Kinetics and Dynamics of Dissolution/Mixing of a High-Viscosity Liquid Phase in a Low-Viscosity Solvent Phase

Transcription

1 J. Phys. Chem. B 2007, 111, Kinetics and Dynamics of Dissolution/Mixing of a High-Viscosity Liquid Phase in a Low-Viscosity Solvent Phase Tomasz Kalwarczyk,, Natalia Ziebacz,, Stefan A. Wieczorek, and Robert Holyst*,, Department of Mathematics and Natural Sciences, College of Science, Cardinal Stefan Wyszynski UniVersity, Dewajtis 5, Warsaw, Poland, and Institute of Physical Chemistry PAS, Dept. III, Kasprzaka 44/52, Warsaw, Poland ReceiVed: May 27, 2007; In Final Form: August 10, 2007 We studied mixing in the initially separated binary mixture of polystyrene/5cb liquid crystal and ternary mixtures of water/surfactant C 12 E 5 /polymer PEG system. In both systems the domains of one phase were characterized by a much higher viscosity than the solvent matrix. We demonstrated experimentally that during mixing these domains decrease their size linearly with time without a visible change of the optical contrast (i.e., without a rapid change of their compositions). Computer simulations and a theoretical model explain quantitatively our experimental observations. Introduction Mixing is very important in modern technologies for processing of materials composed of many different chemical components. 1 Whether two or more substances mix together depends on thermodynamic conditions. Usually two different substances mix only at certain conditions and separate at others. For a binary mixture characterized by the upper consolute point, the system is homogeneous at high temperatures and phase-separates below the consolute point. A mixture, characterized by a lower consolute point, is homogeneous at low temperatures and phaseseparates above this point. The system can be separated into two phases (above or below its consolute point), with one of the phases forming separated domains in the matrix of the other phase. By a suitable change of thermodynamic conditions (usually temperature), we can quench the system back into the homogeneous region and initiate the mixing of the domains with the matrix. In this paper we present such experiments as described above for two mixtures: (1) surfactant (C 12 E 5 )/water or surfactant (C 12 E 5 )/polymer (PEG)/water, characterized by the lower critical (consolute) point (Figure 1), 2 and (2) polystyrene/liquid crystal (5CB) characterized by the upper critical (consolute) point. We note that such studies are very rare 3-13 since for a long time it has been generally accepted that mixing proceeds by a simple diffusion described by Fick s law. More attention was focused on mixing with external force (agitator) or on mixing in granular fluids with shared flow. 14,15 Here we show that our understanding of the mixing process is far from being complete. From Fick s law we expect that domains of one phase diffusively spread during mixing in the second phase with the characteristic growth of their size in time t given by t 1/2. Here we observe different behavior. First of all, domains decrease their size in time during mixing, and second, their size decreases linearly with time. The quantitative experimental observations are supplemented in this paper by computer simulations and theoretical modeling. We emphasize two aspects of dissolution/ * To whom correspondence should be addressed. holyst@ ptys.ichf.edu.pl. Cardinal Stefan Wyszynski University. Institute of Physical Chemistry PAS. Figure 1. Phase diagram of the surfactant (C 12E 5)/water system (after ref 2). mixing. There is a kinetic aspect, i.e., the dependence of the size of dissolving objects on time. There are also dynamic aspects, namely, the thermodynamic forces responsible for dissolution/mixing. The first aspect is determined experimentally and compared to the theoretical predictions based on thermodynamic forces. Here we do not consider the external forces like mixing in a shear flow or agitators and refer the reader to excellent papers done by the Ottino group. 14,15 The paper is organized as follows. In the following section we present the materials and methods used in our study. Then we discuss the phase diagram for the C 12 E 5 /PEG/water system. We will also discuss the phase diagram for the PS/5CB mixture. We next present the results of the mixing kinetics measurements for both systems, followed by a discussion of the results of computer simulation for mixing via direct diffusion. We also present the theoretical model of kinetics and dynamics of mixing processes based on the evolution of the Gibbs free energy during the mixing process. We then show the experiment in which we invert the morphology of the separated phases (from domains of high viscosity in the low viscosity matrix to domains of low viscosity in the high viscosity matrix) in the system with fixed composition. We conclude this paper with a summary of the results and a short discussion /jp074087n CCC: $ American Chemical Society Published on Web 09/22/2007

