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Simulation in electrochemistry using the finite element method part 2: scanning electrochemical microscopy
Electrochimica Acta 48 (2003) 3975 /3980 www.elsevier.com/locate/electacta
Simulation in electrochemistry using the finite element method part 2: scanning electrochemical microscopy Thomas Nann 1, Ju¨rgen Heinze * Institut fu ¨ r Physikalische Chemie, Universita ¨ t Freiburg, Albertstraße 21, D-79104 Freiburg, Germany Received 3 December 2002; received in revised form 26 February 2003; accepted 19 April 2003
Abstract The previously introduced adaptive finite element (AFE) algorithm for use in electrochemistry is applied to the simulation of selected multidimensional problems: steady state simulation, chronoamperometric simulation, cyclic voltammetry at microelectrodes, and simulation of arbitrarily shaped scanning electrochemical microscope (SECM) tips. It is shown that the algorithm is suitable for this kind of problems and can be easily extended to the simulation of many types of electrochemical experiments. # 2003 Elsevier Ltd. All rights reserved. Keywords: Simulation; SECM; Adaptive finite element method
1. Introduction Electrochemical experiments can be modeled mathematically by Fick’s second law with suitable initial- and boundary conditions. Boundary conditions for bulk homogeneous reactions and the heterogeneous chargetransfer reactions have to be taken into account. In case of the examples discussed in this study, the homogeneous boundary conditions represent the chemical reactions, while the non-homogeneous boundary conditions take into account the conditions at the phase boundaries, including the electron transfer reactions. Since the advent of digital computers, numerical solutions of these equations have been used to interpret electrochemical experiments. Cyclic voltammetric experiments in particular have been evaluated by simulation for many years [1 /4]. This was possible because, among other things, cyclic voltammetric experiments can be easily simplified to one dimension when conducted at planar (macroscopic) electrodes. In this case, the differential equations can be solved by finite
* Corresponding author. Tel.: /49-761-203-6202; fax: /49-761203-6237. E-mail address: [email protected] (J. Heinze). 1 Present address: Freiburg Materials Research Center (FMF), Stefan-Meier-Str. 21, D-79104 Freiburg, Germany. 0013-4686/03/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0013-4686(03)00312-8
difference methods. Recently, commercial computer programs for this purpose have become available (cf. [5 /7]). In recent years, both electrochemistry and numerical mathematics made substantial progress. In electrochemistry, new methods have been developed, of which the scanning electrochemical microscope (SECM), introduced by Engstro¨m and Bard [8,9], is one of the most important. This new method requires the simulation of multidimensional problems. Simulations are needed not only to interpret experimental data, but also to improve tip-geometries and other experimental conditions (e.g. the mediator). There have been many attempts to simulate the electron-flux over an SECM-tip or related problems. Besides Finite Element approaches with rectangular elements and fixed grids [11 /13] and adaptive grids [14 /16], further research has been done on boundary element methods [17 /21], analytical approximations [22 /25] and well-known finite difference algorithms, such as the ADI algorithm [26 /29]. Finally, some groups have used an integral equation approach to solve the problem [30 /32]. Meanwhile, numerical methods have been developed to solve multidimensional problems easily. The Finite Element Method (FEM) is particularly popular because of its versatility. Commercial packets are already avail-
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able, but usually fail to meet the special needs of electrochemistry*/particularly the modeling of connected heterogeneous reactions. Part 1 of this series introduced an adaptive finite element (AFE) algorithm for electrochemical simulation [33]. Ref. [33] describes all the computational details of the algorithm. The application of this program to electrochemical experiments will be demonstrated in the following. Chronoamperometric model predictions at microelectrodes and SECM-experiments will be discussed, and special attention is given to the further development of SECMexperiments and equipment.
with equation (1). Within the AFE algorithm, the final accuracy is directly related to the simulation termination error. Therefore, the accuracy can be defined arbitrarily before the simulation. Thus, the results obtained with the AFE algorithm are significantly better then the results obtained with a fixed grid. The positive feedback at a disc electrode with RG / 100 and stot /c */10 2 has been simulated at different distances L from the substrate. Fig. 1(a) shows a comparison between simulated positive feedback data and the analytic formula as presented in [9]:
i 2. Results and discussion In the following, it is shown that the AFE algorithm is suitable for the simulation of many electrochemical problems. The SECM is an excellent method to demonstrate this. We will discuss examples based on classical steady state measurements, kinetic measurements (chronoamperometry), cyclic voltammetry at microelectrodes, and electrochemistry at arbitrarily shaped electrodes. On this basis, it should be possible to adapt the algorithm to any electrochemical model.
