Anal. Chem. 1997, 69, 1045-1053
Voltammetric Response Accompanied by Inclusion of Ion Pairs and Triple Ion Formation of Electrodes Coated with an Electroactive Monolayer Film Masayuki Ohtani,† Susumu Kuwabata, and Hiroshi Yoneyama*
Department of Applied Chemistry, Faculty of Engineering, Osaka University, Yamada-oka 2-1, Suita, Osaka, 565, Japan
Potential distribution at the electrode/solution interface is discussed for metal electrodes coated with a redoxactive self-assembled monolayer, taking inclusion of ion pairs and triple ion formation into consideration. Theory on current-voltage curves obtained by potential sweep experiments is derived for these special cases. Explicit expressions derived in this study for the peak current and the peak potential are similar in their forms to those published previously using the Frumkin adsorption isotherm. A graphical method to evaluate the average local potential in the double layer of real systems is proposed. Electrochemistry of a self-assembled monolayer (SAM) having redox-active groups has been getting great interest in recent years.1-12 Theory on cyclic voltammograms of such surfaceconfined redox species has been developed with use of the Langmuir adsorption isotherm and the Nernst equation.13 However, the experimentally obtained voltammetric waves are often much broader1,2,7,9 than those expected by the theoretical predictions, or in some cases, spiky redox waves appear.5,11 Furthermore, it has been found that the apparent formal potential is apt to shift on increasing the coverage of the surface-confined redox species.1,2,7 Such nonideal voltammetric behaviors were first theoretically treated by Laviron, who introduced empirical adjustable parameters to make good matches of experimentally obtained voltammograms to the theoretically obtained ones.14-16 Recently, Smith and White analyzed the voltammetric behavior of metal electrodes coated with redox-active SAMs and showed that the broadening of voltammetric waves and shifts of their apparent † Present address: Kanazawa College of Art, Kodatsuno 5-11-1, Kanazawa, Ishikawa, 920, Japan. (1) Chidsey, C. E. D.; Bertozzi, C. R.; Putvinski, T. M.; Mujsce, A. M. J. Am. Chem. Soc. 1990, 112, 4301-4306. (2) Acevedo, H. C.; Abrun ˜a, H. D. J. Phys. Chem. 1991, 95, 9590-9594. (3) Rowe, G. K.; Creager, S. E. Langmuir 1991, 7, 2307-2312. (4) Chidsey, C. E. D. Science 1991, 251, 919-922. (5) Uosaki, K.; Sato, Y.; Kita, H. Langmuir 1991, 7, 1510-1514. (6) Finklea, H. O.; Hanshew, D. D. J. Am. Chem. Soc. 1992, 114, 3173-3181. (7) Rowe, G. K.; Creager, S. E. J. Phys. Chem. 1994, 98, 5500-5507. (8) Creager, S. E.; Rowe, G. K. J. Electroanal. Chem. 1994, 370, 203-211. (9) Ravenscroft, M. S.; Finklea, H. O. J. Phys. Chem. 1994, 98, 3843-3850. (10) Rowe, G. K.; Carter, M. T.; Richardson, J. N.; Murray, R. W. Langmuir 1995, 11, 1797-1806. (11) Shimazu, K.; Yagi, I.; Sato, Y.; Uosaki, K. J. Electroanal. Chem. 1994, 372, 117-124. (12) Long, H. C. D.; Buttry, D. A. Langmuir 1990, 6, 1319-1322. (13) Bard, A. J.; Faulkner, L. R. Electrochemical Methods; J. Wiley: New York, 1980; Chapter 12. (14) Laviron, E. J. Electroanal. Chem. 1974, 52, 395-402. (15) Brown, A. P.; Anson, F. C. Anal. Chem. 1977, 49, 1589-1595. (16) Matsuda, H.; Aoki, K.; Tokuda, K. J. Electroanal. Chem. 1987, 217, 15-32.
