J. Phys. Chem. 1980, 84,2025-2032
2025
A Theoretical Investigation of Electrode Oscillations Joel Keirer" Chemistry Department, University of California, Davis, California 956 16
and Daniel Scherson Department of Chemical Engineering, University of California, Berkeley, California 94720 (Received: November 7, 1979) Publication costs assisted by the National Science Foundation
When certain electrodes are driven away from equilibrium by application of an external voltage, undamped oscillations of the voltage or current are observed. We develop here a simple theory for describing transport at the boundary of an electrode which is suitable for examining these oscillations. The theory involves the coupling between the electric field and diffusion of a reducible species near the electrode surface and includes certain effects of double layer relaxation. A simple reaction scheme which involves the reduction of an adsorbed cation is then used to produce electrochemical oscillations. The reduction rate is modeled by using the full Butler-Volmer equation and, when plausible values of transport coefficients and other parameters are used, electrochemical oscillations are obtained. The steady state and dynamical behavior of this model are examined by using graphical, analytical, and numerical methods. Some comments are made regarding the relationship of this work to experimentally observed oscillations.
Introduction Electrochemical oscillations have been observed in a variety of electroche.mica1~ystems.l-~ The simplest of this sort of oscillation involves a coupling to the impedance of the external circuit: and the resulting oscillations are easily understood on the basis of elementary circuit theory. A more complicated and picturesque oscillation occurs in the beating mercury heart5 In that system oscillations are triggered by the reduction of a variety of electron acceptors adsorbed on a mercury surface. The switching mechanism, which leads to voltage and shape oscillations of the mercury, involves a periodic short circuit caused by the electrocapillary effect. These oscillations have been examined in some detail both experimentally and theoretically,6i7and a molecular mechanism, which reproduces the oscillations quantitatively, has been proposed. These oscillations are driven by a corroding metal electrode but do not otherwise depend on the external circuit. A qualitatively different type of oscillation has been observed in a number of oxidation and reduction proce~ses.l-~ These systems are also driven by a source which can operate at a fixed external voltage (potentiostatic) or fixed external current, (galvanostatic). During oscillations periodic changes in the potential of a electrode or the current through the electrode or a surface ion concentration are observed.8 [n these cells there is no means to provide a short circuit and, consequently, the only mechanism for the oscillations can be the physical chemical processes occurring at or near the electrode surface. It is these intrinsic electrochemical oscillations which interest us in this communication. There have been a number of previous theoretical treatments of electrode oscillations.1~4~9-11 Of these some have concentrated on linear approximations to kinetic mechanisms and, consequently, have been unable to verify the existence of nonlinear oscillations. This is a serious limitation since it is well known that unstable steady states in electrochemical systems are often associated with stable states which are attraetors and possess no limit cycles. For example, de Levieg has used the empirical admittance function for the In3+/SCN- oscillator to give a linear circuit analysis of the oscillations. This treatment includes both diffusion and charge transfer effects but is useful only for 0022-3654/80/2084-2025$0 1.OO/O
assessing the stability of steady states. Other treatments, like that of Degn,lohave focused on the nonlinear aspects of adsorption processes, while the electrochemical kinetics are treated in a linear approximation. Degn's treatment also neglects the effect of mass transport at the electrode surface. The work of Franck and Fitz-Hughll has focused on the change of surface coverage and the pH dependence of the Flade potential in passivation processes. These latter two investigations have both lead to a mathematical description of oscillations, although neither is firmly based on the differential equations of electrochemical transport theory. A more complete set of transport equations to describe the oxidation of H2 on Pt has been outlined by Conway and Novak,13but no mathematical treatment of the equations was given. It is our purpose here to outline a simple treatment of electrode oscillations which is based on the equations of electrochemical transport. We begin with Maxwell's equation for charge conservation in the neighborhood of the electrode. Both diffusion and charge transfer fluxes are included for the ionic species which is involved in the charge transfer process. Other species in solution are assumed to be in excess and to follow a Gouy-Chapman distribution. If we assume that the electric field is dominated by the nonparticipating ions, a differential equation for the voltage of the electrode is obtained. We retain in this equation a rapid term referring to the relaxation of the double layer, which turns out to be important for describing oscillations in our model. The voltage equation also depends on the concentration of the reducible or oxidizable species near the electrode surface. To complete the differential transport equations, a model of the electrodiffusional transport of ions is used which is based on the theory of nonlinear, nonequilibrium thermodynamics. For the case of a single reducible ion the two coupled nonlinear differential equations are written out explicitly. To illustrate the use of these equations for treating oscillations, we use as a model an electrocatalytic process: A reducible cation XZ+from solution reacts with two anions Y-which are adsorbed on an inert electrode. The resulting product, XYz(z-2)+,forms a surface film which is then reduced regenerating 2Y-. This model is similar to one proposed by de Levie to explain the oscillations of the 0 1980 American Chemical Society
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The Journal of Physical Chemistty, Vol. 84, No. 16, 1980
-'T
I
+j;y -----_-
-E
1 7 L
0 DE -
2
ANODE
o
Figure 1. The circuit dlagram used for analyzing Maxwell'- equation of charge conservation.
