Charge Neutralization Process of Mobile Species at Any Distance from

Sep 1, 2006 - ... load: https://cdn.mathjax.org/mathjax/contrib/a11y/accessibility-menu.js .... Charge Neutralization Process of Mobile Species at Any...
0 downloads 4 Views 140KB Size
Download PDF
Anal. Chem. 2006, 78, 6740-6746

Charge Neutralization Process of Mobile Species at Any Distance from the Electrode/Solution Interface. 2. Concentration Gradients during Potential Pulse Experiments Fernando Garay*,† and Cesar A. Barbero‡

INFIQC, Departamento de Fı´sico Quı´mica, Fac. de Ciencias Quı´micas, UNC, Pab. Argentina, Ala 1, 2° piso, Ciudad Universitaria, 5000 Co´ rdoba, Argentina, and the Departamento de Quı´mica, Universidad Nacional de Rı´o Cuarto, 5800 Rı´o Cuarto, Argentina

The theoretical model presented in part 1 of this work is employed to simulate and fit experimental probe beam deflection (PBD) data of Fe(CN)63-/Fe(CN)64- and Fe3+/ Fe2+ couples. Current and beam deviation dependency on time at constant potential (chronoamperometry and chronodeflectometry) is analyzed via a new treatment based on the migration and diffusion properties of all the species involved. The diffusion coefficients of electroactive species are obtained by fitting chronoamperometric curves. Those coefficients are then employed to simulate the respective chronodeflectometric profiles. The experimental data and the theoretical function are fitted by the minimum squares Simplex algorithm. The effect of working with systems in which both electroactive species are charged is discussed in detail. Specifically, the possibility of quantitative analysis of nonspecific techniques data is analyzed when a relative high concentration of supporting electrolyte is used. Such analysis widens the scope of techniques as PBD since in many cases the effect of supporting electrolyte species could be negligible as compared to the response of electroactive species. The variation of the refraction index with the concentration gradient of each soluble species is also discussed. In electrochemical systems, the current flux can be controlled by the transport of electrons across the electrode/solution interface or by the mass transport of electroactive (and nonelectroactive) species in the solution.1 While the measurement of current under various potential perturbations could provide information about the mass transport of species in solution, such information is indirect and modulated by the charge transfer at the interface. Then, it is desirable to have direct ways to measure the mass transport. In principle, any measuring technique for concentrations in solution could be used to measure the concentration gradients * Cirresponding author. Tel: +54 351 4334169/80. Fax: +54 351 4 334188. E-mail: [email protected]. † INFIQC. ‡ Universidad Nacional de Rı´o Cuarto. (1) Bard, A. J.; Faulkner, L. R. Electrochemical Methods, Fundamentals and Applications, 2nd ed.; J. Wiley, Inc.: New York, 2001.

