PHYSICAL REVIEW E DECEMBER 1999VOLUME 60, NUMBER 6

Two-dimensional model of phase segregation in liquid binary mixtures

Natalia Vladimirova,1 Andrea Malagoli,2 and Roberto Mauri1

1Department of Chemical Engineering, The City College of CUNY, New York, New York 100312Department of Astronomy and Astrophysics, University of Chicago, Chicago, Illinois 60637

~Received 6 August 1998; revised manuscript received 26 July 1999!

The hydrodynamic effects on the late stage kinetics of phase separation in liquid mixtures is studied usingthe modelH. Mass and momentum transport are coupled via a nonequilibrium body force, which is propor-tional to the Peclet numbera, i.e., the ratio between convective and diffusive molar fluxes. Numerical simu-lations based on this theoretical model show that phase separation in low viscosity, liquid binary mixtures ismostly driven by convection, thereby explaining the experimental findings that the process is fast, with thetypical size of single-phase domains increasing linearly with time. However, as soon as sharp interfaces form,the linear growth regime reaches an end, and the process appears to be driven by diffusion, although thecondition of local equilibrium is not reached. During this stage, the typical size of the nucleating dropsincreases liketn, where 1

3 ,n,12 , depending on the value of the Peclet number. As the Peclet number

increases, the transition between convection- and diffusion-driven regimes occurs at larger times, and thereforefor larger sizes of the nucleating drops.@S1063-651X~99!10611-1#

PACS number~s!: 64.70.Ja, 64.60.Cn, 64.60.Ht, 64.75.1g

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I. INTRODUCTION

The main objective of this work is to determine wheththe linear growth regime of phase separating liquid mixtucan continue indefinitely with time or if it reaches an enfinding the connection between the morphology of the stem and its growth rate. When a binary mixture is quenchfrom its single-phase region to a temperature belowcomposition-dependent spinodal curve, it phase separ@1#. This process, called spinodal decomposition, is chaterized by the spontaneous formation of single-phasemains which then proceed to grow and coalesce. Unnucleation, where an activation energy is required to initithe separation, spinodal decomposition involves the groof any fluctuations whose wavelength exceeds a critvalue. Experimentally, the typical domain size is describby a power-law time dependencetn, where n'1/3 whendiffusion is the dominant mechanism of material transpowhile n'1 when hydrodynamic, long-range interactions bcome important@2–4#.

Dimensionally, all possible scaling of the process wesummarized by Furukawa@5#, showing in particular thatwhen hydrodynamic forces are taken into account,growth laws R(t);(s/h)t and R(t);(s/r)1/3t2/3 can beobtained, depending on whether the surface tensions is bal-anced by viscous or inertial forces, withh and r denotingthe typical viscosity and density of the mixture, respectiveA growth exponentn;5/3 is also associated with inertia, bit seems less relevant, as it has never been observed inperiments or in numerical simulations.

In previous large scale molecular dynamics simulatio@6,5#, it was shown that the late time dynamics of spinoddecomposition reaches a viscous scaling regime withgrowth exponentn51, while n52/3 is obtained in the inertial regime@7,8#. Similar results were obtained using latticBoltzmann simulations@9#. Molecular dynamics and latticeBoltzmann simulations, however, can only model systemvery small sizes and/or for very short times, while in th

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work, we intend to study the behavior of phase separasystems at later times, that is when the typical size ofsingle-phase domains is ofO(1 mm). Therefore, we mususe the third numerical approach that is generally appliedstudy the phase separation of liquid mixtures, namely,merical integration of the coarse-grained conservation eqtion, generally referred to as modelH, in the taxonomy ofHalperin and Hohenberg@10#.