2 11908 J. Phys. Chem. B, Vol. 111, No. 41, 2007 Kalwarczyk et al. Experimental Section For one mixture we used nonionic surfactant C 12 E 5 (pentaethylene glycol monododecyl ether) of molecular mass 400 purchased from Fluka of purity better then 98% and a polymer polyethylene glycol (PEG) of M n ) 5800 and M w /M n ) For the second mixture we used polystyrene (PS) from Fluka Chemical Co. of molecular weight M w ) and M w /M n ) 1.03 (M n is the average molecular weight) and liquid crystal 4-cyano-4 -n-pentyl-biphenyl (5CB). The first mixture was prepared as follows: Water for surfactant solutions was distilled and degassed on an ultracentrifuge. The C 12 E 5 /PEG/water mixtures were prepared at room temperature in a humid atmosphere in order to avoid evaporation of water. We made samples in the following way: First, two circular glass plates were cleaned in plasma cleaner. Between these glasses we put 20 µm thick spacers made of aluminum foil. We sealed the plates at their edges with an epoxy resin glue, but left two openings in the glue in order to fill a resulting chamber with our solution. We placed a drop of solution on the edge of our sample, and this solution was sucked inside the chamber by the capillarity forces. Finally we sealed all holes. Samples of PS/5CB mixtures were made in the following way: 5CB and PS were dissolved in toluene and then stirred mechanically for 48 h at a temperature of 60 C, just above a cloud point temperature. Thin films were prepared by casting from 40% toluene solution onto circular glass plates of 1 cm in diameter. The uncovered films were dried and annealed for h at a temperature of 60 C. Next, the films were covered by a second glass plate. The distance between the glass plates was set by copper spacers of thickness 10 µm. We performed optical microscope measurements using the Nikon Eclipse E400 microscope, equipped with the LINKAM THMS 600 heating/cooling stage. Temperature was controlled up to 0.1 C. The kinetics of mixing processes was determined by a quantitative study of sizes and shapes of domains of different phases using commercially available software LUCIA (purchased together with the Nikon Eclipse E400 microscope) for an optical image analysis. C 12 E 5 /PEG/Water Diagram. Optical thermomicroscopy experiments were done to obtain a phase diagram of 10% C 12 E 5 / PEG/water system for different PEG concentrations. Samples were heated up above the cloud point temperature. The measurements were done for seven concentrations of PEG in C 12 E 5 /water mixtures. The binary C 12 E 5 /water mixture has a lower critical (consolute) point (see Figure 1). Figure 2 shows the phase diagram of the 10% C 12 E 5 /PEG/ water system. A decrease of the cloud point temperature with addition of PEG was induced by depletion interactions. 16,17 These interactions dominate over the van der Waals interactions and strongly reduce a separation temperature in the PEG/ surfactant/water mixtures in comparison to the surfactant/water mixtures. The origin of the depletion interactions can be traced to the conformational entropy of polymer chains. The surfactant in water mixtures forms micelles. When we add PEG polymer we find that there is a zone around each micelle that cannot be penetrated by the center of mass of a polymer molecule. The size of the zone is proportional to the radius of gyration of a polymer. The center of mass of a polymer cannot penetrate the zone because it would result in a decrease of its conformational entropy. If two zones overlap, there is an imbalance of osmotic pressures (induced by a polymer outside the zones) which pushes the micelles together and leads to the phase separation into two phases. One of them is the surfactant-rich phase; the other is the polymer-rich phase. 18 Figure 2. Phase diagram of the 10% C 12E 5/PEG/water system showing the homogeneous region and the two-phase region. Addition of PEG decreases the cloud point temperature. Data were collected using the optical microscope. Figure 3. Phase diagram of PS (M w ) g/mol)/5cb: temperature versus the concentration of 5CB. The symbols in this diagram represent experimental data, and lines have been added as guides to the eyes. Circles represent the transition temperature from isotropic 5CB + isotropic PS (I + I) to homogeneous, isotropic mixture (I), squares represent the transition temperature from nematic 5CB + isotropic PS (N + I) to isotropic 5CB + isotropic PS (I + I), triangles represent the transition temperature from crystalline state 5CB + isotropic PS (C + I) to nematic 5CB + isotropic PS (N + I). 