Consequences of Kinetic Dispersion on the Electrochemistry of an

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Langmuir 1995,11, 1797-1806

1797

Consequences of Kinetic Dispersion on the Electrochemistry of an Adsorbed Redox-Active Monolayer Gary K. Rowe, Michael T. Carter, John N. Richardson, and Royce W. Murray* Kenan Laboratories of Chemistry, University of North Carolina, Chapel Hill, North Carolina 27599-3290 Received September 27, 1994. I n Final Form: February 15, 1995@ Mixed monolayers of (ferroceny1carboxy)alkanethiol+ n-alkanethiol have been investigated electrochemically in 2:l (v:v)ch1oroethane:butyronitrile solvent in the temperature range of 120- 150 K. Cyclic voltammetry(CV)ofthese monolayers showslarge oxidation-reduction peak potentialseparationsindicative of electron transfer rate control. The voltammetric wave shapes are also broadened; this and curved log i us time transients observed in potential step experiments are interpreted as a dispersion in the reaction rates of the ferrocene sites. This paper considers origins and three models for such kinetic dispersion: (i)Using simulations, the observed kinetic dispersioneffects can be successfully represented by a Gaussian distribution among the formal potentials (EO')of the surface redox sites. While only an apparent kinetic dispersion (having a thermodynamic origin), we show by simulations that its presence affects potential step log kMp,,, us overpotential ( 9 )plots, depressing the apparent reorganizational barrier energies (A)and elevating the apparent rate constants ( K O ) , consistent with previous experimental observations. Similarly, cyclic voltammetric simulations with a Gaussian distribution of Eo' give excellent fits to experimental voltammograms with midpoint average rates (that with voltammograms can be simulated to fit both the that ) are roughly 6-fold smaller than the average rate (determined experimental wave shape and ~ E P E M ~ a homogeneous population). The temperature and chain from a fit to the experimental A E P Eassuming length dependence of CV simulations are also consistent with experimental observationsand indicate that the dispersion has little effect on the accurate determination of 1(from an activation analysis) or the electronic coupling coefficient (p) (from a plot of log ko us chain length). (ii) A Gaussian distribution of reorganizational energies, which is a real kinetic dispersion, has consequences on the appearance and the analysis of data quantitatively equivalent to those of a distribution of formal potentials. (iii) A kinetic dispersion model based on a Gaussian distribution of tunneling distances (or equivalently the electronic coupling parameter) from the electrode surface is also evaluated. This model predicts curved potential step log i us time plots and, in analysis of log kApp,,, us q plots, undistorted results for 1 but alteration of the apparent KO.

Introduction The extraordinary order of electroactive self-assembled alkanethiol monolayers (SAMs)chemisorbed on gold has provoked intense intere~tl-~' in using them to study effects of distance, order, and chemical environment on heteroAbstract published in Advance ACS Abstracts, April 15,1995. (1) (a) Chidsey, C. E. D. Science 1991,251,919. (b) Chidsey, C. E. D.; Bertozzi, C. R.; Putvinski, T. M.; Mujsce, A. M. J.Am Chem. SOC. 1990,112,4301. ( c )Dubois, L. H.; Nuzzo, R. G.Annu.Rev. Phys. Chem. 1992, 43, 437. (2) (a) Finklea, H. 0.;Hanshew, D. D. J.Am. Chem. SOC.1992,114, 3173. (b) Finklea, H. 0.;Ravenscroft, M. S.; Snider, D. A. Langmuir 1993, 9, 223. (c) Finklea, H. 0.; Ravenscroft, M. S. J . Phys. Chem. 1994,98,3843. (3) Curtin, L. S.; Peck, S. R.; Tender, L. M.; Murray, R. W.; Rowe, G. K.; Creager, S . E. Anal. Chem. 1993, 65, 386. (4) (a)Tender, L. M.; Carter, M. T.; Murray, R. W. Anal. Chem. 1994, 66, 3173. (b) Creager, S. E.; Weber, K. Anal. Chem. 1994, 66, 3164. (5) Nahir, T. M.; Clark, R. A.; Bowden, E. F. Anal. Chem. 1994,66, 2595. (6) (a)Rowe, G. K.; Creager, S. E. J . Phys. Chem. 1994,98,5500. (b) Creager, S. E.; Rowe, G. K. Langmuir 1993,9,2330. ( c ) Creager, S. E.; Rowe, G. K. Anal. Chim.Acta 1991,246,233. (d)Rowe, G. K.; Creager, S . E. Langmuir 1991, 7,2307. (e) Creager, S. E.; Hockett, L. A.; Rowe, G. K. Langmuir 1992,8, 854. (7) Katz, E.; Itzhak, N.; Willner, I. Langmuir 1993, 9, 1392. (8)Song, S.;Clark, R. A.; Bowden, E. F.; Tarlov, M. J. J.Phys. Chem. 1993,97,6564. (9) Shimazu, K.; Yagi, I.; Sato, Y . ;Uosaki, K. Langmuir 1992, 8, 1385. (10)Collard, D. M.; Fox, M. A. Langmuir 1991, 7, 1192. (11) Hong, H.; Mallouk, T. E. Langmuir 1991, 7, 2362. (12) Hickman, J. J.;Ofer, D.; Zou, C.; Wrighton, M. S.; Laibinis, P. E.; Whitesides, G. M. J . Am. Chem. SOC.1991, 113, 1128. (13) Tsutsumi, H.; Furumoto, S.; Morita, M.; Matsuda, Y. J. Electrochem. SOC.1992, 139, 1522. (14) Creager, S. E.; Rowe, G. K. J . Electroanal. Chem. 1994, 370, 203. (15) De Long, H. C.; Donohue, J. J.; Buttry, D. A. Langmuir 1991, 7, 2196. (16)Duevel, R. V.; Corn, R. M. Anal. Chem. 1992, 64, 337. @

0743-7463/95/2411-1797$09.00/0

geneous electron-transfer events. Among the kinetic investigations of these systems, those of Chidseyl and FinMea2 stand out. Both groups interpreted log k M p us overpotential ( 9 ) plots with the Marcuszs free energyrate relation, obtained rate constants (ko)and reorganizational energy barriers (A)to electron transfer, and found similar electronic coupling coefficients (/3 = 1.02-1.1 per CH2) and symmetrical Tafel plots supporting a throughbond tunneling mechanism. Nearly ideal, kinetically homogeneous behavior was o b ~ e r v e d lin , ~aqueous ~ acid. Electroactive SAMs on gold in aqueous electrolyte have been studied over a limited temperature range, and investigations in nonaqueous solvents have encountered monolayer i n ~ t a b i l i t y . ~ ~This , ~ " ,is~ ~unfortunate since (17)Hickman, J. J.; Ofer, D.; Laibinis, P. E.; Whitesides, G. M.; Wrighton, M. S. Science 1991,252, 688. (18) Popenoe, D. D.; Deinhammer, R. S.; Porter, M. D. Langmuir 1992, 8, 2521. (19) (a) Uosaki, K.; Sato, Y.; Kita, H. Electrochim. Acta 1991, 36, 1799. (b) Uosaki, K.; Sato, Y.; Kita, H. Langmuir 1991, 7, 1510. (20) Clark. B. J.: Cleland. W. E.: Hussev. C. L. J . Electrochem. SOC. 1992,139, L107. ' (21) De Long, H. C.; Buttry, D. A. Langmuir 1992,8,2491. (22) Bunding-Lee, K. A. Langmuir 1990, 6, 709. (23) Katz, E.; Borovkov, V. V.; Evstigneeva, R. P. J . Electroanal. Chem. 1992,326,197. (24) Zhang, L.; Lu, T.; Gokel, G. W.; Kaifer, A. E. Langmuir 1993, 9, 786. (25) Redepenning, J.;Tunison, H. M.; Finklea, H. 0.Langmuir 1993, 9, 1404. (26) Obeng, Y. S.; Bard, A. J. Langmuir 1991, 7, 195. (27)(a)Caldwell, W. B.; Chen, K.; Mirkin, C. A.; Babinec, S. J. Langmuir 1993,9, 1945. (b) Kwan, W. S. V.; Atanasosoka, L.; Miller, L. L. Langmuir 1991, 7, 1419. ( c ) Kwan, W. S. V.; Penneau, J. F.; Miller, L. L. J . Electroanal. Chem. 1990, 291, 295. (28) (a)Marcus, R. A. J.Phys. Chem. 1963,67,853. (b) Marcus, R. A. J . Phys. Chem. 1965,43,679. (c)Sutin, N. Annu. Rev. Phys. Chem. 1984, 35, 437. (29) Groat, K. A.; Creager, S. E. Langmuir 1993, 9, 3668. "

