Dielectric Relaxation, Ion Conductivity, Solvent Rotation, and

11028

J. Phys. Chem. B 2008, 112, 11028–11038

Dielectric Relaxation, Ion Conductivity, Solvent Rotation, and Solvation Dynamics in a Room-Temperature Ionic Liquid Youngseon Shim† and Hyung J. Kim*,†,‡ Department of Chemistry, Carnegie Mellon UniVersity, Pittsburgh, PennsylVania 15213, and School of Computational Sciences, Korea Institute for AdVanced Study, Seoul 130-722, Korea ReceiVed: March 25, 2008

Dielectric susceptibility and related conductivity of the neat ionic liquid 1-ethyl-3-methylimidazolium hexafluorophosphate (EMI+PF6-) are studied via molecular dynamics computer simulations. Both ion translations and reorientations contribute to dielectric relaxation, while their cross-correlation does not play any significant role. Interestingly, ion translational dynamics are found to enhance the static dielectric constant ε0. The increment in ε0 is attributed to rapid development of large anticorrelation in the autocorrelation function of the ionic current, i.e., hindered ion translations of strong librational character. One consequence of hindered translational dynamics is that the real part of conductivity has a maximum in the terahertz region and decreases with diminishing frequency. This in turn yields significant dielectric absorption in the far-IR region, consonant with recent terahertz time-domain spectroscopy measurements. Reorientational dynamics of cations show a marked deviation from diffusion. The well-known relation in the diffusion regime for reorientational correlation -1 fails completely for EMI+PF -, where l is the order of Legendre polynomials used times τ(l) R ∝ [l(l + 1)] 6 in the expansion of reorientational time correlation functions. It is found that dielectric continuum theory generally does not provide a reliable framework to describe solvation dynamics in EMI+PF6- even though the inclusion of ion conductivity in dielectric relaxation tends to improve the continuum description. This is ascribed mainly to electrostrictive effects absent in many continuum formulations. 1. Introduction Solvent dynamics in room-temperature ionic liquids (RTILs) have been the subject of intensive scrutiny recently.1–22 One of the most extensively studied is solvation dynamics, couched usually in terms of time evolution of the Franck-Condon energy gap associated with a fluorescent probe solute.1,2,7–12,16–22 According to many molecular dynamics (MD) computer simulations, short-time dynamics play an important role in solvation and related processes in RTILs despite their high viscosities.23–32 From the perspective of the well-known Grote-Hynes theory,33 the presence of ultrafast solvent dynamics has an important implication in connection with rapid reaction kinetics34,35 observed in RTILs. Our recent MD study36 indicates that barrier crossing for small eletron transfer systems occurs indeed via fast solvent dynamics and resulting kinetics proceed considerably more rapidly than predicted on the basis of the Kramers theory.37 Detailed analysis of underlying molecular dynamics shows that translational motions of solvent ions play a central role in solvation dynamics in RTILs, in contrast to rotational motions in normal polar solvents.26,28,31 Another interesting finding is that long-time nonexponential solvent relaxation is often close to a stretched exponential decay and shows a 1/f spectrum,38 implying that RTILs are dynamically heterogeneous.6,18,35,39–46 Another dynamic process in RTILs that has drawn significant attention is dielectric relaxation. Dielectric permittivity for various RTILs has been measured both in the microwave region47–51 up to 20-40 GHz and in the far-IR region52 up to * Corresponding author. Permanent address: Carnegie Mellon University. E-mail: [email protected]. † Carnegie Mellon University. ‡ Korea Institute for Advanced Study.

∼4 THz in frequency. Their collective and single-molecule dipole reorientation dynamics have been analyzed via MD.53–55 Reorientational dynamics of solvent ions in neat RTILs were also studied via NMR51,56–58 and via MD.27,43,44,59 Potential links between dielectric relaxation and solvation dynamics in RTILs have also been examined in the continuum context.19–21 For many decades, dielectric relaxation measurements have been applied extensively to polar solvents and ionic solutions. In theoretical analysis of the latter systems, the constituents are usually separated into ionic and dipolar species and the dielectric response arising from dipole reorientations of the latter species under the influence of conducting ions is investigated.60–66 In the case of polarizable ions, their induced moments are treated as point dipoles at the location of pointlike ions.67 The decrements in the static dielectric constant of electrolyte solutions at finite ionic concentrations, compared to pure polar solvents, were found to arise from the decrease in the solvent density.66 While analysis of dielectric susceptibility based on the separation of ionic and dipolar species is conceptually straightforward for ionic solutions (or at least for their model descriptions), RTILs are not amenable to this approach because they consist of only ionic species and their individual dipole moments arising from extended charge distributions are not uniquely defined. This could lead to differing and potentially inaccurate interpretations of dielectric permittivity for RTILs. This is one of the main motivations of the present work. In this article, we investigate dielectric and related conductivity behaviors of 1-ethyl-3-methylimidazolium hexafluorophosphate (EMI+PF6-), paying attention to the roles played by the solvent translational and reorientational motions. Relaxation of dipole moments of cations defined with respect to their center of mass is mainly governed by reorientation of their long principal axis. This yields dielectric absorption in the microwave

10.1021/jp802595r CCC: $40.75  2008 American Chemical Society Published on Web 08/09/2008

Solvation Dynamics in an RTIL

J. Phys. Chem. B, Vol. 112, No. 35, 2008 11029 EMI+PF6-. Due to the united-atom representation employed for PF6-, only the cations undergo rotations in our simulation study. We examined their reorientational time correlation functions (l,Ω) (TCFs) CΩ defined as l (t) and associated correlation times τR

C lΩ(t) ≡ 〈Pl[cos θΩ(t)] 〉 ;

Figure 1. Principal axes of EMI+ ion. The moments of inertia about the principal x-, y-, and z-axes are about 420, 360, and 90 amu Å2.

