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Performance of amperometric sensors based on multiple microelectrode arrays
Sensors and Actuators B 44 (1997) 538 – 541
Performance of amperometric sensors based on multiple microelectrode arrays Werner E. Morf *, Nicolaas F. de Rooij Uni6ersity of Neuchaˆtel, Institute of Microtechnology, CH-2000 Neuchaˆtel, Switzerland Accepted 8 April 1997
Abstract The steady-state current response and the dynamic characteristics (response to potential or concentration changes) of multiple microelectrode arrays are discussed. As a rule, an array with a low packing density of electrodes yields the ideal multiple response of a single microelectrode. In contrast, a closely packed array simply mimics the behavior of a macroelectrode of the same total surface area. The theoretical results are confirmed by experimental data. © 1997 Elsevier Science S.A. Keywords: Microelectrode array; Steady-state current; Dynamic response
1. Introduction Microelectrodes [1] or ultramicroelectrodes [2] for amperometric and voltammetric measurements have found a considerable interest because these devices exhibit partly unique response characteristics. For example, they show a very fast current response to changes of the applied potential. In addition, the resulting steady-state signal is nearly independent of the thickness of the Nernstian stagnant layer, and is therefore not influenced by changes of the natural or forced convection in the sample solution [3]. On the other hand, the current output is very low for microelectrodes because of their extremely small size. The current signal of amperometric microsensors can be amplified by the design of multiple electrode arrays that consist of N identically constructed microelectrodes [4–7] (see Fig. 1). Ideally, these arrays should yield a current amplification by a factor N relative to a single microelectrode. It is also expected that the dynamic behavior of microelectrode arrays, e.g. the current versus time response after a potential change, should be comparable to that of a single electrode. In practice, however, these expectations are fulfilled only if certain requirements concerning the design of the microelectrode array are met (see below).
* Corresponding author. 0925-4005/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved. PII S 0 9 2 5 - 4 0 0 5 ( 9 7 ) 0 0 1 5 9 - 7
In the past, a series of theoretical papers dealt with the steady-state response and the dynamic behavior of microelectrode arrays (for a review, see [8–10]). Most of these studies did not account for convection influences in the sample solution, however, since no limitations concerning the maximum thickness of the
Fig. 1. Schematic representation of an array of hemispherical or disc-shaped micro-electrodes (top view, with specification of geometric parameters).
W.E. Morf, N.F. de Rooij / Sensors and Actuators B 44 (1997) 538–541
diffusion layer between sample bulk and electrode surface were imposed. Extended theories, accounting for the influence of the Nernstian stagnant layer, were offered only recently [11 – 13]. In this paper, we summarize and discuss results obtained for the steady-state current response and the dynamic characteristics of multiple microelectrode arrays.
2. Amperometric response at steady state The ideal current response of individual microelectrodes and of macroelectrodes, respectively, is given by [3,11,12]: i( )micro =FnkDkckAel/a0
(1)
i( )macro = FnkDkckAel/d
(2)
with a0 =r0 for hemispherical microelectrodes and a0 = (p/4) r0 for disc-shaped microelectrodes, where i( ) is the steady-state current of the indicated electrode, F is the Faraday constant, nk is the charge number, Dk the diffusion coefficient, and ck the bulk concentration of the analyte k, Ael is the active surface area of the electrode, d is the average thickness of the Nernstian diffusion layer between sample bulk and electrode, a0 is the apparent curvature radius of the electrode surface [11,12], and r0 is the actual radius of the electrode. For multiple microelectrode arrays, an intermediate behavior is found in general cases [11,12]: i( )array = FnkDkck /[(d/Atot)+{1 −(NAel/Atot)1/2}2 (a0/NAel)] : FnkDkckAtot/[d +(d −2r0)2 a0/Ael]
(3)
