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Is cyclic voltammetry above a few hundred kilovolts per second still cyclic voltammetry?
335
J. Electroanal. Chem., 296 (1990) 335-358 Elsevier Sequoia S.A., Lausanne
Is cyclic voltammetry above a few hundred per second still cyclic voltammetry? * Christian
Amatore
** and Christine
Ecole Normale SupPrieure, Laboratoire 75231 Paris Cedex 05 (France) (Received
kilovolts
Lefrou
de Chimie, URA CNRS 1110, 24, rue Lhomond,
12 June 1990; in revised form 30 July 1990)
Abstract The development of ultramicroelectrodes and fast potentiostats in very recent years has allowed the investigation of electrochemical kinetics in the submicrosecond time scale by means of cyclic voltammetry at scan rates in the megavolt per second time scale. Extraction of kinetic data from such cyclic voltammograms, after their correction for distortions arising from effects due to instrumental bandpass, ohmic drop and capacitive charging currents, has been performed using classical theories for cyclic voltammetry. However, application of these classical theories supposes that the diffusion layer and the double layer can be separated theoretically. It is shown here that this classical assumption is no longer valid when scan rates in the megavolt per second range are considered, since the two layers, being then of similar sixes, are no longer physically separable. The ensuing distortions of cyclic voltammograms, with respect to the classical theories, are examined for a single electron transfer mechanism. It is shown that the cyclic voltammograms should normally be extremely dependent on their location with respect to the point of zero charge.
INTRODUCTION
Cyclic voltammetry has proven to be one of the most versatile and most sensitive methods for the investigation of electrochemical kinetics. Its time scale domain, which extended a few years ago to a few tens of microseconds, has been greatly improved by the recent availability of ultramicroelectrodes [l]. Thus, cyclic voltammetry at scan rates in the kilovolt per second [2-51, and then in the hundreds of kilovolt per second range [6-81 has allowed this method to be used for the generation and detection of intermediates with lifetimes of a fews tens of a nanosecond [6]. In a pursuit of shorter and shorter time scales, several authors have
l l
Presented at the J. Heyrovsky Centennial Congress * To whom correspondence should be addressed.
0022-0728/90/$03.50
0 1990 - Elsevier Sequoia
on Polarography,
S.A.
Prague,
20-25
August
1990.
336
reported the experimental feasibility of cyclic voltammetry at scan rates in the range of a few megavolts per second [9,10]. However, at such scan rates, even at ultramicroelectrodes, the cyclic voltammograms are distorted experimentally by instrumental and resistive phenomena. The recent design of ultrafast potentiostats [11,12] with on-line compensation of ohmic drop has allowed the recording of virtually undistorted cyclic voltammograms at scan rates exceeding the hundreds of kilovolts per second range. However, above this limit, instrumental limitations require that the faradaic information be extracted from the experimental voltammogram, in a similar way as has been done at lower scan rates, i.e. via simulation [7-10,131 or deconvolution procedures [13,14]. To the best of our knowledge, these simulation or deconvolution procedures have been based upon classical theories of cyclic voltammetry [15-171 even at scan rates of several megavolts per second. These classical theories for cyclic voltammetry suppose that the concentration profiles of the reactants, intermediates and products are controlled mainly by pure diffusion and kinetics. This amounts to considering (see below) that the sizes of the two layers, the double layer, &, and the diffusion layer, Sdir, are sufficiently different for the double layer to be vanishingly small with respect to the diffusion layer, i.e.: S,, +C &,. Under these conditions, the effects due to the double layer are taken into account via a simple correction of the electrode boundary condition, viz. the Frumkin correction [l&23]. However, such a situation is clearly not met when the diffusion layers are made thinner and thinner, i.e. when the scan rates are made larger and larger. Within the classical theory of cyclic voltammetry, the forward peak of a cyclic voltammogram recorded at a scan rate u corresponds to a diffusion layer of the order of (DRT/Fu)‘/~ [17], i.e. at room temperature and for an average diffusion coefficient of ca. 5 x lop6 cm2 s-‘, 6 = 3.5~“~ pm when u is expressed in V s-l. Therefore at e.g. u = l,OOO,OOOV s-l, the diffusion layer at the forward peak is of the order of 3.5 nm, that is comparable to the thickness of the diffuse layer under the usual electrochemical conditions in organic solvents. The transport equations in the two layers, i.e. the diffuse layer and the diffusion layer, obviously cannot be separated theoretically under such circumstances. This is clear evidence that the classical theories for cyclic voltammetry are no longer valid at such scan rates. It is the purpose of this preliminary work to evaluate theoretically the experimental effects on cyclic voltammograms due to the intimate coupling between the two layers. This will be worked out in the text for a single electron transfer reduction mechanism of a neutral molecule (n = 1, I, = 0, z,, = -l), i.e. for a case corresponding to the usual conditions found in organic electrochemistry: A=a
+
n
e-+
B’b=‘.-”
(E”, k;, a)
whereas Appendix 1 gives the corresponding analysis for the cases corresponding to any value of z, and n (zr, = z, - n). The analysis presented in the text will consider the two limiting situations: (i) fast electron transfer and (ii) quasi-reversible electron transfer. Although the limiting situation (i) is clearly not realistic at the scan rates
331
considered in this study owing to the exceedingly high values of kp require, we wanted to examine it with the purpose of demonstrating that effects predicted hereafter originate from concentration profile distortions not related only to electron transfer kinetics, as is the case for the correction.
it would the main and are Frumkin
RESULTS
Formulation of the problem At electrodes of radii in the micrometer range, and for the large scan rates considered in this study, diffusion can be considered as being planar and semi-infinite. Therefore in the following we will not consider any effect of radial diffusion that will possibly arise at smaller scan rates [l]. Similarly, the double layer, whose size is considerably smaller than that of the electrodes considered here, will be treated within a one-dimension approximation. It will be described by a classical version of the Gouy-Chapman-Stern model [24]. It is also assumed that the resulting electrical potential distribution within the diffuse layer is not affected by the faradaic phenomena [24]. Owing to the fast relaxation times of the double layer [25], this hypothesis is normally valid for any scan rate accessible experimentally today (u < 10’ V s-l). Therefore the electrical potential considered in this study is given as a function of distance, x, from the electrode surface by: x = 0;
@=‘J+,=E-E,,
(2)
o
‘p=~O+w~,)(%-%)
(3)
x > x2:
tanh( FO/4RT)
= tanh( F@,/4RT) X
exp (-(x
- x~)(~c~F~/~~,RT)~‘*)
(4)
where x2 is the location of the OHP, cs the 1: 1 supporting electrolyte bulk concentration, cs the relative dielectric constant of the medium *, E the electrode potential, Epzc the potential at the point of zero charge, and 9, the electrical potential in the OHP, as given by the solution of: a, = Q0 - x~(~RTc”/E~Q)~‘*
sinh( F@,/2RT)
(5)
Owing to the dimensions of organic molecules with respect to x2, no diffusionmigration process is considered within the compact layer, the OHP being taken as the site of the electron transfer. Therefore in the following, x2 is chosen as the origin of the distance from the electrode, the combination of the electrode and the compact layer being referred to as the “electrode surface”. Introducing thus x* =
* C, is supposed to be constant and equal to its value in the bulk of the solution. For all the figures and numerical applications given in the text, E, = 37.5 was taken to represent acetonitrile. Similarly no possible effect arising from the “discreteness-of-charges” [26] will be considered.
338
x - x2 allows one to describe the concentration profiles of the reactant and the product by the following partial derivative equations, valid for any values of z, and
‘b = z, - n: ~[A]/St=D${S[A]/6x*
+ (z,F/RT)[A](i3@/Gx*)}/Sx*
(6)
and : S[B]/Gt
=D,6{
6[B]/6x*
+ (zbF/RZ-)[B](S@/Gx*)}/6x*
associated to the following boundary values at x* = 0, i.e. at the OHP):
conditions
(7)
(where
the subscript
0 refers
to
[A] = co, [B] = 0
(8)
t>O,x*-+m:
[A] + co, [B] + 0
(9)
t>o,
4{S[A]/Sx*
t=o,
x* >o:
x*=0:
+ (z,F/RT)[A](Scp/Gx*)}o
+Db{ 6[B]/6x* t>o,
x*=0:
Da{
+dA]/aX* =
+ +
(Z,F/RT)[B](6@/6X*)},
(lo)
}
(11)
(Z,F/RT)[A](69/6X*)},
k: exp[ - anF( E - 9, - E”)/RT] x { [Alo - PI0 exp[ nF( E - a2 - E’)/RT]
For the particular case where a Nemstian electron when kp B (DFu/RT)“~), condition (11) is replaced that can be expressed at the OHP as: t>o,
= 0
transfer is considered by a modified Nernst
[Alo = [Blo exp[ nF( E - Q2 - E “)/RT]
x’=O:
The Faradaic
current
i, = nFAD,{ G[A]/Sx*
(i.e. law,
(11’)
is then given by: + (z,F/RT)[A](6@/6x*)},
(12)
In the following we will assume that the diffusion coefficients of A and B are identical (Da = D, = D), and that they remain insensitive to the variation of charge density and ionic strength in the diffuse layer. Let us introduce the following parameters: K = F(~c”/c~~, RT)‘12, and P = K( DRT/Fv)“‘, which represent respectively the reciprocal Debye length and the ratio between the diffusion layer and diffuse layer thicknesses. Introducing then the following dimensionless variables: concentrations
:
a = [Al/c’,
b = [ B]/c”
electrode
potential:
E = - nF( E - E o )/RT
electrical
potentials:
C#J = FQ/RT,
C/J,= F@,/RT,
space :
y=x*(nFu/DRT)“‘,
time :
r = t(nFv/RT)
Faradaic
current :
rate constant:
$,,,, = F( E o - E,,,)/RT
y*=Kx*,
y:=Kx2
4 = iF/[ nFAc’( nFuD/RT)“2] A = kf( RT/nFuD)“2,
A* = kf/KD
= A/r
339
allows one to rewrite the system of partial conditions in eqns. (6)-(121, as follows: &l/67=
I%{
6a/6y*
6b/67=r%{Sb/Gy*
derivative
equations
and
boundary
+ zaa(6~/6y*)}/6y*
(13)
+ (z,-n)b(S@y*)}/Gy*
(14)
with: 7=0,
y* >o:
a=l,
b=O
(15)
7>0,
y*-,cc:
u+l,
b+O
(16)
7>0,
y*=o:
{Way*
+z,4W~Y*)}O
= +{6b/6y*+(z,-n)b(6~/6y*)},=O 7>0,
y* =o:
{WSY*
+z,4%/~Y*)},
= A* exp[ 45 or, when 7>0,
+ nG2)] { a0 - b0 exp[ - (t + M2)3 }
A* x=- 1, i.e. when the electron
y* =o:
The Faradaic
a,=& current
(17)
transfer
exp[-(5+W2)1
step is considered
(18)
as Nernstian: (18’ )
is then given by:
rCI=r{6a/6Y*+z,a(S~/GY*)},
(19)
The resolution of the above system and boundary conditions requires the independent formulation of the dimensionless electrical potential, +, as a function of the distance, y *, from the electrode. This is obtained by solving the following equation valid for y * > 0: tanh( G/4) = tanh( +,/4)
exp( -Y * )
where (p2 is given by the implicit +2
=
‘ppzc
-
(5/n
> -
2~;
(20)
equation:
sirW (P2/2)
(21)
In the following we will focus on the special case where z, = 0 and n = f 1, since it corresponds to the most general situation encountered in organic electrochemistry. Transposition to an oxidation being obvious, the following presentation will be restricted to the case of a single electron transfer reduction (n = 1) of a neutral molecule (za = 0, zt, = - 1). However, Appendix 1 gives the corresponding results for any values of I, and n. Although all the equations given in the text or in Appendix 1 are valid for any electrochemical method provided that the corresponding relationship between 5 (or 4) and 7 is introduced, we will restrict our presentation of the different effects related to I to the case of cyclic voltammetry. Also, it should be emphasized that the various simulated voltammograms presented in this text correspond to a strict triangular variation of the potential with time. This supposes that ohmic drop and resulting phenomena are negligible or have been compensated electronically or via any appropriate treatment of the experimental cyclic voltammograms.
340
In the general case, i.e. for any value of I, the whole set of equations and boundary conditions (eqns. 13-21) can be solved via an explicit finite difference method. However, we want to establish first that the above formulation results in a classical one, as soon as l? + cc, i.e. as soon as the dimension of the diffusion layer greatly exceeds that of the diffuse layer. This will serve also as a demonstration of the validity of the classical theories - which are based upon the theoretical separation of the two layers (see above) - under such conditions. Limiting situation for r + 00 Since l? = K( DRT/Fv)‘12 compares the dimension of the diffusion layer, i.e. ca. ‘I2 , to that of the diffuse layer, i.e. ca K-’ [22], the condition I + cc, (DRT/Fv) simply expresses that the diffuse layer is a vanishingly small fraction of the diffusion layer. On the other hand, (6a/ST) = (RT/Fvc”)(8[A]/&) and (G/87) = (RT/ Fuc”)(6[B]/8t), as given in eqns. (13) and (14) have to remain finite since they represent the dimensionless time variations of [A] and [B], respectively. Therefore when I+ -+ cc, the partial derivatives with respect to y*, in eqns. (13) and (14) have to be vanishingly small for any value of y * comparable to unity, i.e. in the region of space where the electrical potential, $I, varies. When considerably larger values of y * are considered, y = y */I = x * ( Fv/DRT)‘j2 is introduced to yield: (&a/ST)
= 6’a/6y2
(22)
= 6%/Sy2
(23)
and : (Sb/&)
since then (&/a~*) = 0 for y* z+ 1. This shows that when I’ + cc, the system of partial derivative equations in eqns. (13) and (14) can be replaced by a set of two simplified systems, each one being valid within a given distance from the electrode. In the region corresponding to the diffuse layer (y * of the order of a few units or less) one has: S2a/6y
*2 = 0
(24)
and: 6{(6b/6y*)
-b(Gcp/Gy*)}/Sy*
= 0
(25)
whereas the classical diffusion equations (22) and (23) apply outside the diffuse layer, i.e. when y* x=- CL*= 1. Classical integration of the latter shows that a, and 4 at any point, y = Z.L = p*/T -=K1, of the diffusion layer are given by: ap = 1 - Z, = 1 - Z+
(26)
where:
zp= 7e2jT[( Sa/Sy),].(~-n)-“2 0
dn
(27)
341
On the other hand, because of eqns. conditions (17) and (19), one has: (W&J),=
-W/W,
= W/r)
As shown in Appendix electrode concentrations, a..,.. = a,, - #p
and
which obey the equality 1 - a..,..= b..,w= r-l’*
(24)
and
(25)
and
of the
boundary
(28)
= J/
2, one can therefore introduce a..,,, and b..,,., such that *:
the following
fictitious
b..,.. = b,, + I+!+
(29)
1 = aao” + b..,,.. and the following $,(v /0
n)-I’*
integral
equation:
dn
Therefore the voltammogram can be determined by the resolution equation (30) provided that CZ..~ or b..,,. are determined. This is done their relationship to the true electrode concentrations, a, and b,. eqns. (24) and (25), taking into account eqn. (28) (see Appendix l), finite value of y * :
(30) of the integral by establishing Integration of yields for any
~=~o+wnY*
(31)
and:
b = 2(rcl/r)ev(M) + [P-
(W’b*l exd+>
(32)
where j3 is an integration constant. Because of the result in eqn. (26) (a + b) as obtained from eqns. (31) and (32) must tend toward unity when y* = p* s=- 1, i.e. in the region where $J = 0. Therefore one obtains: /I = 1 - a0 - 2($/r). Taking into account the boundary condition in eqn. (18) then yields: a,=
{I+(~~)exp[(I-~)5-~~2)-2(~/r)[I-exp(-~2/2)1}/[I+exp(5)1 (33)
where A = A* I = kp/( DFv/RT)‘/*. On the other hand, from eqn. (31) owing to eqn. (29) where p= p*/P a..
one has a, = a, + ($/r)p*,
o..=a,-~p=[~o+(~/r)~*l-44~*m=u,
which
gives,
(34)
Introducing this latter value into eqn. (30) and noting AarP = [ Aexp( a+*)], which corresponds to the definition of the rate constant uncorrected for the Frumkin effect, i.e. k” = [k” t exp(aG2)] such that Aar’P= k”/( DFu/RT)‘/*, affords the
l Note that since p =8*/r, a,, (or b,) in the y space and ap. (or b,,,) in the y * space refer to the same concentration value at the same physical distance, x = x2 + ~/(~Fu/RT)‘/~ = x2 + y*/K, from the electrode surface. For the sake of clarity in the following, the concentrations will be noted CZ,,or b, independently of the space variable, y or y * ,used,whereas the distance will be noted p or p* depending on the space variable considered.
