Bromate electroreduction in acidic solution inside rectangular channel under flow-through porous electrode conditions

Electrochimica Acta 323 (2019) 134799

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Bromate electroreduction in acidic solution inside rectangular channel under flow-through porous electrode conditions Mikhail A. Vorotyntsev a, b, c, d, **, Anatoly E. Antipov a, b, * a

D. I. Mendeleev University of Chemical Technology of Russia, Moscow, Russia M. V. Lomonosov Moscow State University, Moscow, Russia c Institute of Problems of Chemical Physics, Russian Academy of Sciences, Chernogolovka, Moscow Region, Russia d ICMUB, UMR 6302 CNRS-Universit
e de Bourgogne, Dijon, France b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 15 March 2019 Received in revised form 15 August 2019 Accepted 30 August 2019 Available online 5 September 2019

In view of applied prospects of the bromate electroreduction in acidic medium this process has been analyzed for the solution flow through a channel filled in with porous conducting material playing the role of 3D electrode. The process passes via mediator redox cycle where the bromine is reduced at the electrode surface to bromide which is subject to the comproportionation reaction with bromate inside the solution phase to regenerate bromine. Inside the entering solution protons are in excess compared to bromate ions while the bromine concentration is very low. Ohmic losses across the channel are disregarded. Perpendicular size of pores is assumed to be sufficiently small so that the diffusion of solute components suppresses concentration gradients in the transversal direction. Solute species are transported along the channel by the convective mechanism. Numerical and approximate analytical methods of solving coupled transport equations have provided distributions of solute components inside the porous medium and of the current density along the flow coordinate, y. Quite unusual features of the process have been discovered, in particular non-monotonous variation of the bromine concentration and of the current density as functions of y, with narrow maxima at an intermediate distance from the entrance, dependent on the flow velocity and rate constants of the electrochemical and chemical steps of the redox cycle. All these surprising features of the bromate reduction process originate directly from the autocatalytic character of this cycle where each passage of the cycle increases the amount of the catalytic bromine/bromide couple in solution, thus accelerating the bromate transformation. As a result one can reach simultaneously both high efficiency of the bromate transformation and strong current for an optimal value of the channel length-to-flow velocity ratio, even for extremely small ratio of the incoming bromine and bromate concentrations. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Bromate reduction Bromine/bromide redox couple Redox mediator cycle Autocatalysis Redox flow cell 3D electrode

1. Introduction Systems where an electrode (or both electrodes) represents a porous medium through which the solution of reagents is pumped (“flow-through porous electrodes”) are widely used for numerous applications [1e28]. Key advantage of such 3D electrodes is their very large surface area which can be reached by reacting species via

* Corresponding author. D. I. Mendeleev University of Chemical Technology of Russia, Moscow, Russia. ** Corresponding author. D. I. Mendeleev University of Chemical Technology of Russia, Moscow, Russia. E-mail addresses: [email protected] (M.A. Vorotyntsev), [email protected] (A.E. Antipov). https://doi.org/10.1016/j.electacta.2019.134799 0013-4686/© 2019 Elsevier Ltd. All rights reserved.

molecular diffusion mechanism without a long-distance transport. Flow-through porous electrodes have been employed for efficient extraction of trace concentrations of metal ions [1e3,6] or mixtures of such ions [4], transformations of organic molecules [8], Br2 oxidation [17], vanadium redox flow batteries [18e24]. Both theoretical description of such systems and their experimental studies were mostly carried out for the simplest reactional scheme where a solute reagent is subject to the electron-transfer process at the electrode surface [1,2,4,11e13]. More complex schemes included passage of several such reactions in parallel (deposition of two metal ions, or a metal ion in the presence of side reaction) [4,11,12] or accompanying chemical steps: ЕСЕ mechanism [4] or Br
oxidation [17]. In almost all these papers the porous electrode was located

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M.A. Vorotyntsev, A.E. Antipov / Electrochimica Acta 323 (2019) 134799

List of symbols

Latin characters A(y) BrO
3 concentration distribution along the channel, [mol dm
3]
3 A0 Entering BrO
3 concentration [mol dm ] a1, a2 Dimensionless parameters, Eq (B7) a, a(u) Dimensionless BrO3 concentration a(u) ¼ A(y)/A , Eq. (23) aa aa ¼ 1 - a, Eq. (B4) B(y) Br
concentration distribution along the channel, [mol dm
3] b(u) Dimensionless Br concentration b(u) ¼ B(y)/A , Eq. (23) b0 Dimensionless parameter, Eq (B7) [Br
]eq Equilibrium Br
concentration, [mol dm
3] [Br2]eq Equilibrium Br2 concentration, [mol dm
3]
3 [BrO
] Equilibrium BrO
3 eq 3 concentration, [mol dm ] C(y) Br2 concentration distribution along the channel, [mol dm
3] C0 Entering Br2 concentration [mol dm
3] ci(y) Concentration distribution of solute component i along the channel, [mol dm
3] c(u) Dimensionless Br2 concentration c(u) ¼ C(y)/A , Eq. (23) c0 Ratio of incoming Br2 and BrO3 concentrations c0 ¼ C0/A0, Eq. (23) c1, c2 Dimensionless parameters, Eq (B7) D Diffusion coefficient of reagent, [cm2 s
1] d Channel width, Fig. 1b [cm] dz Channel width along the z-coordinate dz ¼ So d
1, [cm] F Faraday's constant, [C mol
1] f1, f2 Dimensionless factors of exponential variation of concentrations, Eq (B7) H0 Proton concentration, [mol dm
3] I Total (cathodic) current, Eq (21), [A] I∞ Total current for infinitely long channel, [A] iCC, iCC(y) Local value of electronic current density via current collector, Eqs (4),(20), [A cm
2] ie(x) Local value of current density across the electron conducting 3D material, [A cm
2] is(x) Local value of current density across the solution phase, [A cm
2] J(u) Local dimensionless current density, Eq. (28) j(y) Local current density per unit length of the channel, Eq. (20), [A cm
1] K Dimensionless ratio of electrochemical and chemical rate constants, K ¼ kc*/(k A0), Eq (27) k Rate constant of chemical step, Eqs (12) and (2), [mol
1 dm3 s
1] kc Potential-dependent rate constant of Br2 reduction per unit surface area,Eq. (16), [cm s
1] kc* Effective rate constant of Br2 reduction per unit channel length, Eqs (17) and (1), [s
1] keff Effective rate constant of BrO
3 -to-Br transformation via redox cycle, Eq (30), [s
1]

L Lmin

Channel length, Fig. 1b, [cm] Minimal channel length for almost complete transformation of incoming BrO
3 species, [cm] N Number of layers of electron conducting elements to level out concentration difference across pores Pe Peclet number, Pe ≡ Re Sc, Eq (8), (A8), (A10) Q Volume flux of solution, Eq (21), [cm3/s] < Universal gas constant, [J mol
1 K
1] R Characteristic size of pore's cross-section and distributed conducting elements, App.A, [cm] Re Reynolds number, Re ¼ 2 R 〈Uy〉/n, App.A rchem Rate of chemical step, Eqs (12) and (2), [mol dm
3 s
1] relchem Effective rate of electrochemical step per unit channel length, Eqs (13) and (1) 0 [mol dm
3 s
1] relchem Rate of electrochemical step per unit electrode/pore interface area, Eqs (16) and (1), [mol cm
2 s
1] ri Total consumption rate of i species, Eq (10) So Cross-section area of channel, Eq (20), [cm2] Sc Schmidt number, Sc ¼ D/n s Specific electrode/pore interface area (per unit volume), Eq (13), [cm2 cm
3] T Absolute temperature, [K] ttrans Characteristic “residence time” for fluid particle to spend inside channel to reach the range of most intensive redox transformation, Eq. (B22), [s] U Average fluid velocity along channel inside pores, U ≡ 〈Uy〉 ¼ Q/ε So, Eq (10) and (A9), [cm s
1] u Dimensionless spatial coordinate, Eq. (23) uL Dimensionless u coordinate at exit of channel (y ¼ L), uL ¼ k A0 U
1 L, Eq. (29) u1/2 Half-transition coordinate for a(u), Eq. (B19) a u1/2 Half-transition coordinate for b(u), Eq. (B19) b umax Coordinate of c(u) function maximum, Eq. (B19) c umax Coordinate of J(u) function maximum, section 3.1 j v0 Superficial flow velocity, v0 ¼ ε U, [cm s
1] x Spatial coordinate along current flow, Fig. 1b, [cm] y Spatial coordinate along fluid flow, Fig. 1b, [cm] y0 Characteristic size of spatial region at upstream electrode's edge of 2D diffusion, Eq (A7), [cm] ycmax Coordinate of Br2 concentration (C(y) function)) maximum, Figs. 2c and 3c, Eq (30), [cm] yk Kinetic length, Eq (A12), [cm] yR Transitional region length to level out concentration differences across pore, Eq (A9), (A11) and (A16), [cm] Greek characters Diffusion layer thickness, Eq (A6), [cm] Diffusion layer thickness at upstream electrode's edge within 2D diffusion region, Eq. (A7), [cm] ε Porosity, i.e. fraction of the volume of porous matrix occupied by catholyte, dimensionless ke, ks Effective conductivities of electron conducting matrix or of catholyte in porous medium, [S/cm] 4e, 4s Local potential distribution in electron conducting matrix or in catholyte inside porous medium, [V]

d,d(y) d0

M.A. Vorotyntsev, A.E. Antipov / Electrochimica Acta 323 (2019) 134799

Fig. 1. Schematic configurations of the cell [14]: (a) “flow-through geometry”; (b) “flow-by geometry”. 3D porous electrode (1), separating membrane (2), counterelectrode (3), current collector of the porous electrode (4). U-arrows and i-arrows show directions of the solution and current flows, respectively.

inside a channel of either cylindrical or rectangular shape, with the solution flow along its axis. Two principal geometries of the cell (Fig. 1) were mostly considered: (a) “flow-through porous electrode” where the counter electrode is located upstream or downstream with respect to the working (porous) one so that the solution and the current flow along the same Y-direction [13,14]; (b) “flow-by porous electrode” (which is also called frequently “flow-through one”) where the directions of the solution and current flows are perpendicular, i.e. along Y- or X-axes, respectively [13,14]. In practical terms each of these configurations has its own advantages and drawbacks [13,14]. As for their theoretical description the distributions of the potential inside the electronconducting matrix and inside the solution are strongly different for these two geometries in the case of significant ohmic losses. On the contrary, for high conductivities of both interpenetrating phases and sufficiently small potential differences the concentration distribution(s) are described by identical transport equation(s). There have been as yet no publications on flow-through porous

