Stochastic Processes in Electrochemistry - American Chemical Society

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Stochastic Processes in Electrochemistry Pradyumna S. Singh, and Serge G. Lemay Anal. Chem., Just Accepted Manuscript • DOI: 10.1021/acs.analchem.6b00683 • Publication Date (Web): 27 Apr 2016 Downloaded from http://pubs.acs.org on April 28, 2016

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Analytical Chemistry

Stochastic Processes in Electrochemistry Pradyumna S. Singh† and Serge G. Lemay* †

Intel Labs, Intel Corporation, 2200 Mission College Boulevard, Santa Clara, CA 95054, USA and *MESA+ Institute for Nanotechnology, University of Twente, 7500 AE Enschede, The Netherlands

ACS Paragon Plus Environment

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ABSTRACT. Stochastic behavior becomes an increasingly dominant characteristic of electrochemical systems as we probe them on the smallest scales. Advances in the tools and techniques of nanoelectrochemistry dictate that stochastic phenomena will become more widely manifest in the future. In this Perspective, we outline the conceptual tools that are required to analyze and understand this behavior. We draw on examples from several specific electrochemical systems where important information is encoded in, and can be derived from, apparently random signals. This Perspective attempts to serve as an accessible introduction to understanding stochastic phenomena in electrochemical systems, and outlines why they cannot be understood with conventional macroscopic descriptions.

KEYWORDS. Stochastic chemistry, noise, nanoelectrochemistry, random walks, mesoscopic chemistry

Randomness is ubiquitous on the molecular scale. Thermal fluctuations cause individual molecules to undergo Brownian motion, associate and dissociate, adsorb and desorb from surfaces, and undergo conformational changes. In most experiments, this randomness is completely hidden from the observer due to massive averaging: so many molecules are involved that only their mean behavior is observable and a probabilistic description is unnecessary. However, in some systems, especially very small ones, averaging of random molecular motions is no longer completely effective and the underlying randomness is made manifest. Such systems are collectively referred to as mesoscopic: larger than the microscopic scale of individual molecules, but small enough compared to the macroscopic world that some aspects of the


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underlying microscopic graininess become observable. With the development of methods for fabricating increasingly sophisticated nanometer-scale structures, the family of experimental systems that fall in the mesoscopic category has grown rapidly and this trend can only be expected to continue. Dealing with systems with a random component requires some adjustments. How far can we scale down the size of a detection system before microscopic randomness prevents its functioning? Conversely, when microscopic fluctuations show up as noise in experimental data, can we learn something about microscopic processes by analyzing the noise? These questions are not frequently encountered in conventional electrochemistry, where mass transport and heterogeneous kinetics can be described adequately in terms of local concentrations, flow velocities and reaction rates, all of which are macroscopic concepts. With the advent of a growing number of nanoscale electrochemical systems, however, we expect that these considerations will become increasingly relevant. Through this Perspective we aim to introduce stochastic systems to the reader in the context of electrochemistry. We begin with an overview of how the information encoded in stochastic signals can be described and quantified. We then illustrate how these general concepts appear in the description of specific electrochemical systems where stochastic behavior is inherent. Our purpose here is not to exhaustively review the literature on stochastic processes in electrochemistry, but rather to illustrate through a few key examples how stochastic processes manifest themselves in electrochemical systems and outline the conceptual tools that are needed to understand them.

Stochastic signals and their representation


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A stochastic or random signal is a time-dependent quantity, such as the electrochemical current in an amperometric measurement, which fluctuates in a random fashion. This need not correspond to a very exotic situation, a simple example of a stochastic process being the irreversible breaking of a chemical bond according to the reaction

k AB  →A + B

(1)

This reaction is characterized by a rate constant , which indicates, on average, how quickly the reaction will take place. If we start with a mole of AB molecules, we know that the number of such molecules remaining after time is simply given by AB = > 0

(2)

where is Avogadro’s number. That is, all molecules are initially associated, and as a function of time more and more of them dissociate; the average lifetime of an AB molecule is 1/. But let us now consider what happens if we monitor a single molecule AB as a function of time. In our simple reaction scheme, the molecule cannot be partly dissociated: it is either AB or A+B. How long will this particular molecule take to dissociate? We cannot say! We know that there is a fairly good chance that it will be dissociated after a time 1/ has elapsed, but our molecule might also choose to dissociate much faster or be one of the few that survive far into the exponential tail described by Eq 2. This is the essence of a stochastic process: we know what must occur on average, but we cannot make deterministic predictions about what will happen to any specific microscopic subset of the system. If we are dealing with a random system, can we make any sort of quantitative prediction? The answer is an emphatic yes. But to do so, we must first change our viewpoint. An alternative way of phrasing Eq 2 that is more suitable for thinking about a single AB molecule is to say that the probability of our molecule still being associated after time is given by


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AB = > 0.

