Electrochimica Acta 54 (2009) 3298–3302
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Electrochemical detection of l-cysteine using a boron-doped carbon nanotube-modified electrode Chunyan Deng, Jinhua Chen ∗ , Xiaoli Chen, Mengdong Wang, Zhou Nie ∗ , Shouzhuo Yao State Key Laboratory of Chemo/Biosensing and Chemometrics, College of Chemistry and Chemical Engineering, Hunan University, Lushan Road, Changsha, 410082, PR China
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Article history: Received 6 September 2008 Received in revised form 11 December 2008 Accepted 25 December 2008 Available online 31 December 2008 Keywords: Electrochemical detection l-Cysteine Boron-doped carbon nanotubes Edge plane sites Glassy carbon
a b s t r a c t A boron-doped carbon nanotube (BCNT)-modified glassy carbon (GC) electrode was constructed for the detection of l-cysteine (L-CySH). The electrochemical behavior of BCNTs in response to l-cysteine oxidation was investigated. The response current of L-CySH oxidation at the BCNT/GC electrode was obviously higher than that at the bare GC electrode or the CNT/GC electrode. This finding may be ascribed to the excellent electrochemical properties of the BCNT/GC electrode. Moreover, on the basis of this finding, a determination of L-CySH at the BCNT/GC electrode was carried out. The effects of pH, scan rate and interferents on the response of L-CySH oxidation were investigated. Under the optimum experimental conditions, the detection response for L-CySH on the BCNT/GC electrode was fast (within 7 s). It was found to be linear from 7.8 × 10−7 to 2 × 10−4 M (r = 0.998), with a high sensitivity of 25.3 ± 1.2 nA mM−1 and a low detection limit of 0.26 ± 0.01 M. The BCNT/GC electrode exhibited high stability and good resistance against interference by other oxidizable amino acids (tryptophan and tyrosine). © 2008 Elsevier Ltd. All rights reserved.
1. Introduction The study of thiols (e.g., homocysteine, l-cysteine, and glutathione) provides critical insight into the proper physiological function and diagnosis of disease states. This is because the thiol compounds are of special significance in biochemistry and environmental chemistry [1–3]. l-Cysteine (L-CySH) is a sulfur-containing ␣-amino acid. It is one of about 20 amino acids commonly found in natural proteins [4]. It plays an important role in biological systems and has been used widely in the medicine and food industries [5]. Moreover, the coupling of l-cystine/l-cysteine is used as a model of the disulfide bond. It also serves as a model for the thiol group of proteins in a variety of biological media [4,6]. Considering this, effort has been taken to develop sensitive methods for the detection of L-CySH in bodily fluids, pharmaceuticals, and food samples [7]. In the past, electrochemical methods have been the most sensitive/favored methods for the determination of thiols (and of cysteine in particular) [6,8]. This is because they are easily automated, have a high sensitivity, and are capable of being readily integrated with other techniques for multi-analysis [9,10]. However, the electrochemical method for analysis of thiols remains very challenging. For example, at the conventional electrodes (e.g., glassy carbon and gold electrodes), the electrochemical response of thiols is not satisfactory due to their sluggish electrochemical processes [8,11–13]. High overpotentials of elec-
∗ Corresponding authors. Tel.: +86 731 8821961; fax: +86 731 8821848. E-mail address:
[email protected] (J. Chen). 0013-4686/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.electacta.2008.12.045
trochemical oxidation are needed for thiols at the conventional electrodes. This results in surface oxide formation and fouling of these electrodes [7,14]. In order to overcome these drawbacks, many strategies have been employed. These strategies include the use of mercury electrodes, diamond electrodes [4,15,16], enzyme-based biosensors [3,9,17] and chemically modified electrodes [2,18,19]. Although the analysis of thiols has been improved greatly, there are still some disadvantages that hinder thiol detection. These challenges include low sensitivity, complexity and expense. Considering all this, it is critical to find a suitable material for the effective and simple determination of thiols. In recent years, novel nanomaterial (foreign atoms, B or N atom) doped carbon nanotubes have been prepared and used for many applications [20–23]. This interest has stemmed from the finding that the electrochemical properties of CNTs can be improved by