3

electrodes for reactional schemes where a non-electroactive species (which represents a dominant solute component of the bulk solution) is transformed via mediator redox cycle, owing to the presence of a very low concentration of the component of the catalytic redox couple in the bulk solution. Particular example of such a process is given by the bromate anion (BrO
3 ) electroreduction. This reaction has recently become an object of attention since it had been proposed by Tolmachev [30e32] as a prospective oxidant for redox flow batteries (RFBs) [33e37]. Its key attractivity is in very high values of the “theoretical” redox-charge and energy densities owing to a high solubility of its lithium salt (over 8 M at room temperature) and the multi
electron reduction mechanism (6 electrons for the BrO
3 to Br transformation) [30,31,38]. As a way to overcome the problem of the non-electroactivity of BrO
3 it has been proposed [30,31] to perform this reaction via a combination of two steps: (quasi) reversible transformation of the Br2/Br
redox couple at the electrode surface:

Br2 þ 2 e
%2 Br


(1)

and comproportionation reaction inside the solution phase regenerating Br2 species:
þ BrO
3 þ 5 Br þ 6 H /3 Br2 þ 3 H2 O

(2)

Kinetics of the bromate process under steady-state conditions described by Eqs (1) and (2) has been considered theoretically for both the rotating disc electrode (RDE) [38e45] and microelectrode [46] geometries. Results of this analysis turned out to be quite unusual, compared to those for other earlier studied reaction mechanisms. In particular, the maximal current density, jmax, corresponding to transport-limited current conditions (i.e. for sufficiently large cathodic overpotentials) increases drastically (by several orders of magnitude) if the convection intensity diminishes within a certain range of RDE rotation frequencies, or for larger microelectrode radius. These astonishing features of the bromate process are directly related to the autocatalytic character of its redox-mediator cycle formed by steps (1) and (2). It means that each passage of the cycle in the cathodic direction results in replacement of 5 Br
species by

0
0 0 0
1 Fig. 2. Distributions of dimensionless concentrations of BrO
y, proportional 3 , a ¼ A/A , of Br , b ¼ B/A and of Br2, c ¼ C/A , as functions of the dimensionless parameter, u ¼ k A U to the coordinate, y, along the channel, for various values of the ratio of the reaction rate constants, K ¼ kc*/k A0 (indicated at each graph), between 1 and 32. Numerical integration of kinetic equations (Appendix B1).

Fig. 3. See Legend to Fig. 2. Values of K between 1 and 0.1.

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M.A. Vorotyntsev, A.E. Antipov / Electrochimica Acta 323 (2019) 134799

6 Br
ones since an additional Br
ion is added from the consumed BrO
3 species. This disbalance of the catalytic components of the redox couple in the course of the redox cycle distinguishes crucially the bromate process from reactions corresponding to the well known EC0 mechanism [47e66] (even though the latter is also based on redox catalysis by mediator cycle) since the total amount of the redox couple components does not change in the course of the EC0 process. All predictions of our analysis for the maximal current density of the bromate reduction have been confirmed by experimental evidences for steady-state currents for both the RDE [67] and microelectrode [68] configurations. Recently this conclusion on a drastic difference of predictions for the bromate system based on Eqs (1) and (2) from those for the conventional EC0 mechanism has been supported by Cho and Razaulla [69] in their analysis of its chronopotentiometric response after a current step. Similar results were earlier obtained by Beran and Bruckenstein [70,71] for non-stationary characteristics of the iodate reduction. Maximal current density of the bromate reduction along the wall of 2D flow channel has been related to the local value of the diffusion layer thickness, this dependence being non-monotonous [72]. This prediction in terms of the effect of the solution flow rate has been confirmed experimentally [73]. Preliminary experimental study of the bromate electroreduction has been performed in H2 - BrO
3 discharge flow cells where the acidized bromate solution passes through rectangular channel filled in with porous carbonaceous material which plays the role of 3D distributed cathode [73]. The actual paper proposes a simplified theoretical description of the bromate-to-bromide transformation inside such a channel in order to determine principal factors affecting the kinetics of this process.

2. Theoretical analysis 2.1. Description of the system Similar to Refs. [2e11,13e19,22,23,25e28] it is assumed that reagent's solution passes through rectangular (or cylindrical) channel filled in with porous electron-conducting material (e.g. carbon paper or cloth). Schematic configuration of the cell is shown in Fig. 1b. Parameters of the porous channel in this figure are denoted below as channel length, L, and its width, d. y-axis is directed along the flow; cross-sections, y ¼ 0 and y ¼ L, correspond to the entrance and the exit sections of the channel, respectively. Electric current passes in the direction perpendicular to the fluid flow (along xaxis). Extension of the porous electrode in the third direction perpendicular to both the solution flow and the electric current is much larger than the one along the x-axis (d) so that the local characteristics (concentration and current density distributions) are independent of this coordinate. The electronically conducting porous material (region 1 in Fig. 1b) plays the cathode role. It is electrically connected with current collector which represents one of the walls of the porous space and the solution flow (line 4 in Fig. 1b). From the opposite side the porous material and the flow are confined by cation-exchange membrane, e.g. Nafion (line 2 in Fig. 1b) that allows the current passage through it owing to the transport of protons generated at anode 3. Entering solution contains bromate anions, BrO
3 (concentration: A0) and molecular bromine, Br2 (its concentration, C0, is very low compared to A0; typical ratios of these concentrations, A0/C0,

are within the range of 100 to 106) as well as protons, H3Oþ, and background ions which are in excess with respect to bromate. Inside the channel the reactive components of catholyte are subject to the transformations described by Eqs (1) and (2) above. Namely, Br2 is reduced to Br
at the surface of the porous material inside the channel (“electrochemical step”). Product of this reaction, Br
, reacts inside the solution phase with BrO
3 , regenerating Br2 species (“chemical step”). Passage of these reactions leads to a progressive change of the composition of each fluid particle of electrolyte solution in time in the course of its displacement along the channel (“effective contact time” by Bennion & Newman [1] or “residence time” by Alkire & Ng [2,5]). On the other hand, the process as a whole is considered as being in steady-state regime, i.e. the composition of the solution in each particular spatial point of the channel does not vary in time. Theoretical model of the bromate process inside the porous electrode has been carried out within the framework of the same basic assumptions (given below) as those in numerous previous papers considering simpler reaction mechanisms [2e11,13e19,22,23,25e28]. Process proceeds under isothermal conditions. High concentrations of protons and background ions allows one to neglect the migration contribution to the transport. 3D electrode possesses a uniform porosity and specific surface area. Its material is characterized below by its specific surface area, s (total area of the solid phase/solution interface per unit volume, [cm
1]) and ε fraction of the volume occupied by catholyte (dimensionless). Porous electrode is described as two 3D interpenetrating media: set of electronically conducting elements (3D electrode) and set of pores filled in by electrolyte solution. Characteristic sizes of spatial elements of both media belong to the microscopic scale (typically within (sub)micrometer range). Local characteristics of each medium (in particular, solution flow velocity and electrolyte concentrations) vary in complicated 3D manner at this microscopic scale. Similar to previous studies of porous flow electrodes [2e11,13e19,22,23,25e28] our theory will deal with these quantities averaged over spatial regions where the region's scale exceeds strongly the microscopic one but it is very small compared to the range of these quantities' variation because of chemical and electrochemical transformations. As a result of this averaging procedure the rate of the electrochemical stage of the process, Eq (1), enters into the balance relations within the spatial region of the porous electrode as quasihomogeneous reaction, despite its basically heterogeneous character [5]. Solution in any point inside the porous space within the channel moves along the y-axis with constant average velocity, U (plug flow), so that the total volume flux of the solution ([cm3/s]) is equal to ε U So where So is the cross-section area of the channel. Passage of a sufficiently strong current through the porous electrode leads generally to appearance of significant differences of the electric potential in both the 3D electron conducting matrix and inside the solution phase (even after the averaging procedure) [1,2]. This potential distribution (mostly for the diffusion-limited current regime) was studied for several electrochemical mechanisms, mostly for simple electrode reaction [1,14] (or several reactions of this type [3]) as well as for ECE mechanism (sequence of electrochemical-chemical-electrochemical steps) [4], electroorganic synthesis [8], bromide oxidation [17] and vanadium redox batteries [24e28]. Bromate process corresponds to a much more complicated reactional mechanism where the mediator redox cycle possesses autocatalytic features [38,39], i.e. its passage leads to increase of the amounts of catalytic components, Br2 and Br
, inside the solution, owing to the transformation of BrO
3 species, Eq (2). Therefore, the goal of this study has been to carry out the analysis

M.A. Vorotyntsev, A.E. Antipov / Electrochimica Acta 323 (2019) 134799

under conditions where potential differences due to ohmic effects inside the porous electrode may be disregarded (inside both the solid matrix and the solution phase). Then, the rate constant of the electrochemical step, Eq (1), has the same value for any point of the matrix/solution interface while variation of the average concentrations of solution components along the y-coordinate (Fig. 1b) is identical for all points inside the cross-section of the channel, i.e. independent of x-coordinate: ci(y). It is evident that this approximation imposes restrictions to the value of passing current (which may be regulated by the proper choice of the reagents' concentrations). For more quantitative evaluation of the potential differences in both phases (see e.g. Ref. [29]) one can use the Ohm's law, e.g. for the solution phase at a fixed value of the y-coordinate (in view of excessive background electrolyte):

is ðxÞ y
ks d4s =dx

(3)

where is is the local current density inside this phase along the xcoordinate, ks is its effective conductivity, 4s is its local potential, x is the coordinate perpendicular to the solution flow (Fig. 1b). Since the rate of the electrode reaction, Eq (1), is the same for all values of the x-coordinate it implies that the value of the derivative of the current density, dis/dx, is constant (for a fixed value of y) for all x values, 0 < x < d [6]. Local value of the total current density of two phases, ie þ is, is also independent of x and equal to the electronic current density passing through the porous electrode/current collector interface, iCC ¼ ie(x ¼ 0) (dependent on the y-coordinate) since is ¼ 0 at x ¼ 0. It gives expression (4) for the distribution of the current density inside the solution phase, is(x):

is ðxÞyiCC x=d

(4)

Combination of Eqs (3) and (4) provides after integration the value for the potential difference inside the solution phase across the porous electrode for any value of y:

4s ð0Þ
4s ðdÞy½iCC d=ks ; i:e: iCC y2 ½4s ð0Þ
4s ðdÞks =d

(5)

Eq (5) allows one to estimate the current density passing through the current collector, iCC, if the absolute value of the potential difference inside the solution phase of the porous electrode, 4s(0) - 4s(d), is sufficiently small (below
iCC < ðks =dÞ 0:05 V

(6)