342
following
integral
{I + (+/A) = I - r-‘/z
equation:
exp[(I J
- a)E - o&,)J - 2(MXI
a’Gn( 7 - n)-rj2
- exp(-&/2)l}/[I
dn
(35)
which gives the equation of the dimensionless reduces to the classical one [16]: [l-($/AaPP)
+ exp(01
exp(-at)]/[l+exp(-5)]
voltammogram.
=V1/2&7$,,(7-n)p1’2
For I + cc, eqn. (35)
dn
(36)
Equation (36) features a quasi-reversible voltammogram. When Aapp -+ co, it simplifies to the classical integral equation corresponding to a Nernstian voltammogram [16]: l/[l
+ exp( -.$)I
= C1/2iT$n(
7 - n)-“2
dn
(37)
i.e. such that #p = 0.446, tp = 1.110, and .$p/2 = - 1.094 [16]. On the other hand when A”rP -+ 0, by introducing [* = ct.$+ ln(Aa%-‘/2), +* = $cu-“~, and 7* = (~7 in eqn. (36), one obtains the classical equation featuring an irreversible electron transfer forward wave: I - $,* exp( -,$*)
= 7~-‘/~
‘*$z 7* - n)-“2 /0 (
dn
(38)
which corresponds to $*p = 0.496, ,$*p = 0.783, [*p/2 = - 1.076 [16]. The above eqns. (36)-(38) establish then the validity of the classical theoretical approaches to the determination of the equations describing cyclic voltammograms, as soon as the diffuse layer is an infinitely small fraction of the diffusion layer, i.e. as soon as I + co, Indeed, they show that under such conditions the effect of the double layer is simply to affect the respective concentrations, [A], and [B],, of A and B at the electrode surface with respect to the fictitious ones, [A]..,,.. and [B]..,,,, used in classical theories. When I = k( DRT/Fu)‘/’ + co, as seen from eqn. (36), this correction amounts only to a thermodynamical correction, i.e. such that: [A]..,.{[B]..,..=
Wlo/PlO~
exd(z,
-+-JF+2/RTl
(39)
which is at the basis of the Frumkin correction [27]. As recognized earlier and demonstrated by eqns. (36) or (38), this effect can be easily be accounted for by using an apparent rate constant of electron transfer, k” = kf exp{ -[(l - (Y)z, + CXZ~]FC#J~/RT}, instead of the true rate constant k,.’ Therefore, as also previously recognized and shown by eqn. (37) this correction has no effect on a Nernstian wave. Limiting behaviour at large but not infinite values of r As shown in Appendix 2, the approximation that the space near the electrode is separated into two regions, viz. y* < p* and y* > EL*,remains valid as soon as the
343
concentration profiles of A and B which would be obtained by integration of eqns. (22) and (23) are almost linear for 0 < y* -C II*. Therefore, provided that I = remains sufficiently large for the diffuse layer to be a very small K( DRT/Fv)“’ but not negligible - fraction of the diffusion layer, it is expected that eqn. (35) will retain its validity. The exact limits of this validity will be determined in the following through the study of the general case corresponding to any value of I. However, the interest of considering eqn. (35) under conditions where the last term in the bracketed expression on its left-hand-side, i.e.: 2 [l - exp( -~$~/2)]/r, is small but does not cancel, is that one can thereby examine under which circumstances the “classical” eqns. (36)-(38) are approached. Let us first examine the case of a Nernstian electron transfer. Although “unrealistic” under experimental conditions where I is not infinite, since most electron transfers are not sufficiently fast for a Nernstian behaviour to be observed at the scan rates considered (see below), this situation will serve the purpose of demonstrating that the effects due to the double layer are not related only to electrode kinetics (compare above for the Frumkin correction when I + cc). Figures la and lb represent the variations of the dimensionless peak current, $p, and peak potential, [P, as a function of I, i.e. of the scan rate (upper abscissa scale) for two typical experimental situations: supporting electrolyte concentration cS = 0.1 M in acetonitrile (cS = 37.5), D = 5 X 10e6 cm2 s-i, and eppzC= - 10, i.e. E o = E,,, - 0.25 V or +pzC= -40, i.e. E” = Epzc - 1.0 V at room temperature. It is seen from these plots that the classical equation for a Nernstian voltammogram is no longer valid as soon as u exceeds a few hundred kilovolts per second. These deviations require, however, a high experimental precision in order to be detected on the forward peak. Yet, as illustrated by Figs. lc and d or 2a-c *, they are more easily observed when the shape of the whole voltammogram is considered. For example, it is seen from Fig. 2a that when u = lo7 V s-i, the anodic peak is no longer observable and is replaced by a broad anodic current. The same trend is observed in Fig. 2b, when the supporting electrolyte concentration is decreased, or in Fig. 2c, [,P cannot be defined when +ppzcis made more and more negative. As a consequence for some values of I, which is the reason for the “interruptions” in the plots of AtP or of E,,2 presented in Figs. lc and d. This arises because for the case considered in this study, the reactant A is a neutral molecule. Its concentration profile is then affected only slightly by the electrical potential within the double layer, through the boundary condition at x = 0. This is particularly clear when one considers that the descending branches of the forward peaks in Figs. 2a-c are not affected. Indeed, when nF( E - E “)/RT -=z 0, one has a, = 0 at the electrode surface, and the concentration profile of the neutral molecule A is not affected at all by the electric field of the diffuse layer, and therefore the current, $, is not affected at all either. This is obviously not the case for the charged molecule B, as shown by the
l Figure 2c also shows that the voltammograms are greatly affected by the exact value of E” - E,,, which may then afford an “easy” way of evaluating this parameter at solid electrodes.
log 2
0.49
a ~ :
qp
0.45
”
6
10 .”
ii,,
El ‘1
,;’
.:i,.
i
___
\
?,
0.4
1
-3
0
3
-log r
-3
0
-log r
r
-log
3
-3
0
3
-log r
Fig. 1. Variations of (a) the dimensionless peak current, #P = ip/[nFAc"(nFuD/RT)"'], (b) dimensionless peak potential, [P = - nF( Et’ - E o )/RT, (c)peak-to-peak potential difference, Alp = nF( E,Pa functionof E,J')/RT, and of (d) the dimensionless half-wave potential, [,,a = nF( E,P+ E,P)/(2RT),as r = K( DRT/Fu)'/~, for two typical experimental situations: cS = 0.1 M, es = 37.5 (acetonitrile), D = 5 X 10-6cm2s-‘and(i): $,,=-10,i.e. Ep,=Eo+0.25V,or(ii): ep,,=-40,i.e. Ep,=Eo+l.OV,in General case; (- - -) the case of a Nemstian electron transfer with z, = 0 and n = 1. ( -) asymptotic behaviour for r infinite (classical theories (15,161); (. . . . .) asymptotic behaviour for r B- 1 (eqn. 35, with AaPt’+ co); (------) asymptotic behaviour for r -=z1 (eqns. 40-45, 46’, 47); (----) limiting behaviour for r = 0 (eqn. 48, with A + co). Note that in (c. d) the potential scan is inverted at tr = 20, i.e. E, =E”--OSVforn=l.
concentration profiles of A and B in the diffuse layer represented in Figs. 3a-c. It is seen that although the diffuse layer is a very small fraction of the double layer for the scan rates considered, the concentration profiles, as obtained from the analytical eqns. (31) and (32), are greatly affected. Figure 4 shows that these results remain essentially identical when a quasi-reversible electron transfer is considered. Note that in Fig. 4, the voltammograms determined according to eqn. (35) correspond to a constant value of A for each set a-c. Therefore according to the classical theories of cyclic voltammetry, even when considering the Frumkin effect, the voltammograms represented on each set as a function of the scan rate should have been exactly superimposable. The obvious
345
e G 6 Fig. 2. Distortions observed for Nemstian voltammograms (z, = 0, n = 1) as a result of the coupling between the diffuse and diffusion layers. (a) Effect of scan rate, u, for D = 5 X lob6 cm2 s-‘, cS = 0.1 M, .) 105, (------) lo6 and (----) 10’ V r,=37.5 and cp,,=-10i.e. E,,=E’+0.25 V. v: (..... s-r (b) Effect of the 1: 1 supporting electrolyte concentration, c’, for u = lo6 V SC’, D = 5 X 10m6 cm2 s-‘, es = 37.5 and Eplr = E o + 0.25 V. P: (- - -) 0.5, (. +. . .) 0.1 and (- - - - -) 0.05 M. (c) Effect for u=105 V s-r, D=5X10-6 cm* s-l, c,=37.5 and c’=O.l M. of +rJpu:=F(E”-E,)/RT, 4 =iF/[nFAco(nFuD/RT)“*] and [= - nF(E - E”)/RT. GppLc:(- --) -10, (......) -30 and (- - - - -) - 50. (Note that in each set the voltammogram in solid line corresponds to I? infinite, i.e. that predicted by the classical theories (15,161.)
v,s-
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et.4 ,/-q
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0.04
Y’
Fig. 3. Dimensionless concentration profiles of A and B, and normalized electrical potential, $+ = G,/+~, near the electrode surface (y* = Kx* = (x - .x~)F(~c~/~~~RT)“~), for a Nemstian voltammogram, as a function of v for the following conditions: t, = 0, n =l, $,= = - 10 i.e. Epze = E o +0.25 V, D = 5 X 10m6 cm2 s-r, rS = 37.5 and cs = 0.1 M. These concentration profiles are determined at 6 = 15, i.e. E = E o -0.375 V, through the finite difference resolution of the general case, except for a and b, where eons. (31) and (32) could be used.
a
1
C
:il 1 I
I
I’
r;
:
-1 0
1020
cFig. 4. Distortions observed for a quasi-reversible voltammogram (a = 0.5, z, = 0, n = 1) as a result of the increasing coupling between the diffuse and diffusion layers when the scan rate, U, is increased, for D = 5 x 10m6cm* s-t, cS = 0.1 M, c, = 37.5 and $_= = - 10, i.e. EPlc = E” +0.25 V. To facilitate the comparison in each set, A = /c~(RT/~FuD)'/~ was kept constant: A =lO (a), 1 (b) and 0.1 (c). $= i,/[nFAc"(nFuD/RT)"*]and .$= -nF(EE')/RT. u:() 103,(-- -) 105,(. . . . . .) lo6 and (-----) 10’ V s-r.
differences layer. Limiting
thus show the effect of variations
in r, i.e. of the role of the diffuse
case for r -+ 0
The condition r + 0 corresponds to a situation where the diffusion layer is an extremely small fraction of the diffuse layer. For usual experimental conditions in liquid solvents, where a 1 : 1 supporting electrolyte at ca. decimolar concentration is used, this corresponds to an “unrealistic” situation, since it would correspond to diffusion layer dimensions of the order of molecular size. Therefore the limit r + 0 is considered here more in a mathematical sense than in an experimental one *, and for the purpose of checking independently the validity of the general solution at small r values (see below). When r = 0, the concentration of A and B differ from their bulk values only in a thin layer adjacent to the electrode where r$ varies almost linearly with the distance, y, from the electrode. Then, neglecting the terms in T2, eqns. (13) and (14) can be approximated by: Sa/Sr
= fY2a/6y2 + z,e (Ga/Sy)
(4)
Sb/ST=62b/6y2+(z,-n)e(Gb/Sy)
(41)
with, as deduced
from eqns. (20) and (21):
e = r(6+p3y*)0
= -2r
sinh($,/2)
-+ 0,
when
r + 0
(42)
l Note, however, than under conditions where low supporting electrolyte concentrations are used, the condition r + 0 would regain an experimental validity, provided that sufficiently high scan rates can be used, or provided that the size of the electrode is small enough (compare e.g. ref. 28).