6740 Analytical Chemistry, Vol. 78, No. 19, October 1, 2006

produced during electrochemical experiments. However, most techniques lack the necessary spatial resolution to measure the concentration gradient profiles; only interferometry2 and probe beam deflection3-6 (PBD) have been used to monitor concentration gradients. Also, spatially resolved UV-visible spectroelectrochemistry has been found by McCreery et al. to be suitable to measure the concentration gradients of absorbing species.7 In electrochemical experiments, a relatively large amount (at least 10 times the concentration of the electroactive species) of a nonelectroactive salt is usually added to the solution. This reduces the solution resistance since it carries out most of the ionic charges in order to suppress the migration of the electroactive species.1 However, a long-standing question in experimental design is how to ensure that the measured profiles are not affected by the presence of the added salt.8,9 Thus, in most cases, it would be useful to examine mass transport of electroactive and nonelectroactive species by specific or nonspecific techniques. The so-called in situ techniques are experiments coupled to an electrochemical system and controlled by electrochemical equipment. Additionally, the in situ technique used has to give some physical parameter that provides extra information of the electrode process.3,4,10-13 In part 1 of this work, several in situ techniques have already been summarized pointing out that all of them have different drawbacks and advantages depending on the experimental problem, the time scale used, the technique (2) Muller, R. H. Advances in Electrochemistry and Electrochemical Engineering; John Wiley & Sons: New York, 1973; Vol. 9, p 281. (3) Rudnicki, J. D.; Brisard, G. M.; Gasteiger, H. A.; Russo, R. E.; McLarnon F. R.; Cairns, E. J. J. Electroanal. Chem. 1993, 362, 55. (4) Henderson, M. J.; Hillman, A. R.; Vieil, E. J. Electroanal. Chem. 1998, 454, 1. (5) Barbero, C. Phys. Chem. Chem. Phys. 2005, 7, 1885. (6) Correia, J. P.; Vieil, E.; Abrantes, L. M. J. Electroanal. Chem. 2004, 573, 299. (7) Jan, C.-C.; McCreery, R. L.; Trevor Gamble, F. Anal. Chem. 1985, 57, 1763. (8) Planes, G.; Miras, M. C.; Barbero, C. Polym. Int. 2002, 51, 429. (9) Rooney, M. B.; Darren, C. C.; Bond, A. M. Anal. Chem. 2000, 72, 3486. (10) Ba´rcena Soto, M.; Kubsch, G.; Scholz, F. J. Electroanal. Chem. 2002, 528, 18. (11) Ba´rcena Soto, M.; Scholz, F. J. Electroanal. Chem. 2002, 528, 27. (12) Bard, A. J.; Abrun ˜a, H. D.; Childsey, C. E.; Faulkner, L. R.; Feldberg, S. W.; Itaya, K.; Majda, M.; Melroy, O.; Murray, R. W.; Porter, M. D.; Soriaga, M. P.; White, H. S. J. Phys. Chem. B 1993, 97, 7147. (13) Ryan, M. D.; Bowden, E. F.; Chambers, J. Q. Anal. Chem.1994, 66, 360R. 10.1021/ac0603680 CCC: $33.50

© 2006 American Chemical Society Published on Web 09/01/2006

sensitivity.14 For example, despite being one of the most widely used in situ techniques for the study of ionic exchange, quartz crystal microbalance cannot discriminate between the individual fluxes of ions when both anionic and cationic fluxes are mixed together with a coupled flux of solvent during the exchange of species.15 Moreover, it is almost impossible to determine the mass exchanged if film swelling effects are involved. Conversely, PBD provides a fast and localized measurement of the dynamic behavior of anionic and cationic fluxes.5,14-16 The latter technique is based on the refraction property: when a light beam crosses a transparent medium with a nonhomogeneous index of refraction, it deviates from its initial trajectory in the direction of the increasing index. Yet, there is still no general theory that establishes a connection between the current and the deflection profiles of soluble species. Owing to this, it has been generally assumed that only measurements performed in binary (one anion and one cation) electrolytes could be interpreted univocally.5 The diffusion of neutral salts, however, could increase the ionic force locally, but it will not neutralize the charge developed during the redox process.1,9 Recently, Correia et al. have employed convolution to study the multifluxes of several ionic species that may be involved in the highly complex electropolymerization process of 3-methylthiophene.6 Nevertheless, neither the diffusion coefficients nor the changes in the refraction index corresponding to each species were considered in that publication. In the first part of this series,14 both the theoretical framework necessary to simulate the charge neutralization process and the simulation of concentration profiles were described. The couples Fe(CN)63-/Fe(CN)64- and the aqueous species Fe3+/Fe2+ reacting on a gold electrode could be the most widely used examples of outer-sphere redox reactions and perhaps the simplest systems to validate our model.9,17-19 Yet, it is necessary to add suitable supporting electrolytes (SEs) to these redox couples in order to work with proper conditions of pH, resistance, and reversibility of the charge-transfer reaction. Nevertheless, the presence of supporting electrolyte complicates the analysis of PBD signals since a relationship between the fluxes of every ionic species and the changes of the refraction index of these solutions is required.3,14 Perhaps for this reason, there are only few works related to PBD experiments of soluble redox couples.3,17 In this work, the simulations presented in part 1 are extended to fit PBD experimental data of the couples Fe(CN)63-/Fe(CN)64and the aqueous species Fe3+/Fe2+ under chronoamperometric conditions. It is important to remark, however, that the aim of this work is not to study the reaction mechanisms of those species at the electrode surface19,20 but to describe their ionic fluxes at some micrometers from the electrode surface. EXPERIMENTAL SECTION All solutions were prepared with ultrapure water (18 MΩ cm-1) from a Millipore MilliQ system. The solution of potassium (14) Garay, F.; Barbero, C. A. Anal. Chem. 2006, 78, 6733-6739. (15) Vieil, E.; Lopez, C. J. Electroanal. Chem. 1999, 466, 218. (16) Grumelli, D. E.; Wolosiuk, A.; Forzani, E.; Planes, G. A.; Barbero, C.; Calvo, E. J. Chem. Commun. 2004, 24, 3014. (17) Barbero, C.; Miras, M. C.; Kotz, R. Electrochim. Acta 1992, 37, 429. (18) Bard, A. J.; Denault, G.; Lee, C.; Mandler, D.; Wipf, D. O. Acc. Chem. Res. 1990, 23, 357. (19) Orellana, M.; Arriola, P.; Del Rı´o, R.; Schrebler, R.; Cordova, R.; Scholz, F.; Kahlert, H. J. Phys. Chem. B 2005, 109, 15483. (20) Campbell, S. A.; Peter, L. M. J. Electroanal. Chem. 1994, 364, 257.

