1842
Langmuir 1999, 15, 1842-1852
Influence of Adsorption/Diffusion Coupling on Surface Voltammetric Waves. First Stages of 2-Mercaptoethanesulfonate Oxidative Adsorption on Gold Juan-Jose Calvente, Marı´a-Luisa A. Gil, Rafael Andreu,* Emilio Roldan, and Manuel Domı´nguez Department of Physical Chemistry, University of Sevilla, 41012 Sevilla, Spain Received August 3, 1998. In Final Form: December 1, 1998 The influence of reactant adsorption/diffusion coupling on the voltammetric behavior of redox processes that lead to the formation of a passivating product monolayer was investigated. The orthogonal collocation technique was used to compute cyclic voltammetric waves under reversible Langmuirian conditions. Systematic analysis of the peak parameter dependence on reactant adsorption, concentration, and scan rate allows the characterization of five distinct adsorption/diffusion regimes, which can be rationalized in terms of a zone diagram. Convoluted voltammograms provide directly the reactant concentration in the vicinity of the electrode surface in the absence of reactant adsorption. An approximate method is proposed to extend the semiintegral approach in the presence of moderate reactant adsorption. For scan rates in the range 0.5-5 V/s and low enough thiol concentrations (e0.25 mM), it is shown that oxidative adsorption of 2-mercaptoethanesulfonate on gold can be described by a fast adsorption/electron-transfer step in equilibrium. Under these conditions, the maximum surface concentration becomes smaller than expected for a hexagonal closed-packed monolayer of adsorbed product. For higher concentrations and/or lower sweep rates, anodic waves are narrower and higher than predicted for a simple adsorption/electrontransfer mechanism. These results suggest that high-order kinetics are involved in a following surface rearrangement process leading to the formation of a more compact product monolayer.
Introduction Electrochemical reactions leading to the formation of well-defined chemisorbed products are currently being exploited in order to modify electrode surfaces.1,2 As long as the chemisorbed products do not polymerize to build 3-D aggregates, the modification process results in a monolayer-covered electrode surface. Eventually, such monolayers may rearrange spontaneously into long-range ordered stuctures of technological interest.3-5 Oxidative (or reductive) adsorption involves a strong electronic interaction between adsorbate and substrate. Such an interaction can be described as a change in the redox state of the adsorbate, and it can be driven by an externally applied potential, as in the case of a simple electron-transfer process. Therefore, a series of diffusion, adsorption, and electron-tranfer steps gives an appropriate framework to discuss the electrochemical properties of these systems.6 Under linear scan voltammetric conditions, analytical solutions have been reported in the case of strong adsorption of both reactant and product6 and in the presence of high reactant concentrations and low scan rates to prevent a departure of the surface concentration from its bulk value.7 Diffusion has no effect on the * Corresponding author. (1) Murray, R. W. In Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker: New York, 1984; Vol. 13, p 192. (2) Finklea, H. O. In Electroanalytical Chemistry; Bard, A. J., Rubinstein, I., Eds.; Marcel Dekker: New York, 1996; Vol. 19, p 110. (3) Laibinis, P. E.; Whitesides, G. M. J. Am. Chem. Soc. 1992, 114, 9022. (4) Rojas, M. T.; Koniger, R.; Stoddart, J. F.; Kaifer, A. E. J. Am. Chem. Soc. 1995, 117, 336. (5) Wolf, M. O.; Fox, M. A. J. Am. Chem. Soc. 1995, 117, 1845. (6) Laviron, E. In Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker: New York, 1982; Vol. 12, p 54. (7) Hatchett, D. W.; Stevenson, K. J.; Lacy, W. B.; Harris, J. M.; White, H. S. J. Am. Chem. Soc. 1997, 119, 6596.
