Analysis of Multiple Reactions on a Bipolar Electrode - American

Ind. Eng. Chem. Res. 2009, 48, 9441–9456

9441

Analysis of Multiple Reactions on a Bipolar Electrode V. A. Juvekar,* Rajkumar S. Patil, Anand V. P. Gurumoorthy, and Asfiya Q. Contractor Department of Chemical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India

Open bipolar electrolysis is a useful technique for conducting electrolytic processes in low concentrations of electrolytes or in nonaqueous solutions. In bipolar electrolysis, both cathodic and anodic reactions occur on the same electrode. In this paper, we analyze the case of multiple reactions occurring in a bipolar manner on a single metal spheroid. Simulations are conducted for various cases involving two redox reactions. Effects of the parameters such as intensity of the imposed electric field, the Nernst potentials, exchange current densities, and transfer coefficients on the current density and the extent of coupling between the reactions have been investigated. The asymptotic cases of low and high imposed electric fields are also considered, for which simple analytic solutions are derived. Together, they cover a major operating region of practical importance. Some important insights into coupling of the reactions on the bipolar electrode have been obtained. These provide useful criteria for design of bipolar electrolytic systems. Introduction In open bipolar electrolysis, a number of small-sized isolated electrodes are confined in an electrolyte between two current feeding electrodes. Electric field, produced by the current flowing through the system, polarizes each isolated electrode into cathodic and anodic regions. Thus both anodic and cathodic reactions can occur on different sides of the same electrode. The advantages of bipolar electrolysis are that it packs a large electrode area in a given space and since cathode and anode are in the vicinity of each other, Ohmic drop is low. In fact, the bipolar electrode provides an alternative pathway of lower resistance. Also, by fluidizing the electrodes, it is possible to obtain high rates of heat and mass transfer and thereby achieve high limiting current densities. The disadvantages are 2-fold; first, since the cathode and anode cannot be separated from each other, the technique is unsuitable in cases where the products of the cathodic reaction interfere with the anodic reaction or vice versa. Moreover, the reaction can reverse. The ideal situation is one in which each reaction occurs only in one direction. This is not normally achieved in practice with at least a part of the reaction being reversed. Second, for efficient bipolar electrolysis, the electrolyte solution should have low electrical conductivity in order that the feed current density, needed to achieve the required polarization, is small. The process is therefore not suitable for solutions having high concentrations of electrolytes. In general, bipolar electrolysis is most suitable for those processes, which involve dilute solutions, for example, electrooxidation of organic pollutants from wastewater, electrowinning of metals at low concentrations, disinfection of water by electrochemical generation of a disinfectant, etc. It can also be advantageously used for electroorganic synthesis in nonaqueous media, where high intensity of electric field could be achieved due to low conductivity of the medium. Examples of bipolar electrolysis are found in such fields as organic electrosynthesis,1,2 oxide film thickness control,3 and supported catalyst metal particle technology.4 Recently Bradley’s group has introduced the concept of spatially coupled bipolar electrolysis (SCBE) for forming metallic interconnects.5-7 The use of bipolar particulate electrodes (dispersion of conducting * To whom correspondence should be addressed. Tel: +91-2225767236. Fax: +91-22-25766895. E-mail: [email protected].

particles) in packed bed, trickle bed, and fluidized bed reactors8-10 has been investigated. For instance, Goodridge et al.,11 experimentally investigated a case of bipolar electrolysis wherein bromine evolution occurs at the anodic face and hydrogen evolution and hydroxyl ion generation at the cathodic face. The bromine further reacts with styrene to form bromohydrin, which is subsequently saponified by the hydroxyl ions generated at the cathodic face. Yen and Yao12 have carried out a mathematical analysis of bipolar electrolysis on a sphere of finite conductivity on which a single electrochemical redox reaction is occurring. Both the linear as well as Butler-Volmer kinetics have been studied. Electrical interactions among multiple bipolar spheres have been studied by Keh and Li.13 These authors take into account differences in radius, electrical conductivity, and electrochemical properties of systems with two or three spheres. The analysis is however limited only to the linear kinetics. Both Yen and Yao and Keh and Li use the boundary collocation technique. Eardley et al.8 analyze the case of fixed arrays of particles dispersed in a continuous electrolyte medium. The work cited in the last paragraph is focused on the applications in which dispersed conducting particles are used to improve the conductivity of a solution having poor inherent conductivity. However, in practice (especially in industrially relevant processes), it is common to find coupled redox reactions, in which one or more auxiliary reactions act either as electron sources or sinks required for conducting the desired reaction. The interest is to find how the reactions are coupled and how to enhance the efficiency of a particular reaction. Fleischmann and Pons have considered multiple reactions on spherical ultramicroelectrodes.2 They have presented a model to compute the current contributions of individual reactions. Duval et al.14 have modeled the dynamics of dissolution of aluminum, coated on a planar electrode. Both these models are approximate. From the aforementioned analysis it clear that there is a need to provide a general mathematical treatment for multiple reactions. In the present communication we analyze the case where multiple reactions occur on a single particle. We have ignored mass transfer limitation in this analysis. Numerical simulations have been conducted for a variety of cases. Asymptotic limits of low and high imposed electric field have also been analyzed. Analytical solutions have been obtained for

10.1021/ie900437n CCC: $40.75  2009 American Chemical Society Published on Web 07/20/2009

9442

Ind. Eng. Chem. Res., Vol. 48, No. 21, 2009

The potential on the equatorial plane, (θ ) π/2), at a location far away from the sphere (r f ∞) is considered as the datum, and all potentials are measured with respect to this datum unless otherwise mentioned. At the interface between the sphere and the electrolyte, the inward current density, is(θ) is related to the normal potential gradient by the following equation m

is(θ) )

∑i

k

k)1

Figure 1. Schematic diagram of a bipolar electrolysis cell.

these cases. Although, the analysis mostly focuses on a spherical bipolar electrode, we have also included the effect of the shape of the electrode on its performance. Some insights, obtained from this work, would be useful for designing systems for bipolar electrolysis. Theoretical Analysis The schematic diagram of the problem considered is depicted in Figure 1. A spherical metal electrode of radius a is placed in an electrolyte, midway between two parallel-plate feeder electrodes. The distance between the two electrodes is L, such that L . a. The solution contains several dissolved species and is well agitated so that the concentrations of the species can be assumed to be spatially uniform. A potential difference is applied between the two electrodes so that the upper plate is at a higher potential than the lower plate. The magnitude of the potential difference is such that a current of magnitude i∞ flows in the negative z-direction in the region far removed from the sphere. If κ is the electrical conductivity of the solution, this current produces an electric field E∞ given by κE∞ ) i∞

(1)

We consider the case of multiple reactions, that is “m” simultaneous electrode processes occurring at the surface of the sphere. Ox1 + n1e S R1 l Oxj + nje S Rj l Oxm + nme S Rm The potential in the electrolyte, denoted by φ, satisfies the Laplace equation in spherical coordinates, with symmetry in the azimuthal plane ∇2φ ) 0

(2)

∂ ∂ 1 ∂ 1 ∂ + 2 sin θ ∇ ) 2 r2 ∂θ r ∂r ∂r r sin θ ∂θ

( )

(

Far away from the sphere, the potential distribution should match the linear potential field, that is, at r f ∞,

φ)

i∞ r cos θ ) E∞r cos θ κ

(3)

|

(4) r)a

{ [ ( ) [ ( )

]

nkF (φ* - φs(θ) - Uk - φr) RT nkF exp Ra,k (φ* - φs(θ) - Uk - φr) RT

ik(θ) ) i0,k exp -Rc,k

]}

(5)

where Uk is the Nernst potential of the kth reaction, measured with respect to the reference potential φr, and φ* is the potential of the metal sphere. The term i0,k represents the exchange current density and Ra,kand Rc,k are the anodic and cathodic transfer coefficients, respectively, for the kth reaction. Ra,k + Rc,k ) 1

(6)

It is important to distinguish between the potential of the electrode φ* and the potential φs(θ) on the electrode surface. The electrode being metallic, is an equipotential body, that is, its potential φ* is spatially uniform. On the other hand, φs(θ) is the potential of the solution at the outer boundary of the electric double layer surrounding the electrode surface. Since the double layer thickness is very small we can consider φs(θ) to be the potential in the solution in contact with the surface of the electrode and hence it is termed the surface potential. The electrode is in the polarized state and hence has spatially nonuniform surface charge density. The potential distribution φs(θ) is consistent with this charge distribution. The difference φ* - φs(θ) provides the electrical driving force for the reaction. If it is greater than the Nernst potential, Uk + φr, the reaction is driven in the anodic direction. Thus φ* - φs(θ) - Uk - φr provides the overpotential for the reaction. Since the overpotential varies with θ, the rate of electrode reactions also varies with the location on the electrode. Thus in some regions on the surface of the electrode, the overpotential is positive and those regions act as anode, whereas on the remaining surface the overpotential is negative and it forms the cathode. Such a drastic variation in the overpotential distinguishes a bipolar electrode from a monopolar electrode. The latter is designed so as to minimize the variation of the overpotential on its surface. The Nernst potential Uk can be related to the concentration of the redox species by the Nernst equation Uk ) U0,k ′ -

)

∂φ ∂r

Here we consider the inward (reduction current) as positive; ik is the contribution to is by the kth reaction and is given by the Butler-Volmer equation:

in which 2

)κ

( )

[Rk]s RT ln nkF [Oxk]s

(7)

′ is the formal potential for the kth reaction. The square U0,k bracket represents the concentration of the species and the subscript s corresponds to the surface of the sphere. We neglect the mass transfer resistance, that is, we assume the concentration of each species at the surface of the sphere to be equal to that in the bulk.