it i
pr0 4d
p 4
L1
(2)
where d is the distance between the tip apex and the substrate and L /d/r0 the dimensionless distance. The simulation shows good agreement with the theory given for L B/0.1. However, Ref. [9] also suggests that as L
2.1. Steady state simulation Essentially, three cases have to be considered in the case of steady state simulations of the SECM experiment: the diffusion limited steady state current, the positive feedback current, and the negative feedback current. The diffusion limited steady state current is of particular interest, because the currents in SECM experiments are usually normalized against the steady state current of the tip electrode. The analytical solution of the diffusion limited steady state current for disc microelectrodes is well established: i 4nFDcr0
(1)
where n is the number of electrons, F Faraday’s Constant, D the diffusion coefficient of the species in question, c * the bulk concentration, and r0 the radius of the electrode. Because this case has already been discussed in detail in another publication [34], the results will be presented very briefly. In a first step, the AFE algorithm is compared with recently published simulation algorithms for accuracy [11]. The comparison of calculation durations is difficult, because these simulation programs are not publicly available. Nevertheless, it is obvious that with an increasing number of nodes, i.e. increasing accuracy, the computational effort increases. The simulation of a disc electrode with rglass /r0 /RG /100 and a simulation termination error of stot /c */102 (cf. [33,35] for details) reaches an accuracy of 99.94% in comparison
Fig. 1. (a) Simulation of a positive feedback at a micro disc electrode with RG/100 (boxes) in comparison with analytic formula (2) (dashed line). (b) Negative feedback simulation with RG/100 and analytic approximation according to equation (3). Initial concentrations were 10 mM, diffusion coefficients 10 5 cm2 s 1; all other variables in dimensionless forms according to [6].
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tends to infinity the current should approach unity. The same is true for the negative feedback. An analytic formula for the approximation of the negative feedback current is also given in [10]: i
it i
p
d
2 ln(rglass =r0 ) r0
p 2 ln(RG)
L
(3)
with all parameters defined as above. The comparison of formula (3) with the simulated data for a disc electrode (RG /100, stot /c */102) is depicted in Fig. 1(b). In the range of ‘‘normal’’ experimental conditions a good agreement can be observed. For shorter distances (approximately L B/101) the simulated currents are lower than expected. At the moment, we do not have experimental confirmation of this effect. One possible explanation is that the linear approximation (3) does not hold for very small distances, because the situation resembles that of a thin-layer cell. Hence, we call this effect ‘‘thin-layer loss’’. 2.2. Simulations of chronoamperometric experiments As second example of the application of the AFE algorithm in electrochemistry, chronoamperometric simulations of SECM experiments are discussed. These simulations were motivated by the observation that the steady state current is reached faster in the case of a positive feedback then in the case of a negative feedback. Hence digital simulation supports the physical model. In case of a positive feedback, the concentration gradients of the participating reactants are completely located between the tip’s apex and the substrate. So the time needed to reach the steady state should be very small. Fig. 2(a) shows a chronoamperometric simulation of a positive feedback at a disc electrode with the same parameters as in Section 2.1. A quantitative discussion of the time needed to reach the steady state current should take into account the diffusion coefficients of the mediators and the distance between tip and substrate. Fig. 2(b) shows the simulation of a negative feedback with the same parameters as in Fig. 2(a). As can be seen, the negative feedback takes much longer to reach the steady state. This is in agreement with the experimental findings and theoretical predictions. It was observed that the AFE algorithm shows oscillations at the beginning of the simulation, when the time grid was not fine enough. This problem could be solved in a future version of the simulation program by introducing an adaptive time grid. 2.3. Cyclic voltammetry at micro disc electrodes Simulations of cyclic voltammetric experiments at disc microelectrodes are normally carried our using the
Fig. 2. (a) Chronoamperometric simulation of a positive feedback at a micro disc electrode; (b) chronoamperometric simulation of a negative feedback. All simulations with RG /100 and L/2; all other parameters like in Fig. 1. Dashed lines, steady state currents according to (2) and (3).
