Accurate Analytical Expressions for Stripping Voltammetry in the

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Accurate Analytical Expressions for Stripping Voltammetry in the Henry Adsorption Limit Juan Jose Calvente* and Rafael Andreu Departamento de Química Física, Facultad de Química, Universidad de Sevilla, 41012, Sevilla, Spain ABSTRACT: A strategy is developed to derive accurate analytical expressions for low-coverage cathodic stripping voltammetry. The procedure relies on the observation that diffusion affects the location of simulated voltammetric waves but not their shape, provided that physisorption of the analyte is negligible. As a proof of the generality of the proposed approach and having in mind the stripping of thiols, analytical solutions are derived for the cathodic stripping of monomers, dimers, and a mixture of monomers and dimers, whose reliability is proved by their comparison with numerically simulated voltammograms. Application to the deposition and reductive desorption of mercaptoacetic acid at a mercury electrode demonstrates that these approximate solutions can be used to get insights into the interfacial organization of incipient films. For this particular system, a transition from monomeric to dimeric behavior is identified upon increasing the thiol surface concentration. Further generalization of the proposed methodology is achieved by deriving an approximate analytical solution for thin-layer anodic stripping voltammetry, which is satisfactorily compared to the existing summation series solution.

S

tripping voltammetry has become a popular technique for electrochemical detection of species that can be accumulated onto (or into) an electrode.1,2 Detection limits as low as 1011 M have been achieved by using its pulse versions. Anodic stripping voltammetry (ASV), the most widely used form of stripping analysis, has been traditionally applied to detect amalgam-forming metals, whereas cathodic stripping voltammetry (CSV) is mostly used to detect organic and inorganic substances whose oxidized form produces an insoluble salt upon adsorption onto an anodically polarized electrode.35 In the former case, the reduced form of the target metal diffuses within the electrode, so that it is accumulated in a small volume; whereas in the second case, the adsorbed species is accumulated at the electrode surface. Cathodic stripping voltammetry can also be used to gain mechanistic information on the growth and dissolution of organic films.6 However, this type of application is hampered by the lack of simple analytical expressions that can describe quantitatively the stripping response. Among the factors that complicate the theoretical description of CSV are the ubiquitous steric restriction, due to the limited number of available surface sites, the frequently encountered intermolecular interactions, and the presence of chemical reactions coupled to the redox process. The theory for simulating CSV, including dimerization6 and reorientation7 of the adsorbate, has been developed in relation to the growth of thiol self-assembled monolayers. To take full advantage of this technique as a tool to characterize the state of adsorbed molecules, it would be desirable to have simple, albeit approximate, analytical expressions describing the entire voltammetric wave, at least for certain limiting cases. One limit of interest is the ideal dilute film, where the adsorbate is surrounded by solvent molecules and steric restrictions are negligible (i.e., the Henry adsorption limit). Then, the theory of Henry cathodic stripping voltammetry (H-CSV) can be envisaged as a yardstick, r 2011 American Chemical Society

whose comparison with experimental results will provide new insights into the interfacial organization of incipient films. From a mathematical point of view, H-CSV is similar to thinlayer anodic stripping voltammetry (TL-ASV), since in both cases no concentration gradient exists inside the electrode and steric restrictions are negligible. Theoretical description of TL-ASV started with the work of De Vries and van Dalen8,9 and Brainina10 who derived close-form equations for some characteristic features of the stripping voltammetric wave (peak potential, peak current, and full width at half-maximum). In a more recent work, Schiewe et al.11 have revisited this problem and derived an analytical expression for TL-ASV as a summation series. Several diagnostic criteria to assess its agreement with experimental results were also reported. The only drawback of their approach is the need to use a high number of significant digits in order to avoid the loss of precision at the positive potential side of the wave. Theory for convective anodic stripping voltammetry has also been developed by several research groups.1216 On the basis of their similarities, solutions for TL-ASV can be straightforwardly converted into their H-CSV analogues and vice versa. However, equations developed hitherto for TL-ASV at stationary electrodes are restricted to a first order redox process, whereas cathodic stripping voltammetry frequently requires consideration of additional chemical reactions.6,7 In the present work, a general strategy to derive approximate analytical solutions for H-CSV is outlined. The rationale behind this strategy is the transitional nature of CSV between diffusive voltammetry (DV), where both redox forms are freely diffusing, and film voltammetry (FV), where the two forms of the redox Received: June 10, 2011 Accepted: July 7, 2011 Published: July 07, 2011 6401

dx.doi.org/10.1021/ac201473z | Anal. Chem. 2011, 83, 6401–6409

Analytical Chemistry

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Figure 1. Schematic illustration of the relationship between voltammetric protocols involving freely diffusing and adsorbed electroactive species and the flow diagram of the strategy developed to derive approximate analytical solutions for low coverage stripping voltammetry. Red solid and brown dashed lines stand for deductive and intuitive steps, respectively.

