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Hydrodynamic voltammetry at channel electrodes
247
J. Electroanal. Chem., 209 (1986) 247-258 Elsevier Sequoia S.A., Lausanne - Printed
HYDRODYNAMIC
in The Netherlands
VOLTAMMETRY
AT CHANNEL
ELECIRODES
PART VIII. THEORY OF REVERSIBLE VOLTAMMOGRAMS FOR CHRONOAMPEROMETRY AND LINEAR SWEEP VOLTAMMETRY
KOICHI
AOKI,
KOICHI
TOKUDA
and HIROAKI
MATSUDA
Department of Electronic Chemistry, Graduate School a? Nagatsuta, Nagatsuta, Midori-ku, Yokohama 227 (Japan) (Received
19th March
Tokyo Insrrtute of Technology,
1986; in revised form 25th April 1986)
ABSTRACT An expression for the reversible transient current at a channel flow electrode was derived when a potential with any time variation was applied to the electrode. First, the expression was adapted to chronoamperometry. As the electrolysis time elapses, the current density distribution varies from a Cottrellian uniform distribution to a non-uniformly steady-state distribution. Second, the expression for the reversible linear sweep voltammogram was derived. For small values of a dimensionless potential sweep rate, the voltammograms are in a sigmoidal form similar to the steady-state curve. As the values become large, the voltammograms have a peak which is often observed in quiescent solution. The dependence of the peak current and the potential at the half peak current on the dimensionless potential sweep rate was examined.
INTRODUCTION
The main advantage of hydrodynamic electrodes is to exhibit current-potential curves under the steady-state condition. On the other hand, transient behaviour at hydrodynamic electrodes is significant from the following viewpoints: (A) A transient technique is useful for controlling the amounts of deposited or adsorbed species on the electrode. (B) It is necessary to confirm that the current approaches the steady state. (C) Analytical sensitivity is enhanced in a transient domain. Furthermore, it is interesting to examine the time variations of the current density distribution at a non-uniformly accessible electrode because the current distribution shows Cottrellian behaviour with uniform accessibility at the beginning of electrolysis and then approaches a distribution with non-uniform accessibility for long-term electrolysis. 0022-0728/86/$03.50
0 1986 Elsevier Sequoia
S.A.
248
A popular hydrodynamic electrode with non-uniform accessibility is the channel flow electrode, which has been widely applied to a detector of liquid chromatography [l-6] and flow injection analysis [7-91. Work on the time dependence at a channel flow electrode includes chronoamperomet~ [lo], chronopotentiomet~ [ll-131, linear sweep voltammetry [14], ac voltammetry [15] and the galvanostatic transient at a double channel flow electrode [16,17]. In the studies on chronoamperometry [lo] and linear sweep voltammetry [14], however, the behaviour on a short time scale has not yet been fully discussed. Further, we are not satisfied with the derivation of the equation for chronoam~romet~ given by Compton and Daly
IW* In this paper we derive an expression for the cbronoamperometric curve in the entire time domain. The derivation is extended to linear sweep voltammetry in the reversible process for any value of the dimensionless potential sweep rate. TRANSIENT
CURRENT
RESPONDING
TO A POTENTIAL
WITH ANY TIME VARIATION
We consider the following anodic charge-transfer reaction: Red + Ox + n e- at the channel flow electrode. It is assumed that: (a) The convection of the solution is the fully developed Poiseuille flow. (b) The thickness of the convective diffusion layer is smaller than half the height of the channel so that the parabolic velocity profile can be linearized about the electrode surface. (c) The width and length of the rectangular channel electrode are so large that edge effects can be neglected. (d) Species 0 is absent initially in the solution. (e) The diffusion coefficients of species 0 and R have a common value, D. (f) The electrode reaction takes place so rapidly that the Nernst equation holds at the electrode surface. Then the following time-dependent convective diffusion equation holds in the vicinity of the electrode surface: ac,/at+(3U,/b)y(ac,/ax)=o(a2c,/ay2)
(j=OorR)
(I) where c, denotes the concentration of the electroactive species Ox (j = 0) or Red ( j = R), U, is the mean flow velocity, b is the half height of the chamlel, and c is the time elapsed since the beginning of electrolysis. The initial condition is given by t =
0,
x>o,
C R=C*,
y>o:
where c* is the buIk concentration by t>o, x=0, y>o: CR=C*, t>o,
x>o,
t > 0,
OdX
y-,cc: y=o:
0
co=
CR+?,
(2)
of species R. The boundary conditions are given co
=
(3) (4)
0
co-‘0
co/+ = exp[ (nF/RT)( i i/RF= D(l&A3y)
=
E - EO’)]
-D(ac,/ay)
0) (6)
249
where E denotes the potential which is a function of the time, E O’ is the formal potential, i is the current density, and the sign for the anodic current is taken to be positive. Introducing new variables: 5 = x/x,
(7)
TJ= (3U,/bxlD)“3y
(8)
6 = (3Um&‘bx,)2’3r
(9)
we obtain ac,/af3 + q( aqag)
( j = 0 or R)
= a%,/aq2
(IO)
with the initial condition: e=o,
t>o,
r]>o:
cn=c*,
co=0
(II)
and with the boundary conditions at 0 > 0 [=O,
?l>o:
a$>o,
?I-+cc:
cR=c*,
co=0
ca+c*,
(12)
co + 0
(13)
c,/c,=exp[(nF/RT)(E-E”‘)]
O<[
i/nF
?j=o: i
(14)
= (3U,,,D2/bx,)1’3(
&,/as)
(15)
5 - (3u,D2/bx,)1’3(aco/a11)
on applying a double Laplace transformation with respect to 7 and 13 to eqn. (10) by taking into account conditions (11) and (12), eqn. (10) is reduced to an ordinary differential equation, of which the solution is given by Ai( PS-“~ + ?v), where Ai is the Airy function [18], and p and s are the Laplace-transformed variables with respect to 8 and E, respectively. Inserting this solution into the double Laplace transform for eqn. (15) and following the same procedure as that described before [13], we obtain the relations between the Laplace-transformed concentrations at the electrode surface and the transformed current density, [QP,
