Vibrational dephasing of liquid carbondisulfide investigated by picosecond CARS P. Aechtner and A. Lmbereau Physikalisches
Institut, Universitiit Bayreuth, D-8580 Bayreuth. Germany
Received 17 July 1990
Using time-resolved CARS the dephasing rate 1/T2 of the Y, mode of CS2 is investigated in the neat liquid and in the solution of n-pentane. A large pressure effect (factor of < 3) is reported for the range O-4 kbar similar to spontaneous Raman results. A non-monotonic temperature dependence is reported for an extended interval of 160-450 K and explained by the contributions of vibration-translation and vibration-rotation coupling to 1/ T,. Estimates of the intermolecular collision frequency from reorientation relaxational data lead to a phenomenological description of vibrational dephasing that accounts semiquantitatively for the experimental data.
In recent years vibrational dephasing was studied by numerous investigations using spontaneous Raman spectroscopy and time-resolved techniques [ l61. The significance of different physical processes was outlined for the measured dephasing times and spectroscopic bandshapes for a variety of experimental examples. The justification for the simplified discussion in terms of one (or two) processes was often intuitive. The qualitative insight provided by the comparison of theoretical models and experimental results suggests that dephasing phenomena represent a local probe of dynamical and structural properties of liquids. A detailed understanding, however, is still lacking. More valuable experimental information is required to test theoretical ideas and promote a quantitative theoretical description. In this letter picosecond CARS  is used for a study of vibrational dephasing under varying thermodynamic conditions. The dephasing time T,of the v, mode at 656 cm-’ of CS2 is investigated in an enlarged temperature range of 160-430 K compared to recent work with spontaneous Raman spectroscopy. The pressure effect is inspected in the range O-4 kbar. A non-monotonic temperature dependence of T, is reported for neat CS2 and in solution of n-pentane and CCL and compared with theoretical expecta0301-0104/91/$03.50
tions. Our work applying a time domain technique avoids the problems connected with the bandshape analysis of frequency spectroscopy.
2. Theoretical models The decay of the coherent excitation of a molecular vibration is caused by external perturbations affecting the amplitude and/or phase of the collective excitation of the molecular ensemble. Energy dissipation and loss of phase correlation result. In simple cases the various kinds of processes may be statistically independent leading to a total dephasing rate:
Here T, is the energy (or population) lifetime while the time constants T2,a to T2,y represent different contributions of the so-called pure dephasing not changing the vibrational energy of the specific vibrational mode under consideration. The following interactions have been considered: (Y- dephasing induced by the translational motion via both the attractive and repulsive parts of the intermolecular potential, B - dephasing induced by the orientational motion via rotation -vibration coupling,
0 1991- Elsevier Science Publishers B.V. (North-Holland)
P. Aechtner,A. Laubereau / Vibrationaldephasingof liquidcarbondisulfde
y - resonance energy transfer via short-range and/ or long-range parts of the intermolecular interaction. The role of processes y is well established in the literature studying isotopic dilution effects [8,9]. By proper choice of concentration and solvent the contribution of T2,7can be made negligible and only the first three terms in eq. ( 1) have to the retained. A further simplification may be introduced by time-resolved techniques directly measuring the population lifetime T, [ 7, lo]. Unfortunately experimental data of this sort are still scarce and have to be taken with care since quite often effective (or apparent) lifetimes are provided because of limited time resolution and rapid energy redistribution (bottleneck situations). The relative role of dephasing processes cy and /3 is unclear in general and shall be investigated in the following for the example CS2. Some additional problems are mentioned that will not be investigated here: (i ) to what extent is the assumption of statistically independent processes correct; (ii ) what is the role of ternary and even higherorder interactions; (iii) what inhomogeneous contribution to vibrational bandshapes exists in simple liquids? As to the last question we note that various results suggest fast modulation of vibrational modes in liquids at least on the time-scale of a few picoseconds; i.e. homogeneous broadening [ 111. The first two points are still open for discussion. With regard to the complexity of the problem of vibrational dephasing in a liquid, it is not surprising that no unique answer has been attempted in the past. 2.1. Pure dephasing via translational motion Several models are discussed in the literature assuming binary interactions and fast modulation. A recent version is the model of Schweizer and Chandler [ 5 ] predicting 1
= const.(T) TLa
Similarly to earlier theoretical attempts the model predicts a fixed relation between the pure dephasing rate 1 / T,,, and the binary collision rate: the latter may be expressed in terms of the Enskog time rE between COlliSiOIISof hard spheres. 7, = 7E(n, T) is known to
depend on molecular number density n and temperature T. The temperature dependence of the constant factor is small and may be often neglected. The factor Tin eq. (2) accounts for the fact that the individual dephasing event becomes more efficient with increasing T. The dominant temperature effect in eq. ( 2 ), however, comes in via the Enskog collision rate
[I21 no2g(a) .
