J. Phys. Chem. 1995, 99,
17311-17337
17311
ARTICLES Subpicosecond Measurements of Polar Solvation Dynamics: Coumarin 153 Revisited M. L. Homg, J. A. Gardecki, A. Papazyan,7 and M. Maroncelli* * Department of Chemistry, 152 Davey Laboratory, The Pennsylvania State University, University Park, Pennsylvania 16802 Received: June 15, 1995; In Final Form: September 5, 1995®
Time-resolved emission measurements of the solute coumarin 153 (C153) are used to probe the time dependence of solvation in 24 common solvents at room temperature. Significant improvements in experimental time resolution (MOO fs instrument response) as well as corresponding improvements in analysis methods provide confidence that all of the spectral evolution (including both the inertial and the diffusive parts of the response) are observed in these measurements. Extensive data concerning the steady-state solvatochromism of C153, coupled to an examination of the effects of vibrational relaxation, further demonstrate that the spectral dynamics being observed accurately monitor the dynamics of nonspecific solvation. Comparisons to theoretical predictions show that models based on the dielectric response of the pure solvent provide a semiquantitative understanding of the dynamics observed.
I. Introduction
such continuum-based predictions have served as a useful benchmark, it is not clear that they should be very accurate when it comes to describing the behavior of molecular solutes. Thus, in more recent years a number of molecularly based theories of polar solvation dynamics have emerged. Whereas most of these more recent theories still rely on the solvent’s dielectric response as empirical input, they account in various ways for the effects of the discrete nature of the solvent. They therefore predict departures from the purely continuum solvation models. Com55’58 puter simulations have tended to confirm these predictions,38 from behavior also show that the continuum but they departures need not be large.50,56
Understanding the time dependence of solvation has been the subject of a wide variety of studies over the past decade.1-8 The primary focus of these studies has been the solvation of ionic and dipolar solutes by strongly dipolar solvents, since it is the dynamics of such “polar solvation” that is expected to be important in determining the kinetics of solution-phase chargetransfer reactions.9-12 (Interesting results are also beginning to appear on solvation dynamics in various “nonpolar” solutesolvent combinations,13-16 but this work is still in the earliest stages.) In the polar solvation case, a combination of results from experimental,3-517-37 simulation,3’8,38-52 and theoretical studies3'6'7'53-59 provides a relatively unified picture of the basic dynamics involved. Subsequent to a perturbation of the charge distribution of a solute (such as occurs upon electronic excitation), solvent molecules reorganize so as to lower the free energy of solvation. For a polar solute in a strongly dipolar solvent, the time-dependent interactions between the electrostatic field of the solute’s charge distribution and the dipoles of the surrounding solvent are the main determinant of this energy relaxation. Molecular dynamics simulations have also shown that the relaxation involved arises primarily from reorientational motions of solvent molecules.16'39’40’44,50’51 The dynamics of polar solvation are therefore closely related to the dielectric response of the solvent, which measures the polarization response of a fluid due to collective dipole reorientation. This connection was exploited over 25 years ago in the pioneering theoretical treatments of dynamic solvation effects on electronic spectra by Bakshiev, Mazurenko, and co-workers.61 These authors employed a spherical cavity solute/dielectric continuum solvent description of solvation comparable to those commonly used in modeling solvatochromic shifts. Application of such models leads to the prediction that very fast (~ 1 ps) solvation times are to be expected in room temperature solvents.3-5 While
One of the qualitative features of polar solvation dynamics first brought to light by computer simulations is the prominent role played by inertial solvent motions.3 29’56 A large number of studies have now shown that in small-molecule solvents like acetonitrile,50’51 water,42-46 and methyl chloride62 well over half of the solvation energy relaxes via an ultrafast (
-1
(3_9)
7(0
=
n
1/2
4ln(2)
h(t) A(0 exp|
.4
ln(2)
(3.13)
IV. Results and Discussion IVA. Steady-State Spectroscopy and the Solvatochromism of C153. Representative steady-state absorption and emission spectra of C153 are shown in Figure 6. In the nonpolar solvent 2-methylbutane (our reference for “time-zero” analysis), both the absorption and emission bands exhibit vibronic structure with a spacing of ~ 1300—1400 cm-1. Analysis in terms of displaced harmonic oscillators shows that more than two Franck—Condon-active modes contribute to the complex band shape observed.92 The absorption and emission spectra show approximate mirror symmetry, but the vibronic spacing in absorption is larger than that in emission. This near symmetry, as well as excitation anisotropy measurements in low-temperature propylene glycol,93 provides convincing evidence that there is only a single excited electronic state contributing to the absorption and emission spectra in the region shown here. As illustrated in Figure 6, both the absorption and emission bands red-shift as a function of increasing solvent polarity. The vibronic structure present in the 2-methylbutane spectrum is obscured in even weakly polar solvents such as diethyl ether. For this reason, a log-normal line shape provides a very good approximation to the absorption and emission line shapes observed in nearly all solvents. We note that, in the case of absorption, the changes in the spectra can be accurately reproduced using the inhomogeneous spectral model described in section IH.A. That is, for a given polar solvent, the absorption line shape is well represented by a convolution of the reference 2-methylbutane spectrum with a Gaussian function. The width
17318
Homg et al.
J. Phys. Chem., Vol. 99, No. 48, 1995
25
20.
15. O'
(
103
cm3
)
6. Representative steady-state absorption (top) and emission spectra (bottom) of Cl53 in a range of solvents of different polarity. The solvents shown are 2-methylbutane, diethyl ether, ethyl acetate, acetone, and methanol in order of decreasing peak frequency. The arrow in the top panel indicates the excitation wavelength typically employed in the time-resolved experiments.
Figure
interactions do not play a significant role in the spectral shifts of C153. Thus, as will be discussed below, the dynamics observed in hydrogen-bonding solvents do not appear distinct from those in aprotic solvents (see also refs 3 and 4). Furthermore, comparable solvation dynamics in 1-propanol have been observed for a wide variety of probe solutes including C153.75 Such similarity would be unlikely if specific solutesolvent interactions were important. Finally, the results of molecular dynamics simulations of C153 in methanol50 show that although there is hydrogen bonding between the solute and solvent, the changes in these specific interactions account for only a small fraction of the changes in solvation energy that accompany electronic excitation. Thus, it appears that the deviations displayed in Figure 7 for hydrogen-bonding solvents are more a matter of imperfect calibration of the n* values of these solvents than a reflection of the importance of specific interactions with Cl53. We now turn to more theoretically based measures of solvent polarity. Figure 8 is the counterpart of Figure 7 wherein we have replaced the n* polarity measure by a dielectric measure of solvent polarity,
of the Gaussian distribution required increases with increasing solvent polarity. In the case of emission, Figure 6 shows that spectra are in fact slightly narrower in high-polarity solvents than in 2-methylbutane. Thus, the line shape cannot be accounted for as simply in the case of emission. We will discuss these width/shape changes further after first considering the shifts in more detail. A complete characterization of the solvatochromic behavior of C153 is provided in Table 1 and in Figures 7—9. For purposes of analysis we have grouped solvents into three categories: (i) “simple” aprotic solvents (1—23, circles), (ii) hydrogen bond donating solvents (31—41, squares), and (iii) aromatic solvents (51—56, triangles). We begin the discussion, partly for historical reasons, by considering correlations with of solvent polarity. In Figure 7 the an empirical measure average (first spectral moment94) frequencies T of the absorption and emission spectra are plotted as functions of solvent polarity, measured by the n* solvatochromic scale.95-96 The 7t* scale was originally constructed in order to represent the nonspecific polarity/polarizability of solvents, exclusive of the influence of solvent—solute hydrogen-bonding and other specific interactions. It has therefore been employed in previous studies of coumarins76-97 as an aid to sorting out what types of interactions give rise to the solvatochromism of these molecules. As found previously, Figure 7 shows that there is a very good correlation between the 7t* polarity of the solvent and the frequencies of Cl53, especially in the case of simple aprotic solvents. (The lines drawn in these figures are the fits to the data in solvents 1—23 which yield correlation coefficients of 0.98 and 0.96 in absorption and emission, respectively.) As also noted in past studies,76-97 solvents capable of donating hydrogen bonds to the solute are found to deviate from the correlations defined by the aprotic solvents. Alcohols and other hydrogen-bonding solvents cause an additional red shift of the spectrum, and this additional shift is greater in emission than in absorption. One can interpret this difference between hydrogen-bonding and aprotic solvents as indicating that the spectral shifts of Cl53 at least partly reflect specific hydrogen-bonding interactions between the solute and solvent. This was the interpretation offered in prior studies.76-97 However, as noted recently,7-3 such a conclusion rests on the assumption that the ji* scale correctly partitions nonspecific and hydrogen-bonding interactions in such solvents, and this assumption is open to question. In fact, a number of observations tend to support the conclusion that specific hydrogen-bonding
fo__
F(e0,n)
eo
+
rf
2
—
n2
+
1
(4.1) 2
where e0 is the static dielectric constant and n the optical refractive index of the solvent. This reaction field factor comes from dielectric continuum theories of solvatochromic shifts of the type originated by Ooshika, Lippert, and McRae98 and developed by many others.99 Assuming a spherical cavity/point dipole solute and assuming that the solvent interacts with the electronic transition 1 2 mainly by virtue of the change of the solute’s dipole moment (wi ^2), the transition frequency is expected to vary with solvent dielectric properties as100 —*
—
1
71—2
=
V0
F Av
eo
+
1
2
n2
+
where the solute-dependent factors ~
At-
=
hca}
fi2)
Ul
2.
and
are
n
2
+ 02 ,
(4.2)
given by
*u
=
B\~
~Pi
hca3
(4.3)
with a denoting the solute radius (assumed spherical). The Au term in eq 4.2 accounts for contributions due to polarizability of solvent nuclear coordinates (i.e., those observable in timeresolved emission measurements) while the Be term involves the electronic polarizability of the solvent. In Figure 8 we have plotted the regression lines obtained by fitting the polar aprotic solvents (minus solvents 9 and 14) to the first two terms of eq 4.2. A summary of the fit parameters and characteristcs for these fits (“set A”), as well as others, is contained in Table 2. As can be seen from Figure 8 and Table 2, there is a reasonably good correlation between both the absorption and emission frequencies of C153 in these selected aprotic solvents (filled circles) and the reaction field factor F(eo,n). In absorption the quality of the fit is significantly improved upon the addition of the 5u term in eq 4.2 (the correlation coefficient changes from 0.85 to 0.94), but there is little change in the quality of the fit to the emission data. In both cases the value of Au is not changed significantly when the Be term is added, but the value of To is. We note that the fits of eq 2.4 including the Be term provide values of To that are close to the gas phase values, which, using the data of ref 92, we estimate to be To(abs) == 27 600 cm-1 and To(em) ==
Polar Solvation Dynamics of Coumarin 153
TABLE
J. Phys. Chem., Vol. 99, No. 48, 1995
1: Solvent Properties and Steady-State Spectral Characteristics
solvent properties'’ solvent"
acetonitrile dimethyl sulfoxide propionaldehyde propylene carbonate dimethylformamide nitromethane
no.