(3)

Eq 3 represents a subtle yet profound departure from Eq 2. On the one hand, the use of probabilities is an admission that we cannot say anything specific about what will happen in any particular run of our hypothetical experiment. On the other hand, it makes experimentally testable predictions. To see this, imagine that we repeat the experiment a large number of times, each time noting the time that elapsed before the molecule dissociated. We can then construct a histogram of these survival times, as illustrated in Fig. 1a. The histogram quantifies the likelihood that our molecule survives for time . As such, it can be compared directly to Eq 2, to which it is equivalent for a sufficiently large number of molecules. There is an additional level of subtlety: the values of any such experimentally generated histogram include in themselves a random component. Any particular time bin in the distribution contains on average a certain number of events, but randomness will ensure that the value of each bin is scattered around this average. A sufficiently large number of events must therefore be monitored before macroscopic information can be reliably extracted.


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Figure 1. (a) Probability distribution of survival times for AB based on the reaction of Eq 1. (b) Fluctuations in the number of associated AB pairs, AB , as a function of time based on the reaction of Eq 4. (c) Probability distribution of finding AB pairs corresponding to (b).

The above example concentrated on an irreversible reaction from all AB to all A + B. Let us now consider an equilibrium situation, AB ⇌ A B

(4)

We know that, on average, the number of associated pairs, AB , to that of dissociated molecules, A and B, will be AB / A B = / . There is however no guarantee that that this relationship holds exactly at any given moment in time: if an AB molecule dissociates at some given instant, there is no mechanism to ensure that a corresponding pair of A and B


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simultaneously associates. Therefore, the exact number of AB pairs fluctuates around its mean value, sometimes being slightly higher and sometimes being slightly lower, as illustrated in Fig. 1b. Should one be able to sample the number of AB pairs with infinite precision, one would therefore be able to reconstruct the probability of having exactly AB pairs at any given instant. A sketch of how the results of such an experiment might look is shown in Fig. 1c (for the experimental realization of a variation on this thought experiment, see Shon and Cohen1). The distribution is sharply peaked around the average, but there is a significant probability of finding the system in a state that slightly deviates from the average. These fluctuations are not some sort of deviation from equilibrium; they are an intrinsic property of the equilibrium state. How far does the system deviate from the average? Intuitively, we expect that randomness will be much less relevant in a system consisting of many AB pairs since some “averaging out” takes place. Conversely, in the limit of a single pair, the system can only randomly switch between two discrete states in an all-or-nothing way and the randomness will be maximal. This intuitive notion of a law of large numbers can be proven quite rigorously and generally.2 For the sake of illustration it is however more interesting to describe a specific example in detail. Imagine that we observe a quantity that varies randomly over time. Here could represent an electrochemical current, a voltage across a membrane, or any other quantity of interest. We assume that the statistical characteristics of do not change with time (a socalled stationary random process, such as the equilibrium system represented by Eq 4). For ease of notation, we can further separate into an average value and a fluctuation component: = ̅ . Suppose further that this signal is itself the sum of a large number of identical sub-components: = ∑

! ,

where = " represents the

contribution from each sub-component. It is then straightforward to show that the average signal


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# obeys ̅ = " and that the root-mean-square fluctuations around the mean obey rms = # , rms , independent of the exact properties of the fluctuators . Finally, further assume

that each fluctuator can only exist in two states: at any given instant, there is probability % that a given oscillator is in an “on” state with = & , where & is a constant, and a probability 1 − % that the state is in an “off” state with = 0. For this particular form of , the equations further simplify to ̅ = %& # = %1 − %&# ()*

(5) (6)

# Given a measurement of ̅ and ()* , equations 5 and 6 link the three microscopic parameters ,

% and & ; if any of these is independently known, then the other two can be deduced from the measurement. Furthermore, if % is very small such that we can approximate 1 − % ≈ 1, equations 5 and 6 can be re-expressed in two unknown variables: the on-state signal, & , and the ", = %. The two equations average number of fluctuators in the “on” state at any given time, are then easily solved to yield explicit expressions for these quantities: & =

2 -./01 .̅ 2 ̅

", = .2 -.