doping with foreign atoms [22–26]. In this paper, the electroanalysis of thiols (l-cysteine as a model) at a boron-doped carbon nanotube (BCNT)-modified glassy carbon (GC) electrode (BCNT/GC) was investigated. The borondoping of CNTs introduces more edge plane sites on the surface of CNTs and may thus lead to more facile electron transfer [27,28] because edge plane sites are the predominant sites of electron transfer [27–30]. It is expected that this system will be particularly attractive for electrochemical applications. In particular, it may be useful in electrocatalysis and electrochemical determinations [31,32]. On the other hand, more functional groups were produced at the surface of BCNTs. This finding is useful in resisting the fouling of the modified electrodes. The functional groups at the defective sites of CNTs may block the adsorption of species onto the CNTs [32,33]. Using the BCNT/GC electrode for electroanalysis of L-CySH
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eliminates fouling of the electrodes. The intrinsic properties of the BCNT allow this resolution. Moreover, the BCNT/GC electrode shows excellent performance in that it is sensitive, exhibits a rapid response, and has good long-term stability. These features should prove favorable in the practical analysis of thiols. 2. Experimental 2.1. Apparatus and reagents Electrochemical measurements were performed on a CHI660A electrochemical workstation (Chenhua Instrument Company of Shanghai, China) with a conventional three-electrode cell. A BCNT/GC electrode (3 mm in diameter) was used as the working electrode. A saturated calomel electrode (SCE) and a platinum wire were used as the reference and auxiliary electrodes, respectively. All measurements were carried out in a phosphate buffer solution (PBS, 0.067 M, pH 7.4) at room temperature (25 ± 2 ◦ C). All the potentials in this paper were referenced to the SCE. Non-doped carbon nanotubes with about 95% purity were obtained from Shenzhen Nanotech Port Company. Further purification was accomplished by sonication in a mixture of concentrated sulfuric acid and nitric acid (3:1, v/v) for about 3 h. The treated CNTs were then rinsed with double-distilled water several times. Finally, they were filtered and dried in an oven. The resulting black powders were sonicated in double-distilled water for about 1 h at a concentration of 0.5 mg mL−1 . Boron-doped carbon nanotubes were synthesized as described in the references [21,34]. The content of boron in the BCNTs was estimated to be ca. 2 wt.% by Inductively Coupled Plasma-Atom Emission Spectroscopy (ICP-AES). BCNTs were treated the same as the undoped CNTs. The concentration of the aqueous suspension of the BCNTs was also 0.5 mg mL−1 . L-CySH, tryptophan and tyrosine were purchased from Sinopharm Chemical Reagent Co., Ltd. All other chemicals were of analytical grade. Double-distilled water was used throughout. 2.2. Electrode preparation The GC electrode was successively polished to a mirror finish using 1.0- and 0.3-m alumina slurry followed by a thorough rinse with doubly distilled water. The electrode was pretreated electrochemically in 0.5 M H2 SO4 aqueous solution by potential cycling. The potential was ranged from −0.15 to 1.0 V at a scan rate of 50 mV s−1 until a stable cyclic voltammogram was obtained. The procedure for the preparation of the BCNT/GC electrode is described here. Five microliters of BCNT aqueous suspension was dropped on the surface of the pretreated GC electrode. It was dried under an infrared lamp to form the BCNT-modified electrode. For comparison, the CNT/GC electrode was also prepared with the same procedure. 3. Results and discussion 3.1. Electrooxidation of L-CySH at the BCNT/GC electrode Fig. 1 presents the cyclic voltammograms of the bare GC (A), CNT/GC (B), and BCNT/GC (C) electrodes in 0.067 M phosphate buffer solution (PBS, pH 7.4) with and without 2 mM L-CySH at a scan rate of 50 mV s−1 . It is clear that the background current of the BCNT/GC electrode is higher than that of the CNT/GC and GC electrodes. This is due to the rough surface of the BCNT caused by boron doping [35]. On the other hand, the oxidation peak of L-CySH can be observed at ca. +0.50 V using the bare GC electrode and ca. +0.47 V using the CNT/GC and BCNT/GC electrodes. The peak potential of L-CySH oxidation using the CNT/GC and BCNT/GC elec-
Fig. 1. Cyclic voltammograms of the bare GC (A), CNT/GC (B) and BCNT/GC (C) electrodes in 0.067 M phosphate buffer (PBS, pH 7.4) without (a) and with (b) 2 mM L-CySH. Scan rate, 50 mV s−1 .
