Letter pubs.acs.org/JPCL
Rate and Amplitude Heterogeneity in the Solvation Response of an Ionic Liquid Sachin Dev Verma,† Steven A. Corcelli,*,‡ and Mark A. Berg*,† †
Department of Chemistry and Biochemistry, University of South Carolina, Columbia, South Carolina 29208, United States Department of Chemistry and Biochemistry, University of Notre Dame, Notre Dame, Indiana 46556, United States
‡
S Supporting Information *
ABSTRACT: In contrast with conventional liquids, ionic liquids have solvation dynamics with more rate dispersion and with average times that do not agree with dielectric measurements. A kinetic analog of multidimensional spectroscopy is introduced and used to look for heterogeneity in simulations of coumarin 153 in [Im12][BF4]. Strong heterogeneity is found in the diffusive solvation rate. An unanticipated heterogeneity in the amplitude of the inertial solvation is also seen. Both heterogeneities exchange at the same rate. This rate is similar to the mean diffusive solvation time, putting it in the intermediateexchange region. Overall, there are multiple violations of the assumptions usually invoked in the theory of reaction dynamics.
T
Solvation dynamics in ionic liquids have been extensively studied by 1D methods, both experiments 1,18−20 and simulations.21−23 As in nonionic liquids, the dynamics have two phases: a subpicosecond inertial phase and a slower diffusive phase. However, the diffusive solvation has strong rate dispersion: the relaxation is not single-exponential but rather appears to have a range of rate constants. One explanation for the rate dispersion is that multiple processes are involved. For example, cation diffusion, anion diffusion, cation and anion rotation, shell exchange, and so on may act independently and have different rate constants. This explanation is homogeneous: every molecule undergoes the same set of process and experiences the same broad range of relaxation rates. Alternatively, the liquid dynamics may be heterogeneous. This type of heterogeneity is well documented in supercooled liquids approaching the glass transition.24 However, the established methods for detecting rate heterogeneity in simulations only apply to pure liquids, not to an isolated solute. In fact, the very definition of heterogeneity in a system where local environments are exchanging has been unclear and can lead to potential paradoxes. For example, how can one learn about the spatial heterogeneity between molecules by simulating a single solute? In the context of single-molecule measurements, defining heterogeneity in terms of multidimensional correlation functions solves these problems.25 Here, those methods are extended to simulations. New methods for interpreting these
o a surprising degree, a nonionic solvent can be modeled as a spatially uniform continuum, even for dynamics on a molecular length scale. In contrast, this letter shows that a prototypical ionic liquid is not uniform: it has strong spatial variation in its solvation dynamics, both in the average rate of the diffusive phase and in the amplitude of the inertial phase. These local environments are short-lived, surviving only long enough for one solvation event to occur. Overall, reaction dynamics in the ionic liquid violate multiple simplifications that apply to conventional liquids. These complexities can explain the failure of continuum solvation theories in ionic liquids1−4 and suggest that new theoretical approaches will be needed to understand chemical kinetics in these fluids. Many types of heterogeneity have been discussed in connection with ionic liquids. These fluids are viscous, which leads to long-lived fluctuations in the local electric field. As a result, heterogeneity can be induced in any fast solute process, for example, its fluorescence frequency5,6 or reaction rate,7−9 even though the solvent itself is behaving