i 2 3
4 5
6
acetone
7
HMPA DMC
8
methyl acetate cyclohexanone ethyl acetate tetrahydrofuran 1,4-dioxane diethylamine dichloromethane 1-chlorobutane diethyl ether chloroform diisopropyl ether TCTFE 2-methylbutane cyclohexane methanol ethanol 1-propanol 2-propanol 1-butanol 1-pentanol 1-decanol
1,2-propanediol A'-methylformamide ethylene glycol formamide benzyl alcohol benzonitrile benzene 1,4-xylene hexafluorobenzene toluene
9 10 11
12 13
14 15 16 17 18 19
20
P (D) 3.5 4.1 2.5
4.9 3.2 3.6 2.7 4.3 1.0 1.7 3.1 1.8 1.8
0.5 1.2 1.1
1.9 1.2 1.2 1.4
22 23
0.0 0.0
31
1.7 1.7 1.7 1.7 1.8 1.7 1.7
35
36 37 38 39 40 41 51
52 53 54 55
56
0.66 1.00 0.71 0.83 0.88 0.75
0.62 0.87 0.47 0.49 0.68 0.45 0.55 0.49 0.35 0.73 0.40 0.24 0.69 0.19 0.01
21
32 33 34
JT*
2.3 3.9 2.3 3.4 1.7
4.0 0.0 0.0 0.3 0.3
fo
35.94 46.45 18.50 64.92 36.71 36.15
20.56 29.30 3.17 6.68 16.10 6.02 7.58 2.21 3.89 8.93 7.39
4.20 4.81 3.88 2.41
-0.15
1.83
0.00 0.60 0.54 0.52 0.48 0.47
2.02 32.66 24.55 20.45 19.92
0.92 0.97 0.98 0.88 0.55 0.45 0.27 0.49
17.51 13.90
7.20 32.00 182.40 37.70 111.00 11.92 25.20 2.27 2.27 2.05 2.38
1.342 1.478 1.359 1.420 1.428 1.380 1.356 1.457 1.386 1.359 1.450 1.370 1.405 1.420 1.383 1.421 1.400 1.350 1.443 1.366 1.356 1.351 1.424 1.327 1.359 1.384 1.375 1.397 1.407 1.435 1.431 1.430 1.431 1.447 1.538 1.526 1.498 1.493 1.375 1.494
of C153 (22 °C)
spectral characteristics"
time-zero analysis4
F(e0,n)
Vabs
Tabs
^em
T err.
do
T inh
v,o
rt0
0.71
24.38 23.74 24.31 24.05 23.95 24.10 24.41 23.82 24.91 24.73 24.22 24.75 24.69 24.96 25.01 24.21 24.57 25.18 24.37 25.26 25.61 26.13 25.98 24.02 24.08 24.13 24.16 24.12 24.13 24.33 23.66 23.99 23.54 23.49 23.43 23.96 24.77 24.87 25.20 24.83
3.89
18.35 18.03 18.48 18.26 18.24 18.37 18.69 18.41 19.38 19.18 18.78 19.36 19.35 19.74 19.89 19.15 19.91
3.27 3.32 3.36 3.27 3.26 3.18 3.30 3.28 3.47 3.37 3.29 3.47 3.36 3.35 3.43 3.20 3.35 3.47 3.20 3.41 3.46 3.38 3.48 3.07 3.18 3.17 3.37 3.18 3.15 3.25 3.29 3.24 3.12 3.16 3.08 3.16
1.74 2.21 1.69 1.89
1.70
20.58 19.94 20.48 20.25 20.08 20.20 20.41 19.95 20.89 20.67 20.25 20.76 20.54 20.90 20.91 20.13 20.72 21.05 20.18 21.12 21.38 21.74 21.67 20.36 20.16 20.13 20.30 20.28 20.28 20.45 19.95 20.03 19.49
3.64 3.69 3.67 3.69 3.68 3.65 3.66 3.63 3.62
0.66 0.63 0.70 0.67 0.69 0.65 0.63 0.18 0.43 0.57 0.40 0.44 0.03 0.26 0.47 0.44 0.30 0.29 0.27 0.10 0.00 0.00 0.71 0.67 0.63 0.63 0.61
0.57 0.41 0.65
0.72 0.67 0.71 0.47 0.58 0.01 0.01 0.03 0.02
17319
3.91
3.87 3.90 3.89 3.83 3.87 3.84 3.84 3.83 3.79 3.81
3.79 3.82 3.73 3.68 3.69 3.72 3.83 3.69 3.66 3.56 3.57 3.97 3.88 3.84
20.32 19.52 20.60 21.02 21.74 21.75 17.99 18.10 18.30
3.81
18.41
3.84 3.83 3.79 3.89 3.89 3.93 3.93 3.79 3.68 3.65 3.67
18.32 18.35 18.68 17.85 17.99 17.62 17.72 18.02 18.75 20.00 20.07 20.46 20.20
3.71 3.65
3.31
3.48 3.36 3.35
1.71
1.67 1.70 1.67
2.03 1.99 1.65 2.25 1.15 1.32 1.76 1.27 1.39 1.02 1.02 1.77 1.22
1.61
1.64 1.54 1.48 1.50 1.49 1.43 1.40 1.36 1.16 1.21 1.13
0.84
1.06 1.07
1.67
0.92 0.78 0.00 0.00
0.75 0.44 0.00 0.09 2.04
1.85 1.72 1.53 1.56 1.63 1.60 1.57 1.73 1.69 1.80 1.71 1.51 1.24 1.00
1.95 1.92 1.89 1.90 1.80 1.70
2.34 2.10 2.50 2.60 2.57 2.06 1.24 1.09 0.83
0.99 1.07
0.95
1.15
19.57 19.67 19.88
20.57 20.80 21.07 20.73
3.61
3.62 3.60 3.59 3.59 3.54 3.54 3.53 3.52 3.51
3.49 3.46 3.38 3.38 3.74 3.69 3.63 3.64 3.66 3.65 3.64 3.70 3.68 3.73 3.69 3.62 3.55 3.50 3.50 3.52 3.49
Av" 2.23 1.91
2.00 2.00 1.85 1.83 1.72 1.55 1.51 1.49 1.48 1.41
1.19 1.16 1.02
0.98 0.81 0.74 0.66 0.52 0.36 0.00
-0.07 2.37 2.06 1.82 1.88 1.96 1.83 1.77
2.10 2.04 1.87 1.85 1.65 1.14
0.76 0.73 0.61 0.53
“ HMPA denotes hexamethylphosphoramide and TCTFE 1,1,2-trichlorotrifluoroethane. b Values of the empirical polarity measure .t* are mostly from the recent compilation provided in ref 96. However, hydrogen bond donating solvents were not provided in that work and for solvents 31-42 the values are from ref 95. Dipole moments (mostly in benzene solution) pi, static dielectric constants Co. and refractive index values nD are for (25 °C) and are taken from the tabulations in: Riddick, J. A.; Bunger, W. B.; Sakano, T. K. Organic Solvents', Wiley: New York, 1986. The values listed in the column labeled “F(e0,n)" are the reaction field factor defined by eq 4.1." First moment frequencies v94 and bandwidths T (full width at half-maximum) of the absorption and emission spectra. All frequencies here and elsewhere are in units of 103 cm-1. 4 Values obtained from “time-zero” analysis, do and T^ are the shift and width (fwhm) of the p(d) distribution (eqs 3.5 and 3.6) required to fit the steady-state absorption spectrum using the spectrum in 2-methylbutane as the nonpolar reference. i>,0 and Tto are the first moment frequency94 and width of the estimated “time-zero” emission spectrum generated according to eq 3.2 (for 385 nm excitation). "The difference Av = (vt0 vem), which is the estimate for the frequency shift due to solvation dynamics that should be observed in time-resolved emission experiments. -
23 200 cm-1. This agreement with the gas-phase frequencies lends some credence to the validity of eq 4.2 in the present context. The values of Au and flu so obtained indicate that nuclear polarization of the solvent (Au) accounts for between 0 and 50% of the overall magnitude of the absorption shifts and 0—70% of the emission shifts of C153 in the selected solvents. Thus, nuclear and electronic polarization play roughly comparable roles in producing the gas to liquid shifts in Cl53. However, the differences between various solvents in this class are dominated by the nuclear polarization, which accounts for ~70 and 90% of the variations with solvent observed in absorption and emission, respectively.
We now consider solvents other than those in the “select” grouping. Figure 8, as well as the data in Table 2, shows that, with the possible exception of 1-decanol (no. 37), alcohols and other hydrogen bond donating solvents fit reasonably to the same correlations established by the aprotic solvents. Thus, unlike the n* correlations, analysis via eq 4.2 again suggests that
specific solute—solvent hydrogen bonding plays a minor role in the observed spectral shifts.101 In contrast to this agreement with the hydrogen-bonding solvents, aromatic solvents deviate significantly from the polar aprotic correlations. Especially in the case of the “nonpolar” (F(eo,n) as 0) aromatics such as benzene, Figure 8 shows the shifts in both absorption and emission are much larger than anticipated based on the behavior in other solvents. One possible reason for this deviation could be the formation of special ground- and/or excited-state complexes of C153 with such solvents. However, the fact that these solvents do not appear anomalous when viewed with the ti* polarity scale (Figure 7), as well as the fact that the fluorescence lifetimes in aromatic solvents show no unusual behavior, argues against such a possibility. The source of this deviation is discussed in more detail in ref 14. Basically, aromatic solvents offer a different “solvation mechansim” than other solvents. This different mechanism appears to be that most aromatic molecules, even if nondipolar, possess large
Homg et al.
1995
J. Phys. Chem., Vol. 99, No. 48,
17320
TABLE 2: Summary of Regression Analyses of Steady-State Frequency Data" N vo(103cm ') Au data set set
A
set B
all data
21 21 31 31
40 40
(103 cm
')
Bu (103 cm
Absorption Frequencies 25.8(2) -2.7(5)
-2.5 (4) -2.8 (4) -2.7 (3)
27.9 (8) 25.9 (2)
27.8(8)
-9 (4)
-8(3)
-2.1(4)
25.3 (4) 27.9 (6)
-2.2 (3)
-10(2)
')
R
0.85 0.94 0.85 0.93 0.68 0.88
Emission Frequencies set
A
21 21
0.
.2
.4 71
.8
.6
1.
*
Figure 7. Average frequencies (first spectral moments) of the steadystate absorption and emission spectra of C153 in a range of solvents plotted versus the 7t* solvent polarity parameter. Circles indicate “simple” aprotic solvents, squares are hydrogen bond donating solvents, and triangles are aromatic solvents. (See Table 1 for a complete listing of solvents.) The lines drawn through the points are the fits to the 2.066rr* (R = 0.981) simple aprotic data: vabs [103 cm'1] = 25.774 3.505;r* {R = 0.962). and vem [103 cm'1] = 21.217 -
-
25.
31 31
all data
40 40
21.6 23.0 20.8 23.3
-4.7 -5.2 -5.0 -3.8 -3.9
(3) (13) (3) (14)
-6(6)
(5) (5) (5) (6) (6)
-6(5) 10(5)
0.94 0.95 0.94 0.95 0.79 0.85
Regressions are of the average absorption and emission frequencies (v, Table 1) to eq 4.2. Data set A includes all of the aprotic solvents (1-23) with the exception of dioxane and dimethyl carbonate. Data set B includes all of those in set A plus all of the hydrogen-bonding solvents (31—41) except for 1-decanol. N is the number of data in each set, and R is the correlation coefficient of the fit. The values in parentheses are the 95% confidence limits in the last digits listed. 0
2.5
27. -
set B
-4.9(6)
21.6(3) 23.0(14)
-
&
b 23. £
2 i.
I
19. /
r
0.0 0.
.2
.4
F(£0 .n) Figure 8. Average frequencies (first spectral moments) of the steadystate absorption and emission spectra of C153 in a range of solvents plotted versus the reaction field factor F(eo,n) defined in eq 4.1. Circles indicate “simple” aprotic solvents, squares are hydrogen bond donating solvents, and triangles are aromatic solvents. The lines drawn through the points.are the fits to the simple aprotic data shown as filled circles: = 2.691F (R = 0.926) and t>em [103 cm'1] = 25.842 i>abs [103 cm'1] 4.896F (R = 0.968). Numbers denote the identity of several 21.557 solvents which deviate from the correlation shown by other members of their grouping. Solvents 14, 9, and 37 are respectively 1,4-dioxane, dimethyl carbonate, and 1-decanol. (See Table 1 for a complete listing -
-
of solvents.)
quadrupole moments. While these quadrupole moments or (local bond polarities) cannot lead to long-range polarization and thus appreciable reaction field factors in continuum-based models, they can be quite effective in solvating molecular solutes.14102 Similar “local” solvation effects are also what give rise to the large deviations observed for the two aprotic solvents dimethyl carbonate (no. 9) and 1,4-dioxane (no. 14). The dioxane case is similar to the benzene case in that its dipole moment is also approximately zero due to geometric constraints. In dimethyl carbonate, highly polar groups are “present but unaccounted for” due to the internal flexibility of the molecule (compare u and eo with propylene carbonate). The common thread that links all of the “anomalous” cases is that the true “polarity” relevant for solvation is inadequately accounted for by dielectric continuum models that focus only on the average dipole moment of the molecules via its link to eo-103
1
--- ---
.2
0.