model of a hemispherical electrode. The application of a hemispherical electrode considerably facilitates the computational effort because a one-dimensional grid with r as spherical variable is sufficient. However, as shown by Amatore et al. [36], such simplifications are only legitimate in some specific cases: In a first step, the electrode geometries are projected to a unique fictitious space; in a second step, the boundary conditions of the fictitious space are compared for various reaction mechanisms. Only if these boundary conditions are independent of the electrode shape, can simplifications to one-dimensional systems be made. Agreement of the cyclic voltammograms of a discand a hemispherical electrode is predicted for a Nernstian electron transfer in the absence of homogeneous kinetics. Both of these cases have been simulated using our AFE-technique for disc-electrodes and the commercial program DigiSim† for hemispherical electrodes. Fig. 3(a) depicts the results of the simulations of an E-
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Amatore et al. distinguishes between four further electrochemical mechanisms. All these mechanisms have been simulated with the AFE and the Finite Difference Method (DigiSim† ) and it was found that equivalence of the voltammograms can only observed for fast heterogeneous charge transfer reactions and additional fast homogeneous kinetics taking place far from the electrode surface. In conclusion it can be stated that the considerations of Amatore et al. have been validated by simulation and that the AFE-method allows the simulation of any mechanism at disc electrodes, which cannot be performed by a one-dimensional algorithm. 2.4. Arbitrarily shaped tip electrodes
Fig. 3. (a) Simulation of a single charge transfer reaction (k0 /104 cm s 1, v/100 mV s 1) at a micro disc electrode with RG /100, calculated with the AFE method (solid line) and a commercial Finite Difference package (DigiSim† ) for a hemispherical electrode (dashed line) with the same scaling factor s/A /d . (b) The same calculations as (a) for a CE-mechanism (Homogeneous chemical reaction: kf /300 s 1, K /0.1; heterogeneous charge transfer: k0 /104 cm s 1). All other parameters like in Fig. 1.
mechanism with k0 /104 cm s 1. As expected a good agreement is observed. For a heterogeneous charge transfer reaction with a fast preceding homogeneous chemical reaction no equivalence of the voltammograms is expected. Fig. 3(b) shows the calculations for a CE-mechanism with kf /300 s 1, K /0.1 for the chemical reaction and k0 / 104 cm s 1 for the following charge transfer reaction. It is observed that the limiting current density of the hemispherical electrode is smaller than that of the disc electrode. This can also be understood intuitively, because the electroactive material available in the volume fraction around the electrodes has more ‘‘surface’’ available in the hemispherical case then at the disc electrode and the preceding chemical reaction introduces a ‘‘non-linearity’’ in the proportions.
A substantial advantage of the AFE method is its ability to simulate arbitrary geometrical conditions. In connection with the SECM, there is a special interest in the simulation of new tip geometries [37]. These investigations can be motivated by considerations concerning process engineering and imaging mechanisms (enhancement of the resolution). Three cases will be discussed in the following: hemispherical electrodes, conical electrodes, and capillary electrodes. Because of the prevalent interest in capillary electrodes, this case will be discussed in greater detail in connection with the feedback behavior of these electrodes. Fig. 4(a) shows the triangular grid, adapted to a hemispherical tip electrode, and Fig. 4(b) the isoconcentration lines calculated for a positive feedback. Figs. 5 and 6 depict the simulation of the same situation for a conical and a capillary electrode. In all three cases it is shown that the grid adapts well to the electrode geometry, where critical regions (e.g. the tip-apex of the sheath electrode transition) are discretized more accurately than regions somewhere in the bulk. According to
Fig. 4. Positive feedback simulation of a spherical SECM-tip with RG /2.0 and L/1.0; all other parameters like in Fig. 1. (a) Adapted grid; (b) isoconcentration lines of the reduced species. The grey areas in scheme represent the electrode material, while the white areas depict the glass sheath, respectively, insulation.
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Fig. 5. Positive feedback simulation of a conical tip electrode with RG /2.0, L/1.0 and the apex angle a/908; all other parameters like in Fig. 1. (a) Adapted grid; (b) isoconcentration lines of the reduced species. Grey and white areas have the same meanings as in Fig. 4.
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balance between too small an h , leading to nonreproducible signals, and too big an h , leading to nondetectable signals. The strategy to find this optimal capillary depth is straightforward. The current density is calculated along the metal coating inside the electrode under steady-state conditions. In the steady state all the substance inside the capillary has reacted, so the whole flux comes from diffusion-effects at the apex. Related parameters, such as distances, modes, etc., are permuted and the current contributions are analyzed. The simulated current density data is evaluated by setting a termination contribution of 0.1% of the total current. Under this condition, an optimal capillary depth of h/39/0.5 times the capillary radius results. The second question concerning simulation was how capillary electrodes behave in the feedback mode of the SECM. Fig. 7 shows the comparison between disc electrodes and capillary electrodes. In the case of a positive feedback (Fig. 7(a)) there is a significant difference between disc electrodes and capillary electro-
Fig. 6. Positive feedback simulation of a capillary tip electrode with RG /4.0, L/2.0 and h/2.0; all other parameters like in Fig. 1. (a) Adapted grid; (b) isoconcentration lines of the reduced species. Grey and white areas have the same meanings as in Fig. 4.
i nFAD
dc dx
(4)
the feedback current for these cases can be calculated from the isoconcentration lines. Therefore, the normals of the concentration gradient at the electrode are used as dc /dx in equation (4). It is possible to make extremely small capillary electrodes, which makes them interesting candidates for new SECM-tips. The properties of these electrodes have been studied through simulation. The first question concerning simulation is the optimal capillary depth h (cf. Fig. 6(a)). The capillary depth is the distance between the lower apex of the capillary and the inner bottom of the insulation or the end of the metal coating, respectively, as shown in Fig. 6. The inside of the capillary needs to be insulated to prevent the signal at the apex of the capillary being overshadowed by the signal from ‘‘inside’’ the electrode in potentiodynamic experiments. One has to find a
Fig. 7. Comparison between capillary electrodes (squares, RG /100, h/3; all other parameters like in Fig. 1) and disc electrodes (crosses, RG /100). (a) Simulation of a positive feedback situation; (b) simulation of a negative feedback.
