Article

pubs.acs.org/JPCC

Two-Electron Transfer Reactions in for Solution- Soluble and Surface-Confined : A Common Approach Manuela Lopez-Tenes, Joaquin Gonzalez, and Angela Molina* Departamento de Química Física, Facultad de Química, Regional Campus of International Excellence “Campus Mare Nostrum”, Universidad de Murcia, 30100 Murcia, Spain

ABSTRACT: In this paper, the general characteristics of the normalized voltammetric response for reversible two-electron transfer reactions (EE mechanism) is analyzed and particularized to the application of derivative normal pulse voltammetry (dNPV) using electrodes of any geometry and size, and cyclic voltammetry (CV), when the undergoing the process is soluble in solution and surface-confined, respectively. The analysis is based on the close relationships between the electrochemical response and the theoretical values of surface concentrations/excesses, and has led to the voltammetric signal − − − of the EE mechanism being interpreted in terms of the percentage of E1e E1e , and E2e character, as a function of the difference between the formal potentials of both electron Δ 0′ 0′ − 0′ − “ ff transfers, E = E2 E1 . In line with the percentage of E2e character, the term e ective ” electron number , neff, has been introduced and related to the probability of the second electron being transferred in an apparently simultaneous way with the first one and a direct 0′ 0′ Δ 0′ method to obtain the values of E1 and E2 for any E has been proposed. The key role of ΔE0′ (in mV, 25 °C) = −142.4, −71.2, −35.6, and 0 values in the behavior of the peak parameters of the voltammetric curves is explained in terms of the usual terminology (transition 2 peaks−1 peak, repulsive− attractive interactions, anticooperativity−cooperativity, and normal-inverted order of potentials). The EE mechanism is also compared with two independent E mechanisms (E+E).

1. INTRODUCTION to/from the electrode surface and when the molecules are fi Electrode processes consisting of two-electron transfers (EE surface-con ned, so the transport is avoided. Traditionally, mechanism) have been widely treated in the literature, both in these two fundamental problems in electrochemical science are − ff their theoretical and applied aspects.1 9 This high productivity treated as completely di erent problems. measures in some way the great presence and relevance of these In this paper, both possibilities are tackled in the case of a processes in many fields, and hence the importance of reversible EE mechanism, in order to establish the correspond- understanding them. This behavior is very common in ences between their electrochemical responses and the electrochemical reactions of alkylviologens and metallocenes, conditions under which a common study of these situations in the reductions of several metallic ions, of polyoxometallates can be made. Thus, it has been shown that, with the and of a number of aromatic species like derivatives of appropriate normalization in each case, the electrochemical − tetraphenylethylene.1,2,4,8 13 In the specific case of biological signal obtained when normal pulse voltammetry (NPV) is molecules, such as oligonucleotides, , enzymes, applied in the case of solution soluble species for electrodes of any geometry and size is coincident with the charge transferred- etc., the application in recent years of techniques such as fi protein film voltammetry (PFV), combined with scanning potential one for surface-con ned systems in any electro- probe microscopic techniques, has made it possible to chemical technique. Thus, their derivatives lead to a common − normalized peak-shaped voltagram in derivative normal pulse characterize the biomolecule electrode interface and elec- 1,2,4,14−30 tron-transfer processes in great detail, which is fundamental to voltammetry (dNPV), for solution soluble molecules, ffi and in cyclic voltammetry (CV), for surface-immobilized exploit the naturally high e ciency of these biological systems 1,3,5−9,31−36 in modern biotechnology (selective last-generation biosensors, ones. These normalized responses are only environmentally sound biofuel cells, heterogeneous catalysts, potential-dependent as a consequence of the reversible 3,5−7,9 behavior of the process, which is reflected in the independence biomolecular electronic components). Many of these − of time of surface concentrations (solution phase case)15,24 26 systems present a reversible behavior (or it can be reached 31−36 acting on the adequate experimental parameter in the particular and excesses (immobilized species). electrochemical technique used), which simplifies the study of Thus, the parallel characteristics of the common normalized multielectron transfer processes by not having to consider the voltammetric curve in dNPV and CV and the surface kinetics of the electron-transfer reactions. When studying the EE mechanism, two situations for the Received: March 14, 2014 molecules undergoing the process can be encountered: when Revised: April 30, 2014 they are soluble in the solution and then need to be transported Published: May 12, 2014

