Finite-size effects on Soret-induced nonequilibrium concentration ... · Finite-size effects on Soret-induced nonequilibrium concentration ﬂuctuations in binary liquids Jan V. Sengers

REVISTA MEXICANA DE FISICA 48 SUPLEMENTO 1, 14 - 25 SEPTIEMBRE 2002

Finite-size effects on Soret-induced nonequilibrium concentration fluctuations inbinary liquids

Jan V. SengersInstitute for Physical Science and Technology and Department of Chemical Engineering. University of Maryland

College Park, MD 20742, USAe-mail: [email protected]

Jose M. Ortiz de ZarateDepartamento de Fısica Aplicada 1, Facultad de Ciencias Fısicas, Universidad Complutense

E28040 Madrid, Spain

Recibido el 19 de marzo de 2001; aceptado el 25 de abril 2001

We consider a horizontal binary-liquid layer that is subjected to a stationary temperature gradient while still being maintained in a stablequiescent state without any convection. In such a thermal nonequilibrium state the Soret effect induces concentration fluctuations. In thepresent paper we show how the finite height of the liquid layer will affect the long-range nature of these nonequilibrium concentrationfluctuations at very small wave numbers. Estimates of the wave numbers where light-scattering or shadowgraph experiments will be affectedby finite-size effects are presented.

Keywords:Binary Boussinesq Equations; light scattering; nonequilibrium fluctuations; Rayleigh-Benard convection

Estudiaremos una capa horizontal de un lıquido binario sometido a un gradiente estacionario de temperatura mientras el estado de reposo dellıquido es estable, lejos de cualquier inestabilidad convectiva. En dicho sistema, el efecto Soret inducira fluctuaciones de la concentracion.En el presente trabajo mostraremos como la altura finita de la capa lıquida afectara a la naturaleza de largo alcance de las fluctuaciones de noequilibrio de la concentracion, cuando el numero de onda de dichas fluctuaciones es muy pequeno. Ademas presentaremos valores estimadosdel numero de onda para el que los efectos de tamano finito afectaran a los experimentos de dispersion de la luz o shadowgraph.

Descriptores:Ecuaciones de Boussinesq binarias; dispersion de luz; fluctuaciones fuera del equilibrio; conveccion de Rayleigh-Benard

PACS: 05.40.+j; 05.70.Ln; 66.50.Cb

This paper is dedicated to Professor Leopoldo Garcıa-Colın Scherer on the occasion of his 70th birthday.

1. Introduction

In this paper we consider a binary liquid system located be-tween two horizontal plates with different but constant tem-peratures. A temperature gradient in a liquid mixture inducesa concentration gradient through the Soret effect [1–5]. Manystudies reported in the literature have been concerned withthe effect of Soret-induced concentration gradients on theconvective Rayleigh-Benard instability [6–11]. However, itturns out that there are also interesting fluctuation phenomenapresent when the liquid mixture is in a hydrodynamicallyquiescent stable state, far away from any convective instabi-lity.

A central feature in the development of nonequilibriumstatistical mechanics during the past decades has been thediscovery that long-range dynamic correlations are presentin fluids that consist of molecules with short-range inter-actions [12, 13]. Originally it was thought that a fluid in anonequilibrium state would proceed to thermodynamic equi-librium in two distinct stages: first a microscopic kineticstage with time scale of the order of the interval betweensuccessive molecular collisions, which for liquids is of thesame order as the duration of the molecular collisions, af-ter which local equilibrium is established; second a macros-copic stage during which the fluid evolves in accordance withthe hydrodynamic equations [14]. In this picture no mecha-

nism exists for the appearance of long-range dynamic corre-lations away from a macroscopic convective instability. Thefirst indication that long-range dynamic correlations can ap-pear was provided when experiments showed that the ther-mal conductivity of fluids diverges at the vapor-liquid crit-ical point [15, 16]. This observation contradicted the classi-cal theory of Van Hove who had assumed that the slowingdown of the fluctuations near the critical point had only athermodynamic origin [17, 18]. In evaluating the randomiz-ing nature of molecular collisions one must make a distinc-tion between quantities like mass, momentum and energy thatare conserved in a binary collision on the one hand and non-conserved quantities on the other hand. The fast modes asso-ciated with nonconserved quantities do indeed decay duringa short kinetic stage, but the slow modes associated with con-served quantities persist well beyond the times between suc-cessive collisions; a coupling between these modes can re-sult in long-range (mesoscopic) dynamic correlations [19].The same mode-coupling effects, responsible for the dy-namics of the long-range critical fluctuations [20–22] turnedout to account also for the presence of long-time tails inthe Green-Kubo correlation functions for the transport coef-ficients [23–25] earlier noticed in computer simulations ofmolecular dynamics [26, 27].

In the earlier80’s it became apparent that fluids in non-equilibrium states would exhibit dramatic long-range fluctua-

FINITE-SIZE EFFECTS ON SORET-INDUCED NONEQUILIBRIUM CONCENTRATION FLUCTUATIONS IN BINARY LIQUIDS 15

tions due to mode-coupling effects. Specifically, Kirkpatrick,Cohen and Dorfman derived the appropriate expressions forthe temperature and the viscous fluctuations in a one-component fluid subjected to a stationary temperature gra-dient using mode-coupling theory [28], indicating thatnonequilibrium fluctuations would enhance and modify theRayleigh spectrum of scattered light [29]. Subsequently, itbecame apparent that the same expressions could be obtainedon the basis of Landau’s fluctuating hydrodynamics, whichis also based on the assumption that only the nonconservedquantities decay rapidly over a molecular time scale [30–35].Physically, the presence of a temperature gradient∇T leadsto nonequilibrium fluctuations because the stationary tem-perature gradient causes a coupling between the componentof the velocity fluctuations parallel to the gradient and thetemperature fluctuations with a wave vector perpendicularto the gradient. Velocity fluctuations parallel to the gradientare probing regions with different local equilibrium tempera-tures, thus causing a nonequilibrium contribution to the tem-perature fluctuations. Since both the temperature and the vis-cous (transverse velocity) fluctuations do not propagate, butdecay diffusively, the nonequilibrium contributions cannotdepend on whether the temperature gradient is in the positiveor negative direction and, hence, they depend on the square ofthe temperature gradient. Moreover, each mode contributes afactor q−2, so that the intensity of the nonequilibrium fluc-tuations becomes inversely proportional to the fourth powerof the wave numberq of the fluctuations. The dependenceof the intensity of the nonequilibrium fluctuations on(∇T )2

and onq−4 has been confirmed by light-scattering experi-ments [36–39]. The presence of such long-range fluctuationsin fluids in nonequilibrium states causes a serious difficulty inthe further development of irreversible thermodynamics as,e.g., studied by Garcıa-Colın and coworkers [40–43].