.4
.6
.8
F(£0 .n) Figure 9. Predicted time-resolved shifts plotted versus the reaction field factor F(eo,n). The predicted shifts are computed as the difference between the (average) frequencies of the estimated time-zero spectrum and the steady-state emission spectrum. Circles indicate “simple” aprotic solvents, squares are hydrogen bond donating solvents, and triangles are aromatic solvents. The lines drawn through the data (slope = 2965 cm-1, R = 0.940) is the fit to all of the open symbols, i.e., excluding the aromatics and the “deviant” solvents 14, 9, and 37. The deviations from continuum dielectric predictions exhibited by these latter solvents become more pronounced when one considers the differences between absorption and emission frequencies. One representation of such differences is provided in Figure 9, where we plot the predicted values of the timedependent shifts (Av) versus F(eo,n). The predicted shifts are the differences between the average frequencies of the estimated time-zero emission spectra (section III.A) and the steady-state emission spectra. These shifts provide the most accurate measures of the effect that repolarization of solvent nuclear of freedom has on the electronic spectrum. (A simpler degrees approximation to these quantities is obtained from the differences between Po(abs) To(em) in a given solvent and this same quantity in the reference solvent 2-methylbutane). If a continuum dielectric description of the solvent is valid, the nuclear repolarization should be directly proportional to the reaction field factor F(eo,n). Given the uncertainties in the timezero estimates (~200 cm-1), there is a reasonably good correlation between Av and F(eo,«) for all of the “well-behaved” —
Polar Solvation Dynamics of Coumarin 153
J. Phys. Chem., Vol. 99, No. 48, 1995
17321
solvents discussed above. However, the anomalous cases stand out quite clearly in this representation, indicating that the dielectric factor F(eo,n) is a poor measure of the effective nuclear polarizability of these solvents. To the extent that a dielectric continuum model is appropriate for the well-behaved solvents, the proportionality between Av and F(€o,n) should be related to the change in solute dipole moment between the ground and excited states, A// = |/Je —
jng|, as
d(Av)
2(A/i)2
_
hca3
(4.4)
Fitting the “B” solvent set (Table 1, open symbols in Figure 9) yields a proportionality factor of [2.90 ± 0.13 (95%)] x 103 cm-1 with a correlation coefficient of 0.91. (Fitting set A or allowing for a nonzero intercept does not improve the quality of the fit.) In order to calculate Afi from this proportionality constant, one must estimate the effective cavity radius a of the solute. The simplest choice is to assume a3 ss 3V/4tt, where V is the van der Waals volume of C153.104 The value of a = 3.9 A so obtained, when combined with the observed slope, yields a dipole moment change of 4.1 D. This value is smaller than the values determined from a combination of dielectric82 and electrochromic84 measurements, A/u = 7.5—9.5 D, or from semiempirical calculations, Afi 7—8 D.80-83’85 One can readily ascribe this difference to the approximate nature of the spherical solute/continuum solvent representation assumed in eqs 4.2— 4. Simulation studies have shown that even in the case of spherical solutes, continuum calculations based on a cavity radius equal to the actual solute radius severely overestimate the solvation energies.38’506 More reasonable estimates are obtained from continuum models if instead of the actual solute radius f?soiute one chooses a value of a as Rs0iute + Solvent-105 For the polar solvents examined here the ratio Rsoiute/f?soivem averages about 1.5 (Table 5). If one increases the initial estimate of a by this factor, the prediction for Ap increases to 7.5 D, which brings it into line with other estimates. Thus, while it is not possible to uniquely define a cavity radius and thus reliably determine Am for a molecule as complex as C153, appropriate choice of this parameter does lead to agreement between simple continuum predictions and observed spectral shifts. We take this agreement as evidence for the basic soundness of continuum dielectric models for describing solvation of C153, at least in strongly dipolar solvents. A final aspect of the steady-state spectra we will consider are the spectral widths. In the case of a linear solvation response, a simple relationship should exist between the spectral broadening and shift that result from solute—solvent interactions.106 Specifically, if a represents the inhomogeneous broadening of either the absorption or emission spectrum (as in the sense of eq 3.5), it should be related to the shift of the spectrum due to nuclear repolarization by
Figure 10. Test of the linear response relationship (eq 4.5) between the shift (Av) and broadening (o) of the spectrum caused by polar solvation. The shift shown here is the predicted time-resolved shift (difference between the average frequencies of the estimated time-zero spectra and the steady-state emission spectra), and the breadth measure a is from the width of the p(d) function required to fit the absorption spectrum. Circles indicate “simple” aprotic solvents, squares are hydrogen bond donating solvents, and triangles are aromatic solvents. The solid line represents the expected relationship between a and Av (slope of unity), and the dashed line is the best fit of all of the data. It has a slope of 1.25 ± 0.01 (N = 40, 95% confidence limit).
—
Av
=
(P’lk.T
(4.5)
The validity of this relationship is independent of any particular solvation model such as the continuum models on which eqs 4.2—4 depend. It requires only that the solvation free energy be a quadratic function of some measure of the solute polarity (“,«”). In Figure 10 we examine the applicability of eq 4.5 using the inhomogeneous widths determined by fitting the absorption spectra as described in section III.A. While the relationship embodied in eq 4.5 is reasonably well known, the data in Figure 10 provide one of the most comprehensive tests of its validity performed to date.107 As illustrated by the dashed line in Figure
10, o2 is proportional to Av in all solvents examined (including those not well described by continuum models), but the proportionality factor is 1.25 ± 0.03 (95%) rather than unity (solid line). This deviation from the expected factor is relatively small. In the most polar of solvents, the inhomogeneous width (Tinh, eq 3.6) that best fits the absorption spectrum is larger by approximately 200 cm-1 (10%) than what would be expected on the basis of eq 4.5. The origin of this small discrepancy is not clear. It might signal the fact that solvation of C153 is not a purely linear process as is required for the validity of eq 4.5. However, molecular dynamics simulations of C153 in acetonitrile and methanol50 show that this relationship is upheld to much better than the 25% level for the sorts of charge redistributions that accompany the So ** Si transition.108 Perhaps a more likely explanation for the lack of perfect agreement is that small changes other than those due to inhomogeneous broadening/shifting take place in the spectra as a function of solvent polarity. Changes in either the extent of homogeneous broadening of the spectrum or the underlying vibronic structure (g(v), fiv) in section III.A) could readily account for the behavior illustrated in Figure 10. Evidence for the latter effect is illustrated in Figure 11, where we show the absorption and emission bandwidths (fwhm) plotted versus spectral frequency. While the absorption spectra, from which the above inhomogeneous widths were derived, show the expected broadening with decreasing absorption frequency (or increasing solvent polarity), the emission spectra actually show the opposite trend. This unusual behavior was previously noted in our earlier work on C153.76 However, contrary to what we suggested in that original study, analysis of the emission band shapes, especially on the blue spectral edges, shows that it is not due to the fact that the inhomogeneous broadening in a given solvent is substantially smaller in emission than in absorption. Such a difference between absorption and emission would require the solvation of C153 to be significantly nonlinear, much more so than would be implied by the deviation shown Figure 10, and computer studies58 108 now render this interpretation implausible. Rather, what appears to give rise to this narrowing in more polar solvents is a reduction in the width of the underlying vibronic structure with increasing
17322
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1995
J. Phys. Chem., Vol. 99, No. 48,
'
Abs.
4.01
-
-
3.8)
-
A -
£
-
O
A
O
ca-.
1
3.6-
=
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25.
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°"
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3.5 f l'“'
7$)
3.3) —
:
3.117.
®
C
0
0
^
A
Em.
A
18. v
19.
(103
20. cm
21. 1
f
22
)
Figure 11. Spectral bandwidths (full width at half-maximum) of steady-state spectra plotted versus the first moment frequencies. Circles indicate “simple” aprotic solvents, squares are hydrogen bond donating solvents, and triangles are aromatic solvents. The data indicate a single correlation between the width and frequency of the spectrum rather than distinct behavior for different solvent groupings.
solvent polarity. The total change amounts to 300—400 cm-1 between extremes of polarity examined here, which could result from relatively small changes in the frequencies and/or displacements of the Franck-Condon-active modes of the spectrum. Such changes are not surprising in view of the large changes in solvation energy, especially of the highly polar excited state, which accompany changes in solvent polarity. The fact that only the emission spectrum shows obvious signs of this change in vibronic structure implies that it mainly occurs subsequent to excitation, during the solvation of the Si state. However, it may be that the small deviation from linear response predictions shown in Figure 10 also results from (more subtle) manifestations of this same polarity dependence prior to solvent relaxation.
We can summarize the results concerning the steady-state spectroscopy of C153 as follows. The lowest wavelength absorption and emission bands of C153 arise from a single electronic transition. The frequency shifts that these bands undergo in different solvents mainly reflect a simple coupling between the dipolar So and Si states and the polarity of the solvent. In most cases dielectric continuum models provide a semiquantitative description of the solvatochromism observed. (However, such models fail to predict the magnitudes of the shifts observed in solvents such as benzene and dioxane, due to the fact that their “polarity” on a molecular scale is not adequately represented by their bulk dielectric properties.) Both protic and aprotic solvents exhibit the same correlations with solvent dielectric properties. Thus, there is no evidence to suggest that hydrogen bonding between the solute and solvent plays any significant role in determining the spectral shifts observed. In addition to spectral shifts, there are changes in the spectral bandwidths as a function of solvent that are only partially understandable in terms of simple models. While the solvent-induced broadening of the absorption spectrum appears to result from precisely the same solute—solvent interactions as does the frequency shift, changes in the vibronic structure of the emission also occur as a function of solvent polarity. This latter behavior suggests that solvent polarity has some effect functions of Cl53 in the Si state. on the vibronic wave on the basis of the behavior of the steady-state However, frequency shifts, it seems that such solute changes should have little impact on the time-dependent emission frequencies used to monitor the dynamics of solvation.
i7
(103 cm"1
) Figure 12. Time-resolved emission spectra of C153 in DMSO. The
top panel shows normalized spectra at the times indicated from 0 (open circles) to 200 ps (open squares). The spectra shown here are representative of the quality of the spectra obtained in this study, but the point density is roughly twice as large as typically employed in order to better display the lack of structure in the spectra. (See Figure 5 for an example of a more typical data set.) The bottom panel of this = = figure shows the t 0 and t 200 ps data again, along with the steadystate emission spectrum (continuous curve through the 200 ps data), the estimated time-zero spectrum (dashed curve), and the absorption spectrum (bluemost curve). The continuous curve passing through the t = 0 data set shows the log-normal fit to these data.
IV.B. Characteristics of the Time-Resolved Spectra; Vibrational Contributions. Representative time-resolved emission spectra of C153 in dimethyl sulfoxide (DMSO) are shown in the top panel of Figure 12. In the bottom panel of this figure the observed spectra at 0 and 200 ps are reproduced (points) along with the steady-state absorption and emission spectrum and the estimated “time-zero” spectrum (dashed curve). The data shown here are typical of what is observed in all polar solvents. The spectral evolution involves mainly a timedependent shift of the spectrum between the limits set by the estimated time-zero and the steady-state emission spectra. Apart from some noise at the earliest times, no structure is discernible in the spectra, at least not at the ~6 nm resolution employed. Furthermore, there are only small changes in the width or shape of the spectrum with time. The magnitudes of these timedependent changes are similar to those observed in the steadystate spectra as a function of solvent polarity. The main differences between different solvents lies in the extent of the frequency shift and its temporal characteristics. For highpolarity, nonassociated solvents like DMSO a shift of ~2000 cm-1, which is largely complete in within 5 ps, is typical. It is the normalized dynamics of this spectral shift, as embodied in the spectral response function (eq 1.1), that we will use as a monitor of the solvation energy relaxation. The steady-state results of the last section show that the connection between spectral frequency and solvation energy is straightforward. However, there is one possible complication that needs to be considered when working in the picosecond and subpicosecond time regime. By necessity, we excite the solute with 3000— 4000 cm-1 of excess vibrational energy. It is possible that effects due to relaxation of this excess vibrational energy might be convoluted with the spectral dynamics due to solvation.109 Before discussing the behavior observed in polar solvents further, we first digress to address this concern. In order to examine how vibrational relaxation might “contaminate” the spectral dynamics observed in the subpicosecond time domain, we have measured time-resolved spectra
Polar Solvation Dynamics of Coumarin 153
18.