couple remain strongly adsorbed (Figure 1). The close connection between stripping and thin film voltammetries has been addressed previously. Brett and Olivera Brett13 have shown that the solution for convective anodic stripping voltammetry at mercury film electrodes can be reduced to that characterizing the irreversible redox conversion of surface immobilized species. Under conditions where adsorption of the reduced form of the redox couple is negligible, two observations motivate our search of approximate analytical solutions for H-CSV: first, the invariance of the low coverage cathodic stripping voltammetric waveshape with respect to the adsorption and diffusion parameters (vide infra), and second, the existence of rigorous analytical solutions describing thin film voltammetry analogues. As evidence of the generality of the present approach, we derive analytical solutions for the stripping voltammetry of monomers, dimers, and a mixture of monomers and dimers. Moreover, an approximate solution for TL-ASV is also obtained and compared to the summation series proposed by Schiewe et al.11 Finally, application to the cathodic stripping of a hydrophilic thiol, such as mercaptoacetate, demonstrates that the voltammetric response in the Henry adsorption limit is readily accessible and that mechanistic information can be easily gained from the surface concentration dependence of the stripping voltammetric waves.

whose area (0.0267 cm2) was determined by weighing three sets of 10 drops. Deposition of MAA onto the mercury surface was carried out in situ, under potentiostatic control (typically at 0.3 V), from solutions containing 120 μM MAA and 0.5 M NaOH, so that the deposition rate on each newly grown mercury drop was under diffusion control. Under these conditions, the amount of adsorbed thiol was modulated by varying the deposition time within the 160 s range. The use of a low bulk thiol concentration ensures that freely diffusing thiol molecules do not interfere with the electrochemical signal generated during the stripping step. Cathodic stripping voltammograms were recorded at 1 V s1 with an Autolab PGSTAT30 (Eco Chemie B. V.).

’ RESULTS AND DISCUSSION The Framework. Cathodic stripping voltammetry of organic substances may involve desorption of monomeric and dimeric species. Therefore, for the sake of completeness, the following reaction scheme is considered to illustrate our strategy:

R sol þ Sads R ads 2Oads

’ EXPERIMENTAL SECTION Mercaptoacetic acid (MAA) and sodium hydroxide were purchased from Aldrich and Fluka Chemicals, respectively, and were used as received. Aqueous solutions were prepared from water purified with a Millipore Milli-Q system (resistivity 18 MΩ cm). Mercury was distilled three times under vacuum after treatment with dilute nitric acid and mercurous nitrate. Electrochemical measurements were carried out in a waterjacketed glass cell, thermostatted at 25 ( 0.2 °C with a Haake D8. G circulator thermostat. Solutions were deaerated with a presaturated argon stream prior to the measurements. A sodiumsaturated Ag/AgCl electrode and a platinum foil were used as reference and auxiliary electrodes, respectively. The working electrode was a hanging mercury drop electrode (EG&G PAR 303A)

Ka, R

s f R ads þ Ssol r Es s f Oads þ ne r

ð1Þ

Kd

s f Zads r

where all steps are assumed to be reversible, R stands for the reduced species, O and Z for the monomeric and dimeric oxidized species, respectively, S for the solvent, and subscripts sol and ads indicate whether the species is in solution or in the adsorbed state, respectively. Since cathodic stripping voltammetry involves the reductive desorption of sparingly soluble oxidized species, the following reactions Rsol a Osol + ne, Osol + Sads a Oads + Ssol, and Zsol + Sads a Zads + Ssol have been ignored. The first step, which represents the physisorption of the reduced form of the analyte, is described by a Langmuir isotherm with an equilibrium constant Ka,R: Ka, R ¼ 6402

b θR Γmax R cS csR Γmax S ð1 θR θO θZ Þ

ð2Þ

dx.doi.org/10.1021/ac201473z |Anal. Chem. 2011, 83, 6401–6409


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Figure 2. (Left panel) Simulated cathodic stripping voltammograms for the reaction scheme in eq 1 when the oxidized species consists of (a) monomers or (b) dimers, for the indicated surface coverages. (Right panel) Normalized cathodic traces of the cathodic stripping voltammograms shown in the left panel for (c) monomers and (d) dimers. Simulation parameters are cbR = 10 μM, cbS = 55.5 M, DR = 5 106 cm2 s1, Ka,R = 1 102, Es = 0, n = 1, max max 10 A = 0.03 cm2, Γmax = 2Γmax mol cm2, T = 298 K and (a,c) Kd = 0 and (b,d) Kd = 1 1015 mol1 cm2. R = ΓO = ΓS Z = 8 10

