A unified new analytical approximation for positive feedback currents with a microdisk SECM tip

Journal of

Electroanalytical Chemistry Journal of Electroanalytical Chemistry 592 (2006) 103–112 www.elsevier.com/locate/jelechem

A unified new analytical approximation for positive feedback currents with a microdisk SECM tip C. Lefrou

*

Laboratoire d’Electrochimie et de Physico-chimie des Mate´riaux et des Interfaces, UMR 5631 CNRS-INPG-UJF, ENSEEG, 1130 rue de la piscine, B.P. 75, Domaine Universitaire, 38402 Saint Martin d’He`res Cedex, France Received 12 February 2006; received in revised form 25 April 2006; accepted 2 May 2006 Available online 12 June 2006

Abstract Analytical approximations for steady-state positive feedback currents in scanning electrochemical microscopy (SECM) experiments using a microdisk are discussed. The use of a conformal map transformation leads to the use of new mathematical functions. A general analytical expression for all values of tip–substrate distances and all types of microdisk (with various insulator thicknesses) are thus obtained with only four adjustable parameters. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Scanning electrochemical microscopy; Feedback mode; Disk electrode; Microelectrodes; Steady-state; Simulation

1. Introduction Since the first demonstration of the use of scanning electrochemical microscopy (SECM) by Bard and coworkers [1], modeling of experimental feedback curves has shown to be of great importance due to the lack of exact analytical descriptions of electrochemical responses [2]. One of the most studied electrode configuration is a disk shaped microelectrode, sealed into a glass tube, working as an insulator [3–5]. Steady-state approach curves for different Rg values (Rg = rglass/a with rglass the radius of the disk insulator and a the electrode radius) have been reported by several groups and are now well established [6–8]. A large variety of analytical approximations have been proposed, as it will be discussed below for the positive feedback situation. However, up to now, no attempt was done to propose a single expression valid for any disk–substrate distance and any Rg value. Using a previous work by Amatore and co-workers about electrochemical simulations on microdisk electrodes with con*

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0022-0728/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jelechem.2006.05.003

formal mapping transformations [9–11], this work presents such a single expression for all type of steadystate positive feedback current with a microdisk electrode for SECM use. 2. Mathematical formulation of the SECM positive feedback case The system we are going to discuss is schematically described in Scheme 1. It shows a disk shaped microelectrode of radius a, sealed in an insulator of radius rglass, having a distance from the substrate d. We consider the case where a redox mediator is present in solution (for example here in its reduced form) at a concentration C*, and follows a one step electron transfer (with n electrons exchanged). Only diffusional mass transfer is considered (with D the diffusion coefficient of the redox mediator): convection and migrational effects are neglected. Following dimensionless parameters are identified (where F is the Faraday constant, iT the current through the tip, C the local concentration of the redox mediator and z and r the space coordinates in this cylindrical geometry):

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C. Lefrou / Journal of Electroanalytical Chemistry 592 (2006) 103–112

a

rglass r d

z Scheme 1. Schematic presentation of the system under consideration.

z r d rglass x¼ L¼ Rg ¼ a a a a C iT c¼  w¼ C 4nDC  Fa

Eqs. (2) and (3) formulate mathematically the steady-state positive feedback case, for fast reversible oxidation at the tip and fast reversible reduction at the surface with diffusion controlled current [6–8]:

ever, this will not be the focus of this work. A commercially available program, ComsolÒ [13], which allows numerical resolution of differential equations based on finite elements have been used for all the numerical responses reported in the following discussion. The size of the box used to describe the solution was chosen with adimensional value of 1000 for the infinite distances. The value considered as Rg infinite was also chosen at 1000.

o2 c 1 oc o2 c þ þ ¼ 0 in the solution ox2 x ox oy 2

3. Analysis and discussions of previous analytical approximations

y¼

c ¼ 0 for 0 6 x 6 1 and y ¼ 0   oc ¼ 0 for 1 6 x 6 Rg and y ¼ 0 oy y¼0   oc ¼ 0 for x ¼ 0 and 0 6 y 6 L ox x¼0   oc ¼ 0 for x ¼ Rg and y 6 0 ox x¼Rg

