Journal of
Electroanalytical Chemistry Journal of Electroanalytical Chemistry 596 (2006) 101–108 www.elsevier.com/locate/jelechem
The use of a new twin-electrode thin-layer cell to the study of homogeneous processes coupled to electrode reactions Thiago Regis Longo Cesar Paixa˜o a, Eduardo Mathias Richter b, Jose´ Geraldo Alves Brito-Neto a, Mauro Bertotti a,* a b
Instituto de Quı´mica, Universidade de Sa˜o Paulo, Sa˜o Paulo, SP 05508-900, Brazil Instituto de Quı´mica, Universidade Federal de Uberlaˆndia, Uberlaˆndia, MG, Brazil
Received 22 March 2006; received in revised form 19 June 2006; accepted 10 July 2006 Available online 5 September 2006
Abstract A twin-electrode thin-layer cell (TETLC) was fabricated by a simple and low-cost procedure based on the use of gold CD-Rs and toner masks, providing channels with heights in the range of tens of lm. In this TETLC the solution species is generated at one electrode and diffuses away to the other one. The TETLC was characterized for operation as a generator–collector device with collection efficiency values of 100%. The fast analyte regeneration ensures the attainment of steady-state conditions at very short times for a reversible electrochemical system such as the ferri/ferrocyanide couple. Owing to the fast transport within the thin-layer solution, electrochemical processes involving EC mechanisms can be investigated as demonstrated by using ascorbic acid as an irreversible model system. Digital simulation was used to relate current at both electrodes (generator and collector) to the gap, flow rate and rate constant for a typical EC mechanism. Appropriate working curves were constructed in the range 0–103 s1 at the experimental conditions employed in this study. 2006 Elsevier B.V. All rights reserved. Keywords: Electrochemical micro-device; Generator–collector; Toner masks; Gold electrodes
1. Introduction Electrochemical cells with multiple electrode configurations have been continuously devised with the attempt to enhance the sensitivity and selectivity of analytical determinations [1–11]. Accordingly, the capability of increasing the redox current for electrochemically reversible species owing to the redox cycling occurring at an adjacent electrode constitutes one of the most attractive aspects of closed-spaced electrodes [12–14]. At these cell configurations one of the electrodes is polarized at the limiting current potential (generator) and the potential of the other one (collector)
*
Corresponding author. Fax: +55 11 3815 5579. E-mail addresses:
[email protected] (T.R.L.C. Paixa˜o), mbertott@iq. usp.br (M. Bertotti). 0022-0728/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jelechem.2006.07.007
is fixed at a value corresponding to the reversible analyte regeneration. The situation is analogous to collection experiments performed with rotating ring-disc electrodes (RRDE), even though it should be pointed out that in this case current observed at the disc (generator) is not affected by the material produced at the ring (collector) [15]. The proximity of both electrodes in thin-layer cells gives rise to a redox cycling (or feedback effect [12–14]), which is characterized by the back diffusion of the electrogenerated analyte to the generator. Hence, current responses at both the generator and the collector electrodes reach larger values and steady-state situations are obtained. As expected, the extent of the feedback effect is dependent on the gap between generator and collector. We have recently reported the fabrication of a twin-electrode thin-layer cell (TETLC) by using recordable CDs and toner masks [16]. The gold electrodes fabricated with the
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CDs were separated by toner masks (spacers), generating structures with 12 lm gaps. Because of the very small space between both electrodes, collection efficiency values as high as 100% were obtained in voltammetric experiments performed with ferricyanide in KCl medium. The feedback effect and the coulometric operation of the electrochemical cell were characterized and some analytical potentialities of the TETLC were demonstrated. Charge transfer reactions involving organic molecules frequently are followed by homogeneous chemical steps. In order to extract quantitative information from experimental data, techniques such as cyclic voltammetry or chronoamperometry are typically used, current values being measured as a function of time [17,18]. The effectiveness of these techniques to investigate the chemical system depends on the time window of the experiment, i.e., the ability to obtain data at very short and very long times. Hence, we describe here the potentiality of the TETLC operating at quiescent conditions to characterize EC reactions. The possibility of using this fabricated device at flowing conditions are also presented envisaging applications in the determination of the rate of homogeneous reactions following electrode processes. 