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reactant concentration ratios: what it means and what it does not mean
Journal of Electroanalytical Chemistry 463 (1999) 45 – 52
Steady state voltammetry at low electrolyte/reactant concentration ratios: what it means and what it does not mean Christian Amatore a,*, Laurent Thouin a, M. Fatima Bento b a
Ecole Normale Supe´rieure, De´partement de Chimie, URA CNRS 1679 24 rue Lhomond, 75231 Paris Cedex 05, France b Departamento de Quı´mica, Uni6ersidade do Minho Largo do Pac¸o, 4719 Braga Codex, Portugal Received 29 July 1998; received in revised form 5 November 1998
Abstract Voltammetric measurements performed at low [electrolyte]/[reactant] ratios are affected by migrational transport, as well as by ohmic drop contributions. The latter depend on the current as well as on the charge of the initial electroactive species because the local electrolysis changes the ionic composition in the vicinity of the electrode. Extraction of thermodynamic or kinetic data from wave shapes and positions is thus impossible without correction of these ohmic drop components. This work extends a previous experimental approach for eliminating ohmic drop contributions from experimental voltammograms obtained at low [electrolyte]/ [reactant] ratios, by combining impedance measurements and voltammetric data. The results presented here confirm our previous independent conclusions that when the reactant is neutral, the variation of ohmic drop along the voltammetric curve (which reflects the progressive ionic enrichment of the diffusion layer) cannot be predicted by considering diffusional/migrational transport alone, but also requires consideration of the influence of natural convection. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Cell resistance; Migration; Impedance analysis; Low electrolyte/reactant ratio; Ohmic drop
1. Introduction In a previous series of contributions [1,2] we have established experimentally that half-wave potentials of steady state voltammograms obtained with microelectrodes at low electrolyte-over-reactant concentration ratios are altered by several contributions which are not representative of the intrinsic electrochemical properties of the process at hand but reflect instead physicochemical properties of the medium. These extraneous contributions involve variations of junction potentials when g =[electrolyte]/[reactant] varies and variations of ohmic drop along the voltammetric curve. In these previous works we proposed two independent methods * Corresponding author. Fax: +33-1-4432-3325; e-mail: [email protected].
to correct voltammograms for these effects. This allowed us to examine in detail the difficult problem of the variation of ohmic drop during steady state voltammetry at disk microelectrodes when the supporting electrolyte concentration is much less than that of the reactant (g 1). The uncompensated resistance of a three-electrode cell is that of the solution volume defined by the electrode surface and by the equipotential surface passing through the reference electrode tip. Because the radius of this equipotential surface is considerably larger than the disk electrode radius r0, the cell resistance at disk microelectrodes is given by integration of Ohm’s law along all the current tubes which originate on the electrode surface. For a disk electrode, owing to the cylindrical geometry of the problem at hand, this can be formulated as follows [3]:
0022-0728/99/$ - see front matter © 1999 Elsevier Science S.A. All rights reserved. PII: S 0 0 2 2 - 0 7 2 8 ( 9 8 ) 0 0 4 3 1 - 8
46
1 = Rcell
&
C. Amatore et al. / Journal of Electroanalytical Chemistry 463 (1999) 45–52 r0
0
&
r dr
0
(1)
(dur )/(k(I,ur )(dsur /rdr))
where ur is the curvilinear abscissa of the elementary current tube originating from the elementary ring surface of the electrode located at distance r from its center, dsu r the elementary cross section surface area of this current tube at the abscissa ur, and k(I,ur ) the solution conductivity in the elementary solution volume (dur )/(dsu r) located at the abscissa ur in the current tube when a total current I flows through the electrode. When g 1, the concentration of ionic species is negligibly affected by the current flow [4 – 10] so that k(I,ur ):kb, where kb is the static conductivity of the bulk solution. Under such conditions, Eq. (1) simplifies and its integration affords readily the classical expression [11]: 1 R gcell = 1/(4kbr0)
(2)
When the supporting electrolyte excess is low, this assumption is no longer valid. Indeed, the excess (or deficiency) of ions due to the electrolysis at the electrode has to be compensated by a displacement of inert ions to (or from) the bulk solution to maintain the local electroneutrality of the solution [4 – 10]. This phenomenon, i.e. migration, couples the fluxes of electroactive species and inert ions. Albeit the usual treatment of migration given in most electrochemical textbooks (see Ref. [12] and refs. therein) is wrong [1,2,4 – 10,13,14], migration has been known since the early years of electrochemistry. Its effects on a voltammogram could be fully addressed theoretically, and investigated experimentally with the adequate precision and accuracy only in recent years when microelectrodes became available [4 – 10,13,14]. However, migration has an important effect that has been neglected until very recently despite its obvious implications on the shape and position of voltammetric waves. Indeed, the migratory flux of inert ions to (or from) the electrode surface changes the ionic content of the diffusion layer with respect to the value in the bulk of the solution. In other words, when the supporting electrolyte is not present in large excess, the conductivity of the solution necessarily varies with the distance of the electrode (i.e. k =k(ur ) " kb) and with the current flowing through the electrode (i.e. k= k(I) " kb) since the ionic enrichment or depletion depends on the electroactive species concentration profiles [4 – 10,13,14]. It follows that Eq. (2) cannot be valid as soon as the ratio g is not much greater than unity. k(ur,I) is available from the integration of the coupled Fick’s laws which describe the concentration profiles of electroactive and inert species, since these profiles give access to the local concentration of each ion at each point of a voltammogram [4 – 10,13,14]. However, at present, the evaluation of k(ur,I) has been performed
on the basis of simplified Fick’s laws which consider only diffusional and migrational contributions and neglect convective transport. Under the most usual experimental conditions, when the reactant is neutral and g 1 such an approximation does not yield sufficient precision at large distances from the electrode for the evaluation of the cell resistance, except for electrodes that are much smaller than a micrometer [13,14]. This occurs because under usual experimental conditions, electrodes with radii larger than a very few micrometers cannot develop their diffusion layers within the range of usual stagnant layers (dconv : 100 mm for solvents of usual viscosities, like dimethylformamide, acetonitrile, etc.) [1,2], so that values of k(ur,I) predicted on the basis of diffusion and migration alone overestimate the real solution conductivity when the reactant is neutral. For micrometric electrodes, this overestimation does not affect the predicted current plateaus values at all since these depend exclusively on the values of k(ur,I) near the electrode surface (viz. at ur of the order of a few r0, i.e. ur : r0 dconv) where most of the concentration gradients occur, but affects the predicted values of ohmic drop severely under the most common situation where the reactant is neutral. When the reactant is charged, predicted values of k(ur,I) on the basis of diffusion and migration are also erroneous. Yet, the inherent error Dk(ur,I)= [k(ur,I)]true − [k(ur,I)]predicted should be only small under these conditions, because migration should have a negligible effect on the ohmic drop since the conductivity of the bulk solution does not tend towards zero when g becomes much smaller than unity. Thus, Dk(ur,I)/ [k(ur,I)]true does not tends towards infinity for r0 ur : dconv as it does when the reactant is neutral [1,2]. In our previous articles on this problem [1,2], we proposed two different (albeit somewhat conceptually related) experimental approaches to evaluating the true cell resistance from experimental voltammograms and we were able to demonstrate that the extracted Rcell values were much larger than predicted ones [9,10] as soon as g 51. We could also show that the error between predictions and experimental measurements could be accounted for by a simple and crude model which incorporates a rough treatment of convection effects [2]. Yet, in both works, to determine experimentally the true cell resistance, we had to rely on the crucial assumption that the electron exchange at the electrode surface was sufficiently fast and reversible for the voltammetric waves to be essentially Nernstian in nature so that the only distortion arose from ohmic drop variations. To the best of our knowledge, this assumption is precisely that used in all models of voltammetry at low g values [4–10]. However, it limits the field of application of these previous approaches, since its validity is a pre-requisite of these methods. Thus, these previous approaches were restricted to experimental situations in which electron transfer is fast
C. Amatore et al. / Journal of Electroanalytical Chemistry 463 (1999) 45–52
and the kinetics of any homogeneous or heterogeneous reaction is invariant with the ionic strength. While this limitation is not problematic for our fundamental purposes in the present series of work, it prevents the use of our previous methods for investigating the effect(s) of reduced ionic strength and low electrolyte/reactant concentration ratios on heterogeneous and homogeneous kinetics. Since then, it occurred to us that one could take advantage of the steady-state nature of voltammetry at disk microelectrodes to determine the cell resistance directly through impedance measurement at any point along a voltammetric wave for any g value. Such a direct determination of the resistance is then independent of any mechanistic considerations of the electrochemical system being investigated, so that the method described here can be applied in the future to investigate the effect(s) of reduced ionic strength and low electrolyte/reactant concentration ratios on heterogeneous and homogeneous electrochemical kinetics. Since our purpose here is to establish (i) the validity of the method and (ii) the fact that it produces results that are in full agreement with our previous investigations [1,2], we will restrict this work to the investigation of the chemically reversible one-electron reduction of a neutral reactant, dicyano(fluoren-9-ylidene)methane (noted DCN in the following), since this reaction is known to exhibit a perfect Nernstian behavior [7,15].
