Temperature Dependence of Proton Diffusion in Ih Ice - The Journal of

May 18, 2009 - Raymond and Beverly Sackler Faculty of Exact Sciences, School of Chemistry, Tel Aviv University, Tel Aviv 69978, Israel. J. Phys. Chem...
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J. Phys. Chem. C 2009, 113, 10285–10296

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Temperature Dependence of Proton Diffusion in Ih Ice Anna Uritski, Itay Presiado, Yuval Erez, Rinat Gepshtein, and Dan Huppert* Raymond and BeVerly Sackler Faculty of Exact Sciences, School of Chemistry, Tel AViV UniVersity, Tel AViV 69978, Israel ReceiVed: March 4, 2009; ReVised Manuscript ReceiVed: April 22, 2009

The temperature dependence of the proton diffusion constant, DH+, in methanol-doped ice was studied over the wide temperature range of 80-260 K. For that purpose, we measured the time-resolved fluorescence of a flavin mononucleotide (FMN) and riboflavin in ice doped with HCl or HF. The analysis of the fluorescence quenching provided the value of DH+. We found that the temperature dependence of DH+ at T > 235 K is rather small, whereas at T < 235 K, it is rather large. These temperature-dependence results are similar to previous conductivity measurements. We used a stepwise, two-coordinate qualitative proton transfer model to explain the temperature dependence of DH+ in ice. Introduction

SCHEME 1: FMN 1-4

The physics of ice, which has been studied for a long time, poses numerous questions that even today remain unanswered.2 Recently, ice studies revealed the importance of ionization of hydrochloric acid on stratospheric ice particles as a key step in the depletion of the stratospheric ozone.5 Lattice imperfections give rise to electrical conduction, defects diffusion, and dielectric relaxation phenomena. Pure ice exhibits a high static relative permittivity, which is larger than that of liquid water. There are two types of structural defects that are largely responsible for the electrical properties of ice. The first is ion defects, which are the result of proton motion from one end of the bond to the other, thus creating a H3O+/OH- ion pair.6 Conduction is then carried out by means of successive proton jumps (von Grotthuss mechanism). The second is Bjerrum defects,7 which are orientational defects caused by the rotation of a water molecule to produce either a doubly protonated bond (D-defect) or a deprotonated bond (L-defect). Over the years, ice has challenged many physicists and chemists. Presently, quantum mechanical ab initio calculations and dynamical simulations present an efficient way to study ion defects,8 Bjerrum defects,9 and the mechanism of proton transfer and mobility in ice. In the early 1960s, it was estimated, from the electrical conductivity measurements of Eigen,10,11 that the proton mobility in ice is 10-100 times larger than that in water. In numerous further measurements, it was found that at ∼263 K, the proton mobility in ice (0.8 × 10-4 cm2 V-1s-1) is smaller than that of water12 by about a factor of 2 (when compared to supercooled liquid water13,14 at the same temperature). The large proton conductivity of ice found in Eigen’s experiments was explained as arising from large surface conductivity rather than from bulk conductivity.3 Onsager15 and later on Nagle16 advocated a larger proton diffusion in ice; however, in the 1972 ice conference in Ottawa, Onsager, Engelhardt, and others abandoned the idea of ice as an intrinsic protonic semiconductor.3 For over thirty years17-26 excited-state intermolecular proton transfer (ESPT) to a solvent or a base in a solution has been widely researched in the liquid phase. We applied ESPT to study ice properties such as L-defect mobility, L-defect reactivity, and * Corresponding author. E-mail: [email protected]. Telephone: 9723-6407012. Fax: 972-3-6407491.

solvation dynamics of photoacids in ice. In more recent works,27-29 we reported a very large proton diffusion constant in ice Ih in the high temperature range of 235-270 K. In these studies, we used photoreactive molecules that react with protons in their excited state. Some of the photoreactive molecules used so far were (1) 2-naphthol-6,8-disulfonate (2N68DS),27 (2) flavin mononucleotide (FMN),28 and (3) 1-naphthol-4sulfonate (1N4S).29 These studies concluded that the proton diffusion constant in ice is 10 times larger than that in water. Subsequently, we concluded that the proton diffusion in ice is much larger than that deduced from conductivity measurements in the 1970s.12,13 In the current study, we set our goal to further explore the properties of the proton diffusion constant in ice and to expand the temperature range investigated in our previous studies to the low temperature regimes. For that purpose, we used excited FMN and riboflavin (scheme 1) as proton sensitive molecules. Using a time-resolved emission technique, we measured the fluorescence quenching by protons in methanol-doped ice in the wide temperature range of 80-270 K. The main new findings of this study are, that at temperatures below 235 K, the temperature dependence of the proton diffusion is large, whereas at T > 235 K, it is small. The overall shape of the Arrhenius plot of log DH+ versus 1/T is convex. The activation energy at low temperatures is ∼30 kJ/mol (∼0.3 eV). Similar temperature dependences are also found in the measurements of low-frequency conductivity, σ0, in HCl-doped ice samples taken by Takei and Maeno.30,31

10.1021/jp901971q CCC: $40.75  2009 American Chemical Society Published on Web 05/18/2009

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Figure 1. Time-resolved emission of FMN samples in liquid and ice at temperatures above 235 K. Samples were excited at 425 nm by ∼150 fs pulses at 500 kHz, and the emission was monitored at 550 nm.

In liquid water, two classes of proton transport phenomena are recognized: “ordinary” mass diffusion according to Stokes law and “abnormal” or prototropic proton mobility. Using a random walk description, the abnormal proton in water mobility is characterized with a hopping time, τp, of ∼1.5 ps at room temperature, as deduced from NMR line-narrowing investigations.32-34 This time scale reproduces the abnormal proton mobility with a hopping length, lp, of 2.5-2.6 Å, which is the O-O distance in H9O4+.35 Bernal and Fowler36 proposed that the rate-determining step of prototropic mobility in liquid water is connected to single water rotation.35,37,38 Traditionally, the dielectric relaxation time, τD, of a liquid is associated with the molecular rotation time. Dielectric relaxation and self-diffusion are somewhat slower than the hopping time, τp, and they show stronger temperature dependence than that of τp. In a previous study,39 we calculated the proton transfer rate constant associated with prototropic mobility in liquid water as a function of temperature, where the Arrhenius plot of DH+ in water has a convex shape, which is the same as in ice; as the temperature decreases, the activation energy increases. In liquid water and in supercooled water, we found a strong correlation between the proton transfer rate constant at low temperatures, T < 290 K, and the dielectric relaxation time. Therefore, in the current ice study, we used a similar approach to explain the temperature dependence of DH+ in ice. We found a good correlation between the temperature dependence of DH+ (T) at temperatures below 235 K and that of τD of ice. Both τD and L-defect mobilities in ice are associated with the orientational motion of water molecules about the oxygen-oxygen axis between adjacent water molecules in the hexagonal structure. Experimental Section We used the time-correlated single-photon counting (TCSPC) technique to measure the time-resolved emission of flavin

