THE JOURNAL OF CHEMICAL PHYSICS 139, 164503 (2013)
Stokes shift dynamics of ionic liquids: Solute probe dependence, and effects of self-motion, dielectric relaxation frequency window, and collective intermolecular solvent modes Snehasis Daschakraborty, Tamisra Pal, and Ranjit Biswasa) Department of Chemical, Biological and Macromolecular Sciences, S. N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake, Kolkata 700098, India
(Received 30 July 2013; accepted 30 September 2013; published online 22 October 2013) In this paper we have used a semi-molecular theory for investigating the probe dependence of Stokes shift dynamics in room temperature ionic liquids (ILs) by considering three different but well-known dipolar solvation probes—coumarin 153, trans-4-dimethylamino-4 -cyanostilbene, and 4-aminophthalimide. In addition, effects on polar solvation energy relaxation in ILs of solute motion, frequency coverage (frequency window) accessed by dielectric relaxation measurements and collective IL intermolecular modes (CIMs) at tera-hertz range have been explored. Eleven different ILs have been considered for the above theoretical study. Calculated results show better agreement with the recent (fluorescence up-conversion (FLUPS) + time-correlated single photon counting (TCSPC)) experimental results, particularly at short times, when the CIM contribution to the frequency dependent dielectric function (ε(ω)) is included. This is done via assigning the missing dispersion in an experimental ε(ω) to an IL intermolecular mode at 30 cm−1 . No significant probe dependence has been observed for solvation energy relaxation although the magnitude of dynamic Stokes shift varies with the dipole moment of the excited solute. Calculations using experimental ε(ω) measured with broader frequency window generate solvation response functions closer to experiments. However, average solvation rates predicted by using different ε(ω) for the same IL do not differ appreciably, implying over-all validity of these dielectric relaxation measurements. Results presented here indicate that inclusion of solvent molecularity via wavenumber dependent static correlations and ion dynamic structure factor relaxation improves significantly the comparison between theory and experiments over the continuum model predictions for polar solvation dynamics in these solvents. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4825195] I. INTRODUCTION
Dynamics of ionic liquids (ILs) have been studied intensely in the recent past by using experiments,1–12 theory,13–20 and computer simulations.21–35 A number of interesting features which are markedly different from those found for conventional polar solvents36 have been observed. Recently, Stokes shift dynamics of a number of ILs has been explored using coumarin 153 (C153) as a probe via combining broad-band fluorescence up-conversion (FLUPS) having ∼80 fs resolution and time-correlated single photon counting (TCSPC) providing ∼25 ps resolution.1 This we refer to as C153/(FLUPS + TCSPC) measurements. Solvation response functions reported by these measurements are distinctly bimodal, possessing an ultrafast sub-picosecond Gaussian response (∼10%–40%), followed by a slow stretched exponential component with time constant in the range of ∼0.01–1 ns. It is also found that the ultrafast time constants correlate with ion mass and the slow time constants with IL viscosity, indicating initial ultrafast response arising from solvent inertial motions and the subsequent slow response from the solvent structural relaxation. Solvation response in ILs measured via a) Author to whom correspondence should be addressed. Electronic mail:
[email protected]
0021-9606/2013/139(16)/164503/12/$30.00
three pulse photo echo peak shift (3PEPS) technique using Oxazine-4 has been found to be even faster,10 and fast time constants have been found to be correlated to the anion mass. Interestingly, a recent theoretical study has attempted to explain these 3PEPS results in terms of non-dipolar solute-IL interactions.19 A comparison between the above C153/(FLUPS + TCSPC) data and those obtained previously by using DCS (trans-4-dimethylamino-4 -cyanostilbene) via combining Kerr-gated emission (KGE) and TCSPC techniques2 reveals the following interesting differences: (i) in contrast to C153 data, the DCS/(KGE + TCSPC) solvation response has been found to fit to a sum of a fast exponential and a slow stretched exponential contributions. In both the cases, however, complete detection of the total response has been reported for a number of ILs, although the KGE technique provided a resolution ∼6 times less sharp than the FLUPS technique. This immediately raises the following question: does this different description of the measured initial fast dynamics (Gaussian versus exponential) relate to different solutes used or originate from the difference in experimental techniques employed for detecting the initial fast solvation response? This is an important issue as Gaussian response is connected to non-diffusive solvation mechanism whereas exponential description suggests diffusive solvent
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TABLE I. Time constants and amplitudes of the solvation energy relaxation of three probes, C153, DCS, and Oxazine-4 in three ILs ([Im41 ][BF4 ], [Im41 ][PF6 ], [Im41 ][Tf2 N]) obtained from Stokes shift dynamics1, 2 and 3-PEPS10 measurements. ILs [Im41 ] [BF4 ] [Im41 ] [PF6 ] [Im41 ] [Tf2 N]
Expt.
Probe
(FLUPS +TCSPC) (KGE +TCSPC) 3-PEPS
C153 DCS Oxazine-4
(FLUPS + TCSPC) (KGE + TCSPC) 3-PEPS (FLUPS + TCSPC) (KGE + TCSPC) 3-PEPS
Fita equation
a1
τ 1 (fs)
a2
τ 2 (ps)
β
Gau. + Str. Exp. Exp.+ Str. Exp. Sum of Exp.
0.34 0.19
200 320 110
0.66 0.81
170 130 1.10
0.48 0.31
C153 DCS Oxazine-4
Gau. + Str. Exp. Exp. + Str. Exp. Sum of Exp.
0.33 0.19
240 330 150
0.67 0.81
450 140 2.2
0.50 0.41
C153 DCS Oxazine-4
Gau. + Str. Exp. Exp.+ Str. Exp. Sum of Exp. + Osc.
0.39 0.10
340 740 26
0.61 0.90
190 78 0.54
0.60 0.46
τ 3 (ps)
21
64
“Gau.” refers to Gaussian, “Exp.” to exponential, “Str. Exp.” to stretched exponential, and “Sum of Exp. + Osc.” to sum of exponentials and oscillations as described in the respective references.
