Recent Advances in Voltammetry

Recent progress in the theory and practice of voltammetry is surveyed and evaluated. The transformation over the last decade of the level of modelling and simulation of experiments has realised major advances such that electrochemical techniques can be fully developed and applied to real chemical problems of distinct complexity. This review focuses on the topic areas of: multistep electrochemical processes, voltammetry in ionic liquids, the development and interpretation of theories of electron transfer (Butler–Volmer and Marcus–Hush), advances in voltammetric pulse techniques, stochastic random walk models of diffusion, the influence of migration under conditions of low support, voltammetry at rough and porous electrodes, and nanoparticle electrochemistry. The review of the latter field encompasses both the study of nanoparticle-modified electrodes, including stripping voltammetry and the new technique of ‘nano-impacts’.


Introduction
The last decade or so has realised remarkable progress in the area of voltammetry-the interrogation of an electrode reaction by means of exploring, under dynamic conditions, the current-voltage characteristics of the process of interest so as to reveal, at one level, kinetic and mechanistic detail and, at another,m ore fundamental level,t he validity or otherwise of theories of electron transfer.
Voltammetric measurements are easily (and cheaply!) carried out, and data rapidly accumulated. However,i nterpretation of the latter is often challenging, even to the well initiated especially if quantitative information is sought. In particular,u ntil relativelyr ecently,d ata analysis required the use of analytical equations confined by mathematical necessity to 'model' (or 'toy') systems under well-defined conditions of transport, electrode kinetics, andm echanism. As such, the area was often limited to the study of experimental systems whichw ere amenable to data analysisr ather than being driven by the physicochemicali nteresto ft he system. The switch from this to real systemsh as been triggered by the ability to accurately simulate the voltammetric problems dictated and driven by chemistry rather than constrained by what is theoretically possible. The origins of this essential switch lies in the pioneering work of Rudolph [1] andthe subsequent commercialisation of his software in the form of the package DIGISIM. [2] This initially provided the first general basis of the modelling of 'nonstandard' linear diffusion problems, subsequently developed to more complex problems, but above all providing the impetus for the physical electrochemist to model his or her own systems. [3] It is the rigorous comparison of theoretical and experimental results [3][4] that underpins the approach advocated in this review,highlighting how such methodsenable greater physical insights into the dynamics of complex systemsa nd the underlying operative chemistry.
This article surveys the progress made, not only in simulation, but also in analytical theory,a nd comprises seven main subjecta reas of interest. First (Section 1), the use of the Butler-Volmer equation for modelling electrochemical reactions is considered, specifically in terms of its application to multistep processes;t his area is then further developed through consideration of the voltammetric response of electroactive species in ionic liquids. Ionic liquidsa re of both theoretical andp ractical interestd ue to their large electrochemical windows and the commonly observed altered chemical reactivity.S econd (Section 2), the physicalv alidity of the Butler-Volmer equation is questioned, and the development of the Marcus-Hush theory of electron transfer is probedf rom both at heoretical and experimental standpoint. Importantly,t he presented 'asymmetricM arcus-Hush' theory enables physical reinterpretation of the Butler-Volmer equation which, in many cases, helps to validate its continued and historical use.
Recent voltammetric studies of the theories of electron transfer have, in an umber of cases, been aided and facilitated by the use of pulse techniques. Moreover,s uch pulse techniques are also of distinct importance in the broad area of electro-analysis. Consequently,S ection3 of this review is dedicated to the theory of these often complex but highly important techniques. In particular 'differential double pulse voltammetry',' additive differentialp ulse voltammetry',' reverse pulse voltammetry', and 'square-wave voltammetry' are considered; ab rief discussion of staircasec yclic voltammetry is also provided. In combination with this work, the concept of the diffusion layer is advanced, enabling insight into the concentration pro-file of electroactives peciesa tt he interface and diffusive mass transport as aw hole. Following from this, Sections 4a nd 5 specifically continue in the development of ideas surrounding mass transport. First, Section 4r eviewsd evelopments in the use and modelling of diffusive random walks. Such theoretical modelsa re invaluable for the interpretation of newly developing stochastic single-molecule and single-nanoparticle techniques, facilitating new insights into mechanismso nt he nanoscale. Second, Section 5l ooks at the influence of migration upon the voltammetric response of electroactive speciesu nder conditions of low support. Beyond being ah indrance, low concentrationso fs upporting electrolyte can greatlyf acilitate electrochemical investigations, allowing mechanistic insights to be gained from the differing charges of the intermediates and their subsequent interaction with the electric field, as described by the Nernst-Planck-Poisson system of equations.
Having covered mass transport in relative detail, Section 6 considers the influence of altered electrode morphology upon the voltammetric response, where the local surface structure (rough, porous, etc.) serves to alter the diffusion regime local to the interface. The results of this section are of utmost importance for defining and evidencing authentic electrocatalysis. Finally,i nl ight of the above discussion, Section 7t urns to consider nanoparticle electrochemistry.T he section discusses nanoparticle-modified electrodes and the expanding field of stochastic 'nano-impact' experiments.S ystems in which the nanoparticles mediate an electrochemical process are considered, as is the direct oxidation or reduction of the nanoparticles.
Although this review encompasses av ery significant body of the work available within the literature, the areash ighlighted are necessarilys electivea nd focus upon fields deemed to be of particularc ontemporary importance;h owever,n otable absences include both AC voltammetry [5] and hydrodynamic [6] techniques, amongst others. Section 1: Advancing Butler-Volmer Theory

Multistep electrode processes
Having performed av oltammetric experiment, observed av oltammetricw ave, and correlated its electrochemical presence to ag iven redox speciesi ns olution, the next common scientific line of enquiry is to try andd iscernt he operative electrochemicalm echanism. Af ullr eview of this area of study is beyondt he scope of the current text, andt he interested reader is directed towards the seminall ecture series by Sa-vØant [7] and the in depth reviews by Evans. [8] However,w ith the recent publication of the IUPAC recommendations on the transfer coefficient, the area of multistep electrode processes warrants brief attention.
The transfer coefficient (a)i snow defined as; [9] a ¼AE RT F d ln j jj dE ð 1 Þ where the flux j has been "corrected for any changes in the reactant concentration at the electrode surface with respectt o its bulk value". The sign in Equation 1d epends upon whether the reaction is anodic or cathodic. This IUPAC definition of the transfer coefficient usefully and deliberately does not presuppose anything about the operative electrode reaction mechanism. Mass-transport corrections to attain the flux j relative to its bulk value is relatively facile for steady-state voltammetry. [10] However,f or macroelectrode cyclic voltammetry,c omplete extraction of this information is more involved, but, as demonstrated by Henstridge and Compton, certainly still obtainable. [11] In most literature, experimental measurement of the transfer coefficient from macroelectrode voltammetry is limited to assessment of the Tafel slope at low overpotentials where the concentration of the reactanta tt he electrochemical interface is not significantly altered from that of the bulk value. Note classical literature tends to quote the magnitude of the Ta fel slope (dE / dlog j j j)w hichi sd irectly related to reciprocal of the transfer coefficient. Having experimentally measured the transfer coefficient, its interpretation provides one route by which the electrode mechanism may be elucidated, as will be discussed below.
For ao ne-electron redox process with unit stoichiometry as given by under reversible (quasi-equilibrium 1 )c onditions the transfer coefficient simply reflects the number of electrons transferred, which in this case is one. This value for the reversible case is generally referred to as an 'apparent transfer coefficient'. The term 'apparent' is used to emphasise that the measured transfer coefficient does not reflect the underlying electron transfer kinetics, but originates directly from the system being under Nernstian control. Under irreversible conditions, the transfer coefficient takes av alue between 0a nd 1, but is commonly 0.5 AE 0.2 for ao ne-electron process. As an ote of caution, the transfer coefficient as measured from av oltammogram may be 'artificially' distorted due to ap otentiostats application of staircase as opposed to at rue analogue voltage ramp;t his point is discussed in further detail in Section 3.4. Use of the Butler-Volmer equation to model an electrochemicals ystem implicitly assumes the transfer coefficient to be constant as af unction of potential;d eviations from such linear Ta fel behavior have been experimentally evidenced. Moreover,t he Butler-Volmer equation is periodically damned due to being 'phenomenological' in its description of electrontransfer,t hat is, it has no direct physical meaning. However,a s will be expanded upon in Section 2o ft his review,t he Butler-Volmer equation can be better understood as often being ah ighly precise approximation of electron-transfer rates at low overpotentials 2 and warrants its use due to its mathematical simplicity.M oreover,c omparison of the Butler-Volmer theory with asymmetricM arcus-Husht heory allows ap hysicalu nderstanding of the transfer coefficient (see Section 2f or further details) in terms of the changing force constantsi nt he redox reaction. Before moving on any further, it is highlighted that the terms 'reversible' and' irreversible' when applied to voltammetry refer to the magnitude of the electron transfer rate relative to the prevailing rate of mass transport to andf rom the electrochemical interface. [12] This is an importantpoint that tacitly underpins much of the work discussed within this body of text. It is also ag eneral lack of insight into this problem that regularly leads to misinterpreted and misreported results within the literature (see Section 7f or further discussion).
One of the principal points in the IUPAC recommendation is that the simultaneous transfer of two or more electrons is highly unlikely.C onsequently,m echanistic interpretation of the transfer coefficient must be done in the light of this fact! [9] At this stage, the simplest multistep electrochemical process is considered:t hat of two sequential reversible one-electron transfers (an E rev E rev reactioni nT esta and Reinmuth notation [13] ) as described by In the absence of coupled homogeneous kinetics the second electron transfer is likely to be less favorable purely due to coulombic repulsion. [8] However,t he relative potentials at which these two redox processes occur is highly solvent and electrolyte dependent; [14] notably hydrogen bonding can have ap rofound influence. [15] Experimentally,t his situation is well exemplified by quinone reductions. In non-aqueous systems two one-electron waves are observed;t he addition of water serves to compress the potential difference between the two waves until they merge to yield one voltammetric wave. However,i na queous media (at high pH), it has been experimentally demonstrated that the potential of the first and second electron transfers can be 'tuned' through ion-pairing. [16] Ac onsequence of the first and second electron transfer occurring at similar potentials is that, although only one voltammetric wave is observed, the peak height is suppressed, and the wave is broader than would be anticipated if the electrons were assumed to be transferred simultaneously. [17] In such cases, the use of square-wave voltammetry is advisable for the precise measurement of the associated formal potentials. [18] The presence of coupled homogeneous chemical processes or structural changesi nt he molecular structure may lead to as ituation known as potential' inversion' where the second electron transfer is easier than the first, that is (E f 1 ÀE f 2 )i sn egative. [19] Reportedly,apotential inversion of 400 mV is required such that 'no' intermediate is formed. [20] Figure 1d epicts the simulatedv oltammetric response for at wo-electron reduction as described by the mechanism above. Also shown is the predicted voltammetric response for the hypothetical situation in which two electrons are transferred simultaneously.T his result clearly demonstrates how even for this simplest multistep process, the corresponding voltammetric wave shapec an vary significantly when the two respective formal potentials are comparable in magnitude, as ituation which is commonly encountered. Twoi mportant insights should be taken from this:f irst, ap otential inversion of greater than 100 mV is required for the peak current of the stepwise two-electron voltammetric response to be within 3% (i.e. within experimental error) of that obtained for the 'simultaneous' case. Second, ac ommonr oute to determining the diffusion coefficient of an analyte is to Figure 1. Simulatedv oltammetric response of at wo-electron transfer as af unction of the difference between the formal potentialsfor the two processes. E f 1 ÀE f 2 : + 50 mV (magenta), 0mV(blue), and À50 mV (red). The black line represents the hypothetical limiting case in which the two electrons are transferred simultaneously.S imulations performedusing DigiSim; other simulation parameters: v=100 mV s À1 , D = 10 À5 cm 2 s À1 , r 0 = 1mm. measuret he respective peak current as af unctiono fs can rate, where the peak current is proportional to the squarer oot of the diffusion coefficient. In the case of am ultistepp rocess, as highlighted by Figure 1, this may lead to as ignificant error due to the sensitivity of the peak current to the formal potentials of the electron transfers. [21] As as econdary example, the case where an electron transfer is coupled to ah omogeneous reactiona sg iven by the mechanism: is considered. This scheme is known as an 'EC reaction' where Es ignifies an interfacial redox process and Ci sacoupled homogeneous reaction. For the case in which the electrochemical step is reversible, as the chemicalstep Cismade more thermodynamically favorable (assuming it is not kinetically hindered), the potentialr equired for the redox process decreases in magnitude, with ac orresponding loss of the voltammetric back peak. As the chemicals tep becomes more highly driven,t he redox wave shifts to ap otential at which the rate of electron transfer becomes the rate-determining step (note the rate of electron transfer decreases exponentially,i na ccordance with the Butler-Volmer equation). Hence a' reversible' electron transfer process (i.e. one which has ah igh k 0 )m ay appear electrochemically irreversible when the product is consumed by ah ighly driven chemical step. [22] It can be seen from the above two examples that the complexity of an electron transfer reaction rapidly increases with the number of steps, and hence such systemsa re best understood through simulation (further discussion of multistep electrochemical processes can be found in the work of Batchelor-McAuley and Compton [21] ). This is especially true for situations involving proton transfer where am ultitude of possible mechanistic routes are possible, and the exact pathway (or pathways) taken will depend strongly upon the pH, the electrode potential, the pK a value associated with the reactants, intermediates,a nd products, and the electron-transfer kinetics of individual steps. [17] Relatedt ot his is the investigation of systems in whicht he proton and electron are transferred simultaneously,areaction which is possibly of distinct importance in biology. [23] Although full understanding of an electrochemical process is best achieved through simulation, the rate-determining step of am ultistep mechanism may be readily assessed by experimental measurement of the Tafel slope, noting that the magnitude of the Ta fel slope may also vary as af unction of scan rate due to the presence and influence of coupled homogeneous kinetics. [24] The Ta fel slope should be interpreted as being equal to À(n' + a RDS )F/RT (for ac athodic process) where n' is the number of electrons transferred before the rate-determining step, and a RDS is the transfer coefficient of the rate-determining electron transfer. [21] Noting that again,i na ccordance with IUPAC, assuming that the Ta fel slope is equal to À(an)F/RT, where n is the total number of electrons transferred is invalid and outdated as it implies the possibility of multiple electrons being transferred simultaneously.Aproblem arises upon recognition that many older analytical expressions used for interpreting voltammograms, which are still regularly used, are in error-one prime example being the use of the variation of the peak potentialofasurface-bound redox wave as afunction of scan rate to extract kinetic and mechanistic information. [25] In this model,t he peak position is relatedt oa n;h ence, use of such expressions for analysis of the operative mechanism will be in error.
As commented above in reference to electrochemical reversibility,v oltammetry is inherently the study of interfacial processes;c onsequently,u nderstanding of as ystem's response requires an understanding of the prevailing mass-transport regime and its effect. An example of this for multistep processes relatest ot he influence of an electrode's size. On decreasing the dimensions of an electrode, that is from macro to micro, the mass transport to andf rom the surface becomes more efficient. [26] This is akin to increasing the rotation rate of ar otating disk electrode, however far higher mass-transport rates are attainable with the use of microelectrodes. [27] The decrease in the electrode size resultsi nt he electron transfer process becomingm ore irreversible and increases the probability of intermediates being released. To more succinctly restate this, on changing the electrode size the overall electrochemical mechanism may be altered, resulting in the possible releaseo f 'higher'-energy intermediates. One example of such as ituation is found with oxygen reduction at silver surfaces. At am acroelectrode the process is found to, on average, involve the transfer of 3.3 electrons to each oxygen molecule. [28] This corresponds to roughlyo ne in three oxygen molecules undergoing only at wo-electron reduction to hydrogen peroxide before release from the electrode, as opposed to the full four-electron reduction.H owever,d ecrease of the size of the electrode leads to an increase in this ratio, such that the probability of hydrogen peroxide production is increased, where on the nanoscale, almosta ll formed hydrogen peroxide is released prior to further reduction. [28] This insightt hat reactivity may change on the nanoscale solelyd ue to the alteredm ass transport (i.e. even without considering plausibly alteredn anoparticle thermodynamicso rt he expression of higher-order crystal facets) has two implications:f irst, it implies al imitation on the efficiency of the use of nanoparticles for catalysis in industrially relevant processes. [25] Secondly,a nd perhaps more importantly, the changed reactivity on the nanoscale arising from the altered mass-transport regime may have wide-ranging implications fort he nanotoxicity of these materials towards biological systems. [29] It is this changed reactivity at diffusionally isolated particles that highlightso ne of the driving forces forw ishing to study reactions at individual nanoparticles, as ubject that will be focused upon more within the final section of this review.
The following section focusses on the simulation and understanding of voltammetry in ionic liquids, am ediumi nw hich the mass-transport rates are appreciably slower than found for common aqueous systems.