mononucleotide (FMN) and riboflavin. For sample excitations, we used a cavity-dumped Ti:Sapphire femtosecond laser (Mira, Coherent), which provides short (150 fs) pulses. The laser second harmonics (SHG), operating over the spectral range of 380-430 nm, was used to excite the FMN or riboflavin ice samples. The cavity dumper operated with a relatively low repetition rate of 500 kHz. The TCSPC detection system was based on a Hamamatsu 3809U photomultiplier and Edinburgh Instruments TCC 900 computer module for TCSPC. The overall instrumental response was ∼35 ps (fwhm). The excitation pulse energy was reduced to ∼10 pJ by neutral density filters. FMN (Scheme 1) of analytical grade was purchased from Sigma. Riboflavin was purchased from TCI (Japan). HCl (1 N) and HF were purchased from Aldrich. For transient measurements the sample concentrations were between 2 × 10-4 and 2 × 10-5 M. Deionized water had a resistance of >10 MΩ. Methanol of analytical grade was purchased from Fluka. All chemicals were used without further purification. The temperature of the irradiated sample was controlled by placing the sample in a liquid N2 cryostat, with a thermal stability of approximately 1.5 K. Ice samples were prepared by first placing the cryogenic sample cell at a supercooled liquid temperature of ∼260 K for ∼20 min. The second step involved a relatively rapid cooling (5 min) to a temperature of ∼240 K. Subsequently, the sample froze within a few minutes. To ensure ice equilibration prior to the time-resolved measurements, we kept the sample temperature at ∼240 K for another 10 min. Results Figure 1 shows the time-resolved emission of FMN samples in liquid and ice at temperatures above 235 K. The samples were excited at 425 nm by ∼150 fs pulses at 500 kHz, and the emission was monitored at 550 nm, which is close to the

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Figure 2. Time-resolved emission of FMN ice samples in the intermediate temperature range of 173-235 K.

maximum of the steady-state emission spectrum. Three samples are shown in each panel: acid-free, 1 mM, and 4 mM HCl. A small amount of methanol, 0.2% mole fraction, was added to high performance liquid chromatography (HPLC) water to prevent the FMN molecules from aggregating at the grain boundaries of the polycrystalline ice samples.27-29 The acidfree liquid samples decay nearly exponentially with a lifetime of τf = 5.0 ns. In the ice samples, the short-time decay is nonexponential with subnanosecond components. Furthermore, it seems that the lower the temperature, the larger the amplitude of the nonexponential component. The acidic samples show a nearly exponential decay in the liquid with a decay time that depends on the acid concentration. The excess decay rate is attributed to the reaction of a proton with the nitrogens or oxygens of the isoalloxazine, which leads to fluorescence quenching.27-29 The fluorescence decay rate in the ice samples at temperatures above 235 K is much larger than in the liquid state. We attribute the difference in the decay rate to the fact that the proton diffusion constant, DH+, in ice is much larger than that in water. In our previous studies,27-29 we deduced that at high temperatures (T > 235 K) and low methanol concentraliquid ice tions DH+/DH+ g 10. Ice is therefore a good proton conductor as was previously suggested by Eigen and co-workers almost half a century ago.10,11 Figure 2 shows the time-resolved emission of FMN ice samples at the intermediate temperature range of 173-235 K. The samples shown in each panel are the same as those shown in Figure 1. As seen in Figure 2, the lower the temperature the smaller the decay rate of the acidic samples. Thus, the proton quenching rate strongly decreases as temperature decreases. At high temperatures above 235 K (Figure 1), the decay rate of the acidic samples depends very weakly on the temperature. We attribute this behavior of the acidic samples to the large temperature dependence of DH+ (T) in ice at the intermediate

temperature range; the lower the temperature the smaller the value of DH+. Figure 3 shows the time-resolved emission of the same FMN samples as shown in figures 1 and 2 at temperatures below 173 K. As seen in Figure 3, the decay rates of the acidic samples are only slightly smaller than that of the acid-free sample. The signals of the acid-free and acidic samples are nonexponential, and the amplitude of the signals at longer times is significantly longer than those at higher temperatures above 185 K (Figure 2). In previous studies of photoacids in methanol-doped ice,27-29 we found that at temperatures below 173 K the excited photoacid molecules are incapable of transferring a proton to the ice. Thus, the proton transfer rate constant, kPT, is much smaller than the radiative rate, kr . kPT. In the current study, we find that the proton quenching reaction of FMN is ineffective below 173 K, and thus, the fluorescence decay rate of acidic samples is only slightly smaller than that of the acid-free sample. We attribute these findings to the fact that in this temperature range DH+ has a very small value. Figure 4 shows the time-resolved emission of FMN in methanol-d-doped D2O ice samples at several temperatures in the range of 88-247 K. Each panel shows the signal at a particular temperature of an acid-free sample and four acidic samples with DCl concentrations of 1, 2, 4, and 8 mM. At 235 and 247 K, the decay rate of the acidic samples is large and depends on the acid concentration. The temperature dependence of the decay rate in the high temperature range is small. At intermediate temperatures (197 and 222 K), the decay rate of the acidic samples is much smaller than at high temperatures, and the temperature dependence is large. At low temperatures (88 and 173 K), the fluorescence decay is nonexponential and long. The acidic samples (except for the 8 mM DCl sample) at 88 K exhibit a decay, which is nearly the same as that of the acid-free sample. A comparison of the decay rates of FMN in

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Figure 3. Time-resolved emission of the same FMN samples shown in Figures 1 and 2 at temperatures below 173 K.

Figure 4. Time-resolved emission of FMN in methanol-d-doped D2O ice samples at several temperatures in the range of 88-247 K.

D2O ice samples with those of H2O ice samples at 80 K (Figure 3) shows that the decay rate of the acidic D2O samples is smaller and almost coincides with the acid-free sample. We attribute this finding to a relatively large kinetic isotope effect on DH+ at lower temperatures.

