Theoretical Analysis of the Relative Significance of Thermodynamic

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Theoretical Analysis of the Relative Significance of Thermodynamic and Kinetic Dispersion in the dc and ac Voltammetry of SurfaceConfined Molecules Graham P. Morris,† Ruth E. Baker,† Kathryn Gillow,† Jason J. Davis,‡ David J. Gavaghan,*,§ and Alan M. Bond*,∥ †

Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG U.K. ‡ Physical and Theoretical Chemistry Laboratory, University of Oxford, South Parks Road, Oxford, OX1 3QZ U.K. § Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford, OX1 3QD U.K. ∥ School of Chemistry, Monash University, Clayton, Vic. 3800, Australia S Supporting Information *

ABSTRACT: Commonly, significant discrepancies are reported in theoretical and experimental comparisons of dc voltammograms derived from a monolayer or close to monolayer coverage of redox-active surface-confined molecules. For example, broader-than-predicted voltammetric wave shapes are attributed to the thermodynamic or kinetic dispersion derived from distributions in reversible potentials (E0) and electrode kinetics (k0), respectively. The recent availability of experimentally estimated distributions of E0 and k0 values derived from the analysis of data for small numbers of surface-confined modified azurin metalloprotein molecules now allows more realistic modeling to be undertaken, assuming the same distributions apply under conditions of high surface coverage relevant to voltammetric experiments. In this work, modeling based on conventional and stochastic kinetic theory is considered, and the computationally far more efficient conventional model is shown to be equivalent to the stochastic one when large numbers of molecules are present. Perhaps unexpectedly, when experimentally determined distributions of E0 and k0 are input into the model, thermodynamic dispersion is found to be unimportant and only kinetic dispersion contributes significantly to the broadening of dc voltammograms. Simulations of ac voltammetric experiments lead to the conclusion that the ac method, particularly when the analysis of kinetically very sensitive higher-order harmonics is undertaken, are far more sensitive to kinetic dispersion than the dc method. ac methods are therefore concluded to provide a potentially superior strategy for addressing the inverse problem of determining the k0 distribution that could give rise to the apparent anomalies in surface-confined voltammetry.

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INTRODUCTION

isotherm applies, then the analytical solution for a reversible process predicts the value for the width of the Faradaic current at half its maximum peak height, designated as W1/2, to be given by

In the case of solution-phase voltammetry, agreement between experiments and the output of theoretical models is often obtained to a level where one is not easily able to distinguish between the two. However, for surface-confined voltammetric studies, such a high degree of agreement between theory and experiment is rarely achieved, even under what are considered to be the most ideal conditions. In terms of these theory− experiment comparisons, the best agreement is probably obtained with a highly uniform surface coverage of adsorbed molecules of at most monolayer thickness.1−4 When this condition is satisfied, models of cyclic dc voltammetry for a reversible process that applies in the absence of uncompensated for resistance, Ru, or capacitive current yield analytical solutions derived from the Nernst relationship.5,6 In particular, if the adsorbed molecules are noninteracting and the Langmuir © 2015 American Chemical Society

W1/2 = 3.53

RT nF

(1)

For a single-electron transfer (n = 1) at temperature T = 293 K, and with usual values for the universal gas constant, R, and Faraday’s constant, F, W1/2 works out to be 89 mV. In practice, most of the assumptions used in the derivation of eq 1 are not unreasonable: levels of ItotRu, the ohmic IR drop, Received: November 25, 2014 Revised: April 7, 2015 Published: April 8, 2015 4996