where csR stands for the volume concentration of R in the vicinity of the electrode surface, cbS for the solvent bulk concentration, Γi for the surface concentration of species i, Γmax for the surface i for its concentration of a full monolayer of i, and θi = Γi/Γmax i surface coverage. In the present approach, Ka,R is always kept low enough so that θR ∼ 0. Equation 2 implies that adsorption of an adsorbate displaces a cluster of solvent molecules of the same size, so that the exponent of the free surface coverage factor (1 θR θO θZ) is 1, irrespective of the adsorbate molecular size.1719 The second step, which represents the surface redox conversion, is characterized by the standard formal potential Es, so that ΓO nFðE Es Þ ¼ exp ¼ξ ð3Þ RT ΓR The third step, which represents the surface dimerization of the oxidized product O, is characterized by the equilibrium constant Kd Kd ¼

θZ Γmax Z 2 θO 2 ðΓmax O Þ

ð4Þ

= Γmax Throughout this work, it will be assumed that Γmax S R = max = 2ΓZ . The corresponding boundary value problem associated with the stripping voltammetric response of the reaction scheme shown in eq 1 is solved numerically by using the modified version of the spline orthogonal collocation technique described in a previous work.6 This numerical solution is used as a reference to assess the validity of the analytical solutions. Limiting CSV Waveshapes at Low Surface Coverages. Cathodic stripping voltammograms for a redox couple whose Γmax O

oxidized form O is strongly adsorbed and whose reduced form R remains freely diffusing in solution (Ka,R ≈ 0) is characterized by a Gaussian-like cathodic wave and a diffusion-controlled anodic wave, resulting from the reoxidation of the previously desorbed molecules (Figure 2a). Apart from its use as a detection tool, the cathodic feature has also been shown to provide information on the energetics and surface organization of the adsorbate.7,2022 Normalization of the cathodic wave with respect to the peak current ip and peak potential Ep (Figure 2c) demonstrates that a unique waveshape is predicted for low surface coverages (θ e 0.1), whose full width at half-maximum fwhm is 75/n mV at 25 °C. This wave turns out to be slightly asymmetrical with half widths at half height of 40/n and 35/n mV at the positive and negative sides of the peak, respectively. An increase of the surface coverage above 0.1 results in the progressive broadening of the cathodic voltammetric wave reaching a fwhm of ∼120/n mV for a full monolayer (Figure 3a). At low surface coverages, the shape of the voltammetric wave is independent not only of the amount of the adsorbed molecules but also of the scan rate v, diffusion coefficient DR, and adsorption equilibrium constant Ka,R, provided that the reduced form of the reactant is not significantly adsorbed. These three parameters, however, affect the wave location since an increase of either Ka,R or v results in a shift of the wave toward more negative potentials, whereas an increase of DR results in a shift toward more positive potentials. Qualitatively, a similar behavior is predicted for the reductive desorption of an oxidized dimer Z (Figure 2b), with two differences. Now, the voltammetric wave is narrower, with a limiting value of the fwhm of 55/n mV at very low surface coverages that increases to ∼90/n mV for a full monolayer (Figure 3a). The peak potential shows a clear variation with the amount of deposited molecules in the preconcentration step, shifting toward more 6403



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Table 1. Parameters for H-CSV Ψm ¼ expðnFðE EFV p, m Þ=ðRTÞÞ, Φm ¼ expðnFðE Em Þ=ðRTÞÞ, Λm ¼ λm ¼ 1 þ Hm, j ¼ 1

Φm β1=m

m ¼ 1, 2, 1•2

m ¼ 1, 2

,

g , ð1 þ Φm Þp

Φj g p ln Λm , λj ð1 þ Φj Þ1 þ p Ka,HR ¼ Ka, R a ¼

m ¼ 1, 2

m ¼ 1, 2, 1•2 m ¼ 1, 2 and j ¼ 1, 2, 1•2

Γmax S cbS

nFv RT

β = 1.25, g = 0.32, p = 0.75

Stripping of Monomers. Redox conversion of monomeric species in film voltammetry (FV) is a well-known problem, whose solution is given by23,24 ΘO ¼

