Elucidating Redox-Level Dispersion and Local Dielectric Effects within

Article pubs.acs.org/ac

Elucidating Redox-Level Dispersion and Local Dielectric Effects within Electroactive Molecular Films Paulo R. Bueno*,† and Jason J. Davis*,‡ †

Instituto de Química, Universidade Estadual Paulista, CP 355, 14800-900, Araraquara, São Paulo, Brazil Department of Chemistry, University of Oxford, South Parks Road, Oxford OX1 3QZ, United Kingdom

‡

S Supporting Information *

ABSTRACT: The electron exchange between a redox-active molecular film and its underlying electrode can be cleanly tracked, in a frequency-resolved manner, through associated capacitive charging. If acquired data is treated with a classical (non quantum) model, mathematically equivalent to a Nernst distribution for one redox energy level, redox site coverage is both underestimated and environmentally variable. This physically unrealistic model fails to account for the energetic dispersion intrinsically related to the quantized characteristics of coupled redox and electrode states. If one maps this redox capacitive charging as a function of electrode potential one not only reproduces observations made by standard electroanalytical methods but additionally and directly resolves the spread of redox state energies the electrode is communicating with. In treating a population of surface-confined redox states as constituting a density of states, these analyses further resolve the effects of electrolyte dielectric on energetic spread in accordance with the electron-transfer models proposed by Marcus and others. These observations additionally underpin a directly (spectrally) resolved dispersion in electron-transfer kinetics. analyzing the redox characteristics of molecular films in a manner which is both frequency-resolved and free from additional current-generating or resistive terms which contribute (sometimes prohibitively) to standard amperometric analysis.17,22 These “parasitic” contributions are, in part, related to the ionic relaxation and dielectric features of the film and can be spectrally separated from faradaic activity.30 Classically, capacitance arises when two biased parallel metallic plates or metallic foils are separated by an intervening dielectric. The energy storing static field is, when the plates are in contact with electrolyte, confined within a Helmholtz layer31 in a double layer region (the associated capacitance being known as double layer capacitance, Figure 1a). When these molecular films contain a moiety with orbital states that are energetically accessible (redox-active) the electron transfer that results to/from the underlying metallic electrode generates a new, sensitively potential-dependent,25 charging process at this interface (Figure 1b). This faradaic capacitance is not entirely electrostatic and can be (for high-quality molecular films with associated fast rates of heterogeneous electron transfer) hundreds of times greater than the Helmholtz contribution. Its presence and underlying Gaussian variance with formal surface potential has been reported recently.22,25

O

ne of the most intensively studied areas in electroanalytical chemistry for the past 2 decades has been the modification of electrode properties by the on-electrode assembly of molecular films.1 This assembly is a powerful means of tuning the thermodynamics and kinetics associated with molecular electron transfer1−4 and has underpinned profound developments in molecular sensing,5,6 molecular electronics,7−9 biosensors, and derived diagnostics.10−15 Electroactive self-assembled monolayers (electroactive SAMs) also present an ideal platform within which to study outer-sphere electron-transfer kinetics as investigated using electrochemical methods such as cyclic voltammetry (CV),1,2,16 ac voltammetry (ACV),17−19 electrochemical impedance spectroscopy (EIS),20 and chronoamperometry.21 In many cases experimentally resolved nonideality (such as in cyclic voltammogram wave shape) has been loosely rationalized by reference to a distribution of formal potentials,22−25 reorganization energies,24 and/or electronic coupling effects.24 Numerous previous reports have made reference to a “spread” in electron-transfer kinetics,23,24,26,27 but this has rarely been experimentally resolved28,29 or specifically pinned to either a distribution of tunneling distances or formal potentials. These nonidealities are, of course, inherent in any real molecular system. We show herein that a significant component of this dispersion arises specifically from the quantized character of the molecular orbital states involved in electron exchange. In recent work we have presented electroactive monolayer capacitance spectroscopy,22,25 EMCS, as a means of sensitively © 2014 American Chemical Society

Received: September 16, 2013 Accepted: January 6, 2014 Published: January 6, 2014 1997

dx.doi.org/10.1021/ac4031708 | Anal. Chem. 2014, 86, 1997−2004

Analytical Chemistry

Article

purging for the duration of the experiment. The complex Z* (impedance) function was converted into C* (capacitance) through Z* = 1/jωC* in which ω is the angular frequency.25

