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Articles Late Stage of the Phase-Separation Process: Coalescence-Induced Coalescence, Gravitational Sedimentation, and Collective Evaporation Mechanisms Tomasz Kalwarczyk,†,‡ Natalia Ziebacz,‡ Marcin Fiałkowski,‡ and Robert Hołyst*,†,‡ Department of Mathematics and Natural Sciences, College of Science, Cardinal Stefan Wyszyn´ski UniVersity, Dewajtis 5, 01-815 Warsaw, Poland, Institute of Physical Chemistry PAS, Department III, Kasprzaka 44/52, 01-224 Warsaw, Poland ReceiVed June 11, 2007. ReVised Manuscript ReceiVed April 4, 2008 We study the separation in the binary and ternary mixtures of the water/surfactant C12E5/polymer PEG system. The phase separation in the mixtures at late stages is governed by two distinct mechanisms: the coalescence-induced coalescence and the droplet evaporation mechanism. We show that when the coalescence-induced coalescence process is globally terminated in the sample consisting of a dense system of domains, another mechanism, which we call the collective droplet evaporation, starts to dominate. It manifests itself as a front of “evaporating” domains, which propagates at constant speed in the system. We show that the collective evaporation is induced by the gravitational drift of large droplets.
1. Introduction Phase separation and miscibility are key processes used in the industry for making food, cosmetics, and composite polymer materials. The experimental and theoretical basis of the phaseseparation process has been established, and the nature of the process is well understood.1–8 The separation starts either by the nucleation of small domain of one phase in the homogeneous mixture or by spinodal decomposition. In the first case, fluctuations are of large amplitude (large difference in concentrations) but small spatial extent, whereas in the second case, spatially extended fluctuation of small amplitude induces the process. In the case of similar volume fractions of two phases, the late stage of phase separation is governed by the mechanism of the coalescenceinduced coalescence (CIC) proposed by Tanaka1–3 and the Lifshitz-Slyozov (LS) mechanism.9,10 The coalescence-induced coalescence mechanism manifests itself as a sequence of domain coalescence events that were generated by the earlier coalescence of domains. There are three different types of CIC mechanisms.1–3 First, there is CIC via flow when the coalescence between two domains induces * Corresponding author. E-mail:
[email protected]. † Cardinal Stefan Wyszyn´ski University. ‡ Institute of Physical Chemistry PAS.
(1) Tanaka, H. Phys. ReV. Lett. 1994, 72, 1702. (2) Tanaka, H. J. Chem. Phys. 1996, 105–10099. (3) Tanaka, H. J. Chem. Phys. 1997, 107, 3734. (4) Demyanchuk, I.; Staniszewski, K; Hołyst, R. J. Phys. Chem. B 2005, 109– 4419. (5) Demyanchuk, I.; Wieczorek, S. A.; Hołyst, R .J. Phys. Chem. B 2006, 110–9869. (6) Demyanchuk, I.; Wieczorek, S. A.; Hołyst, R. J. Chem. Phys. 2004, 121– 1141. (7) Beysens, D. Physica A 1997, 239, 329. (8) Nikolayev, V. S.; Beysens, D.; Guenoun, P. Phys. ReV. Lett. 1996, 76, 3144. (9) Langer, J. S. An Introduction to the Kinetics of First-Order Phase Transitions. In Solids Far From Equilibrium; Godre`cheC., Ed.; Cambridge University Press: Cambridge, U.K., 1992. (10) Bray, A.; J; Emmot, C.; L Phys ReV. B 1995, 52, R685.
hydrodynamic flows that induce other coalescence acts. Second is CIC via diffusion, when a few droplets that are close to each other, exhibiting overlapping concentration fields around them, can merge because of the diffusion of molecules induced by the concentration gradients. The overlapping fields also lead to longrange attractive interactions between the domains. The third type of CIC mechanism is called the geometrical CIC. It is observed when two domains during coalescence change their shapes and touch another domain to induce further coalescence. The coalescence-induced coalescence has been observed in many different systems, such as the styrene oligomer/-caprolactone oligomer,1 poly(vinyl methyl ether), and water mixtures of ε-caprolactone oligomer/styrene oligomer,2 hexathylene glycol monododecyl ether (C12E6)/PEG/water,4 a liquid crystal/polymer system (8CB/polystyrene),5 and polymer/polymer (poly(methylphenylsiloxane) /polystyrene)6 mixtures. At a later stage in the phase-separation process, when a minority phase occupies a vanishingly small volume fraction of the system, the LS mechanism occurs. The molecules from small droplets diffuse toward large droplets (small domains decrease and large domains increase in size).9 Typically, the LS mechanism is very slow compared to CIC. Large domains grow at the expense of smaller domains of radii below the critical radius, but the material is transported to larger domains by diffusion only. Droplets changes their sizes with an algebraic dependence on time.10 The critical radius is a function of the supersaturation in the system and may change during the separation process. Generally, the smaller the supersaturation, the larger the critical radius. In the beginning, the undersaturation is small, and many domains have radii larger than the critical radius. As the process proceeds, the undersaturation and the critical radius increase. As a result, some domains stop growing and start to decrease in size, allowing their material to go back into the bulk.