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is possible, potentially giving rise to a statistical distribution in the values of this parameter. Finally, in contrast to the solutionphase studies, each surface-confined molecule may have a slightly different environment for its redox site, which can also result in a variation in E0. The above intuitive justification for the inclusion of dispersion in theoretical models of surface-confined voltammetry is appealing and has been proposed by several authors, particularly in studies related to catalytic processes involving large enzymes adsorbed onto electrode surfaces. For example, when voltammetric experiments are conducted on such enzymes using a rotating disk electrode configuration, potential-dependent limiting currents are almost always observed, rather than the ideal potential-independent limiting current predicted if all enzymes have identical E0 and k0 values. Leger et al.15 suggested dispersion as a cause, citing an argument similar to the one stated here, and proposed an empirical model to incorporate dispersion based on a distribution of distances between the electrode surface and the electroactive site of the catalyst. In this distribution, all values of the distance between the electrode surface and the electroactive site, d, were assumed to occur with equal probability between a minimum and maximum value, and with zero probability outside of this range. This model has been elaborated upon16 and applied to other enzymes in similar studies17 and recently used as part of a substantial experimental study on the effects of dispersion by Fourmond et al.18 Clark and Bowden19 also examined peak broadening under reversible conditions when cytochrome c is adsorbed, assuming a normal (Gaussian) distribution of reversible potentials, and Rowe et al.7 also assumed a normal distribution in their studies with a monolayer of an adsorbed ferrocene derivative. A significant barrier to quantitative theory−experiment comparisons has been the lack of any information on the actual distributions of E0 and k0 values that might be encountered experimentally. However, recent studies by Davis et al.20−22 involving small numbers of fluorophore-modified azurin metalloproteins have allowed the exact distributions of the values of these Cu(II)−Cu(I) reversible potential and electrode kinetic parameters to be reported. Histograms adapted from Salverda et al.20 are shown in Figure 2 and summarize the distributions of (a) E0 and (b) k0. These distributions will be used as the initial basis to incorporate the effects of kinetic and thermodynamic dispersion into models and, subsequently, to examine whether either thermodynamic or kinetic dispersion is capable of accounting for some aspects of the reported differences between experimental results and previously accepted theoretical predictions, particularly the broadening of the Faradaic peak exemplified by increased values of W1/2. It is of course important to note that Figure 2(a) pertains to a distribution derived from an analysis of data involving hundreds of molecules, whereas voltammetric studies are conducted at monolayer, or close to monolayer, coverage, comprising billions of molecules. The assumption must be made that the same distribution applies to both the microscale and the macroscale. Patil and Davis21 have confirmed that this is a reasonable assumption for the distribution of E0. In recent work,23 we have shown that the sensitivity of Fourier transform ac voltammetry to small changes in electrode kinetics is significantly enhanced relative to the dc voltammetric case. For example, differences in wave shapes and current magnitudes arising from the use of Marcus−Hush theory as an alternative to Butler−Volmer theory for electron transfer with

are typically very low in surface-confined experiments because the total current, Itot, is small, background current due to double-layer capacitance can often be subtracted from the total current empirically, and even for a quasi-reversible process, if a sufficiently low scan rate is used, a reaction should approach the kinetically independent reversible limit, with the characteristics described above. However, reported values of W1/2 from experiments are invariably greater than would be expected from this prediction, with a typical example being the low-scan-rate study by Armstrong et al.4 of the quasi-reversible azurin process, where W1/2 is reported as being 18 mV greater than the value predicted by theory. Other discrepancies such as a nonzero separation between oxidation and reduction peak potentials are also found.4,7 Several possible explanations have been proposed for the consistently reported discrepancy between theoretical predictions and experimental results in the voltammetry of surfaceconfined molecules. In this article, we will consider theoretically the oft-cited phenomenon of dispersion.6 Dispersion in this context can be described as the absence of a single value of the reversible potential, E0, (thermodynamic dispersion) and/or heterogeneities in the charge-transfer rate constant, k0, (kinetic dispersion) across the surface-confined species. In solution-phase voltammetry, a distribution of values of E0 is not relevant and k0 variability is not widely encountered. By definition, there can only be a single value of E0 applying to the electron transfer of dissolved species. At an ideal homogeneous electrode such as is provided by mercury, in contact with a homogeneous solution, there also would be only a single value of k0 (cm s−1), although more recent studies have shown that on some heterogeneous electrodes different values of k0 may occur due to, for example, areas of differing composition in boron-doped diamond electrodes as in the study by Unwin et al.8 or locally varying electrode geometry as in studies at carbon electrodes.9−13 Even in these cases, however, overlapping of the diffusion of species between these sites means that it is normally possible to report only an average value of k0 across an experiment.14 In studies solely concerned with surface-confined electron transfer, where there is no rate-limiting diffusion or dissolution and a Langmuir isotherm is assumed along with no interaction between adsorbed species, k0 (now with units of s−1) averaging is unlikely to occur. Figure 1 shows a cartoon form of an