2. Experimental section 2.1. Chemicals All chemicals used were reagent grade. Ascorbic acid, potassium ferricyanide, potassium chloride, sodium acetate, sodium nitrite and acetic acid were purchased from Merck (Darmstadt, Germany). Solutions were prepared by dissolving the reagents in deionized water processed through a water purification system (Nanopure Infinity, Barnstead). The supporting electrolyte in hydrodynamic experiments consisted of a 0.5 mol L1 KCl solution. A solution containing 25 g L1 iodine and 100 g L1 sodium iodide was used for the gold layer etching. Acetonitrile was used to remove toner traces from the gold surfaces. 2.2. Electrodes and instrumentation Electrochemical measurements were carried out by means of an Autolab PGSTAT 30 (Eco Chemie) bipotentiostat with data acquisition software made available by the manufacturer (GPES 4.8 version). Experiments were done in the TETLC, a Ag/AgCl (saturated KCl) electrode and a platinum wire being used as reference and auxiliary electrodes, respectively. The toner masks were printed using a HP Laser Jet printer (1200 Series) and heat-transferred (100 C/1.5 min) with a thermal press (HT 2020, Ferragini, Sa˜o Carlos, SP, Brazil). The gold surfaces were obtained from a recordable compact disc [19] (Mitsui Gold Standard). Solutions were propelled to the electrolytic cell by using an aquarium air pump [20] in hydrody-
namic experiments. The air from the air pump was firstly bubbled into a water column connected to a closed bottle containing the carrier electrolyte. This artifice minimizes the flow pulsation, which is very important in devices with microchannels. The flow rate was controlled by varying the water column height. In order to estimate how fast the solution volume is replenished in the microchannel, flow values should be converted to flow velocities by considering that 1 lL min1 is equal to 0.14 cm s1 for the structure with 12 lm gap. 2.3. Fabrication of the twin-electrode thin-layer cell and scanning electron microscopic examination Recently, a new process for construction of a microdevice denominated twin-electrode thin-layer cell (TETLC) was developed in our laboratory [16]. This process stems from two previous ideas. The first one involves a pioneering method for constructing gold electrodes by using the thin gold film obtained from CD-Rs [19] and the second one for rapid production of single or multiple coplanar gold electrodes utilizing gold surfaces from CD-Rs and toner masks [21,22]. Briefly, the basic steps for the microdevice fabrication are the following: (1) twin gold electrodes were constructed by using a procedure already described in previous works [21,22]; (2) two orifices (solution inlet and outlet) with 0.8 mm of diameter are made in the CD polycarbonate slice where the top gold electrode is situated; (3) two toner masks are heat-transferred (100 C/1.5 min) to the polycarbonate slice containing the orifices and the top gold electrode; (4) a second piece of polycarbonate containing the other gold electrode is heatsealed (120 C for 2.5 min) on the first CD slice to produce the micro-structure. Each toner mask generates a micro channel with a thickness of 6 lm and as two toner masks were used to prepare the electrochemical cell, the unprinted areas generated microchannels with a depth (d) of approximately 12 lm; (5) finally, a micro tip and a reservoir (500 lL) are glued (epoxy glue) in the inlet and outlet sides, respectively. The same glue is used to reinforce the structure of the micro-device. Reference and auxiliary electrodes were positioned in the reservoir of the outlet side. The twin electrodes were positioned next to the outlet side (1 mm) in order to minimize the uncompensated resistance. The final dimensions of the device are the following ones: micro channel height and consequently gap between the electrodes (12 lm); micro channel width (1000 lm) and, electrodes width (500 lm). With these dimensions, the volume between the electrodes was calculated to be 6 nL (12 · 1000 · 500 lm). Fig. 1A and B shows schematic representations of the structure fabricated by the proposed pro- cedure. Microscopic examination of the gap between both gold electrodes in the twin-electrode thin-layer cell was performed with a Cambridge scanning electron microscope (Stereoscan 440) (Fig. 1C). A silver layer was sputtered on the device prior to the microscopic analysis.