Voltammetry in ionic liquids
The last decadeh as seen an explosion of electrochemical interest in the use of room-temperature ionic liquids (RTILs): liquids composed entirely of ions which only solidify at temperatures well below ambient. [30] This is partly because they offer significant advantages for somea pplications,m ostn otably in energy transformation technology [31] andi ng as sensing, [32] but also since they challenge existing theories of electron transfer [33] and interfacial structure. [34] Progress in the area has been regularly reviewed. [30,35] This section focuseso nt he alteredv oltammetry seen in RTIL media.
From an electrochemical perspective, RTILs offer some important contrasts with conventional solvents such as water, acetonitrile, THF, etc..First, the potentialwindow displayed, defined by the onseto fc athodic and anodics olvent decomposition, is unusually wide and, for rigorously dried solvents, can extend to as much as 5o r6V. [36] This reflects the stability towards oxidation and reduction of the component ions which are generally ab ulky organic cation and as mall inorganic anion such that the size mismatch discourages crystallisation except at unusually low temperatures. Indeed the use of the same materials as supporting electrolyte in conventional solvents has been advocated, reflecting the intrinsic inertness of the ions. [37] As econd significant difference lies in the observation [30] that many ionic liquids have viscosities which are larger-often an order of magnitude greater-thanc onventional molecular organic solvents. This is reflected in the magnitude of the diffusion coefficient of the solutes which, except for rather small sized molecules such as O 2 [38] or H 2 S, [39] generally reflect the Stokes-Einstein equation, with diffusion coefficients scaling inverselyw ith the viscosity. [40] The much reduced rates of diffusion in RTIL media have the very important consequence that the transitionf rom linear to fully convergent diffusion as observed at microelectrodes occurs at quite different (much lower) voltage scan rates than are familiar to electrochemists operating in aqueous or non-aqueous media. [41] As ar esult, it is quite common to see peak-shapedr ather than sigmoidal current voltage curves when using microelectrodes in ionic liquid media, and true steady-state diffusion limited currents can be difficult to observe unless unusually if not pathologically slow scan rates are deployed (under which conditions other factors such as slow adsorption or coupled kinetics may undesirably kick in). It follows that the extraction of kinetic and transport parameters from the voltammetry requires the numerical simulation [41] of the voltammetry rathert han the application of the simple analytical equations derived for pure linear diffusion (Randles-Ševčík equations) and for pure convergent diffusion at am icrodisc (I = 4nFDCr,w here n is the number of electrons transferred, F is the Faraday constant, D is the diffusion coefficient, C is the analyte concentration, and r is the radius of the electrode). Moreover,t he fitting of such simulations is challenging, but can be helpfully simplified if the diffusion coefficients of the reactant, A, andp roduct, B, are determined for: independently of the voltammetry,i deally using potential-step or double-step chronoamperometry. [42] At hird but related issue is that the diffusion coefficients of the analytes in RTIL media can be quite sensitive to the presence of dissolved gases not least because of the high solubilities (~m)o fs pecies such as H 2 So rC O 2 in many RTILs, but also because the dissolved gases significantly perturb the solvent structure and hence the transport of the othersolutes. [43] Af ourth issue relating to the different voltammetry in RTIL media as compared to molecular solvents is that the common approximation of assuminge qual diffusion coefficients for most or all species involved in an electrode reactioni su sually adequate for the quantitative simulation of the voltammetry; this approximation holds much less well for RTIL media. The reason for this is that because of the ionic nature of the solvent, the transport properties are sensitiven ot only to the solute size (Stokes-Einstein equation) but also to the solute charge. [44] One extreme example is the one-electron reduction of oxygen, in the RTIL hexyltriethylammonium bis(trifluoromethyl)sulfonyl imide, where at 25 8C which gives ar atio of diffusion coefficient of over 30! [45] The consequence of this marked differencei st hat the voltammetry leads to curiousc urrent-voltage response in which am icrodisc electrode steady-state voltammogram is seen for the forward scan, corresponding to the faster diffusing O 2 reduction, whereas in the reverses canapeak is observed for the reoxidation of O 2 C À because the slownesso fi ts diffusion leads to the accumulation of O 2 C À near the electrode surface, and the transport contains as ignificant component of linear diffusion. [45] The discussion above has focused on measurements made in RTIL media using microelectrodes, which are ap referred methodology for such investigations. [32,46] This is because the deploymento fm icroelectrodes facilitates the use of smallv olumes (~10 mL) of solvent, which is important in the RTIL area since it is vital to properly dry the solvent and to ensure that they are water free. The drying of RTILs is readily undertaken using aT -cell arrangement [32,[46][47] in which the solvent can be exposed to vacuum andr igorously dried before voltammetric study.A ny residual water will greatly reduce the electrochemical window [36b] and markedly alter the diffusion coefficients of the solutes. [46] The drying of larger quantities of solventi s as low process because of the need for the water to diffuse to the liquid surfacea nd evaporate;a so bserved diffusion in RTILs can be av ery slow process, ands ot he purification of the much larger volumes required for macroelectrode experiments www.chemistryopen.org would be time-consuming and possibly incomplete. Indeed the lack of reproducibility of simple data such as diffusion coefficients (even of 'model' compounds such as ferrocene), in early work in the field probablyr eflects the different composition of the solvent used in terms of dissolved water and gases.
Finally,w ec onsider whether Butler-Volmer theory is applicable in RTIL media noting the near-ubiquitous success claimed for the phenomenological approach in molecular solvents. The first consideration is to note that the slowed diffusion in RTILs promotes the apparent electrochemical reversibility of many redox couples. In order for the voltammetry to revealq uasi-or irreversible electrode kineticb ehavior,i ti sr equired that the studied rate constant must fulfill k 0 < m T ,w here m T (cm s À1 )i s the mass-transport coefficient of the electrode. Typically this is approximated by m T~D /r,w here r is the electrode radius, so that even with microelectrodes it can be challenging to extract electrochemical rate constants from the voltammetric data. It follows that measurements made using macroelectrodes in RTIL are unlikely to give reliable data especially since it is noted that the conductivity of many ionic liquidsi ss imilart o that of conventional organic solvents (DMF,C H 3 CN, THF,e tc.) containing about 0.1 m supportinge lectrolyte so that macroelectrode voltammetry in RTILs is also typically as distorted by ohmic losses as is voltammetry in organic media.
Extensive modeling of aw ide diversityo fv oltammetric systems has been undertaken using small microelectrodes in order to provide ab etter possibility of extracting kinetic parameters. Systems studied include O 2 /O 2 C À , [45] Br À /Br 2 , [48] nitrobenzenes, [49] aryl amines, [50] NO 2 /NO 2 À /NO 2 + , [51] I À /I 2, [52] aromatic diamines, [53] arenes, [54] Li/Li + , [55] benzoquinone, [56] hydroquinone, [57] and H + /H 2 . [58] In many cases the values of k 0 obtained correspond to quasi-reversible behaviour and, as such, do not provide ap erfect test of the validity of Butler-Volmer kinetics since the behaviour is approximately Nernstian. To restatet his, under reversible (Nernstian)c onditions, no information regarding the kinetics of the electron transfer process may be inferred from av oltammetric experiment. Hence, as as ystem tends towardsr eversibility,o btaining unambiguous results evidencing the validity or otherwise of the applicabilityo ft he Butler-Volmer equation becomes inherently more challenging. However,i ns ome cases, notably the I À /I 2 system, [52] the oxidation of hydroquinone, [57] Li/Li + , [55] and the H + /H 2 system [58] there is clear electrochemical irreversibility,a nd the accuracy of Butler-Volmer kinetics in reproducing observed experimental behavior is excellent. Note that the follow-up chemistry in these systems 'promotes' the irreversibility of the system and, as such, multistep processes may be preferred for studying electron transfer in RTILs. [59] Very recently, [60] an attempt has been made to apply Marcus-Hush theory to ionic liquids focusing on the O 2 /O 2 .À couple. Solventr eorganisation energies around 0.4-0.5 eV were found and attributed to inner sphere reorganisation with an egligible contribution from solvent reorganisation.
In summary,q uantitative voltammetry in RTILs present special challenges, butw ith the aid of numerical simulation, quantitative understanding of the both kinetics and mechanism is possible. The following section looks at recent advances in the development of models of electron transfer kinetics.

Section 2: Challenging Butler-Volmer Theory
In recent years, ar enewed interesti nt he suitability of the availablekinetic modelsfor heterogeneouselectron transfer reactions has led to critical assessment [61] of the most well-established approaches:t he Butler-Volmer (BV) [62] and the Marcus-Hush (MH) [63] models.
The BV formalism has been preferred over the years (and still in the present) due to its simplicity and satisfactory descriptiono ft he electrode kinetics of many systemsw ith three fitting parameters:t he standard heterogeneous rate constant (k 0 ), the transfer coefficient (a)a nd the formal potential( E 0 f ). Thus, the rate constantsa re given by the followingw ell-known expressions: In spite of being successful in kinetic parameterisation of ag reat majority of redox systems, easy-to-implement,a nd computationally inexpensive, the BV expressions for the rate constantsa re empirical (but see below in connection with the discussion of asymmetric Marcus-Hush theory). Therefore, the adjustable parameters provide limited physical insighti nt erms of the nature of the electroactivem olecules,t he medium, and the electrode, and it is not possible to make predictions. Moreover,e xperimental deviations from the ever-increasing exponentialv ariation of the rate constants with E À E 0 f predictedb y Equations 9and 10 have been reported. [64] The above limitations of BV calls for the use of more realistic modelst hat enableu st of ully describe the experimental data, as well as to connectt he electron transfer kinetics with the nature of the system. With this aim, in recent years the applicability of the Marcus-Hushmodel has been theoretically and experimentally assessed via voltammetry in its symmetric [64b] and asymmetric [65] versions.

The symmetric Marcus-Hush model(sMH)
The symmetric version of Marcus theory [66] considers the parabolas describing the Gibbs energy of the reactants and products to be of equal curvature. The heterogeneouselectrochemical reaction between am olecule andametallice lectrode involves transfer of charge from ad iscrete molecular energy level to ac ontinuumo fs tates (e), with the energy levels in the electrode occupied according to the Fermi-Dirac distribution. This model leads to the following expressions for the heterogeneous rate constants: [64b] k MH red ¼ k 0 S red h; L ðÞ S red 0; L ðÞ ð 11Þ where S red=ox h; L ðÞ are integrals: with DG y sym;red=ox x ðÞbeing the activation energyo ft he electroreduction/oxidation process for each electronic level that according to the symmetric version of the Marcus theory is given by: ,a nd L is the dimensionless reorganisation energy: L ¼ F RT l.W hen two signs appear, the upper sign refers to reduction and the lower sign refers to oxidation. For the calculation of the integrals of the sMH formalism [Eq. (13)],n umerical integration methods can be employed [67] and analytical approximations have also been proposed to make the implementation of the sMH expressions easier. [68] As with the BV model,s MH theory describes the electron transfer kinetics as af unction of three adjustable parameters: the formal potential, the standard heterogeneous rate constant, and the reorganisation energy (l). The latter corresponds to the energy required to distortt he atomicc onfigurations of the reactant molecule (inner-sphere component of l)a nd its solvation shell (outer-sphere component)t ot hose of the product in its equilibrium configuration. Therefore, l enables us to rationalise the electrode kinetics in terms of the microscopic nature of the system such that the larger the structural and solvation changes as ac onsequence of the electron transfer, the larger the l value and the slower the electrode kinetics.