In the case of a liquid mixture, a temperature gradient∇Tis accompanied by a concentration gradient∇c through theSoret effect. Just as a temperature gradient causes a couplingbetween the temperature fluctuations with wave vectorq per-pendicular to the temperature gradient and the transverse-velocity fluctuations in the direction of the temperature gra-dient, so will a concentration gradient cause a coupling be-tween the concentration fluctuations with wave vectorq per-pendicular to the concentration gradient and the transverse-velocity fluctuations in the direction of the concentration gra-dient. Hence, in a liquid mixture subjected to a temperaturegradient not only nonequilibrium temperature and viscousfluctuations will be present, but also nonequilibrium concen-tration fluctuations as first pointed out by Law and Nieu-woudt [44, 45], and as also analyzed by Velasco and Garcıa-Colın [46] and by Segre and Sengers [47]. The intensity ofthe nonequilibrium concentration fluctuations should be pro-portional to the square of the concentration gradient∇c andagain inversely proportional to the fourth power of the wavenumberq. The case of nonequilibrium concentration fluctua-tions in a colloidal suspension in the presence of a concen-tration gradient has been considered by Schmitz [48]. Vailati

and Giglio [49] have extended the theory to treating the ef-fect of time-dependent concentration profiles on the structurefactor of a binary liquid.

The divergence of the intensity of nonequilibrium fluc-tuations for small wave numbers,i.e., for large wavelengthscannot go on indefinitely. For very small wave numbers thedependence of the nonequilibrium fluctuations onq will beaffected by the presence of gravity even when the liquid isin a stable nonconvective state. The effects of gravity onthe nonequilibrium fluctuations, both in one-component li-quids and in liquid mixtures have been analyzed by Segreet al. [47, 50] for the case of negative Rayleigh numbers.For nonequilibrium concentration fluctuations in a binary li-quid mixture, the effects of gravity have been further evaluat-ed by Sengers and Ortiz de Zarate [51]. It turns out thatfor negative Rayleigh numbers gravity has a damping effecton the nonequilibrium fluctuations, causing theq−4 diver-gence in the intensity of these fluctuations to become satu-rated at smallq and to approach a finite plateau in the li-mit q → 0. This gravitationally induced crossover behaviorfor very smallq has been observed experimentally by Vailatiand Giglio [52, 53].

In addition, when the productqL becomes of order unity,one must expect that the nonequilibrium fluctuations willbe affected by the finite heightL of the liquid layer sub-jected to the temperature gradient. Finite-size effects on thenonequilibrium temperature and viscous fluctuations in aone-component liquid have recently been studied by Ortiz deZarateet al. [54, 55]. It is the purpose of the present paper toanalyze finite-size effects on the nonequilibrium concentra-tion fluctuations in a liquid mixture, a problem which so farhas not been treated in the literature.

A complete derivation of the contributions from nonequi-librium fluctuations should be based on the linearized hydro-dynamic equations supplemented with random-noise termsas was done by Law and Nieuwoudt [44] and by Segre andSengers [47]. However, since in liquid mixtures temperaturefluctuations decay much faster than concentration fluctua-tions, one can obtain a very good approximation for the con-tribution of the nonequilibrium fluctuations to the structurefactor by starting from a simplified set of linearized Boussi-nesq equations as we have demonstrated in a recent reviewon the subject [51]. Physically it means that we can neglectany coupling of the concentration fluctuations with the tem-perature fluctuations through the imposed temperature gradi-ent and only consider the coupling between the concentrationfluctuations and the transverse-velocity (viscous) fluctuationsthrough the concentration gradient induced by the Soret ef-fect.

We shall proceed as follows. In Sec. 2 we formulate thelinearized Boussinesq equations for a binary liquid and esta-blish our notation for the fluctuating quantities. In Sec. 3 weintroduce some approximations into the linearized Boussi-nesq equations as earlier proposed by Schechter and Ve-larde [7, 56] for simplifying the analysis of the convectiveinstability in a two-component liquid. In Sec. 4 we show

Rev. Mex. Fıs. 48 S1(2002) 14 - 25

16 JAN V. SENGERS AND JOSE M. ORTIZ DE ZARATE

how the well known expression for the contribution of thenonequilibrium concentration fluctuations in a binary liquidsubjected to a stationary temperature gradient can be reco-vered from the simplified set of linearized Boussinesq equa-tions when supplemented with random-noise terms. In Sec. 5we then evaluate the modifications needed to account for theboundary conditions due to the presence of two horizontalplanes confining the liquid layer to a finite heightL. In Sec. 6we consider the consequences of any finite-size effects on thenonequilibrium concentration fluctuations for the interpreta-tion of light-scattering or shadowgraph experiments. Someconcluding remarks are presented in Sec. 7.

2. The linearized Boussinesq equations for abinary liquid

The system we are considering in this paper is a binary liquidbounded by two horizontal planes separated by a distanceL.Across this liquid layer a stationary temperature gradient isestablished by maintaining the two horizontal planes at diffe-rent temperatures;∇T0 represents the magnitude of the tem-perature gradient which is assumed to act in theZ-directionperpendicular to the bounding planes. As a consequence ofthe Soret effect, a stationary concentration gradient also de-velops in the system with magnitude∇c0. The relationshipbetween the concentration gradient∇c0 and the temperaturegradient∇T0 is given by:

∇c0 = −c(1− c)ST∇T0, (1)

whereST is the Soret coefficient andc is the (average) con-centration expressed as weight fraction (w/w) of component 1of the mixture. Equation (1) defines the Soret coefficientST

of component 1 in component 2. For isotropic mixturesST

is a scalar quantity and the induced concentration gradientis parallel to the imposed temperature gradient but, depend-ing on the sign ofST , can have the same or the opposite di-rection. WhenST is positive, concentration and temperaturegradients have opposite directions, with component 1 migrat-ing to the colder region. WhenST is negative, concentrationand temperature gradients have the same direction, with com-ponent 1 migrating to the warmer region. Note that, becauseof conservation of mass, whenST of component 1 in compo-nent 2 is positive, thenST of component 2 in component 1is negative andviceversa. In this paper,c designates the con-centration of component 1.