Figure 1. Cartoon showing different binding orientations of molecules (red spheres) resulting in a variable distance between the electrode surface (black line) and the electroactive sites (blue squares) for each molecule.

electrode where the inclusion of dispersion into the theoretical description of the voltammetry of surface-confined species is likely to be needed. In this figure, an ideal monolayer is adsorbed onto an electrode surface. However, variation in the orientation of adsorbed molecules relative to the electrode surface, as can occur with metalloproteins and enzymes, will result in variation in the distance that will be traversed by electrons between the redox-active site in each molecule and the electrode surface. These differences in distance can be expected, via Marcus theory,5 to result in variations in the values of k0. In this situation, a continuous variation of k0 values 4997

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be used to consider dispersion at monolayer coverage. We will consider the reaction kox

A HooI B + e− k red

(2)

where A and B, the reactant and product of a one-electron transfer reaction, respectively, will be attached to the electrode surface, and kox and kred (units s−1) are the rate constants for the forward and reverse processes, respectively. ODE Model of Heterogeneous Charge Transfer without Dispersion. As the species involved in the reaction are immobilized on the surface, we adopt the widely used concept that a proportion of the surface is covered by A and B up to a maximum of a monolayer (ref 5 and references therein) and let θ be the proportion of the total surface coverage accounted for by species A. As a result, the proportion of surface coverage by species B is given by 1 − θ. Making the assumption of a Langmuir isotherm, where adsorbed species do not interact with each other, the governing equation for the evolution of θ with time is5,6 dθ = k red(1 − θ ) − koxθ dt

(3)

5

As elsewhere, the rates of reaction kred and kox are defined in terms of the standard rate constant at E0, k0, and the chargetransfer coefficient α via Butler−Volmer kinetics as ⎡ (1 − α)F ⎤ kox = k0 exp⎢ (E (t ) − E 0 )⎥ ⎣ RT ⎦ ⎡ − αF ⎤ k red = k0 exp⎢ ( E ( t ) − E 0 )⎥ ⎣ RT ⎦

Figure 2. Histogram of the distribution of experimentally determined (a) E0 and (b) k0 values for surface-confined modified azurin metalloproteins, adapted from Figure 4 of ref 20.

(4)

where E(t ) = Estart + νt + ΔE sin(ωt )

surface-confined molecules are significantly amplified in the ac method. Principally, the enhanced sensitivity in this case arises because ac current magnitudes are very sensitive to small variations in k0 whereas dc ones are not. Assuming that an analogous situation would prevail in this study, we have for the first time examined theoretically the impact of dispersion in ac voltammetry and confirmed that enhanced sensitivity to this phenomenon is available relative to the situation that prevails when the same distribution of E0 and k0 values is applied to the dc scenario. In the context of our overall theme in recent papers of providing parameter estimations in voltammetry using data optimization methods (ref 24 and references cited therein), we have examined the implications of dispersion for both dc and ac voltammetry with the goal of ultimately producing a statistical method for the recovery of parameters that represent kinetic and thermodynamic dispersion.

(5)

represents the applied potential, neglecting the Ohmic drop associated with uncompensated resistance, Ru. The chargetransfer coefficient, α, will be assumed to be 0.5 in all calculations. The Faradaic current, If, in the absence of mass transport is If = −FS Γ

dθ dt

(6)

where Γ represents the combined surface coverage by A and B per unit area of the electrode and the electrode surface area is given by S. The contribution of the double-layer capacitance current, Ic, is neglected in this study. The model is completed by the initial conditions

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θ(0) = 1 If (0) = 0

THEORY We will present two different mathematical models in order to describe the underlying chemistry, namely, a continuum ordinary differential equation or ODE-based model and a stochastic model. The former neglects the role of noise by considering average properties of the surface-confined population, while in the latter case each molecule is treated individually and the number of molecules in each possible state is tracked as it evolves with time. The reason for considering the two models is that we wish to satisfy ourselves that both scenarios produce indistinguishable results under voltammetric conditions so that the computationally simpler ODE model can

(7)

To obtain a numerical approximation for If, the system consisting of eqs 3−7 is then discretized according to a backward Euler finite difference scheme. It should be noted that both the impact of uncompensated resistance and double-layer capacitance could be included in the model to make it more experimentally relevant. However, to clearly establish the contribution of dispersion with respect to the Faradaic current, contributions from these terms will be neglected in all simulations provided in this work. Stochastic Model in the Absence of Dispersion. The second model we will consider is an individual-based stochastic 4998