5CB/Polystyrene Phase Diagram. The thermomicroscopy studies were performed to obtain a phase diagram of 5CB/PS. Samples were first annealed at a temperature above phase separation (60 C) in the homogeneous isotropic phase of the binary mixture. Second, by cooling the samples we studied optical textures of our samples. The liquid crystalline ordered nematic phase was identified on the basis of these textures. The measurements were done for five compositions of 5CB/PS: 58/ 42; 63/37; 72/28; 84/16; 95/5% by weight and for pure 5CB. Figure 3 presents the equilibrium phase diagram of the 5CB/ PS mixture. The phase diagram exhibits an upper critical (consolute) temperature. It has an asymmetric shape characteristic for polymer/solvent mixtures. There are three phase transition lines separating four different regions, depending on the ordering of liquid crystal or polymer. The liquid crystal can form in the mixture an isotropic phase (I), the nematic phase (N), and crystalline phase (C). PS can only form an isotropic (I) phase in the solvent, although in the pure state in the temperature range studied it forms a solid phase. The phase diagram exhibits a large I + I misibility gap which is

3 Kinetics and Dynamics of Mixing J. Phys. Chem. B, Vol. 111, No. 41, Figure 4. Domain collapse during mixing observed in 10% C 12E 5/ water mixture after the quench in the homogeneous region (a) 0 s, (b) s, (c) s, and (d) 49.5 s. The domain decreased its size without change of the optical contrast, normally seen as a clear boundary between two phases. When the contrast decreases, usually the boundary becomes blurred. These pictures were collected using the optical microscope. The contrast on the pictures has been artificially enhanced to see properly the smallest domains. characteristic for large molecular weight of PS. For a small molecular weight (of the order of 10000), the I + I region disappears. The transition temperatures to the nematic N + I region and crystalline state C + I region phases of the liquid crystal do not depend on a composition of the mixture within the experimental error of 0.2 C. Thus, temperature ranges of N + I and C + I regions are set by the isotropic-to- nematic and nematic-to-crystalline phase transition temperatures in pure liquid crystal, respectively. In principle, a crystallization of the liquid crystal should be affected by the polymer dissolved in the LC matrix, but we find that lowering of the crystallization temperature is insignificant because polymer separates from the LC matrix at temperatures higher than a crystallization temperature of LC. Mixing in Initially Separated Mixtures after a Rapid Temperature Change. We made optical microscope measurements of a mixing process in the nonionic-surfactant/polymer/ water and polymer/liquid crystal (PS/5CB) mixtures. The kinetics of the mixing process was determined by a quantitative study of sizes and shapes of domains of one phase in a matrix of a second phase. At any single picture we marked diameters (L) of the domains and analyzed them statistically. In the first mixture, characterized by the lower critical point, we heated the system about one degree above the cloud point (spinodal), waited a certain time in order to obtain a two-phase state with large domains of one phase in a second phase. Next we cooled the system and observed a pathway of the mixing process (transformation of the two-phase state into the homogeneous one). In the second mixture, characterized by the upper critical point, we cooled the system into the I + I region (Figure 3) to induce phase separation and next heated the system into the homogeneous region. For 10% of C 12 E 5 in water the domain s size decreased during mixing without a visible change of an optical contrast. In general the optical contrast depends on the differences in the index of refraction between two phases. For surfactant solutions it is in general rather weak, and only boundaries of domains are clearly visible. That is why the contrast was artificially enhanced in Figure 4 which shows the process after (a) 0, (b) 11.25, (c) 36.75, and (d) 49.5 s. An identical process (domains collapsing) was also observed for the ternary 10% C 12 E 5 /1% PEG/water mixture and for mixtures