S0003-2700(96)00952-3 CCC: $14.00
© 1997 American Chemical Society
formal potentials can be explained without any empirical adjustable parameters if the potential drop within the SAM is taken into account.17 Their theoretical treatments were further refined with inclusion of “discreteness-of-charge effects”18,19 and applied to other electrochemical problems which involved potential-dependent acid/base equilibrium of the SAM20,21 and potential-induced orientation of amphiphilic molecules.22 In this paper, we make further refinements of the theoretical treatments of Smith and White17 in terms of formation of ion pairs and triple ions between the redox-active SAM and electrolyte ions. The finding that the voltammetric peak potential of the electrode coated with SAM of an alkanethiol having a ferrocenyl group3,7,11 or [4-(aminomethyl)pyridine]Ru(NH3)53+ 23 was shifted with changes in the concentration of electrolyte solution suggests strongly that the electrolyte ions are involved in the redox reactions of the redox-active SAM. Finklea et al. explained the influence of the concentration of electrolyte ions on the redox potential by introducing the Donnan potential which is determined by the relative activities of counterions of the redox species.23,24 It was, however, revealed by in situ measurements using the electrochemical quartz crystal microbalance (EQCM) technique of the electrodes coated with SAM of alkanethiol having a ferrocenyl11 or viologen12 group that the association and the dissociation of electrolyte ions with surface-confined redox species occur during the course of the redox processes. Figure 1A shows a potential distribution profile in which no ion-pair formation is assumed for surface-confined species. This model is similar to that used in the original work by Smith and White,17 but there is one marked difference in the point that the inner and outer Helmholtz planes are introduced in this model as was done by Fawcett for refinement of the treatments by Smith and White.18,19 If the surface-confined cationic species interact strongly with the electrolyte counteranions, such a potential distribution profile as shown in Figure 1B seems valid. The potential drop between the electrode substrate and the solution bulk is almost concentrated in the SAM. If the surface-confined cationic species are excessively compensated by triple-ion formation, the potential drop in the SAM is more increased, as shown in Figure 1C, which is also (17) Smith, C. P.; White, H. S. Anal. Chem. 1992, 64, 2398-2405. (18) Fawcett, W. R. J. Electroanal. Chem. 1994, 378, 117-124. (19) Andreu, R.; Fawcett, W. R. J. Phys. Chem. 1994, 98, 12753-12758. (20) Smith, C. P.; White, H. S. Langmuir 1993, 9, 1-3. (21) Fawcett, W. R.; Fedurco, M.; Kovacova, Z. Langmuir 1994, 10, 2403-2408. (22) Gao, X.; White, H. S.; Chen, S.; Abrun ˜a, H. D. Langmuir 1995, 11, 45544563. (23) Redepenning, J.; Tunison, H. M.; Finklea, H. O. Langmuir 1993, 9, 14041407. (24) Finklea, H. O. In Electroanalytical Chemistry; Bard, A. J., Rubinstein, I., Eds.; Marcel Dekker: New York, 1996; Vol. 19, pp 243-247.
Analytical Chemistry, Vol. 69, No. 6, March 15, 1997 1045
Figure 1. Schematic illustration of potential distributions across solution/SAM/electrode substrate interface. The zig-zag line indicates a dielectric film like an alkyl chain of SAM, and the signs (+, -) denote charges of the electrode substrate surface, surface-confined species, and electrolyte ions: (A) for insufficient compensation of surface-confined positive charges with electrolyte anions; (B) for full neutralization; (C) for excessive accumulation of electrolyte anions. The parameters given in the figure are described in the text.
valid for metal electrodes having specific adsorption of electrolyte anions.25 If such changes in the potential distributions occur in the redox reactions of the surface-confined redox species, not only the peak potential but also the shape of voltammograms will be changed, depending on the degree of interaction between the electroactive moieties of the SAM and the electrolyte ions. In the present paper, explicit solutions for the peak current and the peak potential of the voltammetric waves in such cases are presented, and discussion is made to correlate them with the Laviron treatments.14,15 Furthermore, a graphical method to evaluate the double-layer effect on cyclic voltammogram is proposed. THEORY Model and Potential Distribution across Monolayer and Double Layer. Figure 2 shows an interfacial structure model of a plane electrode coated with a redox-active SAM having thickness d1 and dielectric constant 1. The electrochemically active species are located at the end of the SAM (x ) d1). The distance between the inner and outer Helmholtz planes is d2 in which a medium of the dielectric constant 2 is filled. The dielectric constant of the solution phase is given by 3. The charge density on the metal electrode surface at x ) 0 is given by σM and that on the inner Helmholtz plane at which the redox head groups are presented is given by σPET. The inner Helmholtz plane is denoted here as PET. It is assumed that both σM and σPET are delocalized in each plane, i.e., the discreteness-of-ion effect18 is ignored as assumed by Smith and White.17 This means that if the ion pairs and triple ions are formed, the charges of electrolyte ions that associated with surface-confined redox species are also delocalized on the PET at x ) d1 and are included in σPET. Consequently, it is assumed that there is no net charges in the region of 0 < x < d1 and d1 < x < d1 + d2. The volume charge density in the diffuse layer (x > d1 + d2) is given by F(x). In order to correlate the electric fields to the charge densities of interest, the Gauss law is applied to boxes 1-3 for the inner Helmholtz region, the region covering up to the outer Helmholtz region, and that extending to the solution bulk, respectively, as shown in Figure 2. Application of the Gauss law to boxes 1 and 2 gives (25) Grahame, D. C. Chem. Rev. 1947, 47, 441-501.
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Analytical Chemistry, Vol. 69, No. 6, March 15, 1997
Figure 2. Schematic illustration of a solution/SAM/electrode substrate interface (upper) and the model system used to calculate the potential profile (lower). The zig-zag line given in the upper figure indicates a dielectric film like an alkyl chain of SAM, and circles are redox group (open) and electrolyte ions (closed and slashed). σM and σPET are delocalized on the electrode surface at x ) 0 and the inner Helmholtz plane (PET) at x ) d1, respectively. Boxes 1-3 are given for regions covering from the electrode substrate to the inner Helmholtz layer, to the outer Helmholtz layer, and to electrolyte solution bulk, respectively, and the Gauss law is applied to them to correlate the electric field to the charge density. The parameters given in the figure are described in the text.