Model H is based on the Ginzburg-Landau theoryphase transition@11#, which was applied to model the phasseparation of mixtures by Cahn and Hilliard@12# and latergeneralized to include hydrodynamics by Kawasaki@13#. Inthis model, the equations of conservation of mass andmentum are coupled via the convective term of tconvection-diffusion equation, which is driven bycomposition-dependent body force. As noted by JasnowVinals @14#, when the system is composed of single-phadomains separated by sharp interfaces, this force incorates capillary effects, and plays the role of a Marangforce. Model H shows that during the early stages of tphase separation process~i.e., spinodal decomposition!, ini-tial instabilities grow exponentially, forming, at the ensingle-phase microdomains whose size corresponds tofastest-growing model0 of the linear regime@15#. Duringthe late stages of the process, i.e., for timest.t05l0

2/D,whereD is the molecular diffusivity, the system consistswell-defined patches in which the average concentrationnot too far from~albeit not equal to! its equilibrium value@16#. At this point, material transport can occur eitherdiffusion or by convection. In cases where diffusion is tonly transport mechanism, both analytical calculations@17#and dimensional analysis@18# predict a growth lawR(t);t1/3, due to the Brownian coagulation of droplets. On tother hand, when hydrodynamic interactions among dropbecome important, the effect of convective mass flow resing from surface tension effects can no longer be neglecIn this case, both dimensional analysis@18,19# and numericalsimulation@19–22# indicate that viscous forces determine

6968 © 1999 The American Physical Society

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PRE 60 6969TWO-DIMENSIONAL MODEL OF PHASE SEGREGATION . . .

growth law R(t);t. On the other hand, when inertiadominant, the growth rate is described in terms of an exnent n'2/3 @23–25#. Recently, Tanaka and Araki@26#showed that the scaling exponent for the domain growthnot universal, and depends on the relative importance ofdrodynamic flow and diffusion.

In this work we continue to explore the influence of covection upon phase segregation in fluid mixtures, applyan extension of modelH which leads to an easily integrabset of equations. Adopting a larger system than that of pvious investigators’@19–22#, we showed that the lineagrowth regime does not continuead infinitum in time: assoon as sharp interfaces form, the system reaches a mstable state, where diffusion is the driving force, characized by a much slower growth rate. In addition, the morphogy of the system appears to change drastically whenPeclet number increases from 102 to 104.

II. GOVERNING EQUATIONS

A. Binary mixture at equilibrium

Consider a homogeneous mixture of two speciesA andBwith molar fractionsxA5f and xB512f, respectively,kept at temperatureT and pressureP. For the sake of sim-plicity, in our model we assume that the molecular weighspecific volumes, and viscosities ofA are equal to those ofB,namely,MA5MB5MW , VA5VB5V, andhA5hB5h, re-spectively, so that molar, volumetric, and mass fractionsall equal to each other, and the mixture viscosity is comsition independent. The equilibrium state of this systemdescribed by the ‘‘coarse-grained’’ free energy functionthat is the molar Gibbs energy of mixing,Dgeq,

Dgeq5geq2~gAxA1gBxB!, ~1!

wheregeq is the energy of the mixture at equilibrium, whilgA andgB are the molar free energy of pure speciesA andB,respectively, at temperatureT and pressureP. The free en-ergyDgeq is the sum of an ideal, entropic part and an enthpic part,

Dgeq5RT@f ln f1~12f!ln~12f!#1RTCf~12f!,~2!

whereR is the gas constant, whileC is a function ofT andP. This expression, which is generally referred to as eitthe Flory-Huggins free-energy density@27# or the one-parameter Margules correlation@28#, is generally derived byconsidering the molecular interactions between neaneighbors @29#, or summing all pairwise interactionthroughout the whole system@30#. Equation~2! can also bederived from first principles, assuming that theA-A andB-Bintermolecular forces are equal to each other and largertheA-B intermolecular forces, i.e.,FAA5FBB.FAB , obtain-ing an expression forC which depends on (FAA-FAB) @15#.In the following, we shall assume thatP is fixed, so that thephysical state of the mixture at equilibrium depends onlyT andf.

In order to take into account the effects of spatial inhmogeneities, Cahn and Hilliard@12# introduced the generalized specific free energyg, which is given by the expressio

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g5geq112 RTa2~¹f!2, ~3!

wherea represents the typical length of spatial inhomogeities in the composition. As shown by van der Waals@31#,since the surface tensions is the energy stored in the unarea of the interface separating two phases at local equrium, we obtain

s51

2

rRT

MWa2E ~“f!2dl;

rRTa

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2 AC22, ~4!

where (Df)eq5(f12f2)eq is the concentration drop acrosthe interface, while we have considered that the width ofinterfacel equals the wavelength corresponding to the fasgrowing mode of the linear regime@15#, l;a/AC22. For atypical liquid mixture near its miscibility curve we obtaina;0.1 mm.