ferricyanide was prepared with 1.3 mM K3Fe(CN)6 (Sigma) and 0.1 M KCl (Merck). The solution of aqueous Fe(II) involved 0.33 mM FeCl2 (Sigma) and 0.1 M HCl (Merck). All chemical reagents were of analytical grade and used as received. The electrochemical control of PBD experiments was performed using a potentiostat (AMEL 2049). The setup was controlled with a PC through a LabPC 1200 AD/DA card running in homemade software performed with LabView 5.1 (National Instruments). The deflection and the electrochemical signals were jointly saved in the PC. The PBD arrangement was similar to the one described before.7 The basic components of the PBD system were a 5-mW He-Ne laser (Melles Griot, 05 LHP11) and a bicell position-sensitive detector (UDT PIN SPOT /2D). All parts of the system were mounted on an optical rail, resting on a stable optical bread-board (Melles Griot). The laser beam was focused by a 50mm lens to a diameter of roughly 60 µm in front of the planar electrode. The electrochemical cell, a 2 × 2 cm optical glass cuvette with 2 cm of path length, was mounted on a three-axis tilt table (Newport). The working electrode was a slide of glass covered with a sputtered film of gold. This electrode provides an optically flat surface of 0.897 × 3.5 cm, and it was used without further treatment. The exposed area was measured with a relatively big error A ) 1.8 ( 0.1 cm2, but the interaction length between the beam and the zone of perturbed index was determined with good precision L ) 0.897 ( 0.005 cm. A coiled Pt wire was the counter electrode, and all potentials of experimental data are referred to a Ag|AgCl|Cl- (3 M) reference electrode. The counter and the reference electrodes are situated facing the working electrode, outside of the path of the beam. A micrometric translation stage allowed controlling the positioning of the sample with respect to the laser beam in 10-µm steps. The position sensitive detector was placed 25 cm behind the electrochemical cell and had a sensitivity of 3 mV/µm, giving a deflection sensitivity of 1 mrad/V. The signal of the two photodiodes making the bicell detector was subtracted and normalized to the overall signal in order to minimize the effect of laser intensity fluctuations. All data were collected during 60 s. To ensure stationary state Ein ) 0.15 V was applied during 4 min before starting the oxidation of the aqueous species of Fe2+. After this initial potential, the first pulse E1 ) 0.65 V was applied during 30 s, and then the potential was shifted back to 0.15 V to record the reducing signals during the last 30 s. A similar procedure was used for the solution of K3Fe(CN)6 where the working electrode was kept at 0.4 V during 4 min. During the first 30 s, the potential E1 ) -0.1 V was applied at t0 producing the reduction of Fe(CN)63- and then the potential was shifted back to 0.4 V to collect the data of the oxidation step during the last 30 s. All calculations were performed using a Visual FORTRAN 6.0 software package. RESULTS AND DISCUSSION Effect of Charge on the Migration of SE Species. In part 1, a model was presented where a simple reversible electrochemical redox reaction involving the transfer of ∆z ) (zo - zr) electrons was considered, eq 1. There, the reduced species (Red) was uncharged (zr ) 0) and only the oxidized species (Ox) was present at t ) 0. Analytical Chemistry, Vol. 78, No. 19, October 1, 2006