a
rearrangement at the early stage. Table I summarizes the fit functions for solvation response functions, and the consequent time constants and amplitudes reported by C153/ (FLUPS + TCSPC) and DCS/(KGE + TCSPC) measurements. Time constants obtained by 3-PEPS measurements are also tabulated here to show the qualitative difference in timescales accessed by 3-PEPS measurements and those by the other two measurements mentioned above. An inspection of these solvation parameters indicates that C153/(FLUPS + TCSPC) measurements not only reveal larger amplitudes of fast relaxation with shorter fast time constants (τ 1 ) than those observed in DCS/(KGE + TCSPC) measurements but also report longer slow time constants (τ 2 ) and larger stretching exponents (β). It may therefore be argued that a technique with sharper resolution detects fast components more accurately and thus the observed differences reflect merely the effects of difference in time resolutions. This argument, however, does not provide any answer to the question that how use of different solutes affects results accessed by these different measurement techniques. More precisely, is there a solute contribution to the observed difference in τ 2 and β values for a given IL reported by these two different measurements? This is one of the questions that we would like to investigate by employing a semi-molecular theory which has been found successful in describing experimental Stokes shift dynamics in various ILs13–15, 17–19 and (IL + polar solvent) mixtures.16 The above experimental results1, 2 further lead to the following debate: if the ultrafast component arises from the solute-IL nearest neighbor interactions then a difference in τ 1 may originate either from a difference in solute size or from a change in IL density.19 This is not the case here as the experiments have been carried out in the same IL using different solutes of very similar sizes.2 The slowing down of τ 2 for C153 by a factor of ∼1.3–3.2 is also surprising as this timescale is believed to originate from the structural relaxation (coupled to viscosity) of the IL under study and thus should not be sensitive to solute size. However, solute motion during measurements can modify the relaxation rate.23, 37–40 It should be mentioned here that solute dependence is not expected for polar solvation response where collective po-
larization density relaxation dominates the solvation energy relaxation.38 Also, within such a framework, difference in solute dipole moments should not modify solvation timescales provided excited solute-induced response remains in the linear response regime.38 C153 and DCS are dipolar molecules with excited state dipole moments larger than those in ground state13–15 and fluorescence Stokes shift dynamics of dipolar ILs (ILs with one of the ions possessing permanent dipole moment) measured by these probes is expected to contain a large solute-IL dipolar contribution. This consideration suggests that Stokes shift dynamics in dipolar ILs would be significantly coupled to frequency (ω) dependent dielectric function, ε(ω), measured in dielectric relaxation (DR) experiments. In such a scenario, one would like to investigate the effects on calculated solvation response of frequency range covered (frequency window) in DR measurements. This is because variations in frequency window have led to differing descriptions of measured ε(ω) (for example, Cole-Cole, Cole-Davidson, or Debye)41–46 producing variations in fitted dielectric dispersion amplitudes (Si ), relaxation time constants (τ i, DR ), and stretching exponents (α and β). For example, relaxation parameters obtained from experimental ε(ω) in 1 MHz to 20 GHz window42 show differences in the above fit parameters from those reported by measurements using a frequency coverage, 0.2–89 GHz.41 When the frequency range has been further broadened up to 3000 GHz,44 contributions to ε(ω) from cation collective IL intermolecular modes (CIMs) appear. DR parameters also differ among experiments even when measured in the same frequency window.42, 43, 45 Since ε(ω) has been shown earlier13–19 to be connected to polar solvation response in ILs as well, the variation in DR parameters will have impacts on predicted shifts and dynamics. As a comparison between the measured solvation response function (S(t)) and that from calculations using varied DR parameters (from various experiments for the same IL) can assist in estimating the spread of the calculated response and test the accuracy of available DR data,41–46 the issue regarding the coupling between IL CIM and solvation response contributions requires careful attention. This is because one would like to assess what roles the low frequency collective solvent modes play for solvation
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energy relaxation of an excited dipolar solute in ILs. These modes are known to be important for ultrafast solvation in polar H-bonded liquids such as water,47–49 and amides,40, 50 and may also couple to the initial fast solvation energy relaxation in ILs. We address this issue here by considering 11 different ILs for which experimental ε(ω) from different groups are available. Simultaneously, probe dependence and effects of frequency window have also been explored, and comparison between theory and experiments has been performed wherever possible. Probes considered are C153, DCS, and 4-AP (4-aminophthalimide), and incorporated in the calculations via their diameters and excited state dipole moments. The ILs are 1-ethyl-3methylimidazolium dicyanamide ([Im21 ][DCA]), 1-ethyl-3methylimidazolium tetrafluoroborate ([Im21 ][BF4 ]), 1-ethyl3-methylimidazolium bis (trifluoromethylsulfonyl) imide ([Im21 ][Tf2 N]), 1-butyl-3- methylimidazolium dicyanamide ([Im41 ][DCA]),1-butyl-3-methylimidazolium tetrafluoroborate ([Im41 ][BF4 ]), 1-butyl-3-methylimidazolium hexafluoro phosphate ([Im41 ][PF6 ]), 1-butyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide ([Im41 ][Tf2 N]), 1-hexyl-3methylimidazolium (trifluoromethylsulfonyl)imide ([Im61 ] [Tf2 N]), 1-methyl-3-octylimidazolium bis(triflluoromethyl 1-butyl-1-methylpyrro sulfonyl)imide ([Im81 ][Tf2 N]), lidinium bis(trifluoromethanesulfonyl)imide ([Pr 41 ][Tf2 N]), and triethylsulfonium bis(trifluoromethylsulfonyl)imide ([Et3 S][Tf2 N] represented here as [S222 ][Tf2 N] for uniform description). Note several of these ILs have been studied before13–15, 19 but we reconsider them again along with several unexplored ones in order to test the generality of the present theory and its predictive ability for the measured Stokes shift dynamics. Comparison between theory and experiments assists in revealing effects of probe, IL CIM, frequency coverage, and solute (probe) motion on solvation energy relaxation in these ILs.
Csd (t) = Esd (t)Esd (0) = 2ρd0
kB T 2π
II. THEORETICAL FORMULATION AND CALCULATION DETAILS
Since the molecular theory used here has already been discussed in detail in Refs. 13–19, we briefly outline the equations necessary for subsequent calculations. Following is the expression13–19, 51 for the position (r), orientation (), and time (t) dependent total fluctuating solvation energy for a mobile dipolar solute with distribution function ρs (r, ;t) Etotal (r, ; t)
= −kB T ρs (r, ; t) +
2
dr d csd (r, ; r , )δρd (r , ;t)
dr csα (r, ; r )δnα (r ; t)
α=1
= Esd (r, ; t) + Esi (r, ; t),
(1)
where csd (r, ; r , ) and csα (r, ; r ) are, respectively, the position and orientation dependent solute dipole-solvent dipole (dipole-dipole) and solute dipole-ion (dipole-ion) direct correlation functions and α denotes the type of ions (cation and anion). δρ d and δnα represent, respectively, fluctuations in dipolar density and ion density from bulk values: δρd (r, ) = ρd (r, ) − ρd0 /4π and δnα (r) = nα (r) − n0α . The solvation energy-energy correlation function averaged over space (r) and orientation () is then written as CE (t) = Csd (t) + Csi (t),
(2)
with Csd (t) and Csi (t), respectively, are the contributions from solute-IL dipole-dipole and dipole-ion interactions, assuming the cross-correlations between fluctuating dipolar and ion densities vanish due to widely different timescales.11–15 The dipole-dipole interaction term can be expressed as
⎡ 2 ∞
10 2 10 10 ⎣ dkk 2 Ssolute (k, t) csd (k) Ssolvent (k; t) 0
∞ +2
⎤
2 11 11 11 dkk 2 Ssolute (k; t) csd (k) Ssolvent (k, t)⎦
(3)
0
and the dipole-ion interaction term is
kB T Csi (t) = Esi (t)Esi (0) = 2 2π
2 α,β
lm where csd (k) represents the wave-number (k) dependent (l, m) component of the static correlation function between the lm (k, t) is the same composolute and dipolar ion, and Ssolvent nent of the orientational dynamic structure factor of the dipolm (k) has been obtained from the mean lar species. While csd
