The theory of square wave voltammetry at uniformly accessible hydrodynamic electrodes

www.elsevier.nl/locate/jelechem Journal of Electroanalytical Chemistry 487 (2000) 75 – 89

The theory of square wave voltammetry at uniformly accessible hydrodynamic electrodes Adrian B. Miles, Richard G. Compton* Physical and Theoretical Chemistry Laboratory, Oxford Uni6ersity, South Parks Road, Oxford OX1 3QZ, UK Received 30 March 2000; received in revised form 25 April 2000; accepted 30 May 2000

Abstract An integral equation approach which allows the ready simulation of square wave voltammetry for electrochemically reversible, quasi-reversible and irreversible processes is developed for the case where transport to the electrode occurs across a uniform diffusion layer of finite thickness. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Square wave voltammetry; Hydrodynamic electrodes; Simulation; Mass transport

1. Introduction Pulse techniques are widely adopted in electroanalytical chemistry for the determination of electroactive target substrates because of their high sensitivity particularly in the presence of background currents such as that which might result from dissolved oxygen [1–4]. The square wave voltammetry (SWV) technique involves the application of the general potential waveform defined in Fig. 1, and the currents I(1) and I(2) in each cycle of the waveform at the points shown are measured. The current difference DI = I(1)−I(2)

complex reactions and of those taking place at hydrodynamic electrodes is precluded by the absence of a suitable theory. We have recently developed simulation approaches [5,6] to allow for the extension of the SWV theory to the case of coupled homogeneous kinetics [7]. In this paper we tackle the problem of SWV at hydrodynamic electrodes using a complementary integral equation [8– 10] approach which is applicable to electrochemically reversible, quasi-reversible and irreversible processes.

2. Theory (1)

is used to generate voltammograms in which DI, when plotted against the applied potential, E, gives rise to symmetrical peak-shaped curves. The latter, in addition to their analytical value, contain much mechanistic and kinetic information about the electrode process under investigation. However, while the SWV behaviour can be satisfactorily modelled for electrochemical processes proceeding via relatively simple mechanisms under conditions of semi-infinite diffusion, the investigation of * Corresponding author. Tel.: +44-1865-275413; fax: +44-1865275410. E-mail address: [email protected] (R.G. Compton).

We consider the following electrode process occurring at a uniformly accessible hydrodynamic electrode with finite diffusion layer thickness, d. OX+ ne − = RED We assume that the redox couple has a formal potential, E°%. The mass transport equations for OX and RED can be formulated as follows: #[OX] #2[OX] =DOX #t #x 2

(2)

#[RED] #2[RED] = DRED #t #x 2

(3)

and the corresponding boundary conditions are:

0022-0728/00/$ - see front matter © 2000 Elsevier Science S.A. All rights reserved. PII: S0022-0728(00)00173-X

A.B. Miles, R.G. Compton / Journal of Electroanalytical Chemistry 487 (2000) 75–89

76

t = 0, all x:

[OX] = [OX]BULK [RED]= 0

t\0, x“ d:

[OX]“[OX]BULK [RED]“ 0 #[OX] #[RED] DOX = −DRED #x x = 0 #x x = 0

2.1. E6aluating the concentration integral equations by implementing the step function method

)

t\0, x= 0

)

=

I(t) nFA

where the symbols represent their standard electrochemical definitions. Implementation of the above boundary conditions, as shown in Appendix A, allows the derivation of the following concentration integral equations from Eqs. (2) and (3), respectively: [OX]tx = 0 = [OX]BULK −

&

× nFA pDOX c 2d 2 c I(u) % (−1) exp − t DOX(t − u) c= −



[RED]tx = 0 =



I(u) t





t −u 1

0

&

1



nFA pDRED

<

[OX]tx = 0 = [OX]BULK −

% (− 1)c

c= −

(4)

× c 2d 2 DRED(t − u)



 du

t − u

0

=

(6)

Employing the dimensionless current of O’Dea et al. [8]: c(t)=

<

I(t) pt

(7)

nFA[OX]BULK DOX

[OX]tx = 0 = [OX]BULK −

du

1

× nFA pDOX c 2d 2 I(u)exp − t DOX(t−u)

&



% (− 1)c

&

c= −

% (−1)c exp −

c= −

n

2.1.1. OX species Eq. (4) can be rearranged to

t

[OX]BULK



p t

c(u)exp −

×

c 2d 2 DOX(t−u)



t − u

0

<&

du

=

(8)

Noting the symmetrical nature (about c= 0) of the term inside the sum over c, we can rewrite this as

n du

(5)

t −u These equations are of general applicability and make no assumptions about the nature of the electrode kinetics of the RED/OX system. 0