with other surfactants (C 12 E 6 ). We have analyzed the size of collapsing domains as a function of time. Figure 5 shows a time evolution of the domain s size during mixing for the binary 10% C 12 E 5 /water mixture. At the Figure 5. Plot shows time evolution of the domain s size (for two domains) in the binary 10% C 12E 5/water mixture (see Figure 4). At the beginning of the mixing process both domains decrease their sizes with comparable speed. The inset shows the mixing process at all times including a change of slope for a bigger domain. This change of the slope was probably due to the decrease of the difference between the concentrations of the components inside and outside of the domains After the dissolution of small domains, the concentration increased in a solvent and reduced the thermodynamic forces (chemical potential differences) responsible for mixing. Data was collected using the optical microscope. Figure 6. Time evolution of the domain size in the 10% C 12E 5/1% PEG/water mixture. The solid lines are linear fits L(t) ) L(0) - at. The slopes are given on the plot. The inset shows the mixing process at all times. The change of the slope was probably due to the decrease of the difference between the concentrations of the components inside and outside of the domains. This change reduced the thermodynamic forces (chemical potential differences) responsible for mixing. Data was collected using the optical microscope. beginning of mixing there are a lot of domains of various sizes in the system. In the first 60 s the domains collapsed linearly (L(t) L(0) - at) with a slope a between to [µm/s] (Figure 5). At later times the domain s size decreased linearly with time, but with a different slope which changed from a ) -0.2 [µm/s] to a )-0.05 [µm/s]. The change of the slope was due to the decrease of the difference between the concentrations of the components inside and outside of the domains. This change reduced the thermodynamic forces (chemical potential differences) responsible for mixing. This point will be discussed further in the theoretical section. Figure 6 shows measurements performed in the 10% C 12 E 5 /1% PEG/ water mixtures. At the beginning of mixing we observed a linear decrease of the size for all domains with slopes from -0.3 to [µm/s]. For later times, t > 25 [s] the slopes changed from to [µm/s]. A similar scenario of mixing was observed in the 95% 5CB/ 5% PS mixture. Figure 7 shows a sample of 5CB/PS mixture: (a) under the crossed polarizers, (b) without the crossed polarizers. Inside liquid crystaline matrix we had PS-rich

4 11910 J. Phys. Chem. B, Vol. 111, No. 41, 2007 Kalwarczyk et al. Figure 7. PS-rich domains in 95% 5CB/5% PS mixture are shown: (a) under crossed polarizers in the nematic phase, (b) without crossed polarizers after heating the sample into the I + I region. Inside the liquid crystaline matrix we had polymer (PS-rich) domains (seen as black domains under the crossed polarizers). Pictures were collected using the optical microscope. Figure 9. Plot showing time evolution of the size L(t) of two PS-rich domains in the binary 95% 5CB/5% PS mixture. For all domains L(t) ) L(0) - at. The slope was given by a [µm/s] for small domain and a [µm/s] for large domains. Data were collected using the optical microscope. Figure 10. Evolution of the domain size during simulation steps (particles move by random walk). The growth of the domain could be described by the algebraic form L(t) t β, with β ) 0.5. The diffusion inside the domain is faster than that outside. Figure 8. Collapse of the PS-rich domain during mixing after (a) 600 s, (b) 4200 s, (c) 8400 s, and (d) s. Pictures were collected using the optical microscope. domains (seen as black domains under the crossed polarizers). Figure 8 shows one of the PS-rich domain s size during mixing after (a) 600, (b) 4200, (c) 8400, and (d) s. Figure 9 shows the time evolution of the domain s size L(t) during mixing for the binary 95% 5CB/5% PS mixture. For all domains we found the linear relation L(t) ) L(0) - at. The slope was given by approximately [µm/s] for small domains and approximately [µm/s] for large domains. We close this experimental section by posing three questions: When can we expect an increase of the domains or decrease of the domains sizes during mixing? Can we explain the linear dependence of the domain size on time during mixing? What is a relevant theoretical description of the mixing process? Theoretical Model Toy Model of Mixing via Diffusion. A toy model studied in this section will help to understand the difference in the kinetics and dynamics