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J. Phys. Chem. B 2008, 112, 12976–12984
Polarization-Resolved Ultrafast Polarizability Relaxation in Polar Aromatic Liquids Ismael A. Heisler and Stephen R. Meech* School of Chemical Sciences, UniVersity of East Anglia, Norwich NR4 7TJ, United Kingdom ReceiVed: July 3, 2008; ReVised Manuscript ReceiVed: August 18, 2008
The isotropic part of the polarizability relaxation of three polar aromatic liquids has been measured as a function of temperature using a diffractive optic based heterodyne detection method. The isotropic spectral density, which contains only interaction induced components of the liquid’s collective polarizability, is contrasted with the anisotropic response to unravel the interaction induced and molecular contributions to the latter. Very similar behavior was observed for the three different aromatic liquids. The fast damping of the isotropic response in the time domain confirms the assignment of the prominent slow exponential relaxation in the anisotropic response to diffusive molecular orientational relaxation. The isotropic response in the frequency domain is found to be asymmetric and peaked at low frequency but extended to higher frequencies, thus contributing over a rather wide frequency range. The relatively low mean frequency of the isotropic spectral density supports the assignment of the highest frequency part of the anisotropic response to molecular librational motion. Finally, a number of affinities are noted between the interaction-induced (isotropic) response and the low frequency shoulder in the anisotropic spectral density, which is so characteristic of the dynamics of aromatic liquids. These data are discussed in relation to some existing molecular dynamics simulations. 1. Introduction A knowledge of the structure and dynamics of molecular liquids is central to an understanding of chemical reactivity in solution.1,2 The rate coefficient for reactions as diverse as isomerization, electron transfer, and solvation are all influenced, and even controlled, by solvent dynamics. Thus, numerous experiments have been devised with the aim of elucidating structure and dynamics in liquids, including neutron, X-ray and dynamic light scattering, and infrared and Raman line shape analysis.3-5 Among these methods, one which has attracted particular attention is the ultrafast optically heterodyne detected optical Kerr effect (OHD-OKE).6-9 Essentially OHD-OKE measures the relaxation of a transient polarizability anisotropy induced in a fluid by a linearly polarized ultrafast pump pulse. The great advantage of the method is that it provides data of exceedingly high signal-to-noise which, through a Fourier transform analysis,9,10 yields the low frequency Raman spectral density of the liquid, undistorted by either the finite pulse width of the laser or the thermal population of low frequency modes, which, in the frequency domain, necessitates a rather large correction.11 The Raman spectral density contains information on low frequency Raman active intramolecular modes (which are of particular significance in flexible molecules); molecular orientational relaxation (both diffusive and nondiffusive); and intermolecular (interaction induced) interactions. The latter two are of particular relevance in understanding the dynamics of molecular liquids. Spectral densities have been measured through the OHDOKE method for a large number of fluids, including small molecule liquids as a function of temperature,10,12-16 H-bonding liquids,17-21 water,22-27 aromatic liquids,28-38 ionic liquids,39-46 liquid crystals,47-53 polymers,54-58 and colloids.59-64 For liquids of small rigid polarizable molecules, the low frequency spectrum frequently comprises a single broad asymmetric band peaked * To whom correspondence should be addressed. E-mail: s.meech@ uea.ac.uk.
at a few tens of wavenumbers and a Lorentzian peaked near the zero wavenumber. The latter contribution is surprisingly well modeled by a modified Stokes-Einstein-Debye (SED) equation65 and can therefore be reliably assigned to diffusive molecular reorientation.13,37 This component can be subtracted from the data in either the time or the frequency domain to isolate the reduced (or nondiffusive) spectral density.7 Planar aromatic liquids usually reveal somewhat more complex behavior; like the small molecule liquids, they have a low frequency peak that is well represented by the SED equation, but when this is subtracted, a broad bimodal or flat topped reduced spectral density is recovered.30,38,66,67 The origin of this line shape has been the subject of a number of recent experimental and theoretical investigations.28,68,69 In this article, we shed further light on the origin of this structure through measurements of both the isotropic and anisotopic polarizability relaxation of the aromatic liquids toluene, nitrobenzene, and benzonitrile as a function of temperature. Early efforts to understand the bimodal line shape of aromatic liquids involved analysis in terms of two distinct functions, a generalized Ohmic (or Bucaro-Litovitz, BL70) line shape plus an antisymmetrized Gaussian (ASG).7 Comparing this empirical fit with other physically reasonable choices (e.g., a sum of ASGs, a multimode Brownian oscillator37) it was found that the BL + ASG pair does a surprisingly good job of fitting a wide range of spectral densities of aromatic liquids with a minimum number of fitting parameters. While it is important to realize that such an analysis is an empirical one, and the assignment of distinct dynamical behavior to the two components is unwarranted in the absence of additional data, this analysis does provide a convenient basis for comparing the spectral densities of aromatic liquids. For those studied here and for many others, the higher frequency ASG describes the main part of the spectral density, while the BL function is of lower weight and accounts for the lower frequency shoulder in the spectral density. Further analysis of the frequency shift of the ASG component as a function of dilution and molecular moment of inertia suggested an assign-
10.1021/jp805862z CCC: $40.75 2008 American Chemical Society Published on Web 09/19/2008
Polar Aromatic Liquids ment to a nondiffusive orientational (or librational) motion.7 In a recent detailed study of the spectral densities of four symmetrically substituted benzene derivatives, Loughnane et al. reported that both the ASG and the BL components scaled with molecular moment of inertia suggesting a similar, orientational, origin for both components.28 This result was discussed in connection with librational motion in a range of dimer structures in the liquids. This in turn suggests that the dominant contribution to the Raman spectral density, at least for these liquids, is orientational dynamics and that translational motion, which may contribute through the interaction induced (II) response, is relatively unimportant. An alternative assignment for the low frequency part of the response is that it reflects a contribution from the II part of the polarizability, which might be expected for such polarizable molecules at liquid densities. This is in fact the historical origin of the BL function.70 However, there is no independent experimental evidence to support such an assignment or indeed to assign any part of the spectral density to translational dynamics. One route to understanding the detail of the spectral density is through analysis of molecular dynamics simulations. Indeed, the high quality of the Raman spectral density recovered from OHD-OKE measurements has long provided data against which to test molecular dynamics calculations of liquid state dynamics.71-73 Simulation and analysis of liquids comprising structurally simple rigid molecules, such as carbon disulfide and acetonitrile, were successful both in reproducing the overall shape of the observed spectral densities and in separating distinct molecular and intermolecular contributions to them. It was found that in general both single molecule orientational and II relaxation mechanisms contribute to the spectral density over a wide frequency range. Ryu and Stratt presented molecular dynamics simulations of the bimodal spectral density of liquid benzene.68 They found that translational (II) dynamics contribute to the spectral density at low frequency through rotationaltranslational coupling, while orientational dynamics contribute across the entire frequency range. They also proposed that the unusual bimodal shape of the liquid benzene spectral density (and probably that of other aromatic liquids) compared to, for example, those of CS2 and acetonitrile, could be accounted for by the lower moment of inertia of the aromatics (itself a consequence of their planarity). One consequence of this would be a higher frequency librational component, thus separating orientational and translational contributions.68 This assignment is in contrast to that of Loughnane et al. who presented evidence for orientational motion within a distribution of parallel and perpendicular structures dominating the complex spectral density.28 More recently, Elola and Ladanyi reported a molecular dynamics study of polarizability anisotropy relaxation in benzene and its 1,3,5-trifluoro- and hexafluor-derivatives.69 This work also pointed to the importance of local structure in determining particularly the librational frequencies in the liquid. However, it was not possible to reach unambiguous conclusions on the frequency dependence of II contributions to the spectral density, with different methods of resolving the separate components leading to different conclusions. In molecular dynamics calculations, the separation into single molecule (orientational) and intermolecular (II) components and their cross terms is possible through different projection schemes.71,72,74,75 Unfortunately, such a separation is not possible experimentally. However, some progress beyond the OHD-OKE spectral density is possible through separate measurements of
J. Phys. Chem. B, Vol. 112, No. 41, 2008 12977 the anisotropic and isotropic components of the collective polarizability Π.
