Iterative Absolute Electroanalytical Approach to Characterization of

The iterative approach is based on successive approximations through calculation and minimizing the least-squares error function. The method is fairly...
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Anal. Chem. 2004, 76, 2694-2699

Iterative Absolute Electroanalytical Approach to Characterization of Bulk Redox Conducting Systems Adam Lewera, Krzysztof Miecznikowski, Malgorzata Chojak, Oktawian Makowski, Jerzy Golimowski, and Pawel J. Kulesza*

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

A novel electroanalytical approach is proposed here, and it is demonstrated with the direct and simultaneous determination of two unknowns: the concentration of redox sites and the apparent diffusion coefficient for charge propagation in a single crystal of dodecatungstophosphoric acid. This Keggin-type polyoxometalate serves as a model bulk redox conducting inorganic material for solid-state voltammetry. The system has been investigated using an ultramicrodisk working electrode in the absence of external liquid supporting electrolyte. The analytical method requires numerical solution of the combination of two equations in which the first one describes current (or charge) in a well-defined (either spherical or linear) diffusional regime and the second general equation describes chronoamperometric (or normal pulse voltammetric current) under mixed (linear-spherical) conditions. The iterative approach is based on successive approximations through calculation and minimizing the least-squares error function. The method is fairly universal, and in principle, it can be extended to the investigation of other bulk systems including sol-gel processed materials, redox melts, and solutions on condition that they are electroactive and well behaved, they contain redox centers at sufficiently high level, and a number of electrons for the redox reaction considered is known. There has been a great deal of interest in electrochemical measurements of solid, rigid, and semirigid (nonfluid) systems that can be performed in the absence of liquid electrolyte phase normally at room temperature.1-5 The applicable materials shall serve as electrolyte medium (i.e., they host a sizable population of mobile counterions) and contain electroactive centers. Solid* Corresponding author. E-mail: [email protected]. (1) (a) Hara, M., In Polyelectrolytes, Science and Technology; Hara, M., Ed.; Marcel Dekker: New York, 1993. (b) Armand, M. Solid State Ionics 1983, 10, 1161. (2) (a) Murray, R. W., Ed. Molecular Design of Electrode Surfaces; Wiley: New York, 1992. (b) Surridge, N. A.; Jernigan, J. C.; Dalton, E. F.; Buck, R. P.; Watanabe, M.; Zhang, H.; Pinkerton, M.; Wooster, T. T.; Longmire, M. L.; Facci, J. S.; Murray, R. W. Faraday Discuss., Chem. Soc. 1989, 88, 1. (c) Dalton, E. F.; Surridge, N. A.; Jernigan, J. C.; Wilborn, K. O.; Facci, J. S.; Murray, R. W. Chem. Phys. 1990, 141, 143. (d) Williams, M. E.; Crooker, J. C.; Pyati, R.; Lyons, L. J.; Murray, R. W. J. Am. Chem. Soc. 1997, 119, 10249. (e) Kulesza, P. J.; Malik, M. A., In Interfacial Electrochemistry, Theory Experiment and Applications; Wieckowski A., Ed; Marcel Dekker: New York, 1999. (f) Kulesza, P. J.; Cox, J. A. Electroanalysis 1998, 10, 73.