[OX]tx = 0 = [OX]BULK −

+ 2 % (− 1)c c=1

&

t

[OX]BULK



p t

c(u)exp −

t

0 2

c(u)

c 2d DOX(t−u)

t − u

0



t − u du

=

du

(9)

We note that the first integral term (corresponding to c= 0) is the same as encountered for the semi-infinite diffusion case [8] and by employing the step function method [11] can be approximated as follows:

&

c(u)

t

0

t − u

du = bm

' ' 2t + l

2t m − 1 % bi S %j l i=1

(10)

where t is the period of the square wave, l is the number of subintervals (at which the current is calculated) per half-period, c(t) is approximated to bm at time t given by t=

mt 2l

(11)

and S %j is given by: S %=

j − j − 1 j

(12)

where j= m−i +1 Fig. 1. Potential waveform for the SWV experiment. Ei is the initial voltage, DE the step voltage, ESW the square wave voltage and tp the duration of a pulse.

(13)

We must now evaluate the sum over c term. Consider,

G1 =

&

A.B. Miles, R.G. Compton / Journal of Electroanalytical Chemistry 487 (2000) 75–89



c(u)exp −

t

c 2d 2 DOX(t − u)



t − u

0

+2 du

G1 =

&



&



c d c(u)exp − mp DOX(mp −u) 2

2

mp −u

0

du

(15)

m

G1 = % bi

&

i=1



ip

c 2d 2 DOX(mp −u)

mp − u

(i − 1)p



du

(16)

Defining, y=

c 2d 2 DOX(mp−u)

m

(19) (20)

Defining, fOX = G2 =

&

c 2d 2 pDOX

(21)

fOX/(m − i)

exp( −y) dy y 3/2 f OX/(m − i + 1)

(22)

Now, introducing a= exp(−y) da = −exp(−y) dy

−2 y 1/2

G2 can be solved by integration by parts:

 &

G2 = − 2

n

exp(− y) y 1/2

fOX/(m − i)



exp(− y) y 1/2

n

f OX/(m − i + 1)

fOX/(m − i + 1) f OX/(m − i)



fOX (m−i)



n

(25)





 '

+ 2 p erf



'

fOX − erf (m−i +1)





fOX (m−i)



Ç Ã Ã Ã É

n

(26)



fOX ÁÆexp − ÃÃ (m−i +1) % bi ÍÃ G1 = 2 1/2 fOX

DOX i = 1 ÃÃ ÄÈ (m−i +1) f exp − OX Ç (m−i) Ã fOX − Ã + p erf fOX 1/2 Ã (m−i +1) É (m−i) Â fOX Ã −erf Ì (27) (m−i) Ã Å or m



  ' n







 '

!

"

exp(− fOX) cd bm + p[erf fOX −1] G1 = 2 f 1/2 OX

DOX fOX ÁÆexp − cd m − 1 ÃÃ (m−i +1) +2 % bi ÍÃ 1/2 fOX

DOX i = 1 ÃÃ ÄÈ (m−i +1) f exp − OX Ç (m−i) Ã fOX − Ã + p erf fOX 1/2 Ã (m−i +1) (m−i) Â É fOX Ã −erf Ì (28) (m−i) Ã Å











 '

G1 eventually reduces to

fOX/(m − i)

exp( − y) −2 dy y 1/2 f OX/(m − i + 1) G2 = 2



  ' n

and

(24)

fOX fOX Æexp − exp − Ã (m−i +1) (m−i) G2 = 2Ã − 1/2 fOX fOX 1/2 Ã È (m−i +1) (m−i)



db 1 = dy y 3/2

b=

G2 becomes

'

fOX − erf (m−i +1)

and (18)

c2d2/(m − i)pDOX

exp( − y) G1 = % bi dy y 3/2 2 2

DOXi = 1 c d /(m − i + 1)pD OX m cd % bi G2 G1 =

DOX i = 1

exp(− y) dy y 1/2

 '

= p erf

cd

&

and, cd

fOX/(m − i + 1)

(17)

dy c 2d 2 D = = OX y 2 du DOX(mp− u)2 c 2d 2

exp(− y) dy y 1/2

Noting that

f OX/(m − i)

and approximating c(u) by bi, a constant over a small duration, we find exp −

fOX/(m − i + 1)

f OX/(m − i)

(14)

Letting t=mp (where, from Eq. (11), p = t/2l),

&

77

(23)

cd G1 = 2 bm [P %%OX,1 + pQ %%OX,1]

DOX cd m − 1 % bi [P %%OX, j + pQ %%OX, j] +2

DOX i = 1 by introduction of the following abbreviations,

(29)