uniformly. In ionic liquids with long side chains, there can be spatial separation of ionic and nonionic groups.10−13 As a result, different solute molecules segregate to different spatial regions.14 However, the ionic liquid studied here has alkyl chains much smaller than the solute, so this effect is negligible. These established heterogeneities are inherently different from the heterogeneity in solvent dynamics sought here. Heterogeneity in diffusion has been reported recently but only for very small solutes.15 Two experimental studies have specifically looked for heterogeneity in solvation dynamics, with contrasting results.16,17 Those studies relied on a correlation between solvation rate and mean electric-field magnitude, which is not guaranteed. © 2016 American Chemical Society
Received: December 21, 2015 Accepted: January 14, 2016 Published: January 14, 2016 504
DOI: 10.1021/acs.jpclett.5b02835 J. Phys. Chem. Lett. 2016, 7, 504−508
Letter
The Journal of Physical Chemistry Letters
Figure 1. (a) 1D autocorrelation of the solvation energy C(1)(τ1) (black dots) and a 6-exponential fit (black curve). A single-exponential diffusive region would give the blue curve. The effect of diffusive-rate heterogeneity is illustrated by the red curve. Inset: Structures of the solute molecule and solvent ions. (b) Rate-correlation spectrum at τ2 = 0, Ĉ (3)(T3,0,T1) and its division into diffusive−diffusive (DD), inertial−inertial (II), and inertial− diffusive (ID) regions. (c) Heterogeneous portion of the 3D spectrum at τ2 = 0, Ĉ (3) h (T3,0,T1). (d) Exchange causes decay of the apparent heterogeneity in the diffusive rate CT(τ2) (red) and in the inertial amplitude CA(τ2) (blue). (e) 3D spectrum as a function of τ2, Ĉ (3)(T3,τ2,T1). All contour plots have a contour spacing of 0.1 and use the same vertical scale.
correlation functions are also introduced, in particular, ratecorrelation spectra and dispersion parameters derived from them. The class of organic, room-temperature ionic liquids is broad, and the single example studied here cannot represent all the processes found among them. In the liquid studied here, the cation has no long side chains and few conformations, minimizing structural heterogeneity (Figure 1a). The anion is small and symmetric, reducing its role in the solvation. However, the features seen in this simple system are likely to carry over to many ionic liquids, even when additional processes are added. The data used are from an all-atom simulation with flexible bonds, except to hydrogen, for one coumarin-153 molecule, 256 1-ethyl-3-methylimidizolium cations, and 256 tetrafluoroborate anions.23 The primary quantity of interest is E(t), the difference between the electrostatic-interaction energy of the atomic charges on the solvent ions with those on the coumarin in its ground or excited state. The fluctuation from the mean δE(t) is used to calculate the standard, 1D correlation function C(1)(τ ) =
1 N
(1)
⟨δE(τ )δE(0)⟩
∞
d=
⎛
⎞2
∫1ps dT ⎜⎝ln TT ⎟⎠ Ĉ(1)(T )
(2)
D
where Ĉ (1)(T) is the inverse-Laplace transform of C(1)(τ). This parameter, d = 3.2, gives the standard-deviation squared of the apparent distribution of rates in factors of e. Although the 1D function yields the extent of rate dispersion, it does not determine how much is due to heterogeneity. Measuring heterogeneity requires the 3D correlation function, which is defined by 1
C(3)(τ3 , τ2 , τ1) =
(3)
N (τ2)
⟨δE(τ3 + τ2 + τ1)
× δE(τ2 + τ1)δE(τ1)δE(0)⟩
(3)
where N(3) sets the amplitude of the diffusive−diffusive component to one. It is easier to interpret this function as the rate-correlation spectrum Ĉ (3)(T3,τ2,T1), which is defined implicitly by C(3)(τ3 , τ2 , τ1) =
(1)
∫0
∞
dT3
∫0
∞
(3)
dT1 Ĉ (T3 , τ2 , T1)e−τ3/ T3e−τ1/ T1 (4)
