J. Phys. Chem. 1996, 100, 11325-11335
11325
High Mobility Ions in Cycloalkanes. Transient Dc Conductivity† M. C. Sauer, Jr., I. A. Shkrob, J. Yan, K. Schmidt, and A. D. Trifunac* Chemistry DiVision, Argonne National Laboratory, Argonne, Illinois 60439 ReceiVed: February 7, 1996; In Final Form: April 16, 1996X
Transient dc conductivity is used to observe high-mobility ions in laser photolysis and pulse radiolysis of cyclohexane and decalins. While in decalins the scavenging of the dc conductivity signal from these ions by solutes is a pseudo-first-order reaction, in cyclohexane the behavior is more complex and is indicative of the involvement of two species. This bimodality is rationalized in terms of a dynamic equilibrium between the conformers of the solvent radical cation. A similar equilibrium between the isomer radical cations of decalin is needed to account for the results on photoinduced conductivity in the mixtures of cis- and trans-decalin. In both of these cases, the mechanism of the equilibrium is the reversible charge transfer.
Introduction High-mobility ions (HMI) in liquid cyclohexanes1-10 and decalins11-14 are elusive species with unusual behavior. Though their existence was demonstrated almost 25 years ago,1 there still is uncertainty about their nature and the mechanism of rapid migration. Originally, these ions were believed to be solvent radical cations (holes) involved in rapid resonant charge transfer.5 However, studies on fluorescence,15 fluorescence-detected magnetic resonance (FDMR),16 and transient absorption spectroscopy3a,17 in radiolysis of cyclohexane indicated short lifetime of the species (tens of nanoseconds), whereas the transient conductivity studies indicate that the HMI has the natural lifetime of g0.3-0.5 µs.5,7 Importantly, rapid scavenging of cyclohexane•+ with rate constants ∼(3-4) × 1011 mol-1 dm3 s-1 (vs 1.5 × 1010 mol-1 dm3 s-1 for diffusion-limited ionmolecule reactions) does occur and can be demonstrated in several ways.3,10,17 However, this scavenging was observed only within 50 ns after the ionization event. In the previous work of this series, we estimated that the lifetime of the solvent hole is ∼30 ns.17 There is little doubt that the charge on the radical cation of cyclohexane moves rapidly. On the other hand, there is no evidence that the long-liVed HMI observed by transient conductivity are these same radical cations. Importantly, in decalin we did observe a long-lived solvent radical cation which reacted with perylene ∼11 times faster than normally diffusing ions.18 The lifetime of the solvent holes was determined as 150 ns.18 Though this lifetime is shorter than that observed in the same solution by dc photoconductivity ( ≈ 1 µs), this discrepancy can be accounted for by reactions of the solvent holes with radiolytic products and anions.12,17,18 For cyclohexane, this is not the case: under identical conditions in the same flowing sample we have observed long-lived HMI by dc photoconductivity and the formation kinetics of aromatic ions by pulse radiolysis typical of short-lived precursor holes.17 This seems to suggest that in cyclohexane there are two mobile ions, one of which does not transfer charge to aromatic solutes. Yet we found no spectroscopic evidence that aromatic cations other than radical cations (such as proton adducts and carbonium ions) were formed rapidly. † Work performed under the auspices of the Office of Basic Energy Sciences, Division of Chemical Science, US-DOE under Contract W-31109-ENG-38. * To whom correspondence should be addressed. X Abstract published in AdVance ACS Abstracts, June 15, 1996.
S0022-3654(96)00363-2 CCC: $12.00
So far, transient conductivity is the only technique allowing detection of the long-lived HMI in cyclohexane. It seems impossible that the two HMI (presumably, of different nature) have identical mobilities and scavenging rates. If there were two mobile ions, that would show up in scavenging experiments. Could a single HMI, such as the solvent radical cation, account for the data on transient conductivity in radiolysis and photolysis of cyclohexane? A critical look at the data shows several inconsistencies. First, the scavenging constants obtained for HMI do not follow the pattern expected for reactions of solvent radical cations (see refs 5, 6, 8, and 10 for data on the reaction constants). For example, scavenging of HMI by biphenyl is slower than by alcohols, which have higher ionization potentials (IP) and comparable proton affinities (PA). Cyclohexene12 and other high-PA olefins scavenge the mobile ions as fast as aromatic amines, while scavenging by benzene is slower than by decalins, cyclohexene, and bicyclohexyl. The observed scavenging constants k2 correlate neither with IP nor with PA of the solutes. For several solutes (e.g., ethanol and cyclohexene) scavenging constants determined in the dc photoconductivity experiments8 are 2-3 times less than those determined in the pulse radiolysis microwave conductivity experiments.5,6 In the present work, some of the values of k2 have been redetermined, and the problems with them reexamined. Second, while the model with a single mobile ion is capable of reproducing the decay kinetics of microwave conductivity in pure cyclohexane, this model does not provide equally good simulation of decay kinetics in the presence of hole scavengers. Figure 1 shows a Monte Carlo simulation of the microwave conductivity for solutions of cyclohexene in cyclohexane. Following Warman et al., we assumed that no normally diffusing ions were formed at t ) 0,4c,5,12 the mobility µm+ of fast holes was 9.5 × 10-3 cm2/V s,5 and their natural lifetime was 0.3 µs.12 Though the curves obtained in neat cyclohexane can be simulated with these parameters quite satisfactorily, in the presence of scavenger the simulation does not fit the observed kinetics for any k2, whether the value of Warman et al. (3 × 1011 mol-1 dm3 s-1) 12 or Schmidt and Sauer (1.3 × 1011 mol-1 dm3 s-1) 8 was used. To explain the fast decrease in the conductivity signal at t < 100 ns, one needs to use a value of k2 ∼ (4-5) × 1011 mol-1 dm3 s-1, whereas the “tail” of the kinetics corresponds to much lower rate constant ∼(1-1.5) × 1011 mol-1 dm3 s-1. The origin of discrepancies between k2 determined in different experiments might be a bimodality of the hole scaVenging. © 1996 American Chemical Society
11326 J. Phys. Chem., Vol. 100, No. 27, 1996
Sauer et al. results suggest a reversible charge transfer between the isomer radical cations of decalin. Experimental Section
Figure 1. Monte Carlo simulation of decay kinetics of microwave conductivity in cyclohexane containing 0, 16, and 88 × 10-6 mol dm-3 of cyclohexene and 0.02 mol dm-3 CO2. We assumed that no normally diffusing ions were formed at t ) 0 , the mobility µm+ of HMI is 9.5 × 10-3 cm2/Vs, the lifetime of HMI in pure cyclohexane is 330 ns, the free ion yield is 3% (the electron-hole distribution p(r) ∝ r2 exp[-(r/ bG)2], bG ≈ 6.1 nm), scavenging constant k2 ) 1.3 × 1011 mol-1 dm3 s-1 (solid lines), 8 k2 ) 3 × 1011 mol-1 dm3 s-1 (broken lines).12 The homogeneous recombination of ions was not included. The data taken from Figure 1b in ref 12 are indicated by symbols.
