Mixing of macroscopically quiescent liquid mixtures - DICCISM

PHYSICS OF FLUIDS 18, 044107 共2006兲

Mixing of macroscopically quiescent liquid mixtures Andrea G. Lamorgese Department of Mechanical and Aerospace Engineering, Cornell University, Ithaca, New York 14853-7501

Roberto Mauri Department of Chemical Engineering, DICCISM, Università di Pisa, 56126 Pisa, Italy

共Received 27 September 2005; accepted 16 March 2006; published online 21 April 2006兲 We simulate the mixing process of a quiescent binary mixture that is instantaneously brought from the two to the one-phase region of its phase diagram. Our theoretical approach follows the diffuse interface model, where convection and diffusion are coupled via a body force, expressing the tendency of the demixing system to minimize its free energy. In liquid systems, as this driving force induces a material flux which is much larger than that due to pure molecular diffusion, drops tend to coalesce and form larger domains, therefore accelerating all phase separation processes. On the other hand, convection induced by phase transition effectively slows down mixing, since such larger domains, eventually, must dissolve by diffusion. Therefore, whenever all other convective fluxes can be neglected and the mixture can be considered to be macroscopically quiescent, mixing is faster for very viscous mixtures, unlike phase separation which is faster for very fluid mixtures. In addition, the mixing rate is also influenced by the Margules parameter ⌿, which describes the relative weight of enthalpic versus entropic forces. In the late stage of the process, this influence can approximately be described assuming that mixing is purely diffusive, with an effective diffusivity equal to D关1 ¯ 共1 − ␾ ¯ 兲兴, where D is the molecular diffusivity and ␾ ¯ is the mean concentration. That shows − 2⌿␾ that mixing at late stages is characterized by a self-similar solution of the governing equations, which leads to a t−1 power law decay for the degree of mixing, i.e., the mean square value of the composition fluctuations. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2194964兴 I. INTRODUCTION

Starting with the 1984 pioneering article by Aref,1 fluid mixing has been studied assuming that it consists of the advection of a fluid A into another fluid B, where the instantaneous velocity of a fluid particle 共of either A or B type兲 is the solution of the Navier-Stokes equation 共see Ottino,2,3 Wiggins,4 Fountain et al.,5 and references therein兲. Clearly, this is rigorously true only in the dilute limit and when the fluid mixture is ideal, which means that the enthalpic properties of the fluid mixture are neglected, assuming that the interparticle forces between A and A are equal to those between B and B and those between A and B.6 This approximation is justified when convection dominates all diffusive processes resulting from the thermodynamic properties of the system. In general, however, at low Reynolds numbers this approximation ceases to be valid and the physicochemical properties of the fluid mixture have to be taken into account. In this work, we present a series of simulations of the mixing process occurring after heating a quiescent and initially phase-separated liquid mixture to a temperature T well above its critical point of miscibility. Now, mixing, like any other transport phenomenon, can be driven by either convection or diffusion; the former consists of the collective, coherent motion of the molecules that constitute the fluid system, while the latter is induced by their random thermal fluctuations. As explained by the so-called diffuse interface model 共otherwise called model H, in the taxonomy of Halperin and Hohenberg7兲, in a macroscopically quiescent fluid mixture far from chemical equilibrium, convection arises as the sys1070-6631/2006/18共4兲/044107/11/$23.00

tem tends to minimize its free energy and, in fact, it is induced by a 共nonequilibrium兲 body force that is proportional to the gradient of the chemical potential.8,9 The diffuse interface model provides a description of two-phase systems that is alternative to the classical fluid mechanical approach, where phase interfaces are modeled as free boundaries that evolve in time.9 In fact, Lowengrub and Truskinovsky10 and Jacqmin11 performed a careful matched asymptotic expansion and showed that the motion of sharp interfaces between immiscible fluids, as it is obtained from the outer expansion of the velocity field calculated using the diffuse interface approach, satisfies the usual Marangoni-type boundary conditions at the interfaces. With few exceptions, the diffuse interface model has been applied to describe the phase separation of partially miscible mixtures. As shown by Valls and Farrell,12 Tanaka and Araki,13 and Vladimirova et al.,14,15 this process strongly depends on the relative importance of convection and diffusion, and the enhanced coarsening rate is due to the strong coupling between the concentration and velocity fields. In fact, at the late stages of phase separation, after the system has developed well defined phase interfaces, the nonequilibrium body force reduces to the more conventional surface tension, as shown by Jasnow and Viñals,16 so that the convection driving force can be thought of as a nonequilibrium attractive force among drops.17 In this work, we have applied the diffuse interface model to the opposite process, that is mixing. In a parallel series of numerical simulations, Vladimirova and Mauri18 have shown

18, 044107-1

© 2006 American Institute of Physics

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044107-2

Phys. Fluids 18, 044107 共2006兲

A. G. Lamorgese and R. Mauri

that in macroscopically quiescent liquid mixtures, mixing is slower for less viscous systems, contrary to common thinking. Here, applying a more accurate numerical scheme, we see that this is true only at short and intermediate times, as complete mixing is achieved eventually by diffusion and therefore it takes approximately the same time, regardless of the viscosity of the mixture. In this article, we illustrate in Sec. II how the diffuse interface model predicts that the mixing process of two miscible liquids is driven by convection, which in turn is induced by a nonequilibrium body force, proportional to chemical potential gradients. Then, after describing in Sec. III the numerical method employed in our simulations, in Sec. IV we show our results. Finally, in Sec. V, we present an asymptotic analysis of the mixing process at a late stage. II. THEORY

The motion of an incompressible binary fluid mixture composed of two species A and B can be described through the diffuse interface model.7–10 Here, A and B are assumed to have equal viscosities, densities, and molecular weights, with the composition of the system uniquely determined through the molar fraction ␾ of, say, species A, as a function of position r and time t. In addition, the specific heat of the mixture is assumed to be large enough that all temperature changes can be neglected. If the flow is assumed to be slow enough to neglect the inertial terms in the Navier-Stokes equation, conservation of mass and momentum lead to the following system of equations 共see Appendix兲: 1 ⳵␾ + v · ⵜ␾ = − ⵜ · J␾ , ⳵t ␳

