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Controlled-potential electrolysis at a membrane-coated electrode
Bioelectrochemistry and Bioenergetics, 14 (1985) 293-312 A section of J. Electroanal. Chem., and constituting Vol. 192 (1985) Elsevier Sequoia S.A., Lausarme - Printed in The Netherlands
718-CONTROLLED-POTENTIAL ELECTROLYSIS MEMBRANE-COATED ELECTRODE PART II. NUMERICAL
M. CASELLI Dipartimento
ANALYSIS
293
AT A
AND RESULTS
and M. MAESTRO di Chimica dell’ Universirir, Via Amendola 173, 70126 Bari (Italy)
(Revised manuscript received September 19th 1984)
SUMMARY The numerical treatment of the integral equation system which resolves the problem of a one-sign charge flow followed by the discharge at an electrode across a dielectric membrane is explained. Some representative results are presented and discussed both for the cases of a constant potential and for a triangular sweep.
SPECIAL
an.
cat.
A,,
jl,,
ur.e,1/2 u r.e.,p
6s 70
SYMBOLS
USED IN THIS PAPER
as subscript refers to the voltammetric anodic branch as subscript refers to the voltammetric cathodic branch = n9=Di+j X*/d voltannnetric limiting current density peak current density in cyclic scanning voltammetry voltammetric half-wave potential peak potential in cyclic scarming voltammetry dielectric constant of the solution = d2/Di+j
For other symbols see the preceding paper [2]. INTRODUCTION
In a previous paper [l] we have described a physical situation in which charges of one sign only go through a dielectric coating a metallic electrode and discharge on it; the transport is allowed by a complexing agent confined in the dielectric which operates as a shuttle. Subsequently [2], the differential equation system which describes the phenomenology has been resolved by transforming it into an equivalent integral system. 0302-4598/85/$03.30
0 1985 Elsevier Sequoia S.A.
294
In this paper the numerical treatment which allows a concrete solution of the problem is discussed and some representative numerical results are given. The main literature which is useful in treating this problem has been discussed previously [2]; we refer to the same paper also for the employed notations and symbols. NUMERICAL
PROCEDURE
The integral equation system obtained previously [2] is the following:
(2)
- W’,+(O,$=&r)d, where I(x) = exp(x’) erfc(x) and @,(x/y) 40 in the preceding paper];
X is a Jacobian elliptic function [see Ref.
(3)
xexp
Xexp ( -4 ‘2U)[~+@3(O/i~~)])d~-exp(~)~y([YoX
exp( - $)
r,‘exp(Xr)-Y;]
C3d(O/ilrw)}dr
where the symbols have the following meaning: o = y - 7; h = a* - (b*/4); (b/2)-a and Q,(O/i?rw), Q,(O/’ IQW) are Jacobian elliptic functions.
x
(4) a=
295
(vo- Ff) It=+ : I, exp $ (- $_.)
)
y
y-
h-----&T
3 X
W, =
d+
US
(
r,
-ow 6
avZIV,
1
-(auVoXo -
I0
a,Y,)exp (
’ 4(Y--7)
1 I 6
dr
(6)
J(r
- T)]dT
;~yW;(+34[0,i~(y
(8)
6( Y; - by,) = oyYO- exVoXo
(9) These equations must be resolved together to obtain X0, Y,, Yi, Y, , Y;, V,, VI, W,, and Z,; all the other time-dependent boundary concentrations and their gradients which appear in the problem (more precisely X4, WA, W,, W;, Vi, I’{ and Z;) can be deduced directly by simple algebraic relations already given [2]. The standard way to solve the given system is to resort to a numerical integration procedure; for this, some critical points must be taken into account. Many of the expressions which appear under the integral present a singularity at the upper limit; this is clearly apparent in equations (l), (5), (6) and (7), but the situation is the same in the other equations, since the Jacobian elliptic functions Qi(O/o) show singularities for w --, 0. To overcome this trouble, we divided the integration domain in two parts, confining the singularity to the last step of the numerical integration grid, and only this last term has been integrated by parts. For instance, equation (1) was transformed in the following way: l-1 iFI [G(i) -6&(i)] + [q(Z) -6%(Z)] AT= ) ‘-’ aYYo(i)-c?[Y’(i)-6Yo(i)]
A7+
%v,(i)Jv,-v,
_~YYO(I)-S[Y~(Z)-~Y~(Z)~ G-2G
I
%l/,V)
or, by insulating the present& unknown Y,(Z), Y;(Z) and U,(Z)
o&(i)
-6[Y;(i)
-by,(i)]
(10)
296
The same procedure can be applied to the other equations in which the Jacobian functions ei appear; for these one meets the problem of the explicit evaluation and also that of their singularities. The following series have been employed [4,5]: @,(O/iqnw)=2
f
exp[-n(n+l)+$r2W]
n=O
B,(O/iaw)
= 1+ 2 5
(11)
exp( -n29r2w)
n-l
@d(O/i?rw)=1+2E(-l)“exp(-n2?rw) n-1
In order to eliminate the singularities, we have resorted to an evaluation of some auxiliary functions which appear in the integration by parts. More precisely we have calculated, also by series, the following functions: T2( y ) =
-
iye2(O/ino)dw
(12)
and the analogous T3 and T4. As an example, equation (3) becomes: Y;(Z)Ar-
,,Z)[$
[Y,(i)+
X
I
AT- Ts(Ar)] - Y,(Z) exp( -i)T,(Ar)=
%k@v,)-
:-@,(y,-y,)
x[Y,(i)-
I
YL]
exp - b2(Y;Pyi)] [
Yiexp(a+Xy,)+Y,‘]
-[Y;(i)-aY,‘exp(Xy)]
x
exp
-exp(
exp [
b2(Y,-Yi) 4
b2(y, -Y,)
4
_ f) x
I
@4(Y,-Yi)
+ Y,‘exp(Xy,)
x[~A~+Tj(A7)-exp(-(I)T~(A~)]+Y;[exp(-~)T,(h7)-~A7]
+
X (13)
where
As can be seen, some of the equations (l)-(8) are not linear in the unknown functions; considering the non-linearity of the boundary conditions at 0 and d, this was evident from the very beginning. However, one can see that in the non-linear terms the concentration of complexing agent always appears, and we can suppose that, in physically interesting situations, this component is present in a great excess.