6741

∆ze-

Oxzo(sol) 798 Redzr(sol)

(1)

The boundary conditions also consider the presence of an inert SE composed of a cation K+ and an anion A-. In addition, it was assumed that the mass transport is controlled by diffusion and migration contributions, and therefore, the charge exchanged during the redox process affects the concentration profiles of all ionic species. Because of this, it is important to discuss how the concentration profiles of SE species are affected when both electroactive species are charged. Figure 1 shows theoretical concentration (A) and concentration gradient profiles (B) calculated at 100 µm and where both redox species are charged according to eq 2. Both figures exhibit rather ∆ze-

Ox+2(sol) 798 Red+(sol)

(2)

asymmetrical profiles due to the very different diffusion coefficients that have been considered for the species involved. Since a great excess of SE was assumed during the calculations, the concentration profiles of electroactive species are controlled only by diffusion. Consequently, the excess of charge left by the redox species during the electrochemical reaction will only affect the concentration profiles of SE species. In other words, the current will exhibit the same profile regardless of whether the electrochemical reaction is given by eq 1 or eq 2, if ∆z ) 1. The concentration profiles of Ox species show the expected behavior for an electroactive-oxidized species during successive reduction and oxidation potential pulses, Figure 1A. On the other hand, the concentration of Red species increases during the reduction pulse. However, if the concentration changes of Ox and Red species are compared, it is evident that the increment in the concentration of Red species is larger than the diminution of Ox during the reductive potential pulse. This behavior corresponds to the accumulation of Red species close to the electrode surface since the diffusion coefficient of Ox (Do) is higher than that of Red (Dr). With regard to this last point, it is important to note that the ratio between the diffusion coefficients of the electroactive species of real systems is hardly ever of an order of magnitude. However, these theoretical curves have been calculated by considering that Do was 10 times higher than Dr in order to do the effect associated with the accumulation of Red species more evident. As stated above, the SE species have to cancel the charge left by the redox ions during the electrochemical process. In addition, it is well known that the migration of a given species depends on the concentration, the charge, and the diffusion coefficient of such species.1 Accordingly, the contribution of the cation of the SE to the total migration is bigger than that of the anion because DK+ is 10 times higher than DA-, Figure 1A. However, under these conditions, the magnitude and not the shape of the concentration and concentration gradients of SE species depends on their diffusion coefficients. In fact, the shape of these profiles is determined by the charge and the concentration profiles of electroactive species. Regarding the concentration gradients of electroactive species, they can manifest a maximum or a minimum depending on the electrochemical reaction that takes place at that time, Figure 1B. 6742

Analytical Chemistry, Vol. 78, No. 19, October 1, 2006

Figure 1. Theoretical Cj/C/o -time (A) and ∂Cj/∂x-time (B) profiles calculated for Eini ) E2 ) 0.3 V, E1 ) -0.3 V, [Ox]/[SE] ) 10-3, ∆x ) 100 µm, Do ) DK+ ) 10-4 cm2 s-1, Dr ) DA- ) 10-5 cm2 s-1, ∆z ) 1, zr ) 1.

The faster diffusion of Ox provides a maximum at shorter times than the minimum of Red species. The time of the maximum (tmax) is an important parameter that could be used to estimate the value of Do with the following equation:17

tmax ) ∆x2(4Dj)-1

(3)