共1兲

ⵜp = ␩ⵜ2v + F␾ ;

共2兲

ⵜ · v = 0,

where v is the average local fluid velocity, J␾ is the diffusion flux, and F␾ is a nonequilibrium body force. As shown by Mauri et al.,19,20 J␾ is proportional to the gradient of the chemical potential through the relation, ˜, J␾ = − ␳␾共1 − ␾兲D ⵜ ␮

a⬃

冑

1 2 ␶共⌬␾兲eq

␴M W , ␳RT

共5兲

where ␶ = 共⌿ − 2兲 / 2, 共⌬␾兲eq is the composition difference between the two phases at equilibrium, while M W is the molecular weight of species A and B. This relation can be easily derived considering that ␴ ⬃ ␳ᐉ共⌬g兲eq / M W, where 共⌬g兲eq is the jump in free energy across an interface, which can be estimated from Eq. 共4兲 and ᐉ ⬃ a / 冑␶ is the characteristic interface thickness.19,23 The body force F␾ appearing in 共2兲 equals the generalized gradient of the free energy,7 and therefore it is driven by the chemical potential gradients within the mixture,14–16,24 F␾ =

冉冊

冉冊

␳ ␦g ␳RT ␳RT ˜ ⵜ␾= ˜ −␾ⵜ␮ ˜ 兴, 共6兲 = ␮ 关ⵜp M W ␦r MW MW

˜ is a pressure term which does not play any where ˜p = ␾␮ role. In particular, when the system presents well defined phase interfaces, such as at the late stages of phase separation, this body force reduces to the more conventional surface tension, as shown by Jasnow and Viñals16 and by ˜ = ␮ A − ␮ B, Jacqmin.11 Therefore, being proportional to ␮ which is identically zero at local equilibrium, F␾ can be thought of as a nonequilibrium capillary force. A simple derivation of the equations of motion is reported in the Appendix. Here we show that these equations are consistent with energy conservation. In fact, multiplying Eq. 共2兲 by v and using the incompressibility condition we obtain ⵜi关␩共ⵜiv j兲v j − vi p兴 − ␩共ⵜiv j兲共ⵜiv j兲 = −

冉冊

␳RT ˜ v iⵜ i␾ . ␮ MW 共7兲

共3兲

where ␳ is the density of the system, D is the molecular ˜ is the generalized chemical potential difdiffusivity, and ␮ ˜ ference between the two species21 defined as ␮ = ␦共g / RT兲 / ␦␾. Here g denotes the molar Gibbs free energy, defined as22 g = 关gA␾ + gB共1 − ␾兲兴 + RT关␾ log ␾ + 共1 − ␾兲 1 ⫻log共1 − ␾兲兴 + RT⌿␾共1 − ␾兲 + RTa2共ⵜ␾兲2 , 2

that at the critical point d2g / d␾2 = 0 and ␾ = 1 / 2, we find that ⌿c = 2 is the critical value of ⌿. Therefore, the single-phase region of the phase diagram corresponds to values ⌿ ⬍ 2, while the two-phase region has ⌿ ⬎ 2. At the end of the phase segregation process, a surface tension ␴ can be measured at the interface and from that, as shown by van der Waals,23 a can be determined as

共4兲

where gA and gB are the molar free energies of the pure species A and B, respectively, at temperature T and pressure P, R is the gas constant, a is a characteristic microscopic length and ⌿ is the Margules parameter, which describes the relative weight of enthalpic versus entropic forces. Phase separation occurs whenever the temperature of the system T is lower than the critical temperature Tc. Imposing

At this point, substituting Eq. 共1兲, we obtain ˜ v iⵜ i␾ = − ␮ ˜ ␮

⳵␾ 1 ˜ J i兲 − J iⵜ i␮ ˜ 兴. − 关ⵜi共␮ ⳵t ␳

共8兲

˜ + ␮B, we have In addition, considering that g / RT = ␾␮

⳵␾ ⳵g ˜ , = RT␮ ⳵t ⳵t

共9兲

where we have applied the Gibbs-Duhem relation,

␾

d␮A d␮B + 共1 − ␾兲 = 0. d␾ d␾

共10兲

Finally, substituting Eqs. 共8兲–共10兲 into Eq. 共7兲 and taking a volume integral with no-flux or periodic boundary conditions we obtain


044107-3

␳ MW


Mixing of macroscopically quiescent liquid mixtures

冕

⳵g dV = V ⳵t

冕冋

− ␩共ⵜiv j兲共ⵜiv j兲 +

V

冉冊册

RT ˜ dV, J iⵜ i␮ MW 共11兲

showing that the decrease in the free energy of the system equals the energy dissipation. Since F␾ is driven by surface energy, it tends to minimize the energy stored at the interface driving, say, A rich drops towards A rich regions. The resulting nonequilibrium attractive force between two drops appears to be much larger than any repulsive interaction among drops due to the presence of surface-active compounds,17 thus explaining why the rate of phase separation in deeply quenched liquid mixtures is almost independent of the presence of surfactants.25 The ratio between convective and diffusive mass fluxes defines the Peclet number, NPe = Va / D, where V is a characteristic velocity, which can be estimated through 共2兲 and 共6兲 as V ⬃ F␾a2 / ␩, with F␾ ⬃ ␳RT / 共aM W兲. Finally we obtain NPe = ␣冑␶,

a2 ␳ RT where ␣ = D ␩ MW

冉冊

d 2g 具共⌬␾兲典 = NkT d␾2

−1

1 ¯2 ⯝ ␾ , ␶N

III. NUMERICAL METHOD

Now we restrict our analysis to two-dimensional systems, so that an incompressible flow v can be described in terms of a stream function ␺, i.e., v = ⵜ⬜␺, with ⵜ⬜ = 共⳵y , −⳵x兲. Consequently, the governing equations 共in dimensionless form兲 can be rewritten as follows:28,29 共 ⳵ t − ⵜ 2兲 ␾ = ⵜ ·