10.1021/la704003q CCC: $40.75 2008 American Chemical Society Published on Web 05/29/2008
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CIC, which is a universal mechanism of phase-separation kinetics, can be described as a self-sustaining “chain reaction”.11 The CIC process may locally terminate when the distances between the droplets become excessively large. In such case, the hydrodynamic flow resulting from a coalescence event is extinguished before reaching a neighboring domain. As a result, the slowing down of the average domain size growth is observed. Note also that, in general, the kinetics of the phase-separation process depends on the geometry of the container. If the sizes of the domains are comparable with the size of the container, then the dimensionality of the process changes, and a crossover from three to two dimensions is observed. The coalescenceinduced coalescence via hydrodynamics can be extinguished by the presence of glass boundaries. Note that the glass boundaries may also induce other effects, for example, diffusional ones. The steady concentration state of surfactant molecules around a spherical domain decays as 1/r. For a domain with a cylindrical shape, there is no steady concentration field around it, and the concentration decays logarithmically. Note that the other factor that can affect the CIC process is the presence of a three-phase contact line (surfactant/water/glass). This line is moving when a domain increases its size or is changing its shape and may play an important role during the coalescence event. In our experimental studies, we confirm that the CIC processes occurring in the system are governed by the universal mechanism of growth described by Tanaka,1–3 which was observed in many different systems.1,2,4–6 According to the theoretical considerations,8,9 the average domain size L(t) grows with time according to L(t) ∝ tβ. Our previous experiment4 showed that domains grow with β ) 1 in three dimensions and, after the dimensional crossover from 3D to 2D, the domains grow more slowly with a growth exponent of β ≈ 1/4-1/3. Two-dimensional computer simulations of the CIC process mediated by hydrodynamic flow2,3,7,8 predict the growth L(t) ∝ t, whereas for CIC via diffusion2,3 one gets L(t) ∝ t1/3. Because of the small vertical dimensions of our samples (in which the domain can be described as a flat disk with a diameter/height ratio on the order of 102), our system should be considered to be 2D. As we show in the following sections, the kinetics of CIC via hydrodynamic flow in our system follows the growth law with the exponent β ) 1 predicted by Nikolayev et al.8 for a 2D system. We demonstrate that the kinetics of this process is not universal. The nonuniversality of growth kinetics is caused by the chainreaction character of the CIC process, which may not be a selfsustaining one in all parts of the sample studied. The reason is that in some parts of the sample the distances between the domains can be large enough to extinguish the hydrodynamic flow and the domains cannot touch each other. Also, it is possible that the above mechanism of domain growth may stop operating in the whole sample. In such a case, one expects that the LS mechanism of evaporation-condensation12 starts to dominate, even when the volume fractions of the two phases are comparable. In this article, we also demonstrate experimentally that in a dense system of domains, when CIC is terminated globally, a novel mechanism of growth is observed. We call this mechanism the collective domain evaporation. It is manifested in the system as “waves” of dissolving domains. These waves appear suddenly in different
In our experiments, we used a solution of nonionic surfactant C12E5 (pentaethylene glycol monododecyl ether) of molecular mass 400 purchased from Fluka of purity better than 98% and polyethylene glycol (PEG) of Mn ) 5800, Mw/Mn ) 1.08. Mw and Mn denote here, respectively, weight-average molecular weight and number-average molecular weight. Water for the solutions was distilled and degassed. For the fluorescent particle tracking measurements, we used fluorescent microspheres of diameter 1.74 µm purchased from Bangs Laboratories. The spheres were made of polystyrene and functionalized by carboxylic group (PS-COOH). The solutions were prepared at room temperature and kept in a humid atmosphere to avoid the evaporation of water. The samples were prepared in the following way: First, two circular glass plates were cleaned in plasma cleaner. Next, 20-µm-thick spacers made of aluminum foil were placed between these glasses. The plates were then sealed at the edges with the epoxy resin glue, and two openings in the glue were left to allow us to fill the chamber with the solution. A drop of solution was placed on the edge of sample, and the solution was sucked inside by capillary forces. Finally, all holes were sealed. We also made a measurement of the cloud-point temperature13 for samples prepared in capillaries. One of the capillaries was open on one side, and the second one has two holes sealed. The cloud-point temperatures were identical (with 0.5 °C accuracy). We performed the optical microscope measurements using the Nikon Eclipse E400 microscope equipped with the LINKAM THMS 600 heating/cooling stage. The temperature was controlled to 0.1 °C. The kinetics of the separation process was determined by the quantitative