Density narrowing effect in the collisional cluster scattering of the light by gases

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Density narrowing eﬀect in the collisional cluster scattering of the light by gases

Victor Teboul

^a,*

, Yves Le Duﬀ

^b

aDepartement de Physique, Laboratoire POMA, Unite Mixte de Recherche CNRS 6136, Universite´ dÕAngers, 2 boulevard Lavoisier, FR-49045 Angers Cedex, France

bLRP, 5 rue de la Rouvraie, 49080 Bouchemaine, France Received 18 January 2005; in ﬁnal form 26 January 2005

Abstract

The spectral intensities of a collisional cluster scattering band in CF4gas have been computed using molecular dynamics. Several densities from 20 to 269 Amagat as well as several models of interaction induced polarizability and of intermolecular potential have been studied. A qualitative agreement with experimental results has been obtained showing the narrowing eﬀect of the gas density on the two-body line shape of the collisional cluster Raman band.

1. Introduction

Molecular interactions influence the scattering spectra of any media. Different aspects of this subject have been discussed at a workshop that held at Banff (Canada) in 1993[1]. Recently it has been shown that a new type of bands is present in the scattering spectra of high pressure gases[2,3]that can be attributable to a double incoherent scattering process. These bands do not belong to the Raman scattering spectrum of a molecule of the gas but are specific of temporary clusters that are generated during collisions or more generally during molecular interaction processes. In the first studies of these collisional cluster (CC) bands it was observed that their line width decreases when the gas density goes up[2]. The origin of this effect was not identified.

In this Letter, we study this density narrowing eﬀect and use molecular dynamics simulations to calculate the spectral intensity of these collisional cluster scatter-

ing bands for gaseous tetraﬂuoromethane at several densities. We consider the two-molecule clusters of tetraﬂuoromethane (CF4) that are temporary created during interaction processes in a CF4gas at room temperature. In particular, we study the cluster scattering band observed at twice the Raman frequency of the m1

allowed vibrational mode of CF4. The Raman shift for this cluster band is about 1816.3 cm¹at 269 Amagat.

Several models of interaction induced polarizability anisotropy as well as intermolecular potential are used.

2. Theoretical considerations

In dense media, interaction induced light scattering (IILS) arise from changes of molecular polarizabilities on account of molecular interaction processes. If evi- dences of three-body irreducible contributions have been reported[1], contributions from pairs of molecules in interaction are preponderant [4] in IILS. However interferences between the electric ﬁelds scattered by pairs of molecules in interaction induce two-body, three-body and four-body contributions, leading to cancellation eﬀects [1,5]at high densities.

doi:10.1016/j.cplett.2005.01.090

* Corresponding author. Fax: +33 2 41 73 52 16.

E-mail addresses:victor.teboul@univ-angers.fr(V. Teboul), yleduﬀ@libertysurf.fr(Y. Le Duﬀ).

www.elsevier.com/locate/cplett

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In the particular case of the CC scattering, the eﬀect of the interference between several molecular pairs, on the IILS, collapses leading to pure two-body intensities.

The incoherence of the scattering processes involved in this CC band may explain this behavior [2]. However, although CC scattering intensities are pure two-body intensities with respect to the size of transient clusters that generated them, these intensities clearly are affected by the presence of the surrounding thermal bath, which is constituted of many molecules in interaction. When the density changes, modifications of the dynamics and of the structure of the medium affect undoubtedly the line shape of the CC band.

In this Letter, we have computed the spectral intensities of the CC band of CF4 centered at about 1816.3 cm¹using molecular dynamics simulations with a Verlet algorithm to integrate the equations of motion [6]. The time step was chosen equal to 10 fs. The intermolecular potential functions used were: (i) a Len- nard–Jones 6-12 potential (LJ) [7], U(r) = 4e((r/

r)¹²(r/r)⁶) withe= 152.5 K andr= 4.7 A˚ ; (ii) a sym- metrized version of the Palmer potential (PP) given in Ref. [8]. Concerning the polarizability anisotropy bp(r) of a pair of two molecules of CF₄, two models were

used: (i) the classical asymptotic dipole induced dipole (DID) anisotropy[2,9] with

b_pðrÞ ¼6ðoa=oQ_vÞ²Q²_vr³; ð1Þ whereQvis the normal coordinate corresponding to the vibrational mode m1of the CF4molecule andoa/oQvis the polarizability derivative associated to this mode;

(ii) an empirical anisotropy deduced from Ref.[10,11].

