J . Phys. Chem. 1990, 94, 4929-4935 thermolysis rate for thin polymer films was more strongly time dependent (a = 0.34) than for polymer solutions (a = 0.66). The poorly stirred conditions for these reactions suggest that classical chemical kinetic models are not appropriate.
Acknowledgment. The helpful suggestions of an anonymous
4929
reviewer are gratefully acknowledged. L. A. Good and A. J. Twarowski assisted in collecting and modeling the data presented in this paper. Registry No. DMNE, 35461-84-8; VNE, 126950-45-6; VNE (homopolymer), 126950-48-9; 0, 7782-44-7.
Solvation Dynamics in N-Methylamides Curtis F. Chapman, Richard S. Fee, and Mark Maroncelli* Department of Chemistry, 152 Davey Laboratory, The Pennsylvania State University, University Park, Pennsylvania 16802 (Received: October 19, 1989; I n Final Form: January 9, 1990)
Solvation times in three homologous amides, N-methylformamide, N-methylacetamide, and N-methylpropionamide, have been detd. from measurements of the dynamic Stokes shift of the fluorescence spectra of two-probe solutes, prodan and coumarin 102. Single-particle reorientation times have also been measured in one of these solvents, N-methylformamide, by using NMR methods. The solvation dynamics are compared to two theoretical models, the simple continuum model and the dynamic MSA model. Although neither model predicts the time dependence of the response satisfactorily, the average solvation times observed are close to the solvent longitudinal relaxation time (TL) predicted by the simple continuum model. In contrast, the predictions of the dynamical MSA model are approximately 4 times slower than the observed solvation response. The failure of the latter model appears to result from an overestimation of the single-particle reorientation times of these solvents. Esimates of such single-particle times based on the NMR measurements are within a factor of 2 of TL. This similarity seems to account for the near equality of average solvation times to T L in the amides.
introduction In the past few years quite a number of experimenta11-8 and t h e o r e t i ~ a l ~studies -~l have considered the dynamical aspects of solvation in polar liquids with a view toward understanding the dynamical coupling between polar solvents and charge-transfer reactions.22 In such studies, the central focus has been on measuring how rapidly a solvent responds to changes in the charge distribution of a solute molecule and on understanding what solvent and/or solute attributes determine this response time. Work in this area has been summarized in several recent review article^:^-^^ so here we will only highlight some of the main results that have so far emerged. Theoretical treatments of the dynamics of polar solvation can be roughly divided into two approaches, which differ in the sophistication with which they model the solvent. The earliest treatments, provided by Bakshiev9 and Mazurenko,Io typify the first approach, which we will refer to as "simple continuum" model^.^-^^ Here the most elementary model of the solvent is adopted, that of a continuous, homogeneous fluid whose only relevant property is its bulk, frequency-dependent dielectric response. In such models the dynamics are simply related to solvent dielectric properties and are relatively insensitive to the solute attributes.12J6 All simple continuum models predict that solvation should proceed on a time scale that is approximately equal to the longitudinal relaxation time of the solvent, TL. This longitudinal relaxation time is a bulk dielectric property of the solvent that is much faster than the dielectric relaxation time T D (see Discussion). The T~ prediction of simple continuum models has served as an important benchmark against which to view experimental data. The second group of goes beyond a simple continuum representation by recognizing the molecular nature of the solvent in some way. The first such model was proposed by WolynesI5 and further developed by Rips et al.'8919and Nichols and Calef.20 This theory describes the solvent in terms of a n equivalent dipolar hard-sphere system whose equilibrium structure can be calculated approximately within the mean spherical approximation (MSA). Although this "dynamical MSA" model of Author to whom correspondence should be addressed. 0022-3654/90/2094-4929$02.50/0
is only one of several molecular models for the dynamics,16*20,21 it is representative of them all, and a t present it is the one most easily applied to the experimental data. The dynamical M S A model predicts much more complex dynamics than the simple continuum models. Rather than there being a single relaxation time such as T ~ this , molecular model predicts that a range of relaxation times, roughly spanning the range between T L and TD, should be present in the solvation response. ( I ) Castner, Jr., E. W.; Maroncelli, M.; Fleming, C . R. J . Chem. Phys. Castner, Jr., E. W.; Bagchi, B.; Maroncelli, M.; Webb, S. P.; Ruggiero, A. J.; Fleming, G. R. Ber. BunsenGes. Phys. Chem. 1987, 92, 1987, 86, 1090.
