~ . . - . . ~" . ~:.,-~ i r"

ELSEVIER Journal of Electroanalytical Chemistry 422 (1997) 161-167

dOogl~10. OF

Analytical solution for the steady-state diffusion and migration. Application to the identification of Butler-Volmer

electrode reaction parameters

L. Bortels, B. Van den Bossche, J. Deconinck Department of Electrical Engineering, Vrije Universiteit Brussel, Pieinlaan 2, B-lOS0 Brussels, Belgium

Received 24 April 1996; revised I 1 July 1996

Abstract

An analytical solution for the one-dimensional steady-state transport of ions in an electrolyte between two planar electrodes has been obtained. This electrolyte contains one electroactive species and any number of non-reacting species. The mass and charge transport equations give rise to an implicit form of a set of non-linear algebr~c equations which must be solved numerically. It has been shown that the same set of equations, with only a very small modification, can easily be used to solve the stagnant boundary layer problem. The solution is generally applicable and can deal with any kind of overpotential relation at both anode and cathode.

The analytical solution for the stagnant boundary layer has been used to determine the diffusion coefficient for the reacting ion and the kinetic parameters in the Buder-Volmer overpotential relation for the electrodeposition of copper from a 0.01 M CuSO4 + 0.1 M H2SO 4 solution. The resulting parameters are in good agreement with the values found in the literature. Analytical results obtained with these parameters match very well with the experimental data for current densities ranging from secondary up to limiting current values and for different values of the rotation speed (100, 500 and 1000revmin-t).

Also, it has been shown that neglecting migration can lead to an overestimation of the diffusion coefficient of about 15%.

Keywords: Transport: Planar electrodes; Stagnant boundary layer; Analytical solution: Kinetics

1. Intr~lueficn

The equations describing mass and charge transport in dilute electrochemical systems due to diffusion, convection and migration constitute a set of coupled non-linear differ- ential equations [1,2]. Treatment of the original problem without any simplifications is quite complicated because of the complexity of the governing equations and the strong non-linear boundary conditions which relate the potential and the concentration of the reacting ion(s) with the poten- tial at the electrodes. A substantial number of papers has been dedicated to the analytical determination of concen- tration, potential and current density distributions in one- dimensional geometries. An overview of this work can be found in an interestit~g paper recently published by Sokirko and Bark [3]. In that paper, an exact explicit formula for the polarization curve of a binary electrolyte has been given, together with an approximate formula for a system with a large surplus of supporting electrolyte. Neverthe- less, the authors used a solution procedure that depended

strongly on the problem to be dealt with. Extension to a more general approach is not straightforward.

A more general approach to finding analytic solutions in one-dimensional cells has been worked out by Pritzker [4]. In that work, an analytical solution was found for the steady-state diffusion and migration in the stagnant bound- ary layer near a planar electrode. The method is applicable to conditions of mixed transport-kinetic control as well as to limiting mass transport conditions and does not depend on the particular form of the kinetic rate law.

The work presented here combines the more general approach adopted by Pritzker [4] with the complexity of the two electrode problem treated by Sokirko and Bark [3] and is applicable to any number of ions and any form of kinetic control law. Furthermore, it will be shown that the set of equations obtained can be used to tackle the stagnant boundary layer problem.

The analytical solution for the stagnant boundary layer has been used to determine the diffusion coefficient for the reacting ion and the kinetic parameters in the Butler-

0022-0728/97/¢17.00 Copyright © 1997 Elsevier Science S.A. All rights reserved. Pll S0022-0728(96)04864-4

162 L Bortels et ai. / Journal of Electroanalytical Chemistr)" 422 (1997) 161-167

Volmer overpotential relation for the electrodeposition of copper from a 0.01 M CuSO4 + 0. I M H2 SO4 solution. Analytical and experimental data match very well for current densities ranging from secondary up to limiting current values and this for three different values of the rotation speed (100, 500 and 1900 rev min- ~ ).

2. Problem formulation

2.1. The two electrode problem

An analytical solution for the one-dimensional steady- state transport of ions in an electrolyte between two planar electrodes has been obtained. This electrolyte contains one electroactive species Me:, + which participates in the reac- tion

Me :'+ + zle- ~ Me (1)

and any number of non-reacting species. The metal ion is deposited at the cathode (x = 0) and the metal dissolves at the anode (x = L).

