Yurij I.Kharkats,‡
Artjom V.Sokirko‡#
and
Fritz.H.Bark#
‡A.N.Frumkin Institute of Electrochemistry, Russian Academy of Sciences, 117071 Moscow, Russia
#Department of Mechanics, Royal Institute of Technology,
100 44 Stockholm, Sweden
Abstract
Polarization curves for parallel electrochemical reactions of two different types for the case of no supporting electrolyte in the solution are analyzed. The consideration of the problem is based on an analytic solution, in parametric form of Nernst - Planck equations for electrodiffusion with boundary conditions of the Butler - Volmer type. It is found that the current - voltage curves for such systems clearly display the interaction of ionic components due to migration current exaltation phenomena.
Introduction
The first experimental observation of the important role of electromigration in parallel electrochemical processes was made by Kemula and Michalski [1], who studied increase of the limiting current of polarographic cation reduction in the presence of parallel reduction of oxygen. This phenomenon was called "migration current exaltation" and its first quantitative description was given by Heyrovsky and Bures [2]. They studied polarographic irreversible reduction of Na+ ions from diluted NaCl solutions. First, the polarographic wave corresponding to the discharge of Na ions was recorded, then the solution was saturated with oxygen from the air, and a new polarographic wave was recorded. The limiting current in the second case was higher than the sum of oxygen reduction current io2 and the limiting current of Na+ reduction iNa+ in the absence of oxygen. Authors of [2] called that additional current "exaltation current" iexal:
i = io2 + iNa+ + iexal
The explanation of migration current exaltation effect given in [3,4] was based on the approximate method of calculation of migration current presented by Heyrovski. In this method the limiting current of discharging ions was taken as sum of diffusion and migration currents, the last was assumed to be proportional to the total current with the proportionality coefficient being equal to the transport number of discharging ions. Heyrovski theory gives us a qualitative description of this phenomenon.
In a series of publications [5-18] a theory of migration current effects for parallel electrode reactions was presented which based on exact analytic solutions of coupled Nernst-Planck equations for electrodiffusion for two parallel electrode reactions. In particular, to the contrary to the Heyrovski theory, it was taken into account that as a result of reduction of neutral substance (oxygen) negatively charged reduction products OH- ions) appear in the diffusion layer close to electrode:
O2 + 2 H20 + 4 e- -> 4 OH-
It was shown that exaltation current depends on the mobility of OH- ion uOH- but not on anion mobility uCl-:
iexal = io2
For the case of two parallel reactions, that involve reduction of two different cations, an extended theory of correlational exaltation of migration current describing the mutual influence of ionic transport due to electrodiffusion process was presented in refs. [5, 7].
In the above-mentioned papers, only the relations between the partial limiting diffusion-migration currents were found. In the present paper we describe the behaviour of total current-voltage curves for parallel electrode processes described by Butler - Volmer reaction kinetics with account of electromigration effects. For simplicity only the case of stationary processes at planar electrode with one dimensional Nernst diffusion layer is considered. This analysis can be generalised for corresponding non-stationary problems, such as parallel reactions at growing mercury drop electrode, as it was considered in [9].
Problem formulation
Consider a reaction with reduction of cations with simultaneous reduction of neutral species in the absence of supporting electrolyte in the solution. The concentrations C1 of cations (Na+, for instance), C2 of anions (Cl-, for instance),C3 of negatively charged products (OH-) of reduction of neutral molecules (O2) together with the electric potential F in the diffusion layer can, for dilute solutions, be computed from the Nernst - Planck equations for electrodiffusion
D1 + z1C1 = , (1)
- z2C2 = 0, (2)
D3 - z3C3 = - , (3)
z1C1= z2C2 +z3C3. (4)
Here D1 and D3 is the diffusion coefficient for cations (Na+) and anions (OH-), F the Faraday constant, R the gas constant, and T the absolute temperature.and i1 is the electric current density for discharge of cations, i0 and i1 are the electric current densities for neutral substance (O2) reduction and discharge of cations (Na+), respectively. X is the coordinate perpendicular to the electrode and z1, z2 and z3 the charge numbers for cations (Na+), anions (Cl-) and product of the second electrode reaction (OH-). It is assumed that the product of cation reduction is neutral. Below we shall consider the simple case z1= z2 = z3 = 1.