363. (2) Maroncelli, M.; Fleming, G. R. J . Chem. Phys. 1987, 86, 6221. (3) Su, S.-G.; Simon, J. D. J . Phys. Chem. 1987,91,2693. Simon, J. D.; Su,S.-G. J . Chem. Phys. 1987.87, 7016. (4) Su, S.-G.; Simon, J. D. J . Phys. Chem. 1989, 93, 753. (5) Nagarajan, V.; Brearley, A. M.; Kang, T.-J.; Barbara, P. F. J . Chem. Phys. 1987,86,3183. Kahlow, M. A.; Kang, T. J.; Barbara, P. F. [bid.1988, 88, 2372. Jarzeba, W.; Walker, G. C.; Johnson, A. E.; Kahlow, M. E.; Barbara., P. F. J . Phys. Chem. 1988, 92, 7039. (6) Kahlow, M. A.; Jarzeba, W.; Kang, T.-J.; Barbara, P. F. J . Chem. Phys. 1989, 90, 151. (7) Kinoshita, S.; Nishi, N.; Kushida, T. Chem. Phys. Lett. 1987, 134,605. ( 8 ) Declemy, A.; Rulliere, C. Chem. Phys. Lett. 1988, 146, I . (9) Bakhshiev, N. G. Opt. Spectrosc. (USSR) 1964, 16, 446. (10) Mazurenko, Yu. T. Opt. Spectrosc. ( U S S R ) 1974, 36, 283. (11) Bagchi, B.; Oxtoby, D. W.; Fleming, G . R. Chem. Phys. 1984, 86, 257. (12) Castner, Jr., E. W.; Fleming, G . R.; Bagchi, B. Chem. Phys. Lett. 1988, 143, 270; 1988, 148, 269. (13) van der Zwan, G.; Hynes, J. T. J . Phys. Chem. 1985, 89, 4181. (14) Loring, R. F.; Yan, Y. J.; Mukamel, S. J . Chem. Phys. 1987, 87, 5840. Loring, R. F.; Mukamel, S. Ibid. 1987, 87, 1272. (15) Wolynes, P . G . J . Chem. Phys. 1987, 86, 5133. (16) Freidrich, V.; Kivelson, D. J . Chem. Phys. 1987, 86, 6425. (17) Castner, Jr., E. W.; Fleming, G. R.; Bagchi, B.; Maroncelli, M. J . Chem. Phys. 1988.89. 3519. (18) Rips, 1.; Klafter, J.; Jortner, J. J . Chem. Phys. 1988, 88, 3246. (19) Rips, 1.; Klafter, J.; Jortner, J. J . Chem. Phys. 1988, 89, 4288. (20) Nichols 111, A. L.; Calef, D. F. J . Chem. Phys. 1988, 89, 3783. (21) Bagchi, B.; Chandra, A. J . Chem. Phys. 1989, 90, 7338, and refer-
ences therein. (22) See, for example: Hynes, J. T. J . Phys. Chem. 1986, 90, 3701. (23) Barbara, P. F.; Jarzeba, W. Acc. Chem. Res. 1988, 21, 195. (24) Simon, J. D. Acc. Chem. Res. 1988, 21, 128. (25) Maroncelli, M.; Maclnnis, J.; Fleming, G . R. Science 1989, 243, 1674. (26)
Maroncelli, M.; Fleming, G . R. J . Chem. Phys. 1988, 89,
0 1990 American Chemical Society
875.