Since, at steady state, charge accumulation cannot occur on the sphere, the net current entering the sphere should be zero. This requires the following condition to be satisfied.

∫

π

0

is sin θ dθ ) 0

F φ φˆ ) RT F E a φˆ ∞ ) RT ∞ r rˆ ) a R Rˆ ) a î ) i a F κ RT î0,k ) i0,k a F κ RT

( )( ) ( )( )

ˆ k ) F Uk U RT ˆφe ) F (φ* - φr) RT z zˆ ) a L Lˆ ) a îk ) ik a F κ RT

(9)

In the rest of the paper, we have used both the notations, î ∞ and φˆ ∞, interchangeably. The governing equations and boundary conditions can be expressed in the dimensionless form as follows; eq 2 transforms to (11)

in which ∂ 1 ∂ 2∂ 1 ∂ rˆ + 2 sin θ 2 ∂r ˆ ∂r ˆ ∂θ ∂θ rˆ rˆ sin θ

( )

(

)

The dimensionless boundary conditions (eq 3 and 4, respectively) are φˆ |rˆf∞ ) rˆφˆ ∞ cos θ

|

(12)

m

rˆ)1

∑ î

ˆ - φˆ s - U ˆ k)] -

0,k{exp[-Rc,knk(φe

k)1

ˆ k)]} exp[Ra,knk(φˆ e - φˆ s - U

(13)

where, the kth term in the summation represents the contribution to the dimensionless current density from the kth reaction. The dimensionless form of eq 8 is

∫

π

0

îs sin θ dθ ) 0

(14)

The general solutions for eq 11 is ∞

φˆ )

∑ [A rˆ P (cos θ) + B rˆ n

n

n

n

-(n+1)

Pn(cos θ)]

Pn(cos θ)

(16)

From this equation, we obtain the dimensionless inward (cathodic) current density at the surface of the sphere as

|

∞

rˆ)1

) φˆ ∞ cos θ -

∑ (n + 1)B P (cos θ) n n

n)0

Since ∫0πPn(cos θ) sin θ dθ ) 0, for all n, and also ∫0π cos θ sin θ dθ ) 0, eq 14 is automatically satisfied by eq 17. On the surface of the sphere, eq 16 reduces to ∞

(10)

∇2φˆ ) 0

-(n+1)

n

n)0

φˆ s(θ) ) φˆ ∞ cos θ +

( )( )

)

∑ B rˆ

(17)

( )( )

î∞ ) κE∞ a F ) φˆ ∞ κ RT

ˆ îs ) ∂φ ∂rˆ

∞

φˆ ) rˆφˆ ∞ cos θ +

ˆ îs ) ∂φ ∂rˆ

φˆ e is the dimensionless electrode potential with respect to the reference electrode. We note from eq 1 and 9 that

∇2 )

The boundary condition stated in eq 12, along with our convention that φˆ (∞, π/2) ) 0, reduces eq 15 to the following form

(8)

The following dimensionless variables and parameters are defined:

9443

(15)

n)0

where Pn is the Legendre polynomial of degree n and the coefficients An and Bn are unknowns to be evaluated using the boundary conditions.

∑ B P (cos θ) n n

(18)

n)0

Using eq 18, we can rewrite the boundary condition in eq 13 in terms of Legendre series as follows: ∞

φˆ ∞ cos θ -

m

∑ (n + 1)B P (cos θ) ) ∑ î n)0 ∞

φˆ ∞ cos θ -

ˆ -

0,k{exp[-Rc,knk(φe

n n

k)1

∑ B P (cos θ) - Uˆ )] - exp[R n n

ˆ -

a,knk(φe

k

n)0 ∞

φˆ ∞ cos θ -

∑ B P (cos θ) - Uˆ )]} n n

(19)

k

n)0

The constants Bn must be chosen such that eq 19 is satisfied at every θ between 0 and π. We consider two cases. The first is the case where the transfer coefficients are symmetric, that is, Rc,k ) Ra,k ) 0.5, for all k. We denote it as the symmetric case. We term the other as the asymmetric case where Rc,k * Ra,k * 0.5, at least for one value of k. For the symmetric case, the Butler-Volmer equation is antisymmetric with respect to φs(θ) for all species, that is, replacing φs(θ) by -φs(θ), only changes the sign of the current density and not the magnitude. This requires that both the potential as well the current density be antisymmetric with respect to the equatorial plane. Hence we must retain only the antisymmetric terms in the Legendre series. Since Pn(cos θ) with odd values of “n” are antisymmetric, we rewrite eq 19 as ∞

φˆ ∞ cos θ -

{

∑

m

(n + 1)BnPn(cos θ) )

nodd)1

∑ î

0,k

×

k)1

∞

exp[-Rc,knk(φˆ e - φˆ ∞ cos θ -exp[Ra,knk(φˆ e - φˆ ∞ cos θ -

∑

ˆ k)] BnPn(cos θ) - U

∑

ˆ k)] BnPn(cos θ) - U

nodd)1 ∞ nodd)1

}

(20)

Equations 16 and 17 are modified to ∞

φˆ ) rˆφˆ ∞ cos θ +

∑

nodd)1

Bnrˆ-(n+1)Pn(cos θ)

(21)

9444

Ind. Eng. Chem. Res., Vol. 48, No. 21, 2009 ∞

îs ) φˆ ∞ cos θ -

∑

∞

(n + 1)BnPn(cos θ)

(22)

nodd)1

φˆ ∞ cos θ -

∑


nodd)1

m

From eq 21, we see that φ|π/2 ) 0 for all r, that is, the equatorial plane has zero potential. Moreover, if we substitute θ ) π/2 in eq 20, we get

∑ î

0,k{Rc,knk

∞

∑

nodd)1

ˆm - U ˆ k)][φˆ ∞ cos θ + exp[-Rc,knk(U

k)1

ˆm - U ˆ k)][φˆ ∞cos θ + BnPn(cos θ)] + Ra,knk exp[Ra,knk(U ∞

m

∑

∑ î

ˆ ˆ ˆ ˆ 0,k{exp[-Rc,knk(φe - Uk)] - exp[Ra,knk(φe - Uk)]} ) 0

BnPn(cos θ)]}

(28)

nodd)1

k)1

(23) Equation 23 implies that the electrode potential φˆ e is independent of the externally imposed potential φˆ ∞ but solely depends on the parameters of the electrode reactions. We denote ˆ m and rewrite eq 23 as φˆ e by U

Comparison of the coefficients of cos θ on both sides of eq 28 gives γ-1 B1 ) φˆ ∞ , 2γ + 1

(

)

B3, B5, ... ) 0

(29)

where m

∑ î

ˆ

0,k{exp[-Rc,knk(Um

ˆ k)] - exp[Ra,knk(U ˆm - U ˆ k)]} ) 0 -U

k)1

m

γ)[

ˆ m can be identified as the dimensionless mixed potential,15 U that is, the electrode potential (in the absence of the imposed electric field) at which the cathodic and anodic parts of the current exactly balance each other. We find from eq 24 that for a single reaction, the mixed potential reduces to the Nernst ˆ 1). For a case of two simultaneous ˆm ) U potential (i.e., U ˆ reactions, the value of Um lies between the two individual Nernst ˆ 2. For several electrode reactions, U ˆ m lies ˆ 1 and U potentials U near the Nernst potential of the reaction having the dominant value of exchange current density î 0,k. We can now rewrite the dimensionless form of eq 5 as ˆ m - φˆ s(θ) - U ˆ k)] îk(θ) ) î0,k{exp[-Rc,knk(U ˆ m - φˆ s(θ) - U ˆ k)]} exp[Ra,knk(U

ˆm - U ˆ k)]}]-1 (30) Ra,knk exp[Ra,knk(U

m

∑ î

0,knk]

ˆ

0,k{exp[-Rc,knk(Um

ˆ k)] - φˆ s(θ) - U

k)1

ˆ m - φˆ s(θ) - U ˆ k)]} exp[Ra,knk(U

(26)

φˆ γ - 1 -2 ) rˆ + rˆ cos θ 2γ + 1 φˆ ∞

[ (

) ]


nodd)1 m

ˆ

0,k{exp[-Rc,knk(Um

k)1

[1 + Rc,knk{φˆ ∞ cos θ +

ˆ k)] × -U

∞

∑

which, on the surface of the particle, reduces to φˆ s 3γ ) cos θ 2γ + 1 φˆ ∞

2γ + 1 φˆ ∞ , 3γ

(

BnPn(cos θ)}] -

()

∞

nodd)1

In light of eq 24, we can simplify eq 27 to

)

(33)

(34)

φˆ îk(θ) ) îk π + ∞ 3γ cos θ 2 γk 2γ + 1

ˆ k)][1 - Ra,knk{φˆ ∞ cos θ + ˆm - U exp[Ra,knk(U BnPn(cos θ)}]}

)

Using eq 33, we can rewrite the condition under which the linearization of eq 20 is valid (i.e.,φˆ s , 1):

nodd)1

∑

(32)

We see that when γ f 0, that is, when electrochemical reactions are rapid, inequality 34 reduces to φˆ ∞ , 1/(3γ), and since γ is small, the range of φˆ ∞ over which linearization is applicable extends to large values. The contribution to the dimensionless current density from the kth reaction is obtained by linearizing eq 25 and substituting φs from eq 33:

∞

∑ î

(31)

This is the expression derived by Yen and Yao12 for a single reaction scheme. Substituting the expressions for Bn from eq 29 into eq 21, we obtain the potential distribution as