We want to study small fluctuations around the conduct-ing state,i.e., fluctuations around the quiescent and stablestate of the fluid. We do not consider any fluctuations in thepressurep related to propagating modes that leads to Bril-louin scattering. The appropriate set of equations for ourproblem are the linearized Boussinesq equations for a binarysystem [7, 8, 10, 57], which read

∂

∂t(∇2w) = ν∇2(∇2w) + g

(∂2

∂x2+

∂2

∂y2

)[αθ − βΓ], (2)

∂θ

∂t= Dth∇2θ − w∇T0, (3)

and

∂Γ∂t

= D

[∇2Γ +

α

βψ∇2θ

]− w∇c0, (4)

whereθ(r, t) represents the local fluctuation of the tempe-rature (δT ), w(r, t) denotes the local fluctuations of thez-component of the velocity(δuz), andΓ(r, t) denotes thelocal fluctuation of the concentration of component 1 of themixture (δc). The fluctuationsθ(r, t), w(r, t) and Γ(r, t)depend on the positionr(x, y, z) and on the timet. Thecoefficientν is the kinematic viscosity,Dth the thermal dif-fusivity, ρ the density, andλT the thermal conductivity of themixture, whileD represents the binary diffusion coefficientand g the gravitational acceleration constant. The symbolα = −ρ−1(∂ρ/∂T )p,c represents the thermal expansion co-

efficient and, similarly,β = ρ−1(∂ρ/∂c)p,T represents the

concentration expansion coefficient. Note thatβ is positivewhen component 1 is the heavier component andβ is nega-tive when component 1 is the lighter component. Finally,ψis the dimensionless separation ratio, which is defined by

ψ = c(1− c)ST

β

α. (5)

The parameterψ is the ratio of the density gradientβ∇c0

produced by the concentration gradient to the densitygradient −α∇T0 produced by the temperature gradient;ψ = −(β∇c0)/(α∇T0).

By adopting Eqs. (2)–(4) we assume that all thermophy-sical properties of the binary liquid depend only weakly ontemperature or concentration so that the variation of theseproperties as a function ofz is negligibly small; in practicethis is a very good approximation [37]. For the same reason,in Eq. (5) the concentrationc of component 1 represents anaverage value in the liquid layer; replacing the local concen-tration by an average concentration is only questionable forextremely dilute solutions [7].

The Boussinesq approximation assumes the liquid to beincompressible [58], so that the coefficientDth in Eq. (3)can indeed be identified with the thermal diffusivityλT/ρcP,wherecP is the specific isobaric heat capacity. As usual, theDufour effect has been neglected in Eqs. (2)–(4), since it isrelevant only in binary gas mixtures and in liquids near thevapor-liquid critical point [57]. In addition, and followingthe usual literature [58], to eliminate the hydrostatic-pressuregradient from the equations we find it convenient to considerEq. (2) for∇2w by taking a double curl in the equation forthe fluctuating fluid velocityδu.

In Eqs. (2)–(4) we presented the linearized version of theBoussinesq equations for a binary liquid. Neglecting the non-linear terms is only justified when the liquid layer is in aquiescent conductive stable state. The linear stability of thebinary Boussinesq equations has been studied by many in-vestigators [7, 9, 59]. As is well known, the stability of thequiescent state depends on the values of the dimensionlessRayleigh numberRa, and of the dimensionless separation ra-tio, ψ [7]. In this paper, we shall focus our attention on the

Rev. Mex. Fıs. 48 S1(2002) 14 - 25


finite-size effects on the structure factor of the liquid mixtureand not consider gravity effects, which have been evaluated inother publications [47, 50]. The approximationg = 0 impliesneglecting the bouyancy term in Eq. (2), making the Rayleighnumber of our problem to be zero. As is discussed in the rele-vant literature [7], for Ra = 0, the quiescent state is stable in-dependent of the value of the separation ratio. Consequently,by adopting theg → 0 limit of the Boussinesq equations, theconducting state will always be stable. In this situation theset of differential equations [Eqs. (2)–(4)] describes correctlythe spatial and temporal evolution of the fluctuations aroundthe quiescent state of the liquid.

3. Linearized Boussinesq equations for smallLewis number

In order to determine the contribution from concentrationfluctuations to the structure factor of a binary liquid mixture,it is convenient to make a number of additional approxima-tions to the standard Boussinesq equations [51]. First of all, toa good approximation the decay rates of the viscous fluctua-tions, the temperature fluctuations and the concentration fluc-tuations can be directly identified withνq2, Dthq2 andDq2,respectively [38]. Furthermore, the diffusion coefficientD isin practice much smaller than the kinematic viscosityν andthe thermal diffusivityDth. Thus, the viscous and thermalfluctuations decay much faster than the concentration fluctua-tions. Moreover, because of the dependence of the refractiveindex on the concentration, the intensity of light scattered bythe concentration fluctuations is usually appreciably largerthan that from the viscous or the temperature fluctuations.Hence, in many cases one can readily determine the concen-tration fluctuations with Rayleigh scattering independentlyfrom the much faster decaying viscous and temperature fluc-tuations [38, 39, 60, 61]. As elucidated in a recent review ofthe subject [51], to deal with the concentration fluctuationswe shall retain only the more relevant physical terms in theBoussinesq equations [Eqs. (2)–(4)] by adopting an approxi-mate set of linearized equations earlier considered by Velardeand Schechter [56] for a convective-instability analysis.

The Velarde-Schechter approximation consists of scalingthe distances by the finite height of the layer,L, and the timeby the diffusional timeτD = L2/D, and then perform a se-ries expansion of the full Boussinesq equations in terms ofthe Lewis numberLe = D/Dth. It is worth noting that forregular liquid mixturesLe is a small quantity (' 10−3). Inthis series expansion only terms of order zero inLe are re-tained. To clarify this approximation scheme, we first switchto dimensionless variables, usingL as the length scale andτD

as the time scale [56]

w =wL

D, t =

tD

L2,

r =rL

, θ =θ

L ∇T0

, (6)

Multiplying Eq. (2) byL5/(DDth), Eq. (3) byL/(∇T0Dth),and Eq. (4) byL2/D, we obtain in terms of dimensionlessvariables

Le∂

∂t

(∇2w)

= Pr∇2(∇2w

)

−RaSc

(∂2

∂x2+

∂2

∂y2

)(θ +

ψ

∆c0

Γ)

, (7)

Le∂θ

∂t= ∇2θ − Lew, (8)