/01

(7) (8)

Equations (7) and (8) represent a remarkable result: solely from a measurement of the amplitude of the fluctuations, it is possible to extract underlying microscopic parameters. Below we will discuss explicit examples in real experiments. So far we have concentrated on the size of the fluctuations away from the average value, which is connected to equilibrium populations. But this is only a subset of the information that is available via the stochastic signal. How fast the fluctuations evolve in time contains additional,


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independent information. More specifically, while the distribution contains information about equilibrium properties, the time evolution provides additional information about the kinetic properties of the system being studied. In terms of our AB model above, the ratio / can be extracted from equilibrium measurements, but determining the absolute magnitudes of and requires a kinetic measurement. Conceptually, the easiest way to do this is to start the system in a state that is away from equilibrium and watch how it then returns to equilibrium. There is however a second approach that can deliver the same type of information without perturbing the system: watching how the equilibrium fluctuations around the average evolve in time. Consider once again the AB system of Eq 4. Imagine that we sample the number of associated molecules, AB at some time and again at 3. If we choose 3 to be very long, then we expect that the second sampling will be completely independent of the first since AB pairs will have had the chance to be broken and re-formed many times over. On the other hand, if we pick 3 much shorter than the typical association and dissociation times, then the second sampling will almost always give the same answer as the first. That is, there will be a high degree of correlation between the two measurements. Measuring how fast the correlation disappears as we increase 3 thus reveals something about the rate constants in the system. This concept of correlation between the values of a random signal at two different times can be formalized through the so-called autocorrelation function, which we define as 43 = 〈 3〉

(9)

Here 〈 〉 denotes an average over time . The function 43 expresses the average degree of correlation that we can expect, on average, between two samplings separated by a time delay 3. # 〉 For 3 → 0, it follows from the definition that 43/〈()* → 1, which represents perfect

correlation (simultaneous samplings necessarily return the same value). Conversely, if 3 is so


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large that and 3 are completely independent of each other, then the integral in Eq 9 averages to zero and 43 → 0. Autocorrelation functions, while providing a description of stochastic processes that is relatively easy to visualize, are not the most widely used description of stochastic signals. Rather, such signals are very often described in the frequency domain. The reader will be familiar with the Fourier transform, which is evaluated from a function as 89 = : ;#9 = 〈?89? 〉.

(11)

Here the symbol 〈 〉 represents an average over multiple independent samplings of the stochastic signal. While the exact shape of >9 depends on the dynamics of the system under


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investigation, it is nonetheless possible to obtain a qualitative sense for >9 without detailed modeling. First, the integral of >9 over all frequencies is equal to the mean square value of ; any increase in the area under the >9 curve thus corresponds to an increase in the magnitude of the fluctuations. Second, for a stochastic signal with a well-defined time scale, 3, >9 typically has a constant value at low frequencies and decays abruptly at frequencies larger than 1/2A3. Visual inspection of >9 thus readily provides a sense of the time scale of the dynamics. Of the two descriptions of stochastic signals that we have discussed, namely, autocorrelation functions and power spectral densities, which is better or more complete? They are, it turns out, completely equivalent: 43 and >9 are themselves a Fourier transform pair,. The choice of either description is thus mostly one of convenience. Most of the examples of stochastic electrochemistry that we will encounter below are more commonly discussed in terms of the power spectrum, >9.

Ion channels As a first, conceptually simple example of an experimentally tractable stochastic process, we begin with a system based solely on ion conduction. Electrical signals in nerve cells and between nerve and muscle cells are controlled by the opening and closing of ion channels embedded in the cell membrane, each channel contributing a pathway for ion conduction across the membrane. The opening of channels can be detected experimentally either via changes in ionic current, or through corresponding changes in the potential difference across the membrane. A broad range of mechanisms drives this opening and closing, but here we focus on a case of particular historical importance: channels that control muscle cells, and whose opening can be


