Isotropic and anisotropic interaction induced scattering in liquid argon

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Isotropic and anisotropic interaction induced scattering in liquid argon

Victor Teboul and Yves Le Duff

Citation: J. Chem. Phys. 107, 10415 (1997); doi: 10.1063/1.474205 View online: http://dx.doi.org/10.1063/1.474205

View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v107/i24 Published by the American Institute of Physics.

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Victor Teboul and Yves Le Duff

Laboratoire des Proprie´te´s Optiques des Mate´riaux et Applications, Universite´ d’Angers, 2 boulevard Lavoisier, 49045 Angers, France

~Received 23 July 1997; accepted 17 September 1997!

The collision induced scattering~CIS!spectra have been studied for liquid argon at 130 K and 615 amagat. For the first time, isotropic CIS intensities are measured and the depolarized CIS spectrum is obtained up to 370 cm²¹. Molecular dynamics simulations are performed for several models of polarizabilities and intermolecular potentials. They show that theoretical polarizabilities deduced from self consistent field calculations are in agreement with both depolarized and isotropic CIS experimental spectral shape. © 1997 American Institute of Physics.@S0021-9606~97!50748-3#

I. INTRODUCTION

Collision induced scattering~CIS!in fluids has been ex- tensively used to explore both the polarization distorsions induced by molecular interactions and the dynamics of fluids particles.^1–3 Many works have concerned atomic fluids which are appropriate systems to test specific features of theoretical models. From low density atomic gases experiments, models for the anisotropy of collision induced polarizability have been proposed for pairs of rare gas atoms.^4,5In the case of atomic liquids, CIS have been observed in the vicinity of the Rayleigh line and have been compared with molecular dynamics computer simulation. For liquid argon depolarized CIS spectra have been measured up to about 100 cm²¹ at several temperatures and depolarized low order spectral moments have been studied.^6–8Surprisingly, up to now, no isotropic CIS spectra have been reported for argon in the liquid state.

In this work we study the CIS spectra obtained in liquid argon at 130 K and 615 amagat. For the first time an isotropic CIS spectrum is observed and depolarization ratios are obtained. The depolarized CIS intensities are measured in a new frequency range going up to 370 cm²¹. Experimental depolarized and isotropic intensities are compared with molecular dynamics calculations using several induced polarizability models.

II. EXPERIMENT

The light scattered by argon liquid at 90 degrees was analyzed using a conventional Raman apparatus including an argon-ion laser giving 9 watts for the green line at 514.5 nm and a double monochromator with holographic gratings. The depolarized and the polarized scattering intensities I_d and I_p were obtained with a laser electromagnetic field parallel and perpendicular respectively to the scattering plane.⁹ The liquid sample was contained in a four-window cell which can work with several hundred bars of gas from helium liquid to room temperature. The cell was set in a specially designed continuous flow cryostat from the Air Liquid company. The temperature was monitored using a calibrated platinum resis- tor inside the cell and the accuracy of the sample temperature was estimated at about 0.5 K. The main experimental diffi- culties come from the perturbation due to the parasitic light

as well as those due to the modification of the beam polarization originating from the windows of the cryostat and the pressure cell. In order to minimize this polarization effect a l/2 plate has been used to compensate the rotation of the laser field inside the cell of the cryostat windows.

III. COMPUTER SIMULATION

Our calculations used 4n³ atoms in a cubic box where n is an integer number. The equations of motion were solved using the Verlet algorithm¹⁰ with a time step of 10²¹⁴seconds and all the simulations were carried out on a Hewlett Packard HP 7200/100 work station. In this study the De Broglie thermal wavelength of the argon atom is equal to 0.24 Å, which is relatively small compared to the character- istic length of the system under consideration (s53.4 Å).

Moreover to avoid high computing uncertainties we have limited the calculated spectra to scattering frequencies lower than 200 cm²¹. In these conditions quantum effects are ex- pected to be weak.