© 2014 American Chemical Society 12312 dx.doi.org/10.1021/jp5025763 | J. Phys. Chem. C 2014, 118, 12312−12324 The Journal of Physical Chemistry C Article ̂ Table 1. Expressions for Diffusion Mass Transport Operators, δj (j =O, I, or R), Functions f (q ,t) and f for the Main a G G G,micro Electrode Geometries ff δ̂ electrode di usion operator j f G(qG,t)(Dj = D) f G,micro plane ∂ ∂2 1 − D 1/2 ∂t j ∂x2 ()πDt π 2 sphere (radius rS, AS =4 rs ) ⎛ 2 ⎞ 11 1 ∂ ⎜ ∂ 2 ∂ ⎟ + − Dj + 1/2 ∂t ⎝ ∂r2 rr∂ ⎠ rs ()πDt rs π 2 ⎛ ⎞ ⎞ disk (radius rd, Ad = rd) ∂ ∂2 ∂ ∂2 ⎛ ⎛ ⎞ 41 ⎜ 1 ⎟ 41 rd rd ⎟ − Dj + + ⎜0.7854++− 0.44315 0.2146 exp⎜ 0.39115 ⎟⎟ ∂ ⎝ ∂ 2 ∂ ∂ 2 ⎠ 1/2 1/2 π rd t r rr z π rd ⎝ ()Dt ⎝ ()Dt ⎠⎠ band (height w, length l, A = wl) ⎛ 2 2 ⎞ 11 12π w ∂ ∂ ∂ +<2 − D⎜ + ⎟ 1/2 ifDt / w 0.4 2 ∂t j⎝ ∂xz2 ∂ 2 ⎠ w ()πDt w ln[64Dt / w ] ⎛ ⎞ ⎛ π ⎞1/2 ()πDt 1/2 π 0.25⎜⎟ exp⎜−+ 0.4 ⎟ ⎝ ⎠ ⎛ 1/2 ⎞ Dt ⎝ w ⎠ ()Dt wln⎜⎟ 5.2945+ 5.9944 ⎝ w ⎠ ifDt / w2 ≥ 0.4 cylinder (radius rC, length l, AC = ⎛ 2 ⎞ ⎛ 1/2 ⎞ 12 π ∂ ⎜ ∂ 1 ∂ ⎟ 1 ⎜ ()πDt ⎟ 1 2 rcl) − Dj + exp−+ 0.1 2 ∂ ⎝ 2 ∂ ⎠ 1/2 ⎝ ⎠ ⎛ 1/2 ⎞ rc ln[4Dt / rc ] t ∂r rr ()πDt rc ⎜⎟()Dt rc ln 5.2945+ 1.4986 ⎝ rc ⎠ a 28,29,37 qG is the characteristic dimension of the electrode: rs for spheres or hemispheres; rd for discs; w for bands; rc for cylinders. concentrations/excesses have been highlighted. On the basis of inverted order of potentials.4 Also, it has been discussed the this study, the variation of the voltagrams shape depending values of ΔE0′ for which two and one peaks appear in the upon the relative values of the formal potentials of both charge- voltammetric curve, and the different evolution of one-peak transfer reactions in the EE mechanism, as expressed by ΔE0′ = curves with ΔE0′, in terms of the intermediate stability. 0′ − 0′ E2 E1 , has been interpreted in terms of the percentage of For completeness, the EE mechanism has been compared − − − consecutive (E1e E1e ) and apparent simultaneous (E2e ) with the case of two independent one-electron-transfer characters of the electron-transfer process, which have been reactions with identical initial concentrations/excesses, which intrinsically related to the surface concentrations/excesses has been called an E+E mechanism in this article.1,33 It has been values of the intermediate and the sum of the two extreme shown that whenever two peaks are obtained in the oxidation states of the molecule, respectively, at the average voltammetric response of an EE mechanism (ΔE0′ < −71.2 0′ 0′ 0′ formal potential, E̅ =(E1 + E2 )/2, and, more practically, with mV) it can be considered practically indistinguishable from that the current value at E̅0′ (peak or valley) in the voltagram. From for an E+E process. these concepts, the term “effective number of electrons ” transferred , neff, has been introduced, which varies between 1 2. THEORY Δ 0′ ≪ − for E 0 (0% character E2e , very stable intermediate) and Consider an electrode process in which a molecule reduces Δ 0′ ≫ − 2for E 0 (100% character E2e , very unstable reversibly involving two-electron transfers, according to the intermediate) and has been related to the probability of the following reaction scheme (EE mechanism) second electron being transferred in an apparently simulta- 0 neous way with the first one. The values of n for any ΔE ′ −′ eff Oe+⇄ IE 0 have been discussed and compared with the “apparent number 1 of electron transferred”, n , extensively used in bioelectro- −′0 app Ie+⇄ RE2 (I) chemistry, and defined in the literature for ΔE0′ ≥−35.6 mV values, i.e., for cooperative behavior between the electron 5,32 in which O (oxidized), I (intermediate, or half-reduced) and R transfers. Related to the above ideas, a simple direct method (reduced) refer to the different states of the molecule and 0′ 0′ for obtaining the individual formal potentials, E1 and E2 , E0′ and E0′ are the formal potentials of the first and second Δ 0′ 1 2 regardless of the E value, has been proposed. steps, respectively. The average formal potential, E̅0′, given by In this study of the two-electron transfer reactions the importance of the ΔE0′ (in mV, 25 °C) = −142.4, − 71.2, 0′′0 0′ EE1 + 2 −35.6, and 0 values (K = 1/28, 1/24, 1/22, 1/20, respectively, E̅ = 2 (1) with K being the disproportionation constant and 2 the number of electron transfers) in the behavior of the peak parameters of plays an essential role in the study of the process since it is the the voltammetric response (peak potentials, peak heights, and formal potential for the reaction half-peak widths) is noteworthy. This capital role has also been explained in terms of the surface concentrations/excesses of the O2eR+⇄− (II) species, establishing the relation between them and the different terminologies that are used in the study of processes i.e., the sum of both steps in scheme (I). Also, a key parameter ff Δ 0′ fi with two-electron transfers: 2−1peaksinthere- is the di erence between the formal potentials, E ,de ned as 14,18,31−36 − − sponse, negative positive (repulsive attractive) in- ′′′ 1 5,32 0 0 0 teractions, anticooperativity−cooperativity, and normal- ΔEEE=−2 1 (2)

12313 dx.doi.org/10.1021/jp5025763 | J. Phys. Chem. C 2014, 118, 12312−12324 The Journal of Physical Chemistry C Article ⎛ ⎞ which determines the stability of intermediate I in scheme (I). F 0′ Δ 0′ ≪ J =−=exp⎜⎟ (EE ) i 1, 2 Indeed, species I is totally stable for E 0 and totally i ⎝ RT i ⎠ (9) unstable for ΔE0′ ≫ 0.1,2 A brief outline follows of the most important insights where qsurface is the value of the normal coordinate to the ff ffi involved in the development of the theory concerning the electrode at the surface, Dj is the di usion coe cient of species “ ” reversible EE mechanism in order to establish the correspond- j (O, I, or R), the superscript s refers to the value of ences between the voltammetric responses obtained when the concentrations at the electrode surface, and F, R, and T have species in scheme (I) are soluble in the solution and when they their usual meaning. From eqs 7 and 8 it follows that are immobilized at the electrode surface forming a monolayer, s CO 2 so that a common study of both physical situations is carried s = J ̅ out in the Results and Discussion. CR (10) 2.1. Solution-soluble Species. Normal Pulse Voltam- with metry at an Electrode of Any Geometry. The first case ⎛ F ⎞ considered is that in which the molecule undergoing the ⎜ 0′ ⎟ J ̅ =−exp⎝ (EE̅ )⎠ process is soluble in solution, assuming that it is initially present RT (11) * only in its totally oxidized state, with concentration cO (see and scheme (I)). When a constant potential E is applied to an s s ⎛ ⎞ electrode of any geometry, the following solution-phase CCO R F 0′ − =Δ=exp⎜⎟E K reaction related to process (I)1,4,14 16,18,23,24,27 s2 ⎝ ⎠ ()CI RT (12)

k1 where K gives the value of the equilibrium constant (= k1/k2) 2IOR⇄+ ≤ ≤∞ k2 (III) for the reaction in scheme (III), being 0 K . NPV The current, IEE , can be expressed as the sum of I1 and I2 for can take place in the vicinity of the electrode. k1 and k2 are the steps 1 and 2, respectively, in reaction I rate constants of the disproportionation and comproportiona- NPV tion reactions, respectively. IEE =+II12 (13) ff ff If di usion is the only transport mechanism, the di erential with equation system to be solved for process (I), taking into account the reaction scheme (III) is given by ⎛ ∂ ⎞ *⎜ CO ⎟ I1GOO= FA D c 2 ⎫ ⎝ ∂q ⎠ ̂ q δOOCkCkCC=− 1I 2OR ⎪ surface (14) ⎪ ̂ 2 ⎬ ⎛ ⎞ δIICkCkCC=−22 1I + 2OR ∂C ⎪ I =−FA D c*⎜ R ⎟ ⎪ 2GRO⎝ ⎠ δ̂ CkCkCC=−2 ⎭ ∂q RR 1I 2OR (3) qsurface (15) δ̂ fi where j is the mass transport operator for the geometry where AG is the electrode area for the speci c geometry considered, given in Table 1, and the normalized concentration, considered (see Table 1). fi fi Cj(q, t), is de ned as From eq 3, it is clear that the concentration pro les of species O, I, and R are dependent on the kinetic of the cqtj(, ) disproportionation/comproportionation reaction, so the cur- Cqtj(, )= j = O, I, R − c* (4) rent given by eqs 13 15 will also depend on it. To avoid this O latter dependence, equal diffusion coefficients are assumed for where q and t refer to the values of a set of coordinates the three species, i.e., DO = DI = DR = D, so, taking into account characteristic of the given geometry and of time, respectively. eq 6, it is fulfilled that The boundary value problem is given by NPV ⎛ ∂+⎞ ⎛ ∂+⎞ IEE ⎜ (2CCOI ) ⎟ ⎜ (2CCRI ) ⎟ =≥ ⎫ = =− tqq0, surface⎪ FA Dc* ⎝ ∂q ⎠ ⎝ ∂q ⎠ ⎬ = * == = GO qq ⎪CCOO1, C I 0, C R 0 surface surface tq>→∞0, ⎭ (5) (16) t >0,q = q : Note that for both lineal combinations of concentrations in surface eq 16, i.e., for ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ∂CO ∂CI ∂CR ⎫ D ⎜ ⎟ + D ⎜ ⎟ + D ⎜ ⎟ = 0 Wqt(, )=+ 2 COI (, qt ) C (, qt )(a)⎪ O⎝ ∂q ⎠ I⎝ ∂q ⎠ R⎝ ∂q ⎠ ⎬ qq q ⎪ surface surface surface Xq(, t )=+ 2 C (, q t ) C (, q t )(b)⎭ (6) RI (17) and thus for C s O = J s 1 Y(,qt )=++ COIR (, qt ) Cqt (, ) C (, qt ) (18) CI (7) it is fulfilled that (see eq 3) C s I ̂ ̂ ̂ (19) s = J2 δYWX= δδ= = 0 CR (8) ̂ The boundary value problem for δY = 0 in eq 19 is given by with (see eqs 5 and 6)