The cut-oﬀ radius of the interaction potential was set equal to 2.5r. Simulations began with a cubic box of CF4 with crystalline symmetry, which was warmed up to the desired temperature. This procedure takes roughly 50 000 time steps, which corresponds to 0.5 ns.

When the configuration was well homogenized the calculations of the scattered intensity then began and were performed during 500 000 time steps (5 ns). The cut-off radius of the polarizability was set equal to the half- length of the box. Different simulations using numbers of molecules from 500 to 8000 were performed to verify that the results of the simulations do not change with the cut-off in the conditions used for the calculations[12]. In the following calculations the intermolecular potential was assumed to be pairwise additive. In the classical approximation the two-body induced spectrum is

10 100

0 2 4 6 8 10 12

Scattering intensity . 1060 (cm6 )

Frequency shift (cm^-1)

Fig. 1. Scattering Stokes intensities in absolute unit for the CC Raman band of gaseous CF4centered about 1816.3 cm¹. The CF4gas is at room temperature with a density of 269 Amagat. Full circles are experimental data from Ref.[2]. They are compared with theoretical intensities computed from molecular dynamics simulations using DID polarizability (Eq.(1)) together with either PalmerÕs potential[8](dashed line) or a Lennard–Jones potential[7](continuous line).

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obtained from the Fourier transform of the following correlation function between scattered electric ﬁelds [4,13–17]:

C^xyðtÞ ¼ VBox

NðN1Þ 1 M

X^M

m¼1

X^N

i;j¼1 i6¼j

T^xy_ijðmDtÞT^xy_ijðmDtþtÞ 0

BB

@

1 CC A;

ð2Þ whereDtis the time step (10¹⁴s in our case),VBoxrep- resents the volume of the simulation box which contains Nmolecules,Mis the number of recorded time steps,x andyrepresent the polarization of, respectively, the incident and the scattered beam, and

T^xy_ij ¼b_px_ijy_ij

r²_ij ; ð3Þ

wherexij,yij,zijare the coordinates of the relative posi- tionr_ijbetween the centers of mass of moleculesiandj.

Then the spectral intensity was obtained from the Fou- rier transform ofC(t). We have for the intensity I(m)

IðmÞ ¼ ðk0k³_sÞe^2k^hcm^B^T 1 p

Z 1 0

cosðxtÞCðtÞdt

; ð4Þ

where the exponential term is introduced for detailed balance correction[18]andk0,ksare the wave numbers of the incident and scattered light, respectively.

3. Results and discussion

This section is organized as follows: In a first part, we study the influence of the intermolecular potential and of the polarizability on the spectral scattering intensities; In a second part, we investigate the behavior of the simulated spectral line shape when density changes and finally, we propose a tentative explanation for the observed density narrowing effect.

We display in Figs. 1 and 2 the theoretical depolar- ized spectral intensities of the CC Raman band (Raman shift equal to 1816.3 cm¹) for CF4at 269 Amagat computed with the DID anisotropy (Eq.(1)) and two intermolecular potentials (LJ and PP). As shown inFig. 1the intensities obtained with the two potentials are higher than the experimental ones but of the same order of magnitude similarly to the result previously deduced from the zero moment[2]. On the other hand concerning the line shape there is a nice agreement between theoret-

0.1 1 10 100

0 10 20 30 40 50 60

Fig. 2. Scattering Stokes intensities up to 60 cm¹in absolute unit for the CC Raman band of gaseous CF4centered about 1816.3 cm¹. The CF4gas is at room temperature with a density of 269 Amagat. Scattering intensities are computed from molecular dynamics simulations using DID polarizability (Eq.(1)) together with either PalmerÕs potential[8](dashed line) or a Lennard–Jones potential[7](continuous line).

