Chapter 5 Third-order experiments: The total spectral density of intermolecular

Transcription

1 Chapter 5 Third-order experiments: The total spectral density of intermolecular motions in liquids 5.1 Introduction Light scattering is a very suitable tool for studying intermolecular dynamics in liquids, as was pointed out already in the early days of spectroscopy by Raman [1]. The broadening of the depolarized Rayleigh line was associated with the anisotropy of the molecular polarizability and attributed to several mechanisms such as collision-broadened rotational lines and local density fluctuations []. Following many experiments, a large variety of models were proposed for explaining the shape of this line, located close to zero frequency shift. However, up to now a full understanding of the underlying processes and, hence, of the intermolecular structure and dynamics in liquids is still lacking. A first quantitative theory of Rayleigh scattering in liquids was formulated in 1941 by Leontovich who showed that a diffusion controlled decay of the polarizability anisotropy will lead to a Lorentzian line shape [3]. It was found experimentally by Starunov [4] and by Shapiro and Broida [5] that the depolarized Rayleigh line of simple organic fluids like benzene and carbon disulfide (CS ) deviates from the predicted Lorentzian shape at frequencies above a few wavenumbers. According to Starunov this might be due to inertial corrections in the diffusion process at early times [6] or due to coherent oscillations of the molecule as a whole in the potential defined by its neighbours [7,8]. Litovitz and coworkers reported exponential line shapes at high frequencies (from several tens up to more than hundred wave numbers) for a number of atomic and molecular liquids [9,10]. These features, which are not fully described by Starunov's theory, were attributed to short-range electronic overlap effects and molecular frame distortions [11]. A number of other authors invoked long-range dipole-induced dipole (DID) interactions for the explanation of the depolarized Rayleigh wing [1-]. In addition, the possible influence of 69

2 cooperative effects due to microscopic orientational ordering in the liquid was considered [13-15,3-5]. When laser pulses with durations of the order of 100 fs became available [6], a variety of time resolved spectroscopic techniques were developed to probe the dynamics in the Rayleigh wing [7-31]. Among these methods, which provide information equivalent to the traditionally used spontaneous scattering [3-34], the optical Kerr effect (OKE) [35-4] is very powerful, since it allows for the implementation of optical heterodyne detection and model-independent deconvolution in frequency space [4]. While the early light scattering studies investigated mainly the interplay of the different scattering mechanisms responsible for the depolarized Rayleigh wing, the more recent time resolved studies focused more on the time scale(s) of the fluctuations, which cause the line width [43-45]. The unambiguous separation of the dynamic and static contributions to the line width, often modeled as homogeneous and inhomogeneous dephasing, is crucial for a detailed picture of liquid state phenomena. The line width of the intermolecular vibrations reflects the distribution and evolution of local potentials, but also the possibly frequency [4] or quantum number [46,47] dependent friction due to interactions with the neighbouring molecules. As was shown by Mukamel and coworkers [48,49], all spontaneous and stimulated nonresonant third-order vibrational spectroscopies measure in principle the total spectral density. Because these techniques all depend on the one-time (two-point) correlation function of the polarizability, they are not sensitive to the specific line broadening mechanism. In order to characterize the time scale(s) of the underlying dynamics it was proposed to probe higher-order correlation functions by multiple pulse experiments [48,49]. Examples for this approach are the Raman echo [50-53] and temporally two-dimensional fifth-order Raman scattering [54-61]. Information on the relative importance of the different scattering mechanisms can be obtained from experiments on binary mixtures [5,16,,3,6-66]. Based on trends in their results, Chang and Castner [63] could assign features in the Rayleigh wing spectrum of formamide in a number of polar solvents to hydrogen networks. McMorrow and coworkers [6,64] reported the optical Kerr effect response of CS in a number of alkanes. The observed spectral changes upon dilution were attributed to decreasing intermolecular force constants and a reduced inhomogeneous width. It should be noted, however, that these conclusions are based on indirect evidence. In the case of CS /pentane binary mixtures the interpretation proposed by McMorrow and coworkers [6,64] is not unique, as was shown in Ref. 66. This chapter is organized as follows: In Sec. 5. the OHD-OKE response of neat CS is introduced. It is used as working example, which illustrates the typical information content of the third-order nonlinear optical response of intermolecular dynamics in liquids. The results of OHD- OKE experiments on CS /pentane binary mixtures are given in Sec The single molecule orientational motion of the highly polarizable CS molecules is modeled in Sec. 5.4 by a set of Brownian oscillators. It is shown that the third-order response can be simulated well by different degrees of inhomogeneous broadening of the intermolecular vibrations. Therefore, it is not possible to separate the static and dynamic part of the material response unambiguously. Only higher-order experiments such as the temporally two-dimensional Raman scattering, which is discussed in the next chapters, are sensitive to the time scale(s) of the fluctuations of the intermolecular potential. The spectra of the CS /pentane mixtures show some peculiar trends upon 70