For small values of φˆ s, the exponential term in eq 19 can be linearized to yield

∑

-1

k)1

(

∑ î

φˆ ∞ cos θ -

ˆm - U ˆ k)] + exp[-Rc,knk(U

γ)[

m

îs(θ) )

0,k{Rc,knk

ˆ 1), or in the situation where ˆm)U For the single reaction case (U ˆ k) , 1 for all k (i.e., all reactions have approximately ˆm - U (U equal Nernst potentials), eq 30 simplifies to

(25)

The current density is given by summing eq 25 over all reaction steps:

∑ î k)1

(24)

(27)

(

)

(35)

where î k(π/2)is the value of î k at the equator: ˆ k)] - exp[Ra,knk(U ˆm - U ˆ k)]} ˆm - U îk π ) î0,k{exp[-Rc,knk(U 2 (36)

()


and ˆm - U ˆ k)] + γk(θ) ) [iˆ0,k{Rc,knk exp[-Rc,knk(U ˆm - U ˆ k)]}]-1 Ra,knkexp[Ra,knk(U

(37)

The current density is obtained by summing î k(θ) from eq 35 over all k. îs ) î∞

( 2γ 3+ 1 ) cos θ

(38)

In deriving eq 38, we have used the fact that (see eq 30 and eq 37) 1 ) γ

reaction. The deviation from this antisymmetry is realized only at high magnitudes of the imposed fields as shown later. When the condition for low imposed field (eq 34) is not satisfied, an analytical solution of eq 19 cannot be obtained and we need to resort to a suitable numerical technique. One of the techniques is the boundary collocation. When applying this technique, we first truncate the infinite sums over index n in eq 19 into finite sums involving only the first N terms, thereby yielding the following equation N

φˆ ∞cos θ -

k)1

∑ (n + 1)B P (cos θ) ) n n

n)0 m

∑ î

m

∑ γ1

ˆ - φˆ ∞ cos θ -

0,k{exp[-Rc,knk(φe

(39)

k

k)1

N

∑ B P (cos θ) - Uˆ )] - exp[R

and also that

n n

k


a,knk(φe

n)0

m

∑

N

∑ B P (cos θ) - Uˆ )]}

îk π ) 0 and φˆ ∞ ) î∞ 2 k)1

()

n n

N

φˆ ∞ cos θj -

s φˆ s)0

1 ˆ (φ cos θ + γ ∞

∑ (n + 1)B P (cos θ ) ) n n

j

n)1 m

∑ (n + 1)B P (cos θ) ) î | n n

(45)

There are a total of N + 1 unknown constants in eq 45, viz., B1, B2, ..., BN and φe. To obtain these constants, we solve eq 45 at N + 1 suitably chosen values of θj. Thus we solve N + 1 simultaneous equations:

∞

n)0

k

n)0

We now consider the asymmetric case where Rc,k * Ra,k * 0.5 at least for one value of k. In this case eq 19 cannot be reduced to eq 20. Hence eq 21 and 22 are not valid. The equator, in general, is not the plane of symmetry and its potential is not zero. Nonzero current density exists at the equator. For the asymmetric case, eq 19 does not reduce to eq 28. For small φˆ s(θ) we can linearize eq 19 and present it as φˆ ∞cos θ -

9445

∑ î

+


0,k{exp[-Rc,knk(φe

k)1

∞

∑ B P (cos θ)) n n

N

(40)

n)0

∑ B P (cos θ ) - Uˆ )] - exp[R n n

j

k

ˆ - φˆ ∞ cos θj -

a,knk(φe

n)1

N

where î s|φˆ s)0 is the current density at the location where the surface potential is zero and is given by îs|φˆ )0 ) s

m

∑ î

ˆ -U ˆ k)) - exp(Ra,k(φˆ e - U ˆ k))]

o,k[exp(-Rc,k(φe

k)1

(41) Comparing the coefficients of cos θ on both sides of eq 40, we obtain B0 ) -

γ ˆ i |ˆ γ + 1 s φs)0

(42)

Other coefficients are the same as those given by eq 29. The potential distribution can now be written by substituting the coefficients Bi in eq 18. φˆ s )

( 2γ3γ+ 1 )φˆ

∞

cos θ -

γ ˆ i |ˆ γ + 1 s φs)0

(43)

The current density given by eq 17 simplifies to îs )

( 2γ 3+ 1 )φˆ

∞

cos θ +

γ ˆ i |ˆ γ + 1 s φs)0

(44)

Now iˆs in eq 44 must satisfy eq 14. This is only possible ˆ if i s|φˆ s)0 ) 0. Hence eq 44 and eq 43 become identical to eq 33 and eq 38. This implies that for the case of asymmetric transfer coefficients, the potential and the current distribution is still antisymmetric about the equatorial plane at low values of the imposed field as is the case with the symmetric

∑ B P (cos θ ) - Uˆ )]} n n

j

k

j ) 1, 2, ..., N + 1

(46)

n)1

and once the constants are obtained, the potential distribution and the current distribution can be obtained using the truncated series forms of eq 16 and 17. The accuracy of the solution depends on the order of the approximation (value of N) and also on the choice of θj. Yen and Yao,12 have used the boundary collocation method, for a single reaction case, with equally spaced values of θj over the interval (0,π). There is one drawback in the procedure used by them. We observed that up to a certain upper limit of N, increase in the value of N improved the accuracy of the solution. Beyond this limit, the set of eq 46 becomes ill-conditioned and the solution tends to diverge with increasing N. This problem could be eliminated by using the orthogonal boundary collocation technique in which the nodes are chosen as the N + 1 roots of the Legendre polynomial PN+1(cos θj). Using this technique, the solution was found to progressively converge with increase in the value of N without becoming unstable. In the present work, we did not pursue the boundary collocation technique, but instead, used COMSOL Multiphysics software package (version 3.5a) for solving the relevant equations. The COMSOL Multiphysics software package is based on multidimensional (two-dimensional, in this case) finite element method. Although this technique is computationally more intensive than the boundary collocation technique (because of the higher dimensionality of the problem), the automatic mesh generation program, powerful equation solvers and the graphical

9446


The electrode potential φe was determined by forcing the following condition on the solution

∫ ∂φ ∂nˆ | ˆ

sˆ

dsˆ ) 0

(51)

s

Here dsˆ is the surface element of the normalized sphere. The domain CDEFGHI was discretized into simple triangular elements. Lagrange-quadratic shape functions were used. The number density of the elements was higher near the spherical boundary DEF as compared to the rest of the domain. Results and Discussion As a special case, we consider two redox reactions occurring simultaneously on the surface of the spherical bipolar electrode. Figure 2. Two-dimensional axisymmetric domain for simulation.

interface of the software makes it easy to solve this problem and also assess various details of the solution. For simulation using COMSOL, it is more convenient to cast the problem in the cylindrical coordinates. We consider a cylindrical region of radius R and length L, enclosing the bipolar electrode and assume that the electrode is on the axis of the cylinder, midway between the current feeders, as shown in Figure 2. The domain of analysis is obtained by revolving the plane region, CDEFGHI around the axis CDFG. The curved surface of the cylinder forms an imaginary boundary of the domain. The center of the particle is taken as the origin and z is taken along the axis of the cylinder; F is the radial distance in the azimuthal plane. All dimensions are normalized with the radius a of the particle. The dimensionless potential in the electrolyte, denoted byφˆ (Fˆ ,zˆ), satisfies the Laplace equation in cylindrical coordinates ˆ 2φˆ ) 0 ∇

(47)

Ox1 + n1e S R1 Ox2 + n2e S R2 All the chemical species involved in the reactions are assumed to be soluble and do not deposit on the electrode surface. The solution contains all the chemical species at uniform concentration. We assume that the surface of the sphere has spatially uniform activity so i0,1 and i0,2 are the same at all points on the sphere. We neglect the mass transfer resistance. Our aim is to determine the current density distribution for the two reactions and its dependence on the model parameters. We first consider the symmetric case. The electrode potential is the mixed potential and is given by eq 24, which is written for two reactions as î0,1{exp[-Rc,1n1(U ˆm - U ˆ 1)] - exp[Ra,1n1(U ˆm - U ˆ 1)]} + ˆm - U ˆ 2)] - exp[Ra,2n2(U ˆm - U ˆ 2)]} ) 0 î0,2{exp[-Rc,2n2(U (52) For the special case when î 0,1 ) î 0,2 eq 52 yields

in which

( )

The dimensionless variable φˆ is same as that previously defined. The datum potential is taken as the potential at Fˆ ) Rˆ and zˆ ) 0, that is: φˆ (Rˆ, 0) ) 0

at Fˆ ) Rˆ

∂φˆ ∂nˆ

|

m

s

)

∑ î

ˆ - φˆ s - U ˆ k)] -

0,k{exp[-Rc,knk(φe

k)1

ˆ k)]} exp[Ra,knk(φˆ e - φˆ s - U

Here nˆ is the unit outward normal to the sphere.

fE )

(50)

2 î∞

∫

π/2

0

îs sin θ dθ

(54)

It is the ratio of the total current passing through the sphere to the current passing through the same area of cross section in the absence of the sphere. For the linear case, we use eq 38 for iˆs and obtain fE )

(49)

The boundary condition at the surface of the sphere is written as

(53)

To quantify the results, we define two terms. The first is the current enhancement factor, fE. This is the same term as defined by Yen and Yao12

(48)

We used Rˆ ) 10 and Lˆ ) 20. These values can be considered sufficiently large since the results of the simulation were insensitive to variation of either Rˆ or Lˆ about these values. Since the potential on the curved surface of the cylinder is not influenced by the particle (Rˆ . 1) we can write ∂φˆ )0 ∂Fˆ

ˆ1 + U ˆ2 U 2

ˆm ) U

∂ ∂2 1∂ Fˆ + 2 ∇ ) Fˆ ∂Fˆ ∂Fˆ ∂zˆ ˆ2

3 2γ + 1

(55)

Although eq 55 has the same form as that given by Yen and Yao,12 the definition of γ as given by eq 30 is more general. It reduces to that of Yen and Yao only for the special case involving single reaction. The other term is the reaction enhancement factor for kth reaction defined as fE,k )

2 î∞

∫

π

0

(iˆk - îk|φˆ ∞)0) sin θ dθ

(56)


ˆ1 ) U ˆ2 Figure 3. Surface potential profile parameters: RC,1 ) RC,2 ) 0.5, U ) 10, î01 ) î02 ) 0.1. (Black) φˆ ∞ ) 1, (blue) φˆ ∞ ) 10, (pink) φˆ ∞ ) 100, (red) φˆ ∞ ) 1000, (green) φˆ ∞ ) 10000.