Third, assuming that the maximum scavenging constant k2 is 3 × 1011 mol-1 dm3 s-1, and using µm+ ) 9.5 × 10-3 cm2/V s,5 we obtain a value for the reaction radius Rsq of ca. 1.6 nm. This radius is unusually large, especially for an ion moving by rapid resonant charge transfer. Using the Pilling-Rice equation for electron tunneling reactions19
Rsq ) R0 + λ{1.16 + ln[eK0λ2/kTµm+]}
(1)
where R0 ∼ 0.5 nm is the radius of minimum approach, K0 ) 1014 s-1 and λ ≈ 0.1 nm (K(r) ) K0 exp(-r/λ) is the rate of tunneling) we obtain Rsq ∼ 1 nm. This estimate is close to the radii of typical ion-molecule reactions in cyclohexane (1.01.2 nm). Either the mobility µm+ of radical cations is 1.5-2 times higher than 9.5 × 10-3 cm2/V s or the hole is delocalized, as was proposed by Warman.5 Interestingly, for trans-decalin, which yields a HMI with µm+ ≈8.7 × 10-3 cm2/V s, the maximum scavenging constants obtained in the present work are ∼(1.5-1.75) × 1011 mol-1 dm3 s-1 , which does correspond to Rsq ≈ 1.0 ( 0.1 nm. It can be argued that the estimates made in refs 4-9 were approximate and that the analysis of decay kinetics was oversimplified. In this work, the accuracy of the dc conductivity measurement and the thoroughness of analysis were both improved. These advances allowed us to demonstrate that for cyclohexane the model formulated by Warman et al.4c,5 is indeed incomplete, while for decalins it works well. In cyclohexane, a dichotomy in scavenging rates at different times was observed. Our result implies that either there are two HMI or, more likely, the high-mobility chair form of the solvent radical cation is in equilibrium with a normally diffusing twist form.17 The dichotomy in the scavenging rates reflects the equilibrium dynamics between these two forms. HMI in cis- and trans-decalin have been much less studied than those in cyclohexane. We demonstrate that on dilution of trans-decalin with cis-decalin and n-hexane the mobility of HMI decreases linearly with the mole fraction of trans-decalin. The scavenging constants decrease in the same fashion so that the reaction radii are constant at 0.9-1.1 nm. Once more, these
Materials. Aromatics: anthracene (An) and anthracene-d10 were obtained from Aldrich and twice sublimed in vacuo. Research-grade solutes: ethanol, n-propanol, cyclohexene, thiophene-free benzene, toluene, amines, and bicyclohexyl were used as received from Aldrich. Solvents: cyclohexane, nhexane (Baxter), trans- and cis-decalins (Wiley Organics, 99%+), and two decalin mixtures (graded as 25:75 and 37:63 trans/cis; Aldrich, 99%+) were passed 4-5 times through 1 m columns filled with fresh activated silica gel. The purity of the solvents was checked by their UV absorption and the lifetime of HMI in CO2-saturated liquid. Gases: oxygen (99.999%), ultrazero air ( 10 µs, the kinetics is due to homogeneous recombination of molecular ions, the dc conductivity
κ ) 10-3Fc0µs/[1 + Rc0t]
(3)
where R ) 10-3FNAµs/0 is the Debye relationship giving the rate constant of neutralization (in mol-1 dm3 s-1). In these equations, is the permittivity (for cyclohexane ≈ 2.023), 0 ) 8.85 × 10-12 C V-1 m-1, F ) 9.65 × 104 C/mol is the Faraday constant, c0 is the “initial” concentration of free ions in mol dm-3, µs ) µ+ + µ- is the sum of ion mobilities in cm2/V s.5 At Rc0t . 1, κ ≈ NAo/t is independent of c0 and µs. Formula 3 provides the means for accurate determination of Q from the second-order fits to the tails of κex(t). In radiolysis experiments, we obtained Q ≈ 0.8-0.9, in laser experiments with 4 cm cells Q ≈ 0.25-0.6. In photolysis, Q linearly increases with absorbance of the solution, since the attenuation of the light causes a decrease in effective length L of the cell. Of all nonaromatic solutes studied in this work, only amines and bicyclohexyl exhibited sufficiently high absorbance at 248 nm to observe this effect at solute concentration below 10-4 mol dm-3. For these solutes Q were fitted individually for each set of data. Results and Discussion Isotope Effect. In photolysis of cyclohexane, the HMI can be produced in two ways: (i) in the pure solvent by simultaneous absorption of two 248 nm photons; (ii) in solutions of aromatic sensitizers (such as anthracene, naphthalene, and biphenyl) by consecutive absorption by the aromatic sensitizer molecule of three 248 nm photons.9 As was shown, generation of HMI in anthracene solutions occurs by monophotonic
excitation of anthracene•+. The latter species is produced by biphotonic ionization of the solute. Two reactions of a highly excited anthracene radical cation can be considered: charge transfer
An•+* + c-C6H12 f An + c-C6H12•+*
(4)
and proton transfer
An•+* + c-C6H12 f An• +c-C6H13+
(5)
In terms of reaction 4, the HMI is either the radical cation of cyclohexane or an ion formed on decomposition of the excited radical cation. Alternatively, the mobile ion could be a proton adduct of cyclohexane formed via reaction 5.9 In the past, we tried to demonstrate the occurrence of reaction 5 by observation of the anthracene radical.21 From the product analysis, d9hanthracene is one of the main aromatic products formed on photolysis of anthracene-d10 in c-C6H12. This substitution indicates efficient generation of aromatic radicals in the course of photolysis.21 However, the radicals might be formed in reactions other than reaction 5. Besides, the proton adduct c-C6H13+ can be formed via proton-transfer reaction 6 following reaction 4.