20. v
22.
24.
(103 cm-1
26.
J. Phys. Chem., Vol. 99, No. 48, 1995
28.
)
Figure 13. Time-resolved emission spectra of C153 in cyclohexane. The top panel shows the observed time-resolved emission spectra at times of 0, 2, and 20 ps. Also shown for reference are the steady-state absorption and emission spectra (solid curves) and the excitation frequency (vex, 366 nm) used in this experiment. The bottom panel shows simulations of the time-resolved emission spectra predicted from a single harmonic oscillator description of vibrational relaxation proposed by Loring et al.uo For these calculations So and Si were assumed to have the same harmonic frequency, v = 1260 cm-1, with a dimensionless displacement between the So and S i wells of 1.73 (“D” in the notation of: Henderson, J. R.; Muramoto, M.; Willet, R. A. J. Chem. Phys. 1964, 41, 580). The 0—0 transition frequency was taken to be 23 140 cm-1, and each vibronic transition was assumed to have a homogeneous width of 1260 cm-1 (fwhm). The spectra are shown at times scaled by the vibrational relaxation rate between the v = 1 and v = 0 levels, y. The curves are shown with an alternating dash pattern for clarity. Different times (listed) can be distinguished by the fact that the intensity near 26 000 cm-1 decreases monotonically with increasing time. a solvent for which we anticipate no relaxation due to solvation (see Table 1). The dynamics spectral observed upon excitation of C153 in cyclohexane at 366 nm are shown in the top panel of Figure 13. With this excitation wavelength, on the order of 3500 cm-1 of excess vibrational energy is deposited into the molecule. At times comparable to the time required for this energy to relax from the Franck— Condon-active modes, the emission spectrum should be very different from the vibrationally relaxed (steady-state) spectrum. Some idea of what is to be expected is provided in the bottom panel of Figure 13, which shows a calculation based on the theoretical treatment of Loring et aV10 The calculation assumes that the vibronic structure in the spectrum can be modeled in terms of a single displaced harmonic mode which is coupled to a harmonic bath. The time dependence of the nonequilibrium vibrational distribution created by excitation can be exactly = 1) (v = 0) relaxation rate of expressed in terms of the (v this key mode, denoted by y.110,111 Shown in the bottom panel of Figure 13 are emission spectra calculated for a series of times (fy) using the best single harmonic oscillator fit to the experimental steady-state spectra. Since the vibrational relaxation dynamics of C153 is undoubtedly much more complicated than can be represented in a single-mode description, these simulations should be taken as only suggestive of the real behavior. However, they suffice to show that at times of order y-1 the emission should be much broader than the steady-state spectrum and display significant intensity to the blue of the 0—0 transition (~24 000 cm-1 here). In the experimental spectra we do not observe any evidence for the dramatic departures from the steady-state line shape predicted to exist at sufficiently early times. To the extent that we are able to explore regions of the spectrum bluer than those shown in Figure 13,112 we find no evidence for the large-amplitude, quickly decaying compo-
of C153 in cyclohexane,
—1-
17323
nents implied by the calculations. Rather, at all wavelengths the decays deviate only slightly from a uniform singleexponential decay. The spectral evolution we do observe results from rise and decay components with ~5—15% amplitude. The time constants observed for these small components vary in the range 1—3 ps. We note that this observed relaxation is qualitatively similar to that found in the simulated spectra for t > 2y-1: the additional decay components are negative (rises) over most of the spectrum but positive (decays) in the redmost regions. The comparison made in Figure 13 suggests a picture of vibrational relaxation in C153 that is consistent with the general findings of a variety of studies over the past decade.113 The fact that we see no dramatic changes in the spectrum with time indicates that the vibrational excitation relaxes very quickly out of the high-frequency, Franck—Condon-active modes responsible for the vibronic envelope of the spectrum. Lack of significant intensity in hot bands at early times using a ~100 fs instrument response implies relaxation times of y-1 < 30 fs for these modes. This ultrafast relaxation is comparable to the
subpicosecond intramolecular vibrational redistribution (IVR) times found in other large polyatomic molecules. If the excess energy deposited in the Franck—Condon modes is thermally distributed within the solute, the internal vibrational temperature should be on the order of ~110 K above ambient.114 Molecules comparable to Cl53 with this kind of excess vibrational temperature are found to equilibrate with their solvent surroundings on a 1 10 ps time scale.113 Thus, it seems natural to associate the small residual relaxation that is observed in the spectrum with this slower, intermolecular component of the relaxation. —
The above results suggest that vibrational relaxation processes should not significantly interfere with the use of the dynamic Stokes shift to monitor solvation dynamics in polar solvents. In solvents such as DMSO, the excitation conditions used (see Figure 6) correspond to excess vibrational energies comparable to those pertaining in the cyclohexane case. There is no reason to expect that IVR in these (more strongly interacting) solvents should be slower than in cyclohexane. Thus, as in cyclohexane we expect that the pronounced effects on the emission spectrum anticipated prior to relaxation of the Franck-Condon modes should be too fast to detect with our time resolution. Furthermore, the spectral effects of the subsequent relaxation of the thermalized vibrational energy of the solute are quite modest. What is observed in cyclohexane is a clear narrowing (~300 cm-1) and a more subtle blue shift (~150 cm-1) of the spectrum that takes place on a 10 ps time scale. These small changes should have little effect on the time-dependent frequencies used to construct the spectral response function for the simple reason that in nearly all cases the observed shifts are larger by more than an order of magnitude. On the other hand, the solventdependent changes in the widths and shapes of the steady-state spectra discussed in the previous section are comparable in magnitude to the width changes observed in cyclohexane. Thus, the time-dependent changes in these more subtle aspects of the spectrum will probably involve a combination of effects due to both changes in solvation state as well as to vibrational relaxation. A final concern with respect to vibrational excitation is whether the excess vibrational energy that the solute deposits into the solvent will have any observable effect on the solvation dynamics being measured. We believe that it does not for two reasons. First, the total temperature rise in the solvent is expected to be small. Using typical values for the heat capacities of organic solvents (~100 K-1 mol-1), we estimate that 3500 cm-1 of excess vibrational energy translates into a 10—15 K ~
17324
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Peak ic Avg.
k
Frequencies
20.fr'
-
19.-
18..05
oO.
o.
.o
500.
Time (ps) Figure 14. Time-dependent frequencies of C153 in DMSO determined from log-normal fits to the time-resolved spectra. The two frequency measures shown correspond to the peak (vp) and first moment (v) frequencies described by eqs 3.9-3.11. The results of three separate experiments, evenly spaced over a period of 7 months are shown here. (The solid curves correspond to the spectra shown in Figure 12.) Arrows indicate the predicted time-zero and time-infinity (steady-state) frequencies.
.05
oO.
o.
,o
Time
oOO
(ps)
Figure 15. Four characteristics of the time-resolved spectra of C153 in DMSO determined from log-normal fits. The three curves in each case correspond to results of three separate experiments as in Figure 14.
The various parameters
are
defined in eqs 3.9—3.13.
change in temperature if it is localized to the 30 or so solvent molecules in the first solvation shell of Cl53.114 If such a temperature change were applied to the solvent in equilibrium, one would anticipate a 10—20% change in the solvation response time based on viscosity changes with temperature. In addition, if the energy flow to the solvent occurs on a 10 ps time scale as expected, in most of the solvents studied here the greater part of the spectral response is over before much of this energy actually gets to the solvent. Thus, in the present experiments we expect the effects of local heating of the solvent to be irrelevant. Such effects can be observed only with difficulty in cases of slow solvents (alcohol at low temperatures) and for larger vibrational excess energies.115 We now return to a more complete characterization of the spectral dynamics observed in polar solvents. We will continue to use DMSO as an example of the behavior typically observed in polar solvents. As discussed in section III, we employ lognormal fits of the time-resolved spectra in order to most accurately follow their time evolution. Figures 14 and 15 show a variety of parameters associated with such log-normal fits for three separate experiments in DMSO. The solid curves in
Figures 14 and 15 are from fits to the spectra shown in Figure 12. The remaining curves are from two other data sets that were recorded at widely separated times covering a 7 month period. The set-to-set variability illustrated here is representative of what we usually observe for repeated measurements in the same solvent. Except at the earliest times, the time-dependent frequencies measured from independent experiments typically span a range of less than 200—300 cm-1, with the peak frequency being somewhat more reproducible than the average. The frequencies observed at zero time are close to those predicted (arrows in Figure 14), even though the spectrum observed at early times is usually found to be narrower than the predictions (see Figure 12). Other characteristics of the time-evolving spectra (Figure 15) are subject to greater relative uncertainty than are the frequencies. Nevertheless, a reasonably consistent pattern is observed in most solvents for which we have sufficiently well-averaged data. The width (Tit), eq 3.12) of the spectrum first increases and then decreases with time in nearly all cases. The magnitudes of these changes are not large. As in the DMSO example shown in Figure 15, the width typically increases at early times by ~500 cm-1 before decreasing by about half this amount, such that the net change represents only about 10% of the overall width of the steady-state emission spectrum. The peak height {hit)) is observed to decrease with time by some 10—15% over the first 200 ps subsequent to excitation, and if often shows an “oscillation” as in the case illustrated. The integrated intensity (/(f), eq 3.13) exhibits a maximum corresponding to the maximum in peak width and decreases thereafter. Over the first 200 ps of spectral evolution the net change in lit) tends to be less than that of h{t), but it still averages some 10%. The changes observed in both of these measures of spectral intensity are significantly larger than what is expected for a simple population relaxation with the a lifetime of ~5 ns, which is typical for Cl53 in most solvents. (The latter should result in only a 4% decrease in intensity over 200 ps.) Finally, the asymmetry parameter (y, eq 3.9), which measures the skewness
of the spectrum, on average tends to decrease slightly (by ~0.06) with time. However, as illustrated by the variability shown in Figure 15, this parameter is difficult to measure reliably. All that can be said with confidence is that the spectrum at early times is tends to be slightly more symmetric than the infinite time (steady-state) spectrum. While we have argued that the time-dependent frequency shifts of Cl53 in solvents like DMSO merely reflect the process of solvation dynamics, what interpretation is appropriate for the variation of these other spectral characteristics? It is likely that a significant portion of the variations in spectral width, shape (y), and intensity also result from solvation-induced changes. We note that the appearance of a maximum in the width of the emission spectrum with time has been previously observed in a number of experiments318'21 on much longer time scales (where vibrational relaxation should be unimportant). Attribution of these width changes to changes in the distribution of solvation environments is supported by computer simulation studies,50-62 which show a comparable nonmonatomic behavior. Nonexponential decay of the total emission intensity has also been observed in experiments on longer time scales.3-22116 At least in some cases, this behavior has been shown to be simply connected to the time-dependent changes in the emission frequency that result from time-dependent solvation, via the v3 dependence of radiative rate on frequency.116 In the DMSO example, this effect would imply a decrease of ~'/3 in radiative rate between t = 0 and t = 200 ps, a change which is comparable to the drop in lit) observed subsequent to the
Polar Solvation Dynamics of Coumarin 153
J. Phys. Chem., Vol. 99, No. 48, 1995
maximum at early times. Thus, it appears that the spectral changes illustrated in Figure 15 could be adequately be explained as being further reflections of the solvation dynamics more directly monitored using v(/). However, given the fact that the magnitudes of the width and intensity changes observed in the nonpolar solvent cyclohexane are of a similar magnitude to those observed in polar solvents, these more subtle aspects of the spectral dynamics probably reflect a combination of both vibrational relaxation and solvation dynamics. We will therefore not consider these spectral characteristics further. Henceforth, we confine our attention to the two frequency measures illustrated in Figure 14 and to the response functions derived from them. IV.C. Spectral/Solvation Response Functions. Deriving spectral response functions, Sv(t), from frequency data of the sort shown in Figure 14 involves three steps. First, in cases like DMSO where there are multiple data sets, the data are averaged to provide the best estimates for vp(t) and v(f). The individual data sets are weighted according to their perceived quality as judged by the appearance of the spectra and the instrument function width. The average data sets are then fit to a multiexponential form, 3
v(t)
=
v(°°) + Av^_ai exp(—f/r,)
with a,
>
0 and
(=i 3 =
1
(4-6)
/=i
For these fits we use v(t) data generated at a set of logarithmically spaced times (i.e., times that appear equally spaced on plots such as Figures 14 and 15). The motivation for this choice, as with the multiple-time-step data collection, is to provide adequate representation of dynamics that cover several orders of magnitude in time.117 A compilation of the parameters of these fits for both vp(/) and v(t) is provided as supporting information.118 In all but the slowest solvents the values of v(°°) obtained from these fits are virtually indistinguishable from the corresponding frequencies of the steady-state spectra,119 as is to be expected from the use of the latter in the normalization procedure (eq 3.8). However, the frequencies extrapolated to zero time from such fits, v(0) = v(°°) + Av, are also in excellent agreement with the time-zero predictions made on the basis of steady-state data. The level of agreement is illustrated in Figure 16, where open symbols denote vp(0) and closed symbols v(0) data. For the 2 x 24 frequencies shown in Figure 16, the mean absolute deviation between the observed and predicted frequencies is 130 cm-1 and the mean (signed) deviation is -20 cm-1.120 This remarkable agreement provides a clear confirmation of the method we have adopted for estimating the position of the zero-time spectrum, as well as for the absence of any significant interference from vibrational relaxation in these frequency shifts. The data in Figure 16 also serve to point out that the present time-resolved data are taken with sufficient time resolution so that little, if any, of the spectral dynamics associated with solvation are missed. To generate the final Sv(t) results from these v(t) data first requires assignment of “proper” values of v(0) and v(°°) and then combining information from the peak and first moment frequency measures. With the exception of 1-decanol, where 100 cm-1 we estimate the steady-state emission spectrum to be blue of fully relaxed spectrum,121 we assume that v(°°) is accurately represented by the steady-state value.119 In most cases (see below) we also use the steady-state estimates of the time-zero spectrum for v(0) and with these two limits compute ~
17325
Figure 16. Comparison of observed and estimated time-zero emission frequencies. Open symbols denote peak (vp(0)), and closed symbols average (i>(0)) frequency data. Circles, squares, and triangles denote aprotic, hydrogen bonding, and aromatic solvents. (The particular solvents shown here are those listed in Table 3.)
