13728
J. Phys. Chem. B 2006, 110, 13728-13733
Ion Transport in the Microporous Titanosilicate ETS-10 Ta-Chen Wei and Hugh W. Hillhouse* School of Chemical Engineering, Purdue UniVersity, West Lafayette, Indiana 47907 ReceiVed: February 17, 2006; In Final Form: May 25, 2006
Impedance spectroscopy was used to investigate ion transport in the microporous crystalline framework titanosilicate ETS-10 in the frequency range from 1 Hz to 10 MHz. These data were compared to measured data from the microporous aluminosilicate zeolite X. Na-ETS-10 was found to have a lower activation energy for ion conduction than that of NaX, 58.5 kJ/mol compared to 66.8 kJ/mol. However, the dc conductivity and ion hopping rate for Na-ETS-10 were also lower than NaX. This was found to be due to the smaller entropy contribution in Na-ETS-10 because of its high cation site occupancy. This was verified by ion exchanging Na+ with Cu2+ in both microporous frameworks. This exchange decreases the cation site occupancy and reduces correlation effects. The exchanged Cu-ETS-10 was found to have both lower activation energy and higher ionic conductivity than CuX. Zeolite X has the highest ion conductivity among the zeolites, and thus the data shown here indicate that ETS-10 has more facile transport of higher valence cations which may be important for ion-exchange, environmental remediation of radionucleotides, and nanofabrication.
Introduction Microporous crystalline materials such as zeolites have many important industrial applications that depend on their ion transport properties. The most common application is for ionexchange in laundry detergents.1 Other applications under development that require either facilitating or inhibiting ionexchange or ion transport include the uptake heavy metal ions2,3 or radionucleotides4 for environmental remediation, coatings for corrosion resistance,5,6 additives to proton transport membranes for fuel cells,7-9 and their use as templates for electrochemical deposition of the smallest of nanowires.10 The use of these microporous frameworks in nanowire synthesis is driven by the possibility of fabricating high efficiency thermoelectric devices. Calculations by Hicks and Dresselhaus11 suggest that thermoelectric figure-of-merit’s (ZT) an order of magnitude higher than existing materials may be possible by using 1D quantum confinement. When the effects of thermal and electrical transport though a surrounding framework are added, calculations show that the figure of merit is substantially reduced but still significantly higher than current state-of-the-art thermoelectric materials.10 However, if microporous frameworks are to be used as templates to electrodeposit nanowires, it is necessary that the cationic species (that serve as the reactants for the interfacial electrodeposition reaction) can easily move through the pore system. As a result, the development of microporous materials with facile ion transport is very important, particularly for divalent or trivalent cationic species. Aluminosilicate zeolites, the most well-known microporous ion conductors, have been extensively used for applications in ion exchange, separations, and catalysis. Transport of various cations have been studied in different zeolite structures, such as proton transport in ZSM-5,12,13 sodium and calcium ion transport in phillipsite and stilbite,14 sodium ion in ferrierite,15 and sodium ions and protons in zeolite beta.16 However, because of Lo¨wenstein’s rule which states that adjacent alumina * To whom correspondence should be addressed. E-mail: hugh@ purdue.edu.
