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2 Journal of Electroanalytical Chemistry 645 (010) Contents lists available at ScienceDirect Journal of Electroanalytical Chemistry journal homeage: A formal scan rate in staircase and square-wave voltammetry Dijana Jadreško, Marina Zelić *, Milivoj Lovrić Deartment for Marine and Environmental Research, Ru - der Bošković Institute, PO Box 180, 1000 Zagreb, Croatia article info abstract Article history: Received 18 January 010 Received in revised form 9 March 010 Acceted 19 Aril 010 Available online 4 Aril 010 Keywords: Formal scan rate Staircase voltammetry Square-wave voltammetry Potential ste increment Because of the fact that real measurements are sometimes described in terms of the (formal) scan rate, the functional relationshis between this arameter and the eak current in staircase (SCV) and square-wave voltammetry (SWV) were investigated theoretically. The real SCV eak current is roortional to the square-root of the formal scan rate, v, and inversely roortional to DE : i = nFSc ox (D/ ) ½ v ½ DE In SWV of a reversible electrode reaction the net eak current deends rimarily on the frequency, while the influence of the formal scan rate is of minor imortance: Di = nfsc ox D½ (0.7339f ½ DE v ½ ). On the other hand, in SWV of a totally irreversible electrode reaction, the real net eak current is a stronger function of the formal scan rate: Di = 0.094nFSc ox D½ DE 0.11 v ½. Exerimental results, obtained in erchlorate solutions of Eu 3+, are in relatively good agreement with the theory. Ó 010 Elsevier B.V. All rights reserved. 1. Introduction The main characteristic of modern voltammetric techniques is a stewise change of the electrode otential [1]. A staircase otential time waveform is a basic signal to which otential ulses are suerimosed in ulse, differential ulse (DPV) [] and squarewave voltammetry (SWV) [3]. The staircase excitation signal is defined by the otential ste increment, DE, and the ste duration, s. These two indeendent variables define a formal scan rate, v = DE/ s. The staircase current resonse deends also on samling arameter, c, which is the fraction of ste at which the current is measured (0 < c <1) [4,5]. It is well known that the variation of the scan rate with following of the resulting signals is an imortant method in investigation of mechanisms of electrode rocesses, esecially in the linear scan voltammetry, LSV [6,7]. More recisely, (aarent) reversibility deends on the time scale of exeriment, i.e. on the scan rate in LSV or cyclic voltammetry. Therefore, aearance of a redox rocess can be changed from reversible to quasireversible by increasing the scan rate. However, it is generally acceted that, unlike the scan rate de/dt in LSV, the formal scan rate is not a meaningful arameter in staircase voltammetric (SCV) [4] and square-wave voltammetric exeriments [8] because there are many combinations of DE and s, (or frequency, f) which can give the same articular value of v while at the same time differing in the current resonse. Furthermore, the characteristic time of SWV exeriment deends only on frequency and not on v = f DE [9]. Nevertheless, some authors quote the scan rate as a relevant * Corresonding author. Tel.: ; fax: address: zelic@irb.hr (M. Zelić). arameter in descrition of SWV [10,11] or even DPV exeriments [1]. In a number of studies the effects of variation of DE, s, and c on the SCV current resonse were investigated in order to find the conditions under which it is the same as in LSV for a wide variety of mechanisms [4,5,13 17]. Also, there are many aers in which the effect of kinetic (a) and exerimental arameters (e.g. DE, E sw, f) on the square-wave voltammograms have been investigated [3,8,9,18,19]. However, little is known about the influence of formal scan rate on eak currents in SCV and SWV. In this communication, the functional relationshis between the eak current and formal scan rate (in both techniques) are investigated by numerical simulation of simle, reversible and totally irreversible electrode reactions. The value of otential ste increment