where Dj is the diffusion coefficient of the jth species in solution. Likewise, the value of tmin can be introduced in eq 3 to determine the value of Dr. The maximum of ∂Co/∂x in Figure 1B indicates the moment at which the largest diminution of Co is achieved at 100 µm. Perhaps a more interesting application of eq 3 is to estimate the distance ∆x between the probe and the electrode surface of some relatively simple electrochemical systems.3,17 Usually, the actual value of ∆x is an unknown parameter that can be estimated by repeating the potential pulse experiment at a series of probe offsets. To do this, the position of the sample with respect to the laser beam is changed by a micrometric translation stage. If only one electroactive soluble species is charged and the migrational effects are neglected in the analyzed system, a linear plot of tmax1/2 as a function of the different distances (x) should be obtained. Thus, the value of ∆x could be then calculated from the linear regression data.17 Nevertheless, eq 3 is not right to determine ∆x values when both electroactive species are charged and have different diffusion coefficients.3 With reference to the concentration gradient profiles of SE species, they can exhibit a maximum and a minimum during the oxidation as well as the reduction pulse, Figure 1B. Moreover, the dependence of tmax and tmin on the values of ∆x2 shows straight lines even for the curves of SE species. Nevertheless, those ions are controlled by migration and their tmax and tmin values cannot be introduced into eq 3 to determine their diffusion coefficients. Figure 2 shows concentration gradients of SE species calculated for different Do values, considering a great excess of SE species and assuming that both electroactive species are charged, eq 2. The concentration gradients of redox species are identical to those discussed in part 1 and correspond to a system in which only the oxidized species is charged.14 However, the migration of SE species is strongly dependent on the charges of Ox and Red

and t denote the finite difference form for a given space grid point and time level and (∂n/∂Cj) is the concentration refractivity of the jth species in solution.14,15 Thus, it is possible to calculate the variation of the refractive index along the x-axis where the perturbation of the redox reaction evolves. Furthermore, the theory of probe beam deflection considers that the dependence of the deviation angle of the light beam on the refractive index gradient is given by the following equation:5,15

Θti

Figure 2. Theoretical ∂CK+/∂x-time (A) and ∂CA-/∂x-time (B) profiles calculated for ∆x ) 100 µm, Eini ) E2 ) 0.3 V, E1 ) -0.3 V, [Ox]/[SE] ) 10-3, ∆z ) 1, zr ) 1, DK+ ) 10-4 cm2 s-1, Dr ) DA- ) 10-5 cm2 s-1, Do/cm2 s-1 × 10-6 ) 100 (a), 60 (b), 30 (c), 10 (d), 6 (e), and 3 (f).

species. During the reduction pulses, the dependence of ∂CK+/∂x on time shows not only a maximum but also a minimum because SE ions have to cancel the charge left during the consumption of Ox and the generation of Red. Consequently, the position of the maximum and the minimum of the ∂CK+/∂x and ∂CA-/∂x will be given by the respective values of Do and Dr. Thus, the profile of ∂CK+/∂x will only show a minimum or a maximum when the values of Do and Dr are similar or equal (curves 2d). The same statement is valid for the dependence of ∂CA-/∂x on time. To evaluate the effect of the diffusion coefficients of SE ions, the following relationship DK+ ) 10DA- was employed. Because of this, the variations of ∂CK+/∂x are practically 10 times larger than the ones found for the anion. Yet, it should be noted that the time in which these maximums and minimums appear depends on the values of Do and Dr and not on the diffusion coefficients of SE species. Application of the Model To Fit Probe Beam Deflection Data. To validate the effects predicted by the model, theoretical concentration gradients were employed to fit experimental chronoamperometric and chronodeflectometric profiles. The wellknown relationship between the current and the concentration gradients of redox species at x ) 0 is given by

Jo(0,t) ) -Jr(0,t) ) -I(t)/∆zFA

(4)

where Jo(0,t) and Jr(0,t) are the fluxes of Ox and Red species at the electrode surface, respectively. If we consider the temperature at the electrode surface constant and the variation of the refractive index due to only the changes in the concentration of soluble species, then the concentration gradients of soluble species and the refractive index gradients can be related by eq 5,15 where i

( ) ( ) t dnj,i

dx

)

t dCj,i

∂n ∂Cj dx

(5)

)

