Anal. Chem. 2006, 78, 6733-6739
Charge Neutralization Process of Mobile Species at Any Distance from the Electrode/Solution Interface. 1. Theory and Simulation of Concentration and Concentration Gradients Developed during Potentiostatic Conditions 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 Departamento de Quı´mica, Universidad Nacional de Rı´o Cuarto, 5800 Rı´o Cuarto, Argentina
The theoretical framework of a general model for the simulation of concentration profiles of electroactive and nonelectroactive species, driven by an electrochemical process under potentiostatic conditions, is presented. Based on this analysis, finite differences simulations are performed to calculate the actual profiles under different experimental conditions. Furthermore, the effect of experimental parameters (diffusion coefficients of the ions of the redox couple or the supporting electrolyte, charge of the different species, etc.) on the concentration profiles is also examined. The results obtained when low and high concentrations of supporting electrolyte are compared aid understanding of the effect of the electrolyte on the measurements. The presented model also underlines the role of supporting electrolyte species when nonspecific techniques are employed to measure the concentration changes produced by electroactive species. On the other hand, if a highly specific technique were used to detect changes in the concentration or concentration gradient of a given species, then it would be possible to estimate the respective profiles of the other species. The simulations suggest that techniques measuring concentration gradients are more sensitive to determining concentration changes than those involving a measurable linearly related to concentration.
transport in the solution.2-5 Additionally, the interpretation of the current responses themselves, which are affected by mass transport of all species, could be impaired without detailed knowledge of the fluxes of all mobile species.4,5 On the other hand, most of modern in situ electrochemical techniques (FT-IR, ellipsometry, reflectance, electrochemical quartz crystal microbalance (EQCM), SPR, SERS, DEMS, etc.) are designed to study only the electrode/electrolyte interface and sometimes they share their limited view with the current-potential measurements.6 To overcome those drawbacks, several techniques have recently been developed to study the fluxes or the concentration values of mobile species away from the electrode. These techniques could be divided into two types: (i) techniques in which the concentration changes occurring in the whole solution space in front of the electrode are measured;7 (ii) techniques in which the concentration (Cj) or the concentration gradient (∂Cj/∂x) of a given species is sampled at a point or through an imaginary line that is close to the electrode surface (e.g., some µm).8-10 The former are easier to implement, but the concentration gradients can only be indirectly determined by fitting a theoretical curve to the experimental response, and thus, they are poorly sensitive to local variations of concentration. The advent of new detection schemes has allowed the sampling of a given concentration (or related data) at positions away from the electrode with good spatial resolution.8,11-13
In electrochemical systems, the current can be controlled by the transport of electrons across the electrode/solution interface or by the mass transport of electroactive (and other ionic) species in the solution.1 Pure electrochemical techniques based on measurements of electron fluxes (current) across the electrode/ electrolyte interface only allow indirect measurement of the mass
(2) Henderson, M. J.; Hillman, A. R.; Vieil, E. J. Electroanal. Chem. 1998, 454, 1. (3) Orellana, M.; Arriola, P.; Del Rı´o, R.; Schrebler, R.; Cordova, R.; Scholz, F.; Kahlert, H. J. Phys. Chem. B 2005, 109, 15483. (4) Garay, F.; Solis V. M. J. Electroanal. Chem. 1999, 476, 165. (5) Garay, F.; Solis V. M. Electroanalysis 2004, 16, 450. (6) 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. (7) Komorsky-Lovric´, Sˇ .; Mircˇeski, V.; Kabbe, Ch.; Scholz, F. J. Electroanal. Chem. 2004, 566, 371. (8) Jan, C.-C.; McCreery, R. L.; Trevor Gamble, F. Anal. Chem. 1985, 57, 1763. (9) Engstrom, R. C.; Meaney, T.; Tople, R.; Wightman, R. M. Anal. Chem. 1987, 59, 2005. (10) Miras, M. C.; Barbero, C.; Kotz, R.; Haas, O.; Schmidt, V. M. J. Electroanal. Chem. 1992, 338, 279.
