J . Phys. Chem. 1989, 93, 1393-1396
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Picosecond Geminate Recombination of Phenyithiyl Free-Radical Pairs T. W. Scott* and S . N. Liu Corporate Research Science Laboratories, Exxon Research and Engineering Company, Annandale. New Jersey 08801 (Received: July 1, 1988)
A rapid decay in optical absorption by the photolysis products of diphenyl disulfide is interpreted as geminate recombination of phenylthiyl free-radical pairs. The first half-life for recombination ranges from 11 ps in n-hexane to 36 ps in liquid decalin. Continuum diffusion models describe the solvent and temperature dependence of the decay profiles.
Introduction Kinetic studies of geminate radical pair recombination are expected to provide fundamental information on the solvent cage effect in bond dissociation reactions, as well as on the more general problem of encounter pair rate processes in bimolecular chemistries. Recombination dynamics have been studied most extensively for the iodine photodissociation reaction and for organic radical ions formed as triplet spin pairs. While recombination times for iodine are obscured by competing vibronic relaxation events,' the recombination of triplet radical ions has been shown to occur during the extended lifetime of a Coulomb bound pair and is controlled by the rate of triplet to singlet state intersystem crossing.2 For large organic radicals formed as singlet spin pairs, the time scales for geminate recombination are not well charact e r i ~ e d . Photolysis ~ of tetraphenylhydrazine in octoi14 appears to be the only time-resolved study in this category. It concludes that while diphenylamino radicals are formed rapidly, they are also long-lived and immune to recombination on the time scale of solvent caging. In this report we describe picosecond transient absorption studies of the photolysis of diphenyl disulfide and the subsequent recombination of phenylthiyl free-radical pairs. There is extensive chemical evidence that photolysis of disulfides cleaves the sulfursulfur bond. The quantum yield for formation of fully separated radicals is 0.18 for diphenyl disulfide in isooctane s o l u t i ~ n . Deviation ~ of the free-radical yield from unity is thought to arise in part from nonradiative decay channels which compete with bond cleavage and in part from geminate recombination of initial pairs. Geminate recombination has been demonstrated from the solvent viscosity dependence for scavenging photodissociated aryl disulfides.6 The spin multiplicity of the initial pair has not been previously determined. Photosensitized dissociation studies show that both singlet- and triplet-state sensitization cleaves the sulfur-sulfur bond.' However, the recombination time scales reported here indicate that thiyl radicals formed by ultraviolet photolysis have singlet-spin-state parentage. In previous studies of disulfide photolysis, we reported8 an initial decay of the transient intermediate during the first few hundred picoseconds after photolysis. The absorption spectrum of the photolysis products together with the solvent and temperature dependence of the initial decay suggests that these transients arise from radical pair recombination. The present work examines both the solvent and temperature dependence in more detail and ( I ) (a) Chuang, T. J.; Hoffman, G. W.; Eisenthal, K. B. Chem. Phys. Lett. 1978,68, 3292. (b) Smith, D. E.; Harris, C. B. J. Chem. Phys. 1987,87,2709. (c) Abul-Haj, A. N.; Kelley, D. F. J. Chem. Phys. 1986,84, 1335. (d) Hynes,
compares experimental results with time-dependent solutions of the continuum diffusion equation for geminate pair reactions. Previous picosecond studies have explored the use of diffusion models in describing the kinetics of geminate electron-cation recombination9 and the decay of photoinjected electrons at the metal-liquid interface.1° Ligand recombination kinetics in photolyzed heme proteins should also be mentioned as an illustration of diffusive recombination of neutral geminate pairs."