291
Thus, we can assume I$( v) and V,(y) to be constant or varying very slowly in comparison with other variables. Therefore a negligible error is introduced by taking, at each step of the calculations, these functions as constants equal to the values obtained in the previous step. In such a way, equations (l), (2) and (7) can be linearized while the system is factorized in two blocks of six and two equations [(5) and (6)J respectively. Therefore, for each value of the integration variable y, a linear non-homogeneous system in the unknown functions can be obtained. From equations (10) and (13) it can be seen that, at each step, the values of the same functions obtained already appear in the second member of the equations, so that a step-by-step procedure can be carried out. In the following part of this section we shall examine some points which refer to the choice concretely made in performing the calculations. Owing to the complexity of the obtained formulae, we gave up optimizing the integration grid which has been assumed simply equispaced with only two possibilities of successive multiplication by a factor. Now, we observed that the process was extremely sensitive even to this simple modification, so that only two successive doublings were employed in practice. On the other hand, since our semianalytical method implies only a one-dimensional numerical integration, the optimization of the grid is less critical. As for the determination of the unknown quantities in the bulk X(X*, _JJ), y(x*, Y), U-x*, y), W(x*, y), Z(x*, JJ), the formulae (35)-(39) of Ref. 2 can be treated in an analogous way. In effect, new two-dimensional auxiliary functions Oi[O/(w/x2)] and the related I; must be calculated; this point does not present any particular difficulty but only some tedious programmation effort. The grid on the x variable has also been assumed equispaced; this choice is by no means critical since x is not an integration variable. CASE STUDY
AND CHOICE OF THE INPUT
PARAMETERS
If the formulae discussed up to now are considered, even at first glance, it appears that the results of a calculation depend on a large number of parameters whose values need to be given, namely: (a) physical general data: temperature T; (b) geometrical and electrostatic quantities: membrane thickness d; ion charge n; dielectric constants E,, em; (c) diffusion coefficients: Di+ Di+j, Dj, Dij, Di; (d) the independent initial concentrations: ci+ and cj; (e) the kinetic constants: kF,i+j; ke,i+j; kF,ij; ks,ij; (f) the potential data: q,e,; U,,e,,o; V,; U,; (g) the potential-related quantities: u and t, where u is the sweep rate and t, the time corresponding to the maximum potential in the linear sweep. On the other hand the physical quantities which can be obtained are also rather numerous; in fact they include both the concentrations of the five components which
298
are involved in the process ci+, c~+~, ci, cj, cij as functions of x and t, and the current densities, at least at the electrode (jr) as well as at the membrane 1solution boundary ( j,,), as functions of time. Clearly, collecting a well-ordered and exhaustive set of all the results which can be obtained even through just a limited variation of each of the parameters, is a hopeless task which may turn out to be not interesting. Consequently, the results given below can be considered as representative of the main features of the equation system solution. In selecting these solutions, some rather intuitive considerations were taken into account. First of all we have chosen to explore the more simple physical situation, in which one can hope that the results are more easily interpretable. For that, all the calculations presented in this paper have been obtained under the hypothesis that the whole potential drop is confined to the immediate proximity of the electrode. Therefore, in our formulae we suppose everywhere U, = 0 and consequently b = 0 and Yi= 0 [case (a) of Ref. 21; as far as the time dependence of the potential drop is concerned a step function and a triangular sweep have been considered. A further paper will be devoted to the more complex case in which U, # 0 and the problem of the coupling with the Poisson equation arises. The result’s dependence on some of the parameters, especially T and n, can be presumed as obvious; moreover, while E, appears only in determining the initial charge distribution, E, is implied also in the coupling constant fi with the Poisson equation. Therefore all these data were maintained constant throughout all the calculations presented. In particular it was assumed that T = 298 K, E, = 80, cm = 2, n = 1. The membrane thickness d, which has a relevant role in determining the initial charge distribution [6], appears as a scale factor in the dynamic case except for the problem of the coupling with the Poisson equation. In the calculations of this paper the value d = 300 A was assumed. For the sake of simplicity a unique reasonable value D = 5 X lop6 cm’ s-* was chosen for all the diffusion coefficients but it does not seem to us that there is any serious reason for supposing that any suitable modification of these quantities could critically influence the results obtained. As far as the initial concentrations are concerned, there are some constraints which are to be respected; they result both from the physical meaning of the variables and from the possible arising of critical conditions for the mathematical procedure. A first condition that must be satisfied is the excess of the complexing component with respect to the charged complex in the membrane; in our case the ratio was never < 10. A second condition pertains to the charge concentration in the membrane; this quantity must remain low since, in our calculations, coupling with the Poisson equation has been neglected. On the other hand, when the coupling is taken into account, one can see that this datum rapidly becomes critical from a computational point of view. Practically we have found that for ci+j < 1O-4 M the program works. The concentration of the free ion in watery solution does not appear to be critical; we have chosen values for which 0.1 < ci+//ci+< 1. It is evident that these values have to ‘be realized by fixing suitable values for the kinetic constants (see equations (23), (24) and (26) of Ref. 2).
299
The other parameters listed in points (e) and (f) were actually varied but only some selected values were considered in order to underline the most interesting physical aspects of the solution. For instance the dependence on the kinetic parameters was considered only in the linear sweep calculations. As for the calculated quantities the main focus was on the current density at the electrode j,(t), while only selected calculations are reported for j,(t) and for the concentrations ci+(x, t), c~+~(x, t), ci(x, r), cjj(x, t). Since, in all cases the concentration of complexing agent cj(x, t), was considered to be in excess the calculations related to this quantity are not reported. Finally, as far as the time variable is concerned, the physical scale parameter is y = Di+j/d2 whi ch , in our case, can vary between - lo5 and - 10’ s-l. We found that, if we chose an initial time step - 10m6 s, the program gives reliable results not sensitive to small variations of the grid. On the other hand, we have seen that 2000 steps are quite sufficient to exhaust the more interesting aspects of the time-dependent quantities that can be calculated. We recall that the time step can be redoubled twice during the process; this procedure was automatically inserted when the relative variation for one step of the most sensitive time-dependent function j, felI under some predetermined limit (- 0.01). RESULTS
AND
DISCUSSION
(A) Constant potential
In Fig. 1 the time dependence of ji in adimensional units is given in a bilogarithmic plot, The different curves correspond to different values of the potential drop at the electrode U,,,, which is measured with respect to the standard hydrogen potential. The main features of the diagram are: (a) a drastic decrease in the current when the potential drop becomes more positive; (b) a parallel decrease in the slope; (c) a characteristic irregular form for very low values of t/r,,. This last effect appears only for the more negative values of the potential drop. While point (a) is almost obvious, considering that the current is supported by a reduction process, (b) and, mostly, (c) require further explication. As mentioned before, the flow is controlled by both the kinetic processes at the boundaries and the electron transfer at the electrode. As the drop at the electrode becomes more and more positive, the small current at the electrode allows a gradual and slow depletion of the membrane. Thus, the flow is controlled by the small concentration gradient near the electrode and the current rapidly becomes almost stationary. On the contrary, a sudden rise of the current such as occurs at the more negative values of the potential, gives rise to a transient depletion of the membrane immediately followed by a small increase in the current. In the subsequent phase, the current drop almost follows the normal t- I’* behaviour typical of a non-hindered diffusion flow; this explanation is also supported by the results reported in Fig. 3 (see below).