ð1Þ

ð2Þ ð3aÞ ð3bÞ ð3cÞ ð3dÞ

c¼1

for x P 0 and y ¼ L

ð3eÞ

c¼1 c¼1

for x ! 1 and y 6 L for x P Rg and y ! 1

ð3fÞ ð3gÞ

The dimensionless current response can be formulated as follows: Z   1 1 oc w¼ 2px dx ð4Þ 4 0 oy y¼0 Usual, experimental approach curves give the variation of normalized current with tip–substrate distance. This normalized tip current, NiT, is defined as the ratio of tip current by the tip current at infinite distance from the substrate: NiT ¼

iT iT;1

¼

w wL!1

:

The numerical solution of the outlined problem has been reported using different mathematical tools [6–8,12]. How-

3.1. Analytical expressions of positive feedback approach curve for Rg ! 1 For infinite Rg values, the adimensional tip current tends to 1 for infinite values of L (wL!1 = 1). Therefore in this case, the adimensional tip current w is the same as the normalized tip current NiT. Table 1 summarizes the different analytical approximations proposed by different groups. Fig. 1 illustrates the variations in the relative errors made using these analytical approximations compared to the exact simulated current as a function of log L. The choice of this representation allows a good visualization of the two limiting cases (low and large values of L). From this figure, the results can be divided into three groups considering the limiting values:  The first works by Tranter and Quart [14] and Hale [15] can be used in order to describe the limiting case where the microelectrode is either far away from the substrate or almost in contact with it. While Tranter and Quart [14] gives a satisfactory approximation at large L (Eq. (6), L > 4.2, error <1%), Hale [15] shows a good agreement for low L (Eq. (7), L < 0.3, error <1%). for large L : for small L :

iT 2 ln 2 1þ pL iT;1 iT p w¼ þ ln 2  iT;1 4L

w¼

ð6Þ ð7Þ

C. Lefrou / Journal of Electroanalytical Chemistry 592 (2006) 103–112

105

Table 1 Different analytical approximations proposed in the literature for the steady-state positive feedback approach curve with a large value of Rg(Rg ! 1) Approximate expressions of NiT(L) for infinite Rg  2  3 2 ln 2 2 ln 2 2 ln 2 fð3Þ þ þ  1þ pL pL pL 4pL3

Reference Tranter and Quart [14]

No. (5)

2 ln 2 pL

(6)

Hale [15]

p þ ln 2 4L

(7)

Mirkin et al. [6]

 a 0:78377 1:0672 þ 0:68 þ 0:3315 exp  L L

(8)

Tranter and Quart [14]

1þ

 b 0:77957 1:29077 þ 0:7314 þ 0:26298 exp  L L

Amphlett and Denuault [7]

  p 1:11 þ ln 2 þ ð1  ln 2Þ exp  4L 0:11 þ L

Galceran et al. [16]



Rajendran and Ananthi [17]

a

(10)

  p 1 p  2 lnp 2 þ ln 2 þ ð1  ln 2Þ exp  4 4L L 1  ln 2

Rajendran and Ananthi [17]

b

(9)

1þ

(11)

  1:5647 1:316855 0:4919707 1:1234 0:6263951 þ þ þ 1 þ L L L2 L3 L2

(12)

Numerical fit from simulated current values for Rg = 10. Numerical fit from simulated current values for Rg = 1002.

3

2

1

iT/iT (%)

0

-1

-2

-3 0.01

0.1

1

10

100

L Fig. 1. Variations in the relative errors made using the analytical approximations listed in Table 1 compared to the exact simulated steady-state positive i iT;simulated feedback current with a large value of Rg (Rg ! 1) as a function of log L. The relative errors, DiiTT ¼ Dw ¼ T;approximate , are given in %. The following w iT;simulated symbols are used (numbers are the No. in Table 1): (5) n; (6) h; (7) e; (8) ; (9) none; (10) j; (11) r; (12) m.

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C. Lefrou / Journal of Electroanalytical Chemistry 592 (2006) 103–112

 Mirkin et al. [6] and Amphlett and Denuault [7] obtained a mathematical expression from a simulated feedback curve using a mathematically fit with a combination of polynomial and exponential terms. They do not give an exact expression for both limiting cases. However, the expression by Amphlett and Denuault is in general better adapted as it is inside the 1% limit for all values of L.  The more recent expressions by Galceran et al. [16] or Rajendran and Ananthi [17] are more interesting, not because they are more precise, but as they try to give a physical interpretation to the mathematical expression. In particular it considers the two exact limiting cases.