Experimental Section Phenylthiyl radical pairs were formed by 354.7-nm photolysis of diphenyl disulfide dissolved in liquid alkanes. An actively and passively mode-locked Nd:YAG laser provided photolysis energies of 0.5 mJ per pulse focused to a 2-mm beam diameter with a time duration of 35 ps. Absorption spectra were recorded in the 400-600-nm wavelength region by using a white light continuum generated by 1064-nm irradiation of an H z O / D 2 0 mixture in combination with a vidicon detection system. The kinetics of radical recombination were measured by transient absorption at 435.7 nm by using stimulated Raman shifting of the Nd:YAG second harmonic frequency in hydrogen gas at 300-400 psi. Pump and probe beams entered the cell collinearly, fixed at a relative polarization angle of 5 5 O , even though polarized absorption experiments failed to detect rotational depolarization at the time resolution of our apparatus. Disulfide solutions at a concentration of 5 X mol/L were held in 1-cm cuvettes equipped with magnetic stirring. Recrystallized samples of diphenyl disulfide exhibited the same kinetic behavior as commercially available samples. Results and Discussion The transient absorption spectrum of the photolysis products of diphenyl disulfide is shown in Figure 1. The absorption band is centered at 445 nm. Previous microsecond flash photolysis12 and low-temperature absorption studies13identified an absorption band at 450 nm, which was assigned to the phenyl thiyl radical on the basis of its similarity to the spectrum of photolyzed thiophenol. Absorption maxima shifted to longer wavelengths have also been reported,14 but there is evidence that aggregation of the parent disulfide may be responsible for this feature.15 A survey of disulfide concentrations above 5 X lo-* M in decalin showed that the transient spectrum in Figure 1 was replaced by a new profile with maxima at 427 and 462 nm. To further test a free-radical assignment for the picosecond absorption signals seen at lower disulfide concentrations, the half-life for radical addition to styrene was measured in neat liquid styrene.8 The disulfide
J. T. In Theory of Chemical Reaction Dynamics; Baer, M., Ed.; CRC Press: 1985; VOI. 4, pp 216-218. (2) (a) Werner, H.-J.; Staerk, H.; Weller, A. J . Chem. Phys. 1978, 68, 2419. (b) Schulten, K.; Wolynes, G. J . Chem. Phys. 1978, 68, 3292. (3) Koenig, T.; Fischer, H. In Free Radicals; Kochi, J. K., Ed.; John Wiley 8c Sons: New York, 1973; Vol. 1, pp 157-189. (4) Anderson, R. W.; Hochstrasser, R. M. J . Phys. Chem. 1976, 80, 2155. (5) Burkey, T. J.; Majewski, M.; Griller, D. J. Am. Chem. Soc. 1986,108, 2218. (6) Schaafsma, Y.; Bickel, A,; Kooyman, E. C. Tetrahedron 1960,10,76. (7) Wallace, W. L.; Van Duyne, R. P.; Lewis, F. D. J . Am. Chem. SOC. 1976, 98, 5319. (8) Scott, T. W.; Liu, S.N. In UItrufust Phenomena V; Fleming, G. R., Siegman, A. E., Eds.; Springer-Verlag: Berlin, 1986; pp 338-340.
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(9) (a) Braun, C. L.; Scott, T. W. Rad. Phys. Chem. 1988, 32, 315. (b) Braun, C. L.; Scott, T. W. J. Phys. Chem. 1987,91,4436. (c) Scott, T. W.; Braun, C. L. Chem. Phys. Letr. 1986,127, 501. (d) Scott, T. W.; Braun, C. L. Can. J . Chem. 1985,63,228. (e) Braun, C. L.; Scott, T. W. J. Phys. Chem. 1983, 87, 4776.
(10) Scott, T. W. J . Phys. Chem. 1986, 90, 1739. (11) Friedman, J. M.; Scott, T. W.; Fisanick, G. J.; Simon, S.; Findsen, E.; Ondrias, M.; MacDonald, V. Science 1985, 229, 187. (12) Thyrion, F. C. J . Phys. Chem. 1973, 77, 1478. (13) Russel, P. G. J . Phys. Chem. 1975, 79, 1353. (14) Ito, 0.;Matsuda, M. J . Am. Chem. SOC.1983, 105, 1937. (15) Burkey, T. J.; Griller, D. J. Am. Chem. Soc. 1985, 107, 246.
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Figure 1. Transient absorption spectrum of the 354.7-nm photolysis products of diphenyl disulfide in liquid decalin (5 X M solution). The spectrum was recorded during the 35-ps photolysis light pulse by using the white light continuum generated by 1064-nm irradiation of an HzO/DzO mixture.
