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Two-dimensional model of phase segregation in liquid binary mixtures with an initial concentration gradient

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Two-dimensional model of phase segregation in liquid binary mixtures with an initial concentration gradient
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  * Corresponding author. Tel.: # 1-212-650-6688; fax: # 1-212-650-6660.  E-mail address:  mauri @ che-mail.engr.ccny.cuny.edu (R. Mauri).Chemical Engineering Science 55 (2000) 6109 } 6118 Two-dimensionalmodel of phase segregationin liquid binary mixtureswith an initial concentration gradient Natalia Vladimirova  , Andrea Malagoli  , Roberto Mauri     *   Department of Astronomy and Astrophysics, Uni v ersity of Chicago, Chicago, IL, USA   Department of Chemical Engineering, The City College of CUNY, Con v ent A v e at 138th St., New York, NY 10031, USA   Department of Chemical Engineering, Uni v ersita %   di Pisa, Italy Received 29 November 1999; received in revised form 25 May 2000; accepted 26 May 2000 Abstract We simulate the phase segregation of a deeply quenched binary mixture with an initial concentration gradient. Our theoreticalmodel follows the standard model H, where convection and di !  usion are coupled via a body force, expressing the tendency of thedemixing system to minimize its free energy. This driving force induces a material  # ux much larger than that due to pure moleculardi !  usion, as in a typical case the Peclet number   , expressing here the ratio of thermal to viscous forces, is of the order of 10  .Integrating the equations of motion in 2D, we show that the behavior of the system depends on the values of the Peclet number   andthe non-dimensional initial concentration gradient   . In particular, the morphology of the system during the separation processre # ectsthe competitionbetweenthe capillarity-induceddropmigrationalong the concentrationgradient andthe random # uctuationsgenerated by the interactions of the drops with the local environment. For large   , the nucleating drops grow with time, until theyreach a maximum size, whose value decreases as the Peclet number and the initial concentration gradient increase. This behavior isdue to the fact that the nucleating drops do not have the chance to grow further, as they tend to move towards the homogeneousregions where they are assimilated.    2000 Published by Elsevier Science Ltd. All rights reserved.  Keywords:  Phase separation; convection-induced spinodal decomposition 1. Introduction In this work we simulate numerically and explainphysically a phenomenon which was observed by Gupta,Mauri, and Shinnar (2000): when a low-viscosity liquidbinary mixture with a strong initial concentration gradi-ent is quenched deeply into the unstable region of itsphase diagram it phase separates without the appearanceof large (i.e. larger than 10   m) drops.In general, when a binary mixture is quenched from itssingle-phase region to a temperature below the composi-tion-dependent spinodal curve, it phase separatesthrough a process called spinodal decomposition (fora review on spinodal decomposition, see Gunton, SanMiguel & Sahni, 1983), which is characterized by thespontaneous formation of single-phase domains whichthen proceed to grow and coalesce. Unlike nucleation,where an activation energy is required to initiate theseparation, spinodal decomposition involves the growthof any  # uctuations whose wavelength exceeds a criticalvalue. Experimentally, the typical domain size  R  is de-scribed by a power-law time dependence,  t  , where n & 1/3 when di !  usion is the dominant mechanism of material transport, while  n & 1 when hydrodynamic,long-range interactions become important (Chou& Goldburg, 1979; Wong & Knobler, 1981; Guenoun,Gastaud, Perrot & Beysens, 1987; Gupta, Mauri &Shinnar, 1999).Theoretically, spinodal decomposition in  # uids hasbeen described within the framework of the Gin-zburg } Landau theory of phase transition (see Le Bellac,1984, Chapter 2) by Cahn and Hilliard (1959), showingthat during the early stages of the process, initial instabil-ities grow exponentially,forming, at the end, single-phasemicrodomains whose size corresponds to the fastest-growing mode     of the linear regime (Mauri, Shinnar& Triantafyllou, 1996). During the late stages of the 0009-2509/00/$-see front matter    2000 Published by Elsevier Science Ltd. All rights reserved.PII: S0 0 09 -2 5 0 9 (0 0 ) 0 0 4 12 - 7  process, i.e. for times   "   /  D , where  D  is the moleculardi !  