The asymmetric Marcus-Hush model(aMH)
In the sMH model,t he Gibbs energy parabolas are assumed to have the same curvature, which means that intramolecular vibrations and solvation are, on average, the same for the reduced and oxidised species. This may not hold as ag eneral rule given the different charge of the reduced and oxidised species, and various theoretical approaches have been considered to overcome this limitation of the sMH formalism. Among them, the use of the asymmetricv ersion of the Marcus theory has been recently applied to heterogeneous electron transfer processes by Comptonetal. [61a] As can be observed in Figure 2, different (vibrational and/or solvation)f orce constantsr esult in Gibbs energy curveso fd ifferent curvature and affect the value of the activation energy that, within the asymmetric Marcus theory,c an be written as: [65a, 66] DG y asym;red=ox x ðÞ with the parameter g being defined as: where l i is the inner-sphere reorganisation energy, Dq 0 s is the difference between the equilibrium values for the s-th normal mode coordinate of reactant and product, and k s and l s are symmetric and antisymmetricc ombinations of the force constants of the s-th mode of the oxidised (f ox s )a nd reduced (f red s ) species: The values of the rate constants in the aMH model are calculated from the expressions presented in Section 2.1 by substituting Equation 15 into 13. In the numerical integration of Equation 13, the limits of the integral must be restricted to the x-range where the integrand value is significant,t ypically AE 50.
From the definitions in Equations 16, 17, and 18, it is clear that the g value relatest od ifferences between the vibrational force constantso ft he electroactive species such that it takes ap ositive value when the force constants of the oxidised species are greater (on average), an egative value in the opposite situation, and g = 0w hen the (average) force constants are equal. Note that the last particularc ase coincide with the sym- In the derivation of Equation 15, only the first terms in the expansion of l s hihave been considered so that the aMH formalism above presented accountsf or differences between the vibrational modes of the reduced and oxidised speciesw ith only one additional fitting parameter with respectt oB Va nd sMH (g,[Eq. (16)]). Higher order terms in l s hi [66] would be necessary when the force constants differ significantly (by af actor of more than 2 [61a] ), which would make the model less general and more complex.

Voltammetric assessment of the BV and MH models
As shown in Figure 3the different kinetic models predict different variations of the rate constants with the applied potential. Thus, whereas the reductiver ate constant increases exponentially and continuouslya sEÀE 0 f is more negative, in the MH models k red shows al imiting value at large which is consistent with the curved Ta fel plots and potentialdependentt ransfer coefficients reported in the literature. [64] The divergence from the BV behaviour is more apparent for small l-values and at large overpotentials ( Figure 3b).
Another key point to consideri nF igure 3i st hat, independently of the l-value, the curvesf or the reduction and oxidation rate constants are symmetrical with respectt ot he axis . This reciprocity relation [68a] breaks down in the aMH model when g ¼ 6 0. Thus,w hen the force constants of the oxidised species are greater, g > 0, the cathodic branch is steeper than the anodic one, and the opposite is true for g < 0. Note that the geffect is more significant at large overpotentials and it is analogous to the effect of a in BV.Indeed, at low overpotentials, the k red/ox values calculated from the aMHf ormalism tend to those obtained in BV with the following transfer coefficient: Thus, the case a < 0.5 relates to force constants of the reduced speciesg reater than those of the oxidised one (i.e., g < 0) and the opposite applies for a > 0.5. This enables physical reinterpretation of the a data available from Butler-Volmer analysis, extending overmany years.
The effect of the asymmetric parameter g on the voltammetric response is also analogous to that of a in BV.T his is shown in Figure 4f or the response of diffusionals ystemsi nc yclic voltammetry and reverse scan square wave voltammetry under transientc onditions. In the latter,a sw ell as in the reverses can of cyclic SWV,acathodic peak and an anodic one can be observed on either side of the formal potentiali nt he case of sluggishelectron transfers. [69] In both CV and SWV,the reorganisation energy affects the reductive and oxidative peaks of the voltammograms similarly (Figure 4b), whereas the g-value has an effecto nt he relative anodic/cathodic peak heights and the peak potentials. The reductive peak increases in heighta nd shifts to less negative potentials as g takes more positive values, as occurs in BV for a > 0.5. Thisf act points out the greater flexibilityo ft he aMH model for quantitative fitting of the voltammetry through the new kinetic parameter.
In summary,t he kinetic formalisms discussed above predict different dependence of the reductiona nd oxidation rate constants with the appliedp otential, the divergence between them being more apparent for small values of the reorganisa- www.chemistryopen.org tion energy and at large overpotentials. In order to point out such differences experimentally and assess the suitability of the different kinetic models, various electrochemical methods have been proposed and employedi nt he literature as an alternative (or complement)t oc yclic voltammetry. [70] The use of differential pulse voltammetries (namely,s quare wave voltammetry andd ifferentialm ultipulse voltammetry) has proven very insightful in revealing differences between the kinetic models [71] as well as being very adequate for quantitative studies. Thus, duet ot he subtractive nature of these techniques, well-defined, peak-shaped responses are obtainedand undesirable distortions associated with double layer charging and other possible background processes can be reduced. Differential pulse techniques in reverse or cyclic modes are of particular value given that, as shown in Figure 4, they enable simultaneous examination of the reduction and oxidation processes, which is essential to confirm the consistency of the kinetic parameters obtained from the fitting of experimental data. [61e] Also, the value of the large amplitude Fourier-transformed AC voltammetry has been examined. [70b] The analysis of the higher order harmonic responses and the frequency-dependence of the peak heights of the harmonics is predicted to be very powerful and sensitivei nt he study of the applicability of the different kinetic modelsa nd the extraction of kinetic parameters.
It is also worth highlighting that only the aMH model is compatible with the asymmetric, curved Tafel plots obtained experimentally for surface-bound and diffusionalr edox systems, [64] as demonstrated in work by Henstridge et al. [65b] by the fitting of experimental data available in the literature. [64c] Other contrasting behaviours between the kinetic formalisms have been theoretically described and they potentially allow for criticale valuation of the models, though the experimental conditions necessary are challenging. Thus, the sMH and aMH models predict deviations from the Randles-Ševčík behaviour for irreversible processes. [72] The experimental evidenceo fs uch deviations is not straightforward, particularly in the case of diffusional systems, given that it requires the study of systemsw ith small reorganisation energy (unlikely in the case of slow kinetics) in ab road range of scan rates.
Another striking difference between the BV and the MH modelsi st hat in the latter,t he limiting current (in singlea nd double-step chronoamperometry,a sw ella sa tf ast-flow channel electrodes [70a] )c an be smaller than the mass-transport-controlled limit and depend upon the electrode kinetics. [61, 70a] Again, this phenomenoni sp redicted to be more apparent for small values of the reorganisation energy.G iven that this is in general associated with fast kinetics (i.e.,l arge k 0 values), spe- cial attention has been paid to the use of nanosize (including nanodiscs [61d, e] and impacting nanoparticles [73] ), nanogap, [61b, c] and channel [74] electrodes such that the enhanced mass transport shifts the kinetic-controlled voltammetric response away from E 0 f .T hus, it is theoretically possible to observe kineticallylimited steady-statec urrents at large overpotentials in the above systems when the size of the electrode or the gap distance is reduced to the nanometer scale, thoughi np ractice this requires that the geometry of the electrode is accurately known and, in the case of electrodes of af ew nanometers, to deal with double layer and nonclassical effects.

Experimental assessment of the kineticmodels
Ac riticals tudy of the models presented above has recently been undertaken by studying the voltammetric response of variouss olution-phase systems, including the one-electronr eductions of 2-methyl-2-nitropropane, 1-nitropentane, 3-nitrophenolate, cyclooctatetraene, and europium(III), as well as the electro-oxidation of tetraphenylethylene. [61a, 65c] As concluded from Figure 3, in order to observe differences between the BV, sMH, and aMH models, the experimental system must give akinetically controlled current at appreciable overpotentials. This has been achievedb yC omptone tal. for electrode processes with k 0 0.02 cm s À1 by using microelectrodes of 25-50 mm radius, which also allows for the reduction of undesirable ohmic drop and capacitive effects. The use of nanosize [61d, e, 73] or nanogap [61b] electrodes would be necessary for faster electron transfers, which presents difficulties in terms of electrode fabrication and characterisation as well as modeling of nonconventional effects.
The voltammetric response in different techniques (mainly cyclic and square wave voltammetries) of severalo ne-electron transfer processes withoutc hemical complications and under fully-supported conditions has been analyzed making use of the three kinetic models. The sMH model has been unable to fit the experimental voltammetry of systems with transfer coefficientsn otably different from 0.5 (as those chosen in the experimental studies), which is expected in light of the results discussed in Section 2.4. On the other hand, the BV anda MH formalisms yield satisfactory fittings of similar quality,w ith the correlation between the parameters g and a above-mentioned being found experimentally,s uch that a-values different from 0.5 may be interpreted as an indicator of different force constants in the oxidised andr educed species. Note that such differences can also arise from the interactions with the solvent as theoretically demonstrated in Laborda et al. [75] making use of the nonlinear Matyushov solvation model. [76] According to all of the above,f or solution-phase redox couples, the simpler,3 -parameter BV model can be recommended for the fitting of experimental data, complemented with the physicali nsights derived from the asymmetric Marcus model. On the other hand, the analysiso fs urface-bound redoxc ouples should be performed using the asymmetric Marcus-Hush model,w hich is the only theoretical approach (among those considered here) compatible with all the experimental results reported in the literature.

Section 3: Advances in Voltammetric Techniques 3.1 Double potential pulse techniques at microelectrodes
In recent years the use of doublep otential pulse techniques for the studyo fe lectrode kinetics and reactionm echanisms has been developed both theoretically and experimentally at microelectrodes. [77] The combination of pulse techniques and small-sized electrodes offersi mportant advantages in terms of accuracya saresult of the reduction of distorting effects (mainly ohmic drop and charging current), [26,78] which leads to well-defined signals adequate for electrochemical studies even in media of low conductivity.W ith respectt oe lectrode reactions complicated by coupled (electro)chemical processes ( Figure 5), analytical theory for double pulse techniques at microelectrodes of different geometries has been developed for the study of the (pseudo)first-order CE, [79] EC, [79][80] catalytic [81] and equilibrium square [82] mechanismsa sw ell as multistep electrode processes. [83] Analytical expressions for one-electron transfer processes of solution-phase redox systems of any reversibility degree have also been deduced ford ouble potential pulse techniques at (hemi)spherical microelectrodes. The reversibility criteria and methodologies for kinetic analysis are appropriate for other microelectrodeg eometries and will be discussed in the following sections.