absorption transient decayed with a half-life of 1.5 ns, corresponding to a bimolecular rate constant of 5.5 X lo7 M-I s-l. The rate constant measuredI6 on the microsecond time scale in dilute solution is 5.1 X lo7 M-I SI. In pure liquid hexatriene, the half-life of the absorption transient is 380 ps. Disulfide photolysis in alkane solvents yields a rapid transient absorption decay over the first few hundred picoseconds. Unlike the liquid olefins, these solvents are chemically unreactive with the phenylthiyl radical. In addition, the extent of initial decay is strongly dependent on the choice of solvent and temperature. The temperature dependence, in particular, is quite dramatic when compared to the exothermic addition reactions with liquid olefins, which involve small energies of activation. Figure 2 illustrates this behavior in hexane, dodecane, and decalin near room temperature and in decalin below room temperature. We interpret the initial decay of the disulfide photolysis products as geminate recombination of phenylthiyl radical pairs. The transient absorption spectrum of the photolysis products measured in liquid decalin gave no evidence of time-dependent spectral shifts between 35 and 800 P S . ~ The influence of reaction environment on the recombination process can be examined in more detail by comparing the data shown in Figure 2 with transient solutions of the diffusion equation. Diffusion models of geminate pair recombination connect the time-dependent pair survival probability P(T)with macroscopic properties of the host solvent. Each radical is treated as a spherical particle embedded in a uniformly viscous medium. The pair is assumed to undergo random Brownian movements that ultimately lead to either recombination or escape. These movements are described by a diffusion equation in which the recombination reaction is introduced as an appropriate boundary condition. For the radiation boundary condition,I7 adapted from the theory of heat conduction in solids, the inward flux of particles across a defined reaction boundary is assumed proportional to the concentration of particles at the boundary surface. The proportionality constant has the nature of an intrinsic bimolecular reaction rate constant. The time-dependent pair survival probability P(T)is given byIg
(16) Ito, 0.; Matsuda, M. J . Am. Chem. SOC.1979, 101, 1815. (17) Collins, F. C.; Kimball, G . E. J . Colloid Sci. 1949, 4, 425.
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Figure 2. Transient absorption decay at 435.7 nm of the 354.7-nm photolysis products of diphenyl disulfide. Decay profiles are shown for (A) n-hexane, (B) dodecane, and (C) liquid decalin at 278 K and for liquid decalin at (D) 243 K. The solid lines are solutions of the diffusion equation combined with 35-ps (fwhm) Gaussian light pulses for excitation and probing. The fitting parameters required in eq 1 of the text are as follows: (A) K = 1.26, X = 0.756, 70 = 23.2 ps; (B) K = 1.35, X 4.15, 7 0 = 127 ps; (C) K = 1.23, X = 5.64, 7 0 173 ps; (D) K = 1.23, X = 13.0, io = 524 ps. Relative values of X and ro for each solvent are specified by the viscosity and temperature dependence of the mutual diffusion coefficient according to the Stokes-Einstein relation. Liquid decalin used for these experiments is a mixture of cis and trans isomers.
where K = ro/u is the ratio of the initial pair separation (ro)to the sum of the hard sphere collision radii ( u ) , A = kR/kDis the ratio of the proportionality constant (kR) appearing in the description of the radiation boundary condition to the rate constant (k,) for pair diffusion, and T is the ratio of time elapsed since pair formation to the diffusive time constant (T~).The parameters X and T~ depend on common physical properties according to kD = 4aaD and T~ = u2/D,where D is the mutual diffusion coefficient. The proportionality constant kR is sometimes described as an intrinsic bimolecular rate constant that would be observed in the absence of diffusional limitations. Our use of eq 1 assumes that the chemical anisotropy of each radical is randomized by rapid rotational reorientation. The case of slow rotational lifetimes has been examined at steady state for bimolecular self-termination reaction^,'^ but apparently not for time-dependent geminate pair recombination. Equation 1 also neglects all radical-radical interaction potentials and any spatial dependence of the diffusion coefficient arising from the discrete molecular nature of real liquidsz0 Finally, the time evolution of singlet-triplet spin states for the radical pair has not been included in the development of eq 1. In the absence of an applied magnetic field, spin transitions are induced by differences in local hyperfine fields at the two radical centers.*I These transitions occur on a time scale of to s for organic free radicals. The time scale for the recombination processes seen here must be associated with singlet-born pairs with no spin interconversion barriers to recombination. . (18) Shin, K. J.; Kapral, R. J . Chem. Phys. 1978, 69, 3685. (19) (a) Solc, K.; Stockmayer, W. H. J . Chem. Phys. 1971,54, 2981. (b) Solc, K.; Stockmayer, W. H. Int. J . Chem. Kinet. 1973, 5, 733. (20) Northrup, S . H.; Hynes, J. T. J . Chem. Phys. 1979, 71, 884. (21) Salikhov, K. M.; Molin, Yu.N.; Sagdeev, R. Z.; Buchachenko, A. L. In Spin Polarization and Magnetic Effects in Radical Reactions; Molin, Yu. N., Ed.; Elsevier Science Publishers: 1984; pp 16-22.
