Yurij I.Kharkats‡

Artjom V.Sokirko#‡

Fritz.H.Bark#,

# Department of Mechanics, Royal Institute of Technology, 100 100 44 Stockholm, Sweden

‡ The A.N.Frumkin Institute of Electrochemistry, Russian Academy of Sciences, 117071 Moscow, Russia

Abstract

Theoretical investigation of potentiostatic electrolysis of a metallic salt in three component electrolyte solution is carried out for a cell consisting of two identical parallel electrodes. Analytical and numerical results are given for polarization curves for electrochemical cells with arbitrary values of transfer coefficient a and exchange current density. Theoretical analysis of electrodiffusion problem, based on exact solution of the Nernst-Planck equations with boundary conditions of Butler - Volmer type, leads to a formula for the polarization curve that is similar to the Tafel equation but with an effective transfer coefficient aeff = a(1 - a).. It is shown that, under certain conditions, the polarization curve can have two inflection points.

Introduction

In recently published paper [1], a
theoretical investigation of potentiostatic one-dimensional
electrolysis of a metallic salt in a three component electrolyte
solution was carried out for electrochemical cell having two
identical parallel electrodes. The analysis led to an exact
solution of the system of Nernst - Plank transport equations
accounting ion transfer by diffusion and migration mechanisms
[2-5]. The polarization curves were derived in parametric
analytic form for the special case where the transfer
coefficient *a* in the
Butler - Volmer law is equal to 0.5 and for an arbitrary
concentration of supporting electrolyte.

Qualitatively, the polarization curves found in [1]
for *a* = 0.5 were similar to
those found for wide class of different systems in which ionic
diffusion and migration transport was analysed at the surface of
one electrode [6 -10].In particular polarization curves of such
type were studied in recently published papers [11 - 16]. On the
other hand, for *a* ≠
0.5, one may expect effects on the current - voltage curve due to
the asymmetry of the reaction. In the present paper, this issue
is examined in some detail for arbitrary values of the exchange
current density.

Problem statement

Consider a binary electrolyte with
ionic charge numbers *z*1
and *z*2
for cations and anions, respectively. The
following reduction reaction occurs at the cathode

*Mz*1+ + *z*1*e- ->* M.

At the anode the same reaction proceeds in the
opposite direction. We shall analyse the current - voltage
characteristic of the cell assuming that there is no convective
motion of the electrolyte in the space between electrodes and
that ionic transport takes place due to diffusion and migration
in the *X* - direction, which
is perpendicular to the electrodes surfaces at X = 0 (cathode)
and X = L (anode).

Let the cathode electric potential be equal to 0
and the anode potential equal to *V*. We shall assume that
there is no redistribution of the electric potential *F* in the diffuse double layers during
current flow.

In the case of dilute electrolyte, the Nernst -
Einstein relation for the diffusion and mobility coefficients can
be used. The concentrations of cations and anions
*C*1 and
*C*2,
respectively, and the electric potential *F* can then be computed from the
following system of equations

*D*1 +
*z*1*C*1* = -* , (1)

*- z*2*C*2* = 0, (2)*

*z*1*C*1*-
z*2*C*2*= 0. (3)*

Here *D*1
is the diffusion coefficient for cations,
*F* is the Faraday constant, *R* is the gas constant,
*T* is the absolute temperature, *i* is the electric
current density.

From the Butler - Volmer law, the electric current
density can be expressed in terms of overpotentials for electrode
reactions, which are *F(0)*
and *V -* F(1) for the cathode and
anode, respectively :

*i = i*0

*i = i*0

In these formulae,
*i*0 is the
exchange current density, for a concentration of cations equal to
*C *. a is
the transfer coefficient.

Taking into account that the reactions at anode and cathode
have equal rates, conservation conditions for
*C*1 and
*C*2 may be
written in the form

= *C L, m* = 1,2, (6)

where *C *are the initial (homogeneous)
concentrations of species in the electrolyte.