where N(1) is a normalization factor (Figure 1a). (See Supporting Information (SI) for details of all the calculations.) This simulated function matches experimental results very well.1,23 The first phase of relaxation is the inertial decay, which has a half life of 0.15 ps. The second phase is the diffusive decay, which has a geometric-mean time of TD = 40 ps. The diffusive phase shows strong dispersion: it is spread over a much wider range of times than an exponential. This fact is illustrated by the blue curve in Figure 1a, which replaces the diffusive phase with a single exponential. The dispersion of the diffusive decay is conveniently quantified by (SI)
For more convenient display, the spectrum is defined using inverse rates, T3 and T1. The rate-correlation spectrum is a direct analog of the frequency-correlation spectra used in multidimensional NMR, IR, and visible spectroscopies, where it is used to analyze spectral inhomogeneity. Rate-correlation spectra have been discussed theoretically,26 but have seen little use in practice. To determine the extent of heterogeneity, it is sufficient to look at the τ2 = 0 limit of the spectrum (Figure 1b). As in other correlation spectra, the diagonal reiterates the 1D spectrum: a narrow peak near 0.15 ps due to inertial relaxation and a broad 505
DOI: 10.1021/acs.jpclett.5b02835 J. Phys. Chem. Lett. 2016, 7, 504−508
Letter
The Journal of Physical Chemistry Letters peak over the picosecond region from the dispersed, diffusive dynamics. The new information is in the off-diagonal regions. Intensity there indicates that the corresponding diagonal rates occur on the same molecule. Different types of heterogeneity are reported by each of the three regions of the 2D spectrum: diffusive−diffusive (DD), inertial−inertial (II), and inertial− diffusive (ID) (Figure 1b). For the DD peak, an elliptical shape results from weak off-diagonal intensity (Figure 1b). Thus, not all of the rates seen along the main diagonal occur on every moleculerate heterogeneity plays an important role. In contrast, the II peak is circular. The inertial decay has no rate heterogeneity, even though the inertial decay is not exponential. (The nonexponential shape also leads to early negative regions. See the SI.) Similarly, strong cross-peaks occur in the ID regions because every molecule experiences both phases of relaxation. The off-diagonal width of the DD peak is measured by the apparent homogeneous-dispersion parameter dg(τ2) dg (τ2) =
1 2
⎛ T ⎞2 (3) dT3 dT1 ⎜ln 3 ⎟ Ĉ (T3 , τ2 , T1) DD ⎝ T1 ⎠
∫∫
CT(τ2) =
dg (τ2) − dg (∞) dg (0) − dg (∞)
(6)
The red curve in Figure 1d shows the result. The half life of the decay, Tex,T ≈ 50 ps, is similar to the mean diffusive solvation time, TD = 40 ps. This process is in neither the fast nor the slow exchange limits; it is a case of intermediate exchange.28 The local environment that defines a particular solvation rate lives long enough for a single relaxation cycle. However, that environment is destroyed by the solvation process itself. Being in the intermediate-exchange region means that the full range of instantaneous rates is not observed as rate dispersion. There is a good analogy with motional narrowing of spectra: the full range of instantaneous frequencies is not observed in the line width. Thus, the observed heterogeneous dispersion is a lower bound on the instantaneous variation between different solvent configurations. The rates of fast charge-transfer reactions are sensitive to the rate of diffusive solvation.7 The effects of intermediate-exchange heterogeneity on reaction dynamics are poorly explored.28 Further work will be important for understanding kinetics in ionic liquids. Rate heterogeneity is measured from the shapes of the peaks in the correlation spectra. However, Figure 1a,e also shows large changes in the height of the II peak: large at τ2 = 0, small at τ2 = 0.2 ps, and recovered at τ2 = 100 ps. Changes in peak height and volume are due to heterogeneity in the