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state voltammetric experiments can be typically performed on electroactive organic polymers containing fixed redox sites,2a-c mixed-valence ionically conducting inorganic compounds,2e,f,3 solid solutions,2b,c redox melts,5 sol-gel processed structures,2f,4f,i-k,6 and systems containing reactive centers.3c,4f,l From the mechanistic viewpoint, the materials must be redox conducting:2a-c they contain macroscopically immobile redox centers between which the electrons are transported via hopping (self-exchange), and to satisfy the electroneutrality requirement, the electron transfers are accompanied by displacements of counterions. By analogy to electrodes modified with redox polymer films,2a,3c the overall charge propagation mechanism can be described in terms of effective or apparent diffusion with the same diffusion coefficient for both oxidized and reduced states. This mechanism is somewhat more complex in porous gels,2g,4f,i-k,6 where significant contribution from physical diffusion is expected, or in conducting (3) (a) Kittleson, G. P.; White, H. S.; Wrighton, M. S. J. Am. Chem. Soc. 1985, 107, 7373. (b) Abruna, H. D.; Bard, A. J. J. Am. Chem. Soc. 1981, 103, 6898. (c) Bard, A. J.; Abruna, H.; Chidsey, C. E.; Faulkner, L. R.; Feldberg, S.; Itaya, K.; Majda, M.; Melroy, O.; Murray, R. W.; Porter, M. D.; Soriaga, M. P. White, H. S. J. Phys. Chem. 1993, 97, 7147. (d) Peerce, P. J.; Bard, A. J. J. Electroanal. Chem. 1980, 114, 89. (e) Mirkin, C. A.; Wrighton, M. S. J. Am. Chem. Soc. 1990, 112, 8596. (f) Fritsch-Faules, I.; Faulkner, L. R. J. Electroanal. Chem. 1989, 263, 237. (g) Saveant, J. M. J. Electroanal. Chem. 1988, 242, 1. (4) (a) Kulesza, P. J.; Faulkner, L. R. J. Am. Chem. Soc. 1993, 115, 11878. (b) Kulesza, P. J.; Karwowska, B. J. Electroanal. Chem. 1996, 401, 201. (c) Kulesza, P. J.; Karwowska, B.; Malik, M. A. Colloids Surf., A 1998, 134, 173. (d) Kulesza, P. J. Inorg. Chem. 1990, 29, 2395. (e) Kulesza, P. J.; Cox, J. A. Electroanalysis 1998, 10, 73. (f) Cox, J. A.; Alber, K. S.; Brockway, C. A.; Tess, M. E.; Gorski, W. Anal. Chem. 1995, 67, 933. (g) Kulesza, P. J.; Malik, M. A.; Denca, A.; Strojek, J. Anal. Chem. 1996, 68, 2442. (h) Cox, J. A.; Alber, K. S.; Tess, M. E.; Cummings, T. E.; Gorski, W. J. Electroanal. Chem. 1995, 396, 485. (i) Holmstrom, S. D.; Karwowska, B.; Cox, J. A.; Kulesza, P. J. J. Electroanal. Chem. 1998, 456, 239. (j) Cox, J. A.; Wolkiewicz, A.; Kulesza, P. J. J. Solid State Electrochem. 1998, 2, 247. (k) Alber, K. S.; Cox, J. A.; Kulesza, P. J. Electroanalysis 1997, 9, 97. (l) Kulesza, P. J.; Faulkner, L. R. J. Electrochem. Soc. 1993, 140, L66. (m) Gorski, W.; Cox, J. A. J. Electroanal. Chem. 1992, 323, 163. (n) Sinru, L.; Osteryoung, J.; O'Dea, J. J.; Osteryoung, R. A. Anal. Chem. 1988, 60, 1135. (5) (a) Williams, M. E.; Masui, H.; Long, J. W.; Malik, J.; Murray, R. W. J. Am. Chem. Soc. 1997, 119, 1997. (b) Long, J. W.; Kim, I. K.; Murray, R. W. J. Am. Chem. Soc. 1997, 47, 111510. (c) Richtie, J. E.; Murray, R. W. J. Am. Chem. Soc. 2000, 122, 2964. (d) Kulesza, P. J.; Dickinson, E.; Williams, M. E., Hendrickson, S., M.; Malik, M. A.; Miecznikowski, K.; Murray, R. W. J. Phys. Chem. B 2001, 105, 5833. (6) (a) Howells, A. R.; Zambrano, P. J.; Collinson, M. M.; Anal. Chem. 2000, 72, 5265. (b) Collinson, M. M., Taussig, J.; Martin, S. A. Chem. Mater. 1999, 11, 2594. (c) Opallo, M. Kukulka-Walkiewicz, J. Electrochim. Acta 2001, 46, 4235. (d) Petrovic, S. C.; Zhang, W.; Ciszkowska, M. Anal. Chem. 2000, 72, 3449. 10.1021/ac030358o CCC: $27.50