Figure 3. Dependence of (a) the full width at half-maximum and (b) the peak potential of the stripping voltammetric wave on the logarithm of the surface coverage of the oxidized species: monomers (red), dimers (blue), and a mixture of monomers and dimers in equilibrium (green). (a) (O) cbR = 1 μM, DR = 1 107 cm2 s1; (4) cbR = 5 μM, DR = 1 107 cm2 s1; (0) cbR = 10 μM, DR = 1 106 cm2 s1; and (3) cbR = 10 μM, DR = 1 105 cm2 s1. (b) (O) cbR = 0.5 μM, (4) cbR = 1 μM, (0) cbR = 5 μM, and (3) cbR = 20 μM and DR = 1 106 cm2 s1. Other parameters: cbS = 55.5 M, Ka,R = 1 102, Es = 0, n = 1, A = 0.03 cm2, max max 10 Γmax = 2Γmax mol cm2, T = 298 K, and the R = ΓO = ΓS Z = 8 10 indicated values of Kd. Symbols represent values obtained from simulated cathodic stripping voltammograms, and lines represent the corresponding analytical solution: eq 9 (red), eq 16 (blue), and eq 24 (green).

negative values at a rate of 2.3RT(2nF)1 per decade of adsorbate surface concentration (Figure 3b). In the case of a mixture of oxidized monomers and dimers (i.e., for intermediate values of Kd), the voltammetric features are more involved due to the dependence of the monomer/dimer ratio on the amount of deposited molecules in the preconcentration step. As can be seen from the dependence of both the fwhm and peak potential on the deposited coverage of the oxidized species (Figure 3), a transition from the monomeric to the dimeric behavior is predicted at rather low surface coverages. The invariance of the low coverage limiting waveshape with DR, v, and Ka,R points out the feasibility of finding analytical solutions under these conditions. Our strategy uses as a starting point the overall functionality of the film voltammetry equations. Accurate Analytical Solutions for H-CSV with Linear Diffusion. The procedure to derive approximate analytical solutions for H-CSV is first illustrated for the reductive stripping of oxidized monomers. Then, it is extended to the cases of oxidized dimers and a mixture of monomers and dimers. In a later section, the small correction required for the case of spherical diffusion will be described.

Ψ1 1 þ Ψ1

iFV ¼

ð5Þ

n2 F 2 vAΓTO Ψ1 RT ð1 þ Ψ1 Þ2

ð6Þ

where ΘO = θO/θTO = ΓO/ΓTO is the relative surface coverage of O with respect to that at the end of the deposition step θTO, and Ψ1 is defined in Table 1 with m = 1. Use has been made of the fact that EFV p,1 = Es. Equation 6 predicts a voltammetric fwhm of 90.6/n mV at 25 °C. When the adsorption of the reduced form is negligible (Ka,R f 0), it is convenient to define the following characteristic potential: ( 1=2 ) RT a H ln Ka, R E1 ¼ Es ð7Þ nF DR max b where KH a.R = Ka.R ΓS /cS represents a Henry adsorption coefficient and a = nFv/(RT). We have found that redox conversion during reductive desorption of O in the H-CSV protocol can quantitatively be described by an expression similar to eq 5:

ΘO ¼

Λλ11

ð8Þ

1 þ Λλ11

where Λ1 and λ1 are defined in Table 1. Recalling that i = nFAvΓTO(dΘO/dE), the following expression is derived for the H-CSV wave: iSV ¼

λ1 n2 F 2 vAΓTO Λλ11 H1, 1 RT ð1 þ Λλ11 Þ2

ð9Þ

where H1,1 is defined in Table 1. At low surface coverages, the peak potential values of simulated cathodic stripping voltammograms are quantitatively described by ESV p, 1 ¼ E1 þ

RT lnðβÞ1=2 nF

ð10Þ

where β = 1.25, as illustrated in Figure 3b. 6404



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be observed how the approximate equation reproduces the simulated traces up to a surface coverage of ∼0.1. Stripping of Dimers. This case corresponds to the reaction scheme in eq 1 with a large enough value of the dimerization equilibrium constant Kd, so that the surface population of monomers O is always negligible. As in the case of monomers, the outset to formulate the approximate analytical solution is the homologous solution for film voltammetry:25 ΘZ ¼ 1

2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 1 þ 16Kd ΓTZ ξ2

ð11Þ

Taking into account that EFV p, 2

RT 8Kd ΓTZ pffiffiffi ln ¼ Es nF 1 þ 2

!1=2 ð12Þ

and the definition of Ψ2 (Table 1), eq 11 can be rewritten in terms of Ψ2 as follows ΘZ ¼ 1

