Direct Determination of Diffusion Coefficients of Substrate and

Notation used: t0, initial time; tg, duration time of the generating step; tr, duration ... The labels 1−4 refer to the following: charge production...
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Anal. Chem. 2002, 74, 149-157

Direct Determination of Diffusion Coefficients of Substrate and Product by Chronoamperometric Techniques at Microelectrodes for Any Level of Ionic Support Wojciech Hyk,* Anna Nowicka, and Zbigniew Stojek*

Department of Chemistry, Warsaw University, Pasteura 1, PL-02-093 Warsaw, Poland

A new method for the determination of the diffusion coefficients of both the substrate, DS, and the product, DP, of an electrode process has been developed. The method proposed is based on the analysis of the transient currents and can be applied to some reactions of the type SzS ) PzP + ne and, in contrast to the concept based on the steady-state current, to any ratio of the concentrations of supporting electrolyte and substrate. The diffusion coefficients can be evaluated sequentially from the two parts of the double-potential step chronoamperogram, since the magnitude of the normalized chronoamperometric current of the first step depends on the DS value, while that of the second step is controlled by both DS and DP values. The corresponding, easy-to-use equations and procedures are given in the paper. The equations were derived on the basis of the numerical simulation data. The proposed methods of determination of diffusion coefficients for the substrates and products have been examined experimentally with the charged and uncharged ferrocene derivatives under diffusional and mixed diffusion-migration conditions. Only the chronoamperometric DS values obtained for the substrates could be compared to those determined from the steady-state diffusional current. It was found that for the systems investigated the agreement between these two methods was good. In this limited comparison, the standard deviations for the transient techniques were slightly larger than those obtained for the excess supporting electrolyte steady-state voltammetry. Knowledge of the values of diffusion coefficients of ions and molecules is of great importance for any transport-based problem. Among the methods used to determine diffusion coefficients, the electrochemical ones are particularly useful. They are fast and relatively easy to implement, and therefore, they are widely employed in the studies of the transport phenomena. Unfortunately, the electrochemical methods are limited to the species that undergo transport-controlled electrode processes. The electrochemical evaluation of diffusion coefficients is usually based on the use of the transient- and the steady-state * Corresponding authors: E-mail: [email protected]. E-mail: stojek@ chem.uw.edu.pl. 10.1021/ac0109117 CCC: $22.00 Published on Web 12/01/2001

© 2002 American Chemical Society

techniques, such as chronoamperometry, chronocoulometry, voltammetry, and chronopotentiometry.1 Commonly, the diffusion coefficients of the substrates are determined. For this purpose the knowledge of the electrode area (A), the number of electrons transferred per molecule (n), and the bulk concentration of the substrate (cbS ) is required. Additionally, the system must contain excess supporting electrolyte to eliminate the migrational contribution to the transport of the species studied. There are several published approaches that indicate how to eliminate the necessity of knowing n and cbS prior to the experiment. One of them is proposed by Denuault et al. and is based on the normalization of the diffusional chronoamperometric response at the microelectrode with respect to its steady-state value.2 The alternative approaches include the following: (a) comparing transient data with the steady-state data3 and (b) the generator-collector electrode system.4 These methods require either at least two separate measurements with different sizes of electrodes or different techniques or a rather specialized electrode assembly and stability of the product of the electrode process. Strongly related to the above problem is also the rapid determination of n5 and the determination of diffusion coefficients with unshielded planar stationary electrodes.6 The usefulness of these methodologies is, however, limited due to the fact that they are applicable to the highly supported media (excess supporting electrolyte). This narrows the range of the systems that could be explored. There are some evident advantageous features in the evaluation of diffusion coefficients in the low-supported (a small ratio of bulk concentrations of supporting electrolyte and substrate) media. First, the determined values of diffusion coefficients are not affected by the ionic strength (if the concentration of the electroactive species is on the order of 10 mM or less). This makes the obtained results comparable to those obtained with nonelectrochemical methods, such as light-scattering spectroscopy,7 (1) Galus, Z. Fundamentals of Electrochemical Analysis; Ellis Horwood: New York, 1994. (2) Denuault, G.; Mirkin, M. V.; Bard, A. J. J. Electroanal. Chem. 1991, 308, 27. (3) Amatore, C.; Azzali, M.; Calas, P.; Jutand, A.; Lefrou, C.; Rollin, Y. J. Electroanal. Chem. 1990, 288, 45. (4) Licht, S.; Cammarata, V.; Wrighton, M. S. J. Phys. Chem. 1990, 94, 6133. (5) Baranski, A. S.; Fawcett, W. R.; Gilbert, G. M. Anal. Chem. 1985, 57, 166. (6) Yap, W. T.; Doane, L. M. Anal. Chem. 1982, 54, 1437. (7) Bansil, R.; Pajevic, S.; Konak, C. Macromolecules 1995, 28, 7536.

