Langmuir 1987,3,409-413 cess, was observed. This may be due to a side reaction during the irradiation with 254-nm light. The'photodimer may produce photodimerization by another remaining double bond on prolonged irradiation. Thus, the absorption peak at 270 nm decreased gradually on continued irradiation.
Conclusions Previous studies in the intermolecular reactions of LB films have been limited mainly to photopolymerization and photodimerization. In this study, the intermolecular photoreversible reaction in an LB films has been found for the first time, although the photoreversible reaction of cinnamylideneacetic acid has already been known in polymer film or in low-temperature glassy matrix. Originally, one might expect that these photoreversible systems are more favorable in LB films than in polymers. However, the reversibility was still low for the LB film of
409
2a, similarly to the polymer film of poly(viny1cinnamylideneacetate). This is due to the overlapping of the absorption spectra of 2a and the photodimer of 2a and also to the side reaction arising from the chromophore containing two double bonds. Therefore, further investigations are required to find other chromophores for the improvement of reversibility.
Acknowledgment. We thank Drs. A. Kuboyama, S. Matsuzaki, and H. Tanaka for valuable discussions and Dr. N. Wasada and Dr. K. Hayamizu and her co-workers for spectral data.41 (41) MS IR, and 'H and 13C NMR spectra for the compounds (four acids and their methyl or ethyl esters) in this work are saved in the Spectral Data Bank System (SDBS) constructed by our laboratory (NCLI) in the Research Information Processing System (RIPS) of Tsukuba Research Center. The spectral patterns for the compounds are available on request.
Blocking Oriented Monolayers of Alkyl Mercaptans on Gold Electrodes H. 0. Finklea," S. Avery, and M. Lynch Department of Chemistry, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061
T. Furtsch Department of Chemistry, Tennessee Technological University, Cookeville, Tennessee 38505 Received August 11,1986. In Final Form: January 12, 1987 Alkyl mercaptans with long hydrocarbon chains (Clz,C14, Cia, and c18)spontaneously form organized monolayers on gold during adsorption from solution. The oriented monolayers are stable over a wide potential range in aqueous electrolytes. They strongly block electrochemical oxidation of gold and also electron transfer with redox couples in solution. Tafel plots exhibit anomalously low but nonzero slopes for overpotentials up to 0.6 V and are often nonlinear. Electron tunneling across the full width of the oriented monolayer is contraindicated. Faradaic current appears to be composed of electron transfer at defect sites and electron tunneling at "collapsed" sites in the monolayer.
Introduction We are using oriented monolayers to create interesting structures on electrode surfaces. Oriented monolayers offer the possibility of precise control of spacing and orientation on a molecular level, a feature of current interest in electron transfer reacti0ns.l Our initial monolayers were synthesized by using octadecyltrichlorosilane (OTS), a molecule which successfully forms bonded oriented monolayers on alumina and OTS readily self-assembles into a monolayer on a gold or platinum electrode, but despite apparent lateral siloxane bonds between the heat groups, the monolayer is unstable during faradaic processes.8 We hypothesize that a strong interaction between the head group and the electrode is necessary to prevent monolayer desorption during the intense ion flux that accompanies oxidation or reduction. Our hypothesis is supported by the results reported herein: monolayers formed from alkyl mercaptans are *To whom correspondence should be addressed. Present address: Department of Chemistry, West Virginia University, Morgantown, WV 26506
0743-7463/87/2403-0409$01.50/0
remarkably stable as electrode coatings. The known high affinity of sulfur for goldgaccounts for the increased stability. Furthermore, the alkylmercaptans form sufficiently compact monolayers that electrochemical processes, Le., gold oxidation and electron transfer with solution redox molecules, are strongly suppressed. In such circumstances it becomes possible to investigate electron transfer by tunneling across the monolayer. (1)(a) Calcaterra, L. T.; Closs, G. L.; Miller, J. R. J.Am. Chem. SOC.
1983,105,670-671. (b) Miller, J. R.; Calcaterra, L. T.; Clm, G. T. J. Am. Chem. SOC. 1984. 106. 3047-3049. (2) Polymeropoulos, E. E.; Sagiv, J. J. Chem. Phys. 1978, 69, 1836-1847. (3) Sagiv, J. Isr. J. Chem. 1979,18, 346-353. 1980, 102, 92-98. (4) Sagiv, J. J. Am. Chem. SOC. (5) Netzer, L.; Iscovici, R.; Sagiv, J. Thin Solid Films 1983, 99, 235-241. (6) Maoz, R.; Sagiv, J. J. Colloid Interface Sci. 1984, 100, 465-496. (7) Gun, J.; Iscovici, R.; Sagiv, J. J. Colloid Interface Sci. 1984,101, 201-213.