再冋

⌿共2␾ − 1兲 +

冉冊册

冎再冉冊册冎

a L

2

ⵜ2␾ 共1 − 2␾兲 ⵜ ␾

冋

− NPe␾ⵜ⬜␺ − ⵜ2 ␾共1 − ␾兲 ⌿共2␾ − 1兲 +

共12兲

a L

2

ⵜ 2␾

,

共14兲

and

coincides with the “fluidity” parameter defined by Tanaka and Araki.13 For systems with very large viscosities, ␣ is small and the model describes a diffusion-driven separation process, as in polymer melts and alloys.19 For most liquids, however, ␣ is very large, typically ␣ ⬎ 103, showing that diffusion is important only in the vicinity of local equilibrium, when the body force F␾ is negligible. In general, therefore, for fluid mixtures that are in conditions of nonequilibrium, either phase-separating or mixing, convection dominates diffusion. Finally, note that in this treatment thermal fluctuations have been neglected. Considering that the mole fraction ␾ is defined within a volume a3, containing N = 共NA␳ / M W兲a3 particles, the mean square value of the mole fraction fluctuations reads26 2

mixtures, at least during the phase separation process. This is why we did not add further terms to generalize our model, although they can be derived rather easily 共see Vladimirova et al., Ref. 27兲.

共13兲

¯ is the mean mole fraction. In our case, using reawhere ␾ sonable values of surface tension and molecular weight 共␴ ⬇ 20 dyn/ cm and M W ⬇ 100 g / mol兲, Eq. 共5兲 gives a ⯝ 10−5 cm= 0.1 ␮m, so that N ⯝ 106 and thus, since ␶ ⯝ 10−1, we obtain 具共⌬␾兲2典 ⯝ 10−6. Now, we must assume that such fluctuations are always smaller than any concentration difference resulting from our deterministic, mean field model. For short times, that means assuming that Ginzburg’s criterion26 is satisfied, stating that the mean square concentration fluctuations are much smaller than the average concentration 共clearly, this result was expected, since Ginzburg’s criterion fails to be satisfied only in the vicinity of the critical point, i.e., when ␶ is very small兲. However, as here we follow the evolution of the system well into the late stages of mixing, we must also assume that fluctuations are smaller than any change in composition occurring at that time. As we will see in Sec. V, though, this condition, too, is readily satisfied. Although this approach has been developed for very idealized systems, it seems to capture the main features of real

ⵜ4␺ = ⵜ␾ ⫻ ⵜⵜ2␾ ,

共15兲

where length and time are scaled by L and L2 / D, respectively, with L denoting a macro length scale, which in our case coincides with the periodicity length of the computational domain. Upon assuming periodic boundary conditions, the spatial discretization of the governing equations has been effected by using the spectral collocation 共or pseudospectral兲 method.30 The resulting system of ordinary differential equations has been time integrated using the standard fourthorder Runge-Kutta scheme. The nonlinear term on the righthand side of 共15兲, when formulated in advective or conservation form, would require five FFTs for its pseudospectral evaluation. However, using the identity ⵜ␾ ⫻ ⵜⵜ2␾ = ⳵2xy共␾2x − ␾2y 兲 + 共⳵2y − ⳵2x 兲␾x␾y ,

共16兲

its computation requires only four FFTs. As a result, it is easy to see that a full RK4 step requires the evaluation of at least 34 FFTs. Equation 共15兲 can be seen as a “static” constraint on the stream function field ␺, i.e., ␺ = ␺共␾兲, so that the ␺ dependence on the right-hand side of 共14兲 can be formally dropped. Therefore, the Fourier transformed system 共14兲 and 共15兲 can be written in the form d k2t ˆ 2 关e ␾共k,t兲兴 = ek tNˆ共k,t兲, dt

共17兲

where ˆf 共k , t兲 denotes the Fourier transform of any function ˆ 共k , t兲 is the Fourier transform of the right-hand side f共r , t兲, N 2 of Eq. 共14兲, while the integrating factor ek t has been used to treat exactly the diffusive term on the left-hand side of Eq. 共14兲. As mentioned above, Eq. 共17兲 is time integrated using the standard fourth-order Runge-Kutta scheme, with a variable time step ⌬t determined by the Courant-FriedrichsLewy condition,30


044107-4

⌬t = NC



⌬x , V

共18兲

where ⌬x is the dimensionless grid spacing, NC is the Courant number, while V is a characteristic dimensionless bulk velocity. In our case, considering that convection is induced by concentration gradients, we have assumed that V = max关兩␾x兩 + 兩␾y兩兴, where max indicates the maximum value attained over all collocation points, so that at each time step the advancement scheme is sensitive to the spatial gradients of ␾. The Courant number NC is chosen such that the time advancement scheme is numerically stable and the smallest dynamically significant motions are accurately computed. Unfortunately, the nonlinearity of the equation prevents a rigorous determination of the stability limit and imposes trial-and-error determination of the maximum acceptable Courant number. It is clear however that, having used the integrating factor, there is no concern for the viscous stability

冦

␾d ;

兩r兩艋 R0 − a,

冋

册

of the scheme for any values of the Courant number. As for the choice of grid spacing ⌬x, here we adopted the same choice as in a previous work,29 where we assumed ⌬x = a / n, with n = 2. In that work, an accurate validation of the numerical results was carried out using this grid spacing to simulate the homogeneous nucleation of binary mixtures, where the sharp, O共a兲 thick interface profile was described through three grid points. During mixing 共apart from the very initial stage of the process兲, concentration profiles are much smoother and therefore this choice of grid spacing is very conservative. The following fields have been considered in order to initialize the simulations: 共1兲 A one-dimensional composition profile

␾共x,y兲 = sgn共x − L/2兲. 共2兲 An isolated drop with composition

1 ␲ ␾d共r兲 = ␾b + 共␾d − ␾b兲 1 + cos 共兩r兩 − R0 + a兲 ; R0 − a ⬍ 兩r兩艋 R0 , 2 a ␾b ; 兩r兩 ⬎ R0 ,

where ␾d is the 共homogeneous兲 composition inside the drop, R0 is the radius of the drop, while ␾b is the background composition, calculated so as to satisfy the conˆ 0=␾ ¯ of the comstraint of a prescribed average value ␾ position field. 共3兲 A random distribution of drops,

