum towards lower z_d and correspondingly lower f values (Fig. 8a,b).

Thus, for the RDE configuration both bulk concentrations should be about 1 M or higher, to reach current densities of the level of 1 A/cm², or even higher.

Generalized Nernst layer model

The whole analysis in Sections “Bromate reduction via catalytic redox-mediator (EC′) mechanism”, “Bromate system without added redox catalysts. Novel autocatalytic redox-mediator (EC″) mechanism”, “Effects of pH” is based on the conventional Nernst layer model where the effect of the convective-transfer component in the ion flux is replaced by the shift of the bulk-solution boundary condition from an infinitely far distance from the surface (z → ∞) to the “outer diffusion layer boundary”, z=z_d, eqs. (6), (15zd), (17zd). An evident drawback of this model is the dependence of this key parameter, z_d, on the diffusion coefficient of the transporting species, D, eq. (3), while it is not clear for the bromate system which of the diffusion coefficients, D_A, D_B, D_C or D_H, should be used as D in this definition. Moreover, it is not evident whether any of these option will be able to simulate correctly the results of the convective-diffusion transport for this system since the transition from the dominantly diffusion transfer to the convective one takes place at different distances from the electrode surface, depending on the diffusion coefficient of the species.

It is the reason why our team has proposed a modification of the conventional approach<“Generalized Nernst layer (GNL) model” [29], [34], [35]. Each transporting species, A, B, C or H, is characterized by its individual diffusion layer thickness, respectively z_dA, z_dB, z_dC or z_dH, which are defined by the Levich formula, eq. (3), with the use of D_A, D_B, D_C or D_H instead of D.

Then, the set of transport equations and boundary conditions given in Sections “Bromate system without added redox catalysts. Novel autocatalytic redox-mediator (EC″) mechanism” and “Effects of pH” for the Nernst layer model is modified to the form:

D A d 2 A/dz 2 = V for z < z dA ,  D B d 2 B/dz 2 = 5 V for z < z dB ,

A(z) = A °  for z ≥ z dA ,  B ( z ) = 0  for z ≥ z dB ,

In view of the values of the diffusion coefficients given in legends to Figs. 1 and 8 the individual diffusion layer thicknesses satisfy to the inequality:

This set of equations and boundary conditions has been analyzed for the same cases:

• excess of protons compared to bromate anions in the bulk solution, H°≫A°, so that H(z)≅H° [34], [35],

• excess of bromate anions compared to protons in the bulk solution, A°≫H°, so that A(z)≅A° [29],

In both cases approximate analytical solutions have been derived with the use of the same procedure exposed in Section “Bromate reduction via catalytic redox-mediator (EC′) mechanism”, i.e. for the regime of the weak currents and for the regime of the thin kinetic layer.

All qualitative results exposed in Sections “Bromate system without added redox catalysts. Novel autocatalytic redox-mediator (EC″) mechanism” and “Effects of pH” have been confirmed (lines D in Fig. 9a,b), in particular:

Fig. 9:

Dependence of the maximal current density, j^max, of the RDE frequency, f. Line D is calculated within the framework of the GNL model, i.e. via the procedure given by eq. (15) from Ref. [34] (Fig. 9a) or from Ref. [29] (Fig. 9b) where the transport of each component, A, B, C or H is described with the use of its particular diffusion layer thickness, z_dA, z_dB, z_dC or z_dH, given by eq. (3) containing the corresponding diffusion coefficient, D_A, D_B, D_C or D_H. Lines A, B, C and H are calculated on the basis of the conventional Nernst-layer model [33], i.e. for identical values of the diffusion layer thicknesses of all components, z_dB=z_dC=z_dH=z_d where z_d is determined by eq. (3) for D_i=D_A (line A), D_i=D_B (line B), D_i=D_C (line C) or D_i=D_H (line H). Fig. 9(a) corresponds to A°=1.0 M, proton excess (H°≫A°), Fig. 9(b) corresponds to H°=1.0 M, bromate excess compared to protons (A°≫H°). k=4.8 M⁻¹ s⁻¹, C°=2.5 mM, other parameters are identical to those in Figs. 1 and 8.