(8) Finklea, H. 0.; Robinson, L. R.; Blackburn, A.; Richter, B.; Allara, D.; Bright, T. Langmuir 1986, 2, 239-244. (9) Chambers, J. Q. In Encyclopedia of Electrochemistry of the Elements; Bard, A. J., Ed.; Marcel Dekker: New York; Vol. XII, pp 329-502.
0 1987 American Chemical Society
410
Langmuir, Vol. 3, No. 3, 1987
Table I. Ellipsometric Thickness for Alkyl Mercaptan Monolayers on Gold no. of no. of carbons d,” i% carbons d,” A 12 14
14 (2) 18 (1)
16 18
-p
Finklea et al. ....
,.
18 (1) 20 (2)
* Numbers in parentheses are standard deviations.
Experimental Section Gold electrodes were purchased from Evaporated Metal Films Inc. They consisted of 1000 A of gold on top of 50 A of a TiOz sticking layer. T h e sheet glass substrate was cut into 1 (or 1.5) by 3 cm slides. The electrodes were cleaned by dipping briefly in hot HZSO4/H2Ozand then electrochemically cycling the potential between -0.4 and +1.4 V vs. SCE in 0.5 M HzSO4 until a stable voltammogram was obtained. Alkyl mercaptans (C12H,SH, CI4HmSH,Cl6HS3SH,CI8H3$H; Aldrich) and chloroform (spectro-analyzed; Fisher) were used as received. Highresolution gas chromatography showed the main component of the mercaptans to comprise 92-98% of the total peak area, with only trace levels of the disulfide dimer. Monolayers were formed by immersing the electrode into a 30-100 m M chloroform solution of the mercaptan for 10 min.1° The electrode surface emerged completely dry. The electrode was then dipped briefly in pure chloroform to rinse off any excess mercaptan residue. Neither the solvent nor the concentration of the mercaptan appeared t o be critical in the deposition. Alternative solvents include hexadecane and bicy~lohexyl.~-’ Mercaptan concentrations could be varied from 5 mM to neat liquid. However, the best monolayers in terms of highest wetting contact angles, ellipsometric thicknesses, and blocking behavior were obtained under the above conditions. Liquids with surface tensions equal to or greater than that of hexadecane do not wet the coated electrode. Static contact angles were uniformly high for all the mercaptans: 10&112° for water and 43-47O for hexadecane. Ellipsometric measurements were taken immediately before and after coating each electrode. Average coating thicknesses are given in Table I. The thicknesses are comparable to values expected for a fully extended and vertically oriented monolayer. The accuracy of the method does not permit a conclusion on the packing density or any tilt in the hydrocarbon tails. A grazing angle infrared spectrum of a CIS monolayer” shows four sharp C-H stretching modes a t 2965,2919, 2879, and 2851 cm-’ (relative intensities are 1:3.60.8:1.4). The spectrum is similar to that reported for an OTS monolayer.8 All of the preceding data are consistent with a single layer of mercaptan molecules oriented with the sulfur head groups attached to the gold substrate and the terminal methyl groups forming the exposed surface. Electrochemistry was performed in a conventional threeelectrode cell. Tafel plot data were generated under conditions of constant stirring with both the oxidized and the reduced forms of the redox couple present in equal concentrations.
Results Figure 1illustrates the typical blocking behavior of the alkyl mercaptan in the absence of any added redox couple. The top cyclic voltammogram of clean gold exhibits the characteristic gold oxide formation at +1.1-1.4 V vs. SCE. On the return scan a sharp oxide stripping peak appears at +0.9 V vs. SCE. On the coated electrode the gold oxidation is not visible, and only a small oxide stripping peak can be distinguished from the charging current. A histogram of the oxide stripping peaks for the different mercaptans is shown in Figure 2. The mercaptan monolayer suppresses gold oxidation by 3-5 orders of magnitude. Reimmersing the electrode in the deposition solution has little effect on the stripping peak. The initial potential (10)Nuzzo, R. G.; Allara, D. L. J. Am. Chem. SOC. 1983, 105, 4481-4483. (11) Finklea, H. 0.; Melendez, J. A. Spectroscopy 1986, 1, 47-48.