tetrahedra are not allowed, cation sites in zeolites are relatively well separated even in low Si/Al ratio materials. Here, we study ion transport in a qualitatively different type of crystalline microporous structure, the titanosilicate ETS-10. ETS-10 is a unique microporous crystal that has attracted interest for shapeselective photocatalyst,17 optoelectronics,18 heavy metal ion uptake,2,3 and selective gas adsorption.19 Its unique properties stem from its unique structure. The framework of ETS-10 is an ordered array of linked SiO4 tetrahedra and TiO6 octahedra. The TiO6 octahedra are linked vertex to vertex forming continuous chains that run parallel to 12-membered ring channels with a cross section 7.6 Å × 4.9 Å.20,21 The -O-Ti-O-Ti-Ochains in adjacent layers are perpendicular to each other (Figure 1). Because of the +4 oxidation state of the titanium, every Ti octahedra in the framework needs two monovalent cations or one divalent cation to maintain electroneutrality. The cations are found distributed along the pore wall adjacent to the titania chains to compensate the anionic framework.22,23 This unique structure is expected to result in different, and perhaps enhanced, ion transport compared to zeolites. Ion transport is typically studied by measuring the electric relaxation. The data are then presented and analyzed in terms of the complex conductivity or the electric modulus. For most ion conductors the real part of the conductivity is well-described by the expression developed by Almond and West:24
σ′(ω) ) σDC[1 + (ω/ωp)n]
(1)
where σDC is the frequency independent dc conductivity, ω is the angular frequency equal to 2πf, ωp is the characteristic hopping frequency of the charge carrier, and n is the powerlaw exponent that depends on the mechanism and dimensionality of the transport.25 The dc ion conductivity results from the random hopping of ions between charge balancing sites. In ETS10, the mobile cations hop between crystallographically defined sites that are adjacent to regions of high partial negative charge associated with the titania octahedra in the microporous framework. As result of this random motion, the dc conductivity
10.1021/jp061037u CCC: $33.50 © 2006 American Chemical Society Published on Web 06/27/2006
Ion Transport in Microporous Titanosilicate
J. Phys. Chem. B, Vol. 110, No. 28, 2006 13729
Figure 1. Three orthogonal projections of polymorph B of ETS-10 showing the perpendicular -O-Ti-O-Ti-O- chains between adjacent layers (Ti, red; Si, blue; O, yellow; extraframework cations not shown). Note the 12-ring pores running parallel to the -O-Ti-O-Ti-O- chains. The structure belongs to space group C2/c with unit cell parameters21 a ) 21.00 Å, b ) 21.00 Å, c ) 14.51 Å, R ) 90°, β ) 111.12°, and γ ) 90°.
may be expressed by the Nernst-Einstein relation:24
σDC ) Cqµ )
Nc(1 - c)q2l2γ ωp kT
(2)
where N is the number of cation sites per unit volume, c is the cation site occupancy, q is the charge of the mobile ion, l is the jump distance, and γ is a geometric factor (γ is 1/6, 1/4, and 1/2 for 3D, 2D, and 1D conduction). C (equal to N*c) is the concentration of mobile ions. The hopping rate is thermally activated and is given by
(
ωp ) ω0 exp -
)
( ) ( )
∆Gh Eh ∆Sh ) ω0 exp exp kT k kT
(3)
where ∆Gh, Eh, and ∆Sh are the free energy, activation energy, and entropy of the hopping frequency of ions in the lattice. Similarly, the concentration of mobile ions may be thermally activated and expressed as
(
C ) C0 exp -
)
( ) ( )
∆Gc Ec ∆Sc ) C0 exp exp kT k kT
(4)
where ∆Gc, Ec, and ∆Sc are the free energy, activation energy, and entropy for the creation of mobile carriers. Therefore, plugging (3) and (4) into (2), the ionic conductivity can be expressed as
σDC )
( )
Ea A exp T kT
(5a)
where the prefactor (A) is given by
A)
[
(
)]
C0(1 - c)ω0q2l2γ ∆Sh + ∆Sc exp k k
(5b)
and the activation energy for ion conduction (Ea) is the sum of activation energy for ion migration (Eh) and association (Ec):
E a ) Eh + Ec
(5c)
Thus entropic effects are observed in the frequency factor while energetic effects are contained in the temperature-dependent exponential factor. In the present contribution, we investigate the ion transport in ETS-10 by impedance spectroscopy and analyze the data in