Equations (2)-(6) can be written in dimensionless
form in terms of the coordinate *x* = *X*/*L*, the
concentrations *c*m*=*
Cm/*C ,* (*m*
= 1,2) and the electric potential *f =* *FF*/*RT*:

+ *z*1*c*1* =* - , (7)

+ *z*2*c*2* = 0, (8)*

*z*1*c*1*-
z*2*c*2 = 0. (9)

The boundary conditions (4) - (5) can be written in the form

j = *j*0

*j* =*j*0

Where *j* = - *iL*/*FD*1*C ,* j0 = *i*0*L/*FD1*C and* v =
*V*/*RT.*

For the normalization conditions (6), one obtains the following expressions

= 1, * *= * *. (12)

Integration of Eqs.(7) - (9) after use have been made of conditions (12), gives the following expressions for the concentrations

*c*1(*x*)
= 1 + = . (13)

From e.g. Eq.(8) one then finds the expression for the total potential drop in electrolyte

D*f* = *f*(1) - *f*(0) =
ln = ln , (14)

where

*c*1(0) = 1 -
, (15)

*c*1(1) = 1 +
, (16)

and

*j*l = .
(17)

*j*l is the
limiting diffusion - migration current in the cell, corresponding
to the condition *c*1(0) = 0.

Rewriting the boundary conditions (10) - (11) by
using expressions (14) - (16) one obtains the following two
relations between *f(0),* *v* and *j:*

*j* = *j*0

and

*j* =*j*0

Elimination of *f(0)* from these expressions
leads to an implicit expression for the polarization curve
*j(v*). A convenient form is obtained by solving for
e

*E* = e

Substitution into relation (18) then gives that

*j* =*j*0 *E *. (21)

Expressions (21) and (20) define *j* as an
implicit function of *v*.

Analysis of some limiting cases

1. For small values of *j* and *v,*
expansion of expression (20) gives, to lowest order, that

E ≈ 1 + ). (22)

Substitution of this expression into expression (21) gives the explicit linearized form of the polarization curve

˜ *v* - . (23)

Thus, the slope of *j*(*v*) curve at
small values of *v* depends on the ratio between
*j*0 and
*j*l. In
the case of *j*0
<< *jl* expression (23) may be written in the form

*j* ≈ *j*0
*v, (24)*

and in the opposite case, i.e. *j*0 >> *jl* ,
as

*j* ≈ *j*l
*v. (25)*

2. Consider now the case of small values of
*j*0. Assuming
that the values of *v* such that *j*0 << *j(v) ~* jl are of the order 100
- 101 or larger, one obtains from
expression (20) the approximate relation

E = e . (26)

For E >> 1, one then finds from expression (21) that the polarization curve is given by the following approximate expression

*j* ≈ *j*0* *e *.*
(27)

This result shows that for values of *j* that
are not too close to *j*l, the polarization curve
*j*(*v*) can be represented as a Tafel-like law with
the effective transfer coefficient

aeff = a(1 - a) (28)

The maximum value of aeff is
equal to 0.25 and the dependence of *j*(*v*) on
a is symmetrical with
respect to a = 0.25. Again, it should be stressed that this symmetry property is
valid for small *j*0
values only.

3. For *j*0
<< *j(v)
~* jl formula (27)
simplifies to

*j* ≈ *j*0e *.*
(29)

A comparison between formulas (24) and (29) shows
that in the transition region from the linear behaviour of
*j* (*v*), which is independent of a, to the exponential one, there is
an inflection point for values of a larger than 0.5. This inflection
point is similar to that of the current - voltage curve for the
reaction at one electrode, which is described by Butler - Volmer
law. For *j*0 << 1 and *j*0
<< *j
*jl one then has
that

*j* = *j*0

The position *v* = *vc*, say, of the
inflection point in this case can be readily computed and one
finds that

*v*c = * *

It turns out that this expression can also be used as an estimate of the location of one of the inflection points of the polarization curve for the cell with two electrodes. The closer is a to 1, the higher is the accuracy of Eq.(31).

The second inflection point of polarization curve
is found in the region of high values of *v* where the
current limitation due to diffusion and migration becomes
important.

4. For high values of the exchange current density,
*j* in the left hand side of expression (21) can be
neglected and the polarization curve is approximately determined
by relation

*j* ≈ *j*l
, (32)

where *E*(*v* ) is given by expression (20).This
relation is, of course, invalid when *j* is very close to
the limiting current density.

The role of the supporting electrolyte in a system with three species.

