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Ind. Eng. Chem. Res. 2001, 40, 2004-2010
Phase Separation of Initially Inhomogeneous Liquid Mixtures Gabriella Santonicola,† Roberto Mauri,*,†,‡ and Reuel Shinnar† Department of Chemical Engineering, The City College of New York, New York, New York 10031, and Dipartimento di Ingegneria Chimica, DICISM, Universita` di Pisa, 56126 Pisa, Italy
In previous works, we showed that the spinodal decomposition of low-viscosity liquid mixtures is driven by the convection induced by chemical potential gradients. In this article, we study the impact of this driving force on the phase separation of low-viscosity liquid mixtures in which a strong initial concentration gradient is present within a few-millimeter-thick layer. This region was created by keeping an initially demixed mixture at a higher-than-critical temperature for 0.5 h and allowing it to mix by diffusion. After a deep and rapid quench within the spinodal range, the mixtures at first remained macroscopically unchanged; then micron-sized drops appeared; and finally, after few seconds, a sharp interface suddenly formed, with droplet sizes never exceeding 10 µm. As it took hours for diffusion and only seconds for phase separation, this result shows that the latter process cannot be driven by diffusion. This conclusion was reinforced by adding glass particles to the mixture and observing the velocity pattern, which had speeds exceeding 0.5 mm/s, thus demonstrating that the process is driven by convection. The impact of gravity was ruled out, as the morphology of a density-segregated mixture as it phase separates appears to be the same as that of an isodensity mixture. These effects can be explained considering that the large initial concentration gradient induces a body force that drives small drops toward one or the other of the homogeneous phases, where they are rapidly reabsorbed, thus explaining why larger drops were not observed during the separation process. In addition, the measured bulk velocities correlate with the magnitude of the chemical potential gradients, in agreement with the theoretical model. 1. Introduction In this work, we study the physical mechanism governing the phase segregation occurring after the quenching of an initially nonhomogeneous liquid mixture to a temperature T deeply below its critical point of miscibility. In particular, by visualizing the bulk flow occurring within the phase-separating system, we show that the process is governed by the convection induced by concentration gradients. In most of the previous experimental studies on the phase separation of liquid mixtures, the system prior to the temperature quench was at equilibrium in its single-phase state. Then, the segregation process was critically retarded either by quenching the mixtures to a temperature T only by few millikelvin below the critical value Tc, i.e., to reduced temperature τ ) (T Tc)/Tc < 10-5,1-3 or by studying polymer blends with viscosities hundreds of times larger than water’s.4,5 It was observed that, right after the temperature of the system crosses that of the miscibility curve, an initially homogeneous mixture starts to separate by diffusion only, leading to the formation of well-defined patches with near-equilibrium average concentrations. The observed morphology depends strongly on the composition of the phase-separating mixture: for critical mixtures, it consists of dendritic, interconnected domains, whereas for off-critical systems, it is composed of spherical drops. Then, in the so-called “late” stage of coarsening, these patches grow by diffusion and coalescence, until they * To whom correspondence should be addressed. Telephone: ++39-050-511248. Fax: ++39-050-5111266. E-mail:
[email protected]. † The City College of New York. ‡ Universita ` di Pisa.