78



exp −

fOX j

A.B. Miles, R.G. Compton / Journal of Electroanalytical Chemistry 487 (2000) 75–89



   ' '

P %%OX, j =



exp −

fOX ( j −1)



− fOX 1/2 fOX j ( j −1) fOX fOX Q %%OX, j =erf −erf j ( j −1)



[RED]tx = 0 (30)

1/2

2d

c=1

DOX

% ( −1)cG1 =

!



bm % c( −1)c[P %%OX,1

"

c=1



m−1

c=1

i=1

 '

+

'

m−1

!

p t

=





m−1

c=1

i=1

"n

(33)

bm % c(−1)c[P %%RED,1 + pQ %%RED,1] c=1



m−1

c=1

i=1

p t 4d

'

[OX]tx = 0 = [OX]BULK −

!'

p t

bm ×

[OX]BULK p t

m−1

!'

"

"

(34)

i=1

Equation Eq. (34) can be abbreviated to: [OX]BULK [OX]tx = 0 = [OX]BULK − (bmAOX,1 +AOX,2) p t

AOX,1 =

'

AOX,2 =

'

2t l



' !'



4d

D

"

2t m − 1 % b S% l i=1 i j

m−1

"

% c(−1)c % bi [P %%RED, j + pQ %%RED, j]

RED c = 1

i=1

(39) and finally, by defining ARED,1 and ARED,2 analogously to AOX,1 and AOX,2 above, abbreviates to [RED]tx = 0 =

[OX]BULK p t

'

DOX (b A + ARED,2) DRED m RED,1 (40)

2.2.1. Re6ersible E mechanism We first consider the case when the electrode process is reversible and hence the Nernst Equation holds: [OX]tx = 0 = [RED]tx = 0o where

(35)

by defining AOX,1 and AOX,2 as follows, 2t 4d + % c( −1)c[P %%OX,1 + pQ %%OX,1] l

DOX c = 1 (36) 2t m − 1 4d % bi S %j + % c( − 1)c × l i=1

DOX c = 1

o= exp

(37)

i=1

2.1.2. RED species In an identical manner, the concentration integral equation for the RED species (Eq. (5)) leads to:



nF(E− E°%) RT



(41)

(42)

Substitution into Eq. (41) for the OX and RED species from above, [OX]BULK − =o

m−1

% bi [P %%OX, j + pQ %%OX, j]

!'

2.2. Formulation of the approximate dimensionless current, bm, by combination of the OX and RED concentration integral equations

2t m − 1 4d % bi S %j + % c( −1)c × l i=1

DOX c = 1

% bi [P %%OX, j + pQ %%OX, j]

(38)

Eqs. (35) and (40) are of no less generality than Eqs. (4) and (5).

2t 4d + % c( −1)c[P %%OX,1 + pQ %%OX,1] l

DOX c = 1

−

"n

% c(−1)c[P %%RED,1 + pQ %%RED,1]

which can be rearranged to [OX]BULK

DOX bm DRED

DRED c = 1 DOX [OX]BULK + DRED p t +

+ pQ %%OX,1]+ % c( −1)c % bi [P %%OX, j + pQ %%OX, j]

2t m − 1 % b S% l i=1 i j



DRED

[OX]BULK

+

2t l

2t 4d % bi S %+ bm % c( −1)c[P %%OX,1 j l i=1 c=1

DOX

p t 4d

[RED]tx = 0

Combining Eqs. (9), (10) and (32),

bm

2t + l

DOX bm DRED

or

(32) [OX]BULK

'  ' ' !

+ % c(− 1)c % bi [P %%RED, j + pQ %%RED, j]

+ pQ %%OX,1]+ % c( −1)c % bi [P %%OX, j + pQ %%OX,j]

[OX]tx = 0 = [OX]BULK −

[OX]BULK

+

(31)

where j is as defined above (Eq. (13)) and the notation in Eqs. (30) and (31) is by extension of that used by Nicholson and Olmstead [11] and O’Dea et al. [8]. From Eq. (29),

=

[OX]BULK

[OX]BULK p t

(bmAOX,1 + AOX,2) p t DOX (b A + ARED,2) DRED m RED,1

'

(43)

or p t −(bmAOX,1 + AOX,2) =o

'

DOX (b A + ARED,2) DRED m RED,1

(44)

A.B. Miles, R.G. Compton / Journal of Electroanalytical Chemistry 487 (2000) 75–89



Assuming that (see Appendix B)

exp − (45)

DOX = DRED =D and therefore fOX =fRED = f

(46)

P %%OX, j =P %%RED, j =P %%j

(47)