2. Experimental Measurements were performed using a potentiostat (Autolab type PGSTAT20, Ecochemie), equipped with an Autolab type FRA2 module for impedance measurements. The instrument was fully computerized and the impedance measurements as well as the extraction of real part and phase angle were performed using the FRA software (Ecochemie) provided with the instrument. Conductivity measurements were performed with a Radiometer CDM 210 conductimeter equipped with a Radiometer CDC641T cell, calibrated with aqueous KCl solutions. All experiments were carried out at room temperature in a three electrode cell placed in a Faraday cage. All solutions were degassed by an argon flow prior to experiments and the cell was maintained under an argon atmosphere during the measurements. It was equipped with a ca. 1 cm2 apparent surface area platinum counter electrode. A platinum wire was used as the reference electrode to avoid any release in the solutions of ionic species that may result from the use of classical reference electrodes. The working electrodes were platinum microdisks of 12.5 mm radius made of cross sections of metal wires of appropriate diameter
47
(Goodfellows) sealed into soft glass. The electrodes were polished by successive steps using three abrasive papers (Presi, P600, P1200 and P4000). The final polishing step was performed on wet tissue (Presi NV) with alumina (Presi, 0.3 mm size). The electrode surface was viewed with an Olympus binocular. The electrode radius was determined according to previously reported calibration procedures [1,2]. All solutions were prepared in N,N-dimethylformamide (DMF) heated over barium oxide and distilled under an N2 atmosphere. The supporting electrolyte was tetrabutylammonium tetrafluoroborate (NBu4BF4) synthesized by mixing aqueous solutions of NBu4HSO4 and NaBF4 (Aldrich). The precipitate was extracted with dichloromethane (Merck) recrystalized from ethylacetate+ petroleum ether and dried under vacuum. The dicyano(fluoren-9-ylidene)methane (DCN) was prepared by the Knoevenagel condensation from 9fluorenone and malononitrile and recrystalized from ethylacetate+petroleum ether (40/60) [7]. An approximate (i.e. not taking into account the residual ionic impurities) concentration of the supporting electrolyte was fixed in the cell by adding a small amount (a few microliters) of a millimolar solution of the supporting electrolyte in DMF to a solution of the reactant alone in DMF. The conductivity of the bulk solution was then measured in the cell and the concentration of the inert electrolyte was thus determined precisely using a calibration curve previously established (conductivity vs. concentration of supporting electrolyte). Special care was taken in the preparation of the solutions at low contents of supporting electrolyte to reduce as much as possible the amount of the residual ionic impurities. Thus, the cell was rinsed first with pure DMF several times until the conductivity of the pure DMF solvent measured in the cell reached its nominal value, i.e. the conductivity of the solvent after distillation (k: 0.05 mS). After addition of 2 mM DCN, this solution had the lowest [electrolyte]/[reactant] ratio (g= 6×10 − 4, viz. 1.2 mM of supporting electrolyte) that could be achieved systematically under our conditions since its conductivity was close to the value measured with distilled DMF. This conductivity was equivalent to the conductivity of an effective electrolyte solution whose concentration was precisely determined using the previous calibration curve extrapolated to the lower contents of supporting electrolyte.
3. Results Fig. 1 reports a series of experimental steady state voltammograms obtained at different values of the supporting electrolyte excess (g=6× 10 − 4 to 5× 10 − 2) for the reversible one-electron reduction of 2 mM
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C. Amatore et al. / Journal of Electroanalytical Chemistry 463 (1999) 45–52
Fig. 1. Reduction of 2 mM DCN, in DMF, at various excesses g (values are indicated on each sub-figure) of the supporting electrolyte (NBu4BF4). Pt disk electrode (radius: r0 = 12.5 mm). Symbols: experimental voltammetric waves (6 =10 mV s − 1). Dashed curves: predicted voltammograms evaluated for each set of conditions from Ref. [9]. Note that the predicted voltammograms are arbitrarily made coincident with the experimental ones at E =E 1/2 to simplify the comparison in each case.