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GENERAL THEORETICAL BACKGROUND Classical Cyclic Voltammetric Analysis. The vast majority of electrochemical analyses performed of electroactive molecular layers are generated from linear sweep CV experiments,1,2,16,31 with its associated mix of convenience and limitation.22 Within such analyses the faradaic peak current density, jfp, is directly proportional to the voltage scan rate, s = dV/dt, as described by eq 1 1,24 according to Nernst thermodynamics.31,33 The redox site surface coverage, Γ, can be estimated from the slope of the linear relationship between jp versus s or by a direct integration of charge passed between the electrode and the film.1,24 The former assesses from a finite range of contributing redox energy levels (since the peak is that acquired at a unique potential value) involved and, as such, underestimates, through the Nernstian expression eq 1, the real redox site presence (as shown below).

Figure 1. (a) Schematic representation of a dielectric layer on a metallic electrode with its associated ionic dynamics (ref 30). IHP and OHP are the inner and outer Helmholtz planes, respectively. Black and green/positive spheres illustrate solvent and ionic species, respectively. The ellipses represent the polarized components (or associated dipoles) of a molecular film. Panel b illustrates the potential or energy span (with respect to a reference energy level, Eref) of orbital states that are exchanging electrons with the underlying metallic surface. These constitute a redox density of states, RDOS, as revealed experimentally through measured redox capacitance. f(μe) indicates the Fermi−Dirac distribution of electrons in the electrode and transitions from a value of nearly unity below the zone of interest (in this case around Er, the half-wave potential) to a value of virtually null above. At Er f(e) is ∼1/2.

jf p =

e2 Γ s 4kBT

(1)

A direct integration of the charge acquired by CV, although numerically correct, provides little resolving information about the film. Electroactive Layer Capacitive Analysis. In initially considering film-confined redox sites as being homoenergetic, one may, in the first instance, consider the effects of thermal energy in generating a (Boltzmann-derived) Gaussian distribution of redox site energies. The process of redox exchange between such levels and those of an underlying electrode (whose level occupancy is described by a Fermi−Dirac distribution) depends, of course, on their coupling. The resultant redox-level occupation function (which describes the distribution of electrons between the coupled electrode and redox site levels), f, is then22,25 f = F(Er,μe) = n/Γ, according to

The goal of the present work is to demonstrate that this frequency-resolved redox-specific capacitive charging (Cr), has a potential dependence that reports very directly on the distribution of redox site energies. This distribution, with its (spectrally resolved) quantum and dielectric counter parts, represents a key source of electron-transfer rate dispersion.

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EXPERIMENTAL PROCEDURES Polycrystalline gold disk electrodes (GDE) (Cypress Gold, diameter 1 mm) were cleaned following a published procedure32 and immersed in a 1 mM 6-ferrocenyl-hexanethiol (FcC6) or 11-ferrocenyl-undecanethiol (FcC11) (Sigma-Aldrich) ethanolic solutions (overnight, room temperature). Electrodes were then rinsed with ethanol and water and dried under nitrogen prior to immersion in the electrochemical cell (a 5 mL volume one-compartment cell with the functionalized GDE working electrode, a Ag/AgCl reference, and a platinum gauze counter electrode). CV and EIS measurements were carried out using a PC-controlled Autolab potentiostat PGSTAT30 (Ecochemie NL) equipped with an ADC750 and a FRA (frequency response analyzer) module. The ac frequencies ranged from 1 MHz to 10 mHz, with an amplitude of 10 mV. All the obtained impedance data were checked regarding compliance with the constraints of linear systems theory by Kramers−Kronig using the appropriate routine of the FRA AUTOLAB software. Three different solvents were used as supporting electrolyte, 500 mM NaClO4 aqueous polar solution (AS, with static dielectric constant, εs, ∼ 80, at room temperature) prepared by using ultrapure water (18.2 MΩ on a Milli-Q system from Millipore Corp.) and nonaqueous solutions of 100 mM TBA PF6 acetonitrile (MeCN with εs ∼ 40 at room temperature) and 100 mM TBA PF6 dichloromethane (DCM, with εs ∼ 10 at room temperature). All solutions were deoxygenated with bubbling argon and surface-

f = F(Er , μe ) =

1 1 + exp[(Er − μe )/kBT ]

