Fourier Transform Large-Amplitude Alternating Current Cyclic

Anal. Chem. 2004, 76, 166-177

Fourier Transform Large-Amplitude Alternating Current Cyclic Voltammetry of Surface-Bound Azurin Sixuan Guo, Jie Zhang, Darrell M. Elton,† and Alan M. Bond*

School of Chemistry, Monash University, Victoria 3800, Australia

The technique of Fourier transform large-amplitude alternating current (ac) cyclic voltammetry has been applied to a study of the electrochemistry of surface-bound azurin (Az) at a paraffin-impregnated graphite electrode. With the methodology used, the total current, as well as the dc component, and the ac harmonics can be obtained from a single experiment. For the dc (and fundamental harmonic cases), the background current dominates the voltammetry, so that quantitatively useful data only can be obtained after employment of an empirical background correction procedure. In contrast, the capacitance current does not contribute to the second and higher harmonic voltammograms. Furthermore, the higher harmonic Faradaic currents are greatly amplified when large-amplitude sinusoidal perturbations are employed as an alternative to the traditionally used small-amplitude ac methods. Results obtained from numerical simulations are presented for a quasi-reversible process (Butler-Volmer model) in order to illustrate the advantages of the higher harmonic ac method, relative to dc cyclic voltammetry. The almost complete background current rejection and high sensitivity of large-amplitude techniques are confirmed experimentally in second- and third-harmonic ac studies with an adsorbed azurin thin film and the surface process, Az[Cu(II)](surface) + e- a Az[Cu(I)](surface). However, as is the case with dc cyclic voltammetry, readily detected nonideal behavior places restrictions on the straightforward use of Butler-Volmer theory for quantitative evaluation of the kinetics and thermodynamics of this so-called model surface-confined process. Direct current (dc) cyclic voltammetry probably has become the most widely employed method used to evaluate electrode mechanisms.1 In part, the widespread use is a result of quantitative theories having been extensively developed,2-4 with commercial * Corresponding author. Fax: +61 3 9905 4597. E-mail: alan.bond@ sci.monash.edu.au. † Permanent address: Department of Electronic Engineering, LaTrobe University, Bundoora, Victoria 3086, Australia. (1) Bard, A. J.; Faulkner, L. R. Electrochemical Methods, 2nd ed.; John Wiley & Sons: New York, 2001. (2) (a) Nicholson, R. S.; Shain, I. Anal. Chem. 1964, 36, 706. (b) Nicholson, R. S.; Shain, I. Anal. Chem. 1965, 37, 178. (c) Olmstead, M. L.; Hamilton, R. G.; Nicholson, R. S. Anal. Chem. 1969, 41, 260. (d) Olmstead, M. L.; Nicholson, R. S. Anal. Chem. 1969, 41, 862. (e) Nicholson, R. S. Anal. Chem. 1965, 37, 667.

166 Analytical Chemistry, Vol. 76, No. 1, January 1, 2004

simulation packages now being available that accommodate many reaction schemes.5,6 However, quantitative studies using this dc technique become difficult when the capacitance rather than Faradaic current dominates the total current measured,1 as is the case with protein film voltammetry,7 which is the subject of interest in this paper. Alternating current (ac) voltammetry, which was invented in the 1950s8 and intensively developed by Smith,9 while only rarely used, is in principle a far more powerful technique than the dc method for quantitative evaluation of the mechanisms of electrode processes.1,10 Conventionally, with the ac technique, a smallamplitude sinusoidal potential with a frequency of 10 Hz-100 kHz is superimposed onto the triangular waveform used in dc cyclic voltammetry, and either the total ac response or the dc, fundamental, and higher harmonic are then measured as a function of dc potential and frequency.1 Unfortunately, in contrast to the dc case, theoretical solutions relevant to this technique are very limited.1,10 Furthermore, available analytical solutions have traditionally required that the amplitude of the applied alternating potential is very small and that the dc and ac time scales are well separated. Unfortunately, while achieving significant theoretical simplification, the use of small ac amplitudes gives rise to only extremely small second and higher harmonic Faradaic currents. Consequently, the experimental difficulty associated with the need to extract inherently small higher harmonic Faradaic currents has (3) (a) Feldberg, S. W. In Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker: New York, 1969; Vol. 3, pp 199-296. (b) Feldberg, S. W. J. Phys. Chem. 1971, 75, 2377. (c) Feldberg, S. W.; Jeftic, L. J. Phys. Chem. 1972, 76, 2439. (4) (a) Nadjo, L.; Saveant, J. M. Electrochim. Acta 1971, 16, 887. (b) Saveant, J. M.; Andrieux, C. P.; Nadjo, L. J. Electroanal. Chem. 1973, 41, 137. (c) Andrieux, C. P.; Nadjo, L.; Saveant, J. M. J. Electroanal. Chem. 1973, 42, 223. (5) Rudolph, M.; Reddy, D. P.; Feldberg, S. W. Anal. Chem. 1994, 66, 589A. (6) Gosser, D. K., Jr. Cyclic Voltammetry: Simulation and Analysis of Reaction Mechanisms; VCH: New York, 1993. (7) (a) Armstrong, F. A.; Heering, H. A.; Hirst, J. Chem. Soc. Rev. 1997, 26, 169. (b) Hirst, J.; Armstrong, F. A. Anal. Chem. 1998, 70, 5062. (c) Jeuken, L. J. C.; Armstrong, F. A. J. Phys. Chem. B 2001, 105, 5271. (d) Jeuken, L. J. C.; McEvoy, J. P.; Armstrong, F. A. J. Phys. Chem. B 2002, 106, 2304. (8) (a) Grahame, D. C. J. Electrochem. Soc. 1952, 99, 370C. (b) Breyer, B.; Hacobian, S. Aust. J. Chem. 1954, 7, 225. (c) Tachi, I.; Kambara, T. Bull. Chem. Soc. Jpn. 1955, 28, 25. (d) Senda, M.; Tachi, I. Bull. Chem. Soc. Jpn. 1955, 28, 632. (9) Smith, D. E. In Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker: New York, 1966; Vol. 1. (10) Bond, A. M. Modern Polarographic Methods in Analytical Chemistry; Marcel Dekker: New York, 1980; p 341. 10.1021/ac034901c CCC: $27.50