* Corresponding 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. 10.1021/ac0603678 CCC: $33.50 Published on Web 09/01/2006
© 2006 American Chemical Society
Analytical Chemistry, Vol. 78, No. 19, October 1, 2006 6733
Table 1. Techniques Used To Measure Variables Related to Cj at Nonzero Distances from the Electrode technique
related quantity
measurable
probe
scanning electrochemical microscopy (SECM)13 surface plasmon resonance (SPR)14
current or potential
Cj
ultramicroelectrode
refractive index
Cj
attenuated total reflectance (ATR)
absorbance
Cj
probe beam deflection (PBD)12 differential electrochemical mass spectroscopy (DEMS)15 micro ISE sensor16 photothermal deflection spectroscopy17 spatial beam absorptiometry8
refraction ionic current of volatile substances potential absorbance
∂Cj/∂x dCj/dt
absorbance
Cj
interferometry18
interference fringes
Cj
fiber optic covered with thin gold film diamond prism or IR optical fiber laser beam porous membrane in front of the electrode ISE sensor laser probe beam with a pumped beam laser beam with a spatial detector interfering light beams
Depending on the probe, it is possible to measure concentrations, concentration gradients, or another related parameter (see Table 1). Nevertheless, some of these in situ techniques are highly specific and can monitor one or a few species.8,13,19 While this feature simplifies the analysis, it restricts its use only to studying those few systems in which the species in question are present. On the other hand, there are also less specific (or nonspecific) in situ techniques that could be used in any electrochemical system because the measurable is present in most systems.12,20-23 However, if all ions are measured, the interpretation of the data will be more complex. From Table 1, it can be inferred that the concentration field in front of the electrode surface is a common system to study. Thus, there would be a common calculation scheme allowing us to evaluate the response of those techniques. While analytical solutions have been widely applied in the past for the study of such systems,24 they lack the flexibility required to incorporate the effects of slow charge transfer between the electrode and the electroactive species, the presence of homogeneous chemical reactions, and the presence (or absence) of the so-called “supporting electrolyte” (SE).25-29 One way to overcome such drawbacks implies the use of digital simulation.27-32 When nonspecific techniques (such as PBD, SPR, interferometry, EQCM, etc.) are studied, the contribution of nonelectroactive (11) Engstrom, R. C.; Wightman, R. M.; Kristensen, E. W. Anal. Chem. 1988, 60, 652. (12) Barbero, C. Phys. Chem. Chem. Phys. 2005, 7, 1885. (13) Bard, A. J.; Denault, G.; Lee, C.; Mandler, D.; Wipf, D. O. Acc. Chem. Res. 1990, 23, 357. (14) Wang, S.; Boussaad, S.; Tao, N. J. Rev. Sci. Instrum. 2001, 72, 3055. (15) Baltruschat, H. Interfacial Electrochemistry. Theory, Experiments and Applications; Marcel Dekker: New York, 1999. (16) Sawai, T.; Shinohara, H.; Ikariyama, Y.; Aizawa, M. J. Electroanal. Chem. 1990, 283, 221. (17) Pawliszyn, J. Anal. Chem. 1988, 60, 1751. (18) Muller, R. H. Advances in Electrochemistry and Electrochemical Engineering; John Willey & Sons: New York, 1973; Vol. 9, p 281. (19) Svir, I. B.; Golovenko, V. M. Electrochem. Commun. 2001, 3, 11-15. (20) Bidoia, E. D. Chem. Phys. Lett. 2005, 408, 1. (21) Salavagione, H. J.; Arias-Pardilla, J.; Pe´rez, J. M.; Va´zquez, J. L.; Morallo´n E.; Miras, M. C.; Barbero, C. J. Electroanal. Chem. 2005, 576, 139. (22) Ba´rcena Soto, M.; Kubsch, G.; Scholz, F. J. Electroanal. Chem. 2002, 528, 18. (23) Ba´rcena Soto, M.; Scholz, F. J. Electroanal. Chem. 2002, 528, 27. (24) McCreery, R. L.; Pruiksma, R.; Fagan, R. Anal. Chem. 1979, 51, 749. (25) Hogan, C. F.; Bond, A. M.; Myland, J. C.; Oldham, K. B. Anal. Chem. 2004, 76, 2256. (26) Garay, F.; Lovric, M. J. Electroanal. Chem. 2002, 518, 91. (27) Bieniasz, L. K. J. Electroanal. Chem. 2004, 565, 273.