347
Resolution of the system of limiting eqns. (40) and (41) associated following boundary conditions derived from eqns. (15)-(19):
with
the
7=0,
y>o:
a=l,
b=O
(43)
7>0,
y-,00:
a+l,
b-+0
(44)
7>0,
y=o:
(Way),
- 2f’[sinh(W2)]
+ 2r[sinh(+,/2)](z, 7 > 0, y = 0:
z,ao + (~~PJJ)~ - n)b,
- 2I? [ sinh( +,/2)]
(6a/6y),
= 0
(45)
z,a,
=~exp[~(5+~~,)]{~o-~oexp[-(~+~~,)l} was performed via an explicit current being then given by: # = (~G.JJ)~
- 2r[sinh(W2)]
For a Nemstian r>o,
y=o:
finite
electron ao=bo
difference
numerical
(46) procedure,
the Faradaic
~,a,
transfer,
(47)
i.e. when
A x=- 1, eqn. (46) was replaced
exp[-(d+W,)]
by: (46’)
The corresponding results are presented in Fig. 1 for the case of a Nernstian electron transfer (boundary condition 46’). Note that when r is sufficiently small, i.e. when f’[sinh($,/2)] + 0, all the eqns. (40)-(45) and (47) tend toward their analogs corresponding to a classical formulation of the problem. Yet the diffuse layer still affects the voltammograms, but now only through a modification of the boundary condition between a, and b,, as shown by eqn. (46’) or by eqn. (46) which, under these conditions, simplify respectively to: r>o,
y=o:
(Way)o
y=o:
a0 = b. exp[ - (< + MJ,)]
= A exp[a(5
+ W)l
{
ao-boexp[-(t+W,)l} (46a)
or: T>O,
Therefore (1 -(#/A) =77
the equation
of the limiting
(46’a) voltammograms
at r + 0 is given by:
exp[-~(5+~~~2)1}/{1+exp[-(E+n~,)l}
-1/2 bt/~~(7 - n)-1’2 /
dn
which corresponds in fact to a classical formulation [16] except that the potential of the electrode is referred to E” + c#B~, instead of E”, because the local potential at the end of the diffusion layer is now $J~ instead of 0, as observed when the diffusion layer thickness exceeds that of the diffuse layer. Therefore if c#B~(E) is constant in the potential region where the voltammogram is observed, this amounts simply to an apparent shifts of standard potential, from E” to E&, = (E” + c$~). If the variations of C#B~ cannot be neglected, the resulting voltammogram will be distorted, because under these conditions [E - (E o + cp2)]is no longer a triangular function of the time. The data reported in Fig. 1 correspond to the latter situation.
348
General case for any value of r In the general situation, i.e. when r does not take extreme values, no integral formulation was found. Therefore the system of partial derivative equations (13) and (14) and boundary conditions (15)-(19) was solved by an explicit finite difference procedure. To allow a simple discussion of the results, we will restrict their presentation to the case of a Nernstian electron transfer (i.e. boundary condition 18’ is used instead of 18). For a given experimental situation, the voltammograms then depend only on a single parameter, F, i.e. on the scan rate. The corresponding results are shown in Figs. la-d, in terms of the variations of +P, electron tp, AtP=[,P -5,P and ,$,,2 = ([,P + 5,P)/2. Note that for a Nernstian transfer the latter parameter should remain constant and close to zero, since E,,2, the half-sum of the two peak potentials, is nearly identical to E ’ provided the scan is inverted sufficiently after E o [15]. DISCUSSION
The comparison, in Fig. 1, of the above results based on the general system of eqns. (13)-(19) without any simplification, to those obtained through the approximation leading to eqn. (35), demonstrates that the latter equation remains valid up to ca. lo6 V s-‘. Above 10’ V s-r one observes that the simplified treatment leading to eqn. (35) introduces significant errors. Thus, eqn. (35) can be considered as affording a reasonably good description of any voltammogram obtained within the experimental range of scan rates accessible today. We will therefore restrict the following discussion to the predictions derived from eqn. (35) i.e. to the case of large values of r. Equation (35) shows that the voltammograms observed under these conditions are greatly affected by the relative magnitude of the diffusion layer size with respect to the diffuse layer, i.e. by the value of r = (2FDcS/eS~,v)‘/* = K( DRT/Fv)‘/*, but also by the exact value of the potential at the OHP, i.e. by any experimental parameter that affects &. Among the latter, the most critical is certainly +rpu:= F( E o - E,,)/RT (compare e.g. Figs. 2c and 5a below), since its exact value is generally unknown owing to the general lack of knowledge on Epu: values for organic conditions at solid ultramicroelectrodes. However, this dependence on & must not be confused with that resulting only from the classical Frumkin effect. This is shown by the comparison of the three voltammograms presented in Fig. 5b. Voltammogram I (- . . . . . ) corresponds to the classical theory of cyclic voltammetry [16] (eqn. 36), where the value of $2 at E = E o was taken to evaluate an average value of k” = kf exp[crF$*( E = E “)/RT] which was then used at all points of the voltammogram, as is usually done [15-171. For voltammogram II (), eqn. (36) was used again but now the Frumkin correction, i.e. the value of k” = kf exp[aF+*( E)/RT], was evaluated at each point of the voltammogram. In the determination of the third voltammogram (- - -), k” was determined as for II, but the effects due to the overlap of the diffuse layer and diffusion layer were now taken into account via eqn. (35). Therefore the effects
349 3 $2
-2
-7
Fig. 5. (a) Variations of I& = Fh/RT, as a function of 6 = - F( E - E”)/RT, for different values of as indicated by the number on the curves. (b) Distortion of a “theoretical” Q = F(E” - E,,)/RT, q!&i-reversible cyclic voltammogram as a function of the level of approximation in the “theory”. a=OS, r,=O, n=l, kp =lOcm s-‘, D=5X10-6cm2 s-‘, c”=O.l M, r,=37.5 and s#+= -10, i.e. = E” +0.25 V. (I) (. . . .) “Classical” theory [15-171, i.e. eqn. (36) was used with a constant value E -- 2.33 cm s-‘. (II) () Equation (36) was ot?O , taken at E=E”, i.e. k”= kp exp[a+a(E”)] used as for (I), but k o was evaluated at each potential, i.e. k o (E) = k,“ exp[a&( E)]. (III) (- - -) As for (II) but eqn. (35) was used instead of eqn. (36). # = i,/[nFAc”(nFuD/RT)“*] and 6 = - nF( E E “)/RT.
arising from the fact that r is not rigorously infinite at scan rates in the megavolt per second range can be appreciated by comparison of voltammograms II and III. Note also that a comparison of voltammograms I and II in this figure demonstrates that the use of an average value for the Frumkin correcting term in extracting a rate constant from an experimental voltammogram, e.g. from the peak-to-peak separation as a function of scan rate [l&12], may result in significant errors, independently of any possible additional effects due to ohmic drop or to the fact that at the large scan rates considered, i.e. for u > lo5 V s-l, l? is not infinite. These effects can be appreciated more quantitatively through the working curves presented in Fig. 6, where it is seen that for a given electroactive couple, and a given electrode-solvent-supporting electrolyte system, the characteristics of quasi-reversible voltammograms depend on two parameters, A = kp( DFv/RT)‘/’ and A* = A/r = kp/( KD), instead of only one parameter, viz. A, in the classical theory of cyclic voltammetry [16]. This can be understood if eqn. (35) is rewritten in the following “classical” form: [l - (Jl/A,i,)
exp(-a[)]/[1
+exp(-E)]
=~“‘/~#,,(r-n)-“~ 0
dn
(49)
where AIic, a fictitious dimensionless rate constant related to A and r by: Atic= [Aexp(~~))1/{1-2[Aexp(a+2)/r]
exp[-(1
-a)[][1
-exp(-+,/2)]} (50)
is now used instead of A = kp/( DFv/RT)
1’2 [15-171.
This amounts in fact to
350 0.46 1
1
w
a
0.5
.,
0.39
0.32
-2.50
-1.50
-0.50
-log
0.1
0.50
A
12
1.20
up
c
112
7
2 -2.50
0.50
-0.20 -1.50
-0.50
-log
0.50
-2.50
- 1.50
A
-log
-0.50
0.50
A
Fig. 6. Variations of the dimensionless (a) peak current, ~P=i~/[nFAc”(nFuD/RT)“*], (b) peak potential, IP = - nF( E,P - E o )/RT, (c) peak-to-peak potential difference, AIP = - nF( E,P - E,P)/RT, and (d) half-wave potential, E,,* = nF(E,P + E,P)/(2RT), as a function of A = kp( RT/nFuD)‘/* and I (solid curves; the numbers on the curves are the values of A* = A/r = kF/KD), for a typical experimental situation: (Y= 0.5, I, =O, n=l, c”=O.l M, ~,=37S(acetonitriIe), D=5X10-6cm2 SC’ and E,, = E o +0.25 V. Same data according to classical theories [15,16], without (. . . . . .), or with (- - - - -) taking into account the variations of & with the potential (compare (II) in Fig. 5). The data reported for each case correspond to a potential scan inversion at 5, = 20, i.e. at Et = E o - 0.5 V.
introducing kg,=
a fictitious
[ki’exp(%)]/{1xed-(l
rate constant
kg,, such that k:, = Al,i,( DFU/RT)“‘:
2[kp ev(wh)/(DK)]
-a)El[l -ev(-W2)1}
(51)
i.e.:
kp,=k0/{1-2[ko/(DK)]exp[-(1 -~)t][l -exp(-h/2)]}
(52)
where k” = kp [exp(a&)] is the rate constant including the Frumkin effect. It is noteworthy that eqn. (52) shows that for an irreversible electron transfer with z, = 0 and n = 1, the forward wave is almost unaffected by the effect of the double layer, except for the Frumkin correction, Indeed, under these conditions the forward wave will be observed at extremely large values of the potential, so that eqn. (52) simplifies to k:, = k” = kp[exp( a&)] in the potential range where the forward wave is observed. However, this does not indicate that the voltammogram is not affected, because, since the backward wave is observed at very negative values of the
351
potential 5, the effect on Zcic will be extremely important, as deduced from eqn. (52). This dissymmetric behaviour originates from the fact that in establishing eqn. (52), we considered a neutral molecule as the starting material, A. Therefore, in the potential range where the forward irreversible wave is observed, the concentration profile of A is insensitive to any electrical effect, since it corresponds to the integration of a partial derivative equation associated with a boundary conditions at the electrode surface: r>o,
{6a/6y},=nexp[cw(~+~*,)la,
y=o:
which does not incorporate any electrical component. potential range where the backward wave is observed, eqn. (53) must be replaced by: 7>0,
y=o:
{W~_Y},=
(53)
On the contrary, in the the boundary condition in
-Aexp[(a-I)((++,)]&
(54)
and will introduce a strong coupling with the local electrical field since B is charged. The analysis presented in Appendix 1 shows that the formulation eqn. (49) remains valid for any value of z, and n, provided that A ric = kz,/( nDFu/RT)‘/* is defined on the basis of the following fictitious rate constant of electron transfer: G, = k”/{ I - (2k”/DK)(
f(~,)
exddl +f(z, - n) ed - (I- cf>Sl)}
where k” = kp exp[an - z,)Z+,/RT] effect, and f(z) is defined as follows:
f(z)=
C m=O
is the rate constant
including
{1-exp[(z/IzI)(m+1/2)~~2/RTl}/(2m+1)
(55)
the Frumkin
(56)
with f. = 0. The corresponding variations of k& with the electrode potential should then be experimentally accessible through convolution procedures, provided the experimental accuracy on the voltammograms obtained at the scan rates considered is sufficient. Indeed, from eqn. (49) one obtains [14,29]: k& = D’&/{
Zlim - Z exp[ (nF/RT)(
where Z is the convoluted
fi,(t-n)p1’2
Z=77-‘/2
experimental
E - E’)] current
}
(57)
given by:
dn
/ 0
and Zlim is its plateau potential of the A/B
value at (nF/RT)( E - E,,2) electrochemical wave.
z+ 0, where E,,2 is the half-wave
ACKNOWLEDGEMENTS
This work was partially supported by CNRS (URA 1110: “Activation Moleculaire) and Cole Normale Suptrieure. The authors greatly acknowledge Drs.
352
H.S. White and S.W. Feldberg for communication, before publication, of their results in ref. 28, which deals with the effect of the double layer on steady-state voltammograms at microelectrodes of nanometer radii. APPENDIX
1: DERIVATION
OF EQUATIONS
(49), (55) and (56) FOR ANY VALUE of z, AND n
As explained in the text, when I x=-1 and y* is not larger than a few units, eqns. (13) and (14) reduce to: Sa/Gy* + Z,a(sC#J/sy*)
=4/r
(Al)
6b/6y*+(z,-n)b(69/6y*)=
-(#/r)
(A2)
with, from eqn. (20): (@/Sy
* ) = - 2 sinh( G/2)
(A3) Since eqns. (Al) and (A2) differ only in the z, of (z, - n) factor and the sign of the second member, let us focus on the integration of eqn. (Al). Integration of this equation by the method of variation of the constant gives readily: a = A exp( -z,$)
(A4)
with:
A=
(G/r)/[exp(z,+)l dy*
W)
+A,
where A, is an integration constant. Introducing w = exp( G/2) and taking eqn. (A3) into account shows that: A -A,=2(~/r)l[w2ia/(i-W2)]
dw=2(+/r)W(z,)
(A6)
Noticing that: +/(I
- &) = #.-2
[I - (1 - w’)]/(l
- w’) = ,2ki)/(l
- w’) - &-2
shows that : Wz,)
= Wz,
- 1) - {exp[(2z,
-
1W21}(2z, - 1)
(A?
from which results, since W, = (y*/2) 2,-l
+ cons& that for any z, > 1:
A = (+/I)Y*
+ l/2)1 Mm
- (V7
C {exp[+ m=O
+ l/2)
+A;
(A8)
and for any zaG -1: -(%+I) C
A=(\t/I)y*-(+/I)
m=O
{exp[-~(m-l/2)1}/(m+1/2)+Al,
(A9)
and for z, = 0: A = (+/r)y*
+A;
(AlO)
353
where Ah is a constant. These three relationships can also be rewritten form valid for any value of za, giving, owing to eqn. (A4):
in a unifying
~~~~~~~,9~=~Jl/~~~*+~~~/~~{~~~~~~~,/l~,I~~I~,I+~/~~1}/~~l~,I+~~ -2(~,r)‘~‘{exp[~cZ,/lr,I)(m+l/2)1}/(2m+1)+Ab m=O (All)
Therefore at any p* such that +,,* = 0, a,* is given by (note that since uY,_* = uyCP, this value is noted up irrespective of the space variable y or y* considered):
a, = M~cL*
I Z, I + 1) - w/r)
+ 2WW2
‘2’ W(2m
+ 111 + 4
(A121
m=O
Eliminating the constant Ah by expressing eqn. A(ll) for y* = 0, and introducas defined in eqn. (29) for p = p*/r), finally affords: ing a..,.,=up -
(+/r)p,
~~~~~,~~~+~~~/~~(~-~~~~~~~~,/I~,I~~I~,I+~/~~1}/~~l~,I+~~
a..,,.=a,
- wm
‘5’ {I- exP[+Aza/lzalb m=O
+ l/2)1 )/Pm + 1)
W)
that is: a..,,.= a0 exp(Q,)
- 2(#/r)f(z,)
(AI4)
where f(0) = 0 and for any z, # 0: I~,‘--’ f(z,)
=
C
{I-
exP[+z(za/lzal)(m
+ 1/2)1}/(2m + 1)
(A15)
m=O
The same treatment can be applied to eqn. (A2). Thus by simple permutation of [a, z, and (4/r)] into [b, zi, = (za - n) and -(J//r)], respectively, one obtains readily: b ssos9=bo exp[(z,
- n)h)
+ %VW(z,
- 4
(AW
Because of their definition in eqn. (29) (see also Appendix 2), the fictitious electrode concentrations a..,,. and b..,., in eqns. (A14) and (A16) must fulfill the mass conservation equality, 1 = a..,., + b..,,, i.e.: I = a0 cxp(r.&z)
+ b. expk
- ~>&)I
- 2(W’)[f(r,)
-.%,-
4
(AI7)
On the other hand, a0 and b,, which are the true electrode surface concentrations, are related to the current, 4 in eqn. (19), by the Butler-Volmer rate law in eqn. (18). Introducing Aapp = A exp[( an - z,)&] = k “/(~DFu/RT)‘/~, where k o
354
= kp exp[(cyn - z,)&J,
allows one to rewrite eqn. (18) as follows:
= b. expK z, - n)+J]
a0 exp(z,&)
Introduction of this latter (A16) yields b..,,.: b .so..= { 1 - ( $/Aapp)
exp(-t)
value
into
+ (#/Aa”““)
exp(-&)
eqn. (A17) affords
b,, which
(Al81 from
eqn.