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des. Although the increase of the current is smaller than at disc electrodes, it should be sufficient to be experimentally detectable. The negative feedback behavior of disc electrodes and capillary electrodes (Fig. 7(b)) is very similar. In conclusion, it is expected that capillary electrodes give satisfactory signals in the SECM feedback mode. An optimal capillary depth of h /3 times the capillary radius should be reached.
2.5. The algorithm, accuracy, and efficiency The present paper is intended as the follow-up of part 1 (Ref. [33]), which describes the algorithm in detail. The interested reader is, therefore, referred to Ref. [33,35] for details about the algorithm. A crucial question for every simulation algorithm is the balance between accuracy, efficiency, and complexity. A major advantage of Finite Element algorithms is the ability to realize complex models. This advantage is bought with an increased computational effort. FEM algorithms with fixed grids have limited accuracy. In case of an adaptive grid, the accuracy can be adjusted externally (cf. Section 2.1). With increasing accuracy, more nodes are needed and the computational cost increases. It has been shown that the AFE algorithm enables the simulation of very complex electrochemical situations, but for each problem it is necessary to weigh up the complexity of the problem versus the efficiency of the algorithm. The AFE algorithm is only adaptive concerning the spatial grid in the present version. Therefore, it is necessary to ensure that the time grid is fine enough to prevent oscillations in the case of chronoamperometric simulations.
3. Conclusion It has been shown that the AFE algorithm can be adapted for the simulation of many electrochemical processes. Examples were presented for steady state simulations, chronoamperometric simulations, cyclic voltammetry at microelectrodes, and arbitrarily shaped SECM tip electrodes. All these examples showed the high suitability and flexibility of the algorithm for problems of this type. Although simulation is a well-established method for interpreting cyclic voltammograms, it is nearly impossible to deconvolute SECM data with this technique. The solution of the inverse SECM problem is a challenge for the future.
Acknowledgements The authors wish to thank the Deutsche Forschungsgemeinschaft (DFG) for their generous support of the research discussed in this paper.
References [1] B. Speiser, in: A.J. Bard, I. Rubinstein (Eds.), Electroanalytical Chemistry, vol. 19, Marcel Dekker, New York, 1996, pp. 1 /108. [2] D. Britz, Digital Simulation in Electrochemistry, 2nd ed, Springer, Berlin, 1988. [3] A.J. Bard, L.R. Faulkner, Electrochemical Methods, Wiley, New York, 1980. [4] R.S. Nicholson, I. Shain, Anal. Chem. 36 (1964) 706. [5] M. Rudolph, J. Electroanal. Chem. 314 (1991) 13. [6] M. Rudolph, J. Electroanal. Chem. 338 (1992) 85. [7] M. Rudolph, D.P. Reddy, S.W. Feldberg, Anal. Chem. 66 (1994) 589A. [8] R.C. Engstro¨m, M. Weber, D.J. Wunder, R. Burgess, S. Winquist, Anal. Chem. 58 (1986) 844. [9] A.J. Bard, F.-R.F. Fan, J. Kwak, O. Lev, Anal. Chem. 61 (1989) 132. [10] K. Borgwarth, Oberfla¨chenanalyse und -strukturierung mit Hilfe des Rasterelektrochemischen Mikroskops und des Rastertunnelmikroskops, Shaker Verlag, Aachen, 1996. [11] J. Kwak, A.J. Bard, Anal. Chem. 61 (1989) 1221. [12] N.P.C. Stevens, S.J. Hickey, A.C. Fisher, Ann. Quim. 93 (1997) 225. [13] N.P.C. Stevens, A.C. Fisher, J. Phys. Chem. 101 (1997) 8259. [14] K. Harriman, D.J. Gavaghan, P. Houston, E. Su¨li, Electrochem. Commun. 2 (2000) 150. [15] K. Harriman, D.J. Gavaghan, P. Houston, E. Su¨li, Electrochem. Commun. 2 (2000) 157. [16] K. Harriman, D.J. Gavaghan, P. Houston, E. Su¨li, Electrochem. Commun. 2 (2000) 163. [17] F. Qiu, A.C. Fisher, G. Denuault, J. Phys. Chem. B 103 (1999) 4387. [18] F. Qiu, A.C. Fisher, G. Denuault, J. Phys. Chem. B 103 (1999) 4393. [19] F. Qiu, N.A. Williams, A.C. Fisher, Elec. Commun. 1 (1999) 124. [20] F. Qiu, A.C. Fisher, J. Phys. Chem. B 102 (1998) 9647. [21] F.L. Qiu, A.C. Fisher, Elec. Commun. 3 (2001) 117. [22] J. Galceran, J. Cecilia, E. Companys, J. Salvador, J. Puy, J. Phys. Chem. B 104 (2000) 7993. [23] D.O. Wipf, A.J. Bard, J. Electrochem. Soc. 138 (1991) 469. [24] M.V. Mirkin, A.J. Bard, Anal. Chem. 64 (1992) 2293. [25] M.V. Mirkin, F.-R.F. Fan, A.J. Bard, J. Electroanal. Chem. 328 (1992) 47. [26] P.R. Unwin, A.J. Bard, J. Phys. Chem. 95 (1991) 7814. [27] F. Zhou, P.R. Unwin, A.J. Bard, J. Phys. Chem. 96 (1992) 4917. [28] K.A. Ellis, M.D. Pritzker, T.Z. Fahidy, Anal. Chem. 67 (1995) 4500. [29] Y. Selzer, D. Mandler, Anal. Chem. 72 (11) (2000) 2383. [30] A.J. Bard, G. Denuault, R.A. Friesner, B.C. Dornblaser, L.S. Tuckerman, Anal. Chem. 63 (1991) 1282. [31] M.V. Mirkin, A.J. Bard, J. Electroanal. Chem. 323 (1992) 1. [32] M.V. Mirkin, A.J. Bard, J. Electroanal. Chem. 323 (1992) 29. [33] T. Nann, J. Heinze, Electrochem. Commun. 1 (1999) 289. [34] R. Ferrigno, P.F. Brevet, H.H. Girault, Electrochim. Acta 42 (1997) 1895. [35] T. Nann, Digitale Simulation in der Elektrochemie mit der Methode der Finiten Elemente, Shaker Verlag, Aachen, 1997. [36] C. Amatore, B. Fosset, Anal. Chem. 68 (1996) 4377. [37] Y. Lee, S. Amemiya, A.J. Bard, Anal. Chem. 73 (10) (2001) 2261.