Below a certain critical temperatureTc , corresponding tovaluesC>2, the molar free energy given by Eq.~2! is adouble-well potential, and therefore a first-order phase trsition will take place. Now it is well known that the molafree energy can be written as@28#

geq/RT5mf, ~5!

where m5(mA2mB) denotes the difference between thchemical potential of speciesA and B in solution, respec-tively. This result can be extended@12# defining the general-ized chemical potentialm,

m5d~g/RT!

df5

]~g/RT!

]f2“•

]~g/RT!

]~“f!, ~6!

and substituting Eqs.~1!–~3! into Eq. ~6!, we obtain

m5mo1 lnf

12f1C~122f!2a2¹2f, ~7!

wheremo5(gB2gA)/RT.

B. Equations of motion

Imposing that the number of particles of each specieconserved, we obtain the continuity equations@32#

]cA

]t1“•~cAvA!50, ~8!

]cB

]t1“•~cBvB!50, ~9!

wherecA andcB are the concentrations, whilevA andvB arethe mean velocities of speciesA andB, respectively. For anincompressible mixture composed of species with eqphysical properties, Eqs.~8! and ~9! lead to the followingcontinuity equations in terms of the mass fractionf of theAspecies~which is equal to its mole fraction!:

]f

]t1v•“f52

1

r“• j , ~10!

“•v50, ~11!

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6970 PRE 60VLADIMIROVA, MALAGOLI, AND MAURI

where r is the mixture mass density,j5rf(12f)(vA2vB) is the diffusive mass flux, andv is the average velocityof the mixture,v5xAvA1xBvB . The velocitiesvA andvB arethe sums of a convective partv and a diffusive part,

vA5v2D“mA , vB5v2D“mB , ~12!

whereD is a composition-independent diffusion coefficienand we have assumed that the diffusive parts ofvA and vBare proportional to the gradients of the chemical potent~see Refs.@33,34# for a justification of this assumption!.Consequently, the diffusive flux becomes

j52rf~12f!D“m. ~13!

Finally, substituting~7! into ~13!, we obtain

j

r52D“f1Df~12f!@a2

““

2f12C“f

1~2f21!“C#. ~14!

This expression forj coincides with that used in Ref.@15#.The termD“f in Eq. ~14! represents the regular diffusioflux, while the last term vanishes for small concentrationseither solvents (f→0 or f→1) and for ideal mixtures (a5C50). Note that thea2 term is always stabilizing, and irelevant only at small length scales, whileC is a knownfunction of the temperature, and near the critical temperaTc it is proportional toTc2T.

If the flow is slow enough that the dynamic terms in tNavier-Stokes equation can be neglected, conservatiomomentum leads to the Stokes equation

h“2v2“p52Ff , ~15!

whereh is the mixture viscosity, which, we assume, is coposition independent, whileFf is a body force, which equalthe gradient of the free energy@10#, and therefore is drivenby the concentration gradients within the mixture@25,14,5#

Ff5r

MW“g5S rRT

MWD m“f. ~16!

The assumption of small Reynolds number is supportedthe experimental observation@35# that during phase separation of liquid mixtures with waterlike viscosity 10mm dropsmove at speeds exceeding 200mm/s. At equilibrium, wherethe free energy is uniform, the body forceFf is zero, andtherefore there cannot be any convection. In addition, wthe mixture is composed of well-defined single-phasemains separated by a thin interface located atr5r s , the bodyforce Ff can be interpreted as a capillary force atr s , i.e.@14#,

Ff~r !5@ nsk1~ I2nn!•“s#d„n•~r2r s!…, ~17!

wheres is the surface tension, whilen and k are the unitvector perpendicular to the interface and the curvature atr s ,respectively. Physically,Ff tends to minimize the energstored at the interface, and therefore it drives, say,A-richdrops towardA-rich regions, enhancing coalescence. Nthat Eq.~15! can also be written as

,

ls

f

re

of

-

y

n-

e

h“2v2“p85S rRT

MWDf“m, ~18!

with p85p2(rRT/MW)mf. Equations~10!, ~11!, and~18!constitute the so-called modelH @10#.