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oci Di o2 ci ¼ ot D0 ox2
103
ð2Þ
Charge transfer at the electrode surfaces is assumed to be reversible, thus the relation between the concentrations of the oxidized and reduced species must always obey the Nernst equation for the prescribed potentials. Moreover, mass conservation requires that the diffusive fluxes of the oxidized and reduced species normal to the electrode surfaces cancel each other out. Eq. (2) was discretized using a finite difference scheme. The concentrations were sampled on a uniform grid along the solution layer and the Laplacian operator was discretized using central finite differences. The system of ordinary differential equations thus obtained was integrated in time using the 4th order Runge–Kutta method [23]. To evaluate the effect of uncompensated resistance when only one of the twin electrodes was used, we assumed that the effective potential on this electrode was given by Eeff ¼ E jAR
Fig. 1. Schematic representation (A) and perpendicular cut (B) of the TETLC. Micrographs of the perpendicular cut of the TETLC (C) fabricated with a different number of toner masks: one (left) and two (right) toner masks.
where E is the potential prescribed to the electrode, j is the current density, calculated from the gradients of the concentration distributions at the electrode surface, A is the electrode area and R is the solution resistance. It is assumed that j is positive when oxidation occurs at the electrode. Since j depends on the effective potential, Eq. (3) must be solved iteratively (by the bisection method) at each time step. The simulation program was implemented in C++ and compiled with the GNU C compiler [24]. The source code can be obtained by contacting us. 2.5. Stationary bidimensional model
2.4. Theoretical models 2.4.1. Time-dependent one-dimensional model In this model, we assume that the concentration distributions in the solution layer between the electrodes depend only on the distance from them. Given the small height of the solution layer if compared to the electrode area, one can conclude that this is a good approximation when there is no convective flow of solution from the reservoirs. We assume that the only mechanism available for the transport of electroactive species in the solution layer is diffusion, thus, the time evolution of the concentration distributions is given by oci o2 c i ¼ Di 2 ot ox
ð3Þ
ð1Þ
where ci is the concentration distribution of the ith chemical species being considered and Di is its diffusion coefficient. Following standard practice, Eq. (1) was made zero-dimensional by inserting the variables x* = x/d, t* = D0t/d2, and ci* = ci/c0, where d, D0, and c0 are a standard length, diffusion coefficient and concentration, respectively. This substitution results in the following equation:
The objective of this model is to evaluate the effect of a hydrodynamic flow on the behavior of the twin-electrode system. Moreover, we would also like to consider the effect of the consumption of the electroactive species by chemical reactions in the homogeneous phase. In this case, the concentration distributions must obey the following partial differential equation, already in non-dimensional variables: Di d d2 r r ci þ ~ ð4Þ vci þ k i ci ¼ 0 D0 D0 D0 where ~ v is the fluidic velocity field, ki is the rate constant of the pseudo first-order decomposition of the ith species, and $* denotes that the differentiation is carried out in respect to the zero-dimensional space coordinates. In this work, the convection velocity field is assumed to be parallel to the channel axis and that it has a parabolic profile (laminar flow) along the transverse coordinate. At the boundaries of the simulation domain corresponding to the reservoirs, the concentrations of all species are made constant. The fluxes of all species normal to the insulating borders are made equal to zero. At the electrode surfaces, the quotient between the concentration of oxidized
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and reduced species are made equal to that prescribed by the Nernst equation and their fluxes normal to the electrode surface are made such that they cancel each other out. Eq. (4) was solved by the finite element method. The simulation domain was discretized in an irregular mesh of triangles and the Galerkin method was used to obtain the linear system of algebraic equations to be solved for the concentrations of all species at the nodes of the mesh. A complete description of this standard method of numerical analysis is out of the scope of this paper, thus the reader is referred to the excellent material available about this topic [25]. The sparse linear system obtained was solved by the generalized minimum residual (GMRES) method using the incomplete LU decomposition of the matrix as preconditioner. The simulation program was written in C++ and compiled with the GNU C compiler. Its source coded can be obtained by contacting us. 3. Results and discussion 3.1. Description of the TETLC and its behavior in reversible electrochemical systems Fig. 1C shows micrographs of structures fabricated with different gaps separating both gold electrodes (named W1 and W2), the ability to control the channel height being possible by increasing the thickness of the toner layer. The distance between both electrodes (d) (around 6 and 12 lm by using one and two toner masks, respectively) affects the collection efficiency value when the device operates in the generator–collector mode, as it will be shown later in this article. To clearly understand the behavior of the TETLC when both electrodes are polarized, it is instructive to begin by studying a well-known system such as the ferri/ferrocyanide couple, which has relatively fast electron-transfer kinetics and is stable in both redox forms in aqueous medium. An experiment that resembles the one typically performed with RRDE consists in sweeping the potential of one of the electrodes and inspecting the material electrogenerated at the other electrode, polarized at a convenient potential where the reversible reaction takes place. By filling the gap of the TETLC with a ferricyanide solution, the equations that occur at W1 and W2 are as follows: FeðCNÞ3 6 4 FeðCNÞ6