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polymers, where polarons, or bipolarons, in addition to electron hopping between adjacent defects have both been proposed as charge carriers.7 Electroanalytical characterization of redox conducting materials1-8 involves reliable estimation of both concentration (Co) of redox centers and apparent diffusion coefficient (Dapp) for charge propagation. Typical methods for determination of Dapp, which include potential step methods2b-d and impedance spectroscopy,8j require the knowledge of Co. Unfortunately, direct determination or reliable estimation of this parameter is often not possible. In such situations, a search for absolute analytical approaches that permit simultaneous determination of Co and Dapp is necessary. Recently, there have been published a few such approaches in which the possibility of eliminating the necessity of knowing Co prior to the experiment is indicated.9 One of them9a is based on the normalization of the diffusional chronoamperometric current-time response at the ultramicrodisk electrode with respect to the steady-state current obtained at the end of the potential step experiment. This method has been successfully used to determine Dapp in sol-gel processed silicate gels containing iron complex redox centers.9b Other approaches developed for species dissolved in liquid solutions are based on comparison of transient data with steady-state data9c or utilization of the generator-collector electrode system.9d More recently, a method9e for direct determination of diffusion coefficients has been developed on the basis of the numerical model of migrational chronoamperometry and applied to systems of any level of ionic support including the absence or the excess of supporting ions. This very useful approach still requires the steady-state limiting current conditions to be achieved. An attractive possibility appears when the electrochemical responses of a system characterized by diffusional charge propagation are amenable to experimental time domains that include both limits of the linear diffusion and the spherical charge transport.2e,f4a-e,i,j Then the absolute analytical problem resolves itself into the system of two equations involving two unknowns (Dapp and Co). The approach is applicable to well-behaved redox conducting systems in which no significant migration and ohmic limitations exist. The method requires each limit of linear and (7) (a) Inzelt, G.; Pineri, M.; Schultze, J. W.; Vorotyntsev, M. A. Electrochim. Acta 2000, 45, 2403. (b) Rubinson, J. F.; Mark, H. B. Jr. In Interfacial Electrochemistry; Wieckowski, A. Ed.; Marcel Dekker: New York, 1999; Chapter 37. (c) Inzelt, G. In Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker: New York, 1994; Vol. 18. (8) (a) Rickert, H. Electrochemistry of Solids; Springer-Verlag: Berlin, 1982. (b) Conway, B. E.; Birss, V.; Wojtowicz, J. J. Power Sources 1997, 66 1. (c) Conway, B. E. J. Electrochem. Soc. 1991, 138, 1539. (d) Monk, P. M. S.; Mortimer, R. J.; Rosseinsky, D. R. Electrochromism, Fundamentals and Applications; VCH: Wienheim, 1995. (e) Mitzi, D. B. Chem. Mater. 2001, 13, 3283. (f) O’Reagan, B.; Gratzel, M. Nature 1991, 353, 373. (g) DePaoli, M.-A.; Nogueira, A. F.; Machado, D. A.; Longo, C. Electrochim. Acta 2001, 46, 4243. (h) Oliver, B. N.; Coury, L. A.; Egekeze, J. O.; Sosnoff, C. S.; Zhang, Y.; Murray, R. W.; Keller, C.; Umana, M. X. In Biosensor Technology: Fundamentals and Applications; Buck, R. P., Hatfield, W. E., Umana, M. X., Bowden, E. F., Eds.; Dekker: New York, 1990. (i) Masui, H.; Murray, R. W. J. Electrochem. Soc. 1998, 145, 3788. (j) Levi, M. D.; D. Aurbach, D. J. Phys. Chem. B 1997, 101, 4641. (9) (a) Denuault, G.; Mirkin, M. V.; Bard, A. J. J. Electroanal. Chem. 1991, 308, 27. (b) Collinson, M. M.; Zambrano, P. J.; Wang, H.; Taussig, J. S. Langmuir 1999, 15, 662. (c) Amatore, C.; Azzali, M.; Calas, P.; Jutand, A.; Lefrou, C.; Rollin, Y. J. Electroanal. Chem. 1990, 288, 45. (d) Licht, S.; Cammarata, V.; Wrighton, M. S. J. Phys. Chem. 1990, 94, 6133. (e) Hyk, W.; Nowicka, A.; Stojek, Z. Anal. Chem. 2002, 74, 149. (f) Ikeuchi, H.; Kanakubo, M. J. Electroanal. Chem. 2000, 493, 93.