300
‘/re.(m”J .= 0 A= 50 0 =lOO *=125
n =l50 r=200 q=250
Fig. 1. Time dependence of the cathodic current density at the electrode for differential potential drops. j; = n.FDitjX*/d; 7 = d2/Di+j. In this diagram X* = 10m6 mol/cm3; cj = 10m4 mol/cm3; kF,i+, = lo2 cm4 SC’ mol-‘; k, i +j = lOcms-‘; k,,,j=102cms-‘; ks,ij = 0.1 cm4 s-l mol-’ which correspond to ax = 6 x 10-5; +6; a=60; a,=6xlO-‘. Wh en no different specification is given these parameters were maintained constant.
Figure 2 shows the potential dependence of j,. The plot j, versus U,,,. is semilogarithmic. The crossing of the curves in the upper part of the figure (short time) is a consequence of the transient effect already discussed; in the lower part of the diagram, the limiting current behaviour is clearly attained. On the other hand, a plot of log[(j,,,-j,)/j,] uersus U,,. (Fig. 3) shows a behaviour which can be considered only nearly linear. For the more positive potentials the slope is RT/n.F. In Fig. 4, the time variation of the ratio j,,/j, is reported on a bilogarithmic scale. Different values of j, and j, for the same time entail, of course, a variation of the total charge inside the membrane, i.e. an accumulation or a depletion. For potentials more negative than 100 mV it can be seen that the rapid depletion of the membrane charge reservoir in the first instants of the process gives rise to a subsequent temporary prevalence of j,, with respect to j,; this prevalence is quenched in the time that follows. This effect is shifted towards longer times by lowering the potential drop; it disappears for more positive potentials. In all cases the behaviour of the current in the membrane approaches the steady state for long time-periods asymptotically. Figures 5 and 6 show the influence of the relative concentration of the complexing agent c/X*. The half-wave potential Ur.e,l,Z = U,.,. at (ji =j, J2) could be
301
- 0.01
-0.1
t’= t/T,
l_jl
a = 0.28
b = 1.39 c = 2.78
d = 8.32
e =52.8 F =9x2
g = 208.3
-l.t1
a
Fig. 2. Cathodic current density at the electrode for different values of time.
-5
U,e.(m”) 1
I
125
150
175
200
1
250
Fig. 3. Plot of I@( j,,, - jI)/jI ] versus V,,,. at the electrode for different time values.
302
o= A=
0 50
l =100 q =125 n =150 A=200
Fig. 4. Time dependence
of the ratio &/j,.
expected to show a linear dependence on log cj/X* with a slope of 59 mV for each exponent unit; in effect it can be seen that this trend is attained only as a limit for a great excess of cj. This behaviour can be explained by observing Fig. 6 where the adimensional current density at the electrode for a high value of the reduced time t/~~ is reported uersus cj/X* in a semilogarithmic plot. It can be seen that for high values of cj/X* the current density limit is practically constant. The following relation can now be written for the half-wave potential at the electrode
Fig. 5. Dependence of the half-wave potential on the reduced concentration and in Fig. 6 X* = 10e7 mol/cm3.
of complexing
agent. Here
303
Fig. 6. Dependence of the reduced current density at the electrode on cj/X*
for a high value of t/7.
but one has [2]: cd,ij =
ks,ijCd,iCd,j Di (aC&> d+ k,,ij
-
kF,ij
(15)
In the region where the current is high and almost constant the second term on the right-hand side of equation (15) is markedly prevalent, and cd,ij can be assumed practically constant. Since j, is almost equal to ji (see Fig. 4), Di+(aci+/b),-9 Di@C;/aX)d+; therefore one has [2] kF,ijCd,ij = kF,i+jCo,jCo,;+- kB,i+jCo,i+j where the term ks,ijCd,i has been neglected. It follows that: c,,~+~s (kF,r+jC~,jC~,~+- kF,iiCd,ij)/ks,,+j. In this equation the second term of the right-hand side is negligible so that cO,i+jis almost proportional to cc,+. On the other hand, in an almost stationary regime ( j, ~ji), and when the current is independent of c~,~ a proportionality between c,,~+~and cd,i+j and consequently between cd,i+j and c~,~ can be inferred. The limit behviour of u r.e 1,2 for high cj must thus be a straight line with slope RT/nS. 6igures 7), 8) and 9) show the behaviour of the concentration of the different components as functions of x* and t. In Fig. 7A the rapid depletion of the membrane reservoir is evident. The effect is so drastic since a potential drop at the electrode at its maximum relative value was considered. Figure 7B shows that, for U,,,.= 125 mV, the process is much slower. Figure 8 shows the evolution of the distribution of reduced complex as a function of time. The parameters chosen impart to cij only a very small fraction of the concentration of the main component c~+~. Because of this, cij is an unbuffered quantity which is very sensitive to the rapid
1.0
a
0.8 -
e'
0.2
01
0
I
I
0.2
I
I
0.4
I
1
0.6
I
I
0.8
I
1 1.0
Fig. 7. Time dependence of the concentration of oxidized complex in the membrane. The unit for length x is the membrane thickness. d = 3 x 10e6 cm; Di+j = 5 X low6 cm2 s-‘; (7A) U,,=.= 0; (7B) V,,,,= 125 mV; ms; r/q,; a=O; b=l; c=4; d=20; d’-30; e=240; e’=270. timeunit 7,=5x1O-4
Fig. 8. Time dependence of the concentration of reduced complex in the membrane. b = 2; c = 3; d = 4; e = 5; f = 10; g - 20; h = 40.
U,, = 0; t/7:
a = 1;
variation of c~+~.Hence it shows a characteristic oscillation in the early phase of the process. FinaIly, Fig. 9 reports both the concentrations of i+ in watery solution and of i in Hg. The plot in this figure clearly shows a diffusion-controlled behaviour. (B) Linear sweep
In Fig. 10 typical cyclic triangular wave voltammetric curves ii/j,* uersus U,.,. are reported. The cathodic peak height appears to be very sensitive to the sweep rate, while the anodic one is much less dependent on this quantity. The influence. of the different kinetic constants on the current density at the
10 100 1000 10 100 1000 100 100 100 100 10 10 10 100
10 10 10 10 10 10 0.1 1 10 10 10 10 10 10
III k B.J+J 10 10 10 1 1 1 1 1 1 100 10 100 10 1
k,,
IV, v
a For the dimension of the kinetic constants see Fig. 1.