Π ) Πaniso + Πiso The former component has been studied in great detail because it is directly measured in the OHD-OKE method, while the latter has been accurately determined for only a few liquids.76-78 As a consequence of the data being readily available, it is the anisotropic response that has most frequently been modeled in molecular dynamics simulations. However, the anisotropic response always comprises a mixture of contributions from the single molecule and II polarizabilities.75,78,79 The former contribution relaxes exclusively through rotational motion, while the latter reflects both rotational and translational dynamics. In contrast, the isotropic response reveals exclusively II components of the collective sample polarizability, although it does not distinguish between translational and rotational contributions.78 Clearly, measurements of the isotropic response are potentially very valuable in deconvoluting distinct contributions to the complex line shape observed in aromatic molecules. Unfortunately, Πiso cannot be directly measured through OHD-OKE measurements in the same way as Πaniso, as the combination of beam polarizations required does not readily permit heterodyne detection. Hence, new methods have been developed to recover Πiso. Fecko et al. introduced a spatially masked (SM) OKE method, which makes use of the spatial variation of phase across the beamfront to reintroduce heterodyne detection to Πiso measurements.78 This method was used successfully to measure isotropic and anisotropic polarizability relaxation of a number of small molecule liquids76,78 and some complex fluids.48,62 An alternative scheme based on a diffractive optic (DO) transient grating method was introduced by Goodno et al.80 This method, which is the one used here, allows precise control over the relative phase of the heterodyning beam and has previously been used to measure a high quality isotropic response of CS2.81,82 Here, we apply this technique to resolve components of the polarizability relaxation of some polar aromatic liquids. The remainder of this article is organized as follows. In the next section, the experimental methodology will be outlined, followed by a discussion of the means of recovering the isotropic response in the frequency domain. In the third section, the isotropic and anisotropic response of three polar aromatic liquids will be compared and their temperature dependence measured. The article ends with a discussion of these results and some conclusions. 2. Experimental and Data Analysis 2.1. Experimental Measurements. Experimental measurements of the third order isotropic and anisotropic polarizability relaxation were performed in the time domain as a function of temperature. To access these contributions in a fully heterodyne detected measurement, a DO element based method was employed.81,82 A diagram of the experimental setup is shown in Figure 1. The ultrashort pulses were generated in a commercial Kerr-lens mode-locked Ti:Sapphire laser (Micra 10, Coherent) pumped by a 10 W (CW) intracavity-doubled diodepumped Nd:YVO4 laser (Coherent Verdi). The output had an approximately 800 nm center wavelength, an average power of 800 mW, and a bandwidth of 85 nm. A pair of fused silica prisms was used for compensation of the temporal broadening of pulses directly emitted from the oscillator and also for precompensation for dispersion due to propagation through the optics used in the experiment. Measuring the autocorrelation at
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Figure 1. Schematic diagram of the main details of the experimental setup based on the diffractive optic element. P, polarizer; L, focusing lens; DOE, diffractive optic element; CS, microscope coverslip; CM, concave mirror; D1,2, silicon photodiode detectors. Zoom shows the beam geometry inside the sample. It is important to note that both probe beams pass through the same optics in order to keep intensities balanced.
the sample position (including transmission through an amount of glass equivalent to the cell window) in a 50 µm BBO crystal revealed typical pulse durations of 17 fs. The pump and probe beams were obtained with a broadband beam splitter, which divided the intensity into a 50:1 ratio. Both beams were routed through separate but identical achromatic half-wave plates and polarizers in order to define the relative polarization directions. For the measurements of the anisotropic polarizability relaxation, the probe polarizer was crossed with the analyzer and oriented at 45° with respect to the pump polarization in the conventional OHD-OKE geometry.7 To access the isotropic response, the probe polarizer was aligned parallel with the analyzer and at 54.7° with respect to the pump (known as the magic angle). The pump and probe beams were focused to a common spot with a 150 mm focal-length lens. The diffractive optical element (National Optics Institute, Canada) was positioned at the focal spot. This element, which was specifically designed for efficient diffraction into first order, consisted of a diffraction grating etched in a fused silica substrate with a period of 23.2 µm. The diffraction efficiency for the (1 orders was greater than 75%, and the angle between the diffracted beams was 5.72°. There was virtually no variation in either the diffraction efficiency or the phase of a given diffraction order with respect to input polarization. After the DOE, there were four beams, two diffracted orders for the pump (kpu1, kpu2) and probe (kpr1, kpr2). A concave mirror with a focal length of 150 mm was placed one radius of curvature away from the DOE in order to exactly refocus the four beams to the focal spot. The concave mirror was tilted slightly off-axis to separate the incident and reflected beams so that it was possible to steer the beams into the sample cell. The pump beams (kpu1, kpu2) were crossed at the sample position and created an interference pattern, a transient index grating, from which the probe beam(s) can be diffracted following the phase matching condition: ks ) kpr1 + kpu1 kpu2 (Figure 1). The box geometry of the experiment ensures that this signal is radiated along the same direction as the kpr2 probe beam; therefore, this probe beam may be used as a local oscillator (LO) to heterodyne the signal.80 In order to adjust the phase between LO and the signal for the measurement of the birefringence or dichroism contributions, a pair of microscope coverslips (150 µm thickness) were inserted separately
