Application of a current-time function of the form I(t) = I0tω ω ≥ −12 to hemispherical microelectrodes

ELSEVIER Journal of Electroanalytical Chemistry 428 (1997) 173-183

dOURNAL OF

Application of a current-t ime function of the form I(t) = Io to > - 1 / 2 to hemispherical microelectrodes

Angela Molina *, Joaquin Gonzfilez Departamento de Qufmica-Fisica. Universidad de Murcia, Espinardo 30100, Murcia. Spain

Received 2 August 1996; revised 15 October 1996

Abstract

The general equations corresponding to the application of a programmed current of the form / ( t )= lo t'°, 09 > - 1 / 2 to microhemispheres by supposing different values for the diffusion coefficients of the electroactive couple are presented. When co = - 1/2, it is not possible to reach a transition time under any circumstance. For a reversible process, an independent of time potential is obtained in planar diffusion. This situation is identical to that of chronoamperometry in the same conditions. For usual spherical electrodes the potential depends on time and on ~ i / r o , and it only becomes dependent on time for microhemispheres for a fixed value of lo/id(oO), id(oo) being the steady-state diffusion current for a microhemisphere. On the contrary, for a current step (co = 0) the potential becomes independent of time at microhemispheres independently of the reversibility of the process. We also analyse the influence of the exponent co on the potential-time response.

Moreover, diagnostic criteria for distinguishing between reversible and irreversible processes and methods for calculating the formal standard potential E ° and kinetic parameters of the charge transfer are proposed. © 1997 Elsevier Science S.A.

Keywords: Hemispherical microelectrodes; Current-time function; Power law

1. Introduction

As is well known, the application of chronopotentio- metric techniques to microelectrodes presents great advan- tages compared with their application to electrodes of usual size due to the decrease of the magnitude of the capacitative current with respect to the faradaic component derived from the use of microelectrodes [1-8]. Moreover, controlled current electrolysis experiments are also of interest in the clinical and biological fields [9].

The use of hemispherical microelectrodes has the ad- vantages of not presenting insurmountable mathematical difficulties and of possessing the simplifying feature of 'uniform accessibility'. Disc microelectrodes are easier to construct; nevertheless, they do not present a uniform flux over their surface since the rate of diffusion to the edge is always higher than to the centre. The edge effects become more important and dominate the response of very small discs even at times of I s or less [2-5]. However, when

~" >> r o / V ~ i (with r 0 being the electrode radius and D i being the diffusion coefficient of the oxidised or reduced species), the shape of the electrode becomes irrelevant and the equations deduced under these conditions for z hemisphere correspond to those derived for a polarised inlaid disc electrode [10].

In recent years a current step has been applied to different types of microelectrodes [2,3,6,9,11-13]. How- ever, with respect to power current-time functions, only one current ramp has been applied to microdiscs [6]. There is also an equation deduced by Hurwitz at the beginning of the 1960s for a current proportional to the square root of time which lends itself to microhemispheres [ 14]. The aim of this work is to obtain the general equations corresponding to the application of a power current-time function of the general farm t ( t ) = Io t°', to >_. - 1 / 2 , to microhemispheres. The expressions deduced here are valid for integral and semi-integral values of to and offer great advan-

tages as are shown below. We have analysed the following current-time functions:

* Corresponding author. E-mail: [email protected].

0022-0728/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved. PII S0022-0728(96)05005-X

=tor'/2

174 A. Molina, J. Gonz61ez / Journal of Electroanalytical Chemistry 428 (1997) 173-183

The experimental application of a current-time function of the form l ( t ) = Io t - l / 2 is described in Refs. [15,16], in order to avoid the current being infinite when the time tends to zero.

With this type of current-time function, it is not possible to reach a transition time under any circumstance. In the case of a reversible process, for a plane electrode the situation is identical to that of chronoamperometry and, thus, a time-independent potential is reached (steady state) [17], whereas for a spherical electrode the potential always depends on time and on ~ i / r o . However, the potential becomes independent of the electrode radius for r 0 < 10 -a cm if D a --- D B = 10 -5 c m 2 s - I if the relation Io/id(oO) is held constant. These values of r 0 are higher than those ,.,otained with other values of the exponent to higher thaa - 1/2.

In the case of irreversible electrode processes, the application of this programmed current to microhemispheres leads to an easy determination of ~, kinetic parameters of the charge transfer reaction.

l( t) =/o

This case (current step) has been studied in depth [6,9,12], and has the advantage that, through using it, a time-independent response can be obtained at hemispherical microelectrodes when the transition time is not reached.

l ( t) =Io t°" with t o > 0

The application of these programmed currents (to = 1/2, I, 3/2, etc.) to microelectrodes is interesting because they offer many possibilities for determining kinetic and thermodynamic parameters [18,19] and, contrary to previous cases, transition times can always be reached.