3 dilution, which suggest that the response may be influenced to a large extent by many-body effects [8-3,66]. These mechanisms are discussed in Sec This chapter ends with some concluding remarks. 5. The OHD-OKE response of neat CS In a time resolved OHD-OKE experiment [40,4], an ultrashort pump pulse induces transient birefringence in an isotropic liquid by impulsive stimulated Raman scattering [67-70]. All Raman-active material modes accessible by the ultrashort broadband laser pulses, are excited impulsively. As a consequence of this impulsive excitation only those modes with a period T longer than the pulse duration J P can be observed. For the sub-50 fs laser pulses used here, the!1 spectral region between 0 and 300 cm is measured. The build-up and decay of the birefringence resulting from the excitation of depolarized vibrational modes by the pump pulse is monitored through the polarization rotation of a variably delayed probe pulse [40,4,64]. This process can be described by the theory presented in Sec..4. For pump and probe pulses, which are identical, except for a relative delay T, the thirdorder polarization can be written as: 4 P (3) (t,t)'e(t!t) dj m 1 R (3) (J 1 )E (t!j 1 ). 0 (5.1) Here, as in the remainder of this chapter, the space indices of the polarization, the optical field and the response function are suppressed for simplicity. Since the probe pulse also serves as an out-ofphase local oscillator [40,4], and the detector integrates the signal intensity over all times, the OHD-OKE signal varies with the pulse delay T as: I (3) 4 Kerr (T)' dt Re6P (3) (t,t)e(t!t)>'re6g () (T)¼R (3) (T)>, m!4 (5.) () where G (T) is the second-order background-free intensity autocorrelation of the optical pulses and q denotes a convolution. The result of an OHD-OKE experiment on room temperature liquid CS is presented in Fig It agrees with previously published data of McMorrow et al. [40]. The sharp peak at zero delay is due to the instantaneous electronic hyperpolarizability of the liquid. This four-photon process, which is described by Eq. (.19.a) and Figs...a and.3.a, follows the second-order () autocorrelation G (T) and vanishes when the pump and probe pulse are separated in time. For finite delays, the signal is completely determined by the induced nuclear motion in the liquid. The 71

4 intensity log(intensity) response is then governed by Eq. (.19.b); the corresponding scattering mechanisms are depicted in Figs...a and..b. The maximum of the signal is reached at a finite delay of 00 fs. This delayed response reflects the inertial character of the induced motions: The nuclei cannot follow the impulsive excitation by the pump pulse due to their finite moment of inertia. During the interaction with the laser pulse they hardly change their position but acquire an appreciable momentum. Only after a few tens of femtoseconds this leads to a distinct deviation from their initial positions. After the maximum is reached, the signal rapidly decays in a nonexponential way. For delays longer than ps the decay becomes single exponential, as can be seen in Fig. 5.1.b. In frequency domain this exponential decay corresponds to the Lorentzian line close to zero frequency, which was first discussed by Leontovich [3], while the nonexponential decay at early delay times corresponds to the non-lorentzian line shapes at high frequencies first reported by Starunov [4] and by Shapiro and Broida [5]. The signal of Fig. 5.1.a is not identical to the true material response since the experiment (a) delay (fs) (b) spectral amplitude (c) delay (ps) frequency (cm -1 ) Fig. 5.1: (a) The OHD OKE response of CS (dotted). Three different contributions can be recognized: (i) Instantaneous response due to electronic hyperpolarizability, having the shape of the second-order autocorrelation of the pulses (dashed); (ii) Diffusive reorientation, modeled by an overdamped mode with exponential rise- and decay times (solid); (iii) The remainder of the signal, which dominates at short time (< 1 ps), has a finite rise time of 00 fs. (b) The diffusive tail of the signal (solid) is single exponential (dashed) with a decay constant of 1.6±0.1 ps. (c) Spectra of the nuclear response, derived from the OHD-OKE data. The solid line gives the total spectrum of nuclear motion, while the dashed line is the spectrum of the non-diffusive part only.