Recombination of Phenylthiyl Free-Radical Pairs
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Figure 3. Diffusion model of the time-dependent radical pair survival probability according to eq 1 of the text. Parts A, B, and C show how changing the initial pair separation, the mutual diffusion rate, and the boundary condition for reaction respectively influences geminate recombination dynamics. The solid line in each panel corresponds to K = 1.3, A = 6.7,and T~ = 256 ps. In part A, the reduced initial separation is changed to K = 1.1 ( - - - ) and then K = 1.5 Part B shows the effect of increasing and then decreasing the mutual diffusion rate by a factor (..e).
of 5, giving X = 1.34 and
= 51.2 ps and X = 33.5 and 70 = 1280 ps (- - -), respectively. Part C shows the recombination kinetics in the Smoluchowski limit X = and for a 4-fold decrease in pair reactivity, X = 1.68 ( - - - ) . T~
(e..)
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Figure 3 shows how the radical pair survival probability depends
on the three physical parameters contained in K, X, and 70. The probability that an isolated pair has evaded geminate recombination is plotted against the time elapsed since its creation. The value of the long time plateau gives the yield for cage escape. Figure 3a shows that pairs created with a large initial separation have a greater escape probability than small-radius pairs. Figure 3b illustrates how the radical pair mutual diffusion rate influences the recombination time scale and yield. Accelerated diffusion, caused by either high temperature or low viscosity, decreases the geminate pair lifetime while increasing the yield of separated pairs. Figure 3c shows the influence of the boundary condition for reaction. In the smoluchowski limit, obtained by letting X approach infinity, radical pairs recombine during their first collision. Reducing the intrinsic pair reactivity via the radiation boundary condition introduces nonreactive radical-radical collisions, allowing a larger fraction of geminate pairs to escape. Solutions of eq 1 were compared with the four transient absorption decays shown in Figure 2, starting with the curve for liquid decalin at 278 K. Equation 1 was evaluated for selected values of K , X, and 7,,, and the resulting P(7)was convoluted with 35-ps Gaussian line shapes (fwhm) representing the time profile of the pump and probe light pulses. Trial simulations for the remaining traces were obtained by adjusting X and 70 according to the Stokes-Einstein prescription for the viscosity and temperature dependence of the mutual diffusion coefficient.22 Agreement between the experiments and the simulations can be improved by selecting different initial pair separations in solvents with different molecular structures. Notice, however, that only one independent choice of X and 70is being used to fit all four decay curves, since the six remaining values are specified by the Stokes-Einstein relation and tabulated solvent viscosities. Final values for the three fitting parameters are given in the legend accompanying Figure 2. The time scales for geminate recombination can be separated from the convolution with the laser pulse width by substituting these parameters into eq 1. The first half-life for recombination is 1 1 ps in n-hexane, 30 ps in dodecane, and 21 ps in decalin at 278 K. At 243 K, the first half-life in liquid decalin is 36 ps. Figure 4 shows the temperature dependence of the transient absorption decay in cis-decalin over an extended range. The vertical axis gives the value of the plateau absorption at 2 ns as a percentage of the initial transient absorption signal at zero time delay. The solid line was obtained from eq 1 by adjusting X and T~ according to the known temperature dependence of cis-decalin v i s ~ o s i t yfollowed ~ ~ ~ ~ by ~ convolution with the 35-ps pump and (22) Relative values of A and r0 at different temperatures ( T ) and viscosities (?) are given by (A’/A) = (T,,’/T~) = ( T / T ? ( q ’ / ? ) based on the Stokes-Einstein description of the mutual diffusion coefficient, D.