Simulation in electrochemistry using the finite element method part 2: scanning electrochemical microscopy Thomas Nann 1, Ju¨rgen Heinze * Institut fu ¨ r Physikalische Chemie, Universita ¨ t Freiburg, Albertstraße 21, D-79104 Freiburg, Germany Received 3 December 2002; received in revised form 26 February 2003; accepted 19 April 2003
Abstract The previously introduced adaptive finite element (AFE) algorithm for use in electrochemistry is applied to the simulation of selected multidimensional problems: steady state simulation, chronoamperometric simulation, cyclic voltammetry at microelectrodes, and simulation of arbitrarily shaped scanning electrochemical microscope (SECM) tips. It is shown that the algorithm is suitable for this kind of problems and can be easily extended to the simulation of many types of electrochemical experiments. # 2003 Elsevier Ltd. All rights reserved. Keywords: Simulation; SECM; Adaptive finite element method
1. Introduction Electrochemical experiments can be modeled mathematically by Fick’s second law with suitable initial- and boundary conditions. Boundary conditions for bulk homogeneous reactions and the heterogeneous chargetransfer reactions have to be taken into account. In case of the examples discussed in this study, the homogeneous boundary conditions represent the chemical reactions, while the non-homogeneous boundary conditions take into account the conditions at the phase boundaries, including the electron transfer reactions. Since the advent of digital computers, numerical solutions of these equations have been used to interpret electrochemical experiments. Cyclic voltammetric experiments in particular have been evaluated by simulation for many years [1 /4]. This was possible because, among other things, cyclic voltammetric experiments can be easily simplified to one dimension when conducted at planar (macroscopic) electrodes. In this case, the differential equations can be solved by finite
* Corresponding author. Tel.: /49-761-203-6202; fax: /49-761203-6237. E-mail address: [email protected] (J. Heinze). 1 Present address: Freiburg Materials Research Center (FMF), Stefan-Meier-Str. 21, D-79104 Freiburg, Germany. 0013-4686/03/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0013-4686(03)00312-8
difference methods. Recently, commercial computer programs for this purpose have become available (cf. [5 /7]). In recent years, both electrochemistry and numerical mathematics made substantial progress. In electrochemistry, new methods have been developed, of which the scanning electrochemical microscope (SECM), introduced by Engstro¨m and Bard [8,9], is one of the most important. This new method requires the simulation of multidimensional problems. Simulations are needed not only to interpret experimental data, but also to improve tip-geometries and other experimental conditions (e.g. the mediator). There have been many attempts to simulate the electron-flux over an SECM-tip or related problems. Besides Finite Element approaches with rectangular elements and fixed grids [11 /13] and adaptive grids [14 /16], further research has been done on boundary element methods [17 /21], analytical approximations [22 /25] and well-known finite difference algorithms, such as the ADI algorithm [26 /29]. Finally, some groups have used an integral equation approach to solve the problem [30 /32]. Meanwhile, numerical methods have been developed to solve multidimensional problems easily. The Finite Element Method (FEM) is particularly popular because of its versatility. Commercial packets are already avail-
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T. Nann, J. Heinze / Electrochimica Acta 48 (2003) 3975 /3980
able, but usually fail to meet the special needs of electrochemistry*/particularly the modeling of connected heterogeneous reactions. Part 1 of this series introduced an adaptive finite element (AFE) algorithm for electrochemical simulation [33]. Ref. [33] describes all the computational details of the algorithm. The application of this program to electrochemical experiments will be demonstrated in the following. Chronoamperometric model predictions at microelectrodes and SECM-experiments will be discussed, and special attention is given to the further development of SECMexperiments and equipment.