DCN in DMF, at a Pt disk microelectrode (r0 = 12.5 mm, 6 = 10 mV s − 1). The steady state nature of the waves has been checked by verifying the exact superimposition of the voltammetric traces recorded for the forward and backward voltammetric scans. Thus, the coincidence was better than the precision on current measurements provided that the forward scan was not extended too far on the plateau region where some undesired pollution of the electrode surface occurred as soon as the electrode potential was maintained too long in this range (compare the slight bell-shape of some of the voltammograms presented in Fig. 1 in the plateau range). Fig. 1 compares also the experimental voltammograms to the ones (dashed lines) predicted on the basis of Oldham’s theory which is based on consideration of transport by means of diffusion and migration only [9]. These predicted voltammograms have been evaluated for the conditions used here and were arbitrarily made coincident with the experimental ones at E= E 1/2 to ease the comparison. This arbitrary potential coincidence has been used to eliminate artificially any influence due to variations of junction potentials and possible variations of E 0 when g varies. Note that since these effects have been investigated previously [1,2] in essentially the same conditions, their investigation did not need to be repeated here. The comparisons made in Fig. 1 clearly establish that while the voltammetric shapes are correctly predicted for large g values,
the predictions disagree drastically for low g values, in agreement with our previous reports [1,2]1. For each measurement presented in Fig. 1, the cell resistance was determined by impedance measurements. For such determinations, a small amplitude sinewave perturbation (DEsine = 10 mV) was superimposed on the constant electrode potential E which was set at the value where Rcell was determined. The frequency range (10 Hz 5 w5 50 Hz) was selected so that the impedance was real (i.e. no phase shift) so as to ensure that only the cell resistance was measured. The measurements were not affected by kinetics and were independent of the frequency w within the precision of their determination (data in Fig. 2 are averaged over the frequency range). Except near the foot and the plateau of the waves, the values of Rcell were not affected by the precise value of DEsine, reflecting thus that DEsine was small enough not to alter detectably the concentration profiles created in the diffusion layer at E. However, in 1 Note that all the experimental data presented and used in Refs. [1,2] were obtained exclusively in Braga, while all the data presented here have been exclusively obtained in Paris, so that in each case a coherent set of data was used. Indeed, the magnitude of natural convective is expected to depend on the vibrations transmitted to the cell, as well as on the cell shape and electrode positioning, through their resulting effects on density gradients generated by the redox reaction under study. It is important to note (as we checked previously) that all the trends observed in Fig. 1 are consistent with our previous reports.
C. Amatore et al. / Journal of Electroanalytical Chemistry 463 (1999) 45–52
49
Fig. 2. Variations of the cell resistance Rcell as a function of r =I/Ilim at various excesses g (as indicated on each sub-figure) for the voltammograms shown in Fig. 1. Dashed curves: predicted [9] variations, (Rcell)predicted. Symbols: experimental variations determined by impedance under each condition shown in Fig. 1; (solid squares): total cell resistance, Rcell; (circles): residual cell resistance, viz., DRcell =Rcell −(Rcell)predicted (see text for the difference between solid and open circles).
the foot region of the waves, the values of Rcell were affected, because it is known [1,2,9,10] that in this range Rcell depends drastically on the actual current density (see the nearly infinite branches at I/Ilim 0 in Fig. 2 for experimental and theoretical Rcell values) which varies severely during the sinewave perturbation. However, this effect is apparent only on Rcell and not on the ohmic drop since the current density is extremely small in this range. Near the plateaus of the waves, Rcell measurements were irreproducibly affected by the slight contamination of the electrode surface (vide supra). Therefore, in the following we relied only on Rcell values determined for 0.25 I/Ilim 50.8 (data shown by solid symbols in Fig. 2) to avoid the foot and plateau ranges (data shown by open symbols in Fig. 2) in which the simple method used here is not adequate. Besides, in classical voltammetric experiments, this is the main region of electrochemical waves used to gain kinetic and thermodynamic information. Fig. 2 presents also the theoretical predictions of Rcell evaluated [9] for each set of conditions (dashed lines) and the difference DRcell =Rcell −(Rcell)predicted. In agreement with our previous and independent series of results [1,2], DRcell is almost independent of the point selected on the voltammetric wave but depends drastically on g. In fact, as noted before [1,2] and established through the use of a simple model [2], the dependence of DRcell on g is only apparent since in fact this parameter depends on kb, the bulk solution conductivity. The dependence on g is an artifact introduced by the fact
that at a constant concentration (c 0 = 2 mM in Figs. 1 and 2) of the neutral reactant, variations of g and kb are directly related since the concentration of ionic species in the solution bulk is 2gc 0: kb = 2gc 0(F 2D/RT)
(3)
D being the average diffusion coefficient of the supporting electrolyte ions in the bulk solution. The variations of DRcell with the experimental kb values, which are represented in Fig. 3 are in agreement
Fig. 3. Variations of the residual cell resistance, DRcell = Rcell − (Rcell)predicted as a function of kb, the experimental resistivity of the initial bulk solution measured at different g values. Symbols: (solid circles): average values calculated from Fig. 2; (open squares): data from Fig. 7 in Ref. [2]; (open diamonds): data from Fig. 5 in Ref. [1]. The lines represent the two individual regression lines obtained from data plotted as solid symbols (full line) or open symbols (dashed line).
C. Amatore et al. / Journal of Electroanalytical Chemistry 463 (1999) 45–52
50
Fig. 4. Symbols: dimensionless representations of the experimental voltammograms shown in Fig. 1 at different g values, as indicated on each sub-figure. Solid curves: Predicted voltammograms on the basis of this work, i.e., including the effect of the residual resistance DRcell determined in Fig. 2 (see text). Dashed curves: Oldham’s [9,10] predicted voltammograms (i.e. for DRcell =0).
with our previous reports2 based on independent procedures [1,2] and follow the theoretical expectations [2] of a reciprocal dependence between the two parameters: DRcell =1/(4kbL)
(4)
where L(L=2209 30 mm for the data in Fig. 3) is a geometrical parameter which depends mostly on dconv and on the relative values of r0 and dconv [1,2]. The simple and crude model we developed previously [2] would predict L= 550 9 150 mm for the conditions used here (r0 =12.5 mm) when considering that dconv = 1109 20 mm [1]. Considering the crudeness of our previous model (see discussion in Ref. [2]), the agreement may be considered as excellent since the right order of magnitude is obtained. Similarly, the present L value is very close to that which we measured previously on the basis of our previous approaches [1,2]. Note however that in Ref. [1], L was determined from experimental DRcell values using a different relationship from that given here in Eq. (4), since Eq. (4) has been developed in a follow-up work [2]. Using the DRcell values reported in Fig. 5 of Ref. [1] (which are plotted again as open diamonds in Fig. 3 of the present work) and the relationship in Eq. (4), one obtains L = 2309 60 mm for the value determined in Ref. [1]. 2
In Fig. 5 of Ref. [1] and Fig. 7 of Ref. [2], which represent the 1 −1 variations of DRcell as a function of k − axis scales are not b , the k b correct due to a mistake in computer drawing of the figures. These previous values are therefore plotted again as open symbols in Fig. 3 of the present paper with a correct scale axis.