(2)

where kB is the Boltzmann constant, T is the absolute temperature, Γ is the redox group surface density/coverage, n is the number of occupied redox centers, and μe is the electron chemical potential (or the Fermi level, EF) related to the electrode potential V (with respect to the reference) by −edV = dμe as stated in previous work.22,25 In turn, μe is related to the redox occupation and electron free energy of a single electrontransfer step process through ΔG = e(V − V 0) = Er − μe

(3)

where e is the elementary charge, V is electrode potential, and V0 is the half-wave (reversible) potential of the redox sites. In rearranging eq 2 it is possible to observe its analogy to classical Nernst expressions (where n/Γ and 1 − n/Γ are, respectively, the reductant and the oxidant occupation and n is the number of electrons occupying the redox molecular states): ⎛ f ⎞ ⎛ n/Γ ⎞ ⎟ μe = Er + kBT ln⎜ ⎟ = Er + kBT ln⎜ ⎝ 1 − n/Γ ⎠ ⎝1 − f ⎠

(4)

Furthermore, note that when μe = Er (the electrode Fermi level is resonant with the redox site energy) the electrode 1998



Article

Figure 2. (a) Typical experimental CV analyses of a FcC6 ferrocene thiol film. The current density here was normalized with respect to scan rate leading to, as discussed in the text, capacitance (eqs 1 and 7). These analyses illustrate the effect of electrolyte dielectric on CV shape, where, specifically, an increase in polarity is associated with a corresponding increase in peak width. Potentials are reported with respect to an Ag/AgCl reference, and all plots obtained from averages over three different polycrystalline electrodes. (b) Nyquist capacitive plots of FcC11 films (the analysis applies equally to FcC6 and FcC11 films of course) in DCM from EMCS analysis carried out with the electrode poised at the half-wave potential. The resultant spectrum [a subtraction of red from black is shown in green, where the nonfaradaic charging current contribution is corrected (and only one semicircle remains) (ref 25)]. The inset shows the nonfaradaic contributions to capacitance, here some 50-fold less than that which arises from redox site charging. Note that the value of Cr is given by the diameter of the semicircle, but can be equally obtained from a Bode diagram as illustrated in Figure 3a. Cu is the uncompensated capacitance as discussed in ref 22.

potential is V0 so that forward, kf, and backward, kb, electrontransfer rates have the same value: kf = kb = kr0. At this point, the net current is zero but Cr, defined by Cr = e 2 Γ

df e2 Γ = f (1 − f ) dμe kBT

significant consequence in terms of kinetic dispersion and calculations of molecular coverage). The Meaning of Redox Capacitance in Molecular Layers. In progressing from this classic picture to one which is “more realistic” in nature, the energetic spread of film-confined redox sites is more correctly given by a density of states (DOS) expression that absorbs both thermal and quantum contributions (the DOS expression being eq 9). The expected redox capacitance (itself describing the distributing of electrons over the resultant coupled levels) is then that which arises by integrating over all energy levels coupled contributions of these with electrode states. Equation 5 thus turns into

(5)

is maximal.25 This expression simplifies further at the reversible potential V0 (or Er) because, according to eq 2, f = 1/2 and f(1 − f) = 1/4, and, at this steady state, this maximum Cr is

Cr =

e2 Γ 4kBT

(6) 22

The corresponding faradaic capacitive current density,

Cr(μe ) = e 2

jf, is

df

dμe

e

dqr

dV jf = = Cr = Crs dt dt

∞

∫−∞ gr(μe) dμ

= (7)

e2 kBT

∞

∫−∞ gr(μe)f (1 − f ) dμe

(8)

where gr(μe), the redox density of states, can be written as

where qr is the absolute charge reversibly associated with redox centers, i.e., a purely faradaic charge.22,25 Note the correspondence between jfp (from eq 1) and jf (from eq 7), the latter corresponding to faradaic current obtained through CV analysis and Nernst thermodynamics. In combining eqs 6 and 7, eq 1 can be promptly obtained, and both (EMCS and CV) analyses then match completely. In other words, Cr is the proportional term relating faradaic current density, jf, and scan rate, s.22 To summarize thus far, one may initially consider surfaceconfined redox levels as being thermally dispersed but otherwise monoenergetic in nature.22 The potential-dependent distribution of electrons between these and electrode Fermi− Dirac states dictates redox capacitive charging as resolved through EMCS analyses. We show below that such a picture is incomplete and, like analogous CV analyses, inadequately describes the interface or its energetic dispersion (with