We also demonstrate in this paper that, under certain experimental conditions, the potential-time response corresponding to a reversible process becomes independent of the electrode radius in hemispherical microelectrodes for any value of to while for a totally irreversible one, the difference between two potential-time responses for a given value of k s and deduced for two values of the electrode radius, r~o t) and ro ~2) is precisely (RT/¢nF)ln[r~ol)/r~o2)]. This fact can be used as a diagnostic criterion for distinguishing reversible and irreversible processes and can be applied more easily when a transition time does not exist. Also analysed in this paper is the influence of to in the reversibility of the process and methods for calculating a and k s are proposed. Moreover, the influence of the values of the diffusion coefficients of the electroactive species on the potential-time curves is discussed.

Finally, it is important to point out that, according to the literature [2-4,10], all the conclusions obtained for microhemispheres with a power current-time function could also be applied to microdiscs.

2. Theory

For a charge transfer reaction

kf A + ne- ~ B (I)

kb

when a current-time function of the form l ( t )=lo t to , to > - 1/2 is applied to a hemispherical electrode, the boundary value problem in semi-inf'mite diffusion conditions is given by

aCi [ ~2Ci 2 ~Ci ] 0-7 = Oi["~'r2 + r"~-r I / = A o r B (1)

I , t = 0; r_> r o CA(r,t) = CA. (2)

t > O ; r ~ . CB( r, t) = c B

t > O; r = r o

oA/0c 1 ac8 =-O.( ) (3) r 0 ~ r r = r 0

/ ,o,-} T r j , o , o - T f f

= 21rr~ (4)

Moreover

lo tto anF 2 nFlr r2 k~ = c A ( r o , t ) e - ' ~ -'(E-E%

(l-- a)nF ~ ( E- E °)

_ca(ro,t) e ~r (5)

We have previously studied this problem in the case of the usual electrodes (planar, spherical and cylindrical) by using the dimensionless parameters method. This proce- dure has been also used for the application of a power current-time function to a hemispherical microelectrode, although it has now been necessary to calculate a great number of terms of the series present in the mathematical expression of the surface concentration of species A and B, which are the solutions of the differential equations system in Eq. (1). These calculations have been made in order to obtain a compact, analytical solution applicable to electrodes of very small radius r o.

Thus, by proceeding in the same way as in Refs. [18,19], we obtain

c,.( ro.t) to , = 1 L A ( 6 )

CA id(°°) '

¢B( ro,t) I n 7 2 Jto,B , = / x + ~ ( 7 )

CA ! d

A. Molina, J. Gonzdlez / Journal of E!ectroanalytical Chemisiry 428 (1997) 173-183 175

where

~i 2j

• j=o J !

zo ~[2j- l)

- 2 7 1 i= A o r B • __ ( j -

(8)

for to = 0, 1 , 2 , 3 . . . .

2 t~ ' ( to+ 1 /2 ) ! L~? erfc(~:i) L , , =

' ~iEcap2~aq 1 [o

m- 1/2 ~:i(2J+ 1) oJ- 1/2 ~i2J ]

+ 2 E E j=O J[PEj+ 1 j--O J! J

1 i = A o r B (9)

for to - - 1 / 2 , l / 2 , 3 / 2 . . . .

~i= D~it - g o r B ro

(1o)

( l l )

CB P, = - 7 (12)

c A

ia(oo ) = 2¢rntroDACA (13)

2r(1+ 2) (14) Px= /" ""~(l+x)

id(oO) in Eq. (13) is the steady state diffusion limited current for a microhemisphere [9,12].

The function J,,,.i (with i = A or B) which appears in the expression of the surface concentrations of the electroactive species A and B is applicable for any integral or semi-integral value of the exponent of the current with to > - 1/2. Moreover, this function is also applicable for hemispherical electrodes of any size, including plane electrodes as a limit case when g~ << 1.

So, for plane electrodes (~i << 1) Eqs. (8) and (9) become (see Appendix B)

J.,,i( ~i << 1) = t •+'/2 ~ / r ( ca+ 1) r o U(1 + ( 2 t o + 1 ) /2 ) (15)

On the other hand, for hemispherical microelectrodes (6i >> 1), Eqs. (8) and (9) are simplified to (see Appendix a) J,~,i( ~i >> 1) - t " (16)

The values of ¢-~i/ro for which Eq. (16) is fulfilled depend on the value of to, as will be seen below.

3. Results and d iscuss ion

3.1. Transition times

From Eq. (6) it is clear that transition time for species A, ~'A, will exist when the surface concentration of A becomes much less than the bulk concentration. Therefore, by imposing this condition on Eq. (6),

id (00) [Jto,A]t=rA -~ I0 (17)

is fulfilled. This identity may or may not be reached depending on the behaviour of J,~.A with t. Note that J,~.A behaves in a different way with t when the power current (to) or the electrode radius ( r o or gi, see Eq. (10)) change. Note that lo/id(oO in Eq. (17) is given in (seconds) -~

By introducing Eq. (15) in Eq. (l 7) we deduce that for plane electrodes the transition time of species A is given by

nFAD~/Ec~ F [ l + ( 2 t o + 1) /2] zff + l/ :(plane) = to r(~,+ l)

to> - 1 / 2 (18)

This equation is the 'generalised Sand equation' [17,20,21] for a plane electrode of area A = 2wro 2.

