Anal. Chem. 2007, 79, 4110-4119
General Theory of Cathodic and Anodic Stripping Voltammetry at Solid Electrodes: Mathematical Modeling and Numerical Simulations Sarah E. Ward Jones, Franc¸ ois G. Chevallier, Christopher A. Paddon, and Richard G. Compton*
Physical and Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford OX1 3QZ, United Kingdom
Theory is presented to describe the voltammetric signals associated with the stripping phase of stripping voltammetry at solid electrodes. Three mathematical models are considered, and the importance of the hemispherical diffusion associated with electrochemical dissolution of particles in the micrometer range is investigated. Model A considers a “monolayer” system where the coverage at a specific point cannot exceed a maximum value. Model B considers a thin layer of metal or metal oxide, but in contrast to model A, the maximum surface coverage is not restricted. Model C represents the stripping of a “thick layer” where the deposition is also unrestricted. Stripping voltammetry is widely used in electroanalysis to offer high sensitivity and, in many cases, good selectivity particularly for the detection of metal ions.1,2 Stripping analysis is a two-step technique. The first step involves the electrolytic deposition of the metal ions onto the electrode surface. This has the effect of preconcentrating them. The second step involves the dissolution of the deposit electrochemically via a cathodic sweep (cathodic stripping voltammetry, CSV) or more usually an anodic sweep (anodic stripping voltammetry, ASV), and the charge passed during this step reflects the concentration of the metal in solution. The preconcentration leads to a significant improvement of the detection limit compared to direct solution-phase electroanalysis. Concentrations as low as 10-10 M are readily achieved in favorable circumstances. Hitherto, most ASV has been conducted at mercury electrodes where the reduction of the metal ion leads to the formation of an amalgam. The theory of ASV under these conditions has been formulated.3-6 However, recent concerns about the toxicity of mercury combined with environmental legislation and pressures has seen a major impetus into the use of solid electrodes for * To whom all correspondence should be addressed. E-mail: richard.
[email protected]. Tel: +44 (0) 1865 275 413. Fax: +44 (0) 1865 275 410. (1) Brainina, K. Z.; Neyman, E. Electroanalytical Stripping Methods; J. Wiley & Sons: New York, 1993. (2) Wang, J. Analytical Electrochemistry, 2nd ed.; J. Wiley & Sons: New York, 2000. (3) Powell, M.; Ball, J. C.; Tsai, Y. C.; Suarez, M. F.; Compton, R. G. J. Phys. Chem. B 2000, 104, 8268-8278. (4) Ball, J. C.; Compton, R. G. Electroanalysis 1997, 9, 1305-1310. (5) Ball, J. C.; Cooper, J. A.; Compton, R. G. J. Electroanal. Chem. 1997, 435, 229-239. (6) Ball, J. C.; Compton, R. G. J. Phys. Chem. B 1998, 102, 3967-3973.