L reflects the value of dconv which depend on the intensity and frequency of vibrations transmitted to the solution surrounding the electrode. Since our objective in this work, as in the two previous ones of this series, is to show that natural convection influences the voltammetric wave shape even if its effect on current plateaus is not apparent (see Refs. [2,16] for a thorough discussion of this problem), no forced convection should be used. Controlling the hydrodynamics in the vicinity of a microelectrode is possible as established and used previously by White et al. [17] or by McCreery et al. [18]. However, this involves forced convection (by mounting the microelectrode as a RDE for White et al., or on a vibrating device for McCreery et al.) and therefore increases the contribution of convection in the transport of ions and molecules to and from the electrode surface, vis a vis migrational and diffusional modes. Increasing the convective transport vis a vis its moderate values when natural convection only operates would therefore bias any conclusion on the effect of natural convection on the voltammetric wave shape when g5 1. On the basis of the Rcell values represented in Fig. 2 and taking again advantage of the steady-state nature of voltammetry at ultramicroelectrodes, one can reconstruct ohmic drop-distorted theoretical voltammograms and compare these to the experimental ones. To do so we started from Oldham’s predictions represented in Fig. 1 which incorporate migrational and ohmic drop contributions although the latter is based on an incorrectly predicted cell resistance (viz., Rcell = (Rcell)predicted
C. Amatore et al. / Journal of Electroanalytical Chemistry 463 (1999) 45–52
instead of Rcell =(Rcell)predicted +DRcell). Therefore, more realistic ‘‘theoretical’’ voltammograms can be reconstructed from Oldham’s predicted ones by correcting their potential scale by the residual ohmic drop term IDRcell = r(IlimDRcell). Also, in order to facilitate the comparisons it is better to eliminate the role of variations of junction potential and possible variations of E 0 with g (which have been thoroughly investigated in our previous work [1]) on the measured E 1/2 values. Therefore, we choose to compare experimental and predicted dimensionless voltammograms, i.e. the predicted and experimental variations of r = I/Ilim versus F(E −E 1/2)/ RT, where for any r value E is either the experimental potential (Fig. 1) or that predicted by Oldham [9,10] further corrected by the residual ohmic drop, 1/2 viz. F(E− E 1/2)/RT =F(EOldham −E Oldham )/RT +rF × (IlimDRcell)/RT, when reduction currents are taken as negative. Fig. 4 presents such plots for the data already shown in Fig. 1, together with the uncorrected theoretical predictions, i.e. r = I/Ilim versus F(EOldham −E 1/ 2Oldham)/RT. Note that in Fig. 4, to eliminate the artifacts due to the method used here (vide supra) in the ranges 05 I/Ilim 5 0.2 and 0.85I/Ilim 51 (data shown as open symbols in Fig. 2) the ohmic drop correction r(IlimDRcell) was evaluated in these ranges using DRcell values extrapolated on the basis of a linear regression of DRcell versus I/Ilim over the range 0.25 I/Ilim 5 0.8 (data shown as solid symbols in Fig. 2). It is seen in Fig. 4, that the agreement between experimental and ‘‘predicted+ corrected’’ voltammograms (solid curves) is excellent at all g values. In contrast, uncorrected predictions (dashed curves) based on Oldham’s treatment give an excellent agreement only when g is close to 0.05 or larger but produce voltammograms that are considerably less sluggish than the experimental ones when g is smaller than 0.05. These results confirm our previous ones [1,2]; however we stress that the evaluation of DRcell has been achieved through a direct and totally independent method which furthermore does not require the hypothesis that the system is Nernstian as was required with our two previous methods [1,2]. The present method is based on straightforward impedance measurements using commercially available standard equipment, and can therefore be readily applied under any mechanistic circumstance to correct a voltammogram from its ohmic drop components. Combined with the procedure we reported previously to evaluate the variations of junction potentials [1], the present method allows virtually any experimental steady state voltammogram to be corrected from junction potentials and ohmic drop, so that mechanistic conclusions can be derived from steady state experiments performed at low [electrolyte]/[reactant] ratios in a similar fashion as derived classically from voltammograms performed in the presence of large excesses of
51
supporting electrolyte. In this respect, it is worth emphasizing that the experimental voltammograms used here would have led to the conclusion that the kinetics of the heterogeneous electron transfer were becoming slower [12] and slower upon reducing the ionic strength if compared to predictions for a Nernstian system for low g values but which neglect the effect of the extra resistive component DRcell. Conversely, the extremely good agreement with predictions which do not neglect the effect of the extra resistive component DRcell, shows that the system remains Nernstian over the whole range of g values investigated here.
4. Conclusions In our previous work [1,2] we were able to establish experimentally on the basis of a rigorous analysis of E 1/2 and (E 3/4 − E 1/4) measurements [1], or of the full voltammetric waves [2], that the cell resistance has two components when the excess of supporting electrolyte is low. The first resistive component is associated with the conductivity of the diffusion layer. The conductivity gradient within the diffusion layer is greatly affected by the ionic enrichment/depletion phenomena that are associated with the migration fluxes and may therefore differ considerably from that of the bulk solution [4– 10,13,14]. The second resistive component DRcell, which has been neglected in current general theories of voltammetry at low excesses of electrolyte, is not affected by the enrichment/depletion phenomena occurring in the diffusion layer because its origin is in the bulk solution which conserves its initial concentration because of convection. However, our former conclusions were critically dependent on the fact that the kinetic nature of the electrochemical system was assumed, viz., a Nernstian waves in Refs. [1,2]. In the present work we showed that the cell resistance can be evaluated independently by simple impedance techniques so that the true cell resistance can be determined and the voltammetric waves corrected for ohmic effects without any assumption on the mechanism at hand. Thus, we could confirm and generalize our previous observations and statement on the existence of a resistive component, DRcell, which does not depend on the electrode current, i.e. on the ionic enrichment/depletion phenomena, and show that the existence of DRcell affects the voltammetric shapes severely at low g values when the reactant is neutral.
Acknowledgements This work was supported by CNRS (URA 1679) and MENRT (ENS) for the French side, and by the Univer-
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sity of Minho for the Portuguese side. Partial support from the bilateral exchange program of the French Foreign Affairs and the ICCTI Portuguese research agency is also greatly acknowledged.
References [1] M.F. Bento, L. Thouin, C. Amatore, I. Montenegro, J. Electroanal. Chem. 443 (1998) 137. [2] M.F. Bento, L. Thouin, C. Amatore, J. Electroanal. Chem. 446 (1998) 91. [3] G. Bruhat, Electricite´, cours de physique ge´ne´rale. Masson, Paris, 1967, pp. 235–236 [4] C. Amatore, in: I. Rubinstein (Ed.), Physical Electrochemistry, Chapter 4, Dekker, NY, 1995, pp. 136–208. [5] C. Amatore, M.R. Deakin, R.M. Wightman, J. Electroanal. Chem. 225 (1987) 49.
.
[6] C. Amatore, B. Fosset, J.E. Bartelt, M.R. Deakin, R.M. Wightman, J. Electroanal. Chem. 256 (1988) 255. [7] C. Amatore, M.F. Bento, M.I. Montenegro, Anal. Chem. 67 (1995) 2800. [8] C. Amatore, S.C. Paulson, H.S. White, J. Electroanal. Chem. 439 (1997) 173. [9] K.B. Oldham, J. Electroanal. Chem. 250 (1988) 1. [10] J.C. Myland, K.B. Oldham, J. Electroanal. Chem. 347 (1993) 49. [11] J. Newman, J. Electrochem. Soc. 117 (1970) 198. [12] A.J. Bard, L.R. Faulkner, Electrochemical Methods, Wiley, NY, 1980. [13] J.D. Norton, W.E. Benson, H.S. White, B.D. Pendley, H.D. Abrun˜a, Anal. Chem. 63 (1991) 1909. [14] J.D. Norton, H.S. White, J. Electroanal. Chem. 325 (1992) 341. [15] V.L. Pardini, L. Roullier, J.H.P. Utley, A. Weber, J. Chem. Soc, Perkin Trans. 2 (1981) 1520. [16] B. Fosset, C. Amatore, Anal. Chem. 68 (1996) 4377. [17] X. Gao, J. Lee, H.S. White, Anal. Chem. 67 (1995) 1541. [18] S.A. Schuette, R.L. McCreery, Anal. Chem. 58 (1986) 1778.