⎡ (E − μ )2 ⎤ 1 r e ⎥ exp⎢ − gr (μe ) = 2 ⎢ ⎥⎦ σg 2π 2σg ⎣

(9)

i.e., a Gaussian spread of redox states with a distribution of Er and a σg standard deviation (σg2 variance). In the limit of an idealized absolute zero temperature (integrating a δ Dirac function, i.e., taking the zero temperature approximation for Fermi−Dirac statistics) there is no thermal energy dispersion, and expression 8 then becomes Cr(μe) = e2gr(μe), demonstrating the redox capacitance reports directly on the occupancy of energetically quantized redox density of states (at T > 0 this Cr function will report on the redox site DOS together with thermal energy contributions, see also the Supporting Information, section S.I.1)31,32 The influence of solvent 1999



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Figure 3. Real (a) and imaginary (b) Bode capacitive diagrams of an FcC11 SAM on gold. The nonfaradaic contributions are mapped by acquiring data with the electrode poised outside of the redox potential window. These can subsequently be subtracted from data acquired with the electrode poised inside (at the half-wave potential) the redox region (red) (the resultant “pure” faradaic response is green). As discussed in previous work (refs 22 and 25) from panel b it is possible to obtain the electron-transfer rate directly from the peak as indicated (here ∼10 s−1). The inset shows the time scale of the lower magnitude nonfaradaic charging process. The low-frequency region indicated in panel a depicts the time scales associated with fully charging the capacitor, where sampling frequency does not influence resolved capacitance (which is accordingly a plateau in this region, i.e., the time scale of measurement is longer than the time scale of faradaic exchange).

poised at one fixed potential (here the half-wave potential) reproduce the generic literature analysis (eq 1) of redox coverage where not all redox levels are accessed.1,33 As such they illustrate the charging effects associated with the population of only a confined range of energy levels (note that a full mapping across all surface-confined redox levels is simply achieved by recording Cr across the full potential windowsee later). It is worth noting that the high symmetry of peaks resolved by CV (Figure 2a) is reflective of good (although nonadiabatic) electronic coupling between the redox energy levels and those of the electrode as expected for films of this type.23−25 The wave shapes as resolved are not influenced by the resistance associated with charging the redox centers, i.e., the electrontransfer resistance, Rct [for data acquired at voltage sweep rates lower than the time scale of the redox process.31 Since Cr is similarly assessed in the limit of low frequency (as indicated in Figure 4a), its resolution is similarly unperturbed by electrontransfer resistive effects.] Figures 2 and 3 summarize the charging fingerprint of these films where it is evident first that nonfaradaic (ionic) charging is a comparatively small contribution to the total charging and, second, that its time scale (frequency peak in C″ Bode plots) is fast (500 s−1) compared to that which is faradaic (here 10 s−1). Once these characteristic magnitudes and their time scales (kr = τr−1 = (RctCr)−1 and/or τu−1 = (RuCu)−1)22,30 are mapped, Cr may be directly calculated from either Bode or Nyquist analyses. From this, the redox site surface coverage can be directly calculated (using eq 5 for the classical model (where only thermal effects are considered). As shown in Table 1 (for two films across three different dielectric environments or Figure 4a) this quantification is clearly observed to be a sensitive function of solution composition. This clearly has no physical foundation (calculated coverages here are observed to be reversible if films are switched between different media) and is an observation which arises, very specifically, because this theoretical treatment of redox molecular coverage processes real data through a model which considers only thermal effects

dielectric will impact the dipolar moments of quantized redox states and (Supporting Information, section S.I.2), through this, the experimentally resolved σg. The Cr fingerprint of a molecular film thus reflects the impact of local dielectric on the energetic spread of redox-active states. This energetic spread will, of course, directly translate into kinetic spread as can be seen in below.

