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VOLUME 86, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 5 MARCH 2001

2014Phase Separation in a Chaotic Flow

Ludovic Berthier,1,2 Jean-Louis Barrat,1 and Jorge Kurchan31Dpartement de Physique des Matriaux, Universit C. Bernard and CNRS, F-69622 Villeurbanne, France

2Laboratoire de Physique, ENS-Lyon and CNRS, F-69007 Lyon, France3PMMH, cole Suprieure de Physique et Chimie Industrielles, F-75005 Paris, France

(Received 17 October 2000)

The phase separation between two immiscible liquids advected by a bidimensional velocity field is in-vestigated numerically by solving the corresponding Cahn-Hilliard equation. We study how the spinodaldecomposition process depends on the presenceor absenceof Lagrangian chaos. A fully chaoticflow, in particular, limits the growth of domains, and for unequal volume fractions of the liquids, a char-acteristic exponential distribution of droplet sizes is obtained. The limiting domain size results from abalance between chaotic mixing and spinodal decomposition, measured in terms of Lyapunov exponentand diffusivity constant, respectively.

DOI: 10.1103/PhysRevLett.86.2014 PACS numbers: 47.52.+j, 05.70.Ln, 47.55.Kf, 64.75.+gA system of two immiscible fluids at rest will graduallyphase separate, forming domains whose size grows alge-braically with time. Everyday experience, however, showsthat by continuously stirring or shaking the fluids the do-mains or droplets of the phases (say, oil and vinegar) breakand coalesce, leading to a dynamic stationary state withdomains of finite size.

A first approach consists of modeling this situation bysubjecting the binary fluid to a homogeneous shear veloc-ity field [1]. However, experiments [2], numerical simula-tions [3], and more recently analytical approaches [4] showthat in such a situation infinitely long domains alignedwith the flow are formed. The effect of the velocity fieldis to counter the Rayleigh instability, stabilizing lamellarand (in certain cases) even cylindrical domains [5,6]. Do-main breakup in those situations takes place only at largeReynolds numbers, and is generally attributed to inertialeffects [1,7]. Studying these inertial effects numerically isdifficult, as a realistic description of the feedback of do-main shape on the flow is required [7]. The correspondingsimulations are therefore limited by finite size effects.

In this Letter, we investigate a different mechanism bywhich domains of finite size can be stabilized in a demix-ing system. In particular, we show that a saturation of theaverage length scale takes place even in the absence of in-ertial effects if the flow has Lagrangian chaos (i.e., if thetrajectories of nearby starting points diverge exponentially0031-90070186(10)2014(4)$15.00with time). This is interesting for two reasons: First, withan appropriate time dependence of the velocity field onecan still have Lagrangian chaos in a situation of high vis-cosity in which inertial effects are negligiblethis is howone mixes pastes. Second, it is possible in that case todecouple the hydrodynamic problem from the phase sepa-ration. This problem of a passive, phase separating scalarfield is, of course, much simpler, so that simulations usinglarge systems are possible. Our approach therefore extendsearlier extensive studies of passive scalar advection by pe-riodically driven chaotic flows [8].

Our study is also related in spirit to earlier studies of ad-vection by synthetic velocity fields tuned to model tur-bulent flows [9,10]. Phase separation was studied in thiscontext in Ref. [11]. An essential difference, comparedto our work, is that in such turbulent flows the separationbetween nearby tracer particles appears to increase alge-braically, rather than exponentially, with time.

We consider a two dimensional flow that can be tuned tobe regular, mixed, or fully chaotic. Specifically, the incom-pressible velocity field yx, y, t is a modified version ofthe so-called time-dependent Harper map [12] (related tothe partitioned-pipe mixer, a special case of eggbeaterflow [8]). The dynamics takes place on a square of sideL, with periodic boundary conditions. The velocity fieldis an alternating sequence of shears in the x and in the ydirection with a time period T ,yx 22paLT

sin

2pyL

; yy 0; n ,

tT

, n 112,

yy 2paLT

sin

2pxL

; yx 0; n 1

12

,tT

, n 1 1 . (1)The parameter a controls the chaoticity of the trajecto-ries. If a is small, the two semicycles are composed intothe smooth, laminar velocity field: yx 2

paLT sin

2pyL ;

yy paLT sin

2pxL . For larger values of a the trajecto-

ries stretch and fold, and the flow becomes chaotic. Inorder to visualize this, it is convenient to follow the po-sition of a point at the end of each cycle. This kickedHarper map is shown in Fig. 1 for several values of a.For a 0.2 the flow is a mixture of laminar and chaoticregions, and becomes fully chaotic around a 0.4. In 2001 The American Physical Society

VOLUME 86, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 5 MARCH 2001FIG. 1. The maps obtained from snapshots at intervals T ofthe lines of current with the dynamics (1), starting from variousinitial conditions. Figures for a 0.1 (top left), 0.25, 0.4, and1.0 (bottom right).

the chaotic situation, it is convenient to characterize theflow by the Lyapunov exponent l, defined by the factthat nearby starting points separate as elt . We havecomputed l as in Ref. [10] and found that the relationl 1.96 ln3.35aT is a good approximation through-out the chaotic regime, a * 0.4.