Steady state voltammetry at low electrolyte/reactant concentration ratios: what it means and what it does not mean Christian Amatore a,*, Laurent Thouin a, M. Fatima Bento b a
Ecole Normale Supe´rieure, De´partement de Chimie, URA CNRS 1679 24 rue Lhomond, 75231 Paris Cedex 05, France b Departamento de Quı´mica, Uni6ersidade do Minho Largo do Pac¸o, 4719 Braga Codex, Portugal Received 29 July 1998; received in revised form 5 November 1998
Abstract Voltammetric measurements performed at low [electrolyte]/[reactant] ratios are affected by migrational transport, as well as by ohmic drop contributions. The latter depend on the current as well as on the charge of the initial electroactive species because the local electrolysis changes the ionic composition in the vicinity of the electrode. Extraction of thermodynamic or kinetic data from wave shapes and positions is thus impossible without correction of these ohmic drop components. This work extends a previous experimental approach for eliminating ohmic drop contributions from experimental voltammograms obtained at low [electrolyte]/ [reactant] ratios, by combining impedance measurements and voltammetric data. The results presented here confirm our previous independent conclusions that when the reactant is neutral, the variation of ohmic drop along the voltammetric curve (which reflects the progressive ionic enrichment of the diffusion layer) cannot be predicted by considering diffusional/migrational transport alone, but also requires consideration of the influence of natural convection. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Cell resistance; Migration; Impedance analysis; Low electrolyte/reactant ratio; Ohmic drop
1. Introduction In a previous series of contributions [1,2] we have established experimentally that half-wave potentials of steady state voltammograms obtained with microelectrodes at low electrolyte-over-reactant concentration ratios are altered by several contributions which are not representative of the intrinsic electrochemical properties of the process at hand but reflect instead physicochemical properties of the medium. These extraneous contributions involve variations of junction potentials when g =[electrolyte]/[reactant] varies and variations of ohmic drop along the voltammetric curve. In these previous works we proposed two independent methods * Corresponding author. Fax: +33-1-4432-3325; e-mail: [email protected].
to correct voltammograms for these effects. This allowed us to examine in detail the difficult problem of the variation of ohmic drop during steady state voltammetry at disk microelectrodes when the supporting electrolyte concentration is much less than that of the reactant (g 1). The uncompensated resistance of a three-electrode cell is that of the solution volume defined by the electrode surface and by the equipotential surface passing through the reference electrode tip. Because the radius of this equipotential surface is considerably larger than the disk electrode radius r0, the cell resistance at disk microelectrodes is given by integration of Ohm’s law along all the current tubes which originate on the electrode surface. For a disk electrode, owing to the cylindrical geometry of the problem at hand, this can be formulated as follows [3]:
0022-0728/99/$ - see front matter © 1999 Elsevier Science S.A. All rights reserved. PII: S 0 0 2 2 - 0 7 2 8 ( 9 8 ) 0 0 4 3 1 - 8
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1 = Rcell
&
C. Amatore et al. / Journal of Electroanalytical Chemistry 463 (1999) 45–52 r0
0
&
r dr
0
(1)
(dur )/(k(I,ur )(dsur /rdr))
where ur is the curvilinear abscissa of the elementary current tube originating from the elementary ring surface of the electrode located at distance r from its center, dsu r the elementary cross section surface area of this current tube at the abscissa ur, and k(I,ur ) the solution conductivity in the elementary solution volume (dur )/(dsu r) located at the abscissa ur in the current tube when a total current I flows through the electrode. When g 1, the concentration of ionic species is negligibly affected by the current flow [4 – 10] so that k(I,ur ):kb, where kb is the static conductivity of the bulk solution. Under such conditions, Eq. (1) simplifies and its integration affords readily the classical expression [11]: 1 R gcell = 1/(4kbr0)
(2)
When the supporting electrolyte excess is low, this assumption is no longer valid. Indeed, the excess (or deficiency) of ions due to the electrolysis at the electrode has to be compensated by a displacement of inert ions to (or from) the bulk solution to maintain the local electroneutrality of the solution [4 – 10]. This phenomenon, i.e. migration, couples the fluxes of electroactive species and inert ions. Albeit the usual treatment of migration given in most electrochemical textbooks (see Ref. [12] and refs. therein) is wrong [1,2,4 – 10,13,14], migration has been known since the early years of electrochemistry. Its effects on a voltammogram could be fully addressed theoretically, and investigated experimentally with the adequate precision and accuracy only in recent years when microelectrodes became available [4 – 10,13,14]. However, migration has an important effect that has been neglected until very recently despite its obvious implications on the shape and position of voltammetric waves. Indeed, the migratory flux of inert ions to (or from) the electrode surface changes the ionic content of the diffusion layer with respect to the value in the bulk of the solution. In other words, when the supporting electrolyte is not present in large excess, the conductivity of the solution necessarily varies with the distance of the electrode (i.e. k =k(ur ) " kb) and with the current flowing through the electrode (i.e. k= k(I) " kb) since the ionic enrichment or depletion depends on the electroactive species concentration profiles [4 – 10,13,14]. It follows that Eq. (2) cannot be valid as soon as the ratio g is not much greater than unity. k(ur,I) is available from the integration of the coupled Fick’s laws which describe the concentration profiles of electroactive and inert species, since these profiles give access to the local concentration of each ion at each point of a voltammogram [4 – 10,13,14]. However, at present, the evaluation of k(ur,I) has been performed
on the basis of simplified Fick’s laws which consider only diffusional and migrational contributions and neglect convective transport. Under the most usual experimental conditions, when the reactant is neutral and g 1 such an approximation does not yield sufficient precision at large distances from the electrode for the evaluation of the cell resistance, except for electrodes that are much smaller than a micrometer [13,14]. This occurs because under usual experimental conditions, electrodes with radii larger than a very few micrometers cannot develop their diffusion layers within the range of usual stagnant layers (dconv : 100 mm for solvents of usual viscosities, like dimethylformamide, acetonitrile, etc.) [1,2], so that values of k(ur,I) predicted on the basis of diffusion and migration alone overestimate the real solution conductivity when the reactant is neutral. For micrometric electrodes, this overestimation does not affect the predicted current plateaus values at all since these depend exclusively on the values of k(ur,I) near the electrode surface (viz. at ur of the order of a few r0, i.e. ur : r0 dconv) where most of the concentration gradients occur, but affects the predicted values of ohmic drop severely under the most common situation where the reactant is neutral. When the reactant is charged, predicted values of k(ur,I) on the basis of diffusion and migration are also erroneous. Yet, the inherent error Dk(ur,I)= [k(ur,I)]true − [k(ur,I)]predicted should be only small under these conditions, because migration should have a negligible effect on the ohmic drop since the conductivity of the bulk solution does not tend towards zero when g becomes much smaller than unity. Thus, Dk(ur,I)/ [k(ur,I)]true does not tends towards infinity for r0 ur : dconv as it does when the reactant is neutral [1,2]. In our previous articles on this problem [1,2], we proposed two different (albeit somewhat conceptually related) experimental approaches to evaluating the true cell resistance from experimental voltammograms and we were able to demonstrate that the extracted Rcell values were much larger than predicted ones [9,10] as soon as g 51. We could also show that the error between predictions and experimental measurements could be accounted for by a simple and crude model which incorporates a rough treatment of convection effects [2]. Yet, in both works, to determine experimentally the true cell resistance, we had to rely on the crucial assumption that the electron exchange at the electrode surface was sufficiently fast and reversible for the voltammetric waves to be essentially Nernstian in nature so that the only distortion arose from ohmic drop variations. To the best of our knowledge, this assumption is precisely that used in all models of voltammetry at low g values [4–10]. However, it limits the field of application of these previous approaches, since its validity is a pre-requisite of these methods. Thus, these previous approaches were restricted to experimental situations in which electron transfer is fast
C. Amatore et al. / Journal of Electroanalytical Chemistry 463 (1999) 45–52
and the kinetics of any homogeneous or heterogeneous reaction is invariant with the ionic strength. While this limitation is not problematic for our fundamental purposes in the present series of work, it prevents the use of our previous methods for investigating the effect(s) of reduced ionic strength and low electrolyte/reactant concentration ratios on heterogeneous and homogeneous kinetics. Since then, it occurred to us that one could take advantage of the steady-state nature of voltammetry at disk microelectrodes to determine the cell resistance directly through impedance measurement at any point along a voltammetric wave for any g value. Such a direct determination of the resistance is then independent of any mechanistic considerations of the electrochemical system being investigated, so that the method described here can be applied in the future to investigate the effect(s) of reduced ionic strength and low electrolyte/reactant concentration ratios on heterogeneous and homogeneous electrochemical kinetics. Since our purpose here is to establish (i) the validity of the method and (ii) the fact that it produces results that are in full agreement with our previous investigations [1,2], we will restrict this work to the investigation of the chemically reversible one-electron reduction of a neutral reactant, dicyano(fluoren-9-ylidene)methane (noted DCN in the following), since this reaction is known to exhibit a perfect Nernstian behavior [7,15].