3.2. Analytical expressions of positive feedback approach curve for finite Rg

As in these last two papers, the present work will preserve the same concern of the limiting current values in the search of a new analytical expression.

Considering that the validity of this approximation is questionable, we have chosen in the present work to preserve the true boundary conditions (3a)–(3g), as chosen by

The simulated results presented in the literature do not always correspond exactly to the same modeled situation: certain authors postulated that one could modify some of the boundary conditions [16,17]. They removed the possibility of diffusion along the perpendicular insulating surface of the tip (x = Rg and y 6 0) by replacing the boundary conditions (3d), (3e) and (3g) by the following one: c¼1

for x ¼ Rg and 0 6 y 6 L

ð3hÞ

Table 2 Different analytical approximations proposed in the literature for the steady-state positive feedback approach curve with Rg = 1.5 Approximate expressions of NiT(L) for Rg = 1.5  a 0:67476 1:42897 þ 0:63349 þ 0:36509 exp  L L   0:6677381 1:496865 þ 0:636836 þ 0:3581836 exp  L L

Reference Amphlett and Denuault [7]

Shao and Mirkin [8]

(14)

   1:5647 1:316855 0:4919707 1:1234 0:6263951 þ þ k 1þ þ 1þ 2 3 2 L L L L L with k ¼ 0:9218 if L P 1 and k ¼ 0:8565 if L < 1

Rajendran and Ananthi [17]

a

No. (13)

(15)

Numerical fit from simulated current values for Rg = 1.51.

3

2

1

NiT/NiT 0 (%) -1

-2

-3 0.01

0.1

1

10

100

L Fig. 2. Variations in the relative errors made using the analytical approximations listed in Table 2 compared to the exact simulated steady-state positive Ni NiT;simulated T feedback normalized current with Rg = 1.5 as a function of log L. The relative errors, DNi ¼ T;approximate , are given in %. The following symbols are NiT NiT;simulated used (numbers are the No. in Table 2): (13) j; (14) none; (15) m.

C. Lefrou / Journal of Electroanalytical Chemistry 592 (2006) 103–112

Amphlett and Denuault [7] and Shao and Mirkin [8]. The positive feedback curves are less sensitive to a change of Rg than the negative feedback curves. However, this influence exists and will be illustrated here by the results of the positive feedback approach curve for a value of Rg = 1.5. Table 2 gives the various analytical approximations proposed by different groups. Fig. 2 illustrates the variations, as a function of L, in the relative errors resulting of these analytical approximations compared to the exact simulated normalized current, NiT. Fig. 3 clearly shows that the expression recently published by Rajendran and Ananthi [17] is not accurate compared to previous expressions [7,8]. Moreover, there are no analytical expressions for the influence of the Rg value. The only attempt found in the literature for analytical approximation with Rg as variable concerns one of the significant characteristics of these curves: current values when the substrate is infinitely far (i.e. for large L values). Table 3 summarizes the various analytical approximations proposed by different groups in this specific case. Fig. 3 illustrates the variations in the relative errors made using these analytical approximations compared to the exact simulated current at infinite L. In order to allow an easy comparison for the low values of Rg, the results are plotted in logarithmic scale for the value of Rg  1 (Rg = 1 would correspond to a zero thickness of insulator). From Fig. 3 we observe that the expression proposed by Zoski and Mirkin [20] is the most satisfactory, as already highlighted in recent work of Ciani and Daniele [21]. It presents a maximum error of 0.3% for Rg higher than 1.1 and the only values which are not well described are the one very close to Rg = 1.

107

Table 3 Different analytical approximations proposed in the literature for the steady-state positive feedback current at large L values (L ! 1) Reference

Approximate expressions of wL!1 (Rg)

No.

Zhao et al. [18]

0:379 1 þ 2:342 Rg

(16)

Fang et al. [19]

1þ

Galceran et al. [16]

(17)

0:234 0:255 þ 2 Rg Rg

(18)

1 1  0:5543 Rg

Zoski and Mirkin [20]

1þ

(19)

0:1380 ðRg  0:6723Þ0:8686

4. New analytical approximations: results and discussions 4.1. Conformal mapping transformations In previous works [9–11], Amatore et al. showed that conformal mapping transformations are powerful tools for the modeling of some electrochemical systems. In particular, the transformation suggested for a microdisk (with infinite Rg and infinite L) makes it possible to transform a situation with an infinite half-plane to a geometry registered in a rectangle of finite size. The change of variables proposed for a microdisk is as follows:

3

2

1

iT/iT 0 (%) -1

-2

-3 0.01

0.1

1

10

100

Rg 1 Fig. 3. Variations in the relative errors made using the analytical approximations listed in Table 3 compared to the exact simulated steady-state positive i iT;simulated feedback current with large values of L (L ! 1) as a function of log Rg  1. The relative errors, DiiTT ¼ Dw ¼ T;approximate , are given in %. The following w iT;simulated symbols are used (numbers are the No. in Table 3): (16) m; (17) j; (18) e; (19) none.