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current profiles. This results in an overall current for the system including dispersion, Idisp, given by

model, often referred to as the stochastic formulation of chemical kinetics, where we track only the discrete population numbers of each molecular species and their evolution with time. Each possible reaction is assigned a probability of occurring as a function of time, and the system is allowed to evolve according to these probabilities. As an individual-based model rather than an average-based model, this stochastic formulation is applicable to a wider range of systems than the ODE-type methods that rely on the assumption that the number of molecules can be modeled as varying continuously and deterministically, which may not be the case in systems involving, for example, a small number of molecules. We find, from a comparison of the results of the analysis of both models, that under typical dc and ac voltammetric conditions and where the number of molecules far exceeds a critical number (around 104), both theories provide identical predictions, thereby justifying the use of the computationally simpler ODE model, which is used in the presentation of the results below. The reader is referred to the Supporting Information for details of results and calculations based on the stochastic model. Including Dispersion. Including the effects of dispersion in each of the ODE or stochastic models is in principle straightforward. The key decision for both is how to go about choosing and then sampling from the distributions of the values of E0 and k0. Under the assumption that there are no interactions between surface-confined molecules, the probability of a molecule having given E0 and k0 values is given by the joint probability density function (pdf) p(E0, k0). The overall current is then Idisp(E) =

∬E ,k p(E0 , k0)IE ,k (E) dE0 dk0 0

0

0

q2

Idisp(E) =

∑ wIi E

0

i

, ki 0(E)

(9)

i=1

where IE0i,k0i represents the current time series for each of the q2 combinations of E0 and k0. The weightings wi are obtained by calculating the area under the pdf’s corresponding to the particular E0 and k0 intervals, wEj and wkj (j = 1,..., q), as shown in Figure 3, and then wi = wEjwkS (10) with the property that q2

∑ wi = 1

(11)

i=1

Throughout this work, for both E0 and k0 we binned the data into 15 intervals and used a midpoint approximation method to approximate wEj and wkj. We note that the results do not change significantly as the number of bins is increased. For the general shape of the distributions of E0 and k0, we appeal to experimental data provided in ref 20. The authors report, in panels (a) and (c) of their Figure 4, adapted here as Figure 2, that they have measured pdf’s for both parameters, which we assume are approximated as normal for E0 and normal on a log scale, i.e.. log-normal, for k0. To sample from these pdf’s, we divide the ranges for each E0 and k0 into q bins and calculate the weight for both, as described above.

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0

RESULTS AND DISCUSSION dc Voltammetry. Initially we will consider dc voltammetry, ΔE = 0 in eq 5, and examine the effect of E0 and k0 dispersion on the half-height width, W1/2, as this has been proposed to provide one of the most obvious differences between predictions using a model that does not incorporate dispersion and results obtained experimentally for surface-confined species. The value of α is assumed to be 0.5. In principle, the impact on the peak potential also could be reported in this article, but this reflects just one data point, and hence is not as sensitive as W1/2 to dispersion which contains contributions from a much larger data set. Figure 4(a) shows the results obtained at a scan rate of ν = 0.1 V s−1 by including the E0 dispersion in different amounts but fixing k0 = 0.1 s−1, which is well within the quasi-reversible regime under these conditions. The full results are summarized in Table 1, comparing the amount of dispersion applied to the half-height width, W1/2. The scaling was performed by using the E0 values and weightings given in Table S1 and then multiplying the E0 values by the scaling factor, l, to spread them out further, i.e.,

(8)

where IE0,k0 is the current with given E0 and k0 and E is the applied potential. As a first approximation, we assume that the E0 and k0 are independently distributed so that p(E0, k0) = pE0(E0) pk0(k0). An illustration of the pdf for E0 is shown in Figure 3. In practice, however, one would discretize the distributions for E0 and k0 and weight the contributions of each of the binned values of E0 and k0 by calculating the area under the relevant pdf (as shown in Figure 3). This means that we solve the ODE model for each combination of E0 and k0 and then take a weighted average of the resulting surface concentration or