␾共r兲 = 兺 ␾共m兲 d 共r − rm兲,

共21兲

m

embedded in a background field of composition ␾b so as to meet the requirement of a prescribed average value ¯ of the composition field. The coordinates of the ␾ˆ 0 = ␾ mth drop 共xm , y m兲 are uniform random deviates in 共0 , 2␲兲 subjected to the obvious constraint that the drops cannot overlap with each other. Experimentally, we can realize this configuration by cooling the mixture below its critical point. Finally, it should be stressed that the DNS requirement of accurate resolution of the smallest time and length scales is trivially satisfied in our case, given that the grid spacing is set to be a fraction of a, which is the smallest significant length scale of the problem. In addition, the macro length scale L 共and hence the number N of modes employed in each coordinate direction兲 is chosen based on physical considerations, i.e., the typical size of a microdomain above which gravity becomes significant; it obviously determines the upper time limit of our simulation, occurring when the charac-

共19兲

冧

共20兲

teristic length of the large-scale field becomes comparable to the box size. IV. RESULTS

Our simulations were carried out in a periodic box of size Na, corresponding to about 0.1 mm, since we saw that a ⬇ 10−5 cm. Time was spanned through an L2 / D characteristic time, corresponding to about 10 s, as molecular diffusivity in the liquids is typically D ⬇ 10−5 cm2 / s. We considered three types of symmetric mixtures, i.e., mixtures whose phase diagrams are mirror symmetric with respect to the ␾ = 1 / 2 axis, so that their excess free energy can be expressed through a single ⌿ Margules parameter 共see Appendix兲. Two of these mixtures are nonideal, with ⌿ = 1 and ⌿ = −1, representing for example, approximately, a trimethyl methanebenzene mixture at 0 ° C and a water-hydrogen peroxide mixture at 75 ° C, respectively 共see Sandler,6 Fig. 7.5-2兲. The third is an ideal mixture, with ⌿ = 0, and is well represented, for example, by a hexane-triethylamine mixture at 60 ° C 共see Sandler,6 Fig. 8.1-1兲 or an acetonitrile-nitromethane mixture at 70 ° C. First of all, we analyzed the mixing process when the initial configuration of the system is one-dimensional, as in the case of a plane interface or that of an isolated circular drop. In these cases, as one can easily see from Eq. 共1兲关or Eq. 共15兲兴, the body force is identically zero and therefore there is no convection, i.e., v = 0. In addition, the results of our simulations are in perfect agreement with those obtained


044107-5



FIG. 1. Evolution of eight identical drops with ⌿ = 1.9 placed on a loop at nondimensional times t = 0.01, 0.1, 1 , 2, with NPe = 0 共top兲 and NPe = 103 共bottom兲.

by Vladimirova and Mauri,18 showing that in 1D, Eq. 共14兲 is well approximated by the following dimensional equation: ¯ 兲⳵x兴␾ = 0, 关 ⳵ t − ⳵ xD *共 ␾

共22兲

which corresponds to a purely diffusive process with an effective local diffusivity, ¯ 兲 = D关1 − 2⌿␾ ¯ 共1 − ␾ ¯ 兲兴, D *共 ␾

共23兲

¯ is the 共constant兲 mean concentration. where ␾ In a previous study14 we simulated the mixing process of a collection of drops immersed in a background field, showing that, while a single drop remains still as it is absorbed, two drops tend to attract each other and even coalesce, provided that the Peclet number is large enough and the drops are sufficiently close to each other. This effect was further investigated in the simulations of Figs. 1 and 2, showing the evolution of eight identical drops of radius 2a that are placed within the bulk fluid. When the initial configuration is that of a loop 共see Fig. 1兲, at first the drops coalesce and form a circular crown when NPe = 1000, while they do not move when NPe = 0. Later, however, since a circular crown is a “stable” structure that can be reabsorbed by diffusion only, mixing occurs with the same characteristic time, regardless of the Peclet number. On the other hand, when the eight

drops are initially located on a straight line 共see Fig. 2兲, while, again, when NPe = 0 the drops do not move and are reabsorbed within the same characteristic time as before, when NPe = 1000 they rapidly coalesce and form a larger drop. This latter, though, has to be reabsorbed by diffusion, too, and, being larger than the original drops, will take approximately eight times longer to disappear. Consequently, mixing appears to be faster in the absence of convection. A complete mixing, however, requires that the concentration field be homogeneous within the whole domain, and therefore it will have the same, O共L2 / D*兲 characteristic time, independent of the Peclet number. These results were confirmed in a series of simulations, where the initial concentration field is that resulting from a random distribution of 620 drops with radius 4a. In Fig. 3 we show some snapshots of these simulations where ⌿ = 1, comparing the behavior for NPe = 1000 with that for NPe = 0. Representing the results of these simulations in terms of the degree of mixing ␦m,

␦m共t兲 =

2 具兩␾共r,t兲 − ␾av兩2典 ␾rms 共t兲 , = 2 2 具兩␾0共r兲 − ␾av兩典 ␾rms共0兲

共24兲

we can see from Fig. 4 that ␦m at first decays faster when NPe = 0 than when NPe = 1000.

FIG. 2. Evolution of eight identical drops with ⌿ = 1.9 placed on a straight line at nondimensional times t = 0.01, 0.02, 0.04, 0.1 with NPe = 0 共top兲 and NPe = 103 共bottom兲.


044107-6



FIG. 3. Snapshots of the composition field at nondimensional times t = 10−3 , 10−2 , 10−1 with NPe = 0 共top兲 and NPe = 103 共bottom兲.