5 was performed with pulses of a finite duration/bandwidth. This leads to spectral filter effects [41]: Material modes with higher frequencies are excited less efficiently than those with a low frequency, since the intensity in the wings of the pulse spectrum decreases rapidly. Therefore, the amplitude and also the line shape of the high frequency modes are partly determined by the used laser pulses. For optically heterodyned detected signals it is possible to remove these spectral filter effects by a deconvolution procedure in frequency domain as was shown by McMorrow and (3) Lotshaw [41,4]. The measured OHD-OKE intensity, I (T), can be written as a convolution of (3) () the material response, R (J), and the second-order autocorrelation, G (T), cf. Eq. (5.). Since () G (T) can be measured independently in a cross correlation experiment (see chapter 4) two of the three functions in Eq. (5.) are known. When this expression is Fourier transformed, the convolution in time domain shows up as a product in frequency domain. Dividing the Fourier transform of the OHD-OKE signal by the Fourier transform of the intensity autocorrelation gives a frequency representation of the true material response free of any spectral filter effects. All information on the nuclear motion is in principle contained in the imaginary part of this complex ratio, which is shown by the solid line in Fig. 5.1.c [41,4]. 5.3 The OHD-OKE response of CS /pentane binary mixtures In this section experimental results on binary mixtures of CS and pentane are presented for various volume fractions of pentane [71]. The latter is chosen as cosolvent because it is chemically inert and has, even at the highest concentration, no significant nuclear response compared to that of CS as is shown in the inset of Fig. 5.. The short time part of the OHD-OKE signals are presented in Fig. 5. for volume fractions ranging from 100 % down to 3 %. They show a number of clear trends on dilution, which were also observed by McMorrow cum suis [40,6,64]. The rise and decay times of the sub-ps component monotonically increase as pentane is added. The instantaneous electronic contribution becomes relatively more important compared to the nuclear part. This is because for neat pentane the ratio of the electronic hyperpolarizability to the nuclear response is much larger than for CS as is evident from the inset of Fig. 5.. As found already by McMorrow and coworkers [6], the exponential decay at longer delays is getting faster with increasing pentane fraction, which can be correlated with the bulk viscosity. In order to evaluate the sub-ps nondiffusive component more carefully, McMorrow and Lotshaw proposed to remove the diffusive component prior to the Fourier deconvolution procedure [4]. Here, we model the exponentially decaying part of the signal by an effective overdamped harmonic mode, which can be characterized either by a damping rate ( D and an ½ effective frequency S D=(( D/4!T D), or by exponential rise and decay times J R and J D [46,49,59]: R (3) D (J)'e!( D J/ 1 S D sinh(s D J)' 1 S D (1! e!j/j R ) e!j/j D. (5.3) As evident from Fig. 5.1.b, the decay time J can be determined with high accuracy. The value D 73

6 intensity delay (fs) Fig. 5.: The short-time part of the OHD-OKE response of CS /pentane binary mixtures. The volume fractions of CS are (from bottom to top): 100%, 80%, 67%, 50%, 33%, 0%, 11%, 6% and 3%. Even at the lowest concentration the nuclear signal is mainly due to CS as is demonstrated in the inset where the response of pure pentane (solid) is compared with the response of the 3% solution (dotted). The relative magnitude of the purely electronic contribution at zero delay rises when pentane is added and is dominated by the contribution of pentane at the lowest concentrations. of the rise time J R, which is introduced phenomenologically to account for a delayed onset of the diffusion, is not so easily established. As demonstrated by McMorrow and Lotshaw [7], the particular choice of J R has only a minor influence on the shape of the remaining signal. We removed the diffusive component before the Fourier transformation, assuming a rise time J R=130 fs for all mixtures. The deconvoluted spectra, shown in Fig. 5.3 for the various solutions, directly display the deviations from the Lorentzian line shape, free of the electronic response and uneffected by any spectral filter effects. Upon dilution, the width of the spectral response decreases and the maximum shifts towards lower frequencies. This could, in principle, be inferred directly from the time domain data of Fig. 5.1, but there these modifications are partly masked by the electronic and the diffusive contributions. The main changes in the spectra occur when the volume fraction of CS is decreased from 100% to 33% while no significant changes occur below 0%. This suggests that CS is completely "dissolved" when there are about pentane molecules per CS molecule [71]. This unexpected, rather surprising finding will play a crucial role in the microscopic interpretation of the data, see Sec Single molecule orientational motion For the discussion of the microscopic scattering mechanisms responsible for the non- Lorentzian line shape in the Rayleigh wing, we want to refer to concepts originally introduced in 74

7 spectral amplitude frequency (cm -1 ) frequency (cm -1 ) Fig. 5.3: Frequency domain representation of the coherent part of the deconvoluted OHD-OKE response of all mixtures shown in Fig. 5.. The spectrum becomes narrower and shifts towards lower frequencies when the concentration of CS decreases. From the right to the left the curves correspond to CS volume fractions of 100%, 80%, 67%, 50% and 33%. The curves for 0% and lower concentrations are almost identical. The slow exponential decay that shows up as a Lorentzian line located close to zero frequency has been subtracted prior to the deconvolution. The effect of this procedure is demonstrated for the data of pure CS in the inset, where the spectrum of the total response (solid) and of the coherent part only (dotted) are depicted. depolarized light scattering (DLS). The DLS spectrum I(T), which has essentially the same information content as the time resolved OHD-OKE experiment [3-34], is related to the classical correlation function C(J) of the anisotropic part A of the macroscopic polarizability via the fluctuation-dissipation theorem [73]. C(J) is given by [15,73]: C(J)'V!1 Tr A (0):A (J), (5.4) where A, an irreducible tensor of rank two, is defined by: A'A 0 1%A. (5.5) Here, 1 and A denote the unity tensor and the total macroscopic polarizability tensor, respectively, and A 0 is defined by A 0/aTrA. When we neglect local field and collisional effects for the moment (they will be discussed in the next section), the macroscopic polarizability A is simply the sum of the polarizabilities of all N molecules in the volume V: 75