was changed from 0.1 to 10 mv, and from 0.1 to 30 mv in the case of SCV and SWV, resectively. Some results were comared with exerimentally obtained relationshis.. Exerimental All solutions were reared from the reagent grade chemicals and water urified in a Milliore Milli-Q system. Square-wave voltammograms were recorded using a static mercury dro electrode PAR 303 (Princeton Alied Research) attached to a lautolab System (Eco Chemie, Utrecht). Platinum wire served as a counter electrode whereas all otentials were given with resect to the saturated Ag/AgCl(NaCl) reference electrode. Before recording each new set of voltammograms, the solution in the olarograhic cell was deaerated with high urity (99.999%) nitrogen for 15 min. The room temerature was ket at 5 C /$ - see front matter Ó 010 Elsevier B.V. All rights reserved. doi: /j.jelechem

3 104 D. Jadreško et al. / Journal of Electroanalytical Chemistry 645 (010) The model A simle, reversible reaction on the stationary, lanar electrode was considered: Ox þ ne Red ð1þ It was assumed that both the reactant Ox and roduct Red were soluble in aqueous electrolyte solution. For semi-infinite linear diffusion the following system of differential equations and boundary conditions has to be ¼ c red ¼ t ¼ 0; x P 0 : c ox ¼ c ox ; c red ¼ 0 ð3þ t > 0; x!1: c ox! c ox ; c red! 0 ð4þ x ¼ 0 : x¼0 ðc ox Þ x¼0 ¼ exðuþðc red Þ x¼0 ¼ x¼0 i nfs The alication of the Feldberg s finite difference aroximation [0] to Eq. (5) gives the exression: coxð1þ coxð0þ Dx ¼ credð1þ credð0þ Dx i nfsc ox D where symbols cox(0) and cred(0) denote the concentrations of Ox and Red secies immediately at the electrode surface, resectively, while the symbols cox(1) and cred(1) corresond to concentrations of Ox and Red secies in the middle of the first sace increment. Introducing Eq. (7) into Eq. (6) yields the exression for the oerating dimensionless current: idt nfsc ¼ dd oxdx 1 þ e u ½coxð1Þ eu credð1þš ð8þ where dd DDt/Dx = 0.4 is a dimensionless diffusion coefficient, Dx and Dt are the sace and time increments, resectively, u = nf(e E 0 )/RT is dimensionless otential and n is the number of electrons. The meanings of all symbols are given in the Table 1. The dimensionless SCV current was calculated by using Dt ¼ s=50 and multilying Eq. (8) with 5 ffiffiffiffiffiffiffiffiffi 10= ffiffiffiffiffi dd, where s ¼ DE=v is a stair-case eriod: ffiffiffiffiffiffi U ¼ i =ðnfsc D Þ ð9þ s ox The similar exression for the dimensionless SWV current was obtained by using Dt ¼ s=5 where s ¼ 1=f (i.e. each squarewave half-eriod is divided into 5 time increments) and multilying Eq. (8) with 5 ffiffiffi ffiffiffiffiffi = dd : ffiffiffi U ¼ i=ðnfsc ox Df Þ ð10þ In case of totally irreversible electrode reaction (i.e. slow charge transfer: Ox + ne? Red), the mathematical formulation includes Eqs. () (5), but the Eq. (6) must be substituted by: i=nfs ¼ k s exð auþðc ox Þ x¼0 ðþ ð5þ ð6þ ð7þ ð11þ where k s is the standard rate constant and a is the cathodic transfer coefficient. In this case, the following SW dimensionless oerating current is obtained: idt nfsc ox Dx ¼ j e au coxð1þ 1 þ j ð1þ e au dd ffiffiffiffiffi where j ¼ k dd =5 and k ¼ ks = ffiffiffiffiffiffi Df is a dimensionless kinetic arameter. Again, multilying this exression (Eq. (1)) with 5 ffiffiffi ffiffiffiffiffi = dd, the dimensionless SWV current is obtained (Eq. (10)). Table 1 List of symbols. c ox Concentration of the reactant c ox Bulk concentration of the reactant (c ox ) x=0 Concentration of the reactant at the electrode surface c red Concentration of the roduct dd Dimensionless diffusion coefficient D Diffusion coefficient E Electrode otential E st Starting otential E 0 Standard electrode otential E swv SW amlitude f SW frequency i Current n Number of electrons F Faraday constant R Gas constant S Electrode surface area k s Standard rate constant t Time t int Integration eriod Dt Time increments T Absolute temerature v Scan rate x Sace coordinate Dx Sace increments DE Potential ste increment U Dimensionless current U Dimensionless SCV eak current DU Dimensionless SWV net eak