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triggered by transient binding of the neurotransmitter acetylcholine (ACh).3-4 This opening process is stochastic: under typical conditions, individual channels open and close in a random manner. Consider a simple model of the muscle cell membrane. Let be the number of channels, % the probability that each individual channel is open at any given instant, and & the amount of current carried by each open channel. This model represents a direct realization of Eqs 5-8, and Eq 7 tells us that the amount of current carried by a single channel can be deduced by measuring both the average value and the size of the fluctuations of the current flowing through the membrane. This relation provided one of the first windows into the microscopic properties of ion channels. In early pioneering experiments, the potential3 or current4 response of frog neuromuscular junctions to doses of ACh neurotransmitter was monitored in real time. Together with an average response in the potential or current, increased fluctuations were observed upon exposure to ACh, as illustrated in Fig. 2a. From the amplitude of the noise, it was deduced that the typical conductance per ion channel is of order 3 x 10-11 Ω-1 (corresponding to currents of order pA). Furthermore, it was observed that the spectral power density, shown in Fig. 2b, exhibits a sharp cut-off around 21 Hz, corresponding to a typical time scale for the duration of a binding event of ~7 ms. This microscopic information deduced from ensemble measurements was corroborated a few years later once measuring the ionic current through individual ion channels became possible.5


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(a)

(b)

Figure 2. Noise in ion channels. (a) Upon exposure to Ach, ion channels imbedded in a muscle cell membrane open and the average ionic current through the membrane increases (bottom traces). This is accompanied by an increase in noise in the current (top traces). (b) Power spectral density of the noise. A characteristic frequency fc indicates the typical duration of a channel opening event. Reprinted with permission from ref 4. Copyright (1973) John Wiley and Sons. The above represents only a simplified snapshot of the experimental phenomenology and theoretical analysis of ion channels, which are the subject of a vast literature. Indeed, the original analyses of Katz & Miledi3 and Anderson & Stevens4 were based on more complex assumptions, for instance that each individual “transport event” is characterized by an exponentially decaying


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detector response (the time constant of which determines the critical frequency in Fig. 2b). That our simple two-state model gives basically the same answer (within a numerical factor of order unity) bears testimony to the generality of the underlying principles.

Shot noise and thermal noise A faradaic current can itself be source of stochasticity. To see why, consider first a simple single-electron oxidation reaction in which a neutral species is oxidized at an ultramicroelectrode (UME) poised at a high over-potential and in the presence of a high excess of supporting electrolyte. Under these conditions, redox molecules being transported to the surface of the electrode by diffusion generate a steady-state current. But this average current is itself the result of individual collisions between the surface of the electrode and single redox molecules, each undergoing an independent diffusive random walk. The average number of electrons " = ̅ /, where –e is the charge of the delivered in a given time interval of duration is electron. But because the events occur randomly, in any given sampling attempt there is a chance that either more or fewer electrons will be transferred. The Poisson distribution gives the " is a large number, this distribution is probability of transferring exactly electrons. So long as " with a standard deviation given by " !/# . Therefore, the relative size of sharply peaked around " !/# : while the fluctuations are the fluctuations compared to the average current scales as always present, they become increasingly negligible as the current or the integration time is " ). For the case where the electron transfer events are treated as instantaneous, increased (large it can further be shown that the corresponding current power spectral density is given by >9 = 2 .̅ That is, >9 is independent of frequency (so-called white noise) and its root-mean square amplitude scales as !̅ /# . For a 1 nA current measured with a 1 kHz bandwidth, which is


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typical of measurements at ultramicroelectrodes, this corresponds to a rms noise amplitude of 0.6 pA, or 0.06% of the signal. For most experimental situations this is utterly negligible. Shot noise may however become more relevant in situations where ̅ is much smaller. Strictly speaking, electron transfer events are not instantaneous. In our example, each charged molecule in the oxidised form that leaves the electrode following electron transfer interacts electrostatically with the electrode until it has moved beyond the thickness of the double-layer, resulting in a finite duration for the events. This in turn leads to values of S(f) lower than the white noise value for (very high) frequencies lying beyond the inverse event duration6-7. In addition to shot noise, there is an additional component of noise that originates from thermal fluctuations of the charge carriers in any electrically conducting system. This so-called Johnson noise is an equilibrium phenomenon that is also present in the absence of any net current. Like shot noise, Johnson noise exhibits a white spectrum with current spectral power density >9 = 4G H/I, where G is Boltzmann’s constant, H is the absolute temperature and I is the smallsignal resistance characterizing the solid-liquid interface. Again for typical conditions encountered near the half-wave potential at an ultramicroelectrode (I = =J/= ~ 10K Ω, 1 kHz bandwidth), this corresponds to a noise level of order 0.4 pA, or only 0.04%, of the limiting current. While there has been much theoretical analysis and modelling of shot and thermal noise in electrochemical charge transfer processes6-9, there have been relatively few corresponding measurements. Blanc et al.10, Gabrielli et al.