On the other hand, by substituting Eq. (16) in Eq. (17) we deduce that %, for very small hemispherical electrodes is given by

id(o° ) 2nFwroDACA ~ = to> -- 1 /2 (19) rff(micro) = io i0

In Fig. 1 we have plotted the behaviour of the Jo,.A functio" with t for different values of ~ A / r o and to (to = - 1 /2 in Fig. l(a), to - 0 in Fig. l(b) and to > 0 in Fig. l(c)).

3.1.1. l(t) = lot- 1/2 Fig. l(a) clearly shows that for to = - 1 / 2 , cA can

never be reached since J_ t/E.A is always decreasing with t and, therefore, Eq. (17) is never fulfilled. We can also conclude that for this value of to, Eq. (16) is fulfilled for D ~ A / r 0 >_ 3S -1/2 and J-I/E,A -" t - l~2 (open circles) for any value of t. Therefore when we apply a 'Cottrellian current' ( l ( t ) -Iot- l /E) , the response obtained (surface concentrations) for a hemispherical microelectrode behaves independently of its radius r o for r o < l0 -3 cm if D A = 10-Scm 2 s -1. These values of r o are higher than these obtained with other values of the exponent higher than - 1 /2 as will be seen below.

3.1.2. l(t) = I o In Fig. l(b) the behaviour of J0.a._corresponding to a

current step is shown for different CD A/ ro values. In this

176 A. Molina, J. Gonzdlez / Journal of Electroanalytical Chemistry 428 (1997) 173-183

11 " 1.5 z

1.2

"~ . 9

.6

.3

0 2 4 6 8 10

lb 1.o

,8

.6 o

.4

.2

0.0 0 2 4 6 8 10

t/s

i~ ~--1

-'7 a

-7 4 -

2 } o)=1/2

0 I I I I I

0 2 4 6 8 10

t/s

Fig. I. Dependence of Yo,.i ( i = A) on t (Eqs, (8) and (9)). D A = i0 -5 cm 2 s- i (a) I(t) = lot- i /2 ca ffi - 1/2. The values of ~'l~A / r 0 (in (seconds) - I /2 ) are on the curves. (b) l ( t ) = I o, ca = 0. The values of i/ '~A/ro (in (seconds) - I / 2 ) are on the curves. (c) l ( t )= !o t°'. ca > O.

The values of ca are on the curves. The values of ~ A / r O (in (seconds) - I /2 ) are: (a) 0.50, (b) 1.0, (c) 5.0, (d) 50.0 and (e) 100.0. Open circles correspond to the variation with t of the function J,o.~ ffi t~' in each case.

case (to ffi 0), Eq. (16) for very small electrodes takes the particular form

Jo,,( ~i :~ l) = 1 (20)

The fact that the J0.A function takes, in this special case, a constant value, means that a steady state response can be reached under these conditions [2-6,9,12].

This behaviour is clearly shown in Fig. l(b) for values of ~'~A/ro>25s-~/2 and t>__ l s ( r 0 < 1.3 × 10-4cm if DA = 10 -5 cm 2 s - ~) and it is time-independent for values of r o < 5 × 10-5 cm for the same value of D A.

We can conclude that for very small electrodes the transition time is reached in accordance with Eqs. (17) and (20) when I 0 > id(oO). In these conditions, as is clearly shown in Refs. [9,12], it is evident that for

I 0 < id(°° ) ( 2 1 )

constant surface concentrations (steady state response) are obtained. We have deduced from Eq. (17) that it is possible to reach security as as ,._Za_with long id(oO)/l o < 0.9 and ( ~:A ),^ = ~/OA'rA/ro < 5.6.

3.1.3. l(t) = lot'° with to > 0 In Fig. l(c) we have plotted the behaviour of J ~ w i t h

t for to = 1 /2 and to = 1, and different values of ~/r~:/ro. As can be deduced from this Fig., J,o.A is always ilacreas- ing with time and the faster this is, the greater is to and the smaller is the electrode radius. In this figure it can be also noted that J,,,.a becomes t " (open circles) and, therefore, is independent of DV~ A / r o, for ~ADA/ro -- > 25 s - i / 2 ( r ° _< 1.3 × 10 -4 cm if D A = 10-Scm 2 s-I . From Fig. l(c) it can be seen that the transition time of species A, ~'A, is reached more rapidly the higher is to for a given value of lo/ia(~). Besides, the transition time can be easily obtained from Eq. (17) and this figure, if we sit on the curve corresponding to a given value of ~ A / r o , as the value of the abscissa corresponding to an ordinate value of Jo,.A = i d ( ° ° ) / / 0 .