exp( --at)
+2(~/r)[f(z,)-_f(~,--n)l}/[1+exp(-5)1 +W/W(za-4 (Al9) i.e.: { 1 - ( $/Rfi,)
b YY=
exp( --(yt)}/[l
+ exp( -Z)l
(A20)
where: A,,=AaPP/{I-2(Aapp/r)(f(Z,)
exp[&]
Equation (A21) amounts transfer, k& such that: A riC= k&/(
nDFu/RT
to the definition
+f(z,-n)
exp[(a-1)5])}
of a fictitious
rate constant
(A21) of electron
)112
(A=)
i.e.:
k”/{l -
k& =
exp[&l +,+,
2(k”PW(f(r,)
- n) exp[(a
- IHI)}
(A23)
where: k” =kp
exp[(cyn--z,)&]
(~24)
is the rate constant uncorrected for the Frumkin effect and kto, the true rate constant. Introducing b..,,. as given by eqn. (A20) into eqn. (30) finally yields the following integral equation: [l-
(#/A,,)
exp(-a[)]/[1
+exp(-5)1
=~-~/~j6t./&-n)-“*dn
(A25)
which is sufficient for describing the voltammogram, owing to the definition of AliC in eqns. (A22)-(A24). The case of a Nernstian voltammogram is obtained by using the local Nernst equation in eqn. (A18’), instead of eqn. (A18), i.e.: a0 exp(z,+,)
= b. exp[(z,
which amounts follows that: b sso..= {I+
- n)+2)1
in fact to considering
&vW(d
-f(z,
(~18’)
exp(-5) that
- dl}/[l
A”PP 1s . infinite
in eqn. (AlS).
+ ew(-5)1+ wmfb,
It then
-4 (A19’)
355
i.e.: b s.os.= {I+
W/~)[.fkJ
+fG,
exd-81)A1+ exd-5)l
- 4
With eqn. (30) this finally yields the following integral voltammogram of a Nemstian electron transfer: {I +2(4/IX/a,)
+fk-
4
equation
(A20’)
which describes
the
ev(-01 )A1+exd-t91
=77 -‘I2/ bqn(7- r~-l’~dn
(A25’)
which amounts in fact to considering eqn. (A25) associated with eqn. (A22) and the limiting form of eqn. (A23), for k” + cc. When r + 00, eqns. (A25) or (A25’) tend toward their classical forms in eqns. (36) and (37), respectively. APPENDIX
2: DERIVATION
OF EQUATIONS
(29) AND (30)
In this Appendix we want to establish the validity of eqns. (29) and (30). Let us thus consider an electrochemical system in which A and B undergo pure diffusion. Their dimensionless concentrations then obey the system of partial differential equations (22) and (23). Integration of these in the Laplace plane, taking into account eqns. (15) and (16), gives readily at any value p of the y space variable: a, = (l/s)
- (&z/&&/~
“2 = (l/s)
- Ir/ exp( - j.~‘/~)/s’/~
(A26)
and: 4 = - ( S_~/G~)JS’/~= Ir, exp( - &‘2)/s”2 where s indicates Laplace transform (&J/&Y), When
G%I/CY),=
14%
Laplace
-4+)=($+4+)=~+
gives:
-Ps”2)
(~29) into eqns. (A26) and (A27):
= (_b, +&u> = $/s”2 transformation
7, and _V the
(A2g)
small, so that ~FLS’/’+ 0 *, this equation
then, when introduced
- (Ccl - 94
Inverse
=J/ exp(-l-is”‘)
-(&/&&=~(I
which affords (l/s)
the Laplace variable corresponding to the time of any variable ‘I”. One has therefore:
= -(M/Q),
p is infinitely
(~27)
of eqn. (A30) finally
(A301 yields: (A3I)
l Note that the inverse Laplace transform of this condition is (p/2)(?r/7)‘/’ + 0, i.e. it implies that the distance x,*, corresponding to p in the real space, is such that x: -=K2( Dt/vr)‘/*, i.e. it is considerably smaller than the diffusion layer for the time, 1, considered. Since eqns. (29) and (30) are considered only under conditions where p = p*/I with I5-> 1, the condition ps’/* -=c1 is always fulfilled whenever these equations are used, since it expresses that the time considered is sufficiently large for the diffuse layer to be only a small fraction of the diffusion layer, i.e. x,’ = K-’ -=x( DI)‘/*.
356
where ZJ/ is defined ZI)=T-“~
/0
by:
‘I//,,(T-~)-“~ dn
Therefore, and b..,..:
one can define
b-w
the following
fictitious
electrode
concentrations
a..,..
a..,,.=a,-$+=1-Z+
(A33)
= bP+ $q = Z$ b ‘sow
(A34)
which follow eqns. (29) and (30). Note that in the classical theories of cyclic voltammetry [15-171 these fictitious concentrations are implicitly considered as the real electrode concentrations, the “error” being corrected a posteriori by the fact that a fictitious electrode boundary condition:
i, = nFAc”{ kp exp[( fxn-L,)F$J~/RT]}
exp[-cunF(E--E”)/RT]
X { a..,.. - b..,.. exp[ nF( E - EO)/RT]}
is used instead
of the true one (eqns. 11 and 12):
t,=nFAc”kto
exp[ -mF(E-
(A35)
E” -+2)/RT]
X{a,-b,exp[nF(E-E”-$,)/RT]} which should
be used but applies
(A36) only to the real electrode
concentrations
a, and
b0’ APPENDIX
3: NUMERICAL
COMPUTATIONS
All the numerical computations were performed on either a COMPAQ 386 or an AMSTRAD PC-AT 2386, using Turbo-Pascal programs. All the cases amenable to an integral-equation formulation (see text) were solved according to standard previously described procedures (see e.g. ref. 14). Two cases could not be solved following these procedures and were solved using standard explicit finite difference procedures: (i) the general case valid for any P value, and (ii) the limiting case at P + 0. For these applications the analytical values of (6+/6y) or (S2+/Sy2): (S+/6y)
= - 2 sinh( +/2)
(A37)
and: ( S2+/Sy2)
= sinh( +)
(A38)
were used instead of numerical determinations based on finite differences. However, when considering larger and larger values of P, a classical explicit finite difference resolution of the general case would have required too large computational times and memory occupation. Therefore, for any P values corresponding to a diffusion layer thickness exceeding twice that of the double layer, a double space grid was used. In the space region adjacent to the electrode with
351
y Q pi, CL, being such that CL,= 10 1+2/(6+/Sy)y_o 1, a space increment Ay, = p,/lOO was used together with a time increment Ar, = 0.44(A~,)~. For y >, ~1, a space increment close to Ay, = p2/100, where p2 = 6 - P,, 6 being the diffusion computation, was used thickness at r = Q,~, i.e. at the end of the voltammogram together with a time increment A72 = 0.44(A~,)~. In practice, Ay, and Ar, were adjusted slightly, with respect to the above values, so that Ar2 was a multiple of AT, in order to facilitate the numerical connection between the two finite difference grids. The numerical connection between the two finite difference grids was performed as follows. Let us define N = Ay,/Ay,; then Ar2/Ar, = N 2. Therefore, at any time TV= k Ar2 = kN2 Ar, the two time grids coincide. Let c( kN2, 1) to c(kN’, 100) be the concentration values of the considered species for time rx., at the different mesh points pertaining to the first grid, and c’(k, 1) to c’(k, 100) those for the second grid. From these concentrations, those c(kN2 + 1, 2) to c(kN’ + 1, 99) are obtained through normal algorithms, and c(kN’ + 1, 1) is obtained from the latter by application of the boundary condition(s) at y = 0. Similarly c’( k + 1, 1) to c’( k + 1, 99) are obtained through standard explicit finite difference procedures from c( kN2, 100) and c’(k, 1) to c’( k, 100) (note that in practice only c’( k + 1, 1) to are determined, y,,_, = n maxAy, being the thickness of the diffuc’(k + I, n,,,) sion layer at the time considered, i.e. nmax= lOO[(k + 1) AT~/T,,_]‘/~; for nmax < n d 100, c’(k + 1, n) is kept identical to ~‘(0, n), i.e. to the value at 7 = 0). From c’( k, 1) and c’( k + 1, l), one generates then (N 2 + 1) intermediate values c”( kN 2 + h) such that: c”(kN2+h)=c’(k,1)+h[c’(k+1,1)-c’(k,l)]/N’
(A39)
with h = 0 to N2. Taking into account the corresponding values of c(kN2 + h, 99) and c( kN2 + h, lOO), the latter are used to determine the values of c( kN2 + h + 1, 100) at each time T = (kN 2 + h + 1) A7,. which are needed to complete the first grid. In this procedure the partial first and second derivatives with respect to space, at 7 = (kN 2 + h) AT, and y = p, = 100 A y,, are expressed as follows: (S~/Gy)=((Ay,)~c”(kN~+h)+[(Ay~)*-(Ay~)~]c(kN~+h,100) -(Ay#c(kN’+ (6’c/Sy2)=
(AY,)(AY~)[(AY,)
whereas
the partial
k)/[(A~i) derivative
+ (AY,)~} -2c(kN’+h,
{2(Ay2)c(kN2+h,99)/[(Ay,)+(Ay2)] +2(Ay,)c”(kN’+
respectively,
k, 99))/{
(AdO) 100)
+ (AY~)~}/[(AY,)(AY~)~ with respect
(A4I)
to time is expressed
(S~/&r)=[c(kN~+h+l,lOO)-c(kN*+h,lOO)],’A~, The validity of this procedure was tested by comparing its results to simulation using a single grid (i.e. with a mesh [Ay,, AT,]) extending up of the diffusion layer. For cases where the diffusion layer thickness was times that of the diffuse layer, the two procedures gave identical cyclic
by: (A42)
those of a to the end up to four voltammo-
358
grams, as demonstrated by the identity (relative difference less than 10P5) of their peak current and peak potential values. However, the gain in computational time and memory occupation was considerable when using the “double grid” method, since the latter is approximately N3 times faster and requires N3 times less memory occupation, where N is the relative magnitude of the diffusion layer with respect to the diffuse layer (see above). REFERENCES 1 R.M. Wightman and D.O. Wipf in A.J. Bard (Ed.), Electroanalytical Chemistry, Vol. 15, M. Dekker, New York, 1989, pp. 267-353. 2 J.O. Howell, J.M. Goncalves, C. Amatore, L. Klasinc, R.M. Wightman and J.K. Kochi, J. Am. Chem. Sot., 106 (1984) 3968. 3 (a) J.O. Howell and R.M. Wightman, Anal. Chem., 56 (1984) 524; (b) J.O. Howell and R.M. Wightman, J. Phys. Chem., 88 (1984) 3915; (c) J.O. Howell, W.G. Kuhr, R.E. Ensman and R.M. Wightman, J. Electroanal. Chem., 209 (1986) 77. 4 M.I. Montenegro and D. Pletcher, J. Electroanal. Chem., 200 (1986) 371. 5 A. Fitch and D.H. Evans, J. ElectroanaI. Chem., 202 (1986) 83. 6 C. Amatore, A. Jutand and F. Pfhtger, J. Electroanal. Chem., 218 (1987) 361. 7 D.O. Wipf, E.W. Kristensen, M.R. Deakin and R.M. Wightman, Anal. Chem., 60 (1988) 306. 8 C.P. Andrieux, P. Hapiot and J.M. Saveant, J. Phys. Chem., 92 (1988) 5987. 9 C.P. Andrieux, D. Garreau, P. Hapiot and J.M. Saveant, J. Electroanal Chem., 248 (1988) 447. 10 D.O. Wipf and R.M. Wightman, J. Phys. Chem., 93 (1989) 4286. 11 C. Amatore, C. Lefrou and F. Pfltiger, J. Electroanal. Chem:, 270 (1989) 43. 12 D. Garreau, P. Hapiot and J.M. Savtant, J. Electroanal Chem., 281 (1990) 73. 13 C.P. Andrieux, D. Garreau, P. Hapiot, J. Pinson and J.M. Savtant, J. Electroanal. Chem., 243 (1988) 321. 14 J.M. Saveant and D. Tessier, J. Electroanal. Chem., 77 (1977) 225. 15 R.S. Nicholson and I. Sham, Anal. Chem., 36 (1964) 706 and references therein. 16 L. Nadjo and J.M. Saveant, J. Electroanal. Chem., 48 (1973) 113 and references therein. 17 C.P. Andrieux and J.M. Saveant in C.F. Bemasconi (Ed.), Investigation of Rates and Mechanisms of Reactions, Vol. 6, 4/E, Part 2, Wiley, New York, 1986, Ch. 7, pp. 305-390 and references therein. 18 A.N. Frumkin, Z. Phys. Chem. A, 164 (1933) 121. 19 R. Parsons, Trans. Faraday Sot., 47 (1951) 1332. 20 J.E.B. Randles, Trans. Faraday Sot., 48 (1952) 828. 21 D.M. Mohilner and P. Delahay, J. Phys. Chem., 67 (1963) 588. 22 P. Delahay Double Layer and Electrode Kinetics, Insterscience, New York, 1965, pp. 153-167. 23 Ref. 22, p. 199. 24 Ref. 22, pp. 33-52. 25 For a discussion of this point see ref. 22, pp. 50-51. 26 W.R. Fawcett and S. Levine, J. Electroanal. Chem., 43 (1973) 175. 27 See e.g. ref. 22, pp. 154-158. 28 J.D. Norton, H.S. White and S.W. Feldberg, submitted. 29 J.M. Sadant and D. Tessier, J. Phys. Chem., 81 (1977) 2192.