trodes is lower than that obtained from the bare GC electrode. There was a reduction in the overpotential of L-CySH oxidation to 30 mV. The oxidation peak current of L-CySH using the BCNT/GC electrode was much higher than that of the GC and CNT/GC electrodes. This implies that the oxidation of L-CySH at the BCNT/GC electrode is significantly improved. This improvement can be ascribed to the superior electrochemical properties of the BCNT/GC electrode. The properties are a result of the high proportion of edge plane sites and oxygen-functional groups presented on the BCNT’s surface, as well as the high conductivity introduced by the boron-doping [22–26,27,30,36]. Based on this, it is expected that sensitive electrodetection of L-CySH can be achieved using the BCNT/GC electrode. Upon reverse sweep, no reduction waves of L-CySH were observed using the above electrodes in the potential range studied. This finding is consistent with the oxidized species of cysteine undergoing a chemically irreversible reaction [37,38]. 3.2. Effect of scan rate The effect of the scan rate on the cyclic voltammetric performance during L-CySH oxidation at the BCNT/GC electrode was also investigated. As shown in Fig. 2A, the peak current increased linearly with the square root of the scan rate. This finding indicates that the oxidation reaction is a diffusion-controlled process. Fig. 2B shows the relation between the peak potential of L-CySH oxidation at the BCNT/GC electrode and log . The peak potential of L-CySH oxidation shifts negatively in response to an increase in scan rate (from 0.01 to 0.50 V s−1 ). The resulting regression equation is Ep (V) = 0.62 + 0.096 log . Since it is an irreversible oxidation process, the peak potential (Ep ) can be represented by the following equation [39,40]: Ep = A +
2.3RT log (1 − ˛)nF
A is a constant related to the formal electrode potential (E0 ) and the standard rate constant of E0 . The transfer coefficient (˛) characterizes the effect of the electrochemical potential on the activation energy of an electrochemical reaction. ˛ was calculated to be
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Fig. 2. (A) A plot of peak current against the square root of the scan rate (1/2 ) for L-CySH oxidation at the BCNT/GC electrode. (B) A plot of the peak potential of L-CySH oxidation at the BCNT/GC electrode as a function of the logarithm of scan rate (log ). The L-CySH concentration was 2 mM in 0.067 M PBS (pH 7.4); the range of scan rates was 10–500 mV s−1 .
Fig. 3. Plots of the oxidation peak potential (A) and oxidation peak current (B) of 2 mM L-CySH at the BCNT/GC electrode at different pH values. The scan rate was 50 mV s−1 .
0.69 for the BCNT/GC electrode, according to the slope (0.096) of Ep ∼ log . The value of the transfer coefficient (˛ = 0.69) was lower than that of the Pt/graphite (0.8) and Pt/CNT (0.74) electrodes [6]. This indicates that the oxidation of L-CySH occurred more easily using the BCNT/GC electrode than using the Pt/graphite and Pt/CNT electrodes. 3.3. Effect of pH The electrochemical oxidation of L-CySH on a solid electrode may proceed by the following mechanism [14]: L-CySH ↔ L-CyS− + H+
(1)
L-CyS− → L-CyS∗ads + e−
(2)
2L-CyS∗ads → L-CySSCy
(3)
Therefore, the pH value of the solution influences the electrochemical behavior of L-CySH at the BCNT/GC electrode. Fig. 3 shows the variation in peak potentials and currents during L-CySH oxidation at the BCNT/GC electrode as a function of pH. It is clear that the LCySH oxidation peak shifts negatively with an increase in pH (curve A in Fig. 3). This finding may be ascribed to the fact that alkaline conditions deprotonate the sulfur moiety of the L-CySH species. This makes the thiol moiety oxidize more effectively [7,37]. The variations in the cyclic voltammetric currents at the different pH values were analyzed. It is clear from curve B in Fig. 3 that a maximum in peak current can be observed at pH 7.4. Considering this and the physiological conditions, pH 7.4 was chosen as the most suitable pH value for investigating L-CySH oxidation. This value was used for subsequent experiments.