of mixing between our experimental systems (polymer/liquid crystal and surfactant/water), where the mixing clearly proceeds by the decrease of the domain s size, and the systems where the mixing process proceeds with an increase of the domain s sizes by diffusion which is given by Fick s law. 19 We postulate at this point that this difference is due to a difference of the viscosities of phases. Before we present our toy model, we note that the viscosity is inversely proportional to the diffusion coefficient of objects forming the domain, which follows from the Einstein-Stokes equation: η ) k B T cd Here η is the viscosity, k B is the Boltzmann constant, T is the temperature, D is the diffusion coefficient, and c is a constant which depends on object s size and shape. The domains in our mixtures are composed of two different kinds of objects. In the water/surfactant mixture, surfactant domains are composed of surfactant micelles; and in the polymer/liquid crystal mixtures, polymer-rich domains are composed of polymer chains. In both cases eq 1 applies, and we can model a domain of high viscosity in the low-viscosity matrix by a system of particles which have small diffusion coefficients inside the domain and large diffusion coefficients outside this domain. In order to support qualitatively our postulate, we performed computer simulations of diffusive spreading of particles (initially enclosed inside a domain) in a system where the diffusion coefficient depends on the concentration of particles. The diffusion was studied by simulating a random walk of particles (1)

5 Kinetics and Dynamics of Mixing J. Phys. Chem. B, Vol. 111, No. 41, Figure 11. Change of the particle density in our simulation system (Figure 10) after (a) 1, (b) 10000, (c) 20000, and (d) simulation steps. The density of particles quickly approached the Gaussian distribution. on a one-dimensional lattice. The total size of our system was 2000h, where h was a distance between lattice sites. The typical size of a domain initially occupied by the particles was L 160 in units of h. The domain was placed in the middle of the system. A typical particle s number was The initial distribution of particles inside the domain was uniform. For each particle we chose a random number V such that 0 < V < 1. If V was smaller than a certain probability p, a particle moved one node to the left. If V was higher than 1 - p, a particle moved one node to the right. If V was in the interval (p, 1 - p), the particle did not move. The probability p was different inside and outside of the domain, i.e., it was dependent on the occupation number (number of particles) at a given lattice site. We assigned different probabilities for jumps to the right and to the left inside and outside of the domain. Inside the domain (when the occupancy number was larger than 8 particles per lattice site), the probability was p ) 0.15 for the jump (to the right or left). By decreasing this probability we reduced the diffusion coefficient (the smaller the p, the smaller the diffusion coefficient). Outside the domain (when the occupancy of particles per lattice site was smaller than 8) the probability was p ) 0.5 for a jump to the left and to the right. One simulation step involved drawing V for each particle and performing the random move. When all particles were assigned new positions, we checked the new domain s size and reassigned the probability p inside and outside the domain. The size of the domain (L) was determined after each simulation step. The average occupancy in the simulations was 2.5 particles per lattice site. We arbitrarily set the domain boundary at 8 particles per lattice site. Figure 10 shows the time evolution (smoothed over 500 simulation steps) of the domain s size during simulation steps when p inside. p outside. The growth of the domain was demonstrated to have the algebraic form L(t) t β, with β ) 0.5. Figure 11 shows the time evolution of particle density in our simulation system after (a) 1 step, (b) steps, (c) Figure 12. Domain size as a function of time (particles move faster outside than inside the domain). The initial increase of the domain size was due to the development of the Gaussian shape of the distribution of particles at the domain boundaries. steps, and (d) steps. The particle density quickly approached the Gaussian distribution. When p inside, p outside, the particle density evolved in a different way. Figure 12 (smoothed plot) shows time evolution of the domain s size after more than steps. At the beginning of the evolution the domain grew quickly. Next the size started to decrease