J. Electroanal. Chem., 296 (1990) 335-358 Elsevier Sequoia S.A., Lausanne
Is cyclic voltammetry above a few hundred per second still cyclic voltammetry? * Christian
Amatore
** and Christine
Ecole Normale SupPrieure, Laboratoire 75231 Paris Cedex 05 (France) (Received
kilovolts
Lefrou
de Chimie, URA CNRS 1110, 24, rue Lhomond,
12 June 1990; in revised form 30 July 1990)
Abstract The development of ultramicroelectrodes and fast potentiostats in very recent years has allowed the investigation of electrochemical kinetics in the submicrosecond time scale by means of cyclic voltammetry at scan rates in the megavolt per second time scale. Extraction of kinetic data from such cyclic voltammograms, after their correction for distortions arising from effects due to instrumental bandpass, ohmic drop and capacitive charging currents, has been performed using classical theories for cyclic voltammetry. However, application of these classical theories supposes that the diffusion layer and the double layer can be separated theoretically. It is shown here that this classical assumption is no longer valid when scan rates in the megavolt per second range are considered, since the two layers, being then of similar sixes, are no longer physically separable. The ensuing distortions of cyclic voltammograms, with respect to the classical theories, are examined for a single electron transfer mechanism. It is shown that the cyclic voltammograms should normally be extremely dependent on their location with respect to the point of zero charge.
INTRODUCTION
Cyclic voltammetry has proven to be one of the most versatile and most sensitive methods for the investigation of electrochemical kinetics. Its time scale domain, which extended a few years ago to a few tens of microseconds, has been greatly improved by the recent availability of ultramicroelectrodes [l]. Thus, cyclic voltammetry at scan rates in the kilovolt per second [2-51, and then in the hundreds of kilovolt per second range [6-81 has allowed this method to be used for the generation and detection of intermediates with lifetimes of a fews tens of a nanosecond [6]. In a pursuit of shorter and shorter time scales, several authors have
l l
Presented at the J. Heyrovsky Centennial Congress * To whom correspondence should be addressed.
0022-0728/90/$03.50
0 1990 - Elsevier Sequoia
on Polarography,
S.A.
Prague,
20-25
August
1990.
336
reported the experimental feasibility of cyclic voltammetry at scan rates in the range of a few megavolts per second [9,10]. However, at such scan rates, even at ultramicroelectrodes, the cyclic voltammograms are distorted experimentally by instrumental and resistive phenomena. The recent design of ultrafast potentiostats [11,12] with on-line compensation of ohmic drop has allowed the recording of virtually undistorted cyclic voltammograms at scan rates exceeding the hundreds of kilovolts per second range. However, above this limit, instrumental limitations require that the faradaic information be extracted from the experimental voltammogram, in a similar way as has been done at lower scan rates, i.e. via simulation [7-10,131 or deconvolution procedures [13,14]. To the best of our knowledge, these simulation or deconvolution procedures have been based upon classical theories of cyclic voltammetry [15-171 even at scan rates of several megavolts per second. These classical theories for cyclic voltammetry suppose that the concentration profiles of the reactants, intermediates and products are controlled mainly by pure diffusion and kinetics. This amounts to considering (see below) that the sizes of the two layers, the double layer, &, and the diffusion layer, Sdir, are sufficiently different for the double layer to be vanishingly small with respect to the diffusion layer, i.e.: S,, +C &,. Under these conditions, the effects due to the double layer are taken into account via a simple correction of the electrode boundary condition, viz. the Frumkin correction [l&23]. However, such a situation is clearly not met when the diffusion layers are made thinner and thinner, i.e. when the scan rates are made larger and larger. Within the classical theory of cyclic voltammetry, the forward peak of a cyclic voltammogram recorded at a scan rate u corresponds to a diffusion layer of the order of (DRT/Fu)‘/~ [17], i.e. at room temperature and for an average diffusion coefficient of ca. 5 x lop6 cm2 s-‘, 6 = 3.5~“~ pm when u is expressed in V s-l. Therefore at e.g. u = l,OOO,OOOV s-l, the diffusion layer at the forward peak is of the order of 3.5 nm, that is comparable to the thickness of the diffuse layer under the usual electrochemical conditions in organic solvents. The transport equations in the two layers, i.e. the diffuse layer and the diffusion layer, obviously cannot be separated theoretically under such circumstances. This is clear evidence that the classical theories for cyclic voltammetry are no longer valid at such scan rates. It is the purpose of this preliminary work to evaluate theoretically the experimental effects on cyclic voltammograms due to the intimate coupling between the two layers. This will be worked out in the text for a single electron transfer reduction mechanism of a neutral molecule (n = 1, I, = 0, z,, = -l), i.e. for a case corresponding to the usual conditions found in organic electrochemistry: A=a
+
n
e-+
B’b=‘.-”
(E”, k;, a)
whereas Appendix 1 gives the corresponding analysis for the cases corresponding to any value of z, and n (zr, = z, - n). The analysis presented in the text will consider the two limiting situations: (i) fast electron transfer and (ii) quasi-reversible electron transfer. Although the limiting situation (i) is clearly not realistic at the scan rates
331
considered in this study owing to the exceedingly high values of kp require, we wanted to examine it with the purpose of demonstrating that effects predicted hereafter originate from concentration profile distortions not related only to electron transfer kinetics, as is the case for the correction.
it would the main and are Frumkin
RESULTS
Formulation of the problem At electrodes of radii in the micrometer range, and for the large scan rates considered in this study, diffusion can be considered as being planar and semi-infinite. Therefore in the following we will not consider any effect of radial diffusion that will possibly arise at smaller scan rates [l]. Similarly, the double layer, whose size is considerably smaller than that of the electrodes considered here, will be treated within a one-dimension approximation. It will be described by a classical version of the Gouy-Chapman-Stern model [24]. It is also assumed that the resulting electrical potential distribution within the diffuse layer is not affected by the faradaic phenomena [24]. Owing to the fast relaxation times of the double layer [25], this hypothesis is normally valid for any scan rate accessible experimentally today (u < 10’ V s-l). Therefore the electrical potential considered in this study is given as a function of distance, x, from the electrode surface by: x = 0;
@=‘J+,=E-E,,
(2)
o
‘p=~O+w~,)(%-%)
(3)
x > x2:
tanh( FO/4RT)
= tanh( F@,/4RT) X
exp (-(x
- x~)(~c~F~/~~,RT)~‘*)
(4)
where x2 is the location of the OHP, cs the 1: 1 supporting electrolyte bulk concentration, cs the relative dielectric constant of the medium *, E the electrode potential, Epzc the potential at the point of zero charge, and 9, the electrical potential in the OHP, as given by the solution of: a, = Q0 - x~(~RTc”/E~Q)~‘*
sinh( F@,/2RT)
(5)
Owing to the dimensions of organic molecules with respect to x2, no diffusionmigration process is considered within the compact layer, the OHP being taken as the site of the electron transfer. Therefore in the following, x2 is chosen as the origin of the distance from the electrode, the combination of the electrode and the compact layer being referred to as the “electrode surface”. Introducing thus x* =
* C, is supposed to be constant and equal to its value in the bulk of the solution. For all the figures and numerical applications given in the text, E, = 37.5 was taken to represent acetonitrile. Similarly no possible effect arising from the “discreteness-of-charges” [26] will be considered.
338
x - x2 allows one to describe the concentration profiles of the reactant and the product by the following partial derivative equations, valid for any values of z, and
‘b = z, - n: ~[A]/St=D${S[A]/6x*
+ (z,F/RT)[A](i3@/Gx*)}/Sx*
(6)
and : S[B]/Gt
=D,6{
6[B]/6x*
+ (zbF/RZ-)[B](S@/Gx*)}/6x*
associated to the following boundary values at x* = 0, i.e. at the OHP):
conditions
(7)
(where
the subscript
0 refers
to
[A] = co, [B] = 0
(8)
t>O,x*-+m:
[A] + co, [B] + 0
(9)
t>o,
4{S[A]/Sx*
t=o,
x* >o:
x*=0:
+ (z,F/RT)[A](Scp/Gx*)}o
+Db{ 6[B]/6x* t>o,
x*=0:
Da{
+dA]/aX* =
+ +
(Z,F/RT)[B](6@/6X*)},
(lo)
}
(11)
(Z,F/RT)[A](69/6X*)},
k: exp[ - anF( E - 9, - E”)/RT] x { [Alo - PI0 exp[ nF( E - a2 - E’)/RT]
For the particular case where a Nemstian electron when kp B (DFu/RT)“~), condition (11) is replaced that can be expressed at the OHP as: t>o,
= 0
transfer is considered by a modified Nernst
[Alo = [Blo exp[ nF( E - Q2 - E “)/RT]
x’=O:
The Faradaic
current
i, = nFAD,{ G[A]/Sx*
(i.e. law,
(11’)
is then given by: + (z,F/RT)[A](6@/6x*)},
(12)
In the following we will assume that the diffusion coefficients of A and B are identical (Da = D, = D), and that they remain insensitive to the variation of charge density and ionic strength in the diffuse layer. Let us introduce the following parameters: K = F(~c”/c~~, RT)‘12, and P = K( DRT/Fv)“‘, which represent respectively the reciprocal Debye length and the ratio between the diffusion layer and diffuse layer thicknesses. Introducing then the following dimensionless variables: concentrations
:
a = [Al/c’,
b = [ B]/c”
electrode
potential:
E = - nF( E - E o )/RT
electrical
potentials:
C#J = FQ/RT,
C/J,= F@,/RT,
space :
y=x*(nFu/DRT)“‘,
time :
r = t(nFv/RT)
Faradaic
current :
rate constant:
$,,,, = F( E o - E,,,)/RT
y*=Kx*,
y:=Kx2
4 = iF/[ nFAc’( nFuD/RT)“2] A = kf( RT/nFuD)“2,
A* = kf/KD
= A/r
339
allows one to rewrite the system of partial conditions in eqns. (6)-(121, as follows: &l/67=
I%{
6a/6y*
6b/67=r%{Sb/Gy*
derivative
equations
and
boundary
+ zaa(6~/6y*)}/6y*
(13)
+ (z,-n)b(S@y*)}/Gy*
(14)
with: 7=0,
y* >o:
a=l,
b=O
(15)
7>0,
y*-,cc:
u+l,
b+O
(16)
7>0,
y*=o:
{Way*
+z,4W~Y*)}O
= +{6b/6y*+(z,-n)b(6~/6y*)},=O 7>0,
y* =o:
{WSY*
+z,4%/~Y*)},
= A* exp[ 45 or, when 7>0,
+ nG2)] { a0 - b0 exp[ - (t + M2)3 }
A* x=- 1, i.e. when the electron
y* =o:
The Faradaic
a,=& current
(17)
transfer
exp[-(5+W2)1
step is considered
(18)
as Nernstian: (18’ )
is then given by:
rCI=r{6a/6Y*+z,a(S~/GY*)},
(19)
The resolution of the above system and boundary conditions requires the independent formulation of the dimensionless electrical potential, +, as a function of the distance, y *, from the electrode. This is obtained by solving the following equation valid for y * > 0: tanh( G/4) = tanh( +,/4)
exp( -Y * )
where (p2 is given by the implicit +2
=
‘ppzc
-
(5/n
> -
2~;
(20)
equation:
sirW (P2/2)
(21)
In the following we will focus on the special case where z, = 0 and n = f 1, since it corresponds to the most general situation encountered in organic electrochemistry. Transposition to an oxidation being obvious, the following presentation will be restricted to the case of a single electron transfer reduction (n = 1) of a neutral molecule (za = 0, zt, = - 1). However, Appendix 1 gives the corresponding results for any values of I, and n. Although all the equations given in the text or in Appendix 1 are valid for any electrochemical method provided that the corresponding relationship between 5 (or 4) and 7 is introduced, we will restrict our presentation of the different effects related to I to the case of cyclic voltammetry. Also, it should be emphasized that the various simulated voltammograms presented in this text correspond to a strict triangular variation of the potential with time. This supposes that ohmic drop and resulting phenomena are negligible or have been compensated electronically or via any appropriate treatment of the experimental cyclic voltammograms.
340
In the general case, i.e. for any value of I, the whole set of equations and boundary conditions (eqns. 13-21) can be solved via an explicit finite difference method. However, we want to establish first that the above formulation results in a classical one, as soon as l? + cc, i.e. as soon as the dimension of the diffusion layer greatly exceeds that of the diffuse layer. This will serve also as a demonstration of the validity of the classical theories - which are based upon the theoretical separation of the two layers (see above) - under such conditions. Limiting situation for r + 00 Since l? = K( DRT/Fv)‘12 compares the dimension of the diffusion layer, i.e. ca. ‘I2 , to that of the diffuse layer, i.e. ca K-’ [22], the condition I + cc, (DRT/Fv) simply expresses that the diffuse layer is a vanishingly small fraction of the diffusion layer. On the other hand, (6a/ST) = (RT/Fvc”)(8[A]/&) and (G/87) = (RT/ Fuc”)(6[B]/8t), as given in eqns. (13) and (14) have to remain finite since they represent the dimensionless time variations of [A] and [B], respectively. Therefore when I+ -+ cc, the partial derivatives with respect to y*, in eqns. (13) and (14) have to be vanishingly small for any value of y * comparable to unity, i.e. in the region of space where the electrical potential, $I, varies. When considerably larger values of y * are considered, y = y */I = x * ( Fv/DRT)‘j2 is introduced to yield: (&a/ST)
= 6’a/6y2
(22)
= 6%/Sy2
(23)
and : (Sb/&)
since then (&/a~*) = 0 for y* z+ 1. This shows that when I’ + cc, the system of partial derivative equations in eqns. (13) and (14) can be replaced by a set of two simplified systems, each one being valid within a given distance from the electrode. In the region corresponding to the diffuse layer (y * of the order of a few units or less) one has: S2a/6y
*2 = 0
(24)
and: 6{(6b/6y*)
-b(Gcp/Gy*)}/Sy*
= 0
(25)
whereas the classical diffusion equations (22) and (23) apply outside the diffuse layer, i.e. when y* x=- CL*= 1. Classical integration of the latter shows that a, and 4 at any point, y = Z.L = p*/T -=K1, of the diffusion layer are given by: ap = 1 - Z, = 1 - Z+
(26)
where:
zp= 7e2jT[( Sa/Sy),].(~-n)-“2 0
dn
(27)
341
On the other hand, because of eqns. conditions (17) and (19), one has: (W&J),=
-W/W,
= W/r)
As shown in Appendix electrode concentrations, a..,.. = a,, - #p
and
which obey the equality 1 - a..,..= b..,w= r-l’*
(24)
and
(25)
and
of the
boundary
(28)
= J/
2, one can therefore introduce a..,,, and b..,,., such that *:
the following
fictitious
b..,.. = b,, + I+!+
(29)
1 = aao” + b..,,.. and the following $,(v /0
n)-I’*
integral
equation:
dn
Therefore the voltammogram can be determined by the resolution equation (30) provided that CZ..~ or b..,,. are determined. This is done their relationship to the true electrode concentrations, a, and b,. eqns. (24) and (25), taking into account eqn. (28) (see Appendix l), finite value of y * :
(30) of the integral by establishing Integration of yields for any
~=~o+wnY*
(31)
and:
b = 2(rcl/r)ev(M) + [P-
(W’b*l exd+>
(32)
where j3 is an integration constant. Because of the result in eqn. (26) (a + b) as obtained from eqns. (31) and (32) must tend toward unity when y* = p* s=- 1, i.e. in the region where $J = 0. Therefore one obtains: /I = 1 - a0 - 2($/r). Taking into account the boundary condition in eqn. (18) then yields: a,=
{I+(~~)exp[(I-~)5-~~2)-2(~/r)[I-exp(-~2/2)1}/[I+exp(5)1 (33)
where A = A* I = kp/( DFv/RT)‘/*. On the other hand, from eqn. (31) owing to eqn. (29) where p= p*/P a..
one has a, = a, + ($/r)p*,
o..=a,-~p=[~o+(~/r)~*l-44~*m=u,
which
gives,
(34)
Introducing this latter value into eqn. (30) and noting AarP = [ Aexp( a+*)], which corresponds to the definition of the rate constant uncorrected for the Frumkin effect, i.e. k” = [k” t exp(aG2)] such that Aar’P= k”/( DFu/RT)‘/*, affords the
l Note that since p =8*/r, a,, (or b,) in the y space and ap. (or b,,,) in the y * space refer to the same concentration value at the same physical distance, x = x2 + ~/(~Fu/RT)‘/~ = x2 + y*/K, from the electrode surface. For the sake of clarity in the following, the concentrations will be noted CZ,,or b, independently of the space variable, y or y * ,used,whereas the distance will be noted p or p* depending on the space variable considered.