∂Γ∂t

=(∇2Γ−∆c0∇2θ

)− w∆c0, (9)

where Pr = ν/Dth is the Prandtl number and where∆c0 = L∇c0 is a dimensionless concentration difference. Ifwe now take the limitLe → 0 of the equations, we ob-tain θ(r, t) = 0; thus to zeroth order inLe, the tempera-ture fluctuations can be neglected. Reverting the equations forw(r, t) andΓ(r, t) to dimensional form, taking at this pointtheg → 0 limit (neglecting the buoyancy term), and supple-menting the equations with the corresponding random-noiseterms, we finally obtain [51]:

0 = ν∇2(∇2w

)+

1ρ

{∇× [∇× (∇ · δT)]}

z, (10)

∂Γ∂t

= D∇2Γ− w∇c0 +∇δJ. (11)

The primary physical meaning of Eqs. (10) and (11) is thatfor small values of the Lewis number we can neglect anycoupling between the concentration fluctuations and the tem-perature fluctuations and retain only the coupling betweenconcentration fluctuations and viscous (transverse-velocity)fluctuations through the Soret-induced stationary concentra-tion gradient∇c0 in Eq. (11). As a consequence, the imposedtemperature gradient∇T0 will appear in the final expressionfor the nonequilibrium structure factor, in this small-Lewis-number approximation, only through the induced concenta-tion gradient, Eq. (1). Hence, our derivation will be equallyvalid when the concentration gradient∇c0 does not resultfrom an imposed temperature gradient, but arises from anyother cause, as in the case of an isothermal free-diffusion pro-cess [49].

Equations (10) and (11) are the starting point for the ana-lysis of the concentration fluctuations to be performed in thispaper. For this purpose, the Velarde-Schechter approxima-tion to the full linearized Boussinesq equations have beensupplemented with random-noise terms representing the con-tributions from rapidly varying short-range fluctuations, inaccordance with the standard fluctuating-hydrodynamics ap-proach [44, 62–64]. As discussed in the Introduction, theserandom-noise terms represent the rapidly decaying noncon-served degrees of freedom, to be distinguished from theconserved degrees of freedom described by the hydrody-namic evolution equations. Thus in Eq. (10) we have intro-ducedδT(r, t) as a random stress tensor following [62, 63]

Rev. Mex. Fıs. 48 S1(2002) 14 - 25


and in Eq. (11) we have introducedδJ(r, t) as a random so-lute flux following [44, 64]. Since in the approximation weare using for the hydrodynamics of the system, the tempe-rature fluctuations do not couple to concentration or viscousfluctuations, we do not need to consider a random heat flux.The subscriptz in Eq. (10) indicates that this random-noiseterm is to be identified with theZ-component of the vec-tor between curly brackets. This procedure of supplementingthe random Boussinesq equations for a binary system withrandom-noise terms has previously been used by Schopf andZimmermann to study the influence of noise close to the con-vective instability [65].

Finally, we mention that the small-Lewis-number appro-ximation adopted by us, has also been widely employed inthe literature to simplify the stability analysis of a binarymixture [7, 56, 66, 67]. As discussed in the relevant publica-tions [7, 56], the Le → 0 approximation has several short-comings; in particular, it does not describe correctly the situ-ation withRa > 0 andψ < 0, missing the interesting Hopfbifurcation and codimension-two instability point. But as

as commented earlier, we restrict ourselves here to the limitg → 0, in which case theLe → 0 approximation is adequateto describe the concentration fluctuations, as will be shownbelow.

4. Calculation of the “Bulk” Structure Factor

Before evaluating any finite-size effects on the nonequili-brium concentration fluctuations, we first review the deriva-tion of their contribution to the “bulk” structure factor,i.e.,the structure factor in the absence of any boundary condi-tions [51]. For this purpose we apply a temporal and spatialFourier transform to Eqs. (10) and (11), so as to obtain

(−νq4 0∇c0 iω + Dq2

)[w(ω,q)Γ(ω,q)

]=

[F1(ω,q)F2(ω,q)

]. (12)

In Eqs. (12),F1 and F2 represent Langevin random-noiseterms, which are related to the Fourier transforms of the ran-dom stress tensorδT(ω,q) and of the random solute fluxδJ(ω,q) by

[F1(ω,q)

F2(ω,q)

]=

[iρ−1

{q2

[q · δT(ω,q)

]z− qzq ·

[q · δT(ω,q)

]}

iq · δJ(ω,q)

]. (13)

In this paper we identify the structure factorS(ω, q) asthe function that determines the intensity of the scatteredRayleigh light as a function of the frequencyω and the scat-tering vectorq arising from concentration fluctuations only.Consequently, this structure factorS(ω, q) is related to theautocorrelation function〈Γ∗(ω,q) ·Γ(ω′,q′)〉 of the concen-tration fluctuations by

〈Γ∗(ω,q) · Γ(ω′,q′)〉 =(

∂c

∂n

)2

p,T

S(ω,q)

×(2π)4δ(ω − ω′)δ(q− q′), (14)

where(∂n/∂c)p,T represents the effect of the concentrationdependence of the refractive indexn on the light-scatteringintensity.

The correlation functions between the Langevin noisetermsF1 andF2, needed to calculateS(ω, q) from Eq. (14),can be obtained from the known correlation functions be-tween the different components of the random stress tensorand the random solute flux. In nonequilibrium fluctuating hy-drodynamics it is assumed that the correlation functions ofthese random-noise terms retain their local equilibrium va-lues, which are short-ranged in space and in time [44, 68, 69].As mentioned earlier, the conserved degrees of freedom ac-tually develop long-ranged correlations in nonequilibriumstates, but the random stress tensor and the random solute fluxrepresent the nonconserved degrees of freedom, thus they areexpected to remain short-ranged, even out of equilibrium.Upon substituting into Eq. (13) the expressions for the cor-

relation functions ofδT as, for instance, given by Eqs. (3.12)in Ref. 62, and from the correlation functions ofδJ, as givenby [44, 64], we obtain [51, 54]

〈F ∗1 (ω,q) · F1(ω′,q′)〉 = 2kBT

ν

ρq2‖q

4

×(2π)4δ(ω − ω′)δ(q− q′),

〈F ∗2 (ω,q) · F2(ω′,q′)〉 = 2kBT

D

ρ

(∂c

∂µ

)

p,T

q2

×(2π)4δ(ω − ω′)δ(q− q′),

〈F ∗1 (ω,q) · F2(ω′,q′)〉 = 〈F ∗2 (ω,q) · F1(ω

′,q′)〉=0, (15)

whereµ = µ1 − µ2 is the difference between the chemicalpotentials per unit mass of the two components of the mix-ture,kB is Boltzmann’s constant, andq‖ represents the mag-nitude of the wave vectorq in thexy-plane,i.e., q2