Analytical Chemistry, Vol. 74, No. 1, January 1, 2002 149

pulsed-field-gradient spin-echo NMR spectroscopy,8 and radioactive tracer methods9,10 which are always carried out with no deliberately added inert electrolyte. It should be particularly useful to study the transport properties of the redox species in the systems containing inert electrolytes at unknown concentrations or in the systems where the introduction of excess supporting electrolyte is impossible or not recommended, e.g., in low-polarity solvents, solid materials, environmental samples. Encouraged by the above arguments, we developed a method for the determination of diffusion coefficients of both the substrate (DS) and the product (DP) of an electrode process for any level of ionic support, including the absence and the excess of supporting ions. The work is restricted to the processes not coupled to a homogeneous chemical reaction. The new method employs double-potential step chronoamperometry and requires the use of microelectrodes. Two of the unique features of microelectrodes11-13 were especially helpful for us: the substantial reduction of the ohmic potential drop that allows the electrochemical measurements to be performed at very low ionic strengths, and the enhancement of the transport of the redox species to the microelectrode surface that makes the electrochemical experiment more sensitive to the transport properties of both substrate and product. An attempt to determine diffusion coefficients under the conditions of the deficit of supporting electrolyte has been presented by Zhou et al.14 Their method, however, is based on a very approximate expression for the chronoamperometric current affected by migration and allows one to estimate only the diffusion coefficient of the substrate-counterion pair. The procedure proposed in this paper is simple. It does not require, for example, the use of the nonlinear fitting,15 which was used to determine diffusion coefficients of the electrode reaction products under the conditions of excess supporting electrolyte. It requires only knowledge of the radius of the microelectrode prior to the chronoamperometric experiment and employment of the common tools for linear regression analysis. The method is developed on the basis of the numerical model of migrational chronoamperometry published in our previous work.16-19 That numerical model consists of rigorous solving of the general transport equations, which requires, among other things, calculation of the solution resistance and the ohmic drop after each time increment. Finally, the developed procedure for the determination of diffusion coefficients was examined experimentally with the charged and the uncharged derivatives of ferrocene. The results obtained for the substrate are compared to those determined from the steady-state diffusion-controlled current. (8) Stejskal, E. O.; Tanner, J. E. J. Chem. Phys. 1965, 42, 288. (9) Wang, J. H. J. Am. Chem. Soc. 1952, 74, 1182. (10) Wang, J. H.; Miller, S. J. Am. Chem. Soc. 1952, 74, 1611. (11) Wightman, R. M.; Wipf, D. O. In Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker: New York, 1988; Vol. 15. (12) Bond, A. M.; Fleischmann, M.; Robinson, J. J. Electroanal. Chem. 1984, 172, 299. (13) Ciszkowska, M.; Stojek, Z. J. Electroanal. Chem. 1986, 213, 189. (14) Zhou, H.; Shi, Z.; Dong, S. J. Electroanal. Chem. 1998, 443, 1. (15) Ikeuchi, H.; Kanakubo, M. J. Electroanal. Chem. 2000, 493, 93. (16) Hyk, W.; Palys, M.; Stojek, Z. J. Electroanal. Chem. 1996, 415, 13. (17) Hyk, W.; Stojek, Z. Electroanal. Chem. 1997, 422, 179. (18) Hyk, W.; Stojek, Z. Electroanal. Chem. 1997, 439, 81. (19) Hyk, W.; Stojek, Z. Anal. Chem. 1998, 70, 5237.

150

Analytical Chemistry, Vol. 74, No. 1, January 1, 2002

EXPERIMENTAL SECTION Electrochemical Measurements. Double-potential step chronoamperometry (DPSC) and voltammetry were performed using a model PGSTAT 20, AutoLab potentiostat that was controlled via personal computer. Two pieces of platinum foil were used as the counter and the quasi-reference electrodes to eliminate a possible leak of ions from the bridge. A platinum disk microelectrode of 10.9 ( 0.2 µm in radius (Project Ltd., Warsaw) was used as the working electrode. The disk radius was determined first from the steady-state voltammetric wave plateau recorded for the oxidation of 2 mM ferrocene in acetonitrile solution containing 0.2 M tetrabutylammonium perchlorate at 25 °C (under these conditions, the diffusion coefficient of ferrocene is reported to be 2.15 × 10-9 m2/s).20 Observations with a LEO, model 435 VP, scanning electron microscope confirmed the two first significant figures of the value of the microelectrode radius measured electrochemically. Before each experiment, the working electrode was polished with aluminum oxide powders of various sizes (down to 0.05 µm) on a wet pad and was rinsed with a direct stream of ultrapure water (Milli-Q, Millipore, conductivity of ∼0.056 mS/cm). The electrodes were dried using ethyl alcohol. After the measurements, the electrode surface was inspected optically with an Olympus, model PME 3, inverted metallurgical microscope. To minimize the electric noise, the electrochemical cell was kept in a grounded aluminum foil Faraday cage. For each redox system, at least three replicates of the background-corrected chronoamperograms and voltammograms were recorded. Chemicals. Electroactive species: Ferrocene (Fe(C5H5)2, 98%) and 1,1′-ferrocenedimethanol (Fe(C5H4)2(CH2OH)2, 98%) were purchased from Aldrich, and sodium ferroceneacetate (Fe(C5H5)(C5H4)(CH2COO-)Na+) and ferrocenylmethyltrimethylammonium hexafluorophosphate (Fe(C5H5)(C5H4)(CH2N(CH3)3)+PF6-) were synthesized in our laboratory according to the procedures available in the literature. Solvents: Acetonitrile (Fluka) was of p.a. grade, and water was deionized with the Milli-Q, Millipore system. Supporting electrolytes: Tetrabutylammonium perchlorate (N(C4H9)4ClO4, 99%) and lithium perchlorate (LiClO4, 99%) were purchased from Fluka. All purchased chemicals were used as received. THEORETICAL SECTION Single-Potential Step Chronoamperometry. An analysis of many numerical results obtained according to the procedures published in our previous work16,18 allowed us to find a generalized expression for chronoamperometry. It describes the changes in the transport-controlled limiting current versus time for hemispherical microelectrodes and for any ratio of the concentrations of supporting electrolyte and substrate. The expression has the form

(

IL(ξ,t) ) ILss(ξ) 1 +

re

xπDSt

)

(1)

where ξ is the support ratio defined as the ratio of bulk (20) Baur, J. E.; Wightman, R. M. J. Electroanal. Chem. 1991, 305, 73.