6734 Analytical Chemistry, Vol. 78, No. 19, October 1, 2006
Cj Cj
species could be evident.12,14,33,34 However, these concentration gradients are always present, and many times their effect should be considered in cases where more specific techniques (e.g., SECM) are employed. Moreover, if the concentration of some nonelectroactive ion were measured by a given technique, the simulation of that system would allow monitoring the concentration of different ions without changing the probe. For instance, if a colored ion (e.g., 2,4,6-trinitrophenoxide) were part of the supporting electrolyte, the concentration of the ion could be measured using an optic fiber by means of focalized spectroelectrochemistry. The changes of the concentration of nonabsorbing redox species could therefore be inferred by the simulation of the absorption data of the supporting electrolyte. In addition, the simulation of concentration profiles away from the electrode should also include the gradients of the supporting electrolyte species.27,35,36 Although several mathematical resolutions of the Nernst-Plank equation have been proposed for more than 30 years, they are commonly associated with some kind of limitation27,36 or assumption.35 Most of these theoretical works have been designed to provide some information about the limiting current of UMEs. In consequence, they have analyzed the effects of migration and diffusion on the concentration profiles of electroactive species mostly under steady-state conditions.36 Thus, it is necessary to develop a general model for the simulation of concentration profiles corresponding to electroactive and nonelectroactive ions. This model should allow us to connect the data obtained from in situ techniques to the respective voltammetric results. As stated above, the simulation of concentration profiles away from the electrode could also be used to support the presence of concentration gradients and the electron flux at the electrode surface.19,29,37 Furthermore, other authors have also (28) Miles, A. B.; Compton, R. G. J. Electroanal. Chem. 2001, 499, 1. (29) Thompson, M.; Klymenko, O. V.; Compton, R. G. J. Electroanal. Chem. 2005, 576, 333. (30) Feldberg, S. W. In Electroanalytical Chemistry; Bard, A. J., Ed.; New York, 1969; Vol. 3. (31) Britz, D. Digital Simulation in Electrochemistry, 2nd ed.; Springer: Berlin, 1988. (32) Storzbach, M.; Heinze, J. J. Electroanal. Chem. 1993, 346, 1. (33) Correia, J. P.; Vieil, E.; Abrantes, L. M. J. Electroanal. Chem. 2004, 573, 299. (34) Henderson, M. J.; Hillman, A. R.; Vieil, E. Electrochim. Acta 2000, 45, 3885. (35) Brumleve, T. R.; Buck, R. P. J. Electroanal. Chem. 1978, 90, 1. (36) Hyk, W.; Stojek, Z. Anal. Chem. 2005, 77, 6481. (37) Myland, J. C.; Oldham, K. B. Anal. Chem. 1999, 71, 183.