In3+/SCN- os~illator.~ The switching mechanism for the kinetic step is the cathodic desorption of Y- which is governed in our model by an equation which was suggested by the pH dependence of the Flade potential. We do not present here a systematic investigation of other kinetic mechanisms that might lead to oscillations. However, in other work15we have established that a variety of simple electrochemical chemical mechanisms do not lead to oscillations. In the electrocatalytic mechanism that we treat here, an important feature is the existence of two distinct * x m s of the surface: the covered surface, which is capable of reduction, and the uncovered surface, which is refractory. We do not believe that this is a necessary feature of electrochemical oscillations and predict that other mechanisms which give rise to electrochemical oscillations will be found. We have analyzed our electrocatalyticmodel analytically by use of graphical techniques and linear stability analysis and numerically by stepwise integration of the equations of rnotiona6 Using realistic values of the transport coefficients and equilibrium parameters we find limit cycle oscillations16in the electrode voltage and surface concentration of Xz+ions. The existence, amplitude, and period of the oscillations depends on the resistance in the external circuit. The steady state current-voltage curves exhibit a negative resistance, although the oscillations can appear prior to the region of negative resistance. The oscillations are characterized by a rapid decrease in the voltage when the reduction of XY2(z-2)+occurs, followed by a sequence of rapidly increasing and slowly increasing voltages. An important feature of the oscillations is the sudden switching of the reduction process on and off by the voltage and concentration-dependent adsorption equilibrium of Y-. This leads to oscillations because the electrodiffusional transport of Xz+in solution is significantly slower than the charge transfer and double layer relaxation of the voltage. We give a brief discussion of the relationship of our results to experiment.
Electric Field Near a Reacting Electrode To derive the transport equations which we use to describe oscillations, we begin with Maxwell's equation of charge conservation1' applied to the circuit illustrated in Figure 1. Thus let j be the true charge current density as a function of time and space and jt E j + dD/dt (1) D in eq 1is the displacement current which will be written as D = €E where t is the dielectric constant. Then conservation of charge is given by o.jt = 0
(2)
which is just the continuity equation written in electromagnetic notation. Integrating eq 2 over the volume enclosed by the dashed line in Figure 1 gives 0 = -i + Aj(x,t) + jt.ds + A d E ( x , t ) / d tS A'tdE(O,t)/dt (3)
1
Keizer and Scherson
where the divergence theorem has been used, i is the current just inside the electrode surface (with a minus sign because the current flows against the normal to the surface),A is the area in solution perpendicular to the current flow, and A'is the area of the electrode where the field is nonzero. Since S'is the area parallel to the current flow the term involving its integral is zero. Also if the electrode is a good conductor, the area A'is essentially zero. Thus the electric field in the electrolyte satisfies the equation cdE(x,t)/dt = -j(x,t) + i / A (4) In obtaining this result the dielectric constant is assumed to be constant, which is only an approximation near the electrode, and edge effects of the electric field have been ignored. Equation 4 says that the time rate of change of the electric field in solution is proportional to the difference in current density at that point and just inside the electrode surface. From conservation of charge in the external curcuit the current density just inside the electrode surface is i / A = (G - V)/RA, where R is the resistance of the external circuit. The current density in solution, on the other hand, has three components: the diffusion current density j d , the mobility current density j,, and the reaction current density j,. This latter current is important only in a region of molecular dimension at the electrode. The diffusion and mobility currents can be attributed to the charge carriers in solution independently, so that jd