Figure 1. Potential waveform for double-potential step chronoamperometry and calculated dimensionless chronoamperograms for several electrode processes carried out in either the diffusionmigration (solid lines) or purely diffusional conditions (dashed lines). Equality of the substrate and the product diffusivities was assumed. Chronoamperometric, generating and reverse, currents were made dimensionless with respect to the steady-state generating current for excess supporting electrolyte. The generating currents are plotted in the logarithmic time scale to expose better their changes. Notation used: t0, initial time; tg, duration time of the generating step; tr, duration time of the reverse step; E0/, formal potential of the redox couple; Eg, generating pulse potential; Er, reverse pulse potential. The labels 1-4 refer to the following: charge production process (S° ) P+ + e, ξ ) 0.001), charge increase process (S+ ) P2+ + e, ξ ) 0), charge decrease process (S2- ) P- + e, ξ ) 0), and charge cancellation process (S- ) P0 + e, ξ ) 0), respectively.

concentrations of supporting electrolyte and substrate (cbse/cbS), ILss is the steady-state limiting current for the same value of support ratio, re is the microelectrode radius, DS is the substrate diffusion coefficient, and t denotes time. It is worth noting that eq 1 is given in the textbooks as valid only for the conditions of pure diffusional transport. The generalizations were made on the basis of the calculations performed for a wide range of support ratio and for a large group of one- and two-electron processes involving substrates and products of various charges and of various diffusion coefficients. A few representative examples of the chronoamperograms calculated for the diffusion-migration and purely diffusional conditions, along with the notation used, are shown in Figure 1. They represent four common reaction types: charge production, charge increase, charge decrease, and charge cancellation processes. As is seen in Figure 1, only for uncharged substrates and a small value of support ratio (curve 1) does the course of the currenttime dependence differ substantially from the well-known

Cottrellian form of the chronoamperometric response.16 In contrast to the other curves in Figure 1, which refer to the processes either carried out under the conditions of excess supporting electrolyte or involving charged substrates with no supporting electrolyte, the chronoamperometric current (curve 1) increases to reach the maximum value and then decreases to reach the steady-state value. The first part of the chronoamperogram results from the decrease in the resistance of the system due to the generation of ionic products and the corresponding accumulation of counterions that neutralize the products electrically. In a chronoamperometric experiment with an uncharged substrate, although the potential applied from the potentiostat is constant, the true electrode potential varies and changes accordingly to the changes in IR drop. Due to this effect, the flux of the substrate, and therefore the transient current, become indirectly dependent on support ratio until the steady state is reached. For the reactions involving charged substrates, the ohmic drop is much smaller and the shape of corresponding chronoamperograms resembles those obtained for excess supporting electrolyte. In the presence of excess supporting electrolyte (dashed line in Figure 1), all the cases discussed above give the identical chronoamperometric response. Equation 1 was derived with the assumption that the product is initially absent and is valid for the transport-controlled electrode processes of the general scheme: SzS ) PzP + ne. Two cases are excluded. The first one is that of uncharged substrates reduced or oxidized under the conditions of supporting electrolyte deficiency (ξ < 1), and the second one involves the so-called charge reversal processes21-23 (sgn(zS) * sgn(zP), zS * 0 and zP * 0) for ξ ) 0. The extensive study of the very specific transient behavior of the uncharged substrates was the subject of our previous work.16,17 For the purely diffusional transport (ξ f ∞) to hemispherical electrodes, the chronoamperometric current is predicted exactly by eq 1 with ILss ) 2πnFDScbSre, if the applied potential (Eg) is sufficiently larger than the formal potential of the redox couple (E0/).1 Subscript g is related to the generating step and is introduced here to distinguish between the generating and reverse pulses in the double-potential step chronoamperometry described in the next section. Under the mixed diffusion-migration transport, ξ e 1, the numerical data fit the generalized expression for the chronoamperometric response (within 1%) for Eg larger than E0/ by at least 5 RT/F and for t greater than 0.005re2/DS (i.e., for t > 0.5 ms, if DS ) 1 × 10-9 m2/s and re ) 10 µm). We call the number 0.005re2/DS the lower time limit of eq 1 applicability, tlim g . For t < tlim g , the true electrode potential departs substantially from the applied potential due to the large variation of the uncompensated electrostatic potential. In practice, the lower time limit should be further increased because of the double-layer charging effect, which was not included into the model. We found that for a 1:1 electrolyte of a very small concentration (10-6 M), in a solvent of relative dielectric permittivity in the range from 30 to 80, at T ) 25 °C, with a substrate diffusion coefficient of 1 × 10-9 m2/s, and for sizes of microelectrodes in the micrometer range, the double-layer effects become negligible at times greater (21) Amatore, C.; Deakin, M. R.; Wightman, R. M. J. Electroanal. Chem. 1987, 225, 49. (22) Amatore, C.; Fosset, B.; Bartelt, J.; Deakin, M. R.; Wightman, R. M. J. Electroanal. Chem. 1988, 256, 255. (23) Myland, J. C.; Oldham, K. B. J. Electroanal. Chem. 1993, 347, 49.