Differential double pulse voltammetry (DDPV) and additive differential pulse voltammetry (ADPV)
The subtractive nature of the DDPV and ADPV techniques (introduced in the work of Molina et al. [84] )m aket hem very valuable for quantitative analysis since the influence of background currents is furtherreduced, and peak-shapedr esponses are obtained. The influence of the electrode kinetics on the DDPV and ADPV signals are shown in Figure6.T he single-peak DDPV response and the double-peak ADPV signal shift to higher overpotentials, and the peaks becomes maller and broader (larger half-peak width) as the electrode reaction transitions between the fully-reversible and the fully-irreversible limits. In the latter,t he shape of the DDPV and ADPV signals is independento ft he k 0 -value whereas the positiond oes depend on k 0 .R egarding the influence of the transfer coefficient (a), the peak width increases, the peak height decreases, and the peak potentialt akes more negative values as the a-value is smaller in the case of electroreductionp rocesses. The splitting of the DDPV and ADPV curves of electro-reductions with k 0 -values within the range 10 À3 -10 À4 cm s À1 and very small transfer coefficients( a < 0.3, see Figure6)p redicted at macroelectrodes [85] have also been found at microelectrodes, [77c] thoughitg radually disappears as the electrode size is reduced. Note that in the case of electro-oxidation processes the splitting is predicted for large a-values (a > 0.7). It is also worth mentioning that the splitting is not predictedb yt he symmetric Marcus-Hushk inetic model. [70a] In practice, deviations from the fully-reversible behavior can be detected by comparison of the experimental resultsw ith www.chemistryopen.org those predicted for fast electron transfers with equal diffusion coefficients for the reduced and oxidised species: DDPV with f G (t 2 )b eing at ime function, the form of which depends on the electrode geometry. [86] Note that in Equation 20, it is considered that the arithmetic average (E 1 + E 2 )/2 is chosen for the x-axis potential. [87] With respect to the peak currents, the magnitude of the peaks obtained with positive (DE > 0) and negative (DE < 0) pulse amplitude in DDPV are the same for reversible systems, as well as the heights of the maximum (I M )a nd minimum (I M ) in the ADPV signal ([Eqs. (20), (21)] and Figure 6). [88] These behaviours do not hold forf inite-kinetic electrode reactions. [88] Note that this reversibility criterion may not be conclusive for systemsw here the diffusivities of the oxidised and reduced speciesd iffer significantly.I ns uch cases, even if the electron transfer is reversible,t he position of the DDPV and ADPV signals depends on the double pulse duration and the electrode size andt he values of j DI peak (DE < 0)/DI peak (DE > 0) j and j I M /I m j differ from 1.
The quantification of the electrode kinetics is possible from single-point fitting of the DDPV and ADPV curves. [77a-c] For this, the peak height( in DDPV)a nd maximum current (in ADPV) are more sensitivei nt he case of quasi-reversible processes whereas the DDPV peak potential and the ADPV crossing potential are more appropriate fori rreversible electrode reactions. Thus, from the fitting of the variation of the DDPV peak current and potentialorthe ADPV maximum current and crossingpotential with the duration of the double pulse (t 1 + t 2 ), the kinetics of three electro-reduction processes of different reversibility were successfully determined using mercury micro-hemispheres as workinge lectrodes:3 -nitrophenolate anion in DMSO,3 -nitrophthalate di-anion in DMSO,a nd europium(III) in H 2 O. [77b] The analysiso ft he "first pulse" and "second pulse" components of the DDPV curve has also been recently proposed for the investigation of the electrode kinetics. [89] It is worth notingt hat the influence of the kinetic parameters on the response in differential multi pulse voltammetry (DMPV) is qualitatively analogoust ot hat discussed above for DDPV,a nd that the fitting methodology proposed is also applicable. Nevertheless,q uantitative kinetic analysis of the DMPV curves requires for the use of numerical simulation methods. [90] Thus, although the DMPV method is generally preferred to DDPV given that equilibrium conditions are not recovered after each pair of pulses and so the time of experiments is shorter,t he theoretical modeling and analysis of results are more complex due to accumulative effects. Only for reversible electrode processes or at ultramicroelectrodes are analytical solutionsa vailablef or DMPV. [91] In the case of nonreversible processes at planar electrodes or conventional microelectrodes, the superposition principle is not applicable due to the time-dependence of the surface concentrations, and numerical methods mustb ee mployed to simulate and fit the DMPV signal. [3a] 3.1.2 Reverse pulse voltammetry (RPV) [92] The RPV response under transientc onditions shows ac athodic and an anodic branch (withoutr equiring the initial presence of the product species), and the shape of the RPV curve is greatly affected by the electron transfer kinetics as shown in Figure 7. As k 0 decreases, the RPV voltammogram gradually splits into ac athodic and an anodic wave. Also, when the process is sluggish (k 0 < 10 À3 cm s À1 )a nd the second potentialp ulse is long enough (t 2 % t 1 ), am aximum ("bump") is observed in the anodic wave that is more apparent at large electrodes. [77d] With Figure 5. Illustrationoft he effects of the thermodynamics and kinetics of coupled homogeneous chemical reactionso nd ifferential double pulse voltammetry(DDPV) [79a] (A)and reverse pulse voltammetry (RPV) [80] (B). Grey solid lines correspond to asimple reversible Emechanism.
ChemistryOpen 2015, 4,224 -260 www.chemistryopen.org regard to the transferc oefficient, the a-value affects both the positiona nd slopeo ft he cathodic and anodic branches such that the cathodic wave is steeper and shifts to smaller overpotentials as a takes larger values (Figure 7), the opposite being true for the anodic wave. Therefore, visual inspection of the RPV curve enables us to estimate the electrochemical reversibility of the system, as well as the transfer coefficient. As ummary of the reversibility criteria for DDPV,A DPV,a nd RPV is found in Ta ble 1.
Analogously to the cases of DDPV and ADPV,i ti sp ossible to quantify the electrode kinetics parameters in RPV from the fitting of "singular" points of the curve.T hus, the values of the mid-wave potentials of the cathodic and anodic branches together with their variation with the double pulse duration have been employedw ith success for the kinetic study of the electroreduction processes mentioned in Section 3.1.1.
It is also worth noting that the cathodic and anodic limiting currents of the RPV curves are not affected by the electrode ki-netics, ands ot hey allow for simultaneous determination of the diffusion coefficients of both electroactive species. For this, an electrode of appropriate size mustb ee mployed (in the case of spherical electrodes:2 ffiffiffiffiffiffi ffi Dt 1 p > r 0 > 0:8 ffiffiffiffiffiffi ffi Dt 1 p ), not being possible either at macro-or at ultramicro-electrodes unless the two electroactives pecies are initially present. [93] Therefore, the use of microelectrodes of medium size in combination with the RPV technique enables the determination of the diffusion coefficients and the study of the electrode kinetics in as ingle experiment.

Square wave voltammetry (SWV)
Square wave voltammetry is well-known for its high sensitivity in electroanalysis, and it is also ap owerful technique in the study of electrode kinetics and reaction mechanisms of solution-phase and surface-confined redox systems. [94] SWV includes the benefits of differential pulse techniques along ChemistryOpen 2015, 4,224 -260 www.chemistryopen.org with those of potential sweep methods (fast experiments). The effects of the BV kineticp arameters on the SWV peaks are similar [94a, 95] to those described in Section 3.1.1 for DDPV such that the peaks are smaller,b roader,a nd situated at larger overpotentials as k 0 decreases and a decreases (in the case of electro-reductions) or a increases (for electro-oxidations). Simple and rapid diagnosis criteria for the detection of finite electrode kinetics can be established based on deviations from the SWV signal expected for fully-reversible processes. Thus, the value of the peak current of nonreversible processes will be smaller than that predicted by the following expressions for reversible electrode reactions at disc, (hemi)spherical, band and cylindrical electrodes and microelectrodes under typical SWV conditions (E SW = 50 mV,E s =5mV): Dt p q G ,w ith t being half the square wave period (t = 1/2 f)a nd q G the characteristic dimension of the electrode:t he radius for discs, spheres and cylinders and the half width for bands.
The value of the half-peak width (W 1/2 )i sa lso ap arameter of interestg iven that the W 1/2 -value for one-electron reversible processes is defined only by the SW amplitude (E SW ,[ Eq. (23)], Figure 7. Influence of the Butler-Volmer kinetic parameters on the response of ao ne-electron reduction process in RPV at a( hemi)spherical microelectrode Table 1. Reversibility criteria for the DDPV, [88] ADPV,a nd RPV techniques for electro-reduction processes when the diffusion coefficients of the electroactive species are equal. Note that in DDPV:

Fully reversible
Quasi-reversible Fully irreversible ðÞ ¼gt 1 þt 2 ðÞ at macroelectrodes for agiven t 1 /t 2 value * DIpeak Id;p t2 ðÞ 6 ¼gt 1 þt 2 ðÞ at macroelectrodes for ag iven t 1 /t 2 value at macroelectrodes for agiven t 1 /t 2 value * IM Im 6 ¼gt 1 þt 2 ðÞ at macroelectrodes for ag iven t 1 /t 2 value RPV * One wave * Transition between one and two waves * Twow aves * Bump when t 2 % t 1 ChemistryOpen 2015, 4,224 -260 www.chemistryopen.org W 1/2 being independent of the electrode geometry and frequencyemployed RT is the dimensionless SW amplitude. Thus, in absence of ohmic drop effects, [96] experimental W 1/2 -values larger than those predicted by [Eq. (23)] indicateanonreversible behaviour.T he effect of the step potential( E s )o nt he SWV signal also offersasimple criterion to estimate the degree of reversibility. [97] Thus, whereas the SWV response of reversible systemsi ss carcely affected by E s ,t he SWV response of irreversible reactions varies significantly with E s :t he smaller the E s value, the smaller the SWV peak.
For quantitative analysis of the SWV response of nonreversible electrode processes, numerical simulation methods are necessary, [3a] semi-analytical solutions in the form of as ystem of recursive formulae being also available. [94a] Though frequency-based approaches have been usually considered for the investigation of the electrode kinetics, [94a] Mircevski et al. [94i] have recently proposed an ew approach based on the variation of the SW amplitude rather than the time scale of the scans. According to the new amplitude-baseds trategy,t he electrode kinetics is characterised from the variation of the separation of the peak potentials of the forward and backward components of the potential-corrected SW voltammogram and/ort he peak current of the net response with the SW amplitude (E SW ). This variation is sensitivet ot he kinetic parameters as shown in Figure 8f or the amplitude-normalised peak current (DI peak /E SW ). The amplitude-based methodologies have been applied with satisfactory results to the study of solution-phase and surfaceconfined [3a, 98] redoxsystems.
Cyclic and reverse scan SWV has also been proposed and employed in quantitative kinetic studies. [69,71,99] In the case of quasi-reversible and irreversible diffusional processes (Figure 9), ad ouble peak is observed in the reverse scan at negative (cathodic nature)a nd positive (anodic nature)p otentials with respectt ot he formal potential. The splitting of the peak is more apparent as the electrode size and/or the frequencyi ncreasea nd they separate as the electron transferi s more sluggish (Figure 9). The fitting of the peak potentials, peak widths, and relative peak heighte nables the characterisation of the electrode kinetics.T his approach has been applied to the study of the electroreduction of 2-nitropropane [71] and europium (III) [99a] on mercury electrodes andm icroelectrodes.

Cyclic pulse voltammetries
Softwarep ackages of modern electrochemical instrumentation enable the researcher to "customize" the voltammetric perturbation appliedt ot he system. Within this context, Jadresko et al. [100] have recently proposed two new variants of pulse techniques:c yclic multi pulse voltammetry (CMPV) and cyclic differential multi pulse voltammetry (CDMPV). Thep otentialtime program is analogous to that employed in normal/reverse pulse voltammetry and DDPV,r espectively,w ith the key differences that the perturbation is appliedi na"cyclic mode",a nd that equilibrium conditions are only restored at the end of the experiment. As ar esult, the characterisation of the system is more complete and sound, and the electrochemical measurements are faster.T he resulting signale nables qualitative analysis of the process (including the electrode kinetics) from visual inspection of the voltammograms as well as quantitative analysis from single-point fittings. [100] Figure 8. Influence of the electrode kinetic parameters on the amplitude-based quasi-reversiblemaximum of ao ne-electronreduction at ahemispherical microelectrode corresponding to asolution-phase redox system. w ¼ k 0 = ffiffiffiffi ffi fD p .

Staircase versus analogue cyclic voltammetry
Since the early 90s most commercial potentiostats have been predominantly digital, computer controlled devices. In part due to cost and ease of implementation the basic cyclic voltammetrict echnique provided by these devices involves the application of a'staircase' ramping potential. The use of astaircase waveform for voltammetry was initially proposed as ar oute by which Faradaic and capacitive currents may be more readily experimentally discriminated between (cf. voltammetric pulse techniques). [101] This discriminationisp artially enabled on the basis of the differing time constantsa ssociated with diffusional redox( t À 0.5 ,f or al inear mass-transport regime) and capacitive charging (e Àt ,i nt he heavily simplified RC circuit analogy) processes. However,p rima facie there is no reasont o assume that staircase and analogue cyclic voltammetry are equivalent. Figure 10 depicts the variation of the potentialu sed for 'staircase' (red) and true analogue (black) cyclic voltammetry. For ag iven electrochemical systems tudied via analogue cyclic voltammetry (CV), the measured response is simply af unction of the scan rate (assuming appropriatelyc hosen start, finish, and turning potentials). Conversely, for staircase cyclic voltammetry (SCV), the resulting voltammogram is af unctiono ft he scan rate (step height/step time = Vs À1 ), the step size (E step /V), and the point (or points) at whicht he current is sampled during each step. In the case that the current is sampled once during each step the time at which the current is sampled is expressed as the dimensionless value alpha (a), where an alpha value of one or zero implies the current is sampled at the end or beginning of each step respectively (see inlay of Figure 10). This sampling alpha value bears no relationt ot he transfer coefficient and the two should in no way be conflated!
In the late 80s, Osteryoung published as eries of papers investigating the differences andp ossible equivalences between staircasea nd analogue cyclic voltammetry. [102] Importantly,f or diffusional redox species the use of SCV tends to lead to voltammetricw aves that exhibit larger peak-to-peak separations and suppressed peak heights. It should be recognised that all analytical expressions (the Randles-Ševčík equation for example) and commercially available simulation packages assume the utilisation of an analogue potentialr amp. Consequently, the use of these equations or simulation softwaref or quantification of SCV results can lead to erroneous results.
For ar eversible diffusional process, equivalency (within experimental error;p eak current, I p error < 3%,p eak position within 2mV) between SCV using an alpha value of 1a nd analogue CV reportedlyr equires the use of as tep size of 0.26 mV.
[102a] However,f or diffusional processes this constraint Figure 9. Influence of the electrode kinetic parameters on the reverse scan of the cyclic SWV response of aone-electron reductioncorresponding to asolution-phaseredox system. www.chemistryopen.org may be relaxed through the use of an alpha value of 0.25-0.3, [103] enabling the use of slightly larger step potentials without too significant ad eviation from the results predicted for analogue CV.D epending on the experimental conditions and the equipment used, it mayo rm ay not be possible to select experimental parameters that allow the SCV response to closely approximate that obtained from analog CV. 3 Although the response is improvedb yu sing an alpha of 0.3, for situations in which accuracy is highly pertinent it may be advisable to revert to using true analogue cyclic voltammetry; [104] alternatively one may explicitly simulatet he response accounting for the staircase ramping potential. [105] Recent theoretical studies have investigated the influence of the alpha value in staircase voltammetry for the case in which the diffusion profile at an electrode is transitional between the linear and convergent limits. [105] Moreover,e xpressions for the analytical solution at am icroelectrode for the staircaser esponse of single-, multi-, and catalytic electron transfer processes have also been provided. [81, 86a, 106] The above discussion has focused on the voltammetric response of diffusional redox processes, where for many systems the application of SCV yields qualitative but not quantitative correspondence with the analogue technique. In contrast, for surface-bound species, the obtained voltammetric resultsc an differ profoundly between the techniques. In the mostextreme cases, where the surface speciese xhibits fast electron transfer kinetics, it may arise that the Faradaic charge transfer occurs prior to the measurement point on the step. In this situation the use of SCV may yield av oltammogram that is completely devoid of av oltammetric feature even if the redox speciesi s present.T oe xemplify this point, Figure 11 shows the staircase voltammetric response of cytochrome cp eroxidase on ap yrolytic graphite electrode, where the experimental sampling alpha value has been varied between 0.13-0.9. [107] For situations in which the current is sampled towards the end of as tep, the reversible cyctochromecperoxidase voltammetric wave is not recorded. SCV of as urface-bound feature only becomes equivalent to analogue CV when the scan rate (v)i sg reater than 10 k 0 E step . [107] Subsequently,t he use of staircase voltammetry for the investigation and quantification (in terms of surface coverage) of ar eversibles urface bound process must be approached with caution.T ot his end it is noted that one of the prime examples of such as ystem 'misrepresented' by SCV is encountered with hydrogen UPD on platinum, [108] however other molecular speciesc an encounter similar problems. [109] This is particularly true when investigating surface-bound species as af unctiono ft emperature, where it may be found that at higher temperatures the surface-bound redox-wavei se ssentially 'lost' when using SCV.T his can occur simply due to the increasei nt he electron transfer kinetics as af unction of temperature. One methodb yw hich these problems may be circumvented is by sampling the current continu-ously over the course of ap otential step and averaging the result;f or surface bound species this resultsi navoltammogram closely comparable to that found with the use of analogue CV.D epending on the potentiostat manufacturer,t his technique goes by av ariety of names including 'current integration' and 'surface mode sampling'.F inally,w hen investigating the fundamentals of the electron transfer process of surface-bound species by voltammetric techniques, if as taircase potentialr amp has been used, it is imperative that this is taken into account in the simulations so as to ensure validity of the results. [110] 3.5 Insights into the concept of the diffusion layer thickness The Nernst diffusion layer concept (or linear diffusion layer) provides au seful approacht ot he species concentration profiles andt he diffusive mass transport in electrochemical systems. The thickness of such al ayer, d,i nforms about the extent of the region in solution where concentration changes take place and the efficiency of diffusion under given experimental conditions. This information is essential in digitals imulation of electrochemical experiments, [3a] the evaluation of possible interferences due to convective mass transport [111] and double layer effects, [112] and the design of micro-and nanoelectrode arrays in order to predict the overlapping betweena djacent diffusion domains. [113] Nevertheless, only very recently have the effects of finite electrode kinetics and convergent diffusion on d been investigated. [114] Thus, analytical expressions have been reported for the study of the linear diffusion layer thickness in any voltammetric experiment.