c-C6H12•+* + c-C6H12 f c-C6H11• +c-C6H13+
(6)
Assuming that the HMI is a proton adduct, its motion is a series of rapid proton transfers between the neighboring solvent molecules. As known from other studies, proton transfers exhibit a maximum isotope effect when ∆G ) 0.22 Given that the proton in c-C6H13+ is not in exchange with other hydrogens, 23 one should observe a difference in the mobility of HMI upon the injection of D+ instead of H+. To test whether the injection of D+ via reaction 5 would cause a decrease in the mobility, we photolyzed (separately) 2 × 10-6 mol dm-3 of d10 and h10 anthracene in CO2-saturated (0.066 mol dm-3) cyclohexane. Comparison of conductivity traces for deuterated and protiated anthracene demonstrates that the kinetics are identical. The absence of the isotope effect on deuteration indicates that if HMI is a proton adduct of cyclohexane then it must be formed in reaction 6 rather than reaction 5. Scavenging of Mobile Ions by O2. In our first publication on dc photoconductivity, we reported that addition of O2 to cyclohexane resulted in a faster decay of HMI.7 In this study we pursued this effect further. Saturation with oxygen (0.01 mol dm-3) considerably changes the absorption spectra of unirradiated neat cyclohexane and its anthracene solutions. Because of this effect we concentrated on the decay of the HMI rather than its yield. Several experiments were performed: (i) A given quantity of the O2-saturated solution was mixed with CO2-saturated solution. The mixture was shaken for 1 min and injected into the cell where it was photolysed with 5-6 laser pulses. The concentration of CO2 was sustained sufficiently high to shorten the lifetime of the solvated electrons to 1-10 ns. We used both pure cyclohexane and anthracene solutions, with the same result. In one of the test experiments, cyclohexane was purged with oxygen for 30 min, the oxygen was removed by purging CO2 for another 15 min, and the solution was photolyzed. The kinetics observed in fresh CO2saturated solutions and previously oxygenated and then saturated with CO2 solutions were identical. Therefore, HMI were scavenged by oxygen rather than by an oxidation product.
11328 J. Phys. Chem., Vol. 100, No. 27, 1996
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(ii) To preclude the variation in the concentration of electron scavengers, the same experiment was performed with the liquid electron scavenger n-butyl chloride. The latter is a nonvolatile liquid which does not absorb at 248 nm and, added in millimolar concentration, has no effect on the lifetime of the HMI. Two 0.01 mol dm-3 n-butyl chloride solutions, one saturated with argon and another with oxygen, were mixed. The effect of O2 was identical with the CO2 experiments. (iii) Gaseous oxygen and argon were mixed in a given ratio, bubbled through the 0.01 mol dm-3 n-butyl chloride solution of 2-5 µmol dm-3 anthracene, and used to force the liquid through the flow cell. Another modification of this experiment was to use, separately, pure argon, pure oxygen and air; 60120 laser shots were averaged. Apart from the improved signalto-noise ratio, the results were identical with those obtained without the flow. Apparently, HMI reacts with oxygen rather than a photoproduct formed in oxygenated solution. In another experiment we used 308 nm photons (120 mJ/pulse). The rate constants of decay of HMI were the same as observed at 248 nm photoexcitation. These experiments gave the scavenging constant k2 ) 3 × 108 mol-1 dm3 s-1. For trans-decalin as a solvent, we obtained a similar value, 2 × 108 mol-1 dm3 s-1. Other electrophilic gases: CO2, N2O, and SF6 do not scavenge the mobile ion. Scavenging by O2 is one of the slowest reactions of highmobility ions. The gas-phase IP of O2 is much higher than that of cyclohexane (12 eV vs 9.88 eV).24 The PA of oxygen is less than that of cyclohexane and cyclohexyl (4.4 eV vs 7.35 and 7.8 eV, respectively).25 Therefore, both proton and charge transfers to O2 are endothermic by 2-3 eV. We cannot conceive any plausible reaction of c-C6H12•+ with O2 which does not break the C-H bond. With c-C6H12•+ the following reaction might occur:
c-C6H12•+ + O2 f c-C6H11+ + HO2•
(7)
Using the IP of cyclohexane and cyclohexyl (9.88 and 7.66 eV, respectively24), the energy of the C-H bond in cyclohexane (4.09 eV26), and the enthalpy of reaction H + O2 f HO2 (-2.15 eV27), we estimated that reaction 7 is exothermic by 0.28 eV (using the appearance potential of 11.66 eV for c-C6H11+24 we obtained 0.37 eV). An analogous reaction with c-C6H13+
c-C6H13+ + O2 f c-C6H12•+ + HO2•
(8)