I,l;II,l; 1.2; IV,2 I,3 IIJ; V,l 11,2,111,3;V11,3 II,3 III,1 III,2 IV,l;VII,l IV,3 V,2;VI,3 VP3 VI,2 VII,2
kci+j a
I, II
Kinetic constants changed in series:
1 1 1 100 100 100 100 100 1 1 100 100 10 10
k,,j
VI, VII 0.0873 0.035 0.0645 0.0071 0.034 0.057 0.0976 0.052 0.035 0.036 0.0072 0.0072 0.0072 0.035
(i/.,/j?),.,
Electric quantities
- 62.5 25.0 87.5 - 162.5 - 100.0 - 35.0 27.5 - 37.5 - 37.5 75.0 - 125.0 - 100.0 - 100.0 -62.5
(mv)
fKe.p)ror
- 0.00074 - 0.0027 - 0.0036 - 0.0164 - 0.077 -0.117 - 0.166 -0.112 - 0.0024 - 0.0029 - 0.0244 - 0.0298 - 0.0044 - 0.014
(il., /if).n
S-‘;
187.5 200.0 237.5 - 37.5 -12.5 12.5 62.5 12.5 137.5 250.0 25.0 75.0 100.0 50.0
WV)
f-Man)
Current and potential peak values for cathodic and anodic portions of the triangular wave for different kinetic constants values. u = 250 V mol/cm3; X* = lo-’ mol/cm3
TABLE 1 cj = 10e5
307
Fig. 9. Time dependence of the concentrations of the penetrating i in Hg. The numbers on the curves are the time in ms.
ion
Fig. 10. cyclic voltammograms at different sweep rate’ values. kB,ij = 10; For the units see Fig. 1. c, = 10e5 mol/cm3.
I‘+ in water and of the reduced metal
k,i+,
=KQ
k,,pj
=lO;
k,tj -1;
308
electrode were examined in detail in the linear sweep case. The main features of the j versus U,,, plots, i.e. the cathodic and anodic peak heights with the correspondent UP values are reported in Table 1. The quantities discussed depend on at least five parameters: kF,i+j; kB,i+j; kF,ij; kBsij and the sweep rate u. It is extremely hard to map the corresponding hypersurface; thus only a group of representative patterns on it were selected. The results appearing in Table 1 can be ordered in different terms each of which has three of the four kinetic constants unchanged, while the sweep rate is constant and equal to 250 V s-l; consequently the main influence of the different partial processes on the current can be distinguished. As an example in the first series only kF,i+j is varied and assumes the values 10, 100 and 1000, respectively (rows 1, 2 and 3), while in the fourth series kF,ij is varied assuming the values 1, 10,100(rows 9,2 and 10). So the set of values of the second row appears both in the first and in the fourth series. Many cases of homogeneous chemical reactions coupled with charge transfer have been investigated [3,7-91; generally our results are at least in qualitative agreement with those quoted in the literature; for instance case III and V of Ref. 3. But one must take into account the greater complexity of our problem because of the coexistence of the two boundary conditions neither of which is at cc and of the sequence of chemical, diffusional, electrochemical and chemical steps. As far as the variation of kF,i+j is concerned (series I and II) it can be seen that the main effects are the increase of j,,, both cathodic and anodic and the corresponding positive shift of their Ur.e,p. A plot of l/j,,, versus the parameter
=lOO
kfiiy
/Ql+j =l. kF,,i j =I. k*,ij =lOO u =25O(V/S) $ = low5mol /cm3
I
0.5
I
1.0
0
Fig. 11. Current/time
I
1.5 I
cyclic voltqmograms.
t(ms)
-0.01 -
-0.03-“; -0.04Fig. 12. Same as in Fig. 11 for different values of the kinetic constants;
j, and j, are indistinguishable.
= [ kF,i+jCj/( k,,i+j{+)] - * [ see Ref. 3 equation (73)J shows a linear dependence as should be expected. Besides, the anodic shift of j,,, is - 60 mV for each tenfold increase of CL.The ratio (jl,p)on/(jl,p)cor increases with CL;but in our
CL-l
0.06
0.05
0.04
0.03
0.02
0.01
I
iv (VSP
10 Fig. 13. Sweep-rate dependence
,
20
I
30
of the peak cur&t
I
40 density at the electrode.
310
first series (I) it remains < 1 since the anodic current is essentially controlled by In the second series (II) this ratio is > 1 as,may be expected in the case of a previous chemical reaction (at the membrane 1water boundary); in fact in this case the anodic current is not limited by the recombination reaction. In the third series (III) the second kinetic constant kB,i+j was varied; one can see that, except for the maximum value of k,,i+j/kB,i+j in which the current is rather high, the linear relationship in p-l remains almost true. Typical j, uersus t plots for this case are shown in Figs. 11 and 12 where the characteristic asymmetry between the cathodic and the anodic portions of the curve is clearly shown. In Fig. 11 a small difference between the peaks of j, and j,, which is accompanied by a small time shift, is also evident. In series (IV) and (V) the constant kF,ij was varied. It can be seen that, in this case, the cathodic peak is not influenced since the parameter which was varied is related to a chemical reaction following the charge transfer. On the other hand (U,.,.),,, is shifted by an amount less than 60 mV under the ks,ij.
-_=--+
‘di+j X
kF i’j
=lOO
k*,i+j
=1
kF,ij
=l
k,,ij
= 100
lr = 250 cj =10v5
Fig. 14. Evolution in time of the concentration
(V/s) mol/cm’
of oxidized complex at the two membrane boundaries.
311
5
IO
35
00
50
Fig. 15. Evolution in time of the concentrations
0
50
00
of i+ and i in water and in Hg, respectively.
influence of the kinetic parameter (see Ref. 3, case V) and (U,,,.),, follows it almost regularly. The last series (VI, VII) show the influence of kB,ij; the expected increase of the anodic peak is clearly confirmed. Figure 13 shows the influence of the sweep rate on j,,, for a particular choice of the kinetic constants; the plot shows the expected behaviour with d j,,,/dfi decreasing to zero for high values of the sweep rate. Figures 14 and 15 show the variation in time of the concentrations of some chemical species that take part in the process. In Fig. 14 a very characteristic behaviour of c~+~is shown; practically, during the electrolysis, the membrane reservoir at first depletes and then fills up again achieving a higher level. This process is coupled with the corresponding dynamics of ci+ and ci which are shown in Fig. 15.
312 REFERENCES 1 2 3 4 5 6 7 8 9
M. Caselli, M. Mgestro, D. Pare0 and A. Traini, J. Membr. Sci., 16 (1983) 77. M. Caselh and M. Maestro, Bioelectrochem. Bioenerg., 14 (1985) 275. R.S. Nicholson and I. Shain, Anal. Chem., 36 (1964) 706. A. Erdelyi, Higher Trascendental Functions, McGraw-Hill, New York, 1953. M. Abramowitz and LA. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972. M. Caselli, M. Maestro and D. Pareo, Bioelectrochem. Bioenerg., 13 (1984) 55. C.P. Andrieux and J.M. Saveant, J. Ehxtroanal. Chem., 93 (1978) 63. C.P. Andrieux, J.M. Dumas-Bouchiat and J.M. Saveant, J. Electroanal. Chem., 123 (1981) 171. C.P. Andrieux, J.M. Dumas-Bouchiat and J.M. Saveant, J. Electroanal. Chem., 131 (1982) 1.
