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Drop Size Evolution during the Phase Separation of Liquid Mixtures† Filomena Califano‡ and Roberto Mauri*,‡,§ Department of Chemical Engineering, The City College of CUNY, New York, New York 10031, and Department of Chemical Engineering, DICCISM, University of Pisa, 56126 Pisa, Italy
After quenching a partially miscible, initially homogeneous, critical liquid mixture to a temperature T deeply below its critical point of miscibility, we observed the formation of rapidly coalescing droplets, whose size grows linearly with time, thus indicating that the phase separation process is driven by convection. Eventually, when their size reaches a critical length, which is roughly equal to one-tenth of the capillary length, the nucleating drops start sedimenting and the two phases rapidly segregate by gravity. This behavior was observed for both densitysegregated and quasi-isopycnic systems, showing that gravity cannot be the driving force responsible for the enhancement of the coalescence among the nucleating drops. This result is in line with previous theoretical works based on the diffuse interface model, predicting that the phase separation of low-viscosity liquid mixtures is a convection-driven process, induced by a body force which is proportional to the chemical potential gradients. Finally, at later times, following the evolution of isolated drops of the secondary emulsion, we saw that their size grows in time like t1/3. 1. Introduction When a binary mixture with critical composition is quenched (or heated) from its single-phase region to a temperature below (above) the composition-dependent spinodal curve, it phase separates through a process called spinodal decomposition,1 which is characterized by the spontaneous formation of single-phase domains, which then proceed to grow and coalesce. Unlike nucleation, where an activation energy is required to initiate the separation, spinodal decomposition involves the growth of any fluctuations whose wavelength exceeds a critical value. In the early experiments on spinodal decomposition, Chou and Goldburg2 and Wong and Knobler et al.3 critically retarded the phase separation process by quenching the system to a temperature only by a few millikelvin below the critical value. Similar slowing can be achieved by studying polymer blends,4 with viscosities hundreds of times larger than water’s. While in these early experiments light scattering techniques were used, more recently the dynamics of phase separation has been observed directly as well with, again, either very small quenches5 or using highviscosity systems.6 Most of these experimental studies observed that, right after the temperature of the system has crossed that of the miscibility curve, the solution starts to separate by diffusion and coalescence, leading to the formation of well-defined patches, whose average concentration approaches its equilibrium value.2,3,5 Eventually, these patches become large enough that buoyancy dominates surface tension effects (i.e., the size R of these domains exceeds the capillary length) and the mixture separates by gravity. In general, from a simple dimensional analysis, we expect that the typical †
“A wooden boy?!” (R. Shinnar, freely quoted from Pinocchio, by Collodi, 1864, via Walt Disney). * To whom correspondence should be addresses. E-mail:
[email protected]. ‡ The City College of CUNY. § University of Pisa.
size of these domains should grow with time according to a power law, that is, R ∝ tn, with an exponent n ) 1/3 when diffusion is the dominant mechanism of material transport,7 while when hydrodynamic, long-range interactions become important, we find either n ) 1 or n ) 2/3, depending, respectively, on whether viscous or inertial forces are dominant.8,9 This theoretical analysis is in good agreement with most of the experimental studies2,3,5 of the spinodal decomposition of liquid mixtures, where it is shown that, after a short initial stage where R ∝ t1/3, the size of the single-phase domains grows linearly with time, that is, R ∝ t. Since in these experiments the mixtures completed their phase separation process before reaching the final, inertia-dominated stage, the scaling R ∝ t2/3 was never observed. The linear growth of the typical size of the nucleating droplets is often associated with the convection-driven coalescence predicted by Binder et al.10 The nature of this convective driving force in phase-separating systems is well-explained by the so-called diffuse interface model11,12 (otherwise13 called model H) as the result of the minimization of the interfacial energy, inducing a (nonequilibrium) body force that is proportional to the gradient of the chemical potential. In particular, at the late stages of phase separation, after the system has developed well-defined phase interfaces, this body force reduces to the more conventional surface tension, as shown by Jasnow and Vin˜als14 and by Jacqmin15 so that the driving force can be thought of as a nonequilibrium attractive capillary force16-18 among drops. 2. Previous Results This research program was fueled by an early work, where we studied the possibility to use phase separation of partially miscible solvent mixtures to design a more effective extraction process.19 The idea was to first bring the system into its single-phase region by heating it up and then induce phase separation by cooling it down.20,21 Clearly, fast coalescence was essential so that at the
10.1021/ie030201m CCC: $27.50 © 2004 American Chemical Society Published on Web 08/06/2003
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Figure 1. Phase separation of system I (density-segregated mixture: 58% acetonitrile-38% water-4% toluene) at time t ) 1.8, 2.4, 3, 4, 5, and 7 s.