2 ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 1 þ 1 þ 2ð1 þ 2ÞΨ2 2

ð13Þ

From eq 13, and on the analogy of eq 7, the following characteristic potential can be defined for the stripping of dimers: ( !1=2 1=2 ) T RT 8K Γ RT a d H Z pffiffiffi E2 ¼ Es ln ln Ka, R nF nF DR 1 þ 2 ð14Þ Figure 4. Comparison of the cathodic trace of stripping voltammograms calculated by digital simulation (symbols) and by using the corresponding approximate analytical solution (lines) for the indicated surface coverages of (a) monomers (eq 9), (b) dimers (eq 16 with Kd = 1 1015 mol1 cm2), and (c) for distinct values of Kd (mol1 cm2) = (1) 1 102, (2) 2 1010, (3) 1 1011, (4) 1 1012, (5) 1 1013, and (6) 1 1014 with θox = 0.05 (eq 24). Other parameters: cbR = 5 μM, cbS = 55.5 M, DR = 5 106 cm2 s1, Ka,R = 1 102, Es = 0, n = 1, A = 0.03 cm2, max max 10 Γmax = 2Γmax mol cm2 and T = 298 K. R = ΓO = ΓS Z = 8 10

Figure 4a compares the cathodic trace of simulated cathodic stripping voltammograms (symbols) with that calculated from eq 9 (lines) for distinct values of the surface coverage θox. It may

iSV

Then, a tentative equation for the redox conversion of Z during its reductive desorption can be formulated by replacing Ψ2 in eq 13 by Λλ22 (Table 1), so that ΘZ ¼ 1

2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 1 þ 1 þ 2ð1 þ 2ÞΛ2λ 2

ð15Þ

Equation 15 has been found to reproduce the population of oxidized dimers Z during their reductive desorption up to a surface coverage of ∼0.1. Recalling that i = 2nFAvΓTZ (dΘZ/dE), the following expression is obtained for the stripping of oxidized dimers:

pffiffiffi 2 8λ2 n2 F 2 vAΓTZ ð1 þ 2ÞΛ2λ 2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 H2, 2 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 2λ2 RT 2 1 þ 2ð1 þ 2ÞΛ2λ 1 þ 2ð1 þ 2 ÞΛ þ 1 2 2

ð16Þ

where H2,2 is given in Table 1. Equation 16 closely resembles the solution reported for film voltammetry:25 iFV ¼

pffiffiffi 8n2 F 2 vAΓTZ ð1 þ 2ÞΨ2 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffi pffiffiffi RT 2 2 1 þ 2ð1 þ 2ÞΨ2 1 þ 2ð1 þ 2ÞΨ2 þ 1

The peak potential values of simulated cathodic stripping voltammograms for dimers can be reproduced by ESV p, 2 ¼ E2 þ

RT lnðβÞ1=4 nF

as illustrated in Figure 3b.

ð18Þ

ð17Þ

Figure 4b shows a comparison between the cathodic trace calculated by numerically solving the corresponding boundary value problem (symbols) and that predicted by the approximate solution eq 16 (lines) for distinct surface coverages. Again, the upper limit of applicability of eq 16 is ∼0.1. 6405



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Stripping of a Mixture of Monomers and Dimers. Here we formulate an approximate analytical solution for the general case of the stripping of a mixture of monomers (O) and dimers (Z), which corresponds to the whole reaction scheme in eq 1 with any value of the dimerization equilibrium constant Kd. Let Γox be the surface concentration of the oxidized species (O, Z) in terms of monomeric units (i.e., Γox = ΓO + 2ΓZ), and ΓTox its total surface concentration at the end of the preconcentration step. As in the previous two cases, we start with the analytical solution for the redox conversion in film voltammetry, which according to our previous work is given by25 Θox ¼ 1

2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ξ þ ð1 þ ξÞ2 þ 8Kd ΓTox ξ2

ð19Þ

RT 4Kd ΓTox pffiffiffi ln 1 þ E1•2 ¼ Es nF 1 þ 2

!1=2

( 1=2 ) RT a H ln Ka, R nF DR

ð21Þ Again, a trial solution for the coverage of the oxidized species during its reductive desorption can be formulated by replacing the Ψm terms in eq 20 by Λλm1•2 (Table ): Θox ¼ 1

1 þ

Λλ11•2

2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi λ1•2 2 1•2 þ ð1 þ Λ1 Þ þ 2ð1 þ 2ÞΛ2λ 2

ð22Þ

ð20Þ

It should be recalled that Λ1 and Λ2 are defined in terms of E1* (eq 7) and E2* (eq 14), with the factor 8KdΓTZ being replaced now by 4KdΓTox, whereas λ1•2 is defined in terms of E1•2* (eq 21). A simple equation for the cathodic stripping voltammetric wave can be obtained from the following quadratic equation for the redox conversion: pffiffiffi 1 þ 2 2λ1•2 Λ2 ½1 Θox 2 þ ð1 þ Λλ11•2 Þ½1 Θox 1 ¼ 0 2 ð23Þ