12314 dx.doi.org/10.1021/jp5025763 | J. Phys. Chem. C 2014, 118, 12312−12324 The Journal of Physical Chemistry C Article ⎫ tqq=≥0, ⎪ where the function fG(qG,t), which only depends on time and surface⎬ * ⎪YY= = 1 on the particular electrode geometry, is given in Table 1 and is tq>→∞0, ⎭ (20) the same as that obtained for a one-electron transfer reaction (E mechanism).28,29,37 Thus, eq 28 shows that the current is ⎛ ⎞ ∂Y not influenced by the homogeneous reaction in scheme (III). t >=0,qq :⎜ ⎟ = 0 ff surface ⎝ ∂q ⎠ By taking into account the expression for the di usion- q NPV 37 surface (21) controlled limiting current for a simple E mechanism, IE,lim: and the solution of this problem leads easily to NPV = * IE,lim FAGO Dc fG (,) qG t (29) Y(,qt )=++= COIR (, qt ) Cqt (, ) C (, qt ) 1 (22) Equation 28 can be written as (see eqs 17b and 23) 24,28 regardless of the electrode geometry considered. Therefore, INPV 2KJ1/2 + ̅ NPV ==EE ==s s +s from eq 22 for q = qsurface, and the nernstian conditions 7 and 8 IEE,N NPV 1/2 1/2 2 XCC2 R I (or their equivalents, eqs 10 and 12), the following expressions IE,lim KJKJ+ ̅ + ̅ s for the surface concentrations, Cj (j = O, I, R), written in terms (30) ̅0′ Δ 0′ NPV of the average formal potential, E (eqs 1 and 11), and E Thus, the normalized current, IEE,N, given by eq 30, is NPV − (eqs 2 and 12), are obtained: independent of time and electrode geometry. Since the IEE,N 1/2 2 ⎫ E response is in form of waves, it is more appropriate to use the s KJ̅ ⎪ derivative of the current in order to have a peak-shaped CO = KJKJ1/2+ ̅ + 1/2̅ 2 ⎪ response. So, in derivative normal pulse voltammetry (dNPV), ⎪ ψdNPV ⎪ the normalized EE is given by (see eq 30) s J ̅ ⎬ CI = NPV 3/2 2 2 3/2 3 1/2 1/2 2 RT dIEE,N KJ̅ + 4 KJ̅ + KJ̅ KJKJ+ ̅ + ̅ ⎪ ψ dNPV == ⎪ EE 1/2 2 2 1/2 F dE ()KKJKJ+ ̅ + ̅ s K ⎪ ⎛ s s ⎞ C = ⎪ RT dCR dCI R 1/2 1/2 2 ⎭ =+⎜2 ⎟ KJKJ+ ̅ + ̅ (23) F ⎝ dE dE ⎠ which are independent of the existence of the disproportiona- KJ3/2̅ + 2 KJ 2̅ 2 −KJ3/2̅ + KJ 3/2̅ 3 = + tion/comproportionation reaction. From eq 23, the surface 2 1/2 2 2 1/2 2 2 concentrations at the particular values of the potential E = E̅0′ ()()KKJKJ+ ̅ + ̅ KKJKJ+ ̅ + ̅ (31) ̅ 0′ ̅ 1/2 0′ ̅ 1/2 (J = 1), E = E1 (J = 1/K ), and E = E2 (J =K ) have the ψdNPV − 0′ EE is an even function of (E E̅ ); i.e., the function takes form the same values by changing J̅ for 1/J.̅ Thus, at E = E̅0′ the 1/2 function presents a maximum (peak) or a minimum (valley) s s K s 1 ==00′′= (CCO R)EE̅ 1/2 ;(CI )̅ 1/2 depending on the value of K (see the Results and Discussion), 12+ K 12+ K (24) which has the value (see eq 31 with J̅ = 1 and eq 24)

s s s s 1 1/2 ⎛ s ⎞ (CC====)(0′′ CC )0 ; dNPV 2K RT dCR s O I E1 I R E2 (ψ ) 0′ = ==2 ⎜ ⎟ 2(C ) 0′ 2 + K EE E̅ 1/2 ⎝ ⎠ R E̅ 12+ K F dE E̅0′ s s K ()CCR EE0′′== ()O 0 s s 1 2 2 + K (25) =+()CCO R E̅0′ (32) s 0′ s Once the surface concentrations are known, the expression Note that (dCI/dE)E̅ = 0 since CI is also an even function for the current can be obtained. Indeed, the solution of the of (E − E̅0′) (see eq 23) and presents a maximum at E = E̅0′,as ̂ differential equation δW = 0 (eq 19), with the boundary can be expected, whose value is given by eq 24 (see dotted s conditions (see eqs 17a, 5, and 23) black curves for CI in Figure 2 below). For E = E0′ (J̅ =1/K1/2) and E = E0′ (J̅ = K1/2), ψdNPV takes =≥ ⎫ 1 2 EE tqq0, surface⎪ the same value; thus, from eq 31 ⎬WW= * = 2 tq>→∞0, ⎭⎪ ⎛ s s ⎞ (26) dNPV 15+ K RT dCR dCI (ψ ) 0′ = =+⎜2 ⎟ EE E1 2 ⎝ ⎠ (2+ K ) F dE dE 0′ s s s E1 t >=0,qq : W =+ 2 CO CI surface 3K 1 − K 1/2 2 = 2 + 2KJ̅ + J̅ 22 = (2+ K ) (2+ K ) (33) 1/2 1/2 2 KJKJ+ ̅ + ̅ (27) ⎛ s s ⎞ dNPV 15+ K RT dCR dCI (ψ ) 0′ = =+⎜2 ⎟ leads to the following expression for the current as a EE E2 2 ⎝ ⎠ s 24,28 (2+ K ) F dE dE 0′ consequence of W only depending on the applied potential E2 (see eqs 16, 26, and 27) 12+ K −+1 K = 2 22+ INPV (2+ K ) (2+ K ) (34) EE = ()(,)WWfqt* − s * G G Δ 0′ − FAGO Dc For K =1( E = 0 mV), eqs 32 34 are coincident, being dNPV (ψ ) 0′ 0′ ̅0′ = 2/3. 2KJ1/2 + ̅ EE E1 = E2 = E = 1/2 1/2 2 fqtG (,)G If ultramicroelectrodes are used, when the stationary state is KJKJ+ ̅ + ̅ (28) attained the current is independent of time, and thus, eqs