∞ 10 10 10 ion dkk 2 Ssolute (k; t)csα (k)csβ (−k)Sαβ (k, t),
n0α n0β
(4)
0
lm (k, t) has been spherical approximation (MSA) theory, Ssolvent 41–46 ε(ω), summarized in Taobtained by using experimental ble S1 (see the supplementary material52 ). The solute selflm (k, t), has been approximated dynamic structure factor, Ssolute by its diffusive limit where the rotational and translational
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diffusion coefficients for a spherical solute have been obtained from the solution viscosity using the stick boundary condition. The longitudinal component of the wave-number dependent direct correlation function between the dipolar solute and 10 (k), is taken as,13–15 ions, csα
10 csα (k)
4π 4π iμ1 qα sin(krc ) =− 3 kB T ε0 k krc 4π 4π iμ1 qα sin[kσI L (1 + R)/2] =− , 3 kB T ε0 k kσI L (1 + R)/2
(5)
where μ1 is the dipole-moment of the excited dipolar solute, qα is the charge of αth type ion, ε0 is the static dielectric constant of the IL under investigation, and rc is the distance of the closest approach between the solute dipole and , σ IL the ionic species. R denotes the solute-IL size-ratio, σσsolute IL ion being the effective diameter of ions. Sαβ (k, t) is the partial isotropic ion dynamic structure factor, and has been obtained as previously.13–15 Note that neither the spatial heterogeneity of ILs53, 54 nor shape anisotropy of the constituent ions has been incorporated in the calculations of the solute-solvent and solvent-solvent spatial correlations. The heterogeneity aspect enters in the present theory via the use of experimental ε(ω). 10 (k) and ε0 renIn addition, inverse proportionality between csα ders ion-dipole contribution to be more sensitive to small variations in ε0 . Such a dependence of the dipole-dipole contribulm tion is not expected (see Eq. (3)) as csd (k) has been obtained in our calculations by using IL density and dipole moment. Subsequently, the normalized dipolar contribution is given by Ssd (t) =
Csd (t) , Csd (t = 0)
(6)
and that due to dipole-ion interaction Ssi (t) =
Csi (t) . Csi (t = 0)
(7)
Note solvation response function measured in experiments is composed of contributions described by Eqs. (6) and (7). The total solvation response function (Sss ) and the average solvation time (τ ss ) are then calculated ∞ as follows: Sss (t) = (1 − f)Ssd (t) + fSsi (t) and τss = 0 dt Sss (t). Following our earlier works on ILs,13–19 we set f = 0.1. Note also that the dipole-dipole and ion-dipole interaction contributions to the total magnitude of dynamic Stokes shift have been calculated, respectively, by using Eqs. √ (3) and (4)√at t = 0 as follows:ν ttotal = ν tsd + ν tsi = Csd (t = 0) + Csi (t = 0), the superscript “t” denoting quantity associated with time-dependent measurements. In all calculations dipole moment values for excited solutes have been used. Other necessary parameters for calculations are presented in Table S2 (see the supplementary material52 ). Note here that while calculating the normalized solvation response functions we have allowed 10% of the ion dynamics contribution via setting f = 0.1 for Ssi (t), the total dynamic shift is obtained via summing up the solvation energies due to
√ solute-IL dipole-dipole interaction ( Csd (t = 0)) and solute√ IL dipole-ion interaction ( Csi (t = 0)). This is because the shift magnitude is determined by the total solute-medium interaction while the average rate of solvation energy relaxation is determined by the slowest of the available multiple channels. Even though the slowest channel determines the average rate, major part of the energy will relax through the faster of the channels. Within the framework of the present theory, the solvation energy of an excited dipolar solute in these dipolar ILs relaxes via the orientational rearrangement (angular motion) of the dipolar ions and through the centre-of-mass adjustment (translational motion) of the ions. The latter being slower determines the average rate but carries a smaller weight.
III. RESULTS AND DISCUSSIONS A. Dynamic stokes shift magnitude: Probe dependence
First we present the calculated magnitudes of dynamic Stokes shift of C153 and DCS in 11 ILs considered above and compare with experiments. Table II summarizes for C153 t ), solute-IL dipole-dipole the calculated total shift (νtotal t t (νsd ), and dipole-ion (νsi ) interaction contributions using μ1 = 14 D.13, 14 These calculations have been done by using different experimental ε(ω)41–46 and are referred to in the tat ble. The experimental shift values1 (νexp t. ) for the above ILs are also provided in the same table for comparison. The last column of Table II provides the ratio between the magnitudes t t /νexp of calculated and measured shifts (χ = νtotal t. ) for C153 in these ILs. Data in this table clearly indicate a fair agreement between theory and experiments,1 the deviation between these two being confined within ±20%. This is satisfactory if one considers the complexity of these ILs and the simplicity of the calculation scheme employed. In addition, the separated solute-IL dipole-dipole interaction contributes ∼40%–50% of the total shift in all these ILs, signifying a substantial role for the solute-IL dipolar interaction. This is in general agreement with our earlier predictions for a variety of ILs.12–18 Shift values given in parenthesis exhibit sensitivt to ε0 reported by different measurements using ity of νtotal different frequency windows. However, νsit shows relatively more dependence on ε0 because of the inverse ε0 dependence of the relevant solute-ion static correlations. Table S3 (see the supplementary material52 ) summarizes the calculated shifts for DCS probe in these ILs and compares with the available experimental data.2 Note use of μ1 = 28 D produces13, 14 a total shift in 4000 cm−1 range for these ILs. The predictions for other ILs with DCS should be reexamined in experiments as too high (∼4700 cm−1 for [Im21 ][DCA]) and too low values (∼2800 cm−1 for [Im81 ][Tf2 N]) may suggest either partial break-down of the present scheme or refinement of the existing DR data for these ILs. Note the dipoledipole interaction contribution to shift for this solute is also within ∼40%–50% and similar to calculations with C153. Note that use of μ1 = 14 D for DCS predicts, as for C153, dynamic shift value in 2000 cm−1 range which is nearly half of what has been measured in some of these ILs.2 Therefore,
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TABLE II. Calculated dynamic Stokes shift valuesa for C153 in various ionic liquids at room temperature. The shift values in parentheses have been calculated using the static dielectric constants taken from the second references of the corresponding superscripts. ILs [Im21 ][DCA]b [Im21 ][BF4 ]b,c [Im21 ][Tf2 N]c,d [Im41 ][DCA]b [Im41 ][BF4 ]b,e [Im41 ][PF6 ]b [Im41 ][Tf2 N]c,d [Im61 ][Tf2 N]b [Im81 ][Tf2 N]c [Pr 41 ][Tf2 N]f [S222 ][Tf2 N]f
t νsd (cm−1 )
t νsi (cm−1 )
t νtotal (cm−1 )
t −1 νexp t. (cm )
t t χ (νtotal / νexp t. )
876 850(850) 958(958) 821 822(796) 885 850(850) 750 782 800 844
1473 1136(1217) 1042(1037) 1330 1085(1315) 877 870(950) 882 620 1006 939
2349 1986(2067) 2000(1995) 2151 1907(2111) 1762 1720(1800) 1632 1402 1806 1783
2080 2430 2080 2070 2220 2170 2060 1850 1790 2080 2070
1.13 0.82(0.85) 0.96(0.96) 1.04 0.86(0.95) 0.81 0.83(0.87) 0.88 0.78 0.87 0.86
a
Shifts have been calculated using static dielectric constant (ε0 ) from various measurements. From Ref. 41. c From Ref. 42. d From Ref. 46. e From Ref. 44. f From Ref. 45. b
this solute dependence of shift arises from dependence on μ1 as their sizes are equal. B. Stokes shift dynamics: Probe dependence
The solute probe dependence of solvation dynamics in ILs is explored in Fig. 1 where solvation response functions (Sss (t)) calculated for C153, DCS, and 4AP in [Im41 ][PF6 ] at 298 K are shown as a function of time in a double-logarithmic plot. Experimental results obtained via C153/(FLUPS + TCSPC)1 and DCS/(KGE + TCSPC)2 measurements are also presented in the same figure to facilitate comparison between theory and different experimental results for [Im41 ][PF6 ]. Calculations have been done by using experimental ε(ω) reported in Ref. 41 measured in the frequency window, 0.2 ≤ ν / GHz ≤ 89. Interestingly, the predicted response functions are almost indistinguishable from each
FIG. 1. Solute probe dependence of solvation dynamics in ILs. Three differently color coded lines represent calculated solvation response functions (Sss (t)) for three different solutes: C153, DCS, and 4AP in [Im41 ][PF6 ] at 298 K. Experimental results obtained via C153/(FLUPS + TCSPC) and DCS/(KGE + TCSPC) measurements are presented using two different symbols indicated in the inset of the figure.