E 0 = 0 + 2.5jl ,

j = −7, −6, ..., 7

(12) 0

In particular, we can see that including E dispersion in amounts consistent with those reported by Salverda et al.20 and as summarized in Figure 2(a) with k0 = 0.1 s−1 has almost no noticeable impact on observable and measurable characteristics such as W1/2 when compared to simply fixing E0 at the mean value. To obtain a 10 mV gain in W1/2 by including only the E0 dispersion, we would have to use roughly four times this experimentally measured spread in E0 values, and a 20 mV W1/2

Figure 3. Illustration of the origin of the weights, wi, as being equal to the area under the probability density function between the values at the edge of each bin. 4999

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Table 1. Results of Applying Different Amounts of E0 Dispersion as in Equation 12 with k0 Fixed at 0.1 s−1, Showing the Effect on the Half-Height Width W1/2a scale, l

spread (mV)

W1/2 (mV)

0 1 2 3 4 5 6 7 8 9 10

N/A 35 70 105 140 175 210 245 280 315 350

123 124 126 129 133 138 144 151 159 167 176

For clarity, the column labelled “spread” shows the separation between the minimum and maximum E0 sample values in each case. Other parameters used are T = 293 K, ν = 0.1 V s−1, S = 1 cm2, α = 0.5, k0 = 0.1 s−1, Estart = −0.2 V,and Γ = 1 × 10−11mol cm−2. a

A similar analysis for the k0 dispersion with fixed E0 is displayed in Figure 4(b) and Table 2. E0 was fixed at 0 V, and k0 Table 2. Results of Applying Different Amounts of k0 Dispersion, Included According to Equation 13, with E0 Fixed at 0 mV, Showing the Effect on W1/2a

Figure 4. (a) Comparison of the effects of including different amounts of E0 dispersion, scaled as described in the text. The curves correspond to no dispersion (l = 0), i.e., E0 = 0 V (blue), 105 mV separation (l = 3) between minimum and maximum E0 values (red), and 350 mV separation (l = 10) between minimum and maximum E0 values (black). The inset shows the probability density functions corresponding to the distributions used for the two cases including dispersion. (b) Comparison of the effects of including different amounts of k0 dispersion, scaled as described in the text and eq 13 and weighted according to Table S1. The curves correspond to no dispersion, i.e., k0 = 0.1 s−1 (blue), exponent multipler m = 0.5 (black), and exponent multiplier m = 1 (red). Other parameters used are T = 293 K, ν = 0.1 V s−1, S = 1 cm2, α = 0.5, Estart = −0.2 V, Γ = 1 × 10−11 mol cm−2, and, where relevant, E0 = 0 V and k0 = 0.1 s−1.

exponent multiplier, m

W1/2 (mV)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

123 124 128 134 141 150 161 172 185 198 211

Other parameters used are T = 293 K, ν = 0.1 V s−1, S = 1 cm2, α = 0.5, E0 = 0 V, Estart = −0.2 V, and Γ = 1 × 10−11mol cm−2. a

was scaled using the exponents from Table S1 multiplied by the exponent multipler, m, from Table 2, an exponential base of 2, and an overall multiplier of 0.1, i.e.,

gain would require about a 6-fold increase in spread. Thus, we suggest that the thermodynamic dispersion is unlikely to contribute significantly to dc voltammetry with k0 values of around 0.1 s−1 at a scan rate of ν = 0.1 V s−1, with the spread of E0 values reported in ref 20. The same conclusion is reached with k0 > 0.1 s−1 up to the reversible limit at this scan rate of about k0 = 10 s−1. Of course, if all k0 values are sufficiently large, then broadening under these Nernstian conditions cannot arise from kinetic dispersion. A wider distribution in E0 has been implied to have an impact by Clark and Bowden19 and Rowe et al.7 solely on the basis of modeling. Similar distributions to those used in this article have been shown experimentally by Patil and Davis 21 to be applicable to metalloprotein monolayers. It is of course possible that the distribution of E0 values in modified azurin metalloprotein films is smaller than in other types of films due to the generally buried (environmentally screened) nature of the redox sites, but a direct measurement of distributions on other systems is needed to confirm whether this hypothesis is correct.

k0 = 0.1 × 2mj ,

j = −7, −6, ..., 7

(13)