into the probe beams (one in each beam). The coverslips were mounted on a common holder to avoid relative motion fluctuations, which could deteriorate the phase stability. By rotating one of the coverslips, the effective path length traversed by one of the beams changes such that it is possible to fine-tune the phase difference. In fact, the total phase is a function of pump beam phase difference, probe beam phase difference, and the phase shift attributable to the (generally complex) sample nonlinear susceptibility. Therefore, in order to measure the real or imaginary components of the nonlinear susceptibility unambiguously, the phase between the input fields must be set independently. This is usually done with some reference material with a well-known strong birefringence response.80 To overcome this dependence on some material calibration, the symmetry of the setup was exploited in an approach developed by Miller and co-workers83 and followed here. Instead of measuring only the signal radiated in the kpr2 direction, the signal radiated along direction kpr1 is also measured using a second detector. The intensities measured by detectors 1 and 2 (see Figure 1) can be written as follows:83
In ) |Esig n|2 + |ELO m|2 + 2|Esig n|ELO m|cos(φsig n - φLO m) (1) where Esig n is the diffracted signal field from probe n ) 1 or 2 and ELO m is the local oscillator given by probe beams m ) 2 or 1. If the intensities of both probe beams are equal, ELO1 ) ELO2 ) ELO , the subtraction of the intensities results in
I1 - I2 ) 4|ELO|Re{Esig}sin(φ2 - φ1)
(2)
where (φ2 - φ1) denotes the phase difference between the probe beams. The advantages of balanced detection are immediately apparent. First, the background and homodyne components are automatically subtracted, and consequently, it is possible to both increase the dynamic reserve of the detection electronics and to isolate the pure heterodyne signal. Balanced detection also reduces the effects of random fluctuations of the laser power and achieves a significant improvement in the signal-to-noise ratio. Furthermore, it is possible to isolate the real part (birefringence component) of the signal with a high degree of discrimination from the dichroism component.83
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After the sample, the pump beams were blocked and both transmitted probe beams propagated through a telescope. A spatial filter was placed at the focal point of the telescope to block unwanted scattered light. Subsequently, both the collimated probe beams were passed through a common analyzing polarizer after which they were measured separately with the two detectors in the balanced detection scheme. The transient grating signal measured by detectors 1 and 2 was routed to, and subtracted in, a low noise preamplifier (SR560 Stanford Research) and the resulting background free heterodyne signal was measured with a lock-in amplifier (SR830 Stanford Research). The chopper positioned between the DOE and the sample modulated one of the pump beams, which greatly reduced unwanted pump-probe contributions and other spurious scattered signals. The entire experiment and data collection were under labview control. Liquids were of analytical grade (Sigma-Aldrich) and were injected into the cuvette through a 0.2 µm micropore filter to remove any particulate matter. For measurements of the anisotropic response, a 1 mm path length quartz cuvette was used. Because of the significant signal from the quartz cell walls to the weak isotropic response, a 5 mm path length cuvette was used to ensure that the beams crossed only in the liquid. The cuvette containing the sample was placed in a thermostatted copper block through which flowed a mixture of water/ ethanediol at a temperature controlled with a refrigerated thermostatic bath. 2.2. Data Analysis. For measurements of the transient isotropic response, the whole dynamics (except for oscillating contributions from intramolecular modes) were usually complete in less than 2 ps. Thus, these scans were made with steps of 2.5 fs/point around zero delay time and 5 fs/point for the rest of the data over a complete period of 2 ps. Each point was averaged over 10 samples, and each data set was an average of 20 scans. The baseline established before zero time delay was averaged and subtracted from the data, which were then normalized prior to fitting. To obtain the frequency domain response function, the established Fourier transform (FT) deconvolution procedure was applied utilizing a second order autocorrelation measured at the sample position.9 In the time domain, data analysis was carried out through a fitting of the complete transient dynamics signal. For the isotropic response, the data were fit by a single exponential relaxation function plus a sum of damped harmonic oscillator functions to account for the intramolecular modes, which were damped on a picosecond time scale. The expression used was as follows:
( ( )(
Riso(t) ) A1exp -
( )) ∑ ( )
t t 1 - exp τiso τr
+
Anexp -
n
)
t sin(ωnt) H(t) (3) τn
where H(t) is the Heaviside function required to satisfy the causality condition. The rising component, τr, ensures that the response rises from zero, but in the actual analysis, its value was not extracted and was fixed at a value less than that of the 20 fs pulsewidth. This value has no effect on the subsequent analysis since the fits are mainly used to extract the long time response and in padding the data prior to the FT. The FT analysis is performed on the experimentally measured data recorded to 30 ps. The exponential relaxation time constants, τiso, obtained were quite similar for the three liquids: 302 ( 50 fs for benzonitrile; 260 ( 50 fs for nitrobenzene; and 311 ( 60 fs
for toluene. The number of harmonic modes (n) used was 3 for toluene and 4 for benzonitrile and nitrobenzene. In order to obtain reasonably spaced data points in the frequency domain (after FT deconvolution), it is necessary to measure the time domain data (eq 3) over a period of at least 30 ps. However, to accurately determine the rapidly oscillating (intramolecular) component, the highest time resolution data was measured over a period of only 2 ps and then with a larger step size to 30 ps. The procedure then adopted was to find the best fit over the 2 ps interval and numerically extend the fitted curve to the longer period of 500 ps, including the oscillating harmonic modes. It was ensured that the extended (or padded) response overlapped the data taken to 30 ps. The extended tail (between 2-500 ps) was then added to the original measurement and the result deconvoluted by the FT method. The transient anisotropic response measurements were performed over a time interval of 50 ps, with steps of 2.5 fs/point around zero delay time and 83 fs/point for the rest of the data. Over the whole time interval, data were fit with a model that included a Gaussian term to describe the fast nonexponential relaxation, a sum of two exponential functions to describe the long time diffusive relaxation, and an initial fast relaxation time, τI, and also damped harmonic functions to account for the intramolecular vibrational modes excited. It is useful to separate the contributions to the total response into terms of a longer, Raniso,p(t), picosecond scale contribution and a shorter, Raniso,sp(t), subpicosecond scale contribution as follows:
((
( ))∑ ( ))
Raniso,p(t) ) 1 - exp and
t τr
Anexp -
n
t H(t) (4) τn
( ( )( ( )) ( ) ) ( )
Raniso,sp(t) ) A1exp Agexp -
t t 1 - exp τI τr
+
t2 t sin(ωgt) + Aoexp - sin(ωot) H(t) (5) 2 τ τg o
where the risetimes were again set to an arbitrary ultrafast value. For the three molecules studied, the subpicosecond exponential relaxation component (which is known in the literature as the intermediate component), τI, the Gaussian damping, τg, and frequency could be fit with similar values, as already reported in previous literature. For instance, the numbers obtained for benzonitrile were τI ) 350 ( 50 fs; τg ) 160 ( 20 fs; and ωg ) 54 ( 5 cm-1. In addition, the fit also included only one damped harmonic oscillator mode. The tail extension procedure, as discussed previously, was also applied in this analysis. In both cases, the experimental data were fit using an evolutionary-genetic (EvoAlg) procedure, written in a LabView environment. This algorithm has been used in several applications and has already been described in the literature.84,85 Briefly, it employs several vectors, called individuals, which compose a generation. Each vector is formed by parameters related to the time-constant decays, amplitudes, and parameters of the model discussed above. The first generation is created randomly within a certain range at the first interaction. The randomly generated first population maps all the space formed by all variables with its 196 different individuals. The best individuals, i.e., those with the smallest difference (actually χ2) between the experimental transients and the associated fitting are mutated and recombined to form a new generation. This process is repeated again with the new best individuals. This search converges when a variation of the χ2 smaller than 10-5 between
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successive generations is reached. It is important to note that the EvoAlg does not give error bars for the best individual. The error is calculated after several runs of the algorithm for the same data. The best individuals of each run are averaged, and the error associated with each parameter is calculated. The lack of convergence of a run or a large error bar may indicate that the fitting model is not adequate or that the data quality is not good. 3. Results and Discussion 3.1. Anisotropic Polarizability Relaxation. The anisotropic polarizability relaxation (measured through the OHD-OKE) for benzonitrile, nitrobenzene, and toluene have been discussed in the literature.37,38 A common feature of these liquids is that on the picosecond time scale they show a biexponential relaxation process (on which are superimposed damped harmonic contributions from intramolecular modes). The longer of the two relaxation times extracted from Raniso,p(t) can be accurately associated with a diffusive reorientation, which is well described by the modified SED equation.37,65 Nitrobenzene and benzonitrile are the most viscous of the liquids and thus show the longest diffusive relaxation times of around 20-30 ps. The less viscous toluene has a diffusive relaxation time around 5 ps. These diffusive reorientation times, which have been reported previously, were reproduced here and will not be analyzed further. The shorter of the two biexponential relaxations for all three liquids was fit to a time constant of around 1.5 ps. It is doubtful that such a fast exponential can be directly ascribed to a diffusive orientational relaxation process. Significantly, its value does approximately scale with the longer relaxation time as a function of temperature, in a manner described elsewhere,7,13 suggesting a relationship to orientational relaxation; however, this component remains unassigned here. For subsequent analysis, we followed the established procedure of subtracting both of these components from the time domain data to isolate the reduced response in the time domain. The Fourier transform deconvolution procedure was applied to the reduced and temporally extended response, and the resulting reduced spectral densities are shown in Figure 2a. These spectra are in good agreement with the earlier data and clearly show the bimodal profile characteristic of aromatic liquids. Indeed, the spectral densities for these three liquids are similar. Apart from a differing spectral peak related to intramolecular vibrations around 176 cm-1 for nitrobenzene and benzonitrile and 215 cm-1 for toluene, the overall band shape for the low frequency spectra are remarkably similar. This similarity was discussed by Stratt and co-workers, who noted that it extends even to the case of biphenyl.86 Their analysis rests on the importance of orientational (librational) molecular motion to the reduced spectral density, as is suggested in a number of studies.7 Tao and Stratt proposed that the low frequency spectra for aromatic molecules undergoing librational motion depend upon the ratio of two features: the mean-square torque, related to the librational force constant, and the moment of inertia.86 To a good approximation, these depend on the molecular size and shape in the same way. In the case of planar molecules, both quantities are proportional to the rotational axis distance, leading to a cancelation of the details of molecular shape and size. Thus, the strong similarity between these spectra is not unexpected, provided the librational motion is the origin of the spectral density. The reduced anisotropic spectral densities were studied as a function of temperature. The data for toluene are shown in Figure 2b. The rather broad and featureless spectra are
Figure 2. (a) Reduced Raman anisotropic spectral densities for benzonitrile, nitrobenzene, and toluene obtained from the Fourier transform deconvolution of the OHD-OKE data from which the response represented by eq 4 has been subtracted. The inset figure presents the result for benzonitrile before the exponential tail due to eq 4 was subtracted. (b) Toluene anisotropic spectral density for different temperatures.