10E1 ) σM 20E2 ) σM + σPET
(box 1) (box 2)
(1a) (1b)
where E1 and E2 are the magnitudes of the electric field in the region of 0 < x < d1 and d1 < x < d1 + d2, respectively, and 0 is the permittivity of vacuum (8.854 × 10-12 J-1 C2 m-1). Indefinite
integration of eq 1a with respect to the distance from the electrode surface (x) using the relation of Ej ) -dφ(x)/dx gives
σM dφ(x) dx ) dx dx 10
∫
∫
(2)
Then, the average local potential, φ(x), as a function of x can be given by
φ(x) ) -(σMx/10) + B1
d1 > x > 0
(
)
(d1 - x)σM d2 1 + + (σ + σPET) 10 20 30κ M
d1 + d2 > x > d1 (3b)
φ(x) ) φ(d1 + d2)e-κ(x-d1-d2)
x > d1 + d2
(4)
where κ is the Debye-Hu¨ckel reciprocal length, which is given by κ ) zF(2C/30RT)1/2 for a symmetric z:z electrolyte solution with the concentration of C. The volume charge density in the diffuse layer, F(x), is related to φ(x) by the Poisson equation.
dx
)
d1 + d2 - x 1 + (σ + σPET) 20 30κ M d1 e x e d1 + d2 (7b)
φ(x) )
where B1 and B2 are integration constants whose values are determined to maintain the continuity of the potential distribution at x ) d1 and x ) d1 + d2, respectively. Since most of the potential drop should occur within the organic layer due to its low dielectric constant, the potential at the outer Helmholtz plane, φF, relative to the potential of the bulk electrolyte solution must be equal to or smaller than ∼50 mV, as already discussed by Smith and White.17 Then, the potential distribution in the diffuse layer (x > d1 + d2) can be estimated using the following relation13,26
)-
(
0 e x e d1 (7c)
(σM + σPET)x φ(x) ) + B2 20
2
φ(x) )
(3a)
Similarly, eq 3b is derived from eq 1b
d2φ(x)
of the potential at x ) d1 + d2 obtained by eqs 7a and 3b, and then B1 in eq 3a is determined so as to satisfy the continuity of the potential at x ) d1 obtained by eqs 3b and 3a. In this way, the potential in the region between the PET and the outer Helmholtz plane, and between the electrode substrate and the PET, is given by eqs 7b and 7c, respectively.
F(x) 30
(5)
The application of the Gauss law to box 3 shown in Figure 2 yields the following relation by taking the charge neutrality into consideration.
σM + σPET + σdif ) σM + σPET +
∫
∞
d1+d2
F(x) dx ) 0 (6)
Voltammetric Response of a One-Electron Reversible System Involving Formation of Ion Pairs and Triple Ions. We consider a one-electron reversible redox reaction of surfaceconfined redox couple, O+/R0 like the ferricinium/ferrocene couple.
O+ + e- a R0
(8)
The redox couple gives the following relationship under equilibrium.17
φM - φPET ) E° - Epzc -
( )
ΓR0 RT ln F ΓO+
where φM is the inner potential of the metallic electrode, φPET is the potential at the PET (x ) d1), E° is the standard redox potential of the redox couple, Epzc is the potential of zero charge of the substrate electrode, ΓO+ and ΓR0 are the surface concentrations of O+ and R0, respectively, and R, T, and F have their usual significance. As described above, the EQCM studies on the redox-active SAM of ferrocene11 and viologen12 derivatives showed that when electrochemical oxidation of the redox-active monolayer occurs, electrolyte anions are bound to the redox groups. When the cationic species (O+) is generated on the SAM, an electrolyte anion X- is associated with it at the PET.
O+ + X- a (OX)0 where σdif refers to the net electric charges of the entire diffuse layer. Combining eqs 4-6, the potential in the diffuse layer is described as
φ(x) )
(σM + σPET) exp[κ(d1 + d2 - x)] 30κ
d1 + d2 e x
(9)
(10a)
If the surface-confined cationic species is generated with a high density in the redox reaction, another X- may also be bound to give triple ion (OX2)-.27-29
(OX)0 + X- a (OX2)-
(10b)
(7a) It should be noted that φ(x) gives the potential relative to the potential of the bulk electrolyte solution (φS ) 0). The integral constant of B2 in eq 3b is determined by considering the continuity (26) Rubinstein, I. In Physical Electrochemistry; Rubinstein, I., Ed.; Marcel Dekker: New York, 1995; Chapter 1.
The formation of triple ions would become marked in such a low dielectric environment as the electrode substrate/solution interface, where solvent molecules have a lower dielectric constant (27) Fuoss, R. M.; Kraus, C. A. J. Am. Chem. Soc. 1933, 55, 2387-2399. (28) Fuoss, R. M.; Kraus, C. A. J. Am. Chem. Soc. 1933, 55, 3614-3620. (29) Fuoss, R. M. Chem. Rev. 1935, 17, 27-42.