4930 The Journal of Phjssical Chemistry. C’ol. 94, No. 12, 1990 Physically these different times are related to varying response times as a function of distance from the solute (see Discussion). Further. the precise mix of times observed depends on probe attributes such as size. The time dependence of solvation can be observed experimentally by measuring the time-dependent shift of the fluorescence spectrum of probe solutes after ultrafast e x ~ i t a t i o n . Electronic ~~ excitation of a solute causes a step-function change in its charge distribution and creates a situation in which the solvent surroundings are not in equilibrium with the solute. The subsequent shift in the fluorescence spectrum of the solute directly monitors the process of solvent relaxation about the excited-state probe. Such measurements have now been undertaken with a variety of probe/solvent and the following general observations can be made: (i) The dynamics are mainly a function of solvent. That is, differences observed between different probe solutes in the same solvent are typically much smaller than differences between different solvents or temperatures.1.4-h(It should however be noted that only modest variations in probe size and chemical constitution have so far been explored.) (ii) Relaxation is in general not a simple exponential process. That is, as predicted by molecular theories, no single time such as T L adequately describes the solvation process.2%6(iii) In most cases, however, the average solvation times measured in experiment are close to the T L (continuum) predicti0n.l” This observation is puzzling since more sophisticated theories such as the dynamical M S A model predict slower relaxation. (iv) Finally, in a few instances involving solvents with very high dielectric constants, solvation times much longer than 7L have been reported.2s8 We previously noted in these cases the deviation from T L apparently increases with the static dielectric constant of the solvent and does so i n a manner that is qualitatively in accord with the dynamical M S A predictions.26 Unfortunately. the data on which the latter observation depends are quite limited. To test and refine more realistic models of solvation dynamics, further measurements in high dielectric solvents are needed. The present work was undertaken to provide such data. Through observations of the time-dependent Stokes shift of two probe solutes, we have measured solvation times in the three amides: ;Y-methylformamide ( N M F ) , ,Y-methylacetamide (NMA), and iY-methylpropionamide (NMP). All of these solvents have dielectric constants of -200 a t room temperature. In the course of this work we have discovered our earlier conclusions concerning solvation in N M P to be incorrect. The roughly 10-fold difference between observed solvation times and T,. reported for this solvent arose from the use of erroneous values of T D . ~Our ~ present results show that in fact the solvation times in all three of these amides are approximately equal to T L , in accordance with the dynamics reported for many other solvents. W e analyze the dynamics observed in the amides in terms of predictions of both the simple continuum and the dynamical MSA models. Whereas average solvation times agree reasonably well with the simple continuum predictions, the MSA model yields very poor quantitative agreement with experiment in the present case. This failure of the latter model and the apparent success of the simple continuum model are related to the near equality of T~ and single-particle reorientation times in these solvents, which we have estimated from independent N M R measurements.
Experimental Methods Time-resolved emission data were collected by using a timecorrelated single-photon-counting apparatus. Fluorescence was excited in the region 350-360 nm by using the doubled output of a synchronously pumped dye laser. The laser system was composed of a cavity-dumped dye laser (modified Coherent Model 599) pumped by a mode-locked Nd:YAG laser (Coherent, Antares Model 70). With the pyridine-I dye (670-750 nm) used in these experiments the autocorrelation widths of the visible output were (27) Ware, W. R.; Lee, S. K.; Brant, G . J.: Chow, P. P. J . Chem. Phys. 1971, 54, 4729.
(28) Maronceili. M.; Fleming, G. R. J . Chem. Phys., in press.