Here u= (~,+a,,)/~ denotes the mean value of the hard sphere diameters a,, c,, of the solute and solvent molecules, respectively. g( 6) represents the correction factor of Camaham and Starling generalized to a binary mixture [ 12 ] :
with packing ratio q= xna~/6. The effective hard sphere diameters a,, 4 are temperature but not pressure dependent. A simple parameterization of cz(T) has the form [ 13 ] : a(T)=a,[l-B(T/T,)“2]/(1-B),
introducing the hard sphere diameter o,,, at the melting point T,,, of the medium. B is a measure of the weak temperature dependence ( 0 < B < 0.2 ) . A negative temperature gradient of the dephasing rate via translational motion results from eqs. (2 )(5). This has also been predicted by other models and observed for several examples. Strekalov and Burshtein [ 14 ] have recently pointed out that under more general conditions a non-monotonic temperature behaviour of vibrational dephasing may originate from vibration-translation (VT) coupling; the change of sign of the temperature gradient is predicted, however, for larger temperature intervals than considered in the following. 2.2. Dephasing via vibration-rotation coupling A different dephasing mechanism is provided by rotational collision events that may significantly contribute e.g. for anisotropic molecules. Several authors have attributed an increase of the dephasing rate
P. Aechtner,A. Lmbereau / Vibrationaldephasingof liquidcarbondisulfide
(line broadening) with rising temperature to this relaxation process. An extension of the J-diffusion model by Brueck [ 15 ] yields: -
where 7.,denotes the angular momentum correlation time. The constant factor on the rhs contains the vibration-rotation (VR) coupling. 7, may be expressed using the cell model or be related to the Enskog collision time considering rough hard spheres [ 16 1. The latter picture leads to (7)
with yz 2.3. An increase with temperature of the dephasing rate is predicted by eqs. (3)-( 7). For the discussion intended here, the vastly different temperature dependences of the VT and VR coupling mechanisms should be noted. Comparison with eq. ( 1) may suggest that the varying temperature dependence of vibrational dephasing observed for different examples can be related to the relative weights of the contributions 1IT,, and 1 / Tzs in this expression. 2.3. Phenomenological description The problem of estimating the translational and rotational collision rates may be avoided introducing the orientational relaxation rate l/7& Using the Hubbard relation [ 17 ]
in eq. (6) and similarly in eqs. (2) and (7)) we rewrite eq. ( 1) in the form 1 = T,(P, T)
+aT2TR(p, T) + bTlg(p,
Here a, b are approximately pressure (p) and temperature independent parameters. Explicit expressions for a, b may be derived using available theoretical models, but have only limited quantitative significance. Eq. (9 ) allows a phenomenological discussion of experimental data on T2(p, T) by comparison with results on rn (p, T). Because of the opposite temperature behaviour of the last two terms on the rhs of eq. ( 9 ) , a non-monotonic temperature depen-
dence is expected in general for the dephasing rate l/T,.
3. Experimental The schematic of the experimental system for timeresolved CARS investigations [ 181 on the picosecond time scale is depicted in fig. 1. Single pulses of 4 ps duration are generated at 19000 cm- * by a modelocked Nd: glass laser with electro-optic gain control. After frequency doubling the pulse is split into three portions. A first part is directed into the sample and serves for the excitation of the v, mode of CS2 together with a synchronized Stokes pulse of proper frequency shift. The latter is derived by stimulated Raman scattering from the major portion of the laser pulse using a generator-amplifier device of two liquid cells with 1,2-dibromopropane. A third fraction of the laser pulse is used for the probing process with carefully aligned beam geometry according to the phase-matching requirements of the CARS process. For the sample we use a high pressure cell with fused silica windows in the pressure range of several kbar. Measurements below 290 K are made in a simple liquid cryostat at 1 bar with sample length 5 mm. Spectral grade CS2 and solvents (n-pentane, Ccl,) are used in the investigation. The coherent Raman scattering of the probe pulse is detected with variable time delay in a small solid angle of acceptance with a grating spectrometer set at the anti-Stokes frequency position and a photomultiplier. For background suppression perpendicular polarization planes are adjusted for the excitation and probing processes.