become large enough that buoyancy dominates surface tension effects and the mixture separates by gravity. This occurs when the size of the domains becomes equal to the capillary length, Lmax ) O(σ/g∆F), where σ is the surface tension, g the gravity field, and ∆F the density difference between the two separating phases.6,7 In the case of a typical liquid mixture, that would correspond to Lmax ) O(1 mm). Now, if diffusion were the only driving force, it is well-known, both experimentally1 and theoretically,8 that the typical size L of a phaseseparating domain grows with time as t1/3, and therefore, it would take a time of O(1 h) to form such millimeter-sized drops. Obviously, whereas similar times are needed for phase segregation of polymer melts and alloys, liquid mixtures separate within seconds from the temperature quench, and therefore, diffusion and buoyancy alone cannot explain the segregation process. The other mechanism of growth is convection-driven coalescence, which implies that drops move against each other under the influence of a capillary force. Balancing surface tension with viscous forces, it can be shown6 that the typical domain grows linearly with time, i.e., L(t) ∝ t, which agrees with most experimental measurements.1-3 The nature of this convective driving force in phase-separating systems is well explained by the socalled model H9 as the result of the minimization of the interfacial energy, inducing a (nonequilibrium) surfacetension-driven bulk flow. In fact, Tanaka and Araki10 and Vladimirova, Malagoli, and Mauri11,12 showed by numerical simulation that spinodal decomposition of fluid mixtures strongly depends on the relative importance of convection and diffusion and that the enhanced coarsening rate is due to the strong coupling between concentration and velocity fields. The critical role of convection in the phase segregation of deeply quenched
10.1021/ie000798v CCC: $20.00 © 2001 American Chemical Society Published on Web 03/22/2001
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liquid mixtures was uncovered in the experimental results by Gupta, Mauri, and Shinnar,13 showing that, when an initially homogeneous liquid is deeply quenched across its miscibility curve, single-phase microdomains form immediately and then grow linearly in time within the range of 10-400 µm. In addition, the coarsening 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. Recently, Gupta14 studied the phase segregation of a mixture that, before the quench, was demixed everywhere, with the exception of a few-millimeter-thick region where a concentration gradient was present. This case was studied by Beysens, Jayalakshmi, and Khalil15,16 for weak concentration gradients and very shallow critical quenches, who observed that the morphology of phase separation depends exclusively on the local initial composition of the system. In the central region, interconnected domains form and grow linearly by coalescence, whereas around that region, isolated droplets are observed that grow by diffusion. In addition, for quasicritical systems presenting an adverse density stratification, Beysens and Jayalakshmi16,17 observed the development at later times of patterns of convective rolls, because of a Rayleigh-Taylor instability. On the other hand, for large concentration gradients and deep critical quenches, Gupta14 observed that the system remains unchanged for a few seconds after the quench, until a sharp interface suddenly appears, with no visible microdomains forming; only then do drops start to appear and move toward the interface, where they coalesce. This surprising phenomenon is analyzed in this work by suspending micron-sized particles within the phase-separating system, so that the existence of any fluid velocity can be visualized. In addition, the behavior of a density-segregated system is compared with that of an isopycnic mixture, i.e., a system whose components have the same density, so that the role of gravity can be studied. After describing, in the next section, the experimental setup and the thermodynamic properties of the liquid mixture that was used in our experiments, in section 3, we illustrate the results obtained. Finally, in the last section, we summarize the results and draw a few conclusions. In particular, we intend to compare our results with those obtained when the liquid mixture is initially uniform, showing that, in both cases, phase separation is driven by the convection induced by a nonequilibrium capillary force, as predicted by the model H. 2. Experimental Setup An experimental setup was designed and built to allow for the observation of the phase-separation process in the size range of 10 µm to 7 mm. It consisted of a temperature-regulated, 1-mm-thick, 40-mm-high sample cell; an optical microscope (Nikon Optiphot-2) with a xenon illuminator (Nikon XBO Lamp 75 W); and a video camera (Sony XC-711) and a camera (Nikon F5 with Data Back MF-28) alternately mounted on the microscope. The temperature was regulated by placing the sample cell into an 8-mm-thick water jacket, into which temperature-controlled water was circulated, allowing for a quenching rate of ∼3 °C/s. Temperatures were measured by inserting 350-µm thermocouples, with 0.04-s response times, at various locations inside the
Figure 1. (a) Phase diagrams of system I (water-acetonitriletoluene) and (b) system II (2-propanone-hexadecane). The vertical arrows represent the critical quenches.