In general, the values of 10/id(oV) for which it is possible to reach the transition time depend on the to value. So, for example, leaving aside to -- - 1 / 2 for which it is not possible to find ~'A for any value of /0/ id(oV), for to = 0 condition lo/id(~)>_ 1 must be verified for CA to exist (the lower limit corresponds to the case of small microelectrodes, V~A/rO >> 1, see Fig. l(b)). However, for to = 1 and for a given value of lo/ ,~oo)< 1 s-i CA can always be found for any value of ~/D A / r o (see Fig. l(c)).

3.2. Potential-time curves

The general expression of the potential-time response can be easily obtained by substituting Eqs. (6) and (7) in Eq. (5):

lot 9 2 nFcf r6 k s

-anF e RT (E- E°)[ I0 ]

= 1 Jo,,A id(oO)

2 1o ] -

/d r°° ) ]

(22)

This response is valid for any power current-time function l ( t )=lo t'° (with t o = - 1 / 2 , 0, 1 /2 , 1, 3 /2 , . . . ) and for any value of the electrode radius r o.

A. Molina, J. GonzSlez / Journal of Electroanalytical Chemistry. 428 (1997) 173-183 177

3.2.1. For a reversible process (Nernstian behaviour) Eq. (22) when c~ = 0 (/~ = 0) is transformed into

ia (oo) J.,,g

RT I o E = Eli , + In

nF TJa,.B (23)

with

RT 1 EL2., = E ° + In -- (24)

nF T

For high values of r o (£i << < 1) Eq. (15) is fulfilled and Eq. (23) is transformed into

RT i ~ - 1o t~ E(plane) = E~/2 + nF In lot~ (25)

with

i~ = nFAD~/2cA tl/2

A =2~r02

+

r(a,+ 1) (26)

On the other hand, for microhemispheres (~i >> 1) Eq. (16) is fulfilled and Eq. (23) is drastically simplified since J~,.i (with i = A or B) becomes independent of Vr~g/ro

and , D~a / r o,

R T id(oV ) -- lo tw E(micro) = Ericro + nF In I o t ~'

RT 1 Ericr° = E~ + n-'~ In - ~

(27)

From Eqs. (25)-(27) it is possible to calculate easily E~/2 and Emicr o. Moreover, we can deduce the following.

3.2.1.1. For a current-time function of the form I(t) = Io t - 1/2 (Cottrellian current). In this case i,,, takes the particular form,

! i _ ! /2=i ' d (~ )=nrADkc~ ~ . D A t i/? (28)

which is the potentiostatic diffusion limited current obtained for a plane electrode (Cottrell's equation).

From F~ls. (25) and (27) we deduce the reversible potential-time response which is given, in these condi-

tions, by

RT i'a(°°) - lo t - ' /2 E(plane) = El~, 2 + nF In lot_l/2 (29)

R T id(oO ) -- lo t - l / 2 E(micro) = Ericro q" n/~-- In lot_l~ 2 (30)

Note that id(~)/lo t-i~2 which appears in Eq. (30) corresponds to the relation between the diffusion current id(~) obtained when a constant potential is applied to a hemispherical microelectrode and the applied current Io t- !/2, while i'd(OO)/Iot-1/2 in Eq. (29) is independent of time since it corresponds to the relation between potentiostatic diffusion limited current i~(~) obtained for a plane electrode (Eq. (28)) and the applied current.

id(~) and i~(~) correspond to the two extreme types of behaviour of the well known general expression of the potentiostatic diffusion current i in hemispherical diffusion field, which is given by [10 p] a

ip = 27rnFc~ DA to( r0 ) l + - - - - ~ - A t =id(~) +i'a(~) (31)

Therefore, ip has a Cottrellian behaviour with time for high r 0 values or short electrolysis times (i~(oo)) and is independent of time for electrodes of small radius or long electrolysis times (id(~)) [2-5,9, l 0 12].

From Eqs. (29) and (30) it is oear that the application of a current-time function with "Cottrellial~ behaviour' ( l ( t ) = lo t - l ~ 2 ) tO planar electrodes for a reversible process leads to a constant potential (steady state) in agree- ment with Ref. [17] and Eq. (29). On the other hand (see Fig. 2(a)), for spherical electrodes the potential is always increasing with time according to Eq. (30). The dependence of potential with time is more noticeable the smaller the electrode, down to values of r o for which Vrffi/ro >_ 3s -! /2 and for which the potential-time response becomes independent of v~Di/ro for a given value of the relation lo/id(~).

3.2.1.2. For a current-time function of the form l(t) = I o (current step). It is evident, as is shown in Refs. [9,12] for D A = D a (T = 1), that for very small hemispheres for which Eq. (20) is fulfilled, the surface concentrations of species A and B are independent of time in such a way that for a reversible charge transfer reaction ~e potential- time response is given by

RT id(~ ) --Io (32) E(micro) = E ~ i . o + n f In !0

So, while the potential varies with time when a current-time function with a Cottrellian behaviour is applied to a hemispherical microelectrode (Eq. (30)), the application of a current step leads to a 'steady state' (Eq.