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Figure 4. Temperature dependence of the 435.7-nm transient absorption decay after photolysis of diphenyl disulfide in cis-decalin (99% pure cis isomer). The percent decay plotted on the vertical axis expresses the decrease in optical absorption between 35 ps and 2 ns after photolysis. The solid line shows the temperature dependence predicted soley from the change in solvent viscosity. This curve was constructed from solutions to eq 1, convoluted with the excitation and probe light pulse widths. Relative values of X and r0 at each temperature were calculated from the Stokes-Einstein relation and tabulated values for the temperature dependence of cis-decalin viscosity. The reduced initial pair separation is held constant at K = 1.3. The absolute values of X and T~ at 268 K are 14.0 and 428 ps, respectively.
probe light pulse widths. The value of K was held constant at 1.3. Over the temperature range displayed in Figure 4, cis-decalin viscosity decreases from 16.2 CP at 237 K to 0.54 CPat 429 K, while the calculated mutual pair diffusion rate increases by a factor of 54. The experimental absorption decay falls from 56% to 9.2% over this range, corresponding to a decrease in the radical pair recombination probability of 0.75 to 0.32. The physical parameters appearing in K, A, and 70 can be determined from an estimate of the hard sphere collision diameter. We assign u a value equal to twice the distance from the center of one phenyl ring to the center of the sulfur-sulfur bond in the parent disulfide. This choice appears to satisfy the implicit assumption in eq 1 that, for a rapidly rotating molecule, a single value of u/2 will describe both the distance of closest approach for spherical particles and the dependence of the diffusion coefficient on particle size. X-ray crystallography studies of disulfides” provide a value of u = 7.2 A, leading to a diffusion coefficient cm2 s-l, an intrinsic bifor a single thiyl radical of 1.3 X molecular rate constant of 9.2 X 1OloM-’ s-I, and an initial pair separation of 9.4 8, in cis-decalin at room temperature. The diffusion coefficient is comparable to values estimated for the benzyl radical in sol~tion,~’ and the bimolecular rate constant is of the same order as combination rates measured in the gas phase.26 The displacement of the radical fragments after photolysis is estimated as 2.2 8,. It is interesting to relate the kinetics of geminate pair recombination to the reaction probability of random encounter pairs formed in bimolecular self-termination reactions. If we say that an encounter pair is formed at the first radical-radical collision, A/( 1 + A), can be obtained from eq the reaction pr~bability,~’ 1 by setting K = 1 and T = a. From the data in Figure 4, calculated reaction probabilities per encounter for phenylthiyl ~~~~
(23) Landolt-Bornstein. Zahlewerte und Functionen; Springer: Berlin, 1969; Vol. 11, Part 5a. (24)Sacerdoti, M.;Gill, G. Acta Crystallogr. 1975, 631, 327. (25) Burkhart, R. D.;Wong, R. J. J . Am. Chem. SOC. 1973, 95, 7203. (26) Kerr, J. A. In Free Rodicals; Kochi, J. K., Ed.;John Wiley & Sons: New York, 1973; Vol. 1, pp 5-9. (27) Notice that there is an error in the limiting form of cq 2.41(b) in ref 18. The numerator of the second term should be multiplied by A
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radicals in cis-decalin range from 0.97 at 237 K to 0.42 at 429 K.
Conclusions Picosecond photolysis of diphenyl disulfide generates transient absorption decays, which we interpret as geminate recombination of phenylthiyl free-radical pairs. This assignment is based on the absorption spectrum of the photolysis products and the temperature and solvent dependence of the absorption decay. To the best of our knowledge, this provides the first example of a polyatomic singlet radical pair suitable for picosecond studies of the solvent cage effect. Preliminary experiments on the photolysis of azocumene2* and asymmetrical disulfides also show absorption
transients that respond to viscosity in a manner suggesting that radical pair recombination is a controlling factor. In the work reported here, continuum diffusion models describe the general trend of enhanced pair recombination with increasing solvent viscosity. These simulations also suggest that early recombination events could be examined in more detail by using an apparatus with improved time resolution. Registry No. Diphenyl disulfide, 882-33-7;n-hexane, 110-54-3;dodecane, 112-40-3;decalin, 91-17-8; cis-decalin, 493-01-6; phenylthiyl radical, 4985-62-0. (28) Doubleday, C.; Scott, T. W., unpublished results.