usivity, the system consists of well-de " ned patches inwhich the average concentration is not too far from itsequilibrium value, although the condition of local equi-librium is reached much later (Vladimirova, Malagoli& Mauri, 1998). At this point, material transport canoccur either by di !  usion or by convection.In cases wheredi !  usion is the only transport mechanism, both analyti-cal calculations (Lifshitz & Pitaevski, 1984, Chapter 12)and dimensional analysis (Siggia, 1979) predict a growthlaw  R & t  , due to the Brownian coagulation of drop-lets. On the other hand, when hydrodynamicinteractionsamong droplets become important, the e !  ect of convec-tive mass  # ow resulting from surface tension e !  ects can-not be neglected any more, and dimensional analysisindicates a growth law  R & t  (Siggia, 1979; Furukawa,1994). This linear growth has been obtained recently ina series of simulations by Vladimirova, Malagoli andMauri (1999a,b), showing that phase separation in low-viscosity liquid mixtures is convection-driven, with typi-cal instantaneous  # uid velocity being  O ( D /    ). Here, theprocess is simulated in two dimension, following theso-called model H, in the taxonomy of Hohenberg andHalperin (1977), where the equations of conservation of massand momentumare coupledvia the convectivetermof the convection } di !  usion equation, which is driven bya composition-dependent body force in the Stokes equa-tion. As noted by Jasnow and Vin    als (1996), when thesystem is composed of single-phase domains separatedby sharp interfaces, this force incorporates capillary ef-fects, and plays the role of a Marangoni force. After theinitial, di !  usion-driven stage leads to a non-uniform con-centration " eld, this capillary driving force induces a ma-terial  # ux, which is several orders of magnitude largerthan its di !  usive counterpart. This convective  # ux drivesthe successive process of phase segregation, and canexplainthe linear growth law which was observed experi-mentally by Gupta et al. (1999). The agreement, whichis both qualitative and quantitative, between 2D numer-ical simulations and 3D experimental measurementsseems to indicate the existence of a universal behaviorcharacterizing all phase separation processes. Thisconjecture is further strengthened by the results of thiswork.After a description of the model in Section 2, inSection 3 we present thenumericalresults.In AppendixesA and B we show how the constitutive relation of thematerial  # ux can be derived, stressing that it can be verydi !  erent, depending on whether the mean properties of the  # uid mixtures are de " ned in terms of mass or of moleaverages. Finally, in Appendix C, the driving force of theStokes equation is analyzed, showing that it can beinterpreted as a capillary force, provided that the systemconsists of single-phase domains separated by sharp in-terfaces, as it happens during the late stage of the phaseseparation process. 2. The governing equations 2.1. The binary mixture at equilibrium Consider a homogeneousmixture of two species A andB with molar fractions  x   and  x  " 1 ! x  , respectively,kept at temperature ¹  and pressure  P . For sake of simplicity, in our model we assume that the molecularweights, speci " c volumes and viscosities of A are equal tothose of B, namely  M  " M  " M  , <   M  " <   M  " <   M  and   "   "  , respectively, so that molar, volumetric andmass fractions are all equal to each other, and the mix-ture viscosity is composition-independent.The generaliz-ation to binary mixtures composed of species withdi !  