Suppose the reacting ion has concentration c 1. In the case of a dilute electrolyte, the Nernst-Einstein relation for the mobility is valid. The flux N ! of the reacting ion in the positive x-direction due to diffusion and migration can therefore be written as

z lDtF dU dc 1 N, ffi R----i-c1 - 01 d x (2)

For the non-reacting ions, numbered from 2 to I, with I being the total number of ions involved, there is no net flux, resulting in

z~D kF dU dc k RT c k ~ x + O k " ~ x = 0 f o r k = 2 , 3 . . . . . I (3)

with c k the concentration, zk the charge, U the potential in the solution, D k the diffusion coefficient, and F Faraday's constant.

The mass transport Eqs. (2) and (3) and the electroneu- trality condition constitute a complex set of equations in c I, c., . . . . . c I and U which must first be decoupled in order to be solved. This can be done by obtaining explicit expressions for c I, c 2 . . . . . c t in terms of U and then using these to derive one single differential equation for U [4].

Summation of Eqs. (2) and (3), multiplied by z~ and Zk respectively, yields

dU i dU N I zlklCl ~ x + E zkki, Ck dx = - Z t - - (4)

kffi2 Dt

with k~ -- (z~ F /RT) , or after grouping the terms in the U derivative together:

z ,k , c , + - z, o , (5 )

The concentration of the reacting ion can be eliminated from the above equation by introducing the electroneutral- ity equation:

! dU N t E (kk - -k l ) zkck = - - Z l - - (6)

t,=2 dr D I

Elimination of the concentration c k of the non-reacting ion can be done after integration of Eq. (3) from x -- 0 to L:

ck = AkC 'U for k = 2, 3 . . . . . I (7)

with A~ integration constants. Combination of Eqs. (6) and (7) results in a first-order differential equation for the determination of the unknown potential U:

= - z i - - ( 8 ) ( k k - k t ) z k A ~ e k ' V dx Dj k=2

Finally, after integration of Eq. (8), and introduction of the integration constant K, the following non-linear equation is found:

t k l _ k k Ni ~ k ~ Z k A k e - k ' u _ m + K

k= 2 ~- Z ! DI x

or (9)

' NI ( zl - zk) Ake -k't' = + K

kffi2 DI

which is an implicit solution of U(x). However, A~ (k = 2 to 1), N~ and K are still unknown

and have to be determined by introduction of the proper boundary conditions. Therefore, it is necessary to express the relationship between N I, and c~ and U at the bound- aries of the problem. This relation is given by the current density of the reacting ion at the metal[solution interface and is. in its most general expression, a function of the local concentration c !, the solution potential U and the metal potential V. The current density at the cathode and anode respectively (defined here as being the current den- sity along the inward normal of the domain) are related to the flux N l of the reacting ion as follows:

ziFNI (10)

z, FNI=ja(c , (L ) ,U(L) ,Va) (11)

Here c1(0), U(0) and V~ are respectively the concentration of the reacting ion at the cathode, the potential of the solution at the cathode and the external potential imposed at the cathode, cl(L), U(L) and V~ have exactly the same meaning for the values at the anode. From Eqs. (9)-(11), it can be remarked that the original problem is now reduced to the determination of 1 + 5 unknowns: cI(0), Cl(L), U(0), U(L), A k (k = 2 to I), N l and K, with the first two relations between these unknowns given by Eqs. (10) and (ll).