Figure 5 shows the time-resolved emission of FMN at several temperatures in H2O and D2O samples that contain also 4 mM HCl or DCl. As seen in Figure 5, the decay rate of the D2O sample at a particular temperature is slower than that of the H2O sample. The difference in the decay rate is attributed to

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Figure 5. Time-resolved emission of FMN at several temperatures in H2O and D2O samples that contain also 4 mM HCl or DCl.

the small value of DD+, i.e., DH+ > DD+. In liquid water, the isotope effect is ∼1.4. In this study, we find that in ice the isotope effect is similar. Figure 6a shows the time-resolved emission of FMN, measured at 550 nm in two acid-free methanol-doped samples. Each panel shows the signals of the samples containing a 0.2% or 0.5% mole fraction of methanol at a particular temperature. As seen in Figure 6, the methanol doping level does not influence the decay curves at long times, but at short times, the decay rate of the 0.2% mole fraction sample is somewhat faster. Figure 6b shows the time-resolved emission of FMN in acidic samples of the same methanol composition as those in Figure 6a. As seen in Figure 6b, the decay rate in acidic samples strongly depends on the methanol doping level,28 i.e., the larger the methanol concentration the smaller the fluorescence quenching rate. We attribute this effect to proton scavenging by a methanol molecule in the ice phase. The larger the methanol concentration the larger the scavenging rate. At about a 2% mole fraction of methanol, the time-resolved emission of FMN (not shown) is almost completely unaffected by an HCl concentration of a few millimolars. Figure 7 shows the time-resolved emission of FMN in an acid-free sample and in samples of two different acids at several temperatures. We chose HF (pKa ) 3.2) and HCl (pKa ) -7), which are weak and strong acids respectively in liquid water, in order to compare their effect on the FMN quenching rate in the temperature range of 88-263 K. The degree of dissociation of the weak HF acid is much smaller than that of the strong acid. Therefore, the time-resolved emission decay rate depends on the acid. At temperatures below 173 K, the long time decay rates of an acid-free sample and a sample containing 4 mM HF are nearly identical. We attribute this finding to the large temperature dependence of Ka (equilibrium constant of the acid).

In general, the temperature dependence of a chemical equilibrium constant can be approximated by40

d(ln Keq) ∆H =d(1/T) R

(1)

where ∆H is the enthalpy of the chemical reaction. The temperature dependence of Ka in endothermic reactions (weak acids) is positive, and the equilibrium constant decreases as the temperature decreases. For strong acids, the reaction is exothermic, consequently Ka increases as the temperature decreases. Thus, the degree of dissociation of HF in ice decreases as the temperature decreases, whereas that of HCl increases. Another interesting phenomenon is observed in samples containing 4 mM HF at temperatures below 173 K. The time-resolved emission signal of the HF sample at short times decays slower than that of the acid-free sample. We attribute this finding to the creation of L-defects by the F- ions. The L-defect protects the isoalloxazine ring from reacting with protons other than those originating from the acid, and it exists in acid-free samples as well. The overall effect is that in a 4 mM HF sample at low temperatures the nonexponential short time component diminishes, and consequently, the decay rate is slower than in acidfree samples. Data analysis was carried out using the Smoluchowski model in order to describe the diffusion-assisted irreversible reaction FMN* + H+ fFMNH, where the proton concentration largely exceeds the excited FMN population, FMN*. We assume that the excess proton transport toward the FMN is the rate-limiting step. The mathematical and computational details of the Smoluchowski model are given elsewhere,41-43 and the model is briefly decribed in the Supporting Information. Modeling In a previous study,39 we calculated the proton transfer rate constant associated with prototropic mobility in liquid water as

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Figure 6. Time-resolved emission of FMN, measured at 550 nm in a 0.2% and 0.5% mole ratio of methanol-doped samples. (a) Acid-free samples. (b) Samples with 4 mM HCl or DCl.

a function of temperature. The prototropic mobility in an aqueous solution exhibits non-Arrhenius behavior in the temperature range of 240-373 K. At high temperatures, the activation energy is small, whereas at low temperatures (in supercooled water) it is large. We also found this behavior in the proton transfer reaction from several photoacids to water as well as in other protic solvents such as monols, diols, and glycerol.44 In the liquid state, we found a strong correlation between the proton transfer rate constant at low temperatures and the dielectric relaxation time, τD.

In the current study on ice, we found that at low temperatures the DH+ activation energy is nearly identical to that of τD.30,45 We therefore present a qualitative model based on our previous study on the abnormal proton conductivity in liquid water accounting for the unusual temperature dependence of the proton diffusion constant in ice. The proton transfer reaction depends on two coordinates. The first coordinate depends on whether the molecular hydrogens are properly aligned with the hexagonal structure of the oxygens, according to the ice rule, which states that one hydrogen is between two oxygens. A plausible

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Figure 7. Time-resolved emission of FMN in an acid-free sample and in HF and HCl samples at several temperatures.

coordinate can be the relative orientation of the hydrogen of a water molecule with respect to the hexagonal structure of the oxygens. The second coordinate is the actual proton translational motion (proton transfer) along the reaction path. We propose an oversimplified model calculation for the purpose of obtaining a qualitative description of the temperature dependence of the experimental proton diffusion constant in ice. The model restricts the proton transfer process to be stepwise. The proton moves to the adjacent water molecule only when the hydrogen alignment of the water molecule brings the system to the lowest energy barrier of the proton coordinate. In the stepwise model, the overall proton transfer time is a sum of two times, τ ) τ1 + τ2, where τ1 is the characteristic time for the hydrogen orientation of the water molecule, and τ2 is the time for the proton to pass over the barrier. The overall rate constant k(T) at a given T is

1 1 1 ) + k(T) kH(T) ks(T)

(2)

where ks is the hydrogen reorientation coordinate rate constant, and kH is the proton coordinate rate constant. Similar expressions for an overall rate constant are used for several important phenomena such as the overall rate constant for the electron transfer rate46 and a diffusion-assisted chemical reaction.43,47 Equation 2 provides the overall proton transfer rate constant along the lines of a stepwise process similar to the processes mentioned above. As a solvent coordinate rate constant in liquid water, we used ks ) b(1/τD), where b is an adjustable empirical parameter determined from the computer fit of the experimental

data. In previous studies on liquids, we found that the empirical factor lies between 2 and 6. Thus, the orientational characteristic time τs ) 1/ks lies between the dielectric relaxation and the longitudinal dielectric relaxation time τL < τS < τD. In ice, the diffusion constant for the L-defect, DL, is associated with a hydrogen rotation around the oxygen-oxygen axis along the hexagonal structure of the ice, where the oxygens are at the vertices. The rotation time in ice is associated with the dielectric relaxation time, τD, in a more complicated way than in liquids. In water, τD ≈ 8 ps at ∼20 °C and τs ≈ 1 ps, whereas in ice at 260 K, τD is much slower, ∼10-4 s, and the solvation time is estimated to be τs ≈ 20 ps.48 The value of τD in ice is related to the average rotation time by the following expression4

nLνL 1 ) τD 3N

(3)

where nL and N are the number of L-defects and water molecules per unit of volume, respectively. The nL/N ratio is very small in pure ice (10-7), but may be larger in acid-doped ice. νL is the frequency of a single rotation of a water molecule that is associated with an L-defect hopping to the next O-O bond. The reaction rate constant kH along the proton coordinate is expressed by the usual activated chemical reaction description given by eq 4. At high temperatures, the orientational relaxation