s>],=o=c*/sP-
[~,(P~
&=cl=
(bx,/3UmD2)1’3(
( bx,/3U,,,D2)1’3(
$nF)j(
;/nF)j( p, s)
p, s)
(16) (17)
with j( p, s) = -Ai(
ps-2/3)/[s1/3Air(
ps-2/3)]
(18)
where the double bar represents the double Laplace transformation with respect to 8 and [, and Ai’ denotes the derivative of the Airy function. The inverse Laplace
250
transforms convolution
of eqns. (16) and (17) with respect theorem,
to p become,
[ ~~(0, s)] ,,=,, = c*/s - (bx,/3U,DZ)“3~8(i/nF)j(B [c,(B,
s)],,=,,=
with the aid of the
- u, s) du
(hx,/3U,D2)“‘joe(~/~F)J(B-
(19)
a, s) du
(20)
where the single upper bar represents the Laplace transformation with respect to [. Carrying out the Laplace transformation for the Nernst equation (14) with respect to 5, inserting eqns. (19) and (20) into the resulting equation, solving it for the term J,“(i/nF)f(B - u, s) du, and performing again the Laplace transformation with respect to 8, we obtain i/nF=
(~*/s)(3U,D~/bx,)~‘~p&[(l+
e-l)-‘]
E1( p, s)
(21)
with { = (nF/RT)(
E - E”‘)
TTl(p, s) = l/{
(22)
spj( p, s)}
= -Ai’(
ps-2/3)/[s2/3pAi(
ps-2/3)]
(23)
where L, denotes the operator of the Laplace transformation with respect to 8. The inverse Laplace transform of eqn. (23) has been derived for the unsteady convective heat-transfer problem by Soliman and Chambre [19]. It is given by wi(e,
E) = [31’3/r(1/3)]5-“3
‘fa,-l exp[
+3&‘3~-b’3
- (2/27)+?3g-2]
Ai [3-4’3+2[-4/3]
(24)
where aJ is the j-th zero of the Airy function. When the values of e3Ee2 are small, it is difficult to evaluate H,(8, .$) from eqn. (24). The equation effective for such small values of e3tm2 has been derived in the Appendix by applying the derivation in previous papers [17,20], and is given by q(e,
t) - (qe)-‘/2
+ (2jlr)3-S/2e4t-3
exp( -3ee3t2)
(25)
In order to help evaluation of H,(B, 5) by a simple calculation, approximate equation for H,(e, 0 in the following form: (&)-1’2 H,(e,
6) =
+ o.17484[-3
i { 3”3/l-(1/3)}
5-1’3
exp( -3t2/30-1) (= 0.5384 [-1’3)
we obtained
an
for 0 < &$-2/3 < 1.26
(26a)
for t?5-2’3 > 1.26
(26b)
where the error involved in these equations is less than 0.8%. The first terms in eqns. (26a) and (26b) have been taken from the first terms of eqns. (25) and (24), respectively. The second term in eqn. (26a) was derived by trial and error on the basis of the functional form of the second term in eqn. (25). Carrying out a double inverse Laplace transformation, we obtain i = i(b), 6) = nFc*(3U,D2/~_x,)“3(d/dB)/os{l
+ e-rcU)}-‘H,(B
- u, 6) du
(27)
251
This is the expression for the transient current density when a potential with any time variation is applied to the electrode. The total current, I, i,s given by Z=Z@)=w =
10
?dx
nFc”(3U,D2x~/b)1’3w(d/d8)~~(1
+ e-rcu1)P1EZ2(B - U, 1) du
(28)
where w is the width of the electrode. Here ZY,(B, 5) is defined by the integral of H,(e, 5)
= [34/3/{2r(1/3)}]
<2’3 + 32/3/‘U-‘/3 0
E o;” exp[ - (2/27)o;e3ZC2] 1’1
dn
XAi[3-4~3i$t12~-4~3]
(29)
where eqn. (24) has been employed. The equation valid for large values of t!Z3em2has been derived in the Appendix, and is given by ff,(e,
t) = <(ne)-1/2
+ e/4 + (2 x 31/2n)e7t-4
exp( - 3P3t2)
(30)
We evaluated the integral in eqn. (29) by means of Simpson’s l/3 rule. As a result of numerical computation of eqns. (29) and (30), we found that the first two terms of eqn. (30) and the leading term of eqn. (29) become a good approximation for ZZ2(B, 6). The approximate equation is given by H,(e,
z) =
- 1/2 + e/4 t(Ve) i [34’3/(21’(1,‘3)}]
for 0 < 86-2’3 Q 1.08 (31) C2’3 ( = 0.8076~2’3)
for 8[-2/3 > 1.08
(32)
where the error involved in this approximate equation is less than 0.6%. Equation (28) combined with eqns. (31) and (32) expresses the transient total current responding to any potential-time curve such as chronoamperometry, linear sweep voltammetry, normal pulse and differential pulse voltammetry, square wave voltammetry and ac volta~et~. It will be applied to c~ono~peromet~ and linear sweep voltammetry in the following sections. CHRONOAMPEROMETRY
When the potential is stepped to the domain of the limiting current, eqns. (27) and (28) are reduced to i = i(f?, 4) = nFc+(3u,D2/~xl)1’3H,(e,
6)
(33)
252
f=l(fq
= “Fc*(3u,D2x:/b)1’3wH,(8,
nFc*x,w(D/st)“2(1 =
+
(&/4)83’9
1) = 3’/3nFc*(U,D2X:/b)“3
forO
xw{(rB)-1’2+8/4}
/ [3’/“/(2r(l/3)}]
~~~*~~~~2~~/~)1’3~
(34)
for t9> 1.08
In Fig. 1, the dimensionless current density, i( 8, S)/i(ao, 1) or ZZ,(@,.$)/ZZ,(co, l), is plotted against both 6 and t. At the onset of electrolysis (at small values of 0), the current distribution is uniform except at the upstream edge (I - 0) of the electrode. With the lapse of the electrolysis time, the current density becomes non-uniform and then finally approaches a steady-state dist~bution with a $-‘/‘-dependence [20]. The current density at the upstream edge is infinite at all times. Conversely, the current density at the downstream edge (.$ - 1) decreases with time and then approaches a steady value. The domain of 85-2/3 for which the transient current density, i(r9, 0, can be regarded as the steady-state one, i(co, 0, within 5% errors, is t9.$-2/3 3 1.03 or t 2 2.~~~b~/~~~~ 2’3. This relation indicates that the current density near the upstream edge (4 - 0 or x - 0) promptly approaches the steady-state current and that it gradually spreads to the downstream edge. When t 2 2.15(b~,/U,~)~‘~, the current density all over the electrode surface can be regarded as the steady-state one. In Fig. 2, the values of the dimensionless total current calculated from eqn. (34) are plotted against 8. The Cottrellian curve expressed by Z,,,,(B)/Z(co)
= 21-‘(1/3)3-4’3(a8)-“2
is also plotted in Fig. 2 (---).