Nonplanar diffusion in any voltammetric technique
In the case of uniformly accessible electrodes, such as (hemi)spheres and cylinders, the diffusion problem can be reduced to as ingle spatial coordinate corresponding to the direction normalt ot he electrode surface, q ( Figure 12). The linear diffusion layer thickness is defined as the distance to the electrode surfacew here the linear concentration profile takes the bulk value c* (Figure 12 a). Accordingly,the d value is given by: Ta king into account that for af ully reversible electron transfer A + e À Ð B, with speciesAand Bh aving equal diffusion coefficients, the surfacec oncentrations only depends on the applied potential, E p ,such that when c B * = 0iti sf ulfilledt hat: where h p ¼

RT
,a nd that the surface concentration gradient can be expressed as follows after as equence of p potential pulses: where c 0 A;surf ¼ c Ã A , t mp = (pÀm + 1)t and f(t mp )i satime function, the form of which depends on the shape of the electrode employed. [86] From Equations 24-26, the following expressions are obtained for the linear diffusion layer thickness at planar,( hemi)spherical, and cylindrical electrodes in any voltammetric technique consisting of as equence of p potentialp ulses of the same duration, t: where: At microdiscs and microbands, the mass transport of species in solution towards/fromt he electrode is not uniform over the whole electrode area. Thus, the mass flux is higher at the electrode edge than at the electrode centre, and the linear diffusion layer thickness has an average character.After the application of asequence of p potential pulses of duration t,the average linear diffusion layer thickness for ar eversible process is given by: Equations 27-29a nd 31-32 enablet he study of the behaviour of the linear diffusion layer for very different electrodes  Figure 13 shows the evolution of the (average) linear diffusion layer thickness in chronoamperommetric (Figure 13 a) and linear sweep voltammetry (Figure 13 b) experiments at microelectrodes of different shape (Figure 13 a) and different radii (Figure 13 b). In both experiments, the thickness of the linear diffusion layer increases as the experiment proceeds and so the duration of the perturbation. Regarding the electrode shape (Figure 13 a), for ag iven r 0 ,t he d values coincidef or any geometry at very short times when diffusion is predominantly planar, with differences between them becoming more apparent with time. Thus, d decreasesi nt he order:c ylindrical > band > spherical > disc, which means that the mass-transport efficiency (current density) follows the inverse order.W ith respectt ot he influence of the electrode size (Figure 13 b), the thickness of the linear diffusion layer in absolutet erms decreases as the electrode shrinks, thought he thickness relative to the electrode radius (i.e. d/r 0 ) increases, and it tends to 1a tm icroelectrodes.
It is also important to mention that d must be taken cautiously as an estimation of the real diffusion layer thickness given that these two magnitudesd iverge very significantly at microelectrodes and nanoelectrodes. Thus, whereas the ratio d real /d is found to be about 2a tm acroelectrodes (planar diffusion), it is about 15 at conventional microelectrodes (a few micrometer-radius), and it tends to 100 at ultramicroelectrodes (steady-state conditions). [114b]

Finite electron transfer kinetics
For the evaluation of the impacto ft he electrode kinetics on the linear diffusion layer thickness, the following expression has been deduced for processes of any degree of reversibility in single potential-step chronoamperometry at (hemi)spherical electrodes of any size: [114b] where: The degree of reversibility hasaprofoundi nfluence on the speciess urface concentrations, which are time-dependent for nonreversible processes (unlike for fast electron transfers, [Eq. (25)]): as well as on the behaviour of the linear diffusion layer thickness, which is potential-dependenta sc an be inferred from Equations 33-36.
The resultso btained from Equation 33 shows that the lineardiffusion layer thickness of nonreversible processes is smaller than forr eversible electron transfers (  When it comest ot he experimental validation of reaction mechanisms (such as the mechanisms discussed in Sections 1 and 2), digital simulations are af requently used tool in today's electrochemical and electroanalytical research. Since simulations can be specifically designed to predict experimental data of ac ertaine lectrochemical system based on an umber of different modelso fu nderlying fundamental processes,t hey may provide data for direct comparison with experimentally-obtained results. Herein, employed simulations alwaysc ombine two models:amodel for charge-transfer processes at solidliquid boundaries and am odel for the mass transport of the analyte. Such interface processes may,f or instance, include electrochemical interactions according to kinetic modelss uch as the above-discussed Butler-Volmer or Marcus-Hushm odels, or other physiochemical processes like adsorption and desorption kinetics, which provide the boundary conditions for the mass-transport problem.I nt he common case that convective processes are negligible, mass transport can be modelled through Fick's second law: [115] @cr; t where c is the concentration and D the isotropic diffusion coefficient of the analyte. This equationc an be solved via an umber of different methods, most prominentlyt hrough finite differences [3a] or finite elements, [3c] which both provide solutionsf or the concentration profile cr; t ðÞ .As o-obtained solution for the concentration profiles of reacting species then allows the calculation of the expected average current across all electrochemical interfaces from the concentration gradient at the respective interface, which is the desired result in most applications. However,t he system's intrinsic noise characteristics cannot be directly modelled through finite-difference or finite-elementa pproaches as such noise characteristics are due to the discrete nature of the analyte, which resultsinastochastic charge transfer across the interface, being particularly relevant at low concentrationsors mall structure sizes.
Concentration profiles can rather be interpreted as probability densities of finding ap articlea tacertain position, but do not allow direct insights into the stochastic nature of the charge-transfer process at the interface. One way to overcome this issue is the use of the random walk method. In this approach,t he pathways,a sw ell as all electrochemical interactions, of each analyte molecule are modelled individually.I nitially at t = 0, all analyte molecule i positionsp 0;i are set by randomly distributing molecules according to the initial concentration profile within the simulated space. The concentration profile can then be writtenint he form: where N A represents the Avogadroc onstant and dðÞ the Dirac delta function. For t > 0t he Dirac delta function, which describes the exact initial positions, can be replaced by the Green's function G of the linear differentialo perator d t þ aD with a 2 R,w hich corresponds to the differentialo perator in the diffusion equation. ThisGreen's function is given by: Gt ðÞ¼qt ðÞ 1 4 pat where q t ðÞ is the Heaviside step function. We then obtain ac ontinuous concentration profile as afunction of time: In order to transform this finding into an exact distribution of molecules, the average displacement dr jj of ap article after ag iven time dt is calculated from the Green's function. Via the investigation of the mean squared displacement of an individual particle, we obtain: for the three-dimensional case. In the one-dimensional case, we calculate: from the mean squared displacement. [116] The temporal evolution of exact particlep ositions that fulfil the diffusion equation can hence be found by substituting each particles spatial probability density function by random displacementa fter discrete time steps of the width dt.M athematically,t his approachcan be expressed as: whereẽ ij is arandom unit vector.F or reasonsofc omputational simplicity,h owever,t he distribution of particles in two-or three-dimensional systemsa re often expressed in terms of independento ne-dimensional average displacements, dx i ,i n each dimension. Based on this assumption, the previous equation is transformed to: where e ijk is ao ne-dimensional random unit vector featuring the values + 1o rÀ1. Since this result provides exact stochastic positions of all active molecules at any time, the noise characteristics of the modelled system can be simulated in great detail andi na ddition to the expecteda verage values that can be obtained from finite differences or finite elements. While random walk simulations offer the advantage of the ability of noise modelling, which is not offered by many other methods, the random walk approach features two main disadvantages. First, since every molecules pathway has to be modelled individually,t he computational effort scales with the number of active molecules in the modelled system. The random walk approachi sh ence not suitable to model high concentrations or large systems. Secondly,t he appropriate definition of boundary conditions may be difficult as the above discussed theoretical justification of the random walk approach implies as ignificant limitation:T he Green's function approach chosen in Equation 41 solely describes the temporal evolution of ad iffusing particle in the absence of diffusion boundaries. In order to model ar eal electrochemical set-up including electrodes and inactive surfaces, approximations must be made.
To illustrate this problem,w ef ocus on the common case of ar andom walker on ao ne-dimensional grid in between two reflecting boundaries;t he approach is,h owever,e qually applicable to three dimensions. In the one-dimensionalc ase, the boundary condition at ab oundary can be formulated in multiple ways;t he mostc ommon definition can be seen in Figure 14 A). The closestg rid point to the boundary is separated from it by dx i =2a nd, during each temporal step dt,t he random walker may either remaino nt his position or perform as tep away from the boundary.I nt his case, the reflection at the boundary has to be divided into two independentf irst passage problems: the diffusive movement to the boundary surfacea nd the movement back to the initial position of this step. Mathematically,h owever,t he expected time of such ar eflection, dt 0 ,d oes not equal the time, dt,a si ti sp resumed in the formulation of the boundary condition. Using Equation 43, we obtain: Such ab oundary condition hence induces an error that scales with the spatiotemporal step width of the randomw alk. In order to reduce this error,t he distance of boundary to the closestg rid point of the random walk must then be corrected to dx' as it can be seen in Figure 14 B): as was discussed by Kätelhçn et al. [117] If, however,c omputational effort is not al imiting factor of the simulation, the simplest way to circumvent the problem of the definition of appropriate boundary conditions to choose as ufficiently small spatials tep width for the randomw alker.S ince the induced error scales with the spatiotemporal step width of the simulation, the deviation from the analytical result will scale with dt and dx i .W hen simulating al arge number of molecules or al ong experiment, this leads to as ignificant increasei nc omputational effort and is therefore often not applicable. In recent years, the noise modelling capabilities of the random walk approachh ave been exploited in an umber of different studies focusing on electrochemical systems. These studies include more fundamentala nalyses of stochastic versus statistic descriptions of diffusion processes [118] as well as the description of experimental systems. Cutress et al. for instance used aG PU-based random walk simulation to investigate cyclic voltammetry [119] and potential-step chronoamperometry [120] at low concentrations, while Kätelhçn and Compton focusedo nt he noise-characteristics of the mediated Faradaic current across an anoparticle impactingo naFaradaically inactive electrode surface. [121] Aside from the modelling of systems at low concentrations, random walk simulations can furtherb e employed to investigate noise characteristics of electrochemical sensors in nanofluidic devices. Hereby, applications include the modelling of methods for single molecule detection [122] as well as the simulation [117,123] of spectra obtained from electrochemicalcorrelation spectroscopy (ECS). [124]