voltammetric response in these two cases, either since no mass transport from the solution bulk is required or because the flux demand to form a monolayer of adsorbed products is too low to entail significant changes in the reactant concentration profile. Intermediate diffusioncontrolled situations become relevant whenever the reactant is not strongly adsorbed, as appears to be the case with the adsorption of some thiols on mercury,8,9 silver,7,10 or gold,11,12 and the experimental time scale is determined by the kinetics of the surface process of interest. When the reactant diffusion problem has to be solved, and adsorption is described through nonlinear isotherms, to account for a limited number of sites available on the electrode surface, numerical methods are required to compute the voltammetric wave. The effect of reactant, and/or product, adsorption and diffusion on linear scan voltammograms has been discussed by several authors from different numerical approaches. Shain et al.13-15 solved the integral equations that result after inverse Laplace transformation of the boundary value problem. Numerical simulations based on explicit,16 or implicit,17 finite difference and orthogonal (8) Szulborska, A.; Baranski, A. J. Electroanal. Chem. 1994, 377, 269. (9) Stevenson, K. J.; Mitchell, M.; White, H. S. J. Phys. Chem. B 1998, 102, 1235. (10) Hatchett, D. W.; Uibel, R. H.; Stevenson, K. J.; Harris, J. M.; White, H. S. J. Am. Chem. Soc. 1998, 120, 1062. (11) Calvente, J. J.; Kovacova, Z.; Sanchez, M. D.; Andreu, R.; Fawcett, W. R. Langmuir 1996, 12, 5690. (12) Yang, D. F.; Morin, M. J. Electroanal. Chem. 1997, 429, 1. (13) Wopschall, R. H.; Shain, I. Anal. Chem. 1967, 39, 1514. (14) Wopschall, R. H.; Shain, I. Anal. Chem. 1967, 39, 1527. (15) Hulbert, M. H.; Shain, I. Anal. Chem. 1970, 42, 162. (16) Feldberg, S. W. In Computers in Chemistry and Instrumentation; Mattson, J. S., Mark, H. B., MacDonald, H. C., Eds.; Marcel Dekker: New York, 1972; Vol. 2, p 185. (17) Britz, D.; Heinze, J.; Mortensen, J.; Storzbach, M. J. Electroanal. Chem. 1988, 240, 27.
10.1021/la980966c CCC: $18.00 © 1999 American Chemical Society Published on Web 02/11/1999
Adsorption/Diffusion Coupling
collocation methods18,19 have been performed. Most of this work was aimed at showing the adequacy of a given numerical technique to solve adsorption/diffusion equations under voltammetric conditions, and the emphasis was placed on voltammetric shapes rather than on peak parameter values, which are more amenable to a direct comparison with experiment. Therefore, our first purpose was to explore in a systematic way the behavior of the peak parameters when the chemisorbed product does not diffuse away from the electrode surface and the electrontransfer and adsorption steps remain in equilibrium. We will also discuss the possibility of deriving quantitative information from convoluted voltammograms. Some previous attempts to obtain surface concentrations from convoluted currents were based on the assumption that reactant and product adsorb to a similar extent,20-22 and therefore are not particularly useful when dealing with stongly adsorbed products. On the other hand, Baranski et al.8 have shown the adequacy of the convolutive approach inasmuch as the reactant is not adsorbed at the electrode surface. Finally, theoretical predictions will be applied to the analysis of the first stages of 2-mercaptoethanesulfonate (MES) electrodeposition on a polycrystalline gold electrode. In a previous report,11 it was shown that spontaneously adsorbed (i.e. at open circuit) and electrochemically readsorbed (i.e. under potential control) MES ions display differences of approximately 100 mV in their desorption peak potentials. Though the desorption process seems to be governed by nucleation and growth kinetics,11,12 these experiments indicate the presence of a rearrangement of the adsorbed products following the initial charge-transfer process, in agreement with recent STM23,24 and IR-RAS25 evidence. We will show how an adequate choice of scan rates and MES concentrations allows the gain of some quantitative information on the first stages of the electrodeposition process.
Langmuir, Vol. 15, No. 5, 1999 1843
Figure 1. (A) Square scheme showing the reaction pathways when both reactant (R) and product (O) are adsorbed and may diffuse to or from the solution. (B) Square scheme simplifies into a triangular scheme when the product is not allowed to diffuse away from the electrode surface. electrode were used as auxiliary and reference electrodes, respectively. To avoid adsorption of chloride ions on the gold electrode, connection of the reference electrode was made through a salt bridge filled with the same solution that was used in the electrochemical cell. Prior to measurements, solutions were deareated by passing argon for 15 min.