end of the cooling stages the mixture would be completely separated into two phases and ready to be reheated in the next stage. Understandingly, we were concerned that the phase segregation resulting from fast cooling could be the limiting step of the extraction process, as we thought that nucleation and growth, followed by coalescence, would be strongly retarded by the presence of surface-active compounds within our system. As it turned out, the separation process is very fast, and hardly affected by coalescence-retarding impurities, whereas the same system, when agitated isothermally, forms stable emulsions that are very difficult to break.22 A dramatic example of this phenomenon was the extraction of a fermentation broth that, when mixed with conventional immiscible solvents, formed stable emulsions that took several hours to separate, while it phase-segregated within 20 s after being mixed with a partially miscible solvent and then quenched across its miscibility curve.21 In recent years, we explained the physical mechanism that causes such rapid coalescence and phase segregation. Experimentally, we saw23,24 that after quenching a partially miscible, initially homogeneous, liquid mixture to a temperature deeply below its critical point of miscibility, we observed the formation of rapidly growing single-phase microdomains, reaching a few millimeters within seconds. While these domains were interconnected dendritic structures when the mixture had critical composition, they appeared to be spherical drops in the off-critical case. In both cases, the singlephase microdomains appeared to grow linearly with time, with a growth rate that was larger in the critical than in the off-critical case, with a growth rate of O(100 µm/s), thus proving (although indirectly) that convection is the dominant transport mechanism during the phase separation of liquid mixtures. This conclusion was reinforced by the fact that the nucleating droplets move in random directions at speeds exceeding 1 mm/s. Similar speeds were also measured by Santonicola et al.25 by suspending micrometer-size particles within a phase-separating inhomogeneous liquid mixture. In addition, we also showed experimentally23 that the convection-driven morphology of phase-separating liquid mixtures is almost independent of the presence of surface-active compounds, indicating that the convective
forces that induce drop coalescence are much larger than any surfactant-driven repulsive interactions. The source of this rapid movement cannot be gravity nor molecular diffusion, as these mechanisms would predict drop velocities a few orders of magnitude smaller than those observed experimentally. In addition, the effect of gravity can be ruled out considering that (a) when the temperature of the system is near its critical value, the density difference between the two phases is very small, thus reducing the sedimentation speed, and (b) the observed morphologies did not depend on whether the cell was kept horizontal or vertical during the separation process. These experimental findings confirm the results of our simulations26,27 that (a) phase segregation is driven by convection, and not by diffusion and neither by gravity, and (b) during most of the phase segregation process the system is far from local equilibrium, explaining why we observe bulk motion even when the system is composed of nucleating drops with sharp interfaces (trivially, no bulk flow would exist if they were at equilibrium). Theoretically, we described the process of spinodal decomposition in fluid mixtures within the framework of the Ginzburg-Landau theory of phase transition due to Cahn and Hilliard,28 who generalized van der Waals’ early approach.29 During the early stages of the process, initial instabilities grow exponentially, forming, at the end, single-phase microdomains whose size corresponds to the fastest-growing mode of the linear regime.30 Coupling this thermodynamic approach to fluid mechanics, we simulated phase transition following the socalled diffuse interface model.11-13 Here, the equations of conservation of mass and momentum are coupled via the convective term of the convection-diffusion equation, which in turn is driven by a body force appearing in the Navier-Stokes equation, depending on the chemical potential gradients.31 The results of our simulations show that the enhanced coarsening rate is due to the strong coupling between concentration and velocity fields, while the morphology changes of the system strongly depends on the relative importance of convection and diffusion.32 In cases where diffusion is the only transport mechanism, we found33 that the typical size R of the nucleating droplets grows in time like t1/3, in agreement with both analytical calculations7 and di-
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Figure 2. Phase separation of system II (quasi-isopycnic mixture: 50% 2-propanone, 50% hexadecane) at time t ) 0, 1, 2, 3, 4, 5, 7, 8, 10, and 15 s.