þ e FeðCNÞ4 6 3
FeðCNÞ6 þ e
at W1 ðgeneratorÞ
ð5Þ
at W2 ðcollectorÞ
ð6Þ
Voltammograms shown in Fig. 2 represent the results of two independent experiments performed to confirm the proper working of both gold electrodes when they operate as generator and collector. Voltammetric curves represented by full lines exhibits the response of W1 and W2 in the first experiment and in this case the potential was scanned at W1 (generator, 0.5–0.0 V) and the material electrogenerated (ferrocyanide) was collected at W2, whose potential was maintained at 0.5 V (where the reaction
Fig. 2. Cyclic voltammograms recorded with the TETLC in a 1 KCl solution. Full lines represent 2:0 mmol L1 FeðCNÞ3 6 þ 0:5 mol L experiments performed by scanning the potential at W1 (0.5–0.0 V) and maintaining W2 at 0.5 V. Dashed lines are the result of another experiment where the potential at W1 was scanned again (0.5–0.0 V), but W2 was maintained at 0.0 V. Scan rate = 20 mV s1. Channel height = 12 lm.
shown in Eq. (6) takes place). A main conclusion that should be drawn from this experiment is related to the attainment of a steady-state condition explained by the fast turnover associated with both electrodic processes. This feedback effect also explains why current responses measured at W1 and W2 are the same, hence the collection efficiency value is found to be 1. A similar experiment was repeated by changing the potential of W2, i.e., the dashed line represents the voltammetry in the same chemical system where the potential was scanned again at W1 (0.5–0.0 V) but in this case W2 was polarized at 0.0 V instead of 0.5 V. As a consequence of the large area to volume ratio, the electroactive species is coulometrically reduced at W2 (Eq. (7)) in a few milliseconds, generating a ferrocyanide solution. At this experimental condition, W1 operates now as an anode, converting quantitatively the material electrogenerated at W2 (ferrocyanide) to ferricyanide (Eq. (8)). 3
4
FeðCNÞ6 þ e FeðCNÞ6
at W2 ðgeneratorÞ
ð7Þ
4 FeðCNÞ6
at W1 ðcollectorÞ
ð8Þ
3
FeðCNÞ6
þ e
The high symmetry among all voltammograms shown in Fig. 2 demonstrates that they possess comparable area and that the transport within the channel is very fast, restraining effects of the planar diffusion. The influence of the distance between W1 and W2 on the collection efficiency was investigated by digital simulation. Accordingly, Fig. 3 presents a set of chronoamperometric curves calculated by changing the gap between electrodes, W1 being the generator and W2 the collector. By looking at curves calculated for W1, it is possible to observe that the achievement of a steady-state condition is very fast when the gap is narrow, longer times being necessary when
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and 18 lm, respectively) yielded collection efficiency values close to unity (results not shown), reinforcing the digital simulation data presented in Fig. 4. No experimental data were obtained with structures containing larger gaps because of difficulties to sandwich a high number of toner masks. 3.2. The use of the TETLC to investigate irreversible systems and EC reactions
Fig. 3. Simulated chronoamperometric curves obtained by changing the gap between W1 and W2. Curves were simulated for a solution containing 1 2:0 mmol L1 FeðCNÞ3 KCl: a = 12, b = 24, c = 33 and 6 þ 0:5 mol L d = 48 lm. W1 (generator) = 0 V and W2 (collector) = 0.5 V.
the distance between W1 and W2 increases. Furthermore, the steady-state current is higher at the narrow gap, as expected because of the increased effectiveness of the turnover process occurring at the inter-electrode space. Results from the simulated experiment shown in the upper part of Fig. 3 demonstrate the influence of the gap on the time for electrogenerated ferrocyanide to reach W2. As a result, current response at this electrode starts to increase at longer times as the gap is made larger, lower steady-state responses being also noticed as a consequence of the diminished influence of the positive feedback. This also explains the decrease in the collection efficiency value as the inter-electrode space increases, as the theoretical values in Fig. 4 show. By analyzing this figure one can see that collection efficiency values close to 1 are obtained at gap values up to 30 lm. Chronoamperometric experiments performed with structures fabricated with two and three toner masks between both electrodes (nominal gap values of 12
Fig. 4. Collection efficiency values obtained from digital simulation at the TETLC as a function of the gap between W1 and W2. Curves were simulated for a solution containing 2:0 mmol L1 FeðCNÞ3 6 þ 0:5 mol L1 KCl.