found that relatively simple theoretical models could be applied to predict the consumption or the generation of species in more complex situations.11,12 This is especially relevant when electrochemical ion releasers, ion sensors, or both are being studied. In the present article, the theoretical framework of the charge neutralization process occurring during electrochemical oxidation/ reduction of soluble species at any distance from the electrode/ solution interface is described. The data obtained for concentration and concentration gradient profiles are compared when a potential pulse perturbation is applied. To provide a detailed explanation related to the time and space evolution of the concentrations and concentration gradients of every species, only the Ox species has been considered to be charged in this paper. However, the effects generated by the charges of both electroactive species are discussed in depth through part 2 of this work.38 THEORETICAL MODEL
m
∑z C
j j(x,t)
∆ze-
Oxzo(sol) 798 Redzr(sol)
(1)
Here, zo and zr are the charge numbers of oxidized and reduced species (Red). In addition, a reversible charge-transfer reaction involving ∆z ) (zo-zr) electrons is assumed. If a quiescent solution and a linear mass transfer along the x-axis (normal to the electrode surface) is considered, the problem of reaction 1 can be described by the continuity equation, the Nernst-Planck equation, and the electroneutrality condition:35
∂Cj(x,t)/∂t ) Dj{∂2Cj(x,t)/∂x2+ zj∂[Cj(x,t)(∂Φ(x,t)/∂x)]/∂x}
Jj(x,t) ) -Dj[(∂Cj(x,t)/∂x) + zjCj(x,t)(∂Φ(x,t)/∂x)]
(4)
where zj is the charge, Cj denotes the concentration, Dj is the diffusion coefficient and Jj indicates the flux of the jth species in solution, m corresponds to the total number of soluble species, and Φ(x,t) refers to the electric potential in the electrolyte normalized by RT/F. On the basis of these assumptions, the scheme of Figure 1 can be described according to the following set of boundary conditions:
t ) 0, x g 0:
Co(x,0) ) C/o;
Cr(x,0) ) 0
(5)
CK(x,0) ) C/SE
(6)
CK(x,0) ) -(zo/zK)C/o + C/SE
(6a)
CA(x,0) ) -(zo/zA)C/o + C/SE; CA(x,0) ) C/SE;
Figure 1 presents a scheme of some general characteristics of techniques that, depending on the probe, can measure parameters related to Cj or ∂Cj/∂x at nonzero distances from the electrode. According to the scheme of Figure 1A, a semi-infinite diffusion of involved species and an infinite plane electrode with negligible border effects has been assumed. The finite difference method is a widely used numerical technique to solve the mass transport phenomena in electrochemical systems.27-32,39 To resolve the differential equations describing the system, an array of boxes corresponding to a space/time fraction is considered, Figure 1B. The absence of concentration gradients of electroactive and nonelectroactive species and the electroneutrality condition is assumed into each box. A simple reversible electrochemical redox reaction was evaluated where only the oxidized species (Ox) was present. Additionally, an inert SE composed of a cation K+ and an anion A- can also be in the solution.
)0
j
Φ(x,0) ) 0 t > 0, x f ∞:
(7)
Co(x,t) f C/o;
Cr(x,t) f 0
(8)
CK(x,t) f C/SE
(9)
CK(x,t) f -(zo/zK)C/o + C/SE
(9a)
Φ(x,t) f 0
(10)
CA(x,t) f -(zo/zA)C/o + C/SE; CA(x,t) f C/SE;
t > 0, x ) 0:
Jo(0,t) ) -Jr(0,t) ) -I(t)/∆zFA
(11)
JA(0,t) ) JK(0,t) ) 0
(12)
Co(0,t) ) exp[φt] Cr(0,t)
(13)
boundary conditions 6 and 9 are applied if Ox is a cationic species, while conditions 6′ and 9′ stand if Ox is an anionic species. In consequence, it is assumed that the counterion of Ox is the same species as one of the supporting electrolyte ions. This is not a constraint of the model, but it will diminish the number of species involved to four in order to simplify the discussion. It is important to note that eq 4 is not limited to four ions and that a higher number of soluble species could be considered. As indicated above, fast electron-transfer kinetics is assumed, and therefore, the Nernst equation is applied at the electrode surface, eq 13. The parameter A is the electrode surface area, φt is defined by eq 14, and C/SE denotes the concentration of SE in bulk solution. The other symbols have their usual meaning.26
φt ) zF[Et - E°]/RT
(14)
(2) (3)
where Et and E° are the applied and the standard potentials, respectively. The total current density at any location in solution is given by the following equation:1 m
(38) Garay, F.; Barbero, C. A. Anal. Chem. 2006, 78, 6740-6746. (39) Gavaghan, D. J. J. Electroanal. Chem. 1997, 420, 147.