spherical diffusional patterns to be achieved. Unfortunately, in the case of rigid or semirigid media characterized by the relatively slow dynamics of charge transport,2e,f5h very often impractically long experimental time scales would have to be used for a radial (spherical) diffusional pattern to be observed. In the present work, we pursue the absolute concept of two equations with two unknowns and propose a more general and powerful approach that combines a mixed (liner-radial) diffusional experiment with the result of a measurement under purely linear or radial diffusional model. While using chronoamperometry or normal pulse voltammetry (NPV), precise description of currents under mixed diffusional control is possible.10 By combining chronoamperometric or NPV current with the result of such electrochemical measurement as short-pulse chronocoulometry or longpulse chronoamperometry performed under the control of a purely linear or radial diffusional pattern, we can calculate simultaneously Dapp and Co. The procedure requires common tools of numerical (iterative) minimizing of the error function according to the nonlinear least-squares approach. Special attention is paid to the importance of diagnostic voltammetric and potential step experiments. We demonstrate the usefulness of our analytical approach by considering outer-sphere redox reactions in a model inorganic (Keggin-type polyoxometalate) material,2e,f,4a-c,11 a single crystal of dodecatungstophosphoric acid. It has already been established11a,b that, up to injection of four electrons, the heteropoly-12-tungstate “rigid” array remains capable of fast electron intersite transfers between mixed-valence W(VI,V) redox (ionic) sites. The actual electron transfer originates from 1 of the 12 structurally equivalent tungsten atoms, and it is readily coupled to the displacement of proton (counterion). H3PW12O40‚29H2O contains large amounts of exceptionally mobile, hydrated, protons (H3O+ or H5O2+) within its hydrated so-called “pseudoliquid” structure.11b,c Thus, electroneutrality is maintained during the system’s redox reactions. Although tungsten redox sites are diffusively immobile, the overall mechanism of charge propagation in single crystals of heteropoly12-tungstic acids can be described in terms of effective or apparent diffusion. EXPERIMENTAL SECTION All solutions were prepared from commercially available 12tungstophosphoric acid from Fluka (purity >99%). Tetragonal phospho-12-tungstic acid single crystals, H3PW12O40‚29H2O,4b,c were grown slowly, i.e., over 3-6 days, at ambient temperature in small beakers containing 5-10 cm3 of a highly concentrated solution, ∼0.8 mol dm-3, of the polyacid. As a rule, crystallization was repeated three times. Gold ultramicrodisk electrodes having a diameter of 10 µm were, before use, activated by polishing on a cloth with water slurry of 0.05-µm alumina (from Buehler, Chicago, IL). All measurements were performed at ambient temperature, 20 ( 2 °C. (10) (a) Aoki, K.; Osteryoung, J. J. Electroanal. Chem. 1981, 122, 19. (b) Schoup, D.; Shabo, A. J. Electroanal. Chem. 1982, 140, 237. (c) Schoup, D.; Shabo, A. J. Electroanal. Chem. 1984, 160, 1. (d) Sinru, L.; Osteryoung, J.; O’Dea, J. J.; Osteryoung R. A. Anal. Chem. 1988, 60, 1135. (11) (a) Pope, M. T. In Mixed-Valence Compounds: Theory and Application in Chemistry, Physics, Geology and Biology; Brown, D. B., Ed.; NATO Advanced Study Institute Series C, Vol. 58; Reidel: Dordrecht, 1980; p 365. (b) Pope, M. T. Heteropoly and Isopoly Oxometalates; Springer-Verlag: New York, 1983; Chapters 2 and 3. (c) Misono, M.; Okuhara, T.; Ichiki, T.; Arai, T.; Kanda, T. J. Am. Chem. Soc. 1987, 109, 5535.

