Light Scattering from Thermal Fluctuations in Disparate Mass Gas Mixtures W

LIGHT SCATTERING FROM THERMAL FLUCTUATIONS IN DISPARATE MASS GAS MIXTURES W. Gornall, Chen Wang

To cite this version:

W. Gornall, Chen Wang. LIGHT SCATTERING FROM THERMAL FLUCTUATIONS IN DIS- PARATE MASS GAS MIXTURES. Journal de Physique Colloques, 1972, 33 (C1), pp.C1-51-C1-56. ￿10.1051/jphyscol:1972110￿. ￿jpa-00214900￿

HAL Id: jpa-00214900 https://hal.archives-ouvertes.fr/jpa-00214900 Submitted on 1 Jan 1972

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LIGHT SCATTERING FROM THERMAL FLUCTUATIONS IN DISPARATE MASS GAS MIXTURES (*)

W. S. GORNALL (**) and C. S. WANG Division of Engineering and Applied Physics, Harvard University Cambridge Massachusetts, 02138 U. S. A.

RBsumB. - La diffusion Rayleigh-Brillouin a 6th etudike dans des mklanges gazeux de He et Xe de concentrations diffkrentes. En augmentant la concentration en He, un elargissement prononc6 et une modification (reduction) du deplacement de fr6quence des composantes Brillouin ont 6te observks et attribues a un fort couplage entre les ondes sonores et les modes de diffusion de concen- tration dans le melange. La solution compl&tedes kquations hydrodynamiques dkrit correctement les spectres observ6s et met en evidence l'importance du couplage entre les fluctuations des variables thermodynamiques. Une ktude simplifike de l'interaction entre les fluctuations de pression et de concentration est presentee qui interprete les caractkristiques essentielles du comportement spectral observk.

Abstract. - Rayleigh-Brillouin scattering has been studied in a gas mixture of He and Xe at various concentrations. Pronounced broadening and frequency pulling of the Brillouin components is observed with increasing He concentration due to a strong coupling between the sound wave and concentration diffusion modes in the mixture. The complete solution of the hydrodynamic equations accurately describes the observed spectra and reveals the importance of the coupling between the fluctuating thermodynamic variables. A simplified analysis of the interaction between the pressure and concentration fluctuations is given that describes the main features of the observed spectral behavior.

I. Introduction. - In a two component fluid mix- tration mode and sound propagation in the He-SF, ture, light scattering from thermal fluctuations in the mixtures was reported in a previous communica- isotropic part of the dielectric constant is governed by tion [2]. This coupling was evident from the large variations in pressure, entropy and concentration in damping of the sound wave and reduction of the Bril- the medium. For liquid mixtures in which the mutual louin shift by as much as 20 %below the value derived diffusion coefficient, D, is small, the contribution to the from the adiabatic sound velocity for the mixture. spectrum from the concentration fluctuations is In this paper we present the experimental results narrow and easily separable from scattering due to the mainly for the He-Xe mixtures and some details of the other hydrodynamic modes [I]. However, for gas analysis used in interpreting the light scattering from mixtures at low pressures, but still within the hydro- gas mixtures. These will include illustration of the dynamic regime, D is typically three orders of magni- relative importance of the contribution to the spectrum tude larger than in liquids and the velocity of sound is from the various hydrodynamic modes and a qualitative lower. Consequently the coupling between concentra- explanation of how the coupling between the concen- tion diffusion and the other hydrodynamic modes plays tration diffusion mode and the sound wave results in an important role in determining the scattered light the sound wave damping and velocity dispersion. The spectrum. In the case of the sound wave mode this significance of imperfect gas corrections in this analysis coupling is particularly pronounced if the two gases at higher pressures will also be demonstrated. have very different masses and polarizabilities. Experiments have been carried out for both He-SF, 11. Theoretical. - The linearized hydrodynamic and He-Xe mixtures with essentially identical results. equations that describe a binary fluid mixture are The existence of strong coupling between the concen- derived from expressions for conservation of mass, momentum, energy and number of particles of an (*) Work supported jointly by the Joint Services Electronics individual species, and can be written as follows [3]. Program under Contract No. 44-923-7309-2 and by Advanced Research Projects Agency Interdisciplinary Laboratories Pro- (i) Continuity equation (conservation of mass) gram under Contract No. DAHC 15-67-C-0219. (**) Recipient of a National Research Council of Canada Postdoctoral Fellowship 1970-1971.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972110 C1-52 W. S. GORNALL AND C. S. WANG

