J. Phys. Chem. B 2001, 105, 5833-5838
5833
Electron Self-Exchange Dynamics of Hexacyanoferrate in Redox Polyether Hybrid Molten Salts Containing Polyether-Tailed Counterions Pawel J. Kulesza,*,†,‡ Enders Dickinson V,† Mary Elizabeth Williams,† Susan M. Hendrickson,† Marcin A. Malik,‡,§ Krzysztof Miecznikowski,‡ and Royce W. Murray*,† Kenan Laboratories of Chemistry, UniVersity of North Carolina, Chapel Hill, North Carolina 27599-3290, and Department of Chemistry, UniVersity of Warsaw, Pasteura 1, PL-02-093 Warsaw, Poland ReceiVed: NoVember 30, 2000; In Final Form: March 20, 2001
Hexacyanoferrate(III) is combined with a quaternary ammonium countercation consisting of triethylammonium connected to a poly(ethylene glycol) methyl ether (MW 350) “tail”, to form a highly viscous room-temperature redox polyether hybrid melt (e.g., a molten salt) in which the concentration of hexacyanoferrate centers is 0.82 M. Microelectrode voltammetry and potential step chronoamperometry in the undiluted melt give an apparent diffusion coefficient DAPP ) 2.5 × 10-10 cm2/s at 20 °C that is interpreted as reflecting primarily the rate of electron self-exchange between Fe(II) and Fe(III) centers. A rate constant of kEX ) 1.1 × 105 M-1 s-1 is derived from this DAPP, and from its temperature dependence, an activation energy barrier of 30 kJ/ mol. kEX is in good agreement with results in fluid solutions. At the same concentration (0.82 M), but in aqueous solution, the (potassium salt) hexacyanoferrate species displays a DAPP of 4 × 10-6 cm2/s, which is interpreted as reflecting physical transport of the hexacyanoferrate species. Transport of the hexacyanoferrate species is enormously “plasticized” in aqueous medium as opposed to the highly viscous polyether melt. Electronic spectra and ionic conductivity of the hybrid redox polyether melt are also reported.
Introduction Combinations of well-defined electron donor/acceptor sites with ethylene oxide oligomersscalled redox polyether hybridss lead to highly viscous, amorphous, electroactive melts (both neutral melts and molten salts) that offer unique opportunities to study electron-transfer chemistry and mass transport in semisolids.1 They also offer significant experimental measurement challenges.1,2 The hybrid redox polyethers have characteristics of both polymer electrolytes3 and redox polymers,4 being semirigid, ionic conductors, yet hosting high concentrations (∼M) of electron donor/acceptor sites. Physical diffusion of redox moieties in hybrid redox polyethers occurs, but typically very slowly. Thus, when the electron self-exchange dynamics of the redox centers are facile, charge transport in mixed valent melts (formed in diffusion layers at electrodes during voltammmetry) occurs not by physical diffusion but by serial self-exchange reactions, i.e., electron diffusion or electron hopping. Redox polyether hybrids can also be viewed as molecular semisolids that are free of grain boundaries; in this respect they are analogous to single crystals of model mixedvalent, ionically conducting polynuclear inorganic compounds.5 Our experiments on electron-transfer chemistry in semisolid polyethers began with solutions of redox solutes in polyetherbased polymer electrolytes, but the meager solubility of many redox materials in polyethers posed a major constraint. A solution was to synthetically combine the redox moiety with polyether chains, which boosts solubility in polyethers.6 It quickly became apparent that the new redox polyether hybrids †
University of North Carolina. University of Warsaw. § Present address: Department of Metallurgy and Materials Engineering, Technical University of Czestochowa, Armii Krajowej 19, PL-42-200 Czestochowa, Poland. ‡
are, for electron-transfer studies, best examined1,6d-j in an undiluted (or mainly undiluted) state, with the attached polyether chains acting as the “solvent”. The attached polyether chains are between 2 and ∼15 monomer unitsslong enough to destroy the innate crystallinity of the redox moiety, yet short enough to avoid the onset of innate polyether chain crystallinity. An advantage of using redox polyether hybrids as undiluted materials is that the average distance between redox sites is short and relatively well-defined (at least as an average distance), and in the absence of other unattached (plasticizing) molecular species, physical diffusion of the sites is typically quite slow.1,6j The first redox polyether hybrids were assembled by attaching the polyether chains to the redox moiety itself (e.g., tetrathiafulvalene, viologen, tetra(phenyl) porphyrin, and a wide range of metal bipyridine complexes), but recent work has shown that the polyether can also be usefully attached7 to the counterion of the electroactive ion. We call these ions “polyether tailed counterions”. The concept of