‖ = q2x+q2

y.Note that the correlation functions between the random-noiseterms F1 and F2 are short ranged both in space and intime, as they are represented by delta functions. Now, invert-ing Eqs. (12) and using Eqs. (14) and (15), we obtain forthe nonequilibrium structure factor of the binary liquid, intheLe → 0 approximation

S(ω,q) = SE

[1 +

(∇c0)2

νD

(∂µ

∂c

)

p,T

q2‖

q6

]2 Dq2

ω2 + D2q4, (16)

Rev. Mex. Fıs. 48 S1(2002) 14 - 25


where SE is the equilibrium static structure factor, givenby [70]

SE =kBT

ρ

(∂n

∂c

)2

p,T

(∂c

∂µ

)

p,T

. (17)

Integration of Eq. (16) forSNE(ω,q) over all frequenciesω,yields the static structure factorS(q) = 1/(2π)

∫S(ω,q) dω

S(q) = SE

{1 + S0

NE

q2‖

q6

}, (18)

with q = qL and

S0NE =

(∂µ

∂c

)

p,T

(∇c0)2L4

νD. (19)

In Eqs. (18) and (19) we have introduced the finite heightof the liquid layerL to write the result as a function of di-mensionless wave numbers so as to facilitate the comparisonwith the finite-size calculation to be performed in next sec-tion. We note that both the dynamic structure factor, givenby Eq. (16), and the static structure factor, given by Eq. (18),have an equilibrium and a nonequilibrium contribution. Forthe equilibrium contribution, as expected, we recover the wellknown expression for the concentration fluctuations of a bi-nary liquid in thermal equilibrium [70]. The dimensionlessproductS0

NEq2‖/q6 inside the brackets in Eqs. (16) and (18)

represents the modification (enhancement) to the Rayleighscattering due to the nonequilibrium concentration fluctua-tions resulting from the presence of a concentration gradient.The intensity of the nonequilibrium fluctuations is propor-tional to the square of the concentration gradient(∇c0)

2 andinversely proportional to the fourth power of the wave num-berq−4. This algebraic type of dependence on the wave num-berq is now considered to be a general feature of fluctuationsin system in stationary nonequilibrium states [13, 71, 72].

For a further discussion of the theory of the “bulk”nonequilibrium concentration fluctuations presented here, werefer to a recent review [51]. The important point to be notedfor the purpose of the present paper is that the validity ofEqs. (16) and (18) has been verified by light-scattering expe-

riments in a polymer solution in the dilute and semi-dilute so-lution regime where the theory should be applicable [60, 61].In addition, the theory has also further been confirmed byshadowgraph experiments performed in free-diffusion pro-cesses [73, 74]. As mentioned in Sec. 3, because of theLe → 0 approximation, Eqs. (16) and (18) are also applicableto isothermal diffusion.

5. Modifications of the structure factor due tofinite-size effects

Equations (16) and (18) imply that the intensity of the non-equilibrium fluctuations would diverge asq → 0. However,there are two other effects that have to be considered for verysmallq. First, for fluctuations with very long wavelengths onecan no longer neglect the bouyancy term in Eq. (2). As com-mented earlier, for negative Rayleigh numbers gravity causesa long-wavelength damping of the non-equilibrium fluctua-tions, causing the non-equilibrium enhancement to saturateand reach a finite limit atq → 0 [47, 50–53]. Second, whenthe wavelength of the fluctuations is no longer small com-pared to the heightL of the liquid layer, limitations due to theboundary conditions imposed by the horizontal plates confin-ing the liquid layer need to be taken into account. In a previ-ous publication we have demonstrated how the effect of grav-ity can readily be incorporated in the derivation of the con-tribution of non-equilibrium concentration fluctuations to the“bulk” structure factor described in Sec. 4 [51]. In the presentpaper we focus our attention on the modification of Eq. (18)for the intensity of the non-equilibrium concentration fluctua-tions due to the presence of boundaries atz = 0 and atz = L.In doing so, we follow a procedure previously used for eval-uating finite-size effects on the non-equilibrium temperatureand viscous fluctuations in a one-component liquid [54, 55].

As in Sec. 4, we again Fourier transform in time and spacebut, to accomodate the effect of boundary conditions in thez-direction we restrict the spatial Fourier transformation tothex andy directions, parallel to the plates. We then deducefrom Eqs. (10) and (11) the following set of linear differentialequations

−ν

(d2

dz2− q2

‖

)2

0

∇c0 iω −D

(d2

dz2− q2

‖

)

[w(ω,q‖, z)

Γ(ω,q‖, z)

]=

[F1(ω,q‖, z)

F2(ω,q‖, z)

], (20)

whereF1(ω,q‖, z) and F2(ω,q‖, z) represent now partial transforms of the random-noise termsF1 and F2 in Eqs. (10)and (11). They have complicated expressions as a function of the partial Fourier transformδT(ω,q‖, z) and δJ(ω,q‖, z),but the actual expressions are similar to those obtained for the problem of nonequilibrium fluctuations in a one-component liq-uid presented in a previous publication [55]. As is often done in the literature [7, 9, 59], for the sake of simplicity, we assumehere stress-free boundary conditions for the vertical velocity and permeable walls for the concentration, so that

Rev. Mex. Fıs. 48 S1(2002) 14 - 25


Γ(ω,q‖, z) = 0 at z = 0, L,

w(ω,q‖, z) = 0 at z = 0, L,

d2

dz2w(ω,q‖, z) = 0 at z = 0, L. (21)

Note that these boundary conditions imply the absence ofany possible fluctuations in the concentration and velocity ofthe fluid adjacent to the walls. The set of boundary condi-tions [Eqs. (21)] corresponds to a liquid bounded by two freeand permeable surfaces, which is rather unrealistic [59, 75].For the case of a liquid confined between two rigid and imper-meable surfaces, no-slip boundary conditions in the velocityand zero value of thez derivative of the concentration aremore appropriate [9, 59]. Nevertheless, for the sake of math-ematical simplicity, we adopt here the boundary conditionsgiven by Eqs. (21). We expect that the conclusions of thiswork will be modified only quantitatively, not qualitatively,for the more realistic no-slip impermeable boundary condi-tions, as is the case for the nonequilibrium fluctuations in aone-component liquid [55].