where Atot is the total surface area of an array of N microelectrodes (including active and inactive zones), and d=(Atot/N)1/2 is the mean distance between the centers of neighbouring electrodes. This result shows that the response behavior of arrays largely depends on the packing density of microelectrodes. Loosely packed arrays (d2r0) yield the expected current signal, which ideally corresponds to the N-fold output of a single microelectrode, whereas closely packed arrays (d: 2r0) simply mimic the response behavior of a macroelectrode of the same total area: i( )array =N · i( )micro =N · FnkDkckAel/a0 (loosely packed)
(3a)
i( )array =i( )macro =FnkDkckAtot/d (closely packed)
(3b)
An analysis of experimental results on the basis of Eq. (3) attests a very good agreement between theory and experiment. An example is given in Table 1 where
539
Table 1 Steady-state response of Pt micro-disc electrode arrays (electrode radius: r0 =1 mm) to oxygen in air-saturated solution at 25°C Current densities [i( )/Atot in mA cm−2] for arrays with d (mm) of:
Calc. (d= 50 mm)b Exp. (quiescent)c Calc. (d =15 mm)b Exp. (stirred)c
20
15
10
5
2a
0.14 0.12 0.19 0.18
0.20
0.28
0.36
0.33
0.60
1.09
0.38 0.40 1.25 1.25
Current values per unit total area are given for arrays with different inter-electrode distances and for natural and forced convection (varying diffusion layer thickness). a In analogy to the behavior of a macroelectrode. b Values calculated from Eq. (3) using nk =4, Dk =1.8 · 10−5 cm2 −1 s and ck =2.7 · 10−7 mol cm−3 [14,15]. c Values given in [7] for a 10×10 microelectrode array with d= 20 mm and for a macroelectrode in quiescent and stirred samples, respectively.
calculated and observed current data for multiple microelectrode arrays are compared. The results document that the apparent current density of an array (per unit total surface area) increases with decreasing interelectrode distance d. However, the intrinsic current density (per unit active electrode area) decreases at the same time, and the influence of the flow-dependent parameter d is found to increase (see Table 1).
3. Chronoamperometric response to a potential change Single microelectrodes and macroelectrodes exhibit widely different current versus time responses in experiments where a voltage well above the respective halfwave potential is applied at the time t= 0 [11]: i(t)micro = i( )micro · [1+a0/(pDkt)1/2]
(4)
i(t)macro = i( )macro · [1+2% exp(− m 2p 2Dkt/d 2)]
(5)
m
where the current for t is given by Eqs. (1) and (2), respectively, and where m are all positive integers. Obviously, Eq. (4) is characterized by the time constant tmicro = a 20/pDk, whereas Eq. (5) depends on the time constant tmacro = d 2/p 2Dk. Since it holds that a0 d, it follows that microelectrodes yield a considerably faster response than macroelectrodes. The corresponding behavior of multiple microelectrode arrays turns out to be much more complicated [13]. Fortunately, the general result can be greatly simplified for systems with a low and a high packing density of microelectrodes, respectively. In these cases, the chronoamperometric response function becomes formally identical to the relationship given in Eqs. (4) and (5), respectively:
W.E. Morf, N.F. de Rooij / Sensors and Actuators B 44 (1997) 538–541
540
i(t)array =i( )array · [1 + a0/(pDkt)1/2]
(loosely packed) (6a)
i(t)array =i( )array · [1 + 2% exp( − m 2p 2Dkt/d 2)] m
(closely packed)
(6b)
where the steady-state current is given by Eqs. (3a) and (3b), respectively. Evidently, the inherent benefits of microelectrodes, namely their extremely fast response, can be exploited only in arrays where the inter-electrode distance is kept sufficiently large (see Fig. 2). Closely packed microelectrode arrays, on the other hand, show exactly the same behavior as macroelectrodes of the same total surface area. For intermediate systems (Fig. 2), the initial course of the chronoamperometric response follows an inverse square-root time dependence as in Eq. (6a), whereas the later current versus time profile is characterized by an exponential function similar to Eq. (6b) [13]. Due to their very fast response to potential changes, microelectrode arrays with d 2r0 tend to yield cyclic voltammograms that are nearly identical to the respective steady-state current versus voltage curves [7].
Fig. 3. Current vs. time response of arrays of hemispherical or disc-shaped microelectrodes to a sample concentration change at t =0. Dynamic response curves were calculated from theory [13] using the same parameters as in Fig. 2.