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Uritski et al. temperatures, we invoke an additional proton reaction. We suggest that it arises from a small fraction of protons that are trapped next to the FMN molecules. These protons are probably trapped in shallow traps found in zones, which have become disordered due to the presence of the FMN molecule. The rate constant for this reaction is denoted in the Figure 8 as k3. The total reaction rate constant, ktotal, is given by

ktotal ) k(T) + k3

Figure 8. An Arrhenius plot of log DH+ (b) in methanol-doped ice as a function of 1/T, along with computed values of DH+ (solid lines). (a) H2O samples. Note that the L-defect diffusion constant data taken from ref 49 (9) are added. (b) D2O samples.

of the hydrogen alignment is fast, and thus, the rate-determining step is the actual proton transfer coordinate

( )

kH ) kH0exp -

GHa RT

(4)

where kH0 is the preexponential factor determined by the fit to the experimental results, and GHa is the activation energy determined by the slope of the Arrhenius plot in the high temperature region of DH+ (T) above 235 K, where ks > kH, and the rate determining process is kH. Figure 8a shows, on a semilogarithmic scale, an Arrhenius plot of log DH+ in methanol-doped ice as a function of 1/T. The values of DH+, which were calculated by fitting the experimental time-resolved emission curves of FMN in acidic ice samples of 4 mM HCl, are shown in the plot as full circles. At high temperatures, T > 235 K, the temperature dependence of DH+ is small, whereas in the intermediate temperature range of 175-235 K the temperature dependence of DH+ is large. At low temperatures of T < 175 K, DH+ is almost independent of the temperature. From the fit of the fluorescence quenching of FMN by an excess proton in methanol-doped ice and using the Smoluchowski model, we find that the proton diffusion constant drops by ∼2 orders of magnitude in the studied temperature range of 80-260 K. The straight solid lines in Figure 8, kH (small slope) and ks (large slope), are determined by the best fit to the experimental data using eq 2. The activation energies for kH and ks are 0.5 and 30 kJ/mol, respectively. At temperatures below 175 K, we find a residual proton reaction with FMN in ice doped with 4 mM HCl. In order to fit the data at low

(5)

where k(T) is given by eq 2. The activation energy of k3 is ∼1 kJ/mol. The low-temperature time-resolved emissions of FMN in an acid-free sample and a sample doped with 1 mM HCl are nearly identical (Figure 3). As a first-order approximation, we assume that the process denoted by k3 is largely suppressed at low acid concentrations. For such a low acid concentration sample, we were unable to attain quantitative values of DH+ at temperatures below 175 K because the radiative lifetime limits the time window of the experimental observation of the proton reaction with FMN. Figure 8b shows, on a semilogarithmic scale, an Arrhenius plot of log DD+ in deuterated methanol-doped D2O ice as a function of 1/T. Because the isotope effect of proton/deuteron diffusion is not large at all of the studied temperatures (Figure 5), graphs a and b in Figure 8 are similar. In a previous study,49 we measured the proton transfer rate from an excited photoacid (2N68DS) ROH* to an L-defect, created in ice by doping it with a few mM KF. F- is incorporated in ice by replacing a water molecule in the hexagonal structure.4 As a consequence, an L-defect is produced for each F- ion introduced into the ice. The L-defects are mobile proton acceptors and, therefore, may react with the ROH*: ROH* + L f RO- * + LH+. We used the Smoluchowski model incorporating diffusive and reactive steps to fit the timeresolved emission of the ROH* of 2N68DS in the presence of KF. The values of DL given in Table 2 of ref 49 are plotted as full squares in Figure 8a. As seen in Figure 8a, DL shows a similar temperature dependence as DH+ below 235 K, which is studied in the current work (solid line ks). DH+ and DL show a similar activation energy (slope of the Arrhenius plot, Figure 8a), which is also comparable to that of τD.30,45 To summarize, at temperatures below 235 K, the activation energies of three processes, τD, DH+, and DL, are nearly identical. Therefore, we propose that the orientational motion of water molecules in ice controls these processes. Discussion In trying to explain the temperature effect on the proton diffusion constant in ice, we have to consider the main observations and to compare the current results with other previous conductivity studies of HCl-doped ice, especially those of Takei and Maeno,30,31 who extended the measured temperature range to ∼150 K.30,31 These observations are as follows: (1) The temperature dependence of the reaction rate of a proton with FMN or riboflavin can be divided into three temperature regions. The first region is the high temperature region, T > 235 K, in which the temperature dependence is small, and the rate is fast. The proton diffusion constant in this region is ∼10-3 cm2/s, which is 10 times larger than that in water at 295 K, and the activation energy is relatively small, i.e., Ea < 1000 J/mol (∼0.01 eV). The second region is the intermediate temperature range, 175-235 K, in which the proton quenching rate strongly depends on the temperature. The proton diffusion constant deduced from the diffusion-controlled rate constant decreases by about a factor of 100 from its value of

Temperature Dependence of Proton Diffusion in Ice ∼10-3 cm2/s at 260 K to ∼10-5 cm2/s at 175 K. In this case, the activation energy of the proton diffusion constant is large, i.e., Ea ) 30 kJ/mol (0.3 eV). The third region is the low temperature region, 80-175 K. In this temperature range, the proton quenching reaction of FMN by a proton is smaller than the radiative rate of FMN. The 4 mM HCl sample shows a shorter decay time of the emission signal than that of an acidfree sample. The proton diffusion constant derived for that sample is ∼10-5 cm2/s. It is almost temperature independent throughout the low temperature region. (2) At temperatures below 175 K, the time-resolved emissions of FMN ice samples in acid-free ice and ice doped with relatively small HCl concentrations, c e 1 mM, are almost identical at all temperatures studied until ∼80 K (Figure 3). According to our estimation, this result indicates that the proton reaction rate with FMN at low acid concentrations is ∼10 times smaller than the radiative rate. The proton diffusion constant is smaller than 10-5 cm2/s and roughly 2 orders of magnitude smaller than DH+ at its maximum value of 260 K. Thus, the radiative rate limits the time window of the experimental observation and consequently limits the lowest value of DH+ we can evaluate from the experiment. (3) At all temperatures, the quenching rate of an FMN ice sample doped with 4 mM HF, a weak acid in water (pKa ≈ 3.2), is slower than the quenching rate of a sample doped with 4 mM HCl (pKa ≈ -9). The FMN proton fluorescence quenching rate in HF ice samples indicates that HF is a weaker acid than HCl. The temperature dependence of the quenching rate of the sample with 4 mM HF is larger than that of the sample with 4 mM HCl. For endothermic reactions, ∆H > 0, the acid dissociation equilibrium constant decreases as the temperature decreases, and therefore, the free proton concentration decreases with a decrease in the temperature. We explained that the low quenching rate of FMN in HF-doped ice at low temperatures arises from a further decrease of the free proton concentration as the temperature decreases. Such a temperature dependence of the proton concentration is expected for an endothermic reaction (eq 1). (4) Takei and Maeno30,31 found a similar temperature dependence for the ionic defect conductivity, σ(, as found in the current study for DH+. At high temperatures above 235 K, σ( is almost temperature independent, whereas below 235 K the value of σ( strongly decreases as the temperature decreases. The strong temperature dependence of σ( at T < 235 K obeys an Arrhenius behavior with an activation energy of 0.31 eV to 150 K, the lowest temperature measured. (5) Takei and Maeno found that the dielectric relaxation time of the Debye dispersion below 235 K corresponds to activation energies, whose values of ∼0.29 eV are comparable to the activation energy of σ( (0.31 eV). The dielectric relaxation times, τD, are associated with a rotation of the Bjerrum orientational defects. The similar activation energies of σ( and τD may suggest that orientational defects limit the proton mobility at temperatures below 235 K and, consequently, also limit the proton diffusion constant at low temperatures. Similarity of the Temperature Dependence of Proton Diffusion in Water and Ice. In a previous study, we correlated the temperature dependence of DH+ in water, especially in supercooled water, with the dielectric relaxation time, τD. We explain the temperature dependence of DH+ results in ice in similar terms as that of DH+ in water. In ice at high temperatures above 235 K, a von Grotthuss mechanism apparently prevails.50 According to the concerted von Grotthuss mechanism, excess proton motion along a proton wire of properly aligned water molecules in