(32)). Fig. 2(b) shows the potential-time response obtained

when a current step is applied to hemispherical electrodes of different radii. In this case, for a fixed lo/id (~) the reversible potential-time response becomes practically independent of time and of r 0 for t > _ l s ( r 0_<1.3× 10 -4 cm if D A = Da = 10-Scm 2 s - i ) and totally independent of time for a value of r 0 < 5 × 10-5 cm for the same


o.!

80 0.2

~ 4 0 ~ ~ O' 102" lO'

o 1[ 2a

-20 ~

I I I I / o 0 2 4 6 8 1

t/s 100

80

;> 40

20 ,J.,

o

6o ~ 2b

-20 5.0

-40 ~ 0 :

I I !

0 2 4 6 8 10

t/s

Fig. 2. Potential-time curves for a reversible process (Eq. (22)) with k~=103cms -I or Eq. (23)). DA=10-Scm2s -I , / z=0 , y = l , n = l , T = 298K. (a) i ( t )= !o t - I /2 , o~ = - 1/2, IoJ_id(oQ = 0.2s I/2. (b) l(t) = ! o, to = O, i o/ia(o~) = 0.9. The values of ~/D A / r o (in (seconds) -1/2) are on the curves.

values of D A and D a. Fig. 2 clearly shows the opposite behaviour, concerning the electrode size, of a Cottrellian current-time function (Fig. 2(a)) and a current step (Fig. 2(b)) for a reversible process.

3.2.1.3. For a current-t ime function o f the form l(t) = Iot'° with to > 0. Fig. 3 shows the influence of the electrode radius on the potenti,~! -time response obtained for a reversible process and for different to values (to > 0) (see Eqs. (23) and (27)). This figure shows that the transition time is reached for a fixed lo/id(ao) and so much the faster, the smaller the electrode radius rg...~d the greater to are, until a given value of the ratio ~/Di/r o >__ 25 s-~/2, above which the potential-time response becomes independent of the electrode radius since Eq. (27) is fulfilled.

Regarding the influence of the different values of the diffusion coefficients of the electroactive species on the potential-time response, it is interesting to point out that

this influence is exerted through y (= ~ ) and through J,,,,A and J.,,B functions (see Eqs. (6)-(10), (22) and (23)). For usual hemispherical electrodes, the effect of D A and D B decreases when the exponent of the current- time function (to) increases. However, in the case of microhemispheres, these effects are dependent on y only because Eq. (16) is fulfilled and, therefore, the Jo,.i ( i = A or B) function does not depend on ~ i / r o (see Eq. (27)). In Fig. 4, these effects are shown for two values of the electrode radius (r o = 1.25 × 10 -3 c m (dashed line) and r o = 1.25 × 10 -4 cm (solid line)) and different values of y when a current step is applied. From this figure, it can be observed that the potential-time response is shifted to more positive potentials when y diminishes.

3.2.2. For a totally irreversible process Eq. (22), when /z = 0, becomes

E E ° RT id(oO) r o ks RT - = I n + ~

ot nF Io D A a nF

× I n ( t - ' [ 1 id(i°°°) JtoA, ) (33)

From Eq. (33) it is evident that the potent.!al becomes more and more negative for a given k s valv.e when the electrode radius diminishes, if the ratio lo/ia(oo) is held

200

150

100

° . °

o _::: ....... V

; ( : . . . . , o o .. a ':b " C b " a

-150 I I I I I 0 3 6 9 12 15

t/s Fig. 3. Influence of the electrode radius on the potential-time curves for a reversible process (Eq. (22) with k s = 103cms - ! or Eq. (23)). (" ~ ) l ( t ) = loft~2 , ca = 3 / 2 , (---) l ( t )= lot, ca = 1 and ( . . . ) l ( t ) = Io tm/2, to = 1 / 2 , l o / i d ( ~ ) = 0.3s -~°. The values of 1 /~A / ro (in (seconds) - I /2 ) are: (a) 1.5, (b) 5.0, (c) 10.0 and (d) 50.0. Other conditions as in Fig. 2.

A. Molina, J. Gonz6lez / Journal of Electroanalytical Chemistry 428 (1997) 173-183 179

50

40

30

> 2o

-10

-20

'\ \

\

\ \

~ _ _ _ . i i

0 2 4 i i

6 8

} 7=0.71

7=I .00

t 7=1.41

]

10

t/s Fig. 4. Influence of the different values of the diffusion coefficients of the electroactive species on the potential-time curves for a reversible process (Eq. (22) wid" k s = 103cms - I or Eq. (23)). i ( t ) = ! o, ~o=0, lo/id(~) =0.5. ( ) V~A / ro= 25s - ' /2 and Vf-~A / ro= 25/Vs -'/2, (---) ~ A / ro = 2.5 S- I/2 and V/-~-A / r o = 2 .5 /7 s- i/2. "l~e values of

7( = V/-~A/'DB ) are on the curves. Other conditions as in Fig. 2.

constant. It is also possible to deduce accurate values of a and k s by representing E - E o vs. In go, with

go,= t-'°[ 1 id(10j=~) .A] (34)

since these graphs are straight lines from whose slope and intercept it is possible to determine accurate values of a and k~ [18,19].