+ j, = Cjd,i + j,,i i
(5)
where the sum is over all ions in solution. Of these, only some small number will react at the electrode at an appreciable rate. To simplify matters a single reactive ion with a charge ze, diffusion constant D , mobility p, and density denoted by p is treated. To further simplify matters it is assumed that the spatial density distributions of th.: nonreactive species rapidly relax to the GouyChapman18density profile. For the systems treated here this is a good approximation, as we detail in the following section. Consequently, for the nonreactive species in eq 5, the sum of the diffusion and the mobility currents vanish leaving a contribution only from the reactive species. This then giveslg
j = j,
+j,
= -zeDdp/dx
+ z2eNpE
(6)
for the current density in the solution. At the surface of the electrode, that is a few angstroms inside or outside, the current density is dominated by reaction and double layer effects. Thus in this region of length d* the current can be written as the sum of the electrochemical reaction current and a capacitive current j = j, + c dV/dt (7) where c is the capacitance per unit area. To correspond to the sign convention in solution the reaction current must be written as positive for a cathodic current.
The Physical Picture In order to simplify electrochemical eq 4, 5, and 7 we utilize a physical picture which comes from earlier work on relaxation of the double layer.2't22 Recall that we have in mind a single reducible species which is present in low concentration with respect to nonreducible ions. For example, the total In concentration for the In3+/SCN- oscillator is about M, while Na+ and SCN- are present at 5 M. This means that the electric field in the double layer will be dominated by the nonreactive ions. Because of this, to the lowest order of approximation the field can
Electrode Osclllatllons
The Journal of Physical Chemistry, Vol. 84, No.
be taken as the Gouy-Chapman fieldsmFurthermore, the distributions of the nonreactive ions can be taken to be the usual Gouy-Chapman distributions. Actually it is known that these distributions are established rapidly on ideal polarized electrodes because of the rapid change of the local electric field.I8 The appropriate relaxation time for this process TF = rc, where r is the local resistance and c is the capacitance of the surface, and for 1 M NaCl Graharnel8 has estimated rF = 3 X s. The electrodiffusive motion of the ions themselves is a much slower process. The relaxation time for this process is approximately TD = d2/D e 10-100 s, where d is the order of the diffusion layer length. This slow process plays only a minor role in determining the distributions of the nonreactive ions, because of the rapid readjustment of the field. However, diffusion is important for the ion which is being reduced at the electrode since a diffusive flux from the bulk will be set uip as these ions are removed at the electrode surface. Our physical picture, then, is that the local GouyChapman field and distributions of the nonreactive ions provide a background in which the reactive ion moves. The primary differences between the background ions and the ion being reduced are that the latter diffuses in from the bulk and reacts a t the surface. As the reducible ion moves in, it perturbs the local electric field, which readjusts itself rapidly to a new quasisteady state value. These effects are contained in eq 6 and the term i / A in eq 4. When the reducible ions reach the surface, they will cause an additional change in the field by discharging electron density at the electrode. This involves capacitive and reactive currents, localized at the electrode surface, and is shown in eq 7. In our deviation in the next section, we make use of this physical picture and eq 4, 5, and 7 to obtain a tractable version of the transport equations near the electrode surface.