study of sizes and shapes of the domains with the help of the LUCIA software for optical image analysis. To visualize the hydrodynamic flow, we conducted fluorescent particle tracking measurements using a Nikon Eclipse 50i microscope with a fluorescence camera. Coalescence-Induced Coalescence in Water/C12E5/PEG Mixtures. Phase separation in a binary surfactant/water mixture is due to the dehydration of the surfactant heads and the progressive increase of the influence of the van der Waals attraction between surfactant micelles. Adding PEG to the binary mixture induces additional attractive interactions between the micelles (depletion interactions14,15). These interactions dominate the van der Waals interactions and strongly reduce the separation temperature in the PEG/surfactant/ water mixtures in comparison to that in the surfactant/water mixtures.16 The origin of the depletion interactions is the conformational entropy of polymer chains. Namely, there is a zone around each micelle that cannot be penetrated by the center of mass of polymer molecules. The center of mass of the polymer cannot penetrate the zone because it would result in a large decrease in its conformational entropy. The size of this zone is proportional to the radius of gyration of the polymer. If these two zones around the micelles overlap, then there is an imbalance of osmotic pressure (induced by the polymers outside the zones) that pushes the micelles
(11) One event of the coalescence of two domains induces further coalescences mediated either by geometry (proximity of the domains) or by the hydrodynamic flow, which arises when the coalescing domains change their shape during the event. Usually, three or four domains are involved in the CIC process. Thus, the spatial scale of the process is on the order of a few domain diameters. The time scale of the process depends strongly on the viscosity of system. In our experiments, at late stages, the length of time between the coalescence events was on the order of few hundred seconds (cf. Figure 3a). (12) Bray, A. J. AdV. Phys. 1994, 43, 357.
(13) The cloud-point temperature corresponds to the temperature when in the initially homogeneous system the phase-separation process starts. Macroscopically, the mixture in the vial becomes nontransparent. This phenomenon is caused by small domains that appear in the system. The light scatters on those domains, causing the opacity of the mixture. (14) Asakura, S.; Oosawa, J. J. Chem. Phys. 1954, 22, 1255. (15) Vrij, A. Pure Appl. Chem. 1976, 48, 471. (16) Kalwarczyk, T.; Ziebacz, N.; Wieczorek, S.; A.; Holyst, R. J. Phys. Chem. B 2007, 111, 11907.
places and propagate across the system with a sharp front line determined by the evaporating domains. The article is organized as follows. In the next section, we present the materials and methods used in our study. In section 3, we investigate the kinetics of the phase-separation process proceeding through the CIC mechanism. In section 4, we discuss the collective evaporation process that takes place in the late stages of the phase separation. The article ends with a summary and discussion of the results.
2. Experimental Section
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Figure 1. Coalescence-induced coalescence observed in a 10% C12E5/ water mixture after (a) 300, (b) 315, (c) 330, and (d) 345 s from the beginning of the process. The sample was placed between two glass plates. The distance between glass plates was 20 ( 2µm. The samples were heated to about 1 °C above the cloud-point temperature.
Figure 2. Coalescence-induced coalescence observed in a 1% C12E5/1% PEG/water mixture after (a) 1350, (b) 1470, (c) 1590, and (d) 1710 s.
together and leads to phase separation into the surfactant-rich phase and the polymer-rich phase.17 In our experiments, we studied five different solutions: 10% C12E5 and 0% PEG, 10% C12E5 and 1% PEG, 10% C12E5 and 2% PEG, 1% C12E5 and 0% PEG, and 1% C12E5 and 1% PEG. The samples were prepared at room temperature and then heated to 1 °C above the cloud-point temperature. The temperature at which the phaseseparation process starts depends on the concentration of PEG added. Figure 1 illustrates the CIC mechanism of growth in a 10% C12E5/ water mixture (a) 300, (b) 315, (c) 330, and (d) 345 s after the increase in temperature above the spinodal. The sequence of coalescence was the following: 1 + 2 w A f A + 3 w B f B + 4 w C, where the double arrow, w, denotes an outcome of the coalescence act that could be induced by the previous collision of domains. The single arrow, f, refers to the coalescence event generated by the preceding coalescence. Figure 2 shows the coalescence of domains in a 1% C12E5/1% PEG/water mixture (a) 1350, (b) 1470, (c) 1590, and (d) 1710 s after the increase in temperature. The sequence of the coalescence acts proceeds as follows: 1 + 2 w A f 3 + 4 w B f B + 5 w C f C + A + 6 w D. The coalescence processes described above were observed in all of the samples that we studied. According to the theoretical predictions2,3,7,8 a single domain undergoing the CIC process mediated by hydrodynamic flow grows linearly with time. To study the growth laws in our system, we investigated the morphology (sizes and shapes) of the domains. In our experiments, we focused on one selected domain and recorded (17) Anderson, V. J.; Lekkerkerker, H. N.W. Nature (London) 2002, 416, 811.