8 A' j N i'1 " i. (5.6) For molecules with axial symmetry, such as CS, the single molecule polarizability " of molecule i i can be decomposed into an isotropic and an anisotropic part [19]: " i '" 1% 3 ( Q i, (5.7) where the tensor Q i can be expressed in terms of the unit vector û i along the symmetry axis: Q i ' 3 ûi ûi! 1 1. (5.8) Here, û û denotes the dyadic product of the vector û with itself. i i i When the correlation function C(J) is expressed in terms of the single molecule orientational coordinates via the unit vectors û, one obtains: i C(J)' ( V j N i,j'1 +û i (0)û i (0):û j (J)û j (J),. (5.9) It is convenient to split the correlation function C(t) into two terms, which describe single molecule orientational motion and dynamic orientational order effects of pairs of molecules, respectively: C(J)'C S (J)%C P (J), (5.10) where the corresponding correlation functions C S(J) and C P(J) are given by: C S (J)'N ( V +û i (0)û i (0):û i (J)û i (J),, (5.11) C P (J)' ( V j i j +û i (0)û i (0):û j (J)û j (J),. (5.1) 76

9 For the interpretation of light scattering data one often assumes a certain functional form for these two functions to fit the spectra. However, the analysis in terms of spectral moments is also a powerful and popular concept [16,,3,73]. The results of time resolved OHD-OKE experiments, on the other hand, are usually described by time dependent quantum mechanical perturbation theory, which was reviewed in chapters and 3. The optical Kerr effect signal is governed by the relevant tensor element of the third-order nonlinear optical response function [3,46,49]: R (3) xyxy (J)'! i S Tr [C (J),C (0)]D(!4). (5.13) Here, C (J) is the interaction picture representation of the quantum mechanical operator C, associated with the classical observable A, while D(!4) denotes the density matrix in thermal equilibrium. The trace is to be taken over the matrix representation of the operator. In the classical limit the quantum mechanical response function is related to the classical correlation function, independent of the specific form of the Hamiltonian H 0, the polarizability operator C and the correlation function C(J), via [74]: R (3) xyxy (J)'! 1 k B T d dj C(J). (5.14) Usually, a certain functional form is assumed for the Hamiltonian H 0 of the system and for the operator C, which allows for a closed form calculation of the response function. The most common assumption is that the optical response is dominated by a discrete number or a continuous distribution of harmonic modes q that change the polarizability with a linear (1) coordinate dependence, according to P =P q. However, anharmonic potentials and nonlinear coordinate dependence of the polarizability have been treated as well [49,75]. The most straightforward way to relate this approach to the DLS theory, is to identify the coordinate q with the single molecule orientational variable. The physical process described by this model is then completely analogous to rotational Raman scattering in gases, except for the fact that the rotations in liquids are strongly hindered [4,7,8]. When it is assumed that the molecule is harmonically bound to an equilibrium position, this yields a spectrum distinctly different from the rotational Raman spectrum: The series of separate lines corresponding to different values of the rotational quantum number J merge to a single line for a harmonic librator. The position of this line depends on the force constant and the reduced moment of inertia of the molecule under consideration [64]. The shape of the line reflects all relevant line broadening processes due to, e.g., an inhomogeneous distribution of force constants, homogeneous dissipation of energy, or phase relaxation. The observation of oscillatory behavior at low temperatures, which is attributed to librational motion in rigid cages [44,45,47], supports this assignment. Due to the excitations of the librations the cage configurations and, hence, the 77