the context of the model given above. However, proper interpretation of impedance data depends on obtaining an accurate impedance response over a wide range of frequencies. To dehydrate these porous materials and measure the electrical properties in-situ across a range of temperatures, a special apparatus with low capacitance and inductance was designed and constructed. In addition, the impedance data were collected at various temperatures to obtain the activation energy of ion conduction, migration, and association. Experimental Details Synthesis and Ion Exchange. The Na,K-ETS-10 samples used in this study were obtained from Engelhard Corporation with Na+/K+ equal to 3. Pure sodium form Na-ETS-10 was synthesized on the basis of a method reported by Rocha and co-workers.26 Pure sodium form zeolite X (NaX) was obtained from Aldrich (molecular sieve 13X). Cu-ETS-10 samples were prepared by mixing Na,K-ETS-10 with a 0.4 M Cu(NO3)2 solution at room temperature for 3 days. Then the mixture was washed and centrifuged several times and dried at 120 °C overnight.27 CuX samples were prepared by mixing 2 g of NaX in 0.05 M copper nitrate solution at room temperature for 24 h, and then the mixture was washed and centrifuged several times and dried overnight. The same procedure repeats three times to reach the maximum degree of exchange. Then the composition of sample was determined by atomic absorption spectroscopy (AAS) from samples dissolved in concentrated hydrofluoric acid and diluted to a concentration of several ppm. A Perkin Elmer AAnalyst 300 atomic absorption spectrometer was then used to determine the content of metal ions in each sample. The atomic adsorption spectroscopy data were first calibrated by using four standard solutions with known concentration that spanned the range of the samples (typically between 1 and 20 ppm). After Cu exchange, the ratios of ions in CuX were 1 Al: 0.493 Cu: 0.005 Na. This shows that 98.6% of sodium has been exchanged by copper (II). Impedance Spectroscopy. The design of our impedance spectroscopy probe was based in part on that reported by Mason.28 However the present apparatus incorporates control of the gas-phase environment. The probe consists of spring loaded platinum electrodes attached to MACOR machinable ceramic (55% fluorophlogopite mica and 45% borosilicate glass). Electrical contact is established by a platinum wire attached to the platinum disk electrode (diameter 12.7 mm,
13730 J. Phys. Chem. B, Vol. 110, No. 28, 2006
Figure 2. Equivalent circuit representation of the impedance from the measuring probe determined from open circuit and short circuit measurements. These circuit elements were then removed from the measured data.
thickness 0.05 mm, obtained from Ladd Research) by spot welding. An outer stainless steel tube was used to provide electrical shielding to control the atmosphere around the sample. Grounding is accomplished by means of a BNC coaxial feedthrough connected between the outer stainless steel tube and the shields of the coaxial cables from the frequency response analyzer. The impedance contribution from the apparatus may be modeled as a resistor and an inductor connected in series (Rs and Ls) and a resistor and a capacitor connected in parallel (Rp and Cp). The values for these elements were determined by fitting the measured impedance from short-circuit and opencircuit impedance measurements on an empty cell. The values determined for these parasitic elements are shown in Figure 2 and are comparable with the apparatus constructed by Mason et al.28 The impedance measurements were carried out using this probe and a Solartron 1260 frequency response analyzer connected to the apparatus by coaxial cables. Powder samples of as-synthesized or ion-exchanged powders of ETS-10 and zeolite X were placed between the Pt electrodes in the probe and dehydrated at 573 K and 1 Torr for 12 h before each measurement. After dehydration, the impedance spectra were collected in the frequency range of 1 Hz to 10 MHz and a temperature range from 293 to 573 K with an alternating voltage of 50 mV (rms). To ensure the samples were equilibrated, impedance spectra were collected every 30 min until no changes were observed. Results and Discussion NaX was chosen for comparison with Na-ETS-10 because of their similar cation density. The complex admittance was calculated from the complex impedance data by Y* ) 1/Z*.
Figure 3. Conductivity spectra of (a) NaX and (b) Na-ETS-10.