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− fi dQ 28 34 are ful lled for any potential-time waveform CV RT EE,N dE applied.28,29 In this case, the function f (q ,t) (eqs 28 and ψ = G G EE vF dE dt 29) becomes f (see Table 1). G,micro 3/2 2 2 3/2 3/2 3 2.2. Surface-Immobilized Species. Cyclic Voltamme- KJ̅ + 2 KJ̅ −KJ̅ + KJ̅ = 2 + try. In this case, it will be assumed that there are no ()()KKJKJ+ 1/2̅ + ̅ 2 2 KKJKJ+ 1/2̅ + ̅ 2 2 interactions between the immobilized molecules, that hetero- (41) geneity of the electroactive monolayer can be ignored, and that ψCV no desorption takes place in the time scale of the experiment, where v =dE/dt is the sweep rate. Equation 41 for EE for so the total excess Γ is constant and independent of the surface-immobilized molecules has, logically, the same ex- T ψdNPV potential during the whole experiment for any potential-time pression as that for EE for solution-soluble species (eq 31). − fi waveform applied to the electrode. Thus, if the molecule is Thus, parallel expressions to eqs 32 34 are also ful lled for CV CV CV ψ 0′ ψ 0′ ψ 0′ initially present only in its totally oxidized state (see eq I), with ( EE )E̅ ,( EE )E1 , and ( EE )E2 , respectively. Γ* Γ fi excess O = T, the surface coverages, f j (j = O, I, or R), de ned Γ Γ Γ fi as f j = j/ T, where j is the surface excess of species j, ful ll 3. RESULTS AND DISCUSSION that After discussion of electrochemical techniques for a reversible f ++ff =1 EE mechanism with species soluble in solution for electrodes of OIR (35) any geometry and size, and immobilized at the electrode surface, the characteristics of the common normalized peak- Equation 35 is equivalent to eq 22, obtained for solution- shaped response for CV, ψCV, in surface-confined processes (eq soluble species. Thus, by combining eq 35 with the nernstian EE 41) and for dNPV, ψdNPV,indiffusion-controlled processes (eq conditions f /f = J and f /f = J (see eqs 7 and 8), the EE O I 1 I R 2 31), are now revised and summarized. Also, several aspects of expressions for surface coverages f , f , and f are obtained, O I R the reversible behavior have been analyzed. In the following which are obviously identical to Cs , Cs, and Cs , respectively, in O I R discussion, the superscript CV and dNPV in ψCV and ψdNPV eq 23. EE EE have been removed, and we refer only to ψ . From this result, the charge Q transferred for the EE EE EE It is well known that the EE mechanism behaves as two mechanism is straightforwardly obtained as the sum of Q1 and 0 33 independent one-electron E mechanisms for ΔE ′ ≪ 0(K → Q for steps 1 and 2, respectively, in the scheme (I) 2 − − − 0) E1e E1e , see scheme (I), with two peaks centered at the 0′ 0′ − individual formal potentials, E1 and E2 , and as a two- Q EE =+QQ12 (36) Δ 0′ ≫ → ∞ − − electron E mechanism for E 0(K ) E2e , see with scheme (II), with a single peak centered at the average formal potential, E̅0′ −.1,2 Hence, in this paper, the case of two simple τ ⎛ df ⎞ independent E mechanisms has also been considered for =− Γ⎜ O ⎟ = Γ − Q1 FA T ∫ dt FAT(1 fO ) comparison, i.e. 0 ⎝ dt ⎠ (37) −′0 Oe111+⇄ RE τ ⎛ ⎞ −′ dfR 0 =Γ⎜ ⎟ =Γ Oe222+⇄ RE (IV) Q2 FA T ∫ dt FAT fR 0 ⎝ dt ⎠ (38) k1 where A is the area of the electrode and τ is the duration of the OR21+⇄ OR 12 + k (V) potential pulse applied. 2

Therefore (see eqs 23 and 35) with the equilibrium constant, K (= k1/k2), for reaction V given by (see eq 12): 1/2 Q EE 2KJ+ ̅ Q == =+2ff ⎛ ⎞ CCs s EE,N 1/2 1/2 2 R I F 0′ O12R Q F KJKJ+ ̅ + ̅ (39) K =Δ=exp⎜⎟E ⎝ RT ⎠ CCs s O21R (42) where Note that in this case, due to the independence of the two Δ 0′ ≤ Q F =ΓFA T (40) electrochemical steps in reaction IV, only the values E 0 (eq 2) must be considered, that is (0 ≤ K ≤ 1, eq 42). In the − Note that the QEE E curve for a reversible EE process following we refer to process (IV) as E+E mechanism. taking place in a monolayer is independent of time (i.e., has a Thus, if O1 and O2 are initially present with identical stationary character), and therefore, eq 39 is fulfilled when any excesses/concentrations values, and all species have equal potential−time waveform is applied to the electrode. It is also diffusion coefficients in the solution-soluble case (such that 1 important to highlight that the normalized charge, QEE,N, has an reaction V has no effect on the current), the expression for the ψ 1,33 identical expression to that for the normalized transient current normalized current, E+E, is (see eqs 31 and 41) NPV IEE,N obtained for solution-soluble species when the NPV KJ1/2 ̅ KJ3/2 ̅ technique is applied to an electrode with any geometry (eq 30) ψ = + EE+ 1/2 2 1/2 2 and also to the normalized stationary current obtained for (1+ KJ̅ ) (KKJ+ ̅ ) (43) solution-soluble species when any potential−time waveform is applied for ultramicroelectrodes with any geometry. with J̅ and K given by eqs 11 and 42, respectively. In eq 43, the − ψCV fi The normalized current potential curve in CV, EE , can be rst and second addends refer to the contributions to the total 31−36 easily obtained from eq 39, being current of O1/R1 and O2/R2 reactions in eq IV, respectively. As

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Table 2. Analytical Expressions for the Roots with Physical Meaning (E , E ) of the Derivative of the Normalized Current, peak valleya ψ, for the EE and E+E Mechanisms (eqs 31 or 41, and 43, Respectively)33,36,

aK is given by eqs 12 and 42 for the EE and the E+E mechanisms, respectively. Note that peak potentials in dNPV (solution-soluble species) and CV (surface-confined species) correspond to the half-wave potentials in NPV1,14 and charge-based techniques,33 respectively. T = 298 K.

ψ − 0′ for an EE mechanism, E+E is an even function of (E E̅ ), Figure 1c). In Figure 1a, the peak potentials are referred to the and at E = E̅0′ takes the form (see eq 43 with J̅ = 1 and ref 1.) average formal potential value, E̅0′. Thus, since the (ψ − E) curves for a reversible process are symmetrical with respect to ⎛ ⎞ 0 1/2 KC1/2 s + Cs the vertical axis located at E = E̅ ′ (see Figure 2 below), it is K ⎜ R12R ⎟ ′ fi (ψ )2E̅0 = = ⎜ ⎟ ful lled that EE+ (1+ K1/2 ) 2 ⎝ 1 + K1/2 ⎠ E̅0′ (44) |ΔEpeak | |EEEE− ̅0′′|=| − ̅0 |= that is, at the average formal potential both single-electron peak,1 peak,2 2 (45) transfers have the same contribution to the current, as expected. with ΔE (= E2 − E1 ) being the difference between peak Table 2 shows the analytical expressions for the roots with peak peak peak potentials. The horizontal black line at E = E̅0′ and the two physical meaning of the derivative of the normalized current, ψ, symmetrical red branches represent, respectively, that only one for the EE and E+E mechanisms (eqs 31 or 41 and 43, peak and two peaks are obtained in the corresponding range of respectively), which correspond to the peaks and valley ΔE0′ values in abscissas, as is indicated in Figure 1. The two potentials in the response, given as a function of K (eqs 12 oblique dashed black lines with slope |0.5| represent that and 42 for the EE and the E+E mechanisms, respec- tively).14,33,36 The values of K (i.e., of ΔE0′) for the transition 0′ 00′′00 ′′|ΔE | 2 peaks−1 peak in each mechanism, and the limit value of K for |EE1 − ̅ |=| EE2 − ̅ |= 2 (46) a response with two peaks centered at the individual formal potentials are also indicated. Hence, for the values of ΔE0′ at which the red and the These results can be seen in Figure 1, which shows the dashed black lines are overlapped, the peak potentials coincide evolution with ΔE0′ of the peak parameters of the (ψ − E) with the individual formal potentials. Figure 1b shows that the ψ response, for the EE and E+E mechanisms (calculated from eqs values of the peak height, peak, for an EE process, change with Δ 0′ Δ 0′ ≪ Δ 0′ ≫ 31 or 41 and 43, respectively): peak potentials (Epeak, Figure E between 1/4 (for E 0) and 1 (for E 0),as ψ 1/2 1a), peak height ( peak, Figure 1b), and half-peak width (W , expected, since the normalized current for an E mechanism