with equation (1). Within the AFE algorithm, the final accuracy is directly related to the simulation termination error. Therefore, the accuracy can be defined arbitrarily before the simulation. Thus, the results obtained with the AFE algorithm are significantly better then the results obtained with a fixed grid. The positive feedback at a disc electrode with RG / 100 and stot /c */10 2 has been simulated at different distances L from the substrate. Fig. 1(a) shows a comparison between simulated positive feedback data and the analytic formula as presented in [9]:
i 2. Results and discussion In the following, it is shown that the AFE algorithm is suitable for the simulation of many electrochemical problems. The SECM is an excellent method to demonstrate this. We will discuss examples based on classical steady state measurements, kinetic measurements (chronoamperometry), cyclic voltammetry at microelectrodes, and electrochemistry at arbitrarily shaped electrodes. On this basis, it should be possible to adapt the algorithm to any electrochemical model.
it i
pr0 4d
p 4
L1
(2)
where d is the distance between the tip apex and the substrate and L /d/r0 the dimensionless distance. The simulation shows good agreement with the theory given for L B/0.1. However, Ref. [9] also suggests that as L
2.1. Steady state simulation Essentially, three cases have to be considered in the case of steady state simulations of the SECM experiment: the diffusion limited steady state current, the positive feedback current, and the negative feedback current. The diffusion limited steady state current is of particular interest, because the currents in SECM experiments are usually normalized against the steady state current of the tip electrode. The analytical solution of the diffusion limited steady state current for disc microelectrodes is well established: i 4nFDcr0
(1)
where n is the number of electrons, F Faraday’s Constant, D the diffusion coefficient of the species in question, c * the bulk concentration, and r0 the radius of the electrode. Because this case has already been discussed in detail in another publication [34], the results will be presented very briefly. In a first step, the AFE algorithm is compared with recently published simulation algorithms for accuracy [11]. The comparison of calculation durations is difficult, because these simulation programs are not publicly available. Nevertheless, it is obvious that with an increasing number of nodes, i.e. increasing accuracy, the computational effort increases. The simulation of a disc electrode with rglass /r0 /RG /100 and a simulation termination error of stot /c */102 (cf. [33,35] for details) reaches an accuracy of 99.94% in comparison
Fig. 1. (a) Simulation of a positive feedback at a micro disc electrode with RG/100 (boxes) in comparison with analytic formula (2) (dashed line). (b) Negative feedback simulation with RG/100 and analytic approximation according to equation (3). Initial concentrations were 10 mM, diffusion coefficients 10 5 cm2 s 1; all other variables in dimensionless forms according to [6].