4. Results and discussion Examples for the signal transients are shown in fig. 2 for the neat liquid and the solution in n-pentane with molar fraction 0.02 (tig.s 2a and 2b, respectively ). The signal overshoot around t,=O is an artifact due to the nonresonant part of the third-order nonlinear susceptibility ; it is especially pronounced for the diluted system (see fig. 2b). The scattering signal for large tn represents the resonant contribution due to the excitation of the v, mode
P. Aechtner. A. Laubereau / Vibrational dephosing of liquid carbondisulfide
Fig. 1. Schematic of the experimental setup for picosecond CARS.Variable delay VD, fixed delay FD; photodiode PD; apertures A 1, A& polarizers GP 1, GP2; filter F; spectrometer SP, photomultiplier PM.
: Pentano = 0.02 -I
s v) x 10-2 k i7i d 10-3 e a E 10-4 t r” 10-t 6
Fig. 2. Time-resolved CARS of the resonantly driven v1 vibration (v=656 cm-‘) of liquid CS2. The coherent scattering signal is plotted versus delay time between pump and probe pulses at various pressures; experimental points, calculated curves. (a) Neat CS2 at T=353 K, (b ) CS2 dissolved in n-pentane (m.f.= 0.02) at T= 368K.
which decays exponentially with time constant TJ2 (note semi-logarithmic plot). The latter point provides experimental evidence for homogeneous broadening; i.e. dephasing close to the fast modulation limit. The significant pressure dependence of the dephasing times in fig. 2 and the measuring accuracy of several percent should be noted. The solid curves
in figs. 2a and 2b are calculated using the theory of picosecond CARS [ 7,19,20]. The dashed line (fig. 2b) indicates the instrumental response function as measured by nonresonant CARS in neat n-pentane. The measured dephasing rates 2/T2 of the vI mode at 293 K are plotted in fig. 3a versus pressure in the range O-4 kbar for neat CS2 (full points) and the so-
P. Aechtner, A. Lmbereau / Vibrational dephusing of liquid [email protected]
0.05 ' ' 0
Fig. 3. Dephasing rate 2/T, of the u, mode of CS1 versus pressure in the range O-4 kbar. (a) Neat CSz (full points) and CSz: npentane (open circles) at T=293 K. (b) Neat CS2 at T=353 K (full points). Theoretical curves; see text.
lution in n-pentane with molar fraction 0.02 (open circles). Increasing the number density via pressure accelerates the relaxation rate by a factor of approximately 2.5 from 2/1”,=0.096+ 0.004 ps-’ to 0.220+0.014 ps-’ for the neat liquid and from 2/ T2= 0.065 f 0.003 ps- ’ to 0.185 f 0.006 ps- * for the solution. Similar data are shown in fig. 3b for neat CS2 at 353 K. The curves in fig. 3 are calculated and will be discussed below. Our results on the temperature dependence are presented in figs. 4a and 4b. The data extend from a low temperature close to the melting point to higher temperatures beyond the boiling point under normal conditions (vertical broken lines); the extended temperature range of our investigation is due to a minor pressure of 20 bar applied to the sample above room tem~rature. A non-monotonic temperature dependence is observed for neat CS2 and for the solution in n-pentane with molar fraction 0.01 (fti points in figs. 4a and 4b respectively). Similar results were also obtained for the solution in Ccl, [ 2 11. The dephasing results of Angeloni et al. [ 221 for solid CSz are also
I . . . ..-- .... 200
Fig. 4. &phasing rate 2/T, of the Yemode of CSI versus temperature in an extended existence range of the liquid state. (a) Neat CSI (full points); data for crystalline CS2 are also shown for comparison (open circles, from ref. [ 221. (b) Solution ofn-pentane ( mol fraction 0.01). A minor pressure of 20 bar is applied above 300 K. Theoretical curves accounting for the non-monotonic temperature behaviour, the estimated ~nt~butions of VT coupling (dashed lines) and VR coupling (dotted curves) are indicated; a third contribution by energy relaxation is also included in the data analysis.
included in fig. 4a (open circles). A drastic change of the dephasing rate is noticed at the liquid-solid phase transition. For a subsequent compari~n with expe~mental dephasing results some data on the orientational relaxation of neat CS2 are shown in fig. 5. Using spontaneous Raman spectroscopy, the pressure dependence of 1/zR was measured by Ikawa and Whalley 1231 at constant temperature (293 K) in the range 0- 10 kbar (see fig. 5a). The drastic drop of the relaxation rate be a factor of 8 with increasing pressure i.e. number density is notewo~hy. Corresponding data by Cox et al. [ 24 ] on the temperature effect are reproduced in fig. 5b. The orientational relaxation rate is seen to increase by a factor of approximately 9 in the interval 165-3 10 K. The solid curves in fig. 5 are calculated from eqs.