Experimental Section The sodium salt of 2-mercaptoethanesulfonic acid was purchased from Aldrich and used without further purification. Solutions of 0.01 M HClO4 and 1 M NaOH were prepared from redistilled 99.999% pure HClO4 (Aldrich), NaOH (Merck p.a.), and water purified with a Millipore Milli-Q system. The working electrode was a disk of 5 N polycrystalline gold (Metal Crystals and Oxides) with a geometric area of 0.724 cm2. Contact with the electrolyte solution was made by the hanging meniscus technique. The electrode was first polished with different sizes of alumina and diamond paste on a polishing wheel (Buehler), then electropolished in perchloric acid solution, and finally flame annealed as described in ref 26. The quality of the electrode was checked by inspecting a 20 mV/s cyclic voltammogram in 0.01 M HClO4 before each set of experiments. Linear sweep cyclic voltammetric measurements were performed with a CAEM PGA1210 potentiostat/generator coupled with a digital acquisition system. All measurements were carried out in a conventional three-electrode electrochemical cell at 25 ( 0.2 °C. A gold wire of large area and a NaCl-saturated Ag/AgCl (18) Leverenz, A.; Speiser, B. J. Electroanal. Chem. 1991, 318, 69. (19) Schulz, C.; Speiser, B. J. Electroanal. Chem. 1993, 354, 255. (20) Freund, M. S. J. Electrochem. Soc. 1992, 139, 8C. (21) Freund, M. S.; Brajter-Toth, A. J. Phys. Chem. 1992, 96, 9400. (22) Dana, S. M.; Jablonski, M. E.; Anderson, M. R. Anal. Chem. 1993, 65, 1120. (23) Poirier, G. E.; Pylant, E. D. Science 1996, 272, 1145. (24) Yamada, R.; Uosaki, K. Langmuir 1997, 13, 5218. (25) Truong, K. D.; Rowntree P. A. J. Phys. Chem. 1996, 100, 19917. (26) Fawcett, W. R.; Fedurco, M.; Kovacova, Z.; Borkowska, Z. Langmuir 1994, 10, 912.
Theory Electrochemical processes involving adsorption and electron-transfer steps can be represented in terms of the square scheme in Figure 1A, in which vertical lines stand for the redox processes whereas horizontal lines represent adsorption/desorption steps.6 The subscripts h and s denote heterogeneous and surface electron-transfer steps, respectively. If all steps remain at equilibrium, they are characterized by their adsorption equilibrium constants (Ka,R, Ka,O) or their formal standard potentials (E°h, E°s). Under conditions where the product is strongly adsorbed (Ka,O . Ka,R) and the heterogeneous electron-transfer process does not contribute to the voltammetric response (E , E°h), the square scheme reduces to the triangular scheme depicted in Figure 1B. The voltammetric features corresponding to this oxidative adsorption process will be obtained under the following assumptions: (i) diffusion transport is spherical; (ii) reactant and product adsorption equilibria are described by Langmuir isotherms; and (iii) Ka,R and Ka,O are potential independent. The corresponding boundary value problem is described in terms of Fick’s second law for the reactant:
∂cR ∂2cR 2DR ∂cR ) DR 2 + ∂t r ∂r ∂r
(1)
subjected to the following initial and boundary conditions: t ) 0
1844 Langmuir, Vol. 15, No. 5, 1999
Calvente et al.
cR(r,0) ) cbR Ka,R )
ΓR(0)cbs [Γmax - ΓR(0) - ΓO(0)]cR(r0,0)
ΓO(0)
nF (E - E°s) ) ξi ) exp RT i ΓR(0)
(
)
(3)
and the voltammetric current is given by6
(4)
i)
t > 0, r ) ∞
cR(∞,t) ) cbR
(5)
t > 0, r ) r0
( )
DR
Ka,R )
∂cR ∂r
) r)r0
dΓO dΓR + dt dt
ΓR(t)cbs [Γmax - ΓR(t) - ΓO(t)]cR(r0,t)
ΓO(t) ΓR(t)
) exp
nF (E - E°)) ) ξ (RT s
dΓO i ) nFA dt
r)r0
cji )
θi )
(8)
X)
K/a,R ) Ka,Rξ )
[Γmax - ΓO(t)]cR(r0,t)
(14)
cbs
Γi Γmax
(15)
r - r0
(16)
L(t) t a
(17)
r0xπDR(t - ts) L(t) ) f r0 + xπDR(t - ts)
(18)
where f is a numerical factor (6 in our case) that allows the extension of simulation space further than the Nernst diffusion layer. For t < ts, the boundary value problem is solved using a constant simulation space, whereas, for t > ts, simulation space becomes time dependent. From a computational point of view, it is convenient to define θT ) θO + θR and to include eq 6 in the system of partial differential equations to be solved. According to the above definitions, the boundary value problem is transformed into
∂2cjR ∂cjR ∂cjR ) βR 2 + λR ∂T* ∂X ∂X