Analytical Chemistry, Vol. 69, No. 6, March 15, 1997
1047
σPET ) FΓT{zOf + zR(1 - f)}
than that in the bulk solution.30 The equilibrium constants of the formations of (OX)0 and (OX2)-, which are denoted here as Ko1 and Ko2, respectively, are defined by
Ko1 ) Γ(OX)0/ΓO+[X-]S
(11a)
Ko2 ) Γ(OX2)-/Γ(OX)0[X-]S
(11b)
where zO is the net ionic charge of the oxidized states, which is given by
zO )
where Γi is the surface concentration of species i at the PET and [X-]S is the concentration of X- in the bulk solution that is equal to the concentration of the supporting electrolyte (C). Note that Ko1 and Ko2 are tacitly assumed to be independent on the potential. The surface concentration of the redox active groups in the oxidized states is then given by
ΓO+ + Γ(OX)0 + Γ(OX2)- ) ΓO+(1 + CKo1 + C2Ko1Ko2) (12)
ΓO+ + Γ(OX)0 + Γ(OX2)-
{
}
{
[ (
1 + K′ exp
φM - φPET ) E° - Epzc RT RT 1 - f ln(1 + CKo1 + C2Ko1Ko2) ln (15) F F f
( )
(20)
φ(0) ) φM - φS ) (φM - φref) - (φS - φref)
Solving eq 15 for f with the use of eqs 7b and 7c (i.e., φM ) φ(0), φPET ) φ(d1)) gives
1 ) F {E° - Epzc - (φM - φPET)} 1 + K′ exp RT
]
(21)
) E - Epzc
where φref is the potential of the reference electrode. Substituting eqs 20 and 21 into eq 7c yields
(
)
(
)
d2 d1 d2 1 1 + + σM + FΓT + × 10 20 30κ 20 30κ
{
zO - zR
[ (
zR +
1 + K′ exp
}
)]
σMd1 F E° - Epzc RT 10
-
E + Epzc ) 0 (22)
Then, the faradaic current is described in the following way.
1 σMd1 F 1 + K′ exp E° - Epzc RT 10
[ (
}
)]
σMd1 F E° - Epzc RT 10
Since the applied electrode potential (E) is the sole parameter that can be externally controlled, it is useful to combine the above descriptions with E using the following relation.
(14)
where ΓT is the total surface concentration of the redox species (ΓT ) ΓO+ + Γ(OX)0 + Γ(OX2)- + ΓR0). Substituting eq 14 into eq 13 yields the Nernst equation, which includes Ko1 and Ko2.
[
(19)
1 + CKo1 + C2Ko1Ko2
(13)
The fraction of the redox-active species in the oxidized states (f) is given by
f ) (ΓO+ + Γ(OX)0 + Γ(OX2)-)/ΓT
)
zO - zR
σPET ) FΓT zR +
ΓR0(1 + CKo1 + C2Ko1Ko2) RT ln F ΓO+ + Γ(OX)0 + Γ(OX2)-
f)
1 - C2Ko1Ko2
ΓO+ - Γ(OX2)-
and zR is the net ionic charge of the reduced state. Although zR is 0 in the present case, it is consciously included in the equations for the purpose of general discussion. If the redox-active species in the reduced states also have any charges, the formation of its ion pairs and triple ions must be taken into account, using the same procedures as those used for obtaining zO. Combining eqs 16 and 18 yields the relation between σPET and σM.
Combining eqs 9 and 12 yields
φM - φPET ) E° - Epzc -
(18)
)]
(16)
if ) FAΓT
( )( ) ( )( )
df df dσM dE ) FAΓT dt dσM dE dt
) FAνΓT
df dσM dσM dE
(23)
where
K′ ) 1/(1 + CKo1 + C2Ko1Ko2)
(17)
where A is the electrode area and ν is the potential sweep rate. df/dσM and dσM/dE are obtained by differentiation of eqs 16 and 22, respectively
The charge density at the PET (σPET) is given by (30) Bockris, J. O’M.; Devanathan, M. A. V.; Muller, K. Proc. R. Soc. 1963, A274, 55-79.
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Analytical Chemistry, Vol. 69, No. 6, March 15, 1997
F(d1/10)K′eη df ) dσM RT(1 + K′eη)2
(24)
dσM dE
) 1
(
)
d2 d1 1 + + + 10 20 30κ
(zO - zR)F2ΓT
(
)( )
d2 d1 1 + K′eη 20 30κ 10
RT(1 + K′eη)2
(25)
E, one must calculate σM for a given E. Though there is a nonlinear relationship between E and σM as given by eq 22, the computational calculations can be carried out using the NewtonRaphson method.31 For this purpose, let the whole left-hand side of eq 22 be represented by a function of g(σM). The problem is to find a σM value for a specified E by iterating calculations using the following equation until σM(j+1) eventually becomes equal to σM(j).31
where
η)
(
)
σ(j+1) ) σ(j) M M -
σMd1 F E° - Epzc RT 10
(26)
The faradaic current is obtained as a function of σM by substituting eqs 24 and 25 into eq 23.