This overall multiplier was added to the simulation study, as the highest k0 values encountered otherwise (26, 27) are in the reversible range where their impacts are almost indistinguishable from one another. However, we can see that using these lower values of k0 compared to those reported experimentally by Salverda et al.20 results in significant increases in W1/2 values. This result is expected: when k0 exceeds the reversible limit, the actual value is insignificant in terms of the impact on the peak current and hence features such as the half-height width. With the scan rate of ν = 0.1 V s−1 used in Figure 5(a), this limit, where results are independent of k0, applies at around k0 ≥ 10 s−1. This situation applies to about half of the molecules in the distribution given in Figure 2(b). If some of the k0 values for the dispersion distribution within a simulation are at, or exceed, the reversible limit that applies to the scan rate 5000

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Table 3. Values of W1/2 Obtained by Applying Different Combinations of E0 and k0 Dispersion, Weighted According to the Distributions Adapted from Reference 20 and Displayed in Figure 2, with E0 Rescaled To Be Centered on 0 V E0 dispersion

k0 dispersion

W1/2 (mV)

no yes no yes

no no yes yes

90 91 99 100

limit, we still predict a 10 mV increase in W1/2 from including the effects of both kinetic and thermodynamic dispersion, although the majority of the contribution is derived from the kinetic dispersion alone. The result of this analysis is that only the kinetic dispersion is significant when E0 and k0 distributions are assumed to be consistent with those resolved experimentally in ref 20. ac Voltammetry. Experimental ac voltammetric studies with azurin also reveal that W1/2, or their equivalent, are larger than predicted theoretically.25 Differences in other characteristics have also been noted.25 In dc voltammetry, attention is usually focused on apparently anomalous W1/2 values. However, in ac voltammetry, current magnitudes for quasireversible processes are very sensitive to k0 variations, unlike dc voltammetry, where the peak current varies only slightly with changes in k0. Consequently, ac methods should be far more sensitive to kinetic dispersion. To test this hypothesis, we first present a brief discussion of the sensitivity of harmonics in ac voltammetry to k0 in the absence of dispersion and then consider the implications of including dispersion on these harmonics. For a full description of the isolation and analysis of harmonics in ac voltammetric experiments, the reader is referred to ref 26. As noted above, dc voltammetry at a scan rate of ν = 0.1 V s−1 is sensitive to changes in k0 only up to the reversible limit of around k0 = 10 s−1. With our distribution of k0 values in Figure 2(b) having a mean close to this limit, the effect of dispersion on, for example, W1/2 is relatively small under conditions of dc voltammetry with a scan rate of ν = 0.1 V s−1. Two methods of increasing sensitivity to k0 would be either using a faster scan rate, ν, in dc voltammetry, which has problems associated with high background currents and requiring multiple experiments, or using ac voltammetry with a suitably high frequency perturbation. When using Fourier-transformed ac voltammetric techniques, only one experiment is needed and access to higher harmonics provides enhanced sensitivity to k0 and an almost complete absence of the double-layer charging current in an experimental setting. This increased sensitivity to k0, particularly in higher-order ac harmonics and again in the absence of dispersion, is shown in Figure 6, where we see that, even using the same scan rate of ν = 0.1 V s−1, an ac perturbation of amplitude ΔE = 80 mV, and a frequency of ω = 72 Hz, we are able to detect significant variations in current magnitude even with k0 values of up to k0 = 10 000 s−1 if the fifth harmonic is used (Figure 6(a)), though only up to about k0 = 1000 s−1 in the second harmonic (Figure 6(b)). Having illustrated the increased sensitivity of the current magnitude to changes in k0 in the frequency domain, we now return to the dispersion case, again using the levels of dispersion observed in the experiments reported in ref 20. The results of including either, both, or neither E0 nor the k0

Figure 5. (a) Effect of k0 on dc voltammetry in the absence of dispersion at a scan rate of ν = 0.1 V s−1. In this example, the reversible limit lies somewhere between k0 = 10 and 100 s−1. (b) Comparison between the effects of including different combinations of E0 and/or k0 dispersion on a dc voltammogram for a surface-confined process at a scan rate of ν = 0.1 V s−1 and k0 = 10 s−1. The blue and green curves, representing simulations without k0 dispersion, lie almost on top of one another, as do the red and black curves, representing simulations including the k0 dispersion. However, these two cases are clearly separated from each other, which shows the significantly greater impact of the k0 dispersion on the resulting current, compared to the impact of the E0 dispersion. Other parameters used include T = 293 K, S = 1 cm2, α = 0.5, E0 = 0 V, Estart = −0.2 V, Γ = 1 × 10−11 mol cm−2, and, where relevant in (b), E0 = 0 V and k0 = 1 s−1.