342
following
integral
{I + (+/A) = I - r-‘/z
equation:
exp[(I J
- a)E - o&,)J - 2(MXI
a’Gn( 7 - n)-rj2
- exp(-&/2)l}/[I
dn
(35)
which gives the equation of the dimensionless reduces to the classical one [16]: [l-($/AaPP)
+ exp(01
exp(-at)]/[l+exp(-5)]
voltammogram.
=V1/2&7$,,(7-n)p1’2
For I + cc, eqn. (35)
dn
(36)
Equation (36) features a quasi-reversible voltammogram. When Aapp -+ co, it simplifies to the classical integral equation corresponding to a Nernstian voltammogram [16]: l/[l
+ exp( -.$)I
= C1/2iT$n(
7 - n)-“2
dn
(37)
i.e. such that #p = 0.446, tp = 1.110, and .$p/2 = - 1.094 [16]. On the other hand when A”rP -+ 0, by introducing [* = ct.$+ ln(Aa%-‘/2), +* = $cu-“~, and 7* = (~7 in eqn. (36), one obtains the classical equation featuring an irreversible electron transfer forward wave: I - $,* exp( -,$*)
= 7~-‘/~
‘*$z 7* - n)-“2 /0 (
dn
(38)
which corresponds to $*p = 0.496, ,$*p = 0.783, [*p/2 = - 1.076 [16]. The above eqns. (36)-(38) establish then the validity of the classical theoretical approaches to the determination of the equations describing cyclic voltammograms, as soon as the diffuse layer is an infinitely small fraction of the diffusion layer, i.e. as soon as I + co, Indeed, they show that under such conditions the effect of the double layer is simply to affect the respective concentrations, [A], and [B],, of A and B at the electrode surface with respect to the fictitious ones, [A]..,,.. and [B]..,,,, used in classical theories. When I = k( DRT/Fu)‘/’ + co, as seen from eqn. (36), this correction amounts only to a thermodynamical correction, i.e. such that: [A]..,.{[B]..,..=
Wlo/PlO~
exd(z,
-+-JF+2/RTl
(39)
which is at the basis of the Frumkin correction [27]. As recognized earlier and demonstrated by eqns. (36) or (38), this effect can be easily be accounted for by using an apparent rate constant of electron transfer, k” = kf exp{ -[(l - (Y)z, + CXZ~]FC#J~/RT}, instead of the true rate constant k,.’ Therefore, as also previously recognized and shown by eqn. (37) this correction has no effect on a Nernstian wave. Limiting behaviour at large but not infinite values of r As shown in Appendix 2, the approximation that the space near the electrode is separated into two regions, viz. y* < p* and y* > EL*,remains valid as soon as the
343
concentration profiles of A and B which would be obtained by integration of eqns. (22) and (23) are almost linear for 0 < y* -C II*. Therefore, provided that I = remains sufficiently large for the diffuse layer to be a very small K( DRT/Fv)“’ but not negligible - fraction of the diffusion layer, it is expected that eqn. (35) will retain its validity. The exact limits of this validity will be determined in the following through the study of the general case corresponding to any value of I. However, the interest of considering eqn. (35) under conditions where the last term in the bracketed expression on its left-hand-side, i.e.: 2 [l - exp( -~$~/2)]/r, is small but does not cancel, is that one can thereby examine under which circumstances the “classical” eqns. (36)-(38) are approached. Let us first examine the case of a Nernstian electron transfer. Although “unrealistic” under experimental conditions where I is not infinite, since most electron transfers are not sufficiently fast for a Nernstian behaviour to be observed at the scan rates considered (see below), this situation will serve the purpose of demonstrating that the effects due to the double layer are not related only to electrode kinetics (compare above for the Frumkin correction when I + cc). Figures la and lb represent the variations of the dimensionless peak current, $p, and peak potential, [P, as a function of I, i.e. of the scan rate (upper abscissa scale) for two typical experimental situations: supporting electrolyte concentration cS = 0.1 M in acetonitrile (cS = 37.5), D = 5 X 10e6 cm2 s-i, and eppzC= - 10, i.e. E o = E,,, - 0.25 V or +pzC= -40, i.e. E” = Epzc - 1.0 V at room temperature. It is seen from these plots that the classical equation for a Nernstian voltammogram is no longer valid as soon as u exceeds a few hundred kilovolts per second. These deviations require, however, a high experimental precision in order to be detected on the forward peak. Yet, as illustrated by Figs. lc and d or 2a-c *, they are more easily observed when the shape of the whole voltammogram is considered. For example, it is seen from Fig. 2a that when u = lo7 V s-i, the anodic peak is no longer observable and is replaced by a broad anodic current. The same trend is observed in Fig. 2b, when the supporting electrolyte concentration is decreased, or in Fig. 2c, [,P cannot be defined when +ppzcis made more and more negative. As a consequence for some values of I, which is the reason for the “interruptions” in the plots of AtP or of E,,2 presented in Figs. lc and d. This arises because for the case considered in this study, the reactant A is a neutral molecule. Its concentration profile is then affected only slightly by the electrical potential within the double layer, through the boundary condition at x = 0. This is particularly clear when one considers that the descending branches of the forward peaks in Figs. 2a-c are not affected. Indeed, when nF( E - E “)/RT -=z 0, one has a, = 0 at the electrode surface, and the concentration profile of the neutral molecule A is not affected at all by the electric field of the diffuse layer, and therefore the current, $, is not affected at all either. This is obviously not the case for the charged molecule B, as shown by the
l Figure 2c also shows that the voltammograms are greatly affected by the exact value of E” - E,,, which may then afford an “easy” way of evaluating this parameter at solid electrodes.
log 2
0.49
a ~ :
qp
0.45
”
6
10 .”
ii,,
El ‘1
,;’
.:i,.
i
___
\
?,
0.4
1
-3
0
3
-log r
-3
0
-log r
r
-log
3
-3
0
3
-log r
Fig. 1. Variations of (a) the dimensionless peak current, #P = ip/[nFAc"(nFuD/RT)"'], (b) dimensionless peak potential, [P = - nF( Et’ - E o )/RT, (c)peak-to-peak potential difference, Alp = nF( E,Pa functionof E,J')/RT, and of (d) the dimensionless half-wave potential, [,,a = nF( E,P+ E,P)/(2RT),as r = K( DRT/Fu)'/~, for two typical experimental situations: cS = 0.1 M, es = 37.5 (acetonitrile), D = 5 X 10-6cm2s-‘and(i): $,,=-10,i.e. Ep,=Eo+0.25V,or(ii): ep,,=-40,i.e. Ep,=Eo+l.OV,in General case; (- - -) the case of a Nemstian electron transfer with z, = 0 and n = 1. ( -) asymptotic behaviour for r infinite (classical theories (15,161); (. . . . .) asymptotic behaviour for r B- 1 (eqn. 35, with AaPt’+ co); (------) asymptotic behaviour for r -=z1 (eqns. 40-45, 46’, 47); (----) limiting behaviour for r = 0 (eqn. 48, with A + co). Note that in (c. d) the potential scan is inverted at tr = 20, i.e. E, =E”--OSVforn=l.
concentration profiles of A and B in the diffuse layer represented in Figs. 3a-c. It is seen that although the diffuse layer is a very small fraction of the double layer for the scan rates considered, the concentration profiles, as obtained from the analytical eqns. (31) and (32), are greatly affected. Figure 4 shows that these results remain essentially identical when a quasi-reversible electron transfer is considered. Note that in Fig. 4, the voltammograms determined according to eqn. (35) correspond to a constant value of A for each set a-c. Therefore according to the classical theories of cyclic voltammetry, even when considering the Frumkin effect, the voltammograms represented on each set as a function of the scan rate should have been exactly superimposable. The obvious
345
e G 6 Fig. 2. Distortions observed for Nemstian voltammograms (z, = 0, n = 1) as a result of the coupling between the diffuse and diffusion layers. (a) Effect of scan rate, u, for D = 5 X lob6 cm2 s-‘, cS = 0.1 M, .) 105, (------) lo6 and (----) 10’ V r,=37.5 and cp,,=-10i.e. E,,=E’+0.25 V. v: (..... s-r (b) Effect of the 1: 1 supporting electrolyte concentration, c’, for u = lo6 V SC’, D = 5 X 10m6 cm2 s-‘, es = 37.5 and Eplr = E o + 0.25 V. P: (- - -) 0.5, (. +. . .) 0.1 and (- - - - -) 0.05 M. (c) Effect for u=105 V s-r, D=5X10-6 cm* s-l, c,=37.5 and c’=O.l M. of +rJpu:=F(E”-E,)/RT, 4 =iF/[nFAco(nFuD/RT)“*] and [= - nF(E - E”)/RT. GppLc:(- --) -10, (......) -30 and (- - - - -) - 50. (Note that in each set the voltammogram in solid line corresponds to I? infinite, i.e. that predicted by the classical theories (15,161.)
v,s-
IO'
1.oo
bl-_.. \ ;--Ic
et.4 ,/-q
0.50
!
\,_...‘.~..~’
,:\
,..:A ‘,
,:’
_:’
I
:.
,:’
0.00
0
0
Y'
IO"
1
.oo
,.I’
,...
1.00
:’ 0.00
@N
‘1\1 ;\
0.50
-:
-;A’
e
,:A
v.s-
0.50
@;
ij
‘\ :: ‘\,B
‘\B
‘\_
0.00
0.40
0.00
Y*
1,
0.00
0.04
Y’
Fig. 3. Dimensionless concentration profiles of A and B, and normalized electrical potential, $+ = G,/+~, near the electrode surface (y* = Kx* = (x - .x~)F(~c~/~~~RT)“~), for a Nemstian voltammogram, as a function of v for the following conditions: t, = 0, n =l, $,= = - 10 i.e. Epze = E o +0.25 V, D = 5 X 10m6 cm2 s-r, rS = 37.5 and cs = 0.1 M. These concentration profiles are determined at 6 = 15, i.e. E = E o -0.375 V, through the finite difference resolution of the general case, except for a and b, where eons. (31) and (32) could be used.
a
1
C
:il 1 I
I
I’
r;
:
-1 0
1020
cFig. 4. Distortions observed for a quasi-reversible voltammogram (a = 0.5, z, = 0, n = 1) as a result of the increasing coupling between the diffuse and diffusion layers when the scan rate, U, is increased, for D = 5 x 10m6cm* s-t, cS = 0.1 M, c, = 37.5 and $_= = - 10, i.e. EPlc = E” +0.25 V. To facilitate the comparison in each set, A = /c~(RT/~FuD)'/~ was kept constant: A =lO (a), 1 (b) and 0.1 (c). $= i,/[nFAc"(nFuD/RT)"*]and .$= -nF(EE')/RT. u:() 103,(-- -) 105,(. . . . . .) lo6 and (-----) 10’ V s-r.
differences layer. Limiting
thus show the effect of variations
in r, i.e. of the role of the diffuse
case for r -+ 0
The condition r + 0 corresponds to a situation where the diffusion layer is an extremely small fraction of the diffuse layer. For usual experimental conditions in liquid solvents, where a 1 : 1 supporting electrolyte at ca. decimolar concentration is used, this corresponds to an “unrealistic” situation, since it would correspond to diffusion layer dimensions of the order of molecular size. Therefore the limit r + 0 is considered here more in a mathematical sense than in an experimental one *, and for the purpose of checking independently the validity of the general solution at small r values (see below). When r = 0, the concentration of A and B differ from their bulk values only in a thin layer adjacent to the electrode where r$ varies almost linearly with the distance, y, from the electrode. Then, neglecting the terms in T2, eqns. (13) and (14) can be approximated by: Sa/Sr
= fY2a/6y2 + z,e (Ga/Sy)
(4)
Sb/ST=62b/6y2+(z,-n)e(Gb/Sy)
(41)
with, as deduced
from eqns. (20) and (21):
e = r(6+p3y*)0
= -2r
sinh($,/2)
-+ 0,
when
r + 0
(42)
l Note, however, than under conditions where low supporting electrolyte concentrations are used, the condition r + 0 would regain an experimental validity, provided that sufficiently high scan rates can be used, or provided that the size of the electrode is small enough (compare e.g. ref. 28).
