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RAYLEIGH-BRILLOUIN SCATTERING FROM FLUID MIXTURES
R. Desai
To cite this version:
R. Desai. RAYLEIGH-BRILLOUIN SCATTERING FROM FLUID MIXTURES. Journal de Physique Colloques, 1972, 33 (C1), pp.C1-27-C1-35. �10.1051/jphyscol:1972106�. �jpa-00214896�
RAYLEIGH-BRILLOUIN SCATTERING FROM FLUID MIXTURES
R. C. DESAI
Department of Physics, University of Toronto, Toronto 181, Ontario, Canada
Resume. — Nous appliquons le formalisme de la fonction memoire a un melange binaire de fluides en vue d'analyser le spectre Rayleigh-Brillouin. En limitant le choix des variables dynamiques dans l'equation de Langevin generalisee a l'ensemble de variables orthogonales conservatives A = { £i(k, 0, 4(k, 0, 0(k, t), /(k, t)}, et en faisant l'approximation de Markov dans la matrice de la fonction memoire, nous retrouvons les resultats de Phydrodynamique macroscopique qu'ont obtenus recemment Mountain et Deutch. Dans l'ensemble indique ci-dessus, £,\ est reli6 auxfluctua- tions de concentration, £, aux fluctuations orthogonales de densite, et 9 aux fluctuations orthogonales de densite d'energie. /est la densite de quantite de mouvement longitudinal.
Nous etendons ensuite cet ensemble en incluant les premieres derivees par rapport au temps de la meme facon que dans le traitement de Desai et Tong pour les fluides purs. En plus du couplage entre le flux de chaleur q et le flux de diffusion Ja, qui est present dans la limite hydrodynamique, nous trouvons aussi des couplages possibles entre le tenseur des viscosites et q et Ja pour les valeurs finies de k. Nous discutons les resultats de l'extension de cette analyse, en particulier l'influence de ces couplages dans les experiences de diffusion de la lumiere et des neutrons.
Abstract. — We apply the memory-function formalism to a binary fluid mixture with a view to analyse the Rayleigh-Brillouin spectrum. By limiting the choice of dynamical variables in the generalized Langevin equation to the set of conserved orthogonal variables,
A ^ { 6(k, t), |(k, t), 0(k, 0, /(k, /)} ,
and making the Markov approximation in the memory function matrix, we recover the macroscopic hydrodynamic results previously obtained by Mountain and Deutch. In the above set, £i is related to concentration fluctuations, <J to orthogonal density fluctuations, and 6 to orthogonal energy density fluctuations, / i s the longitudinal momentum density.
We then extend the above set to include the first order time derivatives in a manner analogous to the previous treatment of Tong and Desai for one component fluids. In addition to the coupling between the heat flux, q and the diffusion flux, Ja which is present in the hydrodynamic limit, we also find possible couplings of the viscosity stress tensor a with q and J a for finite values of k. We discuss the results of this extended analysis, especially the relevance of these couplings to the light and neutron scatterine extieriments.
1. Introduction. — In recent years, since the inven- tion of the laser, light scattering technique has been extensively used to probe dynamical correlations in fluids. In this paper, we describe a microscopic ana- lysis appropriate to the Rayleigh-Brillouin scattering from a binary fluid mixture. The analysis is based on the memory function formalism and extends our previous work [1] for a one component fluid. The memory function formalism which was initiated by Mori [2] and Zwanzig [3], derives the generalized Langevin equation (GLE) for an arbitrary set of dynamical variables A [2], GLE is formally exact and equivalent to the equations of motion for the system ; it is also amenable to tractable and physical approximations.
From a macroscopic point of view, light scattering from a binary fluid mixture has been studied by Mountain and Deutch [4] who start their analysis from the appropriate hydrodynamic equations. By choosing a related set of microscopic variables for the GLE and making a Markov approximation in the memory functions, we rederive their results in
Section II. Mountain and Deutch have analysed these results in terms of approximate modes, from which the effect of the hydrodynamical coupling between the heat flux and diffusion flux on the spectral function S(k, w) can be seen. The coupling between the concentration and pressure fluctuations is also present in their exact solution; it was necessary to take this coupling into account to analyse the recent expe- riments of Gornall et al. [5] on SF6-He mixture.