logarithmically, i.e., L(t) L(t ) t 0 ) - L 1 ln(t/t 0 ). Figure 13 shows the evolution of the particles density after: (a) 1 step, (b) steps, (c) steps, (d) steps, (e) steps, and (f) steps. The results of our simulations show that the difference in the diffusion coefficients inside and outside of the domain (modeled by varying p inside and p outside ) is responsible for the growth or decrease of the domains during mixing. However the model does not explain the kinetics of mixing since in the experiment we observe the linear change of the domain s size with time while in the simulation we have either a square root growth (for p inside. p outside ) or a logarithmic decrease (for p inside

6 11912 J. Phys. Chem. B, Vol. 111, No. 41, 2007 Kalwarczyk et al. Figure 13. Particle density for the system shown in Figure 12 after (a) 1, (b) , (c) , (d) , (e) , and (f) simulation steps., p outside ). It is clear that we need a more elaborated model to explain the experimental results. Such a model is presented in the next section. Gibbs Free Energy Time Evolution during Mixing. The Gibbs free energy of a domain of one phase in the second phase when this system is quenched to the homogeneous region is higher than the energy of the thermodynamically stable homogeneous state. That is why the domain dissolves (by growing or decreasing its size) until the homogeneous state is reached. The difference between the bulk Gibbs free energy of the twophase thermodynamic state has the following form: G b ) k (µ k n k ), where k denotes the k-th component of the mixture, n k is the number of moles of the component inside the domain, and µ k is the chemical potential difference for the k-th component between the value of µ k inside and outside of the domain. Apart from the bulk contribution we have the surface contribution arising from the surface tension. If we further assume that during dissolution the domain does not change its composition (which is approximately true for the case of domains which decrease their size without changing the optical contrast as we observed in our experiments) the free energy difference is given by G ) 4πL 2 γ ( 4/3πL 3 (Fµ), where L is the radius of the domain, 4πL 2 γ is the Gibbs surface free energy, γ is the surface tension and (Fµ) ) k F k µ k, where r k is the number density of the k-th component in the domain, which is fixed. In general the surface tension can change during mixing because it is proportional to the concentration difference squared, and also the chemical potential and the number density could change during mixing. All additional complications of the model can be taken into account. Here we want to discuss the simplest possible case where only the radius changes during mixing. This model should adequately describe the case when the domains decrease their size during mixing without visible change of the optical contrast. During mixing, the whole free energy difference G has to be dissipated. The free energy dissipated per unit time is given by the following equation (with the aforementioned assumptions): W ) dg dl )-8πLγ dt dt - 4πL2 (µf) dl dt This dissipation is caused by the movement of the interface. We propose a dissipation function following the standard analysis in the irreversible thermodynamics: 20 (2) F ) 4πL 2 ω ( dl dt ) 2 (3)

7 Kinetics and Dynamics of Mixing J. Phys. Chem. B, Vol. 111, No. 41, Here ω is the phenomenological coefficient. We assume that W ) Φ, i.e., the free energy dissipated per unit time is due to the interface motion. To obtain the motion of the domain interface we minimize dissipation Φ with respect to the speed of the interface. 21 This is equivalent to the minimum entropy production at each instant of time. 20 Minimization of Φ with respect to V)dL/dt under the condition that W ) Φ leads us to the following equation: 21 Solving this equation gives L(t): dφ dv - 2 dw dv ) 0 (4) L(t) - L(0) + A ln( L(0) + A ) Bt (5) L(t) + A) where A ) 2γ/ (Fµ) and B )- (Fµ)/ω. A is comparable to the critical size of the nucleus. Typically A should be of the order of few nanometers, i.e., 3 orders of magnitude smaller than the domain size L observed under the microscope. A can only assume micrometer size for thermodynamic conditions very close to the coexistence of two phases. Consequently in eq 5 the term associated with B dominates and thus eq 5 reduces to the following equation: L(t) - L(0) ) Bt (6) Since B < 0 the model describes the decrease of the domain size. Experimentally the size of the domain changes linearly with time t as in eq 6 (see Figures 5 and 6). However, during mixing, not only