Q

%% OX, j

=Q

%% RED, j

=Q

%% j

(48)

AOX,1 =ARED,1 = A1

(49)

AOX,2 =ARED,2 = A2

(50)

Eq. (44) simplifies to: p t = (1 + o)[bmA1 + A2]

(51)

(52)

k=

bm = p t − (1+ o)

' '

m−1 4d 2t m − 1 % bi S %− % c(−1)c % bi [P %%j + pQ %%j ] j l i=1 i=1

D c = 1

2t 4d % c(−1)c[P %%1 + pQ %%1] + l

D c = 1

c(t)=

bm =

−[RED]



exp −

nF(E − E°%) exp RT

anF(E −E°%) RT



n

pt [OX]BULK D



k° [OX]tx = 0



− [RED]tx = 0exp

nF(E −E°%) RT

n

×

ko − a pt [OX]BULK



!

[OX]BULK −

[OX]BULK

p t which reduces to ko − a

p

(59)

[OX]BULK

(bmA1 + A2)

p t

n

(bmA1 +A2)

n"

[p t − bmA1(1+o)− A2(1+o)]



n

(60)

(61)

p + A1(1+o) = p t − A2(1+o) ko − a

(62)

Dividing through by (1+ o) and rearranging, p t − A2 (1+ o) bm =

p + A1 −a ko (1+o) and substituting back in for A1 and A2:

(63)

p t − (1+o)

(54)

(58)

ko − a pt [[OX]tx = 0 − o[RED]tx = 0] [OX]BULK

bm =

where k° is the standard electrochemical rate constant and a is the transfer coefficient. Introducing the dimensionless current this becomes: c(t) =

ko − a pt [[OX]tx = 0 − o[RED]tx = 0] [OX]BULK

−o

bm =

×

(57)

Substituting for the OX and RED concentrations into Eq. (59) from Eqs. (35) and (40), given equal diffusion coefficients,

bm





However, the dimensionless current, c(t), is approximated by bm and therefore,

2.2.2. Quasi-re6ersible E mechanism We consider the case where the diffusion coefficients of the OX and RED species are equal, although the general solution for unequal diffusion coefficients is reported in Appendix B. The Butler – Volmer equation describing the current observed for a quasi-reversible E mechanism is:

t x=0

nF(E− E°%) RT

(56)

and therefore,

or





D

o= exp

(53)

I(t) =nFAk° [OX]tx = 0

(55)

k°

bm =

and substituting A1 and A2 back in



which we simplify by introducing k and o:

Rearranging for bm gives p t − A2 (1+o) bm = A1

anF(E− E°%) RT

79

'

m−1 2t m − 1 4d % bi S %− % c(− 1)c % bi [P %%j + pQ %%j ] j l i=1 i=1

D c = 1

p + ko (1+ o) −a

'

2t 4d + % c(− 1)c[P %%1 + pQ %%1] l

D c = 1

(64)

This equation is of a sensible form because the two sums over the c terms tend to zero as the diffusion layer thickness tends to infinity and the equation therefore reduces to the exact formulation of the quasi-reversible E mechanism under semi-infinite diffusion conditions [8]. Further, this equation differs from that for the completely reversible case (Eq. (53)) only by the appearance of the first term in the denominator and this term indeed vanishes in the reversible limit (k“ ).

A.B. Miles, R.G. Compton / Journal of Electroanalytical Chemistry 487 (2000) 75–89

80

2.3. Realisation of the square wa6e 6oltammogram

were analysed using Excel 2000 and the solutions were converged to find appropriate values of both l and cmax.

From Eq. (1) it follows that the dimensionless current difference is given by: DIDL = IDL(1)−IDL(2)

(65)

where IDL(1) and IDL(2) are the dimensionless forms of the current contributions I(1) and I(2), respectively. Further, IDL(i )= c(t)

(66)

where i= 1, 2 and t is a time corresponding to the end of a forward or backward pulse. Finally, c(t) is approximated to bm and consequently, IDL(i ): bm

(67)

where m takes appropriate values. Therefore, the individual dimensionless current contributions, IDL(1) and IDL(2), and the dimensionless current difference, DIDL, can be calculated for the reversible and quasi-reversible E mechanisms via Eqs. (53) and (64), respectively.