region. Ion translational motions are found to enhance the static dielectric constant ε0. This increment is attributed to the fact that the ionic current becomes anticorrelated rapidly in time before its fluctuations dissipate completely. This anticorrelation also leads to a peak in the ion conductivity and strong dielectric absorption, both in the far-IR region. The contribution of crosscorrelation between the ion translations and reorientations to dielectric relaxation is small, indicating that these motions are largely decoupled in EMI+PF6-. Another topic we consider is the applicability of dielectric continuum description to solvation dynamics in RTILs. In accord with previous studies,19–21 dielectric relaxation via ion reorientations provides a poor continuum framework to describe RTIL solvation dynamics. The inclusion of ion conductivity in dielectric relaxation improves the continuum description substantially. This confirms the earlier finding that ion translations play a key role in solvation dynamics in RTILs.26,28,31 Nevertheless our analysis indicates that there is a significant discrepancy between continuum and molecular descriptions of solvation dynamics in RTILs. The outline of this paper is as follows: In section 2, we give a brief description of the models and methods employed in this study. Dielectric relaxation and ion conductivity as well as solvent reorientational dynamics in EMI+PF6– are analyzed in section 3. Comparison between dielectric continuum theory predictions for solvation dynamics and MD results is also made there. Section 4 concludes F6–. 2. Models and Methods The simulation cell is comprised of 212 pairs of rigid EMI+PF6-. We use the united-atom representation of earlier MD studies of Kim and co-workers23,30,31,36,68–70 to describe interionic interactions in the present MD simulation. To be specific, the Et and Me moieties of EMI+ in Figure 1 as well as PF6- are treated as united atoms. The Lennard-Jones parameters and partial charge assignments of EMI+ are taken from the AMBER force field71 and ref 72, respectively. For PF6-, σ ) 5.6 Å and ε/kB ) 200 K are employed. For details on the solvent model description, the reader is referred to ref 68. We have performed simulations in the canonical ensemble at 400 K using the DL_POLY program.73 We carried out two independent simulations, each with 10 ns equilibration, followed by a 20 ns trajectory, from which equilibrium averages were computed. 3. MD Results and Discussion 3.1. Solvent Reorientational Dynamics. We begin by considering reorientational dynamics of solvent ions in

τR(l,Ω) )

∫0∞ ClΩ(t) dt

(1)

where Pl is the lth Legendre polynomial, θΩ(t) is the angle between the molecular orientation Ω of interest at time t and its initial direction, and 〈...〉 denotes an equilibrium ensemble average. The results for CΩ l (t) associated with three principal axes, Ω ) x, y, and z, of EMI+ are displayed in Figure 2. We first consider reorientational dynamics of x- and y-axes. Their CΩ l (t) show a bimodal character, viz., relatively fast relaxation of a small amplitude for t j ps and ensuing slow nonexponential decay. To a good approximation, the long-time relaxation of Cx,y l (t), in particular, for l ) 1, is given by a stretched exponential function exp[-(t/τl)βl]; the parameters τ1 ) 230 and 215 ps together with β1 ) 0.42 reasonably reproduce, respectively, Cx1(t) and Cy1(t) for 1 ps j t j 5 ns. One salient feature is that the long-time decay of Cx2(t) is slower than that of Cx1(t), whereas the former relaxes faster than the latter at short times. A similar trend obtains for reorientations of the y-axis. As a consequence, Ω CΩ 1 (t) and C2 (t) intersect at t ≈ 0.29 and 0.67 ns for Ω ) x and y, respectively. The reorientational correlation times are τ(1,Ω) R ≈ 0.7 ns and τ(2,Ω) ≈ 1.1 ns for the x-axis and 0.65 and 0.85 ns R for the y-axis; thus τ(1,x) < τ(2,x) and similarly for τ(1,y) and τ(2,y) R R R R . This finding is in good accord with an earlier study by Kim and co-workers on rotational dynamics of a nondipolar solute.45 There these interesting behaviors of reorientational TCFs were attributed to intermittent large-amplitude rotations (“hopping” motions),74,75 present in RTILs (see below). For additional insight, we have examined the conditional probability, viz., the Green function GΩ(θ;t), associated with solvent reorientation

GΩ(θ;t) ) 〈δ(θΩ(t) - θ)〉 2l + 1 ) Pl(cos θ) sin θ ClΩ(t); 2 l

∑

ClΩ(t) )

∫0π Pl(cos θ) GΩ(θ;t) dθ (2)

The MD results for GΩ(θ;t) associated with x- and y-axes of EMI+ are exhibited in Figure 3a and Figure 3b, respectively. There are two noteworthy features: First, Gx,y(θ;t) develops a shoulder structure around θ ) 160° with increasing t. A structure similar to this was observed previously in MD studies of different ionic liquids76,77 as well as of small nondipolar solutes in RTILs.45 This is due to the presence of above-mentioned large-amplitude rotations, which are strongly suggestive of dynamic heterogeneity.6,18,35,39–42 Hopping motions of a large magnitude (viz., θ > 90°) lead to the crossing of CΩ 1 (t) and Ω CΩ 2 (t) because they tend to accelerate the relaxation of Cl (t) of odd l compared to that of even l.45 Second, Gx,y(θ;t) show a fixed point at θ ≈ 55° for t J 1 ps. This means that, after initial relaxation, the population of solvent cations with their x- or y-axis at this orientation does not vary with time. In ref 45, it was conjectured that the existence of a fixed point in GΩ(θ;t) is closely related to the stretched exponential behavior of CΩ l (t) at long times. We turn to reorientational dynamics of the principal z-axis, which corresponds to the long axis of EMI+ (Figure 1). We note that charge anisotropy of EMI+ is the largest along the z-direction. For instance, projections of the dipole moment µcm of EMI+ defined with respect to its center of mass (see eq 8

11030 J. Phys. Chem. B, Vol. 112, No. 35, 2008

Shim and Kim

Ω Figure 2. Reorientational TCFs for three principal axes and dipole moment µcm of cations in EMI+PF6- at 400 K: CΩ 1 (t) (s) and C2 (t) (- -). Reorientational correlation times are τR(1,Ω) ≈ 0.7, 0.65, 2.3, and 2.0 ns and τR(2,Ω) ≈ 1.1, 0.85, 1.9, and 1.6 ns for Ω ) x, y, z, and µcm, respectively. µcm is defined with respect to cation center of mass (see eq 8).

Figure 3. Rotational Green function GΩ(θ;t) (θ in degrees) for three principal axes and dipole moment µcm of cations in EMI+PF6- at 400 K. After initial relaxation, GΩ(θ;t) for x- and y-axes is characterized by a fixed point at θ ≈ 55°.

below) on to the principal x-, y- and z-axes are 0.24, 0.84, and 1.47 D, repsectively. Therefore µcm is approximately in the direction of the z-axis. With this in mind, we consider its reorientational TCFs in Figure 2c. Analogous to Cx,y l (t) associated with the principal x- and y-axes, Czl (t) show a bimodal behavior. However, there are two major differences: First, Cz1(t) and Cz2(t) do not intersect, at least during the time window accessible via

our simulations. Cz2(t) decays faster than Cz1(t). Their correlation times are τ(1,z) ≈ 2.3 ns and τ(2,z) ≈ 1.9 ns, so that τ(1,z) > τ(2,z) R R R R . In addition, reorientational motions of the z-axis are considerably slower than those of x- and y-axes.27,43,44,59 Second, their longtime behavior shows a substantial deviation from a stretched exponential decay. Czl (t) are closer to a single-exponential Ω function than Cx,y l (t) even though Cl (t) of all three axes are