Wei and Hillhouse From this, the real part of the admittance was used to calculate the ac conductivity by σ ) (d/A) Y′ where d is the thickness of the sample and A is the cross-sectional area of the sample. The conductivity spectra of both samples exhibit two distinct regions of low and high frequency dispersion which are usually found in ionic conductors (Figure 3). However, at high temperature the hopping frequency is beyond our measurable frequency range, and thus we were not able to obtain accurate values of dc conductivity and hopping rate by using the Almond-West method.29 To circumvent this limitation, we obtained the dc conductivity from equivalent circuit fits of impedance data in the complex plane (z′ versus z′′) and obtained the hopping frequency from the modulus transformed data. Characteristic impedance data for Na-ETS-10 and NaX are shown in an Argand diagram in Figure 4. All impedance spectra show a depressed semicircle at high frequency and a tail extending to low frequency. These spectra may be understood in terms of equivalent circuit models. For instance, an ideal RC parallel circuit element would yield a perfect half-circle in an Argand diagram intersecting the z′ axis at 0 and R, with the center of the circle lying on the z′ axis (z′′ ) 0). However, semicircle fits to the data shown in Figure 4 yield a circle center below the z′ axis. This deviation (depression of the semicircle) results from a relaxation time that is not single-valued but is distributed around a mean value. The angle (θ) by which such a semicircle arc is depressed below the real axis is related to the width of the relaxation time distribution. The larger the depressed angle θ, the wider the width of relaxation time distribution.30 As a result, the Na-ETS-10 has a wider distribution of relaxation time compared to NaX which is due to the correlation effect. Similar behavior was observed in modulus spectra. When the temperature is decreased, the semicircle of Na-ETS-10 and NaX becomes larger reflecting the decreased ionic conductivity. The low-frequency tail is related to the polarization of interface between the sample and electrode.31 The dc conductivities were then calculated from the value of the resistance (R) determined by fitting a depressed semicircle to the impedance data in the high-frequency region of the observed semicircle. The dc conductivity can be calculated from σ ) (1/R)*(d/A), where d is the thickness of sample and A is the area of electrode. The results were plotted as a function of temperature to estimate the activation energy for both samples (corresponding to the activation energy in eq 5a). The Arrhenius plots of ionic conductivities for ETS-10 and NaX are shown in Figure 5. NaX shows a straight line an yields an activation energy of 66.8 kJ/mol, which is similar to the result reported by Simon.32 Na-ETS-10 has a lower activation energy of 58.5
Ion Transport in Microporous Titanosilicate
J. Phys. Chem. B, Vol. 110, No. 28, 2006 13731
Figure 4. Argand diagram (plot of imaginary part of the impedance Z′′ versus real part of the impedance Z′ plotted parametrically as a function of frequency) of (a) dehydrated NaX and (b) Na-ETS-10.
formed to complex modulus data by applying the following transformation:
M′′ ) 2π f C0Z′
Figure 5. Arrhenius plot of conductivity for NaX and Na-ETS-10. The infinite temperature intercept yields prefactors of A ) 2633 Ω-1 cm-1 K for NaX and A ) 19 Ω-1 cm-1 K for Na-ETS-10.
kJ/mol. It is interesting to note that ETS-10 has lower ionic conductivity and also lower activation energy. To further examine this phenomena (and to gain better resolution at high frequency), the impedance data were trans-
(6)
where C0 is the vacuum capacitance of the cell, C0 ) 0 S/d (where S is the area, d is the thickness of the cell, 0 is the permittivity of the vacuum). In the modulus spectra (M′′ vs log f) of NaX and Na-ETS-10, peaks due to separate relaxation processes are clearly observed and shift to lower frequency when the temperature is decreased (Figure 6). When the temperature is decreased to 100 °C, a second peak is clearly observed at high frequency in NaX. The modulus spectra of Na-ETS-10 (Figure 6b) shows wider relaxation peaks indicating the wider distribution of relaxation times. The modulus spectra of NaX can be fit using two RC parallel circuits connected in series as shown in Figure 7. The capacitances for both relaxation processes are quite similar and are on the order of picofarads. Although the widths of the peaks in the model are smaller than the observed peaks in the modulus spectra owing to the distribution of relaxation times, the vertex of peaks match and gives an accurate estimate of the mean relaxation time for each
Figure 6. Modulus spectra of the imaginary part M′′ versus log f of (a) NaX and (b) Na-ETS-10.