current a Cathodic charge transfer coefficient k = k s (Df) 1/ Dimensionless kinetic arameter s Ste duration c =[1+(1 t int /s)]/ Samling arameter 4. Results and discussions 4.1. Staircase voltammetry (SCV) The theoretical treatment of SCV for the reversible system was resented by Christie and Lingane [13], more than 40 years ago. In the meantime SCV and LSV were comared (theoretically and exerimentally) in many aers. They resented a combination of exerimental (e.g. DE) and samling (c) arameters which gave equal voltammograms in SC and LS voltammetry [4,14 17,1]. On the other hand, little is known about the effect of scan rate on the SC eak current. According to Christie and Lingane, the eak current deends linearly on s 1/ and aroximately linearly on DE 1/ [13]. In this aer, the influence of the formal scan rate on dimensionless eak current, U, was tested theoretically by changing the otential ste increment, DE, for reversible electrode reaction (1). In the first tye of calculations, the current was measured at the end of each ste (c = 1). The effect of DE on the dimensionless SCV eak current is resented in Fig. 1A and 1B. It is shown that the absolute value of dimensionless eak current increases with increasing otential ste increment (Fig. 1A). The circles are results of numerical simulation and the interconnecting line is the best fit: U = DE The recision of fitting rocedure can be seen in Fig. 1B, in which a linear relationshi between dimensionless eak current and DE is shown. In a recently ublished aer [] the same deendence was described by an equation: U = /( { 0.5 ), where X = nfde/rt. At first glance it does not seem to be in agreement with our results. The oint is that U, in comarison with Eq. (9), was defined in a somewhat different way. The values from Fig. 1A, however, could be erfectly fitted (R = ) by a function of this tye ( U = A/(B +CDE D ), although the constants (and their hysical meanings) differ significantly from those in the literature examle. The resulting curve, with D = is given in Fig. 1A by a red line.

4 D. Jadreško et al. / Journal of Electroanalytical Chemistry 645 (010) ku i ¼ c ffiffiffi s 1 þ c ¼ r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 t int s ð17þ ð18þ The samling arameter c decreases from c = 1, for t int =0, to c = 0.5 for t int = s. This rocedure was simulated by the summation of dimensionless currents during the last m time increments of each ulse and by calculating the average dimensionless current: U ¼ 1 m X m U ð19þ The eriod of integration was changed from t int =s ¼ 0:0 to t int =s ¼ 0:4, which corresonded to the variation of c from to 0.887, resectively. Fig. shows linear relationshi between the average dimensionless SCV eak current and the recirocal of the samling arameter: U ¼ 0:681 þ 0:133 1 c ð0þ Fig. 1. Deendence of the dimensionless staircase voltammetric (SCV) eak current on otential ste increment, DE, (A) and on DE (B). The best fit of calculated data (d) to theoretical function U = DE is given (black line) together with the best fit to the function U = A/(B + CDE D ) (red line). The current samling fraction c = 1, the initial otential E st = 0.3 V vs. E 0 and n = 1. (For interretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) By introducing Eq. (9) into the current function: U = DE , we observe that the real SCV eak current is roortional to the square-root of the formal scan rate and inversely roortional to DE : ffiffiffiffiffiffi i ¼ 0:1453nFSc ox D= v DE 0:0541 ð13þ where v = DE/s. Eq. (13) alies for the constant DE and variable ste duration. If DE is varied and s is constant, the eak current can be exressed as a function of these two variables: If c = 1, the average eak current is equal to , which is the value of instantaneous eak current corresonding to DE = 10 mv (see the last oint in Fig. 1A). The current-samling rocedure has no significant influence on the relationshi between the real SCV eak currents and the otential ste increment. Fig. 3 shows the deendence of the average dimensionless SCV eak currents on the otential ste increment for three integration eriods. These relationshis can be described by the following functions: U ¼ 0:1453DE 0:4467 for t int /s = 0.04, U ¼ 0:1461DE 0:4507 for t int /s = 0., and U ¼ 0:1473DE 0:4565 for t int /s = 0.4. For the last integration eriod the real eak current deends on the square-root of the formal scan rate and on DE So, the longer integration eriod is, the smaller is the difference between the staircase voltammetry and linear scan voltammetry. 