10 equilibrium positions of the molecules change, leading to a net anisotropy of the molecular orientation, which finally decays by rotational diffusion. In the itinerant oscillator model [76] the cage degrees of freedom are projected onto a second effective harmonic coordinate, which for strong damping results in a signal according to Eq. (5.3). This procedure relies on a time scale separation between the two involved modes: The collective reorientation has to be much slower than the coherent inertia-limited response. The coherent librational motion of harmonically bound molecules can be easily described by quantum mechanical perturbation theory [46,77]. Damping can be introduced in this formalism at different levels of sophistication as was discussed in chapter 3 [46]. Coupling with Brownian oscillators is among the most important of these approaches, since it accounts for both the frequency shift of a damped oscillator and for the continuous change from underdamped to overdamped motion [49,78]. For an underdamped libration with frequency T L and homogeneous damping ( L the third-order response function reads [46,49,59]: R (3) L (J,T L )'e!( L J/ 1 S L sin(s L J), (5.15) ½ where the effective librational frequency is S L=(T L!( L/4). Additional loss of phase coherence can be caused by the fact that there is a distribution of different cage configurations. When the cages change very rapidly this leads to an exponential phase relaxation rate, which adds to the homogeneous decay rate ( L. In the other limiting case, where the cage configurations change very slowly on the time scale of the librations, the resulting inhomogeneous broadening due to the static (slowly fluctuating) distribution of librational frequencies g(t L) can be taken into account by integrating the response function over these frequencies: R (3) L 4 (J)' m 0 dt L g(t L ) R (3) L (J,T L ). (5.16) This equation allows for a continuous change from underdamped (T L>( L/) to overdamped (( L/>T L) motion over the inhomogeneous distribution. Eq. (5.15) then changes without discontinuity into the form of Eq. (5.3). When the cage fluctuations occur on similar time scales as the librational damping itself, the homogeneous and inhomogeneous limits do not apply, and, for instance, stochastic approaches to the dynamics have to be applied [79,80]. Here, we will discuss the response in terms of (in-)homogeneous dynamics, as described by Eqs. (5.15) and (5.16). For any spectrum it is possible to find an inhomogeneous distribution g(t L) such that it perfectly fits the data [48]. To facilitate the numerical simulations, we assume here the presence of a modified Gaussian distribution that assures zero spectral amplitude as the frequency approaches zero [70]: 78

11 g(t L )'T L exp! (T L!T 0 ) F. (5.17) In this expression, the central frequency and the width of the distribution are denoted as T 0 and F, respectively. In Fig. 5.4 the best fits to the deconvoluted nuclear response functions are shown for mixtures of 100 % down to 0 vol. % CS. Both the single molecule librations and the diffusive nuclear response are taken into account here, according to the formula: R (3) (J)'c L R (3) L (J)%c D R (3) D (J), (5.18) where c L and c D denote the fit parameters for the relative amplitude of the librational and the (3) (3) diffusive response, and R L (J) and R D (J) are given by Eq. (5.3) and Eq. (5.16), respectively. As mentioned already in connection with the itinerant oscillator model, this description of the diffusive and librational motions by two independent modes relies on the time scale separation between these two types of motions, which for neat CS is fairly valid [81]. The corresponding parameters for the various mixtures are given in Table 5.1. Some care should be taken here, since a smaller (larger) inhomogeneous width can to some extent be compensated for by a larger (smaller) homogeneous width. The quality of the fits then is only slightly affected, as is shown for neat CS in Fig. 5.5 [59]. Also, modifications of the functional form of g(t L) may yield equally good fits, with different parameters. These uncertainties are due to the temporally one dimensional character of the OHD-OKE experiment. Only higher-order experiments can, in principle, give conclusive information about the nature of the line broadening mechanisms. However, despite these uncertainties we may summarize the following overall trends: Upon dilution of CS from 100 to 0 vol. %, the centre frequency T 0 seems to become smaller and the inhomogeneous width F seems to decrease, whereas the homogeneous damping rate ( L is not much affected. The collective decay time J D decreases as well. Similar features were found by McMorrow and coworkers [6,64]. However, it is not straightforward to compare the numerical results, since their parameters were reported for a classical model with three independent modes and a somewhat different frequency distribution. At a first glance, it seems that the changes of the spectrum can be easily explained by the lower interaction energy between CS and pentane, compared to that of two CS molecules. The reduced interaction energy accounts for (i) a lower viscosity, hence a shorter collective decay time J D; (ii) a decrease of the libration frequency T 0 of the CS molecules; (iii) a narrower distribution g(t L) of these libration frequencies. Although these conclusions seem logical, some surprising aspects needed further explanation and these, in fact, induced us to look for alternative explanations for the microscopic origin of the optical response [66]. In general, the mixing of two species is governed by both entropy and interaction energies between the two species. For rather weakly interacting molecules 79