108

pffiffiffiffiffiffiffiffiffiffiffiffiffi 1  h2   x¼ cos pC 2

C. Lefrou / Journal of Electroanalytical Chemistry 592 (2006) 103–112

 y ¼ h tan

 pC 2

values at large L and small L allows the determination of the parameters of the approximate expression. A first very simple approximation A1 is thus built starting from the equivalents (i.e. only the first term) of the two limiting cases:

ð20Þ

Figs. 4a and b illustrate the effect of the change of variables on the geometry and on the isoconcentration lines for a microdisk with infinite values of Rg and L. Using an identical change of variables, our work investigates the geometries corresponding to the SECM approach curves in the comformal map, as illustrated in Figs. 5a and b for Rg = 2 and L = 1. This highlights two new geometrical parameters for the SECM situation:   2 2 1 arctanðLÞ and arccos ð21Þ p p Rg

1 p A1ðLÞ ¼ þ 2 4 arctanðLÞ

ð22Þ

In order to improve the precision, another expression A2 including a third term proportional to p2 arctanðLÞ is considered. Once again, the parameters are obtained from current values at large L (7) and small L (6). One thus obtains an expression without any adjustable parameter:   p 1 2 A2ðLÞ ¼ ln 2 þ þ  ln 2 arctanðLÞ ð23Þ 4 arctanðLÞ 2 p

4.2. New analytical expressions of positive feedback approach curve for Rg ! 1

Fig. 6 illustrates the variations in the relative errors compared to the exact simulated normalized current, using these new analytical approximations (same scales as in Fig. 2). It immediately appears on this figure that the new expression A2 constitutes an excellent analytical

The idea implemented in this work is to seek an analytical expression as a combination of mathematical functions using arctan(L) rather than L. Evaluation of the current x

(a) 0

5

10

15

20

0

c = 0.8 c = 0.9 -5

c = 0.95

y -10

-15 1

c = 0.95 c = 0.9 c = 0.8

0 0

1

(b) Fig. 4. Geometry and isoconcentration lines for a microdisk with infinite values of Rg and L in regular variables x and y (a) and in conformal map (C and h variables) (b). The isoconcentration lines are shown for the following values of concentration: 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 and 0.95.

C. Lefrou / Journal of Electroanalytical Chemistry 592 (2006) 103–112

109

Fig. 5. Geometry corresponding to the SECM approach curve for Rg = 2 and L = 1 in regular variables x and y (a) and in conformal map (C and h variables) (b).

3

2

1

iT/iT 0 (%) -1

-2

-3 0.01

0.1

1

10

100

L Fig. 6. Variations in the relative errors made using some of the analytical approximations listed in Table 1 and the new A1 and A2 approximate expressions compared to the exact simulated steady-state positive feedback current with a large value of Rg (Rg ! 1) as a function of log L. The relative i iT;simulated errors, DiiTT ¼ Dw ¼ T;approximate , are given in %. The following symbols are used: (9) none; (11) r; A1(22) n; A2(23) e. iT;simulated w

approximation over the entire positive feedback approach curve. The maximum error being 0.06%, the expression A2 is better than any previous one (cf. Fig. 2). The expression A1 on the other hand is not very powerful since the maximum relative error reaches 6.2% for a value of L = 0.8. 4.3. New analytical expressions of positive feedback approach curve for finite Rg

obtained through a numerically fit with a polynomial of degree 2 in p2 arccosðR1g Þ, whose term of degree 0 is fixed by the value for infinite Rg. Adjusting the two parameters allows to obtain the following approximate expression:    2 1 BðRg Þ ¼ 1 þ 0:639 1  arccos p Rg   2 ! 2 1 arccos  0:186 1  ð24Þ p Rg