4.. Square-wave voltammetry (SWV) The influence of the formal scan rate (v = f DE) on the dimensionless net eak current, DU, was tested by changing the ste increment, DE, for reversible and totally irreversible electrode rocesses. The dimensionless current was calculated at the end of each SW half-eriod using Eqs. (8) and (10), and DU > 0 was arbitrary i / DE 0:4459 s 0:5 ð14þ In SCV exeriments the current is integrated during a certain interval at the end of each otential ste and the average current is reorted: i ¼ 1 t int Z s idt s t int ð15þ where t int is the integration eriod. Considering Eq. (9), the average current is defined by the following equation: ffiffiffiffiffiffiffiffiffiffiffiffi i ¼ kuð s s t int Þt 1 int ð16þ ffiffiffiffiffiffi where k ¼ nfsc ox D= and U is a certain function of electrode otential. By multilying both numerator and denominator in Eq. (16) ffiffiffiffiffiffiffiffiffiffiffiffi by ð s þ s t int Þ one obtains: Fig.. Deendence of the average dimensionless SCV eak current on the recirocal of the samling arameter. The results of simulation are oints and the line is a linear relationshi U ¼ 0:681 þ 0:133c 1. E st = 0.3 V vs. E 0, DE = 10 mv, n =1.

5 106 D. Jadreško et al. / Journal of Electroanalytical Chemistry 645 (010) Fig. 3. Deendence of the average dimensionless SCV eak current on the otential ste increment for three times of integration: t int /s = 0.04 (N), 0. (s) and 0.4 (x). The lines are the best fittings by the function y = ax b. All arameters are as in Fig.. chosen for reduction reaction. (The consequences of current samling at different ulse fractions were described reviously [3,3].) The difference in otential of the forward and backward ulses is E sw = 100 mv. Deendence of the dimensionless net eak current on the otential ste increment is shown in Fig. 4A and 4B for reversible electrode reaction (1). The solid line in each instance is the best fitting curve. It is shown that for DE P 0.1 mv this relationshi is a ower function: DU ¼ 0:7339 þ 0:006DE 0:8118 ð1þ Fig. 4B shows that the linear relationshi between DU and DE exists in the wide range DE/mV The intercet is an extraolation of the function (1) to DE = 0 mv and has no hysical meaning because DU = 0 for DE = 0 mv (see the first oint in Fig. 4A). However, for all ractical uroses (DE > 0.1 mv) the dimensionless net eak current is defined by Eq. (1). This offset is a reason that the influence of DE on the DU is not significant. It was found that the value of DU increased by only 5% (from to ) with the increasing of DE from 0.1 mv to 30 mv. The intercets of these curves deend on the value of SW amlitude, E sw, as can be seen in Fig. 5. The relationshis in this figure are the following: DU ¼ 0:4407 þ 0:005DE 0:774 for E sw = 5 mv, DU ¼ 0:6374 þ 0:0035DE 0:7709 for E sw = 40 mv, and DU ¼ 0:8787 þ 0:0011DE 0:936 for E sw = 75 mv. Introducing Eq. (10) into the current function (1) gives the real net eak current as a function of frequency, the formal scan rate and the otential ste increment: ffiffi Di ¼ nfsc ox D ð0:7339 f þ 0:006DE 0:3118 v Þ ðþ where v = fde. The reason for this relationshi is the regeneration of the reactant by the reoxidation of the roduct during the backward, anodic ulse. So, in the vicinity of the working electrode surface the reactant is not exhausted by the slow scan rate as it is in staircase voltammetry. This exlanation is confirmed by the lack of the offset in SWV of totally irreversible electrode reactions, in which there is no reoxidation of the roduct. The effect of DE on the dimensionless SWV net eak current, for irreversible electrode reaction, is resented in Fig. 6A and 6B. The dimensionless