From molecules to nanoparticles
Aside from its application in modelling diffusion of molecular probes, the random walk approach can be used to simulate the Brownian movement of nanoparticles. [125] Here, random walks are particularly helpful,s ince experiments focusing on the mass transport or electrochemistry of nanoparticles, such as nano-impacts, are usually performed at concentrationst hat are sufficiently low to resolve the reactiono fi ndividual nanoparticles. Currents recorded in such experiments are hence www.chemistryopen.org strongly influenced by the stochastic nature of the system, which can be modelled through random walks.
When modelling the diffusion of nanoparticles, the above discussed approach has to be slightly modified in order to account for the nanoparticles' distinct diffusional characteristics that differ from molecular diffusion:D ue to their greater size, nanoparticles are affected by the effect of near-wall hindered diffusion when they approach ad iffusional boundary.T his effect leads to an anisotropic diffusion coefficient distinguishing between diffusion perpendicular and in parallel to the boundary.T he diffusion coefficient D ? in the perpendicular case is then given by: where a is the radius of the particle, h the particle's elevation from the surface, and D 1 the bulk diffusion coefficient. [126] The latter can be approximated well via the Stokes-Einstein equation: where k B is the Boltzmann constant, T the temperature, and h the viscosity of the solvent. [26] The parallel component of the diffusion coefficient D k can be described through: [127] D j The distance-dependency of both diffusion coefficients can be found in Figure 15.
As it can be seen in the plot, perpendicular diffusion slows down near the boundary and eventually vanishes at the surface. Diffusing particles that are located in this area hence spend on average as ignificantly longer time in this zone than in any other zone of equal size in the bulk reservoir.T his effect of hydrodynamic adsorption was recently discussed generally and with respect to the average time of residence that ac atalyticallya ctive particle spends within the zone of electron transfer near an electrode. [125b] Section 5: Modelling Migration and Diffusion The vast majority of electrochemical experiments are carried out in the presence of al arge excess of inert, fully dissociated electrolyte. [78] Thep urpose of this electrolyte is to generate ah igh ionic strength in solution, which will efficiently dissipate the excess charge necessarily introduced into solutionv ia electrolysis,a nd suppress the resultant electric field. There are two main reasonsthat this is normally the case.
The first reason is to prevento hmic drop. [26] Thed riving force behind electron transfer is the potential differenceb etween the electrode and the point in solution where electron transfer takes place, 0 m þ 0 s .T he bulk solution, far from the electrode, has some fixed potential 0 bulk .I ft he potentiald rop between 0 m and 0 bulk occurs over ad istance greater than the electron tunneling distance for electron transfer (outside the zone of electron transfer,Z ET) then the full driving force will not be felt;t he potential difference is lowered as ar esult of ohmic drop. If al arge amount of excesse lectrolyte is added to efficiently dissipate excess charge, the distance over which the drop between 0 m and 0 bulk occurs is compressed to ad istance much smaller than the ZET.T his being the case, the electron transfer is then driven by the maximum potentiald ifference, 0 m À 0 bulk .T his is exemplified schematically in Figure 16.
The solid line shows the potential profile under conditions of high support (a large excess of supporting electrolyte), which is compressedt os hort distances and does not extend very far into the ZET.T he dashedl ine shows low support conditions (small amountso fs upporting electrolyte),w here the  www.chemistryopen.org electric field extends out beyondt he ZET,r esulting in as maller driving force for electron transfer.
Secondly,acompressed electric field will eliminate migratory effects from the mass transport of solutionp hase species. [128] If an electric field extends far beyond the electrode surface, the resultingpotentialgradientinsolution will induce electrical migration of charged species, in addition to diffusion. The majority of analytical theory in electrochemistry assumes diffusiononly conditions, and the presence of migration complicates matters. For these two reasons, excesss upporting electrolyte is usually added to an electrochemical experiment. However, migration effects can offer extra kinetic and mechanistic information unavailable at high support levels. [129] For this reason, it may be desirable to carry out electrochemical experiments where as mall amount,o rz ero, supporting electrolyte is added to solution. For such experimental cases new theoretical modelsa re needed to describe the experiments and allow the experimental electrochemist to interpret results.

The Nernst-Plank-Poisson equations
If migration effects are present in an experiment, Fick's second law alone becomes inadequate to describe mass transport. Instead, the Nernst-Planck equation is used: where c i is the concentration of speciesi(mol m À3 ), t is time (s), D i is the diffusion coefficient of species i( m 2 s À 1 ), z i is the charge on species i( multiples of the electronic charge), F is the Farday constant, R is the gas constant, T is temperature (K) and 0 s is solution potential( V). This equation describes the mass transport of as olution-phase speciesi nt he presence of an electric field.
The solution potential, f s ,i sd escribed using the Poisson equation: where e 0 is the vacuum permittivity (F m À1 )a nd e s is the relative permittivity of the solvent. These two equations together constitute the Nernst-Planck-Poisson system of equations, and subjectt oa ppropriate boundary conditions may be used to model electrochemical experiments in the absence of excess supporting electrolyte. The Poisson equation can be applied to the simulation of weakly supported electrochemical experiments in different ways. The simplest models assume that the electrical double layer is negligible in extent beyondt he electrode, and thus completely exclude it. There are then two main approaches: the electroneutrality approximation and the zero fielda pproximation.

Theoretical treatments neglecting the doublelayer
The electroneutrality approximation simply assumes the solution to be electroneutral at all points: [130] Ap otentialp rofile which satisfies this condition is the mass transport equation and the appropriate electron transfer boundary condition (for cyclic voltammetry this will be the Nernst equation, the Butler-Volmer equation, or Marcus-Hushk inetics, as discussed above). This approximation greatly facilitates analytical solutiono ft he Nernst-Planck equation. [130] Where no such analytical solutions exist and numerical simulation is required, in the case of transient voltammetry for example, this approximation offers little advantage over other methods which do not make the ap riori assumption that the solution is electroneutral at all points.
Am ore rigorous methodi st he zero field approximation. [131] By assuming that the electrical doublel ayer is infinitesimally small, the charge on the electrode surface is completely cancelled in an egligibly small layer of solution immediately adjacent to it.T he electrode surfacea nd this infinitesimall ayer of solution taken as aw hole then has zero charge, and hence zero electric field exists at the electrode surface: This boundary condition may then be used with the Nernst-Planck-Poisson system of equations to numerically generate ap otentialp rofile and species concentration profiles across the solution. This method has been shown to be successful in simulating diversee xperimental data, at both micro and macro electrodes, [104, 129b-g, 132] and is used to obtain theoretical results discussed in sections 5.2 to 5.4.
Neither of these approaches modelst he electrical double layer,a nd both assume it is small enough compared to the depletion layer aroundt he electrode to neglect. This approximation is generally valid for electrodes of radius greater than approximately 10 mm. For electrodes smaller than this, the depletion layer around the electrode will approach the size of the electricald ouble layer,w hichc an then no longer be neglected.

Theoretical treatments including the double layer
Various modelse xist to simulate voltammetry when as ignificant double layer is present.T he simplest assume that electron transfer takes place at ap lane located at af ixed distance from the electrode surface, the plane of electron transfer (PET). This approachl eads to the identification of the Levich and Frumkin effects.
Levich predicted [133] that for aw eakly supported system,i f z A n(where z A is the chargeo nthe speciesundergoing electron transfer and n is the number of electrons transferred from the speciest ot he electrode, which is positive for an oxidation and www.chemistryopen.org negative for areduction)isgreater than zero, then the reacting speciesi se xcluded from the electrode at large overpotentials if as ignificant doublel ayer is present, lowering the current. This leads to the prediction of peak-shaped steady-state voltammetry.Aloweringo ft he current is also predicted by the Frumkine ffect, [134] which predicts ar educed electrochemical rate constant inside ad ouble layer at large overpotentials, if z A n is greater than or equal to zero.
The absence of experimentally observed peak-shaped steady-state voltammetry in many microelectrode systems suggests these pictures are not complete. Rather than assuming electron transfer to occur solely at the plane of electron transfer,D ickinson and Compton [135] developed am odel where electron transfer occurs via ad istance-dependent tunneling mechanism across the diffuse double layer.T his wass hown to dramatically mitigate the Levich and Frumkin effects, and sigmoidal steady-state voltammetry is regained.

How much suppportingelectrolyte is needed?
For steady-state voltammetry,i th as been proposed that ar atio of supporting electrolyte to electroactive specieso f2 6 is sufficient to ensure full support. [136] Dickinson et al. [104] demonstrated that form acroelectrode systems, this is not enough supporting electrolyte and significantly more is needed to avoid the effects of ohmic drop and migration becoming apparent. Ar atio in excess of 100 is shown to be required in some cases. As wella sl arge electrodes, it was shown that fast scan rates and slow diffusion of the electroactive species( all of which lead to more transient voltammetry), as well as slow diffusion of the supporting electrolyte all necessitate ah igher supporting electrolyte concentration than is sufficientf or steady state.
Dickinson's [104] results are summarised in Figure 17, which shows simulated cyclic voltammograms for the reduction of some neutrals peciesAa ta1mmr adius hemispherical electrode and as can rate of 0.5 Vs À1 at various levels of support. The zero fielda pproximationd escribed briefly above was used in the simulations it is seen that even 100 times as much supporting electrolyte as electroactive species is not sufficienti n this case to exactly reproducet he fully supported result. Figure 17 also usefully demonstrates some key features of weakly supported voltammetry.A st he amount of supporting electrolyte is lowered, the ohmic drop results in al arger peakto-peaks eparation and ar educed current.

The effects of weak support
The effect of the chargeb orn by the electroactive species was investigated thoroughly by Belding and Compton. [137] Over the course of ar eduction, negative charge is necessarily introduced into the solution,a nd withoutalarge amount of supporting electrolyte to dissipate this charge it builds up around the electrode. Hence, positively charged species will be attracted towards the electrode, and negative species repelled away from it. This is seen in Figure 18, where the positive species has ah igherl imiting flux than the neutral species due to its electricalmigration towards the electrode. Conversely, the negatively charged species has as maller limiting flux since it is repelled away from the build-upo fn egative chargea roundt he electrode. It is worth noting that if the electroactives peciesi s charged, then it and its counter ion can act as supporting electrolyte,r esulting in a" self-supported" system.T he effect of analyte charge is shown in Figure 18. This figure shows cyclic voltammograms for af ully reversible reduction of some species with ac harge of 0, + 1, and À1a ta2 5m mr adius hemispherical electrode andascan rate of 1mVs À1 ,w ith 1mm monovalent supporting electrolyte present.

Applicationsofw eakly supported voltammetry
As alluded to above, the effects of migration in weakly supported voltammetry may be used to extract valuable kinetic and mechanistic data unobtainablei fe xperiments are carried

Comproportionation in the reduction of anthraquinone
If two successive electron transfers occur as in the following mechanism: then, providing E f 1 > E f 2 E 1 f > E 2 f for reductionsa ss hown (or,i f the electron transfers were oxidations, E f 1 < E f 2 )t hen comproportionation between species Aa nd Cb ecomes thermodynamically (but not necessarily kinetically) favourable: Andrieux and SavØant [138] showed that, under diffusion only conditions, if both electron transfers are fully reversible and all diffusion coefficients are equal, then cyclic voltammetry will be completely insensitivet ot he presence of comproportionation. Hence diffusion only cyclic voltammetry will be inadequate to establish whether or not comproportionation takes place.
Belding et al. [129d] investigated the stepwise two-electron reductiono fa nthraquinone in nonaqueouss olvents at both high and low concentrations of supporting electrolyte to determinei ft he added effects of migration would allow the presense or absence of comproportionation to be determined. Experimental data collected at high support was simulated, and good agreement was seen. Simulations were carriedo ut both in the absence of comproportionation and in the presence of fast comproportionation. The same experimentsa nd simulations werec arried out at low concentrations of supporting electrolyte. When comparing simulatedd ata in the presence and absence of comproportionation, differences were seen, with the second reductive peak significantly reduced in size when comproportionation wasf ast. Comparison of these simulations to experimental data confirmed the presence of fast (k comp > 10 8 dm 3 mol À1 s À1 )c omproportionation between anthraquinone and its dianion, which is impossible to detect using conventional diffusion-only voltammetry.

Ion pairing in electrochemical mechanisms
It is known [22,139] that the reduction of the diphenylpyrylium (DPP) cation follows an EC 2 type electrochemical mechanism, with ion pairing of the DPP radicalo ccurring after reduction: The literaturer eports al arge range of values for k dim from 2.5 10 7 dm 3 mol À1 s À1 to 2.5 10 9 dm 3 mol À1 s À1 . [139] Barnes et al. [129b] investigated this reduction at ar ange of concentrations of tetrabutylammonium tetrafluoroborate supporting electrolyte, and using simulation, were unable to reproduce experimental data across the whole range of supporting electrolyte concentration used. It was found that the experimental peak heights, insteado fl evelingo ff at high support (as in Figure 17), continued to decrease (the peak heighti ncreased with reducing concentration of supportinge lectrolyte due to migration of the positive DPP + ion;see above).
To account for this, af urther mechanistic step was recognised:afast ion-pairing equilibrium of the DPP + ion and the tetrafluoroborate anion,BF 4 À : where K IP (dm 3 mol À1 )i st he equilibrium constant, defined as the ratio of forwarda nd reverse rate constants, k f (dm 3 mol À1 s À1 )a nd k b (s À1 )r espectively.W ith this step included in the mechanism,e xperimental voltammetry wass uccessfully simulated across the whole range of supporting electrolyte concentrations and scan rates used. The concentration of free DPP + wasd ecreased when al arge amount of supporting electrolyte waspresent, and the peak currents therefore decreased. This resolved the literature controversy over the rate of dimerisation, and av alue of k dim = 510 5 dm 3 mol À1 s À1 was established, with an equibibrium constant for ion pairing of K IP = 35 dm 3 mol À1 .T hese observations were only made possible by the ability to simulate cyclic votlammetry at varying concentrations of supporting electrolyte.

The ECE/DISP1mechanism
Am echanism often encountered in electrochemistry,e specially in the reduction of aromatic halides, is the ECE/DISP1 mechanism: [140] A þ e À Ð B À E f 3 ð61Þ B À ! C À k C ð62Þ If E 4 f > E 3 f for the reductions as written, then the disproportination step 64 is thermodynamically viable. The source of the second electron transfer may then be either step 63 (the ECE mechanism) or step 64 (the DISP1 mechanism if step 62 is rate limiting, or the DISP2 mechanism if step 64 is rate limiting). While DISP2 is able to be distinguished from the other mechanisms with relative ease, [141] the discrimination between ECE and DISP1 is difficult. [142] At high concentrations of supporting electrolyte, careful analysiso fv oltammetric wave shapeo ver ar ange of scan rates can in some circumstances be used. [129c] ChemistryOpen 2015, 4,224 -260 www.chemistryopen.org At steady state, however,i ti si mpossible to distinguish the two mechanisms.
If, however,m igration is included as am ode of mass transport through addition of only as mall amount of supporting electrolyte, then discrimination may become much easier through analysiso fp eak heights as af unction of supporting electrolyte concentration, as shown by Barnes et al. [129a] If the chemical step 62 is such that species Ba nd Ch ave the same charge, as written above (for example, an isomerisation), then species Aa nd C, the two possible sourceso ft he second electron in the mechanism,w ill undergo migration to different extents( as written in the mechanism above, speciesAwill not migrate at all, and species Cw ill migrate away from the electrode as negative charge is introduced via reduction). This means that if the mechanism is ECE, and speciesC is the source of the second electron, the peak current will be reduced, since species Ci sr epelled from the electrode. If the mechanism is DISP1, and speciesAi st he source of the second electron, migration will not reduce the current sinces peciesA is electrically neutral. It is therefore possible, in theory,t o assign the mechanism as ECE or DISP1 by carrying out experiments in the presence of both high and low concentrations of supporting electrolyte, and simulating both cases. Only one mechanism shouldg ive ac onsistentvalue of k c over the whole range of supporting electrolyte concentrations used.
Understanding of weakly supported voltammetry has been greatly enhanced in recent years through the use of numerical simulation. Such simulations have been used not only to gain am ore fundamentali nsight into electron transfer processes occurring inside ad ouble layer,b ut have also been appliedt o model experimental data. The latter in particular has allowed for greater insight into chemical processes and the discovery of new,sometimes unexpected, information.This demonstrates the power of this technique.