employed, then the effect is the same as narrowing the upper end of the distribution and increasing the weighting closer to the center, which we would expect to result in a smaller impact on W1/2. Indeed, if we repeat the simulations with weights adapted from the k0 distribution reported in Figure 2(b),20 with E0 still fixed, then we recover a half-height width value of W1/2 = 99 mV, contrasting with W1/2 = 90 mV with no dispersion included or with the value W1/2 = 89 mV predicted for a fully reversible process. Thus, kinetic dispersion under the conditions assumed has a much larger impact than thermodynamic dispersion. To conclude our investigation of the effects of dispersion on dc voltammograms, we present a comparison of the effects of including the kinetic (k0) dispersion and/or the thermodynamic (E0) dispersion on the half-height width, W1/2, using distributions adapted from ref 20 and reproduced in Figure 2. The results of these simulations are shown in Figure 5(b) and Table 3. As we can see, even with this combination of scan rate and k0 range, with many k0 values lying at or near the reversible 5001

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Figure 6. Effects of varying k0 on (a) the second harmonic and (b) the fifth harmonic of an ac voltammetric experiment using a sine wave with ω = 72 Hz and ΔE = 80 mV in the absence of dispersion. A strong dependence of the current on k0 is predicted until k0 lies above about (a) 1000 and (b) 10 000, when the process approaches reversible behavior. Note that in (b) the envelopes swept out by the blue and red curves are hard to distinguish as the data from k0 = 10 000 and k0 = 100 000 are very similar.

dispersion in a simulation of ac voltammetry with amplitude ΔE = 80 mV and frequency ω = 72 Hz are shown in Figure 7 (time domain (a), second harmonic (b), and fifth harmonic (c)). Importantly, even when we consider all four combinations of kinetic and thermodynamic dispersion, when the distributions adapted from Figure 2 are used, the inclusion or exclusion of thermodynamic dispersion makes little difference to the resulting current magnitude. Thus, in Figure 7, the curves which, respectively, include or exclude dispersion in E0, with k0 dispersion included are almost indistinguishable, as are curves derived from the inclusion and exclusion of the E0 dispersion, respectively, when the k0 dispersion is excluded from simulations. These pairs of curves are, however, distinct from each other, and it is the presence or lack thereof of kinetic dispersion which has the greater impact. This is consistent with the results from the dc simulations given in Table 3. While differences in simulations with and without kinetic dispersion are more pronounced in the case of ac voltammetry in the time domain format when compared to the dc case, it is in the frequency domain (harmonics) that we see the greatest potential for the development of a heuristic method for parameter estimation that includes the effect of dispersion. Now not just W1/2 or equivalent parameters reflecting shape may be used, as in dc voltammetry, but in addition the current magnitude variation is highly significant. Each harmonic is

Figure 7. Effect of including E0 and k0 dispersion, in quantities consistent with the distributions taken from Figure 2,20 in (a) an ac time-domain voltammogram, (b) an ac second harmonic, and (c) an ac fifth harmonic, using a sine wave of amplitude ΔE = 80 mV and frequency ω = 72 Hz. Simulations reveal that the k0 dispersion effects dominate, with the black and green curves representing simulations including the k0 dispersion both with and without the E0 dispersion, respectively, being almost indistinguishable. This is also the case with the red and blue curves, representing simulations without the k0 dispersion and again with and without the E0 dispersion, respectively. Note that in all cases the black and green, and red and blue, curves are hard to distinguish due to the similarity of the results with and without the E0 dispersion.

sensitive to different ranges of k0, so by resolving the total current into harmonic components, we can gain additional insight into which values of this parameter would produce a high level of agreement between theory and experiment. In the example presented, we see that the second harmonic, shown in Figure 7(b), exhibits not only a larger magnitude of current for 5002