[
(
if ) F2AνΓTK′eη/ RT(1 + K′eη)2 1 +
(
(zO - zR)F2ΓT
)
1d2 1 + + 2d1 3κd1
) ]
d2 1 + K′eη (27) 20 30κ
Although it is necessary to calculate σM as a function of E using eq 22 to obtain whole current-voltage curves, the peak current and the peak potential are obtained straightforwardly at f ) 0.5. In that case, σM is given by eq 28 from eq 16.
{
10 RT σM ) ln(K′) E° - Epzc + d1 F
}
(28)
Substituting eq 28 into eq 27 yields the peak current, if,peak.
[ (
if,peak ) F2 AνΓT/ 4RT 1 +
) (
1d2 1 + + 2d1 3κd1
(zO - zR)F2 ΓT
d2 1 + 20 30κ
)]
(29)
Substituting eq 28 into eq 22 yields the peak potential relative to the potential of zero charge of the substrate electrode, Epeak Epzc, as given by
Epeak - Epzc )
{
E° - Epzc +
RT ln(K′) F
}(
1+
)
1d2 1 + + 2d1 3κd1
(
)
(zO + zR)FΓT d2 1 + (30) 2 20 30κ It should be noted that the appearance of the peak current at f ) 0.5 is recognized under the assumption of the exponential potential distribution through the diffuse layer. However, if the potential distribution through the diffuse layer is governed by another model, such as the Gouy-Chapman model13 (see Appendix), the peak current does not always appear at f ) 0.5. Numerical Calculation for Whole Current-Voltage Curves and Potential Distribution. The faradaic current of if given by eq 27 is a function of η, which varies linearly with σM as given by eq 26. In order to investigate the variation of if as a function of
g(σ(j) M) (∂g(σM)/∂σM)σM)σ(j)
(31)
M
where superscript (j) is the iteration number. The calculation was carried out using (10E)/d1 as the initial value σM(0). Once σM is obtained for a specified E, then, if, σPET, and f at E are obtained by using eqs 27, 20, and 16, respectively. Potential distribution for a specified E is, in turn, obtained by substituting the obtained σPET and σM into eqs 7a-c. Another method adopted in the work by Smith and White17 is also applicable to draw a current-voltage curve. If one arbitrarily chooses a value of f, the corresponding σM value is obtained by eq 16. Then, E and if can be evaluated by using eqs 22 and 27, respectively. It was confirmed that both methods gave completely the same voltammetric waves. RESULTS AND DISCUSSION Current-Voltage Curves. To examine the effects of the charges at the PET on voltammetric behavior, calculations were carried out using the following common parameters: E° ) 0.2 V vs Epzc, T ) 298 K, 1 ) 3, 2 ) 12, 3 ) 78.5, d1 ) 0.9 nm, and d2 ) 0.1 nm. The dielectric constants of 1 and 3 chosen are typical values of aliphatic monolayers32 and the bulk water,30 respectively, and the 2 used is that reported for the secondary water layer.30 The value of d1 ) 0.9 nm corresponds to the thickness of the SAM of CH3(CH2)7SH.33 The length of d2 ) 0.1 nm is used as the typical size of a water molecule as a dipole.34 The calculations were repeated until g(σM(j))/g′(σM(j)) became smaller than 10-9 C m-2, which was of convergence tolerance. If the potential in the diffuse layer is given by an exponential function, the charging current is not large enough to distort the voltammogram. Therefore, the contribution of the charging current is not considered here. The expected voltammograms in that case are symmetric with respect to the peak potential. However, it is noteworthy that the shape of voltammogram is asymmetric with respect to the peak potential if the charging currents cannot be ignored, as already discussed by Smith and White17 with the use of eq A-1 given in the Appendix. Figure 3 shows normalized voltammograms in 1 mol dm-3 1:1type electrolyte solution, which were obtained by dividing the current value by the product of electrode area and potential scan rate (Aν). An ideal Nernstian-type voltammogram without any consideration of both ionic association and double-layer effect is (31) Ralston, A.; Rabinowitz, P. A First Course in Numerical Analysis; McGrawHill: New York, 1965; Chapter 8. (32) Miller, C.; Cuendet, P.; Gra¨tzel, M. J. Phys. Chem. 1991, 95, 877-886. (33) Bain, C. D.; Troughton, E. B.; Tao, Y.-T.; Evall, J.; Whitesides, G. M.; Nuzzo, R. G. J. Am. Chem. Soc. 1989, 111, 321-335. (34) Barger, V. D.; Olsson, M. G. Classical Electoricity and Magnetism a Contemporary Perspective; Allyn and Bacon, Inc.: Boston, MA, 1987; Chapter 4.