cell and connecting them with a data acquisition system. The F-5 Nikon camera was chosen for its continuous high speed, shooting up to 7.4 frames per second. All pictures were taken with a high shutter speed, in the range 1/6400-1/2000 s, and with a 5-µm-field-depth Meteor frame grabber, and Matrox Inspector software was used to analyze images, make movie files, and enhance the image quality. 2.1. Liquid Mixtures. In our experiments, we used two liquid mixtures. The first (system I) was composed of water, acetonitrile, and toluene, whose thermodynamic properties were determined in previous works.18,19 This mixture has a critical volumetric composition of 38% water, 58% acetonitrile, and 4% toluene, and it undergoes phase transition at Tc ) 35 °C (see Figure 1). The second mixture (system II) is an isopycnic system (i.e., a system that separates into two phases having the same density) with a 50% 2-propanone/50% hexadecane critical volumetric composition and a critical temperature of Tc ) 27 °C.20 The phase diagrams of both systems are shown in Figure 1. All solvents were HPLCgrade, and the water was double-distilled. In addition, 50 ppm of Oil Red O dye were added to system I to enhance the visualization of the separation process, as this dye dissolves preferentially in the upper, acetonitrile-rich phase. For the same reason, 10 ppm of Crystal Violet dye were added to system II, as this dye dissolves preferentially in the lower, 2-propanone-rich phase. When dissolved in such small amounts, these dyes do not change the characteristics of the phase-separation process.
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One of the most important stages of our work was the determination of the most appropriate particles to suspend in the mixture for direct observation of the bulk flow. These particles had to have a small enough buoyancy to minimize sedimentation and not interfere with the phase-separation process. In particular, they could not denature at 50 °C, nor could they interact with the solvent mixtures or flocculate. After careful testing, we found that, for system I, the best choice was S60/ 10000 3M Scotchlite glass bubbles, consisting of hollow glass microspheres, 5-50 µm in size, with a density of 0.8-0.9 g/cm3. Of these particles, we selected those whose density lies between the densities of the two separating phases of system I, so that, during the phase transition, they would remain confined in the interface region, while the heavier or lighter particles would either sediment or float out, respectively. 2.2. Experimental Procedure. In all of our experimental studies, the mixtures were initially phasesegregated at a constant below-critical temperature Tl ) Tc - 10 °C. Then, they were heated to a temperature of Th ) Tc + 5 °C and kept at that above-critical temperature for a given, so-called diffusion time of 30 min without any mixing. At the end of this time, the mixture was still mostly demixed, with the exception of a few-millimeter-thick region around the phase interface where the composition, which was initially discontinuous, changed gradually from one to the other of its equilibrium values. Finally, the mixture was quenched back to Tl, which corresponds to a reduced temperature quench t ) (Tc - Tl)/Tc ≈ 0.03, at a quenching rate of dT/dt ≈ 3 °C/s. When an initially phase-segregated mixture is heated above its miscibility curve, the spontaneous mixing process is governed by diffusion alone, and therefore, it progresses slowly. In our case, the thickness of the diffusion layer could easily be estimated, as the color of the mixture at the interface region changed gradually from red to white. The observed thickness of the mixing layer was about 1.8 mm for both systems I and II, which is in agreement with the measured value of the solvent diffusivity (D ≈ 4 × 10-5 cm2/s), obtained using both GC techniques and laser absorption, and which also coincides with the theoretical prediction of the WilkeChang equation. That corresponds to an initial concentration gradient of ∼0.1φA+T/mm for system I and ∼0.3φP/mm for system II, where φA+T and φP are, respectively, the volume fraction of acetonitrile plus toluene and that of 2-propanone. Two other diffusion times were considered, namely, 10 min and 2 h, corresponding to approximately 1 and 3.5 mm thicknesses of the diffusion layer, respectively. However, the results that were obtained in these cases are not reported here, as they were very similar to those obtained with 30 min of diffusion time and did not add any new element to our understanding of the phenomenon. Finally, the convection field of system I was studied by inserting glass particles into the cell with a syringe; then, after waiting until the particles had reached the diffusion region, we quenched the system and monitored the particle motion. On the other hand, the convection field of system II was studied by measuring the velocity of the glass particles during the phase transition and then subtracting their constant sedimentation velocity. As it turned out, the particle velocity was mostly almost horizontal, and therefore, the influence of sedimentation was small.
Figure 2. Phase separation of system I during a critical quench in the presence of an initial concentration gradient. Pictures are taken using a 1.5-mm field of view. Time is measured from the moment when T ) Tc.
Figure 3. Phase separation of system II during a critical quench in the presence of an initial concentration gradient. Pictures are taken using a 1.5-mm field of view. Time is measured from the moment when T ) Tc.