Di(t)= Di( ) · [1 + 2% (− 1)m exp(− m 2p 2Dkt/d 2)] m
(7) 4. Dynamic response to a concentration change Surprisingly, microelectrodes and macroelectrodes exhibit formally the same dynamic response to a concentration change in the sample bulk at t = 0 [11]:
where Di(t) is the current change obtained at the time t relative to the initial value, and Di( ) is the difference between the final and the initial steady-state current. According to a generalized theory [13], Eq. (7) is also valid for loosely packed or closely packed microelectrode arrays and, as an approximation, for intermediate systems (see Fig. 3). It appears that all these devices respond to sample changes with similar response times. A comparison of the results in Figs. 2 and 3 clearly shows that, for microelectrode arrays with relatively low packing densities, the current versus time response to sample changes is considerably slower than the chronoamperometric response observed after potential steps.
5. Conclusions Results for the steady-state current response and the dynamic characteristics of microelectrode arrays were summarized and discussed. As a rule, arrays with a low packing density of electrodes were found to yield the multiple response of single microelectrodes, whereas closely packed arrays were found to mimic the behavior of macroelectrodes of the same total area. Fig. 2. Current vs. time response of arrays of hemispherical or disc-shaped microelectrodes to a potential change at t =0. Chronoamperometric response curves were calculated from theory [13] using Dk =10 − 5 cm2 s − 1, a0 = 1 mm, d= 100 mm, and different values of N Ael/Atot. For arrays of the type depicted in Fig. 1, packing densities N Ael/Atot of 1, 0.01, 0.0004, and 0.0001 correspond to inter-electrode distances d of around 2, 24, 120, and 240 mm, respectively, depending slightly on the electrode geometry.
Acknowledgements The present work was supported in part by the Swiss Federal Commission for Technology and Innovation (KTI).
W.E. Morf, N.F. de Rooij / Sensors and Actuators B 44 (1997) 538–541
References [1] M.I. Montenegro, M.A. Queiros, J.L. Daschbach (eds.), Microelectrodes: Theory and Applications, Kluwer, Dordrecht, 1991. [2] M. Fleischmann, S. Pons, D.R. Rollison, P.P. Schmidt, Ultramicro-Electrodes, Datatech Systems Inc., Morgantown, NC, 1987. [3] C.M.A. Brett, A.M. Oliveira Brett, Electrochemistry: Principles, Methods and Applications, Oxford University Press, Oxford, 1993. [4] P. Arquint, M. Koudelka-Hep, N.F. de Rooij, H. Bu¨hler, W.E. Morf, J. Electroanal. Chem. 378 (1994) 177. [5] G.C. Fiaccabrino, M.-L. Tercier, J. Buffle, N.F. de Rooij and M. Koudelka-Hep, Proc. Transducers ’95, Eurosensors IX, Stockholm, 1995, vol. 2, p. 936. [6] B. Ross, K. Cammann, W. Mokwa, M. Rospert, Sensors and Actuators B 7 (1992) 758.
.
541
[7] H. Meyer, B. Naendorf, M. Wittkampf, B. Gru¨ndig, K. Cammann, R. Kakerow, Y. Manoli, W. Mokwa, M. Rospert, in A. van den Berg, P. Bergveld (eds.), Micro Total Analysis Systems, Kluwer, Dordrecht, 1995, p. 245. [8] J.L. Anderson, E.F. Bowden, P.G. Pickup, Anal. Chem. 68 (1996) 379R. [9] M.D. Ryan, E.F. Bowden, J.Q. Chambers, Anal. Chem. 66 (1994) 360R. [10] M.D. Ryan, J.Q. Chambers, Anal. Chem. 64 (1992) 79R. [11] W.E. Morf, N.F. de Rooij, Sensors and Actuators A 51 (1995) 89. [12] W.E. Morf, Anal. Chim. Acta 330 (1996) 139. [13] W.E. Morf, Anal. Chim. Acta 341 (1997) 121. [14] CRC Handbook of Chemistry and Physics, 68th edn., CRC Press, Boca Raton, FL, 1987. [15] J.C. Myland, K.B. Oldham, J. Electrochem. Soc. Electrochem. Sci. Technol. 131 (1985) 1815.