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and Mészáros et al.12 describe experimental

configurations that permit observing these sources of stochasticity. Their observations however require accounting quantitatively for the electrostatic environment at the solid-liquid interface (double-layer capacitance and mass transport, as illustrated in Fig. 3) as well as for additional


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sources of noise introduced by the measurement apparatus itself. The difficulties inherent in these proof-of-concept observations highlight the fact that shot noise and Johnson noise are very small effects under conventional experimental conditions.

Electrode

Nernst diffusion layer

Bulk solution

Figure 3. Half-cell noise circuit model of a faradaic electrode according to ref 6. Ingredients relevant to the fundamental noise spectrum include two main sources of noise, namely, Johnson thermal noise from the access resistance (L"2M ) and both thermal and shot noise from the Faradaic process (L"2= ). The latter requires modeling the dynamics of charge transport at the electrode, which is here schematically represented by the frequency-dependent function NO. Accounting for the measured full noise response further requires knowledge of equivalent circuit components including the bulk resistance, IM , the interfacial capacitance, C, the electron-transfer resistance, I , as well as the non-ideal Warburg impedance contribution to the interfacial impedance, PQ . This highlights the inherent challenge in experimentally probing the fundamental limits to electrochemical detection established by thermal and shot noise. Adapted from ref 6.

Corrosion We now turn to stochastic processes where the state of the electrode itself contributes to the randomness. Corrosion involves the formation of soluble or precipitated salts (usually oxides) on metal surfaces while they are in contact with aqueous electrolytic media. Though the overall kinetics of the cathodic and anodic electrochemical reactions that occur on a corroding electrode can be described by average rates, at the microscopic level these reactions are composed of many individual, elementary half-reactions, each of which may have different characteristic rates that may also fluctuate in time due to heterogeneities in the local microenvironment of each reaction site (differences in the local metal structure, local pH variations, mass transport effects, etc.).


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Fluctuations in the rates of these elementary electron-transfer processes, which manifest as fluctuations in the measured voltage or current, can reveal much about the dynamics of corrosion processes as was first described by Iverson.13-14 He observed differing fluctuation characteristics in the electrical voltage of metals and alloys in acidified solutions of potassium ferro- and ferricyanide. Since then, observations of electrochemical noise attending corrosion have been reported for a variety of electrodes in diverse media.10,

15-17

and the measurement of

electrochemical noise is a widespread technique for the study of corrosion. A key area where noise analysis has superseded the more conventional approaches like impedance measurements is the study of localized corrosion phenomena like pitting. At the heart of localized corrosion are two processes: 1) the random breakdown of local passivation barriers and initiation of pitting and 2) the propagation of these metastable pits to wider areas of the electrode surface. Thus, pitting can be considered as comprised of local microscopic processes distributed randomly in space and time. This makes it appropriate for a stochastic description. A first phenomenological model of the pitting process was offered by Williams et al.18, which accounted for current transients originating from the random initial formation of pits (which were Poisson distributed) followed by a deterministic evolution of these pits depending on the particular geometry of the pit. Each of the pit-forming events was assumed to give rise to a local contribution to the total current. Fig. 4a shows current-time transients obtained for steel electrodes in NaCl solutions under potentiostatic control. The transients show a slow rise followed by a sharp decay. The transients are themselves superimposed on a gradually rising background.


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(a)

(b)

Figure 4. (a) An ensemble of potentiostatic current-time traces, obtained at a 5 cm2 stainless steel electrode in NaCl solutions. (b) Corresponding power spectral densities with an f -2 decay. Trace 1 corresponds to chained current-time records, while Trace 2 corresponds to detrended records. Reproduced from ref 20. Reproduced by permission of The Electrochemical Society. Spectral analysis of the currents (Fig. 4b) by comparing the experimental spectra to theory allowed the estimation of important parameters like the probability of death of unstable pits via re-passivation, as well as the rate of birth of stable pits (i.e., pits that have grown to a critical size). This led to the development of a microscopic model of pitting corrosion. The shape or form of the power spectral density is directly correlated with the shape of the potentiostatic