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stripping voltammetry. In addition, the desire to make portable instruments and carry out field measurements has further encouraged this trend. The use of boron-doped diamond7-13 and edge plane pyrolytic graphite electrodes14,15 has been promoted as well as the use of screen-printed electrodes.16,17 In the case of metal (or metal oxide for the case of CSV) deposition on solid electrodes, the current-voltage characteristics of the stripping signals used analytically are relatively unexplored except for the work of Brainina18,19 on carbon paste electroactive electrodes, work on abrasively modified electrodes,20 and the work of Mouhandess21-23 and Grygar24,25 on the electrodissolution of metal oxide particles. All of these are highly approximate theories. The purpose of the present paper is to develop a rigorous basis for a general theory of stripping voltammetry at solid electrodes. The information generated from this kind of theory would allow for the optimization of experimental conditions to maximize sensitivity and also to look at cases were there is a deviation from linearity. (7) Goeting, C. H.; Marken, F.; Gutierrez-Sosa, A.; Compton, R. G.; Foord, J. S. New Diamond Front. Carbon Technol. 1999, 9, 207-228. (8) Compton, R. G.; Foord, J. S.; Marken, F. Electroanalysis 2003, 15, 13491363. (9) Manivannan, A.; Kawasaki, R.; Tryk, D. A.; Fujishima, A. Electrochim. Acta 2004, 49, 3313-3318. (10) Manivannan, A.; Tryk, D. A.; Fujishima, A. Electrochem. Solid State Lett. 1999, 2, 455-456. (11) Prado, C.; Wilkins, S. J.; Marken, F.; Compton, R. G. Electroanalysis 2002, 14, 262-272. (12) Ward-Jones, S.; Banks, C. E.; Simm, A. O.; Jiang, L.; Compton, R. G. Electroanalysis 2005, 17, 1806-1815. (13) Simm, A. O.; Banks, C. E.; Ward-Jones, S.; Davies, T. J.; Lawrence, N. S.; Jones, T. G. J.; Jiang, L.; Compton, R. G. Analyst 2005, 130, 1303-1311. (14) Banks, C. E.; Compton, R. G. Analyst 2006, 131, 15-21. (15) Wantz, F.; Banks, C. E.; Compton, R. G. Electroanalysis 2005, 17, 655661. (16) Honeychurch, K. C.; Hart, J. P. TrAC-Trends Anal. Chem. 2003, 22, 456469. (17) Wang, J.; Tian, B. M. Anal. Chem. 1992, 64, 1706-1709. (18) Brainina, K. Z.; Lesunova, R. P. Zh. Anal. Khim. 1974, 29, 1302. (19) Brainina, K. Z.; Vydrevich, M. B. J. Electroanal. Chem. 1981, 121, 1-28. (20) Chevallier, F. G.; Goodwin, A.; Banks, C. E.; Jiang, L.; Jones, T. G. J.; Compton, R. G. J. Solid State Electrochem. 2006, 10, 857-864. (21) Mouhandess, M. T.; Chassagneux, F.; Vittori, O.; Accary, A.; Reeves, R. M. J. Electroanal. Chem. 1984, 181, 93-105. (22) Mouhandess, M. T.; Chassagneux, F.; Vittori, O. J. Electroanal. Chem. 1982, 131, 367-371. (23) Mouhandess, M. T.; Chassagneux, F.; Durand, B.; Sharara, Z. Z.; Vittori, O. J. Mater. Sci. 1985, 20, 3289-3299. (24) Grygar, T. J. Solid State Electrochem. 1998, 2, 127-136. (25) Grygar, T.; Marken, F.; Schro ¨der, U.; Scholz, F. Collect. Czech. Chem. Commun. 2002, 67, 163-208. 10.1021/ac070046b CCC: $37.00
© 2007 American Chemical Society Published on Web 05/01/2007
Figure 1. Schematic diagram showing the model for the stripping phase of anodic and cathodic stripping voltammetry.
THEORY We consider electrochemically driven stripping from an electrode with a total surface area Aelec and which initially contains N electroactive metal or, in the case of CSV, metal oxide centers of radius rd, which is assumed constant for all nuclei, as illustrated in Figure 1. The initial metal or metal oxide center distribution reflects that observed by AFM in previously work published by this group.26 The electrode reaction of interest is assumed to follow a one-electron-transfer mechanism where the species of interest is stripped from the electrode surface into the solution using linear sweep voltammetry.