(19)
( )
(20)
cjR(X,0) ) cjbR
(21)
∂cjR dθT ) FR dT* ∂X
(11)
This particular case has been solved numerically by Szulborska and Baranski.8 (b) All the redox conversion takes place from the initial amount of adsorbed reactant. This implies a negligible contribution from the diffusion terms, due either to the short time scale of the experiment (i.e. high scan rate) or to the presence of a monolayer of adsorbed reactant (i.e. ΓR(0) ≈ ΓO(t f ∞)); then,
(13)
where a ) nFv/RT, θi is the surface coverage of species i, and L(t) is the thickness of the simulation space, which is taken to be a multiple of the diffusion layer thickness. For the expanding simulation space version of the orthogonal collocation technique,27 L(t) is given by
(9)
(10)
ci
T* )
Under these conditions, oxidative adsorption can be described as a one-step process (diagonal reaction pathway in Figure 1B) with a potential dependent adsorption constant given by
ΓO(t)cbs
RT(1 + ξ)2
(7)
According to eq 6, two limiting cases can be distinguished: (a) Nonadsorbed R molecules are the only source for the surface redox conversion. Provided Ka,R does not depend on the potential, then dΓR/dt ≈ 0 holds during the whole voltammetric scan and eq 6 transforms into
( )
n2F2vAΓR(0)ξ
(12)
where A stands for the electrode surface area and v is the scan rate. Here, we focus on intermediate cases, and consequently, a numerical approach has to be adopted. For the sake of convenience, the following dimensionless variables are defined
(6)
where cbR and cbS are the bulk concentration of reactant and solvent, respectively, Γi is the surface concentration of species i, Γmax is the surface concentration of a monolayer of reactant or product, and Ei is the initial potential in the voltammetric experiment. Assuming that only the surface redox process contributes to the current and neglecting any nonfaradaic contribution, the voltammetric current can be computed from the temporal change of ΓO:
∂CR dΓO ) DR dt ∂r
dΓR dΓO )dt dt
(2)
X)0
T* ) 0
Ka,R )
θR(0) [1 - θT(0)]cjR(0,T*)
(22)
(27) Urban, P.; Speiser, B. J. Electroanal. Chem. 1988, 241, 17.
Adsorption/Diffusion Coupling
θO(0) θR(0)
Langmuir, Vol. 15, No. 5, 1999 1845
) ξi
(23)
T* > 0, X ) 1
cjR(X)1,T*) ) cjbR
(24)
θT(T*) ) θO(T*) + θR(T*)
(25)
T* > 0, X ) 0
Ka,R )
θR(T*)
(26)
[1 - θT(T*)]cjR(0,T*)
θO(T*) θR(T*)
) ξ(T*)
(27)
where βR, λR, and FR are defined as
βR ) λR )
2DR aL(r0 + XL)
DR
(28)
aL2
+ Xr0
2(T* -
T/s )
{ x
FR )
r0 +
πDR (T* - T/s ) a
DRcbs aLΓmax
}
(29)
(30)
For a voltammetric experiment ξ can be written in terms of T*:
ξ(T*) )
{
ξieT*
0 e T* e T/λ
/ ξie2Tλ-T* Tλ e T* e2T*λ /
(31)
where T/λ stands for the dimensionless switching time. The dimensionless current is defined as
I)
dθO iRT ) dT n F νAΓmax 2
2
(32)
The above boundary value problem was solved by using the orthogonal collocation technique.18,19,27 Space derivatives of the reactant concentration were expanded in terms of Legendre polynomials at nine collocation points. The resulting system of ordinary differential equations in T* was integrated employing the DDEBDF subroutine from the SLATEC program library. In each iteration, values of cjR(0,T*), θO(T*), and θR(T*) were provided by solving eqs 25-27. The code was written in FORTRAN and implemented on a 166 MHz Pentium PC computer. Results and Discussion Influence of Mass Transport and Reactant Adsorption on the Voltammetric Wave. Mass transport of the reduced species R to the electrode surface can be modified conveniently by changing the reactant bulk concentration (cbR) or the scan rate (v). On the other hand, the nature of the redox couple will also influence the rate
Figure 2. Influence of reactant concentration (A), scan rate (B), and reactant adsorption (C) on the voltammetric shape. (A) Ka,R ) 104, v ) 0.01 V/s, cbR ) 10-2 M (s), 10-4 M (- - -), 10-5 M (‚‚‚); (B) Ka,R ) 104, cbR ) 10-3 M, v ) 0.01 V/s (s), 5 V/s (- - -), 100 V/s (- - -); (C) cbR ) 10-4 M, v ) 0.5 V/s, Ka,R ) 107 (s), 106 (- - -), 104 (- - -). Other parameter values: Ka,O ) 1012, r0 ) 0.05 cm, DR) 8 × 106 cm2/s, and Γmax ) 5 × 10-10 mol/cm2. The dimensionless current I is defined in eq 32.