U
A
b
t
t
t 0.1 uA
Figure 1. Suppression of gold oxidation on a coated electrode: (a) clean electrode; 0.5 M H&104,100 mV/s, electrode area 0.8 cm2; (b) electrode coated with cl6H3,SH. The heights of the oxide stripping peaks are shown.
clean electrode
log(istrip)
-3
-4
t
*
e e
. ..
i
e
-9
1 t
e
b
e
‘12
‘14
‘16
‘18
Figure 2. Histogram of oxide stripping peak currents. Conditions are the same as in Figure 1; currents are A/cm2.
scan from 0 to +1.4 V vs. SCE often exhibits increased anodic currents and sometimes charging current spikes, but subsequent voltammograms are essentially unchanged after an hour of continuous potential cycling. Extending the potential scan several hundred millivolts beyond the limits in Figure 1 (-0.4to +1.4 V) results in an abrupt current increase, gas evolution, and destruction of the monolayer. Based on charging currents during cyclic voltammetry, the monolayer causes a dramatic decrease in the electrode/electrolyte capacitance. Typical capacitances in 0.5 M H2S04 are 1-5 pF/cm2, compared to capacitances greater than 100 pF/cm2 on bare gold. Assuming a parallel plate capacitance model with the monolayer serving as the dielectric, one predicts a proportionality between the monolayer thickness and reciprocal capacitance. However, our data show no definite correlation between the ellipsometric thickness and the reciprocal capacitance (correlation coefficient ca. 0.25). Greatly reduced faradaic currents are observed for redox couples in aqueous electrolytes with no visible depletion effects even in unstirred solutions. Typical Tafel plots are shown for Fe2+/3+(Figure 3) and R u ( N H ~ ) ~ ~(Figure +/~+ 4). In Figure 4 only the anodic branch is shown because water reduction coincides with reduction of Ru(NH,),~+. An interesting question is whether the alkyl mercaptan monolayer retains its blocking behavior in acetonitrile since a wider potential range is accessible in that solvent. We
Blocking Oriented Monolayers of Alkyl Mercaptans
fraction of defect sites is remarkably low considering that the gold surface is polycrystalline and visibly rough under a microscope. Potential defect sites, i.e., steps and kinks, may be partially protected by collapse of the monolayer about the defect. The sulfide/disulfide redox reaction is conspicuously absent from the aqueous voltammetry over the entire accessible potential range. Strong adsorption is known to passivate other electrochemically active species, i.e., hydroquin~ne.'~J~ As long as the electrode potential is kept between water reduction and water oxidation, the mercaptan head groups remain passivated. In acetonitrile, the mercaptans remain passivated if the potential is kept positive of 0 V vs. SCE. More negative potentials appear -.5 *", ..5 to desorb the monolayer so that the mercaptan moieties become electroactive. The maximum charge passed during Figure 3. Tafel plots for Fe2+I3+:0.5 M H2S04,50 mM each the oxidation step corresponds to 60% of the monolayer Fe(I1) and Fe(III),Eo' = 0.36 V vs. SCE currents are A/cm2; (a) assuming 20 A2 per molecule and one electron per moleclean electrode; (b) ideal slopes for transfer coefficients of 0.5; (c) electrode coated with C16H,SH. The apparent transfer cule. coefficients are 0.25 at +0.5 V and 0.23 at -0.5 V overpotential. The cyclic voltammogram in Figure 1 is stable during continuous potential cycling for more than an hour. Tafel log i plots generated on three successive days with the same a electrode are reproducible within a factor of 2. The concomitant oxide stripping currents fluctuate irregularly, sometimes increasing and sometimes decreasing. Finally, wetting properties and ellipsometric thicknesses do not change significantly; in fact the apparent thickness tends to increase to ca. 20-25 A. The increase is attributed to -5 b adsorption of organic impurities since a quick dip in chloroform brings the thickness down to normal values. Thus, unlike the OTS monolayers: the alkyl mercaptan monolayers exhibit excellent stability during extensive electrochemical usage. -6 The mercaptan monolayer is effective at blocking the access of a variety of redox molecules to the electrode in water. Permeation through the monolayer is not evident since that mechanism would lead to an increase in the apparent reversibility of the Tafel plots rather than the observed decrease (see below). On the other hand, ace0 +.4 +.0v tonitrile appears to render the monolayer permeable to E -Eo ferrocene. We hypothesize that the strong hydrophobic Figure 4. Tafel plots for Ru(NH&~+/~+: 0.1 M Na2S04,p v 6, repulsion of the hydrocarbon tails from water