TABLE 1: First Moment, 〈ν〉, and Full Width at Half Maximum, ∆ν, for the Reduced Anisotropic Polarizability Spectral Densities for the Three Different Liquids As a Function of Temperature benzonitrile nitrobenzene toluene
temp./K
274
293
323
〈ν〉/cm-1 ∆ν/cm-1 〈ν〉/cm-1 ∆ν/cm-1 〈ν〉/cm-1 ∆ν/cm-1
58.4 90.8 57.4 91.1 61.2 92.8
58.0 89.8 56.9 87.8 61.5 91.5
57.4 85.2 56.0 87.5 60 86.8
characterized for all three liquids by the full width at halfmaximum, ∆ν, and the principal frequency or first moment of the spectral distribution, 〈ν〉, displayed in Table 1. As can be seen, the low frequency spectra are only weakly dependent on temperature, as has been reported previously for nitrobenzene and benzonitrile;37 both the first moment and the full width at half-maximum decrease slightly with increasing temperature. These effects can be understood if the intermolecular dynamics reflected in the spectrum are orientation-librational modes of the liquid. Librational motion is modeled on a (transient) harmonic potential surface created by the intermolecular interactions of the molecule with their nearest neighbors, and characterized by a librational force constant.87 Thus, since the molecular moment of inertia is unchanged, we expect that lower densities create a shallower potential (smaller force constant), which in turn leads to molecular libration at lower frequencies.
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Figure 4. Comparison for liquid benzonitrile of the normalized deconvoluted isotropic (s) and anisotropic (---) spectral densities.
Figure 3. (a) Time domain transient grating isotropic signals measured (O) for benzonitrile. The line is the fit to the function described in the text (eq 3). (b) Representative deconvoluted isotropic spectral density obtained for benzonitrile. The inset shows the same result over a broader frequency scale showing the intramolecular vibrational mode contribution to the measured signal. The modulation in the intensity between 30 and 100 cm-1 is not reproducible and arises from the FT analysis of this rapidly decaying weak signal.
Thus, as the density decreases with increasing temperature, the librational frequency (and hence spectral density) will shift to lower frequency. A second consequence of increased temperature is that the bimodal structure is less well resolved. In summary, all of the present data are entirely consistent with librational dynamics being a major component of the reduced anisotropic spectral density. The same conclusion has been reached by several groups for a number of aromatic liquids.28,37 This result does not, however, shed any new light on the origin of the unusual shape of the spectral density, for which we turn to measurements of the isotropic response. 3.2. Isotropic Polarizability Relaxation. The isotropic polarizability relaxation was measured as a function of temperature for the three aromatic liquids. The isotropic response, which contains information only on interaction-induced contributions to the relaxation, is typically 10 to 20 times smaller compared to the corresponding anisotropic response.76 An example of the time domain data is presented in Figure 3a for benzonitrile. Clearly, the signal is strongly influenced by underdamped intramolecular modes. As the isotropic response is a fully polarized signal, it is only sensitive to totally symmetric vibrations. For benzonitrile, there are at least four relevant A1 symmetric intramolecular vibrations (458 cm-1, 756 cm-1, 1002 cm-1, and 1180 cm-1),88 which contribute to the isotropic response. On application of the Fourier transform deconvolution procedure to the time domain signal, the isotropic spectral density is obtained (Figure 3b). The inset shows the full scale spectrum, where the totally symmetric vibrational modes clearly appear. The intermolecular isotropic response shows up as a
broad asymmetric band below 150 cm-1. It is thus spread over the same frequency range as its anisotropic counterpart. The frequency range is, however, about the only similarity between the two spectral distributions. A particularly striking difference is that very low frequency modes (ν < 10 cm-1) are strongly suppressed in the isotropic spectral distribution (cf. inset to Figure 2a). This is consistent with the isotropic response being insensitive to molecular orientational diffusive motion, which only appears in the anisotropic polarizability relaxation. (Alternatively, this result can be regarded as further confirmation that the origin of the slowest relaxation in the OHD-OKE data arises from molecular orientational diffusion.) The disappearance of the very low frequency component from the isotropic response was also noted in studies of liquid CS2 and acetonitrile76,78 and complex fluids.48 The only other experimental study of the isotropic spectral density of aromatic molecular liquids, which employed the position-sensitive Kerrlens method, reported that it was essentially the same as that of the anisotropic response and retained the low frequency Lorentzian.89,90 The origin of this discrepancy with the present data is unknown, although the position-sensitive methods are less sensitive than the DOE based heterodyne method used here, and the isotropic response is small. Whatever