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tends to infinity the current should approach unity. The same is true for the negative feedback. An analytic formula for the approximation of the negative feedback current is also given in [10]: i
it i
p
d
2 ln(rglass =r0 ) r0
p 2 ln(RG)
L
(3)
with all parameters defined as above. The comparison of formula (3) with the simulated data for a disc electrode (RG /100, stot /c */102) is depicted in Fig. 1(b). In the range of ‘‘normal’’ experimental conditions a good agreement can be observed. For shorter distances (approximately L B/101) the simulated currents are lower than expected. At the moment, we do not have experimental confirmation of this effect. One possible explanation is that the linear approximation (3) does not hold for very small distances, because the situation resembles that of a thin-layer cell. Hence, we call this effect ‘‘thin-layer loss’’. 2.2. Simulations of chronoamperometric experiments As second example of the application of the AFE algorithm in electrochemistry, chronoamperometric simulations of SECM experiments are discussed. These simulations were motivated by the observation that the steady state current is reached faster in the case of a positive feedback then in the case of a negative feedback. Hence digital simulation supports the physical model. In case of a positive feedback, the concentration gradients of the participating reactants are completely located between the tip’s apex and the substrate. So the time needed to reach the steady state should be very small. Fig. 2(a) shows a chronoamperometric simulation of a positive feedback at a disc electrode with the same parameters as in Section 2.1. A quantitative discussion of the time needed to reach the steady state current should take into account the diffusion coefficients of the mediators and the distance between tip and substrate. Fig. 2(b) shows the simulation of a negative feedback with the same parameters as in Fig. 2(a). As can be seen, the negative feedback takes much longer to reach the steady state. This is in agreement with the experimental findings and theoretical predictions. It was observed that the AFE algorithm shows oscillations at the beginning of the simulation, when the time grid was not fine enough. This problem could be solved in a future version of the simulation program by introducing an adaptive time grid. 2.3. Cyclic voltammetry at micro disc electrodes Simulations of cyclic voltammetric experiments at disc microelectrodes are normally carried our using the
Fig. 2. (a) Chronoamperometric simulation of a positive feedback at a micro disc electrode; (b) chronoamperometric simulation of a negative feedback. All simulations with RG /100 and L/2; all other parameters like in Fig. 1. Dashed lines, steady state currents according to (2) and (3).
model of a hemispherical electrode. The application of a hemispherical electrode considerably facilitates the computational effort because a one-dimensional grid with r as spherical variable is sufficient. However, as shown by Amatore et al. [36], such simplifications are only legitimate in some specific cases: In a first step, the electrode geometries are projected to a unique fictitious space; in a second step, the boundary conditions of the fictitious space are compared for various reaction mechanisms. Only if these boundary conditions are independent of the electrode shape, can simplifications to one-dimensional systems be made. Agreement of the cyclic voltammograms of a discand a hemispherical electrode is predicted for a Nernstian electron transfer in the absence of homogeneous kinetics. Both of these cases have been simulated using our AFE-technique for disc-electrodes and the commercial program DigiSim† for hemispherical electrodes. Fig. 3(a) depicts the results of the simulations of an E-
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Amatore et al. distinguishes between four further electrochemical mechanisms. All these mechanisms have been simulated with the AFE and the Finite Difference Method (DigiSim† ) and it was found that equivalence of the voltammograms can only observed for fast heterogeneous charge transfer reactions and additional fast homogeneous kinetics taking place far from the electrode surface. In conclusion it can be stated that the considerations of Amatore et al. have been validated by simulation and that the AFE-method allows the simulation of any mechanism at disc electrodes, which cannot be performed by a one-dimensional algorithm. 2.4. Arbitrarily shaped tip electrodes
Fig. 3. (a) Simulation of a single charge transfer reaction (k0 /104 cm s 1, v/100 mV s 1) at a micro disc electrode with RG /100, calculated with the AFE method (solid line) and a commercial Finite Difference package (DigiSim† ) for a hemispherical electrode (dashed line) with the same scaling factor s/A /d . (b) The same calculations as (a) for a CE-mechanism (Homogeneous chemical reaction: kf /300 s 1, K /0.1; heterogeneous charge transfer: k0 /104 cm s 1). All other parameters like in Fig. 1.
mechanism with k0 /104 cm s 1. As expected a good agreement is observed. For a heterogeneous charge transfer reaction with a fast preceding homogeneous chemical reaction no equivalence of the voltammograms is expected. Fig. 3(b) shows the calculations for a CE-mechanism with kf /300 s 1, K /0.1 for the chemical reaction and k0 / 104 cm s 1 for the following charge transfer reaction. It is observed that the limiting current density of the hemispherical electrode is smaller than that of the disc electrode. This can also be understood intuitively, because the electroactive material available in the volume fraction around the electrodes has more ‘‘surface’’ available in the hemispherical case then at the disc electrode and the preceding chemical reaction introduces a ‘‘non-linearity’’ in the proportions.
A substantial advantage of the AFE method is its ability to simulate arbitrary geometrical conditions. In connection with the SECM, there is a special interest in the simulation of new tip geometries [37]. These investigations can be motivated by considerations concerning process engineering and imaging mechanisms (enhancement of the resolution). Three cases will be discussed in the following: hemispherical electrodes, conical electrodes, and capillary electrodes. Because of the prevalent interest in capillary electrodes, this case will be discussed in greater detail in connection with the feedback behavior of these electrodes. Fig. 4(a) shows the triangular grid, adapted to a hemispherical tip electrode, and Fig. 4(b) the isoconcentration lines calculated for a positive feedback. Figs. 5 and 6 depict the simulation of the same situation for a conical and a capillary electrode. In all three cases it is shown that the grid adapts well to the electrode geometry, where critical regions (e.g. the tip-apex of the sheath electrode transition) are discretized more accurately than regions somewhere in the bulk. According to
Fig. 4. Positive feedback simulation of a spherical SECM-tip with RG /2.0 and L/1.0; all other parameters like in Fig. 1. (a) Adapted grid; (b) isoconcentration lines of the reduced species. The grey areas in scheme represent the electrode material, while the white areas depict the glass sheath, respectively, insulation.