0 1995 American Chemical Society

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1798 Langmuir, Vol. 11, No. 5, 1995

Rowe et al.

grade) were used as received. Ethyl chloride (EtC1,Linde) was temperature variation is another route to reorganizational condensed into a Schlenk tube and stored at room temperature. barrier determination, and being nonadiabatic, these Water was purified with a Barnstead NANOpure system. electron-transfer reactions should depend on dielectric Electrode Fabrication. Working and pseudo-reference energetics not dynamics. Nonaqueous solvents are more electrodes were prepared by soldering Teflon-shrouded silveramenable to measurements at lowered temperatures and coated copper wires to the ends of a 2 mm diameter Ag rod also may alter monolayer self-organization in insightful (Johnson Matthey Electronics, 99.999%) and 0.5 mm diameter ways. Finklea has observed nonideal cyclic voltammetry Au wire, respectively, and encapsulating these side-by-sidein a and nonlinear potential step In i us time plots for cylinder of insulating epoxy (Shell Epon 828, m-phenylenedi[ R u ( N H ~ ) ~ P ~ ]monolayers,2c ~+/~+ which gave relatively amine curing agent, cured overnight at 70 "C). The cylinder was sanded (Buehler 600 grit) to expose the Ag and Au disk surfaces, stable responses in several organic solvents. Reorganiwhich before each use were polished with successively finer zational energy barriers determined from potential-stepalumina powder (Buehler) aqueous slurries down to 0.3 pm grit derived log kApp us 7 plots were slightly lower in less polar and sonicated for 5 min in NANOpure water. solvents, did not correlate well with dielectric continuum ChemisorbedMonolayer Preparation. The Au electrodes theory, and were suggested to be influenced by monolayer were etched in dilute aqua regia (3:1:6 Hz0:HCl:HNOd for 5 disorder. These results are consistent with a dispersion min, rinsed with copious amounts of water then ethanol, of kinetic rates in the organic solvents. immersed in the coating solution for a minimum of 24 h, and As part of a project on electron-transfer kinetics at then rinsed thoroughly with ethanol. Ethanolic coating solutions superconductor electrode^,^^ we have m e a ~ u r e d ~the l , ~ ~ contained 1:3 mole ratios (1mM total concentration) of (ferroceny1carboxy)octanethiol and n-octanethiol, or (ferrocenylcarelectron transfer kinetics of ferrocene-terminated SAM's boxy)dodecanethiol and dodecanethiol, or (ferrocenylcarboxy1)over a 115- to 175 Ktemperature range in a binary organic hexadecanethiol and hexadecanethiol. Following the work of cryoelectrochemical solvent. We, as Finklea? also observe Chidsey,lbcochemisorptionwith n-alkanethiols has the purpose nonideal cyclic voltammetric and potential step behavior of reducing ferrocene-ferrocene interactions and film-defectand have sought to examine the consequences of this based lateral ferrocene-fenicenium electron transfers. The Ag kinetic dispersity in the determination of kinetic parampseudo-reference was scraped to expose a fresh metal surface. eters. (Specifically,the ferrocene-terminatedmonolayers No incubation of chemisorbed films in diluent solutionlb was exhibit broadened cyclic voltammetric wave shapes and employed. Electrochemical Measurements. Because EtCl (bp 285 curved log i us time plots in potential step experiments.) K) is a gas at room temperature and to avoid possible thiol We find that the temperature dependence of bo derived desorption, subsequent manipulations were conducted at dry ice from measured cyclic voltammetric hEp~mvalues yields temperature. The working-reference assembly bide infra) was reorganizational energy barriers (1)and (extrapolated) rinsed with chilled butyronitrile and fitted into a slotted stainless k0273 rate constants that are, for several S A M alkyl chain steel cylinder, the closed Pt-faced end of which served as the lengths, reasonably consistent with the dielectric concounter electrode. The working-counter electrode separation tinuum theory and the aqueous phase results of Chidsey.lC was adjusted to ca. 0.5 mm to minimize cell resistance. The cell Analysis of ferrocene SAM voltammetric waveshapes and was filled with chilled 2:l (v:v) EtC1:PrCN electrolyte solution of potential step log kApp us overpotential plots, on the (0.075 M in BudNPFe) and the three-electrode assembly inserted other hand, gave much lower results for 1 and anomalous into the electrochemical cell which was then placed into a liquid helium cryostat (Janis Model 6CND-NVT, Lakeshore Model 805 variations with temperature and potential sweep rate. temperature controller) and brought t o the desired temperature Hypothesizing that the latter methods fail, a t least in (in the present study, 120-150 k 0.5 K). part, because of aspects of kinetic dispersion, we model Cyclic voltammetry (CV)and potential step experiments were effects of kinetic dispersion on determinations of rate carried out using a potentiostat of local construction and a constants, reorganizational barrier energies, and the microcomputer (Micro Systems Engineering 486) with a homeelectronic coupling term p. This paper presents calculamade program for potential control and data acquisition. A hometions based on a Gaussian distribution of formal potentials built active filte1-3~ was employed to improve the signal to noise of the redox sites in the electroactive SAM. The distriburatio and a positive feedback control was employed to effect instrumental positive feedback uncompensated resistance tion of formal potentials is selected to match the broadened correction. The uncompensated resistance correction could be voltammetric wave shape. The thermodynamicallybased estimated from the setting on the feedback compensationmodule. distribution is shown to lead to an apparent dispersion in Actual uncompensated resistances are measured by ac impedance electron transfer rate constants and, upon analysis by log in each experiment at the temperature of interest with a k a p us overpotential plots, low values of determined 1. Schlumberger-Solartron Instruments Model 1255 frequency We also consider a real kinetic dispersion by modeling a response analyzer and a Model 1286 potentiostat. The quantity Gaussian distribution of electron-transfer tunneling disof electroactive chemisorbed ferrocene, r F c , was measured from tances, again analyzing its effects on log k ~ p pus overpothe charge under the ferricenium reduction wave in a slow tential plots. potential scan (e.g.,5- 10 mV/s) cyclic voltammogram.

Experimental Section Chemicals. CpFeCpCOz(CHz),SH (Cp = cyclopentadienyl;z = 8,12,16)were prepared by a literature method.lb Octanethiol, dodecanethiol, hexadecanethiol (Aldrich), HC1 (Mallinckrodt), HN03 (EM Science), butyronitrile (PrCN, Aldrich, 99+%), absolute ethanol ( M E R Alcohol and Chemical Co.), and tetran-butylammoniumhexafluorophosphate(BQ"F6, Fluka, F'uriss (30) (a) Peck, S. R.; Curtin, L. S.; McDevitt, J. T.; Murray, R. W.; Collman, J. P.; Little, W. A,; Zetterer, T.; Duan, H. M.; Dong, D.: Hermann, A. M. J . Am. Chem. SOC.1992,114, 6771. (b) Peck, S.R.; Curtin, L. S.; Tender, L. M.; Carter, M.; Tenill, R. H.; Murray, R. W.; Collman, J. P.; Little, W. A.; Duan, H. M.; Dong, D.; Hermann, A. M. J . Am. Chem. SOC.1995,117,1121. (31)(a)Richardson, J. N.; Rowe, G. K.; Carter, M. T.; Murray, R. W. J . Phys. Chem. 1995,99,766.(b)Richardson, J.N.; Peck, S. R.; Curtin, L. S.; Tender, L. M.; Terrill, R. H.; Carter, M. T.; Murray, R. W.; Rowe, G. K.; Creager, S. E. Electrochim. Acta, in press.