12317 dx.doi.org/10.1021/jp5025763 | J. Phys. Chem. C 2014, 118, 12312−12324 The Journal of Physical Chemistry C Article Γ excess T, for immobilized surface molecules, or the bulk * 31−33,35,36 concentration, cO, for solution-soluble species. The most singular characteristic exhibited by Figure 1 is that the values of ΔE0′ (25 °C) = −142.4, −71.2, −35.6, 0 mV (K = 1/28, 1/24, 1/22, 1/20, respectively, see eq 12, where the base “2” refers to the number of electrons transferred), play an essential role in the behavior of the peak parameters. Indeed, the following are observed (see also Table 2): (1) For ΔE0′ ≤−142.4 mV (K ≤ 1/28) the two peak potentials in the voltagram (for both EE and E+E mechanisms) correspond to the individual formal 0′ 0′ potentials, E1 and E2 (see the superposition of red ψ and dashed black lines in Figure 1a), and peak = 1/4 (see Figure 1b and eqs 33 and 34), both peak parameters being characteristic of a simple E mechanism, although the value W1/2 = 90 mV is only obtained for ΔE0′ ≤ −200 mV (see Figure 1c).1,33 (2) For ΔE0′ = −71.2 mV (K = 1/24) in the EE mechanism and ΔE0′ = −67.7 mV (K =(7−4√3)) in the E+E mechanism (i.e., with a difference of 3.56 mV), the transition 2 peaks−1 peak takes place (see Figure 1a), ψ − ψ obtaining in both cases identical E curves, with peak = 1/3 (see Figure 1b and eqs 32 and 44) and W1/2 = ≈ |Δ 0′| 14,18,32,33,36 143.8 mV 2 E EE (Figure 1c). For more positive values of ΔE0′, only one peak appears in the response, whose peak potential corresponds to E̅0′ (see Figure 1a) and whose peak heights are given by eqs 32 for the EE mechanism and 44 for the E+E one. (3) When two peaks appear in the response for the EE mechanism (i.e., ΔE0′ < −71.2 mV, see Figure 1a-c), it can be considered as practically indistinguishable from the E+E one and, therefore, can be treated in a simpler manner in this case. (4) For ΔE0′ = −35.6 mV (K = 1/22) in the EE and ΔE0′ =0 mV (K = 1) in the E+E mechanisms, both processes present the same response, whose height is double that ψ for a simple E mechanism, that is, peak = 1/2 (Figure 1b), but W1/2 = 90 mV (Figure 1c).1,17,19,25,32 This is an obvious result in the case of an E+E mechanism. (5) For any value of ΔE0′ for the EE mechanism in the Δ 0′ − 0′ − ≤ Δ 0′ ≤− Figure 1. Evolution with E of the peak potentials, Epeak E̅ (a), interval 71.2 E 35.6 mV, there is always a ψ 1/2 0 the normalized peak-height, peak (b), and the half-peak width, W higher ΔE ′ value for the E+E one in the interval −67.7 ψ − (c), of the E response, for the reversible EE and E+E mechanisms ≤ ΔE0′ ≤ 0 mV for which the voltammetric responses in (calculated from eqs 31 or 41 and 43, respectively). The appearance of both mechanisms are identical (see Figures 1b,c). This one or two peaks in the ψ − E response is indicated by the arrows in (a) (the red lines correspond to the apparition of two peaks). The correspondence can be seen in Figure 1d; for example, an Δ 0′ − curve in Figure 1d shows the correspondence (obtained from (b,c)) EE process with E = 48.0 mV will present the same 0 between the ΔE0′ values for the EE (in ordinates) and E+E (in voltagram as an E+E one for ΔE ′ = −37.0 mV. abscissas) mechanisms, in order to obtain an identical voltammetric (6) The peak height for an EE process (eq 32) presents a response. T = 298 K. linear behavior with ΔE0′ in the interval (see the dashed black line in Figure 1b), − 71.2 mV ≤ ΔE0′ ≤ 0 mV: ψ Δ 0′ ≤ ψ ≤ EE,peak =(( E )/(3(71.2))) + 2/3, i.e., 1/3 EE,peak Δ 0′ − with n electrons transferred is ψ = n2/4;1,2 thus, 1/4 for n = 2/3 that is (see Figure 1a), between E = 71.2 mV, E,peak fi ̅0′ 1 and 1 for n = 2. In the same way, Figure 1c shows that the for which it is ful lled that Epeak,1 = Epeak,2 = E − Δ 0′ half-peak width, W1/2, changes in the extreme negative and (transition 2 peaks 1 peak), and E = 0 mV, for which 0 0 0 0 − − ′ ′ ̅ ′ positive values of ΔE ′ between 90 mV (E ) and 45 mV (E ) E1 = E2 = E (transition normal ordering-inverted 1e 2e 4 for an EE process, but a sharp jump in the half-peak width, potentials ). 0 W1/2, is observed at ΔE0′ ≈−135 mV (∼2 × (−67.7 mV), see (7) For ΔE ′ ≥ 200 mV (K → ∞) for the EE mechanism: ψ 1/2 below), which corresponds to a ψ − E curve with two peaks, peak = 1 (eq 32, Figure 1b), W = 45 mV (Figure 1c); whose central trough is situated just at the half-peak height. therefore, the EE process behaves as an E mechanism of Curves a−c of Figure 1 can be used as working curves to two electrons. Obviously, this response cannot be accurately determine the values of ΔE0′ and the total surface obtained for an E+E process.