3.4. Effect of interferents The major interferents in L-CySH determination are tryptophan and tyrosine. Based on this, their electrochemical behavior at the BCNT/GC electrode was investigated. Fig. 4 shows the cyclic voltammograms of 0.1 mM tyrosine (curve b) and 0.1 mM tryptophan (curve c) in 0.067 M PBS at the BCNT/GC electrode. It is clear that both tryptophan and tyrosine are oxidized at the potential of +0.64 V. The oxidation potentials of tryptophan and tyrosine are higher than that of L-CySH (+0.47 V) at the BCNT/GC electrode. This situation may favor the elimination of interference from the oxidizable species (tryptophan and tyrosine) in the oxidation process of L-CySH. We decided to further verify the effect of interferents on the current response of L-CySH. Therefore, we investigated the
Fig. 4. Cyclic voltammograms of the BCNT/GC electrode in 0.067 M blank PBS (a), 0.1 mM tyrosine (b), and 0.1 mM tryptophan (c). The scan rate was 50 mV s−1 .
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Fig. 6. Amperometric response to different concentrations of L-CySH obtained at the bare GC (a), CNT/GC (b) and BCNT/GC (c) electrodes in 0.067 M PBS. The applied potential was +0.47 V.
Fig. 5. Current response of the BCNT/GC electrode to the addition (indicted by the arrow) of 20 M L-CySH (a), 20 M tryptophan (b) and 20 M tyrosine (c) in 0.067 M PBS (pH 7.4) at different applied potentials. (A) +0.60 V, (B) +0.47 V and (C) +0.35 V.
typical current responses of the BCNT/GC electrode to the two oxidizable amino acids at different operation potentials (Fig. 5). From Fig. 5A, the current response to the addition of 20 M tryptophan is at a potential of +0.6 V. This finding may lead to interference in the determination of L-CySH. However, when the applied potential is fixed at +0.47 V (Fig. 5B) or +0.35 V (Fig. 5C), the current response of the BCNT/GC electrode to 20 M tryptophan (curve b) and tyrosine (curve c) is almost negligible compared to the oxidation of 20 M L-CySH (curve a). This implies that the interference from these oxidizable amino acids can be avoided by controlling the applied potential. In order to heighten the current response and lower the interference, +0.47 V was chosen as the applied potential for the detection of L-CySH. 3.5. Analytical properties Based on the above experimental results, the typical amperometric responses of the different modified electrodes were examined. The electrodes were examined in response to different concentrations of L-CySH at the potential of +0.47 V. The corresponding results are shown in Fig. 6. It is clear that the bare GC electrode (curve a) displays a very low current response to the addition of L-CySH compared to the CNT/GC and BCNT/GC electrodes. This may be due to sluggish electron-transfer of L-CySH at the bare GC electrode [8,11,12]. It is significant that the BCNT/GC electrode displays an amplified current response compared to the CNT/GC electrode (about threefold higher than that of the CNTs/GC electrode). These results are consistent with those in Fig. 1. The response of the BCNT/GC electrode in the detection of L-CySH is fast (within 7 s). The calibration plot for L-CySH detection at the BCNTs/GC electrode was investigated under optimum experimental conditions. The corresponding results are shown in Fig. 7. Curve c in the inset of Fig. 7 shows the linear calibration curve for the detection of L-CySH using the BCNT/GC electrode. The linear range is from 7.8 × 10−7 to 2 × 10−4 M (r = 0.998) with a sensitivity of 25.3 ± 1.20 nA mM−1 . The detection limit was 0.26 ± 0.01 M (S/N = 3). These findings indicate