Section 6: Voltammetry at Rough and Porous Surfaces
Much current experimental electrochemistry is am aterialsbased activity devoted to the search for electrocatalysts that might assist varioust echnologically important electrode processes including, for example, the reduction of oxygen, the oxidation of methanol, the oxidationo ff ormic acid, and the hydrogen evolution reaction. Voltammetricm ethods are widely employed to test the success of the electrocatalysts, generally deployed so as to modify the electrode surface, even though the nature of the cyclic voltammetric experiment is rather different from the conditions in which the catalyst is likelyt ob e employed in, say,afuel cell or battery.
The term 'catalyst' implies ac hange in the rate constant for the process of interest either via ac hange of mechanism or via the lowering of the activation energy within the same mechanism. To explore the role of an electrochemical rate constant, we consider asimple one-electrode process where the net rate of the electrochemical process is given by k 0 is the standard electrochemical rate constant, a is the Butler-Volmer transfer coefficient [9,143] and [X] 0 is the surface concentration of species X. It is well known that as k 0 decreases in size, an overpotential is required to 'drive' the electrode process. In terms of cyclic voltammetry this is revealed by an increasei nt he peak-to-peak separation in terms of the potential as shown in Figure 19 which has been calculated for at ypical macroelectrode, radius 0.15 cm, and av oltage scan rate of 0.1 Vs À1 .T he value of k 0 ranges from 10 À12 to 10 À2 ms À1 ,w hich spans the range from electrochemically irreversible,t hrough quasi-reversible to fully electrochemically reversible, where the term electrochemical reversibility indicates the speed of the electron transfer (k 0 )r elative to the prevailing rate of mass transport (k MT~D /r where Di st he analyte diffusion coefficient). The Randles-Ševčík equation for the voltammetric peak current of ar eversible overall n-electron reduction process is: whilst that of an irreversible process is: where n'i st he number of electrons transferred before the rate-determining step (for which the electron transfer is characterised by a). It is interesting to calculate the ratio of the peak currents for reversible and irreversible processes. For the case of n = 1, a = b = 0.5 and n' = 0, as can be seen from Figure 19 [143] described above. It follows that changing the electrochemical rate constant in this case has ar ather tiny effect on the peak current;t he effects of changed k 0 (and hence electrocatalysis) are best judged by the peak-to-peak separation, although if the reactioni sc hemically irreversible, and the processes for which electrocatalyts are most soughta re typicallyo ft his type, then the absolutep otential of as ingle peak potentialm ight be used but subjectt o the caveats raised below.
The rather low value of a2 7% decrease in current between the electrochemically reversible and irreversible limits relates to the case of n = 1. For the case of an n > 1p rocess, and assuming that the rate-determining step under irreversible condition is the first electron transfer, This is larger (than 27 %) but still relativelym odest.
The insensitivity of the peak current to the rate of electron transfer reflects the fact that the voltammetric peak arises as ac ompetition between the two processes of mass transport (diffusion) and electron transfer.T he calculation above assumed semi-infinite diffusion-only conditions, linear diffusion, and af lat and planar macroelectrode. In the case of convergent diffusion to am icroelectrode, the enhancement of the current, of course,w ill be less since under true steady-state conditions, al imiting current will flow,r eflecting simply the total number of electrons transferred: It is importantt on ext explore what happens if the electrode's surfacer emains that of am acroelectrode but is 'modified' with al ayer of catalysts oa st oc reate ap orous structure on the electrode surface, reflecting many of the electrode modification strategies used in energy research.

Semi-infinitediffusion versus thin layer:qualitative insights
Ap articular important case is when am acroelectrode is modified with ac onductive porous layer made, for example, of carbon nanotubes, graphene, or nanoplatelets. In this situation, the mass transport of solute to the conductive surface (assumed to be both the substrate electrode and the conductive, modifying layer) arises from two components: First, semi-infinite diffusion from the solution bulk to the surfaceo ft he porousl ayer and, second, ac omponent due to diffusion transport within the porousl ayer. [144] If the packing density and thickness of the porous layer is suitably large (but typical) then the transport within the porousl ayer can be approximated as a' thin layer'i nw hich the distance diffused by the solute in order to reach al ocation where it can be electrolyzed is short as compared to that which occurs under semi-infinite diffusion conditions, and hence, the (diffusional) overpotential is markedly reduced reaching, in the limit, apparent reversible 'thinlayer'b ehavior.F igure 20 shows the effect:i tc ompares cyclic voltammetry for at hin-layer system (area 30 cm 2 ,t hickness 1 mm) with that seen for semi-infinite diffusion from as ystem with D = 10 À5 cm 2 s À1 and C bulk = 1mm for as tandard electrochemicalr ate constant of 10 À4 cm s À1 (and a = 0.5). Note the much reduced peak-to-peak separation for the thin-layer case with the contrast from semi-infinite diffusion arising solely from the altered mass transport.
The shift in the peak potential under increasingly thin-layer conditions will ultimately (as the distance required for diffusion is steadily reduced) tend to al imiting value where the potential corresponds to that of the formal potential of the couple of interest. This observation explains the fact that the modification of an electrode with an electrochemically conductive porousl ayer can resolve otherwise overlapping voltammetric peaks. This has potential analytical value [144b] and can simply arise as ac onsequence of the altered mass transport;i ti sn ot necessary to invoke changed electron transfer kinetics.
Finally we point out that under extreme conditions it should be noted that it is possible to see two peaks resulting from as ingle A/B redox couple:o ne is at hin-layer signal andt he other that arising from semi-infinite diffusion to the surfaceo f the porous layer. [144d] Figure 20. Comparison of linear sweep voltammetry using as emi-infinite and thin layer planar diffusion models.F or both models, k 0 = 10 À4 cm s À1 ; D = 10 À5 cm 2 s À1 ; n = 0.1 Vs À1 ; c=10 À6 mol cm À3 .S emi-infinite diffusion electrode area, A = 1cm 2 ;t hin-layera rea, A = 30 cm 2 ;t hickness, l = 1 mm. Reproduced with permission from Ref [144a].C opyright 2008,E lsevier. The effects predicted for thin-layer versus semi-infinite diffusion voltammetry are consistentw ith observations made using ar ange of systems. [144a-c] In particular, there has been very considerable work in using carbon nanotubes (or chemically modified nanotubes) to create porous layers on the surface of electrodes. Theo bserved voltammetry is consistent with the ideas outlined in the previous section (refs. [144a-c] and refs. therein). Figure 21 shows the basic model in which the trapped products of analyte-containing solution act as small thin-layer cells. Ak ey indicator of this behavior is the observation that the peak currents flowing associated with the thin-layer behavior can be significantly larger than those seen for semi-infinite diffusion at an electrode of the same geometric area. Note, as discussed above,s uch large enhancements are not understandable in terms of altered electrode kinetics per se. The work of Henstridge et al. [144b] contains tables of examples of CNT and other modified electrodes in whicht he thin-layer behavior may operate. Similarly Kozub et al. [143] report 'electrocatalytic' systems developed allegedly for the detection of nitrite.

Effective heterogeneous rate constants for rough and poroussurfaces
Numerical simulation [145] has been explored to identify the effective standard electrochemical rate constant for both rough and porouss urfaces using surfacem orphologies such as those shown in Figure 22 and2 3. The former models ad ense, but less than monolayer array of nanoparticles whilst the latter approximates ap orous surface.I nt he former case, the apparent electrochemical rate constant (inferred from the peak potential of the voltammetry) was seen to vary accordingt o where k o is the true electrochemical rate constant,a nd Y is the ratio of the electroactives urfacea rea to the geometric surface area. [145b] In this case the peak potential is given by for ao ne-electronp rocess. Note that this relationship assumes that the adjacent particles are sufficiently close together,t hat on the timescale of the experiment the diffusion field is normalt ot he place of the bulk electrode surface; that is to say,t he diffusion fields of the particles are heavily overlapped, which is usually the case except for very dilute coverages.
In the case of ap orous surface, [145a] where z e is the depth of the pores, r e is their radius and V is the porosity defined by, Figure 21. Schematic of the two types of diffusion that contribute to current at acarbon-nanotube-modified electrode. Reproduced with permission from Ref. [144a]. Copyright 2008,E lsevier.  where r e and r d are shown in Figure 24 as defined by the diffusion domain approximation to the electrode surface. [145] The effects of the roughnessa nd porosity are illustrated in the following subsection but an important general extension relates to the use of rotating disc electrodes to extract the number of electrons (n)t ransferred and the electrochemical rate constant by measuring the transport limited current as af unctiono f disc rotations peed and applying the Koutecky-Levich equation. [146] It was found that this analysis gives correctv alues of n but the apparent, rather than true, rate constanta sd efined above. [146] 6.4 Evaluating oxygen reduction catalysts The resultso ft he preceding section have been applied in particular to ac onsideration of nanoparticle-modified electrodes for oxygen reduction catalysis. [147] Such evaluations often involve simply am easurement of ac urrent at af ixed potential.
Simulations of the type reported in the previous sectionc larified that such currents are sensitive to the surfacec overage of nanoparticles withouta ny change in the fundamentalk inetics or thermodynamic parameters, even if the voltammetry shows that the reduction operates under full diffusional transport control.T he need for caution in the evaluation of catalysts in the manner discussed was evident and an essential need for characterising coverage, porosity,a nd particle size demonstrated for establishing authentic electrocatalytic character.

Voltammetry at thin-layer,nanoparticle-modified electrodes
The previous section concerned electrodes modified with electrochemically conductive layers or films. Am ore general situation hasb een modelled using the scheme summarised in Figure 25, in which the modifying layer itself is nonelectroactive but changes the voltammetric response by virtue of altering the solubilites and diffusion coefficients of the electroactive speciesw ithin the layer as compared to the bulk solution. The electron transfer in this scheme is limited to the surface of the substrate electrode. Both cyclic voltammetry [148] and electrochemicali mpedances pectroscopy (EIS) [149] were modelled;i t was established that the illusion of alterede lectron-transfer characteristics could be generated in each case merely by altered diffusion or solubility.T he difficulty of unambiguously modelling the response was noted.