Derivation of the Transport Equations Since voltages, not electric fields, are most easily accessible to measurements, eq 4 will be integrated to give a differential equation for the voltage of the electrode. Integration of the eleckic field between the electrode and a distance point 1 in solution gives
where 41can be taken as the potential in the bulk solution. For a highly conductive medium, and the circuit in Figure 1, the bulk potential is approximately zero, so that
-x 1
dV/dt =
dx dE/dt
(9)
Consequently the intlegration of eq 4 gives (d*c + E ) dV/dt = -.zeD(pl-
1
Pd*)
+ zZepL*pEdx +
(6 - v) - ezd*j, (10) RA where d = 1 - d* and eq 7 was used for j when 0 Ix Id* and eq 6 when d* Ix -< 1. The second term in eq 10 can be rewritten by using Poisson’s equation1’ zep = €aE/ax - pc (11) where pc is the local charge density of all the charges excepting the reactive species, Le., it is the background charge. Substituting into eq 10 the expression for p in eq 11 we obtain (d + d*)-
16, 1980 2027
(12) where E = -&b/ax, so 4 is the electric potential. Now the number density of the background charges can, as we discussed above, be approximated by the Gouy-Chapman form, i.e. Pi = PiB eXP(-ezid)/kBT) (13)
where piB is the bulk concentration of species i, k~ is Boltzmann’s constant, and T the temperature. Consequently,I8 we can write the following identity for the background charge densityP p(a4/ax) C’ezipi a4/ax = kBTap’/ax (14) i
where the primed sum indicates that the reactive ion is not included and p’ is the number density, excluding the reactive ion. When eq 14 is used, the right hand side of eq 12 simplifies to 1 W E Z2ep L * p E dx = T ( E t - Ed*’) + ZpkBT(pd*I - P i ) (15) and since 1 is large, p[ has been taken as PB’, the bulk value of the background density. The electric field and the densities of the nonreactive ions can be eliminated from eq 15 since, according to our discussion in the previous section, the electric field can be approximated by the Gouy-Chapman theory. This means that E
,(E(x,t))2
~BTCP~B(~XP[-~~~~(~,~))/KBTI - 1) = i
~ B T ( P ~ (-xP ,B~~))(16) where the last equality follows from eq 13, and pT is the total number density and pBTits value in the bulk. Combining eq 15 and 16 we obtain (d*c + E ) dV/dt = zeD(pa*- p ~ -) Z p k g T [ p ~eXp(eZfV/kBT) - p ~ -] ezd*i, (d d*)(G - V ) / R A (17)
+
+
In eq 17 the potential at d* has been written as 4dl = -fV with 0 < f < 1 since d* is only a few angstroms away from the electrode surface. Finally eq 17 can be simplified by using the Nernst-Einstein relationshipz3p = eD/kBT, to F dV/dt = zeD[pd*- PB exp(ezfV/k~T)]- ezd*j, + (d + d*)(€ - V ) / R A (18)
+
with F d*c E , Only two variables are involved in eq 18, the voltage of the electrode, V, and the density of the reactive species, Pd*, at the electrode surface. This suggests a model for completing these equations, that is, a way to obtain an independent differential equation for Pdt. In the region close to the electrode, the number of reactive species changes as a result of reaction and a combination of diffusion and mobility. In developing a model for this sort of mass transport one possibility is to use the diffusionmobility equation combined with certain boundary conditions.z2 These boundary conditions are necessary mathematically, but also physically, since the diffusion equation is not valid close to the electrode surface. Another approach, which we adopt here, is to model the boundary by a hypothetical region next to the electrode of width d, cross sectional area A , and uniform density Pd*.6;24 This region is presumed to transport mass by electrodiffusion to an adjacent region which is at the bulk
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Keizer and Scherson
The Journal of Physical Chemistry, Vol. 84, No. 16, 1980
density, pB. The change in the number of reactive ions in the electrode compartment caused by this process can be described by the appropriate expression from nonlinear, nonequilibrium thermodynamic^,'^^^^ namely (m/dt)diff = -Q[exp(pd*/kBT) - e x P ( p ~ / k ~ T ) ] (19) In eq 19 D is the intrinsic rate of the elementary electrodiffusion process, is the electrochemical potential in the volume near the electrode, and pB is its value in the bulk. For a dilute solution one may write &* = po + kBT hl Pd* (20) PB = po + k,T In pB + ezfV which leads to edp0/kBT)[Pd*- PB exP(ezfV/k~T)I (21) Dividing this by the volume dA gives the contribution of diffusion t o the change of density D (dPd*/dt)diff= --[Pd* - PB exP(ezfV/kBT)] (22) d2 where the expression for Q has been based on that taken from Fick's diffusion law.21 Combining this with losses from the electrochemical reaction gives the desired transport equation for Pd* (dN/dt)diff =