L Bortels et aL/Journal ofEiectroanalytical Chemisto, 422 (1997) 161-167 163

As the reactions at the anode and cathode, by assump- tion, proceed at the same rate, the conservation conditions for the I ions can be expressed as [3]

oLckdx = ckbL for k= 1 / (12)

with Ckb the initial concentrations of the species that have to obey the condition of electroneutrality. Eq. (12) can be rewritten in a more convenient way as follows:

folk dx fV(/.) dx -- c, "77.. dU (13) " U(O) u u

which, after introduction of F.q. (6), results in

1 rV(L) [ D, E (k,,,-k,)ZmC,,, dU=c*t,L

for k - - l . . . . . t (14)

In order to be able to compute the integrals in F_Xl. (14), the concentration of the non-reacting ions are written as a function of the unknown potential U:

D! ! A I rV(L)e-(*m+k')u ZlNI E (k , , , - k , )A . , dU= t m = 2 k.~ U(O) Ck b

for k = 2 . . . . . t (15)

A minor computation gives that !

z i N I C , b L = D l ~.. Z m a , . A k F k m ( 1 6 ) m ~ 2

with

Fk, . = ~Zm - - Z I [ e - ( k , , , + k,)U(L) _ e - ( k m + / q ) V ( O ) ]

Zm + Zk

if z,,, + zk ~ 0 (17)

F k , . = - ( k , , , - k l ) ( U ( L ) - U ( O ) ) i f z , . + z k = O (18)

Introduction of the boundary conditions for the current density at the cathode (x = O) and the anode (x = L) results in two new relations between the unknowns:

!

E ( z, - z,) A,e -k'u(°) = K (19) k=2

I zlNi L E ( z, - z,) A~e -*'u(t, = + K (20)

k--2 DI

A formulation for the concentration distribution of the reacting ion as a function of the potential is found by combination of Eq. (7) with the electroneutrality equation:

c,=- ~, Z*A,e-*'v (21) k=2 Zl

Therefore, the final two relations between the unknowns

are obtained just by substituting (x = O) and (x = L) in Eq. (21):

I Zk _ c,(O) = - E -'~i A* e k,u,o,

k=2

c,( L) = - ~, Z* A,e-*,v(t) k=2 El

(22)

(23)

As a result, the following 1 + 5 equations describe in a unique way the I + 5 relations between the unknowns cl(0), cl(L), U(0), U(L), Ak (k = 2 to 1), N: and K of the problem:

I

Y'. ( zl -- zl,) Ake -k'u(°'- K = 0 (24) k=2

I ziNi L Y'~ (Zi--z,)Ake-k'V(L)+ K = 0 (25)

k=2 Dj

! Zk c,(0) + Y'~ T-A,.e-*,v'°)=0 (26)

k=2 4"i

i c,(L) + Y'~ z--~-* A,e- t"v( t )=o (27)

k=2 ZI

!

zlNic, bL - D I ~.. z,,,A,.AkFk. , = 0 ( I - 1 equations) m=2

z, FN, -Jc(c,(O).U(O).V~ ) = 0

z, FN, +A(c,( L).U( L).V~) =0

(28)

(29)

(30)

Eqs. (24)-(30) constitute a set of / + 5 non-linear equa- tions which can be written in a vector formulation:

{ 1/'(X)} = 0 (31)

The unknowns X in Eq. (31) are determined using a standard Newton-Raphson method. Hence, the following set of linear equations has to be solved:

~( X"){AX} = - { qt( X")} (32)

with ~(X n) the Jacobian of the problem and {AX"} the update of the solution vector {X n} found at iteration n given by

i ~ ( X ) ] (33) f f

{ A X} = { X "+ t} - { X"} (34)

The determination of the Jacobian in Eq. (33) is straightforward and involves the computation of the derivatives of the current density at cathode and anode with respect to the unknowns q(0), q(L), U(O) and U(L).

It is very useful to compute first the limiting current density of the problem to have an idea of the maximum applied voltage difference that has to be imposed between cathode and anode. Suppose therefore that the metal poten-

164 L Borteis et aL / Journal of Electroanalytical Chetnistry 422 (1997) 161-167

tial at the cathode V c is fixed and that the potential at the anode V~ is variable. The limiting current density is reached if the concentration of the reacting ion at the cathode drops below a certain fraction 6~ of the bulk concentration, thus

c,(0) -- 8,C,b (35)

The limiting current is then a result of solving Eq. (32) by introduction of the unknown anode potential V a instead of cl(0), which is now known by virtue of Eq. (35), in the vector {AX"}. This implies, of course, the determination of the derivative of the anode current density with respect to tile unknown potential V a. If 6 ! is small enough (e.g. 10-6), the limiting current situation will certainly be reached.