• weak currents determined by the diffusion-limited one for species C (Br₂) for relatively thin diffusion layers of the components: z_di<z_k,

• abnormal behavior inside the range of the ratios of z_di/z_k from about 2–4 to about 8–15 where increase of the diffusion layer thicknesses (because of diminution of f) leads to a drastic increase of the maximal current,

• passage through a maximum of the dependence, j^max vs. f, where the current is comparable with the diffusion-limited one for BrO₃⁻ anions, j_A^lim, so that it reaches the level of Ampers per cm² for the bromate concentration within the 1 M range,

• diminution of j^max for the further decrease of the frequency, f, in parallel to j_A^lim.

At the same time one can find a marked quantitative difference between predictions of the GNL model (lines D in Fig. 9a,b) and the results of the conventional model shown by lines A, B and C in Fig. 9a,b where each of these lines corresponds to the diffusion layer thickness, z_d, calculated with the use of eq. (3) for D=D_A, D=D_B or D=D_C, respectively. In other words, the Nernst layer model is unable to simulate the conclusions of the GNL one which takes into account the difference in individual diffusion-layer thicknesses for various components, especially for the case of the bromate excess because of a significantly larger value of z_dH, compared to the other z_di thicknesses.

Convective diffusion theory of the bromate reduction process

Similar to the analysis in Section “Bromate system without added redox catalysts. Novel autocatalytic redox-mediator (EC″) mechanism” we consider below the case of proton excess in the bulk solution, compared to bromate anion, eq. (17HA). This condition allows us to neglect variation of the proton concentration inside the diffusion layer: H(z)≅H°. Besides, protons and non-reactive ions play the role of a background electrolyte which reduces the intensity of the electric field so that one can neglect the migrational components of the ion fluxes.

Under such conditions the flux of a species in circulating solution, in particular inside the RDE cell represents the sum of its diffusional and convective components. It results in the set of transport equations [47]:

where c_i=A, m_i=1 for i=A, c_i=B, m_i=5 for i=B, c_i=C, m_i=−3 for i=C, V(z) is given by eq. (18k), normal velocity of the solution is given in the Levich approximation [33]: v_z(z)≅−α z², α≅0.51 Ω^1.5/ν^0.5, Ω and ν_kin are defined in eq. (3).

Asymptotically far from the electrode surface the concentrations approach their bulk-solution values:

Boundary condition (15j) at the electrode surface remains unchanged.

Analytical solutions have been found [47] for the same two approximations described in Section “Bromate reduction via catalytic redox-mediator (EC′) mechanism”.

For the weak current regime one may replace A(z) in expression (18k) for V(z) by A°. It results in a closed linear differential equation for the B(z) concentration distribution accompanied by two boundary conditions which has to be integrated numerically. The other concentration distributions, A(z) and C(z), may be found as integrals of this function, B(z).

Analysis of these expressions has shown again that the shapes of these concentration profiles depend on the same parameters: x_dki=z_di/z_k where the weak-current kinetic layer thickness, z_k, is given by eq. (17zk) while the individual diffusion layer thicknesses for components A, B and C, z_dA, z_dB and z_dC, have been introduced in Section “Generalized Nernst layer model”. If these parameters, x_dki, are of the order of 1 or smaller, then the maximal current density, j^max, is of the order of j_C^lim, i.e. it is very weak. For larger values of x_dki the thickness of the kinetic layer becomes small, compared to the diffusion layer thicknesses of the components: z_k≪z_di so that expressions for the alternative regime are applicable.