Performance of amperometric sensors based on multiple microelectrode arrays Werner E. Morf *, Nicolaas F. de Rooij Uni6ersity of Neuchaˆtel, Institute of Microtechnology, CH-2000 Neuchaˆtel, Switzerland Accepted 8 April 1997
Abstract The steady-state current response and the dynamic characteristics (response to potential or concentration changes) of multiple microelectrode arrays are discussed. As a rule, an array with a low packing density of electrodes yields the ideal multiple response of a single microelectrode. In contrast, a closely packed array simply mimics the behavior of a macroelectrode of the same total surface area. The theoretical results are confirmed by experimental data. © 1997 Elsevier Science S.A. Keywords: Microelectrode array; Steady-state current; Dynamic response
1. Introduction Microelectrodes [1] or ultramicroelectrodes [2] for amperometric and voltammetric measurements have found a considerable interest because these devices exhibit partly unique response characteristics. For example, they show a very fast current response to changes of the applied potential. In addition, the resulting steady-state signal is nearly independent of the thickness of the Nernstian stagnant layer, and is therefore not influenced by changes of the natural or forced convection in the sample solution [3]. On the other hand, the current output is very low for microelectrodes because of their extremely small size. The current signal of amperometric microsensors can be amplified by the design of multiple electrode arrays that consist of N identically constructed microelectrodes [4–7] (see Fig. 1). Ideally, these arrays should yield a current amplification by a factor N relative to a single microelectrode. It is also expected that the dynamic behavior of microelectrode arrays, e.g. the current versus time response after a potential change, should be comparable to that of a single electrode. In practice, however, these expectations are fulfilled only if certain requirements concerning the design of the microelectrode array are met (see below).
* Corresponding author. 0925-4005/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved. PII S 0 9 2 5 - 4 0 0 5 ( 9 7 ) 0 0 1 5 9 - 7
In the past, a series of theoretical papers dealt with the steady-state response and the dynamic behavior of microelectrode arrays (for a review, see [8–10]). Most of these studies did not account for convection influences in the sample solution, however, since no limitations concerning the maximum thickness of the
Fig. 1. Schematic representation of an array of hemispherical or disc-shaped micro-electrodes (top view, with specification of geometric parameters).
W.E. Morf, N.F. de Rooij / Sensors and Actuators B 44 (1997) 538–541
diffusion layer between sample bulk and electrode surface were imposed. Extended theories, accounting for the influence of the Nernstian stagnant layer, were offered only recently [11 – 13]. In this paper, we summarize and discuss results obtained for the steady-state current response and the dynamic characteristics of multiple microelectrode arrays.
2. Amperometric response at steady state The ideal current response of individual microelectrodes and of macroelectrodes, respectively, is given by [3,11,12]: i( )micro =FnkDkckAel/a0
(1)
i( )macro = FnkDkckAel/d
(2)
with a0 =r0 for hemispherical microelectrodes and a0 = (p/4) r0 for disc-shaped microelectrodes, where i( ) is the steady-state current of the indicated electrode, F is the Faraday constant, nk is the charge number, Dk the diffusion coefficient, and ck the bulk concentration of the analyte k, Ael is the active surface area of the electrode, d is the average thickness of the Nernstian diffusion layer between sample bulk and electrode, a0 is the apparent curvature radius of the electrode surface [11,12], and r0 is the actual radius of the electrode. For multiple microelectrode arrays, an intermediate behavior is found in general cases [11,12]: i( )array = FnkDkck /[(d/Atot)+{1 −(NAel/Atot)1/2}2 (a0/NAel)] : FnkDkckAtot/[d +(d −2r0)2 a0/Ael]
(3)