12 intensity such as CS and pentane, none of which has a permanent dipole, one might expect that the mixing will be largely determined by entropy. In the ideal case of identical interaction, the distribution of local configurations for the CS molecules becomes broader upon dilution, until a ratio of 1:1 is reached. Then, a reduction of the number of available configurations sets in. Finding a decrease of the inhomogeneous width of librational frequencies while at the same time expecting an increase of possible local configurations is not necessarily impossible in view of the reduced intermolecular interaction between CS and pentane molecules [64]. However, in that case one would expect a further reduction of the inhomogeneous width for concentrations below 0 vol. %, where CS is not fully surrounded by pentane molecules, yet, and the distribution of local environments changes substantially. The rather narrow range of concentrations, over which the spectral changes of the inertial delay (fs) Fig. 5.4: Fits (solid lines) of the deconvoluted optical Kerr response (dots) to a two mode Brownian oscillator model. The CS concentrations were 100, 80, 67, 50, 33 and 0 vol. % (bottom to top). The parameters of the fits are shown in Table 5.1. Table 5.1: The deconvoluted response function of the CS /pentane binary mixtures is fitted best with the following sets of parameters. The parameters for CS concentrations of 0% and below are almost identical (see Fig. 5.3) and not displayed here. vol. fraction CS / % F / rad ps -1-1 T 0 / rad ps ( L / ps

13 intensity response occur, casts serious doubt on the validity of the approach taken so far. From a physical point of view, this is not at all surprising, considering the crudness of treating the optical response only at the single molecule level. In the analysis outlined above, dynamic pair correlations, local field corrections and the influence of collisions were ignored. In reality, these effects probably determine to a large extent the optical response. In the next section we will discuss whether these processes can provide alternative explanations for the observed spectral changes upon dilution. 5.5 Alternative scattering mechanisms Dynamic correlations between molecules can lead to changes in the polarizability, which are not accounted for in the single molecule model described in the previous section. Such correlations are directly related to the molecular shapes and the strength of the interaction energies. They can be particularly important for strongly interacting molecules, e.g., in the presence of hydrogen bridges, but may show up in CS /pentane mixtures as well. The quadrupole delay (fs) Fig. 5.5: Fit of the deconvoluted optical Kerr response (dots) of neat CS for F=0 (homogeneous limit, dashed-dotted), (long dashed),.00 (dashed-double dotted),.45 (short dashed) and.77 rad ps (solid line). The parameters of -1 the fits are shown in Table 5.. The best fit is obtained for F=.77 rad ps. -1 Table 5.: Fit parameters for neat CS. The best fit is obtained for F=.77 rad ps. The deconvoluted response function is shown in Fig. 5.5 together with the fits. The rise and decay times J R=130 fs and J D=1.65 ps of the diffusive component were kept fixed during the fitting. The inhomogeneous width F was varied systematically. F / rad ps -1-1 T 0 / rad ps ( L / ps

14 moment [8,83] of CS may lead to aggregate-like structures in the liquid, resembling small clusters [7]. Like in gas phase clusters, collective low-frequency vibrations are then expected to contribute significantly to the observed Rayleigh-wing spectra [7,84]. In the DLS theory, the collective local modes are described by the dynamic pair correlation function C (t), defined in Eq. (5.1). At t=0 this quantity is closely related to the static p orientational correlation parameter g, which is a measure for the average local ordering in the liquid [13-15,3-5]. For finite times, C (t) describes the decay of the two-particle correlation in P the liquid. The experimental determination of g is far from simple, because its extraction from experimental data relies on several assumptions about the specific form of the single molecule orientational motion and the contributions due to interaction- and collision-induced effects (see below). For pure CS values of g between 1. and.1 have been reported [4,5]. While for CS and pentane there are, to our knowledge, no data available about the dependence of g on the composition of the mixture, MD simulations for binary mixtures of CS and CCl revealed a 4 maximum of the static orientational order parameter at a CS mole fraction of 0.75 [85-87]. Although it is not straightforward to extrapolate this result to a CS /pentane mixture, it indicates that the local structure may depend significantly on the composition of the mixture. In recent literature, the collective local motions in liquids have also been analyzed in terms of instantaneous normal modes (INM) [88,89]. Compared to the pair correlation function C p(t) discussed above, this alternative concept provides a rather natural formulation of the many-body dynamics since it avoids the description in terms of pairs of molecules. Moreover, there is a direct way to incorporate the results of an INM analysis into the theoretical description of the OKE response of Sec. 5.4, since the instantaneous normal modes are modelled by harmonic oscillators. It should be noted, however, that within the INM picture the modes are undamped which corresponds to pure inhomogeneous broadening. This description holds for very short times, but is expected to break down at longer times. Therefore, additional homogeneous damping has to be introduced [Eqs. (5.15) and (5.16)], while the slow reorientational diffusion was modeled by an overdamped mode, according to Eq. (5.3). Due to our limited knowledge of the intermolecular potentials in a mixture of CS and pentane, it is difficult to estimate how the collective local modes are changed upon dilution. When pentane is added, the average interaction between molecules decreases because pentane, in contrast to CS, is less polarizable and does not have any prominent multipole. Therefore, one might anticipate a decrease of the collective vibrational motions, corresponding to a decrease of C p(t). This change would lead to a narrowing and a net shift to lower frequencies, thus providing a qualitative explanation for the observed modifications of the spectrum. It is, however, not directly evident that the collective modes would change exclusively at concentrations above 0 vol. % of CS. This may indicate the formation of CS dimers or higher aggregates, which do not change their size as more pentane is added. Whether this is indeed true can only be answered by quantum chemical calculations. In view of the rather weak interactions in the CS /pentane mixtures we do not think that this explanation is very plausible, though. Collision induced (CI) changes of the molecular polarizability arise through the fact that neighbouring molecules in liquids temporarilly approach each other so closely that the electron clouds and/or molecular frames are seriously distorted [9-11]. The CI mechanism was first proposed to explain certain features in the Raman and infrared spectrum of carbon tetrachloride 8