Potential steps were initiated from the cyclic voltammetrically determined surface-confined ferrocene0/+formal potential, EO', and are thus equal to the applied overpotential, 7. The measured transients were typically 0.2-20 s long, depending upon the temperature and the size of the potential step. The applied potential was returned to EO' and the film equilibrated for 2-5 min to re-establish the original ferrocene/ferriceniumpopulation before initiation of the next overpotential step. Background current-time transients were obtained in each experiment by carrying out potential steps ofthe same magnitude and sign as 7,in the flat, featureless double layer potential regions adjacent to the ferrocene potential region (at potentials well negative of EO'). The currents shown are thusly backgroundcorrected. Simulations. All simulated voltammograms and currenttime transients were calculated on a 486DX PC with a standard (32)The potentiostat and feedback circuits were designed by S. Woodward, UNC electronics shop.

Consequences of Kinetic Dispersion on a Monolayer

1has significant permeability to ions and/or solvent and probably a n associated level of disorder. The capacitances of (ferroceny1carboxy)octanethiol octanethiol, (ferroceny1carboxy)dodecanethiol dodecanethiol, and (ferroceny1carboxy)hexadecanethiol hexadecanethiol monolayers in EtC1:PrCN all increase with increasing temperature (Table 1). In fact, the capacitances of the C8 and C12 monolayers are both similar and not much smaller than that for a bare the double layer capacitance of which also increase with increasing t e m p e r a t ~ r e .The ~~ capacitance for these monolayers seems mainly dominated by ionic permeability rather than the absolute thickness of the layer anticipated based on the known structure of similar layers.

3

A

135 K 2-

+

1-

TY

0-1

-2

Langmuir, Vol. 11, No. 5, 1995 1799

1

i

-3 0.8

0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0

E vs Eo' (Volts) Figure 1. Cyclic voltammetry of a CpFeCpCOz(CHdd3H+ CHdCH2)11SH mixed monolayer on 2.0 x cm2 polycrystalline gold in 2:l EtC1:PrCN with 0.075 M Bu4NPFs (prepared from a 3:1 CH3(CHz)llSH:CpFeCpCOz(CH2)12SHethanolic solution at 1 mM total thiol concentration). r F e = 1.3 x

i = nF'kbT*R exp

+

+

[

(-( 1R$nF)(?)]

(la)

The more readily analyzed aspect of the Figure 1 voltammogram is its unusually large E m . Calculations show that a wave shape broadening of 117 mV at 135 K is much greater than expected from the degree of slow spreadsheet program andor a homemade Pascal program.4a Simulated voltammograms and potential step current-time electron transfer kinetics indicated by the overpotential transients for a Gaussian distributionofEo'were generated with (hEpEAK/2). Calculations using Butler-Volmer kinetics 21 separate values (enough so that the results are unchanged for oxidative surface voltammetry according to Laviron3' for larger numbers of values; at smaller numbers of values this predict that E m should be only 57 mV a t 135 K and a is not so) of voltammetricpeak potentials and EFWHM and of rate = 0.5, for any value ofv, kb, 77., and r*R. Other compositions constants as a function of charge segments in plots of log i us of experimental and calculatedE-M are shown in Table time. 1. The E Fvalue ~ predicted from eqs l a and l b shows only a weak dependence on temperature (2') (Table 1, Results and Discussion column headed E-M, BV) or a. The experimental E m Cyclic Voltammetric Wave Shape Broadening. A values (Table 1)increase a t lower temperatures and with typical cyclic voltammogram at 135 K from our invesincreasing chain length. We will discuss these results in tigation31a of (ferroceny1carboxy)dodecanethiol dodemore detail later on. canethiol mixed monolayers chemisorbed on polycrystalE m for a surface voltammetric wave can also be line gold is shown in Figure 1. The oxidation and reduction predicted with the Marcus-DOS kinetic model described waves are well-formed and similarly shaped, with an by Tender et al.4aand Creager and Webefib for cyclic average peak width at half-maximum ( E ~ M of 117 ) mV. voltammetry, based on that by Chidsey for potential The large oxidation-reduction peak separation ( ~ E P E A Ksteps.la This model avoids the assumptions ofthe Butler= 310 mV) is characteristic of the slow electron transfer Volmer theory about the relative values of the reaction kinetics of a n adsorbed electroactive species. The ferfree energy (7.7) and reorganizational energy barrier (11, rocene coverage, r F c = 1.3 x mol/cm2, is ca. oneand accounts for the continuum of electronic states in the third of a full monolayer.33 The monolayers are excepmetal electrode by integration of the Marcus relation28a,b tionally stable a t low temperature, with r F c and volover a range of energies about the Fermi level of the metal tammetry quantitatively reproducible for as long as 1week electrode, or more. The monolayers are, however, voltammetrically unstable a t room temperature and desorb q ~ i c k l y . ~ The Figure 1 voltammetry differs from room temperature aqueous observationsla,bin that the double layer charging current is larger and the ferrocene wave shape is broadened ( E m ) . In aqueous e l e ~ t r o l y t ethe , ~ ~double layer capacitance of n-alkanethiol monolayers on Au(ll1) follows simple Helmholtz behavior and for a (ferrocenylThe outer sphere ilestimated from the dielectric continuum carboxy)dodecanethiol dodecanethiol monolayer like that in Figure 1 is3* 1.7 ,uF/cm2. The double layer (36) Cdl at naked gold increases by cu. %fold over the temperature range 120-180 K. The values Of& at 120,130,140,150,160,170and capacitance for the monolayer in Figure 1,estimated from 180 K are 9.8, 12.5, 14.6, 15.5, 16.5, 17, and 29 pF/cm2,respectively. voltammetric charging currents a t potentials more nega(37) (a) Laviron, E. J. Electroanul. Chem. 1974,52,355, 395. (b) tive than the ferrocene wave, is 18 pF/cm2 in the EtC1: Bard, A. J.; Faulkner, L.F. Electrochemical Methods. (c) Albery, W. J.; Boutelle, M. G.; Colby, P. J.; Hillman, A. R. J . Electround. Chem. PrCN solvent, over 10-foldlarger (Table 1). Such increases 1982,133,135. (d) The variables in eq l a and l b are k b , the reverse in double layer capacitances in organic solvents have been electron-transferrate; r*R,the total electroactiveprobe coverage;Y , the seen We infer that the monolayer in Figure scan rate, q , the applied overpotential (E - Eo' = hEPEAK/2);k'b, the mol/cm2.

+

+

(33) Assuming spherical ferrocene with a diameter of 6.6 A, hexmoUcm*. agonally close-packed, gives a maximum r F c of 4.5 x (34) Porter, M. D.; Bright, T. B.; Allara, D. L.; Chidsey, C. E.D. J. Am. Chem. SOC.1987,109,3559. (35) Finklea, H. 0.; Hanshew, D. D. J.Electround. Chem. 1993,347, 327.

standard reverse rate constant; and a,the transfer coefficient. (38) (a) In eq 2, x denotes an energy state of the metallic electrode; p , the electronic coupling between the electrode and the redox sites across the n-alkane chain; and e, the density of states of the metal electrode. (b)Thecouplingtermp has aweakl dependence: F = (IHm12/ h)(?c/Akbnu2where IHml is the coupling matrix. IHmI2= exp(-pd), so changes in the coupling can directly effect p.

Rowe et al.