Sv(t) functions from both vp(/) and v(f) data. In most cases the Sv(t) functions computed from these two frequencies differ only slightly from one another. Since there is no clear reason why one of these measures should be preferred, we have chosen to simply average the two S„(0 functions obtained from them in order to derive the final results. The data so generated for a set of 24 solvents are summarized in Table 3. Since Table 3 is a compilation of the main results of this paper, it is useful to fully explain its contents before discussing the nature of the Sv(t) functions observed. The solvent numbering in Table 3, and the division into aprotic, hydrogen bonding, and aromatic solvents, is the same as in section IV.A. The entries listed under the “comparisons/quality measures” heading serve to characterize how much of the spectral shift is observed, the similarity of the peak and average frequency measures, and the overall quality of the data for a given solvent. The number of independent experiments (N) that went into the final averaging is listed in the first column of this section. The next two columns, labeled “Av0bS” and “Avaif”, report the magnitudes of the time-dependent shifts observed (the average of the shifts of vp and v) and the difference between these values and the shifts predicted from “time-zero” analysis. Upon averaging the vp and v frequency measures, the deviations from the time-zero predictions are typically less than 200 cm-1 or in most cases less than 10% of the observed shift. In almost all cases we therefore used the predicted values of v(0) for computing Sv(t). The few cases where the percent deviation was considered to be too large to use the predicted values in determining Sv(t) are denoted by “n” in the column labeled “v(0)7”. Listed in the columns headed “v/vp” are comparisons of two quantities determined from the fits of vp(f) and v(r) data to eq 4.6: the ratio of the magnitudes of the shifts (“Av”) and the 1/e times of the v(t) decays (“tie”). In most cases the shift magnitudes and the decay times are within 20—30% of one another. (We note that the shifts Av are not predicted to be the same for the v is generally predicted to shift two frequency measures, slightly more than vp, and this is indeed what is observed.) In a few cases, the times determined by these two frequency differ substantially. One such case is 1-propanol, measures where the v/vp time ratio is 2.4. In some of these cases (the three marked by asterisks in the “v(0)7” column), it appeared that the first moment frequencies were overly influenced by changes in the asymmetry and/or width of the spectrum, which appeared to be due to uncertainties in the data rather than to real differences. In these three cases we weighted the vp data
17326
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Phys.
TABLE 3: Summary of Spectral Response Functions" comparisons/quality measures*
Chem.,
multiexponential fit parameters'
v!vf solvent
no.
N
Avobs (103 cm
acetonitrile dimethyl sulfoxide propylene carbonate dimethylformamide nitromethane
i 2
3 3
2.28 2.12
4
5
1.91
acetone
7
HMPA DMC
8
methyl acetate tetrahydrofuran 1,4-dioxane dichloromethane chloroform methanol ethanol propanol -butanol -pentanol -decanol /V-methylfonnamide ethylene glycol formamide benzonitrile 1
1
1
1
benzene
5
3
6
1
16
2 2 2 2 2 2 2
19
1
31
5
32
2 2
9 10 13
14
33 35
1
1.98 1.84 1.80 1.47 1.56 1.43 1.32 1.26 1.22 0.81
')(%)
cm
0.02(1) 0.12(6) -0.12 (-6) 0.04 (2) 0.00 (0)
41
3
2.15 1.84
52 53
2
1.29
3
0.83
1.06 1.14 1.34 1.26
1.37 1.10 1.01
0.80
1.01
0.84 1.21
0.11 (16)
1.02 1.10 1.30 1.19 1.55
(-9)
0.95
-0.12 (-5) 0.15(7) -0.06 (-3) -0.21 (-11) -0.08 (-4) 0.00 (0) -0.14 (-7) 0.08 (7) 0.12(17)
5
fie
1.06
-0.02 (-1) -0.08 (-5) 0.03 (2) -0.12 (-8) 0.02 (1) 0.02(1) 0.14(12)
2.01 2.11 1.92 1.73 2.01
2
Av
1.23 1.17 1.13
-0.23
40
1
Avdii (103
2.34 2.02
36 37 39
1
')
—0.16 (
7)
1.31 1.01
0.98 0.69 0.82
v(0)?
Q
a}
T\ (PS)
02
y
l l
0.686 0.500 0.116 0.508 0.642 0.565
0.089 0.214 0.030 0.217 0.155 0.187 0.030 0.226 0.030 0.228 0.177 0.144 0.285 0.030 0.030 0.030 0.243 0.030 0.030 0.092 0.187 0.030 0.356 0.234
0.314 0.408 0.429 0.453 0.358 0.435 0.307 0.464 0.465 0.553 0.520 0.483 0.644 0.340 0.230 0.167 0.107 0.069
y y y y y y y y y y
2 1
2 1
2 2 2 1
n
2 2
n*
3
y y
1
2.40 0.89
y*
3
n
2
1.31 1.71
y y
2
1.35 1.13 1.08
0.63
y*
1.31
2
1.21
0.73 2.89
y y y n
3
1.08 1.01
1.04 0.91 0.93
1.33
1.35 1.28 1.13
1.83
1
2 2 1
2
0.061
0.470 0.094 0.447 0.456 0.518 0.356 0.101 0.085
0.086 0.159 0.036 0.123 0.301
0.307 0.083 0.382 0.366
0.061
0.158 0.255 0.454 0.530 0.600
T2
(PS)
0.63 2.29 0.18 1.70
03
0.092 0.237 0.039
r, (ps)
characteristic times* Vol.
04
t4 (ps)
0.218
6.57
29.1
0.85 1.09
0.24 2.88 0.21 1.52 2.21 1.02
4.15 0.28 0.39 0.34 5.03 0.67 0.58 0.27 4.98 0.16 5.27 1.89
0.405 0.067 0.441
7.25 81.5
0.024
18.3
0.298 0.182
3.20 5.03 6.57 42.6 21.7 43.4 3.42 32.0 2.94 25.0 24.7
0.231
0.393 0.249 0.181 0.465
0.439 0.399 0.088 0.034
(ps)
0.12 0.40
10.7
2.03
To
0.227
30.4
1.70
0.16(0.35 ) 0.38 0.22 0.29 0.30 (0.70) 0.45
0.18(0.36)
0.261
15.3
0.43 0.35 0.25 0.71 0.21 (0.67)
0.502 0.515
29.6 47.8
0.29(1.4) 0.29(1.7)
0.341 0.647 0.635
0.076 0.064
133. 151.
373.
53.4 57.9
1.45
0.76 (8.0) 0.24 (8.0) 0.25 0.59
0.18(0.31) 0.85 0.53
fle (PS)
0.15 0.90 0.73 0.67 0.28 0.40 5.9 1.25 0.51
0.70 0.92 0.38 2.3 2.3 10.9 18
47 87 205 1.6
9.3 0.82 3.2 1.12
(ps)
99,
0.26 2.0 2.0
No.
48,
2.0(0.91) 0.41
0.58 9.9 6.9 (1.5) 0.85 0.94 1.7 (1.3) 0.56 2.8 5.0
1995
16
26 63 103
245 5.7 (1.8) 15.3
5.0(1.3) 5.1 2.1 (1.3)
" derived from averaging the functions obtained using the peak (vp) and first moment (v) measures of the spectral frequency, as described in The data in this table refer to spectral response functions, the text. All results are for 22 °C. h N is the number of independent experiments averaged in the final results. Av„bs is the magnitude of the observed spectral shift, and Avdif is the difference between the observed shift and that predicted based on the time-zero and steady-state emission spectra. The numbers in parentheses are the percent deviations (observed minus predicted). Under the general heading “i>/vp ratio” are listed values of the ratios of the shifts and 1/e times of these two measures of the spectral frequency. The column “v(0)7” indicates whether the estimates of the time-zero frequencies were (y) or were not (n) used in calculating S,.(t). (If not, the extrapolated frequencies were assumed to give the proper time-zero values.) Q is an indication of the quality of the data for a given solvent with a value of Q = ‘ 1, indicating the highest quality data (see text). Parameters of the multiexponential fits of S,.(t) according to eq 4.7. In cases where a lime faster than the expected temporal resolution (~30 fs) was required in the fits, this time was arbitrarily assigned a value of 30 fs. *The characteristic times listed here are the initial time, the 1/e time, and the average or correlation time defined by eqs 4.8 and 4.9. Values in parentheses indicate the times that would be calculated if the fastest and slowest components were ignored in the calculation of To and (r), respectively. Preferred values are underlined (see text).
Homg
et
al.