On the analogy of eqs 7 and 14, it is convenient to define the following characteristic potential for the stripping of a mixture of monomers and dimers:

By differentiation of eq 23 with respect to E and recalling that i = nFAvΓTox(dΘox/dE), the following expression is obtained for the current:

In terms of the corresponding peak potentials for the redox FV conversion of monomers EFV p,1 = Es and dimers Ep,2 (eq 12 with ΓTZ = ΓTox/2), eq 19 can be rewritten as Θox ¼ 1

2 ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 1 þ Ψ1 þ ð1 þ Ψ1 Þ2 þ 2ð1 þ 2ÞΨ2 2

iSV

pffiffiffi 1•2 λ1•2 n2 F 2 vAΓTox ½1 Θox fΛλ11•2 H1, 1•2 þ ð1 þ 2ÞΛ2λ H2, 1•2 ½1 Θox g ¼ pffiffiffi 2 2λ1•2 λ 1•2 RT f1 þ Λ1 þ ð1 þ 2ÞΛ2 ½1 Θox g

where H1,1•2 and H2,1•2 are given in Table 1. Figure 4c compares the cathodic trace of simulated cathodic stripping voltammograms (symbols) with that calculated from eq 24 (lines), for distinct values of the dimerization equilibrium constant. It may be observed how the approximate solution reproduces the simulated traces for dilute mixtures of monomers and dimers. Extension to Spherical Diffusion. The previous analytical expressions describing planar H-CSV can be straightforwardly extended to the case of spherical diffusion, as long as the electrode radius ro > 10 μm, by introducing the following minor changes: (a) Replacement of the (a/DR)1/2 term in the definition of the characteristic stripping potentials Em* (eqs 7, 14, and 21) by a DR ð25Þ 1=2 1 a þ ro DR (b) Redefinition of the g factor that appears in the λm function (Table 1) as 1=2 1 a 0:7 þ 0:32 ro DR g ¼ ð26Þ 1=2 1 a þ ro DR Comparison with Experiment. Organothiols are good candidates to probe the reliability of the above analytical expressions

ð24Þ

for H-CSV as they can be oxidatively adsorbed and reductively desorbed from a metallic substrate, with the exchange of one electron per thiol molecule.26,27 When the deposition is carried out under potentiostatic conditions, as in the case of stripping voltammetry, the amount of adsorbed thiol molecules can be finely tuned by the duration of the preconcentration stage. In a previous work, we have found that the oxidative adsorption of mercaptoacetic acid (MAA) leads to a mixture of monomers and dimers, so that it is a good candidate to assess the validity of our theoretical expressions.6 Figure 5 depicts the baseline-corrected cathodic wave of MAA in 0.5 M NaOH as a function of its surface concentration. Under these conditions, reductive desorption of MAA leads to a single cathodic voltammetric wave, whose peak potential shifts toward more negative values and whose fwhm passes through a minimum, upon increasing the surface concentration. According to Figure 3, this is the behavior to be expected for an intermediate value of the dimerization equilibrium constant. A close inspection of eq 24 reveals that only three parameters can be determined from a fit of experimental voltammograms to theory, namely, n, Kd, and the characteristic potential E6¼ ! H K RT a, R ln 1=2 E 6¼ ¼ Es ð27Þ nF DR Figure 5 shows a comparison between experimental MAA voltammograms and their best theoretical fit to eq 24. It can be observed how the stripping features can be satisfactorily reproduced up to a surface concentration of ∼40 pmol cm2, by using n = 1, E6¼ = 0.497 V, and Kd = 2.7 1010 mol1 cm2, with the 6406



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the use of a mathematical software (e.g., MAPLE) that must be able to operate with a few hundreds of significant digits, to avoid loss of precision for large nF(E E*)/(RT) values. This roundoff error starts to become problematic for nF(E E*)/(RT) > 2 and becomes unworkable for nF(E E*)/(RT) g 4. Bearing in mind the conceptual similarities between TL-ASV and H-CSV, one would expect that the theoretical framework developed for H-CSV would be also applicable to the anodic stripping protocol. We have found that the TL-ASV response can be obtained by carrying out the following changes in our previous derivation of the stripping response of monomers: (a) replacement of the characteristic cathodic stripping potential by that defined in eq 29, (b) replacement of eq 8 by ΘR ¼