other and each of them agrees semi-quantitatively with the available two sets of experimental results.1, 2 Evidently, our present calculations with these dipolar solutes do not suggest any significant probe dependence for Stokes shift dynamics in [Im41 ][PF6 ]. The calculated responses for these probes, however, differ slightly at a very later stage when the probe translation starts operating.39 Note diameter of 4AP (σ 4AP = 6.2 Å)13, 14 is ∼20% shorter and excited state dipole mo= 6.5 D)13, 14 ∼50% smaller than that of C153. ment (μ4AP 1 = 28 D and σ DCS For calculations with DCS, we used μDCS 1 = σ 153 . Clearly, variation of solute size not of μ1 that affects the calculated decay of the solvation response functions in this IL. Similar results with these solutes have been obtained for other ILs considered here as well. Measurements with a limited set of solutes in ILs have also reflected similar insensitivity to solute identity.1 Further measurements using solutes with broader variation in diameter and dipole moment are therefore required to confirm the generality of the predicted solute independence of Stokes shift dynamics in ILs. The solute insensitivity predicted above motivates us further to explore Stokes shift dynamics in other ILs and compare with the recent (FLUPS + TCSPC) measurements. Such a comparison is likely to contribute to the general understanding of Stokes shift dynamics in ILs. For this, we chose C153 because relevant (FLUPS + TCSPC) measurements have been done with this solute. Fig. 2 displays the comparison between theory and experiments for [Im41 ][DCA], [Im21 ][DCA], and [S222 ][Tf2 N] which are arbitrarily chosen from the list for which (FLUPS + TCSPC) data are available. Both calculated total response function (Sss (t)) and that originating from the coupling between the dipolar solute and IL orientational polarization relaxation (Ssd (t)) are shown in the same figures. The latter highlights the dominance of the solute-IL dipolar interaction contribution to the predicted dynamics. Note that although we have predicted earlier15 the solvation response functions for C153 in [Im41 ][DCA] and [Im21 ][DCA], non-availability of suitable experimental results did not allow a comparison between
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FIG. 2. Comparison between theoretical and experimental solvation response of C153 probe in three representative ILs. Experimental results are from C153/(FLUPS + TCSPC) measurements. Both calculated total response function (Sss (t)) and that originating from solute-IL dipolar interaction (Ssd (t)) are shown by using color-coded lines. For further specifications, follow panel legends.
predictions and measurements. Moreover, similar calculations for [S222 ][Tf2 N] were not done earlier. Recent (FLUPS + TCSPC) measurements have now made such a comparison possible for these ILs, creating an opportunity for testing the applicability of the current theory for understanding Stokes shift dynamics in a wide variety of ILs. Except for [S222 ][Tf2 N], the calculations have been done by using experimental41 ε(ω) that cover the frequency range, 0.2 ≤ ν / GHz ≤ 89. For [S222 ][Tf2 N], we have used ε(ω) measured in the frequency window,45 1 MHz to 20 GHz.
J. Chem. Phys. 139, 164503 (2013)
It is evident from Fig. 2 that the calculated responses using C153 in [Im21 ][DCA] and [Im41 ][DCA] are much slower than those obtained via (FLUPS + TCSPC) measurements.1 Particularly, the ultrafast response detected in (FLUPS + TCSPC) measurements is completely missed in the relevant calculations. Even consideration of Ssd (t) alone, which predicts relaxation due only to the orientational relaxation of the IL dipoles, does not reproduce the experimental response functions for these two ILs. Addition of solute-IL dipole-ion contribution (Ssi (t)) to the solute-IL dipole-dipole contribution (Ssd (t)) further slows down the total calculated response (Sss (t)). These aspects are described quantitatively in Table S4 (see the supplementary material52 ) where the time constants (τ i ) and amplitudes (ai ) have been obtained by fitting the calculated responses to the following form: Ssx (t) = a1 exp [−t / τ 1 ] + a2 exp [−(t / τ 2 )β ], “x” denoting “d,” “s,”or “i” and β the stretching exponent. The frequency windows for the dielectric relaxation data used in this calculation are given in the last column of Table S4 (see the supplementary material52 ). The calculated responses are neither able to reproduce the ultrafast timescales nor reflect the distinct biphasic character (via oscillation at a sub-picosecond timescale except for [Im41 ][PF6 ]) of the measured responses for these systems. In addition, the predicted response for [S222 ][Tf2 N] has been found to be stretched exponential with a single solvation time constant (see Table S4 of the supplementary material52 ). As found earlier for water47–49 and amides,40, 50 the origin of such a variable agreement between the calculations and experiments arises from the difference between the square of the refractive index (n2D ) and ε∞ , that is, ε∞ − n2D reported by a particular experimental ε(ω). The experimentally measured nD for these ILs lies in the range ∼1.4–1.6 (or n2D ∼ 1.96–2.56)1, 55, 56 and hence ε∞ − n2D is appreciable for several of these ILs (see Table S1 of the supplementary material52 ) indicating a missing faster component inaccessible to the relevant DR measurements.41 This argument can explain the better agreement between theory and experiments for [Im41 ][PF6 ] and the disagreement for other ILs but not for [S222 ][Tf2 N] as ε∞ (ε∞ = ε0 − i Si ) is reported to be nearly 2 for [S222 ][Tf2 N].45 This observation may be linked to the inaccuracy in the experimental ε(ω) of [S222 ][Tf2 N] due to its high conductivity45 and thus better DR measurements are warranted. Such an outcome notwithstanding, the ratio between calculated and measured average solvation times (τss theo. /τsolv exp t. ) summarized in the last column of Table S4 (see the supplementary material52 ) suggests that the present theory provides a somewhat better description of the experimental dynamics than that by the continuum model1 which predicted a dynamics 2–4 times faster than in experiments for most of the ILs considered here. Note that the observed improvement has become possible via allowing molecularity of both the solute and solvent particles through the systematic incorporation of solutesolvent and solvent-solvent static correlations. In addition, the agreement between theory and experiments may be further improved upon by the inclusion of high frequency response in these ILs accounting for the missing dielectric dispersion gap, ε∞ − n2D . Next we present results from such an investigation.