preparing ionically conductive polyether/salt hybrids based on polyether-tailed counterions was also reported by Ito et al.8 Despite being somewhat dependent on solution composition and electrode surface preparation, the hexacyanoferrate(III,II) couple is often used as a model redox system.9 The present work uses cationic, polyether-tailed counterions to transform salts of hexacyanoferrate(III), a classical outer-sphere anion,10 into a room-temperature molten salt. The Fe(CN)63- ion is combined with triethylammonium countercations tailed with MW 350 poly(ethylene glycol) monomethyl ether (MePEG, ca. 7 monomer units long); this substance is abbreviated (MePEG-Et3N+)3[Fe(CN)63-]. The redox melt is robust; the hexacyanoferrates are stable toward decomposition to Prussian Blue even during prolonged electrochemical experiments. Placing the hexacyanoferrate(III) moiety in a semisolid environment (in the hybrid redox polyether melt) proffers an
10.1021/jp004339j CCC: $20.00 © 2001 American Chemical Society Published on Web 05/24/2001
5834 J. Phys. Chem. B, Vol. 105, No. 24, 2001 opportunity to measure the response of its electron-transfer dynamics to a greatly reduced physical mobility and rigidified surroundings. We will describe microelectrode voltammetric experiments in undiluted (MePEG-Et3N+)3[Fe(CN)63-] molten salts that lead to electron-transfer self-exchange rate constants and their thermal activation barrier energy. Hexacyanoferrate salts are quite water-soluble, and it turns out (by chance) that aqueous solutions can be prepared at the same concentration (0.82 M) as the hexacyanoferrate site concentration in the undiluted redox polyether hybrid molten salt. In the aqueous medium, voltammetry of the hexacyanoferrate ion is controlled by its physical diffusion, whereas in the hybrid redox polyether molten salt, the 104-fold slower charge transport occurs by electron hopping because, in the melt, the physical diffusion is even slower than electron diffusion. Experimental Section Chemicals. Poly(ethylene glycol) monomethyl ether, MePEG, average molecular weight 350 g/mol (ca. seven repeat units, Aldrich) was dried in a vacuum oven at 60 °C prior to use. p-Toluenesulfonyl (p-tosyl) chloride (Aldrich) was stored in a desiccator under nitrogen. Triethylamine (Aldrich) was dried by passing through an activated alumina column. Water was purified using a Barnstead Nanopure System model 4754. Other chemicals were reagent grade and used as received. Synthesis of (MPEG-Et3N+)3[Fe(CN)63-]. The MePEG starting material was tosylated as described elsewhere.1c,7 The triethylammonium (Et3N+) substituents were introduced by refluxing the tosylated MePEG (under nitrogen, 2 days) in an excess of triethylamine; the product is an amber oil7 whose structure was confirmed with proton NMR (Bruker AC-200 MHz) in CDCl3. The tosylate anion was ion-exchanged for hydroxide using Dowex 1 × 8-400 ion-exchange resin (Aldrich). The dilute solution of (MPEG-Et3N+)(OH-) from the ionexchange column was neutralized (to pH 6-7) with hexacyanoferric acid (obtained by ion exchange of Na4[Fe(CN)6], Fluka). Adding excess acid is avoided since it leads to copious formation of Prussian Blue, which is an insoluble, mixed valent hexacyanoferrate. Removing the H2O by rotary evaporation produced a yellow-brown viscous melt. Since these steps were not done under an inert atmosphere, the melt largely contained hexacyanoferrate(III) rather than hexacyanoferrate(II). The material was redissolved in dichloromethane (Mallinckrodt), causing unreacted hexacyanoferrate to precipitate out as Prussian Blue, which was filtered off. The solution was flash-evaporated, and the melt dried at ca. 60 °C under vacuum for 36 h to yield a clear viscous yellow-brown oil. That the melt is completely the hexacyanoferrate(III) form at this point is confirmed by the absence of anodic currents at initial potentials positive of the Fe(II/III) couple. On the basis of its molecular weight (1514) and measured density of 1.24 g/cm3, the material contains 8.2 × 10-4 mol/cm3 (MPEG-Et3N+)3[Fe(CN)63-] sites. Electrochemical Measurements. Cyclic voltammetric and chronoamperometric potential step experiments were performed using a low-current potentiostat of local design operating under computer control and a three-electrode microcell described earlier.1 The electrode assembly was composed of a microdisk working electrode (Pt; unless otherwise stated, 10 µm diameter), a 24 gauge wire Pt counter electrode, and a 0.5 mm Ag quasireference electrode, sealed in an insulating platform that was thoroughly polished. Thoroughly dried, approximately 1.5-2 mm thick films of (MPEG-Et3N+)3[Fe(CN)63-] redox melt were prepared by casting onto the smooth surface of the microdisk assembly followed by drying in a sealed glass cell under vacuum
Kulesza et al.