L

m

∑

( ) t dnj,i

n j)1 dx

(6)

where L is the interaction length between the beam and the zone of perturbed index, m corresponds to the total number of soluble species, and n is the refraction index of solution. The respective neutral salts can be rebuilt by a simple stoichiometric exercise from any of the simulated concentration profiles since the migration of SE species cancels the excess of charge left by the redox ions. However, there would still be positive and negative concentration gradients for each ionic species. Therefore, it was considered that each soluble species (anions, cations, or neutral molecules) should have its own values of Dj and ∂n/∂Cj in eqs 5 and 6. Thus, individual ionic conductivities were employed to estimate Dj values of SE species, and the ∂n/∂C values of neutral salts were consulted. The latter are constants that can be obtained from values labeled # in Table S-1 (Supporting Information). The ∂n/∂C values of each ionic species were calculated by fitting the deflection curves, but they were not considered as free parameters.6 Each of these constants has to be added to the contribution of other ions to rebuild the global ∂n/∂C value of the respective neutral salt. In this respect, it was assumed that those ∂n/∂C values obtained for ions can be added in the same way as in the case of mixed neutral species, but following the stoichiometric relationship of the former neutral salt.5,21 Analysis of the Couple Fe3+/Fe2+ in Acid Medium. Figure 3 shows experimental (circles) and simulated (full lines) chronoamperometric (A) and chronodeflectometric (B) profiles corresponding to a solution with FeCl2 0.33 mM in HCl 0.1 M. The so-called Simplex algorithm22 was employed in order to fit the experimental data to the theoretical function. This algorithm calculates how close the theoretical equation is to the measured data by the following equation of minimum squares: Ω

χ2 )

∑(Data

exp

- Datatheo)2

(7)

i)1

where Ω denotes the amount of data corresponding to the experimental curve and χ2 is the function that the program tries to minimize. This fitting algorithm requires a set of starting values or “seeds” for the parameters that will be fitted. (21) Rosolen, J. M.; Decker, F.; Fracastoro-Decker, M.; Gorenstein, A.; Torresi, R. M.; Cordoba de Torresi, S. I. J. Electroanal. Chem. 1993, 354, 273. (22) Press, W. H.; Teukolsky, S. A.; Vettrerling, W. T.; Flannery, B. P. Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed.; Cambridge University Press: New York, 1997.

Analytical Chemistry, Vol. 78, No. 19, October 1, 2006

6743

Figure 3. Experimental I-t (A) and θ-t (B) profiles (circles) obtained for a solution with FeCl2 0.33 mM in HCl 0.1 M, Ein ) E2 ) 0.15 V, E1 ) 0.65 V, ∆x ) 70 µm. Full lines correspond to theoretical fits while dotted and dashed lines correspond to the contribution of each species to the total deflection.

Figure 4. Circles are experimental I-t (A) and θ-t (B) profiles (circles) obtained for a solution with K3Fe(CN)6 1.3 mM in KCl 0.1 M, Ein ) E2 ) 0.4 V, E1 ) -0.1 V, ∆x ) 69 µm. Full lines correspond to theoretical fits while dotted and dashed lines correspond to the contribution of each species to the total deflection.

Table 1. Values of Dn/DCj Determined for the Species of Figures 3B and 4Ba

ionic species. Since all the salts employed in this experiment have as anionic species chloride, its contribution to the total deflection was employed to calculate, by a stoichiometric relationship, the contributions of other ionic species to the total deflection. The existence of a maximum and a minimum during the same pulse has previously been observed in systems involving electroactive polymers.6,16 However, this is the first time that an experimental signal of soluble species presenting a maximum and a minimum during each potential pulse is simulated and fitted by considering the diffusion and migration of ionic species. As explained above, this dual behavior can be related to the diffusion coefficient of the ionic species involved. Accordingly, those systems in which both redox species are charged, soluble, and with quite different diffusion coefficients would show a maximum and a minimum during a chronodeflectometric experiment. Analysis of the Couple Fe(CN)63-/Fe(CN)64- in Neutral Medium. Figure 4 shows experimental (circles) and simulated (full lines) chronoamperometric (A) and chronodeflectometric (B) profiles corresponding to a solution with K3Fe(CN)6 1.3 mM, KCl 0.1 M at pH 7. These experimental data were fitted as described above for the case of the couple Fe3+/Fe2+. Nevertheless, it is important to note that the chronoamperometric response of the couple Fe(CN)63-/Fe(CN)64- on gold corresponds now to a reduction and a subsequent oxidation potential pulse, Figure 4A. Fit of these data provides the values DFe(CN)63- ) 7.3 × 10-6 cm2 s-1 and DFe(CN)64- ) 6.4 × 10-6 cm2 s-1. These results were obtained as described above, and they are also in very good agreement with data found in the literature.1,23 It would be worth noting that the diffusion coefficients of the different species of Fe(II) and Fe(III) are quite similar, since the sizes of hexaaqua and hexacyanide iron complexes are comparable. The values obtained for DFe(CN)63and DFe(CN)64- were considered, together with the values of DCland DK+ ) 2.0 × 10-5 cm2 s-1, as parameters to fit the deflection profiles of Figure 4B. As previously described, the diffusion