J. Phys. Chem. C, Vol. 113, No. 23, 2009 10293 hexagonal ice consists of the following steps (see Figure 4.10 in ref 4). The first step in ice involves an excess proton that resides at water molecule 1, H3O+, that is transferred to an adjacent neutral water molecule, H2O, (molecule 2), along the hexagonal structure by tunneling and as a result forms the H3O+ ion at water molecule 2. The second step involves a second proton tunneling but not by the proton involved in step 1. A hydrogen atom of water molecule 2, positioned between water molecules 2 and 3, tunnels to molecule 3 and forms H3O+ at position 3. These concerted tunneling processes may continue as long as the ice structure has no imperfections such as L- or D-defects, OH- and H3O+ ion defects, or dislocations and planar defects,4 which happen to occur along the random walk of the proton. An L-defect or Cl- may serve as proton traps, especially at low temperatures. The concentrations of OH- and the D-defects are low in acidic samples, and therefore, we will limit our consideration to the role of the L-defect and Cl-. The diffusion constant, DL, of the L-defect is associated with a hydrogen rotation around the oxygen-oxygen axis along a hexagonal structure, where the oxygens are at the vertices. The rotation time is associated with the dielectric relaxation time, τD. In water, the value of τD is ∼8 ps at ∼20 °C, whereas in ice at 260 K, it is much slower, i.e., ∼10-4 s. The value of τD, which is related to the average rotation time, is given in eq 3, where nL and N are the number of L-defects and water molecules per unit of volume, respectively. The ratio, nL/N, is very small in pure ice but is probably larger in strong acid-doped ice. νL is the frequency of a single orientational motion of the L-defect. nLνL is the total number of the molecular reorientations occurring in a unit of volume per unit of time. Thus, apart from a numerical factor, τD represents the average between successive reorientations of a single molecule. In liquid water, it was found that the solvation dynamics times, τs, and the dielectric relaxation, τD, are correlated.51 In a previous work,48 we studied the solvation dynamics of probe molecules in methanol-doped ice. In a solvation dynamics experiment, the time-dependent spectral shift of the emission band of an excited probe molecule was measured. We suggested there exists a correspondence between the average solvation correlation time, , and 1/νL of eq 3. The problem arises in finding the ratio nL/N in order to quantitatively correlate τs with τD in ice. In liquid water and alcohols, all of the molecules are free to rotate, and therefore, τs is a fraction of τD (in water, τs ) 0.1τD).51,52 In pure ice, the dielectric relaxation time τD at ∼240 K is very slow, i.e., 1.4 × 10-4s, which is 7 orders of magnitude longer than the relaxation time in water at 294 K and ∼8 orders of magnitude longer than τs, the long component of the solvation dynamics of water. The small fraction of L-defects per unit of volume, nL/N, in pure ice (10-7) accounts for the 7 orders of difference between τD and τs in ice (at 260 K, τs ≈ 20 ps).48 The dielectric relaxation time of pure ice samples depends strongly on temperature. The Arrhenius plot of log τD versus 1/T shows that τD increases almost exponentially but exhibits two different slopes. In the high temperature region of 270 > T > 230 K, the activation energy corresponds to ∼55 kJ/mol,45 whereas the slope below 220 K is about half of that value corresponding to 26 kJ/mol (0.27 eV). Takei and Maeno found a slightly larger activation energy of 0.29 eV of τD in HCldoped ice (c e 10-5 M) at temperatures below 230 K. The Arrhenius plot of the average solvation time of 2N68DS and C343,48 as a function of 1/T in the small temperature range between 235 and 175 K, gives an activation energy of ∼20 kJ/mol, which is slightly smaller than the

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activation energy value of τD in pure ice in the same temperature range (26 kJ/mol). As mentioned before, the temperature dependence of the proton diffusion constant, DH+, in ice shows unusual behavior. Similar to the abnormal proton conductivity in water, the activation energy is not constant. At high temperatures, T > 235 K, the activation energy is low, whereas at low temperatures, T < 235 K, it is high and constant. Furthermore, we found that the proton transfer rate in ice at low temperatures has an activation energy similar to DL (Figure 8a), for which the hopping rate depends on the orientational motion of a water molecule, and the time scale is associated with τD of ice according to eq 3. Previously,39,44 we found that the temperature dependence of the proton transfer rate constant from several photoacids to protic liquids (water and alcohols) and the abnormal proton mobility of liquid water may be explained by a qualitative model based on a continuous transition from a nonadiabatic proton transfer process to a solvent-controlled limit. Our previous attempt to correlate the abnormal proton conductivity in liquid water and the dielectric relaxation time, τD, was based on the nonadiabatic proton transfer theory developed by Kuznetsov and his colleagues.53 The theory is very similar to the nonadiabatic electron transfer in its treatment of the involvement of the solvent. In the solvent-controlled limit,54 the solvent motion needed to reach the generalized curve-crossing solvent configuration is the rate-limiting step of the proton transfer reaction. We found39,44 that the characteristic time for the solvent motion, τs, is a fraction of the dielectric relaxation time, τs ) τD/b, where b is an empirical factor. τs is shorter than τD but longer than the longitudinal dielectric relaxation time, τL ) (ε∞/εS)τD, where ε∞ and εS are the high frequency and static dielectric constants, respectively. In recent years, quantum calculations provided more detailed information on proton mobility in water.55-58 On the basis of the calculations of Voth and co-workers, Agmon38,59,60 suggested that the molecular mechanism behind prototropic mobility involves a periodic series of isomerizations between H9O4+ (Eigen’s complex10) and H5O2+ (Zundel dimer61), where the first is triggered by the hydrogen bond cleavage of a second shell water molecule and the second by the reverse hydrogen bond formation process. Strong and Weak Acids in Low Temperature Ice. Our results on the acidity of HCl (strong acid) and HF (weak acid) at low and high temperatures in ice (Figure 7) nicely fit the results by Minot and co-workers,62 who employed periodic DFT calculations to study an ice system interacting with HCl and HF. In their model, HX substitutes a water molecule in a cubic ice matrix. Their study took thermodynamic considerations into account. Energetic barriers can prevent penetration of HX into the ice crystal structure. According to their calculations, HCl completely dissociates inside the ice bulk structure. Moreover, a hydrogen transfer to a second molecule takes place, resulting in a separate ion pair. On the other hand, HF does not show ionization and remains in its molecular form. The different behaviors of HCl and HF inside the ice can be attributed to the different electron affinities. Minot and co-workers reproduced the experimental observations of HCl and HF in liquid water as strong and weak acids, respectively. Comparison with Conductivity Results and Interpretation. Takei and Maeno30 explained their conductivity measurements of HCl-doped ice as a function of temperature as follows. In HCl-doped ice, an HCl molecule is considered to replace an H2O molecule in the ice lattice, producing an L-defect and an