Figure 3. Time evolution of the diameter, L, of a selected domain in a surfactant C12E5(10%)/water mixture. (a) Staircaselike growth of the domain diameter obtained for the data collected at equal time intervals of 15 s. (b) The data points corresponding to the subsequent coalescence events exhibit a linear dependency of L on time.
its evolution in time. We studied the evolution of the domain size in two different mixtures: 10% C12E5/water and 20% C12E5/water. After the increase in temperature, we collected snapshots at a time interval of 15 s. The snapshots were collected using a digital camera connected to an optical microscope. Figure 3 shows the dependence of the diameter, L, on time for a single domain. In Figure 3a, the data were collected at equal time intervals of 15 s. As seen, the time evolution of the size of a single domain is described by a staircase function of irregular jumps. There are relatively long periods of time when the domain does not grow at all. The best way to investigate the growth exponent is to analyze only the data points corresponding to the jumps in size of the domain (i.e., the coalescence events). Figure 3b shows the dependence L(t) for selected time instants corresponding to the subsequent coalescence acts. The dependence L ≈ t obtained agrees with the hypothesis that single domains grow linearly with time. The linear growth of the domain is observed, in agreement with the theoretical predictions.2,3,7,8 In addition to the observations of single domains described above, we sought to answer the following question: What is the global kinetics of the phase-separation process? Here, the term global refers to the focus on many droplets of a minority phase. In our study, we average the function L(t) over many droplets and investigate its changes with time. To investigate the global kinetics, we took snapshots with a time interval 15 s at the beginning of the experiment (just after the increase in temperature). Next (after about 120 s), we changed the time interval to 120 s. In each picture, we marked diameters, L, computed the averaged size, 〈L〉, along with its standard deviation, ((〈L2〉 - 〈L〉2)/N(N - 1))1/2 (Figure 4) where N is the number of samples. Figure 5 shows the log-log plots of the average domain diameter, 〈L〉(t), as a function of time, t, and changes in the number of growing domains during phase separation for three different mixtures: 10% C12E5/water, 1% C12E5/water, and 10% C12E5/
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Figure 4. Scheme of the technique that we used to measure the diameters of the domains. Snapshots used in measurements were taken under the optical microscope at different instants of time. This technique enables us to trace the diameter of a selected domain or calculate the average diameter for a given spot.
2% PEG/water. As seen, for different samples different kinetics of domain growth is observed despite the fact that in all samples the CIC process was the dominating mechanism of the change in domain diameter. Figure 5a shows 〈L〉 as a function of time for a binary 10% C12E5/ water mixture. As seen, at the beginning of the separation process, the domains grow very slowly. After about 80 s, we observe a rapid increase in the growth rate. This effect is caused by the CIC process, which initiated a hydrodynamic flow inducing further coalescence events. Figure 5b shows the time evolution of the average domain diameter in a binary 1% C12E5/water mixture. As seen, after about 200 s, we observe a fast increase in the domain size. Next, the growth stops, and a plateau is observed for t > 900 s. In Figure 5c, we do not observe any rapid changes in the rate of domain growth. As shown in Figure 5, the global kinetics of the phase-separation process changed from sample to sample, in spite of the universal nature of the CIC mechanism. These deviations from the power law (〈L〉 ≈ tβ; β ) 1) can be attributed to the chain-reaction character of the CIC mechanism described in the previous section. In Figure 5, the evolution of the average domain size, 〈L〉, and the average number of growing domains, 〈n〉 is shown. As seen, there is no simple correlation between the average domain size and the average number of growing domains. As shown in Figure 5a,b, when 〈n〉 reaches a plateau one can observe either the fast grow of the domain size (Figure 5a) or a significant slowing down of the domain growth (Figure 5b). Figure 5c shows, in turn, that 〈n〉 can decrease with increasing 〈L〉. The behavior of 〈n〉 and 〈L〉 presented in Figure 5c can be linked to fact that the domains (and the distances between the domains) are large compared to those in Figure 5a,b. One expects that the coalescence events that increase 〈L〉 also increase the distance between the domains. This successively slows down the rate of the coalescence events and results in the decrease in 〈n〉 observed. One coalescence event can generate a hydrodynamic flow that can induce further coalescence. It may happen however that the distances between the domains become too large for the hydrodynamic flow to support the CIC mechanism. Then, the domain growth via the CIC mechanism stops, and other mechanism of growth, for example, the LS mechanism, starts to dominate. One expects that the LS mechanism drives the phase-separation kinetics for sparse and very viscous systems (e.g., polymer systems).9 In this case, domains grow and collapse algebraically with time, with the exponent10 1/4-1/3. However, in our experimental system, which
Figure 5. Time evolution of the average domain diameter 〈L〉 and the average number of growing domains, 〈n〉, in (a) surfactant (10%)/water, (b) 1% C12E5/water, and (c) 10% C12E5/2% PEG/water mixtures. The thickness of the samples was 20 ( 2 µm. The samples were heated to about 1 °C above the cloud-point temperature. Plots a-c show the dependence of 〈L〉 on time for three different samples. The apparent differences between the growth rates are caused by different intensities of hydrodynamic flow in the samples. In sample a, the hydrodynamic flow is at its highest intensity, whereas in sample b it is at its lowest.