Here î k|φˆ ∞)0 is the current density contribution of the kth reaction in the absence of the imposed electric field. It represents the contribution due to spontaneity of the reaction and is given by îk|φ )0 ) î0,k{exp[-Rc,knk(U ˆm - U ˆ k)] ∞ ˆm - U ˆ k)]} exp[Ra,knk(U

(57)

For the case of two reactions, since the surface integral of î s equals zero and since î s ) î 1 + î 2, it follows that fE,1 + fE,2 ) 0

(58)

The reaction enhancement factor fE,1 is an important measure of the coupling between reactions 1 and 2. fE,1 ) 0, implies that reactions are completely decoupled, that is, reaction-1 occurs in the cathodic direction in the upper hemisphere and reverses itself completely in the lower hemisphere. Nonzero value of fE,1, therefore implies coupling between reaction 1 and reaction 2, that is, reaction 2 now acts a source of electrons for reaction 1. Higher the value of fE,1 is, the greater is the coupling between the reactions. This is desirable from the practical point of view. For the linear case, it follows from eq 35 that î1 - î1|φ )0 ) ∞

φˆ ∞ 3γ cos θ γk 2γ + 1

(

)

(59)

Note that î 1|φ∞)0 ) î 1(π/2) since the potential at the equator is zero for the linear case. After substituting the expression from eq 59 into eq 56, we find that fE,1 ) 0. This implies that under linear kinetics, there is no coupling between reactions 1 and 2 and hence the electrolysis must be conducted under nonlinear case (i.e., high values of φˆ ∞) if the coupling between the reactions is desired. This is an important observation. From eq 34 we see that when the reactions are fast (γ f 0), the linear range extends to higher values of φˆ ∞ and increasingly greater intensity of the electric field is needed in order to couple the reactions. Nonlinear Regime: Symmetric Case. Let us now consider the nonlinear range of operation. We consider the symmetric case first. Figure 3 represents the potential on the surface of the sphere as a function of cos(θ), with φˆ ∞ as the parameter. The potential is found to be antisymmetric about the equatorial plane. This is consistent with eq 27. We also see that as φˆ ∞

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ˆ1 Figure 4. Surface current density profile parameters: RC,1 ) RC,2 ) 0.5, U ˆ 2 ) 10, î01 ) î02 ) 0.1. (Black) φˆ ∞ ) 1, (blue) φˆ ∞ ) 10, (pink) φˆ ∞ ) )U 100, (red) φˆ ∞ ) 1000, (green) φˆ ∞ ) 10000.

increases, the potential of the upper-pole, φˆ up, increases (and the potential of the lower pole, φˆ lp, decreases). Thus the difference in potential between the pole and the equator increases. However, this difference is much lower than φˆ ∞. For example, for φˆ ∞ of 104, the magnitude of φˆ up is only about 23. To obtain greater insight into this behavior, we have plotted, in Figure 4, the ratio î s/iˆ∞ as a function of cos(θ) using φˆ ∞ as the parameter. We see from this plot that at high values of φˆ ∞, the ratio î s/iˆ∞ is independent of φˆ ∞ and is a linear function of cos θ, except for a kink at the origin. The maximum value of the current is 3iˆ∞. We can therefore express îs for high intensities of the impressed fields as îs ) 3iˆ∞ cos θ

(60)

This upper limit of 3iˆ∞ is also the limit for a metal sphere at zero surface potential kept in an electrostatic field acting along the z-axis. For this case, the potential profile around the sphere is given by φˆ ) φˆ ∞(rˆ - 1/rˆ2) cos θ.16 From this, the inward current density at the surface of the sphere is obtained as ∂φˆ / ∂rˆ|rˆ)1 ) 3iˆ∞ cos θ. The condition of zero potential of the sphere in an electrostatic field is equivalent, for the present problem, to a situation where there is no resistance to the flow of current at the metal-solution interface. So this upper limit of the current density is not due to the interfacial resistance, but is set by the amount of current that approaches its surface from the surrounding solution, which is limited by the magnitude of the imposed potential. In the present case, the interfacial resistance is present due to the finite speed of the electrochemical reactions at the surface of the sphere. As we increase φˆ ∞, overpotential increases, the interfacial resistance decreases, and consequently the current density on the surface increases. This supply of the current must arrive from the surrounding solution. At the threshold value of φˆ ∞, the surface reaction current density exactly matches with the limiting value of the current density arriving from the solution. No further increase in the overpotential is possible, and hence the surface potential attains a plateau with respect to φˆ ∞. The maximum value of the current density is 3iˆ∞ and is reached at the upper pole of the bipolar electrode. This knowledge can be used for computing the maximum attainable value φˆ m at the upper pole using the Butler-Volmer equation.

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Figure 5. Potential at the upper pole: Comparison between approximation ˆ and simulation. Parameters: RC,1 ) RC,2 ) 0.5, î01 ) î02 ) 0.1. (Blue) ∆U ˆ ) 10 (γ ) 0.815), (red) ∆U ˆ ) 20 (γ ) 0.0673). ) 0 (γ ) 5.00), (pink) ∆U

Figure 6. Surface current density profile near equator parameters: RC,1 ) ˆ 2 ) 10, φˆ ∞ ) 100. (Black) ∆U ˆ ) 0 (γ ) 5.00), RC,2 ) 0.5, î01 ) î02 ) 0.1,U ˆ ) 5 (γ ) 2.65), (pink) ∆U ˆ ) 10 (γ ) 0.815), (red) ∆U ˆ ) 15 (blue) ∆U ˆ ) 20 (γ ) 0.0673). (γ ) 0.235), (green) ∆U

Thus the following equation must be satisfied at the upper pole at the limiting potential m

∑ î

ˆ k)] - φˆ m - U

ˆ

0,k{exp[-Rc,knk(Um

k)1

ˆ m - φˆ m - U ˆ k)]} ) 3φˆ ∞ exp[Ra,knk(U

(61)

We have solved eq 61 for different values of φˆ ∞ to obtain the corresponding values of φˆ m. A plot of φˆ m versus φˆ ∞ is shown ˆ 2). The solid ˆ () U ˆ1 - U in Figure 5 for different values of ∆U lines are based on the simulation and the points correspond to the solutions of eq 61. ˆ ) 0, the match It is seen from the comparison that when ∆U between the two is good only at high values of φˆ ∞. This is ˆ , the expected. However, with increase in the value of ∆U agreement is good even at lower values of φˆ ∞. To understand the reason behind this agreement, we have computed the values of γ for these curves and listed them in Figure 5. Noting that γ is a measure of the interfacial resistance, we see that as we decrease the value of γ, the interfacial resistance decreases and the limit of 3iˆ∞ is reached at lower values of φˆ ∞. To understand the kink in the current density profile near the equator, we have enlarged a part of the profile near the equator, and presented it in Figure 6. It is seen that the slope of each curve is lower at the equator. The reason is that near the equator, surface potential is low and hence the surface current is governed by the linear asymptote given by eq 38. It has a slope of 3/(2γ + 1), whereas the slope of the curve at locations away from the equator is 3, which is higher by a factor of 2γ + 1. As γ decreases, the two slopes approach each other as is ˆ ) 20 (γ ) 0.0673), there is evident from the figure. For ∆U practically no kink at the origin. Figure 7 plots the current enhancement factor fE as a function of φˆ ∞. The parameters used are the same as those used in Figure 3. It is seen that all the plots converge to an asymptotic value of fE ) 3 at the limit of high φˆ ∞. This follows from eq 54 and 60. fE )

2 î∞

∫

π/2

0

3iˆ∞ cos θ sin θ dθ ) 3

This is the upper limit of fE.