In Section III, we extend the set of variables used in Section II to include the first order time derivatives.
The aim of such an extension is twofold. First is to extend the hydrodynamic calculation into the kinetic regime and second to study how the couplings between various modes change. This calculation is also equi- valent to a calculation up to second order of continued fractions [6] using the original set of Section II.
In the appendix, we give the microscopic expres- sions for various quantities used in the text. This tabulation extends a similar fluctuation analysis for one component systems done by Schofield [7], [8].
We [9] have also performed a similar microscopic
3
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972106
C1-28 R. C. DESAI
analysis for a ternary mixture (Dilute A and B in solvent C) in which a reaction of type A $ B can occur. This will be reported elsewhere.
11. Derivation of hydrodynamic formulae. - The generalized Langevin equation for a set of variables A(t!, as derived by Mori [2], is
- iw.A(t) +
1'
q(t - tl).A(tl) dt' = f(f) (2.1)dt 0
where the frequency matrix w, the memory function matrix 9, and the random force vector f have been defined elsewhere [I], [2]. By a Markov approxima- tion in the GLE, we mean
q(t - t') =
[I:
q(tfl) dtrl] 2 6(t - t') , (2.2)= 2 G(0) 6(t - t') . (2 .3) In particular, when the variables A(t) depend parame- trically on the spatial Fourier transform variable k, and the calculation is done for a finite volume, cano- nical ensemble, then explicitly
In this approximation, the Laplace transformed correlation function,
is given by
In eq. (2.4), < > refers to an equilibrium canonical ensemble average.
The variables A(t) that we choose to derive the hydrodynamic results of Mountain and Deutch are the orthogonal set { 5,, 5, 8, J ) ; 5 , and 5 are the same as those considered by Hill [lo]. A(t) are expli- citly defined as (see appendix for microscopic defi- nitions of various quantities) :
and J@, f ) , the longitudinal momentum density. In the above equations, p, and pb are partial mass density JEuctuations of the two components and E the total energy density JEuctuation ; equilibrium quantities are referred with subscript 0. If we denote the total energy
density by E', then the corres,ponding fluctuation E is E' - < E' > and can be written as
E' - eao P: - e,o pb .
In order to compute the frequency matrix, we need to use the microscopic conserwtion laws,
ia = ikJa , ( 2 . 9 ~ )
i ) b = ikJt., , (2.9b)
E' = ikQ , (2 .9c) and
where Q is the energy flux and 7c3 the longitudinal pressure tensor. < n3 > is no~lzero for k = 0 and is the thermodynamic pressure, p,, i. e.
with pa,, pbo being the partial pressures. We can also show that, as k -+ 0,
where K is the isothermal compressibility and c, the specific heat per unit mass at constant volume. p, is the total equilibrium density ; the concentration c and the chemical potential p are as defined by Landau and Lifshitz [1 11. Using eq. (2.9) and (2.10) we+an show that 12 of the 16 elements of the frequency matrix w are zero. The four nonzero elements are given as
We can also show that, as k -+ 0, ( p , - p') is TOraT/rc where a, is the thermal expansion coefficient,
The random force vector f(t) is given by
Thus we can show that
and
where the longitudinal diffusion flux, Jd(k, t) is defined as
Pao Pbo
Jd@7 t, = (pa;' Ja - PC' Jb) -
Po , (2.14) the nonconserved part [8] of the heat flux, q@, t) is given as
q@, t) = Q - < QJ* > J - < QJd*> J d , (2.150)
< JJ* > < J~ ~ d * >
and the nonconserved part of the longitudinal pressure tensor, a3(k, t ) is given as
< n 3 T * > < n 3 8 * > 8 .
~ 3 @ , t ) = n3 -
< er* > r - <
88" >
Also Cab = G, - i& = (ap/ulap)T,c. The proof of eq.