For acidic solutions of the molar range the effective solution conductivity, ks, may be of the order of 0.1 S/cm (after taking into account the porosity effect [6]). Then, Eq (5) gives an upper estimate for the current density: iCC < 0.25 A/cm2 for d ¼ 200 mm (experimental value of the porous channel's width of the bromate cell [73]) which should be satisfied for all points of the current collector/porous electrode interface. Potential difference across the porous electrode inside the electronically conducting matrix is given by similar expression:

4e ð0Þ
4e ðdÞy½iCC d=ke

(7)

Since its effective conductivity, ke, is usually much higher than that of the solution phase, ks, the potential variation inside the matrix (for a fixed value of y) is much smaller than 25 mV if condition (6) is satisfied. Since the intensity of the passing current depends on the reagents' concentrations one can ensure a practically uniform distribution of the potential inside each phase by the properly chosen initial solution composition on the basis of our results for the

5

current density derived below. Then, the rate constant, k, of the electrochemical step, Eq (1), is practically identical for all points of the matrix/solution boundary inside the whole channel in view of the approximate constancy of the interfacial potential difference, 4e - 4s, which varies parallel to the cathode potential with respect to reference electrode (independent of the local composition of the bromate system) inside catholyte. Distribution of reagent's concentration inside porous material where characteristic size of pores in the perpendicular direction to the flow as well as of solid elements of the electron conducting matrix, R, is within micrometer (or even submicrometer) range is discussed in Appendix A. It is demonstrated there that the key role is played by the value of the (diffusional) Peclet number:

Pe ¼ 2RU=D

(8)

where D is the diffusion coefficient of the reagent, U ≡ 〈Uy〉 is the average velocity of the solution along the channel. If the Peclet number is comparable to 1, or if it is much less than 1, then the intensity of the diffusional transport in the direction perpendicular to the average flow is sufficiently high to level out concentration differences inside pore at the microscopic distance along the flow: length of this transitional region, yR, is comparable to R, Eq (A11), i.e. it corresponds to the micrometer range. For larger values of the Peclet number (Pe » 1) the concentration variation inside pore is more complicated because of formation of thin diffusion layer around each solid matrix element, with progressive increase of its thickness downstream, up to the stage where the merged diffusion layers produce uniform concentration distribution across the pore. Transitional length, yR, in this case depends of the structural features of the conducting matrix, e.g. yR ~ R Pe for well known (but hardly realistic) “straight-pore model”, Eq (A9), or yR ~ R Pe1/2 for the model of chaotically distributed solid elements, Eq (A16). It should be kept in mind that for R ¼ 1 mm, U ¼ 0.1e1 cm/s and D ¼ 10
5 cm2/s values of Pe number are between 2 and 20 so that the transitional length, yR, belongs in any case to the micrometer range. One should note that inside the transitional region, y « yR, in most points of the pore's cross-section the concentration of each species, ci, is still close to its initial value, c0i , at the entrance into the porous space, i.e. it has not been subjected to chemical transformations. If the porous electrode length, L (Fig. 1b), belong to the transitional region, L « yR, then most of the reagent(s) will leave the electrode without reaction. In other words, complete transformation of reagents requires a sufficient length of the electrode, compared to the transitional length: L » yR. The above estimate of yR shows that this condition is satisfied for all channel length of the centimeter (even millimeter) range. These two characteristics of the porous electrode should also be compared with the kinetic length, yk, Eq (A12), i.e. with the distance along the flow needed for passage of the chemical and electrochemical transformations. If these transformations are so fast that yk « yR, then their completion within the whole crosssection of the pore will be limited by the non-uniform distribution of the reagent's concentration across the pore, i.e. the process will be controlled by diffusion in the perpendicular direction (“diffusion-limited regime”) and it is sufficient to satisfy the condition: L » yR. Goal of this study has been to analyze the opposite situation of relatively slow kinetics of (electro)chemical transformation where its completion is controlled by the reaction rate(s) (“kinetic regime of electrochemical step (1)”). It implies that the kinetic length should be much larger than the transitional one:

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M.A. Vorotyntsev, A.E. Antipov / Electrochimica Acta 323 (2019) 134799

yk »yR

(9)

Then, the (electro)chemical reaction(s) changes concentrations of solute components so slowly that the diffusion in the normal direction is able to level out concentration differences inside the pore's cross-section prior to a significant change of the solution composition near the matrix/solution interface. In this case each concentration depends practically only on the y-coordinate, even across each pore, ci(y), its variation along this coordinate being determined by combination of the transport along this axis and of the rates of electrochemical and chemical reactions, relchem and rchem, Eqs (1) and (2), respectively. For conventionally used values of the fluid velocity, U (of the order of 0.1e1 cm/s, or even higher) one can neglect the diffusional transport along the channel, compared to the convective transfer by the flow [7]. As a result, within the framework of the above model the balance relation for species of type i may be written down in the form (see e.g. Refs. [8,17]):

U dci = dy ¼
ri

(10)

where ri is the total consumption rate of species of type i (dependent on the value of y-coordinate) taking into account both chemical and electrochemical transformations. For comparison of Eq (10) with analogous transport equations in previous papers on porous flow-through electrodes one should take into account that they use “superficial flow velocity” v0, which is related to our average flow velocity, U, by the formula: v0 ¼ ε U. Besides, as it has been discussed above, Eq (10) corresponds to the kinetic regime (rather than the diffusion-limited one) which means that the mass-transfer coefficient of previous publications, km, is so large that one can neglect the concentration difference across pores.




  log BrO
3 eq þ 5 log Br eq
3 log½Br2 eq ¼
32; 65 þ 6 pH (11) þ i.e. in the presence of noticeable concentrations of BrO
3 and H the
equilibrium concentration of Br is very low while these species are generated from Br2 by electrochemical step (1). According to Eq (2) the chemical transformation gives contributions to reaction rate terms, ri, in Eq (10), proportional to the
product of the local BrO
3 and Br concentrations where the rate constant, k, depends on the concentration of protons, H0 [75e81]:

 
 BrO
¼ 5rchem ; rchem ðBr2 Þ 3 ¼ rchem ; rchem Br

¼
3rchem ; where rchem ¼ kAðyÞBðyÞ


(13) Two reactional schemes have been considered for electrochemical step (1), Eq (14) or (15) [82e85]:

Br2 þ e
%Brads þ Br
; Brads þ e
%Br


(14)

Br2 %2 Brads ; Brads þ e
%Br


(15)

the former one being observed for carbonaceous materials where the first step is the rate-determining one for the cathodic process [82]. Since the comproportionation reaction, Eq (2), is rapid and irreversible under strongly acidic conditions it consumes quickly Br
species, thus forcing the cathodic direction of the electro0 chemical process, too. It implies that its rate, relchem , may be written down as:

r 0elchem ¼ kc C

(16)

where kc is the potential-dependent rate constant of Br2 reduction. Combination of Eqs (10), (12), (13) and (16) results in the set of equations for variation of concentrations of all components along the channel (see for comparison e.g. Eq (6) in Ref. [17] for Br
oxidation process, with taking into account the identity: v0 ¼ ε U):

¼ 3ðk=UÞA B
ðkc *=UÞC; where kc * ¼ kc s=ε

Because of electrochemical and chemical transformations of
solution components, BrO
3 , Br and Br2, their concentrations vary along the y-coordinate: A(y), B(y) and C(y), respectively. Proton concentration is assumed to be close to its initial value at the entrance of the channel, H0, in view of its excess. Comproportionation reaction, Eq (2), is irreversible under strongly acidic conditions since equilibrium concentrations (more accurately, their activities) satisfy relation (11) derived from stan

dard potentials of BrO
3 /Br and Br2/Br couples [74]:

rchem

  relchem ðBr2 Þ ¼ ðs= εÞr 0elchem ; relchem Br
¼
2ðs= εÞr 0elchem

dA=dy ¼
ðk=UÞA B; dB=dy ¼
5ðk=UÞA B þ 2ðkc *=UÞC; dC=dy

2.2. System of equations and boundary conditions for bromate process



the channel located at y and y þ dy due to this reaction, i.e. in the absence of the chemical transformation of C species, is equal to the total rate of the cathodic process (Br2-to-Br- transformation) inside the volume located between these cross-sections. This rate is given by the product of the total electrode surface inside this volume, s So 0 dy, and the reaction rate per unit surface area, relchem . This balance equality gives expression for variation of the Br2 and Br
concentrations due to the electrochemical step, Eq (1):

(12)

Concentrations of Br (B) and Br2 (C) also change due to the electrochemical reaction, Eq (1). Difference of convective fluxes of species C, ε U S0 C(y) - ε U S0 C(y þ dy), across two cross-sections of

(17) Eq (17) contain new kinetic parameter, kc*, which determines the rate of the Br2-to-Br- transformation via reaction (1) inside porous electrode. Its value depends not only on the cathodic rate constant per unit electrode area, kc, but also on geometrical parameter, s/ε, characterizing both the specific surface area of the porous material and the fraction of its volume occupied by solution. Owing to the latter the value of kc* (also dependent on the electrode potential) may reach high values. Initial conditions for Eq (17) have the form:

A ¼ A0 ; B ¼ 0; C ¼ C 0 aty ¼ 0 where C 0 «A0

(18)

Set of Eq (17) shows that the derivative of the sum, A(y) þ B(y) þ 2 C(y), is equal to zero. It gives a relation between the concentrations reflecting conservation of the total amount of Br atoms in all oxidation states:

AðyÞ þ BðyÞ þ 2CðyÞ ¼ A0 þ 2C 0

(19)

This identity may be used for integration instead of any of three equation (10). Above derivation of expression (13) for the rate of the cathodic process between two cross-sections also provides a result for the cathodic current density at any point of the channel, j(y), which is proportional to the current density across the porous electrode/ current collection interface, iCC(y), for the same point, y, in Eqs (4)e(7):

M.A. Vorotyntsev, A.E. Antipov / Electrochimica Acta 323 (2019) 134799

jðyÞ ¼ iCC ðyÞdz ¼ 2FsSo kc CðyÞ 2F So εkc *CðyÞ

(20)

Here, dz is the porous electrode width along the z-coordinate, i.e. perpendicular to both the solution and current flows; So ¼ d dz. Total current generated by the catholyte discharge inside the whole channel of the length, L, for a chosen value of the cathodic rate constant, kc (dependent on the cathode potential), is given by integration of j(y) over y from 0 to L. It may be expressed via the exit concentrations, A(L) and C(L), with the use of the consequence of Eq (17): 3 dA/dy - dC/dy ¼ (kc*/U) C and its integration over y:

ðL I¼

h i dy ¼ Q 6A0
6AðLÞ þ 2C 0
2CðLÞ ; where Q ¼ So εU (21)

The multiplier in Eq (21), So ε U, is equal to the volume flux of the solution, Q, while the coefficients, 6 and 2, reflect the number of electrons to be transferred for transformation of BrO
3 or Br2 species into one or two Br
ones, respectively. If the length of the channel is sufficiently extended the passage of the irreversible cathodic process leads finally to complete
transformation of BrO
3 (A) and Br2 (C) into Br (B) so that the total current, I, approaches to its upper limit determined by the total redox charge supplied by the entering bromate solution per unit time:

    I∞ ¼ F So εU 6A0 þ 2C 0 ¼ F Q 6A0 þ 2C 0

(22)

Both the value of I∞ and the length of the channel to reach this current depend on parameters of the system, in particular on the cathodic reaction rate constant, kc, Eq (1), i.e. on the cathode potential.