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current-time fluctuation. For example, a series of events comprised of a slow exponential rise in current followed by a sharp drop are consistent with a power spectrum that has a 1/9 # decay at low frequencies.19 1/9 # behavior has been observed in the pitting corrosion of various types of steel.17, 19-21 Early studies also reported a characteristic 1/f component in electrochemical noise,16, 22

although a phenomenological description of its origins was not provided. More generally, however, while many authors have inferred mechanistic details from assessing

the power spectrum (e.g., cross-over frequency, the slope of the power decay etc.), it has been suggested that this cannot be employed as a diagnostic method that is universally applicable to infer mechanisms about all localized corrosion processes.23 Much of this owes to the inherent inhomogeneities in electrodes even when they are made of the same material. Other variability in experimental conditions (mass transport conditions, temperature, electrolyte concentrations, etc.), measurement set-ups, sampling frequencies, duration of measurement etc. muddy the waters further especially in studies of local corrosion, and a comparison of different studies is very complicated. An often-overlooked fact is that many corrosion processes are non-stationary (i.e., the statistical properties of the system are not constant in time) and therefore a spectral analysis can be very misleading. Another complication in noise analysis is that besides the randomness of the molecular events, there are other forms of noise (measurement amplifiers, mechanical noise, leakage currents, etc.) that are present in the data, and it is crucial to minimize interference from these sources. Crosscorrelation analysis has been proposed to distinguish noise arising from true molecular scale processes from other background noise.22,

24

Independent experimental verification of local

corrosion using other surface analytical techniques (e.g., atomic force microscopy) and


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correlation of the surface structure with the corresponding noise data can also be very fruitful in developing a comprehensive framework for interpreting electrochemical noise measurements.

Catalytic Nanoparticles We have seen how the measurement of electrochemical noise, which arises from stochastic microscopic events, can reveal information about the electrode surface in corrosion processes. Are these methods useful for investigating not just the properties of the electrode itself but also molecular entities in solution? While optical and other means of probing single molecules have advanced considerably and become well established, electrochemical methods to probe a few or individual analytes are comparably rare. This is not surprising since faradaic reactions usually contribute only one or a few electrons per encounter with the electrode, and this microscopic amount of charge is still impossible to measure. An ingenious approach pioneered by the Bard group to detect single nanoparticles (NPs) relies on the electrocatalytic properties of metal or metal-oxide nanoparticles towards proton reduction or water oxidation, respectively.25 On contacting a suitably biased electrode in solution, a single NP will be charged by only one or a few electrons, R , at a time which is difficult to detect. However, in the presence of certain reactants in solution whose conversion to products can be catalysed by this NP, cat electrons will be transferred to or from the electrode. Usually, cat ≫ NP , because the electrocatalytic NP can turn over several times per second in the catalytic cycle. The resulting amplification in the current can thus allow the detection of the presence of NPs at the electrode (Fig. 5a). The electrocatalytic conversion at the nanoparticle serves as an ‘indicator’ reaction that enables this amplification, and allows for the detection of nanoparticles at the electrode surface.


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So large is this amplification, that with a low enough concentration of nanoparticles it is possible to even detect individual nanoparticle collisions at an inert substrate UME. This amounts to measurement of the number of spikes (or steps) per unit time. The mean collision frequency scales linearly with the diffusion coefficient and concentration of the NPs as well as the radius of the UME, TUME . Because Brownian motion dictates the transport of particles, they arrive at the electrode in a stochastic fashion. Experimentally, this was first demonstrated by Xiao et al. by using 25 pM Pt-NPs at carbon fibre UMEs with the electrocatalytic reduction of protons (from dihydrogen citrate) serving as the indicator reaction.26 Amperometric traces in the presence of NPs showed random spike-like transients whose amplitude was consistent with the expected steady-state current for a single particle. Further, this amplitude was dependent on the concentration of protons as well as the electrode potential which distinguishes it from spikes due to other non-relevant random processes. Careful experimentation with other water-oxidizing NPs (e.g., iridium oxide27) and analysis and simulations of the collision frequency have offered insights unobtainable by bulk experiments, e.g., the adsorption of the NPs, kinetics of their inhibition and eventual degradation.28-29


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Figure 5. (a) Schematic of a nanoparticle that has electrocatalytic activity for conversion of reactant R to product P at the surface of a UME. This reaction cannot occur when the nanoparticle is not in contact with the electrode. (b) “Staircase” response of Pt-NP at Au UME for hydrazine oxidation. (c) “Blip” response of IrOx-NP at a Pt UME for water oxidation. (b) and (c) are reproduced from ref 28 – Reproduced by permission of The Royal Society of Chemistry.