Equations 18 and 23 are the transport equations which we will use to describe electrochemical oscillations. A review of the derivation of these equations shows that they are easily generalized to involve several concentration variables. Comments on the Transport Equations Two important approximations were used in deriving eq 18 and 23. The first is that the local value of the electric field in eq 15 is approximated by its Gouy-Chapman value. This approximation has been used previously by Rangarajaq20 although near the electrode at d* this is never a truly good approximation since orientational effects and adsorbtion in the Stern layer will determine the exact nature of the field there. Nonetheless, in the framework of a dielectric continuum theory with a fixed dielectric constant, it is a good approximation as long as counterions are present and the reducible ion is in low concentration.21 In the example we treat below, the concentration of the reducible species is less than M, which satisfies this condition. The most important approximation is in treating the electrodiffusion process as if it were occurring between two compartments, one at concentration Pd* and voltage -fV and the other at the bulk concentration. As noted in the discussion of that treatment, some sort of model is necessary to describe diffusion near an electrode since both discrete molecular effects and continuum transport effects are important. The model we use includes the discreteness of an ionic concentration layer at the wall along with an appropriate flux expression for transport into the bulk solution. Although the model is simplified, it has several virtues. First, it will be shown below that the resulting differential equations are consistent with the usual circuit analysis description of steady currents in these systems. And second, it leads to a description involving only the surface concentration of the reducible species and the voltage. This means that the resulting differential equations can be examined both analytically and numerically.
In this manner we hope to gain insight into the coupling which is responsible for electrochemical oscillations. A comment regarding the form and magnitude of the various terms in eq 18 is probably appropriate here. Based on the analogy to simple electrical circuits, one usually writes for the voltage relaxation equation6 c dV/dt = -ezj, + ( E - V)/RA (24) Comparison with eq 18 shows a similarity in form between the two equations, except that eq 18 includes the additional terms E dV/dt - ZeD[pd* - PB e~p(eZfV/k~T)] d ( 8 - V)/RA (25) on the left-hand side. These are terms which come about from including dynamical effects in the diffusion layer. These effects involve only the reducible species and, as mentioned above, dissappear in the steady state analysis. The first term is comparable in magnitude to the capacitive term in eq 18. The second and third terms involve very rapid relaxation effects with relaxation times near equilibrium of the order of 1-10 ps, at least for the dilute solutions of reducible species considered here. They are between six and ten orders of magnitude more rapid than the equilibrium relaxation rates for the diffusion and electrochemical reactions that we consider. Nonetheless farther away from equilibrium the relaxation rates of all these processes become comparable. Since it is in just this regime that oscillations are observed, it is important for our purposes to use the complete eq 18. An Oscillating Electrochemical Mechanism We apply our differential equations to the following chemical mechanism, modeled after de Levie: occurring at an inert electrode:
+
Xz+ 2Y-(ads) + XY2(z-2)+(film)
(26)
+ XY2(z-2)+(film) Xo + 2Y-(ads)
(27)
ze-
F=
The reducible species is Xz+and it forms a film with the adsorbed anion Y-. The film is then reduced to regenerate the Y-.It is assumed that film formation is rapid and in equilibrium. Thus the rate of the reduction process can be written as3 r = k'u+ exp(yV) - Ea? exp(-y'V) = a_2[k'Kpdr exp(yV) - k exp(-y'V)] (28) where the emf has been set equal to zero and the equilibrium assumption a+ = KU?Pd*
(29)
was used, with a, the surface activity of the film and athe surface activity of Y-. Using the maximum surface activity of Y-, we can rewrite this as (30) r = kO[p exp(yV) - PB exp(-r'V)I using the definitions I k(a-"ax)2/pB 8 = (a-/&max)2 k = k'K(a-max)2 Henceforward, for simplicity, Pd* is written simply as P. The function 8 measures the fractional occupancy of the surface by Y- and varies between 1at low (anodic) voltages and 0 at high (cathodic) voltage because Y- is negatively charged. Its general shape is shown in Figure 2, where v d is the desorption voltage. In calculations it is frequently convedient to use for 8 the functional form 8(v)=1 v