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C. Stokes shift dynamics: Effects of collective IL intermolecular mode (CIM)
Before we present results on the effects of collective IL intermolecular modes on calculated solvation response function, let us have a glimpse on the existing debate on assigning mechanism to these modes in ILs because this has a bearing on the interpretation of short time dynamics in terms of underlying molecular motions. Measurements of ε(ω) in the frequency range of 0.1 ≤ ν / GHz ≤ 3000 indicate contributions from IL collective modes around ∼70–120 cm−1 for several imidazolium ILs which has been assigned to cation CIM (restricted rotation or oscillatory motion).44 Terahertz time-domain spectroscopic measurements involving metallocenium ILs57(a) have suggested that inter-ion vibrations between the cations and anions, and cation CIMs are responsible for the observed dynamics in THz region (1 THz ≈ 30 cm−1 ). These studies have also revealed that the bands appearing in the frequency range of ∼20–50 cm−1 show the maximum amplitude. Optical heterodyne-detected Raman-induced Kerr-effect spectroscopic (OHD-RIKES) study of several imidazolium ILs have attributed the collective IL dynamics in this region to CIM of imidazolium ring at three frequencies around 30, 65, and 100 cm−1 corresponding to different anionic configurations around the cation.57(b) Other OHDRIKES studies have suggested that this intermolecular collective dynamics in this region (
Stokes shift dynamics of ionic liquids: Solute probe dependence, and effects of self-motion, dielectric relaxation frequency window, and collective intermolecular solvent modes Snehasis Daschakraborty, Tamisra Pal, and Ranjit Biswasa) Department of Chemical, Biological and Macromolecular Sciences, S. N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake, Kolkata 700098, India
(Received 30 July 2013; accepted 30 September 2013; published online 22 October 2013) In this paper we have used a semi-molecular theory for investigating the probe dependence of Stokes shift dynamics in room temperature ionic liquids (ILs) by considering three different but well-known dipolar solvation probes—coumarin 153, trans-4-dimethylamino-4 -cyanostilbene, and 4-aminophthalimide. In addition, effects on polar solvation energy relaxation in ILs of solute motion, frequency coverage (frequency window) accessed by dielectric relaxation measurements and collective IL intermolecular modes (CIMs) at tera-hertz range have been explored. Eleven different ILs have been considered for the above theoretical study. Calculated results show better agreement with the recent (fluorescence up-conversion (FLUPS) + time-correlated single photon counting (TCSPC)) experimental results, particularly at short times, when the CIM contribution to the frequency dependent dielectric function (ε(ω)) is included. This is done via assigning the missing dispersion in an experimental ε(ω) to an IL intermolecular mode at 30 cm−1 . No significant probe dependence has been observed for solvation energy relaxation although the magnitude of dynamic Stokes shift varies with the dipole moment of the excited solute. Calculations using experimental ε(ω) measured with broader frequency window generate solvation response functions closer to experiments. However, average solvation rates predicted by using different ε(ω) for the same IL do not differ appreciably, implying over-all validity of these dielectric relaxation measurements. Results presented here indicate that inclusion of solvent molecularity via wavenumber dependent static correlations and ion dynamic structure factor relaxation improves significantly the comparison between theory and experiments over the continuum model predictions for polar solvation dynamics in these solvents. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4825195] I. INTRODUCTION
Dynamics of ionic liquids (ILs) have been studied intensely in the recent past by using experiments,1–12 theory,13–20 and computer simulations.21–35 A number of interesting features which are markedly different from those found for conventional polar solvents36 have been observed. Recently, Stokes shift dynamics of a number of ILs has been explored using coumarin 153 (C153) as a probe via combining broad-band fluorescence up-conversion (FLUPS) having ∼80 fs resolution and time-correlated single photon counting (TCSPC) providing ∼25 ps resolution.1 This we refer to as C153/(FLUPS + TCSPC) measurements. Solvation response functions reported by these measurements are distinctly bimodal, possessing an ultrafast sub-picosecond Gaussian response (∼10%–40%), followed by a slow stretched exponential component with time constant in the range of ∼0.01–1 ns. It is also found that the ultrafast time constants correlate with ion mass and the slow time constants with IL viscosity, indicating initial ultrafast response arising from solvent inertial motions and the subsequent slow response from the solvent structural relaxation. Solvation response in ILs measured via a) Author to whom correspondence should be addressed. Electronic mail:
[email protected]
0021-9606/2013/139(16)/164503/12/$30.00
three pulse photo echo peak shift (3PEPS) technique using Oxazine-4 has been found to be even faster,10 and fast time constants have been found to be correlated to the anion mass. Interestingly, a recent theoretical study has attempted to explain these 3PEPS results in terms of non-dipolar solute-IL interactions.19 A comparison between the above C153/(FLUPS + TCSPC) data and those obtained previously by using DCS (trans-4-dimethylamino-4 -cyanostilbene) via combining Kerr-gated emission (KGE) and TCSPC techniques2 reveals the following interesting differences: (i) in contrast to C153 data, the DCS/(KGE + TCSPC) solvation response has been found to fit to a sum of a fast exponential and a slow stretched exponential contributions. In both the cases, however, complete detection of the total response has been reported for a number of ILs, although the KGE technique provided a resolution ∼6 times less sharp than the FLUPS technique. This immediately raises the following question: does this different description of the measured initial fast dynamics (Gaussian versus exponential) relate to different solutes used or originate from the difference in experimental techniques employed for detecting the initial fast solvation response? This is an important issue as Gaussian response is connected to non-diffusive solvation mechanism whereas exponential description suggests diffusive solvent
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TABLE I. Time constants and amplitudes of the solvation energy relaxation of three probes, C153, DCS, and Oxazine-4 in three ILs ([Im41 ][BF4 ], [Im41 ][PF6 ], [Im41 ][Tf2 N]) obtained from Stokes shift dynamics1, 2 and 3-PEPS10 measurements. ILs [Im41 ] [BF4 ] [Im41 ] [PF6 ] [Im41 ] [Tf2 N]
Expt.
Probe
(FLUPS +TCSPC) (KGE +TCSPC) 3-PEPS
C153 DCS Oxazine-4
(FLUPS + TCSPC) (KGE + TCSPC) 3-PEPS (FLUPS + TCSPC) (KGE + TCSPC) 3-PEPS
Fita equation
a1
τ 1 (fs)
a2
τ 2 (ps)
β
Gau. + Str. Exp. Exp.+ Str. Exp. Sum of Exp.
0.34 0.19
200 320 110
0.66 0.81
170 130 1.10
0.48 0.31
C153 DCS Oxazine-4
Gau. + Str. Exp. Exp. + Str. Exp. Sum of Exp.
0.33 0.19
240 330 150
0.67 0.81
450 140 2.2
0.50 0.41
C153 DCS Oxazine-4
Gau. + Str. Exp. Exp.+ Str. Exp. Sum of Exp. + Osc.
0.39 0.10
340 740 26
0.61 0.90
190 78 0.54
0.60 0.46
τ 3 (ps)
21
64
“Gau.” refers to Gaussian, “Exp.” to exponential, “Str. Exp.” to stretched exponential, and “Sum of Exp. + Osc.” to sum of exponentials and oscillations as described in the respective references.