( 2 . 1 3 ~ ) ~ (2.13b) and (2.134 is straightforward. One way to prove eq. (2.13~) is to introduce the micro- scopic definition of entropy fluctuations, S&, t), in a unit mass of the mixture, as
poToS(k,t) = 8 - <On:> < t n ? > - ' t - - ( < O n : > < t l a ; > - ' - <8n:>
x < tnf > < <n; > < el nf >-' } e l , ( 2 . 1 6 ~ ) where
defines the longitudinal diffusion pressure tensor, n,.
This definition ensures that the rate of entropy pro- duction $@) is orthogonal to Gk) and tl(k) initially.
Eq. (2.16) is the microscopic analog of the thermo- dynamic relation
and the orthogonal variables 8, 5 and 5, are like the thermodynamic variables e, u and c respectively. It is easy to show that the rate of entropy production is given by
Using eq. (2.16), (2.17) the random force f3(t) can be put into the desired form given in eq. (2.13~).
The memory function matrix is related to the random force vector by the fluctuation-dissipation theorem :
Thus, in Markov approximation, we can express @(O) in terms of the coefficients of the phenomenological transport coefficient matrix. Such a procedure has been followed by Mori [2] using a different set of dynamical variables. Here we follow a slightly different, but equivalent procedure to express $(O) in terms of experimentally measured transport coefficients. To do this, we use the linear phenomenological transport laws for Jd, q and a,. These are [Ill, [4]
- -
where the fluxes ql(t), Jdlt) and J(t) are assumed to be orthogonal to one another at all times. g'(t) is responsible for thermal conduction and is given by
In eq. (2.19), D is the diffusion coefficient,
the pressure diffusion coefficient and
the thermal diffusion coefficient ; 1 is the thermal conductivity and -I- 4 - qs) the longitudinal visco-
3
sity. The bar over the variables indicates a nonequili- brium hydrodynamic ensemble average. Using eq. (2.19) in eq. (2.13) and (2.18) and noting eq. (2.3),
C1-30 R. C. DESAI we find the following expressions for the various ele- ments of the memory function matrix ;(o) :
Vlll = D k 2 , ( 2 . 2 0 ~ )
D, kZ
(2.20b) a , D k2
~ 3 1 = p a o p b o T O ( ~ ) P,T ( D ~ + - - $ ! ) E ? (2'20c)
P; Dp k 2
4012 = - ( 2 . 2 0 4
P a 0 Pbo K ' D, k2
2 2 - ( ) p , T - 7 P O (2.20e)
) . p
(2.20f)
The results of eq. ( 2 . 5 ) , (2.11) and (2.20) can also be used to directly compute the spectral lineshape for Rayleigh-Brillouin scattering. The scattered inten- sity is proportional to S(k, w), which is related to the scalar dielectric fluctuations as
S ( k , w ) = 2 R e < $ ( k , i w ) ~ ( - k ) > , (2.22) z )
z ) z ) z ) -
where
The thermodynamic derivatives in eq. ( 2 . 2 2 ~ ) can be expressed in terms of measurable quantities as
-
-
P a o Pbo
(&I
<,, = 7 p.l-- Pao P b o P:
0
0 0
-
p 4 . 3 = (P. 3 4 = 0 , j = 1 , 2 , 3 . (2.20k) The results of eq. (2.5), (2.11) and (2.20) are equivalent to the results of Mountain and Deutch [4]
as given in their matrix eq. (3.1)-(3.3). This can be shown by noting that the transformation to their variables is given by the matrix equation
111. Generalized hydrodynamic formulae. - In this section, we generalize the description of the binary mixture in a fashion analogoils to that for the one component fluid [l]. If we make the reasonable assump- tion that the scalar dielectric fluctuations couple only to t l , < and 0, then eq. (2.22) and (2.23) can also be used in conjunction with the analysis to be presented in this section for computing the Rayleigh-Brillouin spectra from binary mixtures.