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Langmuir

function of scan rate and initially attempt to simulate the results by assuming that only single values of E0 and k0 apply at all scan rates examined. On comparing experimental data with theoretical predictions, when broader-than-expected peaks are encountered, the concept of thermodynamic and kinetic dispersion may then be considered. On some occasions, data are recalculated, usually by introducing a distribution of E0 values to provide a good fit to the experimental data. However, in principle it would be better to assume that a distribution in both parameters applies right from the start of the modeling process and use simulations resulting from such a model for comparison with experimental data. ac methods, with enhanced sensitivity in current magnitude to small variations in electrode kinetics offer this prospect, and fully accommodate the dependence of k0 on E0. Nevertheless, the implementation of methods that fully address the inverse problem remains extraordinarily complex, and questions such as whether the Butler−Volmer models for electron transfer should be used instead of Marcus−Hush, along with other possible explanations of the nonideality encountered (see above), may also need to be considered. Ultimately, full automated data optimization approaches probably will need to be introduced to quantify the relative importance of a range of sources of presently so-called nonideality, including thermodynamic and kinetic dispersion along with development of more sophisticated experiments that allow the extent of thermodynamic or kinetic dispersion to be varied systematically.

simulations including kinetic dispersion but a different overall shape as well. The fifth harmonic, shown in Figure 7(c), displays a similar magnitude of current when the k0 dispersion is included, but again the shapes of the curves are also markedly different. The fifth harmonic is relatively insensitive to the lower values included in the k0 distribution, and although it is significantly more sensitive to the higher k0 values present in the distribution, they contribute only a little to the overall current. This results in the variation in shape, due to these higher values, but also the lower current in this harmonic, due to the discrimination against lower k0 values. The second harmonic, on the other hand, is relatively sensitive to all values included in the k0 distribution used, so not only do we see the expected shape due to the highest values, we also do not “lose” current from the inclusion of low k0 values.

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CONCLUSIONS Acknowledged discrepancies reported between theoretical predictions and experimental results for dc voltammetric studies undertaken on species confined at up-to-monolayer coverage on an electrode surface have been constructed in terms of data adapted from ref 20 to establish the significance of the inclusion of kinetic and thermodynamic dispersion. The impact of including either or both of the kinetic and thermodynamic dispersion has also been extended to simulations of ac voltammetry. With the distributions reported in ref 20, we found that a dispersion in the values of E0 is unlikely to be detectable experimentally by dc voltammetry due to its minimal impact at realistic parameter values25 but that dispersion in k0 can have a significant impact and account for at least some of the unexplained variation between results predicted by theory and those obtained from experiments.1−4,6,20−22,25 It is important to note that the field of surface-confined electrochemistry is considerably more complex and hence less well understood than its solution-phase counterpart. Importantly, parameter estimations are dependent on potentially problematic modeling assumptions. Thus, the Butler−Volmer model of electron transfer with α = 0.5 has been used in all simulations. However, not only might it be possible that there could be dispersion in the value of α, but also arguments have been made for preferring the use of the Marcus−Hush model instead of the Butler−Volmer model.4,7,23 Our model is also built on the ideal thin film assumptions of a Langmuir isotherm and noninteracting molecules,27 but alternative isotherms5,28 and the possibility of allowing adsorbed molecules to interact with one another,6 along with other reasons for nonideal behavior (for example, refs 7, 19, and 29−34), have been proposed. In a practical situation, the impact of uncompensated resistance and background current would need to be included in the modeling. Even though the inclusion of some of these alternative parameters in the simulation might also assist in accounting for some of the fundamental differences between theoretical prediction and experimental results, they do not exclude the presence of dispersion. Indeed, the experimental evidence for dispersion reported by Davis et al.20−22 is compelling and suggests that this phenomenon needs to be taken into account in all efforts to model surface-confined voltammetric experiments. In the context of endeavoring to obtain quantitative information on kinetic and thermodynamic dispersion, it is worth noting that commonly the present practice is to collect dc cyclic voltammetric data for surface-confined molecules as a

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ASSOCIATED CONTENT

S Supporting Information *

Outline of the simulation process using the stochastic model, along with comparisons of the results produced by the stochastic and ODE models. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This publication is based on work supported by award no. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST). Financial support from the Australian Research Council is also gratefully acknowledged.

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REFERENCES

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DOI: 10.1021/la5042635 Langmuir 2015, 31, 4996−5004