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151

than 1 and 3 ms, respectively. These numbers, for the concentrations of supporting electrolyte corresponding to its excess, e.g., 10-1 M, decrease to 11 and 30 µs, respectively. The above calculations have been done by comparing the thickness of the diffuse part of the double layer, identified with the Debye length, κ-1,24 with that of the depletion layer, δ. The latter can be approximated, for the transient experiments at short times, by (πDSt)1/2. Since disk microelectrodes are most widely used in electrochemical or electroanalytical experiments, the conditions where the chronoamperometric responses at hemispherical and disk microelectrodes are equivalent must be found. To adopt eq 1 to disk microelectrodes, hemispherical radius has to be replaced with the term 2re,d/π, where re,d is the radius of the microdisk. This 2 substitution enlarges tlim g to the value of approximately re /DS -9 2 (i.e., to 100 ms, if DS ) 1 × 10 m /s and re ) 10 µm) and introduces to the expression for the transient current an error proportional to t-3/2.25 Obviously, such the substitution is absolutely correct and exact for the steady-state conditions. By rewriting eq 1, and normalizing the transient current with respect to the steady-state value, one gets the linear dependence between IL(ξ,t)/ILss(ξ) and 1/t1/2, which for disk microelectrodes looks as follows

IL(ξ,t) ILss(ξ)

)1+

2re,d

1 + O(t-3/ 2) πxπDS xt

(2)

where O(t-3/2) denotes how the error of the approximation depends on time (the error order). If re,d and ILss(ξ) are known, then the slope, a ) 2re,d/π(πDS)1/2, of eq 2 yields DS irrespective of ξ. This valuable feature makes transient current much more useful for the evaluation of DS of charged substrates than the concept based on the steadystate current, since the later may be employed only for ξ . 1 (excess supporting electrolyte) where the analytical expressions are known. To calculate DS from the limiting steady-state current under the mixed diffusion-migration conditions, the exact value of the support ratio, the charge of the substrate, and the number of electrons exchanged have to be known to choose the appropriate coefficient for the expression describing the steady-state current. It should be emphasized that, in experimental practice, the true steady-state conditions are never reached. In real situations, the chronoamperometric experiment is stopped at a specified time, tg (see Figure 1). Using the current assigned to this time, IL(ξ,tg), the normalization of the chronoamperometric current can be done and more general relations are obtained

IL(ξ,t) IL(ξ,tg)

)

(

xπDS + re

) ( -1

1

xtg

)

re 1 + 1+ xt xπDStg

-1

for hemispherical microelectrodes, and

IL(ξ,t)

)

IL(ξ,tg)

1+

Analytical Chemistry, Vol. 74, No. 1, January 1, 2002

1 + xt

-1

2re,d πxπDStg

+ O(tg-3/2)

+ O(t-3/2)

for disk microelectrodes. Equation 4 may be rewritten

IL(ξ,t)

1 ) a + b + O(t-3/2) I (ξ,tg) xt

tlim g < t e tg

L

(5)

where

a)

(

)

πxπDS 1 + + O(tg-3/2) 2re,d xtg

-1

πxπDS 2re,d

and b ) a

are the slope and the intercept of the dependence of IL(ξ,t)/IL(ξ,tg) versus 1/t1/2. For tg greater than 10 s, O(tg-3/2) is negligibly small and can be removed from the final equations. The slope (a) yields, for a microdisk, the following expression for DS

DS )

(

4re,d2 1 1 π3 a xtg

)

2

(6)

Taking the intercept (b) into account one may derive an alternative expression for DS

4re,d2 b π3 a

2

()

DS )

(7)

According to eq 6, the uncertainty (error) in the DS determination depends on the uncertainty in the values of the disk radius, re,d, and the slope, a. If eq 7 is considered, an extra source of uncertainty comes from b. Usually, the uncertainty in a measurement is expressed by the standard deviation that can be found on the basis of the propagation of random errors. The latter is evaluated by employing the commonly used Taylor series expansion about the mean values of the variables of a given function. This procedure applied to eqs 6 and 7 leads to the following estimates of the standard deviation of the substrate diffusion coefficient, s(DS)

s(DS) =

152

)

)

-1

tlim g < t e tg (4)

tlim g < t e tg (3) (24) Norton, J. D.; White, H. S.; Feldberg, S. W. J. Phys. Chem. 1990, 94, 6772. (25) Shoup, D.; Szabo, A. J. Electroanal. Chem. 1982, 140, 237.

( (

πxπDS 1 + + O(tg-3/2) 2re,d xtg

(

8re,d 1 1 π3 a xtg

)x(

)

(

re,d 1 2 1 (s(re,d))2 + 2 s(a) a a xtg

)

2

(8)

for eq 6, and

( ) x( ) ( ) ( )