15 [11]. It is not an easy task to derive an expression for the polarizability change as a function of the mutual distances and orientations of the involved molecules, and then average over all the rapidly changing configurations that are possible. To quantify this effect for molecular systems, Bucaro and Litovitz [9] showed that in a Lennard-Jones liquid the polarizability depends on the thirteenth power of the inverse distance provided that all collisions have a zero impact parameter. Based on these assumptions they derived an exponentially decaying line shape, which indeed was observed in the Rayleigh wing of atomic and some molecular liquids. However, they stressed that their approach is rather suspicious for highly anisotropic molecules such as CS [9]. Within the quantum-mechanical perturbation picture, often used for OKE experiments, the CI effects at the repulsive part of the potential may be accounted for by incorporating a strong dependence of the polarizability on the internuclear separation. From this elementary description of the CI effects, it is clear that we may expect this contribution to the optical response to decrease when CS molecules, with their extended electron clouds on the sulfur atoms, are replaced by pentane where the electrons are more confined to the backbone. Since the CI effects are believed to govern the high frequency part of the spectrum [9], we can anticipate the narrowing of the spectrum as apparent in Fig It is, however, quite surprising that these changes occur exclusively at relatively high concentrations of CS. Intuitively, one would expect a further narrowing of the spectrum until each CS molecule is surrounded by pentane molecules only. Changes of the molecular polarizability can also be accomplished by dipole-induced dipole (DID) interactions, which occur on a much longer range [1-]. It should be noticed that the Lorentz-Lorenz theory [90], which accounts for the change of the local field due to a dielectric continuum around the molecule under consideration, is not capable to deal with a surrounding composed of discrete polarizable matter that is free to move. The relation between the optical response and the probed microscopic dynamics is, in fact, very complicated [91,9]. In general, the electric field E i at molecule i is the superposition of an externally applied Lo field E Ex and the local field corrections E i at this molecule, resulting from the (induced) dipoles Lo µ j and higher multipoles of all other molecules: E=E i Ex+E i. When the contribution of the induced dipoles is dominant, one finds: E Lo i ' jj i T(r ij )@µ j ' jj i T(r ij ) :" E j. (5.19) Here r ij=r!r i j is the distance between the centers of mass of molecules i and j, and 3 T(r)=(3rˆrˆ!1)/r is the dipole tensor expressed in terms of the unit vector rˆ=r/r. Note that the local field correction at molecule i depends on the total field E j at molecule Lo j and, hence, on the local field corrections E j of all other molecules. This implies that, in general, the problem has to be solved self-consistently for all molecules. Often, these local field effects are Lo treated by a perturbative approach, where one first approximates E.E j j on the right hand side Eff of Eq. (5.19). The resulting E i is then used to define an effective polarizability i " via: Eff Ex " i E /" i E i, which can be used as a new starting point for the calculation of the local field correction. It has been shown by Ladanyi [17] that this iterative procedure works well for 83