13732 J. Phys. Chem. B, Vol. 110, No. 28, 2006
Figure 7. Modulus spectra of dehydrated NaX at 353 K and the fitted result. The values of each element in the equivalent circuit show two relaxation processes within the grains.
Figure 8. Arrhenius plot of hopping rate for NaX and Na-ETS-10. The infinite temperature intercepts yield values of the maximum attempt frequencies of 1.3 × 1011 Hz for Na-ETS-10 (HF), 7.1 × 109 Hz for Na-ETS-10 (LF), and 4.7 × 1011 Hz for NaX (LF).
process. Because of the small and similar magnitude of the capacitances, both processes are interpreted to be due to intracrystalline hopping processes. This is rationalized based on Maxwell’s layered-dielectric model.33 If the permittivity of material is assumed to be uniform, the low capacitance can be attributed to the thick layer (C ) 0A/d), which is bulk material, and high capacitance is the thin layer representing grain boundaries and electrical double layer. The capacitance of the bulk material and the boundary layer can have at least 2 orders of magnitudes difference according to the Maxwell’s model. This result corresponds to similar observation for NaX by Simon and Flesch in which they attributed these two relaxation processes to cation movement within the grain.32 We used this modulus data to obtain the activation energy for the ion migration (the activation energy in eq 3) by using the linear relationship between ion hopping rate and the frequency of relaxation peak in modulus spectra (M′′ plot).34 Because the frequency of M′′ peak is proportional to the ion hopping rate, the activation energy of ion migration can be calculated from an Arrhenius plot of this frequency versus inverse temperature. The Arrhenius plots of ion hopping rate for Na-ETS-10 and NaX are shown in Figure 8. The activation energy of ion migration for the low-frequency hopping process (labeled LF in Figure 6) for zeolite NaX and Na-ETS-10 is about 62.9 and 54.6 kJ/mol, respectively. This shows that the activation energy for carrier creation for both structures is about 4 kJ/mol (by eq 5c) and that the activation energy of ion hopping for Na-ETS-10 is indeed lower than that of zeolite NaX.
Wei and Hillhouse
Figure 9. Arrhenius plot of conductivity for CuX and Cu2+-ETS-10. The infinite temperature intercept yields values of A ) 27064 Ω-1 cm-1 K for CuX and A ) 1746 Ω-1 cm-1 K for Cu-ETS-10, respectively. Plots of the hopping frequency obtained from the modulus spectra yield maximum attempt rates of 1.7 × 1013 Hz for Cu-ETS-10 (HF), 1.2 × 1011 Hz for Cu-ETS-10 (LF), and 2.98 × 1012 Hz for CuX.