(ii) Longitudinal Navier-Stokes equation (conserva- can be Fourier-Laplace transformed and written in the tion of momentum) matrix form [3] : a~ M(k, z) .N(k, z) = Ti@, z) .N(k) (5) po - = - Vp + y, v2v + at where N(k, z) is a column rnatrix with elements p(k, z), ~(k;z) and C(k, z), or any other chosen set of (iii) Thermal fluctuating thermodynamic variables that completely energy) describe the system [8]. The scattered intensity per unit frequency at a ap a T - = K V~T. distance R from the scattering volume is Oat (3) I(R, k, o) = I. - k' s1n2 @S&, o) , (6) (iv) Concentration diffusion equation (conservation 16 7c2 R~ of particle number of an individual species) where I. is the incident intensity, k, is the scattered light wave vector and @ is the a.ngle between the inci- KT V'T + 5 Vzp]. (4) dent polarization and k,. The dynarnical structure at PO factor S(k, w) is defined as In these equations, cp is the specific heat per unit mass S(k,o)= 2Re{< ~(k,z)&(-k) >z=iw), (7) at constant pressure, where k and o are the momentum transfer and angular frequency change in the scattering process. The quan- tity in braces in the Fourier-Laplace transform of the is the thermal expansion coefficient, and auto-correlation function of the fluctuation in the local dielectric constant. It can be expressed in terms of similar correlation functions of the dynamical variables Ni(k, z) introduced above, viz.

is the barodiffusion coefficient. The chemical potential < ~(k,z) &(- k) > = per unit mass of the mixture is

PI P2 r-I)=--- ml m2? The latter quantities can be expressed in terms of their provided the mass concentration C = n, m,/p, where equal-time correlation functioru through use of the pi, ni and mi are the chemical potential, particle hydrodynamic equations (eq. (5)) as follows : number per unit volume, and molecular mass of the i-th constituent. The mass and number densities are < Ni(k, 2) N,( - k) > = p = n1 m1 n2 m2 and n = n1 n2- For an ideal bas + + = [det M(k, 2>]- ' Pij(k, z)~.< Nj&) N,(- k) > . mixture j (9) Here Pij(k, z) is the determinaat of the matrix M with the i-th column replaced by the j-th column of where k, is Boltzmann's constant. The zero subscripts matrix T. The equal-time correlation functions are throughout denote equilibrium values. obtained from the equipartition theorem [9] eq. (9) states a very important fact that the Fourier-Laplace In addition the equations include the transport transform of the time correlation functions (non-equili- coefficients : the thermal conductivity, ~c ; the shear brium properties of the system) can be expressed in and volume viscosities, y, and v, ; the mutual diffusion terms of a dynamical factor and the equal time corre- coefficient, D ; and the thermal diffusion ratio KT. In lation functions (equilibrium properties of the system). this macroscopic thermodynamic approach these The coupling coefficients, dt:/aNi, which appear in quantities can be regarded as constants and at low eq. (8) can be derived from the Clausius-Mossotti densities are accurately calculable using the Chapman- relation for a binary mixture : Enskog procedure based on a Lennard-Jones molecular interaction potential 141, [5]. Typically for gases such as those considered here the Chapman-Enskog theory is accurate to within a few percent up to pressures -- 30 atmospheres. where aiis the molecular polarizability of the appro- Following the methods of Kadanoff and Martin [6], priate constituent. For the thermodynamic variables p, and Mountain [7], the above hydrodynamic equations T and C one obtains : LIGHT SCATTERING FROM THERMAL FLUCTU[ATIONS IN DISPARATE MASS GAS MIXTURES C1-53