3. Computational aspects Solutions for Eqs. (53) and (64) were programmed in and run on a Silicon Graphics Origin 2000 server. The infinite sums over c (terms A1 and A2) were approximated by sums to a finite value of c, cmax:

FORTRAN

A1 = A2 =

' '

2t 4d cmax + % c( − 1)c[P %%1 + pQ %%1] l

D c = 1

(68)

2t m − 1 % b S% l i=1 i j 4d

cmax

m−1

% c(−1)c % bi [P %%j + pQ %%j ]

(69) i=1

D c = 1 This approach is justified by noting that Pj and Qj decrease rapidly to zero with increasing c. The results +

4. Results and discussion

4.1. Re6ersible mechanism Solutions to Eq. (53) were computed for a range of values of diffusion layer thickness, d, between 1 and 200 mm. The following parameters were utilised throughout the work: ESW = 50 mV; DE = − 10 mV; tp = 0.01 s (corresponding to a frequency of 50 Hz); E°% = 0 V; Ei = 0.5 V; Ef = − 0.5 V; where Ei and Ef are the initial and final voltages, respectively. The resulting square wave voltammograms were characterised by their three key properties: dimensionless peak current, DIDL,p, peak position, DEp, and peak width at halfheight, W1/2, where: DEp = Ep − E°% and Ep is the potential at which the maximum current is observed in the voltammogram. The voltammetric properties were converged to find suitable values of l and cmax in turn. In both cases DIp and W1/2 were required to agree with their most converged values to within 1% whilst DEp was attained to within 0.01 mV. When converging in terms of l, the voltammetric properties were compared to values observed for l= 320. Similarly, the most converged values with respect to cmax correspond to cmax = 80. For large d (] 5 mm) l= 80 is required to give suitable convergence, as described, whilst l= 40 is adequate for small d (1−4 mm). The required value of cmax varies from 40 for d= 1 mm to 1 for d] 20 mm. The decrease in the required value of cmax as d increases is exactly as expected and is a manifestation of the fact that the terms in the sums over c vanish to zero as the diffusion layer thickness is increased. The results reported below

Fig. 2. Plot of the dimensionless current difference, DIDL,p, as a function of diffusion layer thickness, d, for the reversible E mechanism.

A.B. Miles, R.G. Compton / Journal of Electroanalytical Chemistry 487 (2000) 75–89

were calculated using l =320 and cmax =10 for d] 5 mm whilst l =80 and cmax =80 were used for 1 5 dB5 mm. The results were next compared to converged results obtained via an alternative, finite-difference simulation, technique developed by Brookes et al. [5]. The data obtained from the two complementary approaches dis-

81

plays excellent agreement; the results agree within the convergence criteria mentioned previously. The variations of DIDL,p, DEp, and W1/2 with diffusion layer thickness, d, are shown in Figs. 2–4, respectively. All three properties are independent of d for d\ 50 mm, corresponding to semi-infinite diffusion. At very low diffusion layer thickness the dimensionless

Fig. 3. Plot of the current difference peak position, DEp, as a function of diffusion layer thickness, d, for the reversible E mechanism.

Fig. 4. Plot of the current difference peak width, W1/2, as a function of diffusion layer thickness, d, for the reversible E mechanism.

Fig. 5. Plot of the individual dimensionless current contributions, IDL(1) and IDL(2), and the dimensionless current difference voltammogram, DIDL, for the reversible E mechanism with d = 200 mm.

82

A.B. Miles, R.G. Compton / Journal of Electroanalytical Chemistry 487 (2000) 75–89

peak current decreases slightly before rapidly increasing to a transport limited value at d =1 mm. As the diffusion layer thickness is decreased, the peak width shows a small, gradual decrease whilst the peak position displays a small but non-systematic variation, the basis of which is not immediately obvious. However, further insight can be gained by considering the individual

current components, I(1) and I(2), as given in Eq. (1). The square wave current contributions and voltammogram for the semi-infinite diffusion limit (d=200 mm) is shown in Fig. 5. The individual current contributions themselves exhibit peaks which, in an analogous manner to the current difference, can be described by IDL,p(n) and DEp(n) where n= 1 or 2 corresponding to

Fig. 6. Plot of the dimensionless current contributions, IDL(1) and IDL(2), as a function of diffusion layer thickness, d, for the reversible E mechanism.

Fig. 7. Plot of the current contribution peak positions, DEp(1) and DEp(2), as a function of diffusion layer thickness, d, for the reversible E mechanism.

Fig. 8. Plot of the individual dimensionless current contributions, IDL(1) and IDL(2), and the dimensionless current difference voltammogram, DIDL, for the reversible E mechanism with d = 4 mm.