For microhemispheres (~i >> 1) Eq. (16) is fulfilled and Eq. (33) is transformed into

E E 0 RT id(oO)rok s RT [ Io ] . . . . In + .... In t- "

t~NF toO A anF id(oO ) (35)

For a current-time fu-.ction of the form / ( t ) = Io t-l~2 (Cottrellian current), the potential-time curve given by Eq. (35) takes the particular form

RT roks RT i id(~) ] E - E ° = - I n + ~ l n t ~/2 1

ot nF D A ot nF I o

(36)

for ~ / r o >_ 3s -1/2 Fig. 5 shows the behaviour ef Eq. (33) or Fq. (22) (both

results are coincident) for t o = - 1 / 2 , k~= 10 -3cms -n and D A = 10- 5 cm 2 s- l for different values of D(D-A- A / r 0.

From Eq. (36) we deduce that for microhemispheres, w h e n id(oO)/lo>> I S - 1 / 2 (i.e. f o r lo/id(oO)~0.2S n/2) a plot of E - E ° vs. In t must be |inear and again it is

0.1 150

1.0

o

~ - 1 5 o 10 2

~,~ -300 i 03

-450

-6O0 ; o 0 2 4 8 1

t Is

Fig. 5. Potential-time curves for a totally irreversible process (Eq. (22) or Eq. (33)). l ( t ) = 1o t-I/2, to=- i /2 , k s = 1 0 - 3 c m s - j , D A = 10-5 cm2 s - I , a =0.5, Io/id(~)=0.2s I/2. The values of V/I~/ro (in s - I / 2 ) are on the curves. Other conditions as in Fig. 2.

possible to determine a and k s from the approximate equation

RT id(°°) ro ks RT E - E ° -- In + ~ In t (37)

a nF leD A a nF

In Fig. 6 we have plotted E - E ° vs. in t for different v~ues of k s. From this figure, it can be observed that Eq.

-250

-300

-350

-400

-450

"~" -500

-550

-600

-650

ks/cm s l=

1 1 1 0 "3

/

/ i0 "4

10 "s

.s 1.0 1.5 20 25 a.o 35 4.0 4.5

In(t /s)

Fig. 6. Dependence of E on In t for a totally irreversible process (Eq. (36) or Eq. (37)). l(t)=lo t-I/2, ~ o = - 1 / 2 , ~A/rO =|0~s-1/2" The values of k s / c m s- I are on the curves. Other conditions as in Fig. 5.


-150

>

~-3oo r.TJ '~" -450

-600

-750

0.0

~\~ ~ ...... \\

" ' . . ~ C "°

I I t 1

.2 .4 .6 .8 1.0

t/~ A

Fig. 7. Influence of the reversibility of the electrode process on the normalised potential-t ime curve~ (Eq. (22)). i ( t ) = I~ I/2, oJ= ! / 2 , lO/id(OO)=O.5s-l/:. ( ~ ) ~A/ro-----los -I/2, (---) ~D-'A'A/r o = 10-~s - : / ; and ( . . . ) v/DA/ro = 10~s - I/,.. The values of k s / c m s - I are: (a) 10 3, (b) 0.1 and (c) 5 × 10 -4. Other conditions as in Fig. 2.

(36) behaves as Eq. (37) in these conditions. This behaviour enables an irreversible process in microelectrodes to be completely eharacterised in a very simple way and it can be expected only when we apply a current proportional to a negative power of time.

For a cu~ent-time function of the form l(t) = 1 o (current step), general Eq. (33) is transformed into

E EO RT rok~ RT [id(°°) ] -- -- In + ~ In 1 ( 3 8 )

vtnF D A vtnF I o

for ¢rffAA/r o > 25s - l / z

Eq. (38) is also obtained in chronoamperometry when a constant potential E is applied to a hemispherical microelectrode [2-4,12].

Fig. 7 shows the influence of the reversibility of the electrode process on the normalised potential-time curves (E(t) vs. t / r ) deduced for t o - - 1 / 2 when Io/id(oO) is held constant, by the general F_,q. (22) for different k s values at different electrode radius. As can be noted, the reversible curves (curves (a) with k s -- 103 cm s - i ) are not affected by the electrode radius (for r o _< 1.3 × l 0 -4 c m if D^ -- D a = 10 -s cm 2 s - l). On the other hand, for lower k s values (curves (b) and (c)) these curves are shifted to more negative potemials the lower the electrode radius is. This is a common behaviour for any value of to.