J. Phys. Chem. B, Vol. 112, No. 35, 2008 11031

cm Figure 4. MD results for dielectric relaxation: (a) εrot (ω), (b) εtrcm(ω), and (c) εc(ω) for EMI+PF6- at 400 K. For comparison, ε(ω) for CH3CN at 298 K obtained with the model description of ref 82 is shown in (d).

nonexponential. It is noteworthy that, in contrast to the x- and y-axes, Gz(θ;t) for the z-axis in Figure 3c does not have a fixed point. According to ref 45, an increase in size or charge anisotropy of the probe molecule leads to a growing deviation of its rotational relaxation from the stretched exponential decay. For EMI+, the dipole moment with respect to its center of mass is small (∼1.7 D), while its moment of inertia varies significantly with the principal axes, i.e., Ixx ≈ 420, Iyy ≈ 360, and Izz ≈ 90 amu Å2. We thus ascribe qualitative and quantitative differences + between Czl (t) and Cx,y l (t) of EMI primarily to the difference in the molecular dimension. For completeness, we also examined reorientational TCFs Cµl (t) of the individual dipole moments µcm of cations and its Green function Gµ(θ;t) (Figures 2d and 3d). Since the principal z-axis and µcm are roughly parallel, their rotational relaxation dynamics show a similar behavior, including the absence of a fixed point in GΩ(θ;t). Correlation times associated with Cµl (t) are τ(1,µ) ≈ 2.0 ns and τ(2,µ) ≈ 1.6 ns. R R While the detailed reorientational behavior of the z-axis differs substantially from those of the x- and y-axes, we stress that all three principal axes of solvent cations show a stark deviation from diffusive rotational dynamics. For instance, their τ(1,Ω) and R τ(2,Ω) do not satisfy at all the relation τ(1,Ω) ) 3τ(2,Ω) (Ω ) x, y, R R R z, and µ), which is one of the hallmarks of rotational diffusion.78 Rather, τ(1,Ω) is considerably smaller than 3τ(2,Ω) , regardless of R R the principal axes. This has potentially important implications for the interpretation and application of various experimental results. With the exception of dielectric relaxation, most spectroscopic methods, e.g., polarization anisotropy decay and NMR, access τ(2,Ω) directly. In assessing CΩ R 1 (t) relaxation and its correlation time from these measurements, the relation τ(1,Ω) R ) 3τ(2,Ω) is often invoked. According to our analysis above, R however, this could result in a significant overestimation of τ(1,Ω) R and the relaxation of CΩ 1 (t) thus estimated could be much slower

than the correct CΩ 1 (t) behavior. Furthermore, the concept of a rotational “diffusion” coefficient would be essentially moot because CΩ l (t) is highly nonexponential. Thus care should be taken in applying the relation τ(1,Ω) ) 3τ(2,Ω) to rotational R R dynamics in RTILs. Recently, Weinga¨rtner and co-workers studied reorientational dynamics of cations in several EMI+-based RTILs via NMR and compared with dielectric relaxation.51 They found that the l ) 2 reorientational time jτs2 of the C2sH axis of EMI+ (Figure 1) and dielectric relaxation time jτ1 satisfy jτ1 ≈ 3τjs2. In view of our result above, i.e., τ(1,Ω) , 3τ(2,Ω) in RTILs, this experimental R R finding deserves some scrutiny. If we can neglect collective contributions to dielectric relaxation arising from cross-correlations of different ions (see Figure 5 below), dipole reorientations of ions provide a reasonable description for dielectric relaxation up to the microwave region, accessible in ref 51 (see discussions

Figure 5. MD results for Φcm MM(t) (s) and its self-part (- -) and distinct (---) part, ∑R〈µcm,R(t) · µcm,R〉 and 1/2∑′R,β〈µcm,R(t) · µcm,β〉, for EMI+PF6at 400 K. In the distinct part, ∑′ indicates that R ) β is excluded from the sum. Units for the TCFs are D2. We note that, when normalized, the self-part becomes identical to Cµ1 (t) of cation µcm in Figure 2d.


Shim and Kim

of Figure 4c below). In this regime, we can approximate jτ1 of ref 51 as jτ1 ≈ τ(1,µ) ≈ τ(1,z) τs2 ≈ τ(2,y) from Figure 1. R R , while j R This reveals that jτ1 and jτs2 measure approximately the l ) 1 reorientational time of the z-axis and l ) 2 reorientational time of the y-axis of EMI+, respectively. In other words, the NMR and dielectric relaxation results in ref 51 describe reorientations of different molecular axes, whereas the inequality relation τ(1,Ω) R , 3τ(2,Ω) is for a same molecular orientation Ω. Therefore, the R experimental finding jτ1 ≈ 3τjs2 does not have any bearing on whether or not molecular reorientational times τ(l,Ω) scale as R [l(l + 1)]-1. Since τ(1,z) ) 2.3 ns and τ(2,y) ) 0.85 ns, our MD R R also yields τ(1,z) ≈ 3τ(2,y) for EMI+PF6-, consonant with R R + measurements of other EMI -based RTILs in ref 51. 3.2. Dielectric Relaxation and Ion Conductivity. Here we consider dielectric relaxation of EMI+PF6-. The electric susceptibility χ(ω) is defined as

1 M (t) ) V tot

cm χ(ω) ) χcm 1 (ω) + χ2 (ω);

χcm 1 (ω) ) + χcm 2 (ω) )

∫0∞ dt eiωtχ(t);

∫0∞ dt eiωt

Mtot )

∑ qi,Rri,R (3)

ΦMM(t) ) 〈Mtot(t) · Mtot(0) 〉 (4) where kB is Boltzmann’s constant. The generalized dielectric constant ε(ω) and conductivity σ(ω) are related to χ(ω) as

4π [σ′(ω) + iσ′′(ω)] (5) ω

where prime and double prime represent, respectively, the real and imaginary parts of ε(ω) and σ(ω). Because of the nonvanishing dc conductivity, i.e., σ(0) ) σ′(0) * 0, ε(ω) diverges as ω f 0 for ionic systems (see the Drude model case below). Thus we also consider the nondivergent part εc(ω) of the dielectric constant

εc(ω) ≡ ε(ω) - i

4π σ(0) σ(0) ) 1 + 4π χ(ω) - i ω ω

[

]

(6)

We note that experimentally, the static dielectric constant ε0 is determined as19,47,61,62

[

ε0 ) εc(0) ) lim ε(ω) - i ωf0

4π σ(0) ω

]

(7)

In the calculation of χ(ω), we have decomposed Mtot into the translational and rotational components following ref 54 for convenience. Specifically, we employ cm Mtot ) M rot + M trcm; cm ≡ M rot