Chapman et al. 8-12-ps fwhm. Emission from the sample was spectrally resolved by a 0.25-m monochromator (ISA H-10) prior to detection with a microchannel plate photomultiplier (Hamamatsu Model R2809U-01). The P M T signal was amplified (Hewlett-Packard 8447 F amplifier), conditioned by a constant-fraction discriminator (CFD, modified Tennelec Model 454), and then used as a stop signal for a biased time-to-amplitude converter (TAC. Tennelec Model 864). The start signal for the T A C was provided by observing a fraction of the excitation pulse on a fast photodiode ( T F K BPW28A), the output of which was also conditioned by the CFD. Finally, the T A C output was recorded on a multichannel analyzer (MCA, Nucleus PCA-4000AT). The instrumental response of this system, determined by using a scattering solution, was typically 48-52-ps fwhm. Time-resolved fluorescence spectra were reconstructed from a series of 12-1 6 fluorescence decays recorded at different emission wavelengths. The details of the procedure are given in ref 2. Briefly, the individual decays were first fit to a sum-of-exponentials form by using an iterative-reconvolution algorithm. Such fitting partially deconvolutes the instrumental response from the fluorescence response and provides an effective time resolution of I5 ps in these experiments. The fitted decays were then normalized relative to one another by using the steady-state fluorescence spectrum.2 The spectrum at any time is finally given by the relative intensities of this normalized, fitted decay series. A wavelength resolution of 4 nm for the time-resolved spectra was provided by the emission monochromator. The I3C N M R measurements of rotational correlation times of U M F were made using a Bruker AM300 FT-NMR instrument. Nuclear Overhauser Enhancements (NOES) were determined from the average of three runs with and without proton decoupling. An inversion-recovery pulse sequence was used to determine the spin-lattice ( T I )relaxation times. The fluorescent probes coumarin 102 (Eastman Kodak) and prodan (6-propionyl-2-(dimethylamino)naphthalene;Molecular Probes Inc.) showed no detectable impurities based on spectral,
-
CulO2
Prodan
AC -C2H5
GC, and T L C analysis and were used as received. The solvents N M P (Eastman Kodak) and N M A and N M F (Aldrich) were dried over molecular sieves but not otherwise purified. All three solvents showed slight fluorescence under the -350-nm excitation conditions used here; however, the levels were such as to cause negligible interference in the emission region of interest (>400 nm). Fluorescence samples ( M ) were deoxygenated by bubbling with N 2 and then sealed in I-cm cuvettes. Temperature control for these samples was provided either by a brass sample block through which thermostated coolant flowed or by a liquid nitrogen cooled cryostat (APD cryogenics). In both cases the temperature stability and accuracy were approximately *I K. For the N M R meawrements temperature was controlled by means of a flow of cold nitrogen gas. Here the temperature uncertainty is roughly f 2 K.
Results and Analysis Solcation Times f r o m Time-Resolved Fluorescence Spectra. Examples of typical time-resolved fluorescence spectra are shown in Figure 1. The steady-state as well as the time-resolved fluorescence spectra of prodan and Cu102 are broad and relatively featureless in polar solvents. As a function of time after excitation, a continuous red shift of the fluorescence is observed with both probes. This behavior indicates that we are indeed observing a continuous solvation process rather than any sort of activated charge-transfer process in these molecules. The wavelength shifts
The Journal of Physical Chemistry, Vol. 94, No. 12, I990 4931
Solvation Dynamics in N-Methylamides
TABLE I: Average Solvation Times (picoseconds) solvt (0' probe measd ( 7 ) b N M F (234) prodan 61 f 12 NMF (234) cu102 50 f 6 48 f 16 NMA (302) prodan 31 f 6 NMA (302) cu102 125 f 25 NMP (244)c CUI53 340 f 40 NMP (253) prodan 330 f 30 NMP (253) cu102 550 f 60 CUI53 NMP(273)( 400
440
480
520
560
W a v e l e n g t h (nm!
(T)b
37-6 1 31-50 27-48 18-3 1 106-125 280-340 290-330 460-550
'Temperature in kelvin, *I K. bThe values in column 3 refer to average solvation times and their estimated uncertainties as measured directly from fits to the time-resolved spectra as described in the text. The values in column 4 are our best estimates for the range of times that bracket the actual average solvation times when the finite temporal resolution of our instrument accounted for (see text). CData from ref 2.
--
-1
+J v
20
22
24
c -3
Figure 1. Time-resolved fluorescence spectra. (A) Prodan in NMP at 253 K. Spectra as obtained from reconstruction of a series of temporal fluorescence decays at the indicated wavelengths. The times, from left to right, are 0, 100, 200, 400, and 800 ps. (B) Cu102 in N M F at 234 K. Spectra after conversion to frequency and showing log-normal fits. The times, from right to left, are 0, 20, 50, and 1000 ps. Data points are shown only for the t = 0- and 1000-ps spectra.