the size is changing, since the term B depends on the differences in chemical potential. After dissolution of small domains, the chemical potential outside the domains increases and consequently B, which is proportional to the difference (Fµ), decreases. This effect explains the changes of slopes (proportional to B) during mixing as a function of the domain size. For very small domains, i.e., L comparable to A (the size of few nanometers) the logarithmic term in eq 5 would dominate. However the observation length scale of this effect is beyond the scope of our optical study, which is restricted to the micrometer sizes. Our model does not apply to the case of domain growth, since then (Fµ) changes with time and thus the change of G is not only due to the interface motions as in the previous case. Inverting the Morphology in C 12 E 5 /Water Solution. In this section we will present an experiment in which we change the ratio of diffusion coefficients inside and outside the domain during mixing in the system with fixed composition. Let us assume that we have X-rich domains in Y-rich matrix. By phase inversion we mean that this morphology transforms into the one of Y-rich domains in the X-rich matrix. In the following experiment we induced the morphology inversion in our system. We took a binary 10% C 12 E 5 /water system. At room temperature this solution was in the micellar region (point 0 in Figure 15). We started to increase the temperature. Above the binodal curve (two phase regionsfigure 15), the system separated into two phases: one was an almost pure water (bulk), and the second was the highly concentrated micellar solution (domains) (point 1, Figure 15). Figure 14a shows the separated system in the two-phase region. One of the domains was marked on the photograph by a dashed line. After separation we further increased the temperature up to the lamellar region (point 2, Figure 15). The lamellar phase was characterized by orders of magnitude higher viscosity than the one for the micellar phase. Figure 14. Morphology inversion during the temperature changes in the C 12E 5/water system (see Figure 15). The solid line marked a surfactant-rich domain in the water matrix. It is consequently shown on each figure because it also allows to see the morphology inversion. The black line marks (a) a domain in the two-phase region, (b) after a jump to the lamellar phase, (c, d) quench back to the two-phase region and morphology inversion (water-rich domains in a surfactant-rich matrix, and (e, f) mixing in the homogeneous region. Figure 14b shows the change of the morphology in the solution after heating the system to the lamellar region. Edges of domains changed their shapes into a kind of self-similar (fractal) structure indicating a possible inversion of the domains. We note here that the contrast is small between the domains, and only at the interface we observe sufficient scattering of light which discerns between the domain and the matrix. After few minutes we started to decrease the temperature to the initial temperature in the homogeneous micellar region (point 3, Figure 15). Figure 14c,d shows pictures of the system during the process. The domains started to increase their sizes via coalescence-induced coalescence mechanism (Figure 15). In Figure 14c,d, we observe that what was previously a surfactant-rich domain (marked by a dashed line) became a surfactant-rich matrix. When the temperature reached the homogeneous region, the system started to mix (point 4, Figure 15). Figure 14e,f shows that during mixing the domains increase their sizes with decreasing the optical contrast. The growth of the domains indicated that the diffusion coefficient in the matrix was smaller than that in the domains because of the morphology inversion. This kind of mixing was characteristic for the solutions with a low concentration of surfactants (systems with the domain s viscosity comparable to the viscosity of a matrix). Thus in our system (10% of surfactant) we could obtain surfactant-rich domains in a water-rich matrix and, after the morphology inversion described above, water-rich domains in the surfactant-rich matrix. In the former case, mixing proceeds by the decrease of the domain s size without change of the optical contrast while