The system of Eq. (32) can then be solved for the unknowns for different values of the anode voltage going from V c to the value obtained for V a. The resulting values of N l, K and A t (k = 2 to l ) are then substituted into Eq. (9), from which the solution potential U is determined implicitly for different values in the interval 0 __< x ~ L. Once these values are obtained, the electrolyte concentra- tions for all ions can be calculated using Eqs. (7) and (21) or the electroneutrality condition.

One can easily remark that the set of I + 5 equations described by Eq. (31) can be reduced to a set of only 1 + 3 equations by elimination of the unknowns N ! and K by a combination of Eqs. (24), (25), (29) and (30). Since the number of unknowns is not critical, it was decided to leave the set of equations in its original (and most natural) form.

2.2. The stagnant boundary layer problem

It is common practice in the electrochemical literature, to assume that the thickness L of the diffusion layer is known. Hence the anode at x = L can be replaced by a boundary where the concentration ct of each ion k is fixed as equal to the initial bulk concentration:

ct(L) =C~b for k - I . . . . . I (36)

The unknowns A t (k = 2 to 1) can therefore be related directly to the potential at the boundary layer, since, according to Eqs. (7) and (36), one has

Ak -- Ctb ektv~') for k = 2 . . . . . I (37)

As a result, exactly the same set of equations can be used to solve either the two electrode problem as described above, or the boundary layer problem as stated here. The only difference is that for the fh'st problem one has to use Eq. (28), whereas for the latter problem Eq. (37) has to be taken. In the boundary layer case, one can easily express that the potential U(L) at the boundary has to be equal to the imposed potential V~ by introducing the following current density relation at x = L:

J . - J o ( ~ - U ( L ) ) (38)

with Jo a constant. If Jo is chosen to be very large, say

10 m, then Eq. (38) describes in an implicit way that the potential U(L) in the solution at x = L is equal to the imposed potential V a just outside the diffusion layer.

2.3. The binary electrolyte

In its most general form, Eq. (9) provides an implicit solution only for the potential U(x). For each value of the ordinate x, a non-linear equation for the potential has to be solved. The corresponding concentrations of all ions for that particular value of x are then obtained with Eq. (7).

In the special case of a binary electrolyte (the number of ions 1 being equal to 2), however, combination of Eqs. (7) and (9) results in a linear expression for the concentra- tion of the non-reacting ion (and consequently also for ion 1):

z~N, g c2-- (z2 z,)O, x + (39)

- - Z l - - Z2

and a logarithmic expression for the potential:

1 ( z,N, K ) I n ( A 2 ) U - - ~ 2 1 n (Z 2 z,)D, x + ~ + ~ - z~ - z2 k2

(40)

Hence, the concentration and potential in each point of the domain are no longer an implicit function of the position but can directly be computed from Eqs. (39) and (40).

3. Application

For the determination of kinetic and transport properties of copper ions in a 0.01M CuSO 4 + 0.1M H2SO 4 solu- tion, voltammetric experiments have been performed using a rotating disk electrode (RDE) [5]. The counter electrode is a platinum grid and potentials on the RDE are referred to a Tacussel mercury electrode (HglHg2SO41K2SO4(sat)).

The reaction of copper is assumed to be a one-step reduction process. Hence, the current density can be ex- pressed by the following Butler-Volmer relationship:

e ZlFa, - zlFa, J=Jo -Tf-'~ c,(O) ~ , 1

Clb e (41)

in which the cathodic and anodic charge transfer coeffi- cients a c and a a and the exchange current density Jo have to be determined based on the experimental data and the analytical computations, a c and a a are related to each other through [1]

a c + a a --- I (42)

The overpotential ~ is referred to an equilibrium potential of - 3 9 4 mV (SHE) for bulk concentrations.