∑ µcm,R; R

µcm,R )

M trcm ≡

∑ qRrcm,R; R

∑ qi,R(ri,R - rcm,R); i∈R

1 3VkBT

i cm Φ (t)]; ∫0∞ dt eiωt[-Φcm MJ (t) + ω JJ

qR )

∑ qi,R (8) i∈R

where qR and rcm,R are, respectively, the net charge and position of the center of mass (CM) of solvent ion R, and µcm,R is the

cm cm Φcm MJ(t) ) 〈M rot (t) · Jtr 〉 (9)

where Jtr is the current arising from the CM translational motions of solvent ions

i,R

dΦMM(t) ; dt

4πχ(ω) ) ε′(ω) + iε′′(ω) - 1 ) i

cm ∫0∞ dt eiωt[iωΦcm MM (t) - ΦMJ (t)]};

cm cm Φcm JJ (t) ) 〈Jtr (t) · Jtr 〉 ;

j tot(t) is the average total dipole moment of the system where M at time t induced by a homogeneous external electric field E(t), V is the volume, qi,R and ri,R are, respectively, the charge and position of the site i of ion R, and the sum is taken over charge sites of all ions. According to the fluctuation-dissipation theorem,79 χ(ω) is related to the total dipole moment fluctuations via

1 3VkBT

1 cm 2 〉 {〈M rot 3VkBT

cm cm Φcm MM (t) ) 〈M rot (t) · M rot 〉 ;

∫-∞t dt′ χ(t - t′) E(t′); χ(ω) )

χ(ω) ) -

dipole moment of R with respect to its CM. Thus Mcm rot describes the contribution to Mtot from rotations of individual solvent ions with respect to their CM, while Mtrcm arises from translational motions of their CM. One can show after some algebra that the generalized electric susceptibility in eq 4 can be rewritten as54

J trcm ≡

d cm M ) dt tr

∑ qRr˙cm,R

(10)

R

and Φcm AB(t) are unnormalized TCFs between quantities A and B defined with respect to CM of individual solvent ions. For clarity we make three remarks here: First, because of the united-atom description, i.e., point charge, employed for PF6-, µcm,R ) 0 for solvent anions. Thus only EMI+ contributes cm 80 to Mrot . For Mtrcm and Jtrcm, both cations and anions make nonvanishing contributions. Second, though not shown here, we note that the contributions from the cross-correlation function cm ΦMJ (t) to the susceptibilities are much smaller than those from cm ΦMM (t) and Φcm JJ (t) in the present system. This suggests that translational and rotational motions of solvent ions are generally cm decoupled in EMI+PF6-. As a result, χcm 1 (ω) and χ2 (ω) in eq 9 arise mainly from rotations with respect to, and translations of, CM of individual ions, respectively. For later use, we introduce dielectric constant εcm rot (ω) mainly due to ion rotational motions with respect to CM and its difference εcm tr (ω) from εc(ω): cm εrot (ω) ≡ 1 + 4πχcm 1 (ω);

cm εtrcm(ω) ≡ εc(ω) - εrot (ω)

[

) 4π χcm 2 (ω) - i εtrcm(ω)

σ(0) (11) ω

]

in eq 11 arises primarily from translational motions of ion CM. The decomposition of Mtot and thus χ(ω) via eqs 8–10 has an important advantage; i.e., it enables us to understand the roles played by molecular motions in dielectric relaxation and related phenomena, e.g., solvation dynamics (see below). We, however, make a third remark, viz., this decomposition is not unique at all. One can employ a molecular reference other than the center of mass to define cation dipole moments in analyzing χ(ω). One can even consider the center of charge,81 with respect to which the cation dipole moment vanishes. Thus a prudent view would be that the choice of a decomposition scheme (or no decomposition) is a matter of convenience in investigating specific aspects of dielectric relaxation. Physically measurable quantities, e.g., χ(ω), do not and should not depend on the decomposition scheme employed in the analysis. cm In Figure 4, the MD results for εrot (ω), εtrcm(ω), and εc(ω) cm for EMI+PF6- arising, respectively, from χcm 1 (ω), χ2 (ω), and χ(ω) are shown. For comparison, ε(ω) for polar acetonitrile at 298 K determined with the nonpolarizable potential model of ref 82 is also displayed. The dc conductivity σ(0) needed


J. Phys. Chem. B, Vol. 112, No. 35, 2008 11033

in the calculation of εc(ω) in eq 6 was obtained with the aid of eqs 5 and 9, i.e.

σ(0) ) -i lim ωχ(ω) ) ωf0

1 3VkBT

∫0∞ dt Φcm JJ (t)

(12)

where σ(0) was determined as the “plateau” value83,84 of the time integration of Φcm JJ (t). We notice that while Mrot arising from ion rotational motions makes nonvanishing contributions to σ(ω), it does not contribute to the dc conductivity σ(0). + The frequency dependence of εcm rot (ω) of EMI PF6 in Figure 4a differs from that of normal polar solvent acetonitrile in Figure 4d even though their underlying molecular dynamics are similar, i.e., solvent rotational motions. To see this with clarity, we cast our results in the Cole-Davidson form85

εc(ω) - ε∞ )

∑ k

Ak c

(1 + iωτck)βk

(13)

where the number of terms on the right-hand side of eq 13 and the deviation of their βck values from unity gauge the departure from the single-exponential Debye solvent behavior. Dielectric relaxation in acetonitrile in Figure 4d is well described by a single term with A1 ) 25.5, τc1 ) 3 ps, and βc1 ) 1 in the Cole-Davidson function (ε∞ ) 1). This shows that in the model description of ref 82, acetonitrile is a good Debye solvent. By cm contrast, if we are to fit εrot (ω) using eq 13, at least two terms c with βk * 1 (A1 ) 0.63, βc1 ) 0.62, and τc1 ) 2.9 ns, and A2 ) 0.1, βc2 ) 0.1, and τc2 ) 2.9 ns) will be needed. Thus dielectric relaxation via cation rotations in EMI+PF6- is clearly not Debyelike. It is also worthwhile to note that the dielectric absorption occurs in the ω range of 0.1-1 GHz for EMI+PF6- at 400 K, whereas the corresponding frequency range 10-100 GHz for acetonitrile at 298 K is much higher by nearly two decades. This means that despite the elevated temperature of EMI+PF6-, its solvent reorientational dynamics are much slower than those in acetonitrile. cm For additional insight, simulation results for ΦMM (t) in eq 9 and its self-part and distinct part in EMI+PF6- are shown in Figure 5. Φcm MM(t) is characterized by fast inertial decay, followed by a very slow nonexponential decay. We found that a triexponential function with decay times 19 ps, 0.2 ns, and 2.4 cm ns fits the slow relaxation of ΦMM (t) for t J 5 ps. While the functional form of this fit is different from that used above for cm εrot (ω), it nonetheless confirms the non-Debye character of cm εrot (ω). The decomposition of Φcm MM(t) discloses that its relaxation is governed essentially by the self-part, i.e., single-dipole reorientation. The distinct part, which decays very slowly, cm contributes about 10% to ΦMM (t) initially at t ) 0. Because the + direction of EMI dipole with respect to its CM and its principal z-axis roughly coincide as mentioned above, reorientational dynamics of the long axis of cations are mainly responsible for cm cm ΦMM (t) and εrot (ω) of EMI+PF6-. cm Coming back to Figure 4a, we notice that the εrot (0) value cm for the present EMI+PF6- system is rather small; i.e., εrot (0) ≈ 1.7. This is mainly due to the small cation CM dipole moment (µcm ≈ 1.7 D) resulting from the charge assignments of our 2 model description. Since the amplitude of Φcm MM(t) scales as µcm , an increase in µcm in the model description would raise the εcm (0) rot cm value rapidly. For example, µcm ≈ 3.4 D would yield εrot (0) ≈ 3.8 if we can assume that other properties remain nearly unchanged. Another factor that contributes to the low value cm of εrot (0) in the present study is the elevated temperature T ) 400 K, because ε(ω) generally decreases with T.50 Figure 4b shows that the main dispersion of εtrcm(ω) occurs in the frequency region around 10 THz, which is much higher