JJ>(x,t) ) F
∑[z J
j j(x,t)]
(15)
j
Analytical Chemistry, Vol. 78, No. 19, October 1, 2006
6735
the second terms of the numerator and denominator of eq 18. The second term of the numerator involves only the concentration of Ox species, while the first term has a summation of every concentration multiplied by the square of its respective charge. Accordingly, this last equation gives the limit Jo(0,t) ) -Do(∂Co(0,t)/ ∂x), indicating that, if the condition [SE] . [Ox] stands, the migrational contribution would only affect the fluxes of supporting electrolyte ions but not those of electroactive species. The derivation of eq 17 provides the following form for the continuity equation:
∂Cl(x,t)/∂t ) - Dj
Figure 1. (A) Scheme for the detection of the fluxes of soluble species at nonzero distances from the electrode/solution interface. (B) Array of boxes considered for the implementation of the finite difference method.
In view of eqs 15 and 4, JJ(x,t) does not depend on x and it can be written as JJ(t) ) -I(t)/A. Thus, it can be combined with eqs 3 and 11 to obtain the expression for the unknown potential gradient: m
∑[z D (∂C
∂Φ(x,t)/∂x ) - (Jo(0,t)∆z +
j
j
j(x,t)/∂x)])
×
j
m
(
∑[z
DjCj(x,t)])-1 (16)
2 j
j
The denominator is proportional to the total conductivity at any point of the solution. Therefore, if the conductivity of the solution is very high, the potential gradient will become negligible in eq 3 and the fluxes of electroactive species will be given by Fick’s first law. The combination of eqs 3 and 16 results in
(
{
m
∂2Cj(x,t)
∂x
Jo(0,t)∆z +
∑[z D (∂C j
j
j
Dj zjCj(x,t)
2
j
DjCj(x,t)]
j
Consequently, the flux of Ox at x ) 0 can be calculated from eqs 11 and 17:
(
Jo(0,t) )
∂Co(0,t) ∂x
m
m
∑[D z
2
j j
j
j j
m
∑[D z
Cl(x,t) -
2
∂Cj(x,t)/∂x]
j j
j
m
∂x
∑[D z
2
j j
)}
Cj(x,t)]
j
Cj(x,t)]
j
(19)
where l is one of the m-species in solution. Considering the relevant boundary conditions, the solutions of eqs 16 and 18 and the set of equations represented by eq 19 were solved numerically.31,32 All calculations were performed using a Visual FORTRAN 6.0 software package. The simulations of chronoamperometric experiments were performed considering a total time ttot ) 60 s and that each time increment was equal to 1.2 ms. Thus, a set of 5 × 104 divisions was necessary to simulate every experiment. This high number of divisions can be considered as an overcalculation of the current, but it was required to evaluate the concentration change of soluble species at some micrometers from the electrode surface. All concentrations were normalized with C/o and the final expression of current was normalized with the Cottrell equation:
I(t)xπt zFAC/oxDo
(20)
[
∑ zD
Cj(0,t)] - zoCo(0,t)
j
j
])
∂Cj(0,t) j
∂x
The dimensionless parameter h ) Doδt(δx)-2 was fixed at 0.10. This last value was suitable to keep the stability of our calculations when the value of Do is changed. The variable δx corresponds to the width of each ith spatial compartment, and this parameter will vary between 1 and 10 µm when the values of Do are changed from 10-6 to 10-4 cm2 s-1, respectively. The surface concentration of Co was obtained by the following expression:
Cto,0 ) [1 + exp(-φt)]-1
(21)
m
∑[D z
2
j j
Cj(0,t)] - DozoCo(0,t)∆z
j
(18)
It should be noted that, for [SE] . [Ox], it is possible to simplify 6736
+ 2
(17)
m
∑[z
-Do
j
∑[D z
Ψ(t) )
j(x,t)/∂x)]
∂ Cj(x,t)/∂x2]
m
Jj(x,t) ) -Dj(∂Cj(x,t)/∂x) + m
2 2
j j
-
∂x2
∂Φ(x,t) ∂Cl(x,t) zl
∑[D z
zlCl(x,t)
Analytical Chemistry, Vol. 78, No. 19, October 1, 2006
The concentration gradient of each species at a given distance of the electrode surface (∆x ) iδx) can be calculated as follows:39 t t t t t Cj,i-1 ∂Cj,i - 8Cj,i + 8Cj,i+2 - Cj,i+3 ) ∂x 12δx
(22)
Figure 2. Theoretical concentration-time profiles calculated for ∆x ) 100 µm, Ein ) E2 ) 0.3 V, E1 ) -0.3 V, Dr ) Do ) DK+ ) DA-) 10-5 cm2 s-1, ∆z ) 1, zr ) 0; in (A) [Ox]/[SE] ) 10-3, in (B) [Ox]/[SE] ) 10.