2. Experimental Measurements were performed using a potentiostat (Autolab type PGSTAT20, Ecochemie), equipped with an Autolab type FRA2 module for impedance measurements. The instrument was fully computerized and the impedance measurements as well as the extraction of real part and phase angle were performed using the FRA software (Ecochemie) provided with the instrument. Conductivity measurements were performed with a Radiometer CDM 210 conductimeter equipped with a Radiometer CDC641T cell, calibrated with aqueous KCl solutions. All experiments were carried out at room temperature in a three electrode cell placed in a Faraday cage. All solutions were degassed by an argon flow prior to experiments and the cell was maintained under an argon atmosphere during the measurements. It was equipped with a ca. 1 cm2 apparent surface area platinum counter electrode. A platinum wire was used as the reference electrode to avoid any release in the solutions of ionic species that may result from the use of classical reference electrodes. The working electrodes were platinum microdisks of 12.5 mm radius made of cross sections of metal wires of appropriate diameter
47
(Goodfellows) sealed into soft glass. The electrodes were polished by successive steps using three abrasive papers (Presi, P600, P1200 and P4000). The final polishing step was performed on wet tissue (Presi NV) with alumina (Presi, 0.3 mm size). The electrode surface was viewed with an Olympus binocular. The electrode radius was determined according to previously reported calibration procedures [1,2]. All solutions were prepared in N,N-dimethylformamide (DMF) heated over barium oxide and distilled under an N2 atmosphere. The supporting electrolyte was tetrabutylammonium tetrafluoroborate (NBu4BF4) synthesized by mixing aqueous solutions of NBu4HSO4 and NaBF4 (Aldrich). The precipitate was extracted with dichloromethane (Merck) recrystalized from ethylacetate+ petroleum ether and dried under vacuum. The dicyano(fluoren-9-ylidene)methane (DCN) was prepared by the Knoevenagel condensation from 9fluorenone and malononitrile and recrystalized from ethylacetate+petroleum ether (40/60) [7]. An approximate (i.e. not taking into account the residual ionic impurities) concentration of the supporting electrolyte was fixed in the cell by adding a small amount (a few microliters) of a millimolar solution of the supporting electrolyte in DMF to a solution of the reactant alone in DMF. The conductivity of the bulk solution was then measured in the cell and the concentration of the inert electrolyte was thus determined precisely using a calibration curve previously established (conductivity vs. concentration of supporting electrolyte). Special care was taken in the preparation of the solutions at low contents of supporting electrolyte to reduce as much as possible the amount of the residual ionic impurities. Thus, the cell was rinsed first with pure DMF several times until the conductivity of the pure DMF solvent measured in the cell reached its nominal value, i.e. the conductivity of the solvent after distillation (k: 0.05 mS). After addition of 2 mM DCN, this solution had the lowest [electrolyte]/[reactant] ratio (g= 6×10 − 4, viz. 1.2 mM of supporting electrolyte) that could be achieved systematically under our conditions since its conductivity was close to the value measured with distilled DMF. This conductivity was equivalent to the conductivity of an effective electrolyte solution whose concentration was precisely determined using the previous calibration curve extrapolated to the lower contents of supporting electrolyte.
3. Results Fig. 1 reports a series of experimental steady state voltammograms obtained at different values of the supporting electrolyte excess (g=6× 10 − 4 to 5× 10 − 2) for the reversible one-electron reduction of 2 mM
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C. Amatore et al. / Journal of Electroanalytical Chemistry 463 (1999) 45–52
Fig. 1. Reduction of 2 mM DCN, in DMF, at various excesses g (values are indicated on each sub-figure) of the supporting electrolyte (NBu4BF4). Pt disk electrode (radius: r0 = 12.5 mm). Symbols: experimental voltammetric waves (6 =10 mV s − 1). Dashed curves: predicted voltammograms evaluated for each set of conditions from Ref. [9]. Note that the predicted voltammograms are arbitrarily made coincident with the experimental ones at E =E 1/2 to simplify the comparison in each case.
DCN in DMF, at a Pt disk microelectrode (r0 = 12.5 mm, 6 = 10 mV s − 1). The steady state nature of the waves has been checked by verifying the exact superimposition of the voltammetric traces recorded for the forward and backward voltammetric scans. Thus, the coincidence was better than the precision on current measurements provided that the forward scan was not extended too far on the plateau region where some undesired pollution of the electrode surface occurred as soon as the electrode potential was maintained too long in this range (compare the slight bell-shape of some of the voltammograms presented in Fig. 1 in the plateau range). Fig. 1 compares also the experimental voltammograms to the ones (dashed lines) predicted on the basis of Oldham’s theory which is based on consideration of transport by means of diffusion and migration only [9]. These predicted voltammograms have been evaluated for the conditions used here and were arbitrarily made coincident with the experimental ones at E= E 1/2 to ease the comparison. This arbitrary potential coincidence has been used to eliminate artificially any influence due to variations of junction potentials and possible variations of E 0 when g varies. Note that since these effects have been investigated previously [1,2] in essentially the same conditions, their investigation did not need to be repeated here. The comparisons made in Fig. 1 clearly establish that while the voltammetric shapes are correctly predicted for large g values,
the predictions disagree drastically for low g values, in agreement with our previous reports [1,2]1. For each measurement presented in Fig. 1, the cell resistance was determined by impedance measurements. For such determinations, a small amplitude sinewave perturbation (DEsine = 10 mV) was superimposed on the constant electrode potential E which was set at the value where Rcell was determined. The frequency range (10 Hz 5 w5 50 Hz) was selected so that the impedance was real (i.e. no phase shift) so as to ensure that only the cell resistance was measured. The measurements were not affected by kinetics and were independent of the frequency w within the precision of their determination (data in Fig. 2 are averaged over the frequency range). Except near the foot and the plateau of the waves, the values of Rcell were not affected by the precise value of DEsine, reflecting thus that DEsine was small enough not to alter detectably the concentration profiles created in the diffusion layer at E. However, in 1 Note that all the experimental data presented and used in Refs. [1,2] were obtained exclusively in Braga, while all the data presented here have been exclusively obtained in Paris, so that in each case a coherent set of data was used. Indeed, the magnitude of natural convective is expected to depend on the vibrations transmitted to the cell, as well as on the cell shape and electrode positioning, through their resulting effects on density gradients generated by the redox reaction under study. It is important to note (as we checked previously) that all the trends observed in Fig. 1 are consistent with our previous reports.