It was found that the ionic conductivity and ion hopping rate of Na-ETS-10 is lower than NaX (Figures 5 and 8) despite the fact that the NaX has higher activation energy. The y-intercepts in Figures 5 and 8 show that the prefactor of NaX is much larger than that of Na-ETS-10. Since the concentrations are nearly equal, small differences in the jump distance and geometric factors cannot explain the differences in the prefactors. However, the site occupancy is quite different between NaX and Na-ETS10, determined from crystallographic data to be 47 and 90%, respectively. Thus, there are more possible configurations of cations in NaX than in ETS-10. This yields higher activation entropy for NaX, which from eq 5b, is observed to increase the prefactor. This high cation site occupancy in ETS-10 should also manifest itself as correlation effects. Correlation effects can be examined most easily by constructing modulus master curves35 made by scaling each frequency by the frequency of relaxation peak in M′′ plot and determining the Kohlrausch exponent, β, of the decay function, φ ) exp[-t/τ]β. The decay represented by φ results from the sum of many exponential decays weighted by a distribution of individual relaxation times. β is inversely related to the width of the distribution and ranges from 0 to 1. Smaller values of β indicate higher ion-ion interactions36 and result in wider relaxation peaks in the master plot. The value of β is 0.66 and 0.56 for NaX and Na-ETS-10, respectively, and indicates that ion-ion interactions in ETS-10 are much more significant than in zeolite X. To reduce this correlation effect, we decreased the cation site occupancy by exchanging the monovalent cations in ETS-10 by divalent cations such as Cu2+. The structure of ETS-10 is maintained after Cu2+-exchange as determined by powder X-ray diffraction. The composition was verified by atomic absorption spectroscopy (AAS) showing the near complete exchange of the extraframework cations with Cu2+ (Cu/Ti ) 1:0.99). The β value of Cu2+-ETS-10 is larger than that of Na-ETS-10 which confirms a reduction in the ion-ion interactions (β is 0.62 and 0.56 for Cu2+-ETS-10 and Na-ETS-10, respectively). This reduction in site occupancy and ion-ion correlation, and subsequent increase in the activation entropy, yields an increase in the ion conductivity of ETS-10 relative zeolite X for divalent cations (Figure 9). Also the prefactor for Cu-ETS-10 was found to be about 2 orders of magnitude larger than that of Na-ETS10. Also, the activation energy of ion conduction is 15 kJ/mol smaller than CuX. These data show that ETS-10 can facilitate divalent ion transport.
Ion Transport in Microporous Titanosilicate Conclusions The ion transport in Na-ETS-10 was investigated by impedance spectroscopy. The Arrhenius plots for dehydrated Na-ETS10 show lower activation energy than that for NaX which indicates the ions can hop more easily between sites in ETS10. The broader relaxation peak for Na-ETS-10 in the modulus spectra and the smaller magnitude of the Arrhenius prefactor indicate the existence of smaller entropic contribution which is due to the high site occupancy in ETS-10. However, after replacing divalent cations in ETS-10, we found that the Cu2+ETS-10 has lower activation energy of ion conduction and also higher ionic conductivity than CuX. Accordingly, it suggests the structure of ETS-10 can enhance divalent cation transport. Acknowledgment. The authors thank the financial support from National Science Foundation under the CAREER Award (0134255-CTS). Additionally, the authors thank Richard Jacubinas and Engelhard Corporation for providing the ETS-10. References and Notes (1) Sherman, J. D. Proc. Natl. Acad. Sci.U.S.A. 1999, 96, 3471. (2) Lv, L.; Hor, M. P.; Su, F. B.; Zhao, X. S. J. Colloid Interface Sci. 2005, 287, 178. (3) Lv, L.; Tsoi, G.; Zhao, X. S. Ind. Eng. Chem.Res. 2004, 43, 7900. (4) Al-Attar, L.; Dyer, A.; Harjula, R. J. Mater. Chem. 2003, 13, 2963. (5) Cheng, X. L.; Wang, Z. B.; Yan, Y. S. Electrochem. Solid State Lett. 2001, 4, B23. (6) Mitra, A.; Wang, Z. B.; Cao, T. G.; Wang, H. T.; Huang, L. M.; Yan, Y. S. J. Electrochem. Soc. 2002, 149, B472. (7) Takami, M.; Yamazaki, Y.; Hamada, H. Electrochemistry 2001, 69, 98. (8) Libby, B.; Smyrl, W. H.; Cussler, E. L. AIChe J. 2003, 49, 991. (9) Tricoli, V.; Nannetti, F. Electrochim. Acta 2003, 48, 2625. (10) Hillhouse, H. W.; Tuominen, M. T. Microporous Mesoporous Mater. 2001, 47, 39. (11) Hicks, L. D.; Dresselhaus, M. S. Phys. ReV. B: Condens. Matter Mater. Phys. 1993, 47, 16631. (12) Franke, M. E.; Simon, U. Solid State Ionics 1999, 118, 311. (13) Franke, M. E.; Simon, U. Phys. Status Solidi B2000, 218, 287.