Figure 1. Solid-state cyclic voltammetry of hexacyanoferrate redox melt recorded at 5 mV/s and temperatures of (A) 0, (B) 20, (C) 40, and (D) 80 °C. Microelectrode diameter ) 10 µm. Peak potential separation of oxidation and reduction current peaks ) 65-75 mV.
at 60 °C for at least 24 h before experimentation. The temperature was controlled by immersing the glass cell in an ethylene glycol bath thermostated by a Brookfield EX-100 circulating heater and FTC-350A cooler. Ionic conductivities were measured for ca. 2 mm thick melt films drop-cast onto microlithographically fabricated Pt interdigitated array, IDA, electrodes11 that were donated by Dr. O. Niwa of Nippon Telephone and Telegraph. The films were dried under vacuum at 60 °C for at least 36 h prior to measurement. The IDA consisted of 50 pairs of fingers each 3 µm wide and 2 µm apart; its cell constant was calibrated as 0.117 cm-1 using a solution of known conductivity. Film resistance was determined from the real axis intercept of the complex impedance semicircle recorded with a Solartron model SI 1260 Impedance analyzer equipped with SI 1267 electrochemical interface. Ionic conductivity is the reciprocal of [film resistance/cell constant]. Temperature was varied and controlled by a Lakeshore model 330 autotuning temperature controller. Differential Thermal Analysis. The temperatures of carefully dried, preweighed samples of (MPEG-Et3N+)3[Fe(CN)63-] redox melt were scanned from - 80 to +80 °C at a heating rate of 5 °C/min using a Seiko DSC 220-CU DSC instrument. The glass transition temperature (TG) was determined from the average of the transition region inflection points in heating and cooling curves. Viscosity Measurements. Approximately 1 g of dried redox melt was evenly cast on the plate of a Brookfield model DVIII rheometer and dried in a vacuum at 60 °C for 48 h. Viscosities were determined under dry N2 and at constant temperature with a Brookfield cone (model CP-52) and did not depend on cone rotation rate. Spectrophotometry. UV-visible spectrometry was done with a Unicam UV-vis diode array spectrometer. A sample (redox melt or aqueous concentrated hexacyanoferrate solution) was placed between two quartz slides mounted in a sandwich configuration. The quartz slide spectrum was recorded as a reference. Results and Discussion Solid-State Voltammetry of (MPEG-Et3N+)3[Fe(CN)63-]. Figure 1 shows microelectrode voltammetry of the undiluted, carefully dried molten salt. Although the melt has quite high viscosity (ca. (2-3) × 106 cP at 25 °C), its ionic conductivity is ample to support microelectrode voltammetry. The single, well-defined hexacyanoferrate(III,II) reduction wave seen in the (MPEG-Et3N+)3[Fe(CN)63-] melt is chemically reversible and not far from electrochemically reversible over a very wide temperature range. The facile electron-transfer kinetics exhibited by the hexacyanoferrate(III,II) couple in fluid solutions9 are
Self-Exchange Dynamics of Hexacyanoferrate
J. Phys. Chem. B, Vol. 105, No. 24, 2001 5835
Figure 2. Chronoamperometric responses of hexacyanoferrate redox melt at 20 °C: (A) current vs time; (B) current vs reciprocal square root of time. Potential steps were from +0.4 to -0.6 V. Microelectrode diameter ) 10 µm.