0
1000 T i m e (Psi
0 Y
Frequency ( I O 3 c ~ ' !
2000
Figure 2. (A) Time-dependent frequency shifts u ( t ) and (B) corresponding spectral response functions C ( t ) . The data are for Cu102 in NMP (253 K). The solid line shows the evolution of the peak frequency and the dashed line that of the average (first moment) frequency. Note the difference between the C(t)functions provided by these two measures. that we observe are large and are comparable to the spectral widths, which allow for relatively precise determinations of the dynamics with these probes. The quantity of interest to be extracted from the spectra is the response function, C ( t ) ,defined by
where the v's are some measure of the spectral frequency at the indicated times. It is this spectral response function that provides the experimental equivalent of the time-dependent free energy response calculated by theory. To calculate these C(t)functions, the spectra are first converted into the frequency domain and then fit to a log-normal line-shape f ~ n c t i o n . ~ Examples ~*~ of spectral
1
-5
500
0
1000
Time (ps) Figure 3. Spectral response functions observed with Cu102 in NMF (234 K), NMA (302 K), and NMP (253 K). The average frequencies of the spectra were used to generate these C ( t ) curves. Note their nonexponentiality.
fits are provided in Figure 1 B. Finally, C ( t ) curves are generated by using both the peak frequency and the average frequency (first moment) of the fitted spectrum. As illustrated in Figure 2, these two v ( t ) measures do not show identical time dependence. The difference is due partly to changes in the width and shape of the spectrum with time and partly to experimental errors. However, the two methods of measuring frequency usually agree to within 30%, and in reporting solvation times below we quote the average of the two. Spectral response functions for all three solvents are compared in Figure 3, and average solvation times are listed in Table I. When they are plotted on a semilogarithmic scale, as in Figure 3, it is clear that the C ( t ) curves are not simple exponential functions. Thus, to characterize the time scale for the solvation response, we use the average times defined by (T)
= J m C ( t ) dt
These average times were calculated both analytically, using parameters generated from multiexponential fits of the v ( t ) data, as well as numerically, from direct integration of the C ( t ) curves. The values listed in the third column of Table I are these average values along with error estimates based on repeated measurements and variability of the different v measures. Also listed in Table I are two results obtained previously2 by using the probe molecule coumarin 153 (Cu153). This molecule is identical with Cu102 except for a replacement of the CH3 group at the 4 position with a C F 3 group. The average solvation times obtained as just described do not take into account the limited time resolution of our experiments. While we expect to have observed most of the spectral evolution (29) Siano, D. B.; Metzler, D. E.J . Chem. Phys. 1969, 51, 1856. Fraser, R. D. B.; Suzuki, E. In Spectral Analysis; Blackburn, J. A,, Ed.; Marcel Dekker: New York, 1970; p 171.
4932
The Journal of Physical Chemistry, Vol. 94, No. 12, 1990 TABLE 111: Solvent Properties solvt :g D T , K ~ g h g'
TABLE 11: NMR Data and Correlation Times in NMF T." K TI? s NOE'.' T N M R . PS ~ 1.9 f 0.2 45 f 6 234 1.2 f 0.1 9.9 f 0.6 269 5.9 f 0.2 2 1 f 0.1 298 9.3 f 0.5 1.9 f 0.1 5.9 f 0.4
NMF NMA NMP
'13C TI and NOE data for the carbonyl carbon. CTheNOE referred to here is defined by NOE = (double-resonance intensity)/ (single-resonance intensity) "Equation 3; see text. O f 2 K.