Analytical Chemistry, Vol. 69, No. 6, March 15, 1997
1049
Figure 3. Calculated cyclic voltammograms of surface-confined redox species at (A) ΓT ) 0.3 × 10-10 mol cm-2 and (B) ΓT ) 3 × 10-10 mol cm-2: (a) Ko1 ) Ko2 ) 0, zO ) +1; (b) Ko1 ) 1, Ko2 ) 0, zO ) +0.5; (c) Ko1 ) 10, Ko2 ) 0, zO ) +0.09; (d) Ko1 ) 100, Ko2 ) 0, zO ) +0.01; (e, broken curve) Ko1 ) 100, Ko2 ) 0.25, zO ) -0.19; and (f, dotted curve) ideal Nernstian type without any consideration of both ionic association and double-layer effect. Other parameters are given in the text.
also given by the dotted curve in Figure 3 for comparison, where the full width of half-maximum (fwhm) is 90.6 mV (at 25 °C).13 Since the reversible system gives cathodic and anodic waves having a mirror image of each other, only anodic waves are shown in the figures. When the amount of surface-confined redox species is relatively low (ΓT ) 0.3 × 10-10 mol cm-2), the shape of voltammogram is not greatly different from that of the ideal Nernstian as shown in Figure 3A, but the peak potential is negatively shifted with decreasing the zO values. On the other hand, when the amount of the surface-confined redox species is high (ΓT ) 3.0 × 10-10 mol cm-2), the voltammetric wave largely deviates from the ideal Nernstian in both the shape and peak potential, as shown in Figure 3B. When the net ionic charge at the PET (i.e., zO) is positive, the peak currents decrease with an increase in zO and the width of the wave becomes broad, as shown by curves a-c in Figure 3B. If the positive charges of the redoxactive groups are neutralized with electrolyte anions at the PET, the shape of the voltammetric wave becomes close to that of the ideal Nernstian as shown by curve d in Figure 3B, though the peak potential is different. The formation of triple ion (OX2)giving a negative zO makes the shape of the wave higher and narrower than that of the ideal Nernstian as shown by curve e in Figure 3B. Experimental verifications of such voltammetric behaviors are available for an Au electrode coated with the SAM of alkanethiol having a ferrocenyl group, where spiky voltammograms having fwhm narrower than 90.6 mV were obtained and binding of 3-18% excess of electrolyte anions (ClO4-) to surfaceconfined ferricinium cations was suggested by EQCM studies.11 Figure 4 shows the electrostatic potentials of φM, φPET, and, φF (vs φS) as a function of applied electrode potentials E (vs Epzc) for conditions under which voltammograms b, d, and e in Figure 3B are obtained. As shown in Figure 4B, if positive charges generated at the PET are almost neutralized by the ion pair formation, the magnitude of φM - φPET, which determines the equilibrium state of surface-confined redox species (see eqs 15 and 16), is close to the applied electrode potential (E - Epzc). Then, the shape of the voltammogram is almost the same as that of the ideal Nernstian voltammogram. However, if the positive charges at the PET are insufficiently compensated, the value of φM - φPET is smaller than the applied electrode potential for E > Epzc, where oxidation of the surface-confined redox species takes place, as shown in Figure 4A. Then, an excessive polarization is necessary to complete the oxidation, resulting in enlargement of the width
Figure 4. Electrostatic potentials, φM()φ(0)), φPET()φ(d1)), and φF()φ(d1 + d2)) as a function of the applied electrode potential (vs Epzc): ΓT ) 3 × 10-10 mol cm-2; (A) Ko1 ) 1, Ko2 ) 0, zO ) +0.5, (B) Ko1 ) 100, Ko2 ) 0, zO ) +0.01, and (C) Ko1 ) 100, Ko2 ) 0.25, zO ) -0.19. Other parameters are given in the text. 1050 Analytical Chemistry, Vol. 69, No. 6, March 15, 1997
account but the double-layer effect is not, are also given in this figure with dotted curves for comparison. Considering that the difference in the Epeak between the solid curves and the corresponding dotted curves results from the double-layer effect, deviations of Epeak from the ideal Nernstian behavior (dotted curves) are not great if the surface concentration of the surfaceconfined redox species is relatively small, as shown in Figure 5A. The dependence in that case occurs with the slope of roughly -60 mV/log C. In contrast, as shown in Figure 5B, deviations from the ideal Nernstian behavior are remarkable for a high surface concentration of the redox species unless the charge compensation is completed (curve d in Figure 5B). According to the results obtained by voltammetric studies on the SAM having a ferrocenyl group, shifts of Epeak with a one-decade increase of the concentration of perchlorate were 54-60 mV for both cases of relatively low surface density (2.4 × 10-11 mol cm-2)3 and high density (5.4 × 10-10 mol cm-2).5 The results seem reasonable, because the strong ion pairing must occur between the surfaceconfined ferricinium cation and perchlorate anion in those cases,5 resulting in complete neutralization of surface charges as given by curve d of both Figure 5A and B. Significance of Present Work in Reference to the Preceding Work. As already shown, the peak current (if,peak) and the peak potential (Epeak) are given by eqs 29 and 30, respectively. Both equations contain a common term of 1d2/(2d1) + 1/(3κd1), but this term may be too small to make a significant contribution to results of the calculation. If this term is ignored, eqs 29 and 30 can be simplified
F2AνΓT
if,peak )
4RT + (zO - zR)F2ΓT Epeak ) E° -
Figure 5. Dependence of the voltammetric peak potential (vs Epzc) on the concentration of electrolyte solution. (A) ΓT ) 0.3 × 10-10 mol cm-2 and (B) ΓT ) 3 × 10-10 mol cm-2: (a) Ko1 ) Ko2 ) 0, zO ) +1; (b) Ko1 ) 1, Ko2 ) 0, zO ) +0.5; (c) Ko1 ) 10, Ko2 ) 0, zO ) +0.09; (d) Ko1 ) 100, Ko2 ) 0, zO ) +0.01; (e) Ko1 ) 100, Ko2 ) 0.25, zO ) -0.19. Solid curves are obtained using eq 30, and dotted curves are obtained by calculations with use of Epeak - Epzc ) E° Epzc + (RT/F) ln(K′); i.e., the double-layer effect is taken into account for solid curves but not for dotted curves. Other parameters are given in the text.