Two different pair potentials models were used: The HF- DID1 potential of Aziz,¹¹ which is one of the best realistic pair potentials in the literature and the Lennard-Jones 6-12 potential for argon¹² u_LJ54e⁽⁽s^/r)¹²2(s^/r)⁶^{) with} e^/kB

5120 K and s53.4 Å, which stands for a classical pair potential for these kind of studies. The cutoff radius of the interaction potential was set to 2.5s. For each potential several pair polarizability models were tested. For the anisotropy we used: the Dipole-Induced Dipole~DID!model,¹³the Self Consistent Field model of Dacre¹⁴ ~SCFD!, the Mein- ander empirical model⁵and the recent Self Consistent Field model of Joslin et al.¹⁵ ~SCFJ! which takes into accounts electronic correlation effects. For the trace of the polarizability tensor we tested the second order DID model,¹³the SCF models^14,15and the empirical model of Proffit et al.¹⁶

The equilibrium configurational averages have been performed at up to 10⁶ time steps. We assumed that the intermolecular potential and the intermolecular polarizability are pairwise-additive. The induced spectrum have been obtained in the classical approximation, from the Fourier transform of the following correlation function:^17,18

C^x^~a^!^x^~b^!~t!5^^Ti j

x^~a!x^~b!~0!T_kl^x^~a^!^x^~b!~t!&2^^Ti j

x^~a!x^~b^!~0!!&²^,

~1!

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where the brackets^&denote an ensemble average over atoms i, j , k and l. In equation ~1!we have:

T_{i j}^x^~a!^x^~b!5adab2b

3

S

^d^ab²^3x^{i j}^~^r^a^{i j}²^!^x^{i j}^~^b^!

D

^, ^~²^!

where i and j indicate atoms and x_{i j}^(a) ~a51, 2 or 3!are the components respectively x_{i j}, y_{i j} and z_{i j} of the relative posi- tion vector r_{i j} between atoms i and j. Quantities a^(ri j(t)) andb^(ri j(t)) stand respectively for the trace and the anisotropic part of the interaction induced polarizability tensor.⁴In order to improve the statistical accuracy different geometry contributions were then added as usual. For the depolarized spectrum calculation, the isotropic part of the polarizability a^(ri j(t)) was set to zero, while for the isotropic spectrum calculation it is the anisotropic part of the polarizability b^(ri j(t)) which was set to zero. After the Fourier transform calculation, a detailed balance correction have been applied to the classical intensities. For the sake of completeness, a low density simulation was performed both for the isotropic and anisotropic case and compared to the spectra previously calculated in the binary approximation.^19,20 Both results agreed within a few percent, well within the statistical uncer- tainty of the simulations. In order to check that no cutoff effects were present in the spectra and momenta calculations, different simulations have also been performed using different number of atoms, from 256 to 864 atoms.

IV. RESULTS AND DISCUSSION

We have measured depolarized scattering intensities I_d(n) for liquid argon at 130 K, 615 amagat, up to 370 cm²¹. The results are displayed in Figure 1. These experimental scattering intensities show a quasi exponential spectral shape and decrease over about seven orders of magnitude from 20 to 370 cm²¹. From these intensities and from the measure- ments of polarized intensities I_p(n) we have calculated the depolarization ratio at several Raman frequencies using:

hn~n!5I_d~n!/I_p~n!.

The values of hn(n) are shown in Figure 2. It is clear that the depolarization ratiohn(n) for liquid argon is not constant and decreases from the value 6/7 at low frequencies ~com- pletely depolarized scattering! to values lower than 0.5 for the higher frequency part of the spectrum. This behavior shows off, for the first time, in liquid argon, the presence of an isotropic component in the CIS of an atomic liquid. From I_d(n^{) and} hn(n) measurements we have deduced isotropic intensities I_iso(n⁾ at several frequencies from 50 to 370 cm²¹ using²¹ I_iso(n⁾5(1/hn(n^)-⁷6)I_d(n). These isotropic intensities are plotted in Figure 3a forn.50 cm²¹. For lower frequencies than 50 cm²¹the depolarization ratio values are too close to 6/7 to allow a precise evaluation of the isotropic intensities, when experimental uncertainties are taken into account. In the frequency range studied ~up to 370 cm²¹! the isotropic intensities decrease over less than five orders of magnitude. This corresponds to a decreasing spectral slope weaker than the one observed for the anisotropic spectrum in Figure 1. This behavior could be ex-