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RESULTS AND DISCUSSION Impact of Electrolyte Dielectricity. In standard CV analyses of molecular films the impact of electrolyte is exemplified by a broadening of current−voltage peaks with increasing dielectric constant (Figure 2a). Although this directly corresponds to a resolved impact of solution dielectric on RDOS energetic distribution, it is not directly resolved through such analyses.1,33 To scrutinize the process of redox site charging in more detail we can look to the steady-state and distortion-free EMCS25 approach from which Cr can be mapped as a function of surface potential.22,25 Briefly, standard impedance analyses resolve complex impedance Z*(ω) from which complex capacitance C*(ω) is easily obtained (where the electrochemical components can be isolated and resolved without fitting or any assumptions therein). A spectroscopic (frequency-resolved) analysis of the real (C′) and imaginary (C″) components within this, with the electrode held first at a potential outside (mapping nonfaradaic response) of the faradaic window and then at the reversible potential (mapping total response), separately quantifies the characteristic time scales and magnitudes of all charging contributions.22,25 In reality, there are two capacitive contributions to Cr. These are electrostatic, Ce, and quantum, Cq. In the confines of a nanoscale (molecular) system, the relative contribution of Ce is expected to be negligible.34,35 The dielectric effects that are resolved herein (Figure 4b and Table 2) are, therefore, reflective of the influence of environment on molecular dipole moment (as quantified within the DOS spread). It is important to note that the frequency-resolved capacitive analyses presented in Figure 2b acquired with the electrode 2000



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Table 1. Mean Molecular Coverages, Γ, Calculated from ECS and CV Methodologies in Three Different Dielectric Environments for Two Different Redox Films FcC6 and FcC11a molecular coverage, Γ FcC6 FcC11

methodology ECS CV ECS CV

DCM (pM cm−2) 270 244 496 451

± ± ± ±

32 33 57 50

MeCN (pM cm−2) 132 157 283 281

± ± ± ±

16 20 27 25

(“coupling strength”). The difference between the mean (and solvent-independent) RDOS values FcC6 and FcC11 (0.76 ± 0.3 × 1014 and 1.37 ± 0.3 × 1014 states cm−2) suggests that the primary effect here is physical coverage, i.e., the greater crystallinity/order adopted by the longer alkyl film leads to a greater integrated RDOS despite the weaker electronic coupling expected in this film. It is satisfying, then, that on a normalization for monolayer thickness (using film thickness values as reported in ref 40), these mean RDOS values become equivalent within error (0.95 ± 0.3 × 1019 and 0.97 ± 0.3 × 1019 states cm−3, respectively, to FcC6 and FcC11) meaning the RDOS per volume is the same and independent of local redox center environment as expected from a quantum mechanical perspective. In summary, one of the principal conceptual differences between the classical electrochemical picture and that proposed here is based on how electron occupation is treated in the molecular layer. If the electron-residing redox states are considered as noninteracting, chemically and electronically isolated, Boltzmann particles, the resulting thermodynamic analysis presents an unrealistic and incomplete picture of the interface that, additionally, fails to accommodate Nernst thermodynamics (see eq 4, where the film occupation function is given simply by f = n/Γ). If, alternatively, one considers molecular layer occupancy as being governed by the quantum mechanical coupling of a redox DOS (with its thermal and quantum mechanical dispersion) and underlying electrode states, electron density is reported through n = fgr(μe). This comparatively simple physical picture aligns nicely with the real physical coverage of redox sites. It also accommodates the dispersion suggested but not resolved in prior works.18,22,24,25,27 Kinetics and Redox Capacitance. The previously outlined and resolved redox site energy-level dispersion will, of course, directly influence electron-transfer kinetics and, specifically, will lead to an analogous dispersion in this across a population of film-confined sites. Figure S.1c (see the Supporting Information), for instance, is the simulation of kinetic dispersion due to RDOS broadening (since j = e(dn/dt) = e{d[fgr(μe)]/dt}.

AS, H2O (pM cm−2) 67 86 177 198

± ± ± ±

9 15 28 30

a

The classic thermal model (eq 5) was applied both data sets with good (∼10%) resolved agreement in redox site coverage. Though in the ranges expected for films of this type, the values are, however, highly solvent-dependent (refs 16, 20, 38, and 39).

on a population of otherwise homoenergetic redox states, reproducing standard literature approaches.33,36 In other words, coverage is underestimated because not all redox energetic contributions are being considered and falls significantly short of those made for these films in other work (none of which include a detailed consideration of the important quantum effects described herein).16,36,37 Note Figure 4a shows the agreement between coverage as determined by CV peak current and that obtained from Cr by ECS. Both approaches clearly fail to sensibly report a fixed redox state coverage as solution composition is varied when combined with the classic, nonquantum model. If, however, one fits experimentally determined Cr data (Figure 4b) to an expression accounting for distribution of electrons between electrode states and those specified by the DOS defined spread of eq 9, gr(μe), one can more correctly calculate the molecular “coverage”. This is, in fact, more correctly considered physically as a density of states and pleasingly is resolved to be both solvent-independent, and to quantify, through σg, the energy-level spread in any given medium (Table 2). Notably, this spread increases with increase in solvent dielectric. It is also important here to note that resolved mean RDOS values report on the integrated communication between metal and redox DOS and, because of this, has a sensitivity to distance