amplitude of the inertial relaxation (to be published). Amplitude heterogeneity has no signature in 1D and is only detectable by multidimensional methods. The changes in both the shapes and heights of the peaks can be isolated as follows. A 3D spectrum is constructed that is consistent with the 1D results, but that has no effects from heterogeneity. Subtracting this spectrum from the full spectrum gives the heterogeneous component of the spectrum Ĉ (3) h (T3,τ2,T1) (SI). It is shown at τ2 = 0 in Figure 1c. The DD peak is positive along the diagonal and negative in the offdiagonal region, but has no net volume. This is the pattern generated by a shape change and is caused by rate heterogeneity. In contrast, the II peak has a large volume. It would disappear entirely if the amplitude were the same on every molecule. This amplitude heterogeneity is characterized by the standard deviation σ2A of the inertial amplitude A, which is measured by the volumes of the spectral peaks
(5)
In the absence of exchange between rate subensembles during τ2, this parameter gives the portion of the total dispersion due to homogeneous causes. The value at τ2 = 0, dg(0) = 1.5, excludes exchange to the maximum degree possible. Thus, the dispersion due to heterogeneity is dh(0) = d − dg(0) = 1.7 (SI). More than half of the rate dispersion is from heterogeneity. We have not determined the exact shape of the distribution of mean times. However, we have constructed a simple distribution with dh(0) = 1.7 (SI) and show it in Figure 1a. This curve illustrates the effect of rate heterogeneity isolated from other effects. In nonionic liquids, the solvation response can be accurately predicted from the dielectric response: dynamics on a molecular length scale can be modeled by treating the solvent as a uniform continuum.27 For a conductive fluid, Maroncelli and Ernsting have shown that the mean solvation time is inversely related to the conductivity σ0.2 Thus, the heterogeneity in solvation rates can be envisioned as variation in the conductivity of the local ion structure. At a quantitative level, continuum models systematically underestimate the mean solvation time in ionic liquids.1−4 This failure can be caused by spatial heterogeneity. Even if a continuum model holds locally, different measurements average over space differently. For example, the solvation time is an average of the inverse conductivity, which is not equal to the inverse of the average conductivity when there is a distribution of conductivities. Similar effects are believed to cause continuum models of molecular diffusion and rotation to fail in supercooled liquids.24 Measuring exchange between different environments requires the full 3D spectrum as a function of τ2 (Figure 1e). As τ2 increases, the molecule moves through the heterogeneous distribution, and the apparent homogeneous dispersion increases. The off-diagonal width increases, as is seen for the DD peak in Figure 1e. At long times, the width reaches the value expected for complete exchange (to be published). (The narrowing along the diagonal is also an effect of heterogeneity (to be published).) To quantify the exchange, a rate-exchange correlation function is defined by
σA2(τ2) ⟨A⟩2
(3)
=
∫ ∫II dT1 dT3 Ĉ h (T3 , τ2 , T1) (3)
(∫ ∫ dT1 dT3 Ĉ (T3 , τ2 , T1))2 ID
(7)
From the τ2 = 0 spectrum (Figure 1c), the amplitude heterogeneity is as large as the mean amplitude, σA(0) = 1.3⟨A⟩. The distribution must extend from near-zero to several times the mean. As τ2 increases across the inertial region, the small-amplitude subensembles survive preferentially, and the average peak height drops (Figure 1e, τ2 = 0.2 ps). For longer τ2, exchange causes the peak to recover. The amplitude-exchange correlation function CA(τ2) = 506
σA(∞) − σA(τ2) σA(∞) − max(σA(τ2))
(8) DOI: 10.1021/acs.jpclett.5b02835 J. Phys. Chem. Lett. 2016, 7, 504−508