The variables in the extended description are
{ t d k , 0 7 t ( k , t), 0Y. t), J(k7 t)7 Jd(k7 t), q(k7 0 7a,@, t ) ) with i = I, 2, 3. Except a , and o,, other variables have been defined previously. (See discussion around eq. (2.6)-(2.8) and (2.15).) If' a is the nonconserved part of the pressure tensor then a , , a,, a , are its xx, yy and zz components respectively. Thus a,, a , can be defined by equations analogous to eq. (2.15b). Except a , all the variables are orthogonal to one another.
For simplicity we assume that the fluid mixture is spatially isotropic so that the matrix V - l < o i oj >
as well as the corresponding rnemory function matrix -
0
0
o
- - ik Po-
;,,(0) has the simple structure given as [I],
7
t l ( k 7 4 5(k? z ) e(k, Z ) J(k7 z ) - -
Microscopic expressions for o l l and o , , are given in
the appendix. We can also show that j ( k , t ) - ik [ K - l ((k, t ) + B(k, t ) +
Po To cv Po
With the above choice of variables, the random force vector is of the form { 0,0,0, 0, f,, f,, f ), so that the first four GLE, in which only the frequency matrix is nonzero, are just the restatements of the conserva- tion eq. (2.9). Explicit evaluation of the frequency matrix yields,
+ a 3 ( k , t ) = 0 , ( 3 . 7 ) where
I
- a' + (pa0 Pbo - Pbo P ~ ~ ) / P o
8" = uab P' + P O . ( 3 . 7 ~ )
Pao Pbo
The remaining GLE for Jd, q and o i contain new information when compared with the analysis of Section 11. Due to the symmetry in the two transverse directions, equations for a , and o , are identical in structure. For the memory functions appearing in these equations, we again make the Markov appro- ximation. Then the GLE for Jd, q and o i become,
where former set of GLE which yields zero contributiolr
3 from the time derivatives and the random forces. The
61 = C < nd o; > < on* >,;I , ( 3 . 8 ~ ) simplest illustration is through eq. (3.9). It becomes,
j = 1
and
Eq. (3.8)-(3.11) can be viewed as formal generali- Comparing eq. ( 3 . 1 3 ) with eq. (2.19b), (2.19d), it is zation of the linear phenomenological transport laws appropriate to identify
given in eq. ( 2 . 1 9 ) for J,, q and o 3 and the corres- d Ir
ponding law for a, and a, which is - - - - A ( 3 . 1 4 ~ )
Po co 9 4 4
- -
a ~ ( t ) = oz(t) = i k p , '(7, - 4 ~3 J(t). ( 3 . 1 2 ) and
h
The appropriate way to compare eq. (3.8)-(3.11) with
eq. ( 2 . 1 9 ) and (3.12) is to take an average of the ( T ) - T ( ) ] = , ( 3 . 1 4 b )
C1-32 R. C. DESAI as well as to notice the extra term in eq. (3.13) which exhibits the q-a coupling. This is discussed elsewhere [ I ] , 1121. For molecular fluids and fluid mixtures this coupling remains even in k + 0 limit. For monatomic fluid mixtures the origin of the coupling is kinetic.
It is nonzero only for nonzero k ; the finite k provides a vector for the coupling between q and a.
Similar comparison of eq. ( 3 . 8 ) with eq. (2.19a) requires elimination of q(t) from eq. ( 3 . 8 ) and yields the following identifications :
A
(Pga A (Pdq A 1 , ( 3 . 1 5 ~ )
(Pdd - 7
Po 9 4 4
which is consistent with the usual definition of D,/D, and
A
9 q d
A (Pdq * ] , ( 3 . 1 5 ~ )
' ( ( P d d -
which can be shown to be consistent with eq. (3.14b) by noting that cpd,(t) and qqd(t) are related, as well as the fact that they are neither even nor odd fiinctions of time in general. The comparison also shows the nonzero coupling between Jd and o for nonzero k in addition to the Jd-q coupling present in hydrodynamics.