P. Aechtner,A. Lmbereau / Vibrationaldephming of liquidcarbondisulfide
1.0 ‘z; <
f p: -: -
Fig. 5. Orientational relaxation rate 1/fR of neat CS used to estimate the binary collision rate. (a) Pressure dependence at room temperature. (b) Temperature behaviour. Experimental data from refs. [ 23,241, respectively. Theoretical curves considering a hard sphere model; see text.
(3)-(5), (7) and (8). It is important to note that the theory discussed above accounts well for the observed pressure and temperature dependence. Besides an absolute scaling factor for the orientational relaxation rate 1/rR in eq. (8 ), the interesting parameters are the hard sphere diameter a, of CSI and the temperature coefficient B, (compare eq. (5 ) ). Fitting the calculated curves to the published data of fig. 5 we obtain a,=4.85 A at 293 K and B,=O.O9. The quantities a, and B, (together with a,,, and B,,,, compare eq. ( 5 ) ) are used for a parameterization of the orientational relaxation time rR(p, T). The molecular density numbers entering implicitly the computation (see eqs. (3), (4)) are taken from the literature [ 25 1. When the orientational relaxation is known as a function of pressure and temperature, a comparison with vibrational dephasing data is straightforward by the aid of eq. (9). The situation for neat CS2 is discussed first. The calculated curves are depicted in figs. 3a, 3b and 4a. The constants a, b and TI in eq. (9) for the different mechanisms are treated as fitting parameters. For the sake of simplicity, the temperature dependence of the population relaxation 1/T, is neglected here; a comparatively small temperature variation may be expected for T,. The same pressure dependence as for the VT dephasing mechanism (compare eq. (2) ) is assumed for the population re-
laxation, 1/T, a 1/Q. For neat CS2 a value of T, x 20 ps is suggested fitting the phenomenological theory to the experimental data. This number is consistent with estimates of T, from Fermi resonance observations [ 261 by the help of a semi-quantitative theoretical model [ 27 1. The relative contributions of the VT and VR mechanism (constants a and b) are depicted in fig. 4a (broken and dotted curves, respectively). A resonant contribution to the vibrational relaxation via the attractive part of the intermolecular potential is absent because of the vanishing electric transition dipole moment of the yI mode. Resonant dephasing via the repulsive part of the intermolecular interaction, on the other hand, is of minor importance [ 9 ] and displays a similar pressure and temperature as the VT coupling; the latter effect is included by the fitting procedure in the amplitude of the VT curve. The resulting total dephasing rate (solid curve) displays a non-monotonic temperature dependence and accounts semiquantitatively for the experimental data; it should be noted here that the curve for T> 3 10 K represents an extrapolation of the rR data of fig. 5b using the B, value obtained from rR data also for higher temperatures. The same parameter set also describes the pressure dependence of vibrational dephasing of neat CS2 at 293 and 353 K (full points, solid curves in figs. 3a, b). In the light of the theoretical arguments discussed above the different pressure
P. Aechtner, A. Luubereau / Vibrational dephasing of liquid carbondisulJde
gradients (factor of 1.3 ) originate from the opposite pressure behaviour of VR coupling compared to VT and T, processes that determine the dephasing rate with variable relative weights depending on temperature. Calculated results are presented in figs. 3a and 4b also for the solution CS2: n-pentane (molar fractions 0.02 and 0.0 1, respectively). Unfortunately, corresponding rn ( T,p)data are not available for this system. The computed curves in the figures were obtained using the same parameters a,, B.for CS2 as for the neat liquid and ob= 5.8 A for pentane at 293 K, for the temperature dependence of the hard sphere diameter (eq. (7) ) the same value Bb= B,= 0.09is assumed as for C!&. Superposition of the VT and VR processes in fig. 4b (broken and dotted lines, respectively) again yields a non-monotonic temperature dependence (solid curve) in qualitative agreement with the experimental points. A small contribution by population relaxation is suggested by the calculated data with T,x 100 ps. The same set of parameters accounts for the measured pressure dependence of the dephasing rate at 293 K (open circles, calculated broken line in fig. 3a). The observed pressure dependences are fully consistent with the binary collision approach ( eq. ( 3 ) ) . 5. Conclusions
In summary it is pointed out that a non-monotonic temperature effect is reported here for dephasing of the v, mode of CS2. According to our interpretation, the effect originates from competing dephasing mechanisms with opposite temperature dependence and is believed to possess more general significance. From the quantitative point of view the present theoretical analysis is unsatisfactory. Only qualitative features are outlined. More quantitative descriptions of vibrational dephasing are highly desirable to test some of the implicit assumptions, e.g. the binary collision approach. References [ 1] W.E. Rothschild, Dynamics of Molecular Liquids (Wiley, New York, 1984).