L Bortels et ai./Journal of Electroanalytical Chemistry. 422 (1997) 161-167 165

3.1. Determination of the diffusion coefficient of the cupric ion

The limiting current density at a rotating disk electrode in the absence of migration has been determined by New- man [1]. This expression can be used as a first guess for the computation of the diffusion coefficient of the reacting ion, using an experimental value for the limiting current density obtained from polarization curves:

- 0 . 6 2 zl FD2/3u- I /6~ I/2Clb J l im- - 1 + 0 . 2 9 8 ( - ~ ) 1/3 + 0.145(_.~)2/3 (43)

with u the kinematic viscosity and n the rotation speed. The second and third term in the denominator are New- man's corrections to the asymptotic value for the limiting current for large Schmidt numbers, first obtained by Levich. It should be mentioned that although the analytical method takes only diffusion and migration into account, and that in the RDE experiments convection naturally plays an impor- tant role, both can be cot pared by introducing a stationary diffusion layer thickness (e.g. the Nernst layer thickness [6]), so that convection is considered implicitly.

Experimental data have been obtained for 100, 500 and 1000 rev min- ~. At reasonably low cc, neentrations, H 2 SO4 dissociates mainly into H + and HSO~-. Since the concen- tration of copper sulphate is less than half of that of the sulphuric acid, the species considered are Cu 2+, H + and HSO 4 with bulk concentrations, diffusion coefficients and charge numbers presented in Table 1. The diffusion coeffi- cient for the Cu 2+ ion has to be determined from the experimental limiting current density using Eq. (43). For the H + and HSO 4 ions, the diffusion coefficients have been taken from the literature [1].

The determination of the mass transport and kinetic parameters of the electrolyte have been based on the

qt experimental data performed at 1000rev min- ~, since a ' larger potential zone with kinetic control is present. The

limiting current density at this rotation speed was found to be 92.0 A m - 2 . With the kinematic viscosity at 25°(2 equal tO 10 - 6 m 2 s - ! [5], th i s r e s u l t s i n a d i f f u s i o n c o e f f i c i e n t o f

the cupric ion equal to 6.78 × 10-1o m 2 s - l . The diffusion layer thickness L can then be computed from the experi- mental limiting current density using the formula

zlFDlClb (44) Jlim = L

which leads to L = 1 .42 × 10 -s m. It has to be remarked that Eqs. (43) and (44) are valid

only in the absence of migration, whereas the analytical model takes into account both diffusion and migration. As a result, the analytical value of the current density, based on the above-determined values of Dt and L, is equal to 101.2Am -2 instead of 92.0Am -2. In other words, about 9.1% of the analytical limiting current density is due to

Table I Charge numbers, bulk concentrations and diffusion coefficients for the ions involved in the analytical computations

Ion Ion Ion Bulk conc . / !0 I° diffusivity number name charge mol m 3 coefficient/m 2 s - J

! Cu 2 + + 2 I 0.0 to be determined from Eq. (43)

2 HSO~ - i l l0.0 13.3o[1l 3 H + + ! 90.0 93.12 [l]

migratio,~ of ions. Hence, it is reasonable to assume that the contribution of migration to the experimental limiting current density is also 9.1%, resulting in a current density due to diffusion of only 83.7 Am -2. From Eq. (43) the diffusion coefficient is found to be 5.86 × 10 -~° m 2 s -~ and the corresponding diffusion layer thickness is 1.35 x 10 - sm. The analytical limiting current density based on these latter values is equal to 92.2Am -2, which is very close to the experimental current density. One can con- clude that neglecting migration in a 0.01 M CuSO4 + 0. ! M H eSO4 solution leads to an overestimation of the diffusion coefficient of the cupric ion of about 15%.

The diffusion coefficient of the cupric ion reported by Awakura et al. [7] (with the same electrolyte solution and temperature, using the same RDE method) is equal to 6 .50x 10-~°mes- l . Newman [1] reported a value of D~ = 7.2 × 10 -!° m e s -~ for a infinite dilution of CuSO 4 in water. This value is lowered to D~ = 6.57 × 10 -m° m e s -~ by taking into account a correction of 5.0[Cu2+] j/2 introduced by Quickenden and Jiang [8] for non-zero concentrations of CuSO 4. Both reported values, not corrected for the migration contribution, are close to the non-corrected value (6.78 X 10-1°m2s -~) found in this work, indicating that the experimentally obtained lim- iting current density is reliable.