The spinodal decomposition of the two-component fluidis described by the Cahn-Hilliard equation

fr, tt

1 yr, t ? =fr, t G=2dFfdfr, t

. (2)

Here f is a dimensionless concentration field, the concen-trations of the species are 1 6 f2. We work at T 0,since temperature is irrelevant in this process [13]. Thefree energy functional is of the Ginzburg-Landau form andreads

Ff Z

ddxj2

2=f2 2

12f2 1

14f4

. (3)

Here j is the equilibrium correlation length controlling thewidth of the interfaces, and d is the number of spatial di-mensions. We consider the following two topologicallydifferent situations: (i) f fi 0: a species is less abun-dant than the other and forms disconnected droplets, and(ii) f 0: the two phases are in equal quantity and forma bicontinuous structure. Situation (i) has been studied ex-perimentally [14].

In a chaotic flow, the passive scalar mixes rapidly,whereas in the case of phase separation this tendency isopposed by surface tension. The competition betweenthese two effects can be quantified through two adi-mensional parameters, D GTj2 (the adimensionaltransport coefficient of the Cahn-Hilliard equation) andthe chaoticity parameter a, or alternatively the adimen-sional Lyapounov exponent lT . A large D means thatappreciable diffusive transport will take place duringeach laminar half cycle. A large lT , on the other hand,means that the mixing process is efficient within a fewcycles. Note that l can also be interpreted as an averageelongation or shear rate experienced by the fluid particles.

Equation (2) is integrated numerically with the veloc-ity field (1), using the implicit spectral method developedand discussed in Ref. [15]. The results are presented withtime and length units chosen as the cycle period T and theinterfacial thickness j, respectively. The system size, lat-tice parameter, and time step are L 512j, Dx j, andDt 5 3 1024 T, respectively.

Existence of a stationary state.We first show that apurely chaotic flow does, indeed, stop the domain growth.In Fig. 2, we show the evolution of a phase-separatedsample with f 12, upon turning on a chaotic ve-locity field (a 0.4). The large droplets of the initialconfiguration are broken into smaller droplets, until a sta-tionary state where droplets successively grow and breakis reached. Figure 3 shows the late stages of coarseningof a system with equal concentrations of phases (f 0)in the four velocity fields of Fig. 1. For a 0.1, the ve-locity field is laminar. We observe in that case structuresvery similar to those found in a homogeneous shear flow,but which now follow the winding flow lines. In the mixedcase, a 0.25, large-domain structures form in the lami-nar regions of the flow, and break into very small domainsin the chaotic ones. In the fully chaotic situation, a * 0.4,a dynamical stationary state is reached, with small do-mains continuously breaking and reforming. For a 1.0, the sinusoidal nature of the underlying velocity field

FIG. 2. Evolution of an assembly of large droplets in thechaotic flow for times t 0, 0.65, 1.50, and 11.8.2015

VOLUME 86, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 5 MARCH 2001FIG. 3. Late stages of coarsening in the velocity fields ofFig. 1, in the same order.

becomes apparent. This snapshot nicely illustrates the typi-cal stretch and fold processes characteristic of chaoticadvection [8].

Scaling properties in the chaotic flow. In the station-ary regime, it is clear from Figs. 2 and 3 that there existsa typical length scale L which depends on the parametersD and l: this will be confirmed below by a quantitativeanalysis. As in the pure coarsening case [13], scaling prop-erties are expected in the regime j LD,l L. Thelength scale L may be estimated by the following simpleargument. In the absence of flow, the domains grow asLtj DtT 13, and this growth is stopped by thechaotic flow which introduces a time scale l21. Hence,we estimate L Lt l21 and predict

LD,l jDlT

13. (4)