concn (×10-3 M-1)

concn (×10-3 M-1) ∂n/∂CHCl ∂n/∂CKCl ∂n/∂CFeCl3 ∂n/∂CFeCl2 ∂n/∂CK3Fe(CN)6 ∂n/∂CK4Fe(CN)6 ∂n/∂CCoCl2

7.94# 9.60# 41.14# 28 51.64# 70.55# 26.58#

∂n/∂CH+ ∂n/∂CK+ ∂n/∂CCl∂n/∂CFe3+ ∂n/∂CFe2+ ∂n/∂CFe(CN)63∂n/∂CFe(CN)64-

1.3 3.0 6.6 22 15 42 59

a The sign (#) corresponds to data obtained from Table S-1 (Supporting Information).

The chronoamperometric response of the couple Fe3+/Fe2+ on gold shows a typical profile during the oxidation and reduction pulses, Figure 3A. The fit of these data yields the values DFe3+ ) 4.7 × 10-6 cm2 s-1 and DFe2+ ) 8.9 × 10-6 cm2 s-1. These results were obtained employing eqs 11, 18, and 20 of part 1.14 Furthermore, the values found for DFe3+ and DFe2+ are in very good agreement with others found in the literature, and they were employed as parameters to fit the deflection profiles.1,23 The values of DCl- ) 2.0 × 10-5 cm2 s-1 and DK+ ) 8.3 × 10-5 cm2 s-1 were also used as parameters to fit the deflection profiles of Figure 3B. These last values were calculated from their respective individual ionic conductivities.1,24 The seeds employed to fit the deflection profiles were ∆x, estimated with eq 3 from the values of the minimum deflection, ∂n/∂CFeCl2, ∂n/∂CH+, and a scale factor related to the parameter L. ∂n/∂CFeCl2 was estimated from the value of CoCl2. Table 1 summarizes the concentrative refractivities of the salts used in this experiment and the concentrative refractivities found for each (23) Beriet, C.; Pletcher, D. J. Electroanal. Chem. 1993, 361, 93. (24) Coury, L. Curr. Sep. 1999, 18, 91.

6744 Analytical Chemistry, Vol. 78, No. 19, October 1, 2006

Figure 5. Experimental θ-t (A) profiles (circles) obtained for a solution with K3Fe(CN)6 1.3 mM in KCl 0.1 M, Ein ) E2 ) 0.4 V, E1 ) -0.1 V, ∆x ) 109 µm. Full lines correspond to theoretical fits while dotted and dashed lines correspond to the contribution of each species to the total deflection. Theoretical ∂Cj/∂x-t (B) and Cj-t (C) curves obtained from the fit of the deflection curve.

coefficients of SE ions were calculated from their respective individual ionic conductivities.1,24 This time the seeds employed to fit the deflection profiles were ∆x, estimated with eq 3 from the values of the minimum deflection, a scale factor, and the value of ∂n/∂CK+. The concentrative refractivities of the salts used in this second experiment are listed in Table 1 together with the values found for the couple Fe3+/ Fe2+. The salts included in this second experiment have as a common component the cation K+, whose concentrative refractivity was employed to calculate the contributions of the other ionic species. At this point, it is important to compare the results presented in Figures 3 and 4. Both chronodeflectometric profiles show a prevalent negative signal during the first pulse. In Figure 3, however, the first pulse corresponds to the oxidation of Fe2+, while in Figure 4 it involves the reduction of Fe(CN)63- . Thus, different chemical species of iron, presenting the same redox state, would have opposite deflection signals when they are oxidized or reduced. To explain this, it is necessary to notice the contribution of each species to the total deflection signal, eq 6. The electroactive species, mainly the most charged ones, would be associated with the highest ∂n/∂C values in Table 1. This could also be inferred from the ∂n/∂C values of Table S-1 (Supporting Information). Thus, the major contribution to the deflection signal of the system Fe3+/Fe2+ is given by the Fe(III). Yet, the contribution of ferrocyanide will prevail in the system Fe(CN)63-/Fe(CN)64-.