In the presence of a supporting electrolyte, the system of dimensionless equations for the concentrations distributions of electroactive cations, indifferent cations, anions and the electric potential distribution reads as:

+ *z*1*c*1* =* - , (33)

+ *z*2*c*2* = 0, (34)*

* - z*3*c*3* = 0, (35)*

*z*1*c*1*+* z2*c*2
= *z*3*c*3. (36)

In these equations, the dimensionless concentrations are defined as

*c*i*=* (37)

and the electric current density is normalised
according to the relation *j* = -
*iL*/*FD*1*C It turns out to be convenient to introduce the
parameter* k = *C */*C *. Then, *C */*C *=
*z*3/*z*2 - *kz*1/*z*2, 0 < *k <* *z*3/*z*1.

The case of the aqueous solution
of *CuSO*4 with *H*2*SO*4
as a supporting electrolyte will be considered
below. In this case at relatively low *H*2*SO*4
concentrations *H*2*SO*4
dissociates mainly into
*H*+ and
*HSO*4- and, correspondingly, *z*1 =
2, *z*2 =
*z*3 = 1.

For the three component system under consideration, the dimensionless version of Butler-Volmer laws for the electrode reactions, i.e. formulas (4) and (5), read

*j* = *j*0

*j* =*j*0

The dimensionless conservation conditions, c.f. formula (6), takes the form

= *k* , (40)

= 1-2*k*, (41)

= 1. (42)

In order to compute the polarization curve, some algebraic simplifications result if the solutions of Eqs.(34) - (35) are written in the following form:

*c*2 =
*c*2(0)e ,
(43)

*c*3 =
*Nc*3(0)e ,
(44)

where *c*2(0) and *N* are two constants
of integration. As in ref. [1], it is found expedient to express
these constants in terms of the potential drop in solution
D*f* =*f*(1)
-*f*(0).

Combining Eqs.(33) - (36) we have

* *= . (45)

Substituting of this equation into normalization
condition (42) gives the following relation between
*c*2(0), *N*
and *y* = e :

* *= 1. (46)

A second relation between *c*2(0), *N* and *y* follows
from the second conservation condition (41)

* *= 1 - 2*k*,(47)

Combination of reflations (46) and (47) gives
*N*(*y*):

*N* = (48)

Taking use the identity

* *= 1, (49)

and Eqs.(43) - (44), one can express *c*2(0) and *j* in terms of
*y*

c2(0) = ,(50)

*j* = , (51)

Combination of these expressions with the
electroneutrality condition (36) and formulas (43) - (44), leads
to the following expressions for *c*1(0) and
*c*1(1) in
terms of *y*:

*c*1(0) =
*c*2(0),
(52)

*c*1(1) =
*c*2(0).
(53)

Formulas (48) and (50) - (53) and expressions

*j* = *j*0

*j* =*j*0

which follow directly from expressions (38) - (39)
and the definition of *y*, determine the current - voltage
characteristic *j*(*v*) of the cell.

Equating the right hand sides of expressions (54) and (55) one finds that

e =

Substitution of this expression for e into
expression (54) leads to an implicit relation between *v*,
*j* and *y*. For each value of j, this relation permits
numerical computation of *v* as function of *y*.
Together with formula (51), which determines *j*(*y*)
this gives the polarization curve *j*(*v*) in
parametric form for arbitrary values of the exchange current
density *j*0 and
the transfer coefficient *a .*

Numerical simulation and discussion

Some results of numerical
calculations of polarization curves are presented in Fig.1 - 3.
Fig.1 represents a set of curves for a binary electrolyte,
calculated according to Eqs. (51), (54) and (56) and different
values of *a* . One can
see that the curves corresponding to values of *a* that are different from 0.5 are
shifted towards higher *v* values with respect to the curve
corresponding to *a* equal to 0.5.

The extremum property of the value *a* = 0.5 can be illustrated by the
dependence of the value of the potential
*v*1/2,
which corresponds to the current density equal to one half of the
limiting current density, on the values of parameter
*a.* This relation is shown in
Fig.2. The curve *v*1/2(*a*
) is practically symmetrical with respect
to *a* = 0.5 in good
agreement with theoretical analysis. It should be stressed that
such symmetry appears only for small values of exchange current
densities.