© 2004 American Chemical Society Published on Web 11/25/2003

meant that the advantage of an ideally zero capacitance current has been rarely exploited. Use of large-amplitude sinusoidal perturbations would amplify the nonlinearity of the electrode process and hence simplify detection of the enhanced higher harmonic component. An analytical mathematical solution to the large-amplitude case has been reported by Engblom et al.11 for a reversible electrode process involving solution-soluble species. Complete numerical simulations of ac voltammetry for solutionsoluble reactants and products also have been developed by Gavaghan et al.12-15 for all amplitudes. In these later studies, the Fourier transform (FT) method of analysis was employed to separate the dc signal and the harmonics of the ac signal, and instrumentation was specifically developed that could exploit the attractive features of the large-amplitude method.16,17 Protein film voltammetry7 provides an efficient approach to the evaluation of thermodynamic and kinetic aspects of the electrochemistry of redox-active proteins. In this method, extremely small quantities of protein are immobilized on the electrode surface to provide a thin film of redox-active material. Unfortunately, with dc cyclic voltammetry, the background current detected under thin-film conditions is typically as large as, or even larger than, the Faradaic current that needs to be measured. Background currents may be subtracted in an essentially empirical fashion using baseline-subtraction software.7a,b Recently, Armstrong et al.7d utilized the technique of square wave voltammetry18 to improve the Faradaic-to-background current ratio in a study of the gated electron-transfer kinetics at an electrode-azurin interface. In this method, the currents are sampled twice per cycle (at the end of each pulse). A square wave voltammogram may then be generated in the form of a plot of the forward current, reverse current, and net current as a function of potential. However, large-amplitude ac cyclic voltammetry should provide a better method to examine the electrochemistry of surface-bound molecules because of a superior background-to-Faradaic current ratio. Laviron19 developed a theory for the Faradaic ac voltammetric response for small ac amplitudes under conditions of dc reversibility. A more generally applicable numerical simulation of the FT ac voltammetry of surface-bound molecules was recently described by Honeychurch and Bond.20 In this case, discussion of results obtained for a reversible process also was presented. To date, experimental assessment of the ac method has yet to be described. Azurin (Az), a “blue” copper protein, functions as an electrontransfer agent in the respiratory chain present in bacteria such as Pseudomonas aeruginosa.21 It is a very well characterized protein with a mononuclear copper ion in its active site and is capable of (11) Engblom, S. O.; Myland, J. C.; Oldham, K. B. J. Electroanal. Chem. 2000, 480, 120. (12) Gavaghan, D. J.; Bond, A. M. J. Electroanal. Chem. 2000, 480, 133. (13) Gavaghan, D. J.; Elton, D. M.; Bond, A. M. Collect. Czech. Chem. Commun. 2001, 66, 255. (14) Gavaghan, D. J.; Elton, D. M.; Oldham, K. B.; Bond, A. M. J. Electroanal. Chem. 2001, 512, 1. (15) Gavaghan, D. J.; Elton, D. M.; Bond, A. M. J. Electroanal. Chem. 2001, 513, 73. (16) Ha´zı`, J.; Elton, D. M.; Czerwinski, W. A.; Schiewe, J.; Vicente-Beckett, V. A.; Bond, A. M. J. Electroanal. Chem. 1997, 437, 1. (17) Schiewe, J.; Ha´zı`, J.; Vicente-Beckett, V. A.; Bond, A. M. J. Electroanal. Chem. 1998, 451, 129. (18) Osteryoung, J.; O’Dea, J. J. In Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker: New York, 1986; Vol. 14, p 209. (19) Laviron, E. J. Electroanal. Chem. 1979, 101, 19. (20) Honeychurch, M. J.; Bond, A. M. J. Electroanal. Chem. 2002, 529, 3.

exhibiting a fast one-electron-transfer process under both solutionphase and thin-film conditions (see eq 1 for the surface-confined E0f , ks,R

Az[Cu(II)](surface) + e- {\} Az[Cu(I)](surface)

(1)

case). Azurin may be adsorbed from aqueous solutions to provide up to monolayer coverage of the protein on a paraffin-impregnated graphite (rod) electrode (PIGE).22 Features of the surface-confined Cu(II)/Cu(I) electron-transfer process have then been studied by dc cyclic volammetry7,22,23 when an azurin-modified electrode is placed in contact with an aqueous electrolyte solution. Thus, the azurin system studied under these conditions is considered to represent an excellent model system for experimentally assessing the advantages of large-amplitude sinusoidal ac voltammetry relative to use of procedures employed in the literature that are based on dc cyclic voltammetry with baseline subtraction of the background current. EXPERIMENTAL SECTION Reagents. P. aeruginosa azurin was extracted and purified according to the literature.24 The protein solution, azurin (0.93 mM, 3.0 mg mL-1) in HEPES buffer (20 mM, pH 7.0), was stored in the dark at 4 °C. A buffer solution (pH 8.0) containing 0.02 M tris(hydroxymethyl)methylamine ((HOCH2)3CNH2, 99%; Aldrich Chemical Co., Milwaukee, WI) and 0.1 M NaCl (AnalaR; BDH Laboratory Supplies, Poole, U.K.) was used as the electrolyte. The pH value was adjusted using an aqueous NaOH (AnalaR; BDH) or HCl (32%, AnalaR; BDH) solution and was measured with a Metrohm 744 pH meter, equipped with a Metrohm pH glass electrode (Metrohm Ltd., Herisau, Switzerland). All chemical reagents were used as received. Deionized water from a MilliQ-MilliRho purification system (resistivity 18 MΩ cm) was used to prepare the buffer solution. Apparatus and Procedures. The FT ac voltammetric instrumentation was based on a conventional three-electrode potentiostat driven by a 19-bit delta sigma digital-to-analog converter. The potentials were digitized by separate 18-bit delta sigma analog to digital converters. The system was run synchronously at a sampling rate of 39 062.5 s-1. All signal processing was performed using a desktop computer with a 550-MHz Pentium III processor running Windows 98, and the computer code was written in C++. The sampling rate was selected to ensure that all data sets in the experiments were 214 bytes (16 384) in size in order to avoid aliasing and to optimize the speed of the FT processing. A moving average of eight data points was used for data collection, meaning that the effective sampling rate was 4.88 × 103 s-1. The influence of this averaging was compensated for in the subsequent signal processing. Full details of the instrumentation and signal processing algorithms will be described elsewhere.25 The sinusoidal (21) Gray, H. B.; Solomon, E. I. In Copper Proteins; Spiro, T. G., Ed.; Wiley: New York, 1981. (22) Rooney, M. B.; Honeychurch, M. J.; Selvaraj, F. M.; Blankenship, R. E.; Bond, A. M.; Freeman, H. C. J. Biol. Inorg. Chem. 2003, 8, 306. (23) (a) Friis, E. P.; Andersen, J. E. T.; Madsen, L. L.; Bonander, N.; Moller, P.; Ulstrup, J. Electrochim. Acta 1997, 42, 2889. (b) Fristrup, P.; Grubb, M.; Zhang, J.; Christensen, H. E. M.; Hansen, A. M.; Ulstrup, J. J. Electroanal. Chem. 2001, 511, 128. (24) Clair, C. S. St.; Ellis, W. R.; Gray, H. B. Inorg. Chim. Acta 1992, 191, 149.