We start with a new analytical approximation of the values of the current at infinite L for different Rg as  a combination of mathematical functions using arccos R1g rather than Rg.The only limiting value known is the current value for infinite Rg (equal to 1). The approximation B is

Fig. 7 illustrates the variations in the relative errors compared to the exact simulated current using these new analytical approximations (same scales as in Fig. 4). Since the new expression B gives a good representation of the Rg values very close to one, it is a better approximation

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C. Lefrou / Journal of Electroanalytical Chemistry 592 (2006) 103–112 3

2

1

iT/iT 0 (%) -1

-2

-3 0.01

0.1

1

10

100

Rg 1 Fig. 7. Variations in the relative errors made using one of the analytical approximations listed in Table 3 and the new B approximate expression compared to the exact simulated steady-state positive feedback current with large values of L (L ! 1) as a function of log Rg  1. The relative errors, i iT;simulated DiT ¼ Dw ¼ T;approximate , are given in %. The following symbols are used: (19) none; B(24) h. w iT iT;simulated

than the one proposed by Zoski and Mirkin [20]. The maximum error for the entire Rg range is now lower than 0.3%. We now look at a new analytical approximation of the positive feedback approach curves for finite Rg with a combination of mathematical functions using arctan(L) rather than L. The parameters of the approximate expression are then determined using the current values at large L and small L. It is significant to consider that only the current values for large L evolve when one changes the value of Rg. The current values for small L in positive feedback do not depend on the value of Rg. Thus, one has to consider the two following limiting cases:  for large L : w  wL!1  BðRg Þ

ð25Þ

p  for small L : w  4L

ð26Þ

The approach curves are usually exploited through the values of the normalized current. One thus has: iT  for large L : iT;1 ¼ww 1

ð27Þ

L!1

iT

 for small L : iT;1  w

1 L!1

p 4L



1 p BðRg Þ 4L

ð28Þ

On the model of the expression A2, an approximation C with only one adjustable parameter, K is built: CðLÞ ¼ K þ

  p 1 2 arctanðLÞ þ 1 K 4wL!1 arctanðLÞ 2wL!1 p

ð29Þ

For a value of Rg = 1.5, a numerically fit gives a value of K = 0.52. Fig. 8 illustrates the variations in the relative errors compared to the exact simulated current using this new analytical approximation (same scales as Fig. 2). For

this value of Rg one thus obtains an analytical approximation better than 1.3% over the entire approach curve. Finally, in order to obtain a general analytical expression D, according to Rg and L, one has to use the approximation B(Rg) for the current at infinite distance and to give an analytical approximation of the values of K(Rg) p 4BðRg Þ arctanðLÞ   1 2 arctanðLÞ  KðRg Þ þ 1 2BðRg Þ p

DðL; Rg Þ ¼ KðRg Þ þ

ð30Þ

For the approximation of the values of K(Rg), one proceeds as for B(Rg): one seeks a polynomial of degree 2 in 2 arccosðR1g Þ , whose term of degree 0 is fixed by the value p of K for infinite Rg (equal to ln(2)). One obtains, by adjusting the two parameters, the following approximate expression:    2 1 KðRg Þ  ln 2 þ ln 2 1  arccos p Rg   2 ! 2 1 arccos  ln 2 1  ð31Þ p Rg The corresponding values are also represented in Fig. 8. We should emphasize that for this Rg value, the relative error remains lower than 1.4% over the entire approach curve. As reported in the next figure for Rg = 1.5, the error values lie between 1.35% and +1.19%. Fig. 9 illustrates the maximum and minimum relatives errors for the last approximation D(L, Rg) (Eq. (30)) for different values of Rg and shows that one thus obtains a general analytical expression with an error lower than 2% for all the values of L and Rg.

C. Lefrou / Journal of Electroanalytical Chemistry 592 (2006) 103–112

111

3

2

1

NiT/NiT 0 (%) -1

-2

-3 0.01

1

0.1

10

100

L Fig. 8. Variations in the relative errors made using some of the analytical approximations listed in Table 2 and the new C and D approximate expressions compared to the exact simulated steady-state positive feedback normalized current with Rg = 1.5 as a function of log L. The relative errors, Ni NiT;simulated DNiT ¼ T;approximate , are given in %. The following symbols are used (numbers are the No. in Table 2): (13) j; (14) none; C(29) e; D(30) h. NiT NiT;simulated

3

2

1

ΔNiT/NiT (%)