Isolated droplets, f fi 0.Following Ref. [14], wecharacterize the assembly of droplets by computing thedistribution of droplet surfaces fS, where fS dS is theprobability that the surface occupied by a droplet is be-tween S and S 1 dS. In the scaling regime, we expectthis distribution to be of the form

fS 1S

FSS

, (5)

where S is a typical droplet area. In Fig. 4, data obtainedfor a wide range for the values of D and l are collapsedby using a reduced variable SS, with S Dl0.62.The data collapse is satisfactory, and the result for S

reasonably close to what would be expected from Eq. (4),i.e., S Dl23. Finding an exponent slightly smallerthan the one expected theoretically is not surprising, sincethe typical domain sizes are rather small (S & 50j2), sothat the asymptotic value for the domain growth exponent2016FIG. 4. Surface distribution of the droplets rescaled accord-ing to Eq. (5), where the typical surface S is given by S Dl0.62. The dot-dashed line is a fit to an exponential form,and the data are for a range a [ 0.4, 3.0 and D [ 100, 2000.

in the absence of flow may not be reached. The rescaleddistribution functions exhibit an exponential tail, F y e2y (dot-dashed line in Fig. 4). Such distributions are verysimilar to those found in the experiments of Ref. [14].For the largest droplets, deviations from the exponentialfit are observed, indicating either insufficient statistics or adifferent scaling behavior for the extreme values of S.

Equal concentrations: f 0. In the case of equalconcentrations, the domains are ramified and extendthroughout the sample, so that the area is not a usefulmeasure of domain size. A characteristic domain size cannevertheless be obtained from the two-point correlationfunction Cr, t L22

Rd2xfx, tfx 1 r, t, which

is the Fourier transform of the structure factor measuredin light scattering experiments. Performing a time averageover many configurations shows that the bicontinuousstructure is on average perfectly isotropic, as it is inthe absence of flow. One can therefore average Cr, tover orientations to obtain a one variable function, Cr.The characteristic domain size L can be defined byCL 0.5. Figure 5 displays this domain size forvarious combinations of D and l, as a function of theratio Dl. The data can be fitted by L jDlT 0.27.

FIG. 5. Log-log plot of the typical domain size L as a func-tion of the ratio Dl for the same values for D and a as inFig. 4. The full line has a slope 0.27.

VOLUME 86, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 5 MARCH 2001FIG. 6. Two-point correlation function Cr at fixed a for vari-ous mobility D, as a function of the rescaled variable rD0.27.Main: a 1.0 and D 100, 200, 400, 1000, and 2000. Inset:a 0.4 and D 100, 200, and 400.

Again, this is in reasonable agreement with the scalinganalysis, Eq. (4). Larger simulations, with smaller valuesof l, would be necessary to obtain larger domain sizesand avoid the crossover effects which are well known inspinodal decomposition simulations [13].

More detailed information on the domain structure isobtained from the full correlation function Cr. Here onecould expect from the scaling hypothesis a behavior of theform

Cr CrL

(6)

with C a universal function. This hypothesis is tested inFig. 6, where Cr is represented for fixed l and variousvalues of D. At fixed l, a good collapse of the dataobtained for different D is achieved by using a rescaledvariable rD0.27. The inset of Fig. 6, however, shows thatthe shape of the scaling function slightly depends on l,so that the universal scaling expressed by Eq. (6) is notvalid. We attribute this change of the scaling function withthe flow pattern to the fact that even in the chaotic regimethe flow cannot be considered as being homogeneous andisotropic, but exhibits an underlying sinusoidal structure.This is in contrast with the case of isolated droplets wherethe droplet distribution was not affected by this structure;recall Fig. 4.

We have studied the phase separation in conditions inwhich the species boundaries are passively advected by anincompressible flow. We have shown that a chaotic flowresults in a steady state with domains of finite size result-ing from the balance between spinodal decomposition andchaotic advection, Eq. (4). This should be contrasted withthe situation observed in turbulent flow, where the flowintensity must exceed a threshold in order to stop domaingrowth [11]. Such a difference can be traced back to thefact that Lyapunov exponents for passive scalar advectionare actually 0 in the latter case. The essential approxi-mation in our work, compared to realistic experimentalsituations, is the assumption that the flow pattern isnot modified by the domain growth. This assumption,however, may not be unrealistic if the two fluids havesimilar viscosities and if the capillary stresses are smallcompared to viscous stresses. This is measured by thecapillary number Ca hlgL, where h is theviscosity and g the surface tension. In highly viscousfluids, Ca is expected to be large, so that the decouplingis possible. This decoupling also makes it possible toconsider analytical treatments.

We acknowledge useful discussions with A. J. Bray,B. Cabane, P. Leboeuf, J. F. Pinton, and J. E. Wesfreid.

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