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waveform for the applied potential was generated digitally with the maximum frequency in the series being calculated as 0.45 times the sampling rate. Conventional dc cyclic voltammetric experiments were undertaken with a Bioanalytical systems BAS model 100B electrochemistry workstation (Bioanalytical Systems, West Lafayette, IN). A standard three-electrode cell was employed in all electrochemical measurements, with an Ag/AgCl (saturated KCl) electrode as the reference electrode and a platinum wire as the auxiliary electrode. A PIGE (radius 3 mm)26-28 was used as the working electrode. The procedure for the fabrication of an azurin protein film electrode is based on that described in ref 22. In summary, prior to the protein film deposition, the PIGE electrode was polished with 0.3-µm alumina on a clean polishing cloth (Buehler), rinsed with water, and then sonicated in water to remove alumina. The electrode was then placed in ice with the surface pointed upward until it was ice-cold. Next, a 2-5 -µL drop of azurin solution was placed onto a cold upturned electrode surface. The azurinadsorbed electrode was then immediately transferred into the standard three-electrode cell containing buffer solution, which has been purged with nitrogen for at least 15 min, and cooled to 0 °C with a water/ice bath.

O and R are assumed to strongly adsorb onto the electrode surface and follow a Langmuir isotherm. Any interaction of surface-bound molecules and the heterogeneity of the surface are neglected. Under the above assumptions, the Faradaic current, I, is given by the equation1

THEORY 1. Model Employed. The theory for the Faradaic current in large-amplitude sinusoidal ac linear sweep voltammetry of surfaceconfined species has been described in detail elsewhere.20 In brief, for a one-electron charge-transfer reduction process, the surface confined reaction can be described by eq 2, where O is the

E ) Edc + Eac

I ) FAΓ dθ/dt ) FAΓ[kb(1 - θ) - kfθ]

where A is the area of the electrode, Γ is the total surface excess (mol cm-2) of the bound electroactive species (i.e., sum of surface excess of the oxidized and reduced forms), t is time, and dt is the time taken to change the surface coverage θi to a new value θi+1). If dt is sufficiently small, E and thus kf and kb may be considered to be constants on this time scale, and the following equation can be obtained by integrating eq 4 from t to (t + dt),

dθ ) θi+1 - θi ) (kb/(kf + kb) - θi)(1 - exp[-(kf + kb) dt]) (6) In ac cyclic voltammetry, the waveform used for a reduction process is given in Figure 1 (reduction sweep direction only shown). Thus,

O + e {\ }R k

(2)

b

oxidized form (e.g., Az[Cu(II)](surface)) and R is the reduced form (e.g., Az[Cu(I)](surface)) of the surface-confined species; kf and kb are the forward and backward electron-transfer rate constants, respectively. Butler-Volmer-type kinetics are assumed to apply,1 although more advanced electron-transfer models employing Marcus theory, have been applied to dc studies of thin film voltammetry.7b,d,29 Thus,

(7)

where

Edc ) Estart - vt

(8)

Edc ) Estart - 2vts + vt

(9)

Eac ) ∆E sin(ωt)

(10)

0 < t < ts: ts < t < 2ts:

kf

(5)

0 < t < 2ts:

and where v is the magnitude of the scan rate of the dc ramp, ts is the time required to complete the reductive sweep, Estart is the starting or initial potential, ω is the angular frequency, and ∆E is the amplitude of the sine wave. Use of the following dimensionless terms

Ω ) RTω/Fv

(11)

τ ) (Fv/RT)t

(12)

k0norm ) (RT/Fv)k0

(13)

Inorm ) (RT/F2AvΓ)I

(14)

E0norm ) (F/RT)E0f

(15)

Enorm ) (F/RT)Edc dc

(16)

where is the standard electron-transfer rate constant at the formal reversible potential (E0f ), R is the electron-transfer coefficient, E is the applied potential, R, T, and F have their usual meanings, and the number of electron transferred, n, is one. Both

Enorm ) (F/RT)∆E sin(Ωτ) ac

(17)

+ Enorm Enorm ) Enorm ac dc

(18)

(25) Manuscript in preparation. (26) Scholz, F.; Lange, B. Trends Anal. Chem. 1992, 11, 359. (27) Dostal, A.; Meyer, B.; Scholz, F.; Shro ¨der, U.; Bond, A. M.; Marken, F.; Shaw, S. J.J. Phys. Chem. 1995, 99, 2096. (28) Scholz, F.; Meyer, B. In Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker: New York, 1998; Vol. 20. (29) Chidsey, C. E. D. Science 1991, 251, 919.

enables eq 5 to be cast into a computationally convenient dimensionless form:

kf ) k0 exp

[

(E - E )] [-RF RT 0 f

(3)

]

(1 - R)F (E - E0f ) kb ) k exp RT 0

(4)

k0

168


Inorm ) dθ/dτ ) k0norm exp[(1 - R)(Enorm - E0norm)] × (1 - θ) - k0norm exp[-R(Enorm - E0norm)]θ (19)

Figure 1. Applied waveform employed in large-amplitude ac voltammetric measurements (initial reduction sweep direction only shown).

which can be readily and efficiently solved numerically using a Fortran software program available on request to the authors. For the theoretical calculations, the Fourier transform and inverse Fourier transform analyses required to separate the dc and ac components were also undertaken with the aid of software written in Fortran. In eq 19, Inorm represents the Faradaic current. If required, the charging current, Idl, was calculated on the basis of eqs 20 and 21, normalized via use of eq 14, and then added into eq 19 to give the total dimensionless current