cm than the 0.1-10 GHz region relevant to εrot (ω). Furthermore, the transition from the anomalous to normal dispersion for εtrcm′(ω) near ω ) 20 THz clearly signals that it has a resonance character, which was indeed observed in a recent spectroscopic study52 of imidazolium-based RTILs. For better understanding, we have analyzed the autocorrelation function ΦJJ(t) of the total ionic current86

ΦJJ(t) ) 〈Jtot(t) · Jtot 〉 ;

Jtot(t) )

d M (t) ) dt tot

∑ qiRr˙iR i,R

(14) The MD result in Figure 6a shows that dissipation of ΦJJ(t) is completed in about 600 fs. Therefore, corresponding dielectric relaxation by way of ion translations occurs on the time scale shorter than 1 ps. The oscillatory behavior of ΦJJ(t) indicates the hindered librational character of ion translational motions in EMI+PF6-. This librational nature of translations, also observed in several previous simulations,43,87,88 explains the resonance behavior of εcm tr (ω). We note that hindered translations are also found to make important contributions to optical Kerr effect spectra13–15 of RTILs.88 Another noteworthy feature in Figure 4b is that εtrcm(ω) is finite in the ω f 0 limit. In view of eqs 5, 7, and 11, this is equivalent to the conductivity behavior that σ′′(ω)/ω does not vanish as ω tends to zero. This means that ion conductivity arising from translational motions makes a nonvanishing contribution to the static dielectric constant ε0 as defined in eq 6 for ionic systems. As a result, ε0 is greater cm than its rotational component εrot (0) by ∼1.9 () εtrcm(0)). While this may be unexpected, the static dielectric constant is influenced by translational motions of charge carriers even in the simple Drude model description. To gain better insight into this interesting behavior, we digress here to consider briefly a free electron gas in the classical Drude model. In the Langevin equation description, the equation of motion of an electron of mass m with charge -e in the presence of an external field E(t) and the corresponding dipole moment are89

m(r¨ + γr˙) ) -eE(t);

µ ) -er

(15)

where r is the electron position and γ is the friction associated with its motion. We make a Fourier transform of eq 15 and sum over all electrons to obtain

1 M(ω) ) χD(ω) E(ω); V

χD(ω) ) -

1 ne2 m ω(ω + i/τ) (16)

where n is the electron number density and τ ) 1/γ. Thus the dielectric constant εD(ω) and conductivity σD(ω) in the Drude model are89

εD(ω))1 + 4πχD(ω) ) 1 +i

1 4πne2 m ω2 + 1/τ2

1/τ 4πne2 ; m ω(ω2 + 1/τ2)

σD(ω) ) -iωχD(ω) )

1 ne2 (1/τ + iω) (17) m ω2 + 1/τ2

In the high-frequency limit ω . 1/τ, εD(ω) becomes

εD(ω) ≈ 1 -

ωp2 2

ω

;

ωp2 )

4πne2 m

(18)

where ωp is the well-known plasma frequency. In the opposite low-frequency limit, we have


σD(0) σD′′(ω) ; ∆ε ≡ 4π lim ωf0 ω ω 2 2 dσD′′(ω) ne τ ne2τ ) 4π ) 4π ; σD(0) ) (19) dω 0 m m

εD(ω) ≈ 1 - ∆ε + 4πi

|

Equation 19 reveals two effects conducting electrons have on the dielectric constant in the ω f 0 limit: (i) the divergence of εD′′(ω) as ω-1 due to dc conductivity σD(0) and (ii) the reduction of εD′(0) by ∆ε, which is directly related to σD′′(ω) of the electronic conductivity. This situation is very similar to the RTIL case above except that conductivity lowers ε0 in the Drude model, whereas it enhances the static dielectric constant for EMI+PF6-. To understand the difference in the contributions of conductivity to ε0 between the Drude model and EMI+PF6-, we analyzed the real and imaginary parts of their conductivity. The results in Figure 6b show several interesting features. First, the Drude model and EMI+PF6- show a totally opposite behavior in σ′′(ω) in the low-frequency region. Specifically, while σ′′(ω) is always positive irrespective of ω in the Drude description, it is negative in the region ω j 12 THz for EMI+PF6-. Because of this sign difference in σ′′(ω) as ω f 0+, the static dielectric constant is enhanced by conductivity in the latter, whereas it is reduced in the former. Second, the real part of σ(ω) decreases monotonically with ω in the Drude description, while it has a peak in the far-IR region around ω ) 12 THz in EMI+PF6-. Though electronic (rather than ionic) in nature, a similar peak structure in σ′(ω) was found in poor metals90 and disordered conductors91,92 in the IR and near-IR regions. In ref 93, the deviation of electronic conductivity from the Drude behavior was ascribed to the dynamic correlation effect of the current arising from, in particular, backscattering of electrons, which

Shim and Kim yields a sign reversal and thus a negative value for ΦJJ(t) before its complete relaxation. In the present EMI+PF6-, ΦJJ(t) becomes negative around t ) 100 fs presumably due to strong interionic interactions (Figure 6a). Finally, in view of the Kramers-Kronig relations, we expect that the differences in σ′(ω) and in σ′′(ω) between the Drude model and MD results are closely related. To further illustrate the points above, we consider model autocorrelation functions of ionic current and compare their predictions for σ(ω) with the MD results. For convenience, we introduce