2.3. Dimensionless variables, equations and boundary conditions Let us define dimensionless variables for all concentrations and for coordinate along the channel, y:

. . . u ¼ y k A0 U; aðuÞ ¼ AðtÞ A0 ; bðuÞ ¼ BðtÞ A0 ; . . cðuÞ ¼ CðtÞ A0 ; c0 ¼ C 0 A0 «1

da = du ¼
a b; db=du ¼
5a b þ 2K c; dc=du ¼ 3a b
K c (24) a ¼ 1; b ¼ 0; c ¼ c0 at u ¼ 0

(25)

a þ b þ 2c ¼ 1 þ 2c0

(26)

where a new dimensionless potential-dependent parameter, K, is defined via the ratio of rate constants of the electrochemical and chemical steps:

K ¼ k*c

0

7

.  kA0

(27)

Current density in Eq (20) is expressed via c(u): j(y) ¼ 2 F So ε kc* A0 c(u). Then, one can define the dimensionless (normalized) current density, J(u) (its integral over u from 0 to infinity is equal to 1):


1  JðuÞ ¼ K 3 þ c0 cðuÞ; so that jðyÞ   2  6 þ 2c0 JðuÞ ¼ F So εk A0

(28)

Total current in the whole channel, I, may be normalized by division to its value at the infinitely extended channel, I∞, Eq (22). Then,


1 h i I=I∞ ¼ 3
3aðuL Þ þ c0
cðuL Þ 3 þ c0 ; where uL ¼ k A0 U
1 L

(29)

This analysis shows that all dimensionless quantities (distributions of concentrations and current density as functions of u as well as dependence of normalized total current, i.e. efficiency of the redox charge transformation on the channel length, L) are determined by only two dimensionless parameters: ratio of initial concentrations, c0 ¼ C0/A0, Eq (23) and ratio of the electrochemical and chemical rate constants, K ¼ kc*/k A0, Eq (27). The value of the former parameter, c0, is assumed to be very small (c0 ¼ 0.001 in illustrations below) while the latter one may be both large (up to 100) and small (up to 0.1). 3. Results and discussion

(23)

One should pay attention that all concentrations are divided by the same quantity: initial concentration of the principal reagent 0
(BrO
3 ), A , in particular the Br and Br2 concentrations which are zero or very low at y ¼ 0. As it is shown below this choice of dimensionless variables is justified since the latter quantities can reach large values (comparable to A0) within a relatively short time interval. Then, Eqs (17)e(19) take the form:

3.1. Concentration and current density distributions along the channel Distributions of all dimensionless concentrations (a ¼ A/A0 for 0
0 BrO
3 , b ¼ B/A for Br and c ¼ C/A for Br2) obtained via numerical integration of kinetic equation (24) (Appendix B.1) for various values of K are given in Figs. 2e4. Ratio of the initial concentrations of molecular bromine (C0) and bromate anion (A0), c0 ≡ C0/A0, is kept equal to 0.001. It is clear from the analytic consideration in Appendix B2 that variation of this

Fig. 4. See Legends to Figs. 2 and 3. Values of K between 3 and 0.1. Stretched coordinate, u K, instead of u.

8

M.A. Vorotyntsev, A.E. Antipov / Electrochimica Acta 323 (2019) 134799

parameter, c0 (within the range of its very small values: c0 « 1) modifies graphs of b(u) and c(u) within the interval of relatively small u values (not visible in Figs. 2 and 3) as well as shifts slightly (as f2 ln c0) ranges of the principal variation of all concentrations. Kinetic parameter, K, characterizes the ratio of parameters of the electrochemical and chemical steps, Eqs (1) and (2), respectively; K ¼ kc*/(k A0), Eq (27). One should keep in mind that according to Eq (17) the former parameter, kc*, is determined by both the potentialdependent rate constant of the Br2 electroreduction at the surface of porous electrode, kc ¼ kc(E), and geometrical parameters of the porous system (specific surface area of the porous material, s, and fraction of its volume occupied by solution, ε): kc* ¼ kc s/ε. Diminution of characteristic diameters of “microchannels” inside the electron conducting matrix and increase of the spatial density of such elements results in the growth of the geometrical factor in this formula, s/ε. Denominator in this expression for K, i.e. k A0, depends on the initial bromate concentration, A0, as well as on the proton activity in solution, aH ¼ 10
pH: k ~ (aH)2 [80,86]. In Figs. 2 and 3 the concentrations are plotted vs. u ¼ k A0 U
1 y where y is the distance from the entrance of the channel (Fig. 1b). Values of K vary between 32 and 1 in Fig. 2 and between 1 and 0.1 in Fig. 3. Calculations are performed within the assumption that the channel is sufficiently extended so that the complete transformation into Br
takes place (graphs for a(u) and c(u) tends to zero while b(u) approaches 1 þ 2 c0). Otherwise, the graphs should be limited to the point: uL ¼ k A0 U
1 L related to the channel length, L (Fig. 1b), and the transformation is incomplete. For each fixed value of K the graph for BrO
3 , a(u), varies from 1 to 0, the graph for Br
varies from 0 to 1 þ 2 c0, while the graph for Br2 starts from its small initial value, c0, to pass through a maximum and to approach finally 0. If the K parameter increases due to variation of kc* (e.g. as a function of the electrode potential, E) while other parameters are kept constant one may note a progressive shift of the transitional interval for all concentrations to smaller values of u (or y), i.e. complete transformation is reached within a shorter channel. This change of the interval is especially strong within the range of relatively small K values (below its critical value of about 5, see below), see Fig. 3. It implies that the rate of the whole process is limited by the electrochemical step. To verify this interpretation the same graphs for small K values are plotted in Fig. 4 vs. the product, u K ¼ kc* U
1 y. One can see that graphs for such a choice of the x-axis coordinate approach a limiting line where the transformation takes place in the vicinity of the u K ¼ kc* U
1 y value close to 31, see Eq (30) below for umax . c Comparison of concentration distributions found by numerical integration and via approximate analytical consideration (Appendix B3) shows their very good agreement for sufficiently large (Fig. B1) or small (Fig. B2) values of K. Behavior of the dimensionless Br
and Br2 concentrations, b(u) and c(u), near the entrance of the channel (see Fig. B3) as well as other features of the concentrations are also discussed in Appendix B3. In particular, it

has been established for large K values that Br2 plays the role of a short-living intermediate, i.e. its concentration is very small
compared to those of BrO
3 and Br (see Fig. 2) so that the process proceeds practically as progressive transformation of BrO
3 (concentration a) directly to Br
(concentration b). For small K values the process consists of two separate stages: BrO
3 (a) is transformed firstly into Br2 (c) while Br2 is reduced then into Br
(b). As a result, Br2 concentration which is very low at the entrance of the channel (C ¼ C0 « A0) reaches much higher values within the range between the stages (near its maximum) where C y (0.3e0.4) A0 for K values from 0.5 to 0.1, see Fig. 3b. It means a several hundred times increase if c0 ≡ C0/A0 ¼ 0.001. Analytical approach provides interpolation formula, Eq (B22), for the coordinate of the maximal values of Br2 concentration, ycmax, as well as of the c(u) dependence, umax , as function of K: c

  
1  . ymax U keff  uc max U=k A0 yU k A0 þ 5 ðkc *Þ
1 ln 1=2c0 ; c   i:e:uc max yð1 þ 5=Kln 1=2c0 (30) Fig. B3 in Appendix B3 demonstrates a good agreement of its predictions with numerically obtained values of umax in Figs. 2c and c 3c. For large K values (kc* » k A0) the transformation of bromate is controlled by the slower step, i.e. by comproportionation reaction (2) so that its effective rate (keff y k A0 (- ln 2 c0)
1) and its transformation length (ycmax y (- ln 2 c0) U/k A0) are practically independent of the rate constant of electrochemical step (1). On the contrary, for small K values (kc* « k A0) the process is limited by electrochemical step (1) so that ycmax ¼ umax U/k A0 y (- 5 ln 2 c0) U/ c max 0 kc* (i.e. uc K y 31 for c ¼ 0.001, see Fig. 4c) while the overall effective rate (keff y kc* (- 5 ln 2 c0)
1) is independent of the chemical step rate constant, k. Characteristic multiplier in Eq (30), ln (1/2c0) ¼ 6.2 for c0 ¼ 0.001, depends on the ratio of the initial concentrations of Br2 0 0 0 and BrO
3 at the entrance of the channel, c ¼ C /A . It originates from the necessity to accumulate the needed amount of Br2 inside fluid particle during its passage along the channel, owing to the autocatalytic redox cycle, Eqs (1) and (2). Value of ycmax also depends on the sum of inverse values of the rate constants of the chemical and electrochemical steps of the cycle, k A0 and kc*/5, Eq (30). Fig. 5 provides illustrations for the distribution of the dimensionless current density along the channel, J vs u ¼ k A0 U
1 y, both for large (Fig. 5a) and small (Fig. 5b) K values which may be compared with the concentration profiles in Figs. 2 and 3. In view of proportionality between the normalized current density, J(u), and the dimensionless Br2 concentration, c(u), Eq (28), J(u) demonstrates in Fig. 5a and b the same shape as a function of u, i.e. along the channel, as c(u) in Figs. 2c and 3c, for all K values. In particular, expression (30) for ycmax is also valid for the coordinate, ymax , of the maximum of the current density, j(y): ymax ¼ ycmax At j j

Fig. 5. See Legend to Fig. 2. Distribution of the normalized current density, J, Eq (28), as a function of u ¼ k A0 U
1 y (a,b), or J K
1 as a function of u K ¼ kc* U
1 y (c), for various K values (indicated at each graph).