Now we restrict our analysis to two-dimensional systemso that the velocityv can be expressed in terms of a streafunction c, i.e., v15]c/]r 2 and v252]c/]r 1. Conse-quently, substituting Eq.~16! into Eq. ~15!, we obtain

]f

]t5“c3“f1

1

r“• j , ~19!

h¹4c5“m3“f, ~20!

where

A3B5A1B22A2B1 .

Since the main mechanism of mass transport at the bening of phase segregation is diffusion, the length scale ofprocess is the microscopic lengtha. Therefore, using thescaling

r 51

ar , t 5

D

a2t, c5

1

aDc, ~21!

and substituting Eq.~14! into Eq. ~19!, we obtain

]f

] t5a“c3“f1“•~“f2f~12f!@2C1“

2#“f!,

~22!

“

4c52“~“2f!3“f, ~23!

where

a5a2

D

r

h

RT

MW. ~24!

The nondimensional numbera is the ratio between thermaand viscous forces, and can be interpreted as the Peclet nber, that is the ratio between convective and diffusive mfluxes in the convection-diffusion equation~22!, i.e., a5Va/D. Here V is a characteristic velocity, which can bestimated through Eqs.~15! and ~16! as V;Ffa2/h5aD/a, where Ff;rRT/(aMW). A similar, so called‘‘fluidity’’ parameter was also defined by Tanaka and Ara@26#. For systems with very large viscosity,a is small, sothat the model describes the diffusion-driven separation pcess of polymer melts and alloys@15#. For most liquids,however,a is very large, with typical values ranging from103 to 105. Therefore, it appears that diffusion is importaonly at the very beginning of the separation process, in thcreates a nonuniform concentration field. Then tconcentration-gradient-dependent capillary force inducesconvective material flux which is the dominant mechanifor mass transport. At no time, however, can the diffusterm in Eq. ~15! be neglected, as it stabilizes the interfaand saturates the initial exponential growth. In additionshould be stressed that the stream functionc depends on

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PRE 60 6971TWO-DIMENSIONAL MODEL OF PHASE SEGREGATION . . .

high order derivatives of the concentration and thereforevery sensitive to the concentration profile within the inteface.

III. NUMERICAL RESULTS

The governing equations~22! and ~23! were solved on auniform two-dimensional square grid with constant wid@(xi ,yj )5( iDx, j Dy),i 51,N, j 51,N#, where N5500, andtime discretization@ t5nDt,n50,1,2, . . . #. The physical di-mensions of the grid were chosen such thatDx/a,Dy/a52, while the time step Dt satisfies Dt/(a2/D)'0.1–0.001. The choice of the time stepDt was determinedsemiempirically in order to maintain the stability of the nmerical scheme. Note that the nonlinearity of the equatiprevents a rigorous derivation of the stability constraintsDt, but one can roughly estimate that the size ofDt willscale asO(Dx4,Dy4), which is the order of the highest operator in the discretized system. The space discretizationbased on a cell-centered approximation of both the conctration variablef i j

n (t) and of the stream functionc. Thespatial derivatives in the right-hand side of Eqs.~22! and~23! were discretized using a straightforward second-oraccurate approximation. The time integration fromtn5nDtto tn115(n11)Dt was achieved in two steps. First, wcomputed the stream functionc by solving the biharmonicequation ~23! with the source term evaluated at timetn

5nDt. The biharmonic equations was solved usingDBIHAR routine from netlib@36#. Second, Eq.~22! was ad-vanced in time, using the velocity field computed from tupdated stream function and a straightforward explicit Eurian step. This makes the entire schemeO(Dt) accurate intime, which is acceptable for our problem, since the sizethe time step is kept very small anyway by the stability costraints. The boundary conditions were no flux for the cocentration field and no slip for the velocity field. The dicretization of the derivatives near the boundaries wmodified to use only interior points. In general, our resuare not very sensitive to the precise treatment of boundconditions, since the gradients remain close to zero nearboundaries.