erent physical properties is presented in Appendix A.The equilibrium state of this system is described by the ` coarse-grained a  free energy functional, that is the molarGibbs energy of mixing,   g  ,  g  " g  ! ( g  x  # g  x  ), (1)where  g   is the energy of the mixture at equilibrium,while  g   and  g   are the molar free energy of pure speciesA and B, respectively, at temperature ¹ and pressure  P .The free energy   g   is the sum of an ideal part   g   anda so-called excess part  g  , with  g  " R ¹ [ x   log x  # x   log x  ], (2)where  R  is the gas constant, while the excess molar freeenergy can be expressed as g  " R ¹  x  x  , (3)where  is a function of  ¹ and  P . This expression, whichin Chemical Engineering is generally referred to as theone-parameterMargules correlation(Prausnitz, Lichten-thaler & Gomes de Azevedo, 1986), is generally derivedby considering the molecular interactions between near-est neighbors or summing all pairwise interactionsthroughoutthe whole system(Lifshitz & Pitaevski,1984).As shown by Mauri et al. (1996), Eq. (3) can also bederived from  " rst principles, assuming that the A } A andthe B } B intermolecular forces are equal to each otherand larger than the A } B intermolecular forces, i.e. F  " F  ' F  , obtaining an expression for    whichdepends on ( F  ! F  ). In the following, we shall as-sume that  P  is  " xed, so that the physical state of themixture at equilibrium depends only on ¹ and  x  . Inorder to take into account the e !  ects of spatial in-homogeneities, Cahn and Hilliard (1959) introduced thegeneralized speci " c free energy  g    , which for no- # ux orperiodic boundary conditions is given by the followingexpression: g    " g  !  R ¹ a  (  x  )(  x  ), (4)where  a  represents the typical length of spatial in-homogeneities in the composition. As shown by van derWaals (1979),  a  is proportional to the surface tension 6110  N. Vladimiro v a et al.  /   Chemical Engineering Science 55 (2000) 6109 } 6118  between the two phases [see Eq. (C.4)] and for a systemnear its miscibility curve it is typically of the order of 0.1   m. Below a certain critical temperature ¹  , corre-sponding to values  * 2, the molar free energy given byEqs. (2) and (3) is a double-well potential, and thereforea  " rst-order phase transition will take place. Now, it iswell-known that the molar free energy can be written as(Prausnitz et al., 1986), g  /  R ¹"   x  #   x  , (5)where     and     denote the chemical potential of speciesA and B in solution, respectively, i.e.   " 1 R ¹  ( cg  )  c  ,    " 1 R ¹  ( cg  )  c  , (6)with  c  " cx   and  c  " cx   denoting the mole densities,that is the number of moles per unit volume of speciesA and B, respectively, and  c " c  # c   is the total moledensity. Consequently, we see that the two quantities  " x   and (   !   ) are thermodynamically con- jugated, that is (   !   ) " d( g  /  R ¹ )/d  . This resultwas extended by Cahn and Hilliard (1959), de " ning thegeneralized chemical potential    :   "  ( g    /  R ¹ )    (7)and substituting Eqs. (1) } (4) into Eq. (7) we obtain   "   # log   1 !  #  (1 ! 2  ) ! a     , (8)where    " ( g  ! g  )/  R ¹ . 2.2. The equations of motion Imposingthat the numberof particles of each species isconserved, we obtain the continuity equations (see Bird,Stewart & Lightfoot, 1960, Chapter 16)  c   t #  ) ( c  v  ) " 0, (9)  c   t #  ) ( c  v  ) " 0, (10)where  v   and  v   are the mean velocities of species A andB, respectively. For an incompressible mixture composedof species with equal physical properties, Eqs. (9) and (10)lead to the following continuity equations in terms of themass fraction    of the A species (which is equal to itsmole fraction):    t # v )   "! 1   )  j , (11)  ) v " 0, (12)where   " cM   is the mixture mass density,  j "   (1 !  )( v  ! v  ) is the di !  usive mass  # ux, and  v  isthe average velocity of the mixture,  v " x  v  # x  v  .Thevelocities v   and v   are thesums ofa convectivepart, v , and a di !  usive part: v  " v ! D    ,  v  " v ! D    , (13)where  D  is a composition-independent di !  usion coe $ c-ient, and we have assumed that the di !  usive parts of  v   and  v   are proportional to the gradients of the chem-ical potentials (see Appendix B for a justi " cation of thisassumption, where  D  is de " ned as the Onsager mobilitycoe $ cient relating the molar  # ux to its thermodyn-amically conjugated force). Consequently, the di !  usive # ux becomes  j "!   (1 !  ) D    . (14)Finally, substituting Eq. (8) into Eq. (14), we obtain  j  "! D   # D  (1 !  )  [ a      # 2    # (2  ! 1)   ]. (15)This expression for  j  coincides with that used in Mauriet al. (1996). The term  D    in Eq. (15) represents theregular di !  usion  # ux, while the last term vanishes forsmall concentrations of either solvents (  P 0 or 1) andfor ideal mixtures ( a "  " 0). Note that the  a   term isalways stabilizing and is relevant only at small lengthscales, while    is a known function of the temperature,and near the critical temperature ¹   it is proportional to( ¹  !¹ ). If the  # ow is slow enough that the dynamicterms in the Navier } Stokes equation can be neglected,conservation of momentum leads to the following Stokesequation:   v !  p "! F ( , (16)where    is the mixture viscosity, which, we assume, iscomposition independent, while  F (  is a body force. Thislatter, in turn, equals the gradient of the free energy, andtherefore it is driven by the concentration gradients with-in the mixture (Valls & Farrell, 1993): F ( "  M   g "   R ¹ M        . (17)In Appendix C we derive the generalform of   F (  when thebinary mixture is composed of two species having di !  er-ent physical properties. In addition, it is shown that,when the mixture is composed of well-de " nedsingle-phase domains separated by a thin interfacelocated at  r " r  , the body force  F (  can be interpreted asa capillary force at  r  , i.e. [cf. Eq. (C.6)] F ( ( r ) " [ n  (  # ( I ! n  ( n  ( ) )   ]  [ n  (  ) ( r ! r  )], (18)where    is the surface tension, while  n  (  and    are the unitvector perpendicularto the interface and the curvature at r  , respectively. Physically,  F (  tends to minimize theenergy stored at the interface, and therefore it drives, say,  N. Vladimiro v a et al.  /   Chemical Engineering Science 55 (2000) 6109 } 6118  6111  A-rich drops towards A-rich regions, enhancing coales-cence. Note that Eq. (16) can also be written as    v !  p  "   R ¹ M        , (19)with  p  " p ! (  R ¹ /  M  )    . Eqs. (11), (12) and (19) con-stitute the so-called model H (Hohenberg & Halperin,1977). Now we restrict our analysis to two-dimensionalsystems, so that the velocity  v  can be expressed in termsof a stream function   , i.e.  v  "  /   r   and v  "!  /   r  . Consequently, substituting Eq. (15) intoEq. (11) and Eq. (17) into Eq. (16), we obtain    t "      ! 1   )  j , (20)    "   R ¹ M          , (21)where A  B " A  B  ! A  B  .Since the main mechanism of mass transport at thebeginning of the phase segregation is di !  usion, the lengthscaleof theprocessis the microscopiclength  a . Therefore,using the scaling r    " 1 a r ,  t   I  " Da   t ,     I  " 1  D    (22)and substituting Eq. (15) into Eq. (20) and Eq. (8) intoEq. (21), we obtain    t   I  "     I     I      I   #    I   ) (    I   !  (1 !  )[2  #    I   ]    I   ),(23)    I      I  "!    I  (    I    )     I   , (24)where  " a  D  R ¹ M  . (25)The non-dimensional number    is the ratio betweenthermal and viscous forces, and was interpreted by Jas-now and Vin    als (1996) and Vladimirova et al. (1999a) asthe inverse capillary number. In this work, however, weprefer to denote    as the Peclet number, that is the ratiobetween the di !  usion time scale,  t  " a  /  D , and itsconvective counterpart,  t  " a /  < , or, equivalently, theratiobetween the convectiveand the di !  usive mass # uxesin the convection } di !  usion equation (23), i.e.  " < a /  D . Here,  <  is a characteristic random velocity,which can be estimated through Eqs. (16) and (17) as < & F ( a  /   , where  F ( &  R ¹ /  aM  , and can be thoughtof as the mean value of the random velocity  " eld inducedby the non-homogeneous concentration distribution.A similar, so called  ` # uidity a  parameter was also de " nedby Tanaka and Araki (1998).For systems with very large viscosity,  is small, so thatthe model describes the di !  usion-driven separation pro-cess of polymer melts and alloys. For most liquids, how-ever,    is very large, with typical values ranging from 10  to 10  . Therefore, it appears that di !  usion is importantonly at the very beginning of the separation process, inthat it creates a non-uniform concentration  " eld. Then,the concentration-gradient-dependent capillary force in-duces the convective material  # ux which is the dominantmechanism for mass transport. At no time, however, thedi !  