M.A. Vorotyntsev, A.E. Antipov / Electrochimica Acta 323 (2019) 134799

the same time, because of the definition of J(u) (its integral over u from 0 to infinity is equal to 1 which gives an extra K multiplier in J(u)) its amplitude increases for larger K values since the width of the maximum diminishes. Minimal length of the channel, Lmin, to reach an almost com
plete transformation of the initial oxidants, BrO
3 and Br2, into Br , may be crudely estimated via the formula: Lmin y ycmax since the width of the transitional interval for the bromate process is much smaller than its distance from the entrance of the channel, see Figs. 2, 3 and 5. Therefore, all above conclusions of the value of ycmax as a function of K are also valid for Lmin. In particular, one can use Eq (30) for estimation of both the position of the current density maximum, umax , and the minimal channel's length, Lmin. For a fixed K value j these parameters depend on the ratio of the entrance concentrations, c0 ¼ C0/A0, only via ln (1/2c0). It means that even a strong diminution of the incoming concentration of the catalytically active 0 species, Br2, C0, compared to that of the principal reagent, BrO
3, A , results in quite a moderate increase of these parameters. For example, enormous decrease of the entrance Br2 concentration (from c0 ¼ 0.001 to c0 ¼ 10
6) leads merely to a twofold increase of this coordinate since ln (1/2c0) changes from 6.2 to 13.1. This result is in an drastic contrast to the conventional catalytic processes where the transformation rate is proportional to the catalyst concentration inside the channel (which cannot exceed its entrance concentration, C0) so that a 1000 times decrease of C0 means a 1000 times increase of the channel length for the complete transformation of the principal reagent. The above feature of the bromate process is a direct consequence of the autocatalytic character of the mediator cycle of reactions (1) and (2). 3.2. Efficiency of redox-charge transformation and total current Normalized total current, 0 < I/I∞ < 1, defined by Eq (29) is equal to the fraction of the redox charge of the solution entering the channel which has been transformed into the current inside the channel. Therefore, this quantity characterizes the efficiency of this transformation dependent on the channel length, L, and the fluid velocity, U, in the form of its function of their dimensionless combination: uL ¼ k A0 U
1 L. Fig. 6 provides illustrations of this dependence for various K values. For any K value the normalized current increases monotonously from 0 for u ¼ 0 (entrance of the channel) to 1 for uL / ∞
since complete transformation of BrO
3 and Br2 into Br is always reached for sufficiently extended channels. However, both the characteristic interval of uL where this transition takes place and the width of the transitional interval depend strongly on the K value, in conformity with data in Figs. 2, 3 and 5 and B4. In particular, the transition range is narrow and its position approaches a constant value (close to - ln 2 c0) for large K values (Fig. 6a) while the characteristic transition interval increases as a linear

Fig. 6. Normalized total current, I/I∞, i.e. redox charge transformation efficiency, as a function of the dimensionless channel length, uL ¼ k A0 U
1 L, for various K values (indicated at each graph).

9

function of K
1 due to increase of Lmin, see above, for small K values (Fig. 6b). One should also pay attention to a characteristic shape of all graphs in Fig. 6: there is an induction interval in terms of the uL values, i.e. of relatively small channel lengths where the current is very low, followed by exponential increase of the current if the channel length, L, is extended, its exponent being dependent on the K value, in conformity with predictions of Eq (B6). Fig. 7a provides illustrations for the dependence of the normalized total current, I/I∞, i.e. of the efficiency of the redox charge transformation inside the channel of a fixed length, L, as a function of the solution flow rate, again in terms of the dimensionless parameter, (uL)
1 ¼ U/(k A0 L). Fig. 7b illustrates the dependence of the dimensionless average current density, I/L, on the dimensionless flow rate, (uL)
1 ¼ U/(k A0 L), for fixed values of other parameters (entrance solution composition and reaction rate constants, k and K ¼ kc*/k A0). These graphs are based on Eq (21) which may be rewritten in the form:

h    i I ¼ LjU ðuL Þ
1 6
6a uL þ 2c0
2c uL ; uL ¼ k A0 L=U; (31) where jU ¼ F So ε k (A0)2 is a characteristic current density per unit channel length which is independent of either the channel length, L, or the solution flow rate, U. Dimensionless fluid velocity, (uL)
1 ¼ U/(k A0 L), is proportional to the volume flux of the solution through the channel, Q ¼ S0 ε U. For sufficiently small fluid velocities the efficiency of the process is maximal, i.e. transformation of the oxidants of the bromate system, BrO
3 (a) and Br2 (c), is complete (Fig. 7a), i.e. a(uL) y c(uL) y 0, I y I∞, so that both the total current, I (unlike its dimensionless variant, I/I∞) and the average current density, I/L, increase proportionally to the volume flux (Fig. 7b). Further increase of the velocity, U, diminishes the dimensionless coordinate, uL, up to the range of a very rapid variation of the concentrations (Figs. 2 and 3), uL ~ umax in Eq (30), accompanied by sharp decrease c of the transformation efficiency (Fig. 7a). As a result, the current passes through a narrow maximum followed by its drastic decrease, despite the continuing increase of the redox charge supply into the channel proportionally to the solution flow rate, U (Fig. 7b). It is of importance for RFB applications of bromate system that the current, I, within the initial range of relatively small flow rates (where uL > umax ) is close to its upper limit, I∞, corresponding to the c complete transformation of the non-electroactive species, BrO
3. The value of I∞ given by Eq (22) is proportional to the product of the volume flux of the solution through the channel, Q ¼ S0 ε U, and of the initial bromate concentration, A0. Optimal choice of the flow rate, channel length and bromate concentration ensures a high

Fig. 7. Dependences of the normalized total current, I/I∞, i.e. redox charge transformation efficiency (a) and dimensionless average current density, I/L jU (b) on the dimensionless flow rate, (uL)
1 ¼ (k A0 L)
1 U, for various K values (indicated at each graph).

10

M.A. Vorotyntsev, A.E. Antipov / Electrochimica Acta 323 (2019) 134799

electric energy generation. Another important feature of these graphs in Fig. 7b is the position of this maximum which is located within very small values of the dimensionless flow rate. This characteristic property of the bromate process is directly related to its passage via the autocatalytic redox mediator cycle. It has already been found for the rotating disk electrode configuration [42e45] where strong currents were only observed for relatively weak convection. In the system under consideration this upper limitation for the fluid velocity is a consequence of the requirement for the fluid solution particle to spend a sufficiently extended residence time inside the channel, in order to accumulate (owing to the autocatalytic character of the redox mediator cycle) a high concentration of the catalytic redox-couple components, Br2 and Br
, for rapid transformation of the principal reagent, BrO
3.

4. Conclusions Our theoretical analysis of the bromate reduction process passing via Br2/Br
þ redox-mediator cycle (1), (2) inside 3D flowthrough porous electrode has revealed its highly unusual features, compared to previously studied systems based on simpler reactional schemes. It has been demonstrated that for sufficiently extended channels the principal intermediate catalytic component of the system, Br2, changes along the channel in complicated non-monotonous manner: its concentration, C(y) (which is very low in the incoming solution, C0, compared to the incoming BrO
3 concentration, A0), increases exponentially as a function of the y-coordinate along the flow, with passage of a maximum, accompanied by exponential decrease. This strong increase of the Br2 concentration is a direct consequence of the autocatalytic character of the bromate process where each passage of the redox cycle increases the amount of Br2/Br
couple components. Owing to this feature the transitions between the initial (at the channel's entrance) and final (at the channel's exit) values of the
BrO
3 and Br concentrations take place within relatively narrow spatial regions. As a consequence the current density is distributed along the channel in the same non-monotonous way, with a relatively narrow maximum, its position, ycmax, being determined by the smaller among the rate constants of the electrochemical, Eq (1), and chemical, Eq (2), steps, see Eq (30). Efficiency of the bromate transformation, i.e. consumed fraction of the redox charge of the solution entering the channel increases monotonously as a function of the channel length. It possesses an induction region, followed by a rapid growth within a relatively narrow interval in the vicinity of the ycmax coordinate, to approach finally its upper limit, 1. As a consequence, for a fixed channel length, there exists a significant interval of flow velocities where the efficiency is close to 1, i.e. to complete transformation of the incoming bromate, despite its non-electroactivity at the electrode surface. Average current density for the whole channel passes via maximum as a function of flow velocity. The velocity corresponding to this maximum ensures an optimal combination of relatively high values of both the transformation efficiency and the total generated current.

Acknowledgements The work was supported by Mendeleev University of Chemical Technology of Russia. Project Number Х020-2018.

Appendix A. Model of convective diffusion transport in porous materials Description of transport of solute components inside porous media should be based on fundamental results on the convective diffusion in electrolyte solutions. Les us consider first the system which includes a single electroactive species (с0, its bulk solution concentration) participating in a simple electron-transfer reaction at the electrode surface, Ox þ ne
¼ Red, as well as excess of background electrolyte. The overpotential of the electrode is sufficiently large so that the redox process takes place in the regime where its rate is limited by the transport of the reacting species, i.e. its concentration is close to zero at the electrode surface. The simplest example of the convective-diffusion geometry is given by steady-state flow of the solution inside a semi-space (x > 0) parallel to planar wall (x ¼ 0) along the y-axis, i.e. its velocity vector, U, in each point of the flow has a single non-zero component, Uy. Owing to the viscosity of the fluid the velocity component, Uy, is zero at x ¼ 0 while it changes linearly for nonzero values of x:

Uy ðxÞ ¼ U 0 x; where U 0 ¼ dUy dx at x ¼ 0

(A1)

The wall is insulating at y < 0 (so that dc/dx ¼ 0 at x ¼ 0, y < 0) while it represents electrode surface at y > 0 (so that c ¼ 0 at x ¼ 0, y > 0). Due to consumption of the reacting species at the electrode surface its concentration is lowered in its vicinity, forming diffusion layer, i.e. spatial region of a perturbed concentration. Concentration distribution within this region is described by the convective-diffusion equation:

  Uy vc = vy ¼ D v2 c=vx2 þ v2 c=vy2 for x > 0

(A2)

Its approximate solution is well known since publications of Leveque (see e.g. Refs. [86,87]). One neglects the molecular diffusion term along the flow, thus simplifying Eq (A2) to the form:

Uy vc = vyyDv2 c=vx2 for x > 0; y > 0

(A3)

Exact solution of this approximate equation satisfying to the boundary conditions in the bulk solution and at the electrode surface:

c ¼ c0 at x/∞; c ¼ 0 at x ¼ 0; y > 0

(A4)

represents a function of the combination of two spatial arguments, x/y1/3 (its explicit form may be found in Ref. [87]). Thickness of the diffusion layer, d, which characterizes its extension in the direction normal to the wall, may be defined e.g. via the derivative, (vc/vx)x¼0, at x ¼ 0:

d≡c0

.h

 vc=vx

i x¼0

for any y > 0

(A5)

Then, the local flux density, j ¼ - D (vc/vx)x¼0, is always given by the formula: j ¼ D c0/d. The diffusion layer thickness increases monotonously (proportionally to y1/3) as a function of y:





dyCte Dy U

0

1=3

for any y > 0

(A6)

where dimensionless constant, Cte ¼ 3
1/3 G(1/3) ¼ 1.86, is of the order of 1. For a fixed y value, d depends on the ratio of the diffusion coefficient and the velocity gradient, D/U'. One has to emphasize that this well-known solution is not

M.A. Vorotyntsev, A.E. Antipov / Electrochimica Acta 323 (2019) 134799

applicable in the vicinity of the upstream edge of the electrode (line defined by conditions: x ¼ 0, y ¼ 0) since the transition from Eq (A2) to its approximation, Eq (A3), is incorrect if the distance from the edge, y0, becomes comparable with the diffusion layer thickness, d0 ≡ d(y0), so that y0 ~ d0 ~ (D y0/U0 )1/3, i.e.