B(s) ( e- f A(aq)
(1)
Extension to the case of n electrons is trivial. Each center is approximated as a cylinder as shown in Figure 1b and then approximated to be a flat disk (Figure 1c) by assuming that the height of the center is very small. The local position-dependent coverage on the metal center, Γ, is initially set as uniform across the center and given a value of Γmax. We will consider three different models for the stripping process. The first model, model A, considers a “monolayer” system where the position-dependent coverage at a specific point cannot exceed the initial value of Γmax. Model B considers a thin layer of metal or metal oxide, but in contrast to model A, the maximum surface coverage is not restricted and the solution species can freely be deposited anywhere on the metal center. Model C represents the stripping of a “thick layer”, where the deposition is also unrestricted. A thick layer is considered to be multiple layers of metal on the electrode such that, for the major part of the stripping, when layers of metal are removed there are still more layers of the same underneath rather than the electrode surface. The differences between the models will be clarified further below in terms of the formulation of mathematical boundary conditions to the diffusion problem considered. Simulating Random Arrangements of Metal or Metal Oxide Centers. By employing the diffusion domain approach,27-29 (26) Hyde, M. E.; Banks, C. E.; Compton, R. G. Electroanalysis 2004, 16, 345354. (27) Brookes, B. A.; Davies, T. J.; Fisher, A. C.; Evans, R. G.; Wilkins, S. J.; Yunus, K.; Wadhawan, J. D.; Compton, R. G. J. Phys. Chem. B 2003, 107, 16161627. (28) Davies, T. J.; Brookes, B. A.; Fisher, A. C.; Evans, R. G.; Wilkins, S. J.; Yunus, K.; Wadhawan, J. D.; Compton, R. G. J. Phys. Chem. B 2003, 107, 64316444.
Figure 2. Coordinate system used to model the diffusion domain. The plane to be simulated is highlighted.
we consider the electrode surface as an ensemble of independent diffusion domains of radius r0 with the metal or metal oxide center at the center (see Figure 2). The cylindrical symmetry of the domain allows the behavior of the domain to be simulated by just considering the plane highlighted in Figure 2, simplifying the initial three-dimensional problem down to two dimensions. Essentially, the task is to simulate the voltammogram for every domain present on the electrode surface and then add them together. In the case of a regular arrangement of metal or metal oxide centers, all the diffusion domains on the electrode surface are the same and it is simple to extend the simulation from one metal or metal oxide center to the whole electrode.29 However, in this paper, we consider the more complex case of a random distribution of metal or metal oxide centers as is found in stripping voltammetry. We assume that all the metal or metal oxide centers have the same radius, rd, but are distributed in a random nature giving rise to a distribution of diffusion domains with different domain radii, r0. The microscopic coverage, θ, of a single diffusion domain is equal to
θ ) πrd2/πr02 ) rd2/r02
(2)
and the macroscopic, or global coverage, Θ, of the whole electrode is equal to
Θ ) Ndiskπrd2/Aelec
(3)
When the metal or metal oxide centers are randomly distributed on the electrode surface, each diffusion domain corresponds to a Voronoi cell30 as illustrated in Figure 3. In order to use the diffusion domain approach to simulate a random arrangement, it is necessary to calculate the area distribution of the Voronoi cells. In this paper, we use an analytical function for the normalized (29) Davies, T. J.; Compton, R. G. J. Electroanal. Chem. 2005, 585, 63-82. (30) Davies, T. J.; Moore, R. R.; Banks, C. E.; Compton, R. G. J. Electroanal. Chem. 2004, 574, 123-152.
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Figure 3. Schematic diagram of the diffusion domain approximation used when modeling random arrangements.
Figure 4. Example of the radius distribution obtained Θ ) 0.5, rd ) 1 × 10-3 cm, and Aelec ) 1 cm2.
size distribution of two-dimensional Voronoi cells obtained by Ja´rai-Szabo´ and Ne´da:
f2D(y) )
343 15
x2π7 y
(5/2)
(
exp -
7 y 2
)
(4)
where y ) S/〈S〉 with S being the area of Voronoi cells.31 They validated this function by comparison with computer simulations using up to 3 million Voronoi cells, which were constructed using the “disk growth” method.32 The area distribution can then be calculated and converted into real variables. By assigning each area an equivalent cylindrical domain of radius r0, as illustrated in Figure 3b and c, a radius distribution can be obtained. Figure 4 illustrates an example of the radius distribution obtained. The voltammetric response for the whole electrode can be simulated using the five steps illustrated in Figure 5 with the radial distribution allowing us to weight the individual diffusion domain voltammograms. This approach to random distributions has already been used for simulating many partially blocked electrodes27,28,33-38 as well as (31) Ja´rai-Szabo´, F.; Ne´da, Z. arXiv Condens. Matter e-prints 2004, 6, 116. (32) De Berg, M.; Van Kreveld, M.; Overmars, M.; Schwarzkopf, O. Computational Geometry, Algorithms and Applications; Springer: Berlin, 1997. (33) Chevallier, F. G.; Davies, T. J.; Klymenko, O. V.; Jiang, L.; Jones, T. G. J.; Compton, R. G. J. Electroanal. Chem. 2005, 577, 211-221. (34) Chevallier, F. G.; Davies, T. J.; Klymenko, O. V.; Jiang, L.; Jones, T. G. J.; Compton, R. G. J. Electroanal. Chem. 2005, 580, 265-274. (35) Chevallier, F. G.; Jiang, L.; Jones, T. G. J.; Compton, R. G. J. Electroanal. Chem. 2006, 587, 254-262. (36) Davies, T. J.; Brookes, B. A.; Compton, R. G. J. Electroanal. Chem. 2004, 566, 193-216.