0

-1

-2

-3 0.01

0.1

1

10

100

Rg − 1 Fig. 9. Variations in the maximum and minimum relative errors made using the new D approximate expression compared to the exact simulated steadyNi NiT;simulated T state positive feedback normalized current as a function of log Rg  1. The relative errors, DNi ¼ T;approximate , are given in %. NiT NiT;simulated

5. Conclusion

iT iT;1

The use of a conformal map transformation allows proposing new mathematical functions for analytical approximations of the positive feedback SECM approach curves, using the following geometrical parameters:   2 2 1 arctanðLÞ and arccos ð21Þ p p Rg This work shows that one thus obtains a general analytical expression with only four adjustable parameters whatever the values of L and Rg are:

p 4bðRg Þ arctanðLÞ   1 2 arctanðLÞ þ 1  aðRg Þ  2bðRg Þ p

 aðRg Þ þ

ð32Þ

with:    2 1 aðRg Þ ¼ ln 2 þ ln 2 1  arccos p Rg   2 ! 2 1 arccos  ln 2 1  p Rg

ð33Þ

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C. Lefrou / Journal of Electroanalytical Chemistry 592 (2006) 103–112

   2 1 bðRg Þ ¼ 1 þ 0:639 1  arccos p Rg   2 ! 2 1 arccos  0:186 1  p Rg

ð34Þ

The latter parameter, b(Rg), gives also a good analytical approximation of the value of the tip current at large L values but various Rg: iT;1  bðRg Þ 4nDC  Fa

ð35Þ

Considering the experimental inaccuracies (not rigorously perpendicular approach, uncertainty in the zero L position, etc.), the 2% accuracy of this new analytical expression is good enough: the present expression thus constitutes an operational and easy to use approximation for experimental data. The study of the negative feedback curves as well as the approach curves for various kinetic constants of the interfacial reactions is the normal further development of this work. References [1] A.J. Bard, F.-R.F. Fan, J. Kwak, O. Lev, Anal. Chem. 61 (1989) 132. [2] J. Kwak, A.J. Bard, Anal. Chem. 61 (1989) 1221.

[3] A.J. Bard, F.-R. Fan, M. Mirkin, in: A.J. Bard (Ed.), Electroanalytical Chemistry, vol. 18, Marcel Dekker, New York, 1994, p. 243. [4] A.J. Bard, F.-R. Fan, M. Mirkin, in: I. Rubinstein (Ed.), Physical Electrochemistry. Principles, Methods and Applications, Marcel Dekker, New York, 1995, p. 209. [5] M.V. Mirkin, B.R. Horrocks, Anal. Chim. Acta 406 (2000) 119. [6] M.V. Mirkin, F.R.F. Fan, A.J. Bard, J. Electroanal. Chem. 328 (1992) 47. [7] J.L. Amphlett, G. Denuault, J. Phys. Chem. B 102 (1998) 9946. [8] Y. Shao, M.V. Mirkin, J. Phys. Chem. B 102 (1998) 9915. [9] A.C. Michael, R.M. Whightman, C.A. Amatore, J. Electroanal. Chem. 267 (1989) 33. [10] C.A. Amatore, B. Fosset, J. Electroanal. Chem. 328 (1992) 21. [11] C.A. Amatore, in: I. Rubinstein (Ed.), Physical Electrochemistry. Principles, methods and applications, Marcel Dekker, New York, 1995, p. 131. [12] S.C.B. Abercrombie, G. Denuault, Electrochem. Commun. 5 (2003) 647. [13] http://www.comsol.com/. [14] C. Tranter, J. Quart, J. Mech. Appl. Math. 3 (1950) 411. [15] J.M. Hale, Electrochemistry, Sensors and Analysis, Elsevier, Amsterdam, 1986, p. 323. [16] J. Galceran, J. Cecilia, E. Companys, J. Salvador, J. Puy, J. Phys. Chem. B 104 (2000) 7993. [17] L. Rajendran, S.P. Ananthi, J. Electroanal. Chem. 561 (2004) 113. [18] G. Zhao, D.M. Giolando, J.R. Kirchoff, Anal. Chem. 67 (1995) 2592. [19] Y. Fang, J. Leddy, Anal. Chem. 67 (1995) 1259. [20] C.G. Zoski, M.V. Mirkin, Anal. Chem. 74 (2002) 1986. [21] H. Ciani, S. Daniele, Anal. Chem. 76 (2004) 6575.