Idl ) Cdl (dE/dt)

(20)

which on the basis of eqs 7-10, can be reformulated as

Idl ) Cdl((v + ω∆E cos(ωt))

(21)

It follows from eq 21 that the ac component of the charging current only has the same frequency (ωt) as the applied sine wave. Consequently, charging current will not contribute to second and higher harmonics (2ωt, 3ωt, 4ωt, etc.) measurements. 2. Analysis of Theoretically Derived ac Voltammograms. The theoretical model employed in this study implies that the

parameters that need to be considered in order to understand the principles of ac thin-film voltammetry include Γ, f, ∆E, k0, and R, where f is the frequency (ω ) 2πf ). In particular, it is important to examine the influence that these parameters have on the higher harmonics acquired under large-amplitude conditions. To study the influence of these parameters and also to identify the characteristics of ac voltammograms that might be encountered in the electrochemistry of azurin, values of Γ ) 3.0 × 10-11 mol cm-2, k0 ) 70 s-1, and T ) 273 K were used in the initial theoretical simulations. As will be seen later, these parameters are those determined on the basis of initial dc cyclic voltammetric data. A double layer capacitance of 2 µF cm-2 was included in some initial simulations in order to model the influence of the charging current. Simulated currents presented in this section of the paper are presented in a dimensionless form as Inorm, which is consequently independent of Γ, v, and A (eq 14). 2.1. Simulated FT ac Voltammograms of A Quasi-Reversible Process. Previous theoretical studies focused on the reversible electron-transfer case and ignored the effect of the double layer charging current.20 A typical ac voltammogram for a quasireversible process containing charging current and simulated with Ef0) 0 V, f ) 100 Hz, ∆E ) 50 mV, v ) 500 mV s-1, R ) 0.5, and other parameters given above is presented in Figure 2. The results Analytical Chemistry, Vol. 76, No. 1, January 1, 2004

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Figure 2. Quasi-reversible FT ac voltammograms simulated with k0 ) 70 s-1. The initial and final potentials are 400 and -400 mV, respectively, with the switching potential being -400 mV. Other parameters are defined in the text. (a) Total ac current voltammogram. (b) Dc component. (c) Fundamental harmonic voltammogram. (d) Second-harmonic voltammogram and (e) third-harmonic voltammogram. (Inorm in these figures is the dimensionless current of each component obtained by normalizing the real current of corresponding component according to eq 14.).

clearly show that while charging current contributes significantly to the total (dc plus ac) current (Figure 2a), dc (Figure 2b), and fundamental harmonic (Figure 2c), the second (Figure 2d) and third (Figure 2e) harmonics of the ac component are completely void of from the double layer charging current in these largeamplitude ac voltammograms. It is just this outcome that provides the higher harmonic ac method with a significant advantage in voltammetric studies of surface-confined species because the charging current dominates protein film dc voltammetry (see later). Furthermore, the double layer capacitance, which is the origin of the charging current, is significantly modified when the electrode surface is covered with the hydrophobic protein molecules. This feature renders background correction most problematic (empirical) in dc voltammetric measurements associated with surface-confined species because simple measurement of the background current, followed by its subtraction, is not a viable option. Previous studies on a diffusional system12 suggested that separation of the dc and ac components becomes difficult if the frequency of the sine wave is too low relative to the time scale of the dc scan rate. This problem is also encountered with surfaceconfined systems. As clearly shown in Figure 3, “noise” is encountered when the dc and ac components overlap, as occurs when the parameters chosen above are used with a sine wave 170 Analytical Chemistry, Vol. 76, No. 1, January 1, 2004

frequency of 10 Hz. In contrast, with frequencies greater than 20 Hz, overlap between the ac and dc components in the frequency domain is negligible (not shown). Examination of the power spectrum readily establishes whether overlap of dc and ac time domains is a problem. 2.2. Effect of r and k0 on the Second and Higher Harmonics. Theoretical studies were also carried out to investigate the effect of R and k0 on the second and higher harmonics of ac voltammograms. For a quasi-reversible process, the simulated results in Figure 4 clearly show that the second-harmonic ac voltammograms are very sensitive to R. Thus, a symmetrically shaped second-harmonic voltammogram is obtained when R ) 0.5, whereas second harmonics having two branches of unequal height and incompletely resolved are obtained when R * 0.5. The sensitivity to R increases when either v or f increases because effectively this reduces the reversibility of the redox reaction (results not shown). In contrast, dc voltammograms are not nearly as sensitive to R. The dependence of second-harmonic surface-confined voltammograms on k0 also is very marked as noted in voltammograms presented in Figure 5. Not surprisingly, the third harmonic (not shown) is also inherently very sensitive to changes in R and k0. 2.3. Effect of ∆E and f on the Maximum Normalized Current of the Second Harmonic, [Inorm(2ωt)]max. Theoretical

Figure 3. Quasi-reversible ac voltammogram simulated with f ) 10 Hz. Other parameters are as defined for Figure 2. (a) Total current ac voltammogram. (b) Power spectrum. (c) Dc component.

studies were carried out to investigate the dependence of the maximum normalized second-harmonic peak current, [Inorm(2ωt)]max on ∆E and f, for a simulated quasi-reversible process. The peak current value for only the first branch (see, for example, Figure 4b) is presented in this section. These dependencies are also presented for a reversible case for comparison. Figure 6 shows that the magnitude of this parameter increases dramatically when ∆E increases, especially for amplitude in the region between 75 and 200 mV for both quasi-reversible and reversible processes. Results in Figure 7 show how simulated [Inorm(2ωt)]max depend on f for the parameters chosen for a quasi-reversible process. A maximum [Inorm(2ωt)]max can be obtained when the process is quasi-reversible, in contrast to the reversible case where [Inorm(2ωt)]max linearly depends on f. The results in Figure 7 also show that [Inorm(2ωt)]max decreases when the scan rate increases for a given frequency for a quasireversible process. However, the real current exhibits no significant change in this situation according to eq 14. For the reversible case, the real current is independent of scan rate. The simulated voltammograms presented the second-harmonic case reveal both the absence of charging current and the high sensitivity to the kinetics parameters (k0 and R). The third and higher harmonics also exhibit analogous properties. Consequently, the analysis of the higher harmonic terms with FT large-amplitude ac voltammetry should represent an ideal method for quantitative evaluation of the electrode kinetic of surface-confined processes under thin-film conditions.