C. Amatore et al. / Journal of Electroanalytical Chemistry 463 (1999) 45–52
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Fig. 2. Variations of the cell resistance Rcell as a function of r =I/Ilim at various excesses g (as indicated on each sub-figure) for the voltammograms shown in Fig. 1. Dashed curves: predicted [9] variations, (Rcell)predicted. Symbols: experimental variations determined by impedance under each condition shown in Fig. 1; (solid squares): total cell resistance, Rcell; (circles): residual cell resistance, viz., DRcell =Rcell −(Rcell)predicted (see text for the difference between solid and open circles).
the foot region of the waves, the values of Rcell were affected, because it is known [1,2,9,10] that in this range Rcell depends drastically on the actual current density (see the nearly infinite branches at I/Ilim 0 in Fig. 2 for experimental and theoretical Rcell values) which varies severely during the sinewave perturbation. However, this effect is apparent only on Rcell and not on the ohmic drop since the current density is extremely small in this range. Near the plateaus of the waves, Rcell measurements were irreproducibly affected by the slight contamination of the electrode surface (vide supra). Therefore, in the following we relied only on Rcell values determined for 0.25 I/Ilim 50.8 (data shown by solid symbols in Fig. 2) to avoid the foot and plateau ranges (data shown by open symbols in Fig. 2) in which the simple method used here is not adequate. Besides, in classical voltammetric experiments, this is the main region of electrochemical waves used to gain kinetic and thermodynamic information. Fig. 2 presents also the theoretical predictions of Rcell evaluated [9] for each set of conditions (dashed lines) and the difference DRcell =Rcell −(Rcell)predicted. In agreement with our previous and independent series of results [1,2], DRcell is almost independent of the point selected on the voltammetric wave but depends drastically on g. In fact, as noted before [1,2] and established through the use of a simple model [2], the dependence of DRcell on g is only apparent since in fact this parameter depends on kb, the bulk solution conductivity. The dependence on g is an artifact introduced by the fact
that at a constant concentration (c 0 = 2 mM in Figs. 1 and 2) of the neutral reactant, variations of g and kb are directly related since the concentration of ionic species in the solution bulk is 2gc 0: kb = 2gc 0(F 2D/RT)
(3)
D being the average diffusion coefficient of the supporting electrolyte ions in the bulk solution. The variations of DRcell with the experimental kb values, which are represented in Fig. 3 are in agreement
Fig. 3. Variations of the residual cell resistance, DRcell = Rcell − (Rcell)predicted as a function of kb, the experimental resistivity of the initial bulk solution measured at different g values. Symbols: (solid circles): average values calculated from Fig. 2; (open squares): data from Fig. 7 in Ref. [2]; (open diamonds): data from Fig. 5 in Ref. [1]. The lines represent the two individual regression lines obtained from data plotted as solid symbols (full line) or open symbols (dashed line).
C. Amatore et al. / Journal of Electroanalytical Chemistry 463 (1999) 45–52
50
Fig. 4. Symbols: dimensionless representations of the experimental voltammograms shown in Fig. 1 at different g values, as indicated on each sub-figure. Solid curves: Predicted voltammograms on the basis of this work, i.e., including the effect of the residual resistance DRcell determined in Fig. 2 (see text). Dashed curves: Oldham’s [9,10] predicted voltammograms (i.e. for DRcell =0).
with our previous reports2 based on independent procedures [1,2] and follow the theoretical expectations [2] of a reciprocal dependence between the two parameters: DRcell =1/(4kbL)
(4)
where L(L=2209 30 mm for the data in Fig. 3) is a geometrical parameter which depends mostly on dconv and on the relative values of r0 and dconv [1,2]. The simple and crude model we developed previously [2] would predict L= 550 9 150 mm for the conditions used here (r0 =12.5 mm) when considering that dconv = 1109 20 mm [1]. Considering the crudeness of our previous model (see discussion in Ref. [2]), the agreement may be considered as excellent since the right order of magnitude is obtained. Similarly, the present L value is very close to that which we measured previously on the basis of our previous approaches [1,2]. Note however that in Ref. [1], L was determined from experimental DRcell values using a different relationship from that given here in Eq. (4), since Eq. (4) has been developed in a follow-up work [2]. Using the DRcell values reported in Fig. 5 of Ref. [1] (which are plotted again as open diamonds in Fig. 3 of the present work) and the relationship in Eq. (4), one obtains L = 2309 60 mm for the value determined in Ref. [1]. 2
In Fig. 5 of Ref. [1] and Fig. 7 of Ref. [2], which represent the 1 −1 variations of DRcell as a function of k − axis scales are not b , the k b correct due to a mistake in computer drawing of the figures. These previous values are therefore plotted again as open symbols in Fig. 3 of the present paper with a correct scale axis.