1

ð30Þ

1 þ Λλ11

where ΘR = cR/cbR, with cbR and cR being the uniform concentrations of the electrodeposited metal in the amalgam at the end of the preconcentration step and at any time during the anodic scan, respectively. (c) Redefinition of the λ1 and Λ1 functions as p Φ1 ð31Þ λ1 ¼ 1 þ g 1 þ Φ1 Figure 5. Comparison between experimental (red symbols) and theoretical (lines) cathodic stripping features of a mercury electrode modified with the indicated surface concentrations of mercaptoacetic acid (MAA), in a solution containing 2.5 μM MAA and 0.5 M NaOH at 298 K; scan rate 1 V s1; deposition potential 0.3 V; electrode surface area 0.0267 cm2. The solid green lines represent the best fit to eq 24, as described in the text. Dashed lines in the lower panel represent the theoretical predictions for stripping of monomers (pink) or dimers (blue).

spherical correction contribution being negligible. It is interesting to note how the stripping features evolve from those typical of adsorbed monomers toward those characteristic of adsorbed dimers upon increasing the thiol surface concentration, so that only at very low surface concentrations (70%) in the form of monomers. Extension to Thin-Layer Anodic Stripping Voltammetry. Thin-layer anodic stripping voltammetry (TL-ASV) is the analogue of H-CSV for a freely diffusing oxidized species that can be reductively electrodeposited and oxidatively stripped into the solution. The theory of TL-ASV at a stationary electrode with linear diffusion was first developed by De Vries and Van Dalen8,9 in the form of an integral equation and later by Schiewe et al.,11 who derived the following analytical solution: n2 F 2 VcbR v ∞ jnF j j p ffiffi ffi ðE E Þ ð28Þ ð 1Þ i¼ exp RT j ¼ 1 RT j!

∑

where V is the volume of mercury deposited on the electrode, cbR is the uniform concentration of the electrodeposited metal in the amalgam at the end of the preconcentration step, and E* is the characteristic anodic stripping potential, given by ( ) RT A DO 1=2 ln E ¼ Es ð29Þ nF V a where A is the surface area of the disk electrode and DO is the diffusion coefficient of the metal ion in solution. As noted by the authors, computation of the stripping wave from eq 28 requires

Λ1 ¼ βΦ1

ð32Þ AΓTO

VcbR.

(d) Replacement of by According to these changes and recalling that i = nFvVcbR(dΘR/dE), the following expression is obtained for the thin-layer anodic stripping voltammetric wave: iASV ¼

λ1 n2 F 2 VcbR v Λλ11

2 H1, 1 RT 1 þ Λλ11

ð33Þ

where H1,1 is now given by p

H1, 1 ¼ 1 þ

gp ln Λ1 Φ1 λ1 ð1 þ Φ1 Þ1 þ p

ð34Þ

Figure 6 shows a comparison of the normalized dimensionless anodic stripping voltammetric waves, I = iRT/(n2F2VcbRv) vs nF(E E*)/(RT), predicted by the analytical (eq 28) and approximate (eq 33) solutions. As can be seen, eq 33 reproduces quite satisfactorily the voltammetric waveshape predicted by the analytical solution. Deviations between the approximate and analytical values of I, normalized with respect to the dimensionless peak current Ip, are always smaller than 1% (upper panel of Figure 6). Summary. The strategy developed to derive accurate analytical solutions for low coverage stripping voltammetry, with negligible physisorption of the analyte, is summarized in Figure 1. The starting point is the analytical solution for the analogous redox conversion in film voltammetry in terms of the peak potential. From this, the corresponding expression for the redox conversion in stripping voltammetry is obtained by (a) replacing the film voltammetry peak potential Ep by the characteristic stripping potential E* and (b) introducing the λ function in the exponential term that embodies the dependence of the redox conversion on potential. Then, the expression for the faradaic current associated with the stripping process is obtained from the first derivative of the surface concentration of the oxidized 6407



Figure 6. Theoretical normalized thin-layer anodic stripping voltammograms calculated from the summation series solution (eq 28) derived by Schiewe et al.11 (red symbols) and from the approximate analytical solution (eq 33) derived in the present work (solid green line). The upper panel represents the difference between the approximate and analytical values of I, normalized with respect to the analytical dimensionless peak current.

(for cathodic stripping) or reduced (for anodic stripping) species with respect to potential.