718-CONTROLLED-POTENTIAL ELECTROLYSIS MEMBRANE-COATED ELECTRODE PART II. NUMERICAL
M. CASELLI Dipartimento
ANALYSIS
293
AT A
AND RESULTS
and M. MAESTRO di Chimica dell’ Universirir, Via Amendola 173, 70126 Bari (Italy)
(Revised manuscript received September 19th 1984)
SUMMARY The numerical treatment of the integral equation system which resolves the problem of a one-sign charge flow followed by the discharge at an electrode across a dielectric membrane is explained. Some representative results are presented and discussed both for the cases of a constant potential and for a triangular sweep.
SPECIAL
an.
cat.
A,,
jl,,
ur.e,1/2 u r.e.,p
6s 70
SYMBOLS
USED IN THIS PAPER
as subscript refers to the voltammetric anodic branch as subscript refers to the voltammetric cathodic branch = n9=Di+j X*/d voltannnetric limiting current density peak current density in cyclic scanning voltammetry voltammetric half-wave potential peak potential in cyclic scarming voltammetry dielectric constant of the solution = d2/Di+j
For other symbols see the preceding paper [2]. INTRODUCTION
In a previous paper [l] we have described a physical situation in which charges of one sign only go through a dielectric coating a metallic electrode and discharge on it; the transport is allowed by a complexing agent confined in the dielectric which operates as a shuttle. Subsequently [2], the differential equation system which describes the phenomenology has been resolved by transforming it into an equivalent integral system. 0302-4598/85/$03.30
0 1985 Elsevier Sequoia S.A.
294
In this paper the numerical treatment which allows a concrete solution of the problem is discussed and some representative numerical results are given. The main literature which is useful in treating this problem has been discussed previously [2]; we refer to the same paper also for the employed notations and symbols. NUMERICAL
PROCEDURE
The integral equation system obtained previously [2] is the following:
(2)
- W’,+(O,$=&r)d, where I(x) = exp(x’) erfc(x) and @,(x/y) 40 in the preceding paper];
X is a Jacobian elliptic function [see Ref.
(3)
xexp
Xexp ( -4 ‘2U)[~+@3(O/i~~)])d~-exp(~)~y([YoX
exp( - $)
r,‘exp(Xr)-Y;]
C3d(O/ilrw)}dr
where the symbols have the following meaning: o = y - 7; h = a* - (b*/4); (b/2)-a and Q,(O/i?rw), Q,(O/’ IQW) are Jacobian elliptic functions.
x
(4) a=
295
(vo- Ff) It=+ : I, exp $ (- $_.)
)
y
y-
h-----&T
3 X
W, =
d+
US
(
r,
-ow 6
avZIV,
1
-(auVoXo -
I0
a,Y,)exp (
’ 4(Y--7)
1 I 6
dr
(6)
J(r
- T)]dT
;~yW;(+34[0,i~(y
(8)
6( Y; - by,) = oyYO- exVoXo
(9) These equations must be resolved together to obtain X0, Y,, Yi, Y, , Y;, V,, VI, W,, and Z,; all the other time-dependent boundary concentrations and their gradients which appear in the problem (more precisely X4, WA, W,, W;, Vi, I’{ and Z;) can be deduced directly by simple algebraic relations already given [2]. The standard way to solve the given system is to resort to a numerical integration procedure; for this, some critical points must be taken into account. Many of the expressions which appear under the integral present a singularity at the upper limit; this is clearly apparent in equations (l), (5), (6) and (7), but the situation is the same in the other equations, since the Jacobian elliptic functions Qi(O/o) show singularities for w --, 0. To overcome this trouble, we divided the integration domain in two parts, confining the singularity to the last step of the numerical integration grid, and only this last term has been integrated by parts. For instance, equation (1) was transformed in the following way: l-1 iFI [G(i) -6&(i)] + [q(Z) -6%(Z)] AT= ) ‘-’ aYYo(i)-c?[Y’(i)-6Yo(i)]
A7+
%v,(i)Jv,-v,
_~YYO(I)-S[Y~(Z)-~Y~(Z)~ G-2G
I
%l/,V)
or, by insulating the present& unknown Y,(Z), Y;(Z) and U,(Z)
o&(i)
-6[Y;(i)
-by,(i)]
(10)
296
The same procedure can be applied to the other equations in which the Jacobian functions ei appear; for these one meets the problem of the explicit evaluation and also that of their singularities. The following series have been employed [4,5]: @,(O/iqnw)=2
f
exp[-n(n+l)+$r2W]
n=O
B,(O/iaw)
= 1+ 2 5
(11)
exp( -n29r2w)
n-l
@d(O/i?rw)=1+2E(-l)“exp(-n2?rw) n-1
In order to eliminate the singularities, we have resorted to an evaluation of some auxiliary functions which appear in the integration by parts. More precisely we have calculated, also by series, the following functions: T2( y ) =
-
iye2(O/ino)dw
(12)
and the analogous T3 and T4. As an example, equation (3) becomes: Y;(Z)Ar-
,,Z)[$
[Y,(i)+
X
I
AT- Ts(Ar)] - Y,(Z) exp( -i)T,(Ar)=
%k@v,)-
:-@,(y,-y,)
x[Y,(i)-
I
YL]
exp - b2(Y;Pyi)] [
Yiexp(a+Xy,)+Y,‘]
-[Y;(i)-aY,‘exp(Xy)]
x
exp
-exp(
exp [
b2(Y,-Yi) 4
b2(y, -Y,)
4
_ f) x
I
@4(Y,-Yi)
+ Y,‘exp(Xy,)
x[~A~+Tj(A7)-exp(-(I)T~(A~)]+Y;[exp(-~)T,(h7)-~A7]
+
X (13)
where
As can be seen, some of the equations (l)-(8) are not linear in the unknown functions; considering the non-linearity of the boundary conditions at 0 and d, this was evident from the very beginning. However, one can see that in the non-linear terms the concentration of complexing agent always appears, and we can suppose that, in physically interesting situations, this component is present in a great excess.