(35)
With the lapse of time, the total current varies from
Fig. 1. Current distribution in chronoamperometry.
253
OO
0:4
0:2 e
[ =
0.6
0.8
1.0
1.2 4
(34/6x,Pv”fl
calculated from eqn. (34), the Cottrellian Fig. 2. Dimensionless chronoamperometric curve ( -) curve (---) from eqn. (35), and the curve obtained by Compton and Daly (. . . . .).
the Cottrellian to the steady-state current. The transient region of B from the Cottrellian to the steady-state current is narrow. The total current can be approximated by the Cottrell equation within 5% error when 8 < 0.24 or t < the total O-48(bx,/U,,,fi) 2/3. Conversely, for 13> 0.70 or c > 1.47(bx,/U,@)2’3, current can be regarded as the steady-state one within 5% error. The solid curve in Fig. 2 is in excellent agreement with the ‘curve obtained by digital simulation by Flanagan and Marcoux 1211.Compton and D&y [lo] derived an analytical equation for the transient current by transforming H,(p, s) into a series of rational functions. The dotted curve in Fig. 2 was taken from Fig. 1 of ref. 10. The transient curve obtained by Compton and Daly does not satisfy the Cottrell equation at short times, because their approximation is valid for long times. LINEAR SWEEP VOLTAMMETRY
For the linear potential sweep mode, the potential is expressed by E = E, + vt
(36)
or 5=3i+Ue
(36’)
with si= (nF/RT)(E-
E”‘)
u = ( nFv/RT ) ( bx,/3U,,,fi)2’3
(37) (38)
where v is the potential sweep rate, u is the dimensionless potential sweep rate, E,
254
is the initial potential and si is the dimensionless initial potential. Inserting eqn. (36’) into eqn. (28), carrying out the differentiation and replacing u in eqn. (28) by (3’ - M/o yields Z= I((, u) = (31/3/4)nFc’(U,D2x~/~)1’3w~‘sech2({’/2)ZZ2(({
-‘{‘)/e,
1) d[’ (39)
This is the equation for a reversible linear sweep voltammogram. For extremely slow sweep rates or for small values of (I, ZZ2(({- l’)/a) becomes 34/3/(21(1/3)). Then the current is given by Z(l, 0) = Z,, = [35/3{21?(1/3)}] nFc*(U,D2x:/b)1’3w(l
+ eeS)
(40)
where Z,, denotes the steady-state current of which the waveform is sigmoidal. Conversely, ZZ,(({ - {‘)/u) tends to (ntY)-“2 as u approaches infinity. Then
is essentially the same as the equation for a linear sweep voltammogram at a stationary planar electrode in quiescent solution [22]. Therefore eqn. (41) does not involve any hydrodynamic parameters, U, or 6. For any value of u, eqn. (39) is not reduced to a simple form. We computed the integral in eqn. (39) by Simpson’s l/3 rule. The singularity occurring at l’ = S in eqn. (39) was eliminated by integration by parts. The voltammograms thus calculated are shown in Fig. 3 for several values This
c
c=
[nf/Rn
1E -
Fig. 3. Linear sweep voltammograms 0 = 40, calculated from eqn. (39).
r”)
1
for (a) o < 0.2, (b) 0 = 1, (c) 0 = 3, (4
c = 10, (e) 0 = 20 and (f)
255
2..
<
G ,_ 2 z
2-1.
/'
/'
/'
/'
4'
/'
1'
1'
OO
2 0 l'* c =' inFv/Rn
Fig. 4. Variation of the without convection.