is endothermic by 1.52 eV (we used 13.56 eV for IP of H atoms24). [However, in HF solutions O2 oxidizes proton adducts of aromatic solutes to radical cations28 despite reaction 8 being endothermic by 0.4-0.8 eV in the gas phase.] Mobile Ions in Cyclohexane: Analysis of the Decay Kinetics. In the simplest reaction scheme (called here the single-ion model), high-mobility ion Im (which has mobility µm+) transforms to a normally diffusing ion In (with mobility µn+) following a (pseudo) first-order rate law. Taking into account that these cations also decay via homogeneous recombination and that the concentration of anions [I-] ) [In] + [Im], one obtains
d[Im]/dt ) -k1[Im] - R(µm+ + µ-)[Im][I-]
(9)
d[In]/dt ) +k1[Im] - R(µn+ + µ-)[In][I-]
(10)
At t ) 0 we assumed that [Im] ) fmco and [In] ) (1 - fm)c0, where c0 is the initial concentration of free ions and fm is the initial fraction of mobile ions. The dc conductivity is given by
κex ) κQ ) 10-3FQ{(µm+ + µ-)[Im]gdc(m+,-;t) + (µn+ + µ-)[In]gdc(n+,-;t)} (11) where gdc(t) is the correction function added to account for the dc signal from geminate pairs. As was shown by Schmidt, geminate ions give a smaller contribution to dc conductivity than to microwave conductivity.29 Similarly to Warman’s use of gµw(t) as the concentration of geminate radical ion pairs relative to the free ion yield,4c we use gdc(t) as the equivalent conductance of geminate ions relative to that of free ions. According to Warman, in radiolysis gµw(t) ≈ 1 + β (t/tc)-σ, where σ ≈ β ≈ 0.6 and tc ) Rc2/Ds is the Onsager time {here Rc ) e2/4πkT0 is the Onsager radius (≈56.4/ nm) and Ds ) (kT/e)µs is the sum of diffusion coefficients of ions}.4c,5,30 For µs in cm2/V s tc (ns) ≈ 1.3/2µs. Using µm+ ≈ 10-2 cm2/V s and µs ≈ 10-3 cm2/V s, we obtain tc ≈ 30 ns for HMI and tc ) 300 ns for normally-diffusing ions, respectively. Therefore, on the time scale of our experiments (t > 50 ns) most of HMI are free, while a considerable fraction of normally diffusing ions is geminate. For isolated pairs, gdc(t) and gµw(t) can be simulated using a Monte Carlo model described in ref 29. We found that at t/tc > 0.05, gdc(t) - 1 ≈ 1/2{gµw(t) - 1} (see the window in Figure 2). A function 1 + β(t/tc)-σ was used to fit the gdc(t) curves. The best fits were for σ ≈ 0.75 and β ≈ 0.12 (Figure 2). Our value of σ is close to that obtained by Lipsky et al., σ ≈ 0.70 ( 0.05.31 These σ and β were used to fit the kinetics obtained in laser experiments. Examples of fitting the decay kinetics of dc conductivity signals in CO2-saturated isooctane ( ) 1.94, µs ) 1.7 × 10-3 cm2/V s4) and 10-3 mol dm-3 transdecalin in cyclohexane (in which the lifetime of HMI is 50 ns; at shorter times the signal is distorted by artifacts). These β and σ were used to fit the κex(t) curves obtained in radiolysis (Figure 4); laser data can be fit with these parameters too. It must be stressed that eq 11 is approximate and must be used with caution. Particularly, it underestimates homogeneous recombination of ions at t < tc. Using data from ref 32, we assumed that µn+ ) 3.2 × 10-4 cm2/V s and µ- ) 6 × 10-4 cm2/V s. The multiset least-squares fitting was performed using Levenberg-Marquardt optimization with χ2 criterion. Only points with t > 20-50 ns were fit (the integration of eqs 9 and 10 started at 1 ns), which gave 0.81.6K points per set. Due to the slowness of second-order decay in the laser experiments (in which c0 ≈ (1-2) × 10-8 mol dm-3) two experimental traces (10 µs) were combined for every sample (Figure 3). One set of fitting parameters was used to simulate the data for 4-5 concentrations of a given solute (Figure 5). Using this routine, parameters Q and c0 can be found rather accurately since they wholly determine the long-term kinetics. In radiolytic experiments the initial concentration c0 of free ions can be estimated as
c0 (mol dm-3) ) 10-2FGfi(E) dose (Gy)/F
(12)
where F ) 0.78 g/cm3 (cyclohexane) Gfi(E) is the G value of free ions in the electric field E, ∆Gfi(E)/(EGfi(E ) 0)) ≈ eRc/ 2kT (≈ 1.1 × 10-4/ for E in V/cm). Using Gfi(0) ≈ 0.148 (molecules/100 eV) for cyclohexane and E ≈ 8 kV/cm we obtain
High Mobility Ions in Cycloalkanes
Figure 2. Monte Carlo simulation of gdc(t), the equivalent conductance of geminate ions relative to that of free ions, for isolated pairs in cyclohexane (500 000 pairs were averaged).29 Two traces, for µs ) 10-3 cm2/V s (large dots) and µs ) 10-2 cm2/V s (small dots), were combined; gdc(t) is plotted as a function of (t/tc)-σ, where tc is the Onsager time, σ ) 0.75, t ) 0-100 ns; the electric field is 8 kV/cm. Window: correlation between gdc(t) and gµw(t), the concentration of geminate ions relative to that of free ions.
Figure 3. Dots: decay of transient dc conductivity in 248 nm laser photolysis of CO2-saturated isooctane (a) and cyclohexane (b), pure (i) and with 10-3 mol dm-3 trans-decalin (ii). Solid lines: simulations with Q ) 0.4-0.5, σ ) 0.75, and β ) 0.12. For isooctane and decalin solutions we assumed fm ) 0; µs were taken from ref 4. Trace i was simulated assuming µm+ ) 9.5 × 10-3 cm2/V s; fm ) 0.41, k0 ) 1.8 × 106 s-1, c0 ) 1.3 × 10-8 mol dm-3.