mensional analysis.8 On the other hand, when hydrodynamic interactions among droplets become important, our simulations27 showed that phase separation is convection-driven, as the convective mass flow resulting from nonequilibrium capillary effects accelerates the coalescence among drops so that R ∝ t, as predicted from dimensional analysis.8,9 We can summarize our findings as follows. After the initial, diffusion-driven stage leads to a nonuniform concentration field, the nonequilibrium capillary driving force induces a material flux, which in our case is several orders of magnitude larger than its diffusive counterpart. Then, our simulations showed the formation of microdomains of one phase separated from the other phase by sharp interfaces, with compositions that are far from equilibrium. Finally, at later times, the system approaches local equilibrium, but it can reach it only when the droplet size has exceeded its capillary length, at which point the droplets will rapidly sediment.24 3. Experimental Setup An experimental setup was designed and built to allow the observation of the phase separation process
in the size range of 0.1-10 mm. It consisted of a temperature-regulated, 1-mm-thick, 1-cm-wide, 4-cmhigh sample cell and a digital camera (Fuji FinePix S1 Pro) with high-resolution and high-speed continuous shooting (up to 5 frames/s). The temperature was regulated by placing the sample cell into an 8-mm-thick water jacket, into which temperature-controlled water was circulated, allowing a 3 °C/s quenching rate. In our experiments we used two critical liquid mixtures. The first (system I) is the mixture that was extensively studied in previous works21,22 with 38% water, 58% acetonitrile, 4% toluene critical volumetric composition, Tc,I ) 35 °C phase transition temperature, and µ1 ) 1.2 cP viscosity. At ambient temperature, this mixture separates into two phases, with a density difference ∆FI ) 7 × 10-2 g/cm3 and surface tension σI ) 1.3 dyn/cm, so that its capillary length is rc,I ) x[σI/(g∆FI)] ) 1.3 mm. The second mixture (system II) is the quasi-isopycnic system (i.e., a system that separates into two phases having, approximately, the same density) that was described in Santonicola et al.,25 with 50% acetone, 50% hexadecane critical volumetric composition, Tc,II ) 27 °C critical temperature, and µ1I )
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Figure 4. Left: isolated drops during the secondary nucleation in system I. Right: time evolution of the mean drop size R. The continuous line represents the curve R ) 60t1/3. Figure 3. Drop size as a function of time for system I (triangles) and system II (squares).
2.3 cP viscosity. At ambient temperature, this mixture separates into two phases, with a density difference ∆FII ) 6 × 10-4 g/cm3 and surface tension σII ) 1.2 dyn/cm, so that its capillary length rc,II is about 10 times larger than rc,I, that is, rc,II ) 1.4 cm. In addition, 50 ppm of Oil Red O and 10 ppm of Crystal Violet were added to system I and system II, respectively, to enhance the visualization of the two phases as they separate. When dissolved in such small percents, these dyes do not change the phase diagrams of the mixtures, nor the characteristics of the phase separation process. In all our experiments, we started with the mixtures in their phase-separated state at a constant temperature of 20 °C. Then, the solutions were first heated to 40 °C, then mixed thoroughly, and finally, without agitation, quenched back to 20 °C, with a quench rate of about 3 °C/s. Mixing the solutions before the quench is extremely important because we want to study the behavior of initially homogeneous mixtures. In fact, Santonicola et al.25 showed that when the system is kept at 38 °C without mixing for 2 h, the mixture is still mostly demixed, with the exception of a thin, few millimeters thick layer around the phase interface, where there is a sharp concentration gradient. Experimental Results and Discussion The results of the macroscopic visualization of the phase separation of systems I and II are shown in Figures 1 and 