Polar Solvation Dynamics of Coumarin 153
J. Phys. Chem., Vol. 99, No. 48, 1995
Mill
by a factor of 3 in determining the final Sv(t) averages. The final column in this category is an overall quality designation for each solvent, “Q”. A value of Q = 1 represents the highest and Q = 3 the lowest quality data. These numbers are assigned on the basis of the number of data sets recorded, the appearance of the spectra, and the agreement between the vp(f) and v{t) dynamics for a given solvent. Results with Q = 1 and 2 designations are not expected to change significantly upon further data collection. The data designated Q = 3 might be improved by further measurements. However, we anticipate that the characteristic times listed for these solvents to be accurate to at least the ±50% level (see below). Sv(t) results are reported in Table 3 as multiexponential fits, 4
Sv(t)
=
4
with a,
^
0 and
=
1
Figure 17. Observed spectral response functions (Sv(t), eq 1.1) in representative polar aprotic solvents.
As with the v(t) data, these fits were performed on data sets logarithmically spaced in time, typically using 200 points between 50 fs and 200 ps.117 In some cases, the fits of Sv(t) data obtained using the estimated time-zero frequencies required small-amplitude components with time constants near to our expected time resolution limit (~30 fs). In these cases we arbitrarily set the fastest time constant to be 30 fs. (The choice has little effect except on the To values.) Finally, we list three characteristic times of the fitted Sv(t) functions: the “initial” time constant (the inverse of the decay rate at t = 0),
v1=X^r1
1.0 .8 .6
(4-8)
the time required for Sv(t) to reach 1/e = 0.368, fie, and the average time constant or correlation time,
.05
.5
5.
50.
500.
Time (ps)
(4-9)
Figure 18. Observed spectral response functions in the 1-alkanols
single-exponential function all three of these times are equal. However, for nonexponential functions like the Sv(t) decays observed here, these three times provide different characterizations of the time dependence. The t\e values can be viewed as representing some overall time scale for the decay of Sv(t), whereas to and (r) respectively emphasize the shortand long-time behavior of the function. It should be noted that the uncertainties associated with these three characteristic times are not the same. The initial time constant is the least certain measure, since it is often determined by the behavior of the spectrum at times near to the resolution of our instrument, tq also depends on the accuracy of the zero-time estimates which we use to determine whether or not some small fraction of the spectral shift is unobservably fast. The values listed in parentheses under To are the initial times that would be calculated if these small-amplitude “30 fs” components are removed from Sv(t). Even though the amplitude in such components is usually less than 10%, their presence often makes a factor of 2 difference in the value of to. Thus, it unwise to view these numbers as anything more than a rough indication of the initial behavior of Sv{t). The situation with respect to t\e and (r) is considerably better. Based on repeated measurements in a single solvent, the reproducibility of these times is found to be on the order of ±15—25% (sample standard deviation). The expected accuracy in the reported fie values should be comparable to these figures, at least for the Q = 1,2 solvents. However, the (r) values are less accurate in some cases due to the influence of smallamplitude components with large time constants. In analogy to to, the presence of large time constants even when they have
amplitudes of less than 10% can have a dramatic impact on (t). Cases in which small-amplitude, long-time components appear in Sv(t) that we believe might not reflect solvation dynamics are indicated by the appearance of two values in the (r) column of Table 3. (It is possible that such components are due to the small spectral shifts resulting from vibrational relaxation on the 10—100 ps time scale.) In these cases the values in parentheses are values of (t) obtained if these longest components are ignored. In these cases (and in the analogous To cases) the recommended value is the one underlined. Figures 17 and 18 illustrate some of the spectral response functions observed in a variety of solvents. Several features of the data contained in these figures and in Table 3 are noteworthy. First, for no solvent do we observe the spectral response to be close to an exponential function of time. In many polar aprotic solvents (Figure 17) Sv(t) can be reasonably represented by a biexponential function, while in some aprotics and all hydrogenbonding solvents at least a triple-exponential fit is required. The components of this multiexponential behavior include widely differing times such that the relaxation in some solvents (for example, HMPA and most alcohols) extends over several decades. Finally, there is an impressive spread in the solvation times observed in these room temperature solvents. From Table 3 one finds 1/e times ranging over a factor of more than 103, from less than 200 fs in acetonitrile to more than 200 ps in 1-decanol. In the following section we will show how most features of these spectral/solvation response functions can be closely modeled in terms of the widely varying dielectric properties of the solvents examined.
For
an
=
X^T,
methanol
(“Ci”) through 1-pentanol (“Cj”).
17328
J. Phys. Chem., Vol. 99, No. 48,
1995
Homg et al. n-Alcohols
-2.
-1.
0.
log(r) Figure
solvents. These distributions are derived from the observed spectral response functions as described in the text. The numbers 1, 5, 2, 52, and 8 denote the solvents acetonitrile, dimethylformamide, dimethyl sulfoxide, benzonitrile, and hexamethylphosphoramide, respectively.
Before doing so, it is useful to view the dynamical content of these Sv(t) functions using an alternative to the direct timedomain representation of Figures 17 and 18. In a recent study, Vincent et air1 showed that helpful insight into the nature of such response functions can be provided by considering the
“spectral” decomposition, =
(4.10)
f~A(T)e-'lTdz
where A(z) is the distribution or spectrum of relaxation times contained in Sv(t). While A(z) is related to Sv(t) by an inverse Laplace transform, direct inversion of eq 4.10 generally suffers from numerical instability. However, Livesay and co-workers have shown that a maximum entropy (MEM) analysis provides a numerically sound alternative to direct inversion in a variety of experimental situations.122 The underlying idea behind MEM analysis is that there will in general be an infinite set of A(r) functions that reproduce the experimental data (Sv(f) here) to within its uncertainties. Given this inherent uncertainty in A(r), Livesay and co-workers122 have shown that the best choice of A(t), and the one which should be free of artifacts introduced by numerical inversion, is the one that maximizes the “Shannon—Jaynes” entropy,
£[A]
=
-/0°°A(r) ln{rA(r)} dr
(4.11)
subject to the fit criterion, M
]T (S>bs)
-
Sgcalcfr,)}2
>
Mo2
(4.12)
;=i
where {5v'(obs)} represent the M observed values of Sv(t) and a is the uncertainty in these values (assumed equal). For the present problem we also add the further constraint that the calculated Sv(t) be unity at t = 0; namely, we require f£A(z)
dr
=
2.
3
l°g(T)
19. Relaxation time distributions (A(r), eq 4.10) in polar aprotic
Sv(t)
1.
1.
Representative relaxation time distributions determined in this manner are shown in Figures 19 and 20.123 In nearly all cases, these distributions show two clear maxima, one lying in the range 100—300 fs and the other at longer times, typically r > 1 ps. In a few solvents, such as acetonitrile and dimethylformamide (nos. 1 and 5 in Figure 19), it appears that the longer time peak has move to sufficiently short times that the two peaks are partially merged. Nevertheless, clear shoulders are seen in
Figure 20. Relaxation time distributions (A(r), eq 4.10) derived from the Sv(t) response functions in 1-alkanol solvents shown in Figure 18.
both cases, indicating the same underlying bimodal character of A(t) as in the other solvents. It seems quite natural to associate the two major features observed in these relaxation time spectra with the inertial and diffusive components of the solvation response. Support for this interpretation comes from comparing the times observed for the faster spectral component to the times anticipated for inertial solvation, which can be calculated approximately based on the inertial properties of solvent molecules.56'65 124 Numerous computer simulations have shown3-39-45'50'51 that the initial portion of the solvation response is Gaussian function of time, S(t)
at
exp(-‘/2cusV)
(4.13)
characterized by a solvation frequency w%. In the case of solvation of ionic solutes, this frequency has been shown to be accurately represented by56 t«s2
-
(47rp/t2/3/eff)/(60,n)
(4.14)
where p, u, and /etf are the number density, dipole moment, and effective moment of inertia of solvent molecules, mdJ{eo,n) is a dielectric factor close to unity. (See refs 56 and 124 for further details.) Using the 1/e times for the decay of a Gaussian of the form of eq 4.13 (tie = 1.4cus-1), the inertial times estimated from eq 4.14 for the solvents in Figure 19 are 0.13, 0.20, 0.17, 0.41, and 0.41 ps (for acetonitrile, dimethylformamide, dimethyl sulfoxide, benzonitrile, and HMPA, respectively). The maxima in the observed A(r) distributions occur at ~0.1, ~0.3, 0.20, 0.45, and 0.51 ps. This general agreement provides rather convincing evidence that the origin of the maxima at shorter times is indeed inertial solvent dynamics.125 Figures 19 and 20 thus point to the approximate separability of inertial and diffusive parts of the solvation response. They also serve to highlight the relative importance of inertial solvent motions. In the case of polar aprotic solvents, whenever two peaks could be clearly distinguished in A(r), the relative area of the inertial peak was found to be 50 ± 6%. The only exception to this observation is HMPA, shown in Figure 19, for which the short-time component provides only 37% of the total response. Thus, it is not only in “special” small-molecule solvents that inertial solvation plays an important role. Approximately equal contributions from inertial and diffusive solvation components appear to be the norm among polar aprotic solvents. In the alcohol solvents, on the other hand, inertial solvation typically plays a less important role. The trend shown in Figure 20 is for a smaller and smaller fraction of the solvation response being attributable to inertial dynamics as the length
Polar Solvation Dynamics of Coumarin 153
J. Phys. Chem., Vol. 99, No. 48, 1995
of the alkyl chain increases within the 1-alkanol family. (In the other hydrogen-bonding solvents studied the percentage of the total area in the area A(r) distributions associated with the inertial peak are ~40% in M-methylformamide, ~30% in ethylene glycol, and ~55% in formamide.) A final feature of these distributions that is noteworthy is the fact that even the 1-alkanols exhibit a primarily bimodal response. This is in spite of the fact that accurate fits of Sv(t) require at least a tripleexponential function (as does their dielectric response discussed in the following section). This observation suggests that it may not be necessary (or helpful) to think of solvation/dielectric relaxation in these solvents as resulting from a superposition
of several discrete exponential processes,
as was
originally
proposed.126,127
IV.D. Comparisons to Predictions of Dielectric Theories of Solvation Dynamics. In this final section we compare the 5v(f) results to the results of several theories of solvation dynamics. We will show that, in almost all solvents examined here, the spectral dynamics embodied in Sv(t) can be reasonably understood in terms of the bulk dielectric properties of the solvent, assumed known from other experiments. Theoretical analysis of the relationship between polar solvation and dielectric the past few relaxation has been extensive over years, such that highly sophisticated theories are currently available.53-55’58 Unfortunately, the most accurate of these recent treatments, that due to Friedman, Raineri, and co-
workers,58 requires non-trivial XRISM calculations for its implementation. We will therefore postpone tests of this and related theories to future work. Here we limit ourselves to two of the more elementary theories of solvation dynamics: the simple continuum (SC)65’128 and dynamical mean spherical approximation (DMSA)129-131 models. These two theories, whose predictions are readily calculated for most solvents, suffice to demonstrate the fundamental connection between the dynamics measured in these experiments and the dielectric behavior of the pure solvent.