Typical spectra obtained for the He-Xe mixtures appear in figure 2. Starting with 8 atm of pure Xe, He was added to obtain the He number concentrations that are shown. Because it is symmetric, only one side of the Brillouin spectrum is reproduced. The second Brillouin component in each spectrum shown arises from the adjacent interferometer order centered at a frequency shift of 1 500 Mc/s. The smooth curves in It should be emphasized that for the gas mixture figure 2 were calculated for the He-Xe mixtures as studied in this experiment the scattered light spectrum outlined in Section I1 and convoluted with a narrow can be derived solely from a knowledge of the Lennard- (- 18 Mc/s linewidth) instrumental profile. The lower Jones potential parameters for the constituent mole- curve in each case is the theoretical spectrum ; the cules and their molecular polarizabilities (which can be upper curve includes the overlapping contribution of obtained from refractive indices of the pure gases). No the adjacent interferometer order and agrees very well free parameters are involved. Consequently, the accu- with the experimental spectrum for all concentrations. racy of the hydrodynamic description can be stringently tested by comparing the calculated spectra with those observed experimentally. Xs (8 ATMI + He PURE Ke I 20% He 111. Experimental. - The experimental arrangement ,STRAY LIGHT used is shown schematically in figure 1. Light from a I stabilized single frequency He-Ne laser (0.2 mW at 6 328 A) was focused down the axis of the gas cell and the radiation scattered at an angle of 1780 was collected by a 20 conical lens. The spectrum was analyzed using a piezoelectrically scanned confocal Fabry-Perot inter- ferometer having a free spectral range of 1 500 Mc/s, and was recorded on a 1 024-channel scaler. To compensate for slow thermal drift in the interferometer some of the incident laser light (reflected from the cell window) was passed through the interferometer to trigger the multichannel scaler. The trigger pulse also 0 250 YN M IWO activated a shutter which subsequently blocked the FREQUENCY SHIFT IMcIr) trigger beam for the remainder of the interferometer FIG. 2. - Spectra of scattered radiation from mixtures of He sweep lasting approximately 1 sec. In this manner an and Xe for various He number concentrations. The spectra have instrumental finesse of 100 obtained with 99 % been replotted from digital data for comparison with the theore- reflectivity Fabry-Perot mirrors could be maintained tical curves that are also shown (see text). in spectra accumulated over many hours. IV. Discussion. - a) THE SPECTRUM SCATTERED SINGLE FREQUENCY FROM DISPARATE-MASS GAS MIXTURES. - Three impor- n LASER 16328 A) tant changes occur in the spectra of the He-Xe mixtures SHUTTER as He concentration is increased : (i) the components of

GROUND the Brillouin triplet broaden, (ii) the Brillouin peak GLASS shift increases slightly, and (iii) even at low He concen- PHOTOMULTIPLIER' tration, a broad unshifted component contributes -- additional intensity to spectrum beneath the Brillouin 5 cm CONFOCAL FABRY-PEROT triplet. This additional component is due to concentra- INTERFEROMETER PLATE IPIEZCELECTRICALLY SCANNED) tion fluctuations in the mixture. Its intensity increases with He concentration and dominates the spectrum r PHOTON MULTICHANNEL COUNTING SCALER for 60 % He concentration. These changes are obvious from the spectra shown in figure 2. In fact the sound FIG.1. -Experimental arrangement for studying backward wave damping for 60 % He is an order of magnitude Rayleigh-Brillouin scattering from gases. larger than that in pure Xe. The relative contributions to the 50 % He spectrum The combination of a long pressure cell and a small from the correlation functions appearing in eq. (8) angle conical lens made it possible to collect sufficient weighted with the corresponding coupling coefficients scattered intensity from 8 atm of Xe with very low are plotted in figure 3. Here we have chosen the thermo- incident intensity. The signal level was about dynamic variables [8] used in reference 3 to show the 30 counts/s compared with a dark count of w 11s. contrast between the spectral distribution in gas mix- C1-54 W. S. GORNALL AND C. S. WANG tures and that in other fluid systems [I]. It is seen that the cross-correlation function between pressure and concentration fluctuations is large indicating the strong coupling that exists. Furthermore, figure 3 indicates that the dominant scattering at 50 % He concentration comes from the concentration fluctuations. This contribution increases further at higher He concentra- tions and is responsible for the very strong unshifted component in the last spectrum of figure 2.

Xe PRESSURE = 120 PSI He PRESSURE = 120PSI

-AUTO CORRELATION FUNCTION J. ---CROSS CORRELATION FUNCTION I 0 100 200- 300 PRESSURE (psi) FIG. 4. -Adiabatic sound velocity in SF6 as a function of pressure. The experimental points were derived from measured Brillouin shifts. The theoretical curves were computed employing the second virial coefficient (B) only, and both second and third virial coefficients (B, C).

corrections are expected to be equally appropriate for Xe at 8 atm pressure.