Figure 4. Dependence of simulated quasi-reversible secondharmonic ac voltammograms on R for R ) 0.4 (a), 0.5 (b), and 0.6 (c). Parameters employed in the simulation are f ) 50 Hz, with other parameters being the same as those in Figure 2. The definition of the parameter [Inorm(2ωt)]max used in section 2.3 also is defined.

RESULTS AND DISCUSSION 1. Dc Cyclic Voltammetry. Dc cyclic voltammetric studies on thin films of azurin were undertaken with both the home-built FT ac and the commercially available BAS 100B instruments employing scan rates over the range of 10-1907 mV s-1. Background currents were subtracted in these experiments using the baseline-subtraction software used in the studies by Heering and Armstrong et al.7,22 Since dc cyclic voltammograms obtained with either instrument were essentially the same, only data obtained by the new FT ac instrument are presented in this paper. Figure 8 illustrates a typical dc cyclic voltammogram obtained for azurin adsorbed on a PIGE surface. The very large contribution from charging current is noteworthy. The peak areas, after baseline subtraction, were used to determine the azurin protein coverage via use of the relationship, Γ ) (peak area)/(FAv), where A is the geometric electrode surface area and all other parameters are as defined previously. Γ values calculated on this basis typically lay in the range (4 ( 2) × 10-11 mol cm-2. For each individual experiment, the voltammetric response was stable for several hours (attenuation of the signal was less than 8%/h during the course of experiments undertaken in these studies). Figure 9 contains the experimental results for the separation of the reduction and oxidation peak potentials (∆Ep) as a function of scan rate as well as the simulated response for a given set of parameters with a quasi-reversible mechanism. It is evident that, as the scan rate increases, ∆Ep increases, as expected.30 The Analytical Chemistry, Vol. 76, No. 1, January 1, 2004

171

Figure 5. Second-harmonic ac voltammograms simulated with the following parameters: k0 ) 1 × 1010 (reversible) (a), 50 (b), 10 (c), and 1 (d) s-1. Other parameters are the same as used in Figure 2.

Figure 6. Influence of the magnitude of ∆E on the simulated value of [Inorm(2ωt)]max for a quasi-reversible process having a value of k0 ) 70 s-1 and other parameters defined in Figure 2 (0) and for a reversible process (4).

simulated curves were obtained using a value of R ) 0.5 but are insensitive to R in the range of 0.3-0.7. A value of k0 of 70 ( 10 s-1 is obtained on the basis of ∆Ep versus scan rate data for the reduction of azurin. At the slowest scan rate used (10 mV s-1), ideally the azurin system should exhibit close to reversible behavior, in which case ∆Ep would be predicted to approach zero. However, as noted by Armstrong et al.,7b,d ∆Ep approaches a nonzero value (30 mV) at slow scan rates and the peak width at half-height (W1/2) is 117 mV, which significantly exceeds the Nernstian value of 83 mV at 0 °C that is predicted for a reversible one-electron process. Nonideal behavior detected at slow scan rates for surface-confined processes has been attributed to kinetic or thermodynamic distribution effects31 (not all azurin molecules are in an identical state) to the hysteresis produced by the shape of the reaction coordinate32 or to other sources.33 To correct for (30) Honeychurch, M. J. Langmuir 1999, 15, 5158. (31) Rowe, G. K.; Carter, M. T.; Richardson, J. N.; Murray, R. W. Langmuir 1995, 11, 1797. (32) Feldberg, S. W.; Rubinstein, I. J. Electroanal. Chem. 1988, 240, 1.

172 Analytical Chemistry, Vol. 76, No. 1, January 1, 2004

Figure 7. Influence of f and ∆E on the simulated values of [Inorm(2ωt)]max for a quasi-reversible process having a value of k0 ) 70 s-1 for ∆E ) 80 (4), 50 (0), and 30 (O) mV and with a scan rate v ) 500 (a) or 50 mV s-1 (b), and for a reversible process with ∆E ) 50 mV (b). Other parameters used in the simulation are as defined in Figure 2.

this nonideality, ∆Ep data presented in Figure 9 are compensated for the offset in the manner described by Armstrong.7b Ac voltammetric data would be expected to also exhibit this kind of nonideal behavior. Contributions from kinetics of proton-transfer reactions coupled to the electron-transfer step also may influence

Table 1. Dependence of Peak Potentials, Peak Heights, and Peak Widths at Half-Height on Scan Rate after Background Correction

Figure 8. Dc cyclic voltammogram obtained with and without background subtraction at a scan rate of 50 mV s-1 for azurin immobilized on a PIGE (pH 8.0, 0.1 M NaCl). Potential range: +300 to -300 mV (vs Ag/AgCl, saturated KCl).

Figure 9. Dependence of corrected ∆Ep on scan rate for thin films of azurin under conditions of dc cyclic voltammetry at a PIGE surface: ([) experimental data, (s) curve fitted to experimental data, curve predicted for k0 ) 60 s-1 (- ‚ -), k0 ) 70 s-1 (- - -), and k0 ) 80 s-1 (- ‚‚ -), with R ) 0.5.

the voltammetry of azurin.34 However, at the pH value of 8.0 employed in the present study, this should not be a significant factor. As expected theoretically1 for a quasi-reversible surfaceconfined process, the peak current was found to be almost linearly dependent on scan rate over the range of 10-1907 mV s-1. Table 1 contains the experimental results obtained for the peak potentials, peak heights, and peak widths at half-height as a function of scan rate in dc studies of azurin adsorbed onto a PIGE surface. Trends are consistent with previous results presented at other forms of carbon electrode.7 2. Ac Cyclic Voltammetry. In FT ac voltammetric data presented below, the dc, fundamental, and higher harmonic ac voltammograms are obtained via use of the FT and inverse FT algorithms built into the instrumentation. Figure 10 shows an example of an ac cyclic voltammogram (“total current”) obtained (33) (a) Brown, A. P.; Anson, F. C. Anal. Chem. 1977, 49, 1589. (b) Albery, W. J.; Boutelle, M. G.; Colby, P. J.; Hillman, A. R. J. Electroanal. Chem. 1982, 133, 135. (c) Gerischer, H.; Scherson, D. A. J. Electroanal. Chem. 1985, 188, 33. (d) Rowe, G. K.; Creager, S. E. Langmuir 1991, 7, 2307. (e) Acevedo, D.; Abrun ˜a, H. D. J. Phys. Chem. 1991, 95, 9590. (f) Smith, C. P.; White, H. S. Anal. Chem. 1992, 64, 2398. (34) Jeuken, L. J. C.; Wisson, L.-J.; Armstrong, F. A. Inorg. Chim. Acta 2002, 331, 216.