Both the SC and DMSA theories model the solute as a spherical cavity containing a centered point charge or dipole which is responsible for its coupling to the solvent. In the simple continuum model, the solvent is treated as a continuum fluid characterized entirely by its bulk dielectric response, e(co). This theory is therefore a dynamical extension of the sort of reaction field approaches already discussed in section IV. A. The SC theory is a limiting theory which should become exact Previous as solute/solvent size ratio approaches infinity. 56 studies have shown that SC and simulation50 experimental3'4 predictions can be useful guides to the dynamics observed in many situations. However, the fact that the solvent is not a continuous fluid, but is in reality composed of discrete polyatomic molecules comparable in size to the solute, is expected to cause the behavior of real systems to depart in significant ways from that predicted by SC models. The DMSA theory was the first to account for some of the effects of solvent means of an approximate dipolar hard-sphere of the solvent. By doing so, this theory predicts representation
molecularity, by
generally slower relaxation and greater sensitivity to solute attributes than does the SC theory. Discussions of the nature of the approximations implicit in the SC and DMSA theories are available in the literawjjj therefore only sketch how the calcula{m-g 3,6,7,38,105,132 tions are performed. Both dynamical theories derive from equilibrium counterparts which assume that the solvation energy [/ei of a solute containing a point multipole moment m can be written in the form
=
tfd
-®m2Xm(« 0)
17329
(4-15)
where Xm(€o) represents the susceptibility of the solvent to polarization by the solute’s multipole. %m(€o) is a function of solvent polarity, symbolized by the argument to. and it also depends on solute/solvent size and shape factors that determine the solvation structure. For example, the SC result in the case of an ion of radius a is X0(€Q)
=
l/ae0
(4.16)
Equations 4.15 and 4.16, with 4>o the solute charge, is just the well-known Bom equation for ionic solvation. Whatever the form of the susceptibility, as long as x.m(eo) does not depend on the magnitude of the solute multipole moment, Wolynes showed that a simple prediction for the dynamics of solvation can be obtained simply by introducing the frequency-dependent dielectric response of the solvent e((o) in place of the static dielectric constant 6q.129 In particular, the solvation response S(t) to a step function change in the solute moment m can be expressed
as130
5(0
-
-
x&fpwp}
(4-i7)
= represents an inverse Laplace transform and e(p) ?(im). Thus, given an empirical characterization of the solvent’s dielectric response t(co) and a function £m(eo), one can compute S(t). Within the simple continuum theory £m(eo) can be readily derived for any multipole order m.133 However, for the molecular MSA theory, convenient results are available for only an ionic solute of arbitrary size and for a dipolar solute of the same size as the dipolar hard spheres used to represent the solvent. (Explicit expressions for the ion and dipole susceptibilities are given in ref 38). We therefore confine ourselves to ionic (“i”) and dipolar (“d”) solutes in both theories and compare the experimental results to four different predictions: “SC-i”, “SC-d”, “DMSA-i”, and “DMSA-d”. Before making comparison between experiment and the predictions of these four theories, we must make a lengthy digression in order to describe the dielectric data employed for such comparisons. A summary of these data is provided in Table 4.134-148 In this table we list the parameters derived from fits of dielectric data to functions of the form149
where
m
=
e-mw
+ (€„
-
«-mw)5>/i(P)
(4-18)
where
m
i
=
[1
+
(prtaf
(4.19)
As denoted in the column labeled “fnc” in Table 4, three different types of function are used to represent individual dispersion regimes. The Debye (“D”) function is the simplest; it corresponds to eq 4.19 with a = 0, /? = 1. (“2D” and “3D” denote fits to multiple Debye functions.) The other two representations are generalizations of the Debye form. The Cole—Cole (“CC”) form corresponds to eq 4.19 with a > 0, /3 = 1. In the Cole—Davidson form a = 0, /J < 1. In addition to the parametrizations themselves, under the section labeled “quality indicators”, we have provided some measures of the likely accuracy of these data for our purposes. The columns “vmax” and “v ratio” are the maximum frequency used in the measurements and the ratio of this frequency to the frequency = corresponding to the smallest dielectric time detected, vratl0 vmax/(rmin/27r). Values of this ratio greater than unity should
17330
J. Phys. Chem., Vol. 99, No. 48, 1995
Homg et al.
TABLE 4: Summary of Dielectric Dispersion Data (298 K) Used in Solvation Dynamics Calculations fit parameters"
quality indicators4 solvent"
no.
ratio
i
acetonitrile dimethyl sulfoxide propylene carbonate dimethylformamide nitromethane
4
acetone
7
HMPA dimethyl carbonate
9
1.74 1.48 1.44 1.47 1.69 1.56 1.10 1.47
10
1.51
13
1.44 1.63 1.52
methvl acetate tetrahydrofuran dichloromethane chloroform methanol ethanol 1-propanol 1-butanol (296 K) 1-pentanol 1-decanol
2
5
6 8
16 19 31
32 33 35 36 37 39
iV-methylformamide 40 ethylene glycol form amide 41 benzonitrile 52
1.9 1.67
1.52 1.41 1.33 1.09 1.6 1.59 1.79 1.35
vmax
(GHz) 89 89 89
293 32 89 33 88 89 136
250 300 293 89 89 15
/fir
Q
0.5 0.5 0.5 0.3
2
0.8 “0” 2 0.2 16 0.4 4 0.5 2 0.4
3
3
0.1
2
3
0.2
i
l"’ratio
1.9 11
24 1.4
14 11 1
1.3
“50”
10
2
10
0.3
293 70
1.4
89 88
4.8 0.6 19
“0" 0.4 0.3 0.2 0.3 0.1 0.1
0.4 0.5 0.5 0.5
1 1
1
1
1
1 1
fnc
ID CD CD 2D CC CC CD ID
ID
i
CC CC CD 3D 3D 3D CD 3D 3D 3D 2D CC + D
3
ID*
2
i i i 3
2 2 1
2
€„mw
€0
35.84 46.4 64.88 37.25 36.15 20.67 29.6 3.12 6.66 8.05 9.199 4.719 32.5 24.32 20.43 18.38 14.68 7.201 183.3 41.65 108.8 25.1
3.51
4.16 4.14 2.84
a
2
a3
1 1 1
0.954 0.046
1.913
2.29 3.3 2.35
2.56 2.18 2.436 2.08 2.79 2.69 2.44 2.72 2.137 2.29 3.2 3.95 4.48 4.5
D (ps)
3.37 20.5 43.1 10.36 3.83 3.06 79.7 7.23 3.54
*2 (ps)
T) (ps)
0.895 0.034 0.071 0.917 0.031 0.052 0.928 0.030 0.042 0.917 0.876 0.984 0.905 0.975
0.045 0.038 0.077 0.048 0.008 0.008 0.095 0.025
1
163
329 528.4 754 1340 128
122.5 37.3 35.3
/3
0.888 0.904
0.74 0.07 0.02 0.92
0.065 0.032 0.85
3
1.97* 7.2 51.5
a,
7.09 8.97 15.1
1.12 1.81
142
143/137 135
2.4
135
2.37 3.98 0.78
144 145
11
0.0057
1.16
141
135
0.924 25.8 76.1 7.93
ref 134 135 135 136 137 136 138 139 140
145 146 147 146 148
“Size ratio” indicates the relative solute/solvent (linear) dimensions, calculated from van der Waals volumes104 as {246 ATV(solvent)}1'3. 4 vmax is the maximum frequency at which data were collected. “vra,i0" is the ratio vmax/(rmm/2;r) where rmin is the smallest relaxation time observed. This ratio should provide some indication of how well the dielectric dispersion has been resolved by the reported measurements. (Note, however, that in cases where vmax is small entire dispersion regimes may be missed, leading to artificially large values for this ratio, as in the case of 1-butanol.) /fir is the fraction of the solvation response expected to result from parts of e(w) at FIR and higher frequencies and thus “missing” from the parametrizations listed here. This estimate is based on the approximate continuum calculation, eq 4.20. Q is an indicator of the anticipated quality of the dielectric results as they pertain to solvation dynamics calculations. A value of 1 indicates the highest quality data. Parameters of the fits to the dielectric data according to eqs 4.18—4.19. “Fnc” indicates the type of function used with “nD” indicating n Debye type (eq 4.19 with a = 0, /? = 1) dispersions, “CD” indicating a Cole-Davidson type dispersion (a = 0. jl < 1), and “CC” a Cole—Cole dispersion (a > 0, /? = l),149 In the case of dichloromethane the value for ri at 298 K was interpolated from data at nearby temperatures. In the case of benzonitrile the data were only fit to a Debye distribution even though it was clear that such a fit was inadequate at high frequencies. 0
1
provide some measure of confidence that the range of frequencies employed is sufficient to adequately capture the observed part of e(w). (Some skepticism should be applied when in viewing these numbers. If vmax is too small, fast dispersion regimes may be missed entirely, leading to inflated values of this ratio, as appears to be the case in 1-butanol.) On the basis of this measure, the data in Table 4 appear to be of high quality, with most solvents having values of v ratio greater than two. In fact, we have selected most of the solvents for our dynamical studies based on their having already been characterized by highly resolved dielectric measurements. We note that the bulk of these high-quality dielectric measurements have only become available over the past few years, due mainly to the systematic measurements performed by groups of Barthel and Petrucci. There is, however, one very important difficulty with using only the parametrizations of e(a>) obtained from dielectric measurements for solvation dynamics calculations. Even though most of the data show excellent frequency resolution for the diffusive part of e(w), they do not capture inertial aspects of the solvent dynamics. The latter occur at frequencies higher than 103 GHz, in the far-infrared (FIR) region of the spectrum. As we showed in the previous section, inertial solvent dynamics comprise roughly half of the solvation response in many polar aprotic solvents, and it would be inappropriate to neglect their contributions when making comparison to theory. Some idea of how important the portion of e(co) missed in the dielectric parametrizations of Table 4 can be obtained by considering the SC predictions for ion solvation. When different dispersion regimes are sufficiently distinct in frequency, the relative contribution of each regime “t” to S(t) is, according to SC-i predictions,150
are high- and low-frequency limits of regime in eq 4.18 are proportional to (€o, €»»,).) The correct high-frequency limit (€«,) in eq 4.20 is not the same as the one listed in Table 4, which is only the apparent limit based on measurements in the gigahertz regime. Rather, the limit required is the value of e'(w) at frequencies higher than that of all nuclear motions contributing to the solvation response (i.e., at frequencies faster than all those motions unable to follow the electronic as no2. In transition). This value is approximately given by the column labeled “/fir” in Table 4 we list estimates of the fraction of the solvation response contained in the FIR portion of e(aj) based on eq 4.20. As can be seen from these values, a very sizable portion of the solvation response, especially in polar aprotic solvents, is predicted to come from parts of e(cu) that lie outside of the range of dielectric experiments. Thus, these missing portions of e(w), which come primarily from resonance terms in the FIR region, cannot be neglected if accurate comparisons to theory are to be made. Complete characterizations of e(oj) throughout the microwave, FIR, and higher frequency regions are available for only a few solvents. Thus, in most cases one cannot use such comprehensive data as input to solvation dynamics calculations. Luckily, however, one can still make accurate theoretical predictions based on the expected behavior of e(w) outside of the range of the dielectric parametrizations listed in Table 4. In Figure 21 and Table 5 we use the examples of acetonitrile and methanol, two solvents for which good FIR data are available, in order to illustrate how this can be done. For both solvents we have spliced the microwave data listed in Table 4 together with determinations of e(w) in the FIR region (to 250 cm-1 in
where i.
eo;
(The a,
and
-
Polar Solvation Dynamics of Coumarin 153
J. Phys. Chem., Vol. 99, No. 48, 1995
17331
DMSA-d predictions but pronounced in the SC-i predictions. Such oscillatory behavior of the predicted decays is probably more apparent in acetonitrile than in most other solvents, both because/hr is largest for this solvent and because the librational dynamics observed in the FIR would be expected to be more “coherent” for a small linear molecule compared to an asymmetric top (such as in the cases dimethyl sulfoxide, propylene
in
carbonate, dimethylformamide, etc.) For example, even though is nearly as large in the case of methanol, we find that oscillations in S(t) are hard to discern in a plot such as Figure 21. In addition, even in acetonitrile the oscillatory behavior of 5(f) is expected to be relatively unimportant for a solute like C153. The oscillations predicted by the SC-i model are rather similar to what is observed in molecular dynamics simulations of the solvation of small ions in acetonitrile.50,51 However, simulations of large polyatomic solutes like C153 in acetonitrile show only oscillations in 5(0 that are much smaller than the SC-i prediction and probably not discernible in our experiments.50,155 Thus, we conclude that while the inclusion of FIR resonances in i(a>) should in general lead to some oscillatory behavior in the 5(0 predictions, such oscillations are not likely to be important in the experimental measurements. For this reason, it is not necessary to have an accurate characterization of this portion of the dielectric spectrum in order to make reliable theoretical comparisons. What is mainly needed is to account for the amplitude of the drop in 5(0 contributed by the FIR modes.