The three eq. (3.14a), (3.14b) and (3.15a) can be used to express the three independent memory
A A A
functions qdd, (P,, and cp,, in terms of the transport
A
coefficients. Relation between Gqd and (p,, can be
either directly inferred or through the eq. (3.14b) and (3.1%).
Similar analysis for eq. (3.10) and (3.11) yields the identifications, as before [ I ] .
and
as well as the coupling terms corresponding to q-a, Jd-a couplings.
Eq. (3.4)-(3.11) can be easily written down in the matrix form
[sI - iw + &o)] E(s) = I
as done by Mountain and Deutch for the hydrodynamic equations. It is in this form that the analysis of this section can be used directly to compute the Rayleigh- Brillouin spectra. The various couplings obtained
A . " i
above are related to the quan1,ities p;,, (P,,, a:, af.
Since the calculation of the former two involves in general a difficult many-body calculation, accurate experiments on one component fluids and fluid mixtu- res designed to probe the kinetic to hydrodynamic transition region are desirable. Such experiments involve Rayleigh-Brillouin scattering from dilute and dense gases and inelastic neutron scattering from dense liquids. That the q-a couplings plays a significant role in the dynamical correlatictns in one component fluids is seen in recent light scattering experiments on gases [12] as well as in the recent molecular dynamics calculations on simple liquids 1131. We expect that the evidence for q-a and Jd-o coupling will also be found on experiments on bina.ry fluid mixtures. In closing, we note that the present calculation comple- ments the recent work of Boley and Yip [14] on dilute gas mixtures which uses the classical Boltzmann equation.
Acknowledgements. - This research was supported by the National Research Couricil of Canada. It is a pleasure to acknowledge the assist an^^ of Professor P. Lallemand and C. N. R. S., France, which enabled me to attend the colloquium.
Appendix. - The system under consideration in the main text is a binary mixture of species a and b.
N, atoms of type a and N, atoms of type b are enclosed in volume V . In the thermodynamic limit the ratios
remain fixed. Phase of a particle at time t is denoted by (r,, pi). Also
N = N , + N ,
and
r i j = ri - r j .
V i j ( r i j ) is the interaction potential between ith and jth atoms. The fluctuating variable:^ are then given as
N ,
pa&, t ) = z ma[eikarl - V - I a@)] , a = a, b ,
i = 1 (A. 1)
N l N
~ ' ( k , r) - j = l [=Q- 2 m j + i z j z V j ( r i j ) ] eikarj , ( A . 2)
N" k . p . Jdk, t) = 9 ee"." ,
j = l ( A . 3)
J ( k , t) = J , + J b , ( A . 4 )
N k . p j pj2 1 p. r . . k
eck, t ) = j = 1 1 [- 2 kmj mj ( - + i + j 1 ~ ~ ( r ~ ~ ) ) - - 4 i + j ril. (3 mi + m j r ; k -c p..(t CJ r. C J ] ' .) eZkarj 9 (A 5 )
where
nOt, t ) = na + xb ,
xdOt, t) = ( ~ b o na - Pa0 nb) p i 1
The average partial energy densities and partial pressures are found to be
= kB T, n,[l - + nBo z : ~ ] , for central potentcals ,
B
where
( A . 10) ( A . 11) ( A . 12) ( A . 13)
( A . 14)
< nu > = Pao . ( A . 15)
Various equilibium correlations and thermodynamic derivatives encountered in the text are :
V-' < pa P; > = p a o [ m ~ s a p -I- ppo ~ap(k)I , a, P = a, b ( A . 16) where
~ a p ( U = ) dr eik.'[gaB(r) - 11 , ( A . 17)
( A . 18)
- 1 a P b = +
V b = - < Pa 5: > 1 ( A . 19)
mb T,c < t 1 5: > P J , Pbo
( A . 20)
( A . 22) ( A . 23)
Combining the eq. ( A . 16) with (2.10a) and Specific heat per unit mass at constant volume and (2.10b), we can deduce microscopic formulae for concentration is given as
( d ~ l a c ) ~ , ~ and K. It should be noted that whereas
~c is given by < pp* > (pi k , To)-' for one compo- 1 d e t ( < EE* >
nent fluids, the same relation does not hold for mix- c, = - < Pa E" > < pz P: >
tures and instead one should use eq. (2. lob). Po k~ Ti det < pa p: > ( A . 24)
C1-34 R. C. DESAI where pa is a column vector with pa and pb as the ele- ments. Expression for < EE* > is of the same form (with trivial generalization) as eq. (50) of ref. [8].