To search for a solution of Eqs. (20) we representw(ω,q‖, z) andΓ(ω,q‖, z) as a series expansion in a com-plete set of eigenfunctions satisfying the boundary conditions[Eqs. (21)]. Because of the simplicity of these boundary con-ditions, an appropriate set of eigenfunctions is just given bythe Fourier sine basis in the[0, L] interval [58]. We thus as-sume:[

w(ω, q‖, z)

Γ(ω, q‖, z)

]=

∞∑

N=1

[AN (ω,q‖)

BN (ω,q‖)

]sin

(Nπz

L

). (22)

To deduce the coefficientsAN (ω, q‖) and BN (ω, q‖)from Eqs. (20), we need to represent the random-noise termsF1(ω,q‖, z) andF2(ω,q‖, z) also as a Fourier sine series

[F1(ω,q‖, z)

F2(ω,q‖, z)

]=

∞∑

N=1

[F1,N (ω, q‖)

F2,N (ω, q‖)

]sin

(Nπz

L

), (23)

where we have introduced the set of random functionsF1,N (ω,q‖) andF2,N (ω,q‖), which are the projections ofthe random-noise terms onto the eigenfunction basis. Theyare given by[F1,N (ω, q‖)

F2,N (ω, q‖)

]=

2L

∫ L

0

[F1(ω,q‖, z)

F2(ω,q‖, z)

]sin

(Nπz

L

)dz. (24)

Representing the random-noise terms by Eqs. (23), one read-ily deduces from Eqs. (20) expressions for the coefficients ofthe Fourier seriesAN (ω, q‖) andBN (ω, q‖)

AN (ω, q‖) =F1,N (ω, q‖)

ν

(N2π2

L2+ q2

‖

)2 , (25)

and

BN (ω, q‖) =F2,N (ω, q‖)−AN (ω, q‖)∇c0[

iω + D

(N2π2

L2+ q2

‖

)] . (26)

In this paper we are interested in the autocorrelationfunction of the concentration fluctuations. It follows fromEqs. (22) that this function can be expressed as

〈Γ∗(ω,q‖, z) · Γ(ω′,q′‖, z′)〉 =

∞∑

N=1

∞∑

M=1

〈B∗N (ω, q‖) ·BM (ω′, q′‖)〉 sin

(Nπ

Lz

)sin

(Mπ

Lz′

). (27)

To evaluate the autocorrelation function〈B∗N (ω, q‖) · BM (ω′, q′‖)〉, it is necesary to evaluate the correlation functions

between the different projections of the random-noise terms,F1,N (ω,q‖) and F2,N (ω,q‖). The autocorrelation function〈F ∗1,N (ω,q‖) · F1,M (ω′,q′‖)〉 has been evaluated previously [54]. For the purpose of the present paper we have evaluated

for the first time the autocorrelation〈F ∗2,N (ω,q‖) · F2,M (ω′,q′‖)〉 as well as the cross correlations. However, the method ofevaluation is similar to the one implemented previously for the random heat flow instead of the random solute flow. For furtherdetails of the procedure we refer to previous publications [54, 55]. Our final results are

〈F ∗1,N (ω,q‖) · F1,M (ω′,q′‖)〉 = 2kBTν

ρ

2L

q2‖

(q2‖ +

N2π2

L2

)2

δNM (2π)3δ(ω − ω′)δ(q‖ − q′‖),

〈F ∗1,N (ω,q‖) · F2,M (ω′,q′‖)〉 = 〈F ∗2,N (ω,q‖) · F1,M (ω′,q′‖)〉 = 0,

〈F ∗2,N (ω,q‖) · F2,M (ω′,q′‖)〉 = 2kBTD

ρ

(∂c

∂µ

)

p,T

2L

(q2‖ +

N2π2

L2

)δNM (2π)3δ(ω − ω′)δ(q‖ − q′‖). (28)

In this evaluation we continue to assume, as in Sec. 4, that the correlation functions between the different components of therandom stress tensor and the random solute flux retain their equilibrium values. This assumption remains valid as long asL isa macroscopic distance, much larger than the molecular distances in the liquid [54].

Rev. Mex. Fıs. 48 S1(2002) 14 - 25


Equation (14), relating the autocorrelation function of the concentration fluctuations to their contribution to the structurefactor now becomes [54, 62]:

〈Γ∗(ω,q‖, z) · Γ(ω′,q′‖, z′)〉=

(∂c

∂n

)2

p,T

S(ω,q‖, z, z′)(2π)3δ(ω − ω′)δ(q‖ − q′‖). (29)

Substituting Eqs. (26) and (27) into Eq. (29) and making use of Eq. (28), we obtain for the structure factor

S(ω,q‖, z, z′) = SE

2L

∞∑

N=1

1 +

(∇c0)2

νD

(∂µ

∂c

)

p,T

q2‖(

q2‖ +

N2π2

L2

)3

2D

(q2‖ +

N2π2

L2

)

ω2 + D2

(N2π2

L2+ q2

‖

)2

× sin(

Nπ

Lz

)sin

(Nπ

Lz′

), (30)

with SE again defined by Eq. (17).We note that due to the cylindrical symmetry of the problem, the result depends only on the magnitudeq‖ of the vectorq‖.

As in Sec. 4, we are interested here in the static structure factor representing the total intensity of the contribution from theconcentration fluctuations, which is obtained by integration over the frequencyω: S(q‖, z, z′) = 1/(2π)

∫S(ω, q‖, z, z′) dω,

so that

S(q‖, z, z′) = SE

[δ(z − z′) + S0

NE

2L

∞∑

N=1

q2‖ sin

(Nπz

)sin

(Nπz′

)(q2‖ + N2π2

)3

](31)

with S0NE again defined by Eq. (19). As in Eq. (18) we

have again introduced a dimensionless wave numberq suchthat q = qL. As was the case in the absence of any boun-dary conditions, the structure factor contains a contribu-tion SEδ(z − z′) from short-range equilibrium concentrationfluctuations and a contribution from long-range nonequilib-rium concentration fluctuations, as further discussed below.