where Atot is the total surface area of an array of N microelectrodes (including active and inactive zones), and d=(Atot/N)1/2 is the mean distance between the centers of neighbouring electrodes. This result shows that the response behavior of arrays largely depends on the packing density of microelectrodes. Loosely packed arrays (d2r0) yield the expected current signal, which ideally corresponds to the N-fold output of a single microelectrode, whereas closely packed arrays (d: 2r0) simply mimic the response behavior of a macroelectrode of the same total area: i( )array =N · i( )micro =N · FnkDkckAel/a0 (loosely packed)
(3a)
i( )array =i( )macro =FnkDkckAtot/d (closely packed)
(3b)
An analysis of experimental results on the basis of Eq. (3) attests a very good agreement between theory and experiment. An example is given in Table 1 where
539
Table 1 Steady-state response of Pt micro-disc electrode arrays (electrode radius: r0 =1 mm) to oxygen in air-saturated solution at 25°C Current densities [i( )/Atot in mA cm−2] for arrays with d (mm) of:
Calc. (d= 50 mm)b Exp. (quiescent)c Calc. (d =15 mm)b Exp. (stirred)c
20
15
10
5
2a
0.14 0.12 0.19 0.18
0.20
0.28
0.36
0.33
0.60
1.09
0.38 0.40 1.25 1.25
Current values per unit total area are given for arrays with different inter-electrode distances and for natural and forced convection (varying diffusion layer thickness). a In analogy to the behavior of a macroelectrode. b Values calculated from Eq. (3) using nk =4, Dk =1.8 · 10−5 cm2 −1 s and ck =2.7 · 10−7 mol cm−3 [14,15]. c Values given in [7] for a 10×10 microelectrode array with d= 20 mm and for a macroelectrode in quiescent and stirred samples, respectively.
calculated and observed current data for multiple microelectrode arrays are compared. The results document that the apparent current density of an array (per unit total surface area) increases with decreasing interelectrode distance d. However, the intrinsic current density (per unit active electrode area) decreases at the same time, and the influence of the flow-dependent parameter d is found to increase (see Table 1).
3. Chronoamperometric response to a potential change Single microelectrodes and macroelectrodes exhibit widely different current versus time responses in experiments where a voltage well above the respective halfwave potential is applied at the time t= 0 [11]: i(t)micro = i( )micro · [1+a0/(pDkt)1/2]
(4)
i(t)macro = i( )macro · [1+2% exp(− m 2p 2Dkt/d 2)]
(5)
m
where the current for t is given by Eqs. (1) and (2), respectively, and where m are all positive integers. Obviously, Eq. (4) is characterized by the time constant tmicro = a 20/pDk, whereas Eq. (5) depends on the time constant tmacro = d 2/p 2Dk. Since it holds that a0 d, it follows that microelectrodes yield a considerably faster response than macroelectrodes. The corresponding behavior of multiple microelectrode arrays turns out to be much more complicated [13]. Fortunately, the general result can be greatly simplified for systems with a low and a high packing density of microelectrodes, respectively. In these cases, the chronoamperometric response function becomes formally identical to the relationship given in Eqs. (4) and (5), respectively:
W.E. Morf, N.F. de Rooij / Sensors and Actuators B 44 (1997) 538–541
540
i(t)array =i( )array · [1 + a0/(pDkt)1/2]
(loosely packed) (6a)
i(t)array =i( )array · [1 + 2% exp( − m 2p 2Dkt/d 2)] m
(closely packed)
(6b)
where the steady-state current is given by Eqs. (3a) and (3b), respectively. Evidently, the inherent benefits of microelectrodes, namely their extremely fast response, can be exploited only in arrays where the inter-electrode distance is kept sufficiently large (see Fig. 2). Closely packed microelectrode arrays, on the other hand, show exactly the same behavior as macroelectrodes of the same total surface area. For intermediate systems (Fig. 2), the initial course of the chronoamperometric response follows an inverse square-root time dependence as in Eq. (6a), whereas the later current versus time profile is characterized by an exponential function similar to Eq. (6b) [13]. Due to their very fast response to potential changes, microelectrode arrays with d 2r0 tend to yield cyclic voltammograms that are nearly identical to the respective steady-state current versus voltage curves [7].
Fig. 3. Current vs. time response of arrays of hemispherical or disc-shaped microelectrodes to a sample concentration change at t =0. Dynamic response curves were calculated from theory [13] using the same parameters as in Fig. 2.