Figure 3. Arrhenius plot of DAPP for (MePEG-Et3N+)3[Fe(CN)63-] redox melt. Activation energy EA ) 30 kJ/mol; DoAPP (extrapolation to infinite temperature) ) 5.6 × 10-5 cm2/s.
retained by hexacyanoferrate even when embedded within the semisolid medium of polyether-tailed countercations.12 Reasonable voltammetry can be obtained over a -10 to +100 °C range. Voltammetric peak currents increase with increased temperature, meaning that the dynamics of charge transport become more facile there. Peak currents (20 °C) for Figure 1 hexacyanoferrate(III/II) reduction increase linearly (not shown) with the square root of potential scan rate (V), with negligible intercept, from V ) 2 to 100 mV/s, which is characteristic of linear diffusion conditions; i.e., the Randles-Sevcik relation for a one electron reaction applies12
IPEAK ) 2.69 × 105ADAPP1/2CV1/2
(1)
where IPEAK, A, C, and DAPP stand for peak current, electrode area, concentration, and apparent diffusion coefficient, respectively. The numerical prefactor is proportional to [temperature]-1/2; the value given is for 25 °C. Equation 1 yields DAPP ≈ 2 × 10-10 cm2/s (20 °C) from Figure 1 data. (Larger potential sweep rates cause severe iRUNC distortion and slower onessdown to ca. 0.5 mV/ssgive voltammograms symptomatic of mixed, linear-radial diffusion.) Potential step chronoamperometry gives more exact DAPP values since potential steps can override iRUNC effects. Figure 2A shows a typical potential step chronoamperometric currenttime transient for the redox polyether melt investigated in Figure 1. The current initially decays with time but then levels out as radial diffusion conditions are attained. (Radial diffusion corresponds to the diffusion layer depth 2(Dt)1/2 being larger than the microdisk electrode radius.) The curvature in the (linear diffusion) Cottrell i vs t-1/2 plot13 in Figure 2B is also consistent with a change from linear (short times) to radial (long times) diffusion conditions; this has been reported2a in other semisolid media. In the curved, 100-400 s interval of mixed linear-radial diffusion conditions, currents can be analyzed by fitting currenttime transients to the Shoup-Szabo relation,14
i ) 4nFDAPPrC[0.7854 + 0.8862τ -1/2 + 0.2146 -1/2
exp(-0.7823τ
Figure 4. Arrhenius plot of ionic conductivity for (MePEG-Et3N+)3[Fe(CN)63-] redox melt.
10-10 to 20 × 10-10 cm2/s from 273 to 353 K (Table S-I). These temperatures are all higher than the -61.5 °C glass transition temperature (TG) of the melt. Figure 3 presents a thermal activation plot from which the activation barrier energy for charge transport is 30 kJ mol-1. This barrier energy is similar to others1a,6i,7b found in redox polyether hybrids in which electron hopping (as opposed to physical diffusion) is controlling. The ionic conductivity (σION) of the (MPEG-Et3N+)3[Fe(CN)63-] melt, measured by ac impedance as described in the Experimental Section, produced data as given in Table S-II and Figure 4. The Figure 4 activation plot is slightly nonlinear; the average slope gives an activation barrier energy for ion transport of 50 kJ/mol. Charge transport in a redox polyether hybrid such as the (MPEG-Et3N+)3[Fe(CN)63-] melt might occur concurrently by physical diffusion of redox sites (DPHYS) and by electron hopping between them. Their summation is expressed by the DahmsRuff relation, which based on a cubic lattice model of electron hopping is, in corrected form,15
DAPP ) DPHYS + DE ) DPHYS + kEXδ2C/6
(3)
)] (2)
where r is microdisk radius. In thoroughly dry melts, DAPP was reproducible to within (10% or better, and ranged from 1 ×
where DE and δ are respectively the electron diffusion coefficient and the average (equilibrium) distance between the redox sites in the melt. To relate DAPP to kEX, eq 3 requires an estimate
5836 J. Phys. Chem. B, Vol. 105, No. 24, 2001
Kulesza et al.