The advantage of using the TETLC is entirely based on the possibility to regenerate quickly the electroactive species at the collector electrode, which then diffuses back to the generator at very short times. Obviously, this assumption is valid for reversible electrochemical systems such as the ferri/ferrocyanide couple, as previously demonstrated. On the other hand, nitrite is a well-studied anion whose irreversible anodic oxidation leads to nitrate in a 2-electron process [26], as Eq. (9) shows: þ NO 2 þ H2 O NO3 þ 2e þ 2H
ð9Þ
Fig. 5 presents results obtained with the TETLC by using nitrite as electroactive probe. Experiments were conducted at two different conditions, i.e., by leaving W2 at open-circuit conditions and by polarization at 0 V. The absence of current at W2 even when this electrode is potentiostated at a potential region much more negative than the one associated with the anodic process indicates the irreversibility of the electrodic reaction. Hence, no influence of W2 on the response at W1 is observed when this electrode is polarized (full and dashed lines are almost superimposed). Additionally, a steady-state condition similarly as the one obtained by using ferricyanide as a probe is not achieved, because of the absence of feedback.
Fig. 5. Cyclic voltammograms recorded with the TETLC in a 10.0 mmol L1 NaNO2 + 0.5 mol L1 Na2SO4 solution. Potential scanned at W1 and W2 maintained at open-circuit (dashed line) and at 0 V (full line). Scan rate = 20 mV s1. Channel height = 12 lm.
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The anodic oxidation of ascorbic acid is known to be a 2-electron irreversible process for experiments carried out at conventional time windows [27]. At these conditions ascorbic acid is oxidized to dehydroascorbic acid, Eq. (10), and the carbonyl groups of this acid can undergo hydration, becoming electrochemically inactive [28,29], Eq. (11). Nevertheless, at very short time scales the cathodic signal related to the reverse process can be observed. This can be accomplished, for instance, by performing the electrochemical experiment with microelectrodes at very fast scan rates [30], hence the influence of the following chemical step is minimized. HO
HO O HO HO
O
O
O
HO O
OH
+ 2H+ + 2e-
O
ð10Þ HO
HO O
O
HO O
O
O
O
+ 2H2O
O
HO
ð11Þ
HO OH
Because of the very rapid transit time between generator and collector electrodes, thin-layer cells such as the TETLC are well-suited to investigate homogeneous reactions associated with electrodic processes. Fig. 6 illustrates the electrochemical behavior of ascorbic acid at the TETLC. The inset of this figure shows results of a voltammetric experiment where W2 was maintained at open-circuit conditions, the almost total consumption of the electroactive species indicated by the quasi symmetric wave being explained by the coulometric oxidation at the thin-layer cell. Also, the absence of the cathodic signal confirms the irreversibility of the electrochemical process. On the other hand, a very distinct situation is observed when the device operates in the generator–collector mode, the cathodic signal response seen at W2 confirming that some material pro-
Fig. 6. Cyclic voltammograms recorded with the TETLC in a 1 mmol L1 ascorbic acid + 0.1 mol L1 HAc/NaAc solution. Potential scanned at W1 and W2 maintained at 0 V and at open-circuit (inset). Scan rate = 20 mV s1. Channel height = 12 lm. Inset shows a cyclic voltammogram with W2 at open-circuit.