1800 Langmuir, Vol. 11, No. 5, 1995

Table 1. Summary of Cyclic Voltammetric Results as a Function of Chain Length and Temperature calcd EFWHM - .. monolayer FcC02CsSWC8SH

FcCOzClzSWClzSH FcC02C16SWC16SH

T (K) AE, (mV) exptl E 120 125 130 135 140 145 125 135 150 130 140 150

234 198 164 129 83 93 374 310 234 544 480 437

~ (mV)" M fitted dE"') (mWb BV 123 105 102 94 87 87 120 117 112 137 129 121

81 63 59 50 (470 42 40 77 71 (559 63 91 81 71

M-DOS (mV)d u (mV/s) 50 53 55 57 59 61 53 57 63 55 59 63

60 60 62 62 59 64 68 70 76 82 83 86

10 20 20 20 20 35 10 10 10 5 5 5

@Flcm2)e 11(2.3) 14 17 17 21 30 13 (1.5) 18 29 7.9 (1.1) 11 15

cdl

Determined from cyclic voltammetry with positive feedback compensation employed. Determined from with simulations following a = 0.5, u = 10 mV/s, TR* = 1.3 x 10-lo mol/cm2. Calculated with eqs la, l b , and 4. Calculated with eqs l a and l b with k g = 8 x eq 2 with A = 0.80 eV and k and pe such that AEPES equals the experimental @PEAK. e Determined from the charging envelope negative of ferrocene oxidation. Values in arentheses represent values for a monolayer that acts as a n ideal capacitor of EML = 2.6 and thickness equal to (sin 60")[no. of CHz][1.54 1. f Values determined from best fit to experimental cyclic voltammograms with eqs 2 and 4 as in Figure 2 (I

:

w

equationzs

(3) gives A = 0.75 eV a t 120 K and 0.77eV a t 150 K39in the EtC1:PrCN mixed solvent. (These A values are near the result, 0.89 eV, of a n activation analysis of the (ferroceny1carboxy)dodecanethiol monolayer rate constant reported elsewhere.31a) Using A = 0.80 eV and selecting k o = 1.30 x s-l and pe = 8.54 x lo5 s-l eV-l so that a 10 mV/s voltammogram calculated from eq 2 has the same A&EM (310 mV) as observed in Figure 1gives a predicted value of E F ~=M 70 m v a t 135 K. The value of E ~ calculated by the Marcus-DOS method is, while larger than that predicted by the Butler-Volmer theory, again more narrow than that of the Figure 1experiment (as are other comparisons between eq 2 and experiment, as shown in Table 1). Thus, the E ~ predicted M from both kinetic models (eqs 1 and 2) are significantly smaller than the experimental value as long as a single value of rate constant, A, or EO' is assumed as above. Kinetic Dispersion Modeled in Cyclic Voltammetry as a Distribution of Formal Potentials. The voltammetric wave shape broadening in Figure 1 and nonlinear potential step In i us time plots (videinfra,Figure 4)3l appear to reflect a dispersion of the rates at which the ferrocene sites react. There can be several sources of such kinetic dispersion. We consider first a model in which the kinetic dispersion arises from a spread in the formal potentials, EO', ofthe surface ferrocene sites. Such a spread of thermodynamic potentials could arise from uneven ion pairing (Fc+, PFC-), ferrocene site-site interactions, or from a distribution of monolayer structures that induce uneven interactions or incur a range of solvation shells. Whatever its source, a spread in EO' will lead to an apparent spread or dispersion in electron transfer rates and analyzed rate constants, as we show in this section. (39) (4) Equation 3 contains a , the radius of the electroactivespecies a = 3.8 A); d, its (normal) distance from the electrode surface [cl = 18.7 1.33 &CHz(taken from ref 3411; cop and E., the optical and static dielectric constants of the solvent, respectively;60,the permittivity of a vacuum; and e, the charge of an electron. The inner sphere (0.01 eV) and reorganization, AIS, for ferrocene has been is small compared to 10s in most solvents. The distance per methylene unit was determined by multiplying the number of carbons in the ferrocenealkane chain (plus two to account for the ester group) by the bond length of 1.54A for a carbon-carbon bond in a saturated alkane. This value is then multiplied by sin 60°to account for the 30°tilt angle of the monolayer with respect to the surface normal. (b) Gennett, T.; Milner, D. F.; Weaver, M. J. J.Phys. Chem. 1985,89, 2787.

a,

Definitive proof of a distribution of formal potentials, to distinguish it from other potential sources of kinetic heterogeneity, could be gleaned from a slow scan rate voltammetric experiment so a s to give a reversible response. In the case of a n EO' distribution, the reversible voltammetric peak shape should remain broad, but it is difficult at low temperature to scan slowly enough to attain this condition. However, the same films studied here, when examined4aa t room temperature in aqueous electrolyte, are both close to reversibility and still broad. This observation tends to eliminate other possible sources of wave shape broadening such as double layer effects as discussed later. That is, there is a strong possibility that M a n EO' distribution exists for the monolayer results discussed here. A kinetic heterogeneity based on a thermodynamic distribution (potentially Gaussian) of energy gaps between ground and excited state levels for donor-acceptor molecules undergoing photochemically induced electron transfer has been discussed by Gudow~ka-Nowak~~ to explain experimentally observed inhomogeneous line broadening and nonexponential decay. This is analogous to an electroactive monolayer with a distribution of EO' and thus nonideal voltammetry and potential step behavior. We model a distribution ofEO'with a standard Gaussian expression

where a(Eo')is the standard deviation of the distribution. Voltammetric curves were calculated with the MarcusDOS model (eq 2) to match experimental voltammograms by selecting appropriate values of &O') and summing a minimum of 21 separate voltammograms with Eo' values lying between &3a(E0') and scaled by the Gaussian function. A clarification regarding rate constants determined with eq 2 from cyclic voltammograms exhibiting a Gaussian distribution of EO' is useful a t this point. We consider two kinds of rate constants, the first being the midpoint average rate constant, k o M p A , which is that with which a voltammogram can be simulated using eq 2 and a Gaussian distribution ofEo' to fit both the experimental ~ andMAEPEAK,and the voltammogram's shape and E second is the average rate constant, koAV, which is the rate constant determined from the best fit of AEPEM values (40) Gudowska-Nowak,E. J . Phys. Chem.

1994,98, 5257.

Consequences of Kinetic Dispersion on a Monolayer

Langmuir, Vol. 11, No. 5, 1995

1801

Table 2. Best Fit Kinetic Results for Simulated Cyclic Voltammetltic Waves Exhibiting a Gaussian Distribution of EO‘,as a Function of Temperature” T(K)

EFWHM (mVY

v(V/s)

koAv (S-’)a

0.01 100

2.80 x 10-4 3.20 x 10-4

118 202

1.03x 1.08 x 1.13 x 1.24 x

117 128 166 202

125b 135b

0.01 0.1 10 100

h

0.2

150b

0.0 -0.2

10-3 10-3 10-3 10-3

0.01 100

6.0 x 10-3

6.6 x

115 198

0.01 100

4.15 x 4.0 x

109 194

170b 0.0

0.2

0.4

0.6

E vs E” (Volts) Figure2. Cyclicvoltammogram (anodicwave only)from Figure 1(opencircles),and a calculated voltammogram(open squares, using eqs 2 and 4 with an EO’ distribution of u(Eo’)= 55 mV, kMpA = 1.78 X w4, ,Uc)MPA = 1.48 X io5 ev-l S-l, A = 0.80 ev, v = 10 mV/s, T = 135K, r * R = 1.3 x mol/cm2;21 separate voltammograms were scaled by the Gaussian function and added togetherto generate a smooth simulatedvoltammogram.) Currents are normalized to i,.

(experimental or simulated) to values predicted by eq 2, assuming a homogeneous population of reacting sites. The latter procedure has been e m p l ~ y e din ~ ,our ~ ~ previous assessments of experimental monolayer kinetics and involves no modeling of the kinetic dispersion. Figure 2 shows a calculated voltammogram based on the distribution a(Eo’)= 55 mV, p@MPA= 1.48 x lo5 eV-l s-l, il= 0.80 eV, T = 135 K, and the midpoint average rate constant koMpA = 1.78 x lop4 s-l. The excellent fit to the experimental voltammogram of Figure 1 demonstrates that a n Eo’ distribution is a viable explanation for the observed voltammetric broadening. The minor differences between the two voltammograms in Figure 2 appear to result from the experimental background capacitance ~ , fit ~ the experimental current. While it is p ~ s s i b l eto waveshapes with eq 2 without assuming an EO’ distribution, this requires the use of values of il much smaller (0.40 eVI4than that predicted by eq 3 or obtained from an activation The value Of koAV = 1.03 x s-l for a LIEPEAK (only) best fit to Figure 1 is 5.8-fold larger than the OMP PA determined above (1.78 x s-l). Thus, in this example, the inclusion of a Gaussian Eo’distribution into the kinetic analysis of experimental data produces a “midpoint average” rate constant (koMpA) that is 5.8-fold smaller than an “average))analysis31based on UPEM values alone (koAv) and assuming no EO’ dispersion a t all. We assume that the EO‘ of the “midpoint average” ferrocene site in the Gaussian distribution is equal to the true Eo’(Le.,for no dispersion, and where 7 = 0 a t E = 0.0 V), the value of koMpA thus represents the slowest rate constant of the kinetic distribution since all other sites experience 7 > 0 a t E = 0 and thus have larger rates (see Table 2). In this sense it is unsurprising that the AEpEAK-only evaluation (koAv)seems to moderately favor the “faster”reacting sites in a distribution modeled a s a Gaussian spread of EO’ values. Although the above difference between koMpA and koAV is not very large, it is important to examine whether it is potential scan rate, temperature, or chain length dependent since that would cause the kinetic dispersion to bias the measurement of 1 (from an activation plot) and then ,f3 (from a plot of In (rate constant) us chain length). That is, i t is important to known if koMpA and koAv differ by a constant or a changing factor. Voltammograms simulated