In a separate study30, hydrazine oxidation was used as an indicator reaction, and a more “staircase” like current profile (Fig. 5b) was obtained for individual collision events. The magnitude of these steps was correlated with the dimension of the individual Pt-NPs and this allowed the evaluation of the statistical size distribution of the Pt-NPs, which correlated well with independent measurements using transmission electron microscopy (TEM). Comparisons of the transients obtained with proton reduction at Pt-NPs revealed differences in adsorption and kinetic properties of NPs for these reactions. In particular, the step-like response in contrast to the spike-like response from the earlier study was attributed to nanoparticle fouling. Fast-scan cyclic voltammetry (FSCV) has also been employed to study transient collisions and


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immobilization events of single NPs.31 Other studies have shown that the transition from staircase to blip responses is also potential dependent.32 More recently, an analogous technique33 that relies on the suppression of an electrochemical reaction by large inactive macromolecules has been used to detect individual collision events of antibodies, enzymes, DNA, polystyrene nanospheres34 and phospholipid vesicles.35 Interestingly, in some instances, redox active molecules trapped within individual vesicles35 and emulsion droplets36 were shown to generate electrochemical signals. As with the nanoparticles, an analysis of the stochastic collision frequency of these particles enabled insights into their size and transport properties. Other ingenious schemes which rely on coupling electrochemistry with light,37 magnetic fields,38 nanopores39-41 etc. have been employed to observe the dynamics of single particles at electrodes.

Number fluctuations and redox cycling Several techniques now exist to create electrode systems in which redox molecules are confined to sub-micron lengths in one or more dimension. The number of molecules present in such systems at any given moment is much smaller than is normally encountered in macroscopic measurements, and we can therefore expect stochasticity to appear more readily. An important reason for creating confined systems is to take advantage of redox cycling, the repeated reduction and oxidation of redox molecules at two closely-spaced electrodes. In the optimal case where the spacing between the electrodes, U, is much smaller than the lateral dimensions of the electrodes, the measured redox cycling current is proportional to the number of redox-active molecules present in the space between the electrodes. That is, = ∑ ! L , where is the number of molecules in the active region and L is the current


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carried by the th redox molecule. If is constant, the main source of randomness in the system is the shot noise resulting from discrete electron transfer events (fluctuations in the different L ). As we saw above, however, shot noise is normally small and difficult to detect. But if varies in time, additional fluctuations are introduced. If these fluctuations occur on time scales much slower than the electron shuttling time, they can dominate over the shot noise at low frequencies. In that case, L can be taken as a constant L& for all the molecules and we recover a simple expression for the current, = L& . Why redox cycling leads to enhanced fluctuations compared to normal shot noise at low frequencies can be readily understood qualitatively. We saw above that the amplitude of the shot noise is proportional to ̅ !/# , where is the amount of charge being transferred in each elementary event. is normally one or a few electrons, but the shot noise is increased if the charge is delivered in larger packets. In redox cycling, individual molecules enter the detection region and transfer multiple electrons before leaving again. Insofar as we are only interested in time scales longer than the residence time of each molecule in the detection region, each such event corresponds to a sudden burst of charge. Because the number of cycles per molecule can be quite large (order 102 - 104), the noise is enhanced. Consider a single particle undergoing a diffusive random walk. The probability of finding the WXV,

particle at a given position V at time , %V, , is given by the diffusion equation,

W

=

Y∇# %V, . Armed with this knowledge, it is relatively straightforward to estimate the probability that a particle starting at a given position inside or outside a detection volume at = 0 will be found inside the detection volume at a later . This process can be generalized so as to obtain explicit analytical expressions for the power spectral density42-44 or the autocorrelation function.45


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Amplification based on redox-cycling methods offers an opportunity to measure currents from a small number of molecules. First proposed in the 1960s,46 it is an important modality in scanning electrochemical microscopy (SECM) experiments. The authors’ group developed devices that realize redox cycling within nanofluidic channels. The active region of the device (the region enclosed within the electrodes) has an extremely small volume, [~10 !\ litres, such that even at relatively high concentrations (~ 1 mM) there are

Stochastic Processes in Electrochemistry - American Chemical Society

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