in the slowest solvent, N M P , in N M A and N M F the observed ( 7 ) values are close to our experimental time resolution. For these two cases it is likely that the fastest parts of the response are not detected in our analysis. (One indication of this fact is that the total shifts, v ( 0 ) - .(a), are considerably larger in N M P than they are in N M F or NMA.) The C(t)curves generated as above would lack this fast component, and the times listed in column 3 of Table 1 would thus overestimate the true averages. In the last column of Table I we list a range of ( 7 ) values that approximately account for the effect of such unresolved components in the response. The lower limits on ( 7 ) listed here are derived from estimates of the frequency of the true t = 0 fluorescence spectrum, which is obtained by inverting the steady-state absorption spectrum.30 Comparing this estimate of the t = 0 fluorescence spectrum with values extrapolated from the time-resolved fluorescence data indicates that in N M F and N M A we miss roughly 40% of the total response, whereas in the slower N M P only -20% of the shift is unobserved. The smaller of the ( 7 ) values listed were calculated assuming a time constant of zero for the unobserved part of the response, and thus they provide lower limits to the true ( 7 ) values. The experimentally determined values then serve as upper bounds for the true ( T ) . At this point it is useful to remark that the data in Table I may suggest slightly slower solvation for prodan than for the coumarins. Although the data here are not conclusive on this point, results in other solvents indicate that such small differences between probes are the norm. For the present purposes, however, we emphasize the fact that differences observed with different probes in the same solvent are much smaller than differences among different solvents and temperatures. Here we have directly compared two dissimilar probe solutes merely to demonstrate the generality of our results. N M R Rotation Times. To gain some perspective on the solvation times measured above, we have also used N M R methods to determine single-molecule rotation times in one of these solvents, N M F . Carbon-1 3 spin-lattice relaxation times of the carbonyl carbon of N M F provide an unambiguous measure of overall rotation times for the molecule-fixed H-C(0) bond vector. (Unfortunately, the same is not true for the higher homologues since intramolecular rotations complicate interpretation of their T I relaxation.) In the N M F molecule the dominant source of broadening of the carbonyl I3C resonance is due to dipolar coupling to the directly bonded H . In such a case the correlation time of the II-C(0) bond vector, which we designate T ~ is related ~ ~to the I3C T , relaxation time and its NOE via3',j2 T ~ M R =
(5.678 X lo-'' ~ ~ ) ( N O E " ~ ~ / 1 . 9 8/8T) (I 1) ( 3 )
This expression assumes the extreme narrowing limit, which is appropriate for the present case. The ratio of the observed NOE to the theoretical value of 1.988 appearing here is used to take into account the small fraction of the broadening due to sources other than dipolar coupling.32 The prefactor combines various molecule-independent constants and the sixth power of the C-H bond length, for which we have used a value of I .125 A." Values of the observed NOE's, T,'s, and the resulting T~~~ times are listed in Table 11. As indicated by the near equality of the (30) Maroncelli. M., manuscript in preparation. (31) Doddrell, D.: Glushko, V.; Allerhand, A . J . Chem. Phys. 1972, 56, 3683. (32) Boere. R. T.; Kidd, R. G. Ann. Rep. Prog. NMRSpectrosc. 1982, 13%
319.
( 3 3 ) Kitano, M.; Kuchitsu. K . Bull. Chem. Soc. Jpn. 1974, 47, 631
Chapman et al.
3.82 3.71 3.59
234 302 273 253 244
304 182 216 270 300
c,'
4.4 -10 5.0 -10 6.4 -6 7.3 -6 7.8 -6
T D , ns ~
1.50 0.390 4.02 11.1
18.6
ps 52 22 I10 250 370
TL,
CP 6.7 4.0 10.8 20.7 28.7
rl$
"Gas-phase dipole moments (debye) from ref 34. *Dielectric parameters of a single debye fit to t ( ~ ) .Values listed here were obtained from fits to the temperature-dependent data contained in refs 35 and 36. Kirkwood correlation factor from ref 35. dViscosities obtained from fits to temperature-dependent data contained in ref 37.