3. Experimental Results The morphologies of systems I and II as they phase separate are shown in Figures 2 and 3, respectively, with a 1.5-mm field of view. As mentioned above, the systems were initially demixed, with the exception of a millimeter-thick mixing region, in which their composition varied gradually from that of the lower to that of the upper phase. Note that, for system I, the lower, water-rich phase is white, while the upper, acetonitrilerich phase is red (i.e., dark, in the black and white Figure 2), whereas for system II, the colors are inverted. Focusing the experimental apparatus on the diffusion region, we see that, after the quench, system I remained unchanged for about 6 s, when, first, micron-sized drops appeared and then, within few seconds, a sharp interface suddenly formed. Only then, did we see larger drops appearing and moving toward the interface, where they coalesced. A very similar morphology was observed in the isopycnic system, where, however, the process was much faster and the interface formed within a few tenths of a second. This behavior looks remarkably different from that of an initially well-mixed (i.e., uniform) mixture, where single-phase domains appear immediately after the temperature of the system crosses
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Figure 6. Average velocity of glass particles as a function of time during phase separation. Time is measured from the moment when T ) Tc. Figure 4. Phase separation of system I with glass particles suspended within the diffusion layer. Time is measured from the moment when T ) Tc.
Figure 5. Trajectory of the glass particle shown in Figure 4.
its critical value and then grow rapidly and steadily until, in density-segregated systems, they become large enough that they start sedimenting by gravity,13 indicating that the process is driven by convection. Figures 2 and 3 offer clear evidence of the fact that phase separation in low-viscosity liquid mixtures is too rapid to be due to diffusion alone, as a typical solvent molecule takes 1 h to diffuse and form the interface region but only a few seconds to move back during phase separation. In addition, the fact that the morphology of isopycnic systems during phase separation changes in the same way as the morphology of density-segregated systems shows that gravity does not play any role in the process. Actually, as we can see from Figures 2 and 3, the isopycnic system separates 10 times faster than the density-segregated one. Similar conclusions were reached by Guenoun et al.3 with shallower quenches. Figures 4-6 represent more direct evidence of the convection-driven nature of the phase-separation process. In Figure 4, the convective bulk flow is visualized by following the trajectory of a glass particle in system I prior to the formation of the interface. As shown in Figure 5, as soon as the temperature of the mixture decreases below its critical value, the particle suddenly accelerates vertically toward the center of the diffusion region and then starts drifting horizontally with increasing speed until the interface appears; at this point, the particle stops, and the mixture returns to its macroscopically quiescent state. Finally, in Figure 6, the
average velocity of 100 glass particles is represented, showing that it increases with time up to values of about 350 µm. These results show that convective movements occur even during the first 5 s following the quench, when the system does not display any macroscopic morphological changes (see Figure 2). Note that, unlike phase transitions in homogeneous systems, where convective motion is random,13 in our case, the particle trajectories are deterministic, directed toward the interphase within the xy plane of view. In fact, had particles moved along the transversal z direction, they would have become out-of-focus (which never happened), as the depth of field of our camera was about 5 µm. To demonstrate that the bulk flow was not due to thermal gradients within the cell, we suspended heavier silica particles in system I, measuring a horizontal velocity field identical to that obtained with glass particles, while in the vertical direction a steady sedimentation speed was superimposed on the “clean” phase-transition-induced velocity field of the glass particles. Now, when system I was replaced with water, we saw that the silica particles continued to sediment during the temperature quench, with no horizontal movement and only a slight decrease in sedimentation speed, because of a progressive increase in the water density as the temperature decreased. In addition, we studied the influence of the wetting properties of the cell walls by comparing the experimental results obtained using an untreated (i.e., hydrophilic) glass wall with those obtained after depositing an OTS layer on the glass wall, thus making the surface hydrophobic. In both cases, we found the morphology of phase separation to be identical, demonstrating that wetting did not affect the phase-separation process 4. Discussion Our experimental results can be interpreted in terms of the so-called model H,9 where the motion of an incompressible binary fluid mixture composed of species A and B is described via a generalized Cahn and Hilliard equation of continuity21 coupled to the NavierStokes equation
F F
(
(∂φ∂t + v‚∇φ) ) ∇‚(D∇µ)
)
( )
∂v F + v‚∇v ) ∇p ) η∇2v(φ - φc)∇µ ∂t MW
Here, φ(r,t) is the concentration of one of the species,
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Figure 7. Simulation of phase separation in an isopycnic binary mixture with an initial concentration gradient. The pictures represent the morphology of the system 0.4 and 0.8 s after the (assumed instantaneous) temperature quench.