8re,d2 b π3 a

s(DS) =

2

s(re,d) re,d

2

+

s(b) b

2

+

s(a) a

2

(9)

for eq 7, where s(re,d), s(b), and s(a) are the standard deviations of the microdisk radius, the intercept, and the slope, respectively. In the derivation of eqs 8 and 9, the terms containing the secondand higher-order derivatives were dropped. An analysis of eq 8 reveals that for the electrodes of radius in the micrometer range the term ((re,d/a2)s(a))2 is practically negligible. Therefore, it can be concluded that the precision of the evaluation of DS is mainly determined by the precision of the re,d measurement. Double-Potential Step Chronoamperometry. The simplest reversal method based on the potential control is DPSC in which the first potential pulse (primary or generating step) is used to generate the product and the second one is used to reverse the effect of the first step. The numerical implementation of DPSC was based on the simulation scheme developed by us for chronoamperometry.19 The results for several classes of electrode processes carried out under the diffusion-migration and purely diffusional conditions, along with the notation used, are shown in Figure 1. After an analysis of the calculation data, it was found that the ratio Ir(ξ,t)/IL(ξ,tg) is a simple function of the reciprocal of the square root of the reverse pulse time, 1/(t - tg)1/2. Ir(ξ,t) is the reverse current measured at time t, IL(ξ,tg) is the limiting current of the primary (generating) step measured at tg, and the relation holds for any support ratio and for any class of electrode processes except for the charge reversal case with ξ ) 0. In addition, it was found that for the reverse pulse times greater than re2/DS the migrational contributions to the generating and reverse currents compensate each other. The ratio Ir(ξ,t)/IL(ξ,tg) is thus independent of the concentration of supporting electrolyte, and can be expressed as

Ir(ξ,t) IL(ξ,tg)

)

x

DS DP

(

1

xt - tg

-

1 xt

)(

xπDS + re

tlim r

1

)

-1

xtg

< t - tg e tr (10)

where t - tg is the reverse pulse time measured from the moment of imposition of the second (reverse) potential pulse, tr is the duration time of the reverse potential pulse, tlim r is the lower limit of reverse pulse time for the equation applicability, and DS and DP denote the diffusion coefficients of the substrate and the product of the generating step, respectively. In the process of computational derivation of eq 10, the following simulation conditions were employed: (a) initial absence of the electrode process product, (b) the reversible electrode reaction of the general scheme SzS ) PzP ( e, (c) the quasi-steadystate limiting conditions in the generating step (large generating potential step and tg appropriately large), and (d) the changes in the DS/DP ratio from 5 to 1/5. Using the nonlinear regression, we have found that for one-electron processes carried out in the absence or severe deficit of supporting electrolyte, and for (t -

tg) > re2/DS, the numerical results fit eq 10 with the maximum error of 5%. The deviations are very small (∼1.5%) if both reactants are charged. For the purely diffusional conditions, eq 10 predicts the reverse current exactly with no restrictions on the reverse pulse time and agrees with the expression derived for equal diffusivities of both forms of the redox couple.26,27 To make accurate calculations of the diffusion coefficient from the experimental data, it is necessary to rationalize the lower reverse pulse time limit of the applicability of eq 10 and to find the optimal value of tg. The numerical analysis of the computed double-potential step chronoamperograms and the considerations analogous to those for the single potential step chronoamperometry lead to the should be sufficiently large, since the conclusions that tlim r variation of the uncompensated electrostatic potential in the depletion layer19 and the influence of the double-layer reorganization are the largest after the imposition of the reverse pulse. These two factors are particularly important for the cases of severe deficit of supporting electrolyte. According to the simulation done, eq 10 should be applicable to the experimental treatment for any value of ξ, if t - tg is greater than 1.5re2/DS (i.e., for reverse pulse time greater than 150 ms, if DS ) 1 × 10-9 m2/s and re ) 10 µm). Regarding the choice of the generating pulse time, this time should be long enough to reach the quasi-steady-state conditions in the generating step and must be greater than the reverse pulse time. For the electrode processes involving the charged substrates, tg should not be less than 100re2/DS. For the uncharged substrates, the value of tg depends strongly on the support ratio. For example, for ξ equal to 0.001, tg must be ∼1500re2/DS (if DS and DP are comparable), while tg equal to 100 re2/DS is satisfactory for ξ > 0.01. Equation 10 can be used for the determination of the diffusion coefficient of either the primary substrate, if DS ≈ DP, or the product, if DS is known but not necessarily equal to DP. However, the nonlinear form of this equation is rather unpractical for these purposes. If the duration time of the generating step, tg, is significantly greater than the reverse pulse length, tr, eq 10 can be linearized with respect to the reciprocal of the square root of the reverse pulse time by expanding it in a series as follows

Ir(ξ,t) IL(ξ,tg)

)

x

DS DP

(

re

x

)

-1

xπDS +

1

1

-

xtg xt - tg -1 DS xπDStg + O(t - t ) 1+

DP

(

)

re

g

(11)

for hemispherical electrodes, and

Ir(ξ,t) IL(ξ,tg)

)

x

(

)

DS πxπDS 1 + DP 2re,d xtg

x

(

-1

1

xt - tg

)

πxπDStg DS 1+ DP 2re,d

-

-1

+ O(t - tg) (12)

for disk microelectrodes. (26) Ewing, A. G.; Dayton, M. A.; Wightman, R. M. Anal. Chem. 1981, 53, 1842. (27) Galus, Z.; Schenk, J. O.; Adams, R. N. J. Electroanal. Chem. 1982, 135, 1.