a
rearrangement at the early stage. Table I summarizes the fit functions for solvation response functions, and the consequent time constants and amplitudes reported by C153/ (FLUPS + TCSPC) and DCS/(KGE + TCSPC) measurements. Time constants obtained by 3-PEPS measurements are also tabulated here to show the qualitative difference in timescales accessed by 3-PEPS measurements and those by the other two measurements mentioned above. An inspection of these solvation parameters indicates that C153/(FLUPS + TCSPC) measurements not only reveal larger amplitudes of fast relaxation with shorter fast time constants (τ 1 ) than those observed in DCS/(KGE + TCSPC) measurements but also report longer slow time constants (τ 2 ) and larger stretching exponents (β). It may therefore be argued that a technique with sharper resolution detects fast components more accurately and thus the observed differences reflect merely the effects of difference in time resolutions. This argument, however, does not provide any answer to the question that how use of different solutes affects results accessed by these different measurement techniques. More precisely, is there a solute contribution to the observed difference in τ 2 and β values for a given IL reported by these two different measurements? This is one of the questions that we would like to investigate by employing a semi-molecular theory which has been found successful in describing experimental Stokes shift dynamics in various ILs13–15, 17–19 and (IL + polar solvent) mixtures.16 The above experimental results1, 2 further lead to the following debate: if the ultrafast component arises from the solute-IL nearest neighbor interactions then a difference in τ 1 may originate either from a difference in solute size or from a change in IL density.19 This is not the case here as the experiments have been carried out in the same IL using different solutes of very similar sizes.2 The slowing down of τ 2 for C153 by a factor of ∼1.3–3.2 is also surprising as this timescale is believed to originate from the structural relaxation (coupled to viscosity) of the IL under study and thus should not be sensitive to solute size. However, solute motion during measurements can modify the relaxation rate.23, 37–40 It should be mentioned here that solute dependence is not expected for polar solvation response where collective po-
larization density relaxation dominates the solvation energy relaxation.38 Also, within such a framework, difference in solute dipole moments should not modify solvation timescales provided excited solute-induced response remains in the linear response regime.38 C153 and DCS are dipolar molecules with excited state dipole moments larger than those in ground state13–15 and fluorescence Stokes shift dynamics of dipolar ILs (ILs with one of the ions possessing permanent dipole moment) measured by these probes is expected to contain a large solute-IL dipolar contribution. This consideration suggests that Stokes shift dynamics in dipolar ILs would be significantly coupled to frequency (ω) dependent dielectric function, ε(ω), measured in dielectric relaxation (DR) experiments. In such a scenario, one would like to investigate the effects on calculated solvation response of frequency range covered (frequency window) in DR measurements. This is because variations in frequency window have led to differing descriptions of measured ε(ω) (for example, Cole-Cole, Cole-Davidson, or Debye)41–46 producing variations in fitted dielectric dispersion amplitudes (Si ), relaxation time constants (τ i, DR ), and stretching exponents (α and β). For example, relaxation parameters obtained from experimental ε(ω) in 1 MHz to 20 GHz window42 show differences in the above fit parameters from those reported by measurements using a frequency coverage, 0.2–89 GHz.41 When the frequency range has been further broadened up to 3000 GHz,44 contributions to ε(ω) from cation collective IL intermolecular modes (CIMs) appear. DR parameters also differ among experiments even when measured in the same frequency window.42, 43, 45 Since ε(ω) has been shown earlier13–19 to be connected to polar solvation response in ILs as well, the variation in DR parameters will have impacts on predicted shifts and dynamics. As a comparison between the measured solvation response function (S(t)) and that from calculations using varied DR parameters (from various experiments for the same IL) can assist in estimating the spread of the calculated response and test the accuracy of available DR data,41–46 the issue regarding the coupling between IL CIM and solvation response contributions requires careful attention. This is because one would like to assess what roles the low frequency collective solvent modes play for solvation
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energy relaxation of an excited dipolar solute in ILs. These modes are known to be important for ultrafast solvation in polar H-bonded liquids such as water,47–49 and amides,40, 50 and may also couple to the initial fast solvation energy relaxation in ILs. We address this issue here by considering 11 different ILs for which experimental ε(ω) from different groups are available. Simultaneously, probe dependence and effects of frequency window have also been explored, and comparison between theory and experiments has been performed wherever possible. Probes considered are C153, DCS, and 4-AP (4-aminophthalimide), and incorporated in the calculations via their diameters and excited state dipole moments. The ILs are 1-ethyl-3methylimidazolium dicyanamide ([Im21 ][DCA]), 1-ethyl-3methylimidazolium tetrafluoroborate ([Im21 ][BF4 ]), 1-ethyl3-methylimidazolium bis (trifluoromethylsulfonyl) imide ([Im21 ][Tf2 N]), 1-butyl-3- methylimidazolium dicyanamide ([Im41 ][DCA]),1-butyl-3-methylimidazolium tetrafluoroborate ([Im41 ][BF4 ]), 1-butyl-3-methylimidazolium hexafluoro phosphate ([Im41 ][PF6 ]), 1-butyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide ([Im41 ][Tf2 N]), 1-hexyl-3methylimidazolium (trifluoromethylsulfonyl)imide ([Im61 ] [Tf2 N]), 1-methyl-3-octylimidazolium bis(triflluoromethyl 1-butyl-1-methylpyrro sulfonyl)imide ([Im81 ][Tf2 N]), lidinium bis(trifluoromethanesulfonyl)imide ([Pr 41 ][Tf2 N]), and triethylsulfonium bis(trifluoromethylsulfonyl)imide ([Et3 S][Tf2 N] represented here as [S222 ][Tf2 N] for uniform description). Note several of these ILs have been studied before13–15, 19 but we reconsider them again along with several unexplored ones in order to test the generality of the present theory and its predictive ability for the measured Stokes shift dynamics. Comparison between theory and experiments assists in revealing effects of probe, IL CIM, frequency coverage, and solute (probe) motion on solvation energy relaxation in these ILs.
Csd (t) = Esd (t)Esd (0) = 2ρd0
kB T 2π
II. THEORETICAL FORMULATION AND CALCULATION DETAILS
Since the molecular theory used here has already been discussed in detail in Refs. 13–19, we briefly outline the equations necessary for subsequent calculations. Following is the expression13–19, 51 for the position (r), orientation (), and time (t) dependent total fluctuating solvation energy for a mobile dipolar solute with distribution function ρs (r, ;t) Etotal (r, ; t)
= −kB T ρs (r, ; t) +
2
dr d csd (r, ; r , )δρd (r , ;t)
dr csα (r, ; r )δnα (r ; t)
α=1
= Esd (r, ; t) + Esi (r, ; t),
(1)
where csd (r, ; r , ) and csα (r, ; r ) are, respectively, the position and orientation dependent solute dipole-solvent dipole (dipole-dipole) and solute dipole-ion (dipole-ion) direct correlation functions and α denotes the type of ions (cation and anion). δρ d and δnα represent, respectively, fluctuations in dipolar density and ion density from bulk values: δρd (r, ) = ρd (r, ) − ρd0 /4π and δnα (r) = nα (r) − n0α . The solvation energy-energy correlation function averaged over space (r) and orientation () is then written as CE (t) = Csd (t) + Csi (t),
(2)
with Csd (t) and Csi (t), respectively, are the contributions from solute-IL dipole-dipole and dipole-ion interactions, assuming the cross-correlations between fluctuating dipolar and ion densities vanish due to widely different timescales.11–15 The dipole-dipole interaction term can be expressed as
⎡ 2 ∞
10 2 10 10 ⎣ dkk 2 Ssolute (k, t) csd (k) Ssolvent (k; t) 0
∞ +2
⎤
2 11 11 11 dkk 2 Ssolute (k; t) csd (k) Ssolvent (k, t)⎦
(3)
0
and the dipole-ion interaction term is
kB T Csi (t) = Esi (t)Esi (0) = 2 2π
2 α,β
lm where csd (k) represents the wave-number (k) dependent (l, m) component of the static correlation function between the lm (k, t) is the same composolute and dipolar ion, and Ssolvent nent of the orientational dynamic structure factor of the dipolm (k) has been obtained from the mean lar species. While csd