Gfi(E) ≈ 0.21 which corresponds to the free ion yield φ ≈ 0.043, using Ginitial ions ) 5. The c0 obtained with eq 12 are proportional to the dose (Figure 4b) and are within 20% of the values obtained with eqs 9-11. As far as the reaction scheme eqs 9 and 10 is concerned, k1 can be determined with an accuracy of (15%, though (for reasons discussed later) the fits at t < 5 µs are not of good quality (Figure 5a). Particularly, the dc signal at t < 100 ns
J. Phys. Chem., Vol. 100, No. 27, 1996 11329
Figure 4. Dots: decay kinetics of transient dc conductivity in radiolysis of CO2-saturated cyclohexane. 10 and 20 ns electron pulses from 15 MeV Argonne linac were used for ionization. (a) Kinetics observed in (ii) pure cyclohexane and (i) 10-3 mol dm-3 trans-decalin solution (7.8 Gy/pulse). (b) Kinetics obtained in pure cyclohexane with (i) 15 Gy and (ii) 30 Gy pulses (no averaging). Dotted lines: simulations with fm ) 0.27 ( 0.1, µm+ ) 9.5 × 10-3 cm2/V s, Q ) 0.9 ( 0.1, σ ) 0.6, β ) 0.11, k0 ) (2 ( 0.2) × 106 s-1, c0 ) 2 × 10-7 mol dm-3 (b, trace i), 4.2 × 10-7 mol dm-3 (b, trace ii).
Figure 5. (a) Dots: decay kinetics of transient dc conductivity in 248 nm laser photolysis of CO2-saturated cyclohexane containing (from top to bottom) 0, 10-5, 2 × 10-5, 5 × 10-5, and 10-4 mol dm-3 cyclohexene. Dashed lines: best-fit kinetics obtained in the model of a single HMI (the traces were fitted from t0 ) 50 ns; fm ) 0.45, µm+ ) 9.5 × 10-3 cm2/V s, Q ) 0.46, c0 ) 1.1 × 10-8 mol dm-3; k1 was fit for every trace individually). (b) Concentration plots of first-order decay constants k1 obtained by fitting the traces shown in Figure 5a; t0 ) 50 ns (filled circles) and t0 ) 200 ns (open triangles). The slopes are 2.1 × 1011 and 1.5 × 1011 mol-1 dm3 s-1, respectively.
seems to be stronger than the model can provide, and the decay kinetics of HMI is not first-order. We assumed that k1 ) k0 + k2[A], where [A] is the concentration of scavenger, 1/k0 is the
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TABLE 1: Rate Constants k2 (1011 mol-1 dm3 s-1) of Scavenging Reactions of HMI in Cyclohexane at 25 °C solutea
t0 ) 50 nsc
t0 ) 200 nsc
equilibriumd
cyclohexene triethylamine bicyclohexyl trans-decalin pyreneb
2.1 1.6 1.95 1.7 2.5
1.5
3.3
1.2 1.1 1.3
2.8 4.0
a Transient dc photoconductivity; 248 nm biphotonic laser excitation. Pulsed radiolysis, 10 ns pulses, 15 MeV electrons, 8 Gy/pulse. c The kinetics at t > to were fitted in the single-HMI model; standard deviation is (0.1 × 1011 mol-1 dm3 s-1. d The kinetics were fitted in the equilibrium model with Keq ≈ 2 and τ ≈ 30 ns (to ) 10 ns). b
lifetime of mobile ion in “pure” cyclohexane, and k2 is rate constant of scavenging (we assumed that all molecular cations have equal mobilities). Two approaches were used to determine k0 and k2. In the first, k1 was fit individually for each data set, and parameters k0 and k2 were found by linear regression (Figure 5b). In the second, k0 and k2 were found by global optimization. Both of these methods gave similar results, with k0 ≈ (1.3-2) × 106 s-1. Some scavenging constants k2 are given in Table 1; these values are 20-30% lower than the scavenging constants previously reported by Warman et al.5,6 and Schmidt and Sauer.8 It must be stressed, however, that k2 in Table 1 may not be of much value, since the constants that we found by optimization depend on the starting time t0 of the fit. For instance, for t0 ) 200 ns the optimum k2 are 2-3 times less than the values obtained for t0 ) 50 ns. Such behavior is typical for bimodal decay kinetics. We will return to this later. With our optimization procedure, mobility µm+ and the initial fraction fm of HMI cannot be found separately. The conductivity depends on, approximately, the product of µm+ and fm. As fm decreases from 1.0 to 0.2, the optimum µm+ increases from (23) × 10-3 to (1.5-2) × 10-2 cm2/V s. Using µm+ ≈ 9.5 × 10-3 cm2/V s given by Warman,5 we obtained optimum values of fm ≈ 0.5 ( 0.05 and fm ≈ 0.25 ( 0.02 for laser photolysis and pulse radiolysis, respectively. Warman’s estimate of µm+ was obtained assuming that no normally diffusing ions are formed at t ) 0 in radiolysis.4c,5 However, both radiolytic and photolytic conductivity curves can be fit with fm ≈ 1 only when µm+ ≈ 2.5 × 10-3 cm2/V s. Given that the optimum scavenging constants k2 are (1.5-2.2) × 1011 mol-1 dm3 s-1, this means that the reaction radii for these scavenging reactions are ≈3 nm. No mechanism can account for such large reaction radii. Thus, in radiolysis fm < 1. This result is in agreement with studies on transient absorption in pulse radiolysis of cyclohexane.17 A signal from normally diffusing (olefin) cations was observed from the earliest observation times; this absorption is comparable to that from the solvent hole. Assuming that in radiolytic experiments fm < 1 but in laser experiments fm ≈ 1, the optimum values of µm+ ≈ (5-6) × 10-3 cm2/V s and fm(rad) ≈ 0.5 were obtained. With this mobility, Rsc is still too large. It seems that even in laser photolysis fm must be less than unity. In cyclohexane, the photolytic formation of HMI involves biphotonic ionization of the solvent. Since only single pairs are formed in such events and the initial concentration of pairs is low (c0/φ ≈ 10-6 mol dm-3), it is unclear how a significant fraction of normally diffusing ions can be formed in just 10-50 ns unless by a rapid transformation of the solvent radical cations. If the HMI is the cyclohexane hole, the latter must be stable over several microseconds. Then, the only way to account for fm < 1 is to assume that the transformation occurs with excited radical cations (e.g., reaction 6). It can be argued that fm < 1 may result from (i) faster geminate decay of HMI as compared to normally diffusing ions
Figure 6. Dots: decay kinetics of transient dc conductivity in 248 nm laser photolysis of (i) CO2-saturated decalin (1:3 trans/cis mixture) and (ii) the same solution with 1.2 × 10-2 mol dm-3 triethylamine. Dashed line: best-fit kinetics obtained in the model of a single HMI (trace i was fitted from t0 ) 50 ns; fm ) 1, µm+ ) 3.9 × 10-3 cm2/V s, Q ) 0.46, c0 ) 10-8 mol dm-3; k0 ) 8.9 × 105 s-1).