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for any value of t on condition that an appropriate equation (3 or 4) is used. The above formulations are also applicable to the description of NPV currents with respect to pulse time (t).10d An alternate description of currents10b was based on a fit to the shortand long-time analytical solutions of eqs 3 and 4 for chronoamperometric responses at the ultramicrodisk electrode,

f (τ) ) 0.7854+0.8862τ-1/2+ 0.2146 exp(-0.7823τ-1/2) (5)

Figure 1. Schematic diagram of the three-electrode solid-state electrochemical cell utilizing an ultramicrodisk electrode.

Voltammetric measurements were carried out using a CH Instruments (Austin, TX) model 660 electrochemical analyzer. We utilized two types of all-solid, three-elelctrode cells for solid-state electrochemical measurements: the unit described earlier,4c in which a silver disk served as a semireference system, and a new design illustrated in Figure 1. Two glassy carbon half-disks were separated (with insulator) from each other by ∼50 µm (Figure 1). One of the half-disks was covered with a thin film of silver and served as a semireference; the other one was bare and acted as a counter electrode. A gold ultramicrodisk electrode was mounted opposite the reference/counter plane. The single crystal was positioned in the center of the cell. The whole assembly was enclosed in glass tubing and sealed to protect the crystal from partial dehydration. To ensure good contact with the crystal, a weight (brass ring, ∼30 g) was introduced onto the utramicroelectrode. The fact that the results were reproducible implies that full contact must have always been achieved. THEORETICAL SECTION Fundamental Chronoamperometric Equations. In general, regardless of the diffusion type, chronoamperometric current (I) recorded at a planar disk electrode is expressed using equations10a-c written below:

I ) 4nFDapprCo f (τ)

(1)

where n is a number of electrons involved and

τ ) 4Dappt/r2

(2)

is a dimensionless time variable (parameter). Other parameters have already been mentioned or have usual significance. Depending on τ value, the f (τ) function can be expressed in two forms10a-b

f (τ) ) 1 + 0.71835τ-1/2 + 0.05626τ-3/2+ ...

(3)

f (τ) ) [π/(4τ)]1/2 + π/4 + 0.1τ1/2 + ...

(4)

and

for τ values larger (eq 3) and smaller (eq 4) than 1, namely, for the radial and linear dominating diffusional patterns, respectively. Consequently, chronoamperometric current (I) can be described 2696

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The resulting equation was exact, in comparison to eqs 3 and 4 within 0.6% at all time (τ) values. Currents can be analyzed by fitting current-time transients to this relation. but in practice. the successful determination of Co and Dapp would require at least approximate knowledge of either Co or Dapp. The analogous approach,9b based on numerical solutions of equations describing changes in concentration in discrete space near the electrode surface, can also give good analytical results with respect to the diffusion coefficient determination. But the method requires the use of sophisticated software and the knowledge of Co as well. Other approaches9a,b are based on deleting third terms in eqs 3 and 4 and by linearization of eqs 1-4 as a function of time. Thus, the chronoamperometric responses at ultramicrodisk electrode are normalized with respect to steady-state current, Iss, characteristic of radial diffusion