Figure 4. (a) Molecular coverage trends as a function of εs (a graphical representation of data in Table 1) for two different redox films FcC6 and FcC11. (b) The Cr distribution over potential as a function of εs showing, clearly, the progression to greater spread (σg) with solution polarity for a FcC11 film. Note the single-state energy model is applied here by CV and ECS approaches. Note that panel b corresponds to the experimental analysis of the simulated values. 2001



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Table 2. RDOS Analyses of FcC6 and FcC11 Films Where σg Values Increase with Dielectrica σg and RDOS

parameters

FcC6

σg (meV) RDOS (× 1014) (states cm−2) σg (meV) RDOS (× 1014) (states cm−2)

FcC11

DCM 19 0.8 22 1.4

± ± ± ±

4 0.2 3 0.3

MeCN 51 0.7 55 1.5

± ± ± ±

3 0.3 4 0.3

H2O 80 0.8 82 1.2

± ± ± ±

4 0.3 5 0.4

Errors represent variance across three different films. In associating one electron with each ferrocene moiety, the resolved RDOS equates directly with Γ (0.76 ± 0.3 × 1014 and 1.37 ± 0.3 × 1014 molecules cm−2 for FcC6 and FcC11 films, respectively, being in line with that expected for films of this type but with different physical meaning of that) (refs 41 and 42). If the values of DOS are normalized by coupling distance then the values of the states per volume (an average of 0.96 × 1019 states cm−3) are equivalent and independent of solution dielectric or coupling. Analogous linear sweep analyses of the same films give a less accurate report of RDOS (Figure 2a), the data acquisition being “contaminated” by nonfaradaic contributions (ref 22) to recorded current. a

Figure 5. Bode diagrams of the imaginary components of complex capacitance C″, where the impact of solvent dielectric on faradaic electron-transfer rate (given by the frequency where the imaginary capacitance component is maximum, ref 25) and dispersion (peak width) within the film for both FcC6 (a) and FcC11 (b) alkylferrocene SAMs. All derived kr values in the graphics are obtained with 0.98.

maximum occurs when the electrode potential is perturbed at a frequency resonant with the inverse of redox time scale kr = 1/ τr. This resonant frequency/time scale will show a dependency on solution dielectric as engrained through the shape of gr(μe) (last section and Figure 4b). The Bode plots of Figure 5 were acquired with the electrode poised at the half-wave steady-state potential (i.e., the most populous energy level), and the kinetic change with dielectric reflects directly the change in integrated

As previously noted, the electron-transfer rate associated with a redox molecular film, kr, can be conveniently frequency resolved within Bode plots, shown in Figure 5, where FWHM is a direct reflection of kinetic dispersion (closely following the pattern simulated in Figure S.1c, Supporting Information). The effects manifest in these analyses are divisible into “magnitude of kr” and “dispersion” contributions, with the former, of course, being related to electronic coupling efficacy whose 2002



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Figure 7. (a) The distributed characteristics of redox conductivity, σr/L = 1/ARct = ωC″, as a function of electrode energetic level or potential. (b) The distributed pattern of the characteristic relaxation time, τr, also as a function of electrode energy level. In both it can be noted that dispersion increases as a function of εs, but the impact it has on τr is directly associated with electron-transfer rate distribution occurring over energy levels ultimately dependent and having as primary source the shape of gr(μe) DOS Gaussian function. In summary, panel a shows the impact of solution dielectricity on redox conductivity and panel b the dispersion on the associated time constant showing it increases from red to green to yellow, i.e., as higher is the solution dielectricity higher is the dispersion on conductivity and time scale of the process.

correspond to electron-transfer rate (see remarks in the Supporting Information, section S.I.2).