3
4
1’2uJx1/3um)‘nLYsl
peak or maximum current with 6. The dashed line denotes the peak current
of u. The current in the ordinate is normalized by the steady-state limiting current, I ss,llm7given by
Iss,Lim= [3”‘{2I’(l/3)}]
nFc*( U,D2~:/b)*‘3~
(42)
Curve a is the steady-state voltammogram expressed by eqn. (40). As the values of c increase, the curves shift slightly in the negative direction and then a peak appears (curve c). With further increase in u, the peak is so appreciable that the peak potential can be determined unequivocally (curves d-f). The diffusion tail followed by the peak approaches the steady-state limiting current. These variations have also been seen in the linear sweep voltammograms at a rotating disk electrode [23]. In Fig. 4, the peak or maximum current, Ip, is plotted against 6. When the values of u increase, the peak current increases monotonically from the steady-state limiting current and then approaches the peak current at the stationary planar electrode in quiescent solution, which is given by (43) In Fig. 4, the line for eqn. (43) is shown as a dashed line. The peak current for 6 < 1.5 is the same as the steady-state limiting current within 5% error. When 6 > 2.7, the effect of convection on the peak current is negligible. In Fig. 5, the potential at the half-peak height or half-peak potential, Ep,2 - E”‘, is plotted against 6. {p,2 in the ordinate denotes the dimensionless half-peak potential given by &,2 = (nF/RT)(E,,, - E”‘). With increase in 6, &,,2 varies from the steady-state half-peak potential, Ep,2 = E O’, to the half-peak potential at the stationary planar electrode in quiescent solution, &,,2 = - 1.09 or Ep,2 - E O’ = - 28/n mV at 25°C. When 6 < 0.6, the half-peak potential is in agreement with
256
Fig. 5. Plot of the half-peak (wave) potential against 6. T’he dashed line denotes the half-peak potential without convection. The dotted curve was calculated from eqn. (40). which was derived by Compton and Unwin [14].
E O’, within 5/n mV error. Conversely, for 6 > 1.6, the effect of convection on the half-peak potential is negligible within 5/n mV error at 25 o C. Compton and Unwin derived analytically an expression for the linear sweep voltammogram in series form when the values of u are small [14]. They evaluated the half-peak potential and found that it was expressed approximately by &,z = 0.55 u The curve for eqn. (44) is plotted in Fig. 4 ( * . . . - ). The curve for 6 < 1 is in good agreement with the solid curve. Therefore eqn. (44) provides a good measure of whether the measured voltammograms are under the steady-state condition. We summarize the dependence of the voltammogram on 6 as follows: For 6 < 0.6, both the half-peak (wave) potential and the maximum current are of the steady-state voltammogram. For 0.6 < 6 < 1.5 - 1.6, the half-peak (wave) potential shifts in the negative direction, the maximum current keeping the steady-state value. For 1.5 - 1.6 < 6 < 2.7, the half-peak potential is close to (E O’ - 28)/n mV at 25°C and the peak is appreciable. For 6 > 2.7, both the half-peak potential and the peak current are those at the stationary planar electrode without any convection. ACKNOWLEDGEMENT
We thank Dr. Compton for informing us of ref. 14 in a private communication. APPENDIX
In this Appendix, derived.
the asymptotic
expansions
for Hi(@, 6) and H2(@, i) are
257
gi( p, S) is the same as B(p, s)/s in eqn. (24) of ref. 17. The asymptotic expansion for g(p, [) has been expressed by eqn. (Al-9) of ref. 17, from which @(p, 5) is immediately given by H,( p, 5) -p-1’2
1+
(l/r)
I 641)
2 b, {4,(3[P3/2)}J’2Kj(4fiP3/4/dq J=o
[
where K, is the modified Bessel function of the second kind with the j-th order. Taking the first two terms in eqn. (Al) and expressing K,(4fi~~‘~/fi) by the asymptotic expansion [23] yields H,(PY
t> -p-
‘I2 + (2/n)1’23-‘/4[-3/4p-13’8
exp[ - (4/fi),/$~~‘~]
642)
The inverse Laplace transform of p-1318 exp[ - (4/6)fi~~‘~] was derived by the steepest descent method [24], of which examples have been given in Appendix 3 of ref._l7. Thus eqn. (25) was derived. i?2(p, s) can be expressed by g(si)/s, in eqn. (I-2) ofgf. 25 when sp-3’2 is replaced by sr. Applying eqns. (I-19) and (I-20) of ref. 25 to Hi( p, s) and taking the first three terms yields H,(P>
6) - Ufi
+ 1/(4p2)
- V(7r~‘)Ko(4&~“~/fi)
Inserting the asymptotic expansion for Ka(4fi~~‘~/fi) H,(PY
(A3)
into eqn. (A3) leads to
5b-E/fi+1/(4P2) _ @l/42-
3/2,-1/2~-1,‘4)p-19/8
exp[ - (4/6)jZ~~‘~]
out the inverse Laplace transformation of p-19” use of the steepest descent method [26] yields eqn. (30). Carrying
exp[ - (4/ &)&P~‘~]
644)
by
REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
P.T. Kissinger, Anal. Chem., 49 (1977) 447A and refs. cited therein. R.M. Wightman, E.C. Pa&, S. Barman and A.M. Dayton, Anal. Chem., SO (1978) 1410. W.L. Caudill and RM. Wightman, Anal. Chim. Acta, 141 (1982) 269. D.A. Roston, R.E. Shoup and P.T. Kissinger, Anal. Chem., 54 (1982) 1417A and refs. cited therein. W.L. Cat&l, J.O. Howell and R.M. Wightman, Anal. Chem., 54 (1982) 2532. W.L. Caudill, A.G. Ewing, S. Jones and R.M. Wightman, Anal. Chem., 55 (1983) 1877. C.B. Ranger, Anal. Chem., 53 (1981) 20A and refs. cited therein. K.K. Stewart, Anal. Chem., 55 (1983) 931A. J. Ruzicka, Anal. Chem., 55 (1983) 104OA. R.G. Compton and P.J. Daly, J. Electroanal. Chem., 178 (1984) 45. R.E. Meyer, M.C. Banta, P.M. Lantz and F.A. Posey, J. Electroanal. Chem., 30 (1971) 345. F.A. Posey and R.E. Meyer, J. Electroanal. Chem., 30 (1971) 359. K. Aoki and H. Matsuda, J. Electroanal. Chem., 90 (1978) 333. R.G. Compton and P.R. Unwin, J. Electroanal. Chem., 206 (1986) 57. R.G. Compton and G.R. Sealy. J. Electroanal. Chem., 145 (1983) 35. T. Tsuru, T. Nishimura, K. Aoki and S. Haruyama, Denki Kagaku (J. Electrochem. Sot. Jpn.), 50 (1982) 712. 17 K. Aoki, K. Tokuda and H. Matsuda, J. Electroanal. Chem., 195 (1985) 229.