2, respectively, with the time t ) 0 corresponding to the moment when the temperature of the mixture reaches its critical value and crosses the miscibility curve. For system I, these results complement those reported in Gupta et al.,23 where an optical microscope was used to follow the evolution of the nucleating drops in the 10-100-µm range. Enlarging the central part of these pictures (i.e., omitting the wall regions), we determined the mean radius of the drops, showing (see Figure 3) that for both systems it grows linearly with time, thereby showing that the behavior of the quasi-isopycnic system is analogous to that of the density-segregated mixture. In fact, in both cases droplets form and grow linearly until, when they reach a critical size, they start sedimenting and the mixture separates by gravity. We observe that, for system I, the droplets begin to sediment after ∼1 s when their typical size is ∼0.1 mm, while for system II the droplets begin to sediment after ∼2-3 s, when their typical size is ∼1 mm. Therefore, it appears that the critical size of the droplets is about one-tenth of their capillary length, and therefore, as the capillary length is 10 times larger in
system II than that in system I, it is not surprising that the separation process is slower for system II than for system I, despite the fact that in system II drops grow twice as fast as those in system I. Finally, at the end of the process, we observed that some drops had remained “trapped” in the other phase, constituting what is generally referred to as a “secondary emulsion”. These drops are, thermodynamically, in a metastable state; that is, their composition is very close to its equilibrium value. They should not be confused with those which remain adsorbed at the side walls, where other factors, such as the wettability of the walls, become relevant and which may well be at equilibrium. In Figure 4 we show some of the secondary emulsion drops, which have been trapped within the upper phase of system I, 1 min after the quench and at a distance of 2 cm from the interface and 0.5 mm from the side walls (i.e., we focused the microscope as far as possible from the side walls). Following their evolution in time, we saw that their mean size grows in time as t1/3, confirming that the growth of metastable nucleating drops is dominated by diffusion. We may conclude that since the same linear growth was observed for both density-segregated and quasiisopycnic systems, the enhanced coalescence among the nucleating droplets is not induced by gravity. In contrast, this behavior is well-predicted using the diffuse interface model,23 determining that the typical droplet size R grows linearly in time as dR/dt ) kbσ/η, where σ is the surface tension, η the viscosity of the mixture, and kb a nondimensional constant, which in our case gives kb ≈ 10-4, in good agreement with the experimental results. Acknowledgment This work was supported by the National Science Foundation, Division of Chemical and Transport Systems. Literature Cited (1) Gunton, J. D.; SanMiguel, M.; Sahni, P. S. In Phase Transition and Critical Phenomena; Domb, C., Lebowitz, J. L., Eds.; Academic Press: London, 1983; Vol. 8. (2) Chou, Y. C.; Goldburg, W. I. Phase Separation and Coalescence in Critically Quenched Isobutyric-Acid-Water and 2-6Lutidine-Water Mixtures. Phys. Rev. A 1979, 20, 2105 and references therein. (3) Wong, N. C.; Knobler, C. Light-Scattering Studies of Phase Separation in Isobutyric Acid + Water Mixtures: Hydrodynamic Effects. Phys. Rev. A 1981, 24, 3205 and references therein.