The aqueous solution of 0.01 M CuSO4 + 0.1 M H 2SO4 is clearly a system with a su~lus of supporting electrolyte, since only 9.1% of the current density in the diffusion boundary layer is carried by migration. However, in the bulk solution outside this boundary layer (not taken into account in the analytical solution), there are no concentra- tion gradients and migration has t o carry the to ta l current density. The potential distribution in this bulk region is then governed by Laplace's equation.

3.2. Determination of the electrode kinetics

The kinetic parameters in the Butler-Volmer overpoten- tial relation given by Eq. (41) are now determined by fitting the experimental data with the analytical computa- tions of the current density. The computations are done by imposing a metal potential V¢--0V at the cathode. At a

distance L (equal to 1.35 × 10 -5 m), bulk concentrations are imposed and the artificial potential in the solution is set equal to the potential V a by introducing an overpotential given by the relation in Eq. (38) with Jo equal to 10 ~°.

166 L. Bor:eis et al. / Journal of Electroanalyticai Chemistr)' 422 (1997) 161-167

0.0 . . . . I ' ' . . . . . I . . . . I ' ' ' ' ~

u -80.0

-100.0 . . . . ' . . . . ' . . . . t .... I . . . . 1

-0.5 -0.4 -0.3 -0.2 -0. I 0.0 Ovetpotential/V

Fig. 1. Comparison between the experimental current density at 1000revmin -I (<>) and the analytical current density for ot c equal to 0.15 ( . . . ) , 0.20 ( - - - ) , 0.25 ( ~ ) and the compromise value 0.175 ( ~ ) . The exchange current density Jo is 10.0Am -2.

The cathodic charge transfer coefficient a c determines the curvature of the Butler-Voimer overpotential relation and will be determined first. This parameter has been ranged between 0.15 and 0.25, i.e. between the minimum value found in the work of Milora et al. [9] and the maximum value presented by Bockris and Conway [10]. A comparison between the experimental and analytical cur- rent density as a function of the overpotential for a c equal to 0.15, 0.20 and 0.25 is presented in Fig. 1. Since the exchange current density Jo only shifts the curve (towards the right for increasing values) and does not alter the curvature of the current density with respect to the overpo- tential, this value has been taken equal to 10.0 A m -2 [9] for the moment. It can be seen that ot~ should lie between 0.15 and 0.20. For the first value, the analytical current density profile is not curved enough with respect to the experimental data; for the latter value of ot~, the computed curve is too curved. Therefore, the cathodic charge transfer coefficient a~ has been taken equal to 0.175, which gives

0.0 ' ' ' ' I . . . . t . . . . I

,~ -20.0 E

-40.0 ..~

-oo.o ~ . , . ,

~ - 8 o . 0 -

- 1 0 0 . 0 . . . . I . . . . i . . . . I . . . . I . . . .

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 Overpotential/V

Fig. 2. Comparison between the experimenta', current density at 1000revmin-I (<>) and the analytical current density for Jo equal to 15.0 ( ' ), 20.0 ( - - - ) and 25.0Am -z ( . . . ) . The cathodic charge transfer coefficient a c is 0.175

0.0

-20.0 ? E

-40.0

-60.0

r j -80.0

-100.0 -0.5

i 1 , i I i , , i [ ; i i ; I i i i . I q f I i

-0.4 -0.3 -0.2 -0.1 0.0 Overpotential/V

Fig. 3. Comparison between the experimental current density at 100 (<>), 500 (4) and 1000revmin -I (El), and the corresponding analytical current density ( ~ ) for a c = 0.175 and Jo = 20.0 A m- -'.

sufficient accuracy with respect to the curvature of both experimental and analytical values as can be concluded from the curves presented in Fig. 1.

With the above-determined value of ot c = 0.175, the exchange current density has been increased with steps of 5 A m - 2 to shift the analytical values towards the right and thus towards the experimental values. Results are pre- sented in Fig. 2 for Jo equal to 15, 20 and 2 5 A m -2. For Jo = 20 A m -2, very good agreement between the experi- mental and analytical data can be seen. A slight discrep- ancy at Dw overpotentiais is noticed, which may originate from the copper reduction being a two-step mechanism. At low overpotential, this second step (the reduction of cuprous ions) becomes rate determining [1,5].