plained partly by the range of the electromagnetic interactions inducing the scattering light since they would be shorter for the isotropic intensities than for the depolarized ones as shown in the DID model.¹³For comparison with the gaseous argon case we have plotted also in Figure 2 the experimental depolarization ratio for argon gas at low density calculated from depolarized and polarized CIS intensities previously obtained at room temperature.^20,22In order to compare with the liquid phase data and according to a corresponding state principle the frequencies of the gaseous data have been multiplied by a temperature factor (T₂/T₁)^1/2 where T₂5130 K is the temperature of the liquid experiment and T₁5294.5 K is the temperature of the gaseous argon.

Although the depolarization ratio has similar values at high frequencies in the gas and in the liquid phase, we observe in Figure 2 that the liquid depolarization ratio decreases above 80 cm²¹ while the gaseous ratio decreases sharply around 50 cm²¹. This result means that the isotropic component is relatively smaller in the liquid than in the argon gas. This difference could be related to the structural organization of each medium.

The experimental liquid argon CIS spectra are compared in Figures 3b and 4 with spectra obtained from molecular dynamics ~MD! simulations using several polarizability models, and two intermolecular potentials. In the case of depolarized scattering ~Figure 4! the calculated intensities

FIG. 1. Experimental depolarized intensities obtained in liquid argon for a temperature of 130 K and a density of 615 amagat. Error bars are indicated for several intensities at high frequencies. Forn,200 cm²¹error bars at most equal to the diameter of the data circles.

10416 V. Teboul and Y. Le Duff: Interaction induced in liquid Ar

J. Chem. Phys., Vol. 107, No. 24, 22 December 1997

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are given in absolute unit, while the experimental spectra have been adjusted at 50 cm²¹ to the calculated spectrum obtained with SCFD anisotropy and Aziz potential which yield a spectral shape close to the experimental one. For depolarized intensities, we observe in Figure 4 that the asymptotic DID polarizability leads to spectral shapes wider than the experimental measurements. The agreement is slightly worst with SCFJ model and the empirical model of Meinander et al. ~MTZ!. In the case of isotropic spectra shown in Figure 3b, the intensities have been deduced from the experimental depolarized intensities adjusted according to the procedure used in Figure 4. In these conditions SCFJ and SCFD models yield spectral shapes close to the experimental isotropic CIS data while the empirical model of Prof- fitt et al. gives a wider spectral shape. Concerning the inter- molecular potential, no significant effects are visible at high frequencies. However at low frequencies the intensities calculated with Aziz potential are higher than those obtained with Lennard-Jones potential as shown in Figure 4. This effect was not seen for low density gaseous argon^4,19,20,23nor in high density room temperature ‘‘gaseous’’ argon.²⁴ It would be due to the increase of the number of diatoms or many-atoms trapped at low temperature in the well of the Aziz potential compared to the Lennard-Jones case.

FIG. 2. Experimental depolarization ratiohn5Id/Ipversus frequency shift in cm²¹for liquid argon at 130 K and 615 amagat~full circles or triangles! and for low density room temperature argon gas~empty circles or triangles!. Triangles are estimated values when data were obtained with large uncertainties. In order to compare with the liquid phase data the frequencies of the gaseous data have been multiplied by a temperature factor (T2/T1)^1/2~T2

5130 K is the temperature of the liquid experiment and T15294.5 K the temperature of the gaseous one!. For the sake of completeness the true experimental low density data have been plotted in the upper right of the figure without any temperature factor. The dashed lines correspond to depolarized intensity (hn56/7).

FIG. 3. a: Experimental polarized intensities obtained in liquid argon for a temperature of 130 K and a density of 615 amagat. Typical error bars are indicated for several intensities. Triangles are estimated values. b: Calcu- lated and experimental isotropic spectra of liquid argon versus frequency shift, at 615 amagat and 130 K. Circles: experimental data. Upper continu- ous line: spectrum calculated with Joslin et al. polarizability model~Ref.