the origin of this discrepancy, the present data are consistent with the low frequency Lorentzian arising from pure diffusive molecular orientational relaxation. The second major difference is illustrated in Figure 4, which compares the overall shapes for the normalized isotropic and the reduced anisotropic spectral densities for benzonitrile. Clearly, the shapes are very different, with the former being weighted to lower frequency. This is an important result and shows that the interaction-induced dynamics reflect processes that relax with a different frequency distribution compared to the anisotropic dynamics. A similar result was obtained for all three liquids, as illustrated in Table 2, which reports the first moment and width of the isotropic spectral distribution. For example, benzonitrile has a first moment 〈ν〉iso ) 50 cm-1, compared to the first moment for the anisotropic spectral distribution, 〈ν〉aniso ) 58 cm-1. That the isotropic spectral responses measured for the three aromatic liquids studied have quite similar band shapes is illustrated in Figure 5, a result that mirrors the similarity of the anisotropic spectral densities (Figure 2a). This similarity between molecules extends to the temperature dependence of the isotropic response. Figure 5b compares the temperature dependent spectral densities for benzonitrile, and the values for the first moment and full width at half-maximum for other liquids and temper-
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TABLE 2: First Moment, 〈ν〉, and Full Width at Half Maximum, ∆ν, for the Isotropic Polarizability Spectral Densities for the Three Different Liquids As a Function of Temperature benzonitrile nitrobenzene toluene
temp /K
273
283
293
303
313
323
〈ν〉/cm ∆ν/cm-1 〈ν〉/cm-1 ∆ν/cm-1 〈ν〉/cm-1 ∆ν/cm-1
49.5 58.5 54.4 62.0 51.4 52.0
54.0 79.0 54.3 72.1 52.4 53.0
50.5 64.8 50.5 55.1 54.1 51.5
50.6 52.2 53.7 76.0 57.0 60.3
46.9 50.0 58.0 70.0 55.0 55.2
51.4 56.0 NA NA NA NA
-1
atures are shown in Table 2. It is not possible to identify any significant trend in these data within the current signal-to-noise. 4. Discussion In the foregoing text, it was established that the isotropic and anisotropic spectral densities of aromatic liquids have quite different shapes. The purely II dynamics, although sharing the same frequency range as the anisotropic response, are peaked at significantly lower frequencies. This is at least qualitatively consistent with the analysis by Ryu and Stratt of their molecular dynamics simulations of liquid benzene.68 They reported that the II contributions were larger than the molecular contribution at lower frequency. They also determined that the II part of the polarizability for benzene was largely driven by molecular translations. A scaling rule was proposed, that translational motion would dominate the II polarizability relaxation whenever the isotropic part of the polarizability is greater than the anisotropic part. It was shown (Figures 2 and 5) that both isotropic and anisotropic spectral densities are essentially independent of the molecule studied, despite significant differences in their intermolecular interactions. An argument based on the cancelation of parameters determining the librational frequency was advanced to explain this behavior for the dominant molecular orientational (librational) part of the reduced spectral density.86 It is less obvious that the same cancelation necessarily applies for translational modes. In the simplest case, translational modes are expected to reflect both intermolecular interactions, through some force constant, k, the molecular mass, m, and to a first approximation scale as (k/m)1/2. Thus, if k were assumed common to these aromatic liquids, then mass dictates that toluene should exhibit the highest frequency in the isotropic spectral density, which is not what is observed (Table 2). However, intermolecular interactions are really quite different for the three liquids, and the extent to which this influences k should be taken into account. If the enthalpy of vaporization is employed as a crude measure of k in the liquid, then the molecule with the smallest mass (toluene) also has the smallest force constant, leading to a reordering of the expected frequencies, with benzonitrile being predicted to have the slightly highest frequency. However, as also pointed out by Elola and Ladanyi,69 in reality the local liquid structure is likely to strongly influence the force constant associated with translational modes. Evidently, these competing factors of mass, intermolecular interaction, and local structure combine to yield rather similar isotropic spectral densities. The outstanding question to be addressed is whether or not the II components of the polarizability relaxation are responsible for the shoulder in the anisotropic spectral density. Certainly, the intensity distribution and frequency range is right, and the agreement with at least some MD simulations is quite satisfactory.68 It is difficult to go further by directly comparing the two
Figure 5. (a) Comparison of the isotropic spectral densities at room temperature for the three liquids on a semilog scale. (b) Same as (a), for benzonitrile, comparing the results for different temperatures.