v (mV s-1)

Ered p (mV)

Eox p (mV)

Ema (mV)

∆Ep (mV)

W1/2b (mV)

-Ipc (µA)

10 25 50 102 238 477 1040 1907

12.9 18.0 16.0 20.4 23.0 24.3 14.0 6.9

38.7 46.0 48.4 53.4 57.9 63.3 65.4 73.2

25.8 32.0 32.2 36.9 40.5 43.8 39.7 40.1

25.8 28.0 32.4 33.0 34.9 39.0 51.4 66.3

117.6 131.7 133.3 135.4 138.1 140.3 142.9 150.5

0.3 0.42 0.58 0.92 1.79 3.29 6.65 12.6

a E is the midpoint potential, ) (Ered + Eox)/2. b Peak width at m p p half-height. c Peak height for reduction component of cyclic voltammogram.

when azurin is adsorbed onto the PIGE surface, as displayed in both the conventional I-E (Figure 10a) and more convenient I-t formats (Figure 10b). As with dc cyclic voltammograms, a very large charging current contribution is detected in the total current response. After the application of FT and inverse FT algorithms, the separated dc, fundamental, and higher ac harmonic voltammograms can be obtained, to give the voltammograms displayed in Figure 11. Importantly, the second- and third-harmonic ac voltammograms are essentially void of background current, unlike the dc and fundamental harmonic components. However, some “ringing” is evident in the initial and switching potential regions of the voltammograms where the charging current spikes are present. “Ringing” could be filtered out by zeroing data in this potential region prior to undertaking the FT-inverse FT sequence of operations. However, we have chosen to display raw rather than “manipulated” data. The dc component obtained after application of the FT ac method (Figure 11a) is similar to dc cyclic voltammograms obtained directly in the absence of an ac signal (Figure 8). Simulations revealed that clear wave splitting in the dc component of the kind reported when an ac signal is superimposed onto the dc waveform for processes where the reactant and product are soluble35 is only predicted to be observed for a surface-confined process at larger ac amplitudes than used in the present study. The almost complete absence of a capacitance current contribution to the second (Figure 11c), third (Figure 11d), and higher harmonics (not shown) is consistent with the theoretical predictions. Thus, these second and higher ac harmonics may be analyzed directly without the need to correct for the charging current. In practice, readily evaluated parameters in higher harmonic ac voltammetry are peak heights, peak potentials, and zero current crossover potentials as defined in Figure 12a for the secondharmonic response. Representative data based on measurement of the parameters are given in Tables 2-5. Analysis of the dc data, as noted above, implied that k0 is 70 ( 10 s-1, based on analysis of ∆Ep values as a function of scan rate. However, it was also noted that shapes of the voltammograms are significantly different from predictions based on the Butler-Volmer formalism. If k0 is (35) Gavaghan, D. J.; Myland, J. C.; Oldham, K. B. J. Electroanal. Chem. 2001, 516, 2.


173

Figure 10. Total current ac cyclic voltammograms of azurin immobilized on a PIGE surface (pH 8.0, 0.1 M NaCl) displayed in the (a) I-E and (b) I-t formats. Potential range, +300 to -300 mV; v, 50 mV s-1; ∆E, 80 mV; f, 9.54 Hz.

Figure 11. Dc (a) and FT ac fundamental (b), second (c), and third (d) harmonic cyclic voltammograms of azurin immobilized onto a PIGE surface. Experimental conditions are as for Figure 10.

assumed to be 70 s-1, then the second- and third-harmonic peak current magnitudes obtained in this study are consistent with a value of Γ of (3 ( 1) × 10-11 mol cm-2, which is in excellent agreement with the values obtained from dc measurements of (4 ( 2) × 10-11 mol cm-2. Furthermore, the well-defined separation between each lobe of the second-harmonic experimental voltammograms is consistent with an R value of close to 0.5. However, as the simulations in Figure 12 reveal, it is not possible to obtain both unequal peak heights and well-resolved lobes by simply assigning a value of R * 0.5. More detailed analysis of some data obtained under the conditions relevant to Tables 2-5 have been carried out. Figure 13 summaries the effect of the amplitude and frequency on the peak current of second-harmonic ac voltammograms obtained at a scan rate of 50 mV s-1. The peak current for the second harmonic, [I(2ωt)]max, increases with the frequency up to 38.15 Hz and then decreases, but always increases significantly with amplitude. The results obtained at the scan rates of 10, 25, and 102 mV s-1 show analogous trends. The dependence of these data on frequency and amplitude closely resemble simulated data obtained for a k0 value of 70 s-1 and R ) 0.5 (Figure 6). Figure 14a reveals that, for all three amplitudes considered, and with a scan rate of 238 mV s-1, [I(2ωt)]max decreases with increasing frequency, but increases significantly with ac amplitude. These 174


dependences on amplitude and frequency are again consistent with simulations based on k0 ) 70 s-1 and R ) 0.5 (Figure 7). A similar situation prevails at a scan rate of 477 mV s-1 (Figure 14b) and at a scan rate of 1430 mV s-1 at frequencies over the range of 76.29-305.18 Hz with ∆E ) 80 mV (data not shown). In summary, many aspects of the ac data are consistent with a k0 value of 70 ( 10 s-1, but as is the case with the dc response, the shapes of the voltammogram are not those predicted on the basis of the Butler-Volmer model. Table 2 includes simulated values of potential parameters obtained from the second response on the basis of k0 ) 70 s-1 and R ) 0.5 and confirm that the experimental response is broader than that obtained by simulation. For example, the simulated values of ∆Ep are zero for all the conditions applicable to data presented in Tables 2-5, whereas values of g10 mV always are obtained. Furthermore, as shown in Tables 2-5, the experimental values for ∆Ered, ∆Eox, and ∆Ep are always larger than the theoretical values. At this time, we are unable to obtain exact matching of wave shape for any simple combination of k0 and R values that adequately describes all of the second-harmonic data. Consequently, it is concluded that significantly more sophisticated theoretical modeling that considers thermodynamic and kinetic dispersion and alternative to the Butler-Volmer theory will be ultimately needed. That is, most values of k0 reported to date in the literature based on the dc cyclic