(62)

Figure 7. Current enhancement factor. Parameters: RC,1 ) RC,2 ) 0.5, î01 ˆ ) 0 (γ ) 5.00), (blue) ∆U ˆ )1 ) î02 ) 0.1, φˆ ∞ ) 10000. (Black) ∆U ˆ ) 2 (γ ) 4.44), (red) ∆U ˆ ) 5 (γ ) 2.65), (green) (γ ) 4.85), (pink) ∆U ˆ ) 10 (γ ) 0.815). ∆U

In the lower limit of small φˆ ∞, fE also attains a constant value fE ) 3/(2γ + 1) (see eq 55). The simulated values are compared with the theory and an exact match is found between them. It is also seen that the value of fE in the entire range of φˆ ∞ depends ˆ ) 0, 1, and 2 the difference among on γ. We see that for ∆U the corresponding values of γ is small (γ ) 5.00, 4.85, and 4.44, respectively). The fE curves for these three cases almost coincide with each other. Next we consider the reaction enhancement factor fE,1. Figure ˆ () U ˆ1 - U ˆ 2) as the 8 plots fE,1 as a function of φˆ ∞ with ∆U parameter. It is seen from the figure that for low values of φˆ ∞ (linear case), fE,1 is zero. This is expected from eq 56 and 59. Also, fE,1 ˆ ) 0. This is also expected since in this case, both ) 0 for ∆U the reactions have equal Nernst potentials and hence have the same anodic and cathodic regions. Consequently, they cannot couple with each other. Further, it is seen that fE,1 reaches a plateau at high values of φˆ ∞. The value of fE,1 at the plateau ˆ . The highest value increases with increase in the value of ∆U reached is 3. We have already shown that at high values of φˆ ∞, the current density at the upper pole reaches a constant value


9449

îup,1 ) î0,1{exp[-Rc,knk(U ˆ m - φˆ m - U ˆ 1)] ˆ 1)]} ˆ m - φˆ m - U exp[Ra,knk(U

(65)

ˆ m + φˆ m - U ˆ 1)] îlp,1 ) î0,1{exp[-Rc,knk(U ˆ 1)]} ˆ m + φˆ m - U exp[Ra,knk(U

(66)

The limiting value of fE,1 is obtained from eq 56 as fE1 )

1 ˆ (i + îlp,1 - 4iˆ1|φ∞)0) ˆφ∞ up,1

(67)

This equation is valid only for high values of φˆ ∞. For the linear range of φˆ ∞, fE,1 ) 0. For the intermediate range, we have derived the following approximate expression for fE,1 (see Appendix for the details of the derivation): Figure 8. Reaction enhancement factor parameters: RC,1 ) RC,2 ) 0.5, î01 ˆ 2 ) 10. Lines are simulation results, and points ) î02 ) 0.1, φˆ ∞ ) 10000, U ˆ ) 1, (green) ∆U ˆ ) 2, are the estimated values using eq 68: (black) ∆U ˆ ) 5, (blue) ∆U ˆ ) 10. (red) ∆U

Figure 9. Contribution to current density from reaction 1. Parameters: RC,1 ˆ 2 ) 10. (Black) ∆U ˆ ) 0, ) RC,2 ) 0.5, î01 ) î02 ) 0.1, φˆ ∞ ) 10000, U ˆ ) 1, (pink) ∆U ˆ 2 ) 2, (red) ∆U ˆ ) 5, (green) ∆U ˆ ) 10. (blue) ∆U

fE,1 )

[( ( ) (

φˆ m 1 1φˆ ∞ φˆ ∞

2

2γ + 1 3γ

) ){iˆ 2

up

{

+ îlp} - 4(iˆ1|φ∞)0) ×

1-

φˆ m 2γ + 1 φˆ ∞ 3γ

(

)}

]

(68)

Values of fE,1 computed by eq 68, are plotted as points in Figure 8 and compared with the simulation. The agreement is excellent at high potentials. It is worse in the intermediate potentials and somewhat improves at low potentials. However, considering the simplicity of the correlation, we recommend it as the first estimate of the reaction enhancement factor. Figure 10 shows the effect of exchange current density on fE,1. We first examine the high potential asymptotes. It is seen that the highest value of fE,1 is obtained when î 0,1 ) î 0,2. The greater is the disparity between the exchange current densities, the lower is the asymptote. We also see that the asymptotic value depends only on the relative magnitudes of î 0,1 and î 0,2 and not on whether î 0,1 > î 0,2 or î 0,1 < î 0,2. The reason for this behavior is not difficult to understand. At the high potential asymptote, the total current density is 3iˆ∞ cos θ. It is shared by the two reactions. If î 0,1 ) î 0,2, equal sharing of the current occurs. Thus reaction 1 has the complete share of current in the upper hemisphere and reaction 2 has the full share in the anodic region. Electrons required for reaction 1 are supplied

of 3iˆ∞. This current density is shared by the two reactions. Hence current density contribution, at the poles, from each reaction must also reach a constant value. The share from each reaction ˆ 2 and the respective exchange ˆ )U ˆ1 - U will depend on ∆U ˆ increases, reaction 1 becomes more current densities. As ∆U cathodic and will share the greater fraction of the current density in the upper hemisphere (reaction 2 will have greater share of the current in the lower hemisphere). This is clear from Figure 9, which plots the profile of the contribution to current density from reaction 1. We see from Figure 9 that the current density profiles obey the following relations: î1 ) îup,1 cos θ

0 e θ e π/2

(63)

and î1 ) îlp,1 cos(π - θ)

π/2 e θ e π

(64)

where î up,1 and î lp,1 are the current density contributions from reaction 1 at the upper and the lower poles, respectively. They can be obtained using the Butler-Volmer equation.

Figure 10. Effect of exchange current density on reaction enhancement ˆ 1 ) 20, U ˆ 2 ) 10, î 0,2 ) 1. factor parameters: RC,1 ) RC,2 ) 0.5,U (Black) î 0,1 ) 1, (green) î 0,1 ) 0.1, (red) î 0,1 ) 10, (blue) î 0,1 ) 0.01, (pink) î 0,1 ) 100.

9450


Figure 11. Electrode potential for asymmetric transfer coefficient paramˆ1 ) U ˆ 2 ) 10, Rc,2 ) 0.5. (Blue) Rc,1 ) 0.5, (green) eters: î01 ) î02 ) 0.1, U Rc,1 ) 0.45 and Rc,1 ) 0.55, (black) Rc,1 ) 0.3 and Rc,1 ) 0.7, (red) Rc,1 ) 0.1 and Rc,1 ) 0.9. The lines above φˆ e ) 10 correspond to Rc,1 > 0.5 and those below correspond to Rc,1 < 0.5.

entirely by reaction 2. In this case, coupling between the reactions is complete. As the exchange current density of one of the reactions, say reaction 1, becomes more dominant, it begins to attract a greater share of the total current. In this case reaction 2 cannot provide the entire quantity of the electrons required for reaction 1 and a part of reaction-1 must reverse. This leads to lowering of fE,1. In general, the extent of coupling is governed by the relative exchange current density of the slower reaction. Thus the extent of coupling is identical, when î 0,1/iˆ0,2 ) 100 and when î 0,2/iˆ0,1 ) 100. The larger this ratio is, the lesser is the coupling and the lower is the value of fE,1. Among the pairs of curves which show identical asymptotic value, the curve for which the exchange current density is lower shows higher intermediate values of fE,1. The reason is that low exchange current density corresponds to a high value of γ and hence has a shorter linear region. Consequently, the curve begins to rise at lower values of φˆ ∞, hence it rises above the curve which corresponds to higher value of the exchange current density. Nonlinear Regime: Asymmetric Case. We now consider the asymmetric case. Here we assume that reaction 1 is asymmetric Rc,1 * 0.5, but reaction 2 is symmetric, that is, Rc,2 ) 0.5. For the sake of comparison, the case Rc,1 ) 0.5 is also included in the plots. Figure 11, shows how the electrode potential φˆ e varies with ˆ1 ) U ˆ 2 ) 10 and î 01 ) î 02 ) 0.1. ˆφ∞. Here, we have used U It is seen from the figure that at low values of φˆ ∞, φˆ e is same for all Rc,1. Under these conditions, electrode potential ˆ m as has been previously coincides with the mixed potential U ˆ ) 0) mixed shown, and that, for the present case (∆U ˆ1 ) U ˆ 2 irrespective of the value of Rc,1. ˆm ) U potential is U At high values of φˆ ∞, φˆ e either increases with φˆ ∞ or decreases with φˆ ∞ depending on whether Rc,1 > 0.5 or Rc,1 < 0.5. The greater the deviation of Rc,1 from 0.5 is, the greater is the ˆ m. deviation of φˆ e from U To understand the reason behind this behavior, we have plotted the surface potential profiles in Figure 12 for φˆ ∞ of 10000. It is important to note that the potential profile is antisymmetric about the equator. The effect of Rc,1 on the

ˆ1 ) U ˆ2 Figure 12. Surface potential profile parameters: î01 ) î02 ) 0.1, U ) 10, Rc,2 ) 0.5. (Pink) Rc,1 ) 0.5, (green) Rc,1 ) 0.9, (black) Rc,1 ) 0.1, (blue) Rc,1 ) 0.3, (red) Rc,1 ) 0.7.

potential is small with Rc,1 ) 0.5 yielding the highest magnitudes of the surface potential. We also find that φˆ s|Rc,1 ) φˆ s|(1-Rc,1). We also computed the surface current density and found it to be independent of Rc,1; it is therefore not presented here. The current density is also a linear function of cos θ and reaches the limiting value of 3iˆ∞ at the upper pole. This prompts us to write the following limiting current balance equations for the two poles. î0,1{exp[-Rc,1n1(φˆ e - φˆ m - U ˆ 1)] - exp[Ra,1n1(φˆ e - φˆ m ˆ 2)] ˆ 1)]} + î0,2{exp[-Rc,2n2(φˆ e - φˆ m - U U ˆ ˆ ˆ exp[Ra,2n2(φe - φm - U2)]} ) 3φˆ ∞ (69) ˆ 1)] - exp[Ra,1n1(φˆ e + φˆ m î0,1{exp[-Rc,1n1(φˆ e + φˆ m - U ˆ ˆ 2)] ˆ U1)]} + i0,2{exp[-Rc,2n2(φˆ e + φˆ m - U ˆ 2)]} ) -3φˆ ∞ (70) exp[Ra,2n2(φˆ e + φˆ m - U Equation 69 and 70 could be individually solved and the solutions are combined to obtain both φˆ e and φˆ m. The values of φˆ e are plotted as discrete points in Figure 11 for comparing them with the simulation. The agreement between the two is very good when the deviation from symmetry (i.e., Rc,1 ) 0.5) is small. When the deviation is large, the agreement is not so good. The reason is that small but finite amount of deviation of potential (and current density) from antisymmetry is present as Rc,1 deviates from 0.5. In Figure 13, we have fE versus φˆ ∞ for different values of Rc,1. All the plots overlap with each other indicating that fE is independent of Rc,1. This is consistent with the finding that the total current density is independent of Rc,1. Next, in Figure 14, we have plotted fE,1 as a function φˆ ∞, ˆ ) 0, that with Rc,1 as the parameter. Note that, in this case, ∆U ˆ 2. It is seen that Rc,1has a strong effect on fE,1. When ˆ1 ) U is, U Rc,1 ) 0.5, fE,1 ) 0. This is expected since both reactions are now identical and together act as a single reaction. No coupling can occur between them. When Rc,1 > 0.5, fE,1 > 0 and when Rc,1 < 0.5, fE,1 < 0. In the former case, reaction 1 has more cathodic contribution than anodic contribution. It therefore acts cathodic to reaction 2. On the other hand when Rc,1 < 0.5, reaction 1 is more anodic and hence has negative value of fE,1.