of mass transport through the diffusion and adsorption coefficients of R (the adsorption coefficient of O is always big enough, so that no O molecules diffuse away from the electrode). Figure 2 illustrates the influence of cbR and v on the wave of a redox couple, when the reduced form is weakly adsorbed (Ka,R ) 104). For high cbR values and low scan rates, voltammetric waves are symmetrical (|∆Ep| ) 0, |ip,a/ip,c| ) 1). However, as the scan rate increases (Figure 2B) and/or reactant bulk concentration decreases (Figure 2A), a loss of symmetry is observed. Peaks start to move apart from each other, and the forward wave becomes broader and lower. Eventually, only the backward wave shows the typical features of a surface wave. Under these conditions, an oxidative current is still observed in the first stages of the backward scan. Note that diffusion influences not only the shape of the forward scan by setting a limit to the reactant flux but also that of the reverse scan, since desorption starts in the presence of a steep concentration gradient and, as desorption proceeds, it leads to significant changes of the reactant concentration in the vicinity of the electrode surface. Provided that the electrode has been allowed to equilibrate with the surrounding solution, the value of Ka,R determines the amount of adsorbed R molecules before the onset of the oxidation process. When Ka,R is high enough, a preadsorbed monolayer of R will be converted into a monolayer of O, without requiring consumption of
1846 Langmuir, Vol. 15, No. 5, 1999
Calvente et al.
R molecules from the solution. As Ka,R decreases, a larger amount of dissolved R molecules has to reach the electrode surface, setting up the appropriate conditions to observe a mass-transfer control of the deposition process (Figure 2C). Adsorption/Diffusion Regimes. Before attempting a more detailed discussion of the influence of adsorption and diffusion on the voltammetric behavior, it is convenient to consider the following limiting cases: (1) Reversible charge-transfer isolated from the solution, RVs (corresponding to high scan rates and low reactant concentrations). All the redox conversion takes place from the initial amount of adsorbed reactant. Under these conditions, peak parameters are given by6
RT Ka,O ln nF Ka,R
(33)
n2F2AvΓmax Ka,RcbR 4RT Ka,RcbR + cbs
(34)
(Ep)RVs ) E°s ) E°h (ip)RVs )
the dimensionless peak current being
(Ip)RVs )
RT(ip)RVs n2F2AvΓmax
)
b 1 Ka,RcR 4 K cb + c b a,R R s
(35)
Peak potentials do not depend on the reactant bulk concentration, but peak currents do, as long as KaRcbR e cbs . The peak parameters are the same for cathodic and anodic waves. (2) Reversible charge-transfer equilibrated with the solution, RVsb (corresponding to low scan rates and high reactant concentrations). Redox conversion implies consumption of dissolved R species without significant perturbation of its concentration. Therefore, cR(0) ≈ cbR is expected along the whole voltammetric wave. Under these conditions, the peak parameters are given by7
(Ep)RVsb ) E°h -
Ka,RcbR RT Ka,O RT ln ln (36) nF Ka,R nF Ka,RcbR + cbs (ip)RVsb )
n2F2AΓmaxν 4RT
(37)
and the dimensionless current (Ip)RVsb equals 0.25. When the reactant is weakly adsorbed, so that KaRcbR , b cs , eq 36 simplifies to
(Ep)RVsb ) E°h -
b RT Ka,OcR ln nF cb
(38)
s
and a linear dependence of (Ep)RVsb on the logarithm of cbR is expected. Conversely to the case of the RVs regime, now peak potentials depend on the reactant bulk concentration but peak currents do not. Redox conversion of a reactant monolayer represents the limiting behavior of both RVs and RVsb cases, whenever KaRcbR . cbs , and will be denoted by RV. (3) Strong diffusion (SD) control. Reversible redox conversion leads to extensive depletion of dissolved R species, so that the reactant concentration at the electrode surface becomes very small (cR(0) < 0.2cbR) during the forward scan.
Figure 3. Dependence of peak potential Ep (A) and dimensionless peak current Ip (B) on scan rate v and switching potential Eλ: anodic forward scan (s), cathodic reverse scan (- - -); KaR ) 10-1, v ) 10 V/s. Other parameter values as in Figure 2.