results in 1mM each Ru(I1) and Ru(III),Eo' = -0.26 V vs. SCE;currehts a tightly packed and therefore impermeable coating (which are A/cm2; (a) clean electrode; (b) electrode coated with C&IH,SH, might account for the lack of correlation between the re(c) postulated currents due to defects in the monolayer; (d) ciprocal capacitance and the ellipsometric thickness), while currents corrected for the defect currents. The apparent transfer coefficient is 0.12 at +0.5 V overpotential. the presence of slight solvation, even when the liquid does not visibly wet the coating, is sufficient to loosen the have found that the mercaptan monolayer is not blocking packing and allow penetration by a neutral (ferrocene) or to ferrocene electrochemistry; cyclic voltammograms of a cationic (ferricenium) molecule. These observations clean and coated electrodes are identical in peak height suggest that the permeability of a monolayer can be adand peak position a t sweep rates up to 200 mV/s. If the justed by mixtures of organic and aqueous electrolytes as electrode potential stays positive of 0 V vs. SCE,then the the contacting phase. monolayer is still present after the voltammetry as shown Tafel plots (Figures 3 and 4) defy simple interpretation. by ellipsometry and contact angles. Potentials negative All redox couples show small apparent exchange currents of 0 V vs. SCE must be avoided because the monolayer is and Tafel-like behavior, but the slopes imply low transfer apparently disrupted. Cyclic voltammograms with negacoefficients in both the anodic and the cathodic branches; tive potential limits result in a surface redox wave apthe dashed lines in Figure 3 show the expected slopes for pearing at ca. +0.7 V. The maximum area of the surface transfer coefficients of 0.5. The plots are curved with a wave corresponds to 50 pC/cm2. The surface wave is tendency toward lower slopes at larger overpotentials. tentatively assigned to the sulfide/disulfide redox c ~ u p l e . ~ Consequently the electrochemistry cannot be described by a single electron transfer rate constant and transfer Discussion coefficient. Suppression of gold oxidation implies that water is efIn order to interpret the Tafel plots in terms of tunfectively blocked from the gold surface. This fact and the neling, it is necessary to first consider two other possible stability of the mercaptan monolayer (see below) point to a relatively strong bonding between the mercaptan head (12) White, J. H.;Soriaga, M.; Hubbard, A. T. J.Electroanal. Chem. groups and the gold. The residual oxidation which remains 1985.185. 331-338. is presumably due to defect sites in the monolayer. Hence (i3)Soriaga, M.;Hubbard, A. T. J. Am. Chem. SOC.1982, 104, coverages are estimated to be better than 99.9%. The area 3937-3945.
-4r-
7
Langmuir, Vol. 3, No. 3, 1987 411
412 Langmuir, Vol. 3, No. 3, 1987
phenomena: (1)electron transfer at defects in the monolayers, i.e., exposed gold surfaces, and (2) changes in monolayer morphology with potential. If the oriented monolayer is completely blocking, with redox molecules only exchanging electrons at pinholes and other defects, then the assembly is equivalent to an array of microelectrodes. The voltammetry of a microelectrode array has been treated theoretically by Amatore et al.14and simulated by Reller et al.15 and Shoup and Szabo.I6 Qualitatively, the nonlinear diffusion to the microelectrodes results in higher current densities and therefore a lower apparent reversibility when compared with a solid electrode with the same geometric area as the array. Consequently the Tafel plot of a microelectrode array (or an array of defects in a blocking monolayer) would exhibit a lower apparent exchange current than the solid electrode, but there would be no change in the slopes in the Tafel region. Also, the transition from the Tafel region to the limiting current plateau should occur within a 0.2-V range. The observed low Tafel slopes and absence of limiting currents for overpotentials of 0.2-0.6 V for either branch are therefore inconsistent with the defect model. It is expected (see below) that defect electron transfer may dominate the Tafel plots at low overpotentials. In Figure 4 a sharp knee is present at 0.1 V overpotential. It is attributed to the defect currents becoming diffusion limited at that point. The further increase in current at larger overpotentials indicates that another mechanism is occurring as well. Correction for the defect currents (Figure 4c) still results in an anomalously low transfer coefficient (Figure 4d). If the monolayer morphology changes as a function of electrode