Figure 13. Current enhancement factor (asymmetric case) parameters: î01 ˆ1 ) U ˆ 2 ) 10, Rc,2 ) 0.5. (Pink) Rc,1 ) 0.5, (green) Rc,1 ) ) î02 ) 0.1, U 0.9, (black) Rc,1 ) 0.1, (blue) Rc,1 ) 0.3, (red) Rc,1 ) 0.7.

9451

ˆ * 0) Figure 15. Reaction enhancement factor (asymmetric case, ∆U ˆ 2 ) 10, Rc,1 ) 0.7, Rc,2 ) 0.5. (Green) ∆U ˆ parameters: î01 ) î02 ) 0.1, U ˆ 2 ) 2, (red) ∆U ˆ ) 5, (blue) ∆U ˆ ) 10. ) 1, (black) ∆U

even for the lower values considering the simplicity of the approximation. Lastly we review the model developed by Fleischmann et al.2 They have suggested an approximate expression for computing the current density contribution from a reaction step on a spherical bipolar electrode. We can write it in the present notation as follows: îk(θ) ) î0,k{exp[-Rc,knk(U ˆ m - φˆ ∞ cos θ - U ˆ k)] ˆ m - φˆ ∞ cos θ - U ˆ k)]} exp[Ra,knk(U

(71)

The equation uses φˆ ∞ cos θ in place of φˆ s(θ). For large values of φˆ ∞, eq 71 estimates excessively high values of î k(θ). To check its validity at low values of φˆ ∞, we linearize eq 71, and arrive at the following form

ˆ ) 0) Figure 14. Reaction enhancement factor (asymmetric case, ∆U ˆ1 ) U ˆ 2 ) 10, Rc,2 ) 0.5. (Blue) Rc,2 ) 0.5, parameters: î01 ) î02 ) 0.1, U (pink) Rc,1 ) 0.49 and Rc,1 ) 0.51, (green) Rc,1 ) 0.45 and Rc,1 ) 0.55, (black) Rc,1 ) 0.3 and Rc,1 ) 0.7, (red) Rc,1 ) 0.1 and Rc,1 ) 0.9. The lines above ˆfE,1 ) 0 correspond to Rc,1 > 0.5 and those below correspond to Rc,1 < 0.5.

It is important to note that even for very small deviations from symmetry, there is a significant change in fE,1 as seen from the curves corresponding to Rc,1 ) 0.49 and 0.51. This means that when two reactions have nearly equal Nernst potentials, even a small difference in their transfer coefficients is significant. This is not so when the difference between the Nernst potentials is large. This is seen from Figure 15, which plots fE,1 versus φˆ ∞ ˆ . In this case Rc,1 ) 0.7. If we compare for different values of ∆U this figure with Figure 8, we see that spacing between the curves in Figure 15 is much smaller than that between those in Figure 8. This means that increasing Rc,1 above 0.5 is equivalent to ˆ . The reverse is expected if we increasing the magnitude of ∆U reduce Rc,1 below 0.5. We now estimate fE,1 using eq 68, in which the current density contributions of reaction 1 at the poles are estimated using eq ˆ m by φˆ e in these equations. The limiting 65 and 66. We replace U values of φˆ e and φˆ m are obtained using eq 69 and 70. The estimated values are plotted as points in Figure 15. The agreement between the approximate estimates and the results of the simulation is very good, above φˆ ∞ ) 1000, but acceptable

φˆ îk(θ) ) îk π + ∞ cos θ 2 γk

()

(72)

where î k(π/2) is given by eq 36. We see that eq 68 differs from eq 35, by the factor 3γ/(2γ + 1) in the second term. The two will match only when γ ) 1. Nonlinear Regime: Irreversible Reactions. All reactions considered previously are governed by the Butler-Volmer equation, which incorporates both the forward and the reverse reaction steps. On the bipolar electrode, such reactions always occur in a spontaneous manner, that is, the reaction with a higher reduction potential occurs at the cathode, and that with lower reduction potential occurs at the anode. It is not possible to drive the redox couple in the nonspontaneous direction. In practice, we often encounter cases where the electrochemical reactions are irreversible, that is, where only one branch of the Butler-Volmer equation is applicable for the each reaction. This happens in the cases where the products of the electrode reactions are either absent in the bulk, or are removed from the bulk solution either by a homogeneous reaction or by desorption. Thus we can encounter cases where one of the reactions can occur only in the cathodic direction and the other only in the anodic direction. Such reactions can couple on a bipolar electrode, even when the couple is nonspontaneous as shown below. The equations governing the current density of the cathodic and the anodic irreversible reactions are respectively

9452


î1 ) î0,1 exp[-Rc,1n1(φˆ e - φˆ s - U ˆ 1)]

(73)

ˆ 2)] î2 ) -iˆ0,2 exp[Ra,2n2(φˆ e - φˆ s - U

(74)

ˆ 2 < 0, that ˆ )U ˆ1 - U We consider a redox couple for which ∆U is, where the overall reaction is nonspontaneous. Figure 16 ˆ ) shows the surface potential profile on the electrode for ∆U -100. Comparison of this figure with Figure 3 (spontaneous case) shows that the for the same value of φˆ ∞, the electrode potential at the pole is much higher for the present case. The higher potential is caused by nonspontaneity of the reaction. We can illustrate this by considering high potential asymptote. We can write the following equations for this case, which are analogous to eq 69 and 70: î0,1 exp[-Rc,1n1(φˆ e - φˆ m - U ˆ 1)] ˆ 2)] ) 3φˆ ∞ (75) î0,2 exp[Ra,2n2(φˆ e - φˆ m - U î0,1 exp[-Rc,1n1(φˆ e + φˆ m - U ˆ 1)] ˆ 2)] ) -3φˆ ∞ (76) î0,2 exp[Ra,2n2(φˆ e + φˆ m - U

Figure 17. Surface current density profile parameters: Rc,1 ) Ra,2 ) 0.5, ˆ 1 ) 0, U ˆ 2 ) 100, î01 ) î02 ) 0.1. (Blue) φˆ ∞ ) 10, (pink) φˆ ∞ ) 100, (red) U φˆ ∞ ) 1000, (green) φˆ ∞ ) 10000.

For the present case, these equations have the following solution

[

(

φˆ m ) 2 sinh-1 15φˆ ∞ exp -

ˆ1 - U ˆ2 U 4

)]

≈ 50 + 2 ln(15φ∞) (77)

For φˆ ∞ ) 10000, the value of the potential at the pole is φˆ m ) 73 · 83, which matches with that in Figure 16. The plot of current density is shown in Figure 17, which can be compared with Figure 4. It is seen that no reaction occurs on the electrode at low imposed potentials (say φˆ ∞ ) 10). Even at higher imposed potentials, no reaction occurs over a region near the equator. The surface potential in this region is too small to drive the reaction. The values of the current and the reaction enhancement factors are plotted in Figures 18 and 19, respectively. Except for low values of φˆ ∞, the two plots are identical. At low values of φˆ ∞, the current enhancement factor reaches a constant value given by eq 55. It is seen from Figure 19 that the reaction enhancement factor is zero until a particular threshold value of φˆ ∞ is reached. This

Figure 18. Current enhancement factor. Parameters: Rc,1 ) Ra,2 ) ˆ 1 ) 0. (Green) ∆U ˆ ) -1, (yellow) ∆U ˆ ) -5, 0.5, î 01 ) î 02 ) 0.1, U ˆ ) -10, (pink) ∆U ˆ ) -20, (blue) ∆U ˆ ) -50, (olive) ∆U ˆ ) (red) ∆U ˆ ) -100. -80, (black) ∆U

is also due to the nonspontaneity of the reaction. As expected, ˆ. the effect is more pronounced at higher magnitudes of ∆U Effect of the Shape of the Bipolar Electrode. The previous development was based on spherical shape of bipolar electrode. In this section, we consider electrodes with a nonspherical shape. Specifically we consider a spheroid obtained by rotating an ellipse about one of its axes. The equation of the ellipse in dimensionless cylindrical coordinates can be written as zˆ2 Fˆ 2 + 2 )1 2 a b

ˆ 1 ) 0, Figure 16. Surface potential profile parameters: Rc,1 ) Ra,2 ) 0.5, U ˆ 2 ) 100, î01 ) î02 ) 0.1. (Black) φˆ ∞ ) 1, (blue) φˆ ∞ ) 10, (pink) φˆ ∞ ) U 100, (red) φˆ ∞ ) 1000, (green) φˆ ∞ ) 1000.