 1=2 y0  d0  D U 0

(A7) 2

2

Owing to the last rhs term in Eq (A2), v c/vy , the concentration perturbation propagates both in the normal direction at y ¼ 0 and even upstream of the electrode's edge, i.e. into the upstream region, y < 0 (of course, its extensions in both directions satisfy Eq (A7)). Spatial extension of this entrance region defined by Eq (A7) is very small, if it is compared with perpendicular sizes of macroscopic rectangular channels or pipes since the velocity gradient, U0 , is usually of the order of 1 s
1 or larger while D is about 10
5 cm2/s so that d0 is less than a few tens of micrometer. Similar conclusions may be drawn [87] for the extension of the diffusion layer in the configuration where the insulator wall is absent at y < 0 so that the planar electrode occupies the whole wall surface at y > 0, x ¼ 0. Then, the viscous layer is developing near the wall in parallel to the diffusion one, and one has to take into account not only the velocity component parallel to the wall, Uy, but also the normal one, Ux. However, owing to very large values of the Schmidt number of liquid solutions (Sc ¼ n/D is of the order of 1000) the diffusion layer is located deeply inside the viscous one so that one can use power expansions for these velocity components. As a result the same formulas are valid for the diffusion layer thickness both within the developing layer region, Eq (A6) (with a slightly different value of the constant, Cte [87]), and for the entrance one, Eq (A7). Next useful example is provided by the convective-diffusion transport inside electrolyte solution flowing along circular pipe (similar qualitative results are valid for a rectangular channel filled by flowing electrolyte). This system represents a basis for the description of transport within the framework of the simplest model of porous electrodes as a set of parallel cylindrical pores of identical radii [88] (“straight-pore model” [7,29]). First of all, one should compare the diffusion layer thickness within the entrance region, d0 in Eq (A7), and the pipe diameter, R. For channels of a macroscopic size the former parameter is very small compared to the pore radius:



d0  D U

0

1=2

 1=2  D R= < Uy > «R;

i:e:d0 =R  Pe
1=2 « 1 since Pe≡2R < Uy > =D» 1

(A8)

where 〈Uy〉 is the average flow velocity inside the pipe which is proportional to the velocity gradient at the surface, U': 〈Uy〉 ¼ R U'/4 for the stabilized (Poiseuille) flow. Diffusional Peclet number, Pe ≡ Re Sc, defined in Eq (A8) represents an analog of the Reynolds number, Re ≡ 2R 〈Uy〉/n, for the mass transport processes. Then, the Leveque solution [87] corresponding to a thin diffusion layer (its thickness, d, must be much smaller than R) is valid for sufficiently large y values: y » d0, where d » d0 according to Eq (A6). The value of the diffusion layer thickness, d, increases monotonously as y1/3, Eq (A6), unless it becomes comparable to the pipe radius, d ~ R. The length of this transitional region, yR, may be estimated on the basis of the layer thickness (to become comparable to R) given by Eq (A6): 0

yR  U R3

.

D  R2 < Uy > =D  R Pe»R»d0

(A9)

Within this range where y ~ yR the concentration variation within

11

cross-section of the pipe tends to zero exponentially [87], i.e. diffusion in the direction perpendicular to the flow is able to level out the concentration distribution across the pipe. Because of very large values of the Pe number for macroscopic channels (e.g. Pe ~1000 for R ¼ 1 mm, 〈Uy〉 ¼ 1 mm/s, D ¼ 10
5 cm2/ s) the transitional length, yR ~ R Pe, Eq (A9), is enormous so that the diffusional transport takes place usually inside the transitional region, i.e. across a thin diffusion layer, d « R. Quite different scenario takes place for micropipes where the diffusion layer thickness inside the entrance region, Eq (A7), may be comparable with (or even larger than) the pipe radius:



0



d0  D U 1 = 2  R;

. 0 so that Pe ¼ 2 R < Uy > =D ¼ U R2 2D  1 (A10)

i.e. the Peclet number, Pe, should be sufficiently small in this case. Then, the concentration profile is levelled out already within the entrance region of the pipe. It means that the transitional length, yR, is in this case merely about the pore radius, R:

yR  R

(A11)

This condition, Eq (A10), is fulfilled for sufficiently small radii of the pore. For example, if 〈Uy〉 ~0.1 cm/s and D ~10
5 cm2/s the radius, R, should be of the order of 1 mm or smaller. Then, if the length of the pipe, L, is much larger than R, the concentration distribution, c(x,y), for any value of the flow coordinate, y, outside the entrance region (y » R) is close to its average value within the crosssection of the pore, c(x,y) y 〈c(x,y)〉 ≡ c(y), which is equal to 0 for the above irreversible electrochemical reaction, or to the equilibrium concentrations for the reversible process at the electrode surface, etc. For the model system where the only transformation of the reagent consists in irreversible first-order electrode reaction its rate (per unit surface area of the porous electrode/pore interface) is equal to k сs(y) where k is the potential-dependent reaction rate constant, сs(y) is the concentration at the interface. Inside the part of the pore where the concentration distribution is already levelled out within its cross-section one can estimate the characteristic range of the average concentration variation along the pore from the balance relation relating the fluxes of the species in two neighboring sections, p R2 〈Uy〉 [c(y) - c(y þ dy)], and its consumption by the electrode reaction, 2p R dy k c(y). It gives an estimate for the characteristic kinetic length, yk, of the average concentration variation along the pore due to electrochemical process:

yk  R < Uy > = k

(A12)

This distance should be compared with characteristic scales of the concentration profile variation within the cross-section. For sufficiently thin pores where the Peclet number is small, Eq (A10), the concentration distribution in the normal direction is levelled out much more rapidly, Eq (A11), than the average concentration changes along the y-axis, Eq (A12), if yR ~ R « yk ~ R / k, i.e.

k« < Uy >

(A13)

If the average flow velocity is not especially low (e.g. for 〈Uy〉 ~0.1 cm/s) this condition is satisfied even for relatively rapid electrode reactions. More restrictive condition is imposed on the value of the reaction rate constant for larger Peclet numbers, Eq (A8), i.e. for larger pore radii and/or flow velocities. If the k values are so small that the kinetic length, Eq (A12), is much greater than that of the

12

M.A. Vorotyntsev, A.E. Antipov / Electrochimica Acta 323 (2019) 134799

transitional region, Eq (A9): yk » yR, it means that

k « D=R

(A14)

Then, within the transitional region, y « yR, the redox transformation only takes place inside the thin diffusion layer, x ~ d, while the concentration remains unchanged within the central part of the pore, but the degree of the transformation within whole transitional region is very low because of the slow kinetics, Eq (A14). As a result, the concentration distribution, c(x,y), is close to its average value, 〈c(x,y)〉 ≡ c(y) for any value of y if condition (A14) is satisfied. On the contrary, for larger k values where Eq (A14) does not hold, the kinetic length, Eq (A12), is located inside the transitional region, Eq (A9): yk « yR.Then, strong variation of the surface concentration, cs(y), and consequently of the concentration, c(x,y), across the diffusion layer takes place already within the applicability conditions of Eq (A3)-(A6), with a characteristic variation of the current density as a function of y, due to change of the diffusion layer thickness. As alternative structural model one can consider a 3D set of identical quasi-spherical globules (representing together porous electron conducting matrix) where the characteristic sizes of these solid elements and of pores between them are of the same order of magnitude, R. Similar to the above model of cylindrical pores, the key role is played again by the Peclet parameter, Pe ¼ 2 R /D. If the Peclet number is so small that condition (A10) is fulfilled, then the conventional diffusion layer is not formed, i.e. the concentration distribution in the perpendicular direction to the flow is strongly perturbed already within the first layer of electrode's elements, y ~ R, so that the transitional length is given again by Eq (A11). Therefore, if the reaction rate satisfies condition (A13), i.e. the change of the surface concentration by the reaction after passage of a single layer of conducting elements is small, then the variation of the average concentration inside pores is able to follow the one of the surface concentration, i.e. the concentration gradients within any cross-section are practically absent and the concentration only depends on the y-coordinate along the channel. Different situation takes place for larger Pe numbers, Pe » 1, corresponding to Eq (A9). Then, the solution flow around each element of the front layer (near the entrance plane, y ¼ 0) is accompanied by formation of a thin diffusion layer inside the surface region of the element, its thickness varying along the surface of the element. Downstream of the element a narrow region of the perturbed concentration (“diffusional trace”) is formed which is transported by the flow, with a progressive increase of its width as



d  ðDtÞ1=2  Dy= < Uy >

1=2

(A15)

If the electrode's elements form a regular lattice so that each element of the next layer is located strictly inside the concentration trace of the corresponding element of the upstream layer, then the diffusion layer “touches” subsequently a set of such elements, with a progressive increase of its thickness. Variation of the concentration distribution as a function of the flow coordinate, y, is qualitatively similar to the case of micropipe ensemble considered above, thus providing again condition (A14). On the contrary, if the elements of different layers are chaotically distributed inside the cross-section of the channel, the diffusional trace of a front layer element passes generally next layers without diffusional interaction with their elements. In the absence of concentration perturbations by elements of next layers, the widening traces of neighboring front layer elements, Eq (A15), would merge (d ~ R) at the distance: y ~ R2 /D ~ R Pe, equal to the length of the transitional region inside cylindrical micropipe, Eq