Since we did not incorporate thermal noise into our simlation, we introduced some amount of randomness intosystem through a background noise in the concentrafield, df, with ^df&50 and ^(df)2&1/250.01, which wasuncorrelated both in space and in time. That means thaeach time step a spatially uncorrelated noise was added tconcentration field, and was then subtracted at the nextstep, only to be replaced with another spatially uncorrelabackground noise of the same amplitude@16#. This proce-dure which is equivalent to adding noise to the fluxj on theright hand side of Eq.~10!, is fully conservative in the sensthat the volume integral of the compositionf is not alteredby addition of thead hocnoise~numerically we have veri-fied that the composition is indeed conserved to machaccuracy!. In a separate set of numerical simulations, thermfluctuations were included in the initial conditions only, asFurukawa@5#, obtaining identical results. In other simulations, the physical thermal noise was used, satisfyingfluctuation-dissipation theorem@25#, again obtaining identi-cal results. In fact, since for deep quenches the Ginzb

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inequality is satisfied@11#, the background noise does naffect the behavior of the system, and only determinesinstant of time when the system departs from its initial uform state~see Fig. 7 in Vladimirovaet al. @16#!: once thelinear regime is reached, the presence of the noise becoirrelevant.

Both critical and off-critical quenches were considerewith uniform initial concentration fieldsf050.5 and 0.45,respectively. The boundary conditions were no flux for tconcentration field and no slip for the velocity field. In moof our simulations we usedC52.1, because this is the Margules parameter of the water-acetonitrile-toluene mixtwith 20 °C temperature quench that we used in our expmental study@35#. However, simulations with different values ofC were also performed, obtaining very similar resulNote that, forC52.1, at equilibrium the two phases havcompositionsfeq

A 50.685 andfeqB 50.315. Time was mea

sured ast5(105a2/D)t, wheret is a nondimensional timeSince typical values ofD and a are 1025 cm2/s and1025 cm, respectively, thent;t/(1s).

First, we solved Eqs.~22! and ~23! for a system withcritical uniform initial mole fractionf050.5 and for differ-ent values of the Peclet numbera. The first row of images inFig. 1 represents the results fora50, e.g., for the case whediffusion is the only mechanism of mass transfer, showthat, soon after the first drops appear, they coalesce intodroidlike structures. The mean composition within~andwithout! these structures changes rapidly, as at timet50.05 we already see two clearly distinguishable phawith almost uniform concentrations equal to 0.59 and 0.After this early stage, the structures start to grow, increastheir thickness and reducing the total interface area, whilthe same time the composition within the domains aproaches its equilibrium value. This, however, is a slow pcess, driven only by diffusion, and at timet50.1 the phasedomains still have a dendroidlike geometry with a characistic width which is just twice as large as its initial value.the following, we will denote these slow-changing configrations as metastable states, referring to Refs.@16,37# forfurther informations on their evolution.

For nonzero convection, i.e., foraÞ0, dendritic struc-tures thicken faster, but up toa'102 domain growth stillfollows the same pattern as fora50: first, single-phase domains start to appear, separated from each other by sinterfaces, and only later these structures start to grow, wincreasing growth rate for largera. Whena.102, however,phase separation occurs simultaneously with the growthcess. For example, whena5104, we see the formation oisolated drops of both phases, surrounded by the bulk offluid mixture, which is still not separated. In addition, droappear to move fast and randomly while they grow, absoing material from the bulk, colliding with each other ancoalescing, so that single-phase domains grow much fathan when molecular diffusion is the only transport mechnism. Consider, for example, the morphology of the systat time t50.1 for a5104 and 0: in the first case, singlephase domains have already reached a size comparabthat of the container’s, while, in the absence of convectithe dendroid domains have an approximate width of 20a.Clearly, since the motion of the interface is too quick for tconcentration diffusion to establish a metastable state wi

td tontra-

6972 PRE 60VLADIMIROVA, MALAGOLI, AND MAURI

FIG. 1. Composition of a binary mixture at different timest after an instantaneous quenching withC52.1 andf050.5, when the Peclenumbera is 0, 102, 103, and 104. The size of the system is 400a3400a, with no flux boundary conditions. The snapshots correspont50.04, 0.05, and 0.10, expressed in 105a2/D units. The gray level varies linearly between black and white, corresponding to concetions f5feq

A andf5feqB , respectively.

n

i

n--nlyig.

the microdomains, double, or multiple, phase separatioobserved, in agreement with previous numerical@26,14# andexperimental@35,38# results.