The various correlations connected with pressure tensor are
V-' < E' n: > = (PO + pa0 eao + pbo ebo) k~ TO 3
(A. 25)
V-' < pa n: > = pa, kB To , (A. 26)
(A. 27) V-' < n3 nz > = PO C; kB ,
X < n 3 8 ' > <On:> , (A.28)
< oo* > < JJ* >
where co js the adiabatic sound velocity,
V - I <tin:> = 0 , (A. 29)
V-' < 5n3* > = kB To, (A. 30)
V-' < nz > = kg To, (A. 31)
- kB To
V-' < rnd* > = Pao Pbo Vab -- 9 (A. 32) Po
V-' < 67~; > = - fin pao pbo kB - 9 (A. 34) Po
where p" is defined in eq. (3. ~ L Z ) ,
(A. 35)
= - [P" + (PO - PI) V,b] . (A. 36)
Following the analysis of Schofield [7], we have also connected various elastic moduli with the inter- action potential. The results are very similar to those for one component fluid. For central potentials, we can show that
and similar generalizations for < nl(k) n3(- k) > and < n4(k) n,(- k) >. In k = 0 limit, they reduce to
where I?' is defined in eq. (A. 14), n, = n,,, and
(A. 38) (A. 39) (A. 40)
(A. 41) Again for central potentials with isotropic fluid, the Cauchy relation is exactly satisfied. B,: Gm are also given by relations analogous to those for one component systems. In terms of < ,c3 n: > and < 71, n; >,
we have
(A. 42)
and
The constants 6 f which partially characterise the J,-o coupling can also be related to interaction poten- tial through the integrals I? and Through eq. (3.8a), 6; is related t o < nd of >, j = 1 , 2, 3, and
d k B Pa0 ~ b o @ a b - '% P'I/co)
V - I < ndo: > = c,, -
P o " 7
( A . 44)
( A . 45) where
and
References
[I] TONG (E.) and DESAI (R. C.), Phys. Rev., A 2, 1970, 191 DESAI (R. C.) and KAPRAL (R.), to be published.
2129. [lo] HILL (T. L.), Statistical Mechanics, Principles and
[2] MORI (H.), Progr. Theoret. Phys. (Kyoto), 1965,33,423. Selected Applications (McGraw Hill, New York, [3] ZWANZIG (R.), in Lectures in Theoretical Physics, 1956), pp. 113-121.
edited by W. E- Brittin (Wiley> New 1961). [I 11 LANDAU (L. D.) and LIFSHITZ (E. M.), Fluid Mechanics [4] MOUNTAIN (R. D.) and DEUTCH (J. M.), J. Chem. (Addison-Wesley, Reading, Mass., 1959),
Phys., 1969, 50- 1103 ; see also references [gal Chap. VI.
and [20] of this paper.
[51 GORNALL (W. S.), WANG (C. S.), YANG (C. C.) and [12] MAY (A. D.) and HARA (E. H.), (( Rayleigh-Brillouin BLOEMBERGEN (N.), Phys. Rev. Letters, 1971, 26, Scattering in Compressed Hz, Dz and HD )),
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[6] MORI (H.) Progr. Theoret. Phys. (Kyoto), 1965, 34, Phys., 1970, 49, 420.
399 ; sGe also WEINBERG (M.) and KAPRAL (R.), RAHMAN (A.), ((Number and ~ i n e t i c Energy ens it^
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by H. N. V. Temperley, J. S. Rowlinson and tures D, paper presented at this Conference.
G. S. Rushbrooke (North-Holland, Amsterdam, [I51 AKCASU (A. Z.) and DANIELS (E.), Phys. Rev., A 2.
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