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Figure 5. Scheme for the steps involved in simulation the voltammogram for a random arrangement of metal centers.
microdisk arrays29,39 and has been shown to retain physical accuracy. Modeling Single Diffusion Domains. (1) Mathematical Model. The redox reaction follows Butler-Volmer kinetics according to the following equation where A is the solution species and B is the surface-bound species: ka
A(aq) + e- y\ z B(s) k
(5)
d
where ka is the rate constant for addition to the surface and kd is the rate constant for dissolution:
(
ka ) kf exp kd ) kb exp
(
RF (E - E0f ) RT
)
(6)
)
(1 - R)F (E - E0f ) RT
(7)
where kf and kb are the standard rate constants for deposition and dissolution, respectively, and are linked via the Nernst equation so that kb ) kf[A]*. R is the transfer coefficient and E0f is the formal potential and [A]*) 1 × 10-3 mol cm-3. The mass transport (37) Davies, T. J.; Lowe, E. R.; Wilkins, S. J.; Compton, R. G. ChemPhysChem 2005, 6, 1340-1347. (38) Davies, T. J.; Garner, A. C.; Davies, S. G.; Compton, R. G. J. Electroanal. Chem. 2004, 570, 171-185. (39) Davies, T. J.; Ward-Jones, S.; Banks, C. E.; Del, Campo, J.; Mas, R.; Munoz, F. X.; Compton, R. G. J. Electroanal. Chem. 2005, 585, 51-62.
in a uniform layer equal to the maximum coverage. The initial conditions (t ) 0) of the species taking part in the reaction are
Table 1. Boundary Conditions for the Real Single Diffusion Domain Model equations
boundary region
[A] ) 0 (∂[A])/(∂r) ) 0 (∂[A])/(∂r) ) 0 (∂[A])/(∂z) ) 0 model A D(∂[A])/(∂z) ) ka(1 σ)[A]Z)0 - kdσ model B D(∂[A])/(∂z) ) ka[A]Z)0 - kdσ model C D(∂[A])/(∂z) ) ka[A]Z)0 - kd
[A] ) 0
for for for for
0 e r e r0 r)0 r ) r0 rd < r e r0
z)∞ 0ez ka[A]Z)0 - kd for any time step, dt, in the simulation then the value of kd is adjusted so that kd∂t ) - Γmax∂σ + ka[A]Z)0∂t. Before the electrochemical experiment is started, there is no species A in the solution and B is present on the electrode surface
rd
0
and
(8)
∂σ ) ka(1 - σ)[A]Z)0 - kdσ ∂t
∫
∂A ∂2A 1 ∂A ∂2A ) + + ∂τ ∂R2 R ∂R ∂Z2
model A:
where [A] is the concentration and D is the diffusion coefficient of species A. Three different models A, B, and C will be considered in this paper. The change in the local position-dependent coverage of species B for each metal center takes a different form in each model as shown below.
Γmax
0 e r e rd
(2) Normalized Model. The model presented in the previous section was normalized using the set of dimensionless parameters displayed in Table 2. The mathematical model can then be rewritten as
model B:
model A:
(12)
The boundary conditions completing the mathematical model are given in Table 1. For each individual diffusion domain, the current flowing at the electrode surface can be calculated using the following expression:
parameter
(
for
0ez