These simulations were also conducted in 3D, using N = 100 modes in each coordinate direction, with an initial concentration field consisting of a random distribution of 620 drops with radius 2a. As we see in Fig. 5, these results confirm that the phenomenon that we report 共i.e., the fact that mixing is faster for viscous mixtures兲 is valid in 3D as well. In a series of forthcoming articles, a detailed comparison between 2D and 3D simulations of both mixing and demixing processes will be presented. Our results agree with the experimental findings of Tanaka and Sigehuzi,31 who showed that demixing is faster than mixing in low viscosity, symmetric binary mixtures, where the separation process is driven by convection, while they occur at the same rate for strongly off-symmetric or high viscosity mixtures, when phase transition is dominated by diffusion. Eventually, when ␦m ⬍ 10−2, residual mixing becomes a very slow process, with an O共L2 / D*兲 characteristic time, as the system must iron out the small inhomogeneities existing

FIG. 4. The degree of mixing ␦m as a function of the nondimensional time t for NPe = 0 and NPe = 103, when ⌿ = 1.

within the whole domain by diffusion only. Note that however, in a “real” case, even a small forced convection would greatly change this last part of the curve. Finally, in Fig. 6, we see that mixing becomes faster as we decrease ⌿. Even residual mixing 共i.e., mixing when ␦m ⬍ 10−2兲 is faster when ⌿ is smaller, as it takes place with an O共L2 / D*兲 characteristic time. V. MIXING AT LATE STAGES

In this section, we study the evolution of the system at long times assuming, though, that the concentration differences due to mixing are still smaller than those due to thermal fluctuations. As we saw in Sec. II 关cf. discussion following Eq. 共13兲兴, in most liquid mixtures thermal fluctuations ¯ 兲2 / 共␶N兲 ⯝ 10−6 and therefore we must reare 具共⌬␾兲2典 ⯝ 共␾ strict ourselves to cases where ␦m 艌 10−5. As we see in Fig. 6, this condition is satisfied in our simulations.

FIG. 5. The degree of mixing ␦m from 3D simulations with 1003 modes as a function of the nondimensional time t for NPe = 0 and NPe = 103, when ⌿ = 1.


044107-7

FIG. 6. The degree of mixing ␦m as a function of the nondimensional time t when NPe = 103, for ⌿ = 1 , 0, and −1.

After an initial transient, the mixing process exhibits a ¯ +␾ ˜ , with ␾ ˜ Ⰶ␾ ¯ , it is self-similar evolution. In fact, as ␾ = ␾ easy to see that, as convection is negligible, at leading order terms Eq. 共14兲 reduces to ¯ 共1 − ␾ ¯ 兲兴ⵜ2兲␾ ˜ = 0. 共⳵t − 关1 − 2⌿␾

共25兲

This corresponds to a purely diffusive process with constant effective diffusivity ¯ 共1 − ␾ ¯ 兲兴, D* = D关1 − 2⌿␾

共26兲

¯ = 1 / 2 reduces to D* = D共1 − ⌿ / 2兲. Hence, the which, for ␾ composition fluctuations and their spectral density E admit the simple self-similar solutions: ˜ˆ 共k,t兲 = ␾ ˜ˆ 共k,0兲e−D*k2t , ␾

共27兲

˜ˆ 共k,t兲兩2典 = E共k,0兲e−2D*k2t , E共k,t兲 = 2␲k具兩␾

共28兲

where we have implicitly assumed that the composition field is homogeneous and isotropic. As shown in Fig. 7, the results of our simulations are in very good agreement with the selfsimilar solution 共28兲. Once we know the spectral density, the degree of mixing 共24兲 can be calculated as

␦m共t兲 =



冕冕

⬁

E共k,t兲dk

0 ⬁

0

= E共k,0兲dk

冕

⬁

0

E共k,0兲e

冕

−2D*k2t

dk .

⬁

共29兲

E共k,0兲dk

0

Changing the integration variable to ␨ = k冑2D*t, we obtain

FIG. 7. Spectral densities vs 共rescaled兲 wave number at different nondimensional times for ⌿ = 1 and NPe = 0.

␦m共t兲 = 共2D*t兲−1/2

冕

⬁

E共␨/冑2D*t,0兲e−␨ d␨ 2

0

冕

,

⬁

共30兲

E共k,0兲dk

0

which shows that, for t → ⬁, the dominant contribution to the degree of mixing comes from the infrared component of E共k , 0兲. Now, the infrared behavior associated with an initial white-in-space noise is E共k , 0兲 = C1k; this holds true also for the initial composition profiles that we have considered, consisting of one or more drops embedded in a homogeneous background.32 Consequently, we obtain

␦m共t兲 ⬀

C1 2D*t

冕

⬁

2

␨ e −␨ d ␨ =

0

C1 . 4D*t

共31兲

This result shows that at late stages of the process, i.e., when t ⬎ L2 / D*, the degree of mixing 共i.e., the mean square value of composition fluctuations兲 decays as t−1. Therefore, we can conclude that mixing at late stages is characterized by a self-similar evolution of the spectral density E共k , t兲, which in turn leads to a universal power-law decay for the degree of mixing, with a −1 exponent that does not depend on either the effective diffusivity or the Peclet number.

APPENDIX: DERIVATION OF THE EQUATIONS OF MOTION 1. Equilibrium state

Consider a regular binary solution, that is a mixture in which both volume and entropy of mixing are equal to zero. That means that when we mix the two species, say A and B, 共a兲 the volume remains unchanged, so that the mixture can be considered to be incompressible, and 共b兲 the entropy change is equal to that of ideal mixtures 共see Sandler,6 Sec. 7.6兲. Here, we also assume, for sake of simplicity, that the two species have the same molecular weight, so that volume,


044107-8



weight, and molar compositions are all the same. Generalization to nonregular, even compressible, binary mixtures can be found in Lowengrub and Truskinovsky.10 When the mixture has uniform composition, the molar free energy of mixing, ⌬g = g − 关␾gA + 共1 − ␾兲gB兴, where gA and gB are the free energies of the pure components, can be written as ⌬gth/RT = ␾ log ␾ + 共1 − ␾兲log共1 − ␾兲 + ⌿␾共1 − ␾兲, 共A1兲 with ␾ denoting the mole fraction of species A, R is the gas constant, and T is the temperature. We see that ⌬gth is the sum of an entropic, ideal part, 共this part is the entropy of mixing for an ideal, gas or liquid, solution兲, and an enthalpic, so called excess, part, gex = RT⌿␾共1 − ␾兲, where ⌿ is the Margules parameter, which describes the relative weight of enthalpic versus entropic forces 共in fact, it is proportional to FAA + FBB − 2FAB, that is the difference between the attractive forces between equal molecules and those between unequal molecules6兲. In addition, in general, when ␾ moles of component A are mixed with 共1 − ␾兲 moles of component B, energy can be released or absorbed. This heat, called enthalpy of mixing, hmix, can be derived from the excess free energy as6

⳵共g /T兲 . ⳵T

冕

共A5兲

with the constraint 具␾典 =

冕

␾共x兲dV = const.