Fig.3 shows a set of polarization curves with the
exchange current density equal to 0.1 and different values of
transfer coefficient *a* .
For *a* = 0.5
polarization curve has only one inflection point corresponding to
the transition into the current region, where the diffusion
limitation of electrode reaction becomes important. When the
transfer coefficient *a* increases, a second inflection point appears on the
polarization curves at relatively small current densities. This
inflection point is related to the transition from region of
linear "ohmic - like" behaviour of polarization curves at small
reaction overvoltages, which is independent of *a* , to the exponential "Tafel - like"
region of the curve, which depends strongly on *a.* The inflection point of this type
exists on the polarization curve only for relatively small
exchange current densities *j*0. At very small values of this
parameter the position of this inflection point of the current -
voltage curve corresponds to very low current density
values.

For a binary electrolyte it should be noted, that in some special cases such as a = 2/3, a = 1/3 as well as in the case of a = 1/2, which was considered in [1], the polarisation curve can be found from expressions (18) - (19) in analytic form. Eq.(18) can be easily solved with respect to j:

*j* =

Introducing notations

*S* = exp

*p* = - *S* , *q* = - *S*, (59)

one finds after some algebra, that relation (19) for a = 2/3, can be reduced to the form

*t*3 +
*pt* + *q* = 0, (60)

where *t* = e > 0. The solution of Eq.(60) is,see e.g.
[17],

*t*1 = + ,
(61)

Taking *f(*0) to be an independent parameter
one can find the parametric analytic representation of the
function *j*(*v*) from formulas(57) and (61), the
relation *v* = ln *t*1
and the definitions of S, p and q.

In a similar way the parametric form of current -
voltage curve *j*(*v*) can be found for case of
a = 1/3.Some examples
of analytical treatment of electrode reaction kinetics for
fractional values of a
were considered in [18].

Conclusion

An exact solution of the system of
Nernst-Planck equations for a 1-dimensional electrodiffusion
problem, with boundary conditions of Butler - Volmer type shows
that the polarization curves for a cell having two identical
electrodes have some interesting general properties. At small
values of the exchange current density and current densities that
are not close to the limiting current density
*j*l,the
polarization curve is of a form that is similar to the Tafel
equation. One obtains that

j ≈ j0exp(aeffz1v)

with the effective transfer coefficient aeff related to the physical transfer coefficient a by the formula

aeff = a(1 - a).

Another general property of the
polarization curves for the system under consideration is that,
under certain conditions, two inflection points may appear. One
of these is similar to that of the current - voltage curve for
the reaction at one electrode, which is described by a Butler -
Volmer law for the case of small exchange current density
*j*0 and
current densities *j* , such that
*j*0 << *j
*jl. The second
inflection point can be observed on the *j*(*v*) curve
in the region of relatively high values of *v*, where the
current has values of the order of the limiting current
density.

Acknowledgment

The financial support from the Royal Swedish Academy of Sciences for the participation of Yu.I.K. and A.V.S. in the present work is gratefully acknowledged. One of the authors (Yu.I.K.) would like to acknowledge the Department of Hydromechanics of the Royal Institute of Technology, Stockholm, where this work has been done, for financial support and hospitality as well as partial support from the Fund of Fundamental Researches of Russian Academy of Sciences.

References

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Captions to the figures

Fig. 1. Polarization curves
*j*(*v*) determined by Eqs. (51), (54) and (56)
corresponding to different *a*
values. k = 0.4; *j*l = 6; *j *= 0.001; 1
- *a* = 0.5; 2 -
*a* = 0.4; 3 - *a* = 0.6; 4 - *a* = 0.3; 5 - *a* = 0.7.

**Fig. 2.** Dependence of the potential *v*1/2*value corresponding to current density* j =
*j*l/2 on the
transfer coefficient *a* for binary solution. *j *= 0.001; *j*l = 6.

**Fig. 3**.
Polarization curves *j*(*v*) determined by Eqs. (20)
and (21) corresponding to different *a*
values. *j*l = 1; *j *= 0.01; 1
- *a* = 0.5; 2 -
*a* = 0.6; 3 - *a* = 0.7; 4 - *a* = 0.8; 5 - *a* = 0.9; 6 - *a* = 0.95.