ξ ) -ΦJJ (tm)/ΦJJ (0)

if ΦJJ (tm) < 0;

ξ ) 0 otherwise (20)

where tm is the time at which ΦJJ(t) becomes minimum. The parameter ξ measures the degree of dynamic anticorrelation of ionic currentswhich is closely related to its librational charactersas the maximum magnitude of ΦJJ(t) when ΦJJ(t) < 0. If ΦJJ(t) is positive during the entire relaxation, then ξ ) 0, i.e., no dynamic anticorrelation. The ξ values for models A, B, and C in Figure 6a are 0, 0.05, and 0.17, respectively. Thus the degree of dynamic anticorrelation increases in that order. The EMI+PF6- system is characterized by the highest degree of dynamic anticorrelation with ξ ) 0.42. Results of model calculations for σ′(ω) and σ′′(ω) are presented in Figure 6c and Figure 6d, respectively. Dynamic anticorrelation of ΦJJ(t) exerts a striking influence on both the real and imaginary parts of σ(ω). As ξ grows, the location of the peak in σ′(ω) shifts to a higher frequency and the low-frequency region of σ′′(ω) moves downward. We notice that even a very mild anticorrelation in ΦJJ(t), e.g., model B

Figure 6. (a) Autocorrelation function of total ionic current ΦJJ(t) (s) in EMI+PF6- at 400 K. Three models A, B, and C for ΦJJ(t), respectively, with ξ ) 0, 0.05, and 0.17 are also shown (units for ΦJJ(t), D2 ps-2). (b) Frequency-dependent conductivity σ(ω) for EMI+PF6- at 400 K. σD(ω) with arbitrarily chosen parameters in eq 17 is also displayed for comparison. Results for (c) σ′(ω) and (d) σ′′(ω) in EMI+PF6- and three model systems A, B, and C.

Solvation Dynamics in an RTIL with ξ ) 0.05, results in a peak shift for σ′(ω), and thus the conductivity maximum occurs at a finite frequency. By contrast, there is a threshold in ξ for the appearance of a negative-valued domain in σ′′(ω) in the low-frequency region. Thus σ′′(ω) for model B is always positive despite the presence of dynamic anticorrelation in its ΦJJ(t). As ξ further grows, a negativevalued region appears in the low-frequency domain of σ′′(ω). Since the contribution of ion translations to ε0 is governed by the negative of the slope of σ′′(ω) evaluated at ω ) 0 (eq 19), the extent of enhancement of ε0 via conductivity increases as the degree of dynamic anticorrelation in the ionic current grows. For model A with ξ ) 0, σ′(ω) is a monotonically decreasing function and σ′′(ω) is always positive. Therefore, its ε0 is reduced by conductivity. While the Drude model provides a good description for model A, other models show a departure from the Drude description as exposed by, for example, the peak shift in σ′(ω). In this sense, ξ also measures the degree of deviation from the Drude behavior. Returning to dielectric relaxation, we consider εc(ω) in Figure cm 4c. Because εc(ω) is a sum of εrot (ω) and εtrcm(ω), εc(ω) shows both the resonance decay arising from hindered translations and nonresonant dissipation due to reorientations. We point out that εc′(ω) - ε∞ (ε∞ ) 1.0) is negative; i.e., εc′(ω) < ε∞ for ω J 10 THz. As mentioned above, our result for εc′(ω) is in excellent agreement with recent terahertz time-domain measurements52 of imidazolium-based RTILs. Since εc′(ω) - ε∞ ∝ -σ′′(ω)/ω according to eq 5, the frequency range for εc′(ω) < ε∞ coincides precisely with that for σ′′(ω) > 0. εc′′(ω) has a broad peak in the terahertz region, which is directly linked to the peak of σ′(ω) in the same frequency region. Therefore EMI+PF6- is characterized by significant dielectric absorption via hindered ion translations in the far-IR region,52 in addition to absorption in the microwave region via ion reorientations. As mentioned above, the value of µcm employed in the present study seems to be small. If a larger value is used for µcm, ion rotations can also play a considerably more important role in dielectric relaxation than the results in Figure 4. In this sense, dielectric measurements can provide important insight into charge distributions and CM dipole moments of solvent ions. The Cole-Davidson function (eq 13) consisting of two terms, A1 ) 0.67 and τc1 ) 2.9 ns, and A2 ) 1.9 and τc2 ) 0.6 ps, both with βc ) 0.62, well represent εc(ω) with ε∞ ) 1.0 up to ω ≈ 3 THz despite resonance absorption there. The τc1 ) 2.9 ns and τc2 ) 0.6 ps terms arise, respectively, from reorientational and translational dynamics. Our βc value is in a reasonable accord with those found in experimental studies of various RTILs,47,49,50 which range from ∼0.3 to ∼0.7 under normal conditions. 3.3. Solvation Dynamics. Finally, we consider solvation dynamics in RTILs. It is well established that dielectric continuum theory provides a reasonable framework to describe solvation free energetics and dynamics in normal polar solvents.94 Recently the continuum description has been extended to nondipolar but quadrupolar solvents, such as benzene and dense carbon dioxide.95–97 It was found that kinetics98 and thermodynamics99 of charge transfer reactions as well as solubility100 are captured reasonably well by introducing effective polarity for quadrupolar solvents.95–97 Thus a logical next question would be whether the continuum description can be extended also to RTILs. According to several recent studies,19–21 continuum theory based on dielectric susceptibility with the neglect of conductivity provides a rather poor description for solvation dynamics in RTILs.19–21 The inclusion of conductivity, however, tends to improve the agreement between the two.19 Here we examine this issue by the explicit comparison of the

J. Phys. Chem. B, Vol. 112, No. 35, 2008 11035 dielectric continuum predictions and MD results30 for solvation dynamics in EMI+PF6-. Solvation dynamics of a probe solute is usually described via the normalized dynamic Stokes shift

Safb(t) )

∆Eafb(t) - ∆Eafb(∞)

(21)

∆Eafb(0) - ∆Eafb(∞)

where ∆Eafb(t) is the average Franck-Condon energy associated with electronic transitions from state a to state b of the solute at time t after an instantaneous change in its charge distribution from state b to state a. In eq 21 the average is taken over the initial distribution of solvent configurations in equilibrium with the b-state solute. In the linear response regime, the dynamic Stokes shift can be approximated by the equilibrium time correlation function

Cafb(t) ≡

〈δ∆Eafb(0) δ∆Eafb(t)〉a

〈(δ∆Eafb)2〉a

;