Analytical Chemistry, Vol. 74, No. 1, January 1, 2002

153

Eq 12 may be rewritten

Ir(ξ,t) IL(ξ,tg)

1

)a

xt - tg

where

a)

-

a

xtg

+ O(t - tg)

(

x

(13)

)

DS πxπDS 1 + DP 2re,d xtg

-1

is the slope of the dependence of Ir(ξ,t)/IL(ξ,tg) versus 1/(t tg)1/2 and O(t - tg) indicates that the error of the linear approximation depends linearly on the reverse pulse time. Having the DS value determined from, for example, the generating step (eqs 6 and 7), one can use the slope of eq 12 to evaluate the diffusivity of the product (DP)

DP )

(

1 πxπ 1 + a2 2re,d xDStg

)

-2

(14)

Figure 2. Experimental double-potential step chronoamperograms and steady-state voltammograms obtained for supporting electrolyte deficiency (ξ ) 0.005, solid lines) and excess supporting electrolyte (ξ ) 50, dashed lines), for the oxidation of ferrocene in acetonitrile. Conditions: cbS ) 2 mM; generating potential, Eg ) 0.8 V for ξ ) 0.005 and 0.65 V for ξ ) 50; reverse pulse potential, Er ) 0.2 V for ξ ) 0.005 and ξ ) 50; supporting electrolyte, N(C4H9)4ClO4; T ) 21 °C.

The error of the evaluation of DP is propagated from three sources: a, re,d, and DS. The expression for the estimate of s(DP) is rather complicated, so the numerical implementation of the propagation of random errors was applied in the calculations.28 It is clear that, for DS ≈ DP, the slope of the plot of Ir(ξ,t)/ IL(ξ,tg) versus 1/( t - tg)1/2 yields the diffusion coefficient of the primary substrate. The final expressions for DS and the estimate of s(DS) are identical to those derived for single-step chronoamperometry (eqs 6 and 8, respectively). RESULTS AND DISCUSSION The following reactions, that represent three classes of electrode processes, have been treated experimentally in our studies:

I. charge production processes (zS ) 0, zP ) +1): Fe(C5H5)2 ) [Fe(C5H5)2]+ + e

(solvent, acetonitrile)

Fe(C5H4)2(CH2OH)2 ) [Fe(C5H4)2(CH2OH)2]+ + e (solvent, acetonitrile) II. charge cancellation process (zS ) -1, zP ) 0): Fe(C5H5)(C5H4)(CH2COO)- ) [Fe(C5H5)(C5H4)(CH2COO)]0 + e

(solvent, water)

III. charge increase process (zS ) +1, zP ) +2): Fe(C5H5)(C5H4)(CH2N(CH3)3)+ ) [Fe(C5H5)(C5H4)(CH2N(CH3)3)]+2 + e (solvent, acetonitrile) Each redox system above gives a one-electron reversible anodic response at Pt microelectrodes. The obtained electrochemical results for ferrocene and the ferrocene derivatives studied are presented in Figures 2-5. In 154

Analytical Chemistry, Vol. 74, No. 1, January 1, 2002

Figure 3. Experimental double-potential step chronoamperograms and steady-state voltammograms obtained for a deficiency of supporting electrolyte (ξ ) 0.005, solid lines) and excess supporting electrolyte (ξ ) 50, dashed lines), for oxidation of 1,1′-ferrocenedimethanol in acetonitrile. Conditions: cbS ) 2 mM; Eg ) 0.7 V for ξ ) 0.005 and 0.6 V for ξ ) 50; Er ) 0.05 V for ξ ) 0.005 and ξ ) 50; supporting electrolyte, N(C4H9)4ClO4; T ) 21 °C.

these figures, typical experimental double-step chronoamperograms and voltammograms, for both a severe deficit (or the absence) and an excess of supporting electrolyte, are shown. An inspection of the steady-state voltammograms of the redox systems investigated reveals that the migration affects them in a different way. For the uncharged substrates, such as ferrocene and 1,1′ferrocenedimethanol, the height of the voltammetric wave plateau is independent of the support ratio.16,21,23 The small differences between the limiting currents recorded under the conditions of the deficit of supporting electrolyte and its excess result from the changes in the solution viscosity. However, at a small support ratio, a significant shift of the voltammograms toward positive potentials is observed. This is understandable, since the magnitude of the ohmic drop in this case is significantly larger than that observed for the processes involving charged substrates.23 In the case of the charged derivatives of ferrocene, the steadystate current at ξ ) 0 is either increased (oxidation of ferroce(28) Hyk, W.; Stojek, Z. Statistical Analysis in Analytical Laboratory; Komitet Chemii Analitycznej PAN: Warszawa, 2000 (in Polish,).

Figure 4. Experimental double-potential step chronoamperograms and steady-state voltammograms obtained for the total absence of supporting electrolyte (ξ ) 0, solid lines) and for excess supporting electrolyte (ξ ) 50, dashed lines), for oxidation of sodium ferroceneacetate in water. Conditions: cbS ) 2 mM; Eg ) 0.5 V for ξ ) 0 and ξ ) 50; Er ) -0.2 V for ξ ) 0 and ξ ) 50; supporting electrolyte, LiClO4; T ) 21 °C.