A.B. Miles, R.G. Compton / Journal of Electroanalytical Chemistry 487 (2000) 75–89

the current contribution. The dependencies of IDL,p(1), IDL,p(2), DEp(1) and DEp(2) on diffusion layer thickness are shown in Figs. 6 and 7. Each of the properties displays a dependence on d which can be understood given that the decreasing diffusion layer thickness results in more rapid mass transport to and from the electrode. Indeed, for d B 5 mm the mass transport is sufficiently rapid that the individual current contributions do not display peaks and have the general form shown in Fig. 8. Reducing the diffusion layer thickness from 4 to 1 mm results in a further increase in the rate of mass transport and consequently the dimensionless current difference, DIDL,p, is observed to rapidly increase. The interplay of the dependencies of IDL,p(1), IDL,p(2), DEp(1) and DEp(2) on d clearly dictates the perhaps unexpected behaviour of the characteristic properties of the current difference voltammograms, as is most noticeable in Fig. 3.

83

4.2. Quasi-re6ersible mechanism The dependence of the three voltammetric properties on the extent and nature of the reaction reversibility is illustrated by solution of Eq. (64). In light of work by O’Dea et al. [8] and Brookes et al. [6], the three voltammetric properties were calculated for the following range of values a and k: a=0.1, 0.3, 0.5, 0.7, 0.9 and − 35 log(kt 1/2)5 2. Figs. 9–11, respectively, display the dependence of DIDL,p, DEp, and W1/2 on a and log(kt 1/2) in the limit of semi-infinite diffusion (d=200 mm). The data was calculated using l= 320 and cmax = 5. The results agree with those reported by Brookes et al. [6] to within less than 91% for DIDL,p and W1/2 whilst DEp typically compares to within 9 0.5 mV although deviation can be as great as 9 2.0 mV in some cases. Results of the approximate analytical solutions published by O’Dea et

Fig. 9. Plot of the dimensionless current difference, DIDL, as a function of log(kt 1/2) for the quasi-reversible E mechanism with d= 200 mm and various values of a: a= (A) 0.1; (B) 0.3; (C) 0.5; (D) 0.7; (E) 0.9.

Fig. 10. Plot of the current difference peak position, DEp, as a function of log(kt 1/2) for the quasi-reversible E mechanism with d =200 mm and various values of a: a= (A) 0.1; (B) 0.3; (C) 0.5; (D) 0.7; (E) 0.9.

84

A.B. Miles, R.G. Compton / Journal of Electroanalytical Chemistry 487 (2000) 75–89

Fig. 11. Plot of the current difference peak width, W1/2, as a function of log(kt 1/2) for the quasi-reversible E mechanism with d =200 mm and various values of a: a = (A) 0.1; (B) 0.3; (C) 0.5; (D) 0.7; (E) 0.9.

Fig. 12. Plot of the dimensionless current difference, DIDL, as a function of log(kt 1/2) for the quasi-reversible E mechanism with d =10 mm and various values of a: a= (A) 0.1; (B) 0.3; (C) 0.5; (D) 0.7; (E) 0.9.

Fig. 13. Plot of the dimensionless current difference, DIDL, as a function of log(kt 1/2) for the quasi-reversible E mechanism with d= 5 mm and various values of a: a= (A) 0.1; (B) 0.3; (C) 0.5; (D) 0.7; (E) 0.9.

A.B. Miles, R.G. Compton / Journal of Electroanalytical Chemistry 487 (2000) 75–89

al. [8], which the present paper extends to allow the introduction of a finite diffusion layer thickness, also compare extremely well as expected given that Eq. (64) reduces to the form used by O’Dea et al. as d approaches the semi-infinite diffusion regime. The variation of the voltammetric properties with the introduction of a finite diffusion layer thickness is illustrated by Figs. 12 – 20 which display the three characteristic properties for d = 10, 5 and 1 mm. The values of l and cmax used are as follows: d =10 mm, l= 320 and cmax =5; d =5 mm, l =320 and cmax =10; d=1 mm, l=80 and cmax =80. The agreement with converged results calculated via the alternative finite-difference simulation approach of Brookes et al. [6] is similar to the case of d= 200 mm. The computational efficiency of the present work is strongly dependent on the values of l and cmax required. Further, the values of l and cmax necessary for convergence depend on the diffusion layer thickness and consequently the computational time required varies with

85

d. However, as d is increased the required value of cmax decreases whilst the necessary value of l increases. Correspondingly, the computer time required is always approximately constant and is comparable to that required for the finite difference simulations of Brookes et al. [5,6] for the convergence criteria described above.

5. Conclusions An approach to the calculation of square wave voltammograms is developed for systems where a uniform diffusion layer of arbitrary, adjustable thickness gives rise to diffusive mass transport of electroactive species to the electrode. The technique, based on the solution of integral equations, is applicable to reversible, quasi-reversible and irreversible E mechanisms. The forms of the solutions are seen to reduce to those presented by O’Dea et al. [8] in the limit of semi-infinite diffusion. However, the present work ex-

Fig. 14. Plot of the dimensionless current difference, DIDL, as a function of log(kt 1/2) for the quasi-reversible E mechanism with d= 1 mm and various values of a: a= (A) 0.1; (B) 0.3; (C) 0.5; (D) 0.7; (E) 0.9.