The boundary conditions at X = L, where L is the thickness of the Nernst diffusion layer, are
C1(L) = C ; C2(L) = C ; C3(L) = 0; F (L) = 0. (5)
In terms of the dimensionless coordinate x = X/L, concentrations cm= Cm/C , (m = 1,2,3) and electric potential y = FF/RT, Eqs.(1)-(4) may be written in the form
+ c1 = j1, (6)
- c2 = 0, (7)
- c3 = -j0, (8)
c1= c2 + c3 . (9)
Here j1= i1L/FD1C and j0 = i0L/FD3C .
The solution of Eqs.(6)-(9) is, see e.g. [5-6],:
c1= + 1, (10)
y = ln . (11)
The current density at the electrode surface (x = 0) is assumed to be given by the following form of the Butler - Volmer law
j1 = j
where j is the exchange current density, a transfer coefficient, c1(0) the surface concentration of discharging cations, y(0) the potential drop in the diffusion layer and V - y(0) the reaction overvoltage.
Analysis of migration current exaltation
Substitution of expressions for c1(0) and y(0) that were given in previous section into Butler - Volmer law gives the current - voltage curve for cation reduction process in the presence of a parallel reaction
j1 = j
From this relation, one can in the simplest case with a = 1/2, compute V in terms of j1, which gives that
V = - ln - 2 arcsinh
In the general case with an arbitrary value of the transfer coefficient a , it is possible to express the current - voltage curve in parametric form. Denoting h = V - y(0) and substituting the expression for c1(0) into the Butler - Volmer law (12) gives the following relation between j1 and h
j1 = j
This relation and the definition of h written in the form V =h + y(0), where y(0) is given in terms of j1 by the solution (11), gives the current - voltage curve in parametric form (with h as a parameter).
Figure 1 shows the function j1(V ) for a fixed (small) value of j and different values of j0. All curves have a wave - line shape. The limiting current density j , say, depends on j0 as shown by the following formula
j = 2 + j0 . (16)
For j0 = 0, one recovers j = 2, which, in the present scaling, is the limiting diffusion-migration current in the binary electrolyte.
As can be seen from figure 1, the value of the half-wave potential is a monotonically increasing function of j0.
The total dimensional current density in the system under consideration is
it = i1 + i0 = j1 + j0 (17)
and the limiting total current density is
il = 2 + ( + 1) i0. (18)
From these results, it follows that the current - voltage curve for the cation reduction depends, as expected, on the current of the parallel reduction. The current-voltage curve for the latter reaction is, however, independent of the former reaction.
Analysis of correlational exaltation of migration current
Consider now the parallel reduction of two different cations X+ and Y+ in the absence of supporting electrolyte. For z1= z2 = z3 = 1, the corresponding dimensionless equations for electrodiffusion are
+ c1 = j1, (19)
+ c2 = j2, (20)
- c3 = 0, (21)
c1+ c2 = c3 . (22)
Here j1= i1L/FD1C and j2 = i2L/FD2C are dimensionless exchange current densities and cm= Cm/C , (m = 1,2,3). .
At the edge of the diffusion layer (x = 1) the following boundary conditions are prescribed
c1(1) = c ; c2(1) = c ; c3(1) = 1; y(1) = 0. (23)
The current densities at the electrode surface (x = 0) are assumed to be given by following Butler - Volmer laws:
j1 = j
j2 = j
In these expressions, j and j are the dimensionless exchange current densities for the two reactions, a1 and a2 are the corresponding transfer coefficients and D = y - y is the difference of equilibrium potentials between the two reactions. In what folllows, the simplest case with a1 and a2 are equal to 1/2 is considered.