FREQUENCY SHIFT iMc/s) C) COUPLINGBETWEEN THE SOUND WAVE AND FIG. 3. - Relative spectral contribution of the separate corre- CONCENTRATION DIFFUSION MODI~S.- In our previous lation functions that make up the total scattered spectrum communication [2] we emphasized that strong coupl- S(k, w). ing between pressure fluctuations (sound wave) and concentration fluctuations can occur when DklV, - 1 : b) IMPERFECTGAS CORRECTIONS. - In the theore- a condition usually true for low pressure gas mixtures. tical spectra shown in figure 2 and 3 small imperfect gas In addition to damping the sound wave, this coupling corrections have been applied in deriving the values of reduces the sound velocity so that the Brillouin shift is n,, a,, the isothermal sound velocity significantly lower than that calculated from the adia- batic sound speed for the mixl.ure according to the relation Av = V, k/2 n. In figure 5 the actual Brillouin peak shift derived from the hydrodynamic theory is plotted as a function of He conca:ntration. The concen- and the heat capacities, cp and c, [5]. As an example of trations for which matching expt:rimental spectra were the importance of imperfect gas corrections at these taken are marked by the solid dots with error bars and higher pressures the calculated adiabatic sound indicating the accuracy to which the peak shift could velocity be determined. At 50 % He concentration the sound velocity in the mixture is 20 % lower than the adiabatic value. In order to understand clearly how the coupling and that measured from the Brillouin spectra of pure between the sound wave and the concentration diffu- SF, at high pressures is shown in figure 4. For an ideal sion modes can give rise to Ihe observed spectral gas features consider the following analysis. For simplicity assume that the ratios

xk bk IC - - G 1 where x = - ' vs c, is independent of the pressure. The curves in figure 4 v, Po show that the imperfect gas corrections used are both is the thermal diffusivity and b = (413 y, + y,)/po is necessary and adequate for an accuracy of + 1 % in V, the viscous damping coefficient. By neglecting terms for SF, pressures ranging from 5 to 20 atm. At low including these ratios the sound wave equation pressures, the second virial coefficient (B) produces an (eq. (1) and (2) combined) can be written adequate correction. However, at high pressures both second and third virial coefficients (B, C)are necessary. Although it was not verified experimentally, similar LIGHT SCATTERING FROM THERMAL FLUCTUATION IN DISPARATE MASS GAS MIXTURES C1-55

or in rationalized units

(ii) In the isoconcentration limit (yo % 1) the roots are

and in rationalized units o, = iy, Dk2;

Between these limiting cases the numerical solution of eq. (14) is plotted in figure 6 as a function of the parameter y, for He-Xe mixtures of 20 % and 40 % He. He CONCENTRATION The top two curves in each case show that the sound FIG. 5. - Actual Brillouin shift as a function of He concentra- wave damping is a maximum in the vicinity of y, = 1 tion compared with that calculated from the adiabatic sound and the sound velocity changes at the same point. A velocity in mixtures of He and Xe. greater decrease in the sound velocity is found in the 40 % than in the 20 % mixture. The bottom curve (plotted logarithmically) shows the corresponding The thermal diffusion coupling in eq. (4) is very small behavior of the concentration diffusion rate as a func- and can be neglected so that the equation for the tion of y,. concentration diffusion mode becomes

From the Fourier-Laplace transformation of these equations one obtains the following secular equation : where

FIG. 6. - Graphical solutions of eq. (14) in the vicinity of Since k is fixed in light scattering and the parameters yc yc = 1 showing the effect of the coupling between the sound and y, are constant for a given mixture eq. (14) is a wave and concentration diffusion modes for two He-Xe mixtures. third order algebraic equation for z. The one real root is the diffusion rate constant (T,) for the concentration mode in units of V, k. The remaining two complex For comparison, the 20 % and 40 % He spectra of conjugate roots describe the properties of the sound figure 2 correspond to values of yc indicated by the wave ; the real part is the sound wave damping coeffi- arrows in figure 6. Of course, the frequency shifts cient (rB)in units of V, k and the imaginary part is the implied by the top curves in figure 6 are somewhat less Brillouin shift, or equivalently the sound velocity (V) than those actually observed because we have omitted in units of Vs.For two limiting cases the roots are from eq. (12) and (13) the additional damping effects of easily obtained : thermal conduction and viscosity. (i) In the adiabatic limit (yo 4 1) the roots are The curves in figure 6 represent the behavior of the sound wave and concentration diffusion modes only so far as the hydrodynamic description is appropriate. C1-56 W. S. GORNALL AND C. S. WANG

It is generally believed 1101 that the hydrodynamic and for the major portion of these graphs the hydro- theory is valid as long as the parameter dynamic theory is valid.