12318 dx.doi.org/10.1021/jp5025763 | J. Phys. Chem. C 2014, 118, 12312−12324 The Journal of Physical Chemistry C Article

ψ − − 0′ s − − 0′ Figure 2. EE (E E̅ ) curves (red lines) and f j (= Cj) (E E̅ ) curves (j = O, I, or R: solid, thick dotted and dashed black lines, respectively) Δ 0′ 0′ − 0′ − obtained for a reversible EE mechanism from eqs 31 or 41 and 23, respectively, for six representative values of E (= E2 E1 ), shown in parts a 0′ 0′ 0′ f. The cross points PE1 , PE2 , and P E̅ correspond to the points where the surface coverages/concentrations take the same values, according to the 0′ 0′ 0′ nernstian conditions: f O = f I at PE1 (eq 7), f I = f R at PE2 , (eq 8), and f O = f R at P E̅ (eq 10). T = 298 K. ψ − E 3.1. Analysis of the ( EE ) Response from the Study as in a reversible E mechanism of two electrons (eq 10) 0 0 of the Surface Concentrations/Coverages. In Figure 2, the although only in the absence of species I (f O)E̅ ′ =(f R)E̅ ′ = 0.5) ψ − − 0′ − EE (E E̅ ) curves (red lines) for an EE mechanism (eqs 31 will the process really behave as an E2e mechanism (see Figure s − − or 41) and the surface coverages/concentrations, f j/Cj (j =O 2f). In brief, the gradual conversion of a 100% E1e E1e − Δ 0′ (solid), I (dotted), R (dashed) black curves) (eq 23) have been mechanism into a 100% E2e one when increasing the E plotted for six representative values of ΔE0′ (in mV), in value can be quantified in terms of the presence of the accordance with the results in Figure 1: (a) −200, (b) −142.4, intermediate. As explained in section 2, f I presents, logically, a 0′ (c) −71.2, (d) −35.6, (e) 0, (f) 200. For simplicity in the maximum at E = E̅ and consequently (f O + f R) has a nomenclature, in the following discussion we will refer to the minimum at this potential value (eq 24). Thus, the evolution of surface coverages and all the comments are valid for the surface the process with ΔE0′ can be regarded as a “two against one ” 0′ + 0′ concentrations, since both are given by eq 23. contest , so the values of (fI )E̅ and (ffOR)E̅ are an − − − In Figure 2, three characteristic cross points for the surface indicative of the % character E1e E1e and % character E2e , 0′ 0′ coverages can be observed: PE1 (f O = f I, eq 7), PE2 (f I = f R,eq respectively. For ΔE0′ ≤−142.4 mV (K ≤ 1/256) (see Figure 2a,b), from 8), and P E̅0′ (f O = f R, eq 10), located, respectively (see dotted 0′ − 0′ −Δ 0′ 0′ − eq 25 it is fulfilled that (f = f ) 0′ =(f = f ) 0′ ≈ 0.5 and vertical lines), at E = E1 (E E̅ = E /2), E = E2 (E O I E1 I R E2 ̅0′ Δ 0′ ̅0′ ff 0 0 ≈ E = E /2), and E = E , regardless of the di erence between (f R)E ′ =(f O)E ′ 0(f O + f I + f R = 1, eq 35). In these Δ 0′ − 1 2 the formal potentials, E (see Figure 2a f). Note that when conditions, the intermediate I is very stable; i.e., the value of 0 ΔE ′ increases from −200 mV (Figure 2a) to 200 mV (Figure 0 Ψ − (f I)E̅ ′ is near unity and the peak potentials in the EE E 0′ 0′ 2f), the values of the surface coverages at PE1 and PE2 vary response (red curve) correspond to the individual formal 0′ 0′ between 0.5 (i.e., as corresponds to a simple E mechanism) and potentials; that is, they are coincident with those at PE1 and PE2 0′ Ψ 0, and contrarily, between 0 and 0.5 at P E̅ . (see also EE-red line in Figure 1a). So, the peak height, peak = 0′ 0′ The species O and I, at PE1 , and I and R, at PE2 , are formally 1/4 (Figure 1b), coincides with that obtained from eq 33 or 34. Ψ 0′ Ψ ≈ × related as in two separated E mechanisms through the nernstian Indeed, from eq 33, ( EE)E1 = EE,peak,1 2 0 + 1/4, a result conditions given by eqs 7 and 8, although only if the species I is that corroborates the stability of the intermediate. 0 stable enough for (f ) 0′ =(f ) 0′ ≈ 0.5 (see Figure 2a−c), the 0′ ≈ Δ ′ ≤− I E1 I E1 Note that, strictly, (fI )E̅ 1 for E 200 mV (see 1/2 Ψ − process will actually behave as separated E mechanisms. Figure 2a and eq 24), in this case, W = 90 mV in the EE E Additionally, the species O and R, at E̅0′, are formally related response (see also Figure 1c), as indicated at point 1 in the

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− − discussion of Figure 1. This limit situation corresponds to a the process changes from 66.7% character E1e E1e to 66.7% − − − Δ 0′ genuine (100%) E1e E1e mechanism with a totally stable character E2e . However, for E >0(K > 1), also with one − Δ 0′ − intermediate (0% character E2e ). For E = 142.4 mV, peak in the voltagram, Figure 1a, the intermediate is unstable (fff)8()00′′=+ = 8/9 (see eq 24 and Figure 2b); thus, 00′ <= ′ IOREE̅ ̅ ((fffIOR)(EE̅ )̅ ). − − − 0 the process has 89% character E1e E1e and 11% character E2e , For ΔE ′ ≥ 200 mV (Figure 2f, see points P 0′ and P 0′, which 1/2 Ψ E1 E2 which is manifested in the fact that the value of W in the EE − − have interchanged their relative left right positions with E response is larger than 90 mV (Figure 1c). Δ 0′ For −142 mV < ΔE0′ < −71.2 mV (Figure 2b,c), the relation to those for E < 0; see dashed black lines in Figure 1a) the intermediate practically disappears, and 00′′>+ intermediate remains stable ((fffIOR)2()EE̅ ̅ , eq 24), (ff)()EE̅00′ = ̅ ′ = 0.5. Thus, the typical behavior of surface and the cross points P 0′ and P 0′ are still near the value 0.5 OR E1 E2 coverages for a two-electron E mechanism is observed (100% corresponding to simple E processes (0.485 for ΔE0′ = −71.2 character E −), as indicated in point 7 in the discussion of mV; see Figure 2c and Figure 4). Therefore, for ΔE0′ < −71.2 2e Figure 1. mV the EE mechanism behaves practically like the E+E process, as indicated in points 2 and 3 in the discussion of Figure 1. In In the previous discussion, the percentage of character − − − ψ − this interval of ΔE0′ values, two peaks are still obtained in the E1e E1e and E2e in the EE E response has been discussed in ψ − E signal, although, as shown in Figure 1a, the peak terms of the surface coverage values at the average formal EE 0 ̅ ′ 0′ potentials do not correspond to the individual formal potentials potential, E . Moreover, the relation between (ψEE)E̅ and 0 (unlike that obtained for ΔE ′ ≤−142.4 mV). Note that a + 0′ (ffOR)E̅ given by eq 32 allow us to write visual inspection of the curves of surface coverages in Figure 2b, 0 ̅ ′ −′0 and due to the axial symmetry of the curves at E = E , allows us %characterE2e =× (ψEE )E̅ 100 (47) 0′ 0′ to detect that the curve of f I between PE1 and PE2 reproduces − exactly that of peak potentials given in Figure 1a in the interval Thus, by inserting eq 32 in eq 47 both the percentage of E2e of ΔE0′ considered (−142.4 mV ≤ ΔE0′ ≤−71.2 mV); character and the ΔE0′ can be calculated from the value of − ψ − 0′ therefore, the transition 2 peaks 1 peak in the EE E (ψEE)E̅ , independently of there being a valley or a peak at E = 0 response must take place at ΔE ′ = −142.4/2 mV = −71.2 mV. E̅0′ in the ψ − E response (see Figures 2a-f). Indeed, taking Δ 0′ − EE Indeed, for E = 71.2 mV (Figure 2c), one peak centered into account eq 12 0′ ψ − at E = E̅ is obtained in the EE E curves; thus, from the electrochemical response, it could be inferred that the 0′ 0′ RT ()ψEE E̅ intermediate has “lost” the contest, however it is still stable ΔE = ln 2nF 2(1− (ψ )E̅0′ ) (48) (according to eq 24 it is fulfilled that (fff)2()=+0′ = EE IOREE0̅ ′ ̅ 2/3 (see also Figure 2c); i.e., the process has 66.7% of the 0′ 0′ Thus, the individual formal potentials, E1 and E2 , can be − − Δ 0′ character of E1e E1e . Nevertheless, from this value of E , the determined in any case from the average formal potential, E̅0′, intermediate does not contribute to the peak height, since directly located in the response and the ΔE0′ value calculated df I = 0 (see eq 32). from eq 48 (see eqs 1 and 2). dE ( )E̅0′ 0 In line with the above, it can be introduced the term The value Δ E ′ = − 35.6 mV (Figure 2d, “ ff ” fi e ective electron number , neff,de ned as (1 + ((% character ((ff)2()2()EEE̅000′′′==̅ f̅ , i.e., (fff)(EE̅00′ =+ )̅ ′ =1/ − IOR IOR E2e )/100)), i.e., 2) corresponds to the case for which the f I curve becomes ψ − coincident with the EE E one and has the particular interest − − − − of corresponding to 50 50% of character E1e E1e and E2e . Indeed, the intermediate reaches at E̅0′ the half of its maximum value and hence, this value of ΔE0′ would correspond to the equiprobability of the O/I and I/R conversions. Thus, this ΔE0′ could be considered as the boundary between anticooperative and cooperative behavior of both electron transfer reactions (see below). Note also that the set of curves for coverages of f I and (f O + f R) (dotted blue curve in Figure 2d) presents the shape of a “peculiar” E mechanism, similar to that for the E+E mechanism with ΔE0′ = 0 mV, as indicated in point 4 in the discussion of Figure 1. 0 Δ ′ 0′ 0′ 0′ For E = 0 mV (Figure 2e), PE1 , PE2 , and P E̅ become 00′′== 0′ coincident and (fffOIR)()()EE̅ ̅ E̅ =1/3(i.e., 1 (fff)(00′ =+ )′, 33.3% character E −E −, 66.7% E −). I EE̅ 2 OR̅ 1e 1e 2e This value of ΔE0′ (K = 1, see eqs 12 and scheme (III)) corresponds to the stable-unstable intermediate transition (normal-inverted order of potentials4). Thus, in the interval −71.2 mV ≤ ΔE0′ ≤ 0 mV, with only one peak in the voltagram, in which the peak height presents a Figure 3. Evolution with ΔE0′ of the “effective electron number”, n Δ 0′ eff linear behavior with E (see Figure 1b and point 6 in the (solid line, eqs 49 and 47), and napp (dashed line, eq (A.14) in ref 32), discussion of Figure 1), the intermediate is stable (K < 1) and at E = E̅0′. T = 298 K.