Figure 7. Scavenging kinetics of HMI in decalin mixtures. (a) Dots: decay kinetics of transient dc conductivity in 248 nm laser photolysis of CO2-saturated decalin (1:3 trans/cis mixture) containing (i) 0, (ii) 10-5, (iii) 2 × 10-5, (iv) 10-4, and (v) 2 × 10-4 mol dm-3 cyclohexene. Solid lines: best-fit kinetics obtained in the model of a single HMI (see caption to Figure 6). (b) Plot of k1 vs concentration of cyclohexene and triethylamine (mol dm-3).
or (ii) homogeneous ion recombination. In our view, both these arguments are not supported by results. As is clear from Figures 3 and 4, at t > 50 ns even for normally diffusing ions the contribution from geminate ions is rather small and is largely taken into account by eq 11. As for the homogeneous recombination at t ≈ tc, assuming that the HMI reacts with anions with rate constant of ≈5 × 1012 mol-1 dm3 s-1 (derived from the Debye equation with µm+ ≈ 10-2 cm2/V s), the first lifetime of the second-order decay of HMI is ≈0.2 µs. In just
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J. Phys. Chem., Vol. 100, No. 27, 1996 11331
TABLE 2: Rate Constants k2 of Scavenging Reactions (1010 mol-1 dm3 s-1) of Mobile Cations in Mixtures of cis- and trans-Decalin with Several Solutes (25 °C) fTD 1.00e 0.37d 0.25d 0.00 IP, eVa PA, eVb
cyclohexene (16f)
14.5 6.6 5.1 2.6 9.1 8.23
triethylamine 15h
(17f)
5.8 5.0 2.5 7.5 10.1
n-propanolk
toluene
benzene-h6
0.53 ( 0.03 2.8 ( 0.3 3.0 2.6 10.2 8.3
(21,f
0.54 ( 0.03 (0.5f)
17.5 5.7 5.6 2.8 8.8 8.25
18g)
0.50 ( 0.14 9.245 7.86
a IP ) ionization potential (gas phase),24,36 for trans- and cis-decalin IP ≈ 9.24 and 9.26 eV, respectively.36a b PA ) proton affinity (gasphase).25 c Mole fraction of trans-decalin. d These two mixtures were obtained from Aldrich and certified chromatographically. e If not indicated, the standard deviation is (0.2 (1010 mol-1 dm3 s-1); the kinetics were fit at t > 100 ns, k0 ≈ (4-5) × 105 s. f Reference 11. g Reference 39; the constant was obtained for the (excited?) solvent radical cation within 10 ps after ionization. h For tributylamine (11.4 ( 0.2) × 1011 mol-1 dm3 s-1. k For f 10 mol-1 dm3 s-1 and (1.57 ( 0.06) × 1010 mol-1 dm3 s-1, respectively. TD ) 0.54 and 0.75 the rate constants are (1.88 ( 0.06) × 10
30 ns (≈tc) this lifetime increases 10-20 times (≈φ-1). In radiolysis, the initial concentration of ions was ≈10 times higher than in photolysis, and the homogeneous recombination might cause some decay of the HMI at t < t0. In this case one would observe a dependence of fm on the absorbed dose. Contrary to this prediction, the curves obtained for different doses can be simulated with the same fm by varying c0 linearly with dose. Thus, both presumptions of the single-ion model are incorrect: in cyclohexane, the decay of HMI is biexponential and its yield on radiolysis is much less than 100%. Mobile Ions in Decalins. Neat Decalins. For neat transdecalin and cis-decalin as well as their mixtures, the single-ion model accounts for the dc photoconductivity remarkably well (an example is given Figure 6). The scavenging kinetics of HMI is exactly pseudo-first-order (Figure 7). Assuming that fm ) 1 and the sum, µs, of mobilities of normally diffusing ions in trans- and cis-decalins is 5.2 × 10-4 cm2/V s ( ) 2.15) and 3.4 × 10-4 cm2/V s ( ) 2.19), respectively,5 we obtain µm+(trans) ≈ (8.7 ( 0.2) × 10-3 cm2/V s (vs 9 × 10-3 cm2/V s5), µm+(cis) ≈ (1.7 ( 0.1) × 10-3 cm2/V s (vs 2 × 10-3 cm2/V s5). The rate constant k0 of the first-order decay in CO2saturated liquids was (3-5) × 105 s-1, depending on solvent purity. On scavenging, the decay of dc conductivity is fully consistent with the existence of a single long-lived HMI. The scavenging constants are considerably lower than those for cyclohexane (Table 2). For trans-decalin, the maximum scavenging constant k2 is ≈1.75 × 1011 mol-1 dm3 s-1 (by toluene) which is somewhat lower than the value given by Warman et al.11 For HMI in cis-decalin, the fastest scavenging proceeds with k2 ≈ (2.6-2.8) × 1010 mol-1 dm3 s-1. Thus, in both of these liquids Rsc ≈ 1.0 ( 0.1 nm. On continuous photolysis of decalins, the decay constant k1 of HMI increases linearly with the number of laser pulses. Using the scavenging constants given above, we observed that the concentration of scavenger formed in each laser flash is within a factor of 3 of c0, the concentration of free ions. Mixtures. Warman et al. observed linear decrease in µm+ upon dilution of trans-decalin by cyclohexane.5 Others reported a concave curve which was analyzed in terms of the percolation theory.33 Since cyclohexane has higher IP than trans-decalin, radical cations of cyclohexane are rapidly scavenged by transdecalin. Assuming that the mobile ions are radical cations of trans-decalin involved in resonant charge transfer, the decrease in µm+ can be readily understood: as the number of neighboring trans-decalin molecules decreases the transfer becomes slower. We measured the dc photoconductivity in CO2-saturated mixtures of trans-decalin and n-hexane (Figure 8). n-Hexane was used because of its high IP and low PA. Equations 9-11 were used to analyze the data. Note, that with our technique we measure the ratio µm+/µs rather than the absolute mobility µm+. We assumed that µs linearly decreases from 2.2 × 10-3
Figure 8. (a) Decay of transient dc conductivity in 248 nm laser photolysis of CO2-saturated trans-decalin containing 0, 11, 23, 37, 46, 55, 63, 73, and 80 mol % of n-hexane. The arrows point out the decrease in fTD, the mole fraction of trans-decalin. (b) Mobility µm+ of HMI as a function of fTD.