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ical and experimental spectra. The two potentials yield scattering intensities relatively close. However from 0 to 2 cm¹, intensities obtained with PalmerÕs potential [8] are higher than those obtained with the Lennard–

Jones potential [7]. On the contrary, from 2 to 12 cm¹, Palmer intensities are weaker than the Len- nard–Jones ones. PalmerÕs potential has a deeper well than the Lennard–Jones potential that may explain these results. Since the well is deeper, neighbor particles will stay longer in the vicinity of a molecule, leading to a slowing down of the dynamics of pairs that generated a narrowing of the scattering band.

InFig. 3the theoretical intensities of the CC scattering band are shown for two models of polarizability, the DID polarizability given by Eq. (1) and an empirical polarizability deduced from collision-induced scattering studies[10,11] in the Rayleigh region for gaseous CF4. The empirical model includes corrections to the DID anisotropy in order to take into account short-range overlap eﬀect as well as other intermediate interactions.

The two spectra diﬀer signiﬁcantly at high frequency shifts, the line being slightly narrower with the empirical model. On the other hand, the spectra obtained with these two polarizability models are close at low frequen- cies (610 cm¹) for which experimental data have been

given[2]. However, in this low frequency part small dif- ferences still are visible and the spectrum obtained with the empirical model of polarizability is lower than the DID one and thus is closer to the experimental intensities.

Now, to study the influence of the gas density on the CC scattering band we have displayed in Fig. 4 the computed CC Raman intensities obtained for gaseous CF4 at room temperature with the DID polarizability and the Palmer potential. Four densities are used for the calculations: 20 Amagat (bold dashed line), 92 Amagat (dotted line), 176 Amagat (dashed line) and 269 Amagat (continuous line). When the gas density goes up we observe between 0 and 2 cm¹an increase of scattering intensities and then from 2 to 20 cm¹ a decrease of these intensities. This behavior leads to a narrowing of the spectral width at half of the maximum intensity (WHM) when the density goes up. We give in Table 1 the computed WHM of the CC band for the densities studied. Two types of theoretical data are presented; a first one corresponds to WHM computed without slit effect and the second one results from a convolution with a slit function taking into account the influence of the apparatus. These theoretical values (those with slit effect) are to be compared with

0.01 0.1 1 10 100

0 10 20 30 40 50 60

Fig. 3. Scattering Stokes intensities in absolute unit (cm⁶) for the CC Raman band of gaseous CF4centered about 1816.3 cm¹. Scattering intensities are computed for CF4gas at room temperature and a density of 269 Amagat using a Lennard–Jones intermolecular potential[7]together with either the DID polarizability given by Eq.(1)(continuous line) or an empirical polarizability deduced from Ref.[10](dashed line).

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experimental data and more precisely with corrected experimental widths. Indeed, these corrected values are equal to twice the width of the Stokes wing in order to remove the inﬂuence of a weak monomer band present at the anti-Stokes side of the CF4CC band. In

these conditions, a qualitative agreement of the theoretical widths including slit eﬀect with experimental data named ÔcorrectedÕ width is observed. The imper- fections of theoretical models used for potential and anisotropy could account for a part of the discrepan-

0.01 0.1 1 10 100

0 10 20 30 40 50 60

Fig. 4. Theoretical scattering Stokes intensities in absolute unit for the CC Raman band of gaseous CF4centered about 1816 cm¹. Intensities are computed for room temperature using molecular dynamics with DID polarizability (Eq.(1)) and Palmer potential[8]. Several densities are studied.

At the frequency shift of 60 cm¹these densities are, from top to bottom, 269, 176, 92 and 20 Amagat.