Fig. 8 shows the potential-time curves deduced for a totally irreversible process in microhemispheres (k s = 10 -3 cms- I, see Eq. (35) or Eq. (22)) for different values of the exponent to. These curves have been obtained in all the cases for two microhemispheres of radii ro tl~ -- 1.25 ×

10 -4 c m (solid line) and r~o 2) = 6 X 10 -5 cm (dashed line) for a value of D a = 10 -5 cm 2 s -~. We must proceed in such a way that, for a given to, lo/id(oO) must be held constant, i.e. the value of Io in the programmed current must be changed in both experiments (Io tl) for ro ") and lo (2)

for r~o 2)) so that (ttoi)/rtol))=(Ito2)/rto2)). Under these conditions, it can be observed for an

irreversible process that a decrease in the electrode radius causes a shift in the curves towards more negative potentials. It can be also observed that, for a fixed value of the time (which must be selected from the middle zone of the chronopotentiogram when r a exists), the difference between two curves corresponding to the same value of to is always the same, independently of to and approximately equal to 36 mV for a - - 0 . 5 and n = 1.

This behaviour is due to ~he fact that for microhemispheres, for which Eq. (35) holds instead of Eq. (33), the difference between two potential-time curves deduced for two values of the electrode radius, rt0 ') and r<o 2) is, precisely, for any value of to

RT r~o ~ ) 17.81 - ~ m V ( 3 9 ) a nF In rt02 ) a n

for a fixed value of time. This fact can be also used as a diagnostic criterion to

distinguish between and characterise reversible and irreversible processes, now that the response corresponding to a reversible process is not affected by a variation of microhemisphere radius. The most suitable value of to for the application of this criterion is to = - 1 / 2 since Eq. (35) can be used sooner than for higher to values because

-200 ~ I ' i }o

,-300 12

.,oo !

I I I l 1 5 0 5 10 15 20

t / s

Fig. 8. Potential-time curves for a totally irreversible process (Eq. (22) or Eq. (33)). l(t)=lo t~. The values of ~o are on the curves, k~= 10-3cms - I , .F=I, lo/id(oO)----O.2s-°'. ( ~ ) V~A/ro = 25s - I /2 , (---) V ~ A / r o = 50s - ,/2. Other conditions as in Fig. 5.

A. Molina, J. Gonzfdez / Journal of Electroanalytical Chemistry 428 (1997) 173-183 181

3 0 0 200 1 0 0 \ ¢o=!

-200 (o=0

- 3 0 0

- 4 0 0 t - ~ ~ ~

0.0 .2 .4 .6 .8 1.0

Fig. 9. Influence of the exponent (u of the current-time function on the potential-time response deduced from Eq. (22) ( ~ ) and from

~D_~_D A. (33) ( . . . . ). k ~ = 3 x l 0 - 2 c m s - I , l o / id (=)=l . l s - ' , 7 = 1 , / r 0 = 5 s - I/2. The values of oJ are on the curves. Other conditions

as in Fig. 5.

J~,A becomes independent of the electrode radius for r 0 < 10 -~ cm if D A = 10 --~ c m 2 S - ! . Besides, as a transition time does not exist in any case for co = - 1 / 2 : the application of this criterion is clearer. A current step can be also used but for smaller values of the electrode radius.

It may be of interest to take into account that an electrode process behaves as if it is more irreversible the lower is the exponent of the current-time function oJ. So, in Fig. 9 we ha;~e represented the potential-time curves with the general Eq. (22) (solid lines) and Eq. (33) (dashed lines), corresponding to a totally irreversible behaviour, for k~ = 3 × 10 - 2 eros -1 and several values of ¢o. As is evident, by using low oJ values (oJ < 0), this process behaves totally irreversibly (see curves with oJ = 0) and Eq. (33) can be used instead of Eq. (22) (which has no explicit form) in order to calculate the kinetic pm'ameters.

4. Conclusions

In this paper we have deduced the general equations corresponding to the application of a current--time function of the form l ( t ) = lo t'°, ¢o >__ - 1/2 , to microhemispheres.

These equations are applicable for any values of the diffusion coefficients of the oxidised and reduced species a_n_d are valid for integral and semi-integral oJ values.

From Eq. (17) we deduce that the transition time of A species, r A, can never be reached for ¢o = - 1/2, is reached when lo/id(oO)>__ 1 for oJ= 0 and always exists even though lo/id(oO) < 1S -! for ~o >_ 1.

The use of a current-time function of the form l ( t ) =

I o t -~/2, whose experimental application is described in the literature [15,16], is of great interest due to the fact that for a reversible process a constant potential (steady state) is reached in planar diffusion whereas for the usual spherical electrodes the potential depends on time and on ¢-~A/r o, and it becomes only dependent on time for microhemispheres (of electrode radius r 0 _< 10 -3 cm if D A = D B = 10 -5 cm 2 s - l ) for a fixed value of lo/id(oO). Contrarily, the potential depends on the electrode radiu,~ in the same conditions for an irreversible process. This fact can be used as a criterion for distinguishing between reversible and irreversible processes.

For ~o > - 1 / 2 , i.e. for ¢o = 0, 1/2 , 1 . . . . . the above criterion can also be used for r 0 < 1.3 X 10 -4 em if D A = D a = 10- 5 cm 2 s - I . In the particular case of a current step (oJ---0), for t > 1 s the steady state is reached [2-4,9,12] ind~ ,endently of the reversibility of the process.