258 18 M. Abramowitz and LA. Stegun, Handbook of Mathematical Functions, Dover, New York, 1970, pp. 446-450. 19 M. SoIiman and P.L. Chambre, Int. J. Heat Transfer, 10 (1967) 169. 20 K. Aoki, K. Tokuda and H. Matsuda, J. Electroanal Chem., 76 (1977) 217. 21 J.B. Flanagan and L. Marcoux, J. Phys. Chem., 78 (1974) 718. 22 K. Aoki, K. Tokuda and H. Matsuda, J. EIectroanaI. Chem., 146 (1983) 417. 23 P.C. Andricacos and H.Y. Cheh, J. Electrochem. Sot., 127 (1980) 2385. 24 M. Abramowitz and I.A. Stegun in ref. 18, p. 378. 25 K. Aoki, K. Tokuda and H. Matsuda, J. Electroanal. Chem., 79 (1977) 49. 26 G.N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, London. 1966, pp. 235-241.
J. Electroanal. Chem., 209 (1986) 247-258 Elsevier Sequoia S.A., Lausanne - Printed
HYDRODYNAMIC
in The Netherlands
VOLTAMMETRY
AT CHANNEL
ELECIRODES
PART VIII. THEORY OF REVERSIBLE VOLTAMMOGRAMS FOR CHRONOAMPEROMETRY AND LINEAR SWEEP VOLTAMMETRY
KOICHI
AOKI,
KOICHI
TOKUDA
and HIROAKI
MATSUDA
Department of Electronic Chemistry, Graduate School a? Nagatsuta, Nagatsuta, Midori-ku, Yokohama 227 (Japan) (Received
19th March
Tokyo Insrrtute of Technology,
1986; in revised form 25th April 1986)
ABSTRACT An expression for the reversible transient current at a channel flow electrode was derived when a potential with any time variation was applied to the electrode. First, the expression was adapted to chronoamperometry. As the electrolysis time elapses, the current density distribution varies from a Cottrellian uniform distribution to a non-uniformly steady-state distribution. Second, the expression for the reversible linear sweep voltammogram was derived. For small values of a dimensionless potential sweep rate, the voltammograms are in a sigmoidal form similar to the steady-state curve. As the values become large, the voltammograms have a peak which is often observed in quiescent solution. The dependence of the peak current and the potential at the half peak current on the dimensionless potential sweep rate was examined.
INTRODUCTION
The main advantage of hydrodynamic electrodes is to exhibit current-potential curves under the steady-state condition. On the other hand, transient behaviour at hydrodynamic electrodes is significant from the following viewpoints: (A) A transient technique is useful for controlling the amounts of deposited or adsorbed species on the electrode. (B) It is necessary to confirm that the current approaches the steady state. (C) Analytical sensitivity is enhanced in a transient domain. Furthermore, it is interesting to examine the time variations of the current density distribution at a non-uniformly accessible electrode because the current distribution shows Cottrellian behaviour with uniform accessibility at the beginning of electrolysis and then approaches a distribution with non-uniform accessibility for long-term electrolysis. 0022-0728/86/$03.50
0 1986 Elsevier Sequoia
S.A.
248
A popular hydrodynamic electrode with non-uniform accessibility is the channel flow electrode, which has been widely applied to a detector of liquid chromatography [l-6] and flow injection analysis [7-91. Work on the time dependence at a channel flow electrode includes chronoamperomet~ [lo], chronopotentiomet~ [ll-131, linear sweep voltammetry [14], ac voltammetry [15] and the galvanostatic transient at a double channel flow electrode [16,17]. In the studies on chronoamperometry [lo] and linear sweep voltammetry [14], however, the behaviour on a short time scale has not yet been fully discussed. Further, we are not satisfied with the derivation of the equation for chronoam~romet~ given by Compton and Daly
IW* In this paper we derive an expression for the cbronoamperometric curve in the entire time domain. The derivation is extended to linear sweep voltammetry in the reversible process for any value of the dimensionless potential sweep rate. TRANSIENT
CURRENT
RESPONDING
TO A POTENTIAL
WITH ANY TIME VARIATION
We consider the following anodic charge-transfer reaction: Red + Ox + n e- at the channel flow electrode. It is assumed that: (a) The convection of the solution is the fully developed Poiseuille flow. (b) The thickness of the convective diffusion layer is smaller than half the height of the channel so that the parabolic velocity profile can be linearized about the electrode surface. (c) The width and length of the rectangular channel electrode are so large that edge effects can be neglected. (d) Species 0 is absent initially in the solution. (e) The diffusion coefficients of species 0 and R have a common value, D. (f) The electrode reaction takes place so rapidly that the Nernst equation holds at the electrode surface. Then the following time-dependent convective diffusion equation holds in the vicinity of the electrode surface: ac,/at+(3U,/b)y(ac,/ax)=o(a2c,/ay2)
(j=OorR)
(I) where c, denotes the concentration of the electroactive species Ox (j = 0) or Red ( j = R), U, is the mean flow velocity, b is the half height of the chamlel, and c is the time elapsed since the beginning of electrolysis. The initial condition is given by t =