Fig. 15. Plot of the current difference peak position, DEp, as a function of log(kt 1/2) for the quasi-reversible E mechanism with d = 10 mm and various values of a: a= (A) 0.1; (B) 0.3; (C) 0.5; (D) 0.7; (E) 0.9.

86

A.B. Miles, R.G. Compton / Journal of Electroanalytical Chemistry 487 (2000) 75–89

Fig. 16. Plot of the current difference peak position, DEp, as a function of log(kt 1/2) for the quasi-reversible E mechanism with d = 5 mm and various values of a: a= (A) 0.1; (B) 0.3; (C) 0.5; (D) 0.7; (E) 0.9.

Fig. 17. Plot of the current difference peak position, DEp, as a function of log(kt 1/2) for the quasi-reversible E mechanism with d = 1 mm and various values of a: a= (A) 0.1; (B) 0.3; (C) 0.5; (D) 0.7; (E) 0.9.

Fig. 18. Plot of the current difference peak width, W1/2, as a function of log(kt 1/2) for the quasi-reversible E mechanism with d = 10 mm and various values of a: a= (A) 0.1; (B) 0.3; (C) 0.5; (D) 0.7; (E) 0.9.

tends this theory and facilitates the study of hydrodynamic systems. Further, the results show excellent agreement with an alternative finite-difference simula-

tion approach implemented by Brookes et al. [5,6] and are interpreted to validate the use of both techniques. Hence, the current work supports the extension of the

A.B. Miles, R.G. Compton / Journal of Electroanalytical Chemistry 487 (2000) 75–89

87

Fig. 19. Plot of the current difference peak width, W1/2, as a function of log(kt 1/2) for the quasi-reversible E mechanism with d = 5 mm and various values of a: a = (A) 0.1; (B) 0.3; (C) 0.5; (D) 0.7; (E) 0.9.

Fig. 20. Plot of the current difference peak width, W1/2, as a function of log(kt 1/2) for the quasi-reversible E mechanism with d = 1 mm and various values of a: a = (A) 0.1; (B) 0.3; (C) 0.5; (D) 0.7; (E) 0.9.

finite-difference simulation technique to the study, including the investigation of finite diffusion layer effects, of more complex mechanistic schemes [7] for which the presently developed integral equation technique faces possibly acute mathematical challenges. Finally, variation of the potential waveform employed in the calculations is possible and, therefore, the present theory is applicable beyond the study of square wave voltammetry.

Taking the Laplace Transform gives, s([OX]− [OX]BULK)= DOX

([OX]− [OX]BULK)= Ae

OX

+ Be

− x D s

OX

e

I(s)

d D s

nFA sDOX (e

OX

OX

+e

− d D s

OX

)

(A4)

d D s

e

I(s) d D s

[OX]− [OX]BULK x = 0 = A+B (A1)

(A3)

− d D s

OX

− d

s

D OX OX + e nFA sDOX (e ) Now at x=0, Eq. (A3) reduces to

From Eq. (2), #([OX]−[OX]BULK) #2([OX] −[OX]BULK) =DOX #t #x2

x D s

Implementing the boundary conditions at x=0 and x = d gives,

B= − A.1. OX mass transport

(A2)

the general solution of which is:

A= Appendix A. Derivation of the concentration integral equations

#2([OX]− [OX]BULK) #x 2

Substituting for A and B,

(A5)

(A6)

A.B. Miles, R.G. Compton / Journal of Electroanalytical Chemistry 487 (2000) 75–89

88

[OX]− [OX]BULK x = 0 (e

I(s)

=

Appendix B. Unequal diffusion coefficients

s − d D OX

d D

nFA sDOX (e which simplifies to,

s OX

−e

+e

s d D OX

− d D

s OX

)

(A7)

)

In the text we have considered the case when the OX and RED species have equal diffusion coefficients: DOX = DRED = D

[OX] −[OX]BULK x = 0 = −

I(s)

s tanh(d D OX )

nFA sDOX

(A8)

However, equations can also be formulated for systems where this assumption cannot be made and the results are as follows.

Taking the inverse Laplace transform, [OX]tx = 0 − [OX]BULK 1

!

I−1

s I(s)tanh(d D OX )

A.1. Re6ersible E mechanism

"

(A9) nFA DOX

s Noting the following inverse Laplace transforms [12],

=−

!