of DPHYS. We were unable to devise a scheme that directly measured the DPHYS of the [Fe(CN)6]3-/4- ion. The linearity of the activation plot in Figure 3 suggests, however, that it is reasonable to assume that DPHYS , DAPP. We have observed in cases1,6j where control by DPHYS was unambiguously established, that DPHYS activation plots are curved, whereas DAPP plots are linear when DPHYS , DAPP (i.e., electron hopping is the dominant charge transport mechanism). The curvature of DPHYS activation plots arises from coupling of physical motions of the redox units to the segmental motions of the polyether chains attached to them. Additionally, we note that DPHYS ≈ 8 × 10-13 cm2/s at 25 °C in7c a melt based on a Co(bpy)32+ complex with a polyether tailed counterion. The Co(II/III) couple has a negligible electron hopping rate. The viscosity of this melt, 4 × 106 cP at 25 °C7c (to which diffusion rates should be inversely proportional) is similar to that of the (MPEG-Et3N+)3[Fe(CN)63-] melt, ca. (2-3) × 106 cP at 25 °C. From the above observations, we draw the conclusion that DPHYS , DAPP in eq 3, from which, at 20 °C and with the parameters C ) 8.2 × 10-4 mol/cm3 and δ ) 1.27 nm,16 we obtain an electron self-exchange rate constant kEX ) 1.1 × 105 M-1 s-1 for the solid-state electron-transfer couple Fe(CN)63-/4-. This result is in remarkable agreement with values obtained in fluid solutions. NMR measurements9f in aqueous K3[Fe(CN)6] + K4[Fe(CN)6] solutions give electron self-exchange rate constants between 7.2 × 104 and 1.6 × 105 M-1 s-1 at 32 °C. Other rate constants reported in aqueous solutions containing hexacyanoferrate(III,II) and potassium cations are 3.9 × 105 and 4.2 × 105 M-1 s-1.9f,h The kEX determined above is based on δ ) 1.27 nm, which is the average site separation in the melt. If, by rapid local motion, electron transfer occurs at a shorter, closest approach, such as contact between Fe(CN)63- (radius ) 0.43 nm) and Fe(CN)64- (radius ) 0.44 nm) anions, kEX would instead be 2.3 × 105 M-1 s-1. The modest uncertainty in the distance term δ means that kEX lies between 1 and 2 × 105 M-1 s-1. Comparison to Concentrated Hexacyanoferrate Solution. In 0.82 M aqueous K3Fe(CN)6, the medium is much more fluid, by ca. 106-fold (viscosity is 1.2 cP), than the redox polyether hybrid (where viscosity is 4 × 106 cP), so that physical diffusion assumes the dominant role in charge transport. A microdisk electrode immersed into aqueous hexacyanoferrate(III) solution gives (at sufficiently slow potential scan rates, below 10 mV/ s), steady-state limiting (ILIM) reduction currents,14 as shown in Figure 5B. The waveshape is typical of radial mass transport; the appropriate17 radial diffusion equation is
ILIM ) 4nFrDAPPC
(4)
Based on C ) 0.82 M, DAPP ) 6.5 × 10-6 cm2/s (20 °C). DAPP can be alternatively obtained by comparing currents obtained at slow (i.e., Figure 5B) and fast potential scan rates; such data permit concurrent calculation of C and DAPP4c without assuming knowledge of C. The calculation involves simultaneous solution of the equations5c
DAPP ) [(2.69 × 105)2nVπ2r2ILIM2]/(16F2IPEAK2)
(5)
C ) (4FIPEAK2)/[(2.69 × 105)2ILIMVn2π2r3]
(6)
The results, DAPP ) 6.5 × 10-6 cm2/s (20 °C) and C ) 0.82 M, are identical to those obtained above from Figure 5. Also, the same DAPP was determined from short time (25 ms) potential step chronoamperometric experiments under linear diffusion conditions.
Figure 5. Microelectrode voltammetry of 0.82 M K3[Fe(CN)6] aqueous solution recorded at potential scan rates (A) 10 V/s (linear diffusion) and (B) 4 mV/s (radial diffusion). Microelectrode diameter ) 20 µm.