Figure 12. Simulations showing the predicted effect of R on second- and third-harmonic ac voltammograms. (a) and (b), R ) 0.5; (c) and (d), R ) 0.9. Γ ) 3 × 10-11 mol cm-2, and k0 ) 70 s-1. Other parameters employed in simulations are as for Figure 8. The definition of the potential terms given in Tables 2-5 is shown in (a). Table 2. Comparison of Experimental and Simulateda Peak Potential Values Obtained by Second-Harmonic Ac Measurements at a Scan Rate of 10 mV s-1 ∆E ) 30 mV

∆E ) 50 mV

∆E ) 80 mV

f ∆Epb ∆Ered c ∆Eox d ∆Epb ∆Ered c ∆Eox d ∆Epb ∆Ered c ∆Eox d (Hz) (mV) (mV) (mV) (mV) (mV) (mV) (mV) (mV) (mV) 4.77 12.2 9.54 12.0 19.07 11.7 38.15 11.2 57.22 10.8 76.29 10.5

105.1 107.6 (73) 107.4 110.9 (73) 108.1 111.2 (73) 111.9 116.3 (73) 112.8 118.2 (73) 114.9 119.8 (73)

11.5 11.3 11.2 11.2 11.1 11.0

114.9 119.8 (83) 114.8 119.1 (83) 114.9 119.1 (83) 115.2 125.4 (83) 115.5 126.8 (83) 117.6 127.1 (83)

Table 3. Comparison of Experimental and Simulated Peak Potential Values Obtained by Second-Harmonic Ac Measurements at a Scan Rate of 25 mV s-1a

9.5 9.4 9.5 9.6 9.6 9.5

124.7 124.7 (105) 124.7 126.0 (105) 124.7 126.0 (105) 124.7 127.4 (105) 125.0 128.0 (105) 125.9 129.1 (105)

a Simulation values contained in parentheses are based on k0 ) 70 s-1, R ) 0.5, and Γ ) 3 × 10-11 mol cm-2. b-d Parameters are defined in Figure 12a.

voltammetry values of ∆Ep as a function of scan rate probably reflect an average value that need not apply to an individual azurin or other form of surface-confined molecule. The purpose of the present paper is not to develop sophisticated models, but to illustrate the usefulness of large-amplitude ac methods to reject background current and to provide enhanced sensitivities for mechanistic studies. The data obtained readily support this contention but illustrate the inherent complexity present in even a “model” case of azurin protein film voltammetry. The concluding section of this paper therefore emphasizes the highly advantageous features of the use of large rather than the usual small amplitude ac methodologies. 3. Experimental Evaluation of Limitations of Second and Higher Harmonic ac Voltammetry. 3.1. Second Harmonic. To assess from an experimental perspective, the instrumental

∆E ) 30 mV f (Hz)

∆Ep ∆Ered (mV) (mV)

4.77

15.6

9.54

14.8

19.07

14.1

38.15

13.5

57.22

11.6

76.29

10.3

∆E ) 50 mV

∆Eox ∆Ep ∆Ered (mV) (mV) (mV)

114.7 121.8 (73) 114.7 121.2 (73) 114.7 119.8 (85) 117.3 121.8 (98) 120.9 128.6 (110) 123.9 133.0 (120)

13.5 12.6 12.0 11.5 11.0 10.9

∆E ) 80 mV


117.8 120.3 (83) 118.0 120.7 (83) 119.0 121.3 (86) 121.2 126.1 (98) 123.6 131.6 (108) 124.9 132.5 (114)

10.7 10.6 10.4 10.1 10.0 10.0

∆Eox (mV)

127.7 128.7 (105) 127.8 129.6 (105) 126.4 130.0 (105) 128.6 130.6 (105) 129.1 134.5 (105) 130.0 135.3 (114)

a Simulated parameters and terms are as specified in footnote to Table 2.

limitations of the second-harmonic method, voltammograms for thin film of azurin adhered to a PIGE surface were obtained as a function of scan rate, amplitude, and frequency over a potential range of +150 to -150 mV versus Ag/AgCl (saturated KCl). Experiments were undertaken at scan rates of 10, 25, 50, 102, 238, 477, and 1430 mV s-1 using amplitudes of 10, 30, 50, and 80 mV and frequencies of 4.77, 9.54, 19.07, 38.15, 57.22, and 76.29 Hz. Conditions where excellent signal-to-noise ratios were encountered are represented by data entries in Tables 2-5. Thus, with ∆E ) 10 mV (signal-to-noise ratio unfavorable) and at fast scan rates (ill-defined voltammograms obtained), the quality of the second-harmonic response was poor. Consequently, no data related to these experimental situations are contained in Tables 2-5. That is, data presented within the frequency range of 4.7776.29 Hz are restricted to amplitudes of g30 mV and scan rates of e102 mV s-1. Analytical Chemistry, Vol. 76, No. 1, January 1, 2004

175

Table 4. Comparison of Experimental and Simulated Peak Potential Values Obtained by Second Harmonic ac Measurements at a Scan Rate of 50 mV s-1a ∆E ) 30 mV f (Hz)


4.77

18.4

9.54

17.0

19.07

15.2

38.15

14.9

57.22

14.7

76.29

14.1

∆E ) 50 mV


112.7 128.2 (73) 115.9 128.2 (73) 116.9 127.6 (83) 119.4 129.4 (98) 125.0 130.6 (112) 130.0 132.4 (122)

15.9 15.4 14.1 13.9 13.9 13.8

∆E ) 80 mV


119.0 129.6 (83) 121.1 129.6 (83) 121.0 129.6 (86) 122.6 133.6 (98) 128.4 136.7 (108) 132.7 140.8 (115)

12.7 12.2 12.1 11.9 11.4 11.3

∆Eox (mV)

129.6 133.8 (105) 130.5 133.7 (105) 131.0 133.8 (105) 133.2 133.8 (105) 133.6 134.2 (105) 135.8 138.0 (114)

a

Simulated parameters and terms are as specified in footnote to Table 2.