16 molecules with small polarizabilities. For highly polarizable molecules like CS the difference between the applied and the local field are substantial and the iterative method is not appropriate [19]. Dorfmüller and coworkers showed that for CS /CCl 4 mixtures the difference between the first- and the all-order method becomes smaller with decreasing CS concentration [86]. The fluctuations of the surrounding molecules give rise to changes in the induced dipole moment and polarizability, and hence to absorption or scattering of light. The DID mechanism explains the observation of symmetry forbidden lines, e.g., the far infrared (FIR) absorption of nonpolar molecules and the depolarized Rayleigh scattering of atomic systems [16], but it might also contribute considerably to symmetry allowed transitions. However, it is not an easy task to experimentally separate the DID contributions from the other mechanisms discussed in the previous sections [15,18,19]. The DID scattering is particularly sensitive to local density fluctuations. To illustrate this we note that the induced dipoles of isotropically polarizable molecules in a highly symmetric microscopic structure cancel [1]. Only when the symmetry is partly broken, there is a residual correction to the local field. For anisotropically polarizable molecules the situation is even more complicated because then both the mutual distances and orientations are important [15]. Due to these complexities it is virtually impossible to derive closed form expressions for the line shapes resulting from DID interactions in condensed matter systems. The only way to quantitatively investigate the effect is offered by MD simulations [15,17,18,19]. Because the fields induced by the different molecules nearly compensate for each other, this method is very sensitive to the accuracy of the intermolecular potentials and molecular polarizabilities. In most simulations one typically assumes an effective, gas-phase like molecular polarizability that is independent of the surrounding molecules. Therefore, this approach does not include any CI-effects. In their MD study of the depolarized Rayleigh spectrum of CS /CCl binary mixtures, 4 Dorfmüller and coworkers [86] found that the CS molecules were responsible for the main part of the DID signal. For CS /pentane we foresee a similar, but even more pronounced role of the carbon disulfide molecules in the DID contribution. As a first approximation one can expect that the DID effects scale with the average polarizability, normalized by the average density of the 3 mixture [15]. Both CS and pentane have nearly equal average molecular polarizabilities (8.8 Å 3 and 10 Å, respectively) [93] but the density of CS is two times higher than that of pentane. Therefore, a decrease of the DID contribution to the spectral density is expected when CS is diluted in pentane. It should be noted, however, that the DID contribution does not necessarily scale linearly with the index of refraction of the mixture. As discussed above, at high concentrations of CS small corrections occur because the local fields originating from different molecules partly compensate for each other. This cancellation effect, which in the light scattering literature is often discussed in terms of the so-called three-body term, depends in general on the local symmetry. It has been proposed to quantify these effects by investigating the integrated scattering intensity as function of the composition of the mixture [94]. This method has been applied successfully to the analysis of symmetry-forbidden intramolecular modes of CS in several cosolvents [94]. However, in the low-frequency spectrum investigated here, the character of the intermolecular vibrations may change considerably upon dilution (see the discussion on clustering at the beginning of Sec. 5.5), which also leads to a modification of the integrated intensity. The same holds for the 84

17 coherent orientational motion discussed in Sec. 5.4: As pentane is added it may be expected that the amplitude of the librations becomes larger because the potential is presumably less steep, which also results in a concentration dependence of the integrated intensity. Therefore, the intensity dependence of the signals does not provide a direct measure of the DID contribution to the signal. Despite these difficulties in quantifying the exact size and form of the DID effect, we conclude that this mechanism provides an alternative explanation for the narrowing and the red-shift of the spectrum. This conclusion is supported qualitatively by the MD results on neat CS [19,0] and on CS /CCl 4 mixtures [86], which showed that the DID effects are prominent at the high frequency part of the spectrum. We want to end this discussion with the remark that the projection of CI and DID effects on harmonic oscillators seems to be quite dubious. In addition, interfering cross terms between these interaction induced effects and orientational motion complicate the description even more. Currently, no comprehensive theory is available, which takes all these effects properly into account, and allows for signal analysis in terms of closed form expressions. 5.6 Conclusions In this chapter the OHD-OKE responses of neat CS and CS /pentane binary mixtures were presented as typical examples of the results obtained with nonresonant third-order experiments. The signals are composed of three distinct components, an instantaneous contribution around zero delay due to the electronic hyperpolarizability, a delayed nuclear response with an inertia-limited rise time of the order of 00 fs, and a slow diffusion-controlled decay on a ps time scale. Upon dilution in pentane the relaxation rate of the diffusive component increases, which is attributed to a decrease in the bulk viscosity, while the build-up and decay time of the sub-ps component become slower. The non-lorentzian part of the spectral density narrows and shifts towards lower frequencies as pentane is added. Concentration dependent experiments showed that the main spectral changes occur at intermediate concentrations where there is approximately one CS molecule per pentane molecule. For mixtures with less than 0 vol. % CS no spectral changes are observed. There are at least four different scattering mechanisms, which can, in general, contribute to the observed nuclear response: (i) single molecule orientational motions of the anisotropically polarizable molecules, (ii) low-frequency collective vibrations of transient aggregate structures, (iii) short-range collision-induced effects resulting from polarizability changes due to molecular frame distortions and/or electronic overlap of adjacent molecules, and (iv) long-range interactioninduced effects, which are caused by electrostatic dipole or higher multipole interactions. From the discussion of these scattering mechanisms it is clear that a reliable description of the observed Rayleigh-wing line shapes is extremely difficult. It is far from trivial to derive well justified, closed form expressions that accurately describe the collision and interaction induced effects. From the trends upon dilution the relative importance of the various mechanisms cannot be determined unambiguously. It was proposed by McMorrow and coworkers [6,64] that the narrowing of the spectra at intermediate concentrations should be due to a decrease of the 85

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