Although the dynamics of phase separation in fluids

is

s

mostly driven by convection, for very short times the covective driving forceFf is negligible, as composition gradients did not develop yet, and therefore diffusion is the omechanism of mass transport. In fact, the two pictures in F

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PRE 60 6973TWO-DIMENSIONAL MODEL OF PHASE SEGREGATION . . .

2 show that at timet50.02 the concentration fields fora50 and 104 are almost indistinguishable from each othwith patterns having a characteristic periodl equal to thefastest growing mode in the linear regime for a diffusiodriven process@15#, i.e.,

l52pa

AC22. ~25!

A similar behavior was observed for an off-critical phaseparating mixture withf050.45 ~see Fig. 3!. As in thecritical case, the system tends to form larger single-phdomains as the convection coefficientsa increases. Againwhile for smallera the processes of separation and growoccur successively in time, for largera they occur simulta-neously. However, while for critical mixtures the separatiphases tend to form interconnected domains, for off-critimixtures we observe the formation of isolated, mostly circlar drops, with no detectable double phase separation.largera is, the shorter the relaxation time of a drop aftecollision, so that fora.102 we practically do not observeany noncircular drops.

As a quantitative characterization of the influence ofconvection parametera on the average phase compositiwithin the phase domains, we define the separation deps,measuring the ‘‘distance’’ of the single-phase domains frtheir equilibrium state, i.e.,

s5 K f~r !2f0

feq~r !2f0L , with 0<s<1, ~26!

wheref0 is the initial composition, and the bracket indicatvolume and ensemble average. Herefeq is the steady statecomposition of theA-rich phase,feq

A , or the B-rich phase,feq

B , depending on the local compositionf(r ),

feq~r !5feqA , f~r !.f0 , ~27!

feq~r !5feqB , f~r !,f0 , ~28!

so thats51 indicates that the system is a condition of locequilibrium. In Figs. 4 and 5 the separation depths is plottedas a function of time for both critical and off-critical mix

FIG. 2. Concentration field after an instantaneous quenchwith C52.1 andf050.5 at timet50.023105a2/D, when the Pe-clet numbera is 0 and 104. The size of the system is 400a3400a, with no flux boundary conditions. Black pixels correspoto concentrationsf,f0, and white ones tof.f0.

,

-

se

l-he

e

l

tures. The data points represented in these figures weretained using a domain sizeL51000a; however, identicalresults were obtained withL5400a, indicating that they donot depend on the size of the domain nor on the averagprocedure.

As we see in Fig. 4, after a critical quench no detectaphase separation takes place untilt;0.02, when the firstspinodal decomposition pattern is formed. Then phase sration can take place in two different ways, dependingwhethera,102 or a.103. For smallera ’s, single-phasedomains develop very rapidly, until, att;0.06, they appearto be separated by sharp interfaces. From that point on, sration proceeds much more slowly, as the concentrationdients within the single-phase domains are very small, whthe concentrations of the two phases across any interchange only slowly in time~see the discussion in Ref.@16#!.Although Tanaka@38# denoted these states as ones of loequilibrium, here we prefer to use the term ‘‘metastabstates,’’ considering that at stable equilibrium we shouhaves51, while here we haves<0.8. In the case of largea, with a.103, the growth of the separation depth is mogradual, revealing that separation and growth occur simuneously and with no detectable metastable states, althoobviously, at some later stage, sharp interfaces will evenally appear even in this case. Therefore, we may conclthat ~a! the largera is, the longer it takes for sharp interfaceto form, and~b! local equilibrium ~with s51) is probablynever achieved for low-viscosity liquid mixtures. In thcase, in fact,a;1025 cm, and consequently our domasize corresponds to 100mm, while drops start sedimentinwhen they reach 1-mm sizes, so that the system will becogravity driven and rapidly separate before reaching the sing regime, withs51.