Figure 5. Experimental double-potential step chronoamperograms and steady-state voltammograms obtained for total absence of supporting electrolyte (ξ ) 0, solid lines) and excess supporting electrolyte (ξ ) 50, dashed lines), for oxidation of ferrocenylmethyltrimethylammonium hexafluorophosphate in acetonitrile. Conditions: cbS ) 2 mM; Eg ) 0.8 V for ξ ) 0 and ξ ) 50; Er ) 0.2 V for ξ ) 0 and ξ ) 50; supporting electrolyte, N(C4H9)4ClO4; T ) 21 °C.

neacetate) or decreased (oxidation of ferrocenylmethyltrimethylammonium) with respect to the diffusional conditions. This is due to the migrational contribution to the transport of the electroactive species. Such the behavior is directly indicated by the NernstPlanck expression describing the flux of the species at the electrode surface

( |

JS ) -DS

∂cS ∂r

r)re

+

| )

zSF ∂Φ c RT S ∂r

r)re

(15)

where r is the radial distance from the electrode surface and Φ is the uncompensated electrostatic potential. The magnitude of the term DS(zSF/RT)cS(∂Φ/∂r) is the measure of the migrational contribution to the total flux and thus the current observed. According to the theoretical and numerical predictions,16,18,21-23 the oxidation (∂Φ/∂r|r)re < 0) of an anion (zS < 0) gives a positive contribution of the migration to the total flux. Therefore, the limiting current will be larger than the diffusion-controlled one, if the supporting electrolyte is not present

in sufficient excess. On the other hand, if ξ is not large enough, the oxidation of a cation will generate a negative migrational contribution to the total flux and thus will decrease the limiting current. The electrochemical behavior of the redox systems investigated supports these predictions. The calculated values of the ratio of the limiting currents for the absence and excess of supporting electrolyte are equal to 1.52 and 0.88 for ferroceneacetate and ferrocenylmethyltrimethylammonium, respectively. Interestingly, the first number deviates much from the theoretical predictions (2.00) while the second one agrees very well with the theory (0.85).23 The deviation obtained for the first number might be, in our view, related to the difference between diffusivities of the substrate and its counterion (Na+) and to the existence of the opposite charges within the molecule of the product. The latter certainly makes the behavior of such the dipole-like molecules different from that of the uncharged molecules and, in consequence, makes it questionable to classify this reaction as the charge cancellation process. However, it is important that the unclear mechanism of the migrational transport does not invalidate the application of the proposed approach for the determination of the diffusion coefficients. Tables 1 and 2 contain the regression coefficients of the dependencies of the normalized currents (IL(ξ,t)/IL(ξ,tg) for the generating step (Table 1) and Ir(ξ,t)/IL(ξ,tg) for the reverse step (Table 2)) on 1/t1/2 and 1/(t - tg)1/2, respectively, for both a severe deficit (or the total absence) and an excess of supporting electrolyte. The exemplary set of the linearized chronoamperometric dependencies for ferrocenylmethyltrimethylammonium is shown in Figure 6. The linear regression was used to compute the slope, a, the intercept, b, and the corresponding standard deviations, s(a) and s(b), of these dependencies over the transient time region considered: 0.2-10 s and 0.2-2 s for the first and the second steps, respectively. In most cases, the theoretical value of the intercept is located inside the confidence interval of the experimentally determined intercept, b ( ts(b), where t is the critical value of Student’s t distribution at 95% confidence level. This confirms statistically the applicability of eqs 4 and 12 in the chosen time intervals. In general, the linearity of the experimental relations is very good, since the generalized correlation coefficient (r2) departs from 1 by less than 0.001. The values of the diffusion coefficients of the considered ferrocene derivatives determined chronoamperometrically and those obtained from the steady-state diffusional current are collected in Table 3. The numbers obtained for the substrate demonstrate that, in general, the diffusion coefficients determined by single-step chronoamperometry in the unsupported and fully supported systems are in a good agreement with those calculated from the steady-state diffusional current. The largest differences observed are the following: 1.9, 14.6, 3.9 and 11.8, and -5.0 and -14.4% for ferrocene, 1,1′-ferrocenedimethanol, ferroceneacetate, and ferrocenylmethyltrimethylammonium, respectively. On the other hand, the DS values obtained chronoamperometrically under diffusional and mixed diffusion migration conditions are well reproducible with the coefficient of variation not higher than 5.5%. The DS values obtained were used for the determination of the product diffusivity according to eq 14. The results are included in Table 3. It is not surprising that they are similar to the Analytical Chemistry, Vol. 74, No. 1, January 1, 2002

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Table 1. Regression Coefficients (Slope, Intercept, Correlation Coefficient) of the Linearized Chronoamperometric Relations for the Generating Step (Eq 4) Used for the Determination of Diffusion Coefficients of Some Ferrocene Derivatives ξ)0 substrate (solvent) ferrocene (acetonitrile) 1.1′-ferrocenedimethanol (acetonitrile) ferroceneacetate (water) ferrocenylmethyltrimethylammonium (acetonitrile)

a ( s(a), s1/2

0.168 ( 3.3 × 10-5 0.099 ( 9.5 × 10-5

ξ ) 50

b ( s(b)

0.947 ( 2.0 × 10-5 0.968 ( 5.7 × 10-5

r2

a ( s(a), s1/2

b ( s(b)

r2

0.9998 0.9982

0.083 ( 6.2 × 10-5 0.095 ( 4.7 × 10-5 0.174 ( 5.6 × 10-5 0.095 ( 8.2 × 10-5

0.975 ( 3.7 × 10-5 0.968 ( 2.8 × 10-5 0.940 ( 3.4 × 10-5 0.967 ( 4.9 × 10-5

0.9989 0.9995 0.9994 0.9986

Table 2. Regression Coefficients (Slope, Intercept, Correlation Coefficient) of the Linearized Chronoamperometric Relations for the Reverse Step (Eq 12) Used for the Determination of Diffusion Coefficients of the Oxidation Products of Some Ferrocene Derivatives ξ)0 substrate (solvent)

a ( s(a), s1/2

ferrocene (acetonitrile) (0.084 ( 1.5 × 10-4 1.1′-ferrocenedimethanol (acetonitrile) (0.092 ( 1.7 × 10-4 ferroceneacetate (water) 0.166 ( 1.1 × 10-4 ferrocenylmethyltrimethylammonium 0.097 ( 1.1 × 10-4 (acetonitrile) a