of the voltammetric wave. On the contrary, since the formation of triple ions makes the value of φM - φPET larger than the applied electrode potential as shown in Figure 4C, the oxidation is accomplished with small polarization, resulting in decrease in fwhm of voltammetric waves from 90.6 mV. The peak potential (Epeak) is influenced by the concentration of electrolyte solutions, because eq 30 contains two electrolyte solution concentration-dependent variables of K′ given by eq 17 and the Debye-Hu¨ckel reciprocal length κ()zF(2C/30RT)1/2). Figure 5 shows the dependence of the peak potential on the concentration of electrolyte solution for two different surface concentrations of 3.0 × 10-11 and 3.0 × 10-10 mol cm-2 which were used in Figure 3. Results for the case of the ideal Nernstian, in which the formation of ion pairs and triple ions is taken into
(
d2 1 + 20 30κ
)
(32)
RT ln(1 + CKo1 + C2Ko1Ko2) + F (zO + zR)FΓT d2 1 + (33) 2 20 30κ
(
)
Such simplification is acceptable for aqueous media having high 3 but may cause serious errors for nonaqueous media of low 3. Equations 32 and 33 have forms similar to eqs 34 and 35, which were already derived using adjustable parameters,14,15
if,peak )
F2AΓTν 4RT - ΓTRT(rO + rR)
Epeak ) E° -
RTΓT(rO - rR) 2F
(34)
(35)
where rO and rR are so-called interaction parameters which were originally introduced using the Frumkin isotherm to explain an activity effect due to interactions of adsorbed species. However, these equations were derived by assuming that no electrolyte ions were involved in the redox reactions of the surface-confined species. Furthermore, the nature of the interactions itself is not clear, though it may be imagined that purely electrostatic interactions are operative among the surface-confined redox species. The physical significance of the interaction parameters rO and rR, which are given in eqs 34 and 35, can be clarified by correlating the set Analytical Chemistry, Vol. 69, No. 6, March 15, 1997
1051
of eqs 32 and 33 with that of eqs 34 and 35.
rO ) -
(
)
zOF2 d2 1 1 + + ln(1 + CKo1 + C2Ko1Ko2) RT 20 30κ ΓT (36)
rR )
(
)
zRF2 d2 1 1 + ln(1 + CKo1 + C2Ko1Ko2) RT 20 30κ ΓT (37)
According to the original interpretation of the interaction parameters, positive and negative signs of rO + rR mean attractive interaction and repulsive interaction, respectively, and the peak width of voltammetric waves (i.e., fwhm) is determined by the sign and the magnitude of (rO + rR)ΓT.35 With the use of eqs 36 and 37, (rO + rR)ΓT is given by
(
d2 1 + 20 30κ RT
F2(zR - zO)ΓT (rO + rR)ΓT )
)
(38)
In the case of zR ) 0, only zO determines the polarity of rO + rR, and if zO > 0 at zR ) 0, the charge compensation of the cationic species at the PET by electrolyte anions is insufficient, resulting in operation of repulsive forces between the surface-confined cationic species. On the other hand, if triple ions are formed at the PET, the resulting charges give zO < 0, which favors stabilization of cationic species with high density due to operation of the attractive forces. It is of importance to note that the numerator of the righthand side of eq 38 gives the difference between the electrostatic potential energy of the PET at f ) 0 and f ) 1. In other words, the voltammetric peak width depends on the electrical parts of the electrochemical potential but not on the chemical potential. On the other hand, eq 33 predicts that the voltammetric peak position changes, depending on both changes in the chemical potential due to the formation of ion pairs and triple ions (the second term of eq 33) and the self-atmosphere potential of the PET, which stems from σPET at f ) 0.5 (the third term of eq 33). Estimation of OPET in a Real System. As described in the previous section, the potential difference between φM - φPET and E - Epzc is responsible for nonideal voltammetric behavior. If the nonideal voltammetric behavior could be explained by such double-layer effects alone, the fraction of surface-confined species in the oxidized state or reduced state allows the estimation of an average potential at the PET in real systems in the following way. The fraction of surface-confined species in the oxidized state f (E) is obtained from faradaic current if as a function of the applied electrode potential E (vs Epzc).
f (E) )
∫
1 νFAΓT
E i Ein f
dE
(39)
where Ein is an initial potential in the measurement of voltammograms which must be chosen to be f (Ein) ≈ 0. Computational calculations allow the integration of the right-hand side of eq 39 (35) Smith, D. F.; William, K.; Kuo, K.; Murray, R. W. J. Electroanal. Chem. 1979, 95, 217-227.