∞ 10 10 10 ion dkk 2 Ssolute (k; t)csα (k)csβ (−k)Sαβ (k, t),
n0α n0β
(4)
0
lm (k, t) has been spherical approximation (MSA) theory, Ssolvent 41–46 ε(ω), summarized in Taobtained by using experimental ble S1 (see the supplementary material52 ). The solute selflm (k, t), has been approximated dynamic structure factor, Ssolute by its diffusive limit where the rotational and translational
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diffusion coefficients for a spherical solute have been obtained from the solution viscosity using the stick boundary condition. The longitudinal component of the wave-number dependent direct correlation function between the dipolar solute and 10 (k), is taken as,13–15 ions, csα
10 csα (k)
4π 4π iμ1 qα sin(krc ) =− 3 kB T ε0 k krc 4π 4π iμ1 qα sin[kσI L (1 + R)/2] =− , 3 kB T ε0 k kσI L (1 + R)/2
(5)
where μ1 is the dipole-moment of the excited dipolar solute, qα is the charge of αth type ion, ε0 is the static dielectric constant of the IL under investigation, and rc is the distance of the closest approach between the solute dipole and , σ IL the ionic species. R denotes the solute-IL size-ratio, σσsolute IL ion being the effective diameter of ions. Sαβ (k, t) is the partial isotropic ion dynamic structure factor, and has been obtained as previously.13–15 Note that neither the spatial heterogeneity of ILs53, 54 nor shape anisotropy of the constituent ions has been incorporated in the calculations of the solute-solvent and solvent-solvent spatial correlations. The heterogeneity aspect enters in the present theory via the use of experimental ε(ω). 10 (k) and ε0 renIn addition, inverse proportionality between csα ders ion-dipole contribution to be more sensitive to small variations in ε0 . Such a dependence of the dipole-dipole contribulm tion is not expected (see Eq. (3)) as csd (k) has been obtained in our calculations by using IL density and dipole moment. Subsequently, the normalized dipolar contribution is given by Ssd (t) =
Csd (t) , Csd (t = 0)
(6)
and that due to dipole-ion interaction Ssi (t) =
Csi (t) . Csi (t = 0)
(7)
Note solvation response function measured in experiments is composed of contributions described by Eqs. (6) and (7). The total solvation response function (Sss ) and the average solvation time (τ ss ) are then calculated ∞ as follows: Sss (t) = (1 − f)Ssd (t) + fSsi (t) and τss = 0 dt Sss (t). Following our earlier works on ILs,13–19 we set f = 0.1. Note also that the dipole-dipole and ion-dipole interaction contributions to the total magnitude of dynamic Stokes shift have been calculated, respectively, by using Eqs. √ (3) and (4)√at t = 0 as follows:ν ttotal = ν tsd + ν tsi = Csd (t = 0) + Csi (t = 0), the superscript “t” denoting quantity associated with time-dependent measurements. In all calculations dipole moment values for excited solutes have been used. Other necessary parameters for calculations are presented in Table S2 (see the supplementary material52 ). Note here that while calculating the normalized solvation response functions we have allowed 10% of the ion dynamics contribution via setting f = 0.1 for Ssi (t), the total dynamic shift is obtained via summing up the solvation energies due to
√ solute-IL dipole-dipole interaction ( Csd (t = 0)) and solute√ IL dipole-ion interaction ( Csi (t = 0)). This is because the shift magnitude is determined by the total solute-medium interaction while the average rate of solvation energy relaxation is determined by the slowest of the available multiple channels. Even though the slowest channel determines the average rate, major part of the energy will relax through the faster of the channels. Within the framework of the present theory, the solvation energy of an excited dipolar solute in these dipolar ILs relaxes via the orientational rearrangement (angular motion) of the dipolar ions and through the centre-of-mass adjustment (translational motion) of the ions. The latter being slower determines the average rate but carries a smaller weight.
III. RESULTS AND DISCUSSIONS A. Dynamic stokes shift magnitude: Probe dependence
First we present the calculated magnitudes of dynamic Stokes shift of C153 and DCS in 11 ILs considered above and compare with experiments. Table II summarizes for C153 t ), solute-IL dipole-dipole the calculated total shift (νtotal t t (νsd ), and dipole-ion (νsi ) interaction contributions using μ1 = 14 D.13, 14 These calculations have been done by using different experimental ε(ω)41–46 and are referred to in the tat ble. The experimental shift values1 (νexp t. ) for the above ILs are also provided in the same table for comparison. The last column of Table II provides the ratio between the magnitudes t t /νexp of calculated and measured shifts (χ = νtotal t. ) for C153 in these ILs. Data in this table clearly indicate a fair agreement between theory and experiments,1 the deviation between these two being confined within ±20%. This is satisfactory if one considers the complexity of these ILs and the simplicity of the calculation scheme employed. In addition, the separated solute-IL dipole-dipole interaction contributes ∼40%–50% of the total shift in all these ILs, signifying a substantial role for the solute-IL dipolar interaction. This is in general agreement with our earlier predictions for a variety of ILs.12–18 Shift values given in parenthesis exhibit sensitivt to ε0 reported by different measurements using ity of νtotal different frequency windows. However, νsit shows relatively more dependence on ε0 because of the inverse ε0 dependence of the relevant solute-ion static correlations. Table S3 (see the supplementary material52 ) summarizes the calculated shifts for DCS probe in these ILs and compares with the available experimental data.2 Note use of μ1 = 28 D produces13, 14 a total shift in 4000 cm−1 range for these ILs. The predictions for other ILs with DCS should be reexamined in experiments as too high (∼4700 cm−1 for [Im21 ][DCA]) and too low values (∼2800 cm−1 for [Im81 ][Tf2 N]) may suggest either partial break-down of the present scheme or refinement of the existing DR data for these ILs. Note the dipoledipole interaction contribution to shift for this solute is also within ∼40%–50% and similar to calculations with C153. Note that use of μ1 = 14 D for DCS predicts, as for C153, dynamic shift value in 2000 cm−1 range which is nearly half of what has been measured in some of these ILs.2 Therefore,
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TABLE II. Calculated dynamic Stokes shift valuesa for C153 in various ionic liquids at room temperature. The shift values in parentheses have been calculated using the static dielectric constants taken from the second references of the corresponding superscripts. ILs [Im21 ][DCA]b [Im21 ][BF4 ]b,c [Im21 ][Tf2 N]c,d [Im41 ][DCA]b [Im41 ][BF4 ]b,e [Im41 ][PF6 ]b [Im41 ][Tf2 N]c,d [Im61 ][Tf2 N]b [Im81 ][Tf2 N]c [Pr 41 ][Tf2 N]f [S222 ][Tf2 N]f
t νsd (cm−1 )
t νsi (cm−1 )
t νtotal (cm−1 )
t −1 νexp t. (cm )
t t χ (νtotal / νexp t. )
876 850(850) 958(958) 821 822(796) 885 850(850) 750 782 800 844
1473 1136(1217) 1042(1037) 1330 1085(1315) 877 870(950) 882 620 1006 939
2349 1986(2067) 2000(1995) 2151 1907(2111) 1762 1720(1800) 1632 1402 1806 1783
2080 2430 2080 2070 2220 2170 2060 1850 1790 2080 2070
1.13 0.82(0.85) 0.96(0.96) 1.04 0.86(0.95) 0.81 0.83(0.87) 0.88 0.78 0.87 0.86
a
Shifts have been calculated using static dielectric constant (ε0 ) from various measurements. From Ref. 41. c From Ref. 42. d From Ref. 46. e From Ref. 44. f From Ref. 45. b
this solute dependence of shift arises from dependence on μ1 as their sizes are equal. B. Stokes shift dynamics: Probe dependence
The solute probe dependence of solvation dynamics in ILs is explored in Fig. 1 where solvation response functions (Sss (t)) calculated for C153, DCS, and 4AP in [Im41 ][PF6 ] at 298 K are shown as a function of time in a double-logarithmic plot. Experimental results obtained via C153/(FLUPS + TCSPC)1 and DCS/(KGE + TCSPC)2 measurements are also presented in the same figure to facilitate comparison between theory and different experimental results for [Im41 ][PF6 ]. Calculations have been done by using experimental ε(ω) reported in Ref. 41 measured in the frequency window, 0.2 ≤ ν / GHz ≤ 89. Interestingly, the predicted response functions are almost indistinguishable from each
FIG. 1. Solute probe dependence of solvation dynamics in ILs. Three differently color coded lines represent calculated solvation response functions (Sss (t)) for three different solutes: C153, DCS, and 4AP in [Im41 ][PF6 ] at 298 K. Experimental results obtained via C153/(FLUPS + TCSPC) and DCS/(KGE + TCSPC) measurements are presented using two different symbols indicated in the inset of the figure.