共A6兲

V

Therefore, applying to the system a virtual change in composition ␦␾ from its equilibrium condition, we obtain

␦

冕

˜ 兲␾兴dV = 0, 关g共␾,ⵜ␾兲 − 共RT␮

˜ is a Lagrange multiplier. From here, we obtain where RT␮ the Euler-Lagrange equation, ˜⬅ ␮

冋

冉冊册

1 ␦g 1 ⳵g ⳵g = − ⵜi ␦ ␾ RT RT ⳵␾ ⳵ ⵜ i␾

= ␮共␾兲 −

冉冊

1 ⳵g ⵜi , RT ⳵ ⵜ i␾

RT␮A共T, P, ␾兲 = gth共T, P, ␾兲 +

冉冊

共A9兲

RT␮B共T, P, ␾兲 = gth共T, P, ␾兲 −

冉冊

共A10兲

and

1 ⌬gnl/RT = a2共ⵜ␾兲2 . 2

共A3兲

This term is generally referred to as the Cahn-Hilliard term. Finally, the total free energy can be written as

冋

⌬g = ⌬gth + ⌬gnl = RT ␾ log ␾ + 共1 − ␾兲log共1 − ␾兲

册

1 + ⌿␾共1 − ␾兲 + a2共ⵜ␾兲2 . 2

共A4兲

As shown in Mauri et al.,19 generalization to a nonsymmetric case 共i.e., with two Margules coefficients兲 does not give results that are qualitatively different. At equilibrium, the total free energy is minimized, i.e.,

共A8兲

where ␮ is the difference between the thermodynamical chemical potentials of the two species, ␮A and ␮B. This can be seen subtracting the two equations,6

共A2兲

Now, for a regular mixture, the entropy of mixing is equal to zero, i.e., sex = −共⳵gex / ⳵T兲 P,x = 0, so that gex must be independent of T and therefore ⌿ ⬀ T−1; consequently, we see that in that case hmix = gex. When the composition of the mixture is not homogeneous, the free energy will include also a nonlocal part, i.e., terms depending on the composition gradients, gnl关ⵜ␾ , ⵜⵜ␾ , 共ⵜ␾兲2 , . . . 兴. Considering the isotropy of the system 共which rules out the dependence on ⵜ␾兲, the dominant terms will be proportional to ⵜ2␾ and to 共ⵜ␾兲2. However, considering that the total free energy equal the volume integral of ⌬g and that 兰ⵜ2␾dV = 0, we conclude that 共see Pismen33 for a rigorous derivation starting from the partition function兲

共A7兲

V

ex

hmix = − T2

g共␾,ⵜ␾兲dV = min,

V

dgth 共1 − ␾兲, d␾

dgth ␾, d␾

obtaining

␮ = ␮A − ␮B =

1 dgth . RT d␾

共A11兲

Thus, substituting 共A3兲 into 共A8兲 we obtain ˜ = ␮ − a 2ⵜ 2␾ , ␮

共A12兲

˜ and not ␮ to remain showing that at equilibrium it is ␮ ˜ will be referred to as the generalconstant. Accordingly, ␮ ized chemical potential difference. 2. Equations of motion for nondissipative systems

This derivation is a modification of Lowengrub and Truskinovsky’s.10 Alternative derivations of these results can be found in Jasnow and Vinals,16 Antanovskii,8 and Anderson, McFadden, and Wheeler.9 First, let us consider the reversible, dissipation-free case. Then, there is no diffusion and therefore the concentration and temperature fields can be derived from the initial conditions, knowing the velocity field. In fact, if x共t , x0兲 denotes the trajectory of a material particle which is located at x0 at time t = 0, i.e., with x0 = x共0 , x0兲, then the fluid velocity field is v共x , t兲 = x˙ 共t , x0兲, where the dot denotes the time derivative at constant x0. In addition, the concentration field ␾共x , t兲 = ␾共x0兲 and the temperature field T共x , t兲 = T共x0兲 do not depend explicitly on x and t, and therefore


044107-9

d␾ = 0, dt

冕冕

共A13兲

L共v, ␾,ⵜ␾兲dVdt,

共A14兲

V

where L=␳

冉

冊

1 1 2 g , v − 2 MW

共A15兲

ⵜ · v = 0,

共A16兲

and the mass conservation constraint 共A6兲. Accordingly, starting from a flow field that minimizes the action 共A14兲, let us give a virtual displacement ␦x, corresponding to an infinitesimal change of the fluid flow. Among all the possible virtual displacements, let us choose those such that ␦␾ = ⵜi␾ · ␦xi = 0. Since the action S in 共A14兲 is minimized, we have

冕冕冋

册

␳ ␦g − q␦共ⵜivi兲 dVdt = 0, MW

t

0

␳

dvi + ⵜi p = Fi = ⵜ j P ji , dt

␳ v i␦ v i −

V

Pij = −

t

冕冕 t

␳vi␦vidVdt = −

0

dvi ␳ ␦xidVdt, V dt

共A18兲

where we have assumed that the virtual displacement is equal to zero at the beginning and at the end of the trajectories, i.e., when t = 0 and t = t, and also on the boundary, S, of the volume V of integration. Now, consider that, since g = g共␾ , ⵜ␾兲, with ␦␾ = 0,

␦g =

⳵g ␦共ⵜi␾兲, ⳵ ⵜ i␾

共A19兲

with ␦共ⵜ j␾兲 = 共ⵜiⵜ j␾兲␦xi = −共ⵜi␾兲ⵜ j共␦xi兲. Therefore, the second term in the integral 共A17兲 gives