usive term in Eq. (16) can be neglected, as it stabilizesthe interface and saturates the initial exponential growth(see Vladimirova et al., 1999b).In addition to   , another non-dimensional group,   ,can be de " ned, re # ecting the competition between thedrift experienced by the drops due to the initial concen-tration gradient      and the random  # uctuations due totheir interactions with the local environment. Therefore,   equals the ratio between the di !  usion timescale, t  " a  /  D , and the drift timescale,  t  " a /  <  ,where <  &  D (    ) is the drift velocity of a drop withan  O ( a ) radius (see Vladimirova et al., 1999a), obtaining,  "  ( a    ). (26)We see that when    1 random  # uctuations prevail andthe process is identical to the phase separation of an initially homogeneous system, as described inVladimirova et al. (1999b). On the other hand, when   1, drift prevails and drops migrate by di !  usiophoresisas seen in Vladimirova et al. (1999a). In this article, wewill study how these two processes are interrelated.When it is easier, instead of     we will use the ratio  "  " a    , (27)de " ning the non-dimensional concentration gradient.Clearly, any two of the three non-dimensional numbers  ,   and    completely characterize the phase separationprocess. 3. Numerical results The governing equations (23) and (24) were solved ona uniform two-dimensional square grid with constantwidth (( x  , y  ) " ( i  x ,  j  y ), i " 1, N  ,  j " 1, N  ) and timediscretization ( t " n  t , n " 0,1,2, 2 ). The physical di-mensions of the grid were chosen such that  x /  a ,   y /  a " 2, while the time step   t  satis " ed  t /( a  /  D ) + 0.1 ! 0.001. The choice of the time step   t wasdetermined semi-empiricallyin order to maintain thestability of the numerical scheme. Note that the non-linearity of the equations prevents a rigorous derivationof the stability constraints on   t , but one can roughlyestimate that the size of    t  will scale as  O (  x  ,  y  ), 6112  N. Vladimiro v a et al.  /   Chemical Engineering Science 55 (2000) 6109 } 6118  which is the order of the highest operator in the dis-cretized system. The space discretization was based ona cell-centered approximation of both the concentrationvariable    and of the stream function   . The spatialderivatives in the right-hand side of Eqs. (23) and (24)were discretized using a straightforward second-order-accurate approximation. The time integration from t  " n  t  to  t   " ( n # 1)  t  was achieved in two steps.First, we computed the stream function    by solving thebiharmonic equation (24) with the source term evaluatedat time  t  " n  t . The biharmonic equations was solvedusing the  DBIHAR  routine from  NETLIB  (Bjorstad,1980). Second, Eq. (23) was advanced in time, using thevelocity  " eld computed from the updated stream func-tion and a straightforward explicit Eulerian step. Thismakes the entire scheme  O (  t ) accurate in time, which isacceptable for our problem, since the size of the time stepwas kept very small anyway by the stability constraints.In addition, it should be stressed that the stream function  depends on high-order derivatives of the concentrationand therefore it is very sensitive to the concentrationpro " le within the interface.The boundary conditions were no- # ux for the concen-tration  " eld and no-slip for the velocity  " eld and thediscretization of the derivatives near the boundaries wasmodi " ed to use only interior points. In general, anyway,our results were not very sensitive to the precise treat-ment of the bounday conditions, since all gradients re-mained close to zero near the boundaries. Finally, thebackground noise was simulated generating a randomconcentration  " eld of amplitude    " 0.01, which wasuncorrelated both in space and in time. Changing theform and the intensity of the noise, however, did nota !  ect the coarsening process, as one would expect sincethe system is far from equilibrium (see Vladimirova et al.(1998) for a discussion about this point).Eqs. (23) and (24) were solved with the initial condi-tions that the mixture is quiescent and a given concentra-tion gradient is imposed along the  y -direction, i.e.   " 0.5 #  ( y /  a ), where    is the constant non-dimen-sional concentration gradient de " ned in Eq. (27). Notethat, as the axis  y " 0 is at the center of the cell, the meanconcentration is     " 0.5.Eqs. (23) and (24) were solved for di !  erent values of thePeclet number    and the concentration gradient  G . Inmost simulations, the Margules parameter was kept  " 2.1, corresponding to its value for the critical mix-ture used in the experimental studies by Gupta et al.(1999, 2000). However, simulations with di !  erent valuesof     were also performed, obtaining very similar results.The dependence of the phase separation on the Pecletnumber is represented in Fig. 1, where    is equal to0.3  10  . Here, the  " rst row of images representsthe results for  " 0, e.g., for the case when di !  usionis theonly mechanism of mass transfer, showing that phaseseparation starts to occur at the center of the system,where    & 0.5, and then propagates towards the wallsso that, soon after the quench, the morphology of the system consists of dendroid-like structures near thecenter of the cell and of isolated drops elsewhere. Themean composition within (and without) these structureschanges rapidly, as at time  t " 6  10  a  /  D  we alreadysee two clearly distinguishable phases with almostuniform concentrations equal to 0.59 and 0.41, whileat equilibrium their respective compositions are    " 0.685 and     " 0.315. After this early stage, thestructures start to grow, increasing their thickness andreducing the total interface area, while at the same timethe composition within the domains approaches its equi-librium value. This, however, is a slow process, drivenonly by di !  usion, and at time  t " 8  10  a  /  D  the phasedomains still have a dendroid-like geometry with a char-acteristic width which is just twice as large as its initialvalue. For non-zero bulk  # ow, i.e. for   O 0, single-phasedomains grow much faster than when molecular di !  u-sion is the only transport mechanism. As in the case of systems with homogeneous initial composition, we seethat up to   + 10   domain growth still follows the samepattern as for   " 0:  " rst, single-phase domains start toappear, separated from each other by sharp interfaces,and only later these structures grow, with increasinggrowth rate for larger   . When   ' 10  , however, phaseseparation occurs simultaneously with the growth pro-cess, as circular (due to the e !  ect of surface tension)single-phase domains move fast while they grow, absorb-ing material from the bulk, colliding with each other andcoalescing. The motion of these drops is the e !  ect of thenet attractive force between domains of like compositionwhich derives from the previously discussed non-equilib-rium capillary force (Gupta et al., 1999). As such, thismotion is composed of two parts: a random component,which is observed also for homogeneous mixtures(Vladimirova et al., 1999b), and a drift motion, whereA-rich microdomains tend to move upward, i.e. towardsthe A-rich phase, while B-rich microdomains tend tomove downward (Vladimirova et al., 1999a), where theyare reabsorbed. Now, while up to   + 10   the randomcomponent appears to prevail, a further increase of     de-lays the formation of sharp interfaces, thus, preservingthe initial concentration gradient for a longer time, thusenhancing the reabsorption process. In fact, comparingthe last images of Fig. 1 we see that, at the end of theseparation process, only a small number of tiny A-richdroplets remain entrapped within the B-rich phase (andlikewise B-rich droplets in the A-rich phase) when  " 10  , compared with many more and larger dropletsbeing trapped for smaller   . This result is not surprising,as the drift convective bulk  # ow is proportional to thePeclet number (and to the concentration gradient aswell), so that both the drop size and the thickness of theregion where drops are con " ned tend to decrease as   increases. Therefore, drift tends to prevail over random  N. Vladimiro v a et al.  /   Chemical Engineering Science 55 (2000) 6109 } 6118  6113
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