was dense and not very viscous, the LS mechanism was suppressed by other mechanisms described in the following section.
3. Coalescence via Hydrodynamic Flow vs the Brownian Motion of Droplets In the previous section, we posed a hypothesis that the system is strongly dependent on hydrodynamic flow inside the sample generated by coalescing domains. To confirm this hypothesis,
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Figure 6. Fluorescent particle tracking in a 10% C12E5/2.5% PEG/water mixture with the addition of 0.02% PS-COOH. The time interval between pictures is 30 s. The average distance traveled by particles was about 23.9 ( 6.1 µm. The scale bar corresponds to 20 µm.
we carried out fluorescent particle tracking measurements. We used micrometer-sized particles as markers. By an analysis of the position of a fluorescent particle in every snapshot, we can determine the velocity of the particle and also the direction of flow. This method can be used to visualize the hydrodynamic flows in transparent systems. We prepared a surfactant/polymer/ water solution with the addition of fluorescent particles. Next, samples were observed under a fluorescence microscope. We studied a 10% C12E5/2.5% PEG/0.02% PS-COOH/water mixture. The measurements were made at 30 °C. Figure 6 shows CIC and hydrodynamic flows generated by this mechanism. The time interval between pictures is 30 s. As it is shown in Figure 6, the fluorescent particle traveled about 29 µm in 120 s. The movement of the particles was driven by the hydrodynamic flow, generated by the coalescence of neighboring domains. On the basis of this result, we calculated the diffusion coefficients of the fluorescent particle in pure water and in a ternary surfactant/ polymer/water mixture. In our latest letter,18 we presented a study of viscosity in a similar system (10% surfactant C12E6). The experimentally measured viscosity of 10% C12E6 reported in the letter cited is 8 times larger than the viscosity of water. In the present study, C12E5 is used. Because C12E6 and C12E5 do not differ very much, we expect that the viscosity of the mixture is similar to the viscosity of a mixture used in ref 18. We assumed that the viscosity of our system is about 10 times the viscosity of water.19 The diffusion coefficient of a particle is given by the following equation
D)
kBT 6πηr
(1)
where D is the diffusion coefficient of the particle, kB is the Boltzman constant, T is the temperature, η is the viscosity of the system, and r is the radius of the particle. The average square displacement, 〈S2〉, in the time interval t of an object performing the Brownian motion is written as
〈S2 〉 ) 4Dt
(2)
Using eqs 1 and 2, we calculated the diffusion coefficients for hypothetical domains and fluorescent particles (diameter ) 1.74 (18) Szymanski, J; Patkowski, A; Wilk, A; Garstecki, P; Holyst, R. J. Phys. Chem. B 2006, 110, 25593. (19) The presence of PEG in our system increases its viscosity. Our measurements of the diffusion coefficients support the estimation that the viscosity of the mixture is about 10 times the viscosity of pure water. See also Movie 1 in Supporting Information.
Figure 7. Time evolution of one-domain diameter L in a 10% C12E5/ 2.5% PEG/water mixture.