Letter
The Journal of Physical Chemistry Letters
(8) Sahu, K.; Kern, S. J.; Berg, M. A. Heterogeneous Reaction Rates in an Ionic Liquid: Quantitative Results from 2D-MUPPETS. J. Phys. Chem. A 2011, 115, 7984−7993. (9) Nagasawa, Y.; Miyasaka, H. Ultrafast Solvation Dynamics and Charge Transfer Reactions in Room Temperature Ionic Liquids. Phys. Chem. Chem. Phys. 2014, 16, 13008−13026. (10) Wang, Y.; Jiang, W.; Yan, T.; Voth, G. A. Understanding Ionic Liquids through Atomistic and Coarse-Grained Molecular Dynamics Simulations. Acc. Chem. Res. 2007, 40, 1193−1199. (11) Pádua, A. A. H.; Costa Gomes, M. F.; Canongia Lopes, J. N. A. Molecular Solutes in Ionic Liquids: A Structural Perspective. Acc. Chem. Res. 2007, 40, 1087−1096. (12) Hardacre, C.; Holbrey, J. D.; Mullan, C. L.; Youngs, T. G. A.; Bowron, D. T. Small Angle Neutron Scattering from 1-Alkyl-3Methylimidazolium Hexafluorophosphate Ionic Liquids ([Cnmim][PF6], n = 4, 6, and 8). J. Chem. Phys. 2010, 133, 074510−074517. (13) Russina, O.; Triolo, A.; Gontrani, L.; Caminiti, R. Mesoscopic Structural Heterogeneities in Room-Temperature Ionic Liquids. J. Phys. Chem. Lett. 2012, 3, 27−33. (14) Fayer, M. D. Dynamics and Structure of Room Temperature Ionic Liquids. Chem. Phys. Lett. 2014, 616−617, 259−274. (15) Araque, J. C.; Yadav, S. K.; Shadeck, M.; Maroncelli, M.; Margulis, C. J. How Is Diffusion of Neutral and Charged Tracers Related to the Structure and Dynamics of a Room-Temperature Ionic Liquid? Large Deviations from Stokes−Einstein Behavior Explained. J. Phys. Chem. B 2015, 119, 7015−7029. (16) Jin, H.; Li, X.; Maroncelli, M. Heterogeneous Solute Dynamics in Room Temperature Ionic Liquids. J. Phys. Chem. B 2007, 111, 13473−13478. (17) Kimura, Y.; Suda, K.; Shibuya, M.; Yasaka, Y.; Ueno, M. Excitation Wavelength Dependence of the Solvation Dynamics of 4′N,N-Diethylamino-3-Methoxyflavon in Ionic Liquids. Bull. Chem. Soc. Jpn. 2015, 88, 939−945. (18) Castner, E. W.; Wishart, J. F.; Shirota, H. Intermolecular Dynamics, Interactions, and Solvation in Ionic Liquids. Acc. Chem. Res. 2007, 40, 1217−1227. (19) Headley, L. S.; Mukherjee, P.; Anderson, J. L.; Ding, R.; Halder, M.; Armstrong, D. W.; Song, X.; Petrich, J. W. Dynamic Solvation in Imidazolium-Based Ionic Liquids on Short Time Scales. J. Phys. Chem. A 2006, 110, 9549−9554. (20) Samanta, A. Solvation Dynamics in Ionic Liquids: What We Have Learned from the Dynamic Fluorescence Stokes Shift Studies. J. Phys. Chem. Lett. 2010, 1, 1557−1562. (21) Shim, Y.; Jeong, D.; Manjari, S.; Choi, M. Y.; Kim, H. J. Solvation, Solute Rotation and Vibration Relaxation, and ElectronTransfer Reactions in Room-Temperature Ionic Liquids. Acc. Chem. Res. 2007, 40, 1130−1137. (22) Kobrak, M. N. The Chemical Environment of Ionic Liquids: Links between Liquid Structure, Dynamics, and Solvation. Adv. Chem. Phys. 2008, 139, 85−138. (23) Terranova, Z. L.; Corcelli, S. A. On the Mechanism of Solvation Dynamics in Imidazolium-Based Ionic Liquids. J. Phys. Chem. B 2013, 117, 15659−15666. (24) Dynamical Heterogeneities in Glasses, Colloids and Granular Media; Berthier, L., Biroli, G., Bouchaud, J.-P., Cipelletti, L., van Saarloos, W., Eds.; Oxford University Press: Oxford, 2011. (25) Verma, S. D.; Vanden Bout, D. A.; Berg, M. A. When is a Single Molecule Homogeneous? A Multidimensional Answer and Its Application to Molecular Rotation near the Glass Transition. J. Chem. Phys. 2015, 143, 024110. (26) Khurmi, C.; Berg, M. A. Parallels between Multiple PopulationPeriod Transient Spectroscopy (MUPPETS) and Multidimensional Coherence Spectroscopies. J. Chem. Phys. 2008, 129, 064504. (27) Horng, M. L.; Gardecki, J. A.; Papazyan, A.; Maroncelli, M. Subpicosecond Measurements of Polar Solvation Dynamics: Coumarin 153 Revisited. J. Phys. Chem. 1995, 99, 17311−17337. (28) Zwanzig, R. Rate Processes with Dynamical Disorder. Acc. Chem. Res. 1990, 23, 148−152.