v(r,t) is the divergence-free mean velocity field, p(r,t) is the pressure field, and D is the binary diffusion coefficient, while F, η, and MW represent, respectively, the densities, viscosities, and molecular masses of both A and B (note that, in this simplified model, the two species are assumed to have the same density, viscosity, and molecular mass; if some of these assumptions are removed, however, the results do not appear to change qualitatively22). In addition, µ is the difference (µA µB) between the chemical potentials of the two species and is defined as µ ) δg/δφ, where g is the double-welled molar Gibbs free energy. The applicability of model H to the phase separation of liquid mixture has been discussed in detail by Valls and Farrell23 and by Jasnow and Vin˜al,24 and it was generalized by Vladimirova, Malagoli, and Mauri11,12 to the case where D is a function of the local composition. Assuming that inertia is negligible and expressing the free energy g in terms of the Margules parameter Ψ and a typical length a, which represents the typical thickness of the interface at equilibrium, one obtains
[
g ) RT φ log φ + (1 - φ)log(1 - φ) + Ψφ(1 - φ) +
| |]
1 2 a ∇φ 2 2
Vladimirova et al.11,12 showed that model H in two dimensions reduces to a set of two equations, which can easily be integrated numerically and whose predictions can be compared with experiments, as both Ψ and a can be measured. In fact, Ψ can easily be determined from the phase diagram of the mixture, and a depends on the surface tension σ, i.e., the energy stored at the interface25
a)
σMW FRT
The most important characteristics of these governing equations are the antidiffusive molar flux, J ) -D∇µ, and the body force, F ) -(F/MW)(φ - φc)∇µ, appearing in the continuity and Navier-Stokes equations. The former provides the driving force toward phase segregation, while the latter greatly accelerates the process. As shown by Jasnow and Vin˜als,24 the body force reduces to a (nonequilibrium) capillary force when the system is composed of single-phase domains separated by sharp interfaces. In addition, as it tends to minimize the interfacial energy, the body force can also be interpreted
as a (nonequilibrium) attractive force among phaseseparating drops, which greatly accelerates coalescence.26 A quantitative measure of the importance of convection is provided by the Peclet number, NPe ) Va/D, where V is a characteristic velocity, which can be estimated as V ≈ Fa2/η, with F ≈ σ/a2 ≈ (FRT)/(aMW), yielding
NPe )
a2FRT DηMW
The Peclet number coincides with the “fluidity” parameter defined by Tanaka and Araki.10 For systems with very large viscosities, NPe is small, and model H describes a diffusion-driven separation process, as in polymer melts and alloys.22,27 For most liquids, however, NPe is very large. In our case, for example, we have a ≈ 10-5 cm (which can be obtained by knowing that σ ≈ 10 dyn/cm) and D ≈ 10-5 cm2/s, so that NPe ≈ 105. This shows that the chemical potential gradients within the system induce a strong convection, which, as it happens, becomes the dominant mechanism for mass transport. Using this theoretical model, Vladimirova, Malagoli, and Mauri28 recently simulated the phase separation of a partially demixed liquid mixture and obtained results that, in the absence of convection (i.e., for very shallow quenches and/or very viscous liquid mixtures) are identical to those of Lacasta et al.29 and are in agreement with the experimental results by Beysens and Jayalakshmi.15,16 When convection is taken into account, however, Vladimirova et al.28 showed that the morphology of the system is radically different and obtained pictures (see Figure 7) that are strikingly similar to those obtained in our experimental results. They showed that, at a very early stage, the phase separation of an initially demixed system and that of an initially homogeneous mixture are similar to each other, leading to the formation of small microdomains. At this point, however, driven by the nonequilibrium capillary force F, these domains start moving toward one or the other of the homogeneous regions, where they will be reabsorbed, so that larger domains do not have the chance to form during the separation process. This explains why, in our experiments, we could not see drops larger than 10 µm prior to the formation of the interface. In addition, considering that |F| ∝ (φ-φc)2/a (because, at the end of the temperature quench, |φ-φc| ≈ 0.1 for system I, whereas |φ-φc| ≈ 0.3 for system II; see the miscibility curves in Figure 1), this model also accounts for the fact that system II phase separates about 10
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times faster than system I. The fact that the observed bulk flow is mostly horizontal can be explained by considering that the mixture, being incompressible and confined within a fixed volume, can only move horizontally under the influence of the vertical “pinching” capillary force. In addition, as the diffusion layer thickness decreases with time, the concentration gradient increases correspondingly, thus explaining why the bulk velocity increases as phase separation progresses.