For off-critical mixtures, as shown in Fig. 5, the onsetphase separation occurs at later times than in the critcase. In particular, the closerf0 is to the spinodal concentrationfs ~in our case, withC52.1, fs50.388), the longerit takes for the onset of phase separation. In addition,off-critical mixtures the processes of separation and grotend to occur successively in time, even at high values ofPeclet number. For example, comparing Figs. 4 and 5,see that, fora5103, the two processes occur simultaneousin the critical case, and sequentially in the off-critical onThat means that off-critical mixtures are more likely to reaa metastable state, after which single-phase domains gmuch more slowly.

The behavior of a phase-separating system dependmuch on the driving forceFf as on the Peclet numbera.Consider, for example, the behavior of two systems wPeclet numbersa50 and 103. In Fig. 4 we see that, at timet50.08 and with the sames50.6, the system witha50 isin a metastable state, while that witha5103 is still in thedomain forming, separating stage. In fact, although the cillary driving forceFf is the same in the two cases~as it is afunction of the separation depths), it can induce a strongconvection only for systems with small viscosities~i.e., largea ’s!, while for very viscous systems it has hardly any effe

Finally, the equivalent average radius of the drops,R, isplotted in Fig. 6 as a function of time, with

R5A^A&/p, ~29!

g

ondg to

6974 PRE 60VLADIMIROVA, MALAGOLI, AND MAURI

FIG. 3. Composition of a binary mixture at different timest after an instantaneous quenching withC52.1 andf050.45, when thePeclet numbera is 0, 102, 103, and 104. The size of the system is 400a3400a, with no flux boundary conditions. The snapshots correspto timest50.04, 0.05, and 0.10, expressed in 105a2/D units. The gray level varies linearly between black and white, correspondinconcentrationsf5feq

A andf5feqB , respectively.

hrath

rpce

onaben

toto

n

x-

si

ta

n--

-

sti-

,urlid

es.

ingisical-dyto

rpec-rt,

seed,

ysi-s

t

-

PRE 60 6975TWO-DIMENSIONAL MODEL OF PHASE SEGREGATION . . .

where A is the area of a single-phase domain, while tbracket indicates, as before, volume and ensemble aveThe error bars have smaller widths at short times, whereaverages were performed over;500 drops, than at longetimes, when drops were larger and the averages wereformed over;50 drops. Again, our results are robust, sinidentical data points were obtained using 1000a and 400asize domains. For a given value ofa,104, the equivalentradius grows linearly with time, until it reaches a saturativalue, corresponding to the above-mentioned metaststate, after which it grows more slowly. In particular, wha<102, we saw that metastable states grow liket1/3, whilefor larger a ’s they grow more rapidly astn, with 1

3 ,n, 12

~we do not have enough data to be more specific!. On theother hand, whena.104, the equivalent radius appearsgrow linearly until it attains a value which is comparablethe size of the system@39#. The linear growth follows thecurveR;103at51022Dt/a, and appears to be independeof a. Note that forD;1025 cm2/s anda;1025 cm, weobtaindR/dt;100 m/s, in excellent agreement with the eperimental results@35#.

The growth rate of single-phase domains can be eaestimated using our theoretical model, asdR/dt5u j u/r,wherej is the mass flux at the interface. Far from the me

FIG. 4. Separation depths as a function of timet for C52.1and f050.5, and with different values of the Peclet numbera.Results were obtained using 1000a31000a simulations.

FIG. 5. Separation depths as a function of timet for C52.1and f050.45, and with different values of the Peclet numbera.Results were obtained using 1000a31000a simulations.

ege.e

er-

le

t

ly

-

stable state, the dominant term ofj is the antidiffusive termin Eq. ~14!, that is u j u;r(Df)@2f(12f)C21#(a/l )(D/a), whereDf is the concentration drop across the iterface, whilel;a/AC22 denotes the characteristic thickness of the interface@15#. Therefore, we obtain

dR/dt;b~D/a!, ~30!

with b;(C22)2, where we have considered that (Df)eq

;AC22. Equation~30! is in agreement with both experiments and numerical simulations, whereC52.1.