0,
x>o,
C R=C*,
y>o:
where c* is the buIk concentration by t>o, x=0, y>o: CR=C*, t>o,
x>o,
t > 0,
OdX
y-,cc: y=o:
0
co=
CR+?,
(2)
of species R. The boundary conditions are given co
=
(3) (4)
0
co-‘0
co/+ = exp[ (nF/RT)( i i/RF= D(l&A3y)
=
E - EO’)]
-D(ac,/ay)
0) (6)
249
where E denotes the potential which is a function of the time, E O’ is the formal potential, i is the current density, and the sign for the anodic current is taken to be positive. Introducing new variables: 5 = x/x,
(7)
TJ= (3U,/bxlD)“3y
(8)
6 = (3Um&‘bx,)2’3r
(9)
we obtain ac,/af3 + q( aqag)
( j = 0 or R)
= a%,/aq2
(IO)
with the initial condition: e=o,
t>o,
r]>o:
cn=c*,
co=0
(II)
and with the boundary conditions at 0 > 0 [=O,
?l>o:
a$>o,
?I-+cc:
cR=c*,
co=0
ca+c*,
(12)
co + 0
(13)
c,/c,=exp[(nF/RT)(E-E”‘)]
O<[
i/nF
?j=o: i
(14)
= (3U,,,D2/bx,)1’3(
&,/as)
(15)
5 - (3u,D2/bx,)1’3(aco/a11)
on applying a double Laplace transformation with respect to 7 and 13 to eqn. (10) by taking into account conditions (11) and (12), eqn. (10) is reduced to an ordinary differential equation, of which the solution is given by Ai( PS-“~ + ?v), where Ai is the Airy function [18], and p and s are the Laplace-transformed variables with respect to 8 and E, respectively. Inserting this solution into the double Laplace transform for eqn. (15) and following the same procedure as that described before [13], we obtain the relations between the Laplace-transformed concentrations at the electrode surface and the transformed current density, [QP,
s>],=o=c*/sP-
[~,(P~
&=cl=
(bx,/3UmD2)1’3(
( bx,/3U,,,D2)1’3(
$nF)j(
;/nF)j( p, s)
p, s)
(16) (17)
with j( p, s) = -Ai(
ps-2/3)/[s1/3Air(
ps-2/3)]
(18)
where the double bar represents the double Laplace transformation with respect to 8 and [, and Ai’ denotes the derivative of the Airy function. The inverse Laplace
250
transforms convolution
of eqns. (16) and (17) with respect theorem,
to p become,
[ ~~(0, s)] ,,=,, = c*/s - (bx,/3U,DZ)“3~8(i/nF)j(B [c,(B,
s)],,=,,=
with the aid of the
- u, s) du
(hx,/3U,D2)“‘joe(~/~F)J(B-
(19)
a, s) du
(20)
where the single upper bar represents the Laplace transformation with respect to [. Carrying out the Laplace transformation for the Nernst equation (14) with respect to 5, inserting eqns. (19) and (20) into the resulting equation, solving it for the term J,“(i/nF)f(B - u, s) du, and performing again the Laplace transformation with respect to 8, we obtain i/nF=
(~*/s)(3U,D~/bx,)~‘~p&[(l+
e-l)-‘]
E1( p, s)
(21)
with { = (nF/RT)(
E - E”‘)
TTl(p, s) = l/{
(22)
spj( p, s)}
= -Ai’(
ps-2/3)/[s2/3pAi(
ps-2/3)]
(23)
where L, denotes the operator of the Laplace transformation with respect to 8. The inverse Laplace transform of eqn. (23) has been derived for the unsteady convective heat-transfer problem by Soliman and Chambre [19]. It is given by wi(e,
E) = [31’3/r(1/3)]5-“3
‘fa,-l exp[
+3&‘3~-b’3
- (2/27)+?3g-2]
Ai [3-4’3+2[-4/3]
(24)
where aJ is the j-th zero of the Airy function. When the values of e3Ee2 are small, it is difficult to evaluate H,(8, .$) from eqn. (24). The equation effective for such small values of e3tm2 has been derived in the Appendix by applying the derivation in previous papers [17,20], and is given by q(e,
t) - (qe)-‘/2
+ (2jlr)3-S/2e4t-3
exp( -3ee3t2)
(25)
In order to help evaluation of H,(B, 5) by a simple calculation, approximate equation for H,(e, 0 in the following form: (&)-1’2 H,(e,
6) =
+ o.17484[-3
i { 3”3/l-(1/3)}
5-1’3
exp( -3t2/30-1) (= 0.5384 [-1’3)
we obtained
an
for 0 < &$-2/3 < 1.26
(26a)
for t?5-2’3 > 1.26
(26b)
where the error involved in these equations is less than 0.8%. The first terms in eqns. (26a) and (26b) have been taken from the first terms of eqns. (25) and (24), respectively. The second term in eqn. (26a) was derived by trial and error on the basis of the functional form of the second term in eqn. (25). Carrying out a double inverse Laplace transformation, we obtain i = i(b), 6) = nFc*(3U,D2/~_x,)“3(d/dB)/os{l
+ e-rcU)}-‘H,(B
- u, 6) du
(27)
251
This is the expression for the transient current density when a potential with any time variation is applied to the electrode. The total current, I, i,s given by Z=Z@)=w =
10
?dx
nFc”(3U,D2x~/b)1’3w(d/d8)~~(1
+ e-rcu1)P1EZ2(B - U, 1) du
(28)
where w is the width of the electrode. Here ZY,(B, 5) is defined by the integral of H,(e, 5)
= [34/3/{2r(1/3)}]
<2’3 + 32/3/‘U-‘/3 0
E o;” exp[ - (2/27)o;e3ZC2] 1’1
dn
XAi[3-4~3i$t12~-4~3]
(29)
where eqn. (24) has been employed. The equation valid for large values of t!Z3em2has been derived in the Appendix, and is given by ff,(e,
t) = <(ne)-1/2
+ e/4 + (2 x 31/2n)e7t-4
exp( - 3P3t2)
(30)
We evaluated the integral in eqn. (29) by means of Simpson’s l/3 rule. As a result of numerical computation of eqns. (29) and (30), we found that the first two terms of eqn. (30) and the leading term of eqn. (29) become a good approximation for ZZ2(B, 6). The approximate equation is given by H,(e,
z) =
- 1/2 + e/4 t(Ve) i [34’3/(21’(1,‘3)}]