Figure 5 shows experimental (circles) and simulated (full lines) chronodeflectometric (A) profiles corresponding to the same solution depicted in Figure 4 but recorded when the beamelectrode distance was experimentally increased in 40 µm. Figure 5B exhibits the dependence on time for the concentration gradients, and Figure 5C shows the dependence on time for the concentration profiles of each soluble species. The latter has also been obtained from the fitting of the deflection curve. The fits of experimental deflectometric curves of Figures 4B and 5A required values of ∆x of 69 and 109 µm, respectively. The difference between these results agrees with the experimental increment on ∆x. This last outcome allows some evaluation of the model by comparing experimental and theoretical increments on the ∆x values obtained for a set of chronodeflectometric profiles. In other words, if the suggested model properly describes the analyzed experimental systems, then the curves must be correctly fitted providing information related to the fluxes of species and the actual separation between the electrode surface and the laser beam. With regard to Figure 5A and B, it is noteworthy that, although concentration gradients of SE species are large, their influence on the deflection profiles is negligible due to the high ∂n/∂Cj values of the redox species. This is in agreement with a previous work where the authors concluded that concentration gradients of SE species should be large during the redox process.3 However, it has now became evident why practically the same response of deflection is obtained when different SE are used with the Fe(CN)63-/Fe(CN)64- system3. Figure 5C also shows that the concentrations of SE species at some micrometers from the electrode surface have almost the same values as the bulk. Nevertheless, the concentration gradients of SE species are still particularly important at 100 or 200 µm from the surface, as pointed out above, giving a chance to measure the changes of those relatively inert species. CONCLUSIONS The dependence of a given concentration gradient on time could not only exhibit either a maximum or a minimum, but both features can also be observed during a unique reduction or oxidation profile. The occurrence of additional extreme values will mainly depend on the charges and relative values of the diffusion coefficients of the redox species. Provided only one redox species is charged, it should be possible to estimate the diffusion coefficient of such charged electroactive species by analyzing the position of the maximum or the minimum of concentration gradient profiles. However, when more than one redox species is charged, a fit of experimental data would be necessary to estimate the diffusion coefficients. It is the first time that an experimental signal of soluble species presenting a maximum and a minimum during each potential pulse is simulated and fitted. As noted before, this dual behavior is explained only by considering the diffusion coefficient of each ionic species. In addition, the presence of a maximum and a minimum during a given chronodeflectometric experiment allows the straightforward deconvolution of the contributions corresponding to oxidized and reduced species. Thus, simulation of such profiles would be simpler. Analytical Chemistry, Vol. 78, No. 19, October 1, 2006

6745

Not only the concentration gradients of electroactive species but also the ones of SE ions are significant at some micrometers from the electrode surface. It could be assumed that those species with the highest ∂n/∂C values will have the greatest effect on the deflectometric curves. However, the influence of the different involved species should be considered in order to provide a correct fit of the deflection profiles. In this work, a theory considering the fluxes due to the diffusion and migration of all species involved has been applied for the first time to analyze PBD responses. This theory extends the use of the PBD technique to study the movement of electroactive and nonelectroactive species when an excess of SE is used.

6746

Analytical Chemistry, Vol. 78, No. 19, October 1, 2006

ACKNOWLEDGMENT This work was funded by FONCYT, CONICET, Fundacio´n Antorchas. and Secyt-UNRC. C.A.B. and F.G. are permanent research fellows of CONICET. SUPPORTING INFORMATION AVAILABLE Additional information as noted in text. This material is available free of charge via the Internet at http://pubs.acs.org.

Received for review February 27, 2006. Accepted July 25, 2006. AC0603680