δ∆Eafb(t) ) ∆Eafb(t) - 〈∆Eafb〉a (22) where 〈...〉a denotes an equilibrium ensemble average in the presence of the a-state solute. In the dielectric continuum approach to solvation dynamics, the temporal behavior of ∆Eafb(t) is often assumed to be governed by the dielectric response function. Details of ∆Eafb(t) vary with the continuum descriptions because of differing assumptions and approximations, e.g., monopole vs dipole charge distributions of the solute. In our analysis, we use the functional form derived by Song and Chandler101

∆Eafb(t) )

∫-∞∞ dω e-iωt∆Eafb(ω);

∆Eafb(ω) )

1 1 1ωR ε(ω)

[

1)ε(ω) ]∑ (l (l++1)ε(ω) + l[ R ] R0

2l

(23)

l

which was employed in ref 19. In eq 23, ∆Eafb(ω) is a solvation energy for a unit charge at a distance R0 from the center of a spherical cavity of radius R. Before we embark on model calculations, we briefly consider the limitations of many continuum approaches, including eq 23 for perspective. In addition to the spherical shape of the cavity, R is assumed to be fixed; i.e., it does not depend on ω. This means that cavity size fluctuations102,103 and their contributions to solvent reorganization free energy104–106 are not accounted for in eq 23. Cavity size changes describe in the continuum context the solvent density variations near the solute with its charge distribution, i.e., electrostriction103,104,106–110 and its dynamic analogue,103,104,106,111,112 and with thermodynamic conditions.113–115 According to recent model calculations in a polar solvent, the cavity resizing effect on reorganization free energy can be significant.116 With these restrictions in mind, we turn to model calculations. cm Three different cases were considered: (i) εrot (ω) (eq 5), (ii) εc(ω) (eq 6), and (iii) ε(ω) (eq 11) were employed as the frequency-dependent dielectric constant in eq 23. The contribution of ion translational motions to dielectric response is almost completely ignored in case i, while only the dc conductivity is neglected in case ii. Case iii corresponds to incorporation of full susceptibility χ(ω) in ∆Eafb(ω). We parenthetically remark that we also considered a different functional form of ∆Eafb(ω), employed in refs 20 and 21, based on solvent response to dipolar solutes. We found that this theory yields nearly the same results as eq 23 with a minor numerical difference and thus is not presented here.


Shim and Kim

Figure 7. Dielectric continuum theory predictions for dynamic Stokes shift in EMI+PF6-: εcm rot (ω) ( · · · ), εc(ω) (- -), and ε(ω) (- · -). For comparison, equilibrium MD results (s) for CNPfIP(t) and CIPfNP(t) in eq 22 are also presented.

The results for three cases i-iii obtained with eqs 21 and 23 are exhibited in Figure 7. For comparison, earlier MD results30 for Cafb(t) in eq 22 probed via the Franck-Condon transitions between two different electronic states, neutral pair (NP) and ion pair (IP) states, of a diatomic solute are also presented. IP is characterized by unit charge separation between two constituent atoms of the solute, while there is no charge separation for NP. For details of the IP and NP solute electronic description, the reader is referred to ref 68. cm Our model calculations show that εrot (ω) completely fails to + capture solvation dynamics of EMI PF6-, congruent with earlier findings of refs 19–21. The inclusion of ion conductivity in the continuum theory improves the agreement with MD considerably, in concert with the conclusion of ref 19. These results clearly indicate that not rotational motions of solvent ions but their translational motions play a major role in solvation dynamics in RTILs.26,28,31 The continuum theory prediction for the solvation time τs for the present EMI+PF6- at 400 K is about cm 2 ns with εrot (ω) employed as the dielectric permittivity, while the incorporation of the ion conductivity reduces τs to 1 and 0.9 ns with and without accounting for the dc conductivity, respectively. The equilibrium MD results yield τs ) 11 and 450 ps with the NP and IP solutes employed as a probe, respectively.38 Despite significant improvement with the inclusion of frequency-dependent conductivity, the continuum theory predictions for solvent relaxation are considerably slower than the MD results. One potential reason is the absence of cavity resizing effect in eq 23 mentioned above. In ref 26, smallamplitude inertial translations of ions, especially near the solute, were found to play a major role in ultrafast solvent relaxation in RTILs, in contrast to inertial rotations in polar solvents. In the continuum context, displacement of ions near the solute corresponds to cavity size variations, which could introduce a significant energy change in eq 23. Thus inclusion of ultrafast cavity resizing in continuum theory would accelerate the shorttime relaxation of its predictions for Safb(t). We also emphasize that continuum description considered here and in previous studies19–21 completely misses the dependence of solvation dynamics on the probe charge distribution due to the absence of electrostriction. We thus conclude that one should be cautious in applying continuum theory to RTILs. 4. Concluding Remarks In this article, we have studied solvent rotational dynamics, dielectric relaxation, and related conductivity in EMI+PF6- via MD simulations. We found that reorientational motions of solvent ions show a striking departure from diffusive dynamics. After initial relaxation, orientational relaxation of two short

principal axes (Figure 1) of EMI+ exhibit nearly stretched Ω exponential decay and their TCFs CΩ 1 (t) and C2 (t) intersect. As a result, their reorientational correlation times satisfy an unusual inequality: τ(1,Ω) < τ(2,Ω) . Reorientation dynamics of R R the long principal axis of EMI+ are, relatively speaking, closer to diffusion than those of the two short axes. Nevertheless, the long axis is also characterized by strongly nondiffusive rotation with τ(1,Ω) J τ(2,Ω) . Therefore, the well-known relation for R R reorientational correlation times associated with CΩ l (t) in the diffusion limit, viz., τ(l,Ω) ∝ [l(l + 1)]-1, completely fails to R describe solvent reorientations in EMI+PF6-. In a previous MD study,45 a similar breakdown observed for nondipolar solutes in RTILs was attributed to dynamic heterogeneity,42,76,77 which permits intermittent large-amplitude rotations.74,75 There it was also found that, as the solute size grows, its reorientational dynamics become closer to diffusion; i.e., the effect of dynamic heterogeneity decreases with the increasing probe size. This is in good agreement with the findings of the present study. The present analysis of solvent reorientational dynamics also provides new insight into recent NMR and dielectric relaxation measurements51 of EMI+-based RTILs. To be specific, our study shows explicitly that the two measurements describe rotational motions of two different molecular orientations, viz., principal y- and z-axes of cations to a good approximation. This explains nicely why experimental results yield jτ1 ≈ 3τjs2. Our results indicate that ion translational dynamics plays a significant role in dielectric relaxation in EMI+PF6-. One interesting consequence is that strongly correlated translational motions of ions lead to a negative value for the imaginary part σ′′(ω) of the conductivity in the low-frequency region, which makes a positive contribution to the static dielectric constant ε0 of EMI+PF6-. The rapid development of significant anticorrelation in the autocorrelation function of ionic current before its complete relaxation, viz., strong librational character associated with hindered translations, is mainly responsible for negative σ′′(ω) for small ω. This behavior of ionic current, attributed to strong interionic interactions, leads to a maximum in σ′(ω) around ω ) 12 THz. This in turn yields a peak in εc′′(ω) and thus strong dielectric absorption in the far-IR region. In a recent spectroscopic study of imidazolium-based RTILs, a similar structure was observed in the terahertz domain.52 Solvent reorientational dynamics on the other hand results in absorption in the microwave region. The cross-correlation between ion reorientations and translations plays a negligible role, suggesting that these motions are nearly decoupled from each other. Another topic we investigated is whether dielectric continuum theory provides a reliable theoretical framework to describe solvation dynamics in RTILs. Detailed comparison of continuum theory predictions with explicit MD results on solvation