(78)

The spheroid is obtained by revolving the ellipse about z-axis. The different shapes of the spheroids are obtained by varying the ratio of a and b. Thus for large a/b the shape of the spheroid resembles a spindle and for small a/b it resembles a disk. To facilitate comparison between the spheroid and the sphere considered previously, we constrain the spheroid to have the same volume as that of the unit sphere. The volume of the spheroid is 4/3πa2b and that of the unit sphere is 4/3π. Equating them, we get a2b ) 1. The following values of a are used


Figure 19. Reaction enhancement factor. Parameters: Rc,1 ) Ra,2 ) 0.5, î01 ˆ 1 ) 0. (Green) ∆U ˆ ) -1, (yellow) ∆U ˆ ) -5, (red) ∆U ˆ ) ) î02 ) 0.1, U ˆ ) -20, (blue) ∆U ˆ ) -50, (olive) ∆U ˆ ) -80, (black) -10, (pink) ∆U ˆ ) -100. ∆U

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Figure 21. Effect of the shape on surface potential profile. Parameters: î01 ˆ 1 ) 10, U ˆ 2 ) 20, Rc,1 ) 0.5, Rc,2 ) 0.5. (Black) a/b ) 8, ) î02 ) 0.1, U (red) a/b ) 2/2, (olive) a/b ) 1, (green) a/b ) 1/22, (blue) a/b ) 1/8.

Figure 20. Shapes of the bipolar electrodes. The figures are the projections of the electrodes on F z-plane. The electrodes are spheroids obtained by revolving these curves about z-axis. (i) a/b ) 8, (ii) a/b ) 2/2, (iii) a/b ) 1, (iv) a/b ) 1/22, (v) a/b ) 1/8. The direction of the electric field is shown by the arrow.

2, 2 , 1, 1/2, 1/2, which results in the following set of values of a/b: 8, 22, 1, 1/22, 1/8. The sphere is represented by a/b ) 1. The projected shapes of the spheroids are shown in Figure 20. Comparison is done for the symmetric case with the following ˆ 1 ) 10,U ˆ 2 ) 20, and î 01 ) î 02 ) set of parameters: φˆ ∞ ) 100, U 0.1. Figure 21 compares the surface potential profiles. It is seen from the figure that for a spindle shaped electrode (a/b ) 8), the potential variation is linear except near the poles where the potential rises sharply. The reason for this is clear from the shape and orientation of the electrode shown Figure 20i. We see that a large part of the surface of the electrode is parallel to the direction of the field and hence potential variation is linear on this surface. The poles, having a high curvature, act as current attractors. Potential variation is therefore very high in the polar regions. In the case of the disk shaped electrode (a/b ) 1/8), a major part of the surface lies in the direction perpendicular to the field as seen from Figure 20v. Hence the potential on this part of the surface is constant. Only at the equator, a large variation in the potential occurs, again due to high surface curvature. The potential profiles of the rest of the shapes lie intermediate between these two extremes. Figure 22 compares the limiting current density profiles on the electrode surface. The current is normalized with respect to iˆ∞. The quantity î s/iˆ∞ is purely a function of geometry of the electrode. For the sphere it is a straight line, as given by eq 60, with a maximum î s/iˆ∞ ) 3.

Figure 22. Limiting current density profiles. (Black) a/b ) 8, (red) a/b ) 2/2, (olive) a/b ) 1, (green) a/b ) 1/22, (blue) a/b ) 1/8. (Limiting current density profile for a/b ) 8 is truncated in the main figure, but depicted fully in the inset.)

As the ratio a/b increases, the limiting current density near equator becomes progressively lower, but rises near the poles, the rise becoming sharper with increase in the ratio, a/b. For a/b ) 8, the current density at the poles goes much beyond the range of the scale of Figure 22. The complete profile is therefore depicted in the inset. The reason for the sharp rise of the current density at the poles is that the poles have convex surface and they face the feeder electrodes. The field lines originating from the feeder electrode converge at the poles. The greater the curvature of the pole is, smaller is the area on which the field lines converge and the higher is the current density. On the other hand, the equatorial region of the electrode has low curvature. Moreover, it is parallel to the direction of the field. Both these factors result in a lower current density in this region. Behavior of electrodes with a/b ) 1/8 is exactly opposite to that of the electrode with a/b ) 8. There is a sharp rise in current density in the equatorial region due to the convexity of the surface, but there is no variation in the polar region, because the surface is nearly plane and is perpendicular to the field. The

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ˆ1 ) Figure 23. Current enhancement factor. Parameters: î01 ) î02 ) 0.1, U ˆ 2 ) 20, Rc,1 ) 0.5, Rc,2 ) 0.5. (Black) a/b ) 8, (red) a/b ) 2/2, 10, U (olive) a/b ) 1, (green) a/b ) 1/22, (blue) a/b ) 1/8.

These observations also indicate that apart from the magnitude of the current density, the nature of the current distribution on the electrode is also an important factor in deciding the effectiveness of the bipolar electrode. The effectiveness of the electrode is greater if the current density is more concentrated near the poles compared to when it is more concentrated at the equator. The results derived in this section are valid only when the orientation of the electrodes is fixed with respect to the electric field as shown in Figure 20. When electrodes are suspended in the electrolyte, they can assume any random orientation. For this case, the correct way is to conduct simulation for all orientations of the electrode and average the result on the orientation probability space. The results which we have obtained, therefore, are not strictly valid for suspended electrodes. We can, however, draw the following qualitative conclusion from the present results. We can say for example, that the spindle shaped electrode (a/b ) 8), when lying with its axis in the xy-plane will be nearly as effective as the disk (a/b ) 1/8). The disk shaped electrode (a/b ) 1/8), on the other hand, would be similar in effect to the spindle when it is oriented with its axis perpendicular to z-axis. We therefore expect that both of these electrodes will yield nearly the same average current enhancement and the reaction enhancement factors. We can also conjecture on the basis of Figures 23 and 24 that average enhancement factors for long and slim electrodes (spindle and disk) would be substantially higher than those for the sphere having the same volume. Illustrative Example. In this section, we cite an example, which illustrates use of the theory developed in the previous sections. The data used in this example are approximate and hence the results have only a qualitative significance. The example pertains to production of sodium hypochlorite from dilute solutions of sodium chloride. Hypochlorite ion (OCl-) is a powerful disinfectant, which works efficiently even at low concentrations. It can be produced by electrolysis of aqueous solutions of sodium chloride. The following are the main reactions which occur during the electrochemical production of hypochlorite ions 2H2O + 2e- f 2OH- + H2 (cathode)

ˆ1 ) Figure 24. Reaction enhancement factor. Parameters: î01 ) î02 ) 0.1, U ˆ 2 ) 20, Rc,1 ) 0.5, Rc,2 ) 0.5. (Black) a/b ) 8, (red) a/b ) 2/2, 10, U (olive) a/b ) 1, (green) a/b ) 1/22, (blue) a/b ) 1/8.

current density in the equatorial region of the electrode with a/b ) 1/8 is, however, not as large as that at the poles of the electrodes with a/b ) 8. The reason is that, in the former case, the convex equatorial surface of the electrode is parallel to the direction of the field and is therefore not as effective in concentrating the field lines as the electrode with a/b ) 8 whose polar surfaces are perpendicular to the field. Figure 23 and 24, respectively, show the current enhancement factor and the reaction enhancement factors associated with the electrode shapes. We see that both fE and fE,1 increase with increase in the ratio a/b. This is expected, based on the trends of the limiting current density of Figure 22. The most important observation is that both the current and the reaction enhancement factors are very high for a/b ) 8, which corresponds to a spindle-shaped electrode. The reason is 2-fold. First, the current density is very high at the poles of this electrode. Second, the cathodic and anodic regions are now well separated. The overlap between them is the least. Exactly opposite situation exists for a/b ) 1/8 and this results in lowest values for both fE and fE,1.

2Cl- f Cl2 + 2e- (anode) Cl2 + 2OH- f Cl- + OCl- + H2O (solution) The standard reduction potentials for the anodic and the cathodic steps are respectively, U1 ) -0.828 V and U2 ) 1.359 V. The overall reaction is nonspontaneous (standard cell potential ) -2.187 V). Since Cl2 and OH- ions are consumed by the homogeneous reaction in the solution, these reaction steps can be considered irreversible. We consider the case of production of sodium hypochlorite at a temperature of 25 °C from 0.1 M NaCl solution (κ ) 10.6 mS · cm-1), on platinum-plated bipolar spheres of 1 mm radius. The reactor is a fluidized bed. We consider a feed current density of 200 mA · cm-2. The Butler-Volmer equations for the cathodic reaction can be written as17

(

i1 (A · m-2) ) 0.9272 exp -0.3060

ηF RT

)

(79)

and that for the anodic reaction is given by18

(

i2 (A · m-2) ) -2030 exp 0.1807

ηF RT

)

(80)


where η is the overpotential () φe - φs - U). The dimensionless ˆ 2 ) 53.68. Hence ˆ 1 ) -32.73 and U equilibrium potentials are U ˆ ) -86.41. The Butler-Volmer equations in the dimension∆U less form are î1 ) 0.03425 exp[-0.3060(φˆ e - φˆ s + 32.73)]

(81)

î2 ) -74.94 exp[0.1807(φˆ e - φˆ s - 53.68)]

(82)