monitors this process (Figure 1d). Amplitude and rate exchange are simultaneous. Because diffusive solvation is slow in ionic liquids, many reactions rely on the faster inertial component to facilitate either overall charge transfer or crossing through a chargeseparated transition state. In these cases, the relatively longlived variation in inertial amplitude can directly translate into reaction-rate heterogeneity. In summary, multidimensional correlation methods have discovered and quantified several types of dynamical complexity in a prototypical ionic liquid: heterogeneity in the diffusive solvation time, intermediate rate exchange, and heterogeneity in the inertial amplitude. Each new feature affects the rates of chemical reactions but is usually neglected in conventional solvents. New theoretical approaches are needed for predicting reaction rates in ionic liquids. Corresponding multidimensional experiments do not currently exist but are possible in principle.29 We hope these results will spur their development.
■
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.5b02835. Details of the calculational methods. (PDF)
■
AUTHOR INFORMATION
Corresponding Authors
*S.A.C.: E-mail:
[email protected]. *M.A.B.: E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS We thank Zachary Terranova for providing the trajectories from his previous work. This material is based on work supported by the National Science Foundation under CHE1111530 and CHE-1403027 (M.A.B.) and by the American Chemical Society Petroleum Research Fund under 52648-ND6 (S.A.C.).
■
REFERENCES
(1) Zhang, X.-X.; Liang, M.; Ernsting, N. P.; Maroncelli, M. Complete Solvation Response of Coumarin 153 in Ionic Liquids. J. Phys. Chem. B 2013, 117, 4291−4304. (2) Zhang, X.-X.; Liang, M.; Ernsting, N. P.; Maroncelli, M. Conductivity and Solvation Dynamics in Ionic Liquids. J. Phys. Chem. Lett. 2013, 4, 1205−1210. (3) Shim, Y.; Kim, H. J. Dielectric Relaxation, Ion Conductivity, Solvent Rotation, and Solvation Dynamics in a Room-Temperature Ionic Liquid. J. Phys. Chem. B 2008, 112, 11028−11038. (4) Schmollngruber, M.; Schroder, C.; Steinhauser, O. Dielectric Spectra of Ionic Liquids and Their Conversion to Solvation Dynamics: A Detailed Computational Analysis of Polarizable Systems. Phys. Chem. Chem. Phys. 2014, 16, 10999−11009. (5) Samanta, A. Dynamic Stokes Shift and Excitation Wavelength Dependent Fluorescence of Dipolar Molecules in Room Temperature Ionic Liquids. J. Phys. Chem. B 2006, 110, 13704−13716. (6) Hu, Z.; Margulis, C. J. Heterogeneity in a Room-Temperature Ionic Liquid: Persistent Local Environments and the Red-Edge Effect. Proc. Natl. Acad. Sci. U. S. A. 2006, 103, 831−836. (7) Castner, E. W.; Margulis, C. J.; Maroncelli, M.; Wishart, J. F. Ionic Liquids: Structure and Photochemical Reactions. Annu. Rev. Phys. Chem. 2011, 62, 85−105. 507
DOI: 10.1021/acs.jpclett.5b02835 J. Phys. Chem. Lett. 2016, 7, 504−508
Letter
The Journal of Physical Chemistry Letters (29) Berg, M. A. Multidimensional Incoherent Time-Resolved Spectroscopy and Complex Kinetics. Adv. Chem. Phys. 2012, 150, 1−102.
508
DOI: 10.1021/acs.jpclett.5b02835 J. Phys. Chem. Lett. 2016, 7, 504−508