15!. Dashed line: Dacre SCF model~Ref. 14!and dotted line: Proffit et al.

polarizability model ~Ref. 16!. The three simulated line shapes have been calculated using the Aziz pair potential~Ref. 11!.

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We have also calculated the zero, second and fourth or- der moments respectively M₀, M₂, M₄ for the depolarized CIS from the equation

M_n5

E

2`

1`

I~v!vⁿ^dv^,

using detailed balance correction for I(v). Several polarizability models were used and we have compared theoretical moments with the experimental values of M_n deduced from experimental spectra extrapolated in the low frequency domain (0 – 20 cm²¹) using an adjusted spectral shape given by SCFD polarizability model and Aziz potential. For the high frequency part of the intensity I(v⁾ ~above 200 cm²¹ for theoretical results and above 370 cm²¹ for experimental data! we have postulated a pure exponential shape fitted to the highest frequency shift intensities. We have checked that the addition of this frequency part has almost no effect~less than 1 percent!on the experimental moments. For the theoretical results this procedure affects by a few percents ~less than 3 percent for both SCF models! the spectral moments when compared with the same calculation using a zero value intensity above 200 cm²¹. The results are given in Table I.

We observe that SCF results are close to the experiment while DID moments are higher specially for M₄/ M₀. This confirms that the DID model yields scattering intensities too high at high frequency as observed from calculated spectra

~see Figure 4!. The MTZ model yields moments between DID values and those obtained from SCF anisotropy polarizabilities.

V. CONCLUSION

The liquid argon CIS spectra have been investigated at 130 K and 615 amagat. An isotropic component has been found for the first time in the spectrum of an atomic liquid and the anisotropic spectrum has been obtained in a very large frequency domain not previously investigated, up to 370 cm²¹. These experimental results have been compared with molecular dynamics simulations using different polarizability models and intermolecular potentials as input. From this work several specific results are obtained. Depolarization factors measured for argon in the liquid phase and the low density gas are not related by a corresponding state law. The calculated low frequencies anisotropic CIS spectrum changes with the intermolecular potential. The isotropic and aniso- tropic SCF polarizability models of Dacre et al. and of Joslin et al. are in agreement with experimental spectral shapes.

This result, previously obtained for low density binary spectra of argon at room temperature and for high density gaseous argon, is here extended to liquid phase at low temperature.

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FIG. 4. Calculated and experimental depolarized spectra of liquid argon versus frequency shift, at 615 amagat and 130 K. Circles: experimental data.

Dotted line: calculated spectrum using Meinander et al. polarizability model

~Ref. 5!. Continuous line: DID model~Ref. 13!for the upper spectrum then Dacre SCF model~Ref. 14!, and Joslin et al. SCF model~Ref. 15!. All theoretical curves are calculated with the Aziz potential~Ref. 11!. A zoom of the low frequency shift spectra have been plotted in the lower left of the figure in order to display the potential effect on the calculated low frequency spectra. Calculations using the Lennard-Jones potential ~Ref. 12!are displayed in dashed lines and those using the Aziz potential in continuous lines.

For the sake of clarity the Meinander et al. model is not plotted.

TABLE I. First experimental moments of CIS depolarized spectra compared to theoretical ones using different polarizability models together with the Aziz potential.

M2/ M0

(cm²²)

M4/ M0

(cm²⁴) Polarizability model

5.28310² 1.96310⁶ Experiment

6.72310² 3.31310⁶ DID~Ref. 13!

5.69310² 2.20310⁶ MTZ~Ref. 5!

5.34310² 1.95310⁶ SCFD~Ref. 14! 5.24310² 1.97310⁶ SCFJ~Ref. 15! 10418 V. Teboul and Y. Le Duff: Interaction induced in liquid Ar

J. Chem. Phys., Vol. 107, No. 24, 22 December 1997

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Isotropic and anisotropic interaction induced scattering in liquid argon

Isotropic and anisotropic interaction induced scattering in liquid argon

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