The equations (19) - (22) can be easily integrated [6]. One finds the following expressions for the surface concentrations of cations c1(0), c2(0) and the electric potential drop in the diffusion layer y(0)-y(1):
= , (m = 1, 2), (26)
y(0) = ln , (27)
where j = (j1c + j2 c )/2, (28)
From formula (26) one finds the relation between the partial limiting current j and the current j2, which corresponds to the condition c1(0) = 0 [3-4]:
j2 = . (29)
Analogously, the condition c2(0) = 0 leads to the relation between the partial limiting current j and the current j1:
j1 = . (30)
In terms of y = exp[-V/2] the Butler - Volmer laws (24)-(25) may be written as follows
j1 = j , (31)
j2 = j , (32)
where t = exp[-D/2]. From expression (31), one can solve for y whereby one obtains that
y = + . (33)
A similar expression for y can be computed from expression (32). Thus, expressions (32) and (33), after substitution of c1(0) and c2(0), expressed in terms of j, j1 and j2 from solutions (26) and (28), give a relation between j1 to j2, that does not, remarkably enough, depend on V and y(0). By using simple numerical method, one can compute the "trajectories" j1(j2) in the j1,j2 plane, which depend on the values of parameters j , j , c , c and t.
Numerical simulation and discussion
The resulting current - voltage curves for two parallel processes can be found through use of the following algorithm: For given values of parameters and some value of j1, the corresponding value of j2 can, as was pointed out at the end of previous section, be determined. From expressions (27) and (33) one then finds the corresponding values of y(0) and V. In this way, one obtains j1(V ) and j2(V ) curves and the total dimensionless current density jt
jt = = c j1 + c j2. (34)
Some numerical results are presented in Figs.2, 3 and 4.
Fig.2 shows the effective j1-j2 - trajectories for different values of the shift D between the equilibrium potentials of the reactions. All curves are located inside the region limited by coordinate axes and curves j =j and j =j that are given by expressions (29) and (30). For values of t of order of unity, i.e. for y ~y , these curves are found in the central part of this region. When t >> 1, the first reaction approaches its limiting current while the current of the second reaction is still small. For high potentials, both reactions approach their limiting currents which are j = j = 2. In this case, it is interesting to note that two parallel processes become quasiindependent in the sense that their partial limiting currents become equal to the limiting currents for two independent reactions of cation reduction in two binary electrolytes.
Figs.3 and 4 show the current - voltage curves for the two parallel cations reduction processes. The dashed lines correspond to the current - voltage curves for two separate processes of cation reduction with the same concentrations of cations. One notices that, due to electrodiffusional conjugation of two processes, the limiting currents of both reactions increase. This fact reflects the mutual influence of the two processes. Due to considered correlational exaltation of migration currents, the total limiting current, which is always equal to the sum of both partial limiting currents, can thus substantially exceed the sum of the limiting currents of two independent processes proceeding in solutions with an excess of supporting electrolyte.
Conclusion.
Current - voltage curves for two types of kinetically independent parallel electrode reactions have been computed. The results are based on exact solutions of the Nernst - Planck equations for electrodiffusion with boundary conditions given by Butler - Volmer laws. The one dimensional model of Nernst diffusion layer model was used, which made it possible to find the solutions in closed analytical form. Computed polarization curves show a clearly pronounced dependence of currents of parallel reactions as a result of mutual dependence of transport processes. This dependence is caused by conjugation of electromigration processes in the absence of supporting electrolyte.
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.
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Captions to the figures
Fig. 1. Current - voltage curves determined by Eq.(14).j = 0.001; 1 - j0 = 0; 2- j0 = 1;3 - j0 = 2; 4 - j0 = 3; 5 - j0 = 4.
Fig. 2. Functions j1(j2) determined by Eqs. (31) -(32). c = 0.7; c = 0.3; 1 - t = 1; 2 - t = 2; 3 - t = 20; 4 - t = 200; 5 - function j =j determined by Eq.(29); 6 - function j =j determined by Eq.(30).
Fig. 3. Current - voltage curves for correlational exaltation of migration currents. 1 - curve j1(V); 2 - curve j2(V);c = 0.3; c = 0.7; j = j = 0.0001; D1 = D2; t = 200. 1´ and 2´ - curves j1(V) and j2(V) in the absence of migration currents exaltation. 3 - total current density j(V).
Fig. 4. Current - voltage curves for correlational exaltation of migration currents. 1 - curve j1(V); 2 - curve j2(V);c = 0.7; c = 0.3; j = j = 0.0001; D1 = D2; t = 200. 1´ and 2´ - curves j1(V) and j2(V) in the absence of migration currents exaltation. 3 - total current density j(V).