current was calculated using Eqs. (1) and (10) for k = and a = 0.5. This value of dimensionless kinetic arameter k corresonds to totally irreversible reaction (i.e. to the standard rate constant k s =10 5 cm s 1 if D =10 6 cm s 1 and a tyical value of SW frequency f = 100 Hz is used). As can be seen, unlike in Fig. 4. Deendence of the dimensionless SWV net eak current on the otential ste increment for the reversible electrode reaction. (A) The best fit of the results of simulation (d) to the theoretical function DU = DE , for DE P 0.1 mv ( ). (B) Linear relationshi between dimensionless net eak current and DE The current samling fraction is c =1, E st = 0.3 V vs. E 0, n = 1 and E sw = 50 mv. Fig. 5. Deendence of the dimensionless SWV net eak current on the otential ste increment for three values of SW amlitude: E sw /mv = 5 (1), 40 () and 75 (3). All other arameters are as in Fig. 4. the reversible rocesses, the influence of otential ste increment on the net eak current is ronounced and DU increases consid-

6 D. Jadreško et al. / Journal of Electroanalytical Chemistry 645 (010) Fig. 7. Deendence of the real SW eak current on the otential ste increment (DE) and on DE Solution comosition: 0.5 mmol/l Eu 3+, 0.1 mol/l NaClO 4, 0.01 mol/l HClO 4. SW amlitude: 40 mv; SW frequency: 100 s 1. Fig. 6. Deendence of the dimensionless SWV net eak current on the otential ste increment for irreversible reaction. (A) The best fit of the results of simulation (d) to the theoretical function DU = DE 0.39 ( ). (B) Linear relationshi between dimensionless net eak current and DE The current samling fraction is c =1,E st = 0.3 V vs. E 0, n =1,E sw = 50 mv, a = 0.5 and k = erably with the small increase of DE. This deendence can be described by the following function: DU ¼ 0:094DE 0:39 ð3þ This relationshi was confirmed by real measurements, in which reduction of Eu 3+ to Eu + [4,5] was taken as a model (Fig. 7). The real net eak current can be interreted as a function of the formal scan rate: DE 0:11 Di ¼ 0:094nFSc ox ffiffiffi D ffiffiffi v ð4þ This exression is similar to the Eq. (5) given by Fatouros and Krulic [6]. Although their result was originally written in a somewhat different form, it imlies the deendence of the real eak current on v 1/ DE 0.09 which is similar to our conclusion. The exression (4) is more alike to SCV than to SWV of reversible reaction, which indicates that the influence of formal scan rate on the resonses of these techniques originates from the develoment of dee diffusion layer at the working electrode surface. Exerimental results, obtained with Eu 3+ in acidified 1 mol/l Na- ClO 4, are in qualitative agreement with Eq. (4) (Fig. 8). The eak current (measured at a constant value of the ste otential and gradually increasing SW frequency) is a linear function of v 1/ but the resulting straight line does not ass through the origin. The intercet is without hysical meaning, whereas its origin stays Fig. 8. Deendence of the SW eak height on the formal scan rate for ste otentials of (emty oints) and 5 mv (full oints) at gradually increasing frequency. Solution comosition: 0.5 mmol/l Eu 3+, 1.0 mol/l NaClO 4, 0.01 mol/l HClO 4. Inset (A): a real SW voltammogram (recorded at f =10s 1, DE = mv and E sw = 40 mv) together with its comonents. Inset (B): the same voltammogram (solid line) together with the simulated resonse (dashed line), based on the exerimental conditions and arameters determined at the frequency of 100 s 1 (a = 0.5, D = cm s 1 ). unclear. Two lines that reflect measurements erformed at two different ste otentials and gradually increasing SW frequency show that, for the same scan rate, the eak current obtained with DE = mv is always higher than the eak current obtained with DE = 5 mv, as follows from the uer exression. The difference between the two values is, however, oorly ronounced (Fig. 8). The exerimental results (from Fig. 8) which do not confirm theoretical redictions could reflect the higher electrode reaction rate than assumed or erhas the imortance of sherical diffusion. Comonents of the net current, recorded at the lowest SW frequency (10 s 1 ), are not in agreement with the first assumtion (Fig. 8, inset A) indicating that the second ossibility is more real. Suerimosed theoretical and exerimental voltammograms are in good agreement for f = 100 s 1 (not resented here). At f =10s 1, however, the real signal is significantly higher than the calculated current otential curve (Fig. 8, inset B) when the latter is based on