冕冕 t

0

V

冕冕冉 t

␦gdVdt =

0

共A22兲

冊

⳵g ⵜi␾ ␦xidVdt, ⵜj ⳵ⵜ j␾ V

␳ ⳵g ⵜ j␾ M W ⳵ ⵜ i␾

Pij = −

共A23兲

冉冊

␳RT 2 a 共ⵜi␾兲共ⵜ j␾兲. MW

共A24兲

This equation of motion is to be solved with the incompressibility condition, 共A25兲

ⵜivi = 0,

and the additional equations of conservation of chemical species and energy, Eqs. 共A13兲. The body force Fi in Eq. 共A22兲 can be rewritten as Fi = ⵜ j P ji = −

where q共x , t兲 is a function to be determined through the incompressibility constraint 共A16兲. Considering that ␦共dxi兲 = d共␦xi兲 and integrating by parts, the first integral in 共A17兲 gives

V

共A21兲

is the Cauchy stress tensor which, using the expression 共A4兲 for the free energy, coincides with the Korteweg stresses,34

=

0

共ⵜi p兲␦xidVdt,

V

where p = ⳵q / ⳵t and we considered again that ␦xi 共and ␦vi as well兲 vanishes at the boundary S and for t = 0 , t. Concluding, substituting 共A18兲, 共A20兲, and 共A21兲 into 共A17兲 and considering the arbitrariness of the virtual displacements ␦xi, we obtain the linear momentum equation

共A17兲

冕冕

0

where

is the Lagrangian density of the system, subjected to the incompressibility constraint,

␦S =

冕冕 t

q␦共ⵜivi兲dVdt =

V

0

t

0

冕冕 t

dT = 0, dt

where d / dt = ⳵ / ⳵t + v · ⵜ denotes the material derivative. According to the Hamilton, minimum action, principle, the velocity and concentration field of any conservative system minimizes the following functional: S=



␳ MW

冋冉

ⵜj

冊冉冊册

⳵g ⳵g ⵜ i␾ + ⵜ iⵜ j ␾ ⳵ⵜ j␾ ⳵ⵜ j␾

␳RT ␳ ˜ ⵜ i␾ − ␮ ⵜig, MW MW

共A26兲

where we have summed and subtracted the term 共⳵g / ⳵␾兲ⵜi␾. Therefore, the momentum equation becomes

␳

dvi ␳RT ˜ ⵜ i␾ , + ⵜi p⬘ = ␮ dt MW

共A27兲

where the pressure term has been redefined as p⬘ = p + ␳g / M W. Alternatively, this equation can also be written as

␳

dvi ␳RT ˜, + ⵜi p⬙ = − ␾ ⵜ i␮ dt MW

共A28兲

˜ ␾. This last expression for the with p⬙ = p⬘ − 共␳RT / M W兲␮ body force is quite intuitive; the momentum flux is directed towards regions with smaller chemical potential differences. Finally, note that the body force Fi is not dissipative per se; as shown below, the system becomes dissipative only when there is a diffusive mass flux. 3. Dissipative terms

共A20兲

where the surface integral has been put equal to zero, because ␦xi = 0 at the boundary. Finally, the third term on the right-hand side of Eq. 共A17兲 gives

In general, the stress tensor can be considered as the sum of a pressure term, a Korteweg term and a viscous term, i.e.,

␳

dvi = ⵜ jT ji , dt

共A29兲

where


044107-10



Tij = − p␦ij + Pij + ␩共ⵜiv j + ⵜ jvi兲,

共A30兲

with ␩ denoting the fluid viscosity, that we assume to be independent of the shear rate 共i.e., Newtonian behavior兲. In general, ␩ is a function of the composition of the mixture, i.e., ␩ = ␩共␾兲; however, in our simplified model, we assume that the two components of the mixture have the same viscosity, so that ␩ is a constant. Substituting Eq. 共A30兲 into 共A29兲 at low Reynolds number, we obtain the equation of motion 共2兲. As for the transport of chemical species, when diffusion is taken into account the conservation equation 共A13兲 is modified as follows: d␾ + ⵜ · J␾ = r␾ , dt

共A31兲

where J␾ ⬅ JA is the diffusive flux of component A, while r␾ is the rate of generation of component A due to chemical reaction, which in the following we will assume to be equal to zero, i.e., r␾ = 0. Normally 共see Cussler35兲 when we do not consider the Cahn-Hilliard part of the free energy, the mass flux of component A is proportional to the gradient of its chemical potential as JA = − DxA ⵜ ␮A ,

共A32兲

where D is a function of temperature and pressure, but not of composition. In our case, substituting 共A1兲 into 共A9兲, we obtain RT␮A = gA + RT共ln xA + ⌿xB2 兲,

共A33兲

so that JA = − DxA

d␮A ⵜ xA = − D* ⵜ xA , dxA

共A34兲

where D* = D共1 − 2⌿xAxB兲

共A35兲

is the diffusion coefficient. Inverting the suffices A and B, we see that the flux of species B is opposite to the flux of species A, that is JA = −JB, showing that these are really diffusive fluxes, with no convective components.36 In addition, note that for ideal or dilute mixtures, i.e., when either ⌿ = 0, or xA Ⰶ 1 共or xB Ⰶ 1兲, we obtain that D* = D and therefore Eq. 共A34兲 reduces to Fick’s law. Going back to our notation, with xA ⬅ ␾ and JA ⬅ J␾, the constitutive relation 共A34兲 can also be written as J␾ = − D␾共1 − ␾兲 ⵜ ␮ ,

共A36兲

where ␮ = ␮A − ␮B. At this point, a natural extension of the constitutive relation 共A34兲 is the following one: ˜. J␾ = − D␾共1 − ␾兲 ⵜ ␮