Uritski et al. H3O+ ion through HCl dissociation.63,64 The process can be formulated as follows

HCl · L a HCl + L ka

HCl + H2O {\} Cl- + H3O+

(6) (7)

In a previous study,48 we doped ice with KF and found that F generates L-defects, suggested by Takei and Maeno for HCldoped ice (eq 6). The L-defects react with the excited photoacids, and the proton is removed to produce RO-*. This reaction provides the diffusion constant, DL, for the L-defect. Replacing KF in our experiment with KCl, which releases Cl- ions into the ice, does not affect the photoacid dissociation reaction,45 and therefore, we estimated a rather low yield of mobile L-defects for HCl-doped ice. Takei and Maeno30,31 assumed that HCl in ice is not as strong an acid in liquid water. At high temperatures or low HCl concentrations Ka > c0, and therefore -

[H+] ≈ c0

(8)

where c0 denotes the acid concentrations in ice. This implies that HCl molecules at high temperatures completely dissociate to produce H3O+ ions. Takei and Maeno suggested30 that at low temperatures the HCl acid does not completely dissociate, and the proton concentration is given by

[H+] ≈ √Kac0

(9)

The proton diffusion constant, DH+, deduced in the current study, is related to proton mobility by the Stokes-Einstein relation

µH+ )

e D kBT H+

(10)

and the proton mobility is related to the proton conductivity as follows

σH+ ) nH+µH+e(

(11)

where nH+ is the number density of free protons in the sample, and e( is the effective proton charge in ice, which is ∼0.68. The ionic and orientational defect conductivity (σ( and σDL, respectively) is related to the low and high frequency conductivities as follows4

e(2 eDL2 e2 ) + σ0 σ( σDL

(12)

σ∞ ) σ( + σDL

(13)

The low-frequency conductivity, σ0, is limited by the smaller of the two components, σ( and σDL, because this determines the rate at which chains of bonds are unblocked to allow the majority species to flow.4 This argument is the same as in our explanation of DH+ to the limits of k(T) (eq 2), with ks at low temperatures and kH at high temperatures. The assumption of Takei and Maeno that the dissociation constant of Ka for HCl is much smaller in ice at low temperatures is equivalent to stating that the temperature dependence of Ka of HCl in ice is reversed, i.e., Ka decreases as the temperature decreases. Simple thermodynamic arguments (eq 1) predict that for an exothermic reaction, ∆H < 0 (strong acid), Ka increases as the temperature decreases. Thus, the observed temperature dependence of σ0 (Figure 5 in ref 30 and Figure 2 in ref 31) in the higher temperature region, T > 235 K, can be explained by

Temperature Dependence of Proton Diffusion in Ice SCHEME 2: Riboflavin

the fact that the σ( values are constant because the concentration of extrinsic ionic defects produced by HCl is larger than that of thermally produced intrinsic ones, and HCl molecules at these concentrations and temperatures are completely dissociated in the ice lattice. Moreover, the H+ migration energy of activation is small at high temperatures. Consequently, constant values of σ0 in the higher temperature region are expected to be proportional to c0 according to eq 8. The square root dependence of σ0 at T < 235 K is in good agreement with eq 9 and with the results of Young and Salomon63 and Maeno,64 although Gross et al.65 reported that σ0 ∝(c0)0.4 at -51 °C. According to Takei and Maeno30,31 the migration activation energy of H+ below 235 K is large, Em a ) 0.31 eV, and thus, the temperature dependence of the low-frequency conductivity, σ0, is large at T < 235 K. The values of σ0 and σ∞ decrease by ∼4 orders of magnitude from a high value at 240 K to a low value at 150 K, with a nearly constant activation energy slope of ∼0.3 eV. The DH+ data plotted in Figure 8 at T > 175 K are similar in shape to the shapes of σ( in refs 30 and 31. Takei and Maeno’s articles do not explicitly explain the large increase in the migration activation energy of protons below 235 K. We explain this phenomenon by the two-step model, described in detail in Modeling. At low enough temperatures, the rotation of a hydrogen atom of a misaligned water molecule in the ice lattice prevents the proton from tunneling to the neighboring water molecule. The rotation time is related to τD, the dielectric relaxation time, and the L-defect mobility. The activation energy of the L-defect mobility is large. In pure ice, nDL . n(, and thus, the temperature dependence of σ∞ in pure ice is determined by σDL, the conductivity of the D- and L-defects. The L-defect mobility is much larger than that of the D-defect. Consequently, it is the L-defect that determines σ∞. Takei and Maeno measured σ∞ as a function of temperature. The activation energy is large at high temperatures and somewhat lower at temperatures below 235 K. The activation energies of both σ( and σ∞ at temperatures below 235 K are nearly identical to that of τD.30,45 The low-frequency conductivity expression, given by eq 12, has a form similar to that of eq 2, which provides the overall proton transfer rate constant, k, of our stepwise model. Both quantities (σ0 and k) are determined by a rate-limiting step. In our model, the high-temperature rate constant is determined by the proton tunneling rate and by the hydrogen reorientation rate at low temperatures. Low-frequency conductivity will be determined by σ(, if the proton conductivity is smaller than the L-defect conductivity. Data analyses by Takei and Maeno30,31 of σ0 and σ∞ as functions of temperature provided the values of σ( and σDL as functions of temperature. At temperatures above 235 K, σDL is larger than σ(. Below 235 K, both conductivities exhibit similar values and a similar temperature dependence. Proton Conductance in Thin Film Ice. In recent years, surface scientists conducted low-temperature studies of proton diffusion in thin film, grown by controlled methods at vacuum