'itle parameters E~/2, E~icr o and also E ° (if D a and D B are known), can be obtained from the equations corresponding to the reversible potential-time response in planar (E~/. 2) and spherical (Emicr o) diffusion (fer example, from Eqs. (29) and (30)).

An electrode process behaves as more irreversible the lower is the exponent of the current-time function ¢o, as is shown in Fig. 9.

Criteria for determining kinetic parameters of the charge transfer reaction a and k~ are proposed for any value of o~. In the particular situation in which co = 1 /2 this determination can be easily made, according to Eq. (37), by plotting E - E ° vs. In t.

The influence of the values of the diffusion coefficients on the potential-time response has been taken into account. This influence is exerted through 7 ( = CDA/DB ) and through Jo,.A, and Jo,.B functions (see Eqs. (8)-(10)).

Acknowledgements

The authors greatly appreciate financial support by the Direcci6n General de Investigaci6n Cientffica y Ttcnica (Project No. PB93-1134). We would also like to express our gratitude to Professor R. Parsons and to the referee, who have greatly helped us to improve this paper with their valuable suggestions.

Appendix A. Notation

t(t)

kf and k b

ci(ro,t)

programmed current l ( t ) = l o t'° with ~o= - 1 / 2 , O, 1, 1 /2 . . . . ( / o / A s - ' ) hetergeneous rate constants of the forward and backward electrode processes surface concentrations of the i species (with i = A or B)

182 A. Molina, Z Gonz~lez / Journal of Electroanalytical Chemistry 428 (1997) 173-183

Jto.~ series which appears in the expressions of the surface concentrations of the i species, with i -- A or B and to being the exponent of time in the applied current

c~* bulk concentrations of the i species (with i = A or B)

Di diffusion coefficient of the i species (with i = A or B)

ks,a apparent heterogenous constant and transfer coefficient of the charge transfer reaction re- spectively

A electrode area ( = 2 ~" r0 2) id(~) steady state diffusion limited current for a

microhemisphere i~(o0) (= i_ l/~) potentiostatic diffusion limited cur-

rent for a plane electrode (Cottrell's current) /.L = C C y = F Euler gamma function ,r A transition time of A species. r distance from the centre of the electrode for a

spherical electrode r 0 constant electrode radius E ° formal standard potential of the electroactive

couple E~/2 = E o + ( R T / n F ) ln(l / y) E.~,~ =Eo + ( R r / n F ) l n ( I / Y 2) ip potentiostatic diffusion current in a spherical

diffusion field erfc(x) complementary error function Other definitions are conventional

Appendix B, Limiting values of the J,o,~ function

B. !. ~i :~ 1: hemispherical microelectrodes

When ~:i :~'~ i is fulfilled in a general form [22]:

lira e ~? erfc(~) = 0 (Bl)

B. 1.2. Semi-integral values o f ,0 By substituting Eq. (B 1) in Eq. (9) we obtain

Jto,i( ~i >> 1) = 2/to( `0 +p2to+ 1/2) ! [ , 2 to~,t'2j=o, ~i2J-- 2 to+ l j !P2j+ 1

to- 1/2 ~i2J- 2 to ]

2 _ ~ J! + ( t o - 1/2)[P2~ jffi0 ]

(B4)

As in the previous case, the two sums on the right side of Eq. (B4) tend to zero when ~i >> I. Sa, we obtain

4tto( `0+ 1/2)! J,,,,i( ~i >> 1) = = tto (BS) p to+ p2to(`0- 1/2)!

since,

PxP(x+ 1)= 2( x + 1) (A6)

B.2. ~i << 1: plane electrodes

In this case, we can make a McLaurin series develop- ment of the Jto.i functions given by Eqs. (8) and (9), thus obtaining in a general form for any integral or semi-integral value of to

J .i( << i)

= t to [ ---2 ~i ~i 2 4~i 3 [ P2o,+l ( to+ l) + (2 ,0+ 3)P2to+ !

• • •

(B7)

We can neglect all the powers of 6i higher than the unity, obtaining,

r ( o , + l )

s..,( >> l) = r[1 + (2,0 + i)/2] (as)

References

B.I.I. Integral values o f to By substituting Eq. (B 1) in Eq. (6) we obtain

[ ~ ' so/2(j-'°, 1 L, . , (6 , l ) = tto`0! + - -

LJ~:o j! to!

to ~i2( j - ~a )- 1

--2 E (B2) Sffil ( J - I)[P2s-!

The two sums on the right side of Eq. (B2) tend to zero when sei :~ I due to the fact that the exponent of the ~:i parameter is less than zero i~ all cases. So, we obtain

1 Jto,i( ~:i>> 1) =t°'co! - = t °" (B3)

`0!

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Application of a current-time function of the form I(t) = I0tω ω ≥ −12 to hemispherical microelectrodes