Kinetics of Complexation of Lithium Perchlorate with 18-Crown-6 in Propylene Carbonate Daryl P. Cobranchi, Gregory R. Phillips, David E. Johnson, R. Michael Barton, David J. Rose, Edward M. Eyring,* Department of Chemistry, University of Utah, Salt Lake City, Utah 84112
Licesio J. Rodriguez, and Sergio Petrucci Weber Research Institute, Polytechnic University, Farmingdale. New York I 1 735 (Received: July 8, 1988)
The kinetics of complexation of LiC104 with the macrocycle 18-crown-6 have been studied in propylene carbonate by using ultrasonic relaxation techniques. Two concentration-independent relaxations are observed and explained in terms of the Eigen-Winkler mechanism. The influence of propylene carbonate on the complexation of lithium with crowns is compared with the influence of other aprotic solvents.
Introduction Kaplan et a1.I recently reported that crown ethers added to poly(viny1ene carbonate) containing a lithium salt enhance charge transport in this solid electrolyte. Thus, possible applications in lithium batteries are a justification for studying the kinetics of decomplexation and complexation of lithium ions by crown ethers in nonaqueous, aprotic media. A more fundamental reason for such a kinetic study is the aim of understanding the relative importance of several competing factors that can all play a significant role in the reaction kinetics of nonaqueous solutions of alkali-metal cations and crown ethers. These factors include the Gutmann donor number of the solvent, the flexibility of the macrocyclic ligand, the basicity of the ether oxygens of the macrocycle, and the reorganization of the solvent cage around the cation.2 Whereas in aqueous solution the kinetics of lithium ion complexation are dominated by the rate-limiting loss of a solvent water molecule from the first coordination sphere of the cation, in nonaqueous media the crown ether, cation, associated anion, and solvent form a “complex supramolecular a ~ s e m b l y ” ~ in which each participant plays a significant role in the reorganization accompanying the cation complexation or decomplexation. The present kinetic study of lithium ion and 18-crown-6 (18C6) has been carried out in dry propylene carbonate (PC). This is an aprotic, nonassociated solvent with no hydrogen bond donating capacity. At 25 OC this solvent has a higher relative permittivity ( t = 64.4) and higher coefficient of viscosity ( 7 = 2.53 cP) than
most other solvents used in lithium b a t t e r i e ~ . ~ The choice of the perchlorate anion for these kinetic studies was dictated by the availabilityS of stability constants for lithium perchlorate complexes formed with 12C4, 15C5, and 18C6 in several solvents.
Experimental Section Instrumentation. Ultrasonic absorption measurements were made using the laser Debye-Sears, resonator, and pulse techniques. The Debye-Sears apparatus differs from a previous description6 in two major aspects: ( I ) the stepping motors in the present work were manually controlled, and (2) an argon ion laser previously used was replaced by a 5-mW HeNe laser. The lower power of the HeNe laser did not reduce the accessible (3-240 MHz) frequency range. The Debye-Sears apparatus was housed in a glovebox of local construction to exclude water vapor and was purged continuously with dry nitrogen gas. The resonator and pulse apparatus have been previously described.’ Data from these three ultrasonic techniques were combined and analyzed by using a Levenburg-Marquardt nonlinear least-squares algorithm.8 Reagents. Propylene carbonate (Aldrich) was stored over activated (baked under vacuum) 3A molecular sieves for a least 1 week and then distilled under reduced pressure. Anhydrous lithium perchlorate (G. F. Smith Chemicals) was heated under vacuum at 170 OC until there was no further decrease in pressure. (4) Gabano, J.-P. Lithium Batteries; Academic Press: New York, 1983;
p
M. L.; Rietman, E. A,; Cava, R. J.; Holt, L. K.; Chandross, E. A. Solid State tonics 1987, 25, 37-40. (2) Graves, H . P.; Detellier, C. J . Am. Chem. SOC.,in press. (3) Delville, A.; Stover, H. D. H.; Detellier, C. J . Am. Chem. SOC.1987, ( 1 ) Kaplan,
109, 7293-7301.
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(5) Smetana, A. J.; Popv, A. I . J . Solution Chem. 1980, 9, 183-196. (6) Farrow, M. M.; Olsen, S. L.; Purdie, N.; Eyring, E. M. Reu. Sci. Instrum. 1976, 47, 657-661. (7) Chen, C.; Wallace, W.; Eyring, E. M.; Petrucci, S . J . Phys. Chem. 1984, 88, 2541-2547. ( 8 ) Nash, J . C. J . Inst. Math. Its Appl. 1977, 9, 231.
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