J. Phys. Chem. C, Vol. 113, No. 23, 2009 10295 conditions. These experiments, which produced interesting results, can be performed only at temperatures below 140 K. Experimental results on the mobility of protons in thin film ice indicate that proton transport is a thermally activated process that occurs quite slowly in ice at low temperatures66-72 or does not occur at all.73 Proton mobility in cold thin films indicates that it is a thermally activated process with a large activation energy. The thin film experimental results suggest that protons are mobile through the ice film in the amorphous (T < 140 K) and crystalline phases (T g 140 K) that form during the course of the temperature ramp. In addition, it has been observed74 that Cl- ions produced from HCl dissociation do not migrate to the film surface at T e 140 K, while protons build up a substantial population at the surface. This shows that the upward migration of protons is not caused by the presence of counteranions. Kang71 suggests that the anomalous experimental reports on the mobility of protons in ice films66-69,73,75 can be explained by the affinity of the protons for the ice surface and facile proton transport near the surface at T g 130 K. This result verifies that protons are mobile in an ice film and at favorable temperatures can migrate from the film interior to the surface. This conclusion is unaffected by the changes in ice film morphology and thickness (2-8 BL) and by the presence of counteranions. An extrapolation of Takei and Maeno’s conductivity data to ∼140 K indeed shows lower proton conductivities by roughly 7 orders of magnitude than those at T > 250 K. Summary We used time-resolved emission to monitor the fluorescence quenching of a flavin mononucleotide (FMN) and riboflavin (the latter described in the Supporting Information) in methanoldoped ice by excess protons over the large temperature range of 80-260 K. We analyzed the time-resolved emission data using the irreversible Smoluchowski model that accounts for proton diffusion and reaction with the excited FMN or riboflavin target molecules. The analysis provides the proton diffusion constant, DH+, in ice over the large range of temperatures studied. The plot of DH+ as a function of 1/T has a complex shape. In general, the behavior of DH+ as a function of 1/T can be divided into three regions. The first region is the high temperature region, T > 235 K, in which the temperature dependence is small, and the rate is fast. The proton diffusion constant in this region is ∼10-3 cm2/s, which is 10 times larger than that in water at 295 K, and the activation energy is relatively small, i.e., Ea < 1000 J/mol (∼0.01 eV). The second region is the intermediate temperature range, 175-235 K, in which the proton quenching rate strongly depends on the temperature. The proton diffusion constant deduced from the diffusion-controlled rate constant decreases by about a factor of 100 from its value of ∼10-3 cm2/s at 260 K to ∼10-5 cm2/s at 175 K. The activation energy of the proton diffusion constant is large, i.e., Ea ) 30 kJ/mol (0.3 eV). The third region is the low temperature region, 80-175 K. In this temperature range, the proton quenching reaction of FMN by a proton is smaller than the radiative rate of FMN. The 4 mM HCl sample shows a shorter decay time of the emission signal than that of an acid-free sample. The proton diffusion constant derived for that sample is ∼10-5 cm2/s. It is almost temperature independent throughout the low temperature region. We propose an oversimplified model calculation for the purpose of obtaining a qualitative description of the temperature dependence of the experimental proton diffusion constant in ice. The model restricts the proton transfer process to be stepwise. The proton moves to the adjacent water molecule only when

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the hydrogen alignment of the water molecule brings the system to the lowest energy barrier of the proton coordinate. In the stepwise model, the overall proton transfer time is a sum of two times, τ ) τ1 + τ2, where τ1 is the characteristic time for the hydrogen orientation of the water molecule, and τ2 is the time for the proton to pass over the barrier. Figure 8 shows the data and the fit of the model to the experimental results of the temperature dependence of the proton diffusion constant in ice. The model predicts that at high temperatures proton tunneling is the rate-determining step, while below 235 K the orientational motion of the water is the rate-limiting step. Below 235 K, the activation energies of DH+, σ(, σDL, and τD are nearly identical. We propose that orientational motion of a water molecule to properly align the hydrogens along the O-O bond, according to the ice rule, controls all these important parameters (Scheme 2). Acknowledgment. This work was supported by grants from the Israel Science Foundation and from the James Franck German-Israel Program in Laser Matter Interaction. Supporting Information Available: This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Fletcher, N. H. The Chemical Physics of Ice, Cambridge University Press: London, 1970. (2) Hobbs, P. V. Ice Physics; Clarendon Press; Oxford, U.K., 1974. (3) Walley, E., Jones, S. J., Gold, L. W., Eds. Physics and Chemistry of Ice, 5th ed.; Royal Society of Canada: Ottawa, Canada, 1973. (4) Petrenko, V. F.; Whitworth, R. W. The Physics of Ice; Oxford University Press: Oxford, U.K., 1999. (5) Bolton, K.; Pettersson, J. B. C. J. Am. Chem. Soc. 2001, 123, 7360. (6) Jaccard, C. Ann. N.Y. Acad. Sci. 1965, 125, 390. (7) Bjerrum, N. Science 1952, 115, 385. (8) Kobayashi, C.; Saito, S.; Ohmine, I. J. Chem. Phys. 2001, 115, 4742. (9) Podeszwa, R.; Buch, V. Phys. ReV. Lett. 1999, 83, 4570. (10) (a) Eigen, M. Proton transfer. Angew. Chem., Int. Ed. 1964, 3, 1. (b) Eigen, M.; Kruse, W.; Maass, G.; De Maeyer, L. Prog. React. Kinet. 1964, 2, 285. (11) Eigen, M.; de Maeyer, L. In The Structure of Electrolytic Solutions, Wiley: New York, 1959, p 64 ff. (12) Kelly, I. J.; Salomon, R. R. J. Phys. Chem. 1969, 50, 75. (13) Camplin, G. C.; Glen, J. W. In Physics and Chemistry of Ice, 5th ed.; Walley, E., Jones, S. J., Gold, L. W. , Eds.; Royal Society of Canada: Ottawa, Canada, 1973; p 256. (14) Kunst, M.; Warman, J. M. J. Phys. Chem. 1983, 87, 4093. (15) Onsager, L.; Runnels, L. K. J. Chem. Phys. 1969, 50, 1089. (16) Nagle, J. F. J. Phys. Chem. 1983, 87, 4086. (17) Ireland, J. E.; Wyatt, P. A. AdV. Phys. Org. Chem. 1976, 12, 131. (18) (a) Gutman, M.; Nachliel, E. Biochem. Biophys. Acta 1990, 391, 1015. (b) Pines, E.; Huppert, D. J. Phys. Chem. 1983, 87, 4471. (19) Tolbert, L. M.; Solntsev, K. M. Acc. Chem. Res. 2002, 35, 19. (20) (a) Rini, M.; Magnes, B. Z.; Pines, E.; Nibbering, E. T. J. Science 2003, 301, 349. (b) Mohammed, O. F.; Pines, D.; Dreyer, J.; Pines, E.; Nibbering, E. T. J. Science 2005, 310, 5745. (21) Tran-Thi, T. H.; Gustavsson, T.; Prayer, C.; Pommeret, S.; Hynes, J. T. Chem. Phys. Lett. 2000, 329, 421. (22) Agmon, N. J. Phys. Chem. A 2005, 109, 13. (23) Spry, D. B.; Fayer, M. D. J. Chem. Phys. 2008, 128, 084508. (24) Siwick, B. J.; Cox, M. J.; Bakker, H. J. J. Phys. Chem. B 2008, 112, 378. (25) Buch, V.; Milet, A.; Va´cha, R.; Jungwirth, P.; Devlin, P. Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 7342. (26) Mondal, S. K.; Sahu, K.; Sen, P.; Roy, D.; Ghosh, S.; Bhattacharyya, K. Chem. Phys. Lett. 2005, 412, 228.