L reflects the value of dconv which depend on the intensity and frequency of vibrations transmitted to the solution surrounding the electrode. Since our objective in this work, as in the two previous ones of this series, is to show that natural convection influences the voltammetric wave shape even if its effect on current plateaus is not apparent (see Refs. [2,16] for a thorough discussion of this problem), no forced convection should be used. Controlling the hydrodynamics in the vicinity of a microelectrode is possible as established and used previously by White et al. [17] or by McCreery et al. [18]. However, this involves forced convection (by mounting the microelectrode as a RDE for White et al., or on a vibrating device for McCreery et al.) and therefore increases the contribution of convection in the transport of ions and molecules to and from the electrode surface, vis a vis migrational and diffusional modes. Increasing the convective transport vis a vis its moderate values when natural convection only operates would therefore bias any conclusion on the effect of natural convection on the voltammetric wave shape when g5 1. On the basis of the Rcell values represented in Fig. 2 and taking again advantage of the steady-state nature of voltammetry at ultramicroelectrodes, one can reconstruct ohmic drop-distorted theoretical voltammograms and compare these to the experimental ones. To do so we started from Oldham’s predictions represented in Fig. 1 which incorporate migrational and ohmic drop contributions although the latter is based on an incorrectly predicted cell resistance (viz., Rcell = (Rcell)predicted
C. Amatore et al. / Journal of Electroanalytical Chemistry 463 (1999) 45–52
instead of Rcell =(Rcell)predicted +DRcell). Therefore, more realistic ‘‘theoretical’’ voltammograms can be reconstructed from Oldham’s predicted ones by correcting their potential scale by the residual ohmic drop term IDRcell = r(IlimDRcell). Also, in order to facilitate the comparisons it is better to eliminate the role of variations of junction potential and possible variations of E 0 with g (which have been thoroughly investigated in our previous work [1]) on the measured E 1/2 values. Therefore, we choose to compare experimental and predicted dimensionless voltammograms, i.e. the predicted and experimental variations of r = I/Ilim versus F(E −E 1/2)/ RT, where for any r value E is either the experimental potential (Fig. 1) or that predicted by Oldham [9,10] further corrected by the residual ohmic drop, 1/2 viz. F(E− E 1/2)/RT =F(EOldham −E Oldham )/RT +rF × (IlimDRcell)/RT, when reduction currents are taken as negative. Fig. 4 presents such plots for the data already shown in Fig. 1, together with the uncorrected theoretical predictions, i.e. r = I/Ilim versus F(EOldham −E 1/ 2Oldham)/RT. Note that in Fig. 4, to eliminate the artifacts due to the method used here (vide supra) in the ranges 05 I/Ilim 5 0.2 and 0.85I/Ilim 51 (data shown as open symbols in Fig. 2) the ohmic drop correction r(IlimDRcell) was evaluated in these ranges using DRcell values extrapolated on the basis of a linear regression of DRcell versus I/Ilim over the range 0.25 I/Ilim 5 0.8 (data shown as solid symbols in Fig. 2). It is seen in Fig. 4, that the agreement between experimental and ‘‘predicted+ corrected’’ voltammograms (solid curves) is excellent at all g values. In contrast, uncorrected predictions (dashed curves) based on Oldham’s treatment give an excellent agreement only when g is close to 0.05 or larger but produce voltammograms that are considerably less sluggish than the experimental ones when g is smaller than 0.05. These results confirm our previous ones [1,2]; however we stress that the evaluation of DRcell has been achieved through a direct and totally independent method which furthermore does not require the hypothesis that the system is Nernstian as was required with our two previous methods [1,2]. The present method is based on straightforward impedance measurements using commercially available standard equipment, and can therefore be readily applied under any mechanistic circumstance to correct a voltammogram from its ohmic drop components. Combined with the procedure we reported previously to evaluate the variations of junction potentials [1], the present method allows virtually any experimental steady state voltammogram to be corrected from junction potentials and ohmic drop, so that mechanistic conclusions can be derived from steady state experiments performed at low [electrolyte]/[reactant] ratios in a similar fashion as derived classically from voltammograms performed in the presence of large excesses of
51
supporting electrolyte. In this respect, it is worth emphasizing that the experimental voltammograms used here would have led to the conclusion that the kinetics of the heterogeneous electron transfer were becoming slower [12] and slower upon reducing the ionic strength if compared to predictions for a Nernstian system for low g values but which neglect the effect of the extra resistive component DRcell. Conversely, the extremely good agreement with predictions which do not neglect the effect of the extra resistive component DRcell, shows that the system remains Nernstian over the whole range of g values investigated here.
4. Conclusions In our previous work [1,2] we were able to establish experimentally on the basis of a rigorous analysis of E 1/2 and (E 3/4 − E 1/4) measurements [1], or of the full voltammetric waves [2], that the cell resistance has two components when the excess of supporting electrolyte is low. The first resistive component is associated with the conductivity of the diffusion layer. The conductivity gradient within the diffusion layer is greatly affected by the ionic enrichment/depletion phenomena that are associated with the migration fluxes and may therefore differ considerably from that of the bulk solution [4– 10,13,14]. The second resistive component DRcell, which has been neglected in current general theories of voltammetry at low excesses of electrolyte, is not affected by the enrichment/depletion phenomena occurring in the diffusion layer because its origin is in the bulk solution which conserves its initial concentration because of convection. However, our former conclusions were critically dependent on the fact that the kinetic nature of the electrochemical system was assumed, viz., a Nernstian waves in Refs. [1,2]. In the present work we showed that the cell resistance can be evaluated independently by simple impedance techniques so that the true cell resistance can be determined and the voltammetric waves corrected for ohmic effects without any assumption on the mechanism at hand. Thus, we could confirm and generalize our previous observations and statement on the existence of a resistive component, DRcell, which does not depend on the electrode current, i.e. on the ionic enrichment/depletion phenomena, and show that the existence of DRcell affects the voltammetric shapes severely at low g values when the reactant is neutral.
Acknowledgements This work was supported by CNRS (URA 1679) and MENRT (ENS) for the French side, and by the Univer-
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sity of Minho for the Portuguese side. Partial support from the bilateral exchange program of the French Foreign Affairs and the ICCTI Portuguese research agency is also greatly acknowledged.
References [1] M.F. Bento, L. Thouin, C. Amatore, I. Montenegro, J. Electroanal. Chem. 443 (1998) 137. [2] M.F. Bento, L. Thouin, C. Amatore, J. Electroanal. Chem. 446 (1998) 91. [3] G. Bruhat, Electricite´, cours de physique ge´ne´rale. Masson, Paris, 1967, pp. 235–236 [4] C. Amatore, in: I. Rubinstein (Ed.), Physical Electrochemistry, Chapter 4, Dekker, NY, 1995, pp. 136–208. [5] C. Amatore, M.R. Deakin, R.M. Wightman, J. Electroanal. Chem. 225 (1987) 49.
.
[6] C. Amatore, B. Fosset, J.E. Bartelt, M.R. Deakin, R.M. Wightman, J. Electroanal. Chem. 256 (1988) 255. [7] C. Amatore, M.F. Bento, M.I. Montenegro, Anal. Chem. 67 (1995) 2800. [8] C. Amatore, S.C. Paulson, H.S. White, J. Electroanal. Chem. 439 (1997) 173. [9] K.B. Oldham, J. Electroanal. Chem. 250 (1988) 1. [10] J.C. Myland, K.B. Oldham, J. Electroanal. Chem. 347 (1993) 49. [11] J. Newman, J. Electrochem. Soc. 117 (1970) 198. [12] A.J. Bard, L.R. Faulkner, Electrochemical Methods, Wiley, NY, 1980. [13] J.D. Norton, W.E. Benson, H.S. White, B.D. Pendley, H.D. Abrun˜a, Anal. Chem. 63 (1991) 1909. [14] J.D. Norton, H.S. White, J. Electroanal. Chem. 325 (1992) 341. [15] V.L. Pardini, L. Roullier, J.H.P. Utley, A. Weber, J. Chem. Soc, Perkin Trans. 2 (1981) 1520. [16] B. Fosset, C. Amatore, Anal. Chem. 68 (1996) 4377. [17] X. Gao, J. Lee, H.S. White, Anal. Chem. 67 (1995) 1541. [18] S.A. Schuette, R.L. McCreery, Anal. Chem. 58 (1986) 1778.