a Average rates constants determined from the best fit to the AEPEM with eq 2 and 1 = 0.80 eV for a simulated voltammogram with a Gaussian distribution of EO’(u(EO’)= 55 mV, ~ Q M P A= 1.48 x lo5 eV-l s-l, 1 = 0.80 eV). *Midpoint average rate of the distribution, koMpA = 1.78 x s-’, determined at 135 K by comparison to experiment as in Figure 2. Midpoint average rate constants at other temperatures, using ~ Q M P A= 1.48 x lo5 eV-l and A = 0.80 eV, are 4.18 x s-l s-lC125 K), 1.10 x (150 K), 7.7 x s - l ( l 7 0 K). C E ~ are m for simulations that include the Gaussian Eo’ distribution.

based on the koMpA value but a t much faster scan rates, and then evaluated using only their LIEPEAKvalues, yield results for ~ O A V (Table 2) that differ from koMpA by the same factor as that obtained a t 0.01 VIS. To address the temperature dependence of koAv, comparisons like that made a t 135 K were simulated a t 125,150, and 170 K, the results are shown in Table 2 (see footnote b). As a t T = 135 K, the koAvmethod favors the faster reacting (at the average EO’) sites of the distribution a t these other temperatures, but by nearly the same factor. The magnitude of the nearly temperature independence bias would cause, in this example, only -4% error in 1 and a 3-fold increase inpe values derived for the C 12 monolayer from an activation plot. The former difference, are probably the latter, are well within experimental uncertainty. Thus, we conclude that a n Eo’-based kinetic dispersion has little or no consequence on measurement of 1 from LIEPEAK values, and perhaps a modest increase in the derived [email protected] ca. 6-fold difference between koAV and koMpA can be expected to vary with the o(E0 of the particular monolayer system. Comparisons of the temperature dependencies of experimental EWM (Table 1)with simulations involving EO’ distribution (Table 2) show that both E F ~values M decrease gradually with increasing temperature. The slight differences between them are well within experiment-to-experiment variability, indicating that a(Eo’)is nearly independent of temperature. The trend in both is opposite to the thermal broadening anticipated from Butler-Volmer and Marus-DOS calculations of E F ~ M (Table 1). E ~ changes M less with temperature for the Marcus-DOS than the Butler-Volmer predictions (see Table l ) , owing to increased peak overpotential with decreased temperature and subsequent peak broadening that tends to offset slightly the thermal narrowing. Kinetic dispersity apparently magnifies the latter effect in causing the observed peak narrowing with increased temperature. The Marcus-DOS calculations of EFWMin Table 1 predict an increase in E ~ with Mincreased chain length; this is caused by the decreasing rate constant ko with increased chain length and consequent increased 7 relative to reorganization energy. Similarly, a decreased length should have the opposite effect.

Rowe et al.

1802 Langmuir, Vol. 11, No. 5, 1995 To investigate the potential effects of chain length on the difference between kOMp.4 and koAv (and the side of a Gaussian distribution of EO' required to fit experimental data), a cyclic voltammogram for a (ferrocenylcarboxyloctanethiol octanethiol mixed monolayer a t 135 K (see Table 1)was fit in the manner of Figure 2. The best fit voltammogram produced a midpoint average koMpA = 1.10 x s-l and a(Eo')= 47 mV, and the AEPEM of this voltammogram produced a n average koAv = 6.0 x lo-'. This distribution size is only slightly smaller than that for the C12 monolayer (55 mV),indicating a minimal chain length dependence of the EO' distribution. The difference between koMpA and koAV is 5.5-fold for the C8 monolayer, similar to the 5.8-fold difference for the C12 monolayer at the same temperature. The small difference between these factors would translate into a ,f? that is smaller by ea. 2% than that for homogeneous populations for the two chain lengths. The lack of dependence of the koAV analysis on both potential scan rate (Table 2) and chain length is internally consistent since both produce changes in

+

U P E M .

the electric charge from the redox centers, thus eliminating any double layer effect. We next consider uncompensated resistance (RUNC) as a source of voltammetric broadening. Positive feedback compensation is utilized in these experiments to correct for uncompensated resistance, and it can be expected that the instrumental compensation is always less than 100% of R T J N CTypical . ~ ~ values of R m c that are corrected for with the positive feedback system are between 1 x lo6 and 1 x lo5 S2 in the temperature range of 125 and 150 K. Direct ac impedance measurements of RUNC generally agree with the instrumental settings. It is important to realize that even a t slow potential scan rates, the slow electron transfer rates characteristic of low temperatures (and tunneling barriers) can give large AEPEM values (Table 1)and small peak currents in voltammograms (a few nanoamps or less), so that i R U N C is small relative to AEPEM even though the R m c remaining after feedback compensation is by usual electrochemical standardsrather large. As a result, experimental voltammograms are often quite similar with and without positive feedback. Calculations of cyclic voltammograms with Marcus-DOS kinetics (eq 2) based on the best fit parameters to the data in Figure 1 and including the effect of R m c = 2 MS24a,44 predict no additional wave shape broadening (beyondthat reported in Table 1)a t a potential scan rate of 5 mV/s, and a n increased broadening of 5 mV a t a potential scan rate of 10 mV/s. It is therefore unlikely that the -PEAK or E ~ results M reported in Table 1are caused by uncompensated resistance. Lastly, waveshape broadening can result from a real distribution of standard rate constants, such as caused by a distribution of tunneling distances or of reorganizational barrier energies; this is discussed later. Effects of a Gaussian Distribution of EO' on Potential Step Experiments. In potential step experiments, a Gaussian distribution of EO' values provokes a n apparent distribution of electron transfer rates because the applied overpotential 7 is reckoned with respect to the average EO' of the redox couple. The effective overpotential experienced by any given redox site will vary with its microscopic formal potential, resulting in a distribution ofreaction rates over the ensemble of surface redox sites and a consequent decay of current with time which is not strictly first order. This apparent rate dispersion may of course occur concurrently with a real dispersion of standard rate constants for different reacting sites. The first-order reaction of a surface-confined species is expected2"to display an exponentially decaying current:

Calculations based on fitting eq 2 with a Gaussian E( distribution to experimental voltammograms are quite tedious. Determinations of u(Eo') with Butler-Volmer kinetic model (eqs 1 and l b ) gives values similar (but somewhat larger, a(Eo')= 71 mV for the best fit to Figure 1) to those determined with the Marcus-DOS (eq 2) method and are computationally simpler. (The difference between Butler-Volmer and Marcus-DOS wave broadening occurs even for homogeneous populations of reacting sites.) Since the Butler-Volmer-based calculations of u(EO') are simpler, the experimental u(Eo')results in Table 1for different chain lengths and temperatures are derived in this manner, using eqs l a and lb. Other Wave-Broadening Effects. Other possible sources of voltammetric wave shape broadening include (i) double layer effects, (ii)uncompensated resistance, and (iii)a real dispersion ofrate constants (vide infra). Double layer effects for redox-active monolayers have been described theoretically by Smith and White41and experimentally observed by Rowe and Creager.6a In the present experiments the supporting electrolyte concentration is relatively small (0.075 M) and the actual ion population probably even smaller due to ion pairing a t low temperatures. Also, r F c is moderately high (one-third of a full monolayer). Since such factors have in double layer simulations been shown to cause wave broadening, we calculated voltammograms to explore them.42 The calculations predict E F ~values M (70 mV at 125 K and 83 mV a t 150 K) much smaller than those observed (Table 1)and additionally a skewing of the wave ~ h a p e These .~~,~~ observations suggest that double layer effects are not the dominant wave-shape-broadener in our experiments. where kApp,, (s-l) is the sum of the forward and reverse Additionally, ion pairing of surface-confined ferricenium electron transfer rate constants (k,,, kred,J a t the effective with PFG-is highly probable a t low temperature; double overpotential 7 , and Q is the charge passed in the electron layer effects would vanish on a thus neutralized surface. transfer reaction. It is known, for example, that PF6- ion pairs with Current-time responses to applications of effective electrogenerated ferricenium in monolayers containing overpotential (applied potential relative to the average ferrocenylalkanethiols in aqueous electrolyte at room Eo')and for given values of u(Eo')and r F c were calculated temperature.6c,d Lastly, penetration of ions into the following eq 5 . In these calculations, the rate constant nominally pure hydrocarbon region of the monolayer, as kApp,rldepends on the effective overpotential as in eq 2, suggested by the capacitance data in Table 1,may screen while pc and A are assumed to be constant. To calculate the current-time curves, the total charge for the reacting (41) Smith, C. P.; White, H. S. Anal. Chem. 1992, 64, 2398. monolayer was divided into 21 segments, each with its (42) The double layer calculations are for a reversible electron own EO.' The size of each charge segment is determined transfer; we assume that the results are similar for irreversible systems.

+

Parameters used are electrolyte concentration Celee= 0.075 M, r F c = 1.3 x mol/cm2,T = 125 or 150 K, monolayer thickness d = 15.24 A, dielectric constant for monolayer and solvent EML = 2.6 and E S ~ =I ~ 22, respectively, the charge on the oxidized and reduced redox probe are +1 and 0, respectively, Ea'= 0.OV (arbitrary), and E ~ Z=C-0.2 V.

(43) Milner, D.; Weaver, M. J. J.Electroanal. Chem. 1986,191,411. (44) Typically compensation at T = 135 K is 3 x lo5 Q. Cyclic voltammograms were generated via digital simulation following eq 2 such that the potential is corrected at each point by adding IRUnc.

Consequences of Kinetic Dispersion on a Monolayer -9

j

.(E0‘)

125K

q=ov

100mV A I 80mV

Langmuir, Vol. 11, No. 5, 1995 1803

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Time (s) Figure 4. (A) Experimental current time transient at T = 125 K and q = 0.20 V for a CpFeCpCOz(CH2)lzSH CH3(CH2)11SH mixed monolayer on polycrystallinegold in 2:lEtC1:PrCN with -g 0.075 M Bu4NPF6 (prepared from a 3:l CH3(CH2)11SH: -10 4 ’ CpFeCpCOz(CHz)&H ethanolic solution at 1 mM total thiol concentration). There was a 7.95 x lo5 8 positive feedback 0.0 0.2 0.4 0.6 0.8 1.0 compensationand background current was subtracted. (B) log Time (s) i us time plot of the data in part A. Figure3. logi us time plots generatedfrom simulatedpotential step experiments with several distributions of EO‘ values (e.g. effective overpotential generate more current at shorter dEo’))following eqs 4 and 5. The potential step q is equal t o times and dominate the response there, whereas sites the averageEO‘ minus the applied potential. The rates for each experiencing a smaller effective overpotential dominate transient were calculated from eq 2 with A = 0.80 eV and a t longer times. The faster sites dominate over the time = 1.0 x lo6 eV-l s-l. The quantity of ferrocene consumed in range shown for 11 = 0 and a(Eo’) > 0, and the slope lies mol/cm2. (A) q = 0.OV at T = the reaction is r F c = 1.3 x 125 K,Curves A-E are dEo’) = 100,80,60,40,0mV. (B) q above the actual rate except at the largest times, whereas = 0.25 V at T = 125 K,Curves A-D are u(Eo’) = 100,60,40, for 7 = 0.25 V, the average slope (kave)pass from higher 0 mV. to lower than the actual rate near the middle of the plot. Figure 4 shows, for comparison, a log i us time plot of by multiplying the total charge by the normalized intensity typical experimental data a t 125 K and 7 = 0.2 V for a of the Gaussian function a t each g i ~ e n E O ’ .The ~ ~ objective monolayer of (ferroceny1carboxy)dodecanethiolcoadsorbed was to generate plots of log kMp,q us effective 7, to be with dodecanethiol. The plot in Figure 4 is curved with analyzed by best fits to eq 2 to see whether the determined a 4-fold difference in rate from longest to shortest times. kinetic parameters 11, ,q,and ko differ from the values Other potential step experiments in EtC1:PrCN a t low pertinent to a surface population having a single EO‘value temperature have given similarly curved log i us time equal to the average EO’. responses31 regardless of chain length, temperature, or applied overpotential. Others have seen such curvature.2c Figure 3A,B show calculated 125 K current-time The calculations shown in Figure 3 demonstrate that a responses, plotted as log i us time according to eq 5, for potential steps to 7 = 0.0 and 0.25 V, respectively, with distribution of formal potentials &O’) selected to fit a respect to the average EO’, each assuming r F c = 1.3 x broadened cyclic voltammogram (i.e.,Figure 21, produces mol/cm2,and for u(Eo’)values ranging from 0 to 100 log i us time plots that are curved in a manner similar to mV. kMp,q for each 7 was calculated from eq 2 using 11 = that observed experimentally (Figure 4). 0.80 eV and puQ= 1 x lo6 s-l eV-l. Parts A (curve E) and The more significant issue is whether the curved log i B (curve D) of Figure 3 correspond to no dispersion in EO’ us time plots described above translate into any substantial and yield linear log i us time plots. All others are curved. bias when they are analyzed according to eq 2 for apparent The extent of curvature increases with increasing formal values of ko, 11, and ,uQ.Figure 5 shows three log kMp,,, us potential dispersion a(Eo’),i.e., a n increasing dispersion 7 plots generated from calculated 125 Kcurrent-time plots in the apparent reaction rate. The difference in rate (slope) with o(Eo’)= 45 mV. kMp,?for each 7 was calculated from for u(Eo’)= 60 mV between the shortest and longest times eq 2 using values of 1 and ,qof 0.80 eV and 1 x lo6 is about 10-fold. Clearly the sites that experience a greater respectively. The overpotential range shown in Figure 5 is typical of that currently accessible in our experiment^.^^ Due to the curvature of the log i us time plots, the total (45) Each Gaussian function has been normalized so that the sum Faradaic reaction charge was divided into 10 equal of all the Gaussian functions equals one.

1

I‘

+

Rowe et al.