"

) 

I − 1{I(s)}= I(t) I

−1

tanh(a s)

s

=a

(A10)

t u2 0 2 a

−1

From Eq. (44):



bm AOX,1 + oARED,1

u2(6 x)=



and therefore,

% ( −1) e

xp c = − and employing the convolution theorem [13],

&

t

I(u)



1 nFA DOX





% (− 1)c exp −

c= −

'

c 2a 2 t−u

(A12)

a−1 ×

&



n

c

du

(A13) k=

nFA pDOX

&



n

c 2d 2 I(u) % (−1)exp − t DRED(t − u) c= −

0

t − u

ko − a pt [OX]BULK −

du

!

[OX]BULK

−o



[OX]BULK (bmAOX,1 + AOX,2)

p t [OX]BULK p t

'

bm

n



(B4)

(A15)

'  '

p + AOX,1 + oARED,1 ko − a

= p t − AOX,2 − oARED,2 and finally,

p t − AOX,2 − oARED,2 bm =

n"

DOX (b A + ARED,2) DRED m RED,1

Therefore,

du

n

(A14)

In an identical manner to the derivation of Eq. (A14) from Eq. (A1), the RED concentration integral equation can be derived from Eq. (3): 1 [RED]tx = 0 = × nFA pDRED

(B3)

DOX

×

A.2. RED mass transport



k°

bm =

t −u

0

(B2)

DOX DRED

Retaining the unequal diffusion coefficients on substitution of Eqs. (35) and (40) into Eq. (59):

1



'

DOX DRED

For the case of unequal diffusion coefficients, Eq. (59) remains unchanged and k is redefined as:

c 2d 2 I(u) % (−1) exp − t DOX(t − u) c= −

bm =

'

(B1)

A.2. Quasi-re6ersible E mechanism

in which d a=

DOX Rearranging finally yields [OX]tx = 0 = [OX]BULK −

p t − AOX,2 − oARED,2 AOX,1 + oARED,1

(t− u)p a2

0

DOX DRED

(A11)

c − (6 + c)2/x

[OX]tx = 0 − [OX]BULK = −

DOX DRED

= p t − AOX,2 − oARED,2

where 1

'  '

p + AOX,1 + oARED,1 ko − a

DOX DRED

DOX DRED

' '

DOX DRED DOX DRED

(B5)

(B6)

A.B. Miles, R.G. Compton / Journal of Electroanalytical Chemistry 487 (2000) 75–89

Eqs. (B2) and (B6) provide complete solutions to the introduction of a finite diffusion layer thickness for the E and quasi-reversible E mechanisms. The results reported in the text are illustrative and we have therefore considered only the case where the diffusion coefficients are equal. It is also worth noting, however, that the solution of the equations which allow different diffusion coefficients will be more computationally demanding as a result of the fact that each of the ‘A’ terms involve infinite sums over c.

References [1] J. Osteryoung, J.J. O’Dea, in: A.J. Bard (Ed.), Electroanalytical Chemistry, vol. 14, Marcel Dekker, New York, 1986, p. 209. [2] D.P. Whelan, J.J. O’Dea, J. Osteryoung, K. Aoki, J. Electroanal. Chem. 202 (1986) 23.

.

89

[3] N. Fatouros, D. Krulic, J. Electroanal. Chem. 443 (1998) 262. [4] K. Aoki, J. Osteryoung, J. Electroanal. Chem. 240 (1988) 45. [5] B.A. Brookes, J.C. Ball, R.G. Compton, J. Phys. Chem. B 103 (1999) 5289. [6] B.A. Brookes, R.G. Compton, J. Phys. Chem. B 103 (1999) 9020. [7] A.B. Miles, R.G. Compton, J. Phys. Chem. (in press). [8] J.J. O’Dea, J. Osteryoung, R.A. Osteryoung, Anal. Chem. 53 (1981) 695. [9] D.E. Smith, Anal. Chem. 35 (1963) 603. [10] J.J. O’Dea, K. Wikiel, J. Osteryoung, J. Phys. Chem. 94 (1990) 3628. [11] R.S. Nicholson, M.L. Olmstead, in: J.S. Mattson, H.B. Mark, H.C. MacDonald (Eds.), Electrochemistry: Calculations, Simulation and Instrumentation, vol. 2, Marcel Dekker, New York, 1972 Ch. 5. [12] G.E. Roberts, H. Kaufman, Table of Laplace Transforms, W.B. Saunders Company, Philadelphia, USA, 1966, p. xxvii and 283. [13] J.W. Miles, Integral Transforms in Applied Mathematics, Cambridge University Press, Cambridge, UK, 1971, p. 17.