potential, the size and distribution of the defects could be altered, affecting the defect current. These changes may be expected to be slow. We see no evidence for such changes between 0.0 and +1.0 V vs. SCE in 0.5 M HzS04. The Tafel currents are steady state and reproducible as a function of potential. A t potentials more negative than -0.2 V, there is a slow increase in cathodic current with time for Fe3+reduction. Returning to positive potentials yields a slow decrease in current to a steady value. Therefore morphology changes are believed to occur at the negative extremes of the potential range. Theories for tunneling at oxide-covered predict a linear relationship between the thickness of the tunneling barrier and the log of the kinetic current at a given overpotential: log i = -d/d* + constant (1) where d* is the tunneling constant. Reported values for the tunneling constant include 2.3 A for compact quinoline layers on mercury,211.7 A for cobalt complexes attached to gold electrodes by ligands with pendant mercaptan moieties,22and 2.0 A for aluminum-monolayer-aluminum (14)Amatore, C.; Saveant, J. M.; Tessier, D. J. Electround. Chem. 1983, 147, 39-51. (15) (a) Reller, H.; Kirowa-Eisner, E.; Gileadi, E. J . Electround Chem. 1984, 161, 247-68. (b) Ibid. 1982, 138,65-72. (16) Shoup, D.; Szabo, A. J . Electround Chem. 1984, 160, 19-26. (17) Schmickler, W.; Ulstrup, J. Chem. Phys. 1977, 19, 217-232. (18) Schmickler, W. J. Electroanul. Chem. 1977,82, 65-80. (19) Ulstrup, J. Surf. Sci. 1980, 101, 564-582. (20) Schultze, J. W.; Elfenthal, L. J.Electroanul. Chem. 1986, 204, 153-171. (21) Lipkowski, J.; Buess-Herman, A.; Lambert, J. P.; Gierst, L. J . Electroanal. Chem. 1986, 202, 169-189. (22) Li, T. T.-T.;Weaver, M. J. Am. Chem. SOC.1984,106,6107-6108. (23) Samec, Z.; Weber, J. J. Electroanul. Chem. 1977, 77, 163-180. (24) Scherer, G.; Willig, F. J. Electround Chem. 1977, 85, 77. (25) Iwasita, T.;Schmickler, W.; Schultze, J. W. Ber. Bunsenges. Phys. Chem. 1985,89, 138-142.
Finklea et al.
-6
'.
A
0
A A
A
0
0
-1
..
0
8
0
0
0
0
0
0 0
-8
..
00
0 0
0
0
#
Table 11. Correlation of Currents at Large Overpoential with Apparent Rate Constants log i" at overpotentials of redox couple
log k"
-2.5 Fe(CN):-/*-2.0 R u ( N H ~ ) ~ + / ~ + 0.0 ~~2+13+
ref
+0.5 V
-0.5 V
22 23 24
-7.4 (0.4) -6.5 (0.3) -5.1 (0.5)
-7.3 (0.6) -6.4 (0.4)
Currents (A/cm2) are normalized to 1 mM concentration; numbers in parentheses are standard deviations.
sandwichesS2Tunneling currents between a metal and a redox couple across an insulating monolayer would therefore be expected to show a similar dependence on the thickness of the monolayer. Figure 5 shows that the anticipated behavior is not observed for the ellipsometric thicknesses. The currents were measured at a large overpotential(+O.5 V) to ensure that they would be dominated by tunneling rather than by defects. A similar lack of correlation is found for large negative overpotentialsand for reciprocal capacitances. Thus it is evident that the currents cannot be attributed to tunneling across the full width of the monolayer. A striking feature of Figure 5 is the similar values of currents observed for all the monolayers and one particular redox couple. When currents at fixed overpotential (+0.5 and -0.5 V) are averaged for each redox couple, they correlate with the apparent rate constants of the redox couples at bare gold electrodes (Table 11), indicating that electron transfer occurs directly between the metal and the redox couple. To summarize, faradaic currents at large overpotentials at monolayer-coated gold electrodes appear to involve a mechanism other than electron transfer at defects in the monolayer, and yet they do not fit the thickness dependence anticipated for tunneling. We therefore propose a model based on the following assumption: the monolayer contains sites at which a redox molecule can approach closely to the electrode surface but not come in direct contact with it. Such a site may form when the monolayer collapses or becomes disorganized, such as along the edge of a defect (Figure 6A). This introduces the complexity of an electron transfer rate constant which varies over a wide range across the electrode surface. The largest rate constants are present at the defect sites, somewhat smaller rate constants are found at the "collapsed" sites, and the smallest values occur where
Blocking Oriented Monolayers of Alkyl Mercaptans
A
t