other and each of them agrees semi-quantitatively with the available two sets of experimental results.1, 2 Evidently, our present calculations with these dipolar solutes do not suggest any significant probe dependence for Stokes shift dynamics in [Im41 ][PF6 ]. The calculated responses for these probes, however, differ slightly at a very later stage when the probe translation starts operating.39 Note diameter of 4AP (σ 4AP = 6.2 Å)13, 14 is ∼20% shorter and excited state dipole mo= 6.5 D)13, 14 ∼50% smaller than that of C153. ment (μ4AP 1 = 28 D and σ DCS For calculations with DCS, we used μDCS 1 = σ 153 . Clearly, variation of solute size not of μ1 that affects the calculated decay of the solvation response functions in this IL. Similar results with these solutes have been obtained for other ILs considered here as well. Measurements with a limited set of solutes in ILs have also reflected similar insensitivity to solute identity.1 Further measurements using solutes with broader variation in diameter and dipole moment are therefore required to confirm the generality of the predicted solute independence of Stokes shift dynamics in ILs. The solute insensitivity predicted above motivates us further to explore Stokes shift dynamics in other ILs and compare with the recent (FLUPS + TCSPC) measurements. Such a comparison is likely to contribute to the general understanding of Stokes shift dynamics in ILs. For this, we chose C153 because relevant (FLUPS + TCSPC) measurements have been done with this solute. Fig. 2 displays the comparison between theory and experiments for [Im41 ][DCA], [Im21 ][DCA], and [S222 ][Tf2 N] which are arbitrarily chosen from the list for which (FLUPS + TCSPC) data are available. Both calculated total response function (Sss (t)) and that originating from the coupling between the dipolar solute and IL orientational polarization relaxation (Ssd (t)) are shown in the same figures. The latter highlights the dominance of the solute-IL dipolar interaction contribution to the predicted dynamics. Note that although we have predicted earlier15 the solvation response functions for C153 in [Im41 ][DCA] and [Im21 ][DCA], non-availability of suitable experimental results did not allow a comparison between
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FIG. 2. Comparison between theoretical and experimental solvation response of C153 probe in three representative ILs. Experimental results are from C153/(FLUPS + TCSPC) measurements. Both calculated total response function (Sss (t)) and that originating from solute-IL dipolar interaction (Ssd (t)) are shown by using color-coded lines. For further specifications, follow panel legends.
predictions and measurements. Moreover, similar calculations for [S222 ][Tf2 N] were not done earlier. Recent (FLUPS + TCSPC) measurements have now made such a comparison possible for these ILs, creating an opportunity for testing the applicability of the current theory for understanding Stokes shift dynamics in a wide variety of ILs. Except for [S222 ][Tf2 N], the calculations have been done by using experimental41 ε(ω) that cover the frequency range, 0.2 ≤ ν / GHz ≤ 89. For [S222 ][Tf2 N], we have used ε(ω) measured in the frequency window,45 1 MHz to 20 GHz.
J. Chem. Phys. 139, 164503 (2013)
It is evident from Fig. 2 that the calculated responses using C153 in [Im21 ][DCA] and [Im41 ][DCA] are much slower than those obtained via (FLUPS + TCSPC) measurements.1 Particularly, the ultrafast response detected in (FLUPS + TCSPC) measurements is completely missed in the relevant calculations. Even consideration of Ssd (t) alone, which predicts relaxation due only to the orientational relaxation of the IL dipoles, does not reproduce the experimental response functions for these two ILs. Addition of solute-IL dipole-ion contribution (Ssi (t)) to the solute-IL dipole-dipole contribution (Ssd (t)) further slows down the total calculated response (Sss (t)). These aspects are described quantitatively in Table S4 (see the supplementary material52 ) where the time constants (τ i ) and amplitudes (ai ) have been obtained by fitting the calculated responses to the following form: Ssx (t) = a1 exp [−t / τ 1 ] + a2 exp [−(t / τ 2 )β ], “x” denoting “d,” “s,”or “i” and β the stretching exponent. The frequency windows for the dielectric relaxation data used in this calculation are given in the last column of Table S4 (see the supplementary material52 ). The calculated responses are neither able to reproduce the ultrafast timescales nor reflect the distinct biphasic character (via oscillation at a sub-picosecond timescale except for [Im41 ][PF6 ]) of the measured responses for these systems. In addition, the predicted response for [S222 ][Tf2 N] has been found to be stretched exponential with a single solvation time constant (see Table S4 of the supplementary material52 ). As found earlier for water47–49 and amides,40, 50 the origin of such a variable agreement between the calculations and experiments arises from the difference between the square of the refractive index (n2D ) and ε∞ , that is, ε∞ − n2D reported by a particular experimental ε(ω). The experimentally measured nD for these ILs lies in the range ∼1.4–1.6 (or n2D ∼ 1.96–2.56)1, 55, 56 and hence ε∞ − n2D is appreciable for several of these ILs (see Table S1 of the supplementary material52 ) indicating a missing faster component inaccessible to the relevant DR measurements.41 This argument can explain the better agreement between theory and experiments for [Im41 ][PF6 ] and the disagreement for other ILs but not for [S222 ][Tf2 N] as ε∞ (ε∞ = ε0 − i Si ) is reported to be nearly 2 for [S222 ][Tf2 N].45 This observation may be linked to the inaccuracy in the experimental ε(ω) of [S222 ][Tf2 N] due to its high conductivity45 and thus better DR measurements are warranted. Such an outcome notwithstanding, the ratio between calculated and measured average solvation times (τss theo. /τsolv exp t. ) summarized in the last column of Table S4 (see the supplementary material52 ) suggests that the present theory provides a somewhat better description of the experimental dynamics than that by the continuum model1 which predicted a dynamics 2–4 times faster than in experiments for most of the ILs considered here. Note that the observed improvement has become possible via allowing molecularity of both the solute and solvent particles through the systematic incorporation of solutesolvent and solvent-solvent static correlations. In addition, the agreement between theory and experiments may be further improved upon by the inclusion of high frequency response in these ILs accounting for the missing dielectric dispersion gap, ε∞ − n2D . Next we present results from such an investigation.
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C. Stokes shift dynamics: Effects of collective IL intermolecular mode (CIM)
Before we present results on the effects of collective IL intermolecular modes on calculated solvation response function, let us have a glimpse on the existing debate on assigning mechanism to these modes in ILs because this has a bearing on the interpretation of short time dynamics in terms of underlying molecular motions. Measurements of ε(ω) in the frequency range of 0.1 ≤ ν / GHz ≤ 3000 indicate contributions from IL collective modes around ∼70–120 cm−1 for several imidazolium ILs which has been assigned to cation CIM (restricted rotation or oscillatory motion).44 Terahertz time-domain spectroscopic measurements involving metallocenium ILs57(a) have suggested that inter-ion vibrations between the cations and anions, and cation CIMs are responsible for the observed dynamics in THz region (1 THz ≈ 30 cm−1 ). These studies have also revealed that the bands appearing in the frequency range of ∼20–50 cm−1 show the maximum amplitude. Optical heterodyne-detected Raman-induced Kerr-effect spectroscopic (OHD-RIKES) study of several imidazolium ILs have attributed the collective IL dynamics in this region to CIM of imidazolium ring at three frequencies around 30, 65, and 100 cm−1 corresponding to different anionic configurations around the cation.57(b) Other OHDRIKES studies have suggested that this intermolecular collective dynamics in this region (