/fir
Time (ps) Figure 21. Solvation response functions in acetonitrile computed from the SC-i (top) and DMSA-d (bottom) theories using several different representations
full
e(cu)
of e(a>). The solid curves show calculations using resonance terms from the FIR region of
including
the the
spectrum.15'154 The short dashed curves use an approximate e(co) in which the FIR contributions are subsumed into a single Debye relaxation having a longitudinal relaxation time of 100 fs. The long-dashed curve is the result of completely neglecting the FIR components of the dielectric spectrum. (See text for details.) For sake of comparison the experimental Sv(t) function is shown by the dotted curves in both
panels.
acetonitrile151 and 1000 cm
1
in the
case
of methanol152) to the
form
m
=
nD2
+
«.mw)5>^(p) +
~
(e0
5>//(p)
) are completely neglected. a
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J. Phys. Chem., Vol. 99, No. 48, 1995
Figure 22. Comparison of characteristic times of the observed spectral response functions with those predicted for the solvation response by the SC-i and SC-d theories. Circles denote polar aprotic solvents, and squares denote hydrogen-bonding solvents. (The solvents for which this comparison can be made are those for which dielectric data are listed in Table 4. In addition, data points corresponding to solvation of C343 in water, obtained by Jimenez et al.3' have been included.) times and filled symbols (r) times. The lines Open symbols denote
Horng et al.
Time (ps) Observed spectral response functions (solid curves) compared to the predictions of the DMSA-i theory (dashed curves) for several solvents. The solvents shown are 1-pentanol (36), hexamethylphosphoramide (8). methanol (31), dimethyl sulfoxide (2), and acetone (7). Successive pairs of curves have been vertically displaced by 0.25 for clarity. Zero levels are indicated at the right of the figure.
Figure 24.
indicate agreement between observed and predicted times.
Figure 25. Ratios of characteristic limes of S(t) predicted by the DMSA-i theory to those observed in experiment. Numbers indicate different solvents according to the designations listed in Table 1. For each solvent the left-hand bar is the comparison for the rlc time, and the right-hand bar is that for the (r) time.
Figure 23. Comparison of characteristic limes of the observed spectral response functions with those predicted by the DMSA-i and DMSA-d theories. Circles denote polar aprotic solvents, and squares denote hydrogen-bonding solvents. (The solvents for which this comparison can be made are those for which dielectric data are listed in Table 4. In addition, data points corresponding to solvation of C343 in water, obtained by Jimenez el al.3' have been included.) Open symbols denote fic times and filled symbols (r) times. The lines indicate agreement between observed and predicted times.
point out that complete neglect of the resonance portion of I(fti) (as we and others have done in past comparisons between theory and experiment3-5-11) is unacceptable. As illustrated by the long dashed curves in Figure 21 and the last two columns of Table 5, if one takes for the frequency dependence of l(£t>) only the microwave results (i.e., assume only eq 4.18 and the parameters in Table 4), the predicted dynamics are much slower that those using the full £(a>). In both solvents the values of (r) are in error by more than 50% while the t\t times are 2-3fold larger than the values predicted using the full dielectric response! We now
return to comparisons between the theoretical and the observed Sv(t) data. These comparisons are predictions carried out in Figures 22—25 and in Table 6. Figures 22 and 23 show plots of observed versus predicted values for the characteristic times f|e (open symbols) and (r) (filled symbols) in a total of 23 solvents. One of the most obvious conclusions to be drawn from the comparisons in Figures 22 and 23 is that
there is a reasonably good correlation between the observed times and the times calculated from all of the four theories. The observed times vary by nearly a factor of 2000 over this series of room temperature solvents, and all of the theories provide a reasonable account of this variation. This achievement is by no means trivial. Thus, whereas within a solvent series (such as the n-alkanols) the observed times can be reasonably well correlated to various solvent properties such as viscosity, we have been unable to find any nondielectric property that can adequately explain the enormous variations observed for the whole collection of solvents. The agreement exhibited here demonstrates that (i) the spectral response functions measured in experiment are indeed reporting mainly on the phenomenon of polar solvation dynamics and (ii) theories based on the dielectric response of the pure solvent provide an excellent starting point for understanding such dynamics. There are quantitative differences between the quality of the predictions made by the four theories, which can be most easily seen from the summaries provided in Table 6. In all cases the times predicted by these theories increase in the order r(SC-i) < r(SC-d) < r(DMSA-i) < r(DMSA-d). We note that in most instances the DMSA-d predictions are much slower than the predictions of all of the other three theories while the times predicted by the SC-d and DMSA-i theories are often similar. As shown in Table 6, the relative ranking of calculated times leads to a ranking in the quantitative agreement of these theories
Polar Solvation Dynamics of Coumarin 153
J. Phys. Chem., Vol. 99, No. 48, 1995
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TABLE 6: Average Ratios between Calculated and Observed S(t) Times (log{r(calc)/r(obs)})6 data set0
all solvents
no.
SC-i
SC-d
DMSA-i
DMSA-d
44
-0.25 ± 0.24
-0.14 ±0.22
±0.03 ±0.16
±0.44 ±0.21
(0.56)
polar aprotics
24
-0.21 ±0.18
hydrogen bonding
18
-0.31 ±0.29
(0.61) (0.49)
(0.73)
(1-07)
-0.11 ±0.16
±0.05 ±0.11
(0.78)
(1.11)
-0.18 ±0.27
±0.02 ±0.19
(0.66)
(1.04)
(2.8) ±0.41 ±0.17 (2.6) ±0.52 ±0.19 (3.3)
The data sets examined here include a combination of both the fie and (r) times. The “all solvents” set includes all 22 of the solvents listed in Table 4. The “polar aprotic” set is the subset of these between 1-19 and 52 but excluding 9 (dimethyl carbonate). The “hydrogen bonding” set are those between 31 and 41. b The first entry is the average of the logarithms of the ratios of calculated to observed times and the standard deviation in this value. The values listed in parentheses are the average ratios calculated from these logarithmic averages. 0
with experiment. On average the SC-i and SC-d theories predict times that are respectively ~80% and ~40% too fast, and the DMSA-d theory predicts times too slow by nearly a factor of 3, while the theory/prediction ratio for the DMSA-i theory is within 10% of unity. In addition, there is noticeably less scatter in the correlation between the experimental and calculated times in the case of the DMSA-i theory. From these comparisons we conclude that the DMSA-i theory is clearly the best choice for predicting the experimental dynamics. We note that in fact the quality of the predictions made by the DMSA-i theory is remarkably good. Based on the statistics in Table 6, a calculation of a z by the DMSA-i theory enables one to say that the actual solvation time should be within the range 0.6t—2.0t with approximately 90% confidence. This level of agreement for the calculated times also extends to the prediction of the complete Sv(t) response function. Figure 24 shows comparisons between observed and calculated response functions for five typical cases. While the agreement is imperfect, in nearly all cases we find that the general features of the response functions (or equivalently the A(r) distributions) are reasonably captured by the DMSA-i predictions. The behavior of hydrogen-bonding solvents in these comparisons is also noteworthy. As was found in the case of the steady-state spectral shifts (section IV.A), there is little evidence that the solvation dynamics monitored by Sv(t) is significantly different in polar aprotic versus hydrogen-bonding solvents. (The two classes of solvents are denoted by circles (aprotic) and squares (protic solvents) in Figures 22 and 23.) Thus, the statistics in Table 6 for the average ratios of predicted to observed times show no differences between these two solvent classes for any of the theories. The only difference noticeable is that the ratios for hydrogen-bonding solvents are somewhat more scattered than those for the polar protic solvents. This feature is readily seen in Figure 25 where ratios relative to the DMSA-i predictions are plotted for each solvent. But there are no solvents or solvent groups (such as for example the 1-alkanols) that clearly stand out from the general scatter. For example, although the magnitude of the time-dependent shift of C153 in 1-decanol deviates from the dielectric predictions (see for example Figure 9), the dynamics appears unremarkable. Thus, as with the steady-state spectral shifts, we conclude that specific hydrogen bonding between C153 and hydrogen bond donating solvents has little impact on the dynamics observed. Theories that begin with the pure solvent dielectric response and completely ignore specific solute—solvent interactions are equally successful at predicting the behavior in both hydrogenbonding and non-hydrogen-bonding solvents. Before concluding this section, some comments should be made regarding the expected accuracy of the theories examined. While the good correlations between the predictions of all four theories and the observed results point to the fundamental
“dielectric” nature of solvation dynamics, none of these e(co)based theories can be considered a complete description. Electronic excitation of C153 leads primarily to a change in its dipole moment. One would expect that the dipolar version of the DMSA theory should most closely represent the experimental situation. In fact, the DMSA-d theory yields the poorest quantitative agreement with experiment, generally overestimating the solvation times by a large factor. Some part of this disagreement can be attributed to the fact that a size ratio of unity is assumed for calculational purposes whereas a more appropriate ratio of 1.5 would yield somewhat faster predictions. However, since the DMSA-d theory always predicts significantly slower dynamics than the corresponding DMSA-i calculations (for equal size ratio) and since the DMSA-i times are very close to those observed, this approximation cannot be the main source of the discrepancy. The difficulty probably lies instead in what features of the experimental solute/solvent systems are neglected in the DMSA model. Both the ion and dipole versions of the DMSA theory have been shown to provide highly accurate predictions for solvation dynamics simulated in translationally immobile, point multipole solvents.38 (Simple continuum models, especially the SC-d model, predict dynamics that are far too fast in such systems.38) It is therefore reasonable to conjecture that it is the presence of solvent translational motions and an extended charge distribution in real systems that lead to the poor quantitative performance of the DMSA-d model. Translational solvent motions, while not accounted for in e(co) or the DMSA theory, are expected to hasten the solvation response to an extent that should be greater for a dipolar versus a monopolar solute perturbation.7,55 This neglect of translational contributions to the response is probably the main reason for the poor predictions of the DMSA-d theory. The fact that the solvent (and solute) molecules are not point dipoles but rather contain extended charge distributions will also influence the dynamics, but in a way that is more difficult to predict a priori. Thus, we can view the remarkable agreement shown between the experimental solvation dynamics and the predictions of the “wrong” molecular theory, the DMSA-i theory, or the SC-d theory, as being due to a fortuitous cancellation of errors. Simple continuum models neglect all aspects of the molecular nature of the solvent not already contained in e(tu). Some of these features, in particular the structural aspects of solvation, by way of its influence over intermolecular correlations,38,50,56 result in a slowdown of the response compared to SC predictions. These features are adequately modeled by the DMSA theory. However, other neglected features, such as the operation of translational relaxation mechanisms not represented in e(o») and in either the SC or DMSA theories, serve to bring the behavior of the real system closer to the SC-d or DMSA-i predictions. It will be of considerable interest to see whether more sophisticated e(cu)-based theories,58 which incorporate a greatly improved representation of the solvation structure as well ~
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J. Phys. Chem., Vol. 99, No. 48, 1995
approximate treatment of solvent translations, can lead to more quantitative agreement between experiment and theory. For now we note that the DMSA-i theory, although incomplete, is a useful predictive tool. Finally, we should mention the fact that no e(ai)-based theory, not any of the four discussed here nor more sophisticated theories, is adequate to predict the dynamics in some of the solvents we have studied. In the case of the “nondipolar” solvents dioxane and benzene, for which we observe what appears to be much the same dynamics as are observed in weakly polar solvents like tetrahydrofuran, dielectric data are unavailable. More importantly, as discussed in section IV.A, the absence of a significant dipole moment in these solvents means that %[e(a>)] (the dynamic equivalent of F(eo,n)) is not a useful starting point for considering solvation.156 Calculations that begin with the full wave-vector-dependent dielectric function of the solvent will be necessary, and the latter will need to be obtained theoretically rather than by experiment alone. Such solvents therefore represent a new theoretical challenge that should serve to further enhance our understanding of molecular aspects of the solvation process. as an
V. Summary and Conclusions
In this paper we have presented the results of a systematic study of both the static and dynamic aspects of solvation of coumarin 153 in a variety of common room-temperature solvents. Studies of the steady-state solvatochromism of Cl53 show that this molecule provides an excellent probe of the energetics of solvation, uncomplicated by interference from either multiple excited states or specific solute—solvent interactions. Emission spectra, recorded with