(4) Moderate diffusion (MD) control. Diffusion is able to sustain a significant reactant concentration at the electrode surface along the forward scan, though different from the bulk value (0.2cbR < cR(0) < 0.9cbR). As we have seen, analytical expressions for the voltammetric peak parameters are available for the RV, RVs, and RVsb regimes. Under the SD and MD regimes, the peak parameter behavior can be obtained from the numerical solution of the boundary value problem. Figure 3 illustrates some typical features of the peak parameter dependence on scan rate and switching potential for the MD and SD cases. In the SD regime, peak potentials shift in the same direction (to more negative values) as the scan rate increases, and their separation (∼30 mV) remains nearly independent of scan rate and switching potential. In the MD regime, the anodic and cathodic peak potentials move apart from each other upon increasing v. It is interesting to note the asymmetry of the shift, the anodic peak moves faster than the cathodic one. This behavior can be used as a discriminating factor against the charge-transfer control case, which leads to a symmetrical peak potential shift.6,28 The ratio between the cathodic and anodic peak currents can be used also as a diagnostic criterion. In the MD and SD regimes, the cathodic peak current is greater than the anodic peak current, and their ratio increases as the switching potential is made more positive. The variation of d|Ip|/d log(v) with scan rate can be used to discriminate between the MD and SD regimes. Under the MD regime, this derivative increases upon increasing the scan rate, whereas the opposite is found for the SD case. (28) Bowling, R.; McCreery, R. L. Anal. Chem. 1988, 60, 605.
Adsorption/Diffusion Coupling
Figure 4. Dependence of peak potential Ep (A) and dimensionless peak current Ip (B) on reactant concentration cbR and switching potential Eλ: anodic forward scan (s); cathodic reverse scan (- - -). KaR ) 10-1; v ) 10 V/s; other parameter values as in Figure 2.
The variation of peak parameters with reactant bulk concentration is depicted in Figure 4. Under SD control, no significant dependence of the peak potentials on cbR is observed up to a concentration where the MD regime starts to operate. This transition gives rise to the appearance of the maximum in Figure 4A. Under pure MD control, anodic and cathodic peak potentials become more negative as the reactant bulk concentration increases, but the anodic peak potential moves faster than the cathodic peak potential. Thus peak potential separation approaches zero at higher concentrations, where the system enters the RVsb regime. Figure 4B shows the dependence of the dimensionless peak currents on the logarithm of the reactant bulk concentration. The sigmoidal variation of |Ip| with log(cbR) parallels that with log(v), (see Figure 3), stressing the analogy between high concentrations and low scan rates from the point of view of mass transfer to the electrode. In our previous discussion, concerning the peak parameter behavior depicted in Figures 3 and 4, no reactant adsorption was assumed. In the presence of significant reactant adsorption, the high-reactant-concentration/lowscan-rate limit still corresponds to the RVsb regime, but the low-reactant-concentration/high-scan-rate limit pertains now to the RVs regime, instead of the SD regime. Increasing reactant adsorption implies that the range of cbR and v values in which SD and MD behavior is observed becomes progressively narrower, as will be discussed in the next section. Transition between RV, RVs, RVsb, MD, and SD Regimes. Up to now some simple physical situations, which can be characterized from a distinct dependence of the voltammetric peak current and potential on scan rate
Langmuir, Vol. 15, No. 5, 1999 1847
Figure 5. Transitions between adsorption/diffusion regimes. Dependence of peak potential Ep (A) and dimensionless peak current Ip (B) on scan rate, when reactant adsorption is weak (θR(0) < 0.02): anodic forward scan (s); cathodic reverse scan (- - -); KaR ) 102; cbR ) 10-6 M (1), 10-5 M (2), 10-4 M (3), 10-3 M (4), and 10-2 M (5); the switching potential was fixed at Eλ ) E°h - 0.2 V; all other parameter values as in Figure 2.