291
Thus, we can assume I$( v) and V,(y) to be constant or varying very slowly in comparison with other variables. Therefore a negligible error is introduced by taking, at each step of the calculations, these functions as constants equal to the values obtained in the previous step. In such a way, equations (l), (2) and (7) can be linearized while the system is factorized in two blocks of six and two equations [(5) and (6)J respectively. Therefore, for each value of the integration variable y, a linear non-homogeneous system in the unknown functions can be obtained. From equations (10) and (13) it can be seen that, at each step, the values of the same functions obtained already appear in the second member of the equations, so that a step-by-step procedure can be carried out. In the following part of this section we shall examine some points which refer to the choice concretely made in performing the calculations. Owing to the complexity of the obtained formulae, we gave up optimizing the integration grid which has been assumed simply equispaced with only two possibilities of successive multiplication by a factor. Now, we observed that the process was extremely sensitive even to this simple modification, so that only two successive doublings were employed in practice. On the other hand, since our semianalytical method implies only a one-dimensional numerical integration, the optimization of the grid is less critical. As for the determination of the unknown quantities in the bulk X(X*, _JJ), y(x*, Y), U-x*, y), W(x*, y), Z(x*, JJ), the formulae (35)-(39) of Ref. 2 can be treated in an analogous way. In effect, new two-dimensional auxiliary functions Oi[O/(w/x2)] and the related I; must be calculated; this point does not present any particular difficulty but only some tedious programmation effort. The grid on the x variable has also been assumed equispaced; this choice is by no means critical since x is not an integration variable. CASE STUDY
AND CHOICE OF THE INPUT
PARAMETERS
If the formulae discussed up to now are considered, even at first glance, it appears that the results of a calculation depend on a large number of parameters whose values need to be given, namely: (a) physical general data: temperature T; (b) geometrical and electrostatic quantities: membrane thickness d; ion charge n; dielectric constants E,, em; (c) diffusion coefficients: Di+ Di+j, Dj, Dij, Di; (d) the independent initial concentrations: ci+ and cj; (e) the kinetic constants: kF,i+j; ke,i+j; kF,ij; ks,ij; (f) the potential data: q,e,; U,,e,,o; V,; U,; (g) the potential-related quantities: u and t, where u is the sweep rate and t, the time corresponding to the maximum potential in the linear sweep. On the other hand the physical quantities which can be obtained are also rather numerous; in fact they include both the concentrations of the five components which
298
are involved in the process ci+, c~+~, ci, cj, cij as functions of x and t, and the current densities, at least at the electrode (jr) as well as at the membrane 1solution boundary ( j,,), as functions of time. Clearly, collecting a well-ordered and exhaustive set of all the results which can be obtained even through just a limited variation of each of the parameters, is a hopeless task which may turn out to be not interesting. Consequently, the results given below can be considered as representative of the main features of the equation system solution. In selecting these solutions, some rather intuitive considerations were taken into account. First of all we have chosen to explore the more simple physical situation, in which one can hope that the results are more easily interpretable. For that, all the calculations presented in this paper have been obtained under the hypothesis that the whole potential drop is confined to the immediate proximity of the electrode. Therefore, in our formulae we suppose everywhere U, = 0 and consequently b = 0 and Yi= 0 [case (a) of Ref. 21; as far as the time dependence of the potential drop is concerned a step function and a triangular sweep have been considered. A further paper will be devoted to the more complex case in which U, # 0 and the problem of the coupling with the Poisson equation arises. The result’s dependence on some of the parameters, especially T and n, can be presumed as obvious; moreover, while E, appears only in determining the initial charge distribution, E, is implied also in the coupling constant fi with the Poisson equation. Therefore all these data were maintained constant throughout all the calculations presented. In particular it was assumed that T = 298 K, E, = 80, cm = 2, n = 1. The membrane thickness d, which has a relevant role in determining the initial charge distribution [6], appears as a scale factor in the dynamic case except for the problem of the coupling with the Poisson equation. In the calculations of this paper the value d = 300 A was assumed. For the sake of simplicity a unique reasonable value D = 5 X lop6 cm’ s-* was chosen for all the diffusion coefficients but it does not seem to us that there is any serious reason for supposing that any suitable modification of these quantities could critically influence the results obtained. As far as the initial concentrations are concerned, there are some constraints which are to be respected; they result both from the physical meaning of the variables and from the possible arising of critical conditions for the mathematical procedure. A first condition that must be satisfied is the excess of the complexing component with respect to the charged complex in the membrane; in our case the ratio was never < 10. A second condition pertains to the charge concentration in the membrane; this quantity must remain low since, in our calculations, coupling with the Poisson equation has been neglected. On the other hand, when the coupling is taken into account, one can see that this datum rapidly becomes critical from a computational point of view. Practically we have found that for ci+j < 1O-4 M the program works. The concentration of the free ion in watery solution does not appear to be critical; we have chosen values for which 0.1 < ci+//ci+< 1. It is evident that these values have to ‘be realized by fixing suitable values for the kinetic constants (see equations (23), (24) and (26) of Ref. 2).
299
The other parameters listed in points (e) and (f) were actually varied but only some selected values were considered in order to underline the most interesting physical aspects of the solution. For instance the dependence on the kinetic parameters was considered only in the linear sweep calculations. As for the calculated quantities the main focus was on the current density at the electrode j,(t), while only selected calculations are reported for j,(t) and for the concentrations ci+(x, t), c~+~(x, t), ci(x, r), cjj(x, t). Since, in all cases the concentration of complexing agent cj(x, t), was considered to be in excess the calculations related to this quantity are not reported. Finally, as far as the time variable is concerned, the physical scale parameter is y = Di+j/d2 whi ch , in our case, can vary between - lo5 and - 10’ s-l. We found that, if we chose an initial time step - 10m6 s, the program gives reliable results not sensitive to small variations of the grid. On the other hand, we have seen that 2000 steps are quite sufficient to exhaust the more interesting aspects of the time-dependent quantities that can be calculated. We recall that the time step can be redoubled twice during the process; this procedure was automatically inserted when the relative variation for one step of the most sensitive time-dependent function j, felI under some predetermined limit (- 0.01). RESULTS
AND
DISCUSSION
(A) Constant potential
In Fig. 1 the time dependence of ji in adimensional units is given in a bilogarithmic plot, The different curves correspond to different values of the potential drop at the electrode U,,,, which is measured with respect to the standard hydrogen potential. The main features of the diagram are: (a) a drastic decrease in the current when the potential drop becomes more positive; (b) a parallel decrease in the slope; (c) a characteristic irregular form for very low values of t/r,,. This last effect appears only for the more negative values of the potential drop. While point (a) is almost obvious, considering that the current is supported by a reduction process, (b) and, mostly, (c) require further explication. As mentioned before, the flow is controlled by both the kinetic processes at the boundaries and the electron transfer at the electrode. As the drop at the electrode becomes more and more positive, the small current at the electrode allows a gradual and slow depletion of the membrane. Thus, the flow is controlled by the small concentration gradient near the electrode and the current rapidly becomes almost stationary. On the contrary, a sudden rise of the current such as occurs at the more negative values of the potential, gives rise to a transient depletion of the membrane immediately followed by a small increase in the current. In the subsequent phase, the current drop almost follows the normal t- I’* behaviour typical of a non-hindered diffusion flow; this explanation is also supported by the results reported in Fig. 3 (see below).