The parameters obtained, ot c = 0.175 and Jo = 20 A m - e , are now used to compute the polarization curves for 100 and 500revmin - l . The experimental limiting cur- rent densities for these rotation speeds are equal to 30.3 and 65.2 A m -2, resulting in a diffusion layer thickness of 4.10 × 10 -s m and 1.9 × 10 -5 m respectively. The com- putations done for both values of L have been compared with the experimental data for the corresponding rotation speeds and are presented in Fig. 3 together with the already computed values for 1000revmin- i . Very good agreement between the experimental and analytical values can be seen.

4 . C o n c l u s i o n

An analytical solution for the one-dimensional steady- state transport of ions in an electrolyte between two planar electrodes has been obtained which takes the implicit form of a set of non-linear algebraic equations which must be solved numerically. It has been shown that the same set of equations, with only a small modification, can easily be used to solve the stagnant boundary layer problem. The solution method is very general and can deal with any kind of overpotentiai relation at both anode and cathode.

L. Borteis et at./Journal of Eiectroanalytical Chemistry. 422 (1997) 161-167 167

This analytical solution assuming a stagnant boundary layer has been used to determine the diffusion coefficient of the cupric ion and the kinetic parameters in the Butler- Volmer overpotential relation for the electrodeposition of copper from a 0.01M CuSO 4 + 0.1M H2SO 4 solution. The kinetic parameters obtained are in good agreement with the values found in the literature. Analytical results obtained with these parameters match very well with the experimental data for current densities ranging from sec- ondary up to limiting current values and for different values of the rotation speed (100, 500 and 1000rev rain- I ). Also, it has been shown that neglecting migration can lead to an overestimation of the diffusion coefficient of about 15%.

Finally, the method developed yields exact analytical solutions which can be used to access the validity of numerical codes that can deal with diffusion and migration (two electrode problem) or diffusion, migration and con- vection (stagnant boundary layer problem).

Acknowledgements

This research was sponsored by the European Commu- nity under Brite-Euram project no. BE5187, contract num- ber BRE2-CT92-0 170 (L. Borteis) and the Flemish Insti- tute for Science and Fechnology in Industry (B. Van den Bossche).

The authors wish to thank Lutz Gerth from the Labora- toire des Sciences du G6nie Chimique (Nancy, France) for the nteasurements performed.

u(0) U(L) vc VR

liquid phase potential at cathode (V) liquid phase potential at anode (V) metal phase potential at cathode (V) metal phase potential at anode or outside the bound- amy layer (V)

F Faraday constant (C mol-i ) R ideal gas constant (J mol- l K- I) T temperature (K) Jc cathode current density (A m-2) j.~ anode current density or current density outside the

boundary layer (A m- 2 ) Jlim limiting current density (A m-2) Jo exchange current density (A m-2 ) v kinematic viscosity (m 2 s-~) ,fl rotation speed of the RDE (rad s- ~ ) a c cathodic charge transfer coefficient (dimensionless) a , anodic charge transfer coefficient (dimensionless) n number of electrons exchanged in the electrode

reaction (dimensionless) L distance between the electrodes or diffusion layer

thickness (m) 1 number of ions in solvent (dimensionless) A k integration constant (mol m- 3) K k auxiliary constant (V-1 ) K integration constant (mol m- 3 ) Fk,,, auxiliary function (dimensionless)

References

Appendix A. Nomenclature

N k flux of ion k in the positive x direction (molto -2 s - l )

D k diffusion coefficient of ion k (m 2 s- l) zk charge of ion k (dimensionless) c k concentration of ion k (mol m - 3) C~b bulk concentration of ion k (tool m -3) cl(0) concentration of the reacting ion at the cathode

(tool m- 3) cl(L) concentration of the reacting ion at the anode

(tool m- 3) 81 fraction of the bulk concentration of the reacting ion

(dimensionless)

[I] J. Newman, Electrochemical Systems, Prentice-Hall, Englewood Cliffs, NJ, 2nd edn., 1991.

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