For the thin kinetic layer regime one can solve eqs. (19cd) separately inside the kinetic layer and outside of this layer. The unknown constants in these solutions are determined from their comparison inside the intermediate range. The final result turned out to be simple: inside the kinetic layer all concentration profiles, A(z), B(z) and C(z), are close to the distributions found within the framework of the GNL model, see Section “Generalized Nernst layer model”. On the contrary, in the outer part of the diffusion layer the solutions of eqs. (19cd) demonstrate an expected effect of the convective transfer, with a progressive approach of these functions, A(z), B(z) and C(z), to their bulk-solution value, A°, 0 and C°, respectively. As for the maximal current density, j^max, its values for identical parameters of the system are the same within the framework of the GNL model and of the convective diffusion theory, i.e. the latter predicts the same plot, line D in Fig. 9a. At the same time, a significant difference between this line and lines A, B and C based on the conventional Nernst layer model means that the latter cannot provide correct quantitative results.

Bromate reaction within the cathodic channel of hydrogen–bromate MEB

The theoretical analysis in Sections “Bromate reduction via catalytic redox-mediator (EC′) mechanism”, “Bromate system without added redox catalysts. Novel autocatalytic redox-mediator (EC″) mechanism”, “Effects of pH”, “Generalized Nernst layer model” to “Convective diffusion theory of the bromate reduction process” is related to processes inside the RDE cell under steady-state conditions. Electrical energy sources are based on different configurations of the system. Therefore, our team has initiated modeling of the hydrogen-bromate battery where the hydrogen reaction at anode produces protons which are transported across cation-exchange membrane through porous cathode into catholit containing aqueous bromate solution [48]. As a result, the bromate reaction may take place at cathode via the same cycle formed by steps (14E) and (14C).

The difference in the results for this system originates from the non-zero proton flux across the electrode/bromate solution interface which is directly related to the local current density, j:

D H dH ( z)/dz = − j/F for z = 0

This boundary condition changes not only the distribution of proton flux along the z-coordinate but also the distributions of the concentrations of all components of the system. The same procedure described in Section “Bromate reduction via catalytic redox-mediator (EC′) mechanism” has been applied successfully to find approximate analytical solutions for the weak current and thin kinetic layer regimes. Further analysis of these expressions has demonstrated the existence of a complicated dependence of the local thickness of the diffusion layer, z_d, which includes an anomalous behavior within a certain range of z_d values where the maximal current density grows drastically if this thickness increases. As a result, the current density may reach enormous values.

Conclusions

Estimates of the charge and energy density of the bromate reduction process have revealed their very high levels which exceed strongly those for most actually available alternative electrical energy sources. However, its practical utilization had been impeded by the absence of the direct reaction of bromate anions without unacceptably high overvoltages at any substrate. Our analysis of the way to overcome this difficulty based on the redox catalysis (EC′ mechanism) does not look prospective.

In this context it has been highly surprising for us to reveal that the Br₂/Br⁻ redox couple represents a miracle solution of the problem. Our theoretical analysis has demonstrated that even a tracer amount of Br₂ inside a concentrated bromate solution may result in passage of extremely intensive currents, with the density exceeding 1 A/cm², despite the absence of the bromate reaction at the electrode. This possibility has been provided owing to a specific peculiarity of this system, due to the autocatalytic character of the process where the passage of the combination of the electrochemical and chemical steps leads to transformation of the principal oxidant, BrO₃⁻, into components of the redox-mediator couple. This radical difference in the features of the process has allows us to consider it as a representative of a novel electrochemical mechanism (EC″).

It has been demonstrated that because of the presence of several transporting species (BrO₃⁻, Br⁻, Br₂, protons) having strongly different diffusion coefficients the conventional “stagnant Nernst layer” model cannot be used for quantitative description of the system. A novel approximate approach (“Generalized Nernst layer” model) for analysis of these transport processes has been proposed. Comparison of predictions on the basis of this model and of the convective diffusion equations has demonstrated their practical coincidence, thus making the simpler GNL model a prospective tool for the theoretical description of multi-component systems.

Such extremely high current densities for the bromate reaction, in combination with its relatively high potential vs. hydrogen electrode, allows one to expect that a hydrogen-bromate discharge cell might be able to generate electric power densities exceeding 1 W/cm².