共A37兲

This constitutive relation must be substituted into the equation of mole conservation, obtaining Eq. 共1兲. As shown in Sec. II, our equations of motion are consistent with the equation of energy conservation. Finally, the equation of conservation of energy 共A13兲, too, can be generalized to account for the diffusive heat

fluxes and the enthalpy of mixing. In our case, however, we assume that the process is isothermal and therefore this equation can be disregarded. H. Aref, “Stirring by chaotic advection,” J. Fluid Mech. 143, 1 共1984兲. J. M. Ottino, The Kinematics of Mixing, Stretching and Chaos 共Cambridge University Press, Cambridge, 1989兲. 3 J. M. Ottino, “Mixing, chaotic advection and turbulence,” Annu. Rev. Fluid Mech. 22, 207 共1990兲. 4 S. Wiggins, Chaotic Transport in Dynamical Systems 共Springer, Heidelberg, 1991兲. 5 G. O. Fountain, D. V. Khakhar, I. Mezić, and J. M. Ottino, “Mixing in a bounded three-dimensional flow,” J. Fluid Mech. 417, 265 共1999兲. 6 I. S. Sandler, Chemical and Engineering Thermodynamics, 3rd ed. 共Wiley, New York, 1999兲, Chap. 7. 7 P. C. Hohenberg and B. I. Halperin, “Theory of dynamic critical phenomena,” Rev. Mod. Phys. 49, 435 共1977兲. 8 L. K. Antanovskii, “Microscale theory of surface tension,” Phys. Rev. E 54, 6285 共1996兲. 9 D. M. Anderson, G. B. McFadden, and A. A. Wheeler, “Diffuse-interface methods in fluid mechanics,” Annu. Rev. Fluid Mech. 30, 139 共1998兲. 10 J. Lowengrub and L. Truskinovsky, “Quasi-incompressible Cahn-Hilliard fluids and topological transitions,” Proc. R. Soc. London, Ser. A 454, 2617 共1998兲. 11 D. Jacqmin, “Contact-line dynamics of a diffuse fluid interface,” J. Fluid Mech. 402, 57 共2000兲. 12 O. T. Valls and J. E. Farrell, “Spinodal decomposition in a threedimensional fluid model,” Phys. Rev. E 47, R36 共1993兲. 13 H. Tanaka and T. Araki, “Spontaneous double phase separation induced by rapid hydrodynamic coarsening in two-dimensional fluid mixtures,” Phys. Rev. Lett. 81, 389 共1998兲. 14 N. Vladimirova, A. Malagoli, and R. Mauri, “Diffusio-phoresis of twodimensional liquid droplets in a phase-separating system,” Phys. Rev. E 60, 2037 共1999兲. 15 N. Vladimirova, A. Malagoli, and R. Mauri, “Two-dimensional model of phase segregation in liquid binary mixtures,” Phys. Rev. E 60, 6968 共1999兲. 16 D. Jasnow and J. Viñals, “Coarse-grained description of thermo-capillary flow,” Phys. Fluids 8, 660 共1996兲. 17 R. Gupta, R. Mauri, and R. Shinnar, “Phase separation of liquid mixtures in the presence of surfactants,” Ind. Eng. Chem. Res. 38, 2418 共1999兲. 18 N. Vladimirova and R. Mauri, “Mixing of viscous mixtures,” Chem. Eng. Sci. 59, 2065 共2004兲. 19 R. Mauri, R. Shinnar, and G. Triantafyllou, “Spinodal decomposition in binary mixtures,” Phys. Rev. E 53, 2613 共1996兲. 20 N. Vladimirova, A. Malagoli, and R. Mauri, “Diffusion-driven phase separation of deeply quenched mixtures,” Phys. Rev. E 58, 7691 共1998兲. 21 L. Landau and E. M. Lifshitz, Fluid Mechanics 共Pergamon, New York, 1953兲, Chap. 6. 22 J. W. Cahn and J. E. Hilliard, “Free energy of a nonuniform system. III. Nucleation in a two-component incompressible fluid,” J. Chem. Phys. 31, 688 共1959兲. 23 J. D. van der Waals, “The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density,” reprinted in J. Stat. Phys. 20, 200 共1979兲. 24 H. Tanaka, “Coarsening mechanisms of droplet spinodal decomposition in binary fluid mixtures,” J. Chem. Phys. 105, 10099 共1996兲. 25 R. Gupta, R. Mauri, and R. Shinnar, “Liquid-liquid extraction using the composition induced phase separation process,” Ind. Eng. Chem. Res. 35, 2360 共1996兲. 26 L. Landau and E. M. Lifshitz, Statistical Physics 共Pergamon, New York, 1953兲, Chap. 146. 27 N. Vladimirova, A. Malagoli, and R. Mauri, “Two-dimensional model of phase segregation in liquid binary mixtures with an initial concentration gradient,” Chem. Eng. Sci. 55, 6109 共2000兲. 28 A. G. Lamorgese and R. Mauri, “Phase separation of liquid mixtures,” in Nonlinear Dynamics and Control in Process Engineering—Recent Advances, edited by G. Continillo, S. Crescitelli, and M. Giona 共Springer, Rome, 2002兲, pp. 139–152. 29 A. G. Lamorgese and R. Mauri, “Nucleation and spinodal decomposition of liquid mixtures,” Phys. Fluids 17, 034107 共2005兲. 30 C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics 共Springer, New York, 1989兲. 1 2


044107-11 31



H. Tanaka and T. Sigehuzi, “Periodic spinodal decomposition in a binary polymeric fluid mixture,” Phys. Rev. Lett. 75, 874 共1995兲. 32 It is easy to see that the nonlinear terms of the generalized 2D CahnHilliard equation 共14兲 can only generate k5 and higher k powers in E共k , t兲. Therefore, the infrared behavior associated with the initial white-noise perturbation, E共k , 0兲 = C1k is time invariant, which means that E共k , t兲 = C1k as k → 0, with C1 = const. The same conclusion can be reached directly from the self-similar solution 共28兲. 33 L. M. Pismen, “Nonlocal diffuse interface theory of thin films and moving

contact line,” Phys. Rev. E 64, 021603 共2001兲. D. J. Korteweg, “On the form the fluid equations of motion assume if account is taken of the capillary forces caused by density variations,” Arch. Neerl. Sci. Exactes Nat., Ser. II 6, 1 共1901兲共in French兲. 35 E. L. Cussler, Diffusion 共Cambridge University Press, Cambridge, 1984兲, p. 180. 36 In general, from 共A32兲 and applying the Gibbs-Duhem relation, xA ⵜ ␮A = −xB ⵜ ␮B, we see that it is always true that JA = −JB. 34