cm2/V s to 5.2 × 10-4 cm2/V s5 and linearly increases from 1.885 to 2.15 with the mole fraction fTD of trans-decalin, while fm ) 1. Upon addition of n-hexane, the conductivity signal decreases following the decrease in µm+ and c0. An interesting feature of these kinetics is an isosbestic point implying that parameters µm+, µs, and c0 change linearly with fTD (Figure 8a). Mobilities µm+ were optimized for each set and found to increase with fTD (Figure 8b). Alternatively, we postulated that µm+ and c0 decrease with fTD linearly and optimized the coefficients. Both of these approaches gave good-quality fits, with µm+ and c0 being proportional to the mole fraction of trans-decalin in the mixture. At fTD < 0.45 the first- and second-order kinetics are difficult to separate and we were unable to estimate µm+ from the data. We did not observe concave curves reported in ref 33. Another study was performed on the mixtures of trans- and cis-decalins. The objective was to determine whether the HMI are the same species in all of these mixtures, two species that coexist independently, or two species in an equilibrium. It seems that the first two possibilities can be excluded by our results. First, we did not observe a bimodality of scavenging expected if two types of nonconvertible HMI were present in the solution. In all mixtures, the kinetics can be simulated assuming a single HMI with first-order decay (Figure 9). Second, on addition of trans-decalin to cis-decalin, µm+ and the rate constant of scavenging by cyclohexene were found to increase linearly with the mole fraction fTD of trans-decalin, as shown in Figure 9 and Table 2. Apparently, mixing of decalins causes similar
11332 J. Phys. Chem., Vol. 100, No. 27, 1996
Sauer et al. higher than IP of the solvent molecules. That would also explain the inefficiency of electron transfer from benzene to HMI. HMI in Cyclohexane: Bimodal Kinetics. For cyclohexane, the model of a single HMI is inconsistent with many results, including the present data. Facing this difficulty, we propose that the mobile radical cation is involved in a dynamic equilibrium with a normally diffusing ion. Chair and twist forms of cyclohexane radical cation can be suggested as the two conformers involved in the equilibrium.17 At 25 °C, the equilibrium concentration of the twist form of cyclohexane is ≈10-4 mol dm-3.34 Thus, the majority of solvent radical cations formed by radiolysis or photoionization are in the chair form. Resonant charge transfer between the chair form ions and the chair form solvent molecules must be very fast, since it requires minimum reorganization energy of the solvent:
Assuming that the twist form of cyclohexane has lower IP, the twist form molecules would scavenge the chair form radical cations. Once the twist form of the radical cation is formed, the fast migration via charge transfer is stalled since the ion is surrounded by the chair form molecules with higher IP: Figure 9. Mobility µm+ of HMI (open circles, left axis) and rate constants of scavenging HMI by cyclohexene and n-propanol (right axis) as a function of fTD, the mole fraction of trans-decalin in decalin mixtures.
changes as the addition of a high-IP solvent and does not change Rsc appreciably. Third, our experiments indicate that HMI in cis- and trans-decalin cannot be the same species having different rates of percolation in different mixtures. In such case, k2 would always scale with µm+. At least one solute, n-propanol, shows the opposite trend: k2 decreases with fTD despite the increase in µm+ (Figure 9, Table 2). The only way to explain these results is to assume that in the decalin mixtures the isomeric radical cations rapidly transform into each other, by charge transfer. Sensitization by Benzene. In 248 nm photolysis of decalins, the yield of HMI can be increased up to 5-7 times by addition of 10-5-10-4 mol dm-3 of benzene (IP ≈ 9.245 eV;24 248 ≈ 90 mol-1 dm3 cm-1) without much decrease in the lifetime of HMI (k2 ≈ (5.0-5.5) × 109 mol dm-3 s-1, Table 2). In this concentration range, the increase in the signal from HMI is proportional to the absorbance of solution. As in the 248 nm photoionization of neat decalins, in benzene solutions the HMI is formed in a two-photon process. Almost certainly, the generation of HMI is triggered by ionization of the S1 state of benzene after the absorption of the second 5 eV photon. No such behavior was found for toluene (whose IP ≈ 8.82 eV24 is lower than the IP of decalins, 9.24-9.26 eV36a) and for benzene in cyclohexane. For benzene in trans-decalin, our value of k2 is close to the rate constant given by Warman et al.11 We did not observe the fast scavenging with k2 ≈ 7 × 1010 mol-1 dm3 s-1 reported by Lipsky et al.37 d6 and h6 benzenes were found to scavenge HMI with the same rate constant (to within 6%). In solutions containing benzene, the decay kinetics can be simulated using the same parameters as for neat decalins (except for higher c0 and k1). We found that no normally diffusing ions at t ) 0 are needed to simulate these kinetics. Clearly, the ionization of benzene results in the formation of HMI with ≈100% efficiency. That would be unlikely if the excited states of benzene•+ were involved; i.e., the IP of benzene must be