6. Consequences for light-scattering orshadowgraph experiments

In this section we address the question whether the ef-fects due to the finite height of the liquid layer can be ob-served experimentally. For this purpose we consider thetwo experimental techniques which, thus far, have beensucessfully employed to observe nonequilibrium fluctua-tions, namely, small-angle light scattering and shadow-graphy. Small-angle light-scattering has been employed bySengers and coworkers [38, 39, 60, 61] and by Giglio andcoworkers [52, 53, 76]. A schematic representation of sucha light-scattering experiment is shown in Fig. 1. The scat-tering medium is a thin horizontal liquid layer bounded bytwo parallel plates whose temperatures can be controlledindependently so as to establish an uniform temperature gra-dient across the liquid layer. The temperature gradient can beparallel or antiparallel to the gravity. The horizontal platesare furnished with windows allowing laser light to propagatethrough the liquid in the direction parallel to the gravity andto the temperature gradient. Light scattered over an angleφarises from fluctuations with a wave numberq such that [70]

q = 2q0 sin(

φ

2

), (32)

whereq0 is the wave number of the incident light inside thescattering medium. To observe any nonequilibrium effectsone needs to observe the scattered-light intensity at smallwave numbers and, hence, at very small scattering angles.

From the electromagnetic theory [70] it follows thatthe scattering intensityS(q) is obtained from an integra-tion of the structure factor over the scattering volume, sothat [54, 62]

S(q‖, q⊥) =1L

L∫

0

L∫

0

e−iq⊥(z − z′)S(q‖, z, z′) dzdz′. (33)

In Eq. (33) it is assumed that the scattering volume extendsover the full height of the fluid layer as is the case in small-angle light scattering from a thin liquid layer. In this situationscattered light received in the collecting pinhole of the detec-tor indeed arises from all the points illuminated by the laserbeam over the height of the liquid layer. From the Bragg con-dition [Eq. (32) and the geometrical arrangement shown inFig. 1, we note thatq‖ and q⊥ in an actual light-scatteringexperiment are not independent variables, because they arerelated to the scattering angle,φ, by

q‖ = q cos(

φ

2

)= 2q0 sin

(φ

2

)cos

(φ

2

),

q⊥ = q sin(

φ

2

)= 2q0 sin2

(φ

2

). (34)

Equation (34) shows that for small-angle experimentsq‖ ' qandq⊥ ' 0 is a very good approximation. For actual small-angle nonequilibrium light-scattering experiments, this ap-proximation is always adopted.

Rev. Mex. Fıs. 48 S1(2002) 14 - 25


FIGURE 1. Schematic representation of a low-angle nonequili-brium light-scattering experiment.qi is the wave vector of the in-cident light,qs is the wave vector of the scattered light. The mag-nitudeq = |qi − qs| of the scattering wave vector is related to thescattering angleφ by q = 2q0 sin (φ/2), whereq0 is the magnitudeof the wave vectorqi of the incident light inside the liquid. For cla-rity, the magnitude of the scattering angleφ has been exaggerated.

Recently, nonequilibrium concentration fluctuations havealso been investigated by quantitative shadowgraph tech-niques [73, 74, 77]. The experimental arrangement is similarto the one depicted in Fig. 1, but instead of using a laserbeam, an extended uniform monochromatic light source isemployed to illuminate the sample. Then many shadowgraphimages of a plane perpendicular to the temperature gradientare obtained with a CCD (Charged Coupled Device) detec-tor, which measures a spatial distribution of intensityI(x),wherex is a two-dimensional position vector in the imagingplane. For each image a shadowgraph signal is defined by

i(x) =I(x)− I0(x)

I0(x), (35)

whereI0(x) is the blank intensity distribution, when thereare no fluctuations in the index of refraction of the sample. Inpractice,I0(x) is calculated by averaging over many shadow-

graph images, so that fluctuations cancel out and the re-sulting I0(x) contains only contributions coming from non-uniform illumination of the sample. From physical andgeometrical optics, it can be demonstrated that the modu-lus squared of the two-dimensional Fourier transform of theshadowgraph signal,|i(q)|2, after azimuthal averaging canbe expressed as [73, 74, 78]

|i(q)|2(q) = T (q)S(q‖ = q, q⊥ = 0), (36)

where the overline indicates an azimuthal average, in whichcase the result depends only on the magnitude of the two-dimensional Fourier vectorq. The symbolT (q) representsan optical transfer function, which can be derived from theoptical arrangement used to produce the shadowgraph pic-tures; it includes contributions from the response of the CCDdetector and the dependence of refractive index on concen-tration [73, 74, 78]. Therefore, an analysis of the spectrum ofthe shadowgraph pictures can be employed to experimentallydetermine the structure factor of the fluid in the planeq⊥ = 0.

It is interesting to note the equivalence between small-angle light-scattering and shadowgraph techniques, in thesense that both methods give usS(q‖ = q, q⊥ = 0), wherefor light scatteringq is the scattering wave vector as given byEq. (32), while for shadowgraph techniquesq is the modulusof the two-dimensional Fourier vector in the imaging plane;in both cases the observedSNE(q) depends only on the mag-nitude of the wave vectorq. As noted by Bodenschatzet al.,what it is actually measured in these experiments is a kind ofvertical average of the fluctuations [79].

If we substitute Eq. (31) into Eq. (33), using the small-angle approximationq‖ ' q, q⊥ ' 0, perform the summationof the series in Eq. (31) and the double integration in Eq. (33),we obtain [54]

S(q) = SE

{1 + S0

NESNE(q)}, (37)

where S0NE was defined previously by Eq. (19), and whe-

re SNE(q) is a dimensionless normalized nonequilibrium en-hancement, which includes the finite-size effects, given by

SNE(q) =1q4

{1 +

q2[cosh (q)− 1

]+ sinh (q)

[7q − 15 sinh (q)

]

4q sinh (q)[cosh (q) + 1

]}

. (38)

Equation (38) constitutes our final result for the contribu-tion from the nonequilibrium concentration fluctuations tothe structure factor in the small-angle approximation,q‖ ' q.The term inside the square bracket in Eq. (38) accountsfor the finite-size effects as can be seen by comparing itwith SNE(q) = 1/q4 deduced by applying the small-angleapproximation toEq. (18) for the “bulk” structure factor.