Figure 13. Dependence of experimentally obtained [I(2ωt)]max value on amplitude and frequency. Scan rate: 50 mV s-1. (b) ∆E ) 80 mV, (2) ∆E ) 50 mV, and ([) ∆E ) 30 mV.

Table 5. Comparison of Experimental and Simulated Peak Potential Values Obtained by Second-Harmonic Ac Measurements at a Scan Rate of 102 mV s-1a ∆E ) 30 mV f (Hz)


4.77

18.8

9.54

18.4

19.07

16.0

38.15

14.3

57.22

13.9

76.29

13.7

∆E ) 50 mV


107.5 102.5 (73) 116.8 124.3 (77) 122.5 130.5 (82) 124.6 132.6 (95) 134.7 141.3 (112) 137.8 142.0 (124)

16.0 14.6 13.5 13.4 13.4 13.3

∆E ) 80 mV


130.0 130.0 (83) 130.2 130.2 (86) 132.3 134.8 (86) 133.2 139.0 (94) 135.1 141.2 (110) 140.3 150.0 (117)

14.3 13.2 12.8 12.1 12.0 11.8

∆Eox (mV)

132.8 129.8 (106) 136.8 132.6 (106) 137.5 137.5 (106) 137.5 137.5 (106) 139.9 144.0 (106) 141.3 145.0 (114)

a Simulated parameters and terms are as specified in footnote to Table 2.

No well-defined second-harmonic voltammograms could be obtained at low scan rates and high frequencies because the number of data points that can be collected by the instrumentation is restricted to 32 768. This limitation requires that only a few data points can be collected for each sine wave applied if the scan rate is too low, which leads to aliasing in the data sets when utilizing the FT method. Highly distorted second-harmonic ac voltammograms were obtained at scan rates of g238 mV s-1 and at frequencies lower than 19.07 Hz, because under these conditions the dc and ac components overlap and cannot successfully be resolved by the FT method. However, as shown in data presented in Figure 14, when fast scan rates are employed, use of higher frequencies proved to be tractable. In summary, large-amplitude second-harmonic FT ac voltammetry provides a significant advantage over the conventional smallamplitude ac voltammetric method in that background current is almost entirely absent and the sensitivity to electrode kinetics is very high for surface confined processes. 3.2. Third and Higher Harmonics. No measurable third or higher harmonic ac voltammograms could be detected for azurin 176 Analytical Chemistry, Vol. 76, No. 1, January 1, 2004

Figure 14. Dependence of experimentally obtained [I(2ωt)]max value on amplitude and frequency. Scan rate: (a) 238 and (b) 477 mV s-1. (b) ∆E ) 80 mV, (2) ∆E ) 50 mV, and ([) ∆E ) 30 mV.

at small amplitudes of e10 mV. In general, detection of very well defined third or higher harmonics were found to be highly favored by use of large amplitudes, as expected theoretically. Many aspects of third-harmonic data are again consistent with a k0 value of 70 s-1, but also again, the wave shape is broader than that predicted on the basis of Butler-Volmer theory. However, since the second harmonic clearly is already void of

charging current problems, little advantage is likely to be achieved by employing third or higher harmonic techniques to study the electrochemistry of thin films. CONCLUSIONS The technique of Fourier transform large-amplitude ac cyclic voltammetry has been applied to azurin protein thin-film voltammetry. A major advantage of this method is that, from a single experiment, all of the dc and fundamental and higher ac harmonic data become available. As theoretically predicted, the capacitance current is negligible in large-amplitude second and higher harmonic measurements. Consequently, use of these higher harmonics avoids the need to correct for the capacitance current, which is commonly larger than the Faradaic current in dc and fundamental harmonic forms of protein film voltammetry. The enhanced sensitivity of the large-amplitude second-harmonic method also provides a significant advantage over conventional small-amplitude ac voltammetric methods. Analysis of data obtained with azurin confirms advantages deduced on the basis of simulations of the quasi-reversible process. At a PIGE surface, higher harmonic data confirm that an electron-transfer rate constant of 70 s-1 is consistent with many aspects of the data analysis but that significant forms of nonideal behavior related to kinetic or thermodynamic dispersion, inadequacies of the ButlerVolmer equation used to model the voltammetry, or other phenomena are present, even in this “model” thin-film process. The second and higher harmonic large-amplitude ac voltammetric method described in this paper, as well as square wave7d and pulse methods,36 all have been demonstrated to provide superior Faradaic to charging current ratios than obtained under conditions of linear sweep dc voltammetry for a thin-film process. However, only in the case of the higher harmonic ac method is the charging current theoretically nonexistent for an ideally behaved system. Consequently, the higher harmonic ac method may be expected to be close to optimal with respect to this aspect of thin-film voltammetry. (36) Forster, R. J.; Faulkner, L. R. Anal. Chem. 1995, 67, 1232.

Assuming that the charging current contribution can be adequately separated from the Faradaic component, then the upper limit of k0 that can be determined by the above-mentioned techniques is governed by a complicated combination of effects that include the speed of the electronics and the ability to accommodate distortion arising from uncompensated resistance and RC (R, uncompensated resistance; C, double layer capacitance) time constants. The relatively low k0 value of 70 s-1 for azurin at a PIGE can be determined from slow dc scan rate-low ac frequency ac combinations, which avoid significant uncompensated resistance and RC time constant problems. However, the conservatively designed FT instrument used in the present study actually enables dc scan rates of up to 10 V s-1 and frequencies up to 20 000 Hz to be applied to an electrochemical cell. Input of these high scan rate and high-frequency parameters into the theoretical model based on the Butler-Volmer model and ideal behavior (certainly not exhibited by azurin), implies that measurable departures from a model reversible process would be detectable for k0 values of at least 10 000 s-1, which, as would be expected, is competitive with upper limits available with pulse, square wave, and fast scan rate dc approaches designed when using instruments with equivalent electronic performance. ACKNOWLEDGMENT We are grateful to Dr. H.A. Heering for the provision of the baseline-subtraction software, to Prof. F. Scholz for the provision of the paraffin-impregnated electrodes, and to Dr. A. DiBilio for the provision of high-purity P. aeruginosa azurin. The research was supported by a grant from the Australian Research Council. S.G. thanks Dr. M. B. Rooney for her assistance with the protein film voltammetry at the commencement of this project.

Received for review August 3, 2003. Accepted October 10, 2003. AC034901C


177

Fourier Transform Large-Amplitude Alternating Current Cyclic

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