1804 Langmuir, Vol. 11, No. 5, 1995 5

-2:

-3

y/

,

":7

,:

-4

0.0

0.2

0.4

0.6

q (Volts) Figure 5. log k ~ p p us , ~ 7 plots generated from calculated current-time transients with anEO' distributionof a(Eo')= 45 mV, following eqs 2, 4, and 5. kmp,,,was calculated at each overpotential with T = 125K, 1 = 0.80 eV, and pg = 1 x lo6 eV-l s-l. The quantity of ferrocene consumed in the reaction mol/cm2. Charge segments 1 (squares), was r F c = 1.3 x 4 (triangles), and 7 (diamonds) correspond t o 0-10, 30-40, 60-70% increments of reacting population. Best fits to each segment are shown as lines through the calculated rates: segment 1, ,I=0.55 eV, yg = 5.41 x lo4eV-l s-l, and ko = 5.0 x s-l; segment 4, ,I= 0.67 eV, p p = 1.74 x lo5 eV-' s-l, and ko = 1.0 x s-l, and segment 7 , A = 0.72 eV, ,up = 1.88 x lo5eV-l s-l, and ko = 3.4 x s-l. The ideal curve, dE0') = 0, is shown as circles.

segments and the rate constant evaluated for each segment from the average slope during the time interval over which that charge segment is passed. This procedure allows the determination of rates for the same population of ferrocene sites as a function of applied overpotential. Figure 5 corresponds to the indicated ferrocene charge segments, i.e., the first 0-10% of reacting sites, the 3040% of reacting sites, etc. Figure 5 (0) also shows a plot for a n ideal homogeneous population (a(Eo')= 0 mV) of redox sites. This ideal curve lies below the calculated ones and exhibits less curvature at higher applied values. The calculated log kApp,rlus q plot for each charge segment in Figure 5 was compared to eq 2 to extract a best fit forko,d, andpe. The results at 125 K are summarized in Table 3. The best fit to the first (0-10% reacting) charge segment produces il = 0.55 eV, pug = 5.4 x lo4 s-l eV-l, and ko = 5.0 x s-l. The apparent values ofd andpe are smaller and that for k o larger than values for a kinetically homogeneous population (a(Eo')= 0 mV). The later charge segments give values of A, pug, and ko that are more similar to but still considerably different from the ideal values, especially pp. Similar calculations (Table 3) show that the distortion in the derived kinetic parameters is less a t higher temperature and substantially greater for larger a(Eo'). I t is significant, however, that the average rate constants for later charge segments, i.e., in the second half of the reacting population, differfrom the ideal standard rate constant K O by less than 2-fold. For this reason we regard k o data derived from experimental potential step results that exhibit kinetic dispersity (Le.,Figure 4) in the manner of Figure 5 as reliable to approximately a factor of 2, a s long as the analysis is confinedto the later-reacting surface population. Similarly derived values ofil andpe are more suspect. Acomparison of the homogeneous rate and that derived a t each charge segment for simulated potential step experiments is analogous to comparing koMpA and ~ O A V derived from simulated cyclic voltammograms. Recall that

best fit voltammograms are such that the difference between koMpA and koAv is approximately 6-fold; a similar difference in rates for potential step simulations is found between the rate determined from the third to fourth segment and the average rate (i.e., the rate determined with no distribution). The significance of this observation is tenuous, however, because the rates determined for charge segments are complicated by the unusually low value of d necessary to fit the Tafel plot. Finally, another possible reason for curved log i us time plots is charge effects within the electrical double layer as described by Weber and Creager.46 As previously stated, we consider that double layer effects are probably minimal in our experiments due to ion pairing. Also, log i us time plots exhibit the same magnitude of curvature for oxidative and reductive overpotential steps (data not shown), which is not predicted by the double layer model. Kinetic Dispersion Modeled as a Gaussian Distribution of Reorganizational Energy Barriers or Tunneling Distances. Kinetic dispersion can of course be real and not apparent as in the EO' distribution discussed above. Of the several forms of kinetic heterogeneity in electroactive SAM's, we consider (i) a distribution of reorganizational barrier energies A (such as might result from variable ferrocene/monolayer/solution interactions), and (ii) a distribution of tunneling distances or electronic coupling factors (such as might arise from monolayer disorder). Consider a distribution in 1. A monolayer with ferrocene sites that exhibit a distribution of il values (but a uniform EO')will generate a kinetic dispersion that is real but which in appearance is very similar to the apparent dispersion caused by a distribution in Eo' at constant d. The similarity occurs because EO' (as q ) and d both appear in the exponential of eq 2. EO' a n d l dispersions under conditions approaching reversibility (i.e,, very slow potential scan rates, higher temperatures, shorter alkyl chains) would however produce differing voltammetry; for the former dispersion, broadened wave shapes would persist, whereas for the latter the wave shape would become ideal. It seems likely that in reality d or EO' dispersity will occur together, both contributing to the kinetic behavior, and making their deconvolution difficult. Kinetic dispersion can also appear as a distribution in the electronic coupling parameter p , which depends2s exponentially on distance, d ,

where ,UO represents the coupling38b a t the smallest electrode-redox molecule separation distance, do, and p is the electronic coupling parameter or distance decay constant. Thus a distribution of either the distance over which electron transfer occurs or the coupling term /3 (by chain conformational changes47)would result in a distribution of p and thus a dispersion in the reaction rate. This dispersion was modeled by assuming that a t the moment of electron transfer a Gaussian distribution of distances exists over which electron transfer can occur. (46)Creager, S. E.;Weber, K. Langmuir 1993,9,844. (47)See, for example: (a) Closs, G. L.; Calcaterra, L. T.; Green, N. J.; Penfield, K. W.; Miller, J. R. J . Phys. Chem. 1986,90,3673. (b) Beratan, D. N.;Hopfield, J. J. J . Am. Chem. SOC.1984,106, 1584.(c) Hoffman, R.Acc. Chem.Res. 1971,4,1.(d) Newton, M. D. Chem.Rev. 1991,91,767.(e) Jordan, K.D.;Paddon-Row,M. N. Chem. Rev.1992, 92,395.(0 HofFman, R.;Imamura, A.; Hehre, W. J. J . Am. Chem. SOC. 1968,90,1499. (g) Ratner, M. A. J.Phys. Chem. 1990,94,4877.(h) Koga,N.;Sameshima, K.; Morokuma,K. J.Phys. Chem.1993,97,13117. (i)Haran,A.;Waldeck,D.H.;Naaman,R.;Moons,E.;Cahen,D. Science 1994,263,948. (i) Broo, A,; Laisson, S.Chem. Phys. 1990,148,103.

Consequences of Kinetic Dispersion on a Monolayer

Langmuir, Vol. 11,No. 5, 1995 1805

Table 3." Best Fits to Calculated Tafel Plots Derived from a Gaussian Distribution of EO' T = 150 K, d E V )= 35 mV T = 125 K, dE0')= 60 mV 1 pup (eV-l 9-l) ko (s-l) charge 1 p e (eV-ls-l) ko (s-l) charge 2 pup (eV-ls-') ko (5-l) (eV) x x 103 segment (eV) x x 103 segment (eV) x x 102 0.55 5.4 5 1 0.50 5.1 1.5 1 0.55 4.3 4 0.60 8.6 3.5 2 0.55 5.95 5.5 2 0.57 4.8 3 0.63 10.3 1.5 3 0.61 9.53 2.2 3 0.63 10.1 2 0.67 17.4 1 0.65 4 13.1 1.2 4 0.66 14.4 1.6 0.68 15.3 0.7 5 0.67 13 0.75 5 0.71 28.4 1.2 0.70 15.7 0.45 6 0.69 11.1 0.4 6 0.73 34.8 1 0.72 18.8 0.34 7 0.71 11 7 38.3 0.75 0.25 0.75 0.8 100 0.28 no distrbn 0.8 100 0.28 no distrbn 0.80 100 0.7

T = 125 K, dE0')= 45 mV charge segment 1 2 3 4 5 6 7 no distrbn

a Best fit to simulations with eq 2. Total charge divided into 10 equally sized segments. All current-time transients simulated to generate the values in the table were determined with eqs 2, 4, and 5 with 1= 0.80 eV and pup = 1.0 x lo6 eV-l s-l.