280 310 340 T e m p e r a t u r e (K) Figure 4. Comparison of dielectric relaxation times ( T and ~ T L ) with single-particlereorientation times in NMF. All times are in picoseconds. The dielectric times T~ and T~ were obtained from fits to the temperature-dependent data of refs 35 and 36. The circles are the measured N M R times (Table I I ) , and the curve marked T~~~ is a fit of these data to eq 6. The T , curve is the estimated correlation time of the singleparticle dipole autocorrelation function, (P.P(t) ), and calculated by using eq 6 (see text).
220
250
observed NOE's to the theoretical value, essentially all of the broadening observed here is dipolar in origin.
Discussion Some solvent properties of N M F , N M A , and N M P relevant to the discussion of solvation dynamics are listed in Table 111. The noteworthy feature of this homologous series is that these amides possess some of the largest dielectric constants (to) of all liquid solvents. Such high tovalues result from large dipole moments, *hose effect is enhanced by an exceptional degree of correlation among dipole orientations in the liquid. This correlation, reflected in the Kirkwood g factors, arises due to strong hydrogen bonding between monomers and the fact that this hydrogen-bonding geometry produces alignment of individual dipole^.^^*^* The frequency-dependent dielectric response of the amides can be represented by a simple Debye form: ,
(4)
where e, is the limiting dielectric constant a t infrared frequencies is the Debye (or dielectric) relaxation time. Two features and iD of the dielectric data in Table I11 deserve comment. First, the extreme values of to result in a very large disparity between the longitudinal relaxation times (q = ( € , / t 0 ) 7 ~ ) and the dielectric relaxation times ( T ~ in) these solvents. Second, uncertainties in e,, which are likely to be on the order of f25%, lead to similarly large uncertainties in the r L values listed. To put the observed solvation times into perspective, it is useful to compare the above dielectric relaxation times to the singleparticle rotation times measured by N M R . Such a comparison is shown in Figure 4 for N M F . The curves marked 7 D and T~ are the dielectric times determined from temperature-dependent fits to the data of ref 35. The points are the measured NMR (34) Meighan, R. M.; Cole, R. H. J . Phys. Chem. 1964, 68, 503
The Journal of Physical Chemistry, Vol. 94, No. 12, 1990 4933
Solvation Dynamics in N-Methylamides correlation times, and the curve marked *71n was generated as follows. In the extreme narrowing limit, the N M R experiment measures the integral of the second-order reorientational correlation function of the H - C ( 0 ) bond vector:38
(In this expression P2 denotes the second-order Legendre polynomial and li is a unit vector along the H-C(0) bond direction in N M F . ) Since the dielectric response involves the multiparticle for comparison to coldipole correlation function, ( xjfii.fij(t)), lective response times, it would be most desirable to use the single-particle correlation function ( P I[,L@(t)]),where fi is the dipole direction. The latter can be obtained approximately from TNMR if we assume that the reorientation is both isotropic and diffusive. In this case 71 = 37NMR. T o generate the continuous 7 , curve shown in Figure 4, we further fit the N M R correlation times to the hydrodynamic relation:38
In this expression Vis the molecular volume, 7 the solvent viscosity, and 1 the order of the correlation function. C is a coupling factor adjusted to obtain the best fit between the observed N M R times and times calculated from eq 6 with I = 2 (dashed curve in Figure 4). On the basis of van der Waals increment^)^ the volume of N M F is V = 60.2 A3,which yields a value of 0.25 for the constant C. One might question the applicability of eq 6 to the rotational dynamics of a small molecule like N M F in the neat liquid. The fact that C i s significantly less than unity does mean that the stick boundary conditions under which eq 6 was derived are not appropriate here. However, other studies with small molecules have shown that the variation of 7 with a / T is nearly always linear.@ The more severe assumptions made in applying eq 6 are that the motion is isotropic and that a possible nonzero intercept in the relation between 7 and q / p can be neglected. Because of these uncontrolled approximations, we must regard the 71 values so obtained as only semiquantitative estimates of the correlation times of the single-particle dipole correlation function ( P I[fi.ji(t)] ). Fortunately, such estimates are sufficient for our present purposes. Here we are mainly interested in the comparison 7L < 71