Being endothermic, the transfer to the chair form molecules occurs on the nanosecond time scale. Thus, the migration of the hole can be visualized as a series of periods of very fast migration via resonant charge transfer in the chair form and intermittent molecular diffusion in the twist form. In pure transdecalin and methylcyclohexane the conformation dynamics is arrested, and no such behavior is observed. In the following, we will focus on how the postulated equilibrium accounts for the data on the transient dc conductivity. Consider a mobile ion Ia in equilibrium with a normally diffusing ion Ib. The constant of this equilibrium is K14 and the setting time of the equilibrium is τeq ) (k14[twist] + k-14[chair])-1. We will assume that both of these cations react with solute (A) and an impurity (Im). For Ib, we assume the diffusion-controlled reaction with k2b ) 1.5 × 1010 mol-1 dm3 s-1. In is a normally diffusing cation which does not take part in the equilibrium. Similarly to eqs 9 and 10 we obtain
d[Ia]/dt ) -τeq-1(Keq[Ia] - [Ib])/(1 + Keq) - k2a[Ia][S] R(µm+ + µ-)[Ia][I-] (15) d[Ib]/dt ) +τeq-1(Keq[Ia] - [Ib])/(1 + Keq) - k2b[Ib][S] R(µn+ + µ-)[Ib][I-] (16) d[In]/dt ) k2a[Ia][S] + k2b[Ib][S] - R(µn+ + µ-)[In][I-] (17) where Keq ) K14{[twist]/[chair]}eq and [S] ) [Im] + [A]. At t ) 0 [Ib] ) 0, [Ia] ) fmc0, and [In] ) (1 - fm)c0. Conductivity κ is given by
κ ) 10-3F{(µm+ + µ-)[Ia]gdc(m+,-;t) + (µn+ + µ-)([Ib] + [In])gdc(n+,-;t)} (18)
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J. Phys. Chem., Vol. 100, No. 27, 1996 11333
Figure 10. Dots: decay kinetics of transient dc conductivity in 248 nm laser photolysis of CO2-saturated solutions of cyclohexene in cyclohexane (see caption to Figure 5a). Dashed lines: kinetics simulated in the equilibrium model. The traces were fitted from t0 ) 10 ns; fm ) 1, µm+ ) 1.6 × 10-2 cm2/V s, Keq ) 1.8, τeq ) 30 ns, [Im] ) 1.4 × 10-5 mol dm-3, c0 ) 2 × 10-8 mol dm-3; k2 ) 3.3 × 1011 mol-1 dm3 s-1.
Figure 11. Dots: decay kinetics of transient dc conductivity in pulse radiolysis (i) and 248 nm laser photolysis (ii) of CO2-saturated cyclohexane. Solid lines: kinetics simulated in the equilibrium model. The traces were fitted from t0 ) 10 ns; µm+ ) 1.67 × 10-2 cm2/V s, Keq ) 2, τeq ) 30 ns, [impurity] ) 1.8 × 10-5 mol dm-3, k2 ) 4 × 1011 mol-1 dm3 s-1. For trace (i) fm ) 0.53, c0 ) 3 × 10-7 mol dm-3; for trace (ii) fm ) 1, c0 ) 2.4 × 10-8 mol dm-3.
According to eqs 15-18, in the absence of homogeneous neutralization and reactions with impurity, at t . τeq the fraction of mobile ions decreases as (Keq + 1)-1 and the mobility and scavenging rate of the “equilibrium” ion are given by
have to be an ion which cannot transfer charge to an aromatic solute molecule.]
k2eq ) (k2bKeq + k2a)/(Keq + 1)
(19)
d[Ib]/dt ) -χ[Ib]/τ - k2b[Ib][S] - R(µb+ + µ-)[Ib][I-] (22)
µeq ) (µn+Keq + µm+)/(Keq + 1)
(20)
d[In]/dt ) (1 - χ)[Ia]/τ + k2a[Ia][S] + k2b[Ib][S] R(µn+ + µ-)[In][I-] (23)
Thus, at t > τeq the fraction of mobile ions, their mobility and scavenging rate all decrease by ≈(1 + Keq)-1, so the scavenging radius does not change. To simulate the kinetics we assumed that in laser experiments fm ) 1 and τeq ≈ 30 ns (Figure 10). The latter value is based on the estimate given in ref 17. In these simulations, we either fixed the reaction radius Rsc at 0.9-1 nm or used eq 1. We obtained optimum values of Keq ≈ 1.8-2, R ≈ 0.94 ( 0.02 nm, k2a ≈ (3.5 ( 0.2) × 1011 mol-1 dm3 s-1, and µm+ ≈ (1.8 ( 0.2) × 10-2 mol-1 dm3 s-1. The accuracy of the fitting at t < 5 µs, for the photoionization results improves (compare Figure 10 and trace ii in Figure 11 with Figure 5), and the characteristic shape of the kinetics at t < 500 ns is reproduced. Using these parameters and assuming fm ≈ 0.5, we were also able to fit the conductivity curves obtained in radiolytic experiments (Figure 11, trace i). For Keq ≈ 2, the equilibrium constant K14 ≈ 4 × 104, which corresponds to ∆Gion + ∆Gmol ≈ 0.02 eV, where ∆Gion and ∆Gmol are the free energies of transformation from the chair form to the twist form for cyclohexane•+ and cyclohexane, respectively (∆Hmol ) +0.25 eV34). Thus, the scheme requires that the difference in the conformation energy of the twist and chair forms of cyclohexane•+ approximately equals that of the chair and twist forms of cyclohexane. The EPR data on γ-irradiated low-temperature freon solutions indicate that the most abundant radical cation species is the elongated chair form of cyclohexane•+.35 Given that the concentration of the twist form of cyclohexane in such solutions is