ξ ) 50

b ( s(b)

r2

a ( s(a), s1/2

b ( s(b)

r2

-0.030 ( 1.8 × 10-4 -0.039 ( 2.2 × 10-4 -0.026 ( 1.4 × 10-4 -0.037 ( 1.4 × 10-4

0.9988)a 0.9987)a 0.9992 0.9995

0.085 ( 6.9 × 10-5 0.094 ( 5.4 × 10-5 0.176 ( 9.4 × 10-5 0.100 ( 7.5 × 10-5

-0.034 ( 8.5 × 10-5 -0.033 ( 6.8 × 10-5 -0.018 ( 1.2 × 10-4 -0.037 ( 9.3 × 10-5

0.9998 0.9999 0.9995 0.9998

ξ ) 0.005.

Figure 6. Normalized chronoamperometric currents (i.e., IL(ξ,t)/IL(ξ,tg) for generating step and Ir(ξ,t)/IL(ξ,tg) for reverse step, see the text for details) plotted against 1/t1/2 and 1/(t - tg)1/2, respectively, for ferrocenylmethyltrimethylammonium hexafluorophosphate in acetonitrile containing no or excess supporting electrolyte. Regression coefficients can be found in Tables 1 and 2. Conditions as in Figure 5.

corresponding DS values, since the sizes of both oxidized and reduced forms of the compounds studied are rather large and practically the same. For the systems investigated, the transient techniques give the data of standard deviation slightly larger than those obtained from the excess supporting electrolyte steady-state voltammetry. 156 Analytical Chemistry, Vol. 74, No. 1, January 1, 2002

CONCLUSIONS We have shown that double-potential step chronoamperometry which, in fact, consists of two coupled transient experiments, may provide the complete information on the transport properties of the studied redox couple, irrespective of the level of ionic support present in the system. The diffusion coefficients of both the

Table 3. Diffusion Coefficients of Some Ferrocene Derivatives and Their Oxidation Products Determined Chronoamperometrically for Unsupported and Fully Supported Systems

(DS ( s(DS))a × 109, m2/s DPSC generating step ξ)0 substrate (solvent) ferrocene (acetonitrile) 1.1′-ferrocenedimethanol (acetonitrile) ferroceneacetate (water) ferrocenylmethyltrimethylammonium (acetonitrile) a

eq 6

0.49 ( 0.02 1.46 ( 0.05

(DP ( s(DP))a × 109, m2/s DPSC reverse step ξ)0

ξ ) 50

eq 7

eq 6

ξ ) 50 eq 7

eq 14

eq 14

0.49 ( 0.02 1.46 ( 0.05

2.11 ( 0.08 1.59 ( 0.06 0.45 ( 0.02 1.59 ( 0.06

2.12 ( 0.08 1.58 ( 0.06 0.45 ( 0.02 1.58 ( 0.06

2.06 ( 0.07c 1.71 ( 0.06c 0.50 ( 0.02 1.53 ( 0.05

2.02 ( 0.07 1.64 ( 0.06 0.44 ( 0.02 1.45 ( 0.05

(DS ( s(DS))a × 109, m2/s steady-state conditions b ξ ) 50 2.15 ( 0.05 1.85 ( 0.04 0.51 ( 0.01 1.39 ( 0.03

At T ) 21 °C. b DS ) ILss/(4Fre,dcbS). c ξ ) 0.005.

substrate and the product can be evaluated sequentially from the appropriate parts of the double-potential step experiment. It has been shown that the magnitude of the normalized chronoamperometric current of the first step depends on the DS value, while that of the second step is controlled by both DS and DP values. Having the substrate diffusivity determined from the first step, one can use it for the evaluation of DP from the second step. The computation of the slope of the linearized relation between the appropriate currents ratio and the reciprocal of the square root of time of either the first or the second step, and the knowledge of the microelectrode radius, are just everything that this approach requires. The proper choice of the time region and accurate and precise determination of the electrode radius are the key factors leading to the reliable results. As has been mentioned, the procedures based on the treatment of the generating and the reverse chronoamperometric currents are, in general, applicable to any transport-controlled electrode process carried out under either the purely diffusional or mixed diffusion-migration conditions. The processes involving the uncharged substrates should be rather excluded from the approach based on the single-step chronoamperometry, since in this case, the region where the current decays with 1/t1/2 may be quite small (or even absent) when the support ratio is less than 0.001.

However, these processes can be successfully treated with the double-step method to get the diffusion coefficients of the products. According to the numerical calculations, one should obtain the most accurate results for the determination of both DS and DP for one-electron processes involving both reactants charged. The proposed methods have been examined experimentally with charged and uncharged ferrocene derivatives under diffusional and mixed diffusion-migration conditions. In general, the largest discrepancy between the average values of DS obtained from the chronoamperometric experiments and the excess supporting electrolyte steady-state voltammetric experiments does not exceed 14.6%. ACKNOWLEDGMENT This work was partially supported by Grant 3 T09A 146 19 from KBN, the Polish State Committee for Scientific Research, and by a Warsaw University Grant BW-1483/7/2000.

Received for review August 14, 2001. Accepted October 16, 2001. AC0109117

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