is zero, which is the boundary condition used for an array To shorten computation time, the of mi~roelectrodes.~~J~ radial boundaries are expanded exponentially. The complex convection/diffusion conditions of the Tafel experiment are replaced with diffusion across a stagnant layer of fixed width, i.e., a finite number of volume elements perpendicular to the electrode.27 For the simulation shown, the diffusion layer width is arbitrarily equal to the outermost radius. The electron transfer rate constantzs is separately adjustable for each annulus of electrode surface. Transfer coefficients are set at 0.5, and the concentrations and diffusion coefficients of OX and RED are equal. Consequently the simulated Tafel plot is symmetrical about zero overpotential, and only the anodic branch is shown. With only the electron transfer rate constants changing across the electrode surface, it is possible to choose conditions that yield a Tafel plot which has a slope corresponding to an apparent transfer coefficient of 0.18 (solid line in Figure 7). In general terms, the rate constant must be large for a small area fraction of the electrode, smaller for a larger area fraction, and so on. In terms of the monolayer-coated electrode, the innermost element is a defect site with the rate constant found on the bare electrode (Figure 6B). Next to the defect the monolayer is largely collapsed so that tunneling occurs across a thin layer; the rate constant is decreased by 2 orders of magnitude (Figure 6C). A partially collapsed monolayer at a larger radius leads to a further decrease in the rate constant. Over the remainder of the surface the well-oriented monolayer results in the smallest rate constant. The radii and the consequent area fractions are arbitrarily chosen. Also shown in Figure 7 are approximate current components (dashed line) of each of the four domains. The defect current dominates the total current at low overpotentials but quickly becomes diffusion limited. The other domains successively add to the total current as the overpotential increases. Consequently a true Tafel slope is never observed. The main point of the simulation is that it is possible to account for the Tafel behavior of the mercaptan-coated electrodes by assuming that current flow occurs mainly at defect and "collapsed" sites in the monolayer. The overpotential has not been extended far enough in these experiments for tunneling to be observed across the full width of the monolayer. Assuming a tunneling constant of 1.7 A and a monolayer of 17 A, we estimate that overpotentials in excess of 1V would be required for tunneling currents to approach diffusion control. Clearly, thinner monolayers are needed in order to demonstrate tunneling currents. Finally, the sensitivity of faradaic currents to defects or morphology changes in a blocking layer points to the useful application of electrochemistry to studies of these features in organized monolayers.
I
t5
C
d istame Figure 6. Electron tunneling at defecta and collapsed sites in the monolayer. (A) A defect and the collapsed monolayer around it. The center of the defect is at the origin. (B) A simplified representation of (A) suitable for simulation. (C)Variation of the simulation rate constant around the defect site. log(i/i,)
a
Of
-3'
0
Langmuir, Vol. 3, No. 3, 1987 413
+.
2
E- Eo
+A
+.6
Figure 7. Digital simulation of a Tafel plot with a varying electron transfer rate constant. See text for details. Af = area fraction; K = simulation rate constant." (a) Defect Af = 4.0 X lo4, K = 10.1; collapsed site 1 Af = 4.0 X K = 0.101; collapsed site 2 Af = 1.9 X K = 1.01 X perfect monolayer Af = 0.98, K = 1.01 X loa. (b)-(e) are the approximate currents at the respective sites listed in (a).
the monolayer is well oriented. Because of nonlinear diffusion, such a complex model cannot be readily analyzed theoretically, but it is amenable to digital simulation.26 The results of a simple simulation are shown in Figure 7. The simulation is based on a cylindrical array of volume elements oriented with the radii parallel to the electrode surface. The array is finite in the radial direction. The net flux across the outermost radius (26) Feldberg has simulated the effects of tunneling on electron transfer at an electrode but assumed a uniform rate constant across the surface. Feldberg, S. W. J. Electroanal. Chem. 1986,198, 1-18.
Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for support of this research. (27) Bard, A. J.; Faulkner, L. R. Electrochemical Methods: Fundamentals and Applications; Wiley: New York, 1980; pp 27-37. (28) The rate constants in Figure 7 are dimensionless. In terms of the simulation K = ko WID,where ko is the standard rate constant, W is the width of the diffusion layer, and D is the diffusion coefficient.