-5

I

-6

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6

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4h

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Figure 6. log [i] us time plots from calculated otential step experiments (4 = 0.25Vat T = 125 K)with a 1.54ldistribution of redox probe distances from the electrode surface, following eqs 2,4,and 7. The rates for each transient were determined from eq 2 with 1 =* 0.80 eV, pe8,, = 6.0 x lo6eV-l s-l d eve 14 A, d d ) = 1.54 A, p = 1.08 A-l. The quantity of ferrocene consumed in the reaction is r F c = 1.3 x mol/cm2. 9

(The spread in this distribution is implied t o depend on the degree of monolayer disorder.) Figure 6 shows a log i us time plot for a Gaussian distribution of distances, leading to a distribution o f p as described by eq 6, for a monolayer with other parameters as in Figure 1,an average ferrocene distance of 1 4 A from the electrode surface, and an overpotential of 0.25 V at 125 K. The standard deviation of the distance, a(d) = 1.54 A i s taken as the length of a carbon-carbon spacing in an n-alkane, and a constant value of /3 = 1.08 A-l is assumed. The simulated transient (Figure 6) was determined with e q 4, rFc= 1.3 x mol/cm2, and a rate constant that in eq 2 corresponds t o A = 0.80 eV and pea,, = 6 x lo6 eV-l at an average distance of 14 A. The value of p varies from f3a(d), according to eq 6. Figure 6 shows a pronounced curvature, indicative of kinetic dispersion, as anticipated. The r a t e determined from the slope of Figure 5 is greatest at shorter times and varies by over a n order of magnitude. Tafel plots derived from a charge segment analysis of Figure 6 (similar to that of Figure 5 ) are shown as Figure 7, with best fit parameters for charge segments 1,4, and 7 listed in Table 4. As expected, the results vary with the charge segment. Segment 4 yields values of I , p ~a n, d k o close t o the original (no dispersion) values. Interestingly, I varies only mildly with charge segment, which is a significant difference between this type of kinetic dispersion a n d that d u e t o a Gaussian distribution of EO'. The minor variation in d seems reasonable since t h e dispersion is in the pre-exponential term, which contains only a weak

Figure 7. log k ~ ~ usp 17, plots ~ from current-time transients calculated for a 1.54 distribution of d , following eqs 4 and 7, for three charge segments. k a p p , g calculated at each overpotential with T = 125 K,lo=0.80 eV, ppave= 6.0 x lo6 eV-l s-l d d ) = 1.54 A, d,,, = 14 A, and /3 = 1.08 A-1. The quantity 0; ferrocene consumed in the reaction is r F c = 1.3 x mol/cm2. Best fits to each segment are shown as lines through the calculated rates: for segment 1,1 = 0.75 eV, pp = 9.43 x lo6 eV-l s-l, and ko = 8.5 x s-l; segment 4 , 1 = 0.78 eV, pup = 3.5 x lo6 eV-l s-l, and ko = 1.6 x s-l; and segment 7, 1= 0.78 eV, ,up = 1.0 x lo6 eV-' s-l, and ko = 4.5 x s-l. The ideal curve, d d ) = 0, is shown as circles. Table 4." Best Fit Kinetic Parameters of Tafel Plots Derived from a Gaussian Distribution of Redox Probe Distances 1 2 3 4 5 6 7 8 no distrbn

9.4 4.4 5.1 3.5 2.5 1.6 1

0.56 6

0.75 0.75 0.78 0.78 0.78 0.78 0.78 0.78 0.80

8.5 4 2.3 1.6 1.1

0.7 0.45 0.25 1.7

a Best fit to simulationswith eq 2. Total charge divided into 10 equally sized segments. All current-time transients simulated to generate the values in the table were determined with eqs 2,4, 5, and 7 with d d ) = 1.54 A, 1 = 0.80 eV, pup - 6.0 x lo6 eV-l s-l, T = 125 K, p = 1.08 A-1, and d., = 14

dependence38b on A. The other variables, p~ a n d K O , do however change considerably from one charge segment t o t h e next. It is instructive t o examine the physical aspects of a self-assembled monolayer that could lead to a kinetic dispersion through eq 6. Several studies have deduced similar /3 values, 1.0- l.l/CHz for electron transfers through n-alkanethiol monolayers on gold.1aJc92a,48-50P

Rowe et al.

1806 Langmuir, Vol. 11, No. 5, 1995 was also determined to be independent of the electrode potential, consistent with a through-bond tunneling mechanism.1c,47,51 Similar p values result from kinetic studies with immobilized1a,c,2and with soluble redox probe^^^,^^ at SAM’s,suggesting that the mechanism for electron transfer across the monolayer is similar for these different physical situations, Le., the tunneling barriers are similar for electron transfers through coadsorbed n-alkanethiols and through the chain of the redox probe. In fact, recent results by Finklea52asupport the notion of electron transfer through the diluent alkanethiol in mixed alkanethiolhedox-active alkanethiol monolayers. Since the termini of (ideally, all-trans) SAM alkane chains are known to have gauche defectslc and defects there and in the chain interior may be more pronounced in organic solvents, it is reasonable to suppose that variances can occur in distance d or in the orientation of the chains Gee.,PI. Super-exchange calculation^^^ for the electron transfer of donor-acceptor molecules predict substantial changes in coupling with alkyl orientation. For example, a single gauche bond in the middle of an all-trans n-alkane chain is calculated47hto decrease the coupling matrix (HM)by a factor of 20 (from 76 to 1591 cm-l), which translates into a 2-fold change inp. Gauche defects c ~ n c u r r e n t l yproduce ~~j a change in d (measured normal to the surface), so dispersion through eq 6 requires inter-related d d ) and a(p) dispersions. Experimentally, an alkylsiloxide SAM has been observed47hto exhibit a larger tunneling barrier upon annealing to introduce gauche defects. Correlating monolayer structure (disorder) and coupling and thus electron transfer rate dispersion in SAM’s is an interesting experimental prospect which will require application of other structure-sensitive techniques. A more detailed examination of the consequences of a distribution of tunneling distances is possible, particularly in regard to fitting experimental cyclic voltammograms, but since we have little structural information regarding our system, and since simulated Tafel plots with such distributions did not agree well with our experimental results, we did not pursue this notion. It is clear from the above discussion how one would go about doing so, and this may be more appropriate for other systems. Summary. A Gaussian distribution of EO’ with a(Eo’) of approximately 55 mV is consistent with the experi(48) Becka, A. M.; Miller, C. J . Phys. Chem. 1993, 97, 6233. (49)Xu, J.; Li, H-L; Zhang, Y. J . Phys. Chem. 1993, 97, 11497. (50) Carter, M. T.; Rowe, G. K.; Richardson, J. N.; Murray, R. W. J. Am. Chem. SOC.1995,117,2896. (51) Simmons, J. G. J . Appl. Phys. 1963,34, 1793. (52) (a) Finklea, H. 0. 208th meeting of the American Chemical Society, Washington, DC, COLL #0233, August 1994. (b) Rowe, G. K., University of North Carolina, 1994, unpublished results.

mentally observed wave shape broadening in cyclic voltammetric experiments and the curvature in log (i) us time plots derived from potential step experiments. Interestingly, fitting kinetic parameters to Tafel plots derived from calculated potential step experiments that account for a Gaussian distribution ofEO‘gives low values of 1 and puQand high values of ko as compared to the true values (Le. those expected with a homogeneous population of kinetic sites) a t the earliest charge segments and approach but do not equal the true values at later ones. Thus, analyzing Tafel data derived from potential step experiments with monolayers having kinetic dispersion caused by a Gaussian distribution of EO’ can yield erroneously low values of 1 and puQand high values of ko. Error in ko can be reduced to ca. 2-fold by reliance on later charge segments. Results from cyclic voltammetric simulations give an average rate constant roughly 6-fold larger than the midpoint average rate constant incorporating a Gaussian EO’ distribution. This bias is nearly temperature and chain length independent and does not significantly effect values of A and puQdetermined from activation plots or determination ofp. Finally, a Gaussian distribution of reorganization energies, which is real kinetic dispersion, has consequences on the appearance and the analysis of data qualitatively equivalent to those of a distribution of formed potentials. Kinetic heterogeneity caused by a Gaussian distribution of tunneling distances has also been considered. Tafel plots derived from these simulations result in best fit values of puQand ko that are higher than the true values at early charge segments, nearly the same at intermediate segments, and lower at later ones. The value ofil is nearly the same as the true value at all charge segments, which is a significant difference between this type of kinetic heterogeneity and that of a Gaussian distribution of EO’. Therefore this type of kinetic heterogeneity can, in principle, be distinguished from a Gaussian distribution ofEo’ on the basis of its ability to render values of il that are in accord with less dispersion-sensitive appro ache^^,^,^^ like cyclic voltammetric AEPEM analysis or activation plots. Predictions made with a Gaussian EO‘ distribution appear to give the most consistent agreement with experimental cyclic voltammetric and potential step results and is therefore favored as the likely cause of apparent kinetic heterogeneity.

Acknowledgment. This research was supported by grants from the Office of Naval Research and the National Science Foundation. LA940769N