[ 21 W. Zinth and W. Kaiser, in: Topics in Applied Physics, Vol. 60. Ultrashort Laser Pulses and Applications (Springer, Berlin, 1988).  SF. Fischer and A. Laubereau, Chem. Phys. Letters 35 (1976) 6.  D.W. Oxtoby, Advan. Chem. Phys. 40 (1979) 1; J. Chem. Phys. 70 (1979) 2605. [ 51KS. Schweizer and D. Chandler, J. Chem. Phys. 76 ( 1982) 2296.  A.M. Levine, M. Shapiro and E. Polak, J. Chem. Phys. 88 (1988) 1959.  A. Laubereau and W. Kaiser, Rev. Mod. Phys. 50 ( 1978) 607.  G. Dirge, R. Amdt and J. Yarwood, Mol. Phys. 52 ( 1984) 399.  M. Geinaert and G. Gale, Chem. Phys. 86 (1984) 205. [ lo] A. Seilmeier and W. Kaiser, in: Topics in Applied Physics, Vol. 60. Ultrashort Laser Pulses and Applications (Springer, Berlin, 1988). [ 111 G.M. Gale, P. Guyot-Sionnest and W.Q. Zheng, Opt. Commun. 58 (4986) 58. [ 121 I. Chapman and T.G. Cowling, Mathematical Theory of Non-Uniform Gases (Cambridge Univ. Press, Cambridge 1970); G.A. Mansoori, N.F. Camahan, ICE. Starling and T.W. Leland Jr., J. Chem. Phys. 54 ( 1971) 1523. [ 13 ] P. Protopapas, H.C. Anderson and N.A.D. Parlee, J. Chem. Phys. 59 (1973) 15. [ 141 M.L. Strekalov and A.I. Burshtein, Chem. Phys. Letters 86 (1982) 295. [ 151 S.R.J. Brueck, Chem. Phys. Letters 50 (1977) 516. [ 161 D. Chandler, J. Chem. Phys. 60 (1974) 3500,3508. [ 171 P.S. Hubbard, Phys. Rev. 131 (1963) 1155. [ 181 G.M. Gale and A. Laubereau, Opt. Commun. 44 ( 1983) 273. [ 191 N. Kohles and A. Laubereau, Appl. Phys. B 39 ( 1986) 141. [ 201 N. Kohles, P. Aechtner and A. Laubereau, Opt. Commun. 65 (1988) 391. [ 2 I] P. Aechtner, Thesis, University of Bayreuth, Germany (1989). [ 22 ] L. Angeloni, R. Righini, P.R. Salvi and V. Schettino, Chem. Phys. Letters 154 ( 1989) 432.  S. Ikawa and E. Whalley, J. Chem. Phys. 86 (1987) 1836.  T.J. Cox, M.R. Battagba and P.A. Madden, Mol. Phys. 38 (1979) 1539. [ 25 ] American Institute of Physics, Handbook (McGraw-Hill, New York, 1972) pp. 2-178; Gmelin, Handbuch der anorganischen Chemie, 8th Ed. (Verlag Chemie, Weinheim, 1972) part D4, p. 72; J. Timmermans, Physico-Chemical Constants of Pure Organic Compounds (Elsevier, Amsterdam, 1950) p. 3 1.  J.J. Kondilenko, P.A. Korotkov and G.S. Litvinov, Opt. Spectry. 34 (1973) 265. A. Fendt, S.F. Fischer and W. Kaiser, Chem. Phys. 57 (1981) 55.