for 0 < 86-2’3 Q 1.08 (31) C2’3 ( = 0.8076~2’3)
for 8[-2/3 > 1.08
(32)
where the error involved in this approximate equation is less than 0.6%. Equation (28) combined with eqns. (31) and (32) expresses the transient total current responding to any potential-time curve such as chronoamperometry, linear sweep voltammetry, normal pulse and differential pulse voltammetry, square wave voltammetry and ac volta~et~. It will be applied to c~ono~peromet~ and linear sweep voltammetry in the following sections. CHRONOAMPEROMETRY
When the potential is stepped to the domain of the limiting current, eqns. (27) and (28) are reduced to i = i(f?, 4) = nFc+(3u,D2/~xl)1’3H,(e,
6)
(33)
252
f=l(fq
= “Fc*(3u,D2x:/b)1’3wH,(8,
nFc*x,w(D/st)“2(1 =
+
(&/4)83’9
1) = 3’/3nFc*(U,D2X:/b)“3
forO
xw{(rB)-1’2+8/4}
/ [3’/“/(2r(l/3)}]
~~~*~~~~2~~/~)1’3~
(34)
for t9> 1.08
In Fig. 1, the dimensionless current density, i( 8, S)/i(ao, 1) or ZZ,(@,.$)/ZZ,(co, l), is plotted against both 6 and t. At the onset of electrolysis (at small values of 0), the current distribution is uniform except at the upstream edge (I - 0) of the electrode. With the lapse of the electrolysis time, the current density becomes non-uniform and then finally approaches a steady-state dist~bution with a $-‘/‘-dependence [20]. The current density at the upstream edge is infinite at all times. Conversely, the current density at the downstream edge (.$ - 1) decreases with time and then approaches a steady value. The domain of 85-2/3 for which the transient current density, i(r9, 0, can be regarded as the steady-state one, i(co, 0, within 5% errors, is t9.$-2/3 3 1.03 or t 2 2.~~~b~/~~~~ 2’3. This relation indicates that the current density near the upstream edge (4 - 0 or x - 0) promptly approaches the steady-state current and that it gradually spreads to the downstream edge. When t 2 2.15(b~,/U,~)~‘~, the current density all over the electrode surface can be regarded as the steady-state one. In Fig. 2, the values of the dimensionless total current calculated from eqn. (34) are plotted against 8. The Cottrellian curve expressed by Z,,,,(B)/Z(co)
= 21-‘(1/3)3-4’3(a8)-“2
is also plotted in Fig. 2 (---).
(35)
With the lapse of time, the total current varies from
Fig. 1. Current distribution in chronoamperometry.
253
OO
0:4
0:2 e
[ =
0.6
0.8
1.0
1.2 4
(34/6x,Pv”fl
calculated from eqn. (34), the Cottrellian Fig. 2. Dimensionless chronoamperometric curve ( -) curve (---) from eqn. (35), and the curve obtained by Compton and Daly (. . . . .).
the Cottrellian to the steady-state current. The transient region of B from the Cottrellian to the steady-state current is narrow. The total current can be approximated by the Cottrell equation within 5% error when 8 < 0.24 or t < the total O-48(bx,/U,,,fi) 2/3. Conversely, for 13> 0.70 or c > 1.47(bx,/U,@)2’3, current can be regarded as the steady-state one within 5% error. The solid curve in Fig. 2 is in excellent agreement with the ‘curve obtained by digital simulation by Flanagan and Marcoux 1211.Compton and D&y [lo] derived an analytical equation for the transient current by transforming H,(p, s) into a series of rational functions. The dotted curve in Fig. 2 was taken from Fig. 1 of ref. 10. The transient curve obtained by Compton and Daly does not satisfy the Cottrell equation at short times, because their approximation is valid for long times. LINEAR SWEEP VOLTAMMETRY
For the linear potential sweep mode, the potential is expressed by E = E, + vt
(36)
or 5=3i+Ue
(36’)
with si= (nF/RT)(E-
E”‘)
u = ( nFv/RT ) ( bx,/3U,,,fi)2’3
(37) (38)
where v is the potential sweep rate, u is the dimensionless potential sweep rate, E,
254
is the initial potential and si is the dimensionless initial potential. Inserting eqn. (36’) into eqn. (28), carrying out the differentiation and replacing u in eqn. (28) by (3’ - M/o yields Z= I((, u) = (31/3/4)nFc’(U,D2x~/~)1’3w~‘sech2({’/2)ZZ2(({
-‘{‘)/e,
1) d[’ (39)
This is the equation for a reversible linear sweep voltammogram. For extremely slow sweep rates or for small values of (I, ZZ2(({- l’)/a) becomes 34/3/(21(1/3)). Then the current is given by Z(l, 0) = Z,, = [35/3{21?(1/3)}] nFc*(U,D2x:/b)1’3w(l
+ eeS)
(40)
where Z,, denotes the steady-state current of which the waveform is sigmoidal. Conversely, ZZ,(({ - {‘)/u) tends to (ntY)-“2 as u approaches infinity. Then
is essentially the same as the equation for a linear sweep voltammogram at a stationary planar electrode in quiescent solution [22]. Therefore eqn. (41) does not involve any hydrodynamic parameters, U, or 6. For any value of u, eqn. (39) is not reduced to a simple form. We computed the integral in eqn. (39) by Simpson’s l/3 rule. The singularity occurring at l’ = S in eqn. (39) was eliminated by integration by parts. The voltammograms thus calculated are shown in Fig. 3 for several values This
c
c=
[nf/Rn
1E -
Fig. 3. Linear sweep voltammograms 0 = 40, calculated from eqn. (39).
r”)
1
for (a) o < 0.2, (b) 0 = 1, (c) 0 = 3, (4
c = 10, (e) 0 = 20 and (f)
255
2..