Di(t)= Di( ) · [1 + 2% (− 1)m exp(− m 2p 2Dkt/d 2)] m
(7) 4. Dynamic response to a concentration change Surprisingly, microelectrodes and macroelectrodes exhibit formally the same dynamic response to a concentration change in the sample bulk at t = 0 [11]:
where Di(t) is the current change obtained at the time t relative to the initial value, and Di( ) is the difference between the final and the initial steady-state current. According to a generalized theory [13], Eq. (7) is also valid for loosely packed or closely packed microelectrode arrays and, as an approximation, for intermediate systems (see Fig. 3). It appears that all these devices respond to sample changes with similar response times. A comparison of the results in Figs. 2 and 3 clearly shows that, for microelectrode arrays with relatively low packing densities, the current versus time response to sample changes is considerably slower than the chronoamperometric response observed after potential steps.
5. Conclusions Results for the steady-state current response and the dynamic characteristics of microelectrode arrays were summarized and discussed. As a rule, arrays with a low packing density of electrodes were found to yield the multiple response of single microelectrodes, whereas closely packed arrays were found to mimic the behavior of macroelectrodes of the same total area. Fig. 2. Current vs. time response of arrays of hemispherical or disc-shaped microelectrodes to a potential change at t =0. Chronoamperometric response curves were calculated from theory [13] using Dk =10 − 5 cm2 s − 1, a0 = 1 mm, d= 100 mm, and different values of N Ael/Atot. For arrays of the type depicted in Fig. 1, packing densities N Ael/Atot of 1, 0.01, 0.0004, and 0.0001 correspond to inter-electrode distances d of around 2, 24, 120, and 240 mm, respectively, depending slightly on the electrode geometry.
Acknowledgements The present work was supported in part by the Swiss Federal Commission for Technology and Innovation (KTI).
W.E. Morf, N.F. de Rooij / Sensors and Actuators B 44 (1997) 538–541
References [1] M.I. Montenegro, M.A. Queiros, J.L. Daschbach (eds.), Microelectrodes: Theory and Applications, Kluwer, Dordrecht, 1991. [2] M. Fleischmann, S. Pons, D.R. Rollison, P.P. Schmidt, Ultramicro-Electrodes, Datatech Systems Inc., Morgantown, NC, 1987. [3] C.M.A. Brett, A.M. Oliveira Brett, Electrochemistry: Principles, Methods and Applications, Oxford University Press, Oxford, 1993. [4] P. Arquint, M. Koudelka-Hep, N.F. de Rooij, H. Bu¨hler, W.E. Morf, J. Electroanal. Chem. 378 (1994) 177. [5] G.C. Fiaccabrino, M.-L. Tercier, J. Buffle, N.F. de Rooij and M. Koudelka-Hep, Proc. Transducers ’95, Eurosensors IX, Stockholm, 1995, vol. 2, p. 936. [6] B. Ross, K. Cammann, W. Mokwa, M. Rospert, Sensors and Actuators B 7 (1992) 758.
.
541
[7] H. Meyer, B. Naendorf, M. Wittkampf, B. Gru¨ndig, K. Cammann, R. Kakerow, Y. Manoli, W. Mokwa, M. Rospert, in A. van den Berg, P. Bergveld (eds.), Micro Total Analysis Systems, Kluwer, Dordrecht, 1995, p. 245. [8] J.L. Anderson, E.F. Bowden, P.G. Pickup, Anal. Chem. 68 (1996) 379R. [9] M.D. Ryan, E.F. Bowden, J.Q. Chambers, Anal. Chem. 66 (1994) 360R. [10] M.D. Ryan, J.Q. Chambers, Anal. Chem. 64 (1992) 79R. [11] W.E. Morf, N.F. de Rooij, Sensors and Actuators A 51 (1995) 89. [12] W.E. Morf, Anal. Chim. Acta 330 (1996) 139. [13] W.E. Morf, Anal. Chim. Acta 341 (1997) 121. [14] CRC Handbook of Chemistry and Physics, 68th edn., CRC Press, Boca Raton, FL, 1987. [15] J.C. Myland, K.B. Oldham, J. Electrochem. Soc. Electrochem. Sci. Technol. 131 (1985) 1815.