Uritski et al. (27) Uritski, A.; Presiado, I.; Huppert, D. J. Phys. Chem. C 2008, 112, 11991. (28) Uritski, A.; Persiado, I.; Huppert, D. J. Phys. Chem. C 2008, 112, 18189. (29) Uritski, A.; Persiado, I.; Huppert, D. J. Phys. Chem. C, in press. (30) Takei, I.; Maeno, N. J. Phys. Chem. 1984, 81, 6186. (31) Takei, I.; Maeno, N. J. Phys. (Paris) 1987, 48, colloque C1, 121. (32) Meiboom, S. J. Chem. Phys. 1961, 34, 375. (33) Halle, B.; Karlstro¨m, G. J. Chem. Soc., Faraday Trans. 2 1983, 79, 1031. (34) Pfeifer, R.; Hertz, H. G. Ber. Bunsen-Ges. Phys. Chem. 1990, 94, 1349. (35) Agmon, N. J. Phys. Chem. 1996, 100, 1072 Errata: Agmon, N. J. Phys. Chem. 1997, 101, 4352. (36) Bernal, J. D.; Fowler, R. H. J. Chem. Phys. 1933, 1, 515. (37) Bockris, J. O’M.; Conway, B. E. Modern Aspects of Electrochemistry; Butterworths: London, 1964; Vol. 3. (38) Agmon, N. Chem. Phys. Lett. 1995, 244, 456. (39) Cohen, B.; Huppert, D. J. Phys. Chem. A 2003, 107, 3598. (40) Atkins P. W. Physical Chemistry, 5th ed.; Oxford University Press: Oxford, U.K., 1994. (41) Von Smoluchowski, M. Z. Phys. Chem. 1917, 92, 129. (42) Tachiya, M. Radiat. Phys. Chem. 1983, 21, 167. (43) Szabo, A. J. Phys. Chem. 1989, 93, 6929. (44) Cohen, B.; Huppert, D. J. Phys. Chem. A 2002, 106, 1946. (45) Johari, G. P.; Whalley, E. J. Chem. Phys. 1981, 75, 1333. (46) Rips, I.; Jortner, J. J. Chem. Phys. 1987, 87, 2090. (47) Collins, F. C.; Kimball, G. E. J. Colloid Sci. 1949, 4, 425. (48) Uritski, A.; Huppert, D. J. Phys. Chem. A 2007, 111, 10544. (49) Uritski, A.; Huppert, D. J. Phys. Chem. A 2008, 112, 3066. (50) Pang, X. F.; Feng, Y. P. Chem. Phys. Lett. 2003, 373, 392. (51) Horng, M. L.; Gardecki, J.; Papazyan, A.; Maroncelli, M. J. Phys. Chem. 1995, 99, 17311. (52) Rosental, S. J.; Xie, X.; Du, M.; Fleming, G. R. J. Chem. Phys. 1991, 95, 4715. (53) (a) German, E. D.; Dogonadze, R. R.; Kuznetsov, A. M.; Levich, V. G.; Kharkats, Yu. I. Elektrokhimiya 1970, 6, 350. (b) German, E. D.; Kuznetsov, A. M. J. Chem. Soc., Faraday Trans. 2 1981, 77, 2203. (54) (a) Borgis, D.; Hynes, J. T. J. Phys. Chem. 1996, 100, 1118. (b) Borgis, D. C.; Lee, S.; Hynes, J. T. Chem. Phys. Lett. 1989, 162, 19. (c) Borgis, D.; Hynes, J. T. J. Chem. Phys. 1991, 94, 3619. (55) Marx, D.; Tuckerman, M. E.; Hutter, J.; Parrinello, M. Nature 1999, 397, 604. (56) Tuckerman, M. E.; Laasonen, K.; Sprik, M.; Parrinello, M. J. Chem. Phys. 1995, 103, 150. (57) Tuckerman, M. E.; Laasonen, K.; Sprik, M.; Parrinello, M. J. Phys. Chem. 1995, 99, 5749. (58) Day, T. J. F.; Schmitt, U. W.; Voth, G. A. J. Am. Chem. Soc. 2000, 122, 12027. (59) (a) Markovitch, O.; Agmon, N. Mol. Phys. 2008, 106, 485. (b) Markovitch, O.; Chen, H.; Izvekov, S.; Paesani, F.; Voth, G. A.; Agmon, N. J. Phys. Chem. B 2008, 112, 9456. (c) Markovitch, O.; Agmon, N. J. Chem. Phys. 2008, 129, 1. (60) Agmon, N. J. Chim. Phys. (Paris) 1996, 93, 1714. (61) Zundel, G.; Metzger, H. Z. Phys. Chem. (N. F.) 1968, 58, 225. (62) Calatayud, M.; Courmier, D.; Minot, C. Chem. Phys. Lett. 2003, 369, 287. (63) Young, I. G.; Salomon, R. E. J. Chem. Phys. 1968, 48, 1635. (64) Maeno, N. Can. J. Phys 1973, 51, 1045. (65) Gross, G. W.; Hayslip, I. C.; Hoy, R. N. J. Glaciol. 1978, 21, 143. (66) Uras-Aytemiz, N.; Joyce, C.; Devlin, J. P. J. Phys. Chem. A 2001, 105, 10497. (67) Wooldridge, P. J.; Devlin, J. P. J. Chem. Phys. 1988, 88, 3086. (68) Fisher, M.; Devlin, J. P. J. Phys. Chem. 1995, 99, 11584. (69) Everest, M. A.; Pursell, C. J. J. Chem. Phys. 2001, 115, 9843. (70) Geil, B.; Kirschgen, T. M.; Fujara, F. Phys. ReV. B: Condens. Matter Mater. Phys. 2005, 72, 014304. (71) Kang, H. Acc. Chem. Res. 2005, 38, 893. (72) Moon, E.; Lee, C.; Kang, H. Phys. Chem. Chem. Phys. 2008, 10, 4814. (73) Cowin, J. P.; Tsekouras, A. A.; Iedema, M. J.; Wu, K.; Ellison, G. B. Nature 1999, 398, 405. (74) Lee, C. W.; Lee, P.-R.; Kang, H. J. Chem. Phys. 2007, 127, 84701. (75) Park, S. C.; Jung, K. H.; Kang, H. J. Chem. Phys. 2004, 121, 2765.

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