µm). The diffusion coefficients of microspheres and domains (whose diameters were 50 and 200 µm) in surfactant mixtures were, respectively, 3.18 × 10-13, 1.11 × 10-14, and 2.76 × 10-15 m2 s-1. The diffusion coefficients in water were approximately 10 times larger than those in the surfactant mixture. The average distance traveled in t ) 120 s is also much smaller in surfactant mixtures than in water and equal (fluorescent particle; 50 and 200 µm domains) in water (12.35, 2.30, 1.15 µm) and in surfactant mixtures (3.90, 0.72, and 0.36 µm). The particle undergoing Brownian motion in a surfactant mixture could travel only about 4 µm in time t ) 120 s. In experiments, we observed, however, that the average distance traveled by particles was about 23.9 ( 6.1 µm (Figure 6). This result suggests that the movements of the particles that we observed in our experiments were induced by the hydrodynamic flow. (For more details, see Movies 1, 2, and 3 in Supporting Information.) We also confirmed it by the quantitative results of domain growth inside such a system (Figure 7). These results indicate that in the late stage of the phase-separation process (when the domains reach sizes of tens or hundreds of micrometers) the CIC process proceeds through hydrodynamic flow and not Brownian motion. This conclusion follows directly from eq 1. Namely, the diffusion coefficient decreases with the size of the domain. For small diffusion coefficients, the speed of the coalescence driven by Brownian motion is not sufficient to keep up the CIC mechanism.
4. Collective Evaporation of Droplets in a C12E5/Water Mixture In the late stage of the phase-separation process, apart from CIC, we also observed the phenomenon of thte collective evaporation of droplets. This mechanism was much slower than the coalescence-induced coalescence, which is the dominating mechanism of domain growth. When the CIC mechanism was terminated globally in the system, we observed a novel phenomenon, the collectiVe evaporation mechanism of growth. In our experiment, we used a 20% C12E5/water mixture prepared in the following way. First, the mixture was separated at 45 °C. Next we waited for the termination of the CIC process. After about 6 h (or even longer), we observed a front of collectively dissolving droplets. Gravity-Induced Drift of Large Domains. In our experimental setup, the sample was placed horizontally between two glass plates. Although the plates were oriented perpendicular to
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F1 ≈ 2πrηsolV
(3)
The second contribution to the viscous drag force, which originates from the motion of the fluid inside the domain, is calculated as
F2 ) 2πr2ηmic
δV h
(4)
In the above equation, ηmic is the viscosity of the surfactantrich phase inside the domain. (Note that the factor of 2 appears in eq 4 because in the rolling motion both the top and bottom surfaces of the domain are the friction areas.) The quantity δ has been employed to account for possible slip between the surface of the glass and the domain. The slip can effectively lower the viscous friction and is often observed in complex fluids.20 The product δV represents the characteristic velocity of flows involved in the dissipation. The gravity force, Fg, which makes the domains move, is due the difference in density, ∆F, between the solvent and the domain. (The surfactant-rich domains are more dense than the solvent.) For a cylindrical domain, the gravity force is written as
Fg ) πr2h∆Fg sinε
Figure 8. Drift of a large domain observed in the late stages of the phase-separation process. The domain moves under gravity caused by the small local inclination of the surface of the sample. The domain absorbs all domains and micelles and leaves behind a path free of surfactant. The time interval between the subsequent pictures is 600 s. The average velocity of the domain is 8.0 × 10-7 m s-1. The scale bar corresponds to 500 µm.
the gravitational field, small irregularities in their topography (local inclination of the surface) could induce the movement of large domains under gravity (Figure 8). The observed velocity of the domains corresponds to the balance between viscous resistance and gravity forces. One can assume that the domains have a cylindrical shape. The moving domains do not spread on the plates and preserve their shapes. The viscous resistance force consists of two components: The first contribution originates from the flow of the water-rich phase past the domain, and the second contribution is due to the rolling flow of the surfactantrich phase inside the domain. The first contribution, which is due to the motion of the fluid surrounding the domain, can be decomposed into two contributions according to the characteristic magnitude of the velocity gradients. The first contribution, F1roll, is caused by the rolling motion of the fluid inside the domain, which induces velocity gradients on the order of V/h in the water-rich phase along the contact line. Here, V and h denote, respectively, the velocity and thickness of the domain. The second contribution, F1trans, is related to the translational motion of the domain as a whole and is characterized by the velocity gradient V/r, where r is the radius of the domain. Both the contributions, F1roll and F1trans, are proportional to the area, A, of the water-rich and surfactant-rich interfaces, which is calculated as the side of the cylinder, A ) 2πrh. Thus, the two contributions can be approximated (up to a numerical factor on the order of unity) as F1trans ≈ AηsolV/r ) 2πhηsolV and F1roll ≈ AηsolV/h ) 2πrηsolV, where ηsol is the viscosity of the water-rich phase. One also gets F1trans/F1roll ) h/r. Because the ratio h/r is on the order of 10-2, F1roll dominates F1trans. Thus, one has
(5)