Table 1

Spectral width (in cm¹) at half of the maximum intensity for the CC Raman band of gaseous CF4centered about 1816 cm¹for several gas densities

Density (Amagat) Theoretical width (cm¹) Experimental width (cm¹)

Without slit effect With slit effectâ Observed^b Corrected^c

20^d 10.0 10.2

92^d 7.4 7.6 7.8 7.0

176^d 4.8 5.2 6.4 6.0

269^d 2.5 2.6 5.0 4.2

269^e 2.9 3.0

Theoretical values result from simulations using the DID polarizability model (Eq.(1)) and either the Palmer potential from Ref.[8]or the Lennard–

Jones potential of Ref.[7].

a Computed valuesÔwith slit eﬀectÕinclude the inﬂuence of the apparatus function used in the experiment of the Ref.[2](the width of the apparatus function is between 0.5 and 1.0 cm¹).

b Experimental data from Ref.[2].

c TheÔcorrectedÕexperimental width is equal to twice the width at half of the maximum intensity for the Stokes wing of the CC band. It represents

an evaluation of the WHM of the CC band without any contribution from the weak monomer line, present at its anti-Stokes side, contrary to the case of theÔobservedÕexperimental width.

d Theoretical widths obtained with Palmer potential[8].

e Theoretical widths obtained with Lennard–Jones potential[7].

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cies observed specially at high density. On the other hand, for the highest frequency shifts we observe in Fig. 4 an increase of the scattering intensities with the gas density, an eﬀect that may be explained by an increase of the structural organization of the medium at higher density [4,16].

Another way to investigate the influence of the gas density on the light scattering by CF4 is to study the correlation function of the electric field. In Fig. 5 we show the two-body autocorrelation function C(t) (Eq. (2)) for several densities of the CF4 gas (20, 92, 176 and 269 Amagat). C(t) is normalized to unity at time zero in order to simplify the comparison. These autocorrelation functions are calculated at room temperature using the Palmer potential (PP) and the DID polarizability model. We observe that C(t) broadens when density goes up though the magnitude of this effect is partly hidden in the figure by the logarithm time scale used for the X-axis. Thus, when the density progressively approaches the liquid density the motion of the two interacting molecules in a collisional cluster is hindered by the presence of others molecules and the dynamics slows down. A characteristic lifetime s may be defined for the transient scattering pair from the equation

CðtÞ ¼Cð0Þexpðt=sÞ: ð5Þ

In Fig. 5, we observe that the lifetime sincreases from 1.38 ps at 20 Amagat to 3.31 ps at 269 Amagat leading to the observed density eﬀect. Thus, the broadening of C(t) with the density corroborates the density narrowing of the scattering line shape observed inFig. 4at low frequency shifts.

In conclusion, we have shown that the computed two-body scattering spectra generated by the CC at twice the m1 vibrational mode of the CF4molecule de- pends on the density of the CF4gas. An increasing of the density yields a narrowing of the spectral shape of the CC band. Thus, when density goes up, the dynamics of the colliding pairs is aﬀected by neighboring molecules and molecules stay in proximity to each other longer. So we suggest that the narrowing of the line shape observed for the CC Raman band comes from an increase in the time spent by the molecules of a colliding pair in the strong interaction region corresponding to an intermolecular distance shorter than 2r.

This eﬀect usually is hidden in spectroscopic bands since for most of the bands, contributions of many-body interactions (three-body, four-body,. . .) do not allow to observe pure two-body spectra contrary to the case

0.2 0.4 0.6 0.8 1

1 10 100 1000 10000

C(t) (arbitrary units)

Time (10^-14 s)

Fig. 5. Normalized simulated pair autocorrelation functions of the electric ﬁeldC(t) versus time computed for gaseous CF4at room temperature and several densities. From the right to the left, densities are 269, 176, 92 and 20 Amagat. Simulations use the DID polarizability model (Eq.(1)) and the Palmer intermolecular potential[8].

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of CC spectra. It is one of the unusual and attractive properties of CC scattering bands.

Acknowledgment

We thank Professor A. Kouzov for constructive dis- cussions on this subject.

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