Section 7: Nanoparticle Voltammetry
As discussed in the previoussection, alterationofthe electrode surfaces tructure can lead to apparent changes in the electron transfer kinetics of an electrochemical reaction;t his arises due to changes in the mass-transport regime local to the interface, and alters what would classically be referred to as the 'diffusional overpotential'. [150] In light of this insight, this section focuses on understanding the voltammetry of electrodes modified with submonolayer coverages of nanoparticles, how true electron transfer rates may be extracted via simulation from the experimental data, and how the diffusion field influences the stripping voltammetry of nanoparticles. Using the strategy outlined within this section, it is now possible to rigorously investigate the presence or absence of 'nano-effects' arising from the use of novel nanomaterials in voltammetric experiments.I nt he final part of this section, an alternative technique for studying nanoparticle electrochemistry is highlighted. This new technique referred to as 'nano-impacts' exhibits an umber of advantages over more conventional electrochemical investigative techniques.  For an array of nanoparticles supported upon an electrochemically inert electrode substrate, the mass transport to and from the nanoparticulate surface, and hence the voltammetric behaviour of the electrode, depends upon:t he nanoparticle's size and morphology,t he diffusion coefficient of the analyte and product, the experimental (voltammetric) time scale, and the interparticle separation (nanoparticle surface coverage). [151] Assuming the substrate electrode is macroscopic in dimensions, then the diffusion regime may be categorised into four cases. Importantly,d uring the course of av oltammetric scan, the prevailing diffusion regime will likely transit from one case to another;c onsequently,i nsight into the voltammetry of such systemsi sb est achieved through simulation. Figure 26 schematically outlines the four diffusional cases or categories.
First, at very short times, the diffusion profile at each individual nanoparticle is linear;t his situation occurs for times of the order of r NP 2 /D (commonly~1 ms, where D~10 À10 m 2 s À1 and r NP~1 0 nm). As the experimental time increases, the diffusion layer grows in accordance with the Einstein equation (d( 2 Dt) 0.5 ); overlap between the diffusion layers of adjacent nanoparticles occurs when the diffusion layer thickness d is comparable to the interparticle separation. Case 2a rises for situations where the nanoparticle diffusion layers do not significantly overlap, but the experimental time is greater than r NP 2 /D.H ere, the mass transport to each nanoparticle is convergent and can be considered independently of adjacent particles. For an isolated sphere on as urface, the steady-state diffusion-limited current is given by; [152] At longerexperimental times, the diffusion layers of adjacent particles will overlap strongly.U nder such situations, classified as case 4, the mass transport to the whole electrode surfacei s linear,r esulting in am acroelectrode response with an associated apparent electrochemical rate constant. The magnitude of this rate constant is af unction of the nanoparticle surface coverage. [145b] Transition between isolated nanoparticle diffusion layers (case 2) and strongly-overlapping layers (case 4) leads to case 3. Case 3i sas ituation regularly encountered with modified nanoparticle surfaces where the diffusionall ayers are partially overlapping and can only be approached through simulation. Af ifth diffusional case may also be considered as an extension of the above model;t his case arises for the situation in which the dimensions of the supporting electrode is only of the order of microns. [153] Under this case, the mass transport to the whole array is convergent.T his category may be viewed partially as ab reakdown of the diffusion domain approximation used in macroelectrode simulation modelsa nd is notably of significance in application to some SECM experiments. [154] Due to the complex interplay between the electrode surface geometry and the diffusion profile as outlined above, extraction of physically significant kinetic data from modified electrodes is an onfacile problem. To experimentally evidence this point, the one electron reduction of chromium(III) [155] has been studied at as ilver-nanoparticle-modified electrode and as ilver macroelectrode, demonstrating clearly how the voltammetric response varies as af unctiono fn anoparticle surfacec overage. It should also be commented that normalisation of voltammetric results relative to the total electroactivea rea is also insufficient to, in many cases, allow even a qualitative comparison of data. [145b] Theoretically this conclusion has been specifically validated in relation to the oxygen reduction reaction at nanoparticle-modified surfaces. [147] Moreover,e xperimentally,i th as been demonstrated how for sparse coverages of platinum nanoparticles on an electrochemical interface, the situation is furtherc omplicated by the release of the hydrogenp eroxide as an intermediate in the reduction process. [156] As discussed within Section1,t his alteration of the electrochemical mechanism, arising from the enhancement of the mass transport to and from diffusionally isolated nanoparticles has large implications not just for the industrial use of such nanomaterials as catalysts, but may also be of importance in the context of the toxicityoft heses ubstances within biological systems. Figure 27 outlines ag eneral strategy for the combined experimental and computation study of electrocatalytic processes at electrodes modified with ensembles of nanoparticles. This methodology allows the influence of the mass transport and the interfacial electron-transfer kinetics upon the voltammetry to be clearly delineated. [157] Briefly,the approachrequires characterisation of the electrode in terms of particles ize, aggregation, ands eparation, allowing the voltammetric response to be simulated using the kinetic parameters obtained using am acroelectrode. From comparison of the simulated 'bulk' response and the experimentally recorded data, it is possible to determine if the kinetics have been altered while fully account- ing for the diffusional mass transport of the material. From this analysist hree outcomes are possible. If the nanoparticle array simulation using the 'bulk' kinetics is in good agreement with the experimentally recorded data for the nanoparticle array, then the conclusion must be that there is no evidence of an ano-effect associated with using the nanomaterial. Conversely, if the simulated voltammetric response differs from that found experimentally,t hen the kinetics of the electrochemicalr eaction must have been alteredt hrough the use of the nanomaterial. This alteration in the kinetics mayl ead to an increase in the overpotential required to drive the reaction; hence,f or such situations, the nanomaterial is less catalytic than the bulk material, and one has evidenced a' negative' nano-effect. Alternatively,i ft he overpotential required for the electrochemical reactioni sdecreased as compared to the simulated result then it can be confirmed that the use of the nanomaterialh as led to an authentic nano-effect. This procedure was applied to the experimental studyo fa rrays of gold nanoparticles ranging in size from 20 to 90 nm in diameter.F rom these experiments, it was confirmed that for nitritee lectro-oxidation,n oa lteration in the kinetics is observed between the use of gold nanoparticles and ag old macroelectrode. [157] Conversely, the electro-oxidation of l-ascorbate was shown to exhibit true nanocatalytic effects. The origins of these differences were ascribed as likely being due to the l-ascorbate oxidation involving adsorbed intermediates. [157] Finally,t his same methodology has been applied to the oxygen reduction reaction and the hydrogen evolution reaction, where for small gold nanoparticles (1.9 nm diameter), the electrochemical processes were found to be significantly hindered as compared to the kinetics recorded on the macroelectrode. [158] This is ap rime example of how decreasing the size of the nanoparticles has led to a' negative' electrocatalytic effect, likelyr esulting from the changed reactioni ntermediate adsorption on the gold surface.

Stripping voltammetry:t he direct oxidation and reduction of nanoparticles
The above examples focus on the situation in whicht he nanoparticle-modified electrode is utilised to study the electro-catalytic properties of the nanoparticles towards agiven redox process. However,t he diffusional cases outlined above must also be considered when studying the direct oxidation or reduction of the nanoparticles themselves-asi sc ommonly undertaken during the course of an electrochemical nanoparticle stripping experiment. Such nanoparticle stripping experiments are purportedly af acile route to directly study the possibly altered thermodynamics of the metallic nanoparticles [159] and their interactions with the electrode substrates. [160] However,p roblems arising in such experimental setups relatingt op artial oxidation or reductiono ft he nanomaterial has been previously notedap roblem that is circumvented by the use of nano-impacts experiments (see final part of this section for further discussion). [161] In nanoparticle stripping voltammetry,f or the case in which the product of the nanoparticle oxidation or reduction is soluble, the diffusion layer associated with the redox product must be considered. Both simulations [162] and analytical expressions [163] have been previously provided demonstrating clearly how,f or the stripping of nanoparticles from an electrode surface, the observed peak potential varies as af unction of the total amount of electroactive material on the electrode surface, the inter-particle distance and the voltammetric scan rate. Moreover,d ue to the sensitivity of voltammetryt owards the mass-transport regime local to the interface, stripping voltammetry can also be used as ar oute to indirectly evidence agglomeration of the nanoparticles upon the electrode surface. [163] In accord with nanoparticle electrocatalysis studies, care mustb et aken in the analysis of the voltammetric response, accounting for the influence of diffusion prior to ascribing the alteredp eak positions as relating to changes in the thermodynamics of the nanoparticles. Finally,t he use of microelectrodes in stripping voltammetry (an analogous diffusion regime to Case 5d iscussed above) may provide one route by which true nanoparticle thermodynamic effects may be more readily investigated, aided by the relatively well-defined steady-state diffusion. [164] Figure 28 depictst he stripping voltammetry for two sizes of nanoparticles at two different surface coverages from ac arbon-fibre microelectrode (r 0 = 5.5 mm), where the observed shift is consistent with ac hange in the thermodynamics due to the influence of the altered surfacee nergy.N ote that the observed peak potential also varies in both cases as afunction of the total surface coverage of silver.N ot all nanoparticle www.chemistryopen.org redox reactions result in the dissolution of the nanoparticle (thought here will be ac orresponding change in the nanoparticle morphology). One example of such ac ase is the oxidation of silver in the presence of ah alide. Depending on the halide concentration, the electrochemical oxidation will likely lead to the formation of surface bound silver halide. [165] As econdary example would be the electrochemical formation of metal oxides. [166] The voltammetric response of such systems is highly complicated;f irst and foremost, the formation (or solubility) constant between the formed nanoparticle ion and the solution-phase counter-ion serves to alter the thermodynamicso f the redox species in accordance with the Nernste quation. [167] Second, the reduction or oxidation may exhibit complex behaviour such as following an ucleation growth mechanism. [168] Third, speciation of the products may vary as af unctiono f counter-ion concentration. [169] Finally,i ns ome cases the reaction may be limited by the mass transport of the counter-ion to the electrochemical interface. Consequently,a gain when studying these systems, care must be taken when ascribing any alteration in the stripping peak potentials as relatingt oa ltered thermodynamics or 'nano-effects.' This is especiallyt rue due to the fact that the nanoparticle capping agentsh ave been shown to influence the observed stripping voltammetry. [161] As an interesting aside, the strength of the binding of silver ions to halides and the corresponding Nernstian shift in the stripping peak yield an analytically useful route to their detection via the use of nanoparticle stripping voltammetry. [167] From the above discussion in the last two sections, it is clear that although not unsurmountable, the interpretation and measurement of physically significant values from the electrochemicalr esponse of electrodes modified with ensembles of nanoparticles is inherently challenging. Consequently,t here is ad esire to find methods by which nanoparticles and their properties can be studied individually.O ne route through which this has been achieved is with the use of so-called 'nano-impact' experimentsa sw ill be discussed in the next and final section.

'Nano-impacts'
In 'nano-impact' experiments the nanoparticles are suspended within an electrolyte and randomly,b yv irtue of Brownian motion,c ollide with ap otentiostated microelectrode. [170] Upon impact the nanoparticle makes electrical contact, and assuming as uitable potentiali sh eld upon the electrode, either the direct electrochemistry of the nanoparticle or ac atalytic process of interest may be induced. The resulting current yields direct and significant information regarding the interfacial redox processes occurring at individual particles.
The use of 'mediated' nano-impacts has been demonstrated for av ariety of systems. [171] One important factor in such experiments is that the electrode receiving the nanoparticles is inert, such that the reaction of study does not occur on its surface in the absence of the nanoparticle. This requirement has led some researchers to use mercury as the 'inert' electrode substrate. [172] However,p roblematically in the presence of trace chloride, mercury is readily oxidised to form calomel nanoparticles. [173] Once formed, these calomel nanoparticles are easily reduced at the mercury electrode and can lead to results which may be misinterpreted. Consequently,i nm any cases, carbon electrodes are found to be more suitable candidates fort he supporting electrode material. An otable exception to this is an early example of the use of nano-impacts for study of the oxidation of hydrazine on platinum nanoparticles at ag old electrode. [174] The difference in the electron-transfer rate on platinum versusg old is sufficientt oa llow the process to be solely studied upon the impacting platinum nanoparticle. It should, however,b en oted that for this system,t he presence of hydrazine in solution does cause significant aggregation of the particles. [175] Figure 28. Anodic stripping voltammetry for small (r np = 3.7 nm, dashed line) and large (r np = 13.5nm, solid line) silver nanoparticles supported on am icro carbon-fibre electrode (r 0 = 5.5 mm): A) low surface coverage of silver (1.6 10 5 mol m À2 )and B) highsurface coverage (1.6 10 4 mol m À2 ). Scan rate:5 0mVs À1 and supporting electrolyte of 0.1 m NaClO 4 .R eproduced with permission from Ref. [164].C opyright 2014, Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim.
ChemistryOpen 2015, 4,224 -260 www.chemistryopen.org One significant observation for impact experiments is that the recorded collisionf requency is not uncommonly below that theoretically predicted. [171b] This observation has previously been explaineds olely in terms of solution-phase agglomeration/aggregation. [175] However,o ther causes for such effects need to be considered. In many experimental cases the working electrodes used tend to be micron sized wires sealed in glass. Consequently,i ft he nanoparticles of study also adhere to the inert (glass) substrate surrounding the electrode, then the frequency of observed nanoparticle impacts may be significantly reduced due to diffusional shielding. [176] Af urtheri ssue arises upon consideration of the dimensions of nanoparticle as compared to am olecule. As ar esult of the geometrical constraintso fa ni mpacting nanoparticle, the 'nano-impact' experiment will be inherently more sensitivet ot he presence of adsorbing and blocking organic media than conventional molecular redoxp robes. [177] Figure 29 depictsasimple geometric model showing how the presence of surface-adsorbed species leads to efficient blockingo ft he electrode surface towards nanoparticles due to the magnitude of the minimum distance (d min )o fa ni mpacting nanoparticle to the blocking molecule. The magnitude of this minimum distance of approachdepends upon both the radius of the nanoparticle (r p )a nd the heighto f the adsorbed species( h b ). Hence, the observed decreased impact frequency may not just be related to agglomeration or aggregation of the nanoparticles in solution, but may also arise due to factorsr elatingt ot he electrode and the electrode design itself.
Although each individual impact experiment (chronoamperogram) is studied at afixed potential, performing repeat experiments at differing potentials can yield significant information regarding the kinetics of the catalytic process on the nanoparticle. [178] Comparison of the kinetics for the reduction of protons at an impacting gold nanoparticle compared to that found from more conventional ensemblem easurements led to the conclusion that the process was significantly slower on the individual impacting gold nanoparticle. [179] This apparent decrease in the kinetics and the notable fluctuations ob-served in the current during the course of an anoparticle impact was interpreted as likely indicating ac ontact resistance between the electrode and the impacting nanoparticle. Consequently,a lthough the presence of the nanoparticles in solution can be evidenced from their catalytic impacts at an electrode, the full potentialoft his methodh as yet to be realised.
Apart from the study of the catalytic response of impacting nanoparticles, in analogy with nanoparticle stripping voltammetry,t he directr edox of the impacting nanoparticle maya lso be studied. The first example of such an experiment was the in situ detection of silver nanoparticles. [180] Upon impacting an electrode with as uitably anodic potential, the nanoparticles were oxidized, resulting in small spikes of current. Through Faraday's first law,t he magnitude of these spikes can be related directly to the total number of atoms contained within the individual impactingn anoparticle, hence providing measurement of its size. [161,181] Furthermore, the frequency of the impactings pikes can yield informationr egarding the concentration of the nanoparticles in solution. [182] This new nano-metrology technique has been successfully applied to ahost of different nanoparticle materials including other metals, [183] metal oxides, [184] carbon fullerenes, [185] organic nanoparticles, [186] and liposomes. [187] Work has also demonstrated that the technique is suitable for use in complex media such as sea water,h ighlighting its potential use in environmental analysis. [188] To this end the use of carbon-fibre microcylinder electrodes has been advocated as one route by which subpicomolar concentrations of nanoparticles may be readily detected and sized, [189] the secondary advantage of this methodology is the minimisation of problemsa ssociated with nanoparticle adsorption onto the electrode support.
Beyond being an analytical technique, the direct nanoimpact method also provides ar oute for investigating more fundamental problems. Af irst example is the use of the technique for the study and monitoring of solution-phase nanoparticle agglomeration and aggregation. [190] In as imilarv ein, the magnetic-field-induced agglomeration of Fe 3 O 4 has also been directly evidenced. [191] Figure 30 depicts examples of iron oxide reduction spikes and the associated size distributionso btained in the presence and absence of am agnetic field. Second, the electron-transfer kinetics and mechanism to the nanoparticles can be studied. [183b, 192] Third, the interaction of the nanoparticle with the electrochemical double layer is of utmost importance and the nano-impacts methodology has provided ad irect route by whichthese interactions can be probed. [193] Consequently,w ec onclude that the use of 'mediated' and 'direct' nano-impact experiments have significant potential for future research both as nano-metrology techniques and as novel methodsf or addressing fundamental and technological nanoparticle challenges.

Conclusion
The above survey shows that with the era of numerical simulation, the technique of voltammetry has come of age and has power to contribute very significantly,n ot least in aq uantitative manner,t om any problems in analytical, physical, and bio-