’ CONCLUSIONS A strategy has been developed to derive accurate analytical solutions for low coverage stripping voltammetry with negligible physisorption of the analyte. It is based on a combination of deductive and intuitive arguments that use the analytical solutions for redox conversion in film voltammetry as a starting point. To assess its generality, this strategy has been applied to the stripping of monomers, dimers, and a mixture of monomers and dimers. Analysis of the stripping of mercaptoacetate deposited on mercury demonstrates the usefulness of the theoretical approach to get insights into the interfacial organization of adsorbed molecules. For this particular system, a transition from monomeric (HgSR) to dimeric (Hg2(SR) or Hg2(SR)2) behavior has been identified at low surface coverages. Application of the analytical expression developed for a mixture of monomers and dimers allows for the estimate of the surface dimerization equilibrium constant. Overall, this work may contribute to facilitate the use of stripping voltammetry as a surface characterization tool, in analogy to the widely used temperature-programmed desorption technique in surface science. Although we have focused on the Henry adsorption limit, an extension to langmuirian adsorption is underway and will be the subject of a forthcoming communication. ’ AUTHOR INFORMATION Corresponding Author

*Phone: +34-954557177. Fax: +34-954557174. E-mail: pacheco@ us.es.

’ ACKNOWLEDGMENT This work was supported by the Spanish DGICYT under Grant CTQ2008-00371 and by the Junta de Andalucia under Grant FQM02492.

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’ NOMENCLATURE R reduced monomer O oxidized monomer Z oxidized dimer Ox oxidized species (O + Z) m aggregation mumber of the oxidized species (1 = monomer, 2 = dimer, 1•2 = mixture of monomers and dimers) bulk concentration of species J, mol cm3 cbJ s cJ volume concentration of species J in the vicinity of the electrode surface, mol cm3 ΓJ surface concentration of species J, mol cm2 T ΓJ total surface concentration of species J at the end of the deposition step, mol cm2 max ΓJ surface concentration of a full monolayer of species J, mol cm2 θJ surface coverage of species J (θJ = ΓJ/Γmax J ) fractional surface concentration of species J (ΘJ = Γj/ΓTJ ) ΘJ standard formal potential, V Es FV Ep,m film voltammetry peak potential when the oxidized species has an aggregation number m, V stripping voltammetry peak potential when the oxidized ESV p,m species has an aggregation number m, V characteristic potential for the stripping of an oxidized Em* species with aggregation number m, V diffusion coefficient of the reduced species, cm2 s1 DR Ka,R adsorption equilibrium constant of the reduced species KH Henry adsorption equilibrium constant of the reduced a,R species, cm v potential scan rate, V s1 A electrode surface area, cm2 T temperature, K R universal gas constant, 8.314 J mol1 K1 F Faraday constant, 96 485 C mol1 ’ REFERENCES (1) Wang, J. Stripping Analysis: Principles, Instrumentation and Applications; VCH Publishers: Deerfield Beach, FL, 1985. (2) Lovric, M. In Electroanalytical Methods: Guide to Experiments and Applications; Scholz, F., Ed.; Springer: Berlin, Germany, 2010; pp 201221. (3) Florence, T. M. J. Electroanal. Chem. 1979, 97, 219–236. (4) Fojta, M.; Jelen, F.; Havran, L.; Palecek, E. Curr. Anal. Chem. 2008, 4, 250–262. (5) Gupta, V. K.; Jain, R.; Radhapyari, K.; Jadon, N.; Agarwal, S. Anal. Biochem. 2011, 408, 179–196. (6) Calvente, J. J.; Andreu, R.; Gil, M. L. A.; Gonzalez, L.; Alcudia, A.; Dominguez, M. J. Electroanal. Chem. 2000, 482, 18–31. (7) Calvente, J. J.; Andreu, R.; Gonzalez, L.; Gil, M. L. A.; Mozo, J. D.; Roldan, E. J. Phys. Chem. B 2001, 105, 5477–5488. (8) De Vries, W. T.; Van Dalen, E. J. Electroanal. Chem. 1964, 8, 366–377. (9) De Vries, W. T.; van Dalen, E. J. Electroanal. Chem. 1967, 14, 315–327. (10) Brainina, K. Z. Talanta 1971, 18, 513–539. (11) Schiewe, J.; Oldham, K. B.; Myland, J. C.; Bond, A. M.; VicenteBeckett, V.; Fletcher, S. Anal. Chem. 1997, 69, 2673–2681. (12) Roe, D. K.; Toni, J. E. A. Anal. Chem. 1965, 37, 1503–1506. (13) Brett, C. M. A.; Oliveira Brett, A. M. C. F. J. Electroanal. Chem. 1989, 262, 83–95. (14) Ball, J. C.; Compton, R. G. Electroanalysis 1997, 9, 765–769. (15) Ball, J. C.; Compton, R. G. Electroanalysis 1997, 9, 1305–1310. (16) Ball, J. C.; Compton, R. G.; Brett, C. M. A. J. Phys. Chem. B 1998, 102, 162–166. 6408



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Accurate Analytical Expressions for Stripping Voltammetry in the

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