and reactant concentration, have been isolated. However, a given redox system is expected to move from one regime to another as either concentration or scan rate is varied. Under these circumstances the experimental behavior may be more complex than represented so far. Evolution of the peak parameters as a function of the scan rate is depicted in Figure 5 in the presence of a weak reactant adsorption (θR(0) < 0.02). Curves 1 and 2, corresponding to low reactant concentrations, show the typical features of SD control. At higher reactant concentrations (curves 3-5) the transition between the SD and MD regimes is revealed by the presence of a maximum in the Ep vs log v plot and an inflection point in the Ip versus log v plot. Increasing reactant concentration extends the scan rate range where RVsb behavior is observed and shifts the SD/MD transition toward higher scan rates. In the presence of significant reactant adsorption (θR(0) ) 0.5 in Figure 6) the most salient feature, as compared to the case of no reactant adsorption, is the occurrence of the RVs regime (where Ep ) Es and |Ip| ) 0.125 for θR ) 0.5, see eqs 33 and 34) at high scan rates and low reactant concentrations. Taking the RVs regime as reference, a continuous increase of cbR (curves 1-5 in Figure 6) causes the successive appearance of the SD, MD, and RVsb regimes and the shift of the relevant transitions toward higher scan rates. The use of a zone diagram may help to rationalize the conditions at which the above transitions take place. This zone diagram (Figure 7) is defined in terms of the two following dimensionless numbers:
1848 Langmuir, Vol. 15, No. 5, 1999
Calvente et al. Table 1. Numerical Coefficients Defining Transition Lines in the Zone Diagram According to Eq 41 transition
log(Ω2)tr interval -0.75 [-0.75, 1.5] 0.6 [0.6, 1.5] [1.5, 4.5] [-∞, +∞]
RVsb T MD MD T SD SD T RVs RVsb, MD, SD, RVs T RV
Atr
Btr
-0.75 -1 -0.33 0.89 0.6 -1 -2.33 2.22 3.25 -1.5 1.0 0
Ctr 0 -1 0 -1 -1 -1
time scale (∼aΓmax) and the mean flux supplied by reactant -1/2)). Varying the scan rate, diffusion (DRcbR(r-1 0 + (πDR/a) b at a given cR, translates into a horizontal movement in the zone diagram. On the other hand, varying cbR, at constant scan rate, implies a movement along a straight segment of slope -1. The effective use of the zone diagram requires the numerical limits of each transition line, which have been summarized in Table 1 in terms of the Atr, Btr, and Ctr values that define each transition line according to
Atr + Btr log(Ω2)tr + Ctr log(Ω1)tr ) 0
Figure 6. Transitions between adsorption/diffusion regimes. Dependence of peak potential Ep (A) and dimensionless peak current Ip (B) on scan rate, when the initial reactant adsorption corresponds to half a monolayer: anodic forward scan (s); cathodic reverse scan (- - -); KaR ) 5.5 × 107 (1), 5.5 × 106 (2), 5.5 × 105 (3), 5.5 × 104 (4), and 5.5 × 103 (5); cbR ) 10-6 M (1), 10-5 M (2), 10-4 M (3), 10-3 M (4), and 10-2 M (5); the switching potential was fixed at Eλ ) E°h - 0.2 V; all other parameter values as in Figure 2.
Order-of-magnitude estimates of Γmax and Ka,R can be obtained by a straightforward analysis of the cbR and v values corresponding to changes of regime. From this point of view, three experimental situations can be considered: (a) R is strongly adsorbed (log Ω1 > 1 or Ka,RcbR/cbs > 10). No transitions will be observed for any values of cbR or v, since a monolayer of adsorbed reactant is transformed into a monolayer of product along the voltammetric wave. (b) R is weakly adsorbed (log Ω1 < -1 or Ka,RcbR/cbs < 0.1). RVsb/MD and MD/SD transitions can be detected. They should take place for fixed values of the cbR/a1/2 ratio. Comparison of this ratio with the (Ω2)tr values reported in Table 1 for each transition gives an estimate of Γmax/ D1/2 R . If R is not too weakly adsorbed, and charge-transfer kinetics do not interfere, experiments at high enough scan rates should allow one to determine the (Ω2)tr coordinate corresponding to the SD/RVs transition. KaR is then obtained by inserting these (Ω2)tr values in the expression for the transition line given in Table 1. (c) R is moderately adsorbed (-1 < log Ω1 < 1 or 0.1 < Ka,RcbR/cbs < 10). In this case two or three transition lines should be available. At any transition line, eq 41 can be rearranged to give
log
Figure 7. Zone diagram displaying the relative locations of the adsorption/diffusion regimes. Insets show typical voltammetric shapes to be found within each regime.