7 108 D. Jadreško et al. / Journal of Electroanalytical Chemistry 645 (010) the exerimental conditions (E sw, DE, f) and arameters (D, a) used for simulation at the higher frequency. Moreover, the difference between the eak currents of the exerimental and calculated voltammograms at f =10s 1 is in a very good agreement with the value of intercet on the y-axis in Fig. 8. (0.34 vs la for DE = mv). 5. Conclusions In staircase voltammetry the gradient of reactant concentration at the working electrode surface decreases with time during each otential ste, and its value at the end of the ste continuously decreases from ste to ste until the end of the otential scan. So, the thickness of the diffusion layer at the end of the scan is smaller if the otential ste increment is higher and the ste duration is shorter. This is the reason why the eak current deends on the square-root of the formal scan rate. However, the roortionality factor is a ower function of the otential ste increment, as can be seen in Eq. (13), and this is the main difference between staircase voltammetry and linear scan voltammetry. The formal scan rate v = fde is not significant arameter in square-wave voltammetry of reversible electrode reactions because mathematical exression of the net eak current consists of two comonents, the first one being indeendent of the otential ste increment. This is shown in Eq. (). The first comonent deends on the concentration gradients of the reactant and roduct during the cathodic and anodic ulses, resectively. It is a function of the square-root of ulse duration only. Although the reactant consumed in the cathodic ulse is recreated in the anodic ulse, this rocess is not comlete and there is a certain loss of reactant after each air of ulses. So, the diffusion layer develos during the square-wave scan and its thickness is smaller if the formal scan rate is higher. This is the origin of the second comonent of the net eak current. The second comonent is dominant in square-wave voltammetry of totally irreversible electrode reactions because the reactants of these reactions are continuously consumed at both cathodic and anodic ulses. So, the first comonent vanishes (see Eq. (4)). Our calculations show that in staircase voltammetry the influences of otential ste increment and the ste duration on the eak current are not strictly recirocal, which means that the formal scan rate is not a single variable that determines the resonse. Furthermore, in square-wave voltammetry of reversible electrode reaction the net eak current deends rimarily on the frequency, while the influence of the formal scan rate is of minor imortance. There is, however, an essential difference between reversible and totally irreversible electrode reactions regarding the influence of the otential ste increment, or the formal scan rate on the net eak current in square-wave voltammetry. Taking back to real exeriments (which insired us to erform this study) it seems that much more measurements should be erformed in order to find the conditions under which some discreancy between the theory and ractice aears and to find its origin. Acknowledgement Financial suort of the Ministry of Science, Education and Sorts of Reublic of Croatia, within the roject Electroanalytical investigations of microcrystals and traces of dissolved comounds, is gratefully acknowledged. References [1] A.J. Bard, L.R. Faulkner, Electrochemical Methods, second ed., Wiley, New York, 001. [] H.E. Keller, R.A. Osteryoung, Anal. Chem. 43 (1971) [3] L. Ramaley, M.S. Krause Jr., Anal. Chem. 41 (1969) [4] M. Seralathan, R.A. Osteryoung, J.G. Osteryoung, J. Electroanal. 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