that the BCNT/GC electrode is more sensitive for the detection of LCySH than those reported in the literature. For example, Moore et al. [36] studied the electrocatalytic detection of L-CySH using an edge plane pyrolytic graphite electrode with a detection limit of 2.6 M. Maleki et al. [7] investigated the electrocatalysis of L-CySH oxidation at a carbon ionic liquid electrode and found a detection limit of 2 M. The linear calibration curves obtained at the bare GC (a) and CNT/GC (b) electrodes are also shown in the inset plot of Fig. 7. The sensitivity of L-CySH detection is 10.2 ± 0.56 nA mM−1 using the CNT/GC electrode and 1.16 ± 0.07 nA mM−1 using the GC electrode. These values are lower than that of the BCNT/GC electrode (25.3 ± 1.20 nA mM−1 ). These results demonstrate that the BCNT/GC electrode is a good alternative for the determination of L-CySH since it has a high sensitivity and good selectivity. The operational stability of the BCNT/GC electrode was assessed using five amperometric measurements of 2 mM L-CySH in PBS at +0.47 V. The relative standard deviation obtained was 4.3%. In addition, five freshly prepared BCNT-modified electrodes were used to measure 2 mM L-CySH in PBS. All five electrodes exhibited similar amperometric responses and a relative standard deviation of 3.5% was obtained. These findings indicate that the electrochemical behavior of the BCNT/GC electrode is highly reproducible. The long-term stability of the BCNT/GC electrode was also investigated. It retained 90% of its initial current response after 3 weeks. In an example of a practical use, the BCNT/GC electrode was used to detect L-CySH in syrup (NongFu-Spring, Hunan, China). The recovery of L-CySH was confirmed by standard addition. The
Fig. 7. Calibration curve of the current response to the different concentrations of L-CySH obtained at the BCNT/GC electrode in 0.067 M PBS (pH 7.4). The applied potential was +0.47 V. The inset plot shows the linear calibration curves obtained at the bare GC (a), CNT/GC (b) and BCNT/GC (c) electrodes.
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Table 1 Electrochemical determination of L-CySH in the presence of syrup. Sample
Amount added (M)
Amount found (M) (average of five times)
Nong-Fu syrup
1 5 10 50 100
0.96 4.98 10.2 47.5 97.4
± ± ± ± ±
0.04 0.25 0.5 2.8 2.4
Recovery (%) 96 99.6 102 95 97.4
corresponding results are presented in Table 1. The recovery for 10 samples was found to be in the range of 92–102%. This result suggests that the BCNT/GC electrode is very reliable and sensitive in the determination of L-CySH. 4. Conclusions The electrochemical determination of L-CySH using a BCNT/GC electrode was investigated. The BCNT/GC electrode shows excellent electrochemical properties. These properties are due to more edge plane sites and functional groups on the surface of BCNTs. The BCNT/GC electrode shows striking analytical properties, such as high sensitivity, a low detection limit, good stability and a resistance to interference. These characteristics demonstrate that the BCNT/GC electrode should be suitable for the determination of L-CySH in biological systems or in post-separation detection protocols. BCNT/GC electrodes could provide new platforms for the electroanalysis of thiols. Acknowledgments This work was financially supported by NSFC (20675027, 20575019, 20335020), “973” Program of China (2006CB600903) and the SRF for ROCS, SEM, China (2001-498). References [1] [2] [3] [4]
M.L. Hung, D.M. Stanbury, Inorg. Chem. 44 (2005) 3541. A.K.M. Kafi, F. Yin, H.-K. Shin, Y.-S. Kwon, Curr. Appl. Phys. 7 (2007) 496. T. Inoue, J.R. Kirchhoff, Anal. Chem. 72 (2000) 5755. N. Spãtaru, B.V. Sarada, E. Papa, D.A. Tryk, A. Fujishima, Anal. Chem. 73 (2001) 514.