Figure 3. Theoretical ∂Cj/∂x-time profiles calculated for ∆x ) 100 µm, Ein ) E2 ) 0.3 V, E1 ) -0.3 V, Dr ) Do ) DK+ ) DA- ) 10-5 cm2 s-1, ∆z ) 1, z r ) 0; in (A) [Ox]/[SE] ) 10-3, in (B) [Ox]/[SE] ) 10.
RESULTS AND DISCUSSION Using eqs 16, 18, and 19, it is possible to determine the timedependent concentration profiles of the mobile species in front of the electrode surface. Figure 2 shows the concentration profiles calculated at 100 µm for a redox reaction occurring in the presence of excess of supporting electrolyte (A) and in the presence of an amount of supporting electrolyte smaller than that of the redox species (B). In the latter case, the effects related to the solution resistance have not been considered. According to the proposed model, however, the ohmic drop effect would be negligible if the applied potential is high enough or the ohmic resistance is removed by a positive feedback circuit. In consequence, even under these conditions, the contributions of diffusion and migration corresponding to the electroactive species can be readily evidenced and described. In both cases, the curves were evaluated by assuming a common value for the diffusion coefficients, D ) 1 × 10-5 cm2 s-1, of every species. The concentration profiles of Ox species exhibit the expected behavior for an electroactive-oxidized species during successive reduction and oxidation potential pulses. However, during the reduction pulse, Co shows a significant diminution when the concentration of SE is relatively high due to the fact that the ions of SE afford the whole migration contribution, Figure 2A. On the contrary, if the concentration of SE species is low, an extra amount of Ox has to migrate from the bulk to decrease in the local excess of charge at the electrode, Figure 2B. This is a well-known characteristic of electrochemical systems studied without SE because the flux of current is higher when SE is not present.1 A similar conclusion can be drawn from the behavior of Red species. In this case, however, Red is an uncharged species and the charge balance does not affect it. Therefore, the extra flux of Ox will increase not only the current but also the amount of reduced species at x ≈ 0. On the other hand, SE species will define symmetrical concentration profiles when they are in great excess, Figure 2A. Yet, the symmetry of these profiles would be lost if the concentrations of both ions were very different, because one of the ions is the counterion of the Ox or Red species. This occurs for the case described in Figure 2B, where very different concentrations of anionic and cationic species (A- and K+) are present and where the migration of the most concentrated one prevails.