V. Conclusions. - The present experiment has shown that the spectrum of light scattered from a mixture of He and Xe at a few atmospheres pressure where I is the mean free path between collisions in the undergoes dramatic changes as a function of He gas. In our experiment, since the scattering power of concentration, largely due to coupling between pressure the Xe atoms relative to that of the He atoms is and concentration fluctuations. In the strong coupling (axe/ct,e)2 - 400, this criterion need only be applied to regime (y, - 1) concentration diffusion provides an the motion of the Xe atoms. If the atoms are treated as additional mode of relaxation for pressure gradients smooth elastic spheres one finds 41x, = 1.261/pa and associated with thermal sound waves in the mixture. 1.608/pafor 20 % and 40 % He concentration respecti- The result is a velocity dispersion and increased vely, where pa is the pressure in atmospheres. In the damping of the sound wave analogous to that observed present experiment (k z 2.0 x lo5 cm-l) eq. (15) for other forms of thermal relaxation. In the hydro- specifies that collectively the Xe atoms will behave dynamic regime the observed spectral changes are hydrodynamically for total pressures exceeding 2.4 accurately described by a cornplete solution of the and 1.9 atmospheres respectively for the two concen- hydrodynamic equations for a binary fluid mixture. trations. These pressures correspond to values of y, = 2.3 and 2.5. The portions of the graphs in figure 6 Acknowledgements. -It is a pleasure to acknowledge that, by this criterion, fall outside the hydrodynamic helpful discussions with Professor N. Bloembergen regime are plotted as dash-dotted curves. Based on this and the assistance of Mr. C. C. Yang in the experi- analysis it is apparent that for our experimental spectra mental work reported here.

References

[I] BERGS(P.), CALMETTES(P.), DUBOIS(M.) and [8] For simplicity of discussion we have chosen pressure, LAJ (C.), Phys. Rev. Letters, 1970, 24, 89 ; temperature andconcentration as the thermodyna- DUBOIS(M.) and BERGI?(P.), Phys. Rev. Letters, mic variables describing the state of the mixture 1971, 26, 121. but it is clear that any sxmilar choice of variables [2] GORNALL(W. S.), WANG(C. S.), YANG(C. C.) and which completely describe the system would suffice. BLO~MBERGEN(N.), Phy~.Rev. Letters, 1971, 26, Mountain and Deutch (Ref. 3) use the variables p, 1094. @ and C, where @ = :r - (TOLXT/C~ PO) p is a [3] MOUNTAIN(R. D.) and DEUTCH(J. M.), J. Chem. from of entropy introduced to facilitate separating Phys., 1969, 50, 1103. the hydrodynamic mocles of the mixture: an [4] CHAPMAN(S.) and COWLING(T. G.), (( Mathematical approximation particularly suitable for liquid Theory of Non-uniform Gases 1) (Cambridge mixtures. U. P., Cambridge, England, 1952). [9] LANDAU(L. D.) and LIPSHITZ(E. M.), (( Statistical [5] HIRSHFELDER(J. O.), CURTIS(C. F.) and BIRD (R. B.), Physics )) (Addison-W<:sley, Reading, Massa- (( The Molecular Theory of Gases and Liquids )) chusetts, 1958). (John Wiley and Sons, New York, 1954). [LO] YIP (S.) and NELK~(M.), Phys. Rev., 1964, 135, [6] KADANOFF(L. P.) and MARTIN(P. C.), Ann. Phys., A 1241 ; CLARK(N. A.), Ph. D. Thesis, Massa- 1963,24,419. chusetts Institute of Technology, 3970, (unpu- [7] MOUNTAIN(R. D.), Rev. Mod. Phys., 1966, 38, 205. blished).