J. Phys. Chem. B, Vol. 110, No. 28, 2006 13733 (14) Schaf, O.; Ghobarkar, H.; Adolf, F.; Knauth, P. Solid State Ionics 2001, 143, 433. (15) Yamamoto, N.; Okubo, T. Microporous Mesoporous Mater. 2000, 40, 283. (16) Simon, U.; Flesch, U.; Maunz, W.; Muller, R.; Plog, C. Microporous Mesoporous Mater. 1998, 21, 111. (17) Fox, M. A.; Doan, K. E.; Dulay, M. T. Res. Chem. Intermed. 1994, 20, 711. (18) Lamberti, C. Microporous Mesoporous Mater. 1999, 30, 155. (19) Zecchina, A.; Arean, C. O.; Palomino, G. T.; Geobaldo, F.; Lamberti, C.; Spoto, G.; Bordiga, S. Phys. Chem. Chem. Phys. 1999, 1, 1649. (20) Anderson, M. W.; Terasaki, O.; Ohsuna, T.; Philippou, A.; Mackay, S. P.; Ferreira, A.; Rocha, J.; Lidin, S. Nature 1994, 367, 347. (21) Anderson, M. W.; Terasaki, O.; Ohsuna, T.; Malley, P. J. O.; Philippou, A.; Mackay, S. P.; Ferreira, A.; Rocha, J.; Lidin, S. Philos. Mag. B 1995, 71, 813. (22) Anderson, M. W.; Agger, J. R.; Luigi, D. P.; Baggaley, A. K.; Rocha, J. Phys. Chem. Chem. Phys. 1999, 1, 2287. (23) Grillo, M. E.; Carrazza, J. J. Phys. Chem. 1996, 100, 12261. (24) Almond, D. P.; Duncan, G. K.; West, A. R. Solid State Ionics 1983, 8, 159. (25) Sidebottom, D. L. Phys. ReV. Lett. 1999, 83, 983. (26) Rocha, J.; Ferreira, A.; Lin, Z.; Anderson, M. W. Microporous Mesoporous Mater. 1998, 23, 253. (27) Bordiga, S.; Paze, C.; Berlier, G.; Scarano, D.; Spoto, G.; Zecchina, A.; Lamberti, C. Catal. Today 2001, 70, 91. (28) Edwards, D. D.; Hwang, J. H.; Ford, S. J.; Mason, T. O. Solid State Ionics 1997, 99, 85. (29) Almond, D. P.; Hunter, C. C.; West, A. R. J. Mater. Sci. 1984, 19, 3236. (30) MacDonald, J. Impedance Spectroscopy: Theory, Experiment, and Applications; Wiley-Interscience: Hoboken, NJ, 2005. (31) Bruce, P. Solid State Electrochemistry; Cambridge University Press: New York, 1995. (32) Simon, U.; Flesch, U. J. Porous Mater. 1999, 6, 33. (33) Hodge, I. M.; Ingram, M. D.; West, A. R. J. Electroanal. Chem. 1976, 74, 125. (34) Ngai, K. L.; Leon, C. Phys. ReV. B: Condens. Matter Mater. Phys. 1999, 60, 9396. (35) Moynihan, C. T.; Schroeder, J. J. Non-Cryst. Solids 1993, 160, 52. (36) Patel, H. K.; Martin, S. W. Phys. ReV. B: Condens. Matter Mater. Phys. 1992, 45, 10292.