overlap of RDOS and electrode states (following observations noted in Figure 4b and Figure S.2 in the Supporting Information). It is clear (Figure 6), therefore, that the RDOS function shape, and its sensitivity to environment, directly dictates experimental observations. This is, of course, the same influence as predicted by Marcus with a slope that is independent of molecular film (and distance or coupling effects therein). Note that these Bode plots report on kinetic dispersion at specific electrode potentials. The total kinetic spread as mapped across all relevant surface potentials is resolved in Figure 7b as a function of solvent dielectric. Figure 6b displays the normalized cumulative function, NCF, i.e., the integration of the Gaussian functions of Figure 4b which evaluates the occupancy of redox centers as a function of electrode potential energy. This evaluation is across energy levels from zero, f = 0 (anodic surface potentials) to a maximum of f = 1 (full occupancy at highly cathodic surface potentials). It shows the occupation of energy levels according to Fermi level from a minimum, i.e., f = 0, to a maximum, i.e., f = 1 so that, as expected for DCM, for instance, it reaches a maximum at −0.1 V while this maximum for aqueous solution dielectric environment is only reached at potentials higher than −0.2 V. This is reflective of RDOS dispersion as a function of dielectric and specifically reports the span of potentials over which redox occupation is changing (obviously higher for more dispersed film states). Since a maximum in electron-transfer rate equates to a maximum in associated “redox conductivity”, once heterogeneous electron-transfer rate, kr = 1/τr = (RctCr)−1, is mapped, the spread in Figure 7 thus represents that of coupled Rct (the resistance electrons experience in charging redox sites) and Cr terms. Rct itself directly reports on this redox conductivity σr, (as shown in Figure 7a) and depends sensitively on energetic resonance (and thus redox site level energies as resolved within Cr). Since Cr is engrained with current density, according to eq 7, or j = e(dn/dt) = e{d[fgr(μe)]/dt}, it is possible to find a correlation between Rct and Cr throughout the time scale of the electron-transfer process, i.e., τr. It is then possible to plot the dependence of redox related film conductivity on potential (see Figure 7b). For any given electrode potential this ac driven conductance will also be optimal at ac frequencies that directly

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CONCLUSIONS We have introduced here a means of using standard impedance/capacitance methodologies to resolve a redox film density of states and, from this, the energetic levels that are responsible for electron exchange with a supporting electrode and the consequent electron-transfer kinetics (see also remarks in the Supporting Information, section S.I.2). For any real experimental configuration, classical Boltzmann chemical approaches fail to account for experimental observations of kinetic spread or environmental influence. Electron-transfer kinetics are well-known to be dependent on both the energetic alignment of electrode Fermi states with redox states of the molecular layer and the tunneling distances across which coupling is spatially mediated. The latter spatial effects are, of course, potential sources of a resolved dispersion but appear to have a minimal influence compared to those arising from quantum coupling effects and a resultant RDOS shape. The energetic dispersion engrained within this shape can be mapped through 1/kr = τr = RctCr and Cr. We believe the approaches outlined herein to be both convenient, to report these effects with an unprecedented resolution, and to set a theoretical foundation for their origin.

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ASSOCIATED CONTENT

S Supporting Information *

Additional information as noted in text. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Authors

*Phone: +55 16 3301 9642. Fax: +55 16 3322 2308. E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest. 2003



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(36) Williams, F. J.; Scherlis, D.; de Llave, E.; Leo, L. P. M. D. J. Chem. Phys. 2013, 138, 114707. (37) Chidsey, C. E. D.; Bertozzi, C. R.; Putvinski, T. M.; Mujsce, A. M. J. Am. Chem. Soc. 1990, 112, 4301. (38) Finklea, H. O. Electroanal. Chem. 1996, 19, 109. (39) Ho, M. Y.; Li, P.; Estrela, P.; Goodchild, S.; Migliorato, P. J. Phys. Chem. B 2010, 114, 10661. (40) Porter, M. D.; Bright, T. B.; Allara, D. L.; Chidsey, C. E. D. J. Am. Chem. Soc. 1987, 109, 3559. (41) Chidsey, C. E. D.; Bertozzi, C. R.; Putvinski, T. M.; Mujsce, A. M. J. Am. Chem. Soc. 1990, 112, 4301. (42) Chidsey, C. E. D.; Hill, M.; Murray, R. W. J. Phys. Chem. 1986, 90, 1479.

ACKNOWLEDGMENTS This work was supported by the São Paulo state research funding agency (FAPESP) and UNESP Grants. The authors thank Habibur Rahman, Joshua Ryall, and Flávio Bedatty Fernandes for supplying raw experimental data.

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