It must be noticed that nonnegligible concentration changes can be observed for all soluble species despite the fact that they are at 100 µm from the electrode surface. Such results do not indicate that there are nonequilibrated charges at that position, but that the concentrations of soluble species are not the same as in the bulk. In relation to this, there are several remarkable works pointing out how techniques as SECM,13 ATR ,or spatial beam absorptiometry8 are able to measure the concentration changes of redox species at a given distance from the electrode. Although these techniques are usually very specific to determine concentrations of electroactive species, this is not so for SE concentrations, because they do not have enough precision to measure the very small changes that the ionic species of SE undergo, Figure 2A. The theoretical results presented so far can also be used to evaluate the concentration gradient (∂Cj/∂x) of each species by employing eq 22. Figure 3 indicates the dependence on time for the ∂Cj/∂x of each species analyzed. The calculations assume that the probe is placed at 100 µm when successive reduction and oxidation potential pulses are applied. In Figure 3A, there is a great excess of SE, whereas in Figure 3B the concentration of SE is only 10% of Ox species. In both cases, the curves were calculated by assuming a common value for the diffusion coefficients, D ) 1 × 10-5 cm2 s-1, of all species. Although the concentrations of redox and SE species differ markedly, their profiles show significant dependence on time in both figures. This is a highly significant fact since, unlike the direct study of concentration profiles, the behavior of every ∂Cj/∂x should be easily detectable either in the presence of excess or of a small amount of SE. This is a remarkable feature of the concentration gradients as it would allow indirect monitoring of the concentration gradient of a “silent” jth species (e.g., a nonabsorptive species in spectroscopy) from the profiles of the others. On the other hand, if the technique applied is sensitive to all gradients (e.g., probe beam deflection), a mixed response will be obtained. As can be seen, the profiles of ∂Cj/∂x manifest a maximum, a minimum, or both. This is a very important characteristic that could be used to estimate the value of D (see below). In Figure 3A, a maximum of ∂Co/∂x and a minimum of ∂Cr/∂x are observed during the reduction pulse. These profiles exhibit at the same time their respective maximum and minimum values, but their Analytical Chemistry, Vol. 78, No. 19, October 1, 2006
6737
Figure 4. Theoretical concentration-time profiles calculated for ∆x ) 100 µm, Ein ) E2 ) 0.3 V, E1 ) -0.3 V, [Ox]/[SE] ) 10-3, ∆z ) 1, zr ) 0, Dr ) DK+ ) 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). Panels A-D are associated with species Ox+, Red0, K+, and A-, respectively.
shapes are opposed as if they reflect each other. This is also valid for the curves of SE species; however, in this case, the maximum and minimum values are half of those obtained for the redox species. This is due to the fact that both SE species migrate with the same values of C, D, and |z|, and therefore, each one cancels half of the charge left by the redox process. When only a small amount of SE is used and the anion of SE is also the counterion of Ox species, the concentrations of A- and K+ species will greatly differ from each other, Figure 3B. As in the case of concentration profiles, both the different concentrations of SE species and the enhanced migration of Ox species do not allow the symmetry of the curves of ∂Cj/∂x. The diffusion coefficients of species involved are the parameters that control the mass transport. To simplify the calculations, it is usually considered that all Dj are the same during the simulation of a typical electrochemical experiment.1,26 However, such an assumption should not be used to simulate profiles obtained by nonspecific in situ techniques, since they detect the movement of all ions.40 Moreover, the diffusion coefficients of every species involved (Ox, Red, A-, K+) have been taken into account in these studies. The effects of varying the diffusion coefficient of Ox ions in simulated concentration profiles of electroactive and SE species are shown in Figure 4A,B and Figure 4C,D, respectively. The concentration of SE species was assumed to be in great excess (40) Rudnicki, J. D.; Brisard, G. M.; Gasteiger, H. A.; Russo, R. E.; McLarnon F. R.; Cairns, E. J. J. Electroanal. Chem. 1993, 362, 55.
6738 Analytical Chemistry, Vol. 78, No. 19, October 1, 2006
Figure 5. Theoretical ∂Cj/∂x-time profiles calculated for ∆x ) 100 µm, Ein ) E2 ) 0.3 V, E1 ) -0.3 V, [Ox]/[SE] ) 10-3, ∆z ) 1, zr ) 0, Dr ) DK+ ) DA- ) 10-5 cms-1, Do/cms-1 × 10-6 ) 100 (a), 60 (b), 30 (c), 10 (d), 6 (e), and 3 (f). Panels A-D are associated with species Ox+, Red0, K+, and A-, respectively.