<
G ,_ 2 z
2-1.
/'
/'
/'
/'
4'
/'
1'
1'
OO
2 0 l'* c =' inFv/Rn
Fig. 4. Variation of the without convection.
3
4
1’2uJx1/3um)‘nLYsl
peak or maximum current with 6. The dashed line denotes the peak current
of u. The current in the ordinate is normalized by the steady-state limiting current, I ss,llm7given by
Iss,Lim= [3”‘{2I’(l/3)}]
nFc*( U,D2~:/b)*‘3~
(42)
Curve a is the steady-state voltammogram expressed by eqn. (40). As the values of c increase, the curves shift slightly in the negative direction and then a peak appears (curve c). With further increase in u, the peak is so appreciable that the peak potential can be determined unequivocally (curves d-f). The diffusion tail followed by the peak approaches the steady-state limiting current. These variations have also been seen in the linear sweep voltammograms at a rotating disk electrode [23]. In Fig. 4, the peak or maximum current, Ip, is plotted against 6. When the values of u increase, the peak current increases monotonically from the steady-state limiting current and then approaches the peak current at the stationary planar electrode in quiescent solution, which is given by (43) In Fig. 4, the line for eqn. (43) is shown as a dashed line. The peak current for 6 < 1.5 is the same as the steady-state limiting current within 5% error. When 6 > 2.7, the effect of convection on the peak current is negligible. In Fig. 5, the potential at the half-peak height or half-peak potential, Ep,2 - E”‘, is plotted against 6. {p,2 in the ordinate denotes the dimensionless half-peak potential given by &,2 = (nF/RT)(E,,, - E”‘). With increase in 6, &,,2 varies from the steady-state half-peak potential, Ep,2 = E O’, to the half-peak potential at the stationary planar electrode in quiescent solution, &,,2 = - 1.09 or Ep,2 - E O’ = - 28/n mV at 25°C. When 6 < 0.6, the half-peak potential is in agreement with
256
Fig. 5. Plot of the half-peak (wave) potential against 6. T’he dashed line denotes the half-peak potential without convection. The dotted curve was calculated from eqn. (40). which was derived by Compton and Unwin [14].
E O’, within 5/n mV error. Conversely, for 6 > 1.6, the effect of convection on the half-peak potential is negligible within 5/n mV error at 25 o C. Compton and Unwin derived analytically an expression for the linear sweep voltammogram in series form when the values of u are small [14]. They evaluated the half-peak potential and found that it was expressed approximately by &,z = 0.55 u The curve for eqn. (44) is plotted in Fig. 4 ( * . . . - ). The curve for 6 < 1 is in good agreement with the solid curve. Therefore eqn. (44) provides a good measure of whether the measured voltammograms are under the steady-state condition. We summarize the dependence of the voltammogram on 6 as follows: For 6 < 0.6, both the half-peak (wave) potential and the maximum current are of the steady-state voltammogram. For 0.6 < 6 < 1.5 - 1.6, the half-peak (wave) potential shifts in the negative direction, the maximum current keeping the steady-state value. For 1.5 - 1.6 < 6 < 2.7, the half-peak potential is close to (E O’ - 28)/n mV at 25°C and the peak is appreciable. For 6 > 2.7, both the half-peak potential and the peak current are those at the stationary planar electrode without any convection. ACKNOWLEDGEMENT
We thank Dr. Compton for informing us of ref. 14 in a private communication. APPENDIX
In this Appendix, derived.
the asymptotic
expansions
for Hi(@, 6) and H2(@, i) are
257
gi( p, S) is the same as B(p, s)/s in eqn. (24) of ref. 17. The asymptotic expansion for g(p, [) has been expressed by eqn. (Al-9) of ref. 17, from which @(p, 5) is immediately given by H,( p, 5) -p-1’2
1+
(l/r)
I 641)
2 b, {4,(3[P3/2)}J’2Kj(4fiP3/4/dq J=o
[
where K, is the modified Bessel function of the second kind with the j-th order. Taking the first two terms in eqn. (Al) and expressing K,(4fi~~‘~/fi) by the asymptotic expansion [23] yields H,(PY
t> -p-
‘I2 + (2/n)1’23-‘/4[-3/4p-13’8
exp[ - (4/fi),/$~~‘~]
642)
The inverse Laplace transform of p-1318 exp[ - (4/6)fi~~‘~] was derived by the steepest descent method [24], of which examples have been given in Appendix 3 of ref._l7. Thus eqn. (25) was derived. i?2(p, s) can be expressed by g(si)/s, in eqn. (I-2) ofgf. 25 when sp-3’2 is replaced by sr. Applying eqns. (I-19) and (I-20) of ref. 25 to Hi( p, s) and taking the first three terms yields H,(P>
6) - Ufi
+ 1/(4p2)
- V(7r~‘)Ko(4&~“~/fi)
Inserting the asymptotic expansion for Ka(4fi~~‘~/fi) H,(PY
(A3)
into eqn. (A3) leads to
5b-E/fi+1/(4P2) _ @l/42-
3/2,-1/2~-1,‘4)p-19/8
exp[ - (4/6)jZ~~‘~]
out the inverse Laplace transformation of p-19” use of the steepest descent method [26] yields eqn. (30). Carrying
exp[ - (4/ &)&P~‘~]
644)
by
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