Ind. Eng. Chem. Res., Vol. 43, No. 2, 2004 353 (4) Cumming, A.; Wiltzius, P.; Bates, F. S.; Rosedale, J. H. Light-Scattering Experiments on Phase-Separation Dynamics in Binary Fluid Mixtures. Phys. Rev. A 1992, 45, 885 and references therein. (5) Guenoun, P.; Gastaud, R.; Perrot F.; Beysens, D. Spinodal Decomposition Patterns in an Isodensity Critical Binary Fluid: Direct-Visualization and Light-Scattering Analyses. Phys. Rev. A 1987, 36, 4876. (6) White, W. R.; Wiltzius, P. Real Space Measurement of Structure in Phase Separating Binary Fluid Mixtures. Phys. Rev. Lett. 1995, 75, 3012. (7) Lifshitz, E. M.; Pitaevskii, L. P. Physical Kinetics; Pergamon Press: New York, 1984; Chapter 12. (8) Siggia, E. Late Stages of Spinodal Decomposition in Binary Mixtures. Phys. Rev. A 1979, 20, 595. (9) Furukawa, H. Role of Inertia in the Late Stage of the Phase Separation of a Fluid. Physica A 1994, 204, 237. (10) Binder, K,; Stauffer, D.; Mu¨ller-Krumbhaar, H. Theory for the dynamics of clusters near the critical point. I. Relaxation of the Glauber kinetic Ising model. Phys. Rev. B 1975, 12, 5261 and references therein. (11) Anderson, D. M.; McFadden, G. B.; Wheeler, A. A. DiffuseInterface Methods in Fluid Mechanics. Annu. Rev. Fluid Mech. 1998, 30, 139. (12) Lowengrub, J.; Truskinovsky, L. Quasi-Incompressible Cahn-Hilliard Fluids and Topological Transitions. Proc. R. Soc. London, Ser. A 1998, 454, 2617. (13) Hohenberg, P. C.; Halperin, B. I. Theory of Dynamic Critical Phenomena. Rev. Mod. Phys. 1977, 49, 435. (14) Jasnow, D.; Vin˜als, J. Coarse-Grained Description of Thermo-Capillary Flow. Phys. Fluids 1996, 8, 660. (15) Jacqmin, D. Contact-Line Dynamics of a Diffuse Fluid Interface. J. Fluid Mech. 2000, 402, 57. (16) Valls O. T.; Farrell, J. E. Spinodal Decomposition in a Three-Dimensional Fluid Model. Phys. Rev. E 1993, 47, R36 and references therein. (17) Tanaka, H. Coarsening Mechanisms of Droplet Spinodal Decomposition in Binary Fluid Mixtures. J. Chem. Phys. 1996, 105, 10099. (18) Tanaka, H.; Araki, T. Spontaneous Double Phase Separation Induced by Rapid Hydrodynamic Coarsening in TwoDimensional Fluid Mixtures. Phys. Rev. Lett. 1998, 81, 389. (19) Shinnar, R.; Mauri, R. Extraction Process. Liquid-Liquid. U.S. Patent, 5,628,906, 1997. (20) Ullmann, A.; Ludmer, Z.; Shinnar, R. Novel Continuous Multistage Extraction Column Based on Phase Transition of Critical-Solution Mixtures. Chem. Eng. Sci. 1997, 52, 567.
(21) Ullmann, A.; Ludmer, Z.; Shinnar, R. Phase Transition Extraction Using Solvent Mixtures with a Critical Point of Miscibility. AIChE J. 1995, 41, 489. (22) Gupta, R.; Mauri R.; Shinnar R. Liquid-Liquid Extraction Using the Composition Induced Phase Separation Process. Ind. Eng. Chem. Res. 1996, 35, 2360. (23) Gupta, R.; Mauri R.; Shinnar R. Phase Separation of Liquid Mixtures in the Presence of Surfactants. Ind. Eng. Chem. Res. 1999, 38, 2418. (24) Mauri, R.; Califano, F.; Calvi, E.; Gupta, R.; Shinnar, R. Convection-Driven Phase Segregation of Deeply Quenched Liquid Mixtures. J. Chem. Phys. 2003, in press. (25) Santonicola, G.; Mauri, R.; Shinnar, R. Phase Separation of Initially Non-Homogeneous Liquid Mixtures. Ind. Eng. Chem. Res. 2001, 40, 2004. (26) Vladimirova, N.; Malagoli, A.; Mauri, R. Diffusio-Phoresis of Two-Dimensional Liquid Droplets in a Phase Separating System. Phys. Rev. E 1999, 60, 2037. (27) Vladimirova, N.; Malagoli, A.; Mauri, R. Two-Dimensional Model of Phase Segregation in Liquid Binary Mixtures. Phys. Rev. E 1999, 60, 6968. (28) Cahn, J. W.; Hilliard, J. E. Free Energy of a Nonuniform System. III. Nucleation in a Two-Component Incompressible Fluid. J. Chem. Phys. 1959, 31, 688. (29) van der Waals, J. D. The Thermodynamic Theory of Capillarity Under the Hypothesis of a Continuous Variation of Density. English translation by Rowlinson, J. S. J. Stat. Phys. 1979, 20, 200. (30) Mauri, R.; Shinnar, R.; Triantafyllou, G. Spinodal Decomposition in Binary Mixtures. Phys. Rev. E 1996, 53, 2613. (31) Vladimirova, N.; Malagoli, A.; Mauri, R. Two-Dimensional Model of Phase Segregation in Liquid Binary Mixtures with an Initial Concentration Gradient. Chem. Eng. Sci. 2000, 55, 6109. (32) Lamorgese, A. G.; Mauri, R. Phase Separation of Liquid Mixtures. In Non-Linear Dynamics and Control in Process Engineering-Recent Advances; Continillo, G., Crescitelli, S., Giona, M., Eds.; Springer: Berlin, 2002; pp 139-152. (33) Vladimirova, N.; Malagoli, A.; Mauri, R. Diffusion-Driven Phase Separation of Deeply Quenched Mixtures. Phys. Rev. E 1998, 58, 7691.
Received for review March 3, 2003 Revised manuscript received May 29, 2003 Accepted May 29, 2003 IE030201M