where g stands for the gravitational constant and ε is local inclination angle of the surface of the plate relative to the direction of the gravitational field. (ε ) 0 if the surface is perpendicular to the gravitational field.) In the stationary flow, the gravity force is balanced by the viscous drag, Fg ) F1 + F2, Thus, from eqs 3-5 one obtains the following equation for the drift velocity of the droplet, V, and its radius, r:
V(r) )
a1r 1 + a2r
(6)
In the above equation, a1 ) sin ε∆Fgh/2ηsol and a2 ) δηmic/ ηsolh. The quantities ε and δ are the fitting parameters determined from the fit to the experimental data. We performed the calculations for h ) 20 µm, ∆F ) 1.0 kg m-3 21, and ηmic ≈ 10ηsol with ηsol ) 6.0 × 10-4 kg ms-1. The least-squares fit yielded ε ) 3.0 × 10-2 rad and δ ) 5.2. The dependence of the domain’s velocity on its diameter is plotted in Figure 9. The solid line represents the least-squares fit to the experimental data of the relation V(r) calculated from eq 6. The value of the coefficient δ > 1 suggests that the slippage between the surface of the plate and the domain is not present.22 Importantly, the small value of the inclination angle (of the order of 1°) that we obtained indicates that the gravitational drift of the droplets can be caused by micrometer-sized irregularities of the surface of the top and/or bottom glass plates.23 Collective Evaporation of Droplets. At a certain location in the system, a front of dissolving domains starts to move and continues to propagate with constant velocity. (See Figure 10 and Supporting Information.) We found that the domains in the front dissolve in such a way that their radii decrease linearly with time (Figure 11). In the process, the size of the domain changes as16 (20) Barnes, H. A. J. Non-Newtonian Fluid Mech. 1995, 56 221. (21) Maccarini, M.; Briganti, G. J. Phys. Chem A 2000, 104, 11451. (22) We note that the surfactant-rich phase wets the surface of the glass plates. This indicates adhesive forces between the domain and the plates, which prevent slippage. (23) All types of glass, except those with molecularly smooth surfaces, have irregular surfaces. These irregularities results from technological processes and are large enough to cause the drift of the domains observed. (24) Lo´pez-Esparza, R; Guedeau-Boudeville, M. A.; Gambin, Y; Rodrı´guezBeas, C.; Maldonado, A.; Urbach, W. J. Colloid Interface Sci. 2006, 300–105.
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Figure 9. Dependence the observed drift velocity, V, of domains moving under gravity on their diameter, L. The solid line represents the dependence V(r) obtained from the relation (eq 6) by the least-squares fit to the experimental data.
Figure 11. Time evolution of the size, L, calculated for three domains evaporating after the passage of the front line observed in the system shown in Figure 10. The size decreases linearly with time, according to eq 7. The parameter (the slope) B ≈ 3.0 10-3 µm s-1, obtained from the linear fit, is similar for all evaporating droplets.
Figure 12. Position of the domain evaporation front as a function of time measured along the direction of the front propagation in the system shown in Figure 10. Here, x0 is the initial position of the front. Figure 10. Propagation of the evaporation front. The process of evaporation starts at the edge of the channel free of surfactant created by the large domain, which moved under the gravity force (cf. Figure 8). The dashed lines in a-d mark the subsequent positions of the front. As seen, the domains do not change their sizes until reached by the front line. The average distance between the centers of the domains is about 100 µm. The time interval between successive snapshots is 100 min. The scale bar is 500 µm.
L(t) ) L(0) - Bt
(7)
where L(0) is the initial diameter and B is a constant. The quantity B is proportional to the difference in chemical potential between the components inside and outside the domain. The evaporation front propagates at a velocity Vf, which is calculated as follows. The average time, td, needed for the domain to evaporate is, according to eq 7, tevap ) 〈L〉/B. In this time, the front advances by the average distance between the domains, 〈d〉. Thus, the velocity of the evaporation front is estimated from the relation
Vf ≈
〈d 〉 〈L 〉
B
(8)
The phenomenon of collective evaporation was observed in a dense domain system. In such a case, the quantities 〈d〉 and 〈L〉 do not differ much, and 〈d〉 > 〈L〉 (this inequality is fulfilled for
each system consisting of domains). Thus, in view of eq 8, the velocity of the front is always expected to be slightly larger than the B coefficient. The above mechanism of the propagation of the evaporation front is valid provided the matter released from the dissolving domains is removed sufficiently quickly from the line of the front. Then, the process is limited by the rate of the evaporation of domains. The above condition is satisfied when the characteristic diffusion time, tdiff, needed for the micelles to escape from the stripe of width 〈d〉 located at the front is much smaller than tevap. The diffusion time can be estimated as tdiff ) 〈d〉2/Dmic, where Dmic stands for the diffusion coefficient of the micelles in water. Therefore, the model is valid provided the following condition holds:
tdiff , tevap
(9)
Figure 12 shows the observed time dependence of the size, L, of the domains in the evaporation process triggered by the front line. As seen, the size decreases linearly at all times, according to eq 7. The parameter B obtained from the fit to the data points is B ≈ 3.0 × 10-3 µm s-1. The front propagates at the velocity Vf ) 4.0 × 10-3 µm s-1 (Figure 12). Thus, it follows that the speed of the front line is close to the rate at which the
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and release surfactant molecules, which are later absorbed by large domains through the LS evaporation-condensation mechanism.