8 11914 J. Phys. Chem. B, Vol. 111, No. 41, 2007 Kalwarczyk et al. of mixing, in two chemically different mixtures: one composed of surfactant, polymer, and water and characterized by the lower critical temperature and the second composed of liquid crystal and polymer and characterized by the upper critical temperature. We demonstrate that the mixing process of domains of one phase in a matrix of the other phase depends on the viscosity inside the domains and in the matrix. The domain of a phase of high viscosity in a matrix of low viscosity dissolves by decreasing its size linearly with time L(t) L(0) - at, without a rapid change of its composition (here observed as the optical contrast). In the opposite case (low viscosity domain in a high viscosity matrix), a domain dissolves by increasing its size and decreasing its composition or, in other words, decreasing the optical contrast between the domain and the matrix. The theoretical model (eqs 2-6) predicts the linear decrease of the domains sizes in accordance with experimental observation. However, it does not account for the change of the composition in the matrix during the dissolution of the domains. It is possible to include this effect in a spirit similar to that which was done for phase separation in the Lifshitz-Slyozov model. 23 In this model, when the small domain dissolves, the saturation in the matrix increases and reduces the growth of the bigger domains. Acknowledgment. This work was supported from the budget of the Ministry of Science and Higher Education as a scientific project ( ). References and Notes Figure 15. (a) Phase diagram of the surfactant C 12E 5/water mixtures. (b) A cartoon of the heating/cooling process leading to the morphology inversion in 10% C 12E 5/water system shown in Figure 14. (0) Heating started from the homogeneous region; (1) separation in the two-phase region into an almost pure water matrix and concentrated micellar domains; (2) ordering in the lamellar region; (3) cooling to the homogeneous region; (4) mixing in the homogeneous region (see photographs in Figure 14). in the latter the domain growth is observed. As was explained in previous sections, in the former case the diffusion was faster in the matrix while in the latter it was faster inside the domain. We conclude that freezing up the solution by rapidly increasing and again decreasing its viscosity by orders of magnitude can invert the morphology of phases in the system. The morphology inversion leads to the change of the ratio of diffusion coefficients inside the domains and in the bulk. Consequently it changes the scenario of the mixing process. This method of changing the ratio of diffusion coefficients works for the C 12 E 5 /water solution with less than 30% of surfactant. One of the conditions for the phase inversion is the existence of the high viscosity region (at a given concentration of surfactant). In the C 12 E 5 /water system, there is a lamellar region above the two-phase region (Figure 15a). We expect similar behavior for substances with the phase diagram similar to C 12 E 5. One of the examples could be the other nonionic surfactants C 10 E Summary and Conclusions In our report we present a detailed study of the kinetics (size of domains versus time) and dynamics (thermodynamic forces) (1) Holyst, R. Solubility and mixing in fluids. Encycl. Appl. Phys. 1997, 18, 573. (2) Mitchel, D. J.; Tiddy, G. J. T.; Waring, L.; Bostock, T.; McDonald, M. P. J. Chem. Soc., Faraday Trans , 79, 975. (3) Akcesu,. A. Z.; Bahar, I.; Erman, B.; Feng, Y.; Han, C. C. J. Chem. Phys. 1992, 43, 357. (4) Akcesu,. A. Z.; Klein, R. Macromolecules 1993, 26, (5) Graca, M.; Wieczorek, S. A.; Fialkowski, M.; Holyst, R. Macromolecules 2002, 35, (6) Hammouda, B.; Balsara, N. P.; Lefebvre, A. A. Macromolecules 1997, 30, (7) Ermi, B. D.; Karim, A.; Douglas, J. F. J. Polym. Sci., Part B: Polym. Phys. 1998, 36, 191. (8) Vailati, A.; Giglio, M. Nature (London) 1997, 390, 262. (9) Holoubek, J. Macromol. Symp. 2000, 149, 119. (10) Holoubek, J. Macromol. Symp. 2000, 162, 307. (11) Okada, M.; Tao, J.; Nose, T. Polymer 2002, 43, 329. (12) M. Graca, M, Wieczorek, S. A., Holyst R. Macromolecules 2002, 35, (13) Fialkowski, M.; Holyst, R. J. Chem. Phys. 2002, 117, (14) DeRoussel, P; Khakhar, P. V.; Ottino, J. M. Chem. Eng. Sci. 2001, 56, (15) DeRoussel, P; Khakhar, P. V.; Ottino, J. M. Chem. Eng. Sci. 2001, 56, (16) Asakura, S.; Oosawa, J. J. Chem. Phys. 1954, 22, (17) Vrij, A. Pure Appl. Chem. 1976, 48, 471. (18) Anderson, V. J.; Lekkerkerker, H. N. W. Nature (London) 2002, 416, 811. (19) Atkins, P. Physical Chemistry, 6th ed.; Oxford Univerity Press, (20) Kondepudi, D.; Prigogine, I. Modern Thermodynamics; John Wiley & Sons: New York, (21) Oswald, P.; Pieranski, P.; Picano, F.; Holyst, R. Phys. ReV. Lett. 2002, 88, (22) Laughlin, R.G. The aqueous phase behavior of surfactant; Academic Press: New York, (23) Bray, A. J. AdV. Phys. 1994, 43, 357.