and Do values were varied in order to consider the very fast as well as the relatively slow diffusion of ions in water. With regard to curves a, for the fastest diffusion of Ox, its concentration reaches a plateau analogous to the one found for Ox concentration at x ) 0, Figure 4A. When the value of Do is small, however, the concentration profile of Ox does not reach a plateau during the period analyzed. On the other hand, the profiles of Red species never show a plateau, despite being produced by the diffusion of Ox species, Figure 4B. This results from two facts: first, its diffusion coefficient is 1 × 10-5 cm2 s-1, 10 times lower than the one required by Ox to reach the plateau; second, Red species has no charge and therefore its mass transport is controlled only by diffusion. Consequently, for the cases in which Ox diffusion is very fast, the uncharged Red species will be accumulated close to the electrode surface without altering the migration of other ions. In relation to the profiles of A- and K+ species, they track those of Ox, but their concentration changes are exactly half of the ones of Ox. This is because both species have the same values of C, D, and |z|, and thus, they are equally contributing to migration. It is also interesting to note that both species reach a sort of plateau when Ox diffusion is very fast. The data suggest that an extremely precise technique would be necessary to measure the variations predicted for the concentration profiles of SE ions. The dependence of the simulated ∂Cj/∂x profiles on time and on different values of Do is shown in Figure 5A,B, for electroactive species, and in Figure 5C,D for SE species. In Figure 4, Red species is uncharged and the diffusion coefficients of all species
but Ox are equal to 10-5 cm2 s-1. In addition, the concentration of SE was assumed to be in great excess regarding the concentration of Ox species. Since Ox species is consumed at the electrode surface during the reduction pulses (Figure 5A), all curves of ∂Co/∂x show a maximum at those times. Each maximum, however, occurs at a different time, which depends on the value of Do by the following equation:12,41
tmax ) ∆x2(2Dj)-1
(23)
where tmax is the time at the maximum (or minimum) of the concentration gradient of a given species j when it is analyzed at a distance ∆x from the electrode surface. The presence of the maximum points that the largest diminution of Co at 100 µm occurs when t ) tmax. As regards ∂Cr/∂x, every curve in Figure 5B has a different magnitude, but all of them share the same tmax value. That is so because, for this set of curves, Dr is always the same and the Red species is uncharged. Thus, eq 23 is also applicable to Red species regardless of it having a fast or slow diffusion in relation to that of Ox. As mentioned above, the migration of SE ions cancels the charge left during the consumption of Ox species. The profiles of A- and K+ are, therefore, very similar and have half of the magnitude of the corresponding ∂Co/∂x, Figure 5C,D. Consequently, the profiles of ∂CK+/∂x and of ∂CA-/∂x do not follow eq 23 since these ions are controlled by migration. In fact, these profiles are governed by the charges left by the electroactive species during the redox process. Because of this, the behavior of Ox (and/or Red) species could be studied if a certain probe were employed to measure the profiles of ∂CK+/∂x or ∂CA-/∂x. CONCLUSIONS A general simulation scheme for the time-dependent concentration (Cj(x,t)) and concentration gradient (∂Cj/∂x(x,t)) profiles (41) Barbero, C.; Miras, M. C.; Kotz, R. Electrochim. Acta 1992, 37, 429.
produced during oxidation/reduction of soluble electroactive species and in the presence of different amounts of supporting electrolyte has been presented. The calculations allow predicting features of the system, such as dependence on the relative values of diffusion coefficients or on the charge of the electroactive species. As was expected, when the system involves solutions with great excess of SE in relation to the concentration of redox species, the concentration of SE ions undergoes much smaller relative changes than the redox species does. However, the concentration gradients of SE ions have the same magnitude as those of electroactive species, since they have to migrate in order to cancel the excess of charge left by the redox species. It is important to note, however, that, with respect to the results described above, it would be suitable to study parameters related to the concentration gradients of such species, even in the presence of high concentrations of SE. In addition, it is preferable that the soluble species have different mobility in order to simplify the deconvolution of the global response. Yet, nonspecific techniques involving the measurement of parameters directly related to concentration may not be suitable to detect changes in the concentration of SE species at or close to the electrode surface. On the other hand, variations in the concentrations or concentration gradients could be monitored by highly specific techniques, but they would be limited to the analysis of one or few particular species. This theoretical work provides a general expression for the diffusion and migration contribution of electroactive and nonelectroactive species not only at the electrode surface but also at any distance from it. ACKNOWLEDGMENT This work was funded by Fundacio´n Antorchas, FONCYT, CONICET, and SECYT-UNRC. C.A.B. and F.G. are permanent research fellows of CONICET. Received for review February 27, 2006. Accepted July 25, 2006. AC0603678
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