5. Summary and Conclusions
Figure 13. Positions of the evaporation front as a function of time for the 10% C12E5/water mixture. Here, x0 is the initial position of the front. See also Figure S1 in Supporting Information.
domain dissolve, in accordance with eq 8. We note that the ratio 〈d〉/〈L〉 calculated from eq 8 is about 1.3. In the system analyzed, the average distance between the centers of the domains, 〈d〉, is about 100 µm (Figure 10). For the diffusion coefficient of the micelles Dmic ≈ 2.0 × 10-7 cm2 s-1,24 this value of 〈d〉 gives the characteristic diffusion time tdiff ) 5.0 × 102 s. However, for the average domain size 〈L〉 ≈ 75 µm, the time needed for the dissolution of the domain is tevap ) 〈L〉/B ) 2.5 × 104 s. This means that tdiff is 2 orders of magnitude smaller than tevap and the condition (eq 9) for the validity of the model is satisfied. We found that the relationship (eq 8) between B and Vf was satisfied for all samples that we analyzed. Figure 13 shows the propagation of the front in other system (10% C12E5/water), characterized by 〈L〉 ≈ 30 µm, B ) 9.0 × 10-3 µm s-1, and Vf ) 2.3 × 10-2 µm s-1. The calculated ratio 〈d〉/〈L〉 ) 2.6. Note that this value is larger than the value of 1.3 obtained for the previously discussed system shown in Figure 10. This result agrees with experimental observations because the system from Figure 13 is apparently less dense than that from Figure 10. (For more details, see Figure S1 and Movie 4 in Supporting Information.) In summary, in the late stages of phase separation, domain growth is due to the gravity forces experienced by the largest domains. The gravity induces the domain growth in two ways. First, it causes large domains to migrate across the system. In this processsthe gravity-induced coalescencesthe moving domain absorbs the small domains and micelles that it meets. Second, the domain moving under gravity creates a channel that is completely free of surfactants. This channel triggers the process of the collective evaporation of small domains. They dissolve
In this article, we presented a detailed study of the kinetics of the separation process. The system that we investigated is composed of surfactant, polymer, and water. We studied a dense systems of domains of a minority phase in a matrix of a majority phase. Because of the geometry of the system (very flat domains between glass boundaries), the CIC process studied was twodimensional. We showed that CIC via hydrodynamic flow is the dominating mechanism in the late stages of the phase-separation process. The kinetics of this process, however, may be different for each systems. Locally, when one domain is considered, the kinetics of the CIC process is consistent with the theoretically predicted power law 〈L〉 ≈ tβ; β ) 1. However, when we consider many domains, a strong deviation from the power law is observed. Briefly, the system consists of nonequal numbers of growing and nongrowing domains. When the growing domains outnumber the nongrowing ones, an increase in the growth rate is observed. In the opposite situation, a significant decrease in the growth rate (or even a plateau) is observed. In very late stages of the phase-separation process, we observed the termination of the CIC mechanism. The system then achieves a local minimum in the Gibbs free energy. When the CIC mechanism stops, the collective evaporation of droplets takes place at random in different parts of the system. We found that this phenomenon is induced by gravity. Namely, large domains start to migrate through the system, driven by the gravitational forces, absorbing smaller domains and surfactant micelles in the matrix. As a result, channels free of surfactant molecules are created. These channels trigger waves of collective evaporation of small domains. The small domains dissolve and release surfactant molecules, which are later absorbed by large domains. We hope that our results will induce further interest in the late stages of the phase-separation process. Acknowledgment. This work was supported in part by the Ministry of Science and Higher Education as a scientific project (2006-2008) and grant SONS 2006-2009. R.H. and T.K. acknowledge the Foundation for Polish Science (scholarship MISTRZ). N.Z. acknowledges a Ph.D. scholarship from the President of the Polish Academy of Sciences. Supporting Information Available: Brownian motion of particles. Propagation of the evaporation front. This material is available free of charge via the Internet at http://pubs.acs.org. LA704003Q