(A9). However, each element of the next layers produces an additional diffusional trace inside the unperturbed part of the flow. As a result, each layer of elements diminishes distances between diffusional traces of neighboring elements. Since the average distance changes as R/N1/2 after passage of N layers it gives for the distance along the flow (i.e. for relaxational length, yR ~ R N) where diffusional traces are merging (so that the concentration difference within the pore cross-section is levelled out):

dN  ðDyR = < Uy > Þ1=2  R=N1=2 ; i:e:N  Pe1=2 ; yR  RPe1=2 (A16) Then, the combination of Eq (A12) and (A16) gives the condition for the reaction rate constant, k, that the transformation rate is slow enough to allow the concentration distribution within the crosssection to level out prior to a significant change of the concentration, yk » yN:

k« < Uy > Pe
1=2 ¼ ðD=RÞPe1=2

(A17)

Since in the case under analysis the Peclet number is large, Pe » 1, this condition for k values is of course more restrictive than the one for small Pe numbers, Eq (A13), but less restrictive for the same large Pe number than Eq (A14) for the model of the porous electrode as a set of parallel micropipes. In any case the characteristic transitional length for the concentration profile to become practically uniform within the crosssection of pores, yR, remain very small, compared to the typical lengths of the channel along the flow, L, for a wide range of Pe numbers if the pore radius, R, is within the (sub)micrometer range. Appendix B. Solution of transport equations B1. Solution via numerical integration method Transport equations for the bromate process, Eqs (24)-(26), have been transformed into a set of two coupled equations for two dimensionless variables, a(u) and g(u) ¼ 3 b(u) þ 5 c(u), as well as initial conditions for them:

    da = du ¼
a 2g
5
10c0 þ 5a ; dg=du ¼ K 3 þ 6c0
3a
g (B1) a ¼ 1; g ¼ 5c0 at u ¼ 0

(B2)

The equations have been integrated numerically for c0 ¼ 0.001 and a set of K values. Numerical integration has been carried out with the use of standard symplectic partitioned Runge-Kutta method implemented in Wolfram Mathematica software package with the automatic difference order selection of eight to nine and automatic ranges of tolerances with progressive step number. Then, other concentration distributions have been found from relations: b(u) ¼ 2 g(u) - 5e10 c0 þ 5 a(u), c(u) ¼ 3 þ 6 c0 - 3 a(u) - g(u). Distributions of the dimensionless current density, J(u), and of the normalized total current, I/I∞, have been found from Eq (28) or Eq (29), respectively. B2: Approximate analytical solution The whole analysis below is carried out under the standard condition for catalytic processes: concentration of catalytic species, i.e. of the representative of the redox mediator couple, C, in bulk solution, C0, is very low compared to the concentration of the principal reagent.

M.A. Vorotyntsev, A.E. Antipov / Electrochimica Acta 323 (2019) 134799

. C 0 « : A0 ; i:e:c0 ¼ C 0 A0 «1

(B3)

The goal of this approximate analysis is to provide a qualitative interpretation of results (exposed in chapter 3) obtained by numerical integration of the same transport equations (section B1). 1. Weak current regime We consider first the “entrance range” of the channel where the bromate concentration, A(t), is still close to its initial value, A0, i.e. its dimensionless value, a(u), is approximately equal to 1 so that their difference, aa ¼ 1 - a, is small: aa « 1. Then, Eqs (24)-(26) take an approximate form of linear differential equations with constant coefficients:

(B5)

Its general solution for b(u) may be matched to the one for the weak-current range, Eq (B6):

þ c1 expð
f1 uÞ;

bðuÞ y b0 expðf2 uÞ½1 þ b0 expðf2 uÞ
1 ; where f2 y1

(B6)

where

(B10)

(B11)

Combination of expressions (B6) and (B11) provides an interpolation formula for b(u) within the whole range of u:

bðuÞyb0 ½expðf2 uÞ
expð
f1 uÞ ½1 þ b0 expðf2 uÞ
1

aa ðuÞ ≡ 1
aðuÞy
2c0 þ a2 expðf2 uÞ þ a1 expð
f1 uÞ

(B9)

db = duya byð1
bÞb

bðuÞ y b0 ½expðf2 uÞ
expð
f1 uÞ; cðuÞyc2 expðf2 uÞ

f1;2 ¼ ½

dc = du«3a byK c

(B4)

Their solution for each concentration contains two exponentially varying terms, one of them being a decreasing one, exp (- f1 u), while the second term is an increasing function of u, exp (f2 u):



Thus, the Br2 concentration, c(u), drops rapidly down (relaxation time is about K
1) from its initial value, c0, to a much lower value, c2 ~ (6/K) c0 « c0. Within a more extended scale: u ~1, all three dimensionless concentrations, b, c and aa ≡ 1 - a, increases exponentially as exp (f2 u), the amplitude of c(u) being small compared to two other functions, in particular c(u)/b(u) y c2/b0 ¼ 3/K « 1. As a result, b(u) becomes comparable to 1 in the range: f2 u*y u* y ln (1/2c0) while c(u*) is still of the order of K
1, i.e. c(u*) « 1. Therefore, the Br2 concentration, c(u), is a minor term in the balance relation, Eq (26) or (B4), and one may use the approximation of quasi-stationary concentrations with respect to the kinetic equation for this species, i.e.

so that Eq (24) for b(u) may be rewritten as

db = duy
5b þ 2K c; dc=duy3b
K c; aa ¼ b þ 2c
2c0

aa ¼ 0; b ¼ 0; c ¼ c0 at u ¼ 0

13

(B12)

Eq (26) and (B8) can be used to derive expressions for a(u) and c(u) via b(u):

aðuÞ y ½1
bðuÞ ½1 þ 6bðuÞ=K
1 ; cðuÞyc1 expð
f1 uÞ

1=2 K 2 þ 14K þ 25 ± ðK þ 5Þ ;

þ ð3=KÞaðuÞbðuÞ

.     b0 ; c2 ¼ ½ f
1 b0 ¼ 2Kc0 ðf1 þ f2 Þ; c1 ¼ ½ 1 þ f
1 1 2
1 b0 ; a1
1 ¼ b0 f
1 1 ; a2 ¼ b0 f 2

(B7) For all values of K the f1 factor of the decreasing exponent is much larger than that of the increasing one, f1. Therefore, the concentration distributions are subject to a rapid relaxation to new levels owing to exponentially decreasing terms in Eq (B6). After its termination all concentrations in expressions (B6) grow exponentially, starting from very low levels of the order of c0 which are determined by the corresponding constants, b0, c2 or a2, Eq (B7). In view of the approximate character of Eq (B4) the derived expressions are only valid if the resulting change of the bromate concentration, aa ≡ 1 - a, is small, compared to 1, i.e. if a is close to 1, corresponding to a relatively weak current density within the entrance part of the channel. 2. Strong current regime Further derivation of approximate analytical results is carried out separately for the cases of very large and very small values of the K parameter.

(B13)

2) Very small K values. K « 5 Then, one may simplify results in Eq (B7):

f2 y K=5«1; f1 y5»f2 ; b0 yð2K=5Þc0 «c0 ; c2 yc0 »c1 yð3K=25Þc0 ; a2 y2c0 »a1 yð2K=25Þc0 (B14) This time the coefficient, c1, of the decreasing exponent in c(u) is much smaller than the one, c2, of the increasing exponent. Therefore, the latter term represents a good approximation starting from u ¼ 0 up to the u values where c(u) becomes comparable to 1. On the contrary, the coefficient of the increasing term in b(u) is very small: b0 « c2 so that b(u) « c(u) within this u range. As a result, the balance relation, Eq (26) or (B4), within this u range may be approximated as a(u) þ 2 c(u) y 1. Approximation of quasi-stationary concentrations may be applied this time to the kinetic equation for Br
species:

db = du«5a by2K c

(B15)

so that 1) Very large K values. K » 5

dc = duyð1=5ÞK cyf2 c

Then, one may simplify results for the coefficients in Eq (B7): 0

0

0

f2 y 1; f1 yK»f2 ; b0 y2c ; c2 yð6 =KÞc «c1 yc ; a2 y2c0 »a1 yð2 =KÞc0

(B8)

(B16)

It means that expressions (B6) for all concentrations derived under condition: a(u) y 1, remains valid even if the deviation of a(u) from 1, aa(u), becomes of the order of 1, almost up to the u value equal to u* where expression (B6) for a(u) ≡ 1 - aa(u) vanishes: aa(u*) y a2 exp (f2 u*) y 1.

14

M.A. Vorotyntsev, A.E. Antipov / Electrochimica Acta 323 (2019) 134799

After passing this point, u ¼ u*, the concentration of BrO
3 , a(u), is practically equal to 0. Therefore, for the range, u > u* Eqs (24), (26) may be simplified to the form:

dc = duy
K c; bðuÞy1
2cðuÞ; aðuÞy0

for u > u*

(B17)

Solution of Eq (B17) for c(u) has the form:

cðuÞ y cðu*Þexp ½
Kðu
u*Þfor u > u*

(B18)

where c(u*) can be found from expression (B6) for c(u) at u ¼ u* while solutions for b(u) and a(u) are given by Eq (B17). The above expressions for concentrations are not valid in the close vicinity of point u* where the exact solution deviates from both Eq (B6) and Eq (B17).

concentration, b(u), raises up within the same range from 0 to b0 y 2 c0 (for very large K values). It means that almost all Br2 species which enter the channel are transformed into Br
by very rapid electrode reaction, Eq (1), while the rate constant for Br2 regeneration reaction (2), k, is much smaller. For more extended time range (u » 1/K) concentrations b(u) and c(u) as well as the deviation of a(u) from its initial value, 1, are increasing exponentially, Eq (B6), i.e. proportionally to exp (f2 u), f2 being close to 1 for large values of K (Fig. B3a). In view of the relation between the coefficients: b0 y 2 c0 » c2 y (6/K) c0, the Br
concentration at every value of u is much larger than the Br2 one: b(u) » c(u). Therefore, the consumption of BrO
3 species by chemical step (2), 1 - a(u), leads dominantly to increase of the Br
concentration, b(u), while Br2 plays the role of a very efficient short-living catalytic species assuring the global BrO
3 -to-Br transformation.

B3. Features of concentration distributions along the channel for various K values Numerically found (Appendix B1) concentration distributions of all Br-containing solution components along the channel are shown in Figs. 2 and 3 (chapter 3) for various values of the K parameter. Comparison of these graphs with those predicted by approximate analytical expressions, Eq (B12) and (B13) or (B17) and (B18), is made in Figs. B1 and B2 for large or small values of the K parameter, respectively. Graphs for the same value of K show very good agreement, their proximity becoming even closer with increase of K values in Fig. B1 or its diminution in Fig. B2. It means that one may trust qualitative conclusions based on these analytical expressions for concentrations.

Fig. B3. See Legend to Fig. B1. Concentration distributions, b(u) and c(u), at small values of the dimensionless coordinate, u ¼ k A0 U
1 y, for two values of K: (a) 32; (b) 1. Their global behavior for larger u values is given in Fig. 2b and c.