347
Resolution of the system of limiting eqns. (40) and (41) associated following boundary conditions derived from eqns. (15)-(19):
with
the
7=0,
y>o:
a=l,
b=O
(43)
7>0,
y-,00:
a+l,
b-+0
(44)
7>0,
y=o:
(Way),
- 2f’[sinh(W2)]
+ 2r[sinh(+,/2)](z, 7 > 0, y = 0:
z,ao + (~~PJJ)~ - n)b,
- 2I? [ sinh( +,/2)]
(6a/6y),
= 0
(45)
z,a,
=~exp[~(5+~~,)]{~o-~oexp[-(~+~~,)l} was performed via an explicit current being then given by: # = (~G.JJ)~
- 2r[sinh(W2)]
For a Nemstian r>o,
y=o:
finite
electron ao=bo
difference
numerical
(46) procedure,
the Faradaic
~,a,
transfer,
(47)
i.e. when
A x=- 1, eqn. (46) was replaced
exp[-(d+W,)]
by: (46’)
The corresponding results are presented in Fig. 1 for the case of a Nernstian electron transfer (boundary condition 46’). Note that when r is sufficiently small, i.e. when f’[sinh($,/2)] + 0, all the eqns. (40)-(45) and (47) tend toward their analogs corresponding to a classical formulation of the problem. Yet the diffuse layer still affects the voltammograms, but now only through a modification of the boundary condition between a, and b,, as shown by eqn. (46’) or by eqn. (46) which, under these conditions, simplify respectively to: r>o,
y=o:
(Way)o
y=o:
a0 = b. exp[ - (< + MJ,)]
= A exp[a(5
+ W)l
{
ao-boexp[-(t+W,)l} (46a)
or: T>O,
Therefore (1 -(#/A) =77
the equation
of the limiting
(46’a) voltammograms
at r + 0 is given by:
exp[-~(5+~~~2)1}/{1+exp[-(E+n~,)l}
-1/2 bt/~~(7 - n)-1’2 /
dn
which corresponds in fact to a classical formulation [16] except that the potential of the electrode is referred to E” + c#B~, instead of E”, because the local potential at the end of the diffusion layer is now $J~ instead of 0, as observed when the diffusion layer thickness exceeds that of the diffuse layer. Therefore if c#B~(E) is constant in the potential region where the voltammogram is observed, this amounts simply to an apparent shifts of standard potential, from E” to E&, = (E” + c$~). If the variations of C#B~ cannot be neglected, the resulting voltammogram will be distorted, because under these conditions [E - (E o + cp2)]is no longer a triangular function of the time. The data reported in Fig. 1 correspond to the latter situation.
348
General case for any value of r In the general situation, i.e. when r does not take extreme values, no integral formulation was found. Therefore the system of partial derivative equations (13) and (14) and boundary conditions (15)-(19) was solved by an explicit finite difference procedure. To allow a simple discussion of the results, we will restrict their presentation to the case of a Nernstian electron transfer (i.e. boundary condition 18’ is used instead of 18). For a given experimental situation, the voltammograms then depend only on a single parameter, F, i.e. on the scan rate. The corresponding results are shown in Figs. la-d, in terms of the variations of +P, electron tp, AtP=[,P -5,P and ,$,,2 = ([,P + 5,P)/2. Note that for a Nernstian transfer the latter parameter should remain constant and close to zero, since E,,2, the half-sum of the two peak potentials, is nearly identical to E ’ provided the scan is inverted sufficiently after E o [15]. DISCUSSION
The comparison, in Fig. 1, of the above results based on the general system of eqns. (13)-(19) without any simplification, to those obtained through the approximation leading to eqn. (35), demonstrates that the latter equation remains valid up to ca. lo6 V s-‘. Above 10’ V s-r one observes that the simplified treatment leading to eqn. (35) introduces significant errors. Thus, eqn. (35) can be considered as affording a reasonably good description of any voltammogram obtained within the experimental range of scan rates accessible today. We will therefore restrict the following discussion to the predictions derived from eqn. (35) i.e. to the case of large values of r. Equation (35) shows that the voltammograms observed under these conditions are greatly affected by the relative magnitude of the diffusion layer size with respect to the diffuse layer, i.e. by the value of r = (2FDcS/eS~,v)‘/* = K( DRT/Fv)‘/*, but also by the exact value of the potential at the OHP, i.e. by any experimental parameter that affects &. Among the latter, the most critical is certainly +rpu:= F( E o - E,,)/RT (compare e.g. Figs. 2c and 5a below), since its exact value is generally unknown owing to the general lack of knowledge on Epu: values for organic conditions at solid ultramicroelectrodes. However, this dependence on & must not be confused with that resulting only from the classical Frumkin effect. This is shown by the comparison of the three voltammograms presented in Fig. 5b. Voltammogram I (- . . . . . ) corresponds to the classical theory of cyclic voltammetry [16] (eqn. 36), where the value of $2 at E = E o was taken to evaluate an average value of k” = kf exp[crF$*( E = E “)/RT] which was then used at all points of the voltammogram, as is usually done [15-171. For voltammogram II (), eqn. (36) was used again but now the Frumkin correction, i.e. the value of k” = kf exp[aF+*( E)/RT], was evaluated at each point of the voltammogram. In the determination of the third voltammogram (- - -), k” was determined as for II, but the effects due to the overlap of the diffuse layer and diffusion layer were now taken into account via eqn. (35). Therefore the effects
349 3 $2
-2
-7
Fig. 5. (a) Variations of I& = Fh/RT, as a function of 6 = - F( E - E”)/RT, for different values of as indicated by the number on the curves. (b) Distortion of a “theoretical” Q = F(E” - E,,)/RT, q!&i-reversible cyclic voltammogram as a function of the level of approximation in the “theory”. a=OS, r,=O, n=l, kp =lOcm s-‘, D=5X10-6cm2 s-‘, c”=O.l M, r,=37.5 and s#+= -10, i.e. = E” +0.25 V. (I) (. . . .) “Classical” theory [15-171, i.e. eqn. (36) was used with a constant value E -- 2.33 cm s-‘. (II) () Equation (36) was ot?O , taken at E=E”, i.e. k”= kp exp[a+a(E”)] used as for (I), but k o was evaluated at each potential, i.e. k o (E) = k,“ exp[a&( E)]. (III) (- - -) As for (II) but eqn. (35) was used instead of eqn. (36). # = i,/[nFAc”(nFuD/RT)“*] and 6 = - nF( E E “)/RT.
arising from the fact that r is not rigorously infinite at scan rates in the megavolt per second range can be appreciated by comparison of voltammograms II and III. Note also that a comparison of voltammograms I and II in this figure demonstrates that the use of an average value for the Frumkin correcting term in extracting a rate constant from an experimental voltammogram, e.g. from the peak-to-peak separation as a function of scan rate [l&12], may result in significant errors, independently of any possible additional effects due to ohmic drop or to the fact that at the large scan rates considered, i.e. for u > lo5 V s-l, l? is not infinite. These effects can be appreciated more quantitatively through the working curves presented in Fig. 6, where it is seen that for a given electroactive couple, and a given electrode-solvent-supporting electrolyte system, the characteristics of quasi-reversible voltammograms depend on two parameters, A = kp( DFv/RT)‘/’ and A* = A/r = kp/( KD), instead of only one parameter, viz. A, in the classical theory of cyclic voltammetry [16]. This can be understood if eqn. (35) is rewritten in the following “classical” form: [l - (Jl/A,i,)
exp(-a[)]/[1
+exp(-E)]
=~“‘/~#,,(r-n)-“~ 0
dn
(49)
where AIic, a fictitious dimensionless rate constant related to A and r by: Atic= [Aexp(~~))1/{1-2[Aexp(a+2)/r]
exp[-(1
-a)[][1
-exp(-+,/2)]} (50)
is now used instead of A = kp/( DFv/RT)
1’2 [15-171.
This amounts in fact to
350 0.46 1
1
w
a
0.5
.,
0.39
0.32
-2.50
-1.50
-0.50
-log
0.1
0.50
A
12
1.20
up
c
112
7
2 -2.50
0.50
-0.20 -1.50
-0.50
-log
0.50
-2.50
- 1.50
A
-log
-0.50
0.50
A
Fig. 6. Variations of the dimensionless (a) peak current, ~P=i~/[nFAc”(nFuD/RT)“*], (b) peak potential, IP = - nF( E,P - E o )/RT, (c) peak-to-peak potential difference, AIP = - nF( E,P - E,P)/RT, and (d) half-wave potential, E,,* = nF(E,P + E,P)/(2RT), as a function of A = kp( RT/nFuD)‘/* and I (solid curves; the numbers on the curves are the values of A* = A/r = kF/KD), for a typical experimental situation: (Y= 0.5, I, =O, n=l, c”=O.l M, ~,=37S(acetonitriIe), D=5X10-6cm2 SC’ and E,, = E o +0.25 V. Same data according to classical theories [15,16], without (. . . . . .), or with (- - - - -) taking into account the variations of & with the potential (compare (II) in Fig. 5). The data reported for each case correspond to a potential scan inversion at 5, = 20, i.e. at Et = E o - 0.5 V.
introducing kg,=
a fictitious
[ki’exp(%)]/{1xed-(l
rate constant
kg,, such that k:, = Al,i,( DFU/RT)“‘:
2[kp ev(wh)/(DK)]
-a)El[l -ev(-W2)1}
(51)
i.e.:
kp,=k0/{1-2[ko/(DK)]exp[-(1 -~)t][l -exp(-h/2)]}
(52)
where k” = kp [exp(a&)] is the rate constant including the Frumkin effect. It is noteworthy that eqn. (52) shows that for an irreversible electron transfer with z, = 0 and n = 1, the forward wave is almost unaffected by the effect of the double layer, except for the Frumkin correction, Indeed, under these conditions the forward wave will be observed at extremely large values of the potential, so that eqn. (52) simplifies to k:, = k” = kp[exp( a&)] in the potential range where the forward wave is observed. However, this does not indicate that the voltammogram is not affected, because, since the backward wave is observed at very negative values of the
351
potential 5, the effect on Zcic will be extremely important, as deduced from eqn. (52). This dissymmetric behaviour originates from the fact that in establishing eqn. (52), we considered a neutral molecule as the starting material, A. Therefore, in the potential range where the forward irreversible wave is observed, the concentration profile of A is insensitive to any electrical effect, since it corresponds to the integration of a partial derivative equation associated with a boundary conditions at the electrode surface: r>o,
{6a/6y},=nexp[cw(~+~*,)la,
y=o:
which does not incorporate any electrical component. potential range where the backward wave is observed, eqn. (53) must be replaced by: 7>0,
y=o:
{W~_Y},=
(53)
On the contrary, in the the boundary condition in
-Aexp[(a-I)((++,)]&
(54)
and will introduce a strong coupling with the local electrical field since B is charged. The analysis presented in Appendix 1 shows that the formulation eqn. (49) remains valid for any value of z, and n, provided that A ric = kz,/( nDFu/RT)‘/* is defined on the basis of the following fictitious rate constant of electron transfer: G, = k”/{ I - (2k”/DK)(
f(~,)
exddl +f(z, - n) ed - (I- cf>Sl)}
where k” = kp exp[an - z,)Z+,/RT] effect, and f(z) is defined as follows:
f(z)=
C m=O
is the rate constant
including
{1-exp[(z/IzI)(m+1/2)~~2/RTl}/(2m+1)
(55)
the Frumkin
(56)
with f. = 0. The corresponding variations of k& with the electrode potential should then be experimentally accessible through convolution procedures, provided the experimental accuracy on the voltammograms obtained at the scan rates considered is sufficient. Indeed, from eqn. (49) one obtains [14,29]: k& = D’&/{
Zlim - Z exp[ (nF/RT)(
where Z is the convoluted
fi,(t-n)p1’2
Z=77-‘/2
experimental
E - E’)] current
}
(57)
given by:
dn
/ 0
and Zlim is its plateau potential of the A/B
value at (nF/RT)( E - E,,2) electrochemical wave.
z+ 0, where E,,2 is the half-wave
ACKNOWLEDGEMENTS
This work was partially supported by CNRS (URA 1110: “Activation Moleculaire) and Cole Normale Suptrieure. The authors greatly acknowledge Drs.
352
H.S. White and S.W. Feldberg for communication, before publication, of their results in ref. 28, which deals with the effect of the double layer on steady-state voltammograms at microelectrodes of nanometer radii. APPENDIX
1: DERIVATION
OF EQUATIONS
(49), (55) and (56) FOR ANY VALUE of z, AND n
As explained in the text, when I x=-1 and y* is not larger than a few units, eqns. (13) and (14) reduce to: Sa/Gy* + Z,a(sC#J/sy*)
=4/r
(Al)
6b/6y*+(z,-n)b(69/6y*)=
-(#/r)
(A2)
with, from eqn. (20): (@/Sy
* ) = - 2 sinh( G/2)
(A3) Since eqns. (Al) and (A2) differ only in the z, of (z, - n) factor and the sign of the second member, let us focus on the integration of eqn. (Al). Integration of this equation by the method of variation of the constant gives readily: a = A exp( -z,$)
(A4)
with:
A=
(G/r)/[exp(z,+)l dy*
W)
+A,
where A, is an integration constant. Introducing w = exp( G/2) and taking eqn. (A3) into account shows that: A -A,=2(~/r)l[w2ia/(i-W2)]
dw=2(+/r)W(z,)
(A6)
Noticing that: +/(I
- &) = #.-2
[I - (1 - w’)]/(l
- w’) = ,2ki)/(l
- w’) - &-2
shows that : Wz,)
= Wz,
- 1) - {exp[(2z,
-
1W21}(2z, - 1)
(A?
from which results, since W, = (y*/2) 2,-l
+ cons& that for any z, > 1:
A = (+/I)Y*
+ l/2)1 Mm
- (V7
C {exp[+ m=O
+ l/2)
+A;
(A8)
and for any zaG -1: -(%+I) C
A=(\t/I)y*-(+/I)
m=O
{exp[-~(m-l/2)1}/(m+1/2)+Al,
(A9)
and for z, = 0: A = (+/r)y*
+A;
(AlO)
353
where Ah is a constant. These three relationships can also be rewritten form valid for any value of za, giving, owing to eqn. (A4):
in a unifying
~~~~~~~,9~=~Jl/~~~*+~~~/~~{~~~~~~~,/l~,I~~I~,I+~/~~1}/~~l~,I+~~ -2(~,r)‘~‘{exp[~cZ,/lr,I)(m+l/2)1}/(2m+1)+Ab m=O (All)
Therefore at any p* such that +,,* = 0, a,* is given by (note that since uY,_* = uyCP, this value is noted up irrespective of the space variable y or y* considered):
a, = M~cL*
I Z, I + 1) - w/r)
+ 2WW2
‘2’ W(2m
+ 111 + 4
(A121
m=O
Eliminating the constant Ah by expressing eqn. A(ll) for y* = 0, and introducas defined in eqn. (29) for p = p*/r), finally affords: ing a..,.,=up -
(+/r)p,
~~~~~,~~~+~~~/~~(~-~~~~~~~~,/I~,I~~I~,I+~/~~1}/~~l~,I+~~
a..,,.=a,
- wm
‘5’ {I- exP[+Aza/lzalb m=O
+ l/2)1 )/Pm + 1)
W)
that is: a..,,.= a0 exp(Q,)
- 2(#/r)f(z,)
(AI4)
where f(0) = 0 and for any z, # 0: I~,‘--’ f(z,)
=
C
{I-
exP[+z(za/lzal)(m
+ 1/2)1}/(2m + 1)
(A15)
m=O
The same treatment can be applied to eqn. (A2). Thus by simple permutation of [a, z, and (4/r)] into [b, zi, = (za - n) and -(J//r)], respectively, one obtains readily: b ssos9=bo exp[(z,
- n)h)
+ %VW(z,
- 4
(AW
Because of their definition in eqn. (29) (see also Appendix 2), the fictitious electrode concentrations a..,,. and b..,., in eqns. (A14) and (A16) must fulfill the mass conservation equality, 1 = a..,., + b..,,, i.e.: I = a0 cxp(r.&z)
+ b. expk
- ~>&)I
- 2(W’)[f(r,)
-.%,-
4
(AI7)
On the other hand, a0 and b,, which are the true electrode surface concentrations, are related to the current, 4 in eqn. (19), by the Butler-Volmer rate law in eqn. (18). Introducing Aapp = A exp[( an - z,)&] = k “/(~DFu/RT)‘/~, where k o
354
= kp exp[(cyn - z,)&J,
allows one to rewrite eqn. (18) as follows:
= b. expK z, - n)+J]
a0 exp(z,&)
Introduction of this latter (A16) yields b..,,.: b .so..= { 1 - ( $/Aapp)
exp(-t)
value
into
+ (#/Aa”““)
exp(-&)
eqn. (A17) affords
b,, which
(Al81 from
eqn.