The magnitude of the electric field is, i∞/κ ) 1887 V/m. This yields the value of φˆ ∞ ) 74.6. This is the case of irreversible reactions discussed previously. The simulation yields fE,1 ) 0.841. The reaction current per particle is given by πa2i∞ fE,1 ) π(0.1)2 × 0.2 × 0.841 ) 5.28 × 10-3A. If we use a fluidized bed of 0.3 m height and 0.1 m diameter and if the particle holdup in the bed is 25% v/v, the number of bipolar spheres in the reactor is 1.407 × 105. Hence the total reaction current in the bed is 5.28 × 10-3 × 1.407 × 105 ) 744 A. This corresponds to the production rate of sodium hypochlorite ) 744/(2 × 96487) ) 3.86 mmol · s-1 ) 1.035 kg · h-1. The total potential drop between the feeder electrodes (which are spaced 0.3 m apart) is 1887 × 0.3 ) 566 V and the power consumption of the reactor is 0.2 × (π/4)(10)2 × 566 ) 8.89 × 103 W (8.89 kW). The power consumption per unit weight of sodium hypochlorite is 8.59 kWh · kg-1. Apart from the approximate rate expressions used here, we also neglect the effect of interaction among the electrodes. Moreover, we neglect possible side reactions which can reduce the current efficiency of the process.

reaction is spontaneous. The overall reaction can be nonspontaneous only when the coupling reactions follow irreversible kinetics. (7) Shape of the electrode and its orientation with respect to the direction of the field are important in determining effectiveness of the electrode. Thus a spindle-shaped electrode oriented along the field is far more effective than a disk-shaped electrode. (8) Distribution of the current density on the electrode surface plays an important role in bipolar electrolysis. It is preferable to have a larger current density at the poles than that at the equator, since the overlap of the cathodic and the anodic regions is the least when they occur at opposite poles. We therefore expect greater values of the reaction enhancement factor for this case compared to the case in which the current is more concentrated near the equator. The present study would be useful for selecting a suitable coupling reaction for the desired process. It would also be useful for designing a system involving bipolar electrolysis. Acknowledgment Authors wish to acknowledge Unilever Industries Private Limited for providing the funds for this research. Appendix Approximate Equation for Estimation of Reaction Enhancement Factor. We begin with the definition of the reaction enhancement factor as given by fE,1 )

Conclusions The following conclusions can be drawn from the present study. (1) There is an upper limit to the current density that could be attained on the bipolar electrode. This limit arises due to the finite rate of transport of the current through the solution surrounding the electrode and not due to the kinetics of electrochemical reactions. It is dependent on the geometry of the electrode. For spherical shape, it is 3iˆ∞ (Yen and Yao12 also reported this upper limit). For other shapes, it is substantially different as shown in Figure 22. (2) The aim of the bipolar electrolysis is to couple two redox reactions. For coupling between them to occur, the electrolysis should be conducted under nonlinear regime of the Butler-Volmer kinetics (i.e., surface potential at the poles should be high). A certain critical value of φˆ ∞ is needed before this linear limit is to be exceeded. This is also true for the irreversible kinetics. ˆ , very fast kinetics of (3) For a given value of φˆ ∞ and ∆U electrochemical reactions (very small γ) reduces the potential at the electrode surface and consequently the extent of coupling. Hence the kinetics of electrochemical reactions should be moderately fast. (4) The greater the difference is between the Nernst potentials of the coupling reactions, the greater is the extent of coupling between them. On the other hand, a large difference between the exchange current density reduces the coupling. (5) When the difference between the Nernst potentials of the two reactions is small, deviation of the transfer coefficient from 0.5 significantly enhances the coupling. The effect of the transfer coefficients, however, reduces when the difference between the Nernst potentials increases. (6) When Butler-Volmer kinetics governs the electrode reactions, they couple on the bipolar electrode so that the overall

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2 ˆφ∞

∫

π

0

(iˆ1 - î1|φ∞)0) sin θ dθ

(A-1)

We consider two regions on the surface of the sphere. As shown in Figure A-1. The region near the equator has a lower surface potential and hence the linear regime is valid in this region. fE,1 ) 0 in this region. The rest of the region is nonlinear. Hence we can modify eq A-1 as follows. fE,1 )

2 [ ˆφ∞

∫

θ0

0

(iˆ1 - î1|φ∞)0) sin θ dθ +

∫

π

π-θ0

(iˆ1 - î1|φ∞)0) sin θ dθ]

(A-2)

The first term on the right corresponds to the upper hemisphere. Here, we assume that the î 1 ) î up,1 cos θ, which is the limiting value. The second term corresponds to the lower hemisphere and we assume î 1 ) î lp,1 cos(π - θ). We also note that î 1|φ∞)0 is independent of θ. Hence eq A-2 reduces to

Figure A-1. Schematic diagram of a bipolar electrode.

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fE,1 )

1 [(1 - cos2 θ0){iûp,1 + îlp,1} φˆ ∞ 4(iˆ1|φ∞)0){1 - cos θ0}]

(A-3)

From eq 33 we know that in the linear region, the surface potential profile is given by φˆ s 3γ ) cos θ ˆφ∞ 2γ + 1

(

)

(A-4)

We obtain the linear limit of cos θ by assuming that linear region extends up to φˆ s ) φˆ m. Hence cos θ0 )

φˆ m 2γ + 1 φˆ ∞ 3γ

(

)

(A-5)

Combining (A-3) and (A-5), we obtain fE,1 )

[( ( ) (

φˆ m 1 1φˆ ∞ φˆ ∞

2

2γ + 1 3γ

) ){iˆ 2

up

{

+ îlp} - 4(iˆ1|φ∞)0) ×

1-

φˆ m 2γ + 1 φˆ ∞ 3γ

(

)}

]

(A-6)

Literature Cited (1) Pletcher, D. Industrial Electrochemistry; Chapmann and Hall: London, 1982; p 162. (2) Fleischmann, M.; Ghorogchian, J.; Pons, S. Electrochemical Behavior of Dispersions of Spherical Ultramicroelectrodes. 1. Theoretical Considerations. J. Phys. Chem. 1985, 89, 5530. (3) Nadebaum, P. R.; Fahidy, T. Z. Continuous Activation in a Rotating Bipolar Electrode Cell-I via Control of the Surface Oxide Thickness. Electrochim. Acta 1975, 20, 715. (4) Fleischmann, M.; Ghorogchian, J.; Rolison, D.; Pons, S. Electrochemical Behavior of Dispersions of Spherical Ultramicroelectrodes. J. Phys. Chem. 1986, 90, 6392.

(5) Bradley, J.-C.; Crawford, J.; Ernazarova, K.; McGee, M.; Stephens, S. G. Wire Formation on Circuit Boards using Spatially Coupled Bipolar Electro-Chemistry. AdV. Mater. 1997, 9, 1168. (6) Bradley, J.-C.; Crawford, J.; McGee, M.; Stephens, S. G. A Contactless Method for the Directed Formation of Submicrometer Copper Wires. J. Electrochem. Soc. 1998, 145, L45. (7) Bradley, J. C.; Ma, Z.; Clark, E.; Crawford, J.; Stephens, S. G. Programmable Hardwiring of Circuitry using Spatially Coupled Bipolar Electro-Chemistry. J. Electrochem. Soc. 1999, 146, 194. (8) Eardley, D. C.; Handley, D.; Andrew, S. P. S. Bipolar Electrolysis with Intra-Phase Conduction in Two Phase Media. Electrochim. Acta 1973, 18, 839. (9) Goodridge, F.; King, C. J. H.; Wright, A. R. Performance Studies on a Bipolar Fluidised Bed Electrode. Electrochim. Acta 1977, 22, 1087. (10) Plimley, R. E.; Wright, A. R. A Bipolar Mechanism for Charge Transfer in a Fluidized Bed Electrode. Chem. Eng. Sci. 1984, 39, 395. (11) Goodridge, F.; King, C. J. H.; Wright, A. R. The Behavior of Bipolar Packed-Bed Electrodes. Electrochim. Acta 1977, 22, 347. (12) Yen, S. C.; Yao, C. Y. The Bipolar Analysis of a Single Sphere in an Electrolytic Cell. J. Electrochem. Soc. 1991, 138, 2697. (13) Keh, H. J.; Li, W. J. Interactions among Bipolar Spheres in an Electrolytic Cell. J. Electrochem. Soc. 1994, 141, 3103. (14) Duval, J.; Kliejn, J. M.; Van Leeuwan, H. P. Bipolar Electrode Behavior of the Aluminium Surface in a Lateral Electric Field. J. Electroanal. Chem. 2001, 505, 1. (15) Gray, D.; Cahill, A. Theoretical Analysis of Mixed Potentials. J. Electrochem. Soc. 1969, 11, 443. (16) Griffiths, D. J. Introduction to Electrodynamics, 3rd ed.; PrenticeHall: New Delhi, India, 2000; p 141. (17) Angelo, A. C. D. Electrocatalysis of Hydrogen Evolution Reaction on Pt Electrode Surface-Modified by S-2 Chemisorption. Int. J. Hydrogen Energy 2007, 32, 542–547. (18) Kuhn, A. T.; Mortimer, C. J. The Kinetics of Chlorine Evaluation and Reduction of Titanium-Supported Metal Oxides: Especially RuO2 and IrO2. J. Electrochem. Soc.: Electrochem. Sci. Technol. 1973, 120, 231.

ReceiVed for reView March 17, 2009 ReVised manuscript receiVed June 26, 2009 Accepted June 30, 2009 IE900437N

Analysis of Multiple Reactions on a Bipolar Electrode - American

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