Figure 6. Dependence of diffusion coefficient (DAPP) on concentration of hexacyanoferrate(III) plotted for aqueous solutions containing K3[Fe(CN)6] in the range 0.1-1.5 M.
Using the approach of eqs 5 and 6, DAPP was measured for 0.1-1.5 M (saturated) hexacyanoferrate(III) aqueous solutions with results shown in Figure 6. (The result at C ) 1.5 M (DAPP ) 3.8 × 10-6 cm2/s) is smaller than an older one (7.6 × 10-6 cm2/s) for 1 mM K3Fe(CN)6 in 1.0 M KCl.9g) The Figure 6 decrease in DAPP with concentration can be attributed mainly, if not entirely, to the accompanying increase in viscosity of the solutions; the 1.5 M solution has,18 for example, ca. 40% higher viscosity (1.7 cP) than that at 0.82 M. That these aqueous DAPP values are entirely controlled by physical diffusion (i.e., DAPP ) DPHYS) is confirmed by the fact that, for the electron self-exchange term in eq 3 to equal the measured DAPP at the highest hexacyanoferrate(III) concentration, kEX would have to be of the order of 109 M-1 s-1. This is an unrealistic value given previous studies of electron selfexchange for this couple in aqueous media. Additionally, the activation barrier energy observed (not shown) over a 0-90 °C range for 0.82 M K3Fe(CN)6 is 15 kJ/mol, half that seen for the (MPEG-Et3N+)3[Fe(CN)63-] redox melt. It is fascinating that the physical diffusivities of the same concentration (0.82 M) of the hexacyanoferrate(III) species are so different in the polyether melt (DPHYS , ca. 1 × 10-10 cm2/s
Self-Exchange Dynamics of Hexacyanoferrate
J. Phys. Chem. B, Vol. 105, No. 24, 2001 5837 of differences in ion pairing between the two oxidation states), EA can be identified with ∆G*. KA is near unity1d,e,5j so that κν ≈ koEX ≈ 1011 s-1. Since reactions with κν ) 1013 s-1 are considered adiabatic,20 the Fe(III/II) self-exchange reaction in the (MPEG-Et3N+)3[Fe(CN63-] melt appears to be near but somewhat less than adiabatic. One can inquire as to whether the polyether medium separating the redox centers in the melt is of sufficient thickness to have affected the electron-transfer rate constant. The average edge-edge separation is the order of 4 Å. The effect of electron transfer between the redox sites through a polyether medium of this thickness can be estimated using the exponential relationship
k(x) ) k(edge) exp[-βx]
Figure 7. Spectra of (A) (MePEG-Et3N+)3[Fe(CN)63-] redox melt and (B) 0.82 M K3[Fe(CN)6] aqueous solution.
at 25 °C) and aqueous solutions (DPHYS ca. 4 × 10-6 cm2/s). This diffusivity difference comes about by change, from polyether to water, of the ca. 0.4 nm of medium separating the Fe(CN)63- complexes, which have an average center-to-center spacing of 1.27 nm and radii of 0.44 nm. The water medium can be regarded as having “plasticized” the physical diffusion. The dramatic change in physical diffusivity prompts a change from charge transport by electron hopping, in the redox melt, to more conventional physical transport, in aqueous solution. It does appear that (like metal bipyridines6i) the electronic structure of the hexacyanoferrate(III) species is somewhat perturbed in the melt, as compared to aqueous solution, as shown in Figure 7. In the aqueous solution, the electronic spectrum (Figure 7B) of Fe(CN)63- shows characteristic d-d transitions, 1A 1 1 3 19 (The 1g f T1g and A1g f T1g, at 430 and ca. 300 nm. strong charge-transfer bands for metal-to-ligand (π*) type transitions at 200 nm are not shown). The spectrum of Figure 7A, in the redox melt, is qualitatively similar, but an element of one of the two d-d transitions appears to be shifted to 370 nm. This phenomenon most likely reflects differences in interactions between the hexacyanoferrates and their counterions. The visible colors of hexacyanoferrate(III) in water (orangeyellow) and in the melt (brown-yellow) are correspondingly different. Consideration of Electron-Transfer Results. Last, we briefly consider some implications of the electron self-exchange results in the redox melt. The results are, in summary, that at 20 °C, kEX ) 1.1 × 105 M-1 s-1, activation barrier energy EA ) 30 kJ/mol, and Arrhenius plot intercept koEX ≈ 1011 M-1 s-1. These results are evaluated based on DPHYS , ca. 1 × 10-10 cm2/s at 25 °C and an average center-to-center redox site spacing (i.e., distance traversed by an electron hop) of 1.27 nm. Considering the self-exchange as a bimolecular electrontransfer reaction,10b,20