300
‘/re.(m”J .= 0 A= 50 0 =lOO *=125
n =l50 r=200 q=250
Fig. 1. Time dependence of the cathodic current density at the electrode for differential potential drops. j; = n.FDitjX*/d; 7 = d2/Di+j. In this diagram X* = 10m6 mol/cm3; cj = 10m4 mol/cm3; kF,i+, = lo2 cm4 SC’ mol-‘; k, i +j = lOcms-‘; k,,,j=102cms-‘; ks,ij = 0.1 cm4 s-l mol-’ which correspond to ax = 6 x 10-5; +6; a=60; a,=6xlO-‘. Wh en no different specification is given these parameters were maintained constant.
Figure 2 shows the potential dependence of j,. The plot j, versus U,,,. is semilogarithmic. The crossing of the curves in the upper part of the figure (short time) is a consequence of the transient effect already discussed; in the lower part of the diagram, the limiting current behaviour is clearly attained. On the other hand, a plot of log[(j,,,-j,)/j,] uersus U,,. (Fig. 3) shows a behaviour which can be considered only nearly linear. For the more positive potentials the slope is RT/n.F. In Fig. 4, the time variation of the ratio j,,/j, is reported on a bilogarithmic scale. Different values of j, and j, for the same time entail, of course, a variation of the total charge inside the membrane, i.e. an accumulation or a depletion. For potentials more negative than 100 mV it can be seen that the rapid depletion of the membrane charge reservoir in the first instants of the process gives rise to a subsequent temporary prevalence of j,, with respect to j,; this prevalence is quenched in the time that follows. This effect is shifted towards longer times by lowering the potential drop; it disappears for more positive potentials. In all cases the behaviour of the current in the membrane approaches the steady state for long time-periods asymptotically. Figures 5 and 6 show the influence of the relative concentration of the complexing agent c/X*. The half-wave potential Ur.e,l,Z = U,.,. at (ji =j, J2) could be
301
- 0.01
-0.1
t’= t/T,
l_jl
a = 0.28
b = 1.39 c = 2.78
d = 8.32
e =52.8 F =9x2
g = 208.3
-l.t1
a
Fig. 2. Cathodic current density at the electrode for different values of time.
-5
U,e.(m”) 1
I
125
150
175
200
1
250
Fig. 3. Plot of I@( j,,, - jI)/jI ] versus V,,,. at the electrode for different time values.
302
o= A=
0 50
l =100 q =125 n =150 A=200
Fig. 4. Time dependence
of the ratio &/j,.
expected to show a linear dependence on log cj/X* with a slope of 59 mV for each exponent unit; in effect it can be seen that this trend is attained only as a limit for a great excess of cj. This behaviour can be explained by observing Fig. 6 where the adimensional current density at the electrode for a high value of the reduced time t/~~ is reported uersus cj/X* in a semilogarithmic plot. It can be seen that for high values of cj/X* the current density limit is practically constant. The following relation can now be written for the half-wave potential at the electrode
Fig. 5. Dependence of the half-wave potential on the reduced concentration and in Fig. 6 X* = 10e7 mol/cm3.
of complexing
agent. Here
303
Fig. 6. Dependence of the reduced current density at the electrode on cj/X*
for a high value of t/7.
but one has [2]: cd,ij =
ks,ijCd,iCd,j Di (aC&> d+ k,,ij
-
kF,ij
(15)
In the region where the current is high and almost constant the second term on the right-hand side of equation (15) is markedly prevalent, and cd,ij can be assumed practically constant. Since j, is almost equal to ji (see Fig. 4), Di+(aci+/b),-9 Di@C;/aX)d+; therefore one has [2] kF,ijCd,ij = kF,i+jCo,jCo,;+- kB,i+jCo,i+j where the term ks,ijCd,i has been neglected. It follows that: c,,~+~s (kF,r+jC~,jC~,~+- kF,iiCd,ij)/ks,,+j. In this equation the second term of the right-hand side is negligible so that cO,i+jis almost proportional to cc,+. On the other hand, in an almost stationary regime ( j, ~ji), and when the current is independent of c~,~ a proportionality between c,,~+~and cd,i+j and consequently between cd,i+j and c~,~ can be inferred. The limit behviour of u r.e 1,2 for high cj must thus be a straight line with slope RT/nS. 6igures 7), 8) and 9) show the behaviour of the concentration of the different components as functions of x* and t. In Fig. 7A the rapid depletion of the membrane reservoir is evident. The effect is so drastic since a potential drop at the electrode at its maximum relative value was considered. Figure 7B shows that, for U,,,.= 125 mV, the process is much slower. Figure 8 shows the evolution of the distribution of reduced complex as a function of time. The parameters chosen impart to cij only a very small fraction of the concentration of the main component c~+~. Because of this, cij is an unbuffered quantity which is very sensitive to the rapid
1.0
a
0.8 -
e'
0.2
01
0
I
I
0.2
I
I
0.4
I
1
0.6
I
I
0.8
I
1 1.0
Fig. 7. Time dependence of the concentration of oxidized complex in the membrane. The unit for length x is the membrane thickness. d = 3 x 10e6 cm; Di+j = 5 X low6 cm2 s-‘; (7A) U,,=.= 0; (7B) V,,,,= 125 mV; ms; r/q,; a=O; b=l; c=4; d=20; d’-30; e=240; e’=270. timeunit 7,=5x1O-4
Fig. 8. Time dependence of the concentration of reduced complex in the membrane. b = 2; c = 3; d = 4; e = 5; f = 10; g - 20; h = 40.
U,, = 0; t/7:
a = 1;
variation of c~+~.Hence it shows a characteristic oscillation in the early phase of the process. FinaIly, Fig. 9 reports both the concentrations of i+ in watery solution and of i in Hg. The plot in this figure clearly shows a diffusion-controlled behaviour. (B) Linear sweep
In Fig. 10 typical cyclic triangular wave voltammetric curves ii/j,* uersus U,.,. are reported. The cathodic peak height appears to be very sensitive to the sweep rate, while the anodic one is much less dependent on this quantity. The influence. of the different kinetic constants on the current density at the
10 100 1000 10 100 1000 100 100 100 100 10 10 10 100
10 10 10 10 10 10 0.1 1 10 10 10 10 10 10
III k B.J+J 10 10 10 1 1 1 1 1 1 100 10 100 10 1
k,,
IV, v
a For the dimension of the kinetic constants see Fig. 1.