Solvation Dynamics in an RTIL dynamics indicates that there is generally a gap between the two even though the inclusion of ion conductivity in the dielectric response function for the continuum theory tends to narrow this gap. We believe that the inclusion of electrostriction and its dynamic analogue can further improve the applicability of continuum theory to RTILs. Notes Added. After the submission of this work, a MD study on dielectric relaxation in RTILs by Schro¨der, Haberler, and Steinhauser was published [J. Chem. Phys. 2008, 128, 134501]. We note that their results on the role of conductivity in dielectric relaxation, for example, enhancement of static dielectric constant, are in excellent agreement with our findings here. References and Notes (1) Karmakar, R.; Samanta, A. J. Phys. Chem. A 2002, 106, 4447. (2) Karmakar, R.; Samanta, A. J. Phys. Chem. A 2002, 106, 6670. (3) Hyun, B.-R.; Dzyuba, S. V.; Bartsch, R. A.; Quitevis, E. L. J. Phys. Chem. A 2002, 106, 7579. (4) Giraud, G.; Gordon, C. M.; Dunkin, I. R.; Wynne, K. J. Chem. Phys. 2003, 119, 464. (5) Triolo, A.; Russina, O.; Arrighi, V.; Juranyi, F.; Janssen, S.; Gordon, C. M. J. Chem. Phys. 2003, 119, 8549. (6) Cang, H.; Li, J.; Fayer, M. D. J. Chem. Phys. 2003, 119, 13017. (7) Arzhantsev, S.; Ito, N.; Heitz, M.; Maroncelli, M. Chem. Phys. Lett. 2003, 381, 278. (8) Ingram, J. A.; Moog, R. S.; Ito, N.; Biswas, R.; Maroncelli, M. J. Phys. Chem. B 2003, 107, 5926. (9) Baker, S. N.; Baker, G. A.; Munson, C. A.; Chen, F.; Bukowski, E. J.; Cartwright, A. N.; Bright, F. V. Ind. Eng. Chem. Res. 2003, 42, 6457. (10) Chakrabarty, D.; Hazra, P.; Chakarborty, A.; Seth, D.; Sarkar, N. Chem. Phys. Lett. 2003, 381, 697. (11) Chowdhury, P. K.; Halder, M.; Sanders, L.; Calhoun, T.; Anderson, J. L.; Armstrong, D. W.; Song, X.; Petrich, J. W. J. Phys. Chem. B 2004, 108, 10245. (12) Ito, N.; Arzhantsev, S.; Heitz, M.; Maroncelli, M. J. Phys. Chem. B 2004, 108, 5771. (13) Rajian, J. R.; Li, S.; Bartsch, R. A.; Quitevis, E. L. Chem. Phys. Lett. 2004, 393, 372. (14) Shirota, H.; Castner, E. W., Jr. J. Phys. Chem. A 2005, 109, 9388. (15) Shirota, H.; Castner, E. W., Jr. J. Phys. Chem. B 2005, 109, 21576. (16) Arzhantsev, S.; Jin, H.; Ito, N.; Maroncelli, M. Chem. Phys. Lett. 2005, 417, 524. (17) Lang, B.; Angulo, G.; Vauthey, E. J. Phys. Chem. A 2006, 110, 7028. (18) Samanta, A. J. Phys. Chem. B 2006, 110, 13704. (19) Halder, M.; Headley, L. S.; Mukherjee, P.; Song, X.; Petrich, J. W. J. Phys. Chem. A 2006, 110, 8623. (20) Arzhantsev, S.; Jin, H.; Baker, G. A.; Maroncelli, M. J. Phys. Chem. B 2007, 111, 4978. (21) Jin, H.; Baker, G. A.; Arzhantsev, S.; Dong, J.; Maroncelli, M. J. Phys. Chem. B 2007, 111, 7291. (22) Castner, E. W., Jr.; Wishart, J. F.; Shirota, H. Acc. Chem. Res. 2007, 40, 1217. (23) Shim, Y.; Duan, J.; Choi, M. Y.; Kim, H. J. J. Chem. Phys. 2003, 119, 6411. (24) Margulis, C. J. Mol. Phys. 2004, 102, 829. (25) Kobrak, M. N.; Znamenskiy, V. Chem. Phys. Lett. 2004, 395, 127. (26) Shim, Y.; Choi, M. Y.; Kim, H. J. J. Chem. Phys. 2005, 122, 044511. (27) Bhargava, B. L.; Balasubramanian, S. J. Chem. Phys. 2005, 123, 144505. (28) Kobrak, M. N. J. Chem. Phys. 2006, 125, 064502. (29) Hu, Z.; Margulis, C. J. J. Phys. Chem. B 2006, 110, 11025. (30) Jeong, D.; Shim, Y.; Choi, M. Y.; Kim, H. J. J. Phys. Chem. B 2007, 111, 4920. (31) Shim, Y.; Jeong, D.; Manjari, S. R.; Choi, M. Y.; Kim, H. J. Acc. Chem. Res. 2007, 40, 1130. (32) Kobrak, M. N. J. Chem. Phys. 2007, 127, 184507. (33) Grote, R. F.; Hynes, J. T. J. Chem. Phys. 1980, 73, 2715. (34) Ozawa, R.; Hamaguchi, H. Chem. Lett. 2001, 736. (35) Gutkowski, K. I.; Japas, M. L.; Aramenda, P. F. Chem. Phys. Lett. 2006, 426, 329. (36) Shim, Y.; Kim, H. J. J. Phys. Chem. B 2007, 111, 4510. (37) Kramers, H. A. Physica 1940, 7, 284. (38) Jeong, D.; Choi, M. Y.; Jung, Y.; Kim, H. J. J. Chem. Phys. 2008, 128, 174504. (39) Mandal, P. K.; Sarkar, M.; Samanta, A. J. Phys. Chem. A 2004, 108, 9048.

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