spectra (Figure 4) because the relationship between intensities in the isotropic and anisotropic spectral densities is not a straightforward one. Instead, we attempt a further analysis by progressively reducing the anisotropic spectral density using as an example benzonitrile. The longest exponential component in the time domain can be associated with diffusive orientational relaxation with some certainty and can reasonably be subtracted in creating the reduced spectral density. The status of the shorter picosecond exponential component in Raniso,p(t), which scales with the orientational relaxation but cannot easily be associated with pure diffusive orientational motion, is less clear, but the lack of such a picosecond component in Riso(t) suggests this component to be associated with orientational diffusion. Nevertheless, when both components of Raniso,p(t) are subtracted in the time domain prior to the Fourier transform procedure, the reduced spectral density that results is shown in Figure 6, where it is compared with the isotropic data. This is the familiar bimodal spectrum characteristic of aromatic liquids, and it is possible to see that there is a good agreement between the rising edge of this spectral density and the measured isotropic one. Furthermore, if the intermediate subpicosecond component in Raniso,sp(t), τI, is also subtracted in the time domain, the resulting spectral density is shown again in Figure 6. The match between the low frequency edge and the isotropic response is less good. In fact, the shoulder has essentially disappeared, and a single band remains. Thus, it might be argued that the subpicosecond component, τI, in the aromatic liquids reflects II contributions to the polarizability anisotropy relaxation (although the caveat that the correct intensities for the comparison are unknown must be entered; therefore, the normalization of the isotropic and reduced anisotropic spectral densities in Figure 6 is arbitrary). Interestingly, the result is that this procedure effectively isolates the high frequency part of the
Polar Aromatic Liquids
Figure 6. Comparison between the isotropic (s) and reduced anisotropic spectral densities for benzonitrile. Prior to generation of the spectral density, the time domain data had Raniso, p(t) subtracted (---); the intermediate (subpicosecond) exponential (τI) component has also been subtracted (- · -).
anisotropic response, which can thus be assigned to single molecule librational dynamics. 5. Summary and Conclusions The temperature dependent isotropic and anisotropic polarizabiltiy relaxation has been measured for three aromatic liquids utilizing a diffractive optic based optically heterodyned approach. The isotropic response reflects only II dynamics, while the anisotropic contains a mixture of single molecule and II components. In all cases, the temperature dependence is small, with only the reduced anisotropic response showing a small shift to lower frequency, characteristic of librational dynamics. The isotropic II response was shown to contribute over a wide frequency range but is peaked to lower frequency than the reduced anisotropic response. There is a good match between the low frequency part of the reduced anisotropic and the isotropic spectral densities, suggesting an assignment of the low frequency shoulder in the former to II dynamics. Thus, the picture which emerges from these studies of aromatic liquids is a high frequency response arising mainly from molecular librational dynamics, an intermediate frequency response that reflects intermolecular interactions and a low frequency component, which is well described by collective diffusive orientational dynamics. Acknowledgment. We thank Professor R. J. Dwayne Miller for helpful advice on the selection of the diffractive optic element. We are grateful to EPSRC for financial support. References and Notes (1) Burghardt, I.; Hynes, J. T. J. Phys. Chem. A 2006, 110, 11411. (2) Liu, M.; Ito, N.; Maroncelli, M.; Waldeck, D. H.; Oliver, A. M.; Paddon-Row, M. N. J. Am. Chem. Soc. 2005, 127, 17867. (3) Proceedings of NATO ASI NoVel Approached to Stdies of Structure and Dynamics of Liquids: Experiments, Theories and Simulations; Samios, J.; Durov, V., Eds.; Kluwer Academic: Norwell, MA, 2004. (4) Liquid Dynamics: Experiment, Simulation and Theory; Fourkas, J., Eds.; ACS Symposium Series; American Chemical Society: Washington DC, 2002. (5) Applications of Neutron Scattering to Soft Condensed Matter; Gabrys, B., Eds.; Taylor and Francis: London, 2000. (6) Righini, R. Science 1993, 262, 1386. (7) Smith, N. A.; Meech, S. R. Int. ReV. Phys. Chem. 2002, 21, 75. (8) Kinoshita, S.; Kai, Y.; Ariyoshi, T.; Shimada, Y. I. J. Modern Phys. B 1996, 10, 1229. (9) Lotshaw, W. T.; McMorrow, D.; Thantu, N.; Melinger, J. S.; kitchenhak, R. J. Raman Spectrosc. 1995, 26, 571. (10) McMorrow, D.; Lotshaw, W. T. J. Phys. Chem. 1991, 95, 10395.
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