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Fig. 5. Positive feedback simulation of a conical tip electrode with RG /2.0, L/1.0 and the apex angle a/908; all other parameters like in Fig. 1. (a) Adapted grid; (b) isoconcentration lines of the reduced species. Grey and white areas have the same meanings as in Fig. 4.
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balance between too small an h , leading to nonreproducible signals, and too big an h , leading to nondetectable signals. The strategy to find this optimal capillary depth is straightforward. The current density is calculated along the metal coating inside the electrode under steady-state conditions. In the steady state all the substance inside the capillary has reacted, so the whole flux comes from diffusion-effects at the apex. Related parameters, such as distances, modes, etc., are permuted and the current contributions are analyzed. The simulated current density data is evaluated by setting a termination contribution of 0.1% of the total current. Under this condition, an optimal capillary depth of h/39/0.5 times the capillary radius results. The second question concerning simulation was how capillary electrodes behave in the feedback mode of the SECM. Fig. 7 shows the comparison between disc electrodes and capillary electrodes. In the case of a positive feedback (Fig. 7(a)) there is a significant difference between disc electrodes and capillary electro-
Fig. 6. Positive feedback simulation of a capillary tip electrode with RG /4.0, L/2.0 and h/2.0; all other parameters like in Fig. 1. (a) Adapted grid; (b) isoconcentration lines of the reduced species. Grey and white areas have the same meanings as in Fig. 4.
i nFAD
dc dx
(4)
the feedback current for these cases can be calculated from the isoconcentration lines. Therefore, the normals of the concentration gradient at the electrode are used as dc /dx in equation (4). It is possible to make extremely small capillary electrodes, which makes them interesting candidates for new SECM-tips. The properties of these electrodes have been studied through simulation. The first question concerning simulation is the optimal capillary depth h (cf. Fig. 6(a)). The capillary depth is the distance between the lower apex of the capillary and the inner bottom of the insulation or the end of the metal coating, respectively, as shown in Fig. 6. The inside of the capillary needs to be insulated to prevent the signal at the apex of the capillary being overshadowed by the signal from ‘‘inside’’ the electrode in potentiodynamic experiments. One has to find a
Fig. 7. Comparison between capillary electrodes (squares, RG /100, h/3; all other parameters like in Fig. 1) and disc electrodes (crosses, RG /100). (a) Simulation of a positive feedback situation; (b) simulation of a negative feedback.
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des. Although the increase of the current is smaller than at disc electrodes, it should be sufficient to be experimentally detectable. The negative feedback behavior of disc electrodes and capillary electrodes (Fig. 7(b)) is very similar. In conclusion, it is expected that capillary electrodes give satisfactory signals in the SECM feedback mode. An optimal capillary depth of h /3 times the capillary radius should be reached.
2.5. The algorithm, accuracy, and efficiency The present paper is intended as the follow-up of part 1 (Ref. [33]), which describes the algorithm in detail. The interested reader is, therefore, referred to Ref. [33,35] for details about the algorithm. A crucial question for every simulation algorithm is the balance between accuracy, efficiency, and complexity. A major advantage of Finite Element algorithms is the ability to realize complex models. This advantage is bought with an increased computational effort. FEM algorithms with fixed grids have limited accuracy. In case of an adaptive grid, the accuracy can be adjusted externally (cf. Section 2.1). With increasing accuracy, more nodes are needed and the computational cost increases. It has been shown that the AFE algorithm enables the simulation of very complex electrochemical situations, but for each problem it is necessary to weigh up the complexity of the problem versus the efficiency of the algorithm. The AFE algorithm is only adaptive concerning the spatial grid in the present version. Therefore, it is necessary to ensure that the time grid is fine enough to prevent oscillations in the case of chronoamperometric simulations.
3. Conclusion It has been shown that the AFE algorithm can be adapted for the simulation of many electrochemical processes. Examples were presented for steady state simulations, chronoamperometric simulations, cyclic voltammetry at microelectrodes, and arbitrarily shaped SECM tip electrodes. All these examples showed the high suitability and flexibility of the algorithm for problems of this type. Although simulation is a well-established method for interpreting cyclic voltammograms, it is nearly impossible to deconvolute SECM data with this technique. The solution of the inverse SECM problem is a challenge for the future.
Acknowledgements The authors wish to thank the Deutsche Forschungsgemeinschaft (DFG) for their generous support of the research discussed in this paper.
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