Iss ) 4nFDappCor

(6)

that is developed under long-pulse chronoamperometric conditions or at slow scan rates. In the case of solid-state redox conducting systems that are characterized by fairly low diffusion coeficients, Iss values are often difficult to obtain. Proposed Approach. The procedure proposed here refers to our previous electroanalytical methodology based on clearly defined two time regimes4a-c of linear and radial diffusional patterns. Now, instead of considering two equations in both limits of the radial transport and linear diffusion, we propose to combine the first equation in a well-defined (either radial or linear) diffusional regime with the second general equation describing chronoamperometric current under mixed diffusional conditions (eq 1). The first equation shall provide an unequivocal description at either diffusion limit, e.g., in a form of either chronoamperometric steady-state (Iss) current (eq 6), if the radial diffusion case applies, or as chronocoulometric slope (a) of the charge (Q) versus square root of time (t1/2) according to the integrated Cottrell equation

a ) Q/t1/2 ) 2nFπ1/2r2Dapp1/2Co

(7)

if the linear diffusional limit is considered. Analytical solution of the combination of the respective equation with the general equation involving mixed diffusion requires numerical approach because of the complexity of function f (τ) and its dependence on τ (eqs 1-4). In other words, it is determined from the calculation of τ which equation (3 or 4) must be used. Use of Chronocoulometric Slope from Linear Diffusion. Typical redox conducting materials for solid-state voltammetry are characterized by rather low Dapp (40 V s-1), kinetic limitations seem to appear, and the overall description of volatmmetric peak currents becomes more complex. The voltammetric data of Figures 2 and 3 show that both extreme and mixed diffusional cases are feasible. For quantitative (12) Bard, A. J.; Faulkner, L. R. Electrochemical Methods, Fundamentals and Applications, 2nd ed.; Wiley: New York, 2001.

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Figure 4. Chronocoulometric responses of PW12 single crystal recorded upon application of (A) 16 and (B) 200 ms potential steps from 0.0 to -0.55 V, i.e., over the first, most positive, system’s reduction process.

Figure 5. Chronoamperometric responses of PW12 single crystal recorded upon application of potential steps from 0.0 to -0.55 V lasting for (A) 250 ms and (B) 100 s.

purposes, it is more convenient to utilize potential step techniques. Although the kinetic data can be derived from short-pulse chronoamperograms, or more precisely from the current versus reciprocal of the square root of time responses, we have found that the analogous chronocoulometric dependencies of charge (Q) versus t1/2 (Figure 4) are particularly useful to visualize and to diagnose the case of linear diffusion. Judging from the existence of linear portions in the chronocoulometric plot of Figure 4A, the linear diffusional pattern has been a case here. From the linear portion of the dependence of Q on t1/2, one can determine the slope, a (eq 7). It should be remembered that the Q - t1/2 chronocoulometric plot permits one to diagnose easily possible ohmic and kinetic limitation effects that produce a sizable negative intercept.2e Judging from the data of Figure 4A, the above limitations can be neglected. When, a potential step is sufficiently long to ensure a spherical contribution to linear diffusion, a positive deviation from linearity appears (Figure 4B). Finally, the pattern of radial diffusion can be easily diagnosed using the double potential step long-pulse chronoamperometric experiment (Figure 5B). An important feature of the Figure 5B data is that the forward potential step (100 s) features a steady-state current whereas the current decays rapidly to zero in the reverse pulse because the electrolysis products diffuse away from the surface of an ultramicrodisk electrode. The application of much shorter 250-ms potential step would lead to mixed diffusional pattern (Figure 5A) because, as expected from eq 2, τ ) 1.6, i.e., close to 1. Under such conditions, the chronoamperometric currents can readily be described using eqs 1 and 3 or 4. It is noteworthy that the abovementioned equations are also applicable to the description of NPV plateau currents. Thus, a single chronoamperometric experiment under mixed diffusional conditions (Figure 5A) may be replaced by a series of measurements of the NPV platteau currents at different pulse times (Figure 6). The latter procedure is lengthier,

Figure 6. Solid-state NPV responses of PW12 single crystal recorded upon application of the following pulse times: (A) 5, (B) 10, (C) 15, (D) 30, (E) 100, and (F) 1000 ms.

