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 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. JOURNAL DE PHYSIQUE Colloque Cl, supplkment au no 2-3, Tome 33, Fkvrier-Mars 1972, page (21-51 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).