1052 Analytical Chemistry, Vol. 69, No. 6, March 15, 1997
Figure 6. Illustration of procedure to estimate graphically average local potential in the double layer. (A) Curve a is for f vs (φM - φPET) - (φM - φPET)f)0.5 obtained using eq 16 and curve b is for f vs (E Epzc), which is obtained by converting the experimentally obtained voltammogram using eq 39. (B) Solid curve is for (φM - φPET) vs (E - Epzc) obtained from the one-to-one correspondence of Figure 6A, and the dotted line for (φM - φS) vs (E - Epzc).
to determine f (E). On the other hand, f can also be obtained as a function of φM - φPET - (φM - φPET)f)0.5 from eq 16 without any knowledge of the dielectric constants and the thickness of the electrical double layer, where (φM - φPET)f)0.5 is φM - φPET at f ) 0.5 and is equal to E° - Epzc - (RT/F) ln(1 + CKo1 + C2Ko1Ko2). Of course, the ion pairing of the reduced species must be considered, if necessary. Figure 6A shows f relative to E - Epzc and φM - φPET - (φM - φPET)f)0.5. Since E - Epzc and φM - φPET - (φM - φPET)f)0.5 having the same f value must have one-to-one correspondence, we can obtain the relative relationship between these two parameters as shown in Figure 6B with the unit given by the right axis of the ordinates. The origin of the relationship is determined with use of the relation that φM - φPET ()(10σM)/ d1) is 0 at E ) Epzc. A dotted line in Figure 6B is a line having the slope of 1 drawn through the origin, and one can deduce φPET relative to the potential of bulk solution (φS) at a given E from the difference in the ordinate between the solid curve and dotted line. Furthermore, the value of φM - φPET, which is a function of Epeak - Epzc, gives the formal potential corrected for the double-
layer effect. Thus, the Frumkin effect (or φ2 effect) in the real systems can be evaluated quantitatively from knowledge of the fraction of a surface-confined redox species in either an oxidized or a reduced state.
are obtained.
φ(x) )
(
)
d1 + d2 - x (σM + σPET) + φF 20
d1 + d2 g x g d1 (A-2)
CONCLUSION The theoretical approaches made by Smith and White for cyclic voltammograms of a electroactive self-assembled monolayer electrode17 are refined with inclusion of the potential-independent ion pairing and triple-ion formation. The results derived here does not require the use of adjustable parameters in obtaining the peak current and the peak potential in cyclic voltammograms. It seems desired to refine the present treatments further by taking potentialdependent ion pairing into account. However, one serious problem which we will have in that case is that much more complicated mathematical treatments are required.
φ(x) )
(d1 - x)σM d2(σM + σPET) + + φF 10 20
(A-3) where φF is given by
φF )
[
[
APPENDIX If the potential drop within the organic layer is too small to be approximated by the potential in the diffuse layer, which is given by eq 4, the potential distribution must be governed by the GouyChapman model13
where
]
[
]
(A-4)
The faradaic currents in that case are given by the use of the following relation instead of eq 25.
d2 dσM d1 S ) 1/ + + + dE 10 20 30κ
[
]
zF(σM + σPET) 2RT sinh-1 zF 2RT30κ
ACKNOWLEDGMENT The authors thank Mr. Sinpei Kado at Osaka University for his kind helps in preparing this work. This work was supported by Grant-in-Aid for General Scientific Research No. 08455397 and for International Scientific Research No. 07044150 from the Ministry of Education, Science, Sports, and Culture, Japan.
zF(φ(x) - φS) tanh ) 4RT zF(φF - φS) tanh exp[κ(d1 + d2 - x)] 4RT
d1 g x g 0
{
(zO - zR)F2ΓT
d2 S + 20 30κ
}( )
d1 K′eη 10
RT(1 + K′eη)2
]
(A-5)
S ) [1 + u(u2 + 1)-1/2]/[u + (u2 + 1)1/2] and
u ) zF(σM + σPET)/2RT30κ At the limit of u to 0, eq A-5 becomes eq 25.
x g d1 + d2 (A-1)
If this is the case, the same mathematical treatments as those described in the text can be applied and the following relations
Received for review September 17, 1996. January 8, 1997.X
Accepted
AC960952G X
Abstract published in Advance ACS Abstracts, February 15, 1997.
Analytical Chemistry, Vol. 69, No. 6, March 15, 1997
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