I,l;II,l; 1.2; IV,2 I,3 IIJ; V,l 11,2,111,3;V11,3 II,3 III,1 III,2 IV,l;VII,l IV,3 V,2;VI,3 VP3 VI,2 VII,2
kci+j a
I, II
Kinetic constants changed in series:
1 1 1 100 100 100 100 100 1 1 100 100 10 10
k,,j
VI, VII 0.0873 0.035 0.0645 0.0071 0.034 0.057 0.0976 0.052 0.035 0.036 0.0072 0.0072 0.0072 0.035
(i/.,/j?),.,
Electric quantities
- 62.5 25.0 87.5 - 162.5 - 100.0 - 35.0 27.5 - 37.5 - 37.5 75.0 - 125.0 - 100.0 - 100.0 -62.5
(mv)
fKe.p)ror
- 0.00074 - 0.0027 - 0.0036 - 0.0164 - 0.077 -0.117 - 0.166 -0.112 - 0.0024 - 0.0029 - 0.0244 - 0.0298 - 0.0044 - 0.014
(il., /if).n
S-‘;
187.5 200.0 237.5 - 37.5 -12.5 12.5 62.5 12.5 137.5 250.0 25.0 75.0 100.0 50.0
WV)
f-Man)
Current and potential peak values for cathodic and anodic portions of the triangular wave for different kinetic constants values. u = 250 V mol/cm3; X* = lo-’ mol/cm3
TABLE 1 cj = 10e5
307
Fig. 9. Time dependence of the concentrations of the penetrating i in Hg. The numbers on the curves are the time in ms.
ion
Fig. 10. cyclic voltammograms at different sweep rate’ values. kB,ij = 10; For the units see Fig. 1. c, = 10e5 mol/cm3.
I‘+ in water and of the reduced metal
k,i+,
=KQ
k,,pj
=lO;
k,tj -1;
308
electrode were examined in detail in the linear sweep case. The main features of the j versus U,,, plots, i.e. the cathodic and anodic peak heights with the correspondent UP values are reported in Table 1. The quantities discussed depend on at least five parameters: kF,i+j; kB,i+j; kF,ij; kBsij and the sweep rate u. It is extremely hard to map the corresponding hypersurface; thus only a group of representative patterns on it were selected. The results appearing in Table 1 can be ordered in different terms each of which has three of the four kinetic constants unchanged, while the sweep rate is constant and equal to 250 V s-l; consequently the main influence of the different partial processes on the current can be distinguished. As an example in the first series only kF,i+j is varied and assumes the values 10, 100 and 1000, respectively (rows 1, 2 and 3), while in the fourth series kF,ij is varied assuming the values 1, 10,100(rows 9,2 and 10). So the set of values of the second row appears both in the first and in the fourth series. Many cases of homogeneous chemical reactions coupled with charge transfer have been investigated [3,7-91; generally our results are at least in qualitative agreement with those quoted in the literature; for instance case III and V of Ref. 3. But one must take into account the greater complexity of our problem because of the coexistence of the two boundary conditions neither of which is at cc and of the sequence of chemical, diffusional, electrochemical and chemical steps. As far as the variation of kF,i+j is concerned (series I and II) it can be seen that the main effects are the increase of j,,, both cathodic and anodic and the corresponding positive shift of their Ur.e,p. A plot of l/j,,, versus the parameter
=lOO
kfiiy
/Ql+j =l. kF,,i j =I. k*,ij =lOO u =25O(V/S) $ = low5mol /cm3
I
0.5
I
1.0
0
Fig. 11. Current/time
I
1.5 I
cyclic voltqmograms.
t(ms)
-0.01 -
-0.03-“; -0.04Fig. 12. Same as in Fig. 11 for different values of the kinetic constants;
j, and j, are indistinguishable.
= [ kF,i+jCj/( k,,i+j{+)] - * [ see Ref. 3 equation (73)J shows a linear dependence as should be expected. Besides, the anodic shift of j,,, is - 60 mV for each tenfold increase of CL.The ratio (jl,p)on/(jl,p)cor increases with CL;but in our
CL-l
0.06
0.05
0.04
0.03
0.02
0.01
I
iv (VSP
10 Fig. 13. Sweep-rate dependence
,
20
I
30
of the peak cur&t
I
40 density at the electrode.
310
first series (I) it remains < 1 since the anodic current is essentially controlled by In the second series (II) this ratio is > 1 as,may be expected in the case of a previous chemical reaction (at the membrane 1water boundary); in fact in this case the anodic current is not limited by the recombination reaction. In the third series (III) the second kinetic constant kB,i+j was varied; one can see that, except for the maximum value of k,,i+j/kB,i+j in which the current is rather high, the linear relationship in p-l remains almost true. Typical j, uersus t plots for this case are shown in Figs. 11 and 12 where the characteristic asymmetry between the cathodic and the anodic portions of the curve is clearly shown. In Fig. 11 a small difference between the peaks of j, and j,, which is accompanied by a small time shift, is also evident. In series (IV) and (V) the constant kF,ij was varied. It can be seen that, in this case, the cathodic peak is not influenced since the parameter which was varied is related to a chemical reaction following the charge transfer. On the other hand (U,.,.),,, is shifted by an amount less than 60 mV under the ks,ij.
-_=--+
‘di+j X
kF i’j
=lOO
k*,i+j
=1
kF,ij
=l
k,,ij
= 100
lr = 250 cj =10v5
Fig. 14. Evolution in time of the concentration
(V/s) mol/cm’
of oxidized complex at the two membrane boundaries.
311
5
IO
35
00
50
Fig. 15. Evolution in time of the concentrations
0
50
00
of i+ and i in water and in Hg, respectively.
influence of the kinetic parameter (see Ref. 3, case V) and (U,,,.),, follows it almost regularly. The last series (VI, VII) show the influence of kB,ij; the expected increase of the anodic peak is clearly confirmed. Figure 13 shows the influence of the sweep rate on j,,, for a particular choice of the kinetic constants; the plot shows the expected behaviour with d j,,,/dfi decreasing to zero for high values of the sweep rate. Figures 14 and 15 show the variation in time of the concentrations of some chemical species that take part in the process. In Fig. 14 a very characteristic behaviour of c~+~is shown; practically, during the electrolysis, the membrane reservoir at first depletes and then fills up again achieving a higher level. This process is coupled with the corresponding dynamics of ci+ and ci which are shown in Fig. 15.
312 REFERENCES 1 2 3 4 5 6 7 8 9
M. Caselli, M. Mgestro, D. Pare0 and A. Traini, J. Membr. Sci., 16 (1983) 77. M. Caselh and M. Maestro, Bioelectrochem. Bioenerg., 14 (1985) 275. R.S. Nicholson and I. Shain, Anal. Chem., 36 (1964) 706. A. Erdelyi, Higher Trascendental Functions, McGraw-Hill, New York, 1953. M. Abramowitz and LA. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972. M. Caselli, M. Maestro and D. Pareo, Bioelectrochem. Bioenerg., 13 (1984) 55. C.P. Andrieux and J.M. Saveant, J. Ehxtroanal. Chem., 93 (1978) 63. C.P. Andrieux, J.M. Dumas-Bouchiat and J.M. Saveant, J. Electroanal. Chem., 123 (1981) 171. C.P. Andrieux, J.M. Dumas-Bouchiat and J.M. Saveant, J. Electroanal. Chem., 131 (1982) 1.