Ω1 )
Ω2 )
DRcbR
KaRcbR cbs
aΓmax 1 1 + r0 πDR a
{
(39)
( )
}
(40)
1/2
where Ω1 measures the extent of reactant adsorption and Ω2 denotes the ratio between the mean flux required to form a monolayer of adsorbed product in the experimental
(41)
cbR cbs
[
) Atr log
DRcbR
{
a 1 1 + r0 πDR a
( )
}
1/2
]
+ Btr +
tr ΓAmax (42) log KaR
so that the ratio (Γmax)Atr/KaR can be obtained from the (cbR)tr or (v)tr values, as long as DR is known. The diffusion coefficient can be determined from the limiting convoluted current under the SD regime (see eqs 45 and 48 below). Γmax can also be estimated from the charge under the voltammetric peak, under conditions where a monolayer of product is formed. Convolutive Voltammetry. Convolution provides a straightforward route to isolate the influence of diffusion on the voltammetric response. However, coupling between mass transport and adsorption of electroactive species often hampers a quantitative analysis of convoluted vol-
Adsorption/Diffusion Coupling
Langmuir, Vol. 15, No. 5, 1999 1849
tammograms. Whatever the adsorption extent, and under semiinfinite spherical diffusion conditions, the surface concentration of reactant R can be expressed as
( )
cR(r0,t) ) cbR - D1/2 R
∂cR ∂r
*g(t)
r)r0
(43)
where the asterisk denotes convolution of two functions of time and g(t) is the inverse Laplace transform of 1/2 -1 ((D1/2 R /r0) + s ) . When the oxidized product remains attached to the electrode surface, eqs 6 and 43 can be combined to obtain
cR(r0,t) ) cbR -
(
)
1 dΓ0 dΓ0 + *g(t) dt dt D1/2 R
(44)
This last equation permits the exploitation of the advantages of convolutive voltammetry, as long as the time derivatives of the two surface excesses can be related to the observed current. In the simplest case, R is assumed not to adsorb at the electrode surface, and therefore (see eq 10)8
cR(r0,t) ) cbR -
i(t)*g(t) nFAD1/2 R
) cbR -
II nFAD1/2 R
(45)
where II is the convoluted current. Figure 8A shows the expected convoluted response in the absence of reactant adsorption. It may be noted that II remains always eIIlim ) nFAcbR D1/2 R and that S-shaped curves are characteristic of the SD regime, whereas a maximum develops under the MD regime. The presence of a scan rate independent plateau is strong evidence against reactant adsorption.29,30 Quantitative analysis of these II versus E plots may give some insight into mechanistic aspects of the electrochemical reaction.31 When the reactant is strongly adsorbed (Figure 8C), convoluted voltammograms become peak shaped, and often exceed the IIlim value. On the other hand, weak reactant adsorption (Figure 8B) leads to mixed behavior, in which S-shaped convoluted voltammograms (with a scan rate dependent plateau) are obtained at intermediate scan rates, and they evolve into peaked shapes as the scan rate is made either higher (SD regime) or lower (MD regime). In the presence of significant reactant adsorption, eqs 7-9 allow one to relate dΓR/dt with i(t) and Q(t) at any time. However, comparison with simulated voltammograms has shown that this approach gives a notorious noise amplification and is of little practical value. Instead, a more robust, though approximated, procedure can be adopted by assuming that the faradaic current is the sum of a first component iads, originated in the preadsorbed reactant, and a second component idiff, that accounts for the contribution from diffusing reactant molecules, that is,
i ≈ iads + idiff
Figure 8. Forward scan convoluted voltammograms as a function of scan rate: (A) reactant is not adsorbed; (B) reactant is weakly adsorbed, θR(0) ) 0.018; (C) reactant is strongly adsorbed, θR(0) ) 0.5. Numbers on each curve indicate increasing scan rate in the range 0.01-100 V/s. cbR ) 10-3 M. Other parameter values as in Figure 2. Dotted lines correspond to the diffusion-limited plateau IIlim.
cR(r0,t) ≈ cbR -
δ) iads ΓR(0)aξ dΓR ≈)dt nFA (1 + ξ)2
(47)
and therefore the reactant surface concentration is approximated by (29) Bernard, M. O.; Bureau, C.; Soudan, J. M.; Le´cayon, G. J. Electroanal. Chem. 1997, 431, 153.
nFAD1/2 R
) cbR -
II - IIads nFAD1/2 R
(48)
Implementing eq 48 requires the values of ΓR(0) and E°s, which may not be readily available. Here it is possible to take advantage of the expected S-shaped dependence of II - IIads on the applied potential under SD conditions. In fact by imposing a sigmoidal shape to these plots, E°s and ΓR(0) can be determined iteratively with accuracies somewhat better than (10 mV and (3 × 10-11 mol cm-2, respectively, as is illustrated in Figure 9. Equation 48 is better suited to deal with situations in which the reactant is weakly adsorbed and the scan rate is low; thus, reactant surface concentrations can be obtained with an error