exp( --at)
+2(~/r)[f(z,)-_f(~,--n)l}/[1+exp(-5)1 +W/W(za-4 (Al9) i.e.: { 1 - ( $/Rfi,)
b YY=
exp( --(yt)}/[l
+ exp( -Z)l
(A20)
where: A,,=AaPP/{I-2(Aapp/r)(f(Z,)
exp[&]
Equation (A21) amounts transfer, k& such that: A riC= k&/(
nDFu/RT
to the definition
+f(z,-n)
exp[(a-1)5])}
of a fictitious
rate constant
(A21) of electron
)112
(A=)
i.e.:
k”/{l -
k& =
exp[&l +,+,
2(k”PW(f(r,)
- n) exp[(a
- IHI)}
(A23)
where: k” =kp
exp[(cyn--z,)&]
(~24)
is the rate constant uncorrected for the Frumkin effect and kto, the true rate constant. Introducing b..,,. as given by eqn. (A20) into eqn. (30) finally yields the following integral equation: [l-
(#/A,,)
exp(-a[)]/[1
+exp(-5)1
=~-~/~j6t./&-n)-“*dn
(A25)
which is sufficient for describing the voltammogram, owing to the definition of AliC in eqns. (A22)-(A24). The case of a Nernstian voltammogram is obtained by using the local Nernst equation in eqn. (A18’), instead of eqn. (A18), i.e.: a0 exp(z,+,)
= b. exp[(z,
which amounts follows that: b sso..= {I+
- n)+2)1
in fact to considering
&vW(d
-f(z,
(~18’)
exp(-5) that
- dl}/[l
A”PP 1s . infinite
in eqn. (AlS).
+ ew(-5)1+ wmfb,
It then
-4 (A19’)
355
i.e.: b s.os.= {I+
W/~)[.fkJ
+fG,
exd-81)A1+ exd-5)l
- 4
With eqn. (30) this finally yields the following integral voltammogram of a Nemstian electron transfer: {I +2(4/IX/a,)
+fk-
4
equation
(A20’)
which describes
the
ev(-01 )A1+exd-t91
=77 -‘I2/ bqn(7- r~-l’~dn
(A25’)
which amounts in fact to considering eqn. (A25) associated with eqn. (A22) and the limiting form of eqn. (A23), for k” + cc. When r + 00, eqns. (A25) or (A25’) tend toward their classical forms in eqns. (36) and (37), respectively. APPENDIX
2: DERIVATION
OF EQUATIONS
(29) AND (30)
In this Appendix we want to establish the validity of eqns. (29) and (30). Let us thus consider an electrochemical system in which A and B undergo pure diffusion. Their dimensionless concentrations then obey the system of partial differential equations (22) and (23). Integration of these in the Laplace plane, taking into account eqns. (15) and (16), gives readily at any value p of the y space variable: a, = (l/s)
- (&z/&&/~
“2 = (l/s)
- Ir/ exp( - j.~‘/~)/s’/~
(A26)
and: 4 = - ( S_~/G~)JS’/~= Ir, exp( - &‘2)/s”2 where s indicates Laplace transform (&J/&Y), When
G%I/CY),=
14%
Laplace
-4+)=($+4+)=~+
gives:
-Ps”2)
(~29) into eqns. (A26) and (A27):
= (_b, +&u> = $/s”2 transformation
7, and _V the
(A2g)
small, so that ~FLS’/’+ 0 *, this equation
then, when introduced
- (Ccl - 94
Inverse
=J/ exp(-l-is”‘)
-(&/&&=~(I
which affords (l/s)
the Laplace variable corresponding to the time of any variable ‘I”. One has therefore:
= -(M/Q),
p is infinitely
(~27)
of eqn. (A30) finally
(A301 yields: (A3I)
l Note that the inverse Laplace transform of this condition is (p/2)(?r/7)‘/’ + 0, i.e. it implies that the distance x,*, corresponding to p in the real space, is such that x: -=K2( Dt/vr)‘/*, i.e. it is considerably smaller than the diffusion layer for the time, 1, considered. Since eqns. (29) and (30) are considered only under conditions where p = p*/I with I5-> 1, the condition ps’/* -=c1 is always fulfilled whenever these equations are used, since it expresses that the time considered is sufficiently large for the diffuse layer to be only a small fraction of the diffusion layer, i.e. x,’ = K-’ -=x( DI)‘/*.
356
where ZJ/ is defined ZI)=T-“~
/0
by:
‘I//,,(T-~)-“~ dn
Therefore, and b..,..:
one can define
b-w
the following
fictitious
electrode
concentrations
a..,..
a..,,.=a,-$+=1-Z+
(A33)
= bP+ $q = Z$ b ‘sow
(A34)
which follow eqns. (29) and (30). Note that in the classical theories of cyclic voltammetry [15-171 these fictitious concentrations are implicitly considered as the real electrode concentrations, the “error” being corrected a posteriori by the fact that a fictitious electrode boundary condition:
i, = nFAc”{ kp exp[( fxn-L,)F$J~/RT]}
exp[-cunF(E--E”)/RT]
X { a..,.. - b..,.. exp[ nF( E - EO)/RT]}
is used instead
of the true one (eqns. 11 and 12):
t,=nFAc”kto
exp[ -mF(E-
(A35)
E” -+2)/RT]
X{a,-b,exp[nF(E-E”-$,)/RT]} which should
be used but applies
(A36) only to the real electrode
concentrations
a, and
b0’ APPENDIX
3: NUMERICAL
COMPUTATIONS
All the numerical computations were performed on either a COMPAQ 386 or an AMSTRAD PC-AT 2386, using Turbo-Pascal programs. All the cases amenable to an integral-equation formulation (see text) were solved according to standard previously described procedures (see e.g. ref. 14). Two cases could not be solved following these procedures and were solved using standard explicit finite difference procedures: (i) the general case valid for any P value, and (ii) the limiting case at P + 0. For these applications the analytical values of (6+/6y) or (S2+/Sy2): (S+/6y)
= - 2 sinh( +/2)
(A37)
and: ( S2+/Sy2)
= sinh( +)
(A38)
were used instead of numerical determinations based on finite differences. However, when considering larger and larger values of P, a classical explicit finite difference resolution of the general case would have required too large computational times and memory occupation. Therefore, for any P values corresponding to a diffusion layer thickness exceeding twice that of the double layer, a double space grid was used. In the space region adjacent to the electrode with
351
y Q pi, CL, being such that CL,= 10 1+2/(6+/Sy)y_o 1, a space increment Ay, = p,/lOO was used together with a time increment Ar, = 0.44(A~,)~. For y >, ~1, a space increment close to Ay, = p2/100, where p2 = 6 - P,, 6 being the diffusion computation, was used thickness at r = Q,~, i.e. at the end of the voltammogram together with a time increment A72 = 0.44(A~,)~. In practice, Ay, and Ar, were adjusted slightly, with respect to the above values, so that Ar2 was a multiple of AT, in order to facilitate the numerical connection between the two finite difference grids. The numerical connection between the two finite difference grids was performed as follows. Let us define N = Ay,/Ay,; then Ar2/Ar, = N 2. Therefore, at any time TV= k Ar2 = kN2 Ar, the two time grids coincide. Let c( kN2, 1) to c(kN’, 100) be the concentration values of the considered species for time rx., at the different mesh points pertaining to the first grid, and c’(k, 1) to c’(k, 100) those for the second grid. From these concentrations, those c(kN2 + 1, 2) to c(kN’ + 1, 99) are obtained through normal algorithms, and c(kN’ + 1, 1) is obtained from the latter by application of the boundary condition(s) at y = 0. Similarly c’( k + 1, 1) to c’( k + 1, 99) are obtained through standard explicit finite difference procedures from c( kN2, 100) and c’(k, 1) to c’( k, 100) (note that in practice only c’( k + 1, 1) to are determined, y,,_, = n maxAy, being the thickness of the diffuc’(k + I, n,,,) sion layer at the time considered, i.e. nmax= lOO[(k + 1) AT~/T,,_]‘/~; for nmax < n d 100, c’(k + 1, n) is kept identical to ~‘(0, n), i.e. to the value at 7 = 0). From c’( k, 1) and c’( k + 1, l), one generates then (N 2 + 1) intermediate values c”( kN 2 + h) such that: c”(kN2+h)=c’(k,1)+h[c’(k+1,1)-c’(k,l)]/N’
(A39)
with h = 0 to N2. Taking into account the corresponding values of c(kN2 + h, 99) and c( kN2 + h, lOO), the latter are used to determine the values of c( kN2 + h + 1, 100) at each time T = (kN 2 + h + 1) A7,. which are needed to complete the first grid. In this procedure the partial first and second derivatives with respect to space, at 7 = (kN 2 + h) AT, and y = p, = 100 A y,, are expressed as follows: (S~/Gy)=((Ay,)~c”(kN~+h)+[(Ay~)*-(Ay~)~]c(kN~+h,100) -(Ay#c(kN’+ (6’c/Sy2)=
(AY,)(AY~)[(AY,)
whereas
the partial
k)/[(A~i) derivative
+ (AY,)~} -2c(kN’+h,
{2(Ay2)c(kN2+h,99)/[(Ay,)+(Ay2)] +2(Ay,)c”(kN’+
respectively,
k, 99))/{
(AdO) 100)
+ (AY~)~}/[(AY,)(AY~)~ with respect
(A4I)
to time is expressed
(S~/&r)=[c(kN~+h+l,lOO)-c(kN*+h,lOO)],’A~, The validity of this procedure was tested by comparing its results to simulation using a single grid (i.e. with a mesh [Ay,, AT,]) extending up of the diffusion layer. For cases where the diffusion layer thickness was times that of the diffuse layer, the two procedures gave identical cyclic
by: (A42)
those of a to the end up to four voltammo-
358
grams, as demonstrated by the identity (relative difference less than 10P5) of their peak current and peak potential values. However, the gain in computational time and memory occupation was considerable when using the “double grid” method, since the latter is approximately N3 times faster and requires N3 times less memory occupation, where N is the relative magnitude of the diffusion layer with respect to the diffuse layer (see above). REFERENCES 1 R.M. Wightman and D.O. Wipf in A.J. Bard (Ed.), Electroanalytical Chemistry, Vol. 15, M. Dekker, New York, 1989, pp. 267-353. 2 J.O. Howell, J.M. Goncalves, C. Amatore, L. Klasinc, R.M. Wightman and J.K. Kochi, J. Am. Chem. Sot., 106 (1984) 3968. 3 (a) J.O. Howell and R.M. Wightman, Anal. Chem., 56 (1984) 524; (b) J.O. Howell and R.M. Wightman, J. Phys. Chem., 88 (1984) 3915; (c) J.O. Howell, W.G. Kuhr, R.E. Ensman and R.M. Wightman, J. Electroanal. Chem., 209 (1986) 77. 4 M.I. Montenegro and D. Pletcher, J. Electroanal. Chem., 200 (1986) 371. 5 A. Fitch and D.H. Evans, J. ElectroanaI. Chem., 202 (1986) 83. 6 C. Amatore, A. Jutand and F. Pfhtger, J. Electroanal. Chem., 218 (1987) 361. 7 D.O. Wipf, E.W. Kristensen, M.R. Deakin and R.M. Wightman, Anal. Chem., 60 (1988) 306. 8 C.P. Andrieux, P. Hapiot and J.M. Saveant, J. Phys. Chem., 92 (1988) 5987. 9 C.P. Andrieux, D. Garreau, P. Hapiot and J.M. Saveant, J. Electroanal Chem., 248 (1988) 447. 10 D.O. Wipf and R.M. Wightman, J. Phys. Chem., 93 (1989) 4286. 11 C. Amatore, C. Lefrou and F. Pfltiger, J. Electroanal. Chem:, 270 (1989) 43. 12 D. Garreau, P. Hapiot and J.M. Savtant, J. Electroanal Chem., 281 (1990) 73. 13 C.P. Andrieux, D. Garreau, P. Hapiot, J. Pinson and J.M. Savtant, J. Electroanal. Chem., 243 (1988) 321. 14 J.M. Saveant and D. Tessier, J. Electroanal. Chem., 77 (1977) 225. 15 R.S. Nicholson and I. Sham, Anal. Chem., 36 (1964) 706 and references therein. 16 L. Nadjo and J.M. Saveant, J. Electroanal. Chem., 48 (1973) 113 and references therein. 17 C.P. Andrieux and J.M. Saveant in C.F. Bemasconi (Ed.), Investigation of Rates and Mechanisms of Reactions, Vol. 6, 4/E, Part 2, Wiley, New York, 1986, Ch. 7, pp. 305-390 and references therein. 18 A.N. Frumkin, Z. Phys. Chem. A, 164 (1933) 121. 19 R. Parsons, Trans. Faraday Sot., 47 (1951) 1332. 20 J.E.B. Randles, Trans. Faraday Sot., 48 (1952) 828. 21 D.M. Mohilner and P. Delahay, J. Phys. Chem., 67 (1963) 588. 22 P. Delahay Double Layer and Electrode Kinetics, Insterscience, New York, 1965, pp. 153-167. 23 Ref. 22, p. 199. 24 Ref. 22, pp. 33-52. 25 For a discussion of this point see ref. 22, pp. 50-51. 26 W.R. Fawcett and S. Levine, J. Electroanal. Chem., 43 (1973) 175. 27 See e.g. ref. 22, pp. 154-158. 28 J.D. Norton, H.S. White and S.W. Feldberg, submitted. 29 J.M. Sadant and D. Tessier, J. Phys. Chem., 81 (1977) 2192.