kEX ) KAκν exp[-∆G*/kBT]
(7)
where ∆G* ≈ λ/4 (λ being reorganization energy), KA is the donor-acceptor precursor complex formation constant, and ν and κ are the frequency and electronic factors, respectively. For a symmetrical electron self-exchange, (ignoring the possibility
(8)
where k(x) and k(edge) are the electron-transfer rate constants at distance x and at edge-edge contact and β is the electronic coupling term. Assuming the common value of 1 Å-1 for β gives a rate attenuation of 0.02. This small factor could readily be accommodated in the difference between the experimental value of κν and the ideal adiabatic one. Whether site-to-site electron transfer actually does occur by a tunneling pathway depends, of course, on the extent to which the hexacyanoferrate(III,II) sites are locked in place on the electron-transfer frequency, as opposed to being microscopically translated by thermal motions to a more nearly edge-edge separation at a frequency faster than the electron hopping one. In a previous study of metal polypyridine complexes in which the electrontransfer rate constant was unresponsive to variation of the polyether chain length, we concluded1a that the latter (rapid thermal motions) was more likely the case. Acknowledgment. The research was supported by grants from the Department of Energy and the National Science Foundation. This work was also supported in part by the State Committee for Scientific Research (KBN), Poland. Supporting Information Available: Tables of numerical values of redox melt DAPP and σION as a function of temperature, which are plotted in Figures 3 and 4. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) (a) Williams, M. E.; Masui, H.; Long, J. W.; Malik, J.; Murray, R. W. J. Am. Chem. Soc. 1997, 119, 1997. (b) Masui, H.; Murray, R. W. Inorg. Chem. 1997, 36, 5118. (c) Long, J. W.; Kim, I. K.; Murray, R. W. J. Am. Chem. Soc. 1997, 119, 111510. (d) Williams, M. E.; Crooker, J. C.; Pyati, R.; Lyons, L. J.; Murray, R. W. J. Am. Chem. Soc. 1997, 119, 10249. (2) (a) Longmire, M. L.; Watanabe, M.; Zhang, H.; Wooster, T. T.; Murray, R. W. Anal. Chem. 1990, 62, 747. (b) Porat, Z.; Crooker, J. C.; Zhang, Y.; Mest, Y. Le; Murray, R. W. Anal. Chem. 1997, 69, 5073. (c) Crooker, J. C.; Murray, R. W. Anal. Chem. 2000, 72, 3245. (3) Hara, M. In Polyelectrolytes, Science and Technology; Hara, M., Ed.; Marcel Dekker: New York, 1993. Armand, M. Solid State Ionics 1983, 10, 1161. Hooper, A.; North, J. M. Solid State Ionics 1983, 10, 1161. (4) (a) Murray, R. W., Ed. Molecular Design of Electrode Surfaces; Wiley: New York, 1992. (b) Peerce, P. J.; Bard, A. J. J. Electroanal. Chem. 1980, 114, 89. (c) Sullivan, M. G.; Murray, R. W. J. Phys. Chem. 1994, 98, 4343. (d) Burgmayer, P.; Murray, R. W. J. Electroanal. Chem. 1982, 135, 335. (e) Dalton, E. F.; Murray, R. W. J. Phys. Chem. 1991, 95, 6383. (5) (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.; Cox, J. A. Electroanalysis 1998, 10, 73. (e) Kulesza, P. J.; Malik, M. A. In Interfacial Electrochemistry, Theory Experiment and Applications; Wieckowski, A., Ed.; Marcel Dekker: New York, 1999. (6) (a) Aida, T.; Takemura, A.; Masahiro, F.; Inoue, S. J. Chem. Soc., Chem. Commun. 1988, 391. (b) McKeown, N. B.; Painter, J. J. Mater. Chem.
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