In addition, as the difference in composition between the two phases at the end of the temperature quench is about three times larger in system II than it is in system I, this model also predicts that the corresponding driving force is about 10 times larger, which therefore explains why system II phase separates about 10 times faster than system I.
5. Conclusions
This work was supported by the National Science Foundation, Grant CTS-9978781.
In this experimental work, we studied the phase separation of low-viscosity, partially demixed liquid mixtures by direct observation. Two systems were considered, one density-segregated and the other densitymatched (so-called isopycnic mixture). In both cases, the systems, which were initially phase-separated, were first heated and kept at an above-critical temperature for 0.5 h, so that, at the end of this period, the mixtures were still mostly demixed, with the exception of a thin, millimeter-thick layer around the phase interface, where a finite concentration gradient was present because of diffusion. Finally, the mixtures were quenched back to a reduced temperature t ≈ 0.03 with a quenching rate dT/dt ≈ 3 °C. In our visualizations, we showed that, after the quench, both systems remained macroscopically unchanged for a few seconds, at which point micron-sized drops first appeared, and then, within a few tenths of a second, a sharp interface suddenly formed. Only later did larger drops start to appear and move toward the interface, where they coalesced. This behavior is radically different from that of an initially well-mixed system, where single-phase domains appear immediately after the quench and then grow by convectiondriven coalescence until they become large enough to sediment. As it takes one-half hour to diffuse and a few seconds to segregate, our experimental results indicate that the phase-separation process is not driven by diffusion. In addition, because no significant differences were observed between density-segregated and densitymatched systems, the process is also not driven by gravity. Our experimental results can be explained using the theoretical model denoted as model H, which predicts that the phase separation of low-viscosity liquid mixtures is driven by the convection induced by chemical potential gradients. In the presence of an initial gradient of concentration (and therefore of chemical potential as well), small single-phase domains are rapidly convected toward one or the other of the homogeneous phases, where they are rapidly reabsorbed. This explains why drops do not have the chance to grow, so that larger drops were not observed. The large bulk flows predicted by the theory were visualized by adding glass particles to the phaseseparating systems. We saw that the typical bulk velocity continues to increase in time, reaching values of about 0.4 mm/s for the density-segregated system and even higher values for the isopycnic mixture. Eventually, the interface appears, and at this point, the bulk flow vanishes, and the system resumes its quiescent state. This behavior can be explained, albeit qualitatively, via model H by considering that the increasing convection within the diffusion layer stems from an increase in the concentration gradients with time, as the thickness of the diffusion layer steadily decreases.
Acknowledgment
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(22) Mauri, R.; Shinnar, R.; Triantafyllou, G. Spinodal Decomposition in Binary Mixtures. Phys. Rev. E 1996, 53, 2613. (23) Valls, O. T.; Farrell, J. E. Spinodal Decomposition in a Three-Dimensional Fluid Model. Phys. Rev. E 1993, 47, R36 and references therein. (24) Jasnow, D.; Vin˜als, J. Coarse-Grained Description of Thermo-Capillary Flow. Phys. Fluids 1996, 8, 660. (25) van der Waals, J. D. The Thermodynamic Theory of Capillarity under the Hypothesis of a Continuous Variation of Density. J. Stat. Phys. 1979, 20, 200 (English translation by J. S. Rowlinson). (26) Tanaka, H. Coarsening Mechanisms of Droplet Spinodal Decomposition in Binary Fluid Mixtures. J. Chem. Phys. 1996, 105, 10099.
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Received for review September 5, 2000 Revised manuscript received January 11, 2001 Accepted January 26, 2001 IE000798V