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Δ 0′ s Figure 4. Evolution with E of the surface coverages/concentrations, f j/Cj, obtained for the reversible EE (j = O, I, R (a)) and E + E (j =O1,R1, 0′ 0′ O2,R2 (b)) mechanisms from eq 23 and ref 1, respectively, for E = Epeak,1 and Epeak,2 (green and pink lines), E = E̅ (red and blue lines), and E = E1 0′ fi ′ ′ ′ Δ 0′ and E2 (thick dotted black line), as detailed in the gure. Points A, A ,B,B (=C), and C are cross points at the characteristic values of E = −71.2, − 67.7, − 35.6, and 0 mV. T = 298 K.

0′ “ 2eRT/2 FΔ E (eq 49), is not coincident with the apparent number of =+ψ 0′ = ” neff 1()EE E̅ 0′ electron transferred , napp, which is used mainly in bioelec- + RT/2 FΔ E (49) fi Δ 0′ ≥ 12e trochemistry. In fact, napp is de ned in refs 5 and 32 for E −35.6 mV and obtained, according to eq A.14 in ref 32, from ψ 0′ in such a way that ( EE)E̅ (which corresponds to the lost of the half-peak width, W1/2 (see Figure 1c). − 0′ Δ 0′ stability of the intermediate, i.e., to 1 ()fI E̅ )isthe In Figure 3 the evolution of neff and napp with E has been fi probability of the second electron to be transferred in an plotted. It can be observed that neff is de ned for any value of apparently simultaneous way with the first one. So, for ΔE0′ ≤ ΔE0′ and takes the value 1.5 for ΔE0′ = −35.6 mV, that is, for a − −200 mV, from eq 47 and Figure 2 it is easily deduced that the 50% character E2e process or null cooperativity degree between Δ 0′ probability of an apparent simultaneous transfer of the second the electron transfers. However, for this E value napp =1. Δ 0′ ≥ − electron is practically null; thus, neff = 1 (eq 49); in the For E 200 mV, the 100% character E2e corresponds to a − Δ 0′ − ̑ transition 2 peaks 1 peak ( E = 71.2 mV), neff = 1.3 that is, 100% cooperativity degree, and neff = napp = 2. Note that only in Δ 0′ − 5,32 this probability is 1/3; for E = 35.6 mV, neff = 1.5 (the this limit neff and napp take the same value. That is, napp is not second electron has a probability equal to 0.5 for being related with a cooperativity or anticooperativity degree scale fi transferred in an apparently simultaneous way with the rst); since napp only has a clear physical meaning in the upper limit Δ 0′ ̑ Δ 0′ ≥− Δ 0′ ≥ for E = 0 mV, neff = 1.6; for E 200 mV, neff =2, E 200 mV. s 0′ 0′ logically. In Figure 4, the values of f j/Cj for E = Epeak,1, Epeak,2, E̅ , E1 , fi 0′ Δ 0′ It is important to highlight that neff,dened in this paper and E2 are plotted vs E for the EE mechanism (j =O,I,R, − from the stability of the intermediate and the % character E2e Figure 4a) and for the E+E one (j =O1,R1,O2,R2, Figure 4b)