The above dimensional analysis can be rewritten subtuting Eqs.~24! and ~4! into Eq. ~30!, obtaining

dR

dt5kb

s

h, ~31!

wherekb5AC22/a. Equation~31! was obtained by Siggia@18# and San Miguelet al. @40#, although their predictionskb50.6 and kb50.25, respectively, far overestimate ogrowth rate results, due to the fact that their analysis is vafor shallow quenches, while ours assumes deep quench

IV. CONCLUSIONS AND DISCUSSION

In this work we simulated the phase separation occurrwhen an initially homogeneous liquid binary mixturedeeply quenched into its two-phase region. Our theoretscheme followed the standard modelH, where mass and momentum transport are coupled via a nonequilibrium boforce, expressing the tendency of the demixing systemminimize its free energy. This driving force, which for shainterfaces reduces to capillary interaction, induces a convtive material flux much larger than its diffusive counterpaas in a typical case the Peclet numbera is of order 105.However, as sharp interfaces form delimiting single-phadomains, a condition of metastable equilibrium is reachthe nonequilibrium driving force~almost! vanishes, and theprocess becomes diffusion driven.

The set of equations that we used depend on four phcal, measurable quantities: the typical interfacial thicknesa~proportional to the surface tension!, the diffusion coefficientD, the kinematic viscosityh/r, and the Margules coefficien

FIG. 6. Equivalent average radiusR as a function of timet forC52.1 andf050.45, and with different values of the Peclet number a. Results were obtained using 1000a31000a simulations.

d

r-s

efnn

r-suen-o

a

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nai

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n

src

tion

te,andhatbuthusbyn

ser,rrowmi-s byes

o

rtS-

6976 PRE 60VLADIMIROVA, MALAGOLI, AND MAURI

of the mixture,C. After rescaling velocity, time scales, anlength scales, we obtain a set of two equations@cf. Eqs.~22!and~23!#, that can be solved numerically using finite diffeence techniques. These equations are expressed in termtwo independent parameters, namely, the Margules cocient and the Peclet numbera, expressing the ratio betweeconvective and diffusive molar fluxes. Our main conclusiocan be summarized as follows.

~1! For critical quenching, the formation of sharp intefaces and the growth of single-phase domains are twocessive stages of the phase segregation process wha,102, while for a.103 they occur simultaneously. For offcritical quenching, the transition between the two stagescurs at largera ’s.

~2! Before the formation of sharp interfaces, the equivlent average radiusR grows linearly, withdR/dt;0.01D/afor C52.1 and for alla ’s, in agreement with both experiments and a first-order dimensional analysis. This lingrowth regime ends as sharp interfaces form and the sysreaches a metastable state, where diffusion is the domimechanism of mass transport. As the Peclet numbercreases, the transition from a convection-driven todiffusion-driven process occurs at larger times and larsizes of the nucleating domains.

~3! After the formation of sharp interfaces, the equivaleaverage radiusR grows in time liket1/3 whena,102, whileit grows somewhat faster~albeit still slower thant1/2) forlarger a ’s. The condition of local equilibrium, however, inever reached, showing that the nonequilibrium body fo

.

m

,

R

ett

ev

tt

n,

offi-

s

c-

c-

-

rmnt

n-ar

t

e

does not vanish even at such late stage of the separaprocess.

Compared to previous numerical integration of modelHequations@19–22#, our main contribution is to point out thathe linear growth regime cannot continue indefinitely in timbut it reaches an end as sharp interfaces are formeddiffusion becomes the dominant transport mechanism. Tshows that the asymptotic scaling regime is not linear,instead corresponds to the later diffusion-driven stage, tresolving the apparent contradiction recently pointed outGrant and Elder@41#, who showed that if there exists aasymptotic scaling regime withR;tn, then the growth ex-ponentn must be<1/2, in order to prevent the Reynoldnumber from diverging at long times. Practically, howevfor large values ofa the diffusion-driven regime might nevebe reached, as the nucleating drops would continue to glinearly until they become large enough that buoyancy donates surface tension effects, and the mixture separategravity. This occurs when the size of the domains becomequal to the capillary length,Rmax5O(s/gDr), wheres isthe surface tension,g the gravity field, andDr the densitydifference between the two separating phases@18#, which,for a typical low-viscosity liquid mixture, correspond tRmax5O(1 mm) @35#.

ACKNOWLEDGMENTS

During this work, N.V. and R.M. were supported in paby the National Science Foundation, Grant No. CT9634324.

ze-

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er

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