[5] P.R. Lima, W.J.R. Santos, R.C.S. Luz, F.S. Damos, A.B. Oliveira, M.O.F. Goulart, L.T. Kubota, J. Electroanal. Chem. 612 (2008) 87. [6] S.D. Fei, J.H. Chen, S.Z. Yao, G.H. Deng, D.L. He, Y.F. Kuang, Anal. Biochem. 339 (2005) 29. [7] N. Maleki, A. Safavi, F. Sedaghati, F. Tajabadi, Anal. Biochem. 369 (2007) 149. [8] J.A. Reyanud, B. Maltoy, P. Canessan, J. Electroanal. Chem. 114 (1980) 195. [9] T. Inoue, J.R. Kirchhoff, Anal. Chem. 74 (2002) 1349. [10] J. Lakritz, C.G. Plopper, A.R. Buckpitt, Anal. Biochem. 247 (1997) 63. [11] D.L. Rabenstein, G.T. Yamashita, Anal. Biochem. 180 (1989) 259. [12] J.L. D’Eramo, A.E. Finkelstein, F.Q. Boccazzi, O. Fridman, J. Chromatogr. B 720 (1998) 205. [13] P.J. Vandeberg, D.C. Johnson, Anal. Chem. 65 (1993) 2713. [14] T.R. Ralph, M.L. Hitchman, J.P. Millington, F.C. Walsh, J. Electroanal. Chem. 375 (1994) 1. [15] M.T. Stankovich, A.J. Bard, J. Electroanal. Chem. 75 (1977) 487. [16] F.G. Bãnicã, J.C. Moreira, A.G. Fogg, Analyst 119 (1994) 309. [17] D. Mimica, F. Bedioui, J.H. Zagal, Electrochim. Acta 48 (2002) 323. [18] K.P. Gong, X.Z. Zhu, R. Zhao, S.X. Xiong, L.Q. Mao, C.F. Chen, Anal. Chem. 77 (2005) 8158. [19] N. Sehlothl, T. Nyokong, J.H. Zagal, F. Bedioui, Electrochim. Acta 51 (2006) 5125. [20] C.Y. Deng, J.H. Chen, X.L. Chen, Z. Nie, S.Z. Yao, Electrochem. Commun. 10 (2008) 907. [21] C.Y. Deng, J.H. Chen, X.L. Chen, C.H. Xiao, L.H. Nie, S.Z. Yao, Biosens. Bioelectron. 23 (2008) 1272. [22] N.Q. Jia, L.J. Wang, L. Liu, Q. Zhou, Z.Y. Jiang, Electrochem. Commun. 7 (2005) 349. [23] R.X. Wang, D.J. Zhang, Y.M. Zhang, C.B. Liu, J. Phys. Chem. B 110 (2006) 18267. [24] J.W. Jang, C.E. Lee, S.C. Lyu, T.J. Lee, C.J. Lee, Appl. Phys. Lett. 84 (2004) 2877. [25] J.Y. Lao, W.Z. Li, J.G. Wen, Z.F. Ren, Appl. Phys. Lett. 80 (2002) 500. [26] Y. Miyamoto, M.L. Cohen, S.G. Louie, Solid State Commun. 102 (1997) 605. [27] C.E. Bank, R.G. Compton, Analyst 130 (2005) 1232. [28] L.N. Wu, X.J. Zhang, H.X. Ju, Anal. Chem. 79 (2007) 453. [29] T.J. Davies, R.R. Moore, C.E. Banks, R.G. Compton, J. Electroanal. Chem. 574 (2004) 123. [30] A. Salimi, C.E. Banks, R.G. Compton, Analyst 129 (2004) 225. [31] J. Koehne, J. Li, A.M. Cassell, H. Chen, Q. Ye, H.T. Ng, J. Han, M. Meyyappan, J. Mater. Chem. 14 (2004) 676. [32] M.N. Zhang, K.P. Gong, H.Z. Zhang, L.Q. Mao, Biosens. Bioelectron. 20 (2005) 1270. [33] A. Kuznetsova, D.B. Mawhinney, V. Naumenko, J.T. Yates Jr., J. Liu, R.E. Smalley, Chem. Phys. Lett. 321 (2000) 292. [34] W.Q. Han, Y. Bando, K.J. Kurashima, T. Sato, Chem. Phys. Lett. 299 (1999) 368. [35] Z. Wang, C.H. Yu, D.C. Ba, J. Liang, Vacuum 81 (2007) 579. [36] R.R. Moore, C.E. Banks, R.G. Compton, Analyst 129 (2004) 755. [37] O. Nekrassova, N.S. Laawrence, R.G. Compton, Talanta 60 (2003) 1085. [38] O. Nekrassova, G.D. Allen, N.S. Lawrence, L. Jiang, T.G.J. Jones, R.G. Compton, Electroanalysis 14 (2002) 1464. [39] E. Laviron, J. Electroanal. Chem. 115 (1979) 19. [40] A.J. Bard, L.R. Faulkner, Electrochemical Methods, Fundamental and Applications, Wiley, New York, 1980, p. 226.