duced at W1 reaches W2 before its chemical transformation. However, by measuring limiting current at both electrodes it is also possible to conclude that the collection efficiency value is found to be lower than unity, which is expected owing to the rapid kinetics involving the reaction of dehydroascorbic acid with the solvent, as reported in the literature [29]. 3.3. The operation of the TETLC at flowing conditions Another application envisaged for the proposed TETLC is based on the considerations above described involving both the effective consumption of an electroactive species at W1 and the relatively short time for the transport of the electrogenerated product to W2. Accordingly, if some chemical perturbation occurs during the diffusion to the collector, kinetic information is possible to be obtained by analyzing the effect of the chemical step on the collection efficiency. As the development of strategies to characterize coupled homogeneous reactions with increasingly faster kinetics constitutes a major challenge in electrochemistry, some preliminary considerations on this subject were done. To study chemical systems with such high rate constants, the achievement of enhanced and controlled mass-transport rates is needed [31]. To this end, several approaches have been used and the most promising ones associate microelectrodes with hydrodynamic voltammetric techniques [32,33]. At these experimental approaches masstransfer rates may compete with the reaction kinetics, allowing rate constants to be determined. For instance, Fig. 7 shows the results of experiments performed with
Fig. 7. Influence of the flow rate on the collection efficiency (N) measured 1 KCl solution. at the TETLC in a 2:0 mmol L1 FeðCNÞ3 6 þ 0:5 mol L Potential scanned at W1 in the range 0.5–0.0 V. W2 polarized at 0.5 V. Scan rate = 20 mV s1. The inset represents the digital simulation concentration profiles for a quiescent solution (a) and at flowing condition (1 lL min1) (b). W1 = 0 V and W2 = 0.5 V. Channel height = 12 lm.
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the TETLC in hydrodynamic conditions. By looking at the dependence of the collection efficiency (N) on the flow rate, a decrease of N value is observed owing to the competition between back diffusion to the collector and the convective transport imposed by the flowing solution. The clear establishment of a concentration gradient at quiescent solutions at the TETLC is observed in the simulated diagram shown in the inset of Fig. 7a, whereas a noticeable perturbation is observed at flowing conditions, especially in the region not comprised by W1 and W2 (Fig. 7b). As already discussed, the resulting steady-state flux in the TETLC is directly proportional to the diffusion coefficient of the electroactive species and inversely proportional to the length of the thin layer of solution separating the generator and the opposite electrode. Since in the TETLC this gap is very small, collection efficiency values reach 100% and only systems with very fast kinetics or at flowing conditions (as shown in Fig. 7) will make this value to be lower. Accordingly, Fig. 8 presents results for a digital simulation of current values measured at W2 calculated with a TETLC with gap = 12 lm as a function of the rate constant and at different flow rate values for a pseudo firstorder reaction in a typical EC electrode process as described below: A þ e B k
B!C
Electrochemical reaction ðat W1Þ
Homogeneous coupled reaction
B A þ e
Electrochemical reaction ðat W2Þ
ð12Þ ð13Þ ð14Þ
Working curves presented in Fig. 8 show that current at W2 decreases for chemical systems with high rate constants (the material is chemically consumed before it reaches W2) and at high flow rates (the material is convectively removed by the flowing solution). The absence of changes in the working curve profiles for k values higher than 103 s1 indicates that this superior limit defines the applicability of the TETLC for kinetic studies, at least with the stipulated cell
107
geometry. Electrode processes with faster coupled chemical reactions may also be investigated by using cells with smaller gaps. Accordingly, we are now undergoing studies in an attempt to reduce the gap between generator and collector electrodes. This would enhance significantly mass-transfer rates, contributing to the study of fast electrode kinetics. To this end, two main strategies are envisaged. The first one is based on the reduction of the thickness of the toner mask, which acts as an effective spacer. Other possibility refers to the electrodeposition of a metallic layer which can be thickened until a desired value, hence the interelectrode spacing is narrowed as already reported in the literature [12]. 4. Conclusions Research into low volume solutions confined in miniaturized electrochemical cells is becoming increasingly relevant. Accordingly, two potentiostated closely spaced electrodes constitute a powerful tool for monitoring the chemistry of systems involving reactions following the electron-transfer owing to the diffusion layer overlap. The ability to control the mass-transfer rate by varying the interelectrode separation and the solution flow rate ensures experimental conditions for competition with the homogeneous reaction. As a matter of fact, we have demonstrated in this study that the TETLC can be used as a new approach to investigate fast coupled chemical kinetics in electrodic processes under steady-state conditions. Acknowledgements Authors are thankful to FAPESP (Fundac¸a˜o de Amparo a` Pesquisa do Estado de Sa˜o Paulo) and CNPq (Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico) for the financial support. References
Fig. 8. Working curves simulated at various flow rate values for an EC electrochemical process with different rate constants: 0.1 (a), 0.2 (b), 0.5 (c), 1 (d), 2 (e), 5 (f), 10 (g), 20 (h), 50 (i), 100 (j), 200 (k), 500 (l) and 1000 s1 (m). Channel height = 12 lm.
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