12321 dx.doi.org/10.1021/jp5025763 | J. Phys. Chem. C 2014, 118, 12312−12324 The Journal of Physical Chemistry C Article for comparison. Thus, the behavior given in Figure 1 for the ψ the behavior of an apparent simultaneous two-electron E − E response is reproduced in this figure in terms of surface mechanism being attained with ΔE0′ ≫ 0. coverage values (note that, according to eq 32, the curve for + 0′ (ffOR)E̅ (dashed red line) in Figure 4a is coincident with 4. CONCLUSIONS Ψ Δ 0′ ≥− that for EE,peak in Figure 1b for E 71.2 mV, Δ 0′ − In this paper, a comparative study of the reversible EE corresponding to the valley height for E < 71.2 mV, not mechanism with surface-immobilized and solution-soluble shown in Figure 1b). Also according to the above results, the species has been carried out. In the latter, it has been 0′ + 0′ curves for (f I)E̅ (blue line) and (ffOR)E̅ (dashed red line) considered that the species are transported to/from the surface − − − in Figure 4a represent the character E1e E1e and character E2e , of an electrode with any geometry and size only by diffusion, respectively, of the EE process, as a function of ΔE0′ values (eq with equal diffusion coefficients. 24). Thus, dashed red line is the same as that of neff in Figure 3, The correspondences between NPV and dNPV techniques translated one unity (see eq 49). (for solution-soluble molecules) and charge-transferred based In Figure 4a it can be observed that there are three techniques and CV (for surface-confined molecules), respec- characteristic values of ΔE0′ for which different curves cross tively, have been established. Thus, a common study of the (see points A (double), B and C): corresponding voltammetric response can be made for both For ΔE0′ = −71.2 mV (see points A) the curves for surface situations of solution-soluble and immobilized molecules with coverages of O and R at Epeak,1 and Epeak,2 (green curves) the appropriate normalization in each case. + 0′ become coincident and cross the curve of (ffOR)E̅ (red A parallel analysis of the common normalized peak-shaped voltammetric signal in dNPV (solution-soluble species) and CV line). Also, the curves for (f I)E =(f I)E (pink line) and peak,1 peak,2 (surface-confined systems) and of the surface concentrations/ (f ) 0′ (blue line) cross. This corresponds to the transition 2 I E̅ coverages has been carried out, highlighting the close − ′ peaks 1 peak. Points A in Figure 4a are equivalent to points A relationship between the two behaviors. On the basis of the Δ 0′ − ′ in Figure 4b (for E = 67.7 mV); i.e., points A also surface concentrations/coverages values at the average formal correspond to the transition 2 peaks−1 peak. In this case 0′ 0′ 0′ potential, E̅ =(E1 + E2 )/2, the evolution of the voltammetric ′ ′ 0 0 0 (ff= )E̅0 = 0.79 and (ff= )E̅0 = 0.21. For these values Δ ′ ′− ′ RO12 OR12 response with E = E2 E1 , has been interpreted and of surface coverages it is fulfilledfromeq44that quantified as a gradual conversion of the EE mechanism from − − Δ 0′ ≪ 0′ − 100% character E1e E1e response for E 0 and 100% (ψEE+ )E̅ = 1/3, characteristic of the transition 2 peaks 1 − Δ 0′ ≫ peak (see Figure 1b). character E2e one for E 0, thus characterizing the most 0 interesting intermediate situations, corresponding to transitions For ΔE ′ = −35.6 mV (see point B) the curves for (f ) ̅0′ I E 2 peaks centered-not centered at the individual formal (blue line) and (ff+ ) 0′ (dashed red line) in Figure 4a ORE̅ Δ 0′ − − potentials ( E = 142.4 mV, 11% character E2e ), 2 cross. Note that, morphologically, this cross point is − Δ 0′ − − peaks 1 peak ( E = 71.2 mV, 33.3% character E2e ), comparable (both correspond to the ordinate 0.5) with that repulsive−attractive interactions, anticooperativity−cooperativ- Δ 0′ for E = 0 mV for the E+E mechanism in Figure 4b (see Δ 0′ − − − ′ ψ − ity ( E = 35.6 mV, 50% character E2e ), and normal point B ). The E response for both mechanisms is double inverted order of potentials (ΔE0′ = 0 mV, 66.7% character of that for a simple E mechanism. The origin of this equivalence − E2e ). The percentages indicated can easily be obtained in (which is in agreement with the well-known statistical behavior ̅0′ “ ” Δ 0′ practice from the current height at E (peak or valley) in the as noninteracting centers of an EE mechanism with E = voltammetric curve. Thus, the value of ΔE0′ can also be −35.6 mV1,17,19,25), lies in the relation between the surface 0 obtained in this simple and direct way, and hence, those of coverages at both values of ΔE ′. Indeed, for the EE mechanism 0′ 0′ individual formal potentials, E1 and E2 , regardless of the value 00′ =+ ′ 0 (Figure 4a), (fffI )(E̅ OR )E̅ = 0.5, that is, the surface of ΔE ′. coverages are in the relation f O:f I:f R = 1:1 + 1:1 (K = 1/4, eq As a consequence of this study, the term “effective electron 0′ ” 12); for the E+E mechanism (Figure 4b), (fj )E̅ = 0.5 and the number , neff, has been introduced and related to the relation is f :f :f :f = 1:1:1:1 (K = 1, eq 42). Thus, the probability of the second electron being transferred in an O1 R1 O2 R2 fi Δ 0′ − apparently simultaneous way with the rst one. Indeed, neff value E = 35.6 mV corresponds to the transition Δ 0′ ≪ − − − varies between 1 for E 0 (0% character E2e , very stable repulsive attractive (negative positive) interactions in ref 1 Δ 0′ ≫ − 5,32 intermediate) and 2 for E 0 (100% character E2e , very and to the transition anticooperative−cooperative behavior. − 0 unstable intermediate). Thus, in the transition 2 peaks 1 peak For ΔE ′ = 0 mV (see point C) the curves for (f ) 0′ (blue Δ 0′ − ̑ I E̅ ( E = 71.2 mV), neff = 1.3, that is, this probability is 1/3; for 0 line), (ff= ) 0′ (red line) and (f ) 0′ (dotted sigmoidal black Δ ′ − ORE̅ j Ei E = 35.6 mV, neff = 1.5 (the second electron has a probability equal to 0.5 for being transferred in an apparent line) cross, so (fff)()()00′′== 0′ =1/3.This OIREE̅ ̅ E̅ fi Δ 0′ ̑ simultaneous way with the rst); for E = 0 mV, neff = 1.6. corresponds to the transition normal ordering-inverted “ ff ” 4 ′ The e ective number of electrons transferred is not potential. Note that point C is also equivalent to point B coincident with the “apparent number of electrons transferred”, (= C′) for ΔE0′ = 0 mV for the E+E mechanism in Figure 4b, napp, extensively used in bioelectrochemistry (see Figure 3). in the sense that it is fulfilled that (ff)()00′ = ′, but the j E̅ j Ei It has been shown that, although the intermediate is stable in indicated transition is not possible for the E+E process because the interval −71.2 mV ≤ ΔE0′ ≤ 0 mV, only one peak is normal and inverted order of potential correspond to the same obtained in the voltammetric curve, whose peak height situation in this case (reflected in that the thick dotted black increases linearly with ΔE0′. For ΔE0′ > 0 mV, (unstable line of ordinate 0.5 in Figure 4b is horizontal). The ligature intermediate) the peak height increases nonlinearly with ΔE0′ − between O, I, and R in the EE mechanism leads to the until the process behaves as a 100% character E2e when the sigmoidal thick dotted black line in Figure 4a, and ultimately to intermediate practically disappear.

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From the study of two independent E mechanisms (E+E in (18) Ruzic, I. On the Theory of Stepwise Electrode Processes. J. this paper), it has been concluded that if two peaks appear in Electroanal. Chem. 1974, 52, 331−354. the response both mechanisms behave as practically indis- (19) Flanagan, J. B.; Margel, S.; Bard, A. J.; Anson, F. C. Electron tinguishable. Transfer to and from Molecules Containing Multiple, Noninteracting Redox Centers. J. Am. Chem. Soc. 1978, 100, 4248−4253. (20) Bachman, K. J. Electrochemical Reactions With Consecutive ■ AUTHOR INFORMATION Charge-Transfer Steps - Steady State and Time-Dependent Behavior Corresponding Author Under Charge-Transfer and Diffusion Control. J. Electrochem. Soc. *E-mail: [email protected]. 1972, 119, 1021−1027. (21) Sakai, M.; Ohnaka, N. Kinetics of the Second Charge-Transfer Notes Step in an EE Mechanism by Rotating-Ring-Disk Electrode The authors declare no competing financial interest. Voltammetry. J. Electrochem. 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