The normalized nonequilibrium concentration-fluctua-tions enhancementSNE(q), given by Eq. (38), is plotted inFig. 2 as a solid curve on a double-logarithmic scale for thecaseL = 0.1 cm, which is typical height of fluid layers inexperiments [38, 61, 73, 74]. A simple inspection of the solidcurve in Fig. 2 shows that for large values of the wave num-

ber q, the term inside the brackets in Eq. (38) approachesunity and we recover theq−4 dependence of the nonequi-librium enhancement in the absence of finite-size effects.However, such dependence onq−4 cannot go on indefinitelywith decreasing wave numbers and, as shown in Fig. 2, the fi-nite size of the system causes a crossover to aq2 dependenceof the nonequilibrium enhancement at very small wave num-bers. Forq → 0, the nonequilibrium enhancement becomes

SNE(q)q→0−−−→ 17

20160q2, (39)

Rev. Mex. Fıs. 48 S1(2002) 14 - 25


FIGURE 2. Double-logarithmic plot of the wave-number depen-dence of the normalized nonequilibrium concentration-fluctuationcontributionSNE. The solid line isSNE taking into account finite-size effects, but neglecting gravity, as given by Eq. (38), for a liquidlayer with a heightL = 1 mm. The dashed line isSNE taking intoaccount gravity effects, but neglecting gravity, as given by Eq. (41),for a “roll-off” scattering wave numberqRO = 400 cm−1. q× is thecrossover scattering wave number as defined by Eq. (40).

indeed compensating the divergentq−4 dependence whenfinite-size effects are neglected and causing an asymptotic de-pendence onq2, as shown in Fig. 2. From Eq. (39) we maydefine a characteristic wave number

q× =(

2016017

)1/6 1L' 3.25

L, (40)

which, forL = 0.1 cm, corresponds toq× ' 33 cm−1. FromFig. 2 we see thatSNE(q) exhibits a maximum atq ' q×,so thatq× may be interpreted as a “crossover” wave numberseparating theq−4 dependence and theq2 dependence of thenonequilibrium enhancement.

It is worth observing that the existence of a maximumat' q× (numerically we find the position of the maximumat about' 2.222/L = 0.68q×), indicates that, even in theabsence of gravity, the simple presence of a temperature gra-dient selects a particular length scale in the system for whichnonequilibrium fluctuations are maximally enhanced. Thislength scale becomes macroscopically evident above the con-vective instability, where spatiotemporal patterns with a par-ticular length scale appear.

As mentioned earlier, not only the finite size of the sys-tem, but also gravity will cause deviations from theq−4 de-pendence of the nonequilibrium enhancement. These effectswere first analyzed by Segre and Sengers [47], who predictedthat theq−4 divergence will saturate to a constant value in-dependent ofq for small wave numbers. This prediction hasbeen confirmed experimentally [52, 53]. The most recent pre-sentation of gravity effects can be found in Ref. [51], wherethe problem is studied starting from the linearized fluctuatingBoussinesq equations in the small Lewis number approxima-

tion. From Eqs. (14) and (15) in Ref. [51], we find that thestructure factor of a fluid subjected to a temperature gradient,accounting for the effects of gravity but neglecting finite-sizeeffects, can be rearranged in a form like Eq. (37) such that:

S(q) = SE[1 + S0NESNE(q)] = SE

{1 +

S0NE

q4 + q4RO

}. (41)

Due to the presence of gravity, there is a slight difference be-tween the dimensionless nonequilibrium enhancementS0

NE

in Eq. (41) andS0NE as defined by Eq. (19). But in practice,

this small difference can be neglected. From Eq. (41), we ob-serve that the gravitationally induced saturation of theq−4

divergence occurs at a “roll-off” wave numberqRO which isgiven by [51, 52]

qRO =(

gβ∇T0

νD

)1/4

, (42)

while qRO = qROL. To facilitate the comparison betweengravity and finite-size effects, we have added in Fig. 2, asa dashed curve, the functionSNE(q) as given by Eq. (41).For this plot in Fig. 2 we have employedqRO = 400 cm−1,which is the value calculated by Vailati and Giglio [52] fora critical mixture of aniline and cyclohexane subjected toa temperature gradient∇T0 = 163 Kcm−1. From Fig. 2we conclude that, depending on the size of the system andthe applied temperature gradient, finite-height effects may beequally important as deviations from theq−4 behavior due togravity. Therefore, for the interpretation of actual small-anglelight-scattering experiments both effects should be taken intoaccount simultaneously.

From the plots in Fig. 2 it can be also concluded thatquenching of theq−4 divergence as a result of finite-size ef-fects is even stronger than that caused by gravity, makingthe nonequilibrium contribution to the scattering function tovanish asq → 0. The conclusion that this crossover fromtheq−4 to theq2 dependence as a result of the finite size ofthe system occurs at wave numbers aroundq× ' π/L seemsintuitively plausible. The observation that the nonequilibriumconcentration fluctuations vanish asq → 0 is a consequenceof the imposed condition of the absence of concentration andvelocity fluctuations at the boundaries.

Small-angle nonequilibrium scattering experiments so farperformed in liquid mixtures have probed wave numbersdown toq = 100 cm−1 [52]. At such wave numbers devia-tions from theq−4 divergence are likely caused by a mixtureof gravity and finite-size effects. Shadowgraph experimentsperformed with binary liquids [73, 74] have probed wavenumbers of a similar magnitude. But for one-component li-quids close to the convective instability, shadowgraph expe-riments probing even smaller wave numbers have been re-ported [80] (down to q ' 2). In fluctuations near theconvective instability inclusion of both boundary conditionsand gravity is essential [81]. Although this topic is beyondthe scope of the present paper, we note that such experi-ments [80] do show a maximum enhancement of non-equili-

Rev. Mex. Fıs. 48 S1(2002) 14 - 25


brium structure factor at a nonzero value of the wave num-berq.

7. Concluding remarks

In this paper we have evaluated the structure factor in a hori-zontal layer of a binary liquid with finite heightL, subjectedto a vertical stationary concentration gradient induced by atemperature gradient through the Soret effect. Rayleigh scat-tering or shadowgraph experiments probe index-of-refractionfluctuations that, in the small-Lewis-number approximation,arise from concentration fluctuations. We have elucidatedhow the well knownq−4 dependence of the nonequilibriumenhancement of the structure factor in a nonequilibrium situ-ation is quenched by the finite height of the system yieldinga crossover to aq2 dependence at very small values of thewave numberq of the fluctuations. This means that the simple

presence of a temperature gradient selects a particular lengthscale in the system, for which fluctuations are maximally en-hanced. We find that for liquid mixtures the deviations fromthe q−4 behavior due to the finite size of the system couldbe just as important as deviations caused by the presence ofgravity. Therefore, for a quantitative interpretation of low-angle light scattering or shadowgraph experiments, it is im-portant to account for both gravity and finite-size effects si-multaneously.

Acknowledgements

The research at the University of Maryland is supported bythe Chemical Sciences, Geosciences and Biosciences Divi-sion of the Office of Basic Energy Sciences of the US De-partment of Energy under Grant No. DE-FG-02-95ER14509.

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