Figure 7. Dependencies of error functions on concentration for the combinations of the mixed diffusion chronoamperometric currents with (A) the linear diffusion chronocoulometric slope and (B) spherical diffusion steady-state current.

but it allows visualization and possible separation of consecutively appearing processes. Our absolute analytical approach to calculate Dapp and Co requires combining the chronocoulometric slope, a (determined under linear diffusion), or chronoamperometric steady-state current, Iss (determined under radial diffusion), with mixed diffusional current (see eqs 8-12 and 14-16, respectively). As mentioned in the previous section, the solution is achieved through successive approximations (nonlinear least-squares method) that involve minimizing error function (eq 13). Figure 7 illustrates two dependencies of error function on concentration for the combinations mentioned above with (A) eq 9 and (B) eq 15, i.e., involving linear and spherical diffusions, respectively. The error functions (fe) are minimized over the arbitrary chosen concentration range, and in both cases, only one minimum has been found. Thus, the true concentration (Co) can be determined. The analogous error function dependencies on concentration are obtained (for simplicity not shown here) when, instead of the chronoamperometric data, NPV currents under mixed diffusional control are considered. In the latter case, relatively less experimental points are, in practice, available. Here, we have considered 26 NPV-currents recorded at distinct pulses ranging from 5 to 1000 ms. Once Co is known, the corresponding value of Dapp can be calculated either from eq 8 or 14, depending on availability of pure linear or radial diffusional conditions. The following parameters have been determined from the case involving combination of mixed diffusion chronoamperometry with linear diffusion chronocoulometry, Co ) 1.5 ((0.17) mol dm-3 and Dapp ) 3.4 ((0.5) × 10-7 cm2 s-1, and with spherical diffusion chronoamperometry, Co ) 1.4 ((0.15) mol dm-3 and Dapp ) 3.2 ((0.3) × 10-7 cm2 s-1.

Figure 8. Comparison of experimental and simulated chronoamperometric current values for PW12 single crystal.

The numbers in parentheses refer to standard deviations based on at least 10 independent experiments. Substitution of mixed diffusional chronoamperometry with NPV measurements leads to virtually the same results. On average, we can say that Co ) 1.45 mol dm-3 and Dapp ) 3.35 × 10-7 cm2 s-1. The results are in agreement with the previously reported data,4b,c Dapp ) 3.6 × 10-7 cm2 s-1 and Co ) 1.4 mol dm-3. To verify the correctness of our absolute analytical approach, we compared the simulated chronoamperometric currents (calculated using eqs 1-4 and the Co and Dapp values determined above) with the chronoamperometric experimental points measured at the same time (Figure 8). Almost ideal agreement between these two sets of data shall be noted. We also simulated cyclic voltammograms using the Digisim program (BAS, West Lafayette, IN): the responses are almost identical to those obtained experimentally (e.g., as for Figure 3A and B). The consistency between the experimental and simulated data supports the reliability of the assumptions made and justifies the use of our electroanalytical approaches. Finally, we have evaluated our analytical approach using a standard, highly concentrated, solution of PW12 (from which the single crystals were grown). This solution was prepared by dissolving a known amount of PW12 powder to obtain a concentration of 0.80 mol dm-3. Using the approach based on combination of radial and mixed diffusional patterns, we have determined Dapp ) 8.5 ((0.3) × 10-7 cm2 s-1 and Co ) 0.81 ((0.15) mol dm-3. Since both electron hopping and physical diffusion contribute to the overall charge propagation mechanism, Dapp is now higher than in the case of PW12 single crystal. In conclusion, it is reasonable to expect that the method is fairly universal, and it can be extended to determination of Co and Dapp in various bulk systems including sol-gel processed materials, redox melts, and electroactive polymers. The approach is of potential utility to characterization of materials for batteries and redox capacitors,8a-c electrochromic and luminiscent displays,8d,e charge relays for photoelectrochemical energy conversion,8f,g amperometric sensors for gaseous analytes,3e,4l,8h and molecular electronics systems.3e,8i ACKNOWLEDGMENT This work was supported by State Committee for Scientific Research (KBN), Poland under Grant 3 T09A 05426.

Received for review October 14, 2003. Accepted March 4, 2004. AC030358O Analytical Chemistry, Vol. 76, No. 10, May 15, 2004

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