Crucially important feature of the bromate process as a representative of the EC” mechanism is its autocatalytic character since the reaction product is the Br2/Br
redox mediator couple so that

0
0 0 0
1 Fig. B1. Distributions of dimensionless concentrations of BrO
y, 3 , a ¼ A / A , of Br , b ¼ B / A (a) and of Br2, c ¼ C / A (b), as functions of the dimensionless parameter, u ¼ k A U proportional to the coordinate, y, along the channel, for various values of the ratio of the reaction rate constants, K ¼ kc* / k A0 (indicated at each graph): large K values, 32 or 100 (indicated at graphs). Comparison of results found by numerical integration in Appendix B1 (shown by points) and by analytical calculations in Appendix B2 (shown by lines), Eq (B12) and (B13).

Fig. B2. See Legends to Fig. B1. Very small K values: 0.1 or 0.03 (indicated at graphs).

According to Eq (B6) the behavior of c(u) for very small u values (Fig. B3) is determined by the ratio of the coefficients, c2 and c1, of the increasing and decreasing exponential terms, their sum being equal to c0. Since for large K values one has f2 y 1, f1 y K » f2, c2 y (6/K) c0 « c1 y c0, Eq (B8), the Br2 concentration, c(u), drops rapidly (within the range: u ~ 1/f1 ~ K
1 « 1) from its initial value, c0, at the entrance of the channel to c2 « c0 (Fig. B3a) while the Br


passage of the redox cycle, Eqs (1) and (2), results in accumulation of catalytically active species which accelerate proportionally the rate of the BrO
3 transformation. The autocatalytic scenario leads to the typical exponential growth law for the concentrations, b(u) and c(u), as functions of u. Because of very low relative concentration of Br2 the balance relation may be approximated as a(u) þ b(u) y 1, as it is confirmed by graphs in Fig. B1a. As for the Br2 concentration, c(u), in Fig. B1b it is proportional to the rate of the chemical step, Eqs (2) and (12), i.e. to a(u) b(u). Therefore, c(u) passes through a maximum in the u range where a(u) y b(u) y 0.5, accompanied by its exponential decrease at larger u values. Within the whole range its amplitude is proportional to 1/K, i.e. it diminishes for larger K values, Eq (B13), where the exponentially decreasing term may be disregarded within this u interval. Since the exponential factors for the increasing and decreasing branches of c(u), f2 y 1 and 1, respectively, are close to one another for large values of K the shape of this function is almost

M.A. Vorotyntsev, A.E. Antipov / Electrochimica Acta 323 (2019) 134799

symmetrical in the vicinity of its maximum with respect to the maximum position, u ¼ umax (Fig. B1b). c Evolution of the concentrations has a different form for sufficiently small K values (Figs. B2 and B3b), K « 5, i.e. kc* « 5 k A0. For its interpretation one can use again approximate analytical expressions for concentrations derived in Appendix B2. Similar to the case of large K values, there is a great difference between the factors, f2 and f1, in exponential terms in Eq (B6). f2 y K/5 « f1 y 5. However, the ratios of the b0, c2 and c1 coefficients are quite different: b0 y (2K/5) c0 ~ c1 y (3K/25) c0 « c2 y c0. It implies that the rapidly decreasing term, c1 exp (- f1 u), is very small compared to the increasing one, c1 exp (f2 u), even at u ¼ 0. As a result, no significant change of the Br2 concentration, c(u), takes place at the shortest range (u ~ 1/f1 y1/5) (Fig. B3b), i.e. c(u) y c0 exp (f2 u). The Br
concentration, b(u), is also increasing: b(u) y b2 exp (f2 u) but the coefficient, b2, is much smaller than that for c(u) (Fig. B3b). In other words, contrary to the case of large K values, the diminution of the BrO
3 concentration, a(u), leads dominantly to increase of the Br2 one, c(u), while the Br
concentration, b(u), remains a relatively minor component of the system. This feature is a direct consequence of a much faster chemical step, Eq (2), compared to the electrochemical one, Eq (1). It means the validity of such an approximate variant of the balance relation: a(u) þ 2 c(u) y 1. One should keep in mind that c(u) cannot exceed its upper limit: c(u) < 0.5 (Fig. B2b). Another important feature of this case, K « 1, is a small value of the factor, f2, in the exponential term in Eq (B6) as well as its strong dependence on K: f2 y K/5 « 1. It manifests itself in a much slower increase of the b(u) and c(u) functions (Fig. B3b) and in extension of this transition region towards a larger u range for smaller K values in Fig. 3 and B2. Since the exponential factor, f2, is proportional to K the use of the modified x-coordinate, u K ¼ kc* t, in Fig. 4 compensates this dependence on K, and the graphs for concentrations approach a limiting behavior in the limit of K / 0. Peculiarity of the shapes of the all concentrations for small K values, compared to those for large K ones, is the conservation of the simple exponential dependence, exp (f2 u), even within the range where c(u) and aa(u) ¼ 1 - a(u) become already comparable to 1 (Figs. B2a,b), up to the value, u y u*, where the value of a(u) is becoming small, a(u) ~ K « 1. It leads to a very rapid decrease of the rate of the chemical step, Eq (2), since it is proportional to the product, a(u) b(u). Within this narrow interval of u (u y u*) b(u) and c(u) change abruptly their variation: increase of b(u) becomes much faster while variation of c(u) changes its sign to diminution. Both effects originate from interruption of transformation of Br
to Br2 by the chemical step while the electrochemical reaction continues its action. Therefore, the Br2 concentration passes through a narrow peaklike maximum (Fig. B2b). The further evolution is governed by the balance relation in the form: b(u) þ 2 c(u) y 1, with transformation of Br2 species into Br
ones. The c(u) function diminishes again exponentially but with a different factor: exp (- K u). Significant difference between the factors for the increasing and decreasing branches of c(u), f2 y K/5 and K, results in a pronounced asymmetry of the shape of c(u) (Fig. B2b). As for the maximal value of c(u), it is approaching 0.5 in the limit of infinitely small K values. Thus, contrary to the case of large K values, the evolution of graphs in Fig. B2 (for a sufficiently extended channel) corresponds to a two-stage scenario: first, transformation of the almost whole
amount of BrO
3 into Br2 (with a very low content of Br dependent
on the K values; thus, Br plays the role of a short-living very efficient catalytic species), then to the Br2 transition into Br
at the electrode owing to disappearance of BrO
3 in solution. Each plot in Figs. 2 and 3 (chapter 3) may be characterized by the value of its dimensionless coordinate, u, at the “half-transition” of a

15

concentration, or of its maximum, namely by u1/2 for a(u), a(u1/ a a 1/2 max ) ¼ 0.5, by u1/2 for b(u), b(u ) ¼ 0.5, and by u for the position of b b c the c(u) maximum. These coordinates characterizing the inverse value of the effective rate constant of the whole process, keff, are plotted in Fig. B4a for various K values on the basis of data in Figs. 2 and 3. 1/2 max These parameters, u1/2 , increase within the range a , ub and uc of small K values (Fig. B4a). This behavior is transformed into an approximately linear dependence for each function if it is plotted versus K
1 (lines 1, 2 and 3 in Fig. B4b). 2

1/2 Fig. B4. Half-transition coordinates for a(u) (u1/2 a ) and b(u) (ub ) as well as coordinate of maximum of c(u) (umax ) for concentration distributions in Figs. 2 and 3 as functions c of (a) K or (b) K
1. (a) Horizontal dashed line: u ¼ - ln (2c0). (b) Inclined dashed line: u ¼ - (f2)
1 ln (2c0).

In conformity with remarks on Fig. 2 in chapter 3 and the analytical consideration in Appendix B2, for very large K values (K » 1, i.e. kc* » k A0) this rate constant, keff, approaches a constant value proportional to that of the chemical step, k, while it is practically independent of K, i.e. of the rate constant of the electrochemical step, kc* (Fig. B4a). Thus, the overall rate of the redox transformation inside the channel is limited by the transformation of
BrO
3 and Br into Br2, Eq (2), inside the solution phase while the electroreduction of Br2 at the electrode surface, Eq (1), is much more rapid. As a result, the Br2 concentration, c(u), in Fig. 2c is very low within the whole transitional interval, c(u) « 1, while its maximal value decreases for larger K values. Analysis of Eq (B12) and (B13) provides limiting values of parameters shown in Fig. B4 for K / ∞:

1=2

ub

 h i  1=2 y ua yumax y
ðf2 Þ
1 ln 2c0 f1 f2 =ðf1 þ f2 Þ y
ln 2c0 c (B19)

i.e. these identical values of all three positions are equal to 6.2 for c0 ¼ 0.001 (shown as horizontal dashed line in Fig. B4a). The value for K ¼ 100 in Fig. B4a is already close to this limit. For sufficiently small K values these parameters may be evaluated with the use of expressions (B6) for a(u), (B18) for c(u) and (B17) for b(u). It gives:

1=2

ua

y
ðf2 Þ
1 ln 2a2 y
ðf2 Þ
1 ln 4c0 ; umax y c


ðf2 Þ
1 ln

2y


ðf2 Þ
1 ln c0 ; ub yumax
K
1 ln 2 c 1=2

(B20)

where (f2)
1 y 5 K
1 þ 1.4, ln 4 c0 ¼ - 5.5, ln 2 c0 ¼ - 6.2 for c0 ¼ 0.001, ln 2 ¼ 0.69. For very small K values the transition range varies approximately as 5 K
1 ln (1/2 c0) ¼ 31 K
1 so that it increases as 1/kc*, i.e. it is the electrochemical step, Eq (1), which limits the whole transformation. Combination of Eq (B19) and (B20) provides us with interpolation formulas, in particular, for umax : c

16

M.A. Vorotyntsev, A.E. Antipov / Electrochimica Acta 323 (2019) 134799

umax y
ðf2 Þ
1 ln 2c0 ; whereðf2 Þ
1 y1 þ 5K
1 c

[23]

þ 2ðK þ 5Þ
1 and ln 2c0 ¼
6:2 for c0 ¼ 0:001

(B21)

Predictions for this dependence, umax vs K
1, shown by dashed c line 4 in Fig. B4b are in good agreement with numerically found results. If the last term in expression for (f2)
1, 2 (K þ 5)
1, is disregarded one can get a simple expression for the time, ttrans, which fluid particle is to spend inside the channel to reach the range of the most intensive redox transformation:

  
1  .

0
1 0 ln 1=2c0 t trans ≡ 